CHAPTER 5 COUPLED COILS AND TRANSFORMERS (.I
INTRODUCTION
Thcn the parasiticscanbe ignored,transformersarc ideally suitedfor impedancescaling. :ealandpracticaltransformerswill be consideredin this chapter. A practical transformer differs from the ideal in that it has leakageflux, finite izing inductance, losses, and parasitic capacitance, all of which degrade its
-.rnance.Severalequivalentcircuitsfor practicaltransformers will bepresented here. is required widebandtransformation ofresistance Transformers areoftenusedwhen possible,this is usuallya betteroptionthanusingLC networks).It will be shown of a transformeris mainlydeterminedby thecoupling thatthewidebandperformance . This is alsothereasonwhy stackedtoroidsor baluncoresareusuallyusedto realize a transformer. Although the finite magnetizing inductance and the leakage inductance are in a wideband transformer,they can be put to good use in narrowband *nhing networks. Several narrowbandmatchins networks using transformerswill be :ered in detail.
Becauseit is importantto adjustthe couplingfactor of atransformerto the required when narrowbandmatching networks are used,methodsto measurethe coupling will be alsobe consideredin this chapter.
:
TIIE IDEAL TRANSFORMER
:o uivalentcircuit of the idealtransformeris shownin Fisure 5.I . The numberof turns ..in . primaryand secondarysidesof the transformerare,respectively,n, andn2. The idealtransformerhasthe followine characteristics: The magneticflux in the two windings is the same.Therefore,there is no leakageflux. The voltageinducedby the changingflux in eachwinding is given by Faraday'slaw: 151
F t52
Design of RF and Microwave Amplifien and Oscillaton '
R
r
h I
l
+
I
Irr
:
Figure 5.1
T
v2
R2
The ideal transformer.
V=n,AAlAt
(5.1)
wheren, is thenumberofturns inthe winding underconsideration.Because the flux coupling eachwinding is the same,the ratio of the primary to secondaryvoltageof thetransformeris \ ll/z = n, / n,
(5.2)
This relationshipis moresignificantif it is written in the form V1ln1=Vr/n,
(s.3)
The voltage per turn, therefore, is the same for both sides of thc transformer.
2.
to establishtheflux inthe idealtransformer Theprimarycurrentnecessary is negligible.The input impedanceof the transformerwith the load oper circuited,therefore,is infinite.
3.
Thereareno lossesin theidealtransformer.The averagepowerdissipated in the load,therefore,is exactlythe sameastheaveragepowerenteringtb transformer. Becausethe ideal transformerhas no reactivecomponents,tb powerdissipatedinthe loadis alsoequalto theinstantaneos instantaneous power enteringthe transformer;that is, vlt = v2t2
l
(-i 'l t
By using (5.2) to replace the voltages in this equation, the relationshig between the primary and secondary currents is found to be nrl, = nrl2
(j j I
CoupledCoils and Transformers
153
Thedemagretizingforce(magnetomotiveforce)ofthe currentin the secondarywinding is, therefore,balancedby that of the current in the primarywinding. By using this equationand (5.3), the relationshipbetweenthe primary and secondaryimpedancesis found to be
Zt =Vt I I, =fn, I nrlzZ,
(s.6)
The impedanceratio is, therefore,only a function of the tums ratio of the transformer. 4.
of theflux density Thepermeabilityofthe idealtransformeris independent is a perfectly linear the ideal transformer implies that the core. This in device.
From an impedance-matchingviewpoint, the ideal transformer is very by any factor. usefulin that it canbe usedto scaleimpedances
5.3 EQUIVALENT CIRCUITSFOR PRACTICAL TRANSFORMERS I practicaltransformerdeviatesfrom the ideal in the following ways: 1.
Thereis someleakageflux and,therefore,leakageinductance.
2.
The magnetizinginductanceis finite.
a
Therearelossesin thewindingsof thetransformer(copperlosses),aswell as in the core(hysteresisandeddycunentlosses).
4.
with signallevel Therelativepermeabilityofthe magneticmaterialchanges anddc current(saturation),aswell aswith frequencyandtemperature. Apart from the effect ofthe leakageinductance,the high-frequencyrebetweenthe sponseis degradedby the presenceofparasitic capacitance windingsandtums of eachwinding.
A circuit model for the practical transformer,ignoring the capacitanceand inlinearities,is shown in Figure 5.2 Ul. The two dots indicatethe sidesof the two .rndingsthat havethe samevoltagepolarity. represent the leakageflux, the seriesinductanceto the Thetwo seriesinductances 'rt togetherwith the shuntinductancearethe magnetizinginductances, r1 the resistance
154
Design of RF and Microwave Amplifrers and Oscillators
i lLrr-u/n1 i r,
Figure 5.2
Ir/,
transformer. An equivalentcircuit for a pra.ctical
and 12 representthe copper losses,and Rorepresentsthe lossesin the magnetic material. The mutual inductance Mcan be determined as a function ofthe magnetizing inductances .Lrt and Lrz by using the re\ationship
I : I
L I t f
s
t
M = k(LtrLzz)tt2
whereft is the coupling factor of the transformer. The symboln in Figure5.2canhaveanyarbitraryvalue,but it is usuallychosento be equalto thetumsratio ofthe two windingsofthe transformer.However,a betterchoice for it is
n=-
tr h
ft
l
(s.7\
r lE -
(s.8)
k\ Lu
Ifthe lossesin the magneticmaterialcanbe ignored,the equivalentcircuit for two oorryledcoils (seeFigure 5.3(a) can be usedfor the transformer.The circuit shownin Figure5.3O) is equivalentto the coupledcoil circuit [2]. This canbeprovenby settingup tbe Z-parametermatricesfor the two circuits. The transformationratio shown in Figure 5.3(b)is that for the impedances. of thetransformerareimmediatelyevidentfrom the The following characteristics oquivalentcircuitin Figure5.3(b): l.
The load impedanceis transformedto the primarysideof the transformer as ZL =lLr, / &2 Lz)lfZ, + rrl
(5.9)
that is, as long as
IahzQ- k')l'
