4772 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 11, NOVEMBER 2012

A Dynamic Multivariable State-Space Model forBidirectional Inductive Power Transfer Systems

Akshya K. Swain, Member, IEEE, Michael J. Neath, Student Member, IEEE,Udaya K. Madawala, Senior Member, IEEE, and Duleepa J. Thrimawithana, Member, IEEE

Abstract—Bidirectional inductive power transfer (IPT) systemsfacilitate contactless power transfer between two sides, which areseparated by an air gap, through weak magnetic coupling. Typicalbidirectional IPT systems are essentially high-order resonant cir-cuits and, therefore, difficult to both design and control without anaccurate mathematical model, which is yet to be reported. This pa-per presents a dynamic model, which provides an accurate insightinto the behavior of bidirectional IPT systems. The proposed state-space-based model is developed in a multivariable framework andmapped into frequency domain to compute the transfer functionmatrix of eight-order bidirectional IPT systems. The interactionbetween various control variables and degree of controllability ofthe system are analyzed from the relative gain array and singularvalues of the system. The validity of the proposed dynamic modelis demonstrated by comparing the predicted behavior with thatmeasured from a 1 kW prototype bidirectional IPT system undervarious operating conditions. Experimental results convincingly in-dicate that the proposed model accurately predicts the dynamicalbehavior of bidirectional IPT systems and can, therefore, be usedas a valuable tool for transient analysis and optimum controllerdesign.

Index Terms—Contactless power transfer, inductive powertransfer, power converters, relative gain array, singular valueanalysis.

I. INTRODUCTION

INDUCTIVE power transfer (IPT) is becoming an acceptedtechnology for supplying power to a variety of applications

with no physical contacts. This technology transfers power fromone system to another across an air gap and through weak mag-netic coupling. It offers high efficiency typically between 85–90%, robustness, and high reliability, even when used in hostileenvironments, because it is unaffected by dust or chemicals. Inthe past, many unidirectional IPT systems, with various circuittopologies or compensation strategies and levels of sophisti-cation in control, have been proposed and successfully imple-mented to cater to a wide spectrum of applications, rangingfrom very low-power biomedical implants to high-power batterycharging systems [1]–[7]. Recently, bidirectional IPT systemshave also been proposed and developed for applications such asV2G systems [8], [9].

Manuscript received September 12, 2011; revised December 18, 2011;accepted January 16, 2012. Date of current version June 20, 2012. Recom-mended for publication by Associate Editor C. R. Sullivan.

The authors are with the Department of Electrical and Computer Engi-neering, The University of Auckland, Auckland-1142, New Zealand (e-mail:a.swain@auckland.ac.nz).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPEL.2012.2185712

The power handling capability of IPT systems is usually im-proved through either series or parallel compensations [10]. As aconsequence, these systems invariably become high-order reso-nant networks, which are complex in nature and difficult to bothdesign and analyze, especially when operated at frequencies inthe range of 10–50 kHz. Although the IPT technology has nowestablished itself as a technique for contactless power transfer,both the design and analyses of such systems are still beingcarried out only through relatively simple steady-state modelsmainly because of their complex nature [11]–[17]. Steady-statemodels are incapable of providing an accurate insight into thedynamic behavior of the system and, as such, cannot be re-garded as a tool that facilitates both proper controller synthesisand physical design, without which the system cannot be op-timized. At present, therefore, there is a need for an accuratedynamic model, which can be used as a valuable tool during thedesign stage of any IPT system.

To address this need, this paper proposes a dynamic modelfor bidirectional IPT systems. The proposed model is based onthe concept of state variables, and can easily be modified forunidirectional IPT systems. Using the model, the relative gainarray (RGA) matrix is computed from the transfer function toinvestigate the interactions between various input–outputs ofan eight-order parallel-compensated IPT system. Singular valueanalysis is also carried out to obtain information with regard tothe degree of controllability of the system as such informationis vital for controller design. Measured results, under variousoperation conditions of a 1 kW bidirectional IPT system, arecompared with the predicted behavior to demonstrate that theproposed dynamic model is accurate and can be used as a valu-able tool during controller synthesis and optimization of IPTsystems.

This paper is organized as follows. Section II briefly describesthe principle of bidirectional IPT system and develops a statevariable model of the system. Behavior of the system from fre-quency domain information is studied in Section-III A throughRGA and singular value analysis. In Section IV, the dynamicmodel is validated by comparing its performance under variousconditions using a prototype of a 1 kW bidirectional IPT systemand Section V presents the conclusions.

II. DYNAMIC STATE VARIABLE MODEL OF A TYPICAL

BIDIRECTIONAL IPT SYSTEM

The schematic of a typical bidirectional IPT system proposedin [8] is shown in Fig. 1. The output of the pick-up is con-nected to the load, which is represented as a dc supply to eitherabsorb or deliver power. Analogous to typical IPT systems, a

0885-8993/$31.00 © 2012 IEEE

SWAIN et al.: DYNAMIC MULTIVARIABLE STATE-SPACE MODEL FOR BIDIRECTIONAL INDUCTIVE POWER TRANSFER SYSTEMS 4773

Fig. 1. Bidirectional IPT system.

Fig. 2. Equivalent circuit representation of bidirectional IPT system.

primary supply generates a constant track current iT (t) in LT ,which is magnetically coupled to the pick-up coil. The primaryand pick-up circuits are implemented with virtually identicalelectronics, which include a reversible rectifier and a tuned (res-onant) inductor–capacitor–inductor (LCL) circuit, to facilitatebidirectional power flow between the track and the pick-up.Each LCL circuit is tuned to the track frequency, generated bythe primary supply, and each reversible rectifier is operated atthe same track frequency either in the inverting or rectifyingmode, depending on the direction of the power flow. The mag-nitude and phase the voltages applied to the reversible rectifierswill determine the amount and direction of power flow.

The primary and pick-up system can, thus, be represented bythe circuit model shown in Fig. 2. The instantaneous value ofthe induced voltage vsi(t) of the pick-up coil Lsi due to trackcurrent iT (t) is given by

vsi(t) = MdiT (t)

dt(1)

where M represents the magnetic coupling or mutual inductancebetween the track inductance LT and pick-up coil inductanceLsi .

The pick-up, the output of which is connected to the load,may be operated either as a source or a sink by the pick-up sidereversible rectifier. Despite the mode of operation, the instan-taneous value of the voltage vr (t) reflected back into the trackdue to current isi(t), in the pick-up coil can be expressed by

vr (t) = −Mdisi(t)

dt. (2)

A. Dynamic Model

The dynamic model of this circuit is developed by introducingthe state variables

x = [x1 x2 x3 x4 x5 x6 x7 x8 ]T

= [ ipi vcpi vpt iT iso vcso vst isi ]T

whereipi current through the primary side inductor Lpi ;vcpi voltage across the primary input capacitor Cpi ;vpt voltage across primary side capacitor CT ;iT current through track inductor LT ;iso current through the pick-up side inductor Lso ;vcso voltage across the pick-up output capacitor Cso ;vst voltage across the pick-up side capacitor Cs ;isi current through the pick-up side inductor Lsi .Let the input vector u be denoted as follows:

u = [u1 u2 ]T = [ vpi vso ]T (3)

where u1 = vpi = input voltage applied at the primary side.Note that this voltage is essentially the output voltage of theprimary side converter and u2 = vso = voltage at the pick-upside.

Following the basic principles of circuit theory, the dynamicmodel can be expressed by the following eight differential equa-tions:

x1 = −Rpi

Lpix1 −

1Lpi

x2 −1

Lpix3 +

1Lpi

u1

x2 =1

Cpix1

x3 =1

CTx1 −

1CT

x4

x4 = γ

[1

LTx3 −

RT

LTx4 − βx7 − βRsix8

]

x5 = −Rso

Lsox5 −

1Lso

x6 +1

Lsox7 −

1Lso

u2

x6 =1

Csox5

x7 = − 1Cs

x5 +1Cs

x8

x8 = γ

[βx3 − βRT x4 −

1Lsi

x7 −Rsi

Lsix8

](4)

where

β =M

LsiLT, γ =

11 − Mβ

. (5)

This can be expressed in the standard state space form as follows:

x = Ax + Bu (6)

4774 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 11, NOVEMBER 2012

where the system matrix A is given by

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−Rpi

Lpi− 1

Lpi− 1

Lpi0 0

1Cpi

0 0 0 0

1CT

0 0 − 1CT

0

0 0γ

LT−γRT

LT0

0 0 0 0 −Rso

Lso

0 0 0 01

Cso

0 0 0 0 − 1Cs

0 0 γβ −γβRT 0

0 0 0

0 0 0

0 0 0

0 −γβ −γβRsi

− 1Lso

1Lso

0

0 0 0

0 01Cs

0 − γ

Lsi−γRsi

Lsi

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(7)

and the input matrix B is given by

B =

⎡⎢⎢⎣

1Lpi

0 0 0 0 0 0 0

0 0 0 0 − 1Lso

0 0 0

⎤⎥⎥⎦

T

. (8)

Considering the track current iT = x4 and pick-up currentiso = x5 as outputs, the output equation can be written as

y = Cx (9)

where

y = [ y1 y2 ]T = [ iT iso ]T

C =[

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

]. (10)

The output power is obtained by multiplying the voltage VDC2with the rectified pick-up output current iso . Since VDC2 is keptconstant at VDC , the output power is controlled by controllingthe current iso . This will be explained in Section IV.

After getting the state-space model of the bidirectional sys-tem, analysis is carried out using standard tools of multivariable

Fig. 3. Magnitude plots of system transfer function matrix.

systems. Since the frequency response functions, RGA, and sin-gular values convey important information about the physicalbehavior of the system, these are computed and analyzed in thefollowing section.

III. FREQUENCY DOMAIN ANALYSIS FROM THE TRANSFER

FUNCTION MATRIX

The transfer function matrix G of the bidirectional IPT systemis computed from the system model given in (4) and is expressedas follows:

G(s) =Y (s)U(s)

= C(sI − A)−1B =[

G11(s) G12(s)

G21(s) G22(s)

]

(11)where

G11(s) =IT (s)Vpi(s)

, G12(s) =IT (s)Vso(s)

, G21(s) =Iso(s)Vpi(s)

G22(s) =Iso(s)Vso(s)

. (12)

The magnitude and phase plots of each element of the transferfunction matrix are computed for the parameters given in Table Iand are shown, respectively, in Figs. 3 and 4.

From the magnitude plots of G12(s) and G21(s), shown inFig. 3, it is observed that there exists significant coupling be-tween the primary and secondary sides of the bidirectional IPTsystem. Furthermore, it is observed that peaks in the variousgains of the transfer function matrix occur at the same frequen-cies. At these frequencies, behavior of the bidirectional IPTsystem changes from inductive to capacitive or capacitive toinductive as reflected in the phase response of the system. Theeffective impendence seen by the source at these frequencies isminimum, which causes excessive current to flow in the circuit.The losses in the system will be very high around these frequen-cies and operation of the system should, therefore, be avoidedat these frequencies.

SWAIN et al.: DYNAMIC MULTIVARIABLE STATE-SPACE MODEL FOR BIDIRECTIONAL INDUCTIVE POWER TRANSFER SYSTEMS 4775

Fig. 4. Phase plots of system transfer function matrix.

A. Relative Gain Array of Bidirectional IPT System

The RGA, introduced in [18], is a heuristic method to predictthe degree of coupling or interaction in a multivariable system. Ifuj and yi denote a particular input–output pair of a multivariablesystem with transfer function matrix G(s), then the relative gainλij between input j and output i is defined as follows:

λij =

(∂yi

∂uj

)uk ,k �=j(

∂yi

∂uj

)yk ,k �=i

(13)

where ( ∂yi

∂uj)uk ,k �=j is the gain between input j and output i

with all other loops open and ( ∂yi

∂uj)yk ,k �=i is the gain between

input j and output i with all other loops closed. The RGA ofa nonsingular complex square matrix G is a square complexmatrix, which is given by [19]

RGA(G) = Λ(G) = G × (G−1)T . (14)

The different elements of the RGA matrix can provide im-portant intuitive information about the system. For example,if RGA(i, i) = 1, there is no interaction with other inputs; ifRGA(i, j) = 0, the manipulated input i does not affect the out-put j; and RGA(i, j) = 0.5 implies a high degree of interaction.Further, if RGA(i, j) > 1, this implies that this interaction re-duces the effective gain of the control loop and if RGA(i, j) < 0,this shows that closing the loop will change the sign of the effec-tive gain and should be avoided, if possible. More details aboutinterpreting the RGA elements can be found in [19].

Note that RGA is a function of frequency. Furthermore, thesum of elements of each column or each row of the RGA matrixis 1. Therefore, for a 2 × 2 system, only one element of the RGAfor a given frequency is calculated to find the entire RGA. Thisis denoted by

Λ =[

λ11 λ12

λ21 λ22

]=

[λ11 1 − λ11

1 − λ11 λ11

]. (15)

Fig. 5. RGA of the bidirectional pick-up. Real parts of λ11 and λ12 havebeen plotted.

For the bidirectional pick-up considered, the RGA is computedover a wide frequency range and is shown in Fig. 5. When thepick-up is operated at a frequency of 20 kHz, the relative gainarray is given by

Λ =[

1.0931 − 0.0033i −0.0931 + 0.0033i

−0.0931 + 0.0033i 1.0931 − 0.0033i

]. (16)

The RGA elements λ11 and λ22 at 20 kHz are close to 1. Thisindicates that there exists a strong interaction between y1 andu1 and between y2 and u2 . Therefore, the variable y2 , whichrepresents the current iso in the pick-up, can easily be controlledby controlling u2 . Since the other RGA elements λ12 and λ21are negative, the variables y1 should not be controlled or pairedwith u2 , and y2 should not be paired or controlled using u1 . Thisobservation from the model confirms the physical understandingof the system. For example, if the output current iso i.e., y2 iscontrolled using the input voltage vpi i.e., u1 , the primary trackcurrent iT becomes lower at lower values of input voltage vpi ,i.e., u1 . This causes the induced voltage vsi in the pick-up coilto be low. To maintain the same output voltage vso , the pick-upneeds to operate at high Q-values. This is often undesirable andmay make the system unstable.

B. Singular Value Analysis of Bidirectional IPT System

The eigenvalues of a transfer function matrix, denoted asλi(G(jω)), do not provide any meaningful information unlikethe eigenvalues of single input single output (SISO) system.One of the most useful parameters for analyzing a multiple in-put multiple output (MIMO) system is its singular values andthe condition number [20]. The singular values are the posi-tive square roots of the eigenvalues of GH G, where GH is thecomplex conjugate transpose of G and is denoted as

σi(G) =√

(GH G). (17)

4776 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 11, NOVEMBER 2012

Fig. 6. Maximum and minimum singular values and condition number ofbidirectional pick-up within ±25% of nominal operating frequency of 20 kHz.

The singular values are functions of the frequency. The con-dition number γ of a system is a measure of input–outputcontrollability of a system and is defined as the ratio betweenthe maximum and minimum singular values. This is expressedas

γ(G) =σ(G)σ(G)

. (18)

where σ(G) and σ(G) are, respectively, the maximum and min-imum singular value of the system. The plots of singular valuesand the condition number over ±25% of the normal operatingfrequency of 20 kHz is shown in Fig. 6. The condition numberof the system at 20 kHz is 12.76, which indicates that controlof this system at this frequency is not trivial. However, if theoperating frequency deviates from 20 kHz, the condition num-ber increases sharply, which indicates that the control of thesystem will become increasingly difficult at other frequencies.This confirms the physical understanding of the system.

IV. EXPERIMENTAL VALIDATION OF MODEL

In order to verify that the developed state-space model iscorrect, a 1 kW bidirectional IPT system prototype shown, inFig. 7, was built as a benchmark. The various parameters of theprototype are given in Table I.

A. Model Validation in Frequency Domain

During the first phase, the model is validated in frequencydomain. Since the bidirectional system is usually tuned to oper-ate at a particular frequency, which is 20 kHz in this particularcase, it is essential to understand the frequency domain behavioraround this frequency. The frequency response functions of thesystem were measured in the frequency range of 15–25 kHz,which is ±25% of the nominal frequency of 20 kHz, and werecompared with those computed from the model and shown in

Fig. 7. Experimental prototype of bidirectional IPT system.

Fig. 8. Comparison of magnitudes of frequency response functions computedfrom the model and observed through experiment within ±25% of the nominalfrequency of 20 kHz. The × denotes the experimental observation.

Fig. 8. From Fig. 8, it is observed that the magnitude plots ofG11 and G21 match satisfactorily with the experimentally ob-served response. However, there are some discrepancies in themagnitude plots of G12 and G22 at a specific frequency rangewhere the magnitude of the frequency response functions arevery small. This is expected because of the possible inaccura-cies in the measurement process at such lower scale.

B. Model Validation by Comparing PowerRegulation Performance

The next phase of validation is carried out by comparing thepower regulation performance of both the dynamic model andthe experimental prototype. Note that the amount of bidirec-tional power flow can be controlled either by phase or voltagemodulation. However, in this study, simple proportional andintegral (PI) controller, which exists in the prototype, has beenused to control the voltage on pick-up side at unity power factor,which alternately controls both the magnitude and direction ofpower flow. The basic principle of the method is briefly stated

SWAIN et al.: DYNAMIC MULTIVARIABLE STATE-SPACE MODEL FOR BIDIRECTIONAL INDUCTIVE POWER TRANSFER SYSTEMS 4777

Fig. 9. Simplified pick-up control strategy for bidirectional IPT system.

for sake of completeness. If the proposed model is accurate, theresults of current/power regulation using both should exactly besimilar.

With voltage control, the pick-up side voltage is generatedeither at ±90◦ phase angle with respect to the primary voltage.A lagging or leading 90◦ phase angle dictates whether the powerflow is from pick-up side to the primary side or from primaryside to the pick-up side. Note that this phase lag is in additionto the 90◦ that is required for the usual square-wave operation.The primary converter is driven as a square-wave voltage togenerate a constant voltage and keep the track current constantfor all loads. The voltage generated by the pick-up side converteris regulated as desired by driving both legs of the full bridgewith respect to each other using a phase shift αs . A zero phaseshift generates a square-wave voltage of the maximum possiblevoltage of the converter whereas 180◦ phase shift creates a shortcircuit, giving a zero voltage as in the case of typical IPT control.As the phase shift changes, the effective output current changes,and thus for a given output voltage, the power output can beregulated by varying the phase shift αs .

1) Procedure for Model-Based PI Controller Design: Thebidirectional IPT system is a MIMO system and before de-signing a fully interacting multivariable controller, it is use-ful to check on whether a completely decentralized design canachieve the desired performance. From the RGA analysis of theIPT system carried out in Section III-A, it was observed thata controller can be designed using a decentralized approach.A simplified block diagram, illustrating the control strategy, isshown in Fig. 9, where the desired power Pref is compared withthe measured power Ps and the error is fed to a PI controller witha limiter to yield the required phase angle α for modulating theoutput voltage vso(t) = u2(t). The change in vso(t) changes theeffective output current iso(t) and thereby regulates the outputpower. The output of the PI controller is expressed as

u(t) = Kp

[e(t) +

1Ti

∫ t

0e(τ)dτ

](19)

where Kp and Ti are, respectively, the proportional gain andintegral time of the controller. Many tuning rules for PI andPID controllers utilizing different methodologies have beenproposed in the literature. [21]. In this study, these gains aredetermined using the Ziegler–Nichols method, which requiresknowledge about the ultimate gain and ultimate frequency ofthe system. The steps for designing PI controller are as follows.

1) From the Bode or Nyquist plot of G22(s), determine thegain margin (GM), phase margin (PM), phase cross overfrequency ωGM , and the gain cross over frequency ωPM .

Fig. 10. Primary input voltage vp i , track current iT , phase modulated pick-upvoltage vso , and pick-up output current iso are shown. Controller output αs

and power output ps during power flow in forward direction obtained from thedynamic model under high-power condition.

2) Determine the ultimate gain Ku and ultimate period Tu

from

Ku = GM, Tu =2π

ωGM. (20)

3) Compute the proportional gain Kp and the integral timeTi from

Kp = 0.45Ku, Ti =Tu

1.2. (21)

For the system considered here, the value of the gain pa-rameters become Kp = 0.067 and Ti = 27.07μs. The transientresponse of the system with this choice of controller parameterresults in a 45% overshoot. Note that the tuning rules proposedby Ziegler–Nichols give a rough estimate of the controller gainsand these need to be adjusted to get better quality of response.The proportional gain Kp was, therefore, decreased by 1.5 timesand both the simulation and experiments were performed withKp = 0.0447 and Ti = 27.07μs that results in less than 5% ofovershoot.

2) Results and Discussion: The primary side converter ofthe system, supplied by a 150 V dc supply, was controlled toproduce a 150 V square-wave voltage and maintain a constanttrack current of approximately 40 A at 20 kHz. The pick-up wasmagnetically coupled to the track to either extract power fromthe track or deliver power back to the track. The pick-up sideconverter or reversible rectifier was connected to a 150 V dcsource, and was driven with a phase shift to produce the desiredvoltage of the pick-up side converter and regulate the outputpower.

Figs. 10 and 11 show, respectively, the results from the modeland the prototype when delivering 1 kW to the pick-up. Thefirst and second plots of both Figs. 10 and 11 shows the voltagesand currents of the primary and pick-up. From these plots, it isevident that the pick-up side converter generates a voltage Vso ,which is lagging the voltage Vpi generated by the primary sideconverter. In this situation, the primary side converter operates inthe inverter mode to deliver power to the pick-up side converter,

4778 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 11, NOVEMBER 2012

Fig. 11. Primary input voltage vp i , track current iT , phase modulated pick-upvoltage vso , and pick-up output current iso are shown. Controller output αs

and power output ps during power flow in forward direction obtained from theexperimental prototype under high-power condition.

Fig. 12. Primary input voltage vp i , track current iT , phase modulated pick-upvoltage vso , and pick-up output current iso are shown. Controller output αs

and power output ps during power flow in reverse direction obtained from thedynamic model under high-power condition.

which operates in the rectifier mode. Further, from the plot ofVso , it is observed that the pick-up side converter operates witha phase shift, and hence generates a square wave voltage whosevalue is less than the maximum possible value of 150 V. Thethird and fourth plots of both Figs. 10 and 11 show the controlleroutput and the power output of the pick-up. It is observed thatthe system exhibits satisfactory transient response with less than5% overshoot and settling time of 400 μs.

The effects of changing the direction of power flow on thevoltages and currents under high-power flow conditions areshown in Figs. 12 and 13. From the first and second plots ofFigs. 12 and 13, it is evident that the pick-up side convertergenerates a voltage Vso , which is leading, instead of lagging thevoltage Vpi generated by the primary side converter, as in thecase of power flow in forward direction. In this situation, theprimary side converter operates in the rectifier mode to receivepower from the pick-up side converter. Further, the transient

Fig. 13. Primary input voltage vp i , track current iT , phase modulated pick-up voltage vso , and pick-up output current iso are shown. Controller output αs

and power output ps during power flow in reverse direction obtained from theexperimental prototype under high-power condition.

Fig. 14. Effects of variation in coupling between the primary and the pick-upby 31.25% in forward direction obtained from the dynamic model.

response of the system is quite satisfactory as is evident fromthe fourth plot of both the figures.

After validating the model under high-power conditions at1 kW, experiments were conducted under low-power conditionsat 650 W in both the forward and reverse directions and the re-sults were compared with those obtained from the model. It wasobserved that the results from the model and the prototype arein good agreement with each other. However, the results underlow-power conditions are not shown here due to limitations ofspace.

3) Effects of Variation of Coupling and Primary TuningCapacitance: To further demonstrate that the model is accurate,the mutual inductance between the primary and the pick-up wasvaried over a wide range. The power regulation performance ofboth the model and the experimental prototype when the mutualinductance was changed from 8.0 to 5.5μH, which amounts to achange of 31.25%, is shown in Figs. 14 and 15, respectively. Theresults of power regulation as well as the output of the controllerare shown in Figs. 14 and 15 for forward directions of powerflow. The results in the reverse direction of power flow are very

SWAIN et al.: DYNAMIC MULTIVARIABLE STATE-SPACE MODEL FOR BIDIRECTIONAL INDUCTIVE POWER TRANSFER SYSTEMS 4779

Fig. 15. Effects of variation in coupling between the primary and the pick-upby 31.25% in forward direction obtained from the experimental prototype.

Fig. 16. Effects of variation of primary tuning capacitance CT obtained fromthe dynamic model.

similar, which are not given due to limitations of space. Theresults shows the performance obtained from the model and theprototype agree closely, and therefore, the model is an accuraterepresentation of the system.

During the last phase of this research, the value of the primarytuning capacitor CT was changed over a wide range. The resultsof regulation for power flow in forward direction with a 18%change in capacitance is shown in Figs. 16 and 17. The results inthe reverse power flow condition is similar. Note that the effectsof changing the mutual inductance and the capacitor are studiedunder high-power condition.

V. CONCLUSION

A dynamic model has been developed for an eight-order bidi-rectional IPT system using state variables. This model is an idealtool for both steady state and transient analysis of IPT systemsas well as for the design of controllers. Model-based analysisof the system was carried out using RGA and singular valuesand the results obtained from this analysis confirm the physicalbehavior of the system. The accuracy of the proposed model isvalidated by comparing its performance with the experimental

Fig. 17. Effects of variation of primary tuning capacitance CT obtained fromthe experimental prototype.

results collected from a prototype of a 1 kW bidirectional IPTsystem, which was operated under various conditions. First, thefrequency response functions were computed from the modeland compared with the experimentally observed frequency re-sponse, and they show a satisfactory match. Second, the controlperformance of the model in both forward and reverse direc-tions of power flow, under various conditions, such as high- andlow-power, variations of magnetic coupling, and primary tuningcapacitance, were compared with the performance of the experi-mental prototype. Results of the investigation demonstrated thatthe proposed model can accurately predict the behavior of bidi-rectional IPT system and can be used for design and control ofIPT systems.

TABLE IPARAMETERS OF PROTOTYPE BIDIRECTIONAL IPT SYSTEM

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4780 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 11, NOVEMBER 2012

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Akshya K. Swain (M’97) received the B.Sc. Engi-neering degree in electrical engineering and the M.Sc.Engineering degree in electronic systems and com-munication from Sambalpur University, Sambalpur,India, in 1985 and 1988, respectively. From 1994 to1996, he was a Commonwealth Scholar in the UnitedKingdom and received the Ph.D. degree from theDepartment of Automatic Control and Systems En-gineering, University of Sheffield, Sheffield, U.K., in1997.

From 1986 to 2002, he was a Lecturer, an AssistantProfessor, and a Professor of electrical engineering in the National Institute ofTechnology, Rourkela, India. During 1988–1989, he was an Assistant Directorin the Ministry of Energy for the Indian government. Since September 2002,he has been with the Department of Electrical and Computer Engineering, TheUniversity of Auckland, Auckland, New Zealand. His research interests includenonlinear system identification and control, biomedical signal processing, sen-sor networks, and control applications to power system and inductive powertransfer systems.

Dr. Swain acts as a member of the Editorial Board of the International Jour-nal of Automation and Control and International Journal of Sensors, WirelessCommunications and Control.

Michael J. Neath (S’08) received the B.E. degree(Hons) in electrical engineering from The Universityof Auckland, Auckland, New Zealand, in 2011, wherehe is currently working toward the Ph.D. degree inpower electronics.

His research interests include the fields of powerelectronics, inductive power transfer, wireless electricvehicle charging, and vehicle to grid systems.

Udaya K. Madawala (M’93–SM’06) receivedthe B.Sc. degree (Hons.) in electrical engineer-ing from The University of Moratuwa, Moratuwa,Sri Lanka, in 1987, and the Ph.D. degreein power electronics from The University ofAuckland, Auckland, New Zealand, in 1993.

After working in industry, he joined the Depart-ment of Electrical and Computer Engineering, TheUniversity of Auckland, in 1997, where he is cur-rently an Associate Professor. He is also a Consul-tant to the industry. His research interests include the

fields of power electronics, inductive power transfer and renewable energy.Dr. Madawala is an active IEEE volunteer and serves as an Associate Editor

for both IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND IEEE TRANS-ACTIONS ON POWER ELECTRONICS. He is a member of the Power ElectronicsTechnical Committee and also Chairman of the Joint Chapter of IEEE Indus-trial Electronics Society and Industrial Applications Society in New Zealand(North).

Duleepa J. Thrimawithana (M’09) received theB.E. degree (Hons.) in electrical engineering and thePh.D. degree in power electronics, from The Univer-sity of Auckland, Auckland, New Zealand, in 2005,and 2009, respectively.

From 2005 to 2008, he was a Research Engineer inthe areas of power converter and high voltage pulsegenerator design in collaboration with the Tru-TestLtd., Auckland, New Zealand. In 2008, he joined theDepartment of Electrical and Computer Engineering,University of Auckland, as a Part-Time Lecturer. Dur-

ing this time, he was involved in modeling of microgrids and vehicle to gridinterface systems in collaboration with the Department of Energy Technology,Aalborg University, Aalborg, Denmark. He is currently a full-time Lecturer anda Research Fellow in the Department of Electrical and Computer Engineering,University of Auckland. His main research interests include the fields of induc-tive power transfer systems and power electronic converters that are suitable forgreen energy applications.