Electromagnetic 8

8/13/2019 Electromagnetic 8

http://slidepdf.com/reader/full/electromagnetic-8 1/30

C H A T E

Transmissionines

8-1 GeneralConsiderations

8-2 Lumped-ElementModel

8-3 Transmission-LineEquations

8-4 Wave Propagationon a Transmission ine

8-5 The Lossless iansmission ine

8-6 Input Impedanceofthe Lossless ine

8-7 SpecialCasesof the Lossless ine

8-8 PowerFlow on a Lossless iansmission ine

8-9 The Smith Chart

8-10 ImpedanceMatching

8-11 Transientson Ttansmission ines



TnRrslilssr0NrNEs

3-1 Generalonsideralions

.\tthough he amilyof transmissioninesmayencom-

:J5s llstructuresndmediathatservetotransferenergyr

:brmation betweenwopoints,ncluding erve ibersn

= human ody,acousticwavesn fluids, andmechanical

ressurewavesnsolids,weshall ocus urtreatmentinhis

:3pteron ransmissionines sedorguiding lectromag-

:ericsignals. uch ransmissionines nclude elephone.rres,coaxial ables arryingaudioandvideo nforma-

:.)n o TV sets r digitaldata o computermonitors, nd:rdcal ibers arryingightwavesor the ransmissionf:Jaatveryhighrates. undamentally,transmissionine: a two-portnetwork,with eachportconsisting f twocrminals, s llustratedn Fig.8-1.Oneof thepons sthe<ndingendand heother s the eceivingend.Thesource::,rnnectedo its sendingend may be any circuit with anr.rtput oltage, uchasa radar ransmittet anamplifier,or

I :omputererminal operating n the transrnissionmode.--:om

circuit theory any suchsourcecanberepresented

by a Thdvenin-equlvalenteneratorcircujl consisting fa generatorvoltage V, in serieswith a generatoresis-tanceRr,asshownn Fig.8-1.Thegeneratoroltagemayconsist f digitalpulses, rnodulatedime-varying inu-soidal ignal, r anyother ignalwaveform.n thecase fa-csignals,hegeneratorcircuits representedyavoltage

phasorYgandan mpedance r.Thecircuitconnectedo the eceiving ndofthe rans-

missionine s called he oadcircuit,or simply he ood.This may be an antennan the caseof a radar,a com-

puter erminal peratingn the eceivingmode, he nputterminals f anamplifier, r anyoutput ircuitwhosen-put terminals an be representedy an equivalentoadresistance 1,or a oad mpedance 1 in thea-ccase.

8-1.1 TheRole lWavelength

In low-frequencylectrical ircuits,weusually sewiresto connect he elements f the circuit in the desired

I

r Sending-endI Dortt '

[igurr 8-l: A nansmissionine is a two-portnetworkconnecting generatorircuit at the sending nd o a load at the:eceiving nd,

245



246

configuration.lnthecircuitshowninFig.-2, orexample,

thegenerators connectedo a simpleRC loadvia apair

of wires. n view of ourdefnition in thegeccdingpara-

graphs fwhatconstitutes ransmissionine,weposehe

followingquestion:sthepairof wiresbetwe€nerminals

AA'and erminalsB' a ransmissionine? f so,why s timportant? fter all,weusuallysolveor thecurrentn the

circuitand hevoltageacrosstselementswithoutregard

for thewiresconnectingheelements. heanswero this

questionsyes;indeedhepairofwires onstitutes rans-

missionine,but he mpact f the ineon hecurrent nd

voltagesn thecircuitdependsn he engthoftheine and

thefrequency ofthesignalprovidedbyhegenerator'As

we will seeater, hedetermining actor s theratioof the

lengthto hewavelengthofthewave ropagatingn he

transmissioninebetween A' andBB'.) If thegenerator

voltages cosinusoidaln time, hen hevoltageacrossheinput erminals A' is

VM,=Vs(t) : Yocosot (V) ' (8.1)

wherea.r 2rl is heangularfrequency,nd fweassume

thatthecurrent lowing through hewirestravelsat the

speed f light, c = 3 x 108m/s, hen hevoltageacross

theoutput erminalsBB' will have o be delayedn time

relativetothatacross.AA'bythetraveldelaytime//c.Thus'assumingosignificanthmic ossesn the ransmission

line,

Vtt,G)-- Vtx(t - l/c)

= Vocos[a(t l/c)] (V). (8.2)

I-et us comPareVsp, to Vp' at t : 0 for an

ultralow-frequencylectronic ircuitoperating t a fre-

quency / = I kHz. For a typical wire length

I : 5 cm, Eqs. 8.1)and(8.2) giveV11' = y0 and

Vss, : Vocos(2ltl lc) :0.999999999998V6 Thus,

forallpracticalurposes,heengthofhetransmissionline

maybegnorednderminalAA' maybe reateds denti-

calwithB B'. On heotherhand,ad he inebeen 20-km

long elephoneable arryinga l-kHz voicesignal,henthesame alculation ouldhave ed o Vps' : 0.91V0'

CHAPTER8 TRANSMISSION

The determining factor is the magnitude of r.-rl/c.Eq.(7.91),he velocityof propagationo of a

wave srelatedo heoscillationfrequency and he

length . by

ur: f)' (r/s).

In thepresentcase,rp : c' Hence, hephaseactor

u l 2nf l - I= :

--:--!-:2z - radians.

c c i '

When /l is verysmall, ransmission-lineffectsignored, utwhen /). ] 0.01, t maybenecessaryo

count not only for thephase shift associatedwith the

delay, utalsoor hepreserrcef eflected ignals

havebeenbouncedback by the load toward the

Porver osson the ine anddispersiveeffectsmay

beconsidered swell. A dispersive

on which thewave velocity is not constantasa

the frequency / . This means hat the shapeof arectan;

lar pulse,which throughFourieranalysis s composed

manywaves fdifferentrequencies,ill bedistorted

travelsdown he ine becausets different requency

ponents ill notpropagatet hesame elocity Fig'E-

,j Transmissionine ,tj

|<-l----------{

Figure8-2:Generatoronnectedo anRCcircuit hroqf

a transmissionine of length .



.1 GENERALCONSIDERATIONS 247

rL[LfL* -- Jl_fLlL-D

Dispcrsionlcssine

rLfLfL* -. J\-AJL-Shortdisoersiveine

IJLIL*

acterized yelectricandmagneticields hatareen-thely transverseothedirectionofpropagation, hisis calleda TEM mode.A goodexamFles the coax-ial line shown n Fig. 8-5;the electric ield linesarein the radialdirectionbetweenhe innerand outerconductors,he magnetic ield forms circlesaroundthe nnerconductor, ndhenceneitherhasanycom-ponents long he engthof the ine (thedirectionofwavepropagation). therTEM transmissioninesincludehe wo-wireine and heparallel-plateine,bothshowninFig.-4.Althoughhe ields resenrna microstrip ine donot adhereo theexactdennitionof a TEM mode,he nontransverseieldcomponentsaresufficiendymall n comparisonothe ransversecomponentss o be gnored,hereby llowing heinclusionofmicrostripinesntheTEMclass. com-mon eature mong EM ines sthat hey

consist ftwoparallel onductingurfaces.

Higher-orderyansmis ion ines..Waves ropagat-ingalong heseines ave t east ne ignificantieldcomponentn the direction f propagation.ollowconducting aveguides,ielectricods,andoptichlfibers elongo hisclass flines.

Only TEM-moderansmissionineswill be reatednthischapter. his s becauseessmathematicaligor s re-quiredor treatinghisclass flines han hatrequiredortreating avescharacterizedyhigher-ordermodesnd,naddition, EM inesaremore ommonly sednpractice.Westartour treatment y representinghe ransmissionline n terms fa lumped-elementircuitmodel, nd henweapplyKirchhoff's oltage ndcurrentaws o derivesetof wogoveming quationsnownastheelegrapher'sequations.Byombininghese quations,eobtainwaveequationsor the voltage ndcurrent t anypointon the

line. Solutionof the waveequationsor the sinusoidal

---.,_ryyLLong dispenive line

Figure -3:A dispersionlessine does ot distort ignals:=ssinghrought regardlessf its ength,whereas dis-rersiveine distortsheshape fthe inputpulses ecause

:e differcnt requency omponents ropagate t different:iocities.The degreeof distortion s proportional o the::sth of thedisDersiveine.

ofpulseshapes very mportantn high-speedtransmission,oth between erminalsas well as nspeedntegated ircuits n which ransmission-line

4n and abricationprocessesrean ntegralpartof thersignprocess. t l0GHz, orexample,hewavelength= I cm n airand s on heorderof I cm n a semicon-

:-ormaterial.Hence,evenconnection engthsbetween:ces n heorderofmillimetersbecome ignificant,andpresenceas o be ncorporatedn theoveralldesign

ir circuit.

' .2 Plopagationodes

.:"rexamples fcommon ypesoftransmissioninesare

.- ninFig.8-4.Transmissioninesmaybeclassifidinto: basic ypes:

t Transverse electromagnetic (TEM) transmission

/rnes.'Wavespropagating along these ines arechar-



248 CHAPTER8 TRANSMISSION

p*

(b) Two-wire inea) Coaxial line

metalstrip conductor

metalgroundplane

di€l€ctric spacing

TEM Transmission

(e) Microstrip line

Lines

#-n"",.,r"dielectriclayers

<net8l

L,/

/^"tut grounapunz dieleclric spacing

(h) Coplanarwaveguide(f) Rectangularwaveguide (g) Optical fiber

Higher Order Transmission ines

Figure 8-4: A few examples of transverseelectomagnetic (fEM) and highef-order ransmission lines.

melal

dielectricspacing

(c) Parallel-plate.ine

diel€ctricspacing

(d) Strip ine

steady-stateaseeadso a setofformulas hatcanbeused

forsolvingwide ange fpractical roblems.n he atter

partof thischapter e introduce graphicalechnique

known s heSmitftclrarl, which acilitateshesolutionof

manyransmissionJineroblems ithouthavingoper-form aboriousalculationsnvolving omplex umbers.

8-Z Lumped-ElementModel

When we draw a schematicof an electronic

use specific symbols to represent esrstors,

inductors,diodes,and the like. In eachcase, he

representshe functionalityof the device, ather



i2 LUMPED-ELEMENTMODEL u9

- - - Magnetic ield lines

- Elcctic field lincs

G€nerator

Crosssection

Figurr E-5: n a coaxial ine, the electricfield lines are n the radialdirectionbetweenhe inner andouter conducto$,and hemagnetic ield formscircles aroundhe innerconductor

r.,shape, izeor otherattributes.Weshalldo thesamer:th regard o transmissionines; we shall represent

'rsntissionine b1,a parallel-wireconfguration,asrown nFig.8-6(a),egardlessf hespecifc hape f the

i under onsideration.hus,Fig.8-6(a)may epresenti:oaxial ine,a wo-wire ine,or anyotherTEM ine.

Drawingagain on our familiarity with electroniccir-rts, wheD eanalyzeacircuitcontainingransistotwerpresenthe unctionalityof the hansistorby an equiva-ertcircuit omposedf sources,esiston, nd apacitors.i. will apply hesame pproachothe ransmissionine

:"'orientinghe ine along hez-direction, ubdividingtmodifferential ections achof lengthAz tFig.8-6(b)lr,1 hen epresentingach ection yanequivalent ircuit,r illustratedn Fig. 8-6(c).This representarion,hich slledthe lumped-eementcircuit model, onsistsf four'rsicelements, hichhenceforth ill becalledhe razs-nissionEneparamelers.Theseare

; : Thecombinedesistancefbothconductorserunitlength,n Q/m,

-. : Thecombinednductanceofbothconductorsperunitlength,n Vm,

G': Tlte conductanceofthe insulationmediumoer unitlength, n S/m, and

C': The capacitance of the two conductorsper unitlength, n F/m.

Whereashefour line parametersavedifferentexpres-sions or differentypes nddimensionsf transmissionlines, heequivalent odel epresentedy Fig.8-6(c) sequally pplicableo all transmissioninescharacterized

by TEM-modewave ropagation-heprimesuperscriptis used.sa reminderhat he ineparametersredffir-entialquantitiesvhose nitsareper unit ength.

Expressionsor the ine parametersR' L' , G', andC ,

aregivenn Table -l for he hreeypes fTEMtransmis-sion inesdiagrammednparts a) hroughc)ofFig. 8-4.For eachof theseines, heexpressionsre unctions ftwosets fparameters:1)geometricarametersefiningthe cross-sectionalimensionsf thegiven ine and 2)electromagneticonstitutivearametersharacteristicfthematerials f which heconductorsnd he nsulating

materialbetweenhemaremade.Thepertinentgeometricparametersreas ollows:



CHATTER8 TRANSMISSION

(a) Parallel-wire cprcsentation

(b) Differential ections achAz ong

(c) Each ections representedy an equivalent ircuit

Figur€ E-6: Regardless f its actualshape, TEM transmissionine is representedy theparallel-wireconfigurationshownin (a).To analyz€hevoltage ndcurrcnt elations,he ine s subdividedntosmalldifferential ectionsb),each f which sthen epresentedy anequivalentircuit c).

C'Az G' C'Lz G', C'Az G'

l-<-&--.--.--------- l-<-&________+l<_Az--"."._-------- l-r_Az_______-.'l

Coaxial line IFig. 8-4(a)]

a : outer adiusof inner conductor,m

b : inner radiusofouter conductor,m

Two-wire ine I Fig. 8-4(b) :

a = radiusofeach wire,m

d : spacingbetweenwires'centers,m

Paralle -p ate line I Fie. 8-4(c :

rr,l width ofeachplate,md : thickness finsulationbetweenplates,m

The constitutiveparameters pply to all three inesconsistof two groups:l.[cand o" are the magneticmeability and electricalconductivityof theande, p, and o arethe electricalpermittivity,permeability,andelectricalconductivityof thematerialseparatinghe conductors.AppendixBtabulated alues or these onstitutiveDarametersorious ypesof materials. or thepurposes f thechapter. eneed otconcem urselves irh he eriresponsibleortheexpressionsivennTable - Themulationsecessaryorcomputing' L' G',and,C'

madeavailable n eadierchapters or the generalcaseany wo-conductor onfiguration.



i.2 LTJMPED-ELEMENTMODEL 251

Table 8-1: Transmission-line arametcrsR' L', G', andCt for three ypesof lines.

PNrameter Coaxial Ttvo Wirt Parallel Plate Unit

C '

#(:.;)f;nP1"1

2tt o

l"(b t)

2n e

l"U t)

Rs

TA

2R,

w

ttd

w

q w

d

tu-;a

f,2lm

Lnl<a/utr L l rd/r". . . t ) lVm

,fo

nl lapt+J@lZf-t ]S/m

TE

nl<apt+,/@/r"7-tlF/m

Notes: l ) Refer o Fig. 8-4 or definitions f dimensions.2) p, e, ando p€rtaino theinsulatingmaterialbetween he conductors.3) Rs= JnTpJ\. (4) g," and o" pertain

to rheconductors.5) t (d/2a)2 >> l,t}ler nl@/2al + ,t@p8 11- In(d/a).

The lumped-element odel shown n Fig. 8-6(c)trresents the physical processes ssociatedwith the:Jrents and voltages n any TEM transmissionine.-r-herequivalent models are available also and are-:uallyapplicableswell.All thesemodels, owever,lead: exactly he same etof telegrapher'squations,rom

rtch all our uture esultswill be derived.Hence, nly:e model escribedn Fig.8-6(c)will beexaminedntherlsent treatment. t consistsof two serieselements,R'

rd Z', and wo shuntelements, ' andC' . By way of::,vidingaphysical xplanationor the umped-element:,iel, letusconsider smallsection fa coaxial ine,asrown n Fig.8-7.The ineconsists fan nnerconductorr zdiusa separatedromanouter onductingylinder f-"irusb by a materialwithpermittivity ,permeability ,

adconductivityo. The wo metalconductors remade f.:aterial with conductivityo" andpermeability r"When

. .oltase ources connectedcrosshe woconductorst

thesending ndofthe ine,currents ill flow hroughheconductors, rimarilyalong he outer surface f the nnerconductorandhe nner urface ftheouterconductor.heline resistance ' accountsor the combined esistanceperunit ength fthe nnerandouterconductors.heex-pressionor R' wasderivedn Chapter and s givenby

Eq. 7.151)s

R i : (o/rn), (8.5)

where Rr, which representshe surface esistance f the

conductors,s called he ntinsic resisrarrce nd s given

byEq. (7.147a) s

*,:rE (a). (8.6)

#(j . ;)



CHAPTER8 TRANSMSSION

model. tsexpressionsgiven y Eq. 4.76) s

Figure 8-7: Crosssectionof a coaxial ine with innercon-

ductor of radius c and outer conductor of radius b. The

conductorshavemagneticpermeabilityp., and conduc-

tivity o6, and the spacingmaterial b€tween the conductors

haspermittivity s, permeability p, and conductivity o.

The ntrinsic esistanceependsot only on hematerial

properties f the conductorso" and/."), but on thefre-

quency of thewavehaveling nthe ineaswell. Fora

perlectconiluctorwith o" = oo or a high-conductivity

material uch hat f p./o"\ (( 1, R' approachesero,

and odoesR'.

Next, et usexaminehe nductanceerunit engtha'.Application f Ampdre'saw n Chapter5o hedefinition

ofinductanceed othe ollowingexpressionEq. 5.99)l

for he nductance€runit ength fa coaxialine:

f | _

Theshunt onductanceerunit engthG' accountsor

currentlowbetweenheouter nd nnerconductors,ade

possibley hematerialconductivityof he nsulator.tis

preciselyecausehe urrentlow s romoneconductorto

theotherhatG'is ashunt lementn the umped-element

t:ffi (S/m).

Ifthe materialseparatinghe nnerandouteraperfect dielectic with a = 0, thenG' = 0.

The ast ineparameternourlist s he

unitlength '. Whenequalandoppositecharges

onany wononcontactingonductors,voltage

will be nduced etweenhem.Capacitances

the atioof chargeo voltage ifference.or he

line,C isgiven yEq. 4.17) as

7 r c

C ' : . : ' . , : . , ( F / m ) .ln\b a' )

All TEM transmissioninesshare he following

relations:

and

L 'C ' : pe , (8 .10)

REVIEWUESTIONS

Q8.1What s a transmissionine?Whenshould

missionline ffects econsidered?

+^(:)

If the nsulatingmedium etweenheconductorss

transmissionineiscalledanairize(e.g.,

two-wire ir ine).Foran ir ine,e = eo 8.854 I

( rVm) . (8 .7 )F lm 'p : Fo :4n x 10-7Vm'o :0 'andG ' :O

G ' 6

e(8 . r )



i-3 TRANSMISSION-LINE EQUATIONS ,<1

Q6,2 What is the differencebetweendispersiveandndispersive transmission ines?What is the practicalognificance?

Q63 What constitutes TEM transmissionine?

J6.4What purposedoes he lumped-element

ircuitr.:delserve?Howarethelineparameterst, ',G',and,C'Elated o the physicalandelectromagnetic onstitutiveTrDerties f the ransmissionine?

=CRCISE.1 UseThble8-1 o computehe ineparam-=rs of a two-wire air line whosewires areseparatedy. :stance f2 cm,andeachs I mm n radius. hewires:,r be reated soerfect onductors itho" : m.

r:s. R' : 0, L' : l.2O QtWm), G' = 0,-

: 9.29 pF/m). (SeeC)8.2 Calculatehe ransmissioninenarameters

. MHz or a rieid coaxial ir inewith an nnerconduc-Jiameterof 0.6 cm and an outer conductordiameter1.2cm. The conductorsare made of copper seeAp-

dix B forp" ando" ofcopperl.

R' : 2.O8 l0-2 (Q/m), t' : 0.14 pWm),o)0, C' :80.3 (pF/m). (See

3 Transmission-Linequations:ansmissionine usually onnects source noneendr loadon theotherend.Beforewe considerhecom-: circuit, however,we need o developequationshat

ibe the voltageacross he transmissionine and he

carriedby the ine asa unctionoftime I andspatialionz. Usingthe umped-elementmodeldescribedn8-6(c),we beginby consideringa differential ength

: s shownn Fig.8-8.Thequantities 2, ) and (2, \the instantaneous oltageandcurrent at the left

of the differentialsection nodeN), andsimilarly: - Az, t) and (z + Ae, ,) denotehesame uantities

1<-Aa

FigureE-8:Equivalentircuitofa differentialengthAzof a two-conductorransmissionine.

at the right end(nodeN * l). Application of Kirchhoff'svoltage aw accounts or the voltagedrop acrosshe seriesresistanceR'Az and nductanceL' A.z'.

u(2, ) - R'Lz i (2, )

2 ( " , \- t ' t z -

- r l : " - t ) (z+ Lz, r )=0 . (8 .12 )i l t

Upon dividing all termsby Az and rearrangingerms,weobtain

Iu(z Lz - r \2, t ) t )

AZ

In the imit asAz -+ 0,Eq.(8.13)becomes differential

eouatlon:

: n ' k , , l t ,u f ' ' )

0u(2 .). , o i ( 2 .

)- a , = K t l z ' t t+ L - -T l -

(8.13)

(8.14)



254

Similarly,applicationof Kirchhoff's current aw at node1V+ I in Fig.8-8 eadso

i(2, ) - G' Lzu(z+ Az, )

- C , L ZAUQ Lz, )

Upondividingalltermsbyz andtakingthelimitasz --+

0,Eq. 8.15) rovides second ifferential quation,

\ i ( z . t \ . , } u ( z , t )- ; = G ' u t z , r )+ C ' : : i : : - : . ( 8 . 1 6 )

The irst-order ifferential quationsivenbyEqs. 8.14)

and 8.16)are he ime-domainorm of the ransmission

Iineequations, therwise alledhe elegrapher's qua-

tions.Except or the astsection, urprimary nterestn thischapterisnsinusoidalteady-stateonditions.o hisend,we shallmakeuseof phasors ith the cosine eferencenotation soutlinedn Section -1.4.Thus.wedefine

Line

The wo i rst-ordercoupledequationsivenbyEqs.

-i(z* Lz, ) = 0. (8.15)

and 8.18b)canbecombined ogive wosecond-ordr

coupledwaveequations,neor V(z)andanotherfaThe waveequation or V(x) is derivedby


8-4 Wave lopagationnaTra

bothsides fEq.(8.18a) ithrespectoz,giving

-{:P = rn, .t,tdt!r\ .az ' 4z

+f v2i(z):0, s.21)

s2 i t - t

# - y 2 r ( z ) : 0 .( s . 2 3 )

at

anduponsubstituting q. (8.18b) nto Eq. (8.19)

d (z)/dz,Eq. 8.19) ecomes

a'7e)- (R'+ jaL')G' -t ac' t i tzl -- o,dz2

u(2,) :ne[i (a ej ' ' ] ,

i(2, ) : nef k) ej''),

(8.17a)

(8.17b)

where7(z) and (7) arephasor uantities,ach f whichmaybe realor complex.Uponsubstituting qs. 8.17a)

and 8.17b)ntoEqs. 8.14) nd 8.16) ndutilizingheproperty ivenby Eq. 7 45) hat0 0r in the imedomainbecomesquivalento multiplication y rr-rn thephasor

domain,weobtain he ollowingpairof equations:

where

(R '+ j aL ' ) (G '+ j aC ' ) (8.22)

Application f thesame tepso Eqs. 8.18a) nd 8.

but n reverserder,leadso

dV(z l(R + jaL ' ) I ( z ) . (8 .18a )

az

d i t z t(G + j@c ' ) Y (z ) . (8 .18b )

dzEquations8.21) nd 8.23) re alled apeV(z) and 1(2), respectively,

ndy

is called theThese re he elegrapher'squationsnphasororm. propagationconsrdntof the ransmissionine. As



{ WAVEPROPAGATIONONA TRANSMISSIONLINE , {<

= n . (

--'osistsofa realpadrl, calledtJileancualion constontof

:e line with unitsofNp/m,andan maginaryPartp,called

Ephase const4n!of the inewith unitsof rad/m.Thus,

y - _ u + l f (E.24)

t = Be(y)

wave amplitudes yo-, /o-) of the *z propagatingwaveand Vf , If ) of the-z propagatingwave.Wecaneasilyrelatehc currentwavearnpliodes,1o+ nd f , to thevolt-agewaveamplitudes,Vo+ nd Vf, respectively, y usingEq. 8.26a)n Eq. 8.18a) nd hensolving or thecurrent1(z) oget he esult

iOl : _ J - lvn+e-vz -v;erz). G.271R '* iaL ' ' "

Comparisonf each ermwith hecorrespondingerm ntheexpression given by Eq. (8.26b) leads o the conclusion

that

: Z o = (8.28)

where

is defined as the characteristic impedance of the line- lt

shouldbenoted haitZo s equal o theratio ofthe voltage

amplitude o the curent amplitudeor eachof the avel-

ing waves ndivirlually (with an additional minussign inthe caseof the -z propagating uave),but it is not equal

to the ratio of the total vobageV (z) to the total current

I (z\, unlessone ofthe two v'ayes s absent n termsof 26,

Eq. (8.27)can berewritten n the orm

I : lm(y)

(Np/m),

(8.25a)

(radlm).

(8.2sb)

-: Eqs. 8.25a) nd(8.25b),we choosehe square-root-ues hatgivepositive aluesor a andp. Forpassive

:--rsmissionines, is either eroorpositive.Mosttrans---.sionlines, nd ll hose onsideredn his hapter,re f:c passiveype.Theactive egion fa lasers anexampler :n active ransmissionine with anegative .ThewaveequationsivenbyEqs.8.21) nd 8.23) ave

--.ielingwave olutions f the ollowing orm:

i 127 v; "-" t vo eYz (v), (8.26a)

I1z'1 I;e-t, ,, J;"", (A), (S.26b)

-ere,analogouso theplane-wavecasein ection7-3,the: termrepresents avepropagationn the+z-directionthe e/z term representswavepropagationn the -z-

:::ction. Verification hat hese re ndeed alid solutions

:asily accomplishedby substituting he proposedex-

aswell as heir second erivatives,nto Eqs.I I )and 8.23).n heirpresentform,hesolutions iven

Eqs. 8.26a) nd 8.26b) ontain our unknowns,he

zo=R'+ u)L ': ,W- (or . (8.29)

Y \ G ' + i a ; (

-v;Io

v 0

r

- v^+ vÎ t?\ = --:Le-Yz - --LeYz.

Zs Zo(8.30)

In later ections, ewill applyboundary onditions t heload nd tthe ending ndof he ransmissionine oobtain

expressionsor the remainingwaveamplitudes n+ nd

(R'+ jaL')(G' + jaC'\

(R'+ jaL')(G' + joC')



256

Vf . Ingeneral, achwill beacomplexquantitycomposedof amagniode ndaphase ngle. hus,

vo+- 1v{1"io*

v; : lv ; le io- .

(8.31a)

(8.31b)

Uponsubstitutinghese efinitionsn Eq. 8.26a) nd e-placing withEq. 8.24),wecanconvert ack o he imedomain o obtainan expressionor r(z,t), the nstanta-neous oltage n he ine:

u(2, ) :rie(V k)ej'' )

:nel(vie-", + v;eYz) j.tl

= ne[lv{ leio+ jate-@+ia)z

+ lV; leio-ei 't et"+iq)zl

: lvdle-", cos(e)t pz + O+)

* lVo e"'cos(at+ Fz + Q ). (8.32)

Fromourreview ftravelingwavesnSection - .2, werecognizehe irst erm n Eq. 8.32) sawaveravelingnthe+z-direction thecoefficientsft andzhave ppositesigns) nd hesecondermasa wave ravelingn the z-

directionthe oefficientsft and arebothpositive),othpropagating ithaphaseelocityupgiven y Eq. 7.14):

(t)

; (8.33)

The factor e-"2 accounts for the attenuation of the *z

propagating wave, and he e"z accounts or the attenuation

of the -z propagating wave. The presenceof two waves

on the ine propagatingn oppositedirectionsproduces

standing wave.Togaina physicalunderstanding fwhat

this means,we shall irst examine he relativelysimplebut

importantcaseofa losslessine (a : 0) and henextend

the results o the moregeneral aseof lossy ransmissian

lines (a * 0). In fact, we shall devote he next several

sectionso thestudyoflosslessransmissioninesbecause

inpracticemany inescanbedesigned o exhibit very ow-

losscharacteristics.


ffi ntuntAt ab Ene s a transmissionine for which air

dielectric materialpr€sentbtween the twowhich rendersG' : 0.In addition, hemadeof a materialwith high conductivityso hatNi'For an air line with characteristicmpedance f 50phase onstant f20rad/mat700MHz, findpermeterand he capacitanceermeterof the ine.

Solution: he ollowing uantitiesre iven:

Zo :50Q, f :20 rad lm ,

f :7NMHz:1 x IOB z.

with R' : G' : 0, Eqs. 8.25b) nd 8.29) educe

fl:

Jn \t/(

t'tL'\{ l. C'))

= tn ( .,[rc '\ = .J L'c ,\- ,/

I t a L t LZ n : l - : . / - ."

\ i ' c ' \ c '

The atio sgivenby

2 n x i x 1 0 8 x 5 0

:9.09 x 10-tt(F/m) 90.9 pF/m).

From22Z6JT /C'-,

L'= z1c': (50)2 90.9x 10-12

:2.27 x t0-7(Wm) - 227 nHtm). I

Rc ' = '

aZ o

4

20

-



3.5 THELOSSLESSTRANSMISSIONLINE 257

IIEBCISE.3 Verify hatEq. 8.26a)s ndeed solutioncf hewave quation ivenbyEq. 8.21). (SeeO)

UERCISE.4 A two-wireair ine has he ollowing inera.ameters: ' : 0.404 ms2/m), ' : 2.0 (p.Wm),

G' : O,andC' - 5.56 pF/m).Foroperation t5 kHz,trermine (a) heattenuationonstant.r,b) hephase on-lant p,(c)thephase elocityno,and d) hecharacteristic.npedance6. (SeeO)

rns. (a)a : 3.37x 10-7 Np/m), b)f : 1.05xD-a (radlm), (c) llo : 3.0 x 108 (m/t, (d)

6: (600 j2.qa=$$Q/-o.re"d2.

E-SThe osslessransmissionine

\;cording otheprecedingection, ransmissionine s:ir.rracterized y two fundamental roperties,ts propa-Frionconstant andcharacteristicmpedances,both:i whicharespecified y the angular requencyr-r nd::- lineparameters ' L', G' andC '. In manypracti-zl situations,he ransmissionine can be designedomnimize hmic osses y selecting onductors ith veryrehconductivitiesnd ielectricmaterialsseparatingher ductors) ith negligible onductivities.s a esult,R'

mdG'assume erysmall alues uch hatR' ( rr.rl'andI ( arC'.Theseosslesslineonditions llowus osetr = G' : 0 in Eq. 8.22),which hengiveshe esult

Application f the osslessline onditionso Eq.(8.29)gives hecharacteristicmpedance s

Z o : (losslessline), (8.36)

which s nowa realnumber.Using he ossless-linex-pressionforB iven yEq. 8.35),weobtain he ollowingrelationsor thewavelength, and hephaseelocity o:

2n

p

t -t';-;;- '

I

(8.37)

(8.38)

(8.42)

Juc ' 'UponusingtherelationivenbyEq.8. 0), whichissharedby all TEM ransmissionines,Eqs. 8.35) nd 8.38)maybe ewritten s

y : a + j f = j a { U C ' ,

.hichmeanshat

wherep ande are, espectively,hemagnetic ermeabilityandelectricalpermittivityof the nsulatingmaterialsepa-rating heconductors. aterials sed or thispurpose reusually haracterizedy apermeability t : ps, where

po : 4tr x l0-7 H/m s thepermeability f freespace,and hepermittivity s usuallyspecifiedn termsof therelative ermittivity .defined s

er e eo, (8.4r

wherees 8.854 l0-r2F/m (1 l36n)x 10-eF/misthepermittivity f freespace. ence, q. 8.40) ecomes

l l'

Jpn€,€o ,ltro€o

where : l/Jtloeo : 3 x 108m/s s the velocityof

light n avacuum.f the nsulatingmaterial etwe€nhe

f = roJuc (rad/m), (8.39)

u,--+ (m/s), (8.40)'JPe

(8.34)

a:0 (losslessine),

p - a^[Lrc' (losslessine). (8.35)



25E

conductorss air, then8r = I andap = c. In view of

Eq. 8.41) nd he elationshipetween anduogiven y

Eq. t.33),thewavelcnglhsgivenby

CHAPIER8 TRANSMISSION

EXERCISE.5 For a lossless transmission

I = 20.7cm at I GHz. Findc, ofthe insulating

Ans. e.:2.1. (SeeO)

EXERCISE.6 A losslessransmissionineusestric nsulatingmaterialithe. = 4.Ifitsli

C'= l0 (pF/m),ind a) hephaseelocity p, b)

inductanceL'. and c) thecharacteristicmpedance

Ans. (a)uo 1.5x 106rn/s),b)L' -- 4.45(c)Zo : 667l O. (SeeO)

8-5.1 V0llageRellecti0nCoeflicient

Withy = jp for the osslessine, heexpressions

by Eqs. 8.26a)and 8.30) or the otal voltageand

on the ine become

u^) - - l -

.l++=+ . (8 .43 )! ^/Er J€,

where 6 c// is thewavelengthn aircorrespondingo

a requency. Note lat, becauseoth lp and depend

one,, hechoice fthetypeofinsulatingmaterial sedna ransmissionine s dictated otonlyby its mechanical

properties,utby ts electrical ropertiesswell.

When hephase elocityof a medium s independent

of frequency,he medium s callednondispersivewhich

clearlys the case or a lossless EM transmissionine.

Thiss an mportant eature or the ransmission fdigital

datan he ormofpulses. rectangularulse raseries f

pulsess composedf manyFouriercomponents ith dif-

fercntrequencies.f thephase elocity s the same or all

frequencyomponentsorat eastor thedominant nes),

thepulse hape ill remainhesame s hepulseravelson he ine. In contrast,he shapeof apulsepropagating

in a dispersivemediumgetsprogressively istorted,and

thepulseength ncrcasesstretches ut)asa function of

distancen themedium,herebymposing imitation n

themaximu data ate whichs elatedo he engthofthe

individualulsesndthe pacing etweendjacentpulses)

that anbe ransmittedhroughhe mediumwithout oss

ofinformation.

Table -2provides listoftheexpressionsor ,20, and

uo or hegeneralase fa lossyineand or severalypes

oflosslessines.Theexpressionsor the osslessinesarebasedn heeouationsor L'and C' sivenn Table - .

i (z) : Yo+- io' + v; ejlz

i{d:! i"- iu, -\" '0,.

These expressions contain two unknowns, y0+and

the voltageamplitudes fthe incidentand eflected

respectively.odetermine0+andyt, weneed

theosslessransmissionine n he ontext f he

circuit,ncluding generatorircuitat ts nputandaloadat ts outputterminals,asshown nFig.

line, of length , is terminated n an arbitrary oad

dance Zy. For convenience,he referenceof the

coordinatez is chosensuch that z : 0

the ocationof the ocd At the sendingendatz :

line s connectedtoasinusoidaloltagesourcewith-V, and an ntemal impedanceZB.At the load, he



i.5 THELOSSLESS RANSMISSIONLINE 259

Table 8-2: Characteristic arameters f transmissionines.

PmpagationConstant

y = d + jp

PhaseVelocity

CharacteristicImpedance

Zs

General case

Lmsless(R' = G' :0 )

Lossless oaxial

Lossless

two wire

Losslessparallel plate

u : Q , f l : ( D J € r / c

q = 0 , F : t o J E / c

a : O , f l : < o J e , / c

up= @/p Zo=

up: c/Ji zo: y/L ' /c '

up = c/Jer

u p : c / J d

zo = ($/J-e)rn(b/a)

zo:Qn/J-€).rnt@/24 u4d/23-- r)

zo- (120/.trs)h(d/a),i f d ) > e

q : 0 . F : - J E / c up:c/Je, Zs: \ lZ ln/J-e) (d/w)

(R t+ jaLt ) (Gt j j toC ' )(Rt t jtoL')

(Gt jaCt )

Notes: (l) tt = tto, € = €rtO, c : l/JtLoEo, andJ/ffi - (l20tt) fl, wherce. is the relativepermittivityof insulatingmaterial.2) For coaxial ine,c andb are adii of innerandouterconductors.(3) For two-wire line, a = wire radius andd = sepa.rationetweenwire centers. 4) For parallel-platelilg. u : width of plateand d : separation etweenheplates.

uri.igeacross t, V1,and hephasorcurrent hrough t, iL,

:elated y the oad mpedanceZL as ollows:

Uponusing hese xpressionsn Eq. 8.45),weobtain heresult:

roltage V1 is equal to the total voltageon the line

: given yEq.18.zl4a),nd i isequalo 17;given y3.44b),both evaluated tz : 0:

VZ r , : *

I L

iy:717:s1: yJ + yt,

v: v^-L : I ( z : 0 ) = ; - *

o:(#+)"Solving or Vf gives

,;=(ffi),;

(8.45)

(8.46a)

(8.46b)

(8.47

(8.48)

The ratio oJ he amplitudeof the reflectedvoltage waveto lhe amplitud.e fthe incident voltage waveat the load



zffi

is krtovn as the voltage reflection coefficient l. From

Eq. 8-48), hisdefinitiongives he result

andinviewofEq. (8.28), he atioofthecurrentamplitudes

ls

V; Zr -ZnF - _ _ _ L -' -

vd - Z t+ zo

=t=t/,1: . l (dimensionless).8.49a)Z1-/Zs +


for which Zr = R * jarl. Hence, n generalcomplex lso:

f : lf leie, (S.50)

where f I s themagnitude f f andd. s tsphascNote hat l-l < l.

A load is said to be matched o the line if Zy =

because hentherewillbe no refectionby the oad(l

andv; : 0).Ontheotherhand, hen he oad sancircuit Z; : oo), = | andVf : Vo*. ndwhenshort ircuit .Z1 0), = -l andyt : -y0+.

Relleclionoeffcienlol a Ssries C Load

A 100-(])ransmissionine s connectedo a oadsisting fa 50-Q esistorn series itha 10-pFFind the reflectioncoefficientat the load for a I

sisnal.

Solution:The ollowingquantitiesregiven Fig.&

Rr : 50O, Cr_ l0 pF 10-rrF,

Zo 100O'

The oad mpedances

Zr: Rr- i /aCr

,f : 100MHz : 108Hz.

: 5 0 - i -r2n x 108x l0- l l

: (s0 j159)

t r : - [ = -. . (8.4eb)6 v j

Wenote hat isgovemedy a single arameter,he oadimpedanceL,normalizedtothecharacteristicmpedanceofthe ine,Zo.As ndicated y Eq. 8.36),Zoofa lossless

line sa ealnumber. owever, 1 is n general complexquantity,s n thecase fa seriesRl circuit, or example,

IVLI

{

Vi

_+G€nerator

z = - l z = 0

Figure8-9: Transmissionine of length connectednone end o a generator ircuit and on the other end to aload21. The oad s located t z = 0 and hegeneratorterminals reatz : -r.



!5 THELOSSLESS RANSMISSIONLINE 261

:romEq. 8.49a),thevoltagereflctioncoefficientisiven:v

Z r Z a - lF :

- ' "'

ZL/Zg * |

_0.5 j l .s9 - l

0 . 5 - j 1 . 5 9 + l

_-0.5 - j 1.59_

-1.6'I i72.6': _o 16.,jrle.3"

1.5 i1.59 2.19e-tst"

-risresultmay be convertednto a form with positive

:.rgnitudeor f by replacingheminus ignwithe-l180'.nus,-

= 0.76ei te.3"e-jt8o"

O.76e-6o.7':0.j6//-$.1" ,

(

l f | : 0 .76, 0, : -60.1" . t

trample-3 lf I 0rPrrely eaclivooad

Showhatfl : I for apurelyeactiveoad.

..:lution: The oad mpedance fa purelyreactive oad s

;'. enby

Z y : j X y .

rm Eq.(8.49a), he reflectioncoefficient s

zt - Zo

Zy* Zs

- j X r - Z ojXL* Zs

__ - (Zo - j x )

(Zo+ j Xt)

.:ere 0 : tan-L XLlZo.Hence

l l : | - e-)201: [1e-ze1p- tze1*]1/2 l . t

EXERCISE.7 A5GQ losslessransmissionine s ermi-natedinaloadimpedance -- (3O- j200)Q.Calcularcthevoltage eflectioncoefficientat the oad.

Ans. f _ 0.93,/_27.5..SeeO)

EXERCISE.8 A 150-A osslessine is terminatedn acapacitorwhoseirnpedanceisy : -j30Q.Calculatef .

Ans. f :1,/-157.a' . (SeeO)

8-5.2 Standingaves

Using he elationVo : f Vo+n Eqs. 8.zl4a)nd 8..14b)givesheexpressions

71zy

yo*

1"-

a. +teifz1, (8.51a)

r r +

i e t= l ! 1 " - 1z y " i f z1 . (8 .5 Ib )z"

which now containonly one, yet to be determined,unknown,Vo+.Before we proceed oward that goal,however, et us examine he physicalmeaning repre-sentedby theseexpressions.We begin by deriving anexpressionor ly(z)|, the magnitudeof V(z). Uponugng Eq. (!.50)ln (8.51a)and alplying rhe rclar.ion

lV (z)l : IV-12'y.1111'tz,hereV-(z) is thecomplexconjugate f 712),wehave

1712;1 { vo+1e-oz1 1ypie,ilz1]

' l{vd1'1"i0' lrle-o'e-il')]l/2

: lvo+lIltrl'

+ lllkjQfz+o')a "- i tzOz+e4]L/z

: ly0+ l I l l r l , +2 l f coszpr+q] ' / ' (s .sz)

wherewe haveused he dentity

e ] r + e - J x : 2 c o s - r (8.s3)

, - i20z l+ x le

zl+ x/e ie



CHAPTERS TRANSMISSION

the positionon the line at which the incident and

flected waves are in phase [202 * 0, = -2n*

Eq.(E.52)land hcrefore ddconstructivclyogivce

egualo (l + lf l)lyo*l= 1.3V. Thcminimum alu

lV(z)l conespondso destructivenlerference,hich

curswhen he ncidentand reflectedwavesarc inopgosition 2Fz * 0, - -(2n + l)tr). In this

lv(z)l = (l - lf l)ly0*l 0.7V. whereashe epl

for any realquantity . By applying he same teps o

Eq. 8.5b),asimilarexpressionanbederivedor l1(z)Lthemagnitudefthecxrrent (z)._

Thevariationsf lV(z)l and 1(z)lasa function f z,thepositionon the line relative o the oadat z : 0,

are llustratedn Fig. 8-11 or a line with ly.+l : 1 Y

lf l = 0.3,0r : 30, md Zs : 50 O. The sinusoidalpattern s calle.da standing wave, and t is causedby

the nterlbrencef the wo waves. he maximum alue

of thesianding-waveatternof lfr(z)l conespondso

oeriod sI forthe incidentandreflected

therepetitionperiod ofthe standing-w)areattern is

The standing-waveatterndescribeshespatial

of the magnitude f V(z) as a function of z. If one

to observehevariations f the nstantaneousoltage

functionof timeat any ocationz,correspondingo

themaximain hestanding-wavepattem,orexample,

variation would be as cos artand would have an

equalo 1.3V [i.e.,u(t) wouldoscillate etweenand 1.3Vl. Similarly,he imeoscillation f u(2, )

any ocation will havean amplitude qual o lV(z)lthatz.

Close inspection of the voltage and

standing-wavepatterns shown in Fig. 8-11

that he wopatterns re n phase pposition when

atamaximum,heother s ataminimum, ndvice

This is a consequencef the fact that the second

Eq.(8 5 a) sprecededy aplussign,whereashe

term n Eq. 8.51b)sprecededya negativeign.

The standing-waveattemsshown n Fig. 8-11 are

a typical situationwith f : 0.3 ei30'.The peak

variation of thepatterndepends n lf l, which can

between and 1. Forthe special ase f a matchedwithZs : 20,wehavef | = 0 and i tz l t = l tfor all valuesof z, as shown n Fig. 8-12(a).Mlh

refected u,avepresent, therc will be no interft

and no stundingwcves.The other end of the lfl

at ll.l : 1, correspondso when he load s a

circuit (f = -1) or an opencircuit (f - 1).

standing-waveattems or these wo cases re showtr

Figs. 8-12(b)and(c), both of which havemaxima

to 2 yo+| andminimaequal o zero,but thetwo

fI

q

ti(z)l

lvlln.r1.41.21.0

0.80.6o.4

ltl,nin - -

lll'na,(- - -

l7lr;n - t- - - - - :

-l -).

z 4

) v\z)t versusz

Figure8-ll: Standing-waveattern or (a) ly(z)l and(b) 1(z) for a osslessransmissionine of characteristic

impedance = 50 I, terminated n a load with a re-

flection oeftcient = 0.3ej30'.The magnitudc f the

incidentwavelVo+l

= I V. Tbe standing-waveatio s

s = lY lma,( / lv lnnn1.3/o.7 = 1.86-

-31

(b



TIIE LOSSLESS RANSMISSION INE 263

from he oadat which 7(z)l isa maximum,hen

$rzll - l7l,"-= lyo+ltl lrll, (8.54)

and hisoccurswhen

2 fz *0 , - -2 / l û *Q = - lnn , (8.5s)

with n : 0 or apositiventeger.SolvingEq.(8.55) or1."r,wehave

. 0,* 2nn 0,x n),-: =,mar = --;-;- = ;--r -

2 p 4 n 2 '

I n - 1 . 2 . . . i f 6 .< 0 . , o z \

l n = 0 . 1 . 2 . . . . i f d , 0 .

( 6 ' 1 0 )

{-- shifted in z relative to each other bv a distanceofJ

\ow letusexaminethe aximum ndminimum aluesrirhe oltagemagnitude.romEq. 8.52),l7(z)lisamax-num when heargument f thecosineunctions equalr:zero r multiples f 2n . Noting hat he ocation n here alwayscorresponds

o negative aluesof z(since

he,:ads atZ = 0), f wedenotema* -z as hedistance

wherewe haveused he relationA : 2tr/^. Thephaseangle f thevoltageeflection oefficient, r,is boundedbetweenz andz radians.f 4 > 0, theirst voltagemaximum occursat l^"^ : 0,)./4tt , correspondingon : 0,but f4 < 0, he irstphysicallymeaningful ax-imumoccurs t mo : (0.),/4n) * l,/2, correspondingtoa : I . Negative aluesof /Itl.*correspondo locations"beyond"the oadat heendof the ineand herefore aveno physical ignificance. s wasmentioned arlier,helocationson the line corresponding o vohagemaximaalso correspond o currentminima, and yiceyersa.

Similarly, the minimum valuesof l7(z)l occur at dis-tances .1n : -z conesponding o when he argumentofthe cosine unction n Eq. (8.52) s equal o -(2n * I )2,which gives heresult

l 7 l . i " : l yo+ l t l l r l l ,

when8 - zpl",i') -(2n + t)n, (8.57)

with-n

S 0, < r . The irstminimumorresDondson : 0. The spacing etween maximum r,* and he

lv(z)l

Figure 8-12: Voltage standing-wavepattems for (a) amatched

load,(b)

a shon-circuited line, and (c) anooen-circuitedine.



l-6

adjacent inimumri" is L/4. Hence,lheftrstminimum

occursat

,,"'.

{ffii',1:,i',Xli',| (8 8)

The ratio of lVl.o to lVlmin is called tlte voltage

standing-waveatiaS,which romEqs. 8.54) nd(8.57)

isgivenby

Thisquantity,whichoften s referredo by its acronym,

VSWR,or the shorteracmnymSWR,providesa measure

of the mismatchb€tween he load and the transmissionline;foramatchedoadwith f : 0, wegetS : I, and or

a inewith l-l : I, S : oo.


ffi sbnding-waveRaiio

A 5GO transmissionine is terminatedn a load

ZL : offi + j50) Q. Find hevoltage efle.tion

cient and hevoltagestanding-waveatio(SWR).

Solution:FromEq. 8.49a), isgivenby( 1 0 0 + j s0 ) - 5 0

rw l-h

9 c&*

-nk

Z r - Z nn: --:---------:

Zt* Zo ( 1 0 0 + j s O ) + 5 0

50+ jso

150 jso'

cv@

^ l y l . * l + l f lS= j ; i "* : - (dimensionless).8.59)

l Y l m i n r -l r I

7o'7eits"" :

#;, '*:0.4sei266'.

Using hedefinitionor Sgivenby Eq. 8.59),wehave

l + l F t | + 0 . 4 5

r = f f i = r _ o / s : 2 . 6 .

Converting he numeratorand denominator o polar

and hensimplifyingyields

0.5 llleasuringL

L slofted-lineorobe s an nstrumentused o

unknown mpedance fa load, Zr. A coaxialslotted

contains narrowonsitudinal lit n theouter

A smallprobe ns€rtedn the slit canbe us€d o

the mlgninrdeof the electric ield and,hence, hetudeyl ofthevoltage n he ine Fig. -13). ytheprobealong he eqgthofthe slotted ine, tis possil

measure V | nr,and IV | in and he distances rom the

REVIEWUESTIONS

QE,s The attenuation onstanto represents hmic

losses.n view of the modelgiven n Fig. 8-6(c),what

shouldR' and G' be in order to have no losses?Verify

yourexpectationhrough he expressionor a givenby

Eq.8.25a).

Q8.6 How is the wavelength , of the wave travel-

ing on the transmission ine related to the free-space

wavelength6?

QE.7 Whens a oadmatchedo the ine?Why s t im-

portant?

Q8.8 Whatisastanding-wavepattem?Whysitsperiod

l/2 andnot ?

Q8.9 What s the separation etween he locationof a

voltage aximum nd headjacenturtentmaximumn

the ine? Figure 8-13: Slotted coaxial ine (Example8-5).



i6 INPTTTMPEDANCE OF THELOSSLESS INE 265

1 which theyoccur.Use of Eq. (8.59)then provides hedage standing-waveatio .S.Measur€ments ith a5G;- slottedine connectedo anunknown oad mpedance,demined thatS : 3. Thedistance etween uccessiverttageminimawasound obe30cm,and he irstvoltaeenoimumwas ocated r 12cm from he oad.Determiie

r loadmpedance1.

:ilution: The ollowingquanritiesregiven:

Z o : 5 0 Q , S : 3 , / m i n : 2 c m .

:rce thedistance etweenuccessiveoltaseminima sxualtoL/2,

) , : 2 x 0 . 3 : 0 . 6 m .

i--':

^ 2n 2t lOn

(rad/m).I 0 . 6 3::m Eq. 8.59),olvingor f lin termsof gives

. t _ | 3 _ I/ l _ = _ = n <s + 3 +

N.rr.weuse hecondition ivenby Eq. g.57) or the o_:con of a voltageminimumo finde:

i7 - 2plîn : -v, forn : 0 (firstminimum).

.r;chgives

- :2Qlmin rlOr r

=2x; ' x 0 .12 - r : -O .2 r Iad ) -36" .

EXERCISE.9 If I. : 0.5 -60.and : 24sm,6.6 th,locationsof the voltagcmaxirnumandminimumnearrsttothe oad.

Ans.l,or,, l0cm,lmin4cm. (SeeO)

EXERCISE.10 A tzt0-S2osslessine s terminatedn aload rnpedanceL -- (28O+ j 182)O. f ), : 72 cm,find (a) the reflecrioncoefficienr l, (b) the voltagestanding-waveatioS, c) he ocationsf voltagemaxima,and d) he ocationsfvoltage inima.

Ans. (a) : 0.5 2e.,(b).s 3.0, c)/.", : 2.9 m' 2, d) în= 20.9 m n),2.where = 0,1,2,. . . .

(SeeOt

8-6 Inputmpedancef heLosslessine

Thestanding-waveattemsndicatehatoramismatchedlinethe oltage ndcurrentmagnitudesreoscillatoryithposition n he ineandn phaseoppositionitheachLrher.Hence,hevoltageocurrent atio,cal edthe nput impe-danceZin,mustvary ithposition lso.UsingEqs.g.5la)and 8.51b), ;n sgiven y

i t , tZi"Q) : -=:::

I (z\

- vf 1e-ia' leir ' l - vi le-ipz fei l4Lo

- f | + f e j 2 0 z 1: zoLt- rei ' p,l(sl) (8'60)

Note that Z;nk) is the ratio of the total voltase(incident- and reflected-wave oltages)

o he total(;r-

rent at anypoint z on the ine, in contrastwith thechar_

:fe,

r - -Ekio ' :0.5e- j36":0.405 - j0.294.

' ,'. ngEq. (8.49a) or ZL,vtehaye

- =z^r* r l"L l f J

-^f I +0.405 i0.2941

= )uL : olos l02%l = (8s i67tQ' t



26

acteristicmpedance f the line Zs, which relates he

vohage nd currentof eachof thenuowaves ndividually(zo - vd t; = -v; /6 ).

Of particularnterestn many ransmission-linerob-

lemsistheinputimpedanceattheinputofthelineatz-r,

which sgivenby

f eiPI le- jPIlzaGt):zt l4, _y; :1gi)

f 1 + l e - i 2 q t 1:z" l t : i t ,a,) (8'6r)

By replacing with Eq. 8.49a) ndusinghe elations

CHAPTER8 TRANSMISSION j5

Ja- l&

eiqt cosFl + j sinBl,

e-i?t = cosFI - j sinpl,

Eq. 8.61) anbe ewrittenn terms f ZL as

(8.62a)

(8.62b)

From the standpointof thegeneratorcircuit, the transmis-sion ine can bereplacedwith an mpedanceZin, as shown

in Fig. 8-14.ThephasorvoltageacrossZin s givenby

Equating q. 8.64)o Eq. 8.65) nd hen olvingc

leads o the esult

(8.64) This comDleteshe solutionofthe transmissionline

equations,iven y Eqs. 8.21)and 8.23),or the

caseofa losslessransmissionine. We started ut

general olutionsgivenbyEqs. 8.26a)and(8.26b),

includedourunknown mplitudes, ; , V; , ry ,We hen ound ut h^tzo : vo+ It : -v; lI; ,reducinstheunknownsothe wo

Uponapplyingheboundaryondition t he oad,we

i.z,^V i = I t Z i n : # ,

LA + Li l

butfrom he tandpointf he ransmissionine, he oltage

acrosstat he nputoftheine sgiven yEq. 8.51a) ith

z : - l :

ii: i1-4 : vd[ejtl + re-iPIL (s.6s)

l \

i )i ê I ) :ZD(

: 2 " (

Zlcos Pl * j Zn sinPsinB/

)

Zscosp l * jZ r -

Zt 'f j Zotan ll

Zo* jZutanfl(8.63)

r+vi ztn+ h

t

1ti

_i

Figure 8-14: At thegeneratorend, he t€rminatedrdE

mission ine canbe replacedwith the input mpedance

the ine Zin.

';=(fu)(t"+;-) (866)



}5 INPUTIMPEDANCEOFTIIE LOSSLESS INE 267

eM (2, \

A 1.05-GHz enerator ircuitwith seriesmpedance= | 0 Oandvoltage ource ivenby

ug(t) 10sin(a.rr 30') (V)

_. onnectedo a loadZL : (100+ i50) A throughai,!Q, 67-cmlong osslessransmissionine.Thephase

:locityof the ine s0.7c,where is thevelocity i light: avacuum.indu 2, ) and(z,l) on he ine.

rrlution: From herelationship p : Lf, we find the: velensth:

. u o 0 , 7 x 3 x 1 0 8^ : - - - : - - _ _ n . -

" f t .OS lOe- " ' - " ' '

$le torelateyt to Vo+kough f, and, inally,by applyingie boundary onditionat thesendingendof the ine, we$ained anexpressionor 7o+.

With referenceo Fig. 8-14,the input impedance f theline,givenbyEq. 8.63),s

. , . -o f z t+ jzo.nf l lL ' n - L o L 4 +

i z t t ^ p t l

- . f z t /zo+ j ta t t f l 1-Lt + j(zL/zdtn pr)

: 5o f ( z + i t ) + i * t ' u ' '

L + i(2 + 11)a 126": (21'9 i r7 4) '

Rewriting heexpressionorthegeneratoroltagewiththecosineeference,wehave

ur(t) : l0 sin(a.rr 30')

: 1O oskr2 - al/ - 30")

= 10cos(arr 60.)

= n lt1e-j$'

ei@tl nelisei.,l (V).

Hence,hephasor oltage7, isgivenby

7, : 1g2-ia" 1y)_ t0/:& (V).

Application fEq.(8.66) ives

,r:(#h)G;*llt)

f tqe-i$"et.9+ t7.4t1L t0+2r .9+ j17.4).fe

t26'L"9.45"26.6'e- t26"l-

l

: l1.2eitss' v) : lO.Ztss. (V).

Thephasor oltage n he ine s then

71zy=y ;1 " - i o , te ipz)

: 10.2eise"k- jP.+ 0.45ei26.6'ipz,

CompfeteolulionorD(2,)

= tan126'

( 1 0 0 + l s0 ) - 5 0: 0.45ei26.6"(100 j50) + 50

t "@t): ,""(+r)

:t^(fr"o.et): tan6.7r : tan0.7n

r*re we havesubtractedmultiples of 22. The voltase:lection coefficienrat the oad s

. Z t - ZoI = + :Zt* Zo





i.7 SPECIALCASESOF THE LOSSLESS INE 269

:E rnput mpedancef $e lineatz = -l isgivenbyrrio of V*(-/) to 1*(-l). DenotingZff as he nput

for ashort-circuitedline.wehave

:rt of Zff/lZo versusnegarivez is shown in!r5(d).

h reneral,he nput mpedanceinmayconsistofareal.rtnputresistancein, ndan maginaryart, rinput

Xn :

Zin: R;n jX in . (8.69)

,: drecaseof the short-circuitedosslessine, he nputrgedance spurelyreactive R6 = 0). If tan / > 0, the

j:dappearsnductive, cting ikeanequivalentnductor_-.whosempedances equalo Z,sfThus,

jtltL"o= jZotanpI, if tanBl > 0, (8.70a)

(8.70b)

Since isapositivenumber,he shortestenglh for whichtanpl < 0 correspondso the range /2 < pt < t.Hence,heminimum ine length that wouldresult n aninput impedance iT equivalento thatof a capacitorofcapacitancecos

/ | \ tl - ; - l l ( m ) . ( 8 .7 1 c )\aLqzo, / - )

Theseresultseanthat,hroughproperchoiceofhelengthof a short-circuitedine,wecanmakesubstitutesor ca-pacitors nd nductors ithanydesiredeactance.uchapractices ndeed ommonn thedesignofmicrowave ir-cuitsandhigh-speedntegratedircuits, ecause akineanactual apacitorr nductorsoftenmoredifficult haimaking shortedransmissionine.

$7-l Equivalanteactivetements

Choose he length of a shorted50-O lossless rans_mission ine (Fig. 8-16) such hat ts input impedanceat2.25GHz s equivalento the eactance fa capacitorwithcapacitanceC", : 4 pF. The wavevelocityon the ine is0.75c.

t : ) 1 " - " " '

L"4:z-!g Pt (H).

--minimumine ength thatwould esult n an nput

uedance Z,Tequivalentothatofan inducror find;c-:rte l* is

t : ) tan- ' (+) (m). (8.70c)

rilarly, iftan p/ < 0, the nput mpedanceis apacitive,- : hichcase he ineacts ike anequivalent apacitorCeq-. h hat

I, . . -

= iZotanPI. i f t : ,n l < 0. (8.71a.1l aLeq

/ - - l

" . --

h . t ^ p l (F). (8.71b)

: j Zotan l.

u--f

vf -..-- t7- r|

'jaC,q

---6

FigurcE-16:ShortedineasequivalentapacitorExam-Dle8-7).



270

Solution:Wearcgiven

up=0.75c 0.75x 3 x 108 2.25x l08n/s,

z o = 5 0 Q ,

f = 2.25GHz 2.25 x l}e Hz,

C* = 4pF:4 x 10-12.

Thephaseonstants


8-7.2 0pen-Circuiledine

WithZL = oo.as llustratedn Fig.8-17(a), e| - I, S: oo,and hevoltage, unent,and nput

2nR - -, \ =Zof :2" :=?'25,\" lot 62.8 (rad/m).

up 2.25 106

dance regiven y

ink) : vdt"-iPz+ eiPzfZvd cos z, (8.

i,,<O\f,-jfz

- "iuzl#

rr Ur, r.

-- v*(-l lZ: -- ::-:--- : - jZ6cotpl. (8.73)"'

1*(-/)

Frorn q. 8.71a),

tanPl =ZsaCeq

50 2n x2 .25x lOex4x0- r2

The tangentunction is negativewhen ts argument s in

the second r fourth quadrants.The solution for the second

quadrants

l l t t : 2 . 8 r a dr 11T : # :4 .4 6 cm .

andhesolutionor the ourthquadrants

5 9 4f l z :5 .94 rad or 12 - =9 .46 r^ .

Wealso ouldhaveobtainedhevalueof 11 y applying

Eq. 8.71c).he ength/2sgreaterhan rby exactly"/2.

In act, ny ength = 4.46cm*nl12, wherensapositive

integer,salso solution. I

Plotsof these uantities redisplayedn Fig. 8-17

function fnegative .

8-7.3ApplicationlSh0rt-Circuitnd0pen-Circuileasuremenls

A network analyzer s a radio-frequencyRF)

capablef measuringhempedancef anyoad

to ts nput erminal.Whenusedomeasure ,sfthe

impedancefa losslessine erminatedn a shortandagainZff, the nput mpedancef the ine whenminated n an opencircuit, he combination f the

measurementsanbe used o determine heimpedancef the ine Zs and ts phase onstant .product fEqs. 8.68) nd 8.73) iveshe esult

7 - - (8.74\

and he ratio ofthe sameequationseads o

: -0.354.

'iq

I

q

l . *

r / z *(8.75)



SPECIALCASESOF T}IE LOSSLESS INE 271

,27fi _+

Figurc 8-17: Transmission line terminated in an open cir-

:uit: (a) schematic r€presentation, O) normalized voltage

rn the line, (c) normalized current, and (d) normalized

-:rput mpedance.

Because f thez phase mbiguityassociated ith the an-gent unction, he ength shouldbe ess hanor equal ol/2 toprovideanunarnbiguousesult.

Find Zn md p of a 57-cm-longosslessransmis-sion line whose nput impedancewas measured sZff: j4O.42A when erminatedn a shortcircuitandasZff : - j 121.24Q when erminatedn anopen ircuit.Fromothermeasurements,eknow hatthe ine sbetween3and3.25wavelengthsong.

Solution:FromEqs. 8.74) nd 8.75),

zo: {lzff zff : JQ4o.42)(-r21.24):70e,f - *

r ^ ^ a t - l - L i a -'*'"' -,1 zff-

Since isberween), and3.25),,l : (2zl/t) isbetween6z radians nd 3rrl2) radians. hisplaces l in the ustquadrant 0 to r/2) in apolarcoordinate ystem.Hence,theonly acceptable olution or theabove quationsp/ =

z/6 radians.This value,however,does not include the2z

multiplesassociated ith the nteger multiplesof I .Hence,he ruevalueof 6l is

p l : 6 r *

in which case

l9.4 (rad),

R - : 34 (rad/m). r

1t

o

19.4

057



Microwavevens

Percy Spencer,while workingthe 1940s on the design andmagnetronsor radar,observed

TECHNOLOCYBRIEF: MICROWAVE

bar that had unintentionallyeen exposed ocrowaves ad melted n hispocket. hecooking ymicrowave aspatentedn 1946enthe 1970s.microwave/ens had becomehouseholdtems.

tor Raythoon nconstruction f

that a chocolate



IECHNOLOGY RIEF:MICROWAVE VENS z I t

l l icrowavebsorption

micro$ra'vesan electromagnetic avewhose re-

S.,encyies n the 300 MHz-€00GHz range. SeeFrg.1-9.)Whena material ontainingwater s ex-posedo microwaves,he watermoleculeeac,tsyotating tselfso as io align ts own electricdipolealong he directionof the electric ield of the mi-crowave. he rotationmotioncreatesheat in thenaterial, esultingn the conversion f microwave3nergynto hermalenergy.Microwave bsorptionJywater xhibits spectrum itha peak hatoccurseta resonantfrequenc/ whose rEllueepends nh€ temperaturef the waterand on theconcentra-lonof dissolved altsor sugars resentn hewater.-he

lrequencymostcommonly sed n microwave:venss 2.54GHz,and hestandard ource orgen-

erating nergyat this frequencys the magnetron.rvhereasmicrowavesre readilyabsorbed y wa-'er,fats, and sugars, hey can penetrate hrough-ost ceramics, lass,or plasticswithoutossof en-:rgy, hereby mparting o heat o thosematerials.

Ovenperation

Togeneratehigh-powermicrowaves- 70Owatts)the microwavevenusesa magnetronube,whichrequires he application f a voltageon the or-der ot 4000volts.The ypicalhouseholdoltageof115volts s increasedo the required oltage evelthrough high-voltageransformer.he microwaveenergygenerated y the magnetrons transferredinto a cookingchamberdesigned o contain hemicrowaves ithin t through he use of metalsur-facesandsafety nterlockwitches.Microwavesrereflec'tedby metal surfaces,so they can bouncearound he nterior f the chamber r be absorlredby the food, but not escape o the outside. f theovendoor s madeof a glasspanel,a metalscreenor a layero{ conductivemesh sattachedo it to en-

sure the necessary hielding;microwaves annotpass hrough he metal screen f the mesh widthis muchsmaller han the wavelengthf the mi-crowave- 12cm at 2.5GHz). n hechamber,hemicrowavenergy stablishes standing-waveat-tern,which eads o an uneven istribution. his smitigated y usinga rotatingmetalstirrer hatdis-perses he microwave nergy o differentpartsofthe chamber.

Date post:	04-Jun-2018
Category:	Documents
Upload:	abdi-kurniawan
View:	214 times
Download:	0 times

Electromagnetic 8

Documents