射频微波放大器1

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APPENDIX ATHE UNBALANCED TRANSMISSION LINEThe basicequationsassociated with a transmission whenthe currentsareunbalanced line will be derivedhere.For the sakeof simplicity, it will be assumed thereis no[l] that magneticcoupling betweenthe two conductorsof the line. The equivalentcircuit shown in FigureA.l applies.

1r(0)

tdx

Lrh

Inx

Ldx

Figure A.1

The equivalentcircuit of an unbalanced transmission line.

The currentat positionx on the line will be considered first. The currentcan be expressed termsof balanced unbalanced in and components follows: as Ir(x)=Io@)* I"(x) and I.r(x)=Jo1v)+1"(x) (A.l)

(A.2)

(x) in these equationsis the balancedcomponentof the current,while 1"(x)is the unbalanced component. It follows from thesetwo equations that Ir(x)=fl oQ) - I,(x))+ 21,(x)= l, (x)+ 21"(x) 469

(A.3)

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4l

tf*F *ffl

biCtofMandMicrowaveAmplifiersandOscillators

= Ilx)+/o(:) - *:. ri1x.p G fr*n"e is berween nnocurrents. the

(A.4)

Becausc 6ere is no groundpathon the line itself, the differencecurrent(/s(x) must bethesame all positions ar alongtheline (referto Figure6.17,if necessary), (A.4) can and thereforebe sinplified to

#F

Ir(x)= Ir(r)+ Io With the qrrents defined,it follows that D,' Ir(x)= -sCVn(x) D,'.fr(r)=-sCVn(x) D,'Yr(x)= -sL IJx) D, 'Vr(x)= s[,I2(x)= s1 I,(x)+ sL I,

(A.s)

ltl s

(A.6) (A.7) (A.8)

r

I

II;

(A.e) (A.10)

YnG )= r,@) - Y r ( t )Differentiation of (4.6) yields that

o] .r,1x;=-[D, .rrr@)] sC= - sC ID, .Vr(x)-D,.Vr(x)]= - .rCf- sLI r(x) - sLI,(x) - sLI o'l= 2 s 2L C I l x l+ s 2 L C I o thatis,= (o1 -2r' LC)'1,(x) szLC Io: Thesolution theequation to is Ir(i= as-J'LC''"+ BeW"' - Io/ 2(A.12)

(A.lr)

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Line Transmission Appendix A: The Unbalanced

471

=-Iol2+Ae-r'+Be*t' where

:

1

(A.13)

.s f=J?-LC= il,l'"[ztCfor The equation 1dx) canbe derivedsimilarly andis givenby Ir(x)= Io/2+ Ae-r*+ Be*r'

(A.14)

(A.15)

(A.8) aftersubstitution 1'(r) of by for An expression I/1(r)cannow be obtained integrating (A.13): andby using

=rr(0)- z0I 2'(A - B)+ z0I 2'l.Ae-"- B.t'I Vr(x) l+ s L x . l o2where

(A.16)

(A.17)

followssimilarly: for expression Vr(x) Therequired=+ V2(x) Vr(O) ZoI 2' (A - B) - Z0I 2'lAe-" - B"t'l+ sLx-Il2 o (4.6): by can Vrr(x) beobtained using An expressionfor Yrr(*)= T/r(x) Vr(x)= 701tre-f'- Bet' l below: derivedaresummarized Theequations Ir(t)= -Io l2+ Ae-rr*3"+fx Ir(x)= Io/ 2+ Ae-r'+ Be*r' Vr(x)=Vr(O) ZoI 2'(A - B)+ z0 I 2'1.A"-" - Btt'l+sLx.Iol2(A.13) (A.ls) (A.18)

(A.le)

(A.16)

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Design of RF and Microwave Amplifiers and Oscillators

vr(x )=z r ( 0 ) z 0 1 2.(A -B )- zo1 2 .[l e -t' - B .t' l++ s L x . I o2/=vr(x) - vz@) z of Ae-r'- 6.11= v12(x) 1

(A.18)

(A.le)

REFERENCESl. Abrie, P. L. D., Impedance-Matching Networluand Bandwidth Limitationsof Class B Power Amplifiers in the HF qnd vHF Ranges,Master's Thesis,University of Pretoria, 1982..J: