A measurement-based approach to maximum power transfer for

A MEASUREMENT-BASED APPROACH TO MAXIMUM POWER TRANSFERFOR LINEAR N -PORTS WITH UNCOUPLED LOADS

Edwin Alejandro Herrera Herrera

Dissertacao de Mestrado apresentada aoPrograma de Pos-graduacao em EngenhariaEletrica COPPE da Universidade Federal doRio de Janeiro como parte dos requisitosnecessarios a obtencao do tıtulo de Mestre emEngenharia Eletrica

Orientadores Amit BhayaOumar Diene

Rio de JaneiroAbril de 2015

DISSERTACAO SUBMETIDA AO CORPO DOCENTE DO INSTITUTO ALBERTOLUIZ COIMBRA DE POS-GRADUACAO E PESQUISA DE ENGENHARIA (COPPE)DA UNIVERSIDADE FEDERAL DO RIO DE JANEIRO COMO PARTE DOSREQUISITOS NECESSARIOS PARA A OBTENCAO DO GRAU DE MESTRE EMCIENCIAS EM ENGENHARIA ELETRICA

Examinada por

Prof Amit Bhaya PhD

Prof Glauco Nery Taranto PhD

Prof Vitor Heloiz Nascimento PhD

RIO DE JANEIRO RJ ndash BRASILABRIL DE 2015

Herrera Herrera Edwin AlejandroA measurement-based approach to maximum power

transfer for linear n-ports with uncoupled loadsEdwinAlejandro Herrera Herrera ndash Rio de JaneiroUFRJCOPPE 2015

XIII 65 p il 29 7cmOrientadores Amit Bhaya

Oumar DieneDissertacao (mestrado) ndash UFRJCOPPEPrograma de

Engenharia Eletrica 2015Bibliography p 61 ndash 631 Bodersquos Bilinear Theorem 2 Maximum Power

Transfer 3 Multilinear Transformation 4 Theveninrsquostheorem 5 Moebius Transformation I Bhaya Amitet al II Universidade Federal do Rio de Janeiro COPPEPrograma de Engenharia Eletrica III Tıtulo

To my family and girlfriendJenny

Acknowledgement

Firstly I would like to thank my parents sister and niece whose were always supportingme and encouraging me with their best wishes

I would like to thank my girlfriend She was always there cheering me up and stoodby me through the good times and bad despite the distance

I wish to express my sincere thanks to my advisor Professor Amit Bhaya for hisexcellent guidance and patient for providing me with all the necessary facilities for theresearch I would like to thank Professor Oumar Diene who was always diligent to solveall doubts I am also grateful to the examination commission for their knowledge andcontributions in the revision of this dissertation In addition a thank you to my partnersin the laboratory

Finally I am grateful to the God for the good health and wellbeing that were necessaryto complete this goal

Resumo da Dissertacao apresentada a COPPEUFRJ como parte dos requisitosnecessarios para a obtencao do grau de Mestre em Ciencias (MSc)

UM MEacuteTODO BASEADO EM MEDICcedilOtildeES PARA MAacuteXIMA TRANSFEREcircNCIA DEPOTEcircNCIA EM REDES LINEARES DE N -PORTAS COM CARGAS

DESACOPLADAS

Abril2015

Programa Engenharia Eletrica

Este trabalho revisita trecircs resultados claacutessicos da teoria de circuitos o teorema deThevenin o teorema de maacutexima transferecircncia de potecircncia e o teorema bilinear de Bodebem como da sua generalizaccedilatildeo multilinear Combinando esses resultados propotildee-se umanova abordagem praacutetica baseada em mediccedilotildees ao teorema de transferecircncia ldquomaacuteximardquopara n portas terminadas com cargas desacopladas no sentido de permitir uma descriccedilatildeofuncional algeacutebrica da hipersuperfiacutecie inteira da potecircncia em funccedilatildeo dos paracircmetros dasportas escolhidas Exemplos de sistemas de potecircncia mostram que isto eacute mais uacutetil doque meramente computar os paracircmetros da porta para a maacutexima transferecircncia de potecircn-cia dado que a hipersuperfiacutecie da potecircncia pode ser utilizada junto com restriccedilotildees nosparacircmetros da porta como nas tensotildees para encontrar os paracircmetros oacutetimos da trans-ferecircncia viaacutevel de potecircncia

Abstract of Dissertation presented to COPPEUFRJ as a partial fulfillment of therequirements for the degree of Master of Science (MSc)

April2015

Advisors Amit BhayaOumar Diene

Department Electrical Engineering

This work revisits three classical results of circuit theory the Theacutevenin theorem themaximum power transfer theorem and Bodersquos bilinear theorem as well as its multilineargeneralization Combining these results it proposes a new measurement-based approachthat provides a practical new version of the ldquomaximumrdquo power transfer theorem for n-ports terminated with uncoupled loads in the sense that it allows a functional algebraicdescription of the entire power hypersurface as a function of the chosen port parametersExamples from power systems show that this is more useful than merely computing portparameters for maximum power transfer since the power hypersurface can be used alongwith constraints on port parameters such as voltages to find parameters for optimalviable power transfer

Contents

List of Figures x

List of Tables xiii

1 Introduction 111 The Thevenin theorem 112 The maximum power transfer theorem 213 Bodersquos bilinear theorem 414 Structure of dissertation 5

2 Bodersquos bilinear form theorem and generalizations 621 Bodersquos theorem 6

211 Representation of circles by hermitian matrices 8212 The Moebius transformation 9

22 Generalizations Multilinear transformation 1123 Measurement-based approach 1224 Theveninrsquos and Nortonrsquos theorem are particular cases of Bodersquos bilinear

theorem 1525 Linrsquos theorem applied to two-port networks 16

3 The maximum power transfer (MPT) 1931 Derivation of the maximum power transfer theorem for general n-port 2032 Illustrative examples for MPT with complex load impedances 2533 MPT for the case of uncoupled resistive loads 25

331 Modifications to Desoerrsquos theorem for uncoupled resistive loads 29

4 Application of Linrsquos theorem to MPT 3241 Illustrative examples of MPT for two-port 33

5 Power systems applications 4751 Thevenin equivalent parameters from measurement-based approach review 47

511 Simple linear circuit 49512 IEEE 30-bus system 52

52 Calculating Smax for 2-port 56521 Cross-checking maximizing impedance and power results 57

6 Conclusions 5961 Conclusions and remarks 5962 Future work 60

Bibliography 61

A Guide to use algorithm developed in Matlab 64

List of Figures

21 A simple circuit that shows the nature of the bilinear transformation be-tween the resistance r and the impedance T (r) between points A and B asr varies along a line T (r) traces a circle in the complex plane The dashedsemi-circle corresponds to negative values of the variable resistance r 7

22 A simple circuit that shows the nature of the bilinear transformation be-tween the inductance l and the impedance T (l) between points A and Bas l varies along a line T (l) traces the solid circle in the complex planeThe dashed semi-circle corresponds to the case considered in Fig 21 andit can be shown that the two cirles intersect orthogonally 8

23 Thevenin equivalent circuit for 1-port et corresponds to Thevenin or opencircuit voltage and zt z` are respectively the Thevenin and load impedances 15

24 A two-port network can be modeled by the two dependent source equiva-lent circuit For a non-reciprocal network z12 6= z21 16

31 An n-port N represented in its Thevenin equivalent form connected toa load n-port N` where i isin Cn is the current vector et isin Cn is theThevenin equivalent voltage and Zt isin CntimesnZ` isin Cntimesn are respectivelythe Thevenin equivalent and load impedance matrices 21

32 T-network model for Example 34 in which Zt + Zlowastt is indefinite and thload Z` = infinminusj Intuitively the total power P lowastT takes large valuesbecause there exist active elements in the circuit 26

33 T-network model for Example 311in which P is unbounded with R` = (0 0) 3034 T-network model for Example 312 in which P is unbounded with R` =

(infin 0) 31

41 Two-port network terminated in resistances The resistors r1 r2 are designelements and adjust to draw the maximum total power in two-port Theremaining elements of the circuits are unknown it is a black box 33

42 Surface plot of total power function PT (r1 r2) for resistive reciprocal two-port circuit of example 41 The point marked lsquotimesrsquo represents the maximumpower 35

43 Contour lines of total power function PT (r1 r2) for the resistive recipro-cal two-port circuit of example 41 The point market lsquotimesrsquo represents themaximum power the level sets of PT are nonconvex 35

44 Surface plot of total power function PT (v1 v2) as a function of port voltagesv1 v2 for a resistive reciprocal two-port circuit of example 41 Notice thatthe maximal point occurs (v1 = v2 = 5V ) represented by lsquotimesrsquo 36

45 Surface plot of total power function PT (r1 r2) for non-reciprocal two-portcircuit of example 42 The point marked lsquotimesrsquo represents the maximumpower 37

46 Contour lines of total power function PT (r1 r2) for a non-reciprocal two-port circuit of example 42 The point marked lsquotimesrsquo represents the maximumpower The level sets are nonconvex 37

47 AC circuit with two design elements inductor l and resistor r2 The rest ofthe elements are assumed to be unknown (vin = 10V rms c = 025mF r =3 Ω ω = 2πf) 38

48 Surface plot of total power function PT (l1 r2) for two-port AC circuit ofExample 43 Fig 47 The point marked lsquotimesrsquo represents the maximum power 39

49 Contour lines of total power function PT (l1 r2) for two-port AC circuit ofExample 43 Fig 47 The level set are convex The point marked lsquotimesrsquorepresents the maximum power 39

410 Surface plot of total power function PT (r1 r2) for two-port reactive circuitof Example 44 The point marked lsquotimesrsquo represents the maximum power 41

411 Contour lines of total power function PT (r1 r2) for two-port reactive circuitof Example 44 The point marked lsquotimesrsquo represents the maximum power 41

413 Contour lines of total power function PT (r1 r2) for two-port reactive circuitof Example 45 The level sets are nonconvex The point marked lsquotimesrsquorepresents the maximum power 43

414 Surface plot of total power function PT (r1 r2) for the two-port reactivecircuit of Example 46 The point marked lsquotimesrsquo represents the maximumpower 45

415 Contour lines of total power function PT (r1 r2) for the two-port reactivecircuit of Example 46 The level sets are nonconvex The point markedlsquotimesrsquo represents the maximum power 46

51 Simple linear circuit For example 51 z2 is a variable impedance and therest of the circuit parameters are assumed to be known (z0 = j01 Ω z1 =1 Ω) For example 52 z1 z2 are chosen as z1 = z2 = 4microz0 where micro is thevariable scaling factor 49

52 Magnitude of current and active total power with respect to r2 for Exam-ple 51 50

53 Total active power PT as function of scaling factor micro for Example 52 5254 IEEE 30-bus test system 5355 Maximum apparent total power |ST | for each load buses of the IEEE 30-bus

system with voltage constraints vminkle vk le vmaxk

5556 Illustration of process utilized to validate the multilinear transformation

for any load impedance zi zj 58

A1 Algorithm to find the maximum total power PT and the correspondingimpedances zk with a measurement-based approach 65

List of Tables

51 Load impedances complex and apparent power for each load bus of IEEE30-bus system without voltage constraints 54

52 Load impedances complex and apparent power for each load bus of IEEE30-bus system with voltage constraints vmink

le vk le vmaxk 55

53 The maximizing impedances to draw maximum power for two buses (j k)in the IEEE 30-bus system without voltage constraints 57

54 The maximizing impedances to draw maximum power for two buses (j k)in the IEEE 30-bus system with voltage constraints vminjk

le vjk le vmaxjk

for each bus (j k) 57

A1 Main function implemented in Matlabr to find the maximum total powerPT and the corresponding impedances zk with a measurement-based ap-proach MT Multilinear Transformation 65

Chapter 1

Introduction

We start by recalling three classical results of circuit theory two of which are very wellknown and the third less so These are the Thevenin theorem the maximum powertransfer theorem (MPT) and Bodersquos bilinear theorem The objective of this dissertationis to relate these three results in a novel way in order to propose a practical new version ofthe maximum power transfer theorem for n-port terminated with uncoupled loads Thenext three sections provide brief and somewhat informal recapitulations of these classicalresults which will be formally detailed later

11 The Thevenin theorem

Theveninrsquos theorem provides a two-parameter characterization of the behavior of a linearelectrical network containing only voltage and current sources and linear circuit elementsIt is one of the most celebrated results of linear circuit theory because it provides com-putational and conceptual simplification of the solution of circuit problems The theoremshows that when viewed from two given terminals such a circuit can be described byjust two elements an equivalent voltage source vt in series connection with an equivalentimpedance zt As pointed out by Bertsekas [1] it is fruitful to view these elements as sensi-tivity parameters characterizing for example how the current across the given terminalsvaries as a function of the external load to the terminals The values of these parametersbetter known as equivalent or Thevenin voltage and impedance can be determined bysolving two versions of the circuit problem one with the terminals open-circuited and theother with the terminals short-circuited The theorem was independently derived in 1853by the German scientist Hermann von Helmholtz and in 1883 by Leon Charles Theveninan electrical engineer with Francersquos national Postes et Telegraphes telecommunicationsorganization and is therefore also referred to as the HelmholtzndashThevenin theorem [2]

The usual proof referred to or sometimes explicitly given in textbooks involves two

steps The first step invokes the superposition theorem to write down the affine form of asolution followed by the use of the uniqueness theorem to show that the solution is uniqueAnother viewpoint is that Theveninrsquos theorem is the result of systematic elimination of thecircuit voltages and currents in the linear equations expressing Kirchhoffrsquos laws and thelinear constitutive relations describing circuit elements Based on this multidimensionalversions of Theveninrsquos theorem have been developed the clearest of which in our opinioncan be found in [3]

From a practical viewpoint given any electrical device (black box) with two outputterminals in order to try to determine its properties one could (i) measure the voltageacross the terminals (ii) attach a resistor across the terminals and then measure thevoltage (iii) change the resistor and measure the voltage again Clearly these are allvariations on one simple measurement Theveninrsquos theorem tells us that the minimumnumber of measurements needed to characterize this black box completely from the(external) point of view of the output terminals is just two

The purpose of the brief description above was to contextualize the following questionsthat now arise what are the possible generalizations of Theveninrsquos theorem and howdoes the number of measurements increase for n-port Subsequently answers to thesequestions will be discussed

12 The maximum power transfer theorem

Consider a one-port consisting of sources and linear circuit elements driving a load z`assuming that the circuit is in sinusoidal steady state at a fixed frequency The problemis to determine the load impedance z` so that the average power received by the loadis maximum This kind of problem arises for example in the design of radar antennaSuch an antenna picks up a signal which must be amplified and the problem is to choosethe input impedance of the amplifier (z`) so that it receives maximum average power

This question has also received much attention both in the 1-port and n-port casesand details will be given in Chapter 3 Here we will give a brief overview of the literaturehighlighting the main results as well as the gaps in the literature one of which it is theobjective of this dissertation to fill As we have just seen in the previous section a linear1-port containing independent sources at a single frequency can be characterized by

v = et minus zti

where zt and et are respectively the Thevenin equivalent impedance and voltage sourceand v and i are respectively the port voltage and current Supposing that a passive

impedance is connected at the port to extract power from the 1-port then it is wellknown (see [4]) that (i) an impedance load z` will extract the maximum power when z`

is equal to the conjugate of zt assuming that Re z gt 0 and (ii) that a resistive load r`

extracts maximum power when z` is equal to the magnitude of ztThe extension to n-port is more recent a series of papers starting from 1969 have

led to better understanding of the problem which is inherently more difficult than the1-port problem in part because solutions are no longer unique Nambiarrsquos result [5] forlinear RLC n-port characterized by v = etminusZti where the lower case (resp uppercase)letters have the same interpretation as the 1-port case except that they now denotevectors (resp matrices) is that maximum power transfer occurs when Z` = Zt wherethe overbar denotes complex conjugation Later Desoer [6] gave an elegant demonstrationof the most general condition for nonreciprocal n-port and coupled load impedances (iewithout any assumptions on the structure of the load matrix Z`) Flanders [7] derivedcomplete results for the case of arbitrary resistive load n-port including the uncoupled (ordiagonal) case of a single resistor terminating each port Finally Lin [8] called attentionto the fact that a ldquomuch more difficult problem however is the case when the loads aren uncoupled resistors and the total power delivered to these resistors is to be maximizedrdquo

Despite all these results two issues have not been satisfactorily resolved AlthoughFlanders derives complete theoretical results for the resistive load case the mathematicalexpressions obtained are complicated and as he himself writes [7 p337] ldquoit seems thatfurther investigation will produce more useful expressionsrdquo The other issue is to findexpressions for the dissipated power in the case of complex uncoupled loads for whichno specific results exist since the existing general mathematical results give the form ofthe answer but no method to actually calculate it

Finally there are practical objections to the use of the so-called conjugate matchingcondition of the maximum power transfer theorem These are most eloquently put for-ward by McLaughlin and Kaiser [9] who write that ldquoa religious aura emanates from themaximum power transfer theoremrdquo and argue that it has ldquolimited usefulness in practicerdquoAs one example they mention that ldquowhen a load is connected across a battery rarely isthe load impedance selected to be equal to the (conjugate) impedance of the battery since(the latter) is often low (and) intentionally connecting a low-impedance load across thebattery could excessively load the battery causing its output voltage to fall or even causingan explosionrdquo

This dissertation will provide answers to these questions as its main contribution

13 Bodersquos bilinear theorem

Bode [10 p233ff] in his classic 1945 book writes ldquoIn many network design problemsit is convenient to study the effects of the most important elements on the network char-acteristic individually by assigning them various values while the remaining elements areheld fixedrdquo and a few lines later ldquothe simplest proposition we can use depends merelyupon the general form of the functional relationship between the network characteristicsof the greatest interest and any individual branch impedance For example if Z is eithera driving point or a transfer impedance it follows that it must be related to any givenbranch impedance z by an equation of the type

Z = A+Bz

where ABC and D are quantities which depend on the other elements of the circuitrdquoBode then points out that this bilinear transformation has the interesting geometricproperty of mapping circles to circles and goes on to describe how to use this fact ingeometrical sensitivity analysis This fundamental result was surprisingly little noticedor used in the circuit theory literature and progress in the area was dormant until almostthirty years later when Lin [11] rediscovered and slightly generalized the algebraic partof this result stating his multilinear form theorem in the context of symbolic computa-tion of network functions without realizing that Bode had anticipated the bilinear formof the result and therefore not citing him Lin realized that the fundamental reason formultilinearity stems in the ultimate analysis from the multilinear property of deter-minants and the fact that network functions can be found by applications of Cramerrsquosrule On the other hand since his interest was in symbolic calculation as opposed tonumerical calculation no applications to the latter were contemplated in [11] Howeverin his textbook [12] written another thirty years later examples are given of the use ofLinrsquos multilinear form theorem in measurement-based numerical computation of variousnetwork functions Finally a recent paper by Datta et al [13] rediscovers Linrsquos multilin-ear generalization of Bodersquos bilinear theorem as well as the measurement based idea ofDeCarlo and Lin [12] giving several examples from circuit and control systems

Also within the realm of symbolic or so-called ldquofast analyticrdquo calculation of networktransfer functions Vorperian [14] revisits the so-called extra element method due toMiddlebrook and co-authors [15 16] acknowledging that it is a version of Bodersquos BilinearTheorem The main claim in [14 p61ff] is that while it is ldquono picnicrdquo to use Bodersquosbilinear theorem to calculate transfer functions symbolically (because the coefficientsare given by determinants) his proposed extra element method allows human beingsto quickly and intuitively arrive at so called ldquolow entropyrdquo algebraic expressions for the

desired transfer functionsWhile Vorperianrsquos claim may well be true in a symbolic context it misses the point

that the knowledge that a desired transfer function has a multilinear form allows it tobe calculated numerically in an extremely efficient manner with a small number of mea-surements In fact the main conceptual contribution of this dissertation is to show thatthe correct approach to the maximum power transfer theorem for n-port subjected to un-coupled loads is to use the multilinear form theorem for the port currents (or voltages)in order to derive expressions for power dissipated in the load impedances that are com-putationally tractable in the sense that (i) they have a standard form that is computableby solving a linear system and (ii) are easily maximized using standard optimizationsoftware Item (i) implies that a small number of measurements allows the calculationof port variables as already noticed by Bode and more recently and emphatically byDatta et al [13] Furthermore once port variables have been calculated it is possibleto obtain an expression for the load power that is valid for all loads not merely for themaximizing ones As we will point out later this gives a result that does not have theldquoreligious aurardquo of the maximum power transfer theorem and can be used for practicalapplications

14 Structure of dissertation

This dissertation is organized in six chapters The first chapter presents a brief overviewof state of the art Chapter 2 reviews Bodersquos bilinear theorem as well as Linrsquos multilineargeneralization Chapter 3 presents a simplified derivation of the maximum power transfertheorem for n-port with general loads and also proposes some additional results forthe case of uncoupled resistive loads Chapter 4 presents the main argument of thedissertation namely that the maximum power transfer theorem should be approached byapplying the multilinear form theorem to port variables in order to obtain expressions forpower dissipated in the load impedances Several illustrative examples are also presentedin this chapter Chapter 5 showcases some applications to power systems comparing ourresults with those obtained by other techniques Finally Chapter 6 reviews the mainconclusions obtained in the thesis and suggests directions of future research

Chapter 2

Bodersquos bilinear form theorem andgeneralizations

This chapter recapitulates the properties of the bilinear transformation which seems tohave been first used in the context of circuit theory by Bode [10] Bode uses the wellknown geometric property of the bilinear transformation (mapping circles to circles in thecomplex plane) and describes its use in geometrical sensitivity analysis The algebraicversion of this result was rediscovered and generalized by Lin [11] to multilinear formsSubsequently Middlebrook and coauthors [15 16] introduced the so called extra elementmethod that they acknowledge as a symbolic manipulation-friendly ldquolow entropyrdquo versionof Bodersquos bilinear theorem A recent paper by Datta et al [13] essentially rediscoversLinrsquos multilinear form theorem and reemphasizes its measurement based aspect [12]with examples from circuit analysis and control systems

In the following sections the results cited above will be described in detail empha-sizing the power and elegance of Bodersquos bilinear theorem which provides the basis for ameasurement-based approach to the analysis and design of circuits leading for exampleto clear proofs of the Thevenin and Norton theorems as pointed out in [13]

21 Bodersquos theorem

Bodersquos bilinear form theorem stated and proved in his classic 1945 textbook [10 p223ff]showed that the variations in a network characteristic produced by changes in a singleelement z can always be represented by a bilinear transformation For instance in alinear electrical network the functional relation between any given branch impedance zand a driving point or transfer impedance T in another branch is as follows

T = a+ bz

c+ dz(21)

where a b c and d are quantities which depend on the other linear elements of the circuitThese quantities will vary with frequency which will therefore always be assumed to befixed so that a b c and d are constants

A bilinear transformation has the following fundamental property ldquoif z assumes valueslying on a circle (including as a special case a straight line) then the corresponding valuesof T will also lie on a circlerdquo as detailed in [17] where equation (21) is called lsquothe Moebiustransformationrsquo which is uniquely defined by a Hermitian matrix The next subsectiondescribes all the basic facts about circles and Moebius transformations required for thisdissertation

Bodersquos explanation of the fact that a small number (three) of measurements sufficesto characterize a 1-port from terminal measurements is essentially the geometric fact thatthree distinct points determine a circle Specifically as z varies the circle described byT (z) is determined with the previous knowledge of three points on it For example twoof these points can be found by choosing the particular values of zero or infinity for zand the third point can be found by choosing any adequate intermediate value

This means of course that it is possible find all four parameters a b c and d ofthe bilinear transformation with just three different pairs of values (z T ) where T is themeasurable variable in the circuit or network To exemplify in the case where the variableimpedance is a resistance r Bode [10] considered the circuit in Figure 21 for which itcan be seen by a simple calculation that the magnitude of impedance T between pointsAB is constant ie |T (r)| = 2x for all values of the variable resistance r Evidently byinspection T (0) = 2ix and T (infin) = minus2ix yielding two imaginary axis points belongingto the locus of T (r) Thus by symmetry and the constant modulus property of T thecircle in Figure 21 is obtained as the locus of T (r) as r varies

Figure 21 A simple circuit that shows the nature of the bilinear transformation betweenthe resistance r and the impedance T (r) between points A and B as r varies along a lineT (r) traces a circle in the complex plane The dashed semi-circle corresponds to negativevalues of the variable resistance r

For the same circuit Fig 21 now suppose that the resistance is fixed at value r = x

and the inductance l varies Similar considerations show that the locus of the impedanceT (l) as l varies is the solid circle in Figure 22 In fact from other properties of bilineartransformations it can be shown that the new solid circle intersects the dashed circleorthogonally where the latter corresponds to the case above (Figure 21)

Figure 22 A simple circuit that shows the nature of the bilinear transformation betweenthe inductance l and the impedance T (l) between points A and B as l varies along aline T (l) traces the solid circle in the complex plane The dashed semi-circle correspondsto the case considered in Fig 21 and it can be shown that the two cirles intersectorthogonally

211 Representation of circles by hermitian matrices

For completeness this subsection and the next recapitulate the main algebraic and geo-metric properties of bilinear or Moebius transformations Consider points z = x + iy inthe complex plane on a circle of radius ρ and center γ = α + iβ These points satisfy

|z minus γ|2 = ρ2

zz minus γz minus γz + γγ minus ρ2 = 0(22)

Consider the general equation

C(z z) = Azz +Bz + Cz +D = 0 (23)

In order to match (22) with (23) clearly AD should be real and B and C complexconjugates of each other In matrix notation defining

M = A B

= MH [M is Hermitian] (24)

and defining w = (z 1) the equation of the circle can be written as the quadratic formC(z z) = wHMw Clearly (22) is a particular case of (23) if A 6= 0 and

B = minusAγ C = minusAγ = B D = A(γγ minus ρ2) (25)

Thus every Hermitian matrix is associated with an equation like (23) and defines or isrepresentative of a circle unless A = B = C = D = 0 and by abuse of notation theletter M can be used to denote both the circle and the corresponding Hermitian matrixClearly two Hermitian matrices M1 and M2 represent the same circle iff M1 = λM2λ isin Rminus 0

The number ∆ = det M = AD minus BC = AD minus |B|2 isin R is called the discriminant ofthe circle For (22) a real circle ∆ = minusρ2 For (23) from (25) it follows that

∆ = minusA2ρ2 (26)

Thus the circle C given by (23) is a real circle iff

A 6= 0 ∆ lt 0

and its center γ and radius ρ can be found from (25) and (26) as follows

ρ =radicminus∆A

γ = minusCA

Straight lines are represented by Hermitian matrices M if A = 0 The circle coulddegenerate into a point circle if A 6= 0 and ∆ = 0 implying that ρ = 0

212 The Moebius transformation

This subsection defines and recalls basic properties of the Moebius transformationA Moebius transformationM Crarr C is given by (21) and rewritten for convenience

M z 7rarr T = az + b

cz + d a b c d isin C (27)

is uniquely defined by the hermitian matrix

M = a b

where det M = ad minus bc = δ if δ = 0 then T is a constant Three basic easily provenfacts are as follows

bull All matrices qM where q isin Cminus 0 define the same Moebius transformation M

bull M is a one-to-one transformation between the z-plane and the T -plane

bull The number zinfin = minusdc mapped the function M(z) to infinity and is called thepole of the Moebius function

A Moebius transformation is also called a linear transformation in z for the followingreason Introducing the homogeneous variables z1 z2 T1 T2 such that

z = z1

z2 T = T1

Equation (27) then becomes a linear homogeneous transformation in the variables z1 z2in matrix form T1

forallq isin Cminus 0 (28)

Finally (27) is also called a bilinear transformation because it is derived from a bilineartransformation between z and T

czT + dT minus az minus b = 0 (29)

which is an implicit representation of the functionM in (27) Indeed this representationis seen to be fundamental in the measurement-based approach developed in the sequel

The basic properties of the Moebius transformation are as follows

bull Product Given two Moebius transformations

M1 = a1 b1

M2 = a2 b2

If these transformations are carried out in succession z1 = M1(z) T = M2(z1)we obtain the transformation T = M2(M1(z)) which is called the product ofthe two transformations and is easily seen to be another Moebius transformationT =M3(z) which is associated to the matrix M3 = M2M1

bull Invertible The Moebius transformation (27) is invertible and its inverse is again aMoebius transformation

z =Mminus1(Z) = dZ minus bminuscZ + a

and the matrix associated to Mminus1 is Mminus1 up to the factor 1δ

= 1det M

Mminus1 is represented by d minusbminusc a

bull Identity transformation It is a special Moebius transformation that leaves everyelement unchanged T = z and the matrix I with 1s on the diagonal and 0severywhere else is given by

I = 1 0

A fascinating video [18] first presents four basic types of transformations in two di-

mensions translations dilations rotations and inversions arise from a stereographicprojection and then shows that the most complicated Moebius transformations are infact combinations of the four basic types and correspond to simple movements of thesphere followed by stereographic projection

22 Generalizations Multilinear transformation

This section presents two generalizations of Bodersquos theorem Bodersquos bilinear theoremwas rediscovered and generalized without reference to the work of Bode in the contextof calculation of so called symbolic network functions by Lin [11] who stated it as thefollowing theorem

Theorem 21 [Lin [11]] Let a lumped linear time-invariant network N consist ofimpedances admittances and all four types of controlled sources Let some or all ofthese network elements be characterized by distinct variables (x1 xn) while the re-maining elements are assigned numerical values Then any network function T whichis VoVi VoIi IoVi or IoIi may be expressed as the ratio of two polynomials of degreeone in each variable xi

To exemplify if only two elements are represented by variables then T can always beexpressed as

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

The proof of Linrsquos theorem involves the use of Theveninrsquos theorem and Cramerrsquos rule Ina textbook written several years later DeCarlo and Lin [12 p191] refer to Bodersquos resultas the Bilinear Form theorem attributing it to Lin [11] instead of Bode [10] Theseauthors also give an example that is the algebraic equivalent of Bodersquos observation that

three measurements suffice to determine the parameters (ac bc dc assuming c 6= 0)of a bilinear function thus being amongst the first authors after Bode himself to realizethe importance of Bodersquos bilinear form theorem from a measurement-based viewpointDeCarlo and Lin [12 Example 520p191ff] also carry out the key step of using the implicitform (29) of the bilinear transformation in order to rewrite the problem of determiningthe parameters of the bilinear transformation as that of the solution of a linear systemof equations based on as many measurements as there are unknown parameters

Subsequently Datta and coauthors [13] rediscovered Linrsquos result and stated it as thefollowing multilinear form lemma

Lemma 22 [Datta et al [13]] Let the matrices AK isin Rntimesn and the vectors bf isin Rn

be given Let p = (p1 p2 pn) isin Rn be a parameter vector Denoting the ith row ofa matrix M as (M )i suppose that the matrix A(p) is defined as follows (A(p))i =[ai1 ain] + pi[ki1 kin] and that the vector b(p)i is defined as (b(p))i = bi + pifiThen for the linear system A(p)x(p) = b(p) each component of the solution vectorx(p) is a multilinear rational function of the parameters pi that is for m = 1 2 n

xm(p) =nm0 + Σin

mi pi + Σijn

mijpipj + middot middot middot

dm0 + Σidmi pi + Σijdmijpipj + middot middot middot

where for ` = 0 1 the coefficients nm` depend on A K b f whereas the coefficientsdm` depend only on A K

This lemma was then used in [13] to obtain Bodersquos bilinear theorem as well as toemphasize its use in a measurement-based approach Several circuit examples as well asa novel application to control systems were also given

23 Measurement-based approach

This section recapitulates the details of the measurement based approachFor a linear electrical circuit operating at a fixed angular frequency ω consider the

case in which two elements of the circuit are the design variables The multilinear trans-formation (210) is rewritten here for convenience dividing all parameters by b12 whichis nonzero as will be proved in section 25

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

where m is a measurable variable like current or voltage the impedances z1 and z2 aredesign elements in the network All parameters can be written in terms of their real and

imaginary parts as followsm = p+ jq

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

The multilinear transformation relationship (211) can be rewritten as

a0 + a1z1 + a2z2 + a12z1z2 minus b0mminus b1mz1 minus b2mz2 = mz1z2 (213)

This implies that if measurements of m(n) are available for seven different values of[z(n)

1 z(n)2 ] n = 1 2 7 the constants a0 a1 a2 a12 b0 b1 b2 that depend on the whole

circuit can be found by solving the system of linear equations

1 z(1)1 z

(1)2 z

(1)1 z

(1)2 minusm(1) minusm(1)z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Replacing (212) in (214) and equating the real and imaginary parts we obtain in ab-breviated notation

Ax = p

where the matrix A is given by

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

a19 = minusp(1) a110 = q(1) a111 = x(1)1 q(1) minus r(1)

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

a29 = minusq(1) a210 = minusp(1) a211 = minusr(1)1 q(1) minus x(1)

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

a213 = minusr(1)2 q(1) minus x(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

in the same way the matrix A is built with the other six measurements The otherelements of the system of linear equations are given by

x =[ar0 ai0 ar1 ai1 ar2 ai2 ar12 ai12 br0 bi0 br1 bi1 br2 bi2

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

If A is a square invertible matrix the solution is

x = Aminus1p (215)

For a linear resistive network the functional dependence of any current (resp voltage) onany resistor can be determined with a maximum of 3 measurements of the pair [currentvalue (resp voltage value) corresponding resistor value] while the functional dependencebetween any current and any m resistances can be determined with a maximum of (2m+1minus1) measurements In particular if it is desired to determine the dependence of a branchcurrent on the resistance of the same branch then only two measurements suffice Thisis the well-known Theveninrsquos theorem that we will discuss in the next section

24 Theveninrsquos and Nortonrsquos theorem are particularcases of Bodersquos bilinear theorem

The celebrated Thevenin theorem can be derived as a special case of the bilinear transfor-mation [13] As it is well-known the network is reduced to the simple equivalent circuit in(Figure 23) Applying Bodersquos bilinear theorem (21) with the current i as a measurablevariable yields

Figure 23 Thevenin equivalent circuit for 1-port et corresponds to Thevenin or opencircuit voltage and zt z` are respectively the Thevenin and load impedances

i(z`) = a

c+ dz`(216)

From this expression by inspection i(0) = ac which corresponds to short circuitcurrent isc = i(0) obtained by setting z` to zero (ie short-circuiting the load terminals)The Thevenin voltage et is obtained multiplying both sides of (216) by z` to get

v(z`) = az`c+ dz`

By inspection v(infin) = ad ie the open circuit voltage et = voc = v(infin) Wheneverisc 6= 0 the Thevenin impedance is given by

zt = vocisc

d(218)

so that (216) can be rewritten as

i(z`) = etzt + z`

which is the Thevenin theorem The bilinear theorem approach makes it clear thatany two measurements suffice to find the parameters ad cd which are of course theparameters of the Thevenin equivalent circuit There is thus no need to measure shortcircuit current and open circuit voltage

For the Norton theorem we will analyze the voltage relationship from (21) given by

v(z`) = bz`c+ dz`

Using a procedure similar to the one above we see that v(infin) = bd so that the opencircuit voltage voc = v(infin) The short circuit current is obtained dividing both sides of(219) by z` and letting z` rarr 0 to obtain the short circuit current isc = bc leads to(218) so that (219) can be rewritten as

v(z`) = vocz`zt + z`

25 Linrsquos theorem applied to two-port networks

Consider a linear two-port circuit terminated with an uncoupled resistive load Supposingthat the resistances r1 and r2 are design variables to be chosen in order to transfermaximum total power P lowastT to the loads given by the expression (41) or (42) Current orvoltage mk can be expressed as a multilinear transformation in r1 and r2

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

Figure 24 A two-port network can be modeled by the two dependent source equivalentcircuit For a non-reciprocal network z12 6= z21

After modeling the two-port network as a two dependent source equivalent circuit (seeFigure 24) and following the algorithm introduced in [4 p226] the tableau equationformulation for the circuit can be written as

minusAgt I 00 M N

written in abbreviated notation asT x = b

where T is the tableau matrix on the left hand side of (222) x = (ev i) is thevector of unknown node voltages and branch voltages and currents respectively whileb = (00us) contains the independent source voltages and currents Solving (222) tofind the relationship between the currents i1 i2 and the resistances r1 r2 in Figure 24by Cramerrsquos rule applied to the linear system yields

i1 = e1z22 minus e2z12 + e1r2

z11z22 minus z12z21 + z22r1 + z11r2 + r1r2(223)

equating each expression with (220) and (221) respectively for i1 we obtain

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

for i2 we havec0 = e2z11 minus e1z21 c1 = e2 c2 = 0 c12 = 0

and the denominator is the same for the two currents

b0 = |Zt| = z11z22 minus z12z21 b1 = z22 b2 = z11 b12 = 1 (225)

Since numerator coefficients that correspond to resistor in the same branch where thecurrent was measured are zero This means that currents are given by

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

where ak bk ck are complex parameters and 5 measurements suffice to determine allof them The measurements in question are values of currents i1 i2 measured at fivedifferent points in the (r1 r2) space The system of linear equations is set up using (213)for each measurement and solving (215) all parameters are found (for more details seesection 23)

Using the same procedure to find the relationship between the voltages v1 v2 and theresistances r1 r2 (Figure 24) by Cramerrsquos rule applied to 222 yields

v1 = (e2z12 minus e1z22)r1 minus e1r1r2

equating each expression with the respective multilinear transformation (220) and (221)we obtain the same parameters in the denominator (225) The numerator coefficients inv1 are obtained as

a0 = 0 a1 = e2z12 minus e1z22 a2 = 0 a12 = minuse1

for v2 we havec0 = 0 c1 = 0 c2 = e1z21 minus e2z11 c12 = minuse2

In this case the numerator coefficients that multiply the resistor of same branch wherethe voltage was measured are not zero This means that voltages are given by

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

As before five measurements suffice to determine all parameters as is detailed in sec-tion 23

This solution can be extended for linear two-port circuit terminated by impedancesunder the same considerations above where the uncoupled load matrix Z` = diag (z1 z2)with zk = rk + jxk where rk and xk are the resistance and reactance of impedancerespectively So the multilinear fractional relationship between the current ik voltage vkand the impedances zkj (k 6= j) can be expressed as

ik(zk zj) = a0k + ajzjb0k + bkzk + bjzj + zkzj

vk(zk zj) = akzk + ajkzjzkb0k + bkzk + bjzj + zkzj

Once again the coefficients that occur in (232) and (233) can be determined by makingmeasurements of ikj or vkj for five different values of zkj

Chapter 3

The maximum power transfer(MPT)

The maximum power transfer theorem in circuit theory has a long history being knownfor linear 1-port since the nineteenth century A linear 1-port containing independentsources at a single frequency can be characterized by

v = et minus zti

where zt and et are respectively the Thevenin equivalent impedance and voltage sourceand v and i are respectively the port voltage and current Supposing that a variablepassive impedance is connected at the port to extract power from the 1-port then it iswell known (see [4]) that (i) an impedance load z` will extract the maximum power whenz` is equal to the conjugate of zt assuming that Rez gt 0 and (ii) that a resistive load r`

led to better understanding of the problem which is inherently more difficult than the1-port problem in part because solutions are no longer unique Nambiarrsquos result [5] forlinear RLC n-port characterized by v = etminusZti where the lower case (resp uppercase)letters have the same interpretation as the 1-port case except that they now denotevectors (resp matrices) is that maximum power transfer occurs when Z` = Zt wherethe overbar denotes complex conjugation Later Desoer [6] gave an elegant demonstrationof the most general condition for nonreciprocal n-port and coupled load impedances (iewithout any assumptions on the structure of the load matrix Z`) Flanders [7] derivedcomplete results for the case of arbitrary resistive load n-port including the diagonal (oruncoupled) case of a single resistor terminating each port Finally Lin [8] called attentionto the fact that a ldquomuch more difficult problem however is the case when the loads are

n uncoupled resistors and the total power delivered to these resistors is to be maximizedrdquoThis chapter will revisit both the general case as well as the resistive case with theobjectives of preparing the ground for the application of a multilinear form theorem aswell as obtaining some new results for the uncoupled resistive case

31 Derivation of the maximum power transfer theo-rem for general n-port

In order to motivate the general approach proposed by Desoer [6] and point out a subtletyin this approach the 1-port resistive case is presented first since the nature of the problemis already evident in this simple case For a 1-port represented by its Thevenin equivalentsource et and resistance rt connected to a resistive load r` the following equations hold

r`i = et minus rti (31)

The expression for the power P dissipated in the load can be written as

P = i2r` = ir`i (32)

Substituting (31) into (32) yields

P = iet minus irti (33)

In order to find extrema of the power the partial derivative of the power P with respectto the current is set to zero

parti= et minus 2irt = 0 (34)

which yields the extremizing current i as

i = et2rt

which is of course the current that maximizes the quadratic form (33) in terms of rtSince rt is fixed (and is not the variable of interest r`) it is necessary to substitute (35)into the constraint (31) to solve for r`

r`et2rt

= et minus rtet2rt

which simplifies tor` = rt (37)

recovering the familiar result that the maximizing load r` must equal the Thevenin resis-tance rt

The usual textbook derivation (see for example [12]) calculates i explicitly in orderto eliminate it from the expression for P which becomes P =

)2r` which is then

differentiated with respect to r` to obtain (37) The alternative derivation detailed in thesimple case above shows that the maximum power transfer problem can also be viewedas that of minimizing a quadratic form in the current (33) subject to a constraint(31) Furthermore it is seen that the result can be expressed in terms of the Theveninequivalent resistance from which the load resistance which maximizes power transfer iscalculated This approach is in fact the one generalized by Desoer [6] to obtain thegeneral n-port maximum power transfer theorem which is now stated and derived

Figure 31 An n-port N represented in its Thevenin equivalent form connected to aload n-port N` where i isin Cn is the current vector et isin Cn is the Thevenin equivalentvoltage and Zt isin CntimesnZ` isin Cntimesn are respectively the Thevenin equivalent and loadimpedance matrices

For the circuit in Fig 31 Desoer [6] proved the maximum power transfer theoremfor n-port Some preliminaries are needed in order to state the theorem

The basic equations for the n-port with Thevenin representation given by et Zt andterminated by load n-port Z` are as follows

v = et minusZti (38)

v = Z`i (39)

Equating (38) and (39) givesZ`i = et minusZti (310)

which can also be written as(Z` + Zt)i = et (311)

The so called solvability condition namely Z`+Zt nonsingular that guarantees the well-posedness (unique solvability of current vector) is assumed to hold whenever relevantThe average power is given by (using effective or rms values for phasors)

P = Revlowasti (312)

= 12(vlowasti + ilowastv) (313)

Substituting 39 into 313 the average power dissipated in Z` is given by the expression

P = 12ilowast(Z` + Zlowast` )i (314)

Substituting (310) to eliminate Z` in (314) gives

P = 12[ilowastet + elowastt iminus ilowast(Zt + Zlowastt )i] (315)

The maximum power transfer theorem for n-port can now be stated as follows

Theorem 31 [Desoer [6]] For given etZt

D1 If Zt + Zlowastt is positive definite then there exists a unique maximizing load currentim which is the solution of

et minus (Zt + Zlowastt )im = 0 (316)

Furthermore the (nonunique) load Z` draws maximum average power from the n-port characterized by etZt if and only if

Z`im = Zlowastt im (317)

and the maximum power drawn by the load n-port N` from the n-port N is givenby

P = 12elowastt im or equivalently (318)

P = 12elowastt (Zt + Zlowastt )minus1et (319)

D2 If Zt + Zlowastt is positive semidefinite and et isin C(Zt + Zlowastt ) where C(A) denotes thecolumn space of the matrix A and P attains the same maximum value as in (318)

D3 If Zt + Zlowastt is positive semidefinite and et 6isin C(Zt + Zlowastt ) then P can take arbitrarilylarge values

D4 If Zt + Zlowastt has at least one negative eigenvalue then P can take arbitrarily largepositive values (ie is unbounded)

We will give a simple derivation of the maximum power transfer theorem for linearn-port networks with a general load Z` based on the work of [6 7] since it will motivateour development for a purely resistive load R` which is the focus of this dissertation

Proof The derivation requires the analysis of (315) under all the possible hypotheses onZt + Zlowastt We will start with the proof of item D2 since it will lead to the proof of itemD1 as wellProof of D2 For item D2 the hypothesis is that the matrix Zt + Zlowastt is positive semidef-inite and furthermore et isin C(Zt + Zlowastt ) which means that there exists (at least one)vector k such that

et = (Zt + Zlowastt )k (320)

Substituting this expression for et in (315) and completing the square the expressionfor power P can be written as

P = 12[klowast(Zt + Zlowastt )k minus (iminus k)lowast(Zt + Zlowastt )(iminus k)] (321)

Since Zt + Zlowastt is positive semidefinite the second term is always nonpositive hence

P le 12klowast(Zt + Zlowastt )k (322)

Clearly equality can be attained in (321) if it is possible to choose im such that imminusk isinN (Zt + Zlowastt ) ie

(Zt + Zlowastt )(im minus k) = 0

which in view of (320) can be rewritten as

(Zt + Zlowastt )im = et (323)

Comparing (311) (in which i is replaced by im) and (323) leads to a relation that mustbe satisfied by a maximizing current vector im

Summing up we have just proved that if Zt+Zlowastt is positive semidefinite and there exists

im such that et = (Zt + Zlowastt )im then the maximum power transfer P to the load Z` andthe maximizing current satisfy

P = 12ilowastm(Zt + Zlowastt )im (325)

thus completing the proof of item D2Proof of D1 The proof just given clearly also works if Zt + Zlowastt is positive definite Inthis case im is unique and (325) (326) can be replaced by

im = (Zt + Zlowastt )minus1et (327)

Proof of D3 This is the case in which et 6isin C(Zt + Zlowastt ) From 315 it follows that

parti= 1

2[et minus (Zt + Zlowastt )i] + [et minus (Zt + Zlowastt )i]lowast (329)

Thus if et 6isin C(Zt + Zlowastt ) (329) shows that P cannot have a stationary point In factP can take arbitrarily large values Suppose that rank(Zt + Zlowastt ) = k lt n Then sinceZt + Zlowastt is Hermitian we can find an orthonormal basis (q1 qn) (assume that thesevectors form the columns of a matrix Q) such that the first k vectors span the columnspace of Zt + Zlowastt and the remaining n minus k vectors form a basis for its nullspace Sinceet 6isin C(Zt + Zlowastt ) the representation of et in this new basis denoted (e1 e2) must havee2 6= 0 Furthermore in this new basis

Zt + Zlowastt = Z1 0

This means that if we choose Z` = Zlowastt at least one port ` for ` gt k will be in short circuitmode with a nonzero source (because e2 6= 0) thus proving that P can be unboundedProof of D4 In this case the matrix Zt + Zlowastt is indefinite Let iminus be a unit vectorcorresponding to a negative eigenvalue λminus of Zt + Zlowastt (at least one exists since thismatrix is being assumed indefinite) ie (Zt+Zlowastt )iminus = λminusiminus with λminus lt 0 and iminus = 1The argument made in [6] is that by choosing i = αiminus αrarrinfin P rarrinfin

Remark If we consider resistive loads items D3 and D4 of Theorem 31 no longerhold as we will show in the section 33

32 Illustrative examples for MPT with complex loadimpedances

This section gives 2-port examples for each case that occurs in Theorem 31 above

Example 32 Illustrating item D2 of Theorem 31 given et = (1 0) Zt = 1 j

Zt+Zlowastt ge 0 We obtain P = 14 with Z` = diag 0 1 and im = 1

2(1minusj) For this examplewe show non-uniqueness of the load impedances that draw maximum power by findingother solutions that satisfy (323) Here C(Zt + Zlowastt ) = span(1 0) and N (Zt + Zlowastt ) =span(0 1) are respectively the column space and the null space of Zt + Zlowastt Choosing

η isin N (Zt + Zlowastt ) as η = (α+ jβ) 0

from (323) (Zt + Zlowastt )(im + η) = et is satisfied

On the other hand from (311) (Zt + Z`n)in = et where in = im + η and Z`n =diag (r1 + jx1 r2 + jx2) Solving for in we get

r1 = 2β x1 = minus2α α = x2

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

Choosing z2 = 1 + j2 yields z1 = 25(2 + j) and in =

(12 minus

(1 + j 1

)) which is also a

solution for (311) and draws the maximum power

Example 33 Item D3 of Theorem 31 is exemplified for the specific case when Zt+Zlowastt

is the zero matrix Given et = (1 1) Zt = minusj j

with Z` = diag infinminusj + r we

obtain P rarrinfin as r rarr 0

Example 34 Illustrating item D4 of Theorem 31given et = (1 1) and Zt = j 1minus j1minus j j

for which Zt + Zlowastt = 0 2

is indefinite with eigenvalues minus2 2

We obtain P rarrinfin with Z` = diag infinminusj Figure 32 shows the equivalent T-model forthis example

33 MPT for the case of uncoupled resistive loads

This section applies the results on maximum power transfer due to Flanders [7] Lin [19]and Sommariva [20] in order to compare them with the proposed approach

In [19] Lin gives the following theorem to calculate the maximum power for resistivemultiport loads

-1-2j -1-2ji1

Figure 32 T-network model for Example 34 in which Zt + Zlowastt is indefinite and th loadZ` = infinminusj Intuitively the total power P lowastT takes large values because there existactive elements in the circuit

Theorem 35 [Lin [19]] Assume that the port voltage v is written in terms of the n-portwith Thevenin representation given by the resistance matrix Rt and the equivalent voltageet as

v = Rtiminus et (331)

where Rt is real symmetric and positive semi-definite and suppose that is connectedto an uncoupled resistive load R` = diag (r1 rn) such that rj gt 0 forallj A necessarycondition for the maximum available power P lowast to be obtained with some Rt is that

(RtRminus1` minus 1)e = 0 (332)

Specifically for a resistive 2-port network Lin defines the following sets of so calledboundary load resistances that are candidates for drawing the maximum power P

P1 = maxP |r1 gt 0 r2 = 0P2 = maxP |r1 = 0 r2 gt 0P3 = maxP |r1 gt 0 r2 rarrinfinP4 = maxP |r1 rarrinfin r2 gt 0

The maximum available power P can be obtained in the interior (ie r1 gt 0 r2 gt 0)and in case it occurs on the boundary Lin proves that

P = maxP1 P2 P3 P4 (334)

To determine P for a DC two-port network Lin proposes to solve (332) and then deter-minate which one of the following three cases occurs

Case 1 There is no solution to (332) with r1 gt 0 and r2 gt 0 Thus P must be

obtained on the boundary Hence we calculate P from (334)Case 2 Expression (332) and the power Pi have a unique solution with r1 gt 0 and

r2 gt 0 Therefore we have P = Pi The author mentions that in this case ldquoa rigorousmathematical proof is very complicatedrdquo

Case 3 Expression (332) has multiple solutions with r1 gt 0 and r2 gt 0 Howeverfor a 2-port uncoupled resistive load the solutions of (332) will be limited under one ofthe following circumstances

1 If e1 = 0 e2 = 0 then P = 0

2 If r12 = 0 e1 = 0 or e2 = 0 then the network is reduced to 1-port problem and thepower will be P = e2(2r0)

3 r11 = r22 = r12 = r0 e1 = e2

4 r11 = r22 = minusr12 = r0 e1 = minuse2

For the last two circumstances the expression (331) can be satisfied with a uniqueresistance r0 and one voltage source e1 Consequently we obtain P = e2

1(2r0) andr0 = r1r2

In [7 pp336] Flanders proves the following theorem to calculate the maximal powerfor an uncoupled load

Theorem 36 [Flanders[7]] Suppose Zt+Zlowastt is positive semi-definite Zt is non-singularand et is given Set

P = elowasttCet where C = (R` + Zt)minus1R`(R` + Zlowastt )minus1 (335)

and the matrix for load resistances R` = diag (r1 rn) rj gt 0 Then maximal poweris given by

P = supR`

elowasttCet = supv 6=0

|elowasttv|2

vlowast(Zt + Zlowastt )v + 2sum |vi||(Zv)i|(336)

This result is based on Theorems 1 and 8 proved in the same paper It shouldbe highlighted that equation (336) is quite a complicated expression to calculate themaximal power Furthermore the argument which maximizes P is v from which it isnot easy to determine the value of R` directly which is in fact the variable in which weare interested

Sommariva [20] develops a theorem for MPT for DC linear two-port after carrying outextensive calculations For the case of concern (331) is rewritten here for convenience

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

The following assumptions are made in [20] on the two-port

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

r11 gt 0 det Rt ge r2d rd = |r12 minus r21|

2 (340a)

An alternative formulation of (340a)

r11 gt 0 r2b minus r2

m ge 0 rb = radicr11r22 rm = r12 minus r21

2 (340b)

Assumptions on load in [20] are as follows

1 Uncoupled resistive load ie

R` = diag (r1 r2) (341)

2 Passivity ie0 le r1 le infin 0 le r2 le infin (342)

Unique solvability constraint

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

Under assumption (343) and by Kirchhoffrsquos laws

i`1 = minus(e1r2 + r22e1 minus r12e2)D(r1 r2) (344)

Total power supplied to load resistors is

PT = i2`1r1 + i2`2r2 (346)

Theorem 37 [Sommariva [20]] Under the hypotheses (337)-(343) above if

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

then the load resistances (r1 r2) that globally maximize load dissipated power PT aregiven by

r1 = r11 + r21r11e2 minus rme1

r22e1 minus rme2(350)

PT = r22e21 minus 2rme1e2 + r11e

4r2b minus 4r2

331 Modifications to Desoerrsquos theorem for uncoupled resistiveloads

In this section we show that items D3D4 of Desoerrsquos Theorem 31 do not necessarilyhold for the case of purely resistive loads as was realized by Flanders [7] and give somenew lemmas for this case

Lemma 38 If et 6isin C(Zt) then P is unbounded

Proof [by contradiction] From (312) P can be unbounded if one or both of v i areunbounded From (38) since et Zt are fixed (and bounded) we see that if i is unboundedthen so is v and vice versa We conclude that if P is bounded both i and v must bebounded Now choose R` = diag (r1 rn) and let rk rarr 0 forallk Then (310) can berewritten as

et minusZti = R`i

As i is bounded we have

et minusZti = 0 (353)

et = Zti (354)

In this way et isin C(Zt)

Remark If et isin C(Zt + Zlowastt ) and et isin C(Zt) the power dissipated P may still beunbounded if Lemma 39 holds

Figure 33 T-network model for Example 311in which P is unbounded with R` = (0 0)

Lemma 39 If (Zt)kk = 0 and (et)k 6= 0 then P is unlimited

Proof Choose R` = diag (r1 rn) and let rk rarr 0 rj rarrinfin j 6= k

We summarize this discussion in the following Proposition

Proposition 310 For the n-port given by Zt and terminated by load n-port R` ifet isin C(Zt + Zlowastt ) and neither Lemma 38 or 39 holds then P is bounded

Lemmas 38 and 39 are illustrated in the following examples

Example 311 To illustrate Lemma 38 choose Zt = j minusjminusj j

which is singular

and et = (1 1) We obtain P rarr infin with R` = diag (0 0) Figure 33 illustrates theT-network model where the equivalent reactance is null and each branch has a sourcevoltage with a resistor in series (r1 r2 rarr 0)

Example 312 To illustrate Lemma 39 choose Zt = 1 j

which is nonsingular

and et = (1 1) We obtain P rarr infin with R` = diag (infin 0) Figure 34 illustrates theT-network model where the right branch has a source voltage with a resistor r2 rarr 0 inseries and the equivalent reactance is null

To show that item D4 of Theorem 31 does not hold for an uncoupled resistive load itis enough to revisit Example 34 in which with R` = diag 224 0 P now attains thefinite value of 06W

1-j -ji1

Figure 34 T-network model for Example 312 in which P is unbounded with R` =(infin 0)

Chapter 4

Application of Linrsquos theorem toMPT

In this chapter we present one of the main contributions of this dissertation namely thecalculation for a given n-port N of the total dissipated power when N is terminated byn uncoupled load impedances zi i = 1 n (ie the load n-port is a diagonal matrixdiag (z1 zn) In symbols we calculate total dissipated power PT (z1 zn) whichis a function of the n load impedances zi i = 1 n This is done by using each portvoltage or current and the value of the corresponding load impedance where the formerare calculated as multilinear functions of the load impedances as explained in Chapter2 Since the resulting expressions for PT (z1 zn) have a standard and fairly simpleform we are able to use standard optimization software to maximize PT and indeed tofind the arguments z1 zn that maximize PT Several illustrative examples are given

Exemplifying for a linear 2-port terminated by uncoupled resistive loads (see Figure41) the total power PT dissipated in the loads is given by the expressions

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

where the currents i1 i2 are multilinear functions given by (211) obtained with themeasurement-based approach The maximizing resistive loads (r1 r2) and the corre-sponding maximum total power PT can be found by solving the following optimizationproblem

maxr1r2isinR

PT (r1 r2) (43)

For the examples presented in the next sections the maximization is carried out using

Linear

Two Port

Network

Figure 41 Two-port network terminated in resistances The resistors r1 r2 are designelements and adjust to draw the maximum total power in two-port The remainingelements of the circuits are unknown it is a black box

the fmincon function in the Matlabr Optimization Toolbox

41 Illustrative examples of MPT for two-port

In this section we will explore different 2-port examples represent by the basic equation(38) rewritten here for convenience

v = Ztiminus et (44)

where i isin C2 is the current vector et isin C2 is the Thevenin equivalent voltage and Zt isinC2times2R` = diag (r1 r2) are respectively the Thevenin equivalent and load impedancematrices Some examples are taken from the literature and the results are comparedhere

Example 41 In this example the reciprocal two-port network (z12 = z21) is characterizedby v1

minus 10

Assume that the currents i1 and i2 can be measured while the resistors r1 r2 are variedThen the multilinear transformation can be obtained with five measurements as wasexplained in the previous section So the currents (226) and (227) yields

i1(r1 r2) = minus40minus 10r2

44 + 10r1 + 8r2 + r1r2

substituting i1 i2 in equation (41) leads to

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

Using the Matlabr routine fmincon a maximizer (r1 r2) of the power function is foundThe entire power surface is plotted in Figure 42

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

The contour lines of power function (46) are shown in Figure 43 Additionally inFigure 44 we plot the surface of total power as a function of two variables (v1 v2)where the voltages can be written as a multilinear transformation from (45) becausevk = ikrk Then we have

v1(r1 r2) = minus40r1 minus 10r1r2

44 + 10r1 + 8r2 + r1r2

Note that the maximum available power PT is obtained for positive values of r1 r2 iein the interior of search space (r1 ge 0 r2 ge 0) We have recovered Linrsquos results [19pp386] with the proposed measurement-based method Linrsquos solution consists of findingthe maximizing resistors from (332) and carrying out some additional calculations toobtain the power Finally according Case 2 in section 33 Lin concludes that there is aunique solution for the power The proposed approach obtains the maximizing resistancesand the corresponding power directly without solving equations such as (332) or case bycase analyzes as in (333)ff

Example 42 Given the non-reciprocal two-port network (z12 6= z21) v1

and following the same procedure as in Example 41 above PT is obtained in the interiorFurther calculations give

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistor r2[Ω]Resistor r1[Ω]

Figure 42 Surface plot of total power function PT (r1 r2) for resistive reciprocal two-portcircuit of example 41 The point marked lsquotimesrsquo represents the maximum power

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

Figure 43 Contour lines of total power function PT (r1 r2) for the resistive reciprocaltwo-port circuit of example 41 The point market lsquotimesrsquo represents the maximum powerthe level sets of PT are nonconvex

The total power is given by the next expression and its respective surface plot is shownin Figure 45 Figure 46 shows the contour lines of total power function identifying the

|Pt|[W

Voltage v1[V ]Voltage v2[V ]

Figure 44 Surface plot of total power function PT (v1 v2) as a function of port voltagesv1 v2 for a resistive reciprocal two-port circuit of example 41 Notice that the maximalpoint occurs (v1 = v2 = 5V ) represented by lsquotimesrsquo

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

Solving (43) for (47) we obtain the maximizers resistor for the power

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

We now use Sommarivarsquos Theorem 37 summarized in section 33 Hypotheses (337)-(343) are satisfied and conditions (347)-(349) of the theorem give the following results

r1 = 5 + 12

(15minus 4rm8minus 3rm

)r2 = 2 + 8minus 3rm

15minus 4rmP = 32minus 24rm + 45

4r2b minus 4r2

where rm = 34 rb =

radic10 are given by (340b) Substituting in previous equations the

results is the same that in (48) Hence the proposed method gives the same results

|Pt|[W

Figure 45 Surface plot of total power function PT (r1 r2) for non-reciprocal two-portcircuit of example 42 The point marked lsquotimesrsquo represents the maximum power

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

Figure 46 Contour lines of total power function PT (r1 r2) for a non-reciprocal two-portcircuit of example 42 The point marked lsquotimesrsquo represents the maximum power The levelsets are nonconvex

numerically that in [20] but we can also plot power surfaces as a function of any twoport variables

Example 43 Given the circuit in Figure 47 at fixed frequency of 60Hz suppose thatthe inductance l1 and the resistance r2 are to be chosen to achieve a maximum totalpower PT at the output of two-port Clearly in this specific case the active total powerdepends only on current i2

PT (r2) = |i2|2r2 (49)

Figure 47 AC circuit with two design elements inductor l and resistor r2 The rest of theelements are assumed to be unknown (vin = 10V rms c = 025mF r = 3 Ω ω = 2πf)

As discussed in section 25 five measurements of current i2(ω) corresponding to fivedifferent values of the impedances z1(ω) = jωl1 and z2 = r2 and the solution of a linearsystem of equations (232) yield the following multilinear transformation

i2(l1 r2) = 30minusj3183 + (1061 + j3)ωl1 + 3r2 + jωl1r2

Substituting (410) in (49) we obtain

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

Once again the maximizing loads (l1 r2) are found with Matlabrrsquos fmincon functionThe surface plot in Figure 48 and contour lines in Figure 49 for the total power function(411) show a unique maximum point

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Resistor r2[Ω]Inductance l1[H]

Figure 48 Surface plot of total power function PT (l1 r2) for two-port AC circuit ofExample 43 Fig 47 The point marked lsquotimesrsquo represents the maximum power

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

Figure 49 Contour lines of total power function PT (l1 r2) for two-port AC circuit ofExample 43 Fig 47 The level set are convex The point marked lsquotimesrsquo represents themaximum power

Note that this example in which the load n-port has one inductor and one resistancecannot be treated using the results of Flanders [7] Lin [19] or Sommariva [20] which alltreat the case of a purely resistive load n-port

In order to check these results suppose all circuitrsquos elements are known then fromKirchhoffrsquos law applied to the circuit in Figure 47 the current is given by

i2(l1 r2) = vinr

rr2 + l1cminus1 + j(ωl1(r2 + r)minus r(ωc)minus1) (414)

Substituting all circuit parameter values in (414) and comparing with (410) we canconclude that these expressions are equal

Example 44 The reciprocal complex two-port network is characterized as follows v1

Assuming that this circuit is terminated by uncoupled resistors r1 r2 and following theproposed measurement-based approach the functional dependency of port currents on theload resistors is given by

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

i2(r1 r2) = 1minus j + r1

1 + j + jr1 + r2 + r1r2

Substituting the currents i1 i2 in equation (41) yields

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

Maximizing (416) we obtain PT = 05W for

(r1 r2) = (infin 1 Ω)

Figures 410-411 show respectively the total dissipated power (416) surface and contourplot indicating the maximum point (r1 r2)

In [7] this example is solved and we reproduce all details for comparison with ourmethod Flandersrsquos solution [7 p334ff] is as follows

elowastt (R` + Zt)minus1 = 1∆elowastt

r2 + j minusjminusj r1 + 1

= 1∆(r2 r1)

|Pt|[W

Figure 410 Surface plot of total power function PT (r1 r2) for two-port reactive circuitof Example 44 The point marked lsquotimesrsquo represents the maximum power

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

Figure 411 Contour lines of total power function PT (r1 r2) for two-port reactive circuitof Example 44 The point marked lsquotimesrsquo represents the maximum power

where ∆ = det(R` + Zt) = ((r1 + 1)r2 + 1) + j(r1 + 1) Thus from (335)

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

By the arithmetic-geometric mean inequality

r1 + r2+ r1 + r2

r1r2ge 2

with equality iff r1r2 = r1 + r2 Thus Pminus1 gt 2

Pminus1 rarr 2 + as r1 rarrinfin and r2 rarr 1minus

Thereforer1 rarrinfin r2 = 1 P = 1

2Example 45 Given the following complex two-port network v1

= minusj j

As in the previous examples the uncoupled load matrix is denoted as R` = diag (r1 r2)and following the same procedure we obtain the multilinear transformation with the pro-posed measurement-based approach as

i1(r1 r2) = r2

2 + jr1 minus jr2 + r1r2

i2(r1 r2) = minus2j + r1

Substituting the currents i1 i2 in equation (41) the total power is

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

Using fmincon to maximize (418) it turns out that the maximum total power is obtainedin the boundary

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

The surface plot of total power (418) is shown in Figure 412 and contour lines for thefunction PT (r1 r2) are shown in Figure 413

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

Figure 413 Contour lines of total power function PT (r1 r2) for two-port reactive circuitof Example 45 The level sets are nonconvex The point marked lsquotimesrsquo represents themaximum power

This example is also solved by Flanders [7 p336] who obtains the same solution but

notes that it is hard to prove directly From the previous example (417) can be rewritten

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Flanders says ldquoa hill climbing search suggests Pm = 1 for (r1 r2) = (0 2)rdquo and hisanalyzes is now reproduced By Theorem 36

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

where f is homogeneous of degree 0 and f(0 1) = 12 lt f(1 1) = 1 Hence to maximize f

Flanders takes v1 = 1 and v2 = y so it is necessary to prove

|1 + y|2

2(|y minus 1|+ |y||1 + y|) le 1 that is |1 + y|2 le 2|y minus 1|+ 2|y||1 + y|

with equality only for y = 1 Briefly if |y| ge 1 then

|1 + y| le 1 + |y| le 2|y| |1 + y|2 le 2|y||1 + y| le 2|y minus 1|+ 2|y||1 + y|

If |y| le 1 then |1 + y| le 1 + |y| le 2 and

|1 + y| = |(1minus y) + 2y| le |1minus y|+ 2|y|

|1 + y|2 le |1 + y||1minus y|+ 2|y||1 + y| le 2|1minus y|+ 2|y||1 + y|

This proves the inequality for all zClearly the algebraic-analytic approach requires considerable ingenuity for each prob-

lem as apposed to the proposed numerical approach

Example 46 The following complex two-port network also is studied in [7Ex13p331ff] v1

= 1minus j j

For this reciprocal network we develop the same procedure as in above examples Withthe measurement-based approach the multilinear transformations are given as follows

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

Substituting the currents i1 i2 in equation (41) the total power is given by

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

Maximizing (419) with fmincon the maximal total power PT is obtained in the boundary

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

Surface plot of total power (419) is shown in Figure 412 Additionally the contour plotof the function PT (r1 r2) in (419) is shown in Figure 415

|Pt|[W

Figure 414 Surface plot of total power function PT (r1 r2) for the two-port reactivecircuit of Example 46 The point marked lsquotimesrsquo represents the maximum power

In [7] Flanders uses his result (Theorem 36) to find from (335)

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

Definingf(v1 v2) = |v1 + v2|2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Flanders takes v1 = 1 and v2 = z = x+ jy From (336)

Pm = f(1 1 + j) = 52(1 +

radic10)asymp 060063

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

Figure 415 Contour lines of total power function PT (r1 r2) for the two-port reactivecircuit of Example 46 The level sets are nonconvex The point marked lsquotimesrsquo representsthe maximum power

For this v2 we have

Zv = 0minus1 + j2

By [7 Theorem 8] yields r1 = 0 and r2 =

radic25 which is a typographical error the correct

result is r2 =radic

52 asymp 1581

The presentation of each one of six examples above explored by different authors in[7] [20] and [19] allows the conclusion that proposed approach provides a relatively simpleand standard functional expression for the maximum total power PT in terms of the cor-responding uncoupled load matrix valid for all possible load impedances not necessarilypurely resistive The advantage of finding an algebraic expression for the total dissipatedpower as P (z1 zn) or P (r1 rn) or P (v1 vn) is that design constraints caneasily be incorporated Specifically this addresses the concerns of McLaughlin amp Kaiser[9] if a battery impedance zb is given then the problem of choosing an adequate loadimpedance z` should really be approached by studying the surface P (zb z`) or P (zb v`)in order to impose restrictions on zb z` etc

Chapter 5

Power systems applications

In the previous chapter we discovered that is possible to find the maximum power trans-fer in two-port linear networks calculating the two maximizing resistive uncoupled loadsIn this chapter we will evaluate the performance of the proposed approach on an im-portant application in power systems which involves estimating the Thevenin equivalentparameters as well as the maximum power transfer limits

Voltage instability is an undesirable phenomenon that for complex load impedancesthreatens the operations of many power systems For this reason different methods havebeen developed to determinate the maximum loadability of a bus area or system Thephasor measurement unit (PMU) is a cheap device designed to get real-time measure-ments of system variables such as phase and voltage Corsi et al [21] point out thatThevenin equivalent received considerable attention in the analysis of voltage instabilityFor example an approximate approach for online estimation of maximum power transferlimits is presented in [22 23] This chapter shows that the proposed approach is capa-ble of obtaining the results in [22 23] in a simple manner without making any of theapproximations proposed in the cited papers

51 Thevenin equivalent parameters frommeasurement-based approach review

In this section we will present the results obtained for different kinds of power systemswhere the Thevenin equivalent parameters ldquoseenrdquo from the design element k are deter-mined The measurement-based approach is used to calculate all parameters of the Bodersquosbilinear transformation (21) for a measurable variable mk (current or voltage) which is

rewritten here for convenience

mk(xk) = a0k + akzkb0k + zk

Specifically at a fixed frequency ω the port current can be written in abbreviated nota-tion as

ik(zk) = a0k

b0k + zk (52)

and the port voltage asvk(zk) = akzk

b0k + zk (53)

The implicit forms of (52) and (53) are respectively

ikzk = a0k minus b0kik (54)

vkzk = akzk minus b0kvk (55)

Equation (52) is the same as (216) of section 24 and the Thevenin impedance (218)is rewritten here for convenience as

zt = b0k (56)

The maximum total power is found by solving the following optimization problem

maxzkisinCPTk

(zk) (57)

PTk(zk) = |ik|2 Rezk or (58)

PTk(zk) = |vk|2 Re

is the total active power draw by port k zk is the port impedance and ik or vk are givenby bilinear transformations (51)

In addition for the following examples we will verify the matching condition (37)rewritten here as follows

z` = zlowastt (510)

511 Simple linear circuit

In this subsection we will analyze two different cases for the circuit in Figure 51 Firstall elements of the circuit are known and the impedance z2 is variable In the secondexample there are two variable elements z1 and z2 that vary together in accordance witha common scaling factor micro In both cases the objective to find the design elements thatresult in maximum power transfer in the circuit Additionally in the first example weidentify the Thevenin impedance ldquoseenrdquo by z2

Figure 51 Simple linear circuit For example 51 z2 is a variable impedance and therest of the circuit parameters are assumed to be known (z0 = j01 Ω z1 = 1 Ω) Forexample 52 z1 z2 are chosen as z1 = z2 = 4microz0 where micro is the variable scaling factor

Example 51 To identify all parameters of (52) where k = 2 for this example mea-surements of the current i2 are needed for two different values of the impedance z2 Thenequating the real and imaginary parts on both sides of (54) and solving the linear systemof equations yields

i2(z2) = minus1 + j01001 + j02 + z2

Substituting (511) in (58) where z2 = r2 + jx2 yields

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

We wish to find the maximizing load z2 of the power function ie to solve (57) with PT2

as in (512) and using fmincon we get

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

Considering the impedance to be purely resistive (z2 = r2) we obtain

r2 = 02 ΩPT2 = 236W

Equation (513) shows that the impedance matching condition (510) is satisfied becausefrom (56) we have zt = 001 + j02 Ω Equation (514) shows that for a resistive loadthe power is less than of 10 of the power drawn with the maximizing impedance load z2

Plotting the active power PT2 and the magnitude of current i2 we can identify thepoint of maximum power transfer (see Figure 52) which occurs for r2 = 02 Ω

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

Figure 52 Magnitude of current and active total power with respect to r2 for Exam-ple 51

Comparing with the results in Li et al [23] we have found the same equivalentimpedance to draw maximum power transfer without making any approximation Li etal affirm that their approach is valid only if z1 is more than 3z0

Example 52 In this example we reproduce another result by the same authors [22] Inthis case for the circuit in Figure 51 the load impedances are increasing using a commonscaling factor micro (z1 = z2 = 4microz0) and the goal is to find the optimal scaling factor tomaximize the total active power PT in the network given by (41) (in which r is replaced

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

Observe that for the simple circuit in Fig 51 the choice z1 = z2 = 4microz0 means that wecan work with a bilinear transformation in the single variable micro

The load currents can thus be expressed by the bilinear transformations in z1 = z2 =microz0

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

To find all parameters it is necessary to make measurements of currents i1 i2 for threedifferent values of scaling factor micro Thus the bilinear transformation is given by

i1(micro) = 0487 + 0003microz0

0578 + microz0

i2(micro) = 0293minus 0009microz0

1054 + microz0

Maximizing (515) with above currents and z0 = 1 we obtain (see Figure 53)

micro = 06317 z1 = z2 = 25269

PT = 04855W

This result differs from the one obtained in [22] since there is a typographical errorin equation (23) in [22] In fact the correct load currents are

= 1 + 8micro

4micro

z0((1 + 4micro)(2 + 4micro)minus 1)

Correcting the mistake the same answer is obtained Using our proposed approach thistype of problem is solved by maximizing (515) with the respective currents given in thestandard format (bilinear transformation) (516) that is simpler than solving the follow-ing two expressions given in [22]

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL + ZLL(Zr

L)minus1Z+L + ZLL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

Scaling factor micro

Figure 53 Total active power PT as function of scaling factor micro for Example 52

where + denotes the conjugate transpose and ZrL is the real part of uncoupled load

impedance matrix ZL

512 IEEE 30-bus system

In this section the well-known IEEE 30-bus shown in Figure 54 is analyzed In theIEEE 30-bus system some equality and inequality constraints are imposed because thephysical system must respect bounds on voltages and generator capacities However forthe first example all constraints will be ignored in order to obtain the equivalent circuit(see Figure 23) and the load that achieves maximum power transfer

In general the procedure is very similar to the one developed for the cases abovewith the difference that here we will work with the total complex power ST defined asfollows

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

where PT and QT are respectively the total active and reactive power and |ST | is thetotal apparent power Now we can rewrite the optimization problem for each bus k

Figure 54 IEEE 30-bus test system

which has a load impedance zk as follows

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

and the voltage vk(zk) is given by (53)The system parameters can be found in Matpower [24] which is a Matlabr simulation

package for solving power flow problems In order to calculate the parameters of thebilinear transformation two voltage measurements vk for each bus k are necessary HenceMatpower is used for two different values of the real power demand Pk which is equivalentto varying the real part of the load impedance zk at bus k Equating the real andimaginary parts on both sides of (55) and solving the linear system of equations forbuses 3 4 and 5 we get

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

The Thevenin impedance (56) for buses 3 4 and 5 yield

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

Solving (521) for each expression in (523) and the remaining buses we obtain the resultsshowed in Table 51 Comparing the Thevenin impedances (524) with Table 51 for thebuses 34 and 5 we conclude that the impedance matching condition (510) is satisfied

Table 51 Load impedances complex and apparent power for each load bus of IEEE30-bus system without voltage constraints

Bus No z`(pu) ST (pu) |ST |(pu)3 0019 - j0095 12857 + j65239 664944 0015 - j0091 16533 + j103761 1050705 0035 - j0161 6880 + j31539 322817 0032 - j0140 7298 + j32411 332228 0022 - j0126 9727 + j55429 5627610 0008 - j0231 29342 + j828465 82898412 0011 - j0242 21482 + j475549 47603414 0096 - j0364 2476 + j9344 966715 0033 - j0287 7217 + j62920 6333316 0053 - j0310 4444 + j25837 2621617 0031 - j0270 7444 + j64482 6491118 0084 - j0367 2800 + j12300 1261419 0079 - j0359 2918 + j13248 1356620 0072 - j0347 3275 + j15896 1623021 0005 - j0267 47543 + j2596565 259700224 0052 - j0318 4527 + j27568 2793726 0323 - j0777 0722 + j1739 188329 0157 - j0708 1521 + j6859 702630 0184 - j0750 1291 + j5262 5418

The same procedure shall be used in the case when we impose an inequality constrainton bus voltages vmink

le vk le vmaxk Table 52 which was generated by using fmincon to

solve (521) with (523) and voltage constraints shows the maximizing load impedance z`to draw maximum complex power ST under these constraints As expected the condition(510) is not satisfied and the apparent power in table 52 is lower than for the previouscase Figure 55 shows the results for maximum apparent power |S| of table 52

Table 52 Load impedances complex and apparent power for each load bus of IEEE30-bus system with voltage constraints vmink

le vk le vmaxk

Bus No z`(pu) ST (pu) |ST |(pu) vmin(pu) vmax(pu)3 0048 - j0063 8431 + j11143 13974 095 1054 0045 - j0058 9271 + j11802 15008 095 1055 0081 - j0109 4841 + j6546 8142 095 1057 0069 - j0098 5330 + j7492 9194 095 1058 0062 - j0086 6117 + j8465 10444 095 10510 0115 - j0128 4281 + j4764 6405 095 10512 0120 - j0137 3999 + j4553 6060 095 10514 0183 - j0262 1977 + j2834 3455 095 10515 0143 - j0174 3106 + j3784 4896 095 10516 0154 - j0201 2656 + j3453 4357 095 10517 0134 - j0166 3255 + j4025 5177 095 10518 0183 - j0255 2048 + j2853 3512 095 10519 0178 - j0249 2099 + j2928 3602 095 10520 0173 - j0235 2239 + j3045 3779 095 10521 0133 - j0148 3715 + j4127 5553 095 10524 0158 - j0204 2615 + j3379 4273 095 10526 0396 - j0682 0703 + j1210 1399 095 10529 0354 - j0484 1086 + j1485 1839 095 10530 0375 - j0528 0986 + j1387 1701 095 105

Figure 55 Maximum apparent total power |ST | for each load buses of the IEEE 30-bussystem with voltage constraints vmink

le vk le vmaxk

52 Calculating Smax for 2-port

In this section we will present the results for a scenario in which two load buses arevariables and we wish to find the maximum power transfer to these two buses for theIEEE 30-bus system In chapter 4 all but one example assumed uncoupled resistiveloads but in this section the load is allowed to be a complex load Thus the optimizationproblem (521) can be rewritten here for convenience

maxzj zkisinC

ReST (zj zk) j 6= k (525)

whereST (zj zk) = |vj|

zj+ |vk|

zk(526)

and the voltage vjk is given by a multilinear transformation written as

vjk(zj zk) = a0 + ajzj + akzk + ajkzjzkb0 + bjzj + bkzk + zjzk

As shown in section 512 we make measurements of voltages vjk for seven differentvalues of the real power demand Pjk which is equivalent to varying the real part of theload impedance zjk of each bus Using Matpower to obtain all seven measurements allparameters of the bilinear transformations (527) can be found For load buses 3 and 5we get

v3(z3 z5) = 00026minus j00002 + (00296 + j00123)z3 minus (00001 + j00214)z5 + (0982minus j00235)z3z5

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

v5(z3 z5) = 00028minus j00001 + (00036 + j00387)z3 + (00102minus j00088)z5 + (09814minus j00307)z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

Substituting equations (528) and (529) in (526) and solving (525) with fmincon with-out voltage constraints we obtain the maximizing loads (z3 z5) and the correspondingmaximum total active and reactive power (PT QT )

z`3 = 0013minus j0096 pu z`5 = 0029minus j0193 puPT = 31858 pu QT = 224618 pu

These results can be found in Table 53 as well as other results for different combinationsof buses Table 54 shows the maximizing impedances to draw maximum power in twobuses with voltage constraints (vminjk

le vjk le vmaxjk) already shown in Table 52 for

each bus and was generated by fmincon using the same function as for Table 53 butadding the constraints

Table 53 The maximizing impedances to draw maximum power for two buses (j k) inthe IEEE 30-bus system without voltage constraintsBus j Bus k z`j (pu) z`k(pu) ST (pu) |ST |(pu)

3 5 0013 - j0096 0029 - j0193 31858 + j224618 2268663 6 0032 - j0135 - j0095 61113 + j601105 6042033 7 0015 - j0099 0028 - j0187 30068 + j202497 20471710 15 0 - j0188 (249768 - j554635)times109 608279times109

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

Table 54 The maximizing impedances to draw maximum power for two buses (j k) inthe IEEE 30-bus system with voltage constraints vminjk

le vjk le vmaxjkfor each bus

(j k)Bus j Bus k z`j (pu) z`k(pu) ST (pu) |ST |(pu)

3 5 0033 - j0048 0177 - j0065 16160 + j17750 240043 6 - j0285 0028 - j0026 20861 + j22544 307153 7 0034 - j0050 0183 - j0090 15445 + j17503 2334310 15 0007 - j0007 0011 - j0015 112208 + j115824 16126420 30 - j0078 0469 + j0531 0918 + j11881 11916

521 Cross-checking maximizing impedance and power results

To validate the results given in (530) we assess the performance of functions (528) and(529) Thus evaluating v3(z3 z5) and v5(z3 z5) for any chosen value of impedances z3 z5we calculate the complex power S with (522) for buses 3 and 5 In the same way theactive and reactive power are obtained from (519) for each bus Pk and Qk are inserted inMatpower to get voltage vkm and calculate the impedances in each bus with the followingexpression

zkm = |vkm(zk)|2STk

comparing zkm with zk we can evaluate the performance of multilinear transformationsChoosing the following values of impedances

z3 = 1 + j pu z5 = 2minus j08 pu (532)

from (528) and (529) we get

v3 = 0941minus j0067 pu v5 = 0951minus j0108 pu (533)

Substituting (532) (533) in (522) yields

ST3 = 0445minus j0445 pu ST5 = 0395minus j0158 pu (534)

Inserting the values (534) in Matpower we obtain

v3m = 0995minus j0096 pu v5m = 0945minus j0111 pu (535)

Calculating (531) now using the values in (535) and (534) the impedances obtainedare

z3m = 1123 + j1123 pu z5m = 1979minus j0791 pu (536)

Calculating the relative error between (533) and (535) we have δ3 = 647 for v3 andδ5 = 067 for v5 The error δ3 is significant because small changes of vk are reflectedas large changes in Sk (522) so the error increases We feedback the results obtained in(536) to the multilinear transformation Then reevaluating v3(z3m z5m) and v5(z3m z5m)is that to say solving (528) and (529) with the values of z in (536) yields

Calculating the relative error between (533) and (537) we now have δ3 = 057 for v3

and δ5 = 034 for v5 These errors are significantly lower than those obtained beforeshowing the sensitivity to variation in vk The process utilized is illustrated in Figure 56

Multilinear

TransformationMatPower

Figure 56 Illustration of process utilized to validate the multilinear transformation forany load impedance zi zj

Chapter 6

Conclusions

In this chapter we present a brief review of the results obtained in this dissertation aswell as some concluding remarks Future work that can build on the results obtained isalso mentioned

61 Conclusions and remarks

The main contributions of this dissertation can be highlighted as followsIn section 33 the maximum power transfer theorem for the case of uncoupled resistive

load was studied and compared with theorem 31 (for complex loads) we discoveredsome differences in the cases for which the total power is unbounded For these casestwo lemmas are announced and demonstrated and illustrated with specific examples

Section 41 gave several illustrative examples for two-port networks terminated inresistive loads as well as general uncoupled loads All examples are solved with theproposed measurement-based approach which provides a simple and standard functionalexpression for the total power Notably the proposed approach solves problems where theload n-port has complex impedances or a mix of them (eg resistances and reactances)as opposed to the results of Flanders Lin and Sommariva which all treat the case of apurely resistive load n-port

Chapter 5 was dedicated to exploring power system applications with the proposedapproach Section 51 shows how to find the Thevenin equivalent parameters making twomeasurements without short-circuiting the load terminals This approach is useful inmore complicated power systems as shown in examples

Finally section 52 presented the results to obtain maximum power transfer for two-ports in the well-known IEEE 30-bus power system finding the complex load to drawmaximum power We validated the proposed measurement-based approach

Thus the major conclusion of this dissertation is that the proposed measurement-based

approach provides a practical new version of the ldquomaximumrdquo power transfer theoremfor n-ports terminated with uncoupled loads in the sense that it allows a functionaldescription of the entire power (hyper)surface as a function of chosen port parametersIt can be argued that this is more useful than merely computing port parameters formaximum power transfer since the power hypersurface can be used along with constraintson port parameters to find parameters for optimal viable maximum power transfer

62 Future work

The results presented here are applicable to any system where Kirchhoff like laws applysuch as hydraulic systems and wireless power transfer In addition the results presentedwere for the case of exact measurements It would be of interest to investigate extensionsfor noisy measurements

Bibliography

[1] BERTSEKAS D P ldquoThevenin Decomposition and Large-Scale Optimizationrdquo Jour-nal Of Optimization Theory And Applications v 89 n 1 pp 1ndash15 jan 1900

[2] JOHNSON D H ldquoOrigins of the equivalent circuit concept the voltage-sourceequivalentrdquo Proceedings of the IEEE v 91 n 4 pp 636ndash640 2003

[3] HAJJ I N ldquoComputation Of Thevenin And Norton Equivalentsrdquo Electronics Lettersv 12 n 1 pp 273ndash274 jan 1976

[4] CHUA L O DESOER C A KUH E S Linear and Nonlinear Circuits McGraw-Hill 1987

[5] NAMBIAR K K ldquoA generalization of the maximum power transfer theoremrdquo Pro-ceedings IEEE pp 1339ndash1340 July 1969

[6] DESOER C A ldquoThe maximum power transfer for n-portsrdquo IEEE Transactions onCircuit Theory 1973

[7] FLANDERS H ldquoOn the maximal power transfer theorem for n-portsrdquo InternationalJournal of Circuit Theory and Applications v 4 pp 319ndash344 1976

[8] LIN P M ldquoCompetitive power extraction from linear n-portsrdquo IEEE Transactionson Circuits and Systems v 32 n 2 pp 185ndash191 1985

[9] MCLAUGHLIN J C KAISER K L ldquoldquoDeglorifyingrdquo the Maximum Power TransferTheorem and Factors in Impedance Selectionrdquo IEEE Transactions On Educa-tion v 50 n 3 pp 251ndash255 2007

[10] BODE H W Network Analysis and Feedback Amplifier Design Princeton NJ DVan Nostrand 1945

[11] LIN P M ldquoA survey of applications of symbolic network functionsrdquo IEEE Trans-actions on Circuit Theory v CT-20 n 6 pp 732ndash737 1973

[12] DECARLO R LIN P M Linear circuit analysis 2nd ed Oxford Oxford Univer-sity Press 2001

[13] DATTA A LAYEKS R NOUNOU H et al ldquoTowards data-based adaptativecontrolrdquo International Journal of Adaptative Control and Signal Processingv 27 pp 122ndash135 September 2012

[14] VORPERIAN V Fast analytical techniques for electrical and electronic circuitsCambridge UK Cambridge University Press 2004

[15] MIDDLEBROOK R D ldquoNull Double Injection And The Extra Element TheoremrdquoIEEE Transactions on Education v 32 n 3 pp 167ndash180 1989

[16] MIDDLEBROOK R D VORPERIAN V LINDAL J ldquoThe N extra elementtheoremrdquo IEEE Transactions on Circuits and SystemsmdashPart I FundamentalTheory and Applications v 45 n 9 pp 919ndash935 1998

[17] SCHWERDTFEGER H Geometry of complex numbers 1st ed New York Doverpublications inc 1979

[18] ARNOLD D ROGNESS J ldquoMoebius transformations revealedrdquo Jun 2007Disponıvel em lthttpswwwyoutubecomwatchv=0z1fIsUNhO4gt

[19] LIN P M ldquoDetermination of available power from resistive multiportsrdquo IEEETransactions on Circuit Theory v 19 n 4 pp 385ndash386 1972

[20] SOMMARIVA A M ldquoA maximum power transfer theorem for DC linear two-portsrdquoIEEE International Symposium on Circuits and Systems v 1 pp Indash553 ndash Indash556 2002

[21] CORSI S TARANTO G ldquoA real-time voltage instability identification algorithmbased on local phasor measurementsrdquo IEEE Transactions on Power Systemsv 23 n 3 pp 1271ndash1279 Aug 2008

[22] LI W CHEN T ldquoAn Investigation on the relationship between impedance match-ing and maximum power transferrdquo IEEE Electrical Power and Energy Confer-ence 2009

[23] LI W WANG Y CHEN T ldquoInvestigation on the Thevenin equivalent parametersfor online estimation of maximum power transfer limitsrdquo IET GenerationTransmission amp Distribution v 4 n 10 pp 1180ndash1187 jul 2010

[24] ZIMMERMAN R MURILLO-SANCHEZ C GAN D ldquoMatpower Version 50rdquoFeb 2015 Disponıvel em lthttpwwwpserccornelledumatpowergt

Appendix A

Guide to use algorithm developed inMatlab

In this appendix we will explain the basic algorithm developed in Matlabr for obtainingthe maximum power transfer and the corresponding resistor or impedances We will alsoexplain the main functions used in this implementation

In Figure A1 is shown a flowchart to describe the base algorithm implemented for n-port networks Following the flowchart the algorithm starts choosing m different valuesfor the n-port whose termination can be with impedances zk or resistors rk So make thesame m measurements of currents or voltages The quantity of measurement m is givenby

m = 2n+1 minus 1 (A1)

Then call a function which solves a system of linear equations (Ax = b) given by (213)to obtain all parameters of the multilinear transformation (211) At this point the powerfunction PT is defined with the multilinear transformation of current or voltage given inthe previous step Finally using fmincon function the maximum total power PT and thecorresponding optimal load (zk or rk) are calculated and displayed

The examples presented in this dissertation include one or two-port (n = 1 2) hencefrom (A1) m = 3 7 respectively All measurements are simulated choosing any valuefor the design element depend on the network or system ie for two-port networks areusing respectively (223)-(224) or (228)-(229) to measure current or voltage For thiscase the multilinear transformation is given by (232) or (233) and the respective powerPT from (41) or (42)

In Table A1 the main functions implemented are summarized

READ MEASUREMENTS

FIND ALL PARAMETERSSOLVE

MULTILINEAR TRANSFORMATION

PICK VALUE FOR

FMINCON

DEFINE POWER

Figure A1 Algorithm to find the maximum total power PT and the correspondingimpedances zk with a measurement-based approach

Table A1 Main function implemented in Matlabr to find the maximum total powerPT and the corresponding impedances zk with a measurement-based approach MTMultilinear Transformation

Name file Description Input Output Remarks about namefile

UFRJ funport M Function to find all pa-rameters of MT solv-ing Ax = b

A b x parameters of MT n-port (12) M measurable variable(VI)

UFRJ PfunportL M Function of power usedby fmincon to find themaximum values

x MT parameters n-port (12) M measurable variable(VI) L Type of load(RZ)

UFRJ 2portL Script to find the max-imum power and thecorresponding loads fora two-port network

Zt Theveninimpedance matrixet Thevenin voltage

Maximum valuesPT and R` or Z`

L Type of load (RZ)

UFRJ portZ -MPower

Script to find the max-imum power and thecorresponding loadsusing MatPower

Case ie IEE30-bus(lsquocase30rsquo) b bus nquantity of measure-ments

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction

The Theacutevenin theorem

The maximum power transfer theorem

Bodes bilinear theorem

Structure of dissertation

Bodes bilinear form theorem and generalizations

Bodes theorem

Representation of circles by hermitian matrices

The Moebius transformation

Generalizations Multilinear transformation

Measurement-based approach

Theacutevenins and Nortons theorem are particular cases of Bodes bilinear theorem

Lins theorem applied to two-port networks

The maximum power transfer (MPT)

Derivation of the maximum power transfer theorem for general n-port

Illustrative examples for MPT with complex load impedances

MPT for the case of uncoupled resistive loads

Modifications to Desoers theorem for uncoupled resistive loads

Application of Lins theorem to MPT

Illustrative examples of MPT for two-port


Theacutevenin equivalent parameters from measurement-based approach review

Simple linear circuit

IEEE 30-bus system

Calculating Smax for 2-port

Cross-checking maximizing impedance and power results

Conclusions

Conclusions and remarks

Future work

Bibliography

Guide to use algorithm developed in Matlab

DISSERTACAO SUBMETIDA AO CORPO DOCENTE DO INSTITUTO ALBERTOLUIZ COIMBRA DE POS-GRADUACAO E PESQUISA DE ENGENHARIA (COPPE)DA UNIVERSIDADE FEDERAL DO RIO DE JANEIRO COMO PARTE DOSREQUISITOS NECESSARIOS PARA A OBTENCAO DO GRAU DE MESTRE EMCIENCIAS EM ENGENHARIA ELETRICA

Examinada por

Prof Amit Bhaya PhD

Prof Glauco Nery Taranto PhD

Prof Vitor Heloiz Nascimento PhD

RIO DE JANEIRO RJ ndash BRASILABRIL DE 2015

Acknowledgement

DESACOPLADAS

Abril2015

April2015

Contents

List of Figures x

List of Tables xiii

Bibliography 61

List of Figures

(infin 0) 31

List of Tables

le vk le vmaxk 55

le vjk le vmaxjk

Chapter 1

Introduction

v = et minus zti

Z = A+Bz

Chapter 2

T = a+ bz

c+ dz(21)

|z minus γ|2 = ρ2

C(z z) = Azz +Bz + Cz +D = 0 (23)

M = A B

∆ = minusA2ρ2 (26)

A 6= 0 ∆ lt 0

ρ =radicminus∆A

γ = minusCA

M = a b

z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Acknowledgement

DESACOPLADAS

Abril2015

April2015

Contents

List of Figures x

List of Tables xiii

Bibliography 61

List of Figures

(infin 0) 31

List of Tables

le vk le vmaxk 55

le vjk le vmaxjk

Chapter 1

Introduction

v = et minus zti

Z = A+Bz

Chapter 2

T = a+ bz

c+ dz(21)

|z minus γ|2 = ρ2

C(z z) = Azz +Bz + Cz +D = 0 (23)

M = A B

∆ = minusA2ρ2 (26)

A 6= 0 ∆ lt 0

ρ =radicminus∆A

γ = minusCA

M = a b

z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Acknowledgement

DESACOPLADAS

Abril2015

April2015

Contents

List of Figures x

List of Tables xiii

Bibliography 61

List of Figures

(infin 0) 31

List of Tables

le vk le vmaxk 55

le vjk le vmaxjk

Chapter 1

Introduction

v = et minus zti

Z = A+Bz

Chapter 2

T = a+ bz

c+ dz(21)

|z minus γ|2 = ρ2

C(z z) = Azz +Bz + Cz +D = 0 (23)

M = A B

∆ = minusA2ρ2 (26)

A 6= 0 ∆ lt 0

ρ =radicminus∆A

γ = minusCA

M = a b

z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Acknowledgement

DESACOPLADAS

Abril2015

April2015

Contents

List of Figures x

List of Tables xiii

Bibliography 61

List of Figures

(infin 0) 31

List of Tables

le vk le vmaxk 55

le vjk le vmaxjk

Chapter 1

Introduction

v = et minus zti

Z = A+Bz

Chapter 2

T = a+ bz

c+ dz(21)

|z minus γ|2 = ρ2

C(z z) = Azz +Bz + Cz +D = 0 (23)

M = A B

∆ = minusA2ρ2 (26)

A 6= 0 ∆ lt 0

ρ =radicminus∆A

γ = minusCA

M = a b

z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


DESACOPLADAS

Abril2015

April2015

Contents

List of Figures x

List of Tables xiii

Bibliography 61

List of Figures

(infin 0) 31

List of Tables

le vk le vmaxk 55

le vjk le vmaxjk

Chapter 1

Introduction

v = et minus zti

Z = A+Bz

Chapter 2

T = a+ bz

c+ dz(21)

|z minus γ|2 = ρ2

C(z z) = Azz +Bz + Cz +D = 0 (23)

M = A B

∆ = minusA2ρ2 (26)

A 6= 0 ∆ lt 0

ρ =radicminus∆A

γ = minusCA

M = a b

z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


April2015

Contents

List of Figures x

List of Tables xiii

Bibliography 61

List of Figures

(infin 0) 31

List of Tables

le vk le vmaxk 55

le vjk le vmaxjk

Chapter 1

Introduction

v = et minus zti

Z = A+Bz

Chapter 2

T = a+ bz

c+ dz(21)

|z minus γ|2 = ρ2

C(z z) = Azz +Bz + Cz +D = 0 (23)

M = A B

∆ = minusA2ρ2 (26)

A 6= 0 ∆ lt 0

ρ =radicminus∆A

γ = minusCA

M = a b

z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Contents

List of Figures x

List of Tables xiii

Bibliography 61

List of Figures

(infin 0) 31

List of Tables

le vk le vmaxk 55

le vjk le vmaxjk

Chapter 1

Introduction

v = et minus zti

Z = A+Bz

Chapter 2

T = a+ bz

c+ dz(21)

|z minus γ|2 = ρ2

C(z z) = Azz +Bz + Cz +D = 0 (23)

M = A B

∆ = minusA2ρ2 (26)

A 6= 0 ∆ lt 0

ρ =radicminus∆A

γ = minusCA

M = a b

z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Bibliography 61

List of Figures

(infin 0) 31

List of Tables

le vk le vmaxk 55

le vjk le vmaxjk

Chapter 1

Introduction

v = et minus zti

Z = A+Bz

Chapter 2

T = a+ bz

c+ dz(21)

|z minus γ|2 = ρ2

C(z z) = Azz +Bz + Cz +D = 0 (23)

M = A B

∆ = minusA2ρ2 (26)

A 6= 0 ∆ lt 0

ρ =radicminus∆A

γ = minusCA

M = a b

z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


List of Figures

(infin 0) 31

List of Tables

le vk le vmaxk 55

le vjk le vmaxjk

Chapter 1

Introduction

v = et minus zti

Z = A+Bz

Chapter 2

T = a+ bz

c+ dz(21)

|z minus γ|2 = ρ2

C(z z) = Azz +Bz + Cz +D = 0 (23)

M = A B

∆ = minusA2ρ2 (26)

A 6= 0 ∆ lt 0

ρ =radicminus∆A

γ = minusCA

M = a b

z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


List of Tables

le vk le vmaxk 55

le vjk le vmaxjk

Chapter 1

Introduction

v = et minus zti

Z = A+Bz

Chapter 2

T = a+ bz

c+ dz(21)

|z minus γ|2 = ρ2

C(z z) = Azz +Bz + Cz +D = 0 (23)

M = A B

∆ = minusA2ρ2 (26)

A 6= 0 ∆ lt 0

ρ =radicminus∆A

γ = minusCA

M = a b

z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


List of Tables

le vk le vmaxk 55

le vjk le vmaxjk

Chapter 1

Introduction

v = et minus zti

Z = A+Bz

Chapter 2

T = a+ bz

c+ dz(21)

|z minus γ|2 = ρ2

C(z z) = Azz +Bz + Cz +D = 0 (23)

M = A B

∆ = minusA2ρ2 (26)

A 6= 0 ∆ lt 0

ρ =radicminus∆A

γ = minusCA

M = a b

z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


List of Tables

le vk le vmaxk 55

le vjk le vmaxjk

Chapter 1

Introduction

v = et minus zti

Z = A+Bz

Chapter 2

T = a+ bz

c+ dz(21)

|z minus γ|2 = ρ2

C(z z) = Azz +Bz + Cz +D = 0 (23)

M = A B

∆ = minusA2ρ2 (26)

A 6= 0 ∆ lt 0

ρ =radicminus∆A

γ = minusCA

M = a b

z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Chapter 1

Introduction

v = et minus zti

Z = A+Bz

Chapter 2

T = a+ bz

c+ dz(21)

|z minus γ|2 = ρ2

C(z z) = Azz +Bz + Cz +D = 0 (23)

M = A B

∆ = minusA2ρ2 (26)

A 6= 0 ∆ lt 0

ρ =radicminus∆A

γ = minusCA

M = a b

z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


v = et minus zti

Z = A+Bz

Chapter 2

T = a+ bz

c+ dz(21)

|z minus γ|2 = ρ2

C(z z) = Azz +Bz + Cz +D = 0 (23)

M = A B

∆ = minusA2ρ2 (26)

A 6= 0 ∆ lt 0

ρ =radicminus∆A

γ = minusCA

M = a b

z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Z = A+Bz

Chapter 2

T = a+ bz

c+ dz(21)

|z minus γ|2 = ρ2

C(z z) = Azz +Bz + Cz +D = 0 (23)

M = A B

∆ = minusA2ρ2 (26)

A 6= 0 ∆ lt 0

ρ =radicminus∆A

γ = minusCA

M = a b

z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Z = A+Bz

Chapter 2

T = a+ bz

c+ dz(21)

|z minus γ|2 = ρ2

C(z z) = Azz +Bz + Cz +D = 0 (23)

M = A B

∆ = minusA2ρ2 (26)

A 6= 0 ∆ lt 0

ρ =radicminus∆A

γ = minusCA

M = a b

z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Chapter 2

T = a+ bz

c+ dz(21)

|z minus γ|2 = ρ2

C(z z) = Azz +Bz + Cz +D = 0 (23)

M = A B

∆ = minusA2ρ2 (26)

A 6= 0 ∆ lt 0

ρ =radicminus∆A

γ = minusCA

M = a b

z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Chapter 2

T = a+ bz

c+ dz(21)

|z minus γ|2 = ρ2

C(z z) = Azz +Bz + Cz +D = 0 (23)

M = A B

∆ = minusA2ρ2 (26)

A 6= 0 ∆ lt 0

ρ =radicminus∆A

γ = minusCA

M = a b

z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


|z minus γ|2 = ρ2

C(z z) = Azz +Bz + Cz +D = 0 (23)

M = A B

∆ = minusA2ρ2 (26)

A 6= 0 ∆ lt 0

ρ =radicminus∆A

γ = minusCA

M = a b

z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


|z minus γ|2 = ρ2

C(z z) = Azz +Bz + Cz +D = 0 (23)

M = A B

∆ = minusA2ρ2 (26)

A 6= 0 ∆ lt 0

ρ =radicminus∆A

γ = minusCA

M = a b

z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


∆ = minusA2ρ2 (26)

A 6= 0 ∆ lt 0

ρ =radicminus∆A

γ = minusCA

M = a b

z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


z = z1

z2 T = T1

M1 = a1 b1

M2 = a2 b2

= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


= 1det M

I = 1 0

T = a0 + a1x1 + a2x2 + a12x1x2

b0 + b1x1 + b2x2 + b12x1x2(210)

xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


xm(p) =nm0 + Σin

mi pi + Σijn

m = a0 + a1z1 + a2z2 + a12z1z2

b0 + b1z1 + b2z2 + z1z2(211)

zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


zk = rk + jxk

ak = ark + jaik

bk = brk + jbik

1 z(1)1 z

(1)2 z

(1)1 z

(1)1 minusm(1)z

1 z(2)1 z

(2)2 z

(2)1 z

(2)1 minusm(2)z

1 z(3)1 z

(3)2 z

(3)1 z

(3)1 minusm(3)z

1 z(4)1 z

(4)2 z

(4)1 z

(4)1 minusm(4)z

1 z(5)1 z

(5)2 z

(5)1 z

(5)1 minusm(5)z

1 z(6)1 z

(6)2 z

(6)1 z

(6)1 minusm(6)z

1 z(7)1 z

(7)2 z

(7)1 z

(7)1 minusm(7)z

m(1)z(1)1 z

m(2)z(2)1 z

m(3)z(3)1 z

m(4)z(4)1 z

m(5)z(5)1 z

m(6)z(6)1 z

m(7)z(7)1 z

Ax = p

a11 a12 a114

a141 a142 a1414

a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


a11 = 1 a12 = 0 a13 = r(1)1 a14 = minusx(1)

a15 = r(1)2 a16 = minusx(1)

2 a17 = r(1)1 r

(1)2 minus x

(1)1 x

(1)2 a18 = minusr(1)

1 x(1)2 minus r

(1)2 x

1 p(1) a112 = r(1)1 q(1) + x

(1)1 p(1)

a113 = x(1)2 q(1) minus r(1)

2 p(1) a114 = r(1)2 q(1) + x

(1)2 p(1)

a21 = 0 a22 = 1 a23 = x(1)1 a24 = r

a25 = x(1)2 a26 = r

(1)2 a27 = r

(1)1 x

(1)2 + r

(1)2 x

(1)1 a28 = r

(1)1 r

(1)2 minus x

(1)1 x

1 p(1) a212 = x(1)1 q(1) minus r(1)

1 p(1)

2 p(1) a214 = x(1)2 q(1) minus r(1)

2 p(1)

(r(1)1 r

(1)2 minus x

(1)1 x

(1)2 )p(1) minus (r(1)

1 x(1)2 + r

(1)2 x

(1)1 )q(1)

(r(1)1 x

(1)2 + r

(1)2 x

(1)1 )p(1) + (r(1)

1 r(1)2 minus x

(1)1 x

(1)2 )q(1)

1 r(7)2 minus x

(7)1 x

(7)2 )p(7) minus (r(7)

1 x(7)2 + r

(7)2 x

(7)1 )q(7)

(r(7)1 x

(7)2 + r

(7)2 x

(7)1 )p(7) + (r(7)

1 r(7)2 minus x

(7)1 x

(7)2 )q(7)

x = Aminus1p (215)

i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


i(z`) = a

c+ dz`(216)

v(z`) = az`c+ dz`

zt = vocisc

d(218)

i(z`) = etzt + z`

v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


v(z`) = bz`c+ dz`

m1(r1 r2) = a0 + a1r1 + a2r2 + a12r1r2

b0 + b1r1 + b2r2 + b12r1r2(220)

m2(r1 r2) = c0 + c1r1 + c2r2 + c12r1r2

b0 + b1r1 + b2r2 + b12r1r2(221)

z11 z22

-z21i1

-z12i2+

minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


minusAgt I 00 M N

a0 = e1z22 minus e2z12 a1 = 0 a2 = e1 a12 = 0

i1(r1 r2) = a0 + a2r2

b0 + b1r1 + b2r2 + r1r2 (226)

i2(r1 r2) = c0 + c1r1

b0 + b1r1 + b2r2 + r1r2 (227)

v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


v1(r1 r2) = a1r1 + a12r1r2

b0 + b1r1 + b2r2 + r1r2 (230)

v2(r1 r2) = c2r2 + c12r1r2

b0 + b1r1 + b2r2 + r1r2 (231)

Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Chapter 3

v = et minus zti

P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


P = i2r` = ir`i (32)

i = et2rt

r`et2rt

= et minus rtet2rt

)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


)2r` which is then

v = Z`i (39)

parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


parti= 1

(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


(β minus 1

) β = x2

2 + r22 minus r2

2(x22 + r2

2) gt 0

(12 minus

(1 + j 1

)) which is also a

-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


-1-2j -1-2ji1

P = maxP1 P2 P3 P4 (334)

1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


1 If e1 = 0 e2 = 0 then P = 0

3 r11 = r22 = r12 = r0 e1 = e2

1(2r0) andr0 = r1r2

P = supR`

|elowasttv|2

v = Rti + et (337)

v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


v = v1

Rt = r11 r12

r21 r22

i = i1

et = e1

det Rt 6= 0 (338)

|r12|+ |r21| 6= 0 (339a)

|r12e2|+ |r21e1| 6= 0 (339b)

2 (340a)

2 (340b)

R` = diag (r1 r2) (341)

D(r1 r2) = r1r2 + r22r1 + r11r2 + det Rt 6= 0 (343)

PT = i2`1r1 + i2`2r2 (346)

r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


r11e2 minus rme1 6= 0 (347)

r22e1 minus rme2 6= 0 (348)

(r11 + r21

r11e2 minus rme1

r22e1 minus rme2

)(r22 + r12

r22e1 minus rme2

r11e2 minus rme1

)ge 0 (349)

4r2b minus 4r2

et minusZti = R`i

et = Zti (354)

which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


which is singular

1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


1-j -ji1

Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Chapter 4

PT (r1 r2) = |i1|2r1 + |i2|2r2 or (41)

PT (r1 r2) = |v1|2

r1+ |v2|2

r2(42)

maxr1r2isinR

PT (r1 r2) (43)

Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Linear

Two Port

Network

minus 10

44 + 10r1 + 8r2 + r1r2

PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


PT (r1 r2) = 1600r1 + 400r2 + 1200r1r2 + 100r1r22 + 100r2

1r22 + 20r2

1r2 + 100r21 + 16r1r2

2 + 248r1r2 + 880r1 + 64r22 + 704r2 + 1936

(r1 r2) = (11 Ω 22 Ω)

PT = 3409W

44 + 10r1 + 8r2 + r1r2

i1(r1 r2) = 5 + 4r2

95 + 2r1 + 5r2 + r1r2

i2(r1 r2) = 13 + 3r1

95 + 2r1 + 5r2 + r1r2

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 5 10 15 20 25 30

|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


|Pt|[W

maximal point

PT (r1 r2) = 25r1 + 169r2 + 118r1r2 + 16r1r22 + 9r2

22 + 4r2

1r2 + 4r21 + 10r1r2

2 + 39r1r2 + 38r1 + 25r22 + 95r2 + 361

(r1 r2) = (6043 Ω 2479 Ω)PT = 1563W

r1 = 5 + 12

4r2b minus 4r2

where rm = 34 rb =

|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


|Pt|[W

Resistorr1[Ω

Resistor r2[Ω]0 2 4 6 8 10 12 14 16 18 20

PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


PT (r2) = |i2|2r2 (49)

P (l1 r2) = 900r2

14400(l1r2ω)2 + 6l21r2ω2 + 121572110000 l

2 minus 954950 l1ω + 9r2

2 + 1013148910000

= 900r2

14400(πl1r2)2 + 86400π2l21r2 + 4376595625 π2l21 minus 114588

5 πl1 + 9r22 + 10131489

10000(411)

PT = 4787W (412)

for (l1 r2) = (087mH 1029 Ω) (413)

|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


|Pt|[W

Inductance l1[H]

Resistorr2[Ω]

0 0002 0004 0006 0008 001 0012 0014 0016 0018 0020

i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


i2(l1 r2) = vinr

i1(r1 r2) = r2

1 + j + jr1 + r2 + r1r2

PT (r1 r2) = 2r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + r2

2 + 2r2 + 2 (416)

(r1 r2) = (infin 1 Ω)

= 1∆(r2 r1)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 5 10 15 20 25 300

P = r1r22 + r2r

((r1 + 1)r2 + 1)2 + (r1 + 1)2 (417)

It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


It follows that

Pminus1 = r1r2

r1 + r2+ r1 + r2

r1r2+ 2r1r2

+ 2r1r2(r1 + r2) + 2r2

r1 + r2

r1 + r2+ r1 + r2

r1r2ge 2

= minusj j

i1(r1 r2) = r2

PT (r1 r2) = 4r2 + r1r22 + r2

22 + r2

1 + 2r1r2 + r22 + 4 (418)

PT = 1W

for (r1 r2) = (0 Ω 2 Ω)

|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


|Pt|[W

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


P = r1r22 + r2(r2

1 + 4)(r1r2 + 2)2 + (r1 minus r2)2

Pm = supv 6=0

f(v1 v2) f(v1 v2) = |v1 + v2|2

2(|v1||v1 minus v2|+ |v2||v1 + v2|)

|1 + y|2

= 1minus j j

i1(r1 r2) = r2

2 + j + jr1 + (1minus j)r2 + r1r2

i2(r1 r2) = 1minus 2j + r1

2 + j + jr1 + (1minus j)r2 + r1r2

PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


PT (r1 r2) = 5r2 + 2r1r2 + r1r22 + r2

22 + r2

1 + 2r1r22 + 2r1r2 + 2r1 + 2r2

2 + 2r2 + 5 (419)

PT = 06006W

for (r1 r2) = (0 Ω 15810 Ω)

|Pt|[W

P = r1r22 + r2(r1 + 1)2 + 4

((r1 + 1)r2 + 2)2 + (r1 minus r2 + 1)2

2|v1|2 + 2(|v1||(1minus j)v1 + jv2|+ |v2||v1 + v2|)

Pm = f(1 1 + j) = 52(1 +

Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Resistor r1[Ω]

Resistorr2[Ω]

0 2 4 6 8 10 12 14 16 18 200

For this v2 we have

Zv = 0minus1 + j2

result is r2 =radic

52 asymp 1581

Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Chapter 5

ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


ik(zk) = a0k

b0k + zk (52)

b0k + zk (53)

zt = b0k (56)

maxzkisinCPTk

(zk) (57)

PTk(zk) = |vk|2 Re

z` = zlowastt (510)

i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


i2(z2) = minus1 + j01001 + j02 + z2

PT2 = 101r2

100r22 + 2r2 + 100x2

2 + 40x2 + 401100

z2 = 001minus j02 Ω |z2| = 02PT2 = 25W

r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


r2 = 02 ΩPT2 = 236W

0 05 1 15 2 25 30

Curren

t|i2|[A

Resistor r2

0 05 1 15 2 25 30

Resistor r2

by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


by z)PT (z1 z2) = Re|i1|2z1 + |i2|2z2 (515)

i1(z1) = a0 + a1z1

b0 + z1

i2(z2) = c0 + c1z2

d0 + z2

0578 + microz0

1054 + microz0

micro = 06317 z1 = z2 = 25269

PT = 04855W

= 1 + 8micro

4micro

Ptotal = E+open

(microZL(Zr

L)minus1Z+L + ZL(Zr

L)minus1Z+LL

)minus1

dPtotaldmicro

= 0 (518)

0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


0 1 2 3 4 5 6 7 8 9 100

impedance matrix ZL

ST = PT + jQT (519)

|ST | = |PT + jQT | (520)

maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


maxzkisinCReSTk

(zk) (521)

whereSTk

(zk) = |vk(zk)|2

zk(522)

v3 = (0982minus j0024)z3

0019 + j0097 + z3

v4 = (0979minus j0023)z4

0015 + j0091 + z4(523)

v5 = (0982minus j0032)z5

0035 + j0161 + z5

zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


zt3 = 0019 + j0097

zt4 = 0015 + j0091 (524)

zt5 = 0035 + j0161

le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


le vk le vmaxk

maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


maxzj zkisinC

zj+ |vk|

zk(526)

00006 + j00044 + (00339 + j00623)z3 + (00192 + j00735)z5 + z3z5

minus00009 + j00043 + (00401 + j01214)z3 + (00142 + j00407)z5 + z3z5

20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


20 30 0036 - j0210 0122 - j0586 14300 + j75842 77178

zkm = |vkm(zk)|2STk

z3 = 1 + j pu z5 = 2minus j08 pu (532)

Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Multilinear

Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Chapter 6

Conclusions

62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


62 Future work

Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Bibliography

Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Appendix A

m = 2n+1 minus 1 (A1)

READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


READ MEASUREMENTS

PICK VALUE FOR

FMINCON

DEFINE POWER

L Type of load (RZ)

UFRJ portZ -MPower

Maximum valuesPT Z`

n-port (12)

List of Figures

List of Tables

Introduction






Bodes theorem

















IEEE 30-bus system



Conclusions


Future work

Bibliography


Documents

A measurement-based approach to maximum power transfer for