Transmission Lines-Basic Principles 01515900

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The IEE Measurement, Sensors, Instrumentation and NDTm ProfessionalNetworkm

Transmission Lines-Basic Principles

Richard Collier,University of Cambridge

. 0 he IEEPrinted and published by the IEE, Michael F a r a d a y House, Six Hills W a y ,Stevenage,Herts SG12AY, UK


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TRANSMISSIONLINES -BASIC PRINCIPLESDr R Collier University of Cambridge

1 IntroductionThe.aim of this lecture is to revise the basicprinciples of transmission lines in preparationfor many of the lectures, which follow in thiscourse. Obviou sly this lecture cannot coversuch a wide topic in any depth and at the end ofthese notes are listed some textbooks whichmay prove useful for those wishing to gofurther into the subject.Microwave m easurements involve transmissionlines because many of the circuits used arelarger than the wavelength of the signals beingmeasured. In such circuits, the propagationtime for the signals is not negligible as it is atlower frequencies. So some knowledge oftransmission lines is essential before sensiblemeasurements can be made at microwavefrequencies.For many of the transmission lines, like coaxialcable and twisted pair lines, there are twoseparate conductors separated by an insulatingdielectric. These lines can described usingvoltages and currents in an equivalent circuit.However, another group of transmission lines,often called waveguides, like metallicwaveguid e and optical fibre, have no equiv alentcircuit and these are described in terms of theirelectric and magnetic fields. These notes willdescribe the two conductor transmission linesfirst, followed by a description of waveguides.The notes will end with some general

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comments about attenuation, dispersion andpower. A subsequent lecture will describe theproperties of some of the transmission lines incommon use today.Loss-less Two Conductor TransmissionLines - Equivalent Circuit an d Velocity ofPropagationAll two-conductor transmission lines can bedescribed using a distributed equivalent circuit.In order to simplify the treatm ent, the lines withno losses will be considered first. The lineshave an inductance per meter, L, because thecurrent going along one conductor andreturning along the other produces a magneticflux between the wires. Norm ally, at highfrequencies the skin effect reduces the self-inductance of the wires to zero so that only thisloop inductance is important. The wires willalso have a capacitance per meter, C, becauseany charges on one conductor will induce equaland opposite charges on the other. Thiscapacitance between the wires is the dominantterm and is much larger than any self-capacitance. The equivalent circuit is shown infigure 1.

dVI - CAX - dt

LhxV

Ax4 bFigure 1: The equ ivalent circuit of a short length of transmission line with no losses.


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If a voltage, V, is applied to the left hand side ofthe equivalent circuit, the voltage at the righthand side will be reduced by the voltage dropacross the inductance. In mathematical terms:

81 .V becomes v - Lh x - in a distance hx . Soatthe change AV in that distance is given by:-

dIA V =-LAX-at

ar= -L -V i3Vhx ax at-- -nd Lim

AX+OIn a very similar fashion the current, I, enteringthe circuit on the left hand side is reduced bythe small current going through the capacitor.Again, in mathematical terms:-

avatI becomes I -chx- in a

distance AxSo the change in that distance is given by : -

i?Vhl=-cAx-atdV--e-ence --Ax at* A l

dVC-bl arA x a x at-=-=-and LimAX+O

These equations are called the Telegraphistsequations:-

Differentiating these equations with respect toboth x and t gives:-

Given that x and t are independent variablesthen the order of the differentiation is notimportant, the equations can be reformed intowave equations.

The equations have general solutions of theform of an y function of the variable ( t J ?).VSo if any signal, which is a function of time, isintroduced at one en d of a loss-less transmissionline then at a distance x down the line thefunction will be delayed by L . If the signalwere travelling in the opposite direction thedelay would be the same except x would benegative and the positive sign in the variablewould be needed.

V

XVIf the finction is f ( t - ) substituting in th ewave equation gives:-.

1 X x- t- ) = LCS ( t- )v 2 V vThis shows that for all types of signal - pulse,triangular, inusoidal - there is a unique velocityon loss-less lines, v, given by

1v=- m

This is the velocity of both the cwent andvoltage waveforms since the same waveequation governs both parameters.


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2.1 Characteristic ImpedanceThe relationship between the voltage waveformand the current waveform is derived from theTelegraphists equations.I f the voltage waveform is:-

XVv = V , f ( t - )

then

Using the first Telegraphist equation:-

Integrating with respect to tim e gives:-v o x vI = - - f ( t --) =-.-Lv v Lv

This ratio is called the ch aracteristic impedan ce2, and for loss-less lines.VI- 2,

This is for waves travelling in a positive xdirection. If the wave was travelling in anegative x direction, i.e. a reverse or backwardwave, then the ratio of V to I would be equal to-Zo.

2.2 Reflection CoefficientA transmission line may have at its end animpedance, Z , , which is not equal to thecharacteristic impedance of the line 2 , . Thus,a wave on the line faces the dilemm a of obeyingtwo difference Ohms laws. In order to achievethis a reflected wave is formed. Giving positivesuffices to the incident waves and negativesuffices to the reflected waves the Ohms lawrelationships become:

V-=z,1,

where vL nd 1, are the voltage and currentin the terminating imped ance 2,.

A reflection coefficient, p or r, s defined as theratio of the reflected to the incident wave.Thus:

Since 2, for loss-less lines is real and ZLmaybe complex then in general will be complex.One of the main parts of microwave impedancemeasurement is to measure the value of r andhence ZL,

3.2 Phase Velocity and Phase Constant forSinusoidal WavesSo far the treatment has been perfectly generalfor shapes of waves. In this section, just thesine waves will be considered. In figure 2, asine wave is shown at one instant in time. Sincethe waves move down the line with a velocityof v , the phase of the waves further down theline will be delayed compared with the ph ase ofthe oscillator on the left hand side of figure 2.


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2x radians of phase delay I

Figure 2: Sine waves on transmission lines

The phase delay or -a whole wavelength isequal to 2x. The phase delay per metre is calledp and is given by:-2np= -a

Multiplying top and bottom of the right handside by frequency gives:-

Where v is now the phase velocity, ie thevelocity of a point of constant phase and is thesame velocity that is given in Section 2 if thelines are lossless.2.4 Power Flow for Sinusoidal Waves

If a transmission line is terminated in animpedance equal to z,,hen all the power in thewave will be dissipated in the matchingterminating impedance. For lossless lines asinusoidal wave with an amplitude v,he powerin the termination would be:-

If a transmission line is not matched then part ofthe incident power is reflected (see Section 2.2)and if the amplitude of the reflected wave is V,then the reflected power is

Then ,2 Power reflectedI = Incident Power

Clearly, for a good match the value of Irlshould be near to zero. The return loss is oftenused to express the match:-

1Return loss = 10 log,,-In microwave circuits a return loss of greaterthan 2OdB means that less than 1% of theincident power is reflected.Finally, the power transmitted into the load isequal to the incident power minus the reflectedpower. A transmission coefficient,z, is used asfollows:-

This is also the power in the wave arriving atthe matched termination.


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Transmitted Power 2 2=J f l =i-\rIIncident Power2.5 Standing Waves resulting from SinusoidalWaves

When a sinusoidal wave is reflected by aterminating impedance which is not equal toz,,he incident and reflected waves formtogether a standing wave.If the incident wave is:-

F+= sin(cut -@)and the reflected wave is:

where x=O at a distance D from the termination.Then at some points on the line the two waveswill be in phase and the voltage will be:-

where VMAx is the maximum of the standingwave pattern. At other points on the line thetwo waves will be out of phase and the voltagewill be

V, is the minimum of the standing wavepattern. The Voltage Standing Wave Ratio orVSWR or S is defined as:-

Now

+ ?1- Irio s=-

Measuring S is relatively easy and so a value for(r(an be obtained. From the position of th emaxima and minima the argument or phase ofr ca n be found. For instance, if a minimum ofthe standing wave pattem occurs a distance Dfrom a termination then the phase differencebetween the incident and reflected waves at thatpoint must be nr (n= 1,3,5 ..) . Now thephase delay as the incident wave goes from thatpoint to the termination is PD. The' phasechange on reflection is the argument of r.Finally, the further phase delay as the reflectedwave travels back to D s also BD. So:-

n a = 2pR + arg(r)So, by a measurement of D and a knowledge ofb-the phase of r can also be measured.

3 Two Conductor Transmission .Lines withLosses. Equivalent Circuit and Low-lossApproximation.In many two-conductor transmission lines thereare two sources of loss which cause the wavesto be attenuated as they travel along the line.One source of loss is the ohmic resistance of theconductors. This can be added to the equivalentcircuit by using a distributed resistance, R,whose units are ohms per metre. Anothersource of loss is the ohmic resistance of thedielectric between the lines. Since this i s inparallel with the capacitance it is usually addedto the equivalent circuit using a distributedconductance, G, whose units are Siemens permetre. The full equivalent circuit is shown infigure 3.

s-1or lrl=-+ l .


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Figure3: Th e equivalent circuit of a line with losses.The Telegraphists' equations become:

or

Again, wave equations can be found bydifferentiating with respect to both x and t .

a2V dV-- -L C I ( LG +RC)- + RGV2vax at ata 2 i 8I~ = LC- + (LG +RC ) - +RGI2 1ax at a

These equations are not easy to solve in thegeneral case. H owever, for sinusoidal waves onlines with small losses, i.e. WL>>R ;WC >>G there is a solution of th e form: -

v = V, exp(- m ) ~--[ 3m i ' the same as for loss-here v = 1JLCless lines: -

R GZ,a=-+- nepers m"22, 2

a = 8 . 6 8 6 -+-I&p = w&? radians m" as for loss-less lines

as for loss-less lines

3.1 Pulses on transmission lines with lossesAs well as attenuation a pulse on atransmission line with losses will also changeits shape. This is caused by the fact that all thecomponents of the transmission lines L, C , Gand R are actually different functions offrequency. So, if the sinusoidal components ofthe pulse are considered separately they alltravel at different velocities and with differentattenuation. This frequency dependence iscalled dispersion. For a limited range offrequencies it is sometimes possible to describea group velocity which is th e velocity of thepulse rather that the velocity of the individualsine waves that make up the pulse. One effectof dispersion in pulses is th e r ise time isreduced and often the pulse width is increased.It is beyond the scope of these notes to includea more detailed treatment o f his topic.

3.2 Sinusoidal Waves on Transmission Lineswith Losses

For sinusoidal waves there is a solution of thewave equation and it is: -

a + j p = .\I@+ ~ & L ) ( G j w ~ )


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an d

In general a,p and 2, are all functions offrequency. In particular, Zo at low frequenciescan be complex and deviate considerably fromits high frequency value. Since R;G, L and Calso vary with frequency, a carefulmeasurement of these properties at eachfrequency is required to characterise completelythe frequency variation of 2, .

4. Lossless WaveguidesThese transmission lines cannot be easilydescribed in terms of voltage and current asthey sometimes only have one conductor, e.g.metallic waveguide or no conductor, e.g. opticalfibre. The only way to describe their electricalproperties i s in terms of the electromagneticfields that exist in, and in some cases, aroundtheir structure. This section of the notes willbegin with a revision of the properties of aplane or transverse electromagnetic (T.E.M.)wave. The characteristics of metallicwaveguides will then be described using thesewaves. The properties of other waveguidingstructures will be given in a later lecture.

4.1 Plane (orTransverse) ElectromagneticWavesA Plane (o r Transverse) Electromagnetic Wavehas two fields which are perpendicular ortransverse to the direction of propagation. Oneof the fields is the electric field and thedirection of this field is usually called thedirection of polarisation (e.g. vertical,horizontal, etc.). The other field which is atright angles to both the electric field and thedirection of propagation is the magnetic field.These two fields together fonn theelectromagnetic wave. The electromagneticwave equations for waves propagating in the zdirection are: -

a 2H d 2 Eat2

--& 2 -PE-

where p is the permeability of the rnedium. IfpR is the relative permeability thenp =pRpo and po is the free space

permeability and has a value of 4 ~ . 1 0 - ~m-'.Similarly, E is the permittivity of the mediumand if E~ is the relative permittivity thenE = E,&, and E, is the free space permittivityand has a value of 8.854.10-'*Fm-'. These waveequations are analogous to those in section 2 ofthese notes. The variables V, I,L and C arereplaced with the new variabies E, H, p andE and the same results follow. For a planewave the velocity if the wave in the z directionis Y given by: -

1(see Section 2 where v =-v = -6If p = po and &=E , , then

v0 = 2.99792458.10' ms-'The ratio of the amplitude of the electric field tothe magnetic field is called the intrinsicimpedance and has the symbol q ,

(seeSec tion2 where 2, =

If p = po and E = E,, then q, = 376.61Qor 120~52

As in Section 2 fields propagating in thenegative z direction are related by - 7. Th eonly difference is the orthogonality of the fields,which comes from Maxwell's Equations. Foran electric field polarised in the x direction: -

If E , is a function of ( t - t) as beforethen


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I f an E , field w as chosen; the m agnetic fieldwould be in the negative x direction. Forsinusoidal waves the phase constant is calledthe wave number and given symbol k.

Inside all two conductor transmission lines arevarious sh a p e of plane waves and it is possible

to describe them completely in terms of fieldsrather than voltages and currents. Theelectromagnetic wave description is morefundamental but the equivalent circuitdescription is often easier to use. At highfrequencies, two conductor transmission linesalso have higher order modes and the equivalentcircuit model for these becomes more awkwardto use whereas the electromagnetic wave modelis able to accommodate all such modes.

4.2 Rectangular MetalLic Wav egu idesFigure 4 hows a rectangular metallic waveguide.

Ela b/ a =2 bFigure 4: A rectangular metallic waveguideIf a plane w ave enters the waveguide such thatits electric field is in the y ( or vertical )direction and its direction of propagation is notin the z direction it will be reflected back andforth by the metal walls in th ey direction. Eachtime the wave is reflected it will have the phasereversed so that the sum of the electric fields onthe surfaces of the two walls in th ey direction iszero. This is consistent with the walls beingmetallic and therefore, good conductors andcapable of short circuiting any electric fields.

The walls in the x direction are also goodconductors but are able to sustain these electricfields perpendicular to their surfaces. Now, ifthe wave after two reflections has its peaks andtroughs in the same positions as the originalwave then the waves will add together and forma mode. If there is a slight difference in thephase the vector addition after many reflectionswill be zero and no mode is formed. Thecondition for forming a mode is thus a phasecondition and it can be found as follows.

MetalWaveguideWall ,

0'Direction of propagationor Wave Vector/

0" r / Wall

1 8 0 I


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Figure 5: A plane'wave in a rectangular metallic waveguideFigure S shows a plane wave in a rectangular As this wave is incident on the right wall of th emetallic waveguide with its electric field in the waveguide in figure 5, it .will give reflectiony direction and the direction of propagation at according to the usual laws of reflection asan angle @ to the z direction. . shown in figure 6.

The rate of change of phase in the direction ofpropagation is k, the free space. wave num ber.

A90"-20

Fields cancelJ t metal walls

BFields add

-at thecentreTElomode

Figure6:Two plane waves in a rectangular metallic waveguide. The phases 0" and 180"refer to the lines

On further reflection, to form a mode this'wavemust 'rejoin' the original wave. So , figure 6also shows the sum of all the reflections Th e phase delay can be found from resolvingforming two waves one incident on the rightwall one on the left. The waves form the modeif they are linked together in phase.Consider the line AB. This is a line of constantphase for the wave moving to the right. Part ofthat wave at B reflects and moves along BA toA where it reflects again and rejoins the wavewith the same phase. At the first reflectionthere is a phase shift of n. Then along BAthere i s a phase delay followed by another phaseshift of ai the second reflection. The phasecondition is: -

below each figure.where m = 0 , 1,2,3etc.

the wave number along BA.This is : -k , sin 2 0 radians m"

If the walls in the y direction are separated by adistance a, then: -aA B = -cos 0

acos0So the phase delay is k , sin 2 0- r2~ + phase delay along BA for wave movingto the left = 2 m 9 2k, sin0


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Hence the phase condition is:k , u s i n@ =m x w h e r e m = 0 , 1 . 2 , 3 ..The solutions to this phase condition give thevarious waveguide modes for waves with fieldsonly in th ey direction, i.e. the TE,, modes. ntis the number of half sine variations in the xdirection.

4.3 The Cut-off ConditionwFrom the phase condition since k, =- he nv o

v p l rau s i n 6 =- s the phase condition.

The terms on the right hand side are-constant.For very high frequencies the value of @ tendsto be zero and the tw o waves almosi propagatein the z direction. How ever, if # reduces invalue the largest value of sin @ is 1 and at thispoint the mode is cut off and can no longerpropagate. The cut-off frequency is a, nd isgiven by:-

v omnw, =- Uor f,=- 0m where f, s the cut-off frequency.2u

2aa = - where IC s the cut-off wavelength.mA simple rute for TE,, modes is that at cut-off,the wave just fits in sideways. -Indeed,since0 90 at cut-off the two plane wavesare propagating from side to side with a perfectstanding wave between the walls.

4.4 The Phase Velocity. All waveguide modes can be considered interms of plane waves. Since the simpler modes,

just considered consist of just two plane wavesthey form a standing wave pattern in the xdirection and yet form a travelling wave in the zdirection. Since the phase velocity in the zdirection is reIated .to the rate of change ofphase, i.e. the wave number then: -

wVelocity in th e z direction = wave number in the i: directUsing figure 5 or 6 the wave number in the tdirection is

k, cos0??lENo w from the phase condition sin @ =-oa

Where V , is the free space velocity

Hence the velocity in the z directionwV I = k , c o s 0

As can be seen from this condition when& isequal to the cut-off wavelength (see Section4.3) then v, is infinite. As & gets smallerthan A, then the velocity approaches v, . Th ephase velocity is thus always greater than V o .Waveguides are no t normally operated near cut-off as he high rate of change of velocity meansimpossible design criteria and high dispersion.

4.5 Th e Wave ImpedanceThe ratio of the electric to the magnetic field fora plane wave was discussed in Section 4.1 ofthese notes. Although the waveguide has twoplane waves in it the wave impedance is defined


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as the ratio of the transverse electric andmagnetic fields. For TE modes this is given thesymbol ZTE.

where E,and Ho efer to the plane waves.The electric fields of the plane waves are in they direction but the magnetic fields are at anangle 0 to the x direction.so

4.6

Thus, for the A, 5 A, the value of z isalways greater than 77, . A typical value mightbe 500R.The Group V elocitySince a plane wave in air has no frequencydependant parameters like those of the twoconductor transmission lines, i.e. po and 6,are constant, then there is no dispersion and sothe phase velocity is equal to the group velocity.A pulse in a waveguide.therefore, would travelat V , at an angle of @ to the z axis. The groupvelocity along the z axis is given by

Group velocity, vg = V, COS@

z plane. This will involve the wave reflectingfrom all four walls. If the two walls in the xdirection are separated by a distance b then thefollowing are valid for all modes: -

m = 0, 1,2,n = 0, 1,2,

VThen the velocity v = AvAz, = -

v, = A v oThe modes with the magnetic field in the ydirection - the. dual of TE - are calledTransverse Magnetic Modes, or TM. Theyhave a constraint that neither m or n can be 0 sthe electric field for the8e m odes has to be zeroat all four walls.

The relative cut-off frequencies are shown infigure 7. As can be seen in that diagram mono-mode propagation using the T Elo mode ispossible up to twice the cut-off frequency.However, the full octave bandwidth is not usedas propagation near cut-off is difficult and jus tbelow the next mode can be hampered byenergy coupling into that mode as well.

Th e group velocity is always less than v, and isa function of frequency.For rectangular metallic waveguides: -

Phase Velocity x Group Velocity = V:4.7 General Solution

In order to obtain all the possible modes in arectangular metallic waveguide the plane wavemust also have an angle I to the z axis in th ey


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TE4Figure 7: Relative cut-off frequenciesfor rectangular m etallic wavegu ides5. GeneralReferences1. Fields and Waves in CommunicationElectronics,S . Ramo, J A . Whiney & T.Van Duzer. Wiley, 1994.2. Wav eguide Handbook, N. Marcowitz,Peter Peregrinus Ltd, 1986.

Field and Wave Electromagnetics,D. K.Cheng, Addison Wesley, 1989.3.

4. Electromagnetic Fields, Energy, and5Waves, L. M.Magid, 1972.5 . Electranagnetics,Fifth Edition.J.D.Kraus. McGraw-H ill, 1999.6. ElectromagneticWaves and RadiatingSystems,Second Edition, E.C. Jordan.Prentice-Hall,1968.7. Transmission lines, Schaum9sOutlineSeries. McGraw-Hill, 1968.

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Transmission Lines-Basic Principles 01515900