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376

Theory of Nonlinear Reactance Amplifiers

Abstract-A class of power amplifiers exists that uses a combina-tion of an ac source and anonlinear eactance as he basis foramplification. Included in this class are he varactorparametricamplifiers, the so-called dielectric amplifiers, and he magneticamplifiers. The 3-frequency parametric amplifier is often comparedto the mase r amplifier since both operate on the 3-frequency prin-ciple and both are used or the same application, i.e., low noise frontend in microwave receivers. From a circuit point of view, however,theyare quite different. Th e paper develops a heory of a classof amplifiers termed nonlinear reactance amplifiers, and establishesgain-bandwidth limitationsby means of Bodes[*] ntegral constraintsand Mason~[~lnilateral gain quantity U.

C CONDITIONSOR POWER AMPLIFICATION3 RTAIN necessa,ry conditions or a nonlinear net,-work to be active, i.e., possess power gain, can be de-duced from a result due t,o Du&n.[31He considered a gen-eral nterconnection of linear and nonlinetlr passive re-sistors, capacitors, and induct,ors and proved th at such aninterconnection is passive if dc sources only are present.Thus it is impossible to convert dc t o ac power or to con-struct anamplifier by :m interconncction of such elements.The following condition follows imnlediately rom hisresult:

C ond i t i on 1 : A necessary condition for power amplifica-tion in a nonlinear networlr whose individual elements arepassive is the presence of st least one nonlinea,r reactiveelement and one ac power source.

Fromparametricand magnetic amplifier theory t isknown tha t the use of an ac source (the pump) and onenonlinear capacitor or inductor, logether with other inearpassive lements, canprovide amplification. I n view ofCondition 1, however, one might ask whether the linearpsssive elemenk arenecessary or whether one can achievepower amplification from an int,crconnection o f a singlenonlinear eactiveelement and an ac power source. Tothis endageneral3-terminalnetwork composed o f anonlinear eactiveelement andan ac power source isconsidered,as hown nFig. 1. This network may heconnected between a single source and a load as shown inFig. 2. There are other ways in which a signal source anda load can be connected to he 3-terminal network;however, t8he thcr configurations yield no additionalinsight and herefore will not be discussed.

Manuscript, received March 1, 1967.P. Johannessen a,ndW. KII are with the Applied Research Lab-oratory, Sylvania Electronic Systems, Waltham, Mass.J. Andersen is with th e Massachusetts Insti tute of Technology,Cambridge Mass., and is a consultant to the Applied Research Lab-oratTory,Sylvania EIectronic Systems, Waltham, Mass.

Pig. 2. Basic amplifier elornenI.

t

A simple qualitative way f viewing p o \ v c ~ nq)lificatiotlo f th e amplifier of Fig. 2 is to consider t.he nonlinearelement as a variable impedance wl~oscnmgnitutlc iscontrolled by dhe sigrd source. Thus, n wwinblc: amountof power is made to flow from t,he pump ource to t8he oad.If the otal change in load power is great,cr th m trhcavaihble signal source power, power anlplificat.ion isobtained.When a general nonlinearreactiveclement isconsidered, there is no evidence leading t,o t>hebeliefth at power amplification is possible. A dctjailetldiscussionof these considerations is eyond the scope o f this paper.

For a quantitat ive evaluation of t he amplifier shown inFig. 2, apart,icular type of elenlent rlordirlexriij?;m d

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JOHASXESSEN ET AL.: NONLINEAR REACTANCE AMPLIFIERS 377mode of operation is considered. The particular elementnonlinearity does not imit hegenerality of the result,obt'ained; in fact, t renders th e amplifier linear and givesmaximumdynamic ange. The nonlinearelementchar-acteristic is shown in Fig. 3 . It consists of two regions:a inear region of infinitepermeability and : nonlinearregion of arbitrary permeability. It should be emphasizedthat the analysis t o bepresentedhere is validfor anytype of charact'eristic in the nonlinear region, t,hus makingthe analysis quite general. First, the operatingmode of thenonlinear lementduring teady-state operat,ion is es-ta,blished by assuming that th e operation is periodic suchthat the nonlinear element retur ns to state 1 in Fig. 3,periodically, a t times nT, where n =0, 1, 2 , . . . . TimenT has been selected as the time of transitmion rom th enonlinear to he linear region of t,he nductorcharac-teristic.Second, it isassumed th at duringa period ofoperat'ion [a period of operation is defined as the intervalbetween nT and (n + 1 ) T ] here exists oniy one lineara.nd one nonlinear interva.1, as shown in Fig. .

The operatingmode ust described canbe physicallyrealized using voltage (or current') waveforms t.hat satisfycertain resDrictions. A detailed discussion of these re-strictions is given by Johannessen.[41~[51s shown inthese references, linear equations in terms of half-periodaveragevoltages andcurrentscan be written for thebasic amplifier element shown in Fig. 2 . For convenience,the resistance of the signal source R, is included i n th ebasic model of Fig. 2 . Thus, the power gain is equal t othe ratio of the output power -e2& divided by the avail-ableignal power e12/4R,.1 From Johannessen[51 therelationbetween half-period averagevoltages and cur-rents is given by the matrix equation

where

If a load resistor RL is connected across port 2 of the basicamplifier element, one can write

Ezg = -RLIZ'. (5)Substituting (4) a,nd ( 5 ) in (1) and solvingfor the loadcurrent gives

aI Z C = - l - bR p +RL(1 - 6 ) +R, +aL(l- h ) E, (6)

The first terms on the right-hand sides of (6) an d (7)represent the component of the output , cur rent due to t heinputvoltage; he second termsaredue t,o t'hepumpvoltage. I n a practical circuit,he output component due the pump voltage s usually cancelled and therefore is notconsidered when calculating the ou t'put power. Thus, theoutput power becomes

Since t'heavaihble inputpower can be writt'en ,s

the power gain n 'erms of half-period a,verage voltagesand curyents

Rsb = Rs+R, (3 ) This power g.ain is maximum when Rp= 0 and R, =R,,givingSuperscripts c and g denote half-period average voltagesandcurrentsduring t,hehalf-periods nl' to nT + T / 3 Thus, we conclude that no power gain s possible in thean d nT + T / 2 to (n + l ) T , respectively (see Fig. 3). basicelement of Fig. 2 .It is now trivial to find a sufficient, condition for power

amplification. If an inductor is connected in series withhe designated Kvs . port 1 and t4heignalourcesperatedt, a, frequency

W,a)max =1. (11)

This power gain mag be termed the available power gmn and will

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378 I E E E THSNSACITIONS O S X\GKETICS SEPTEMBER 1967II

CONTROL ICIRCUIT

OUTPUTCiRCUllI

I

eP

Fig. 4. Series-connected sat urab le inductor.

low compared t o the pump frequency, the pump sourcehas been effectively isolated from he signal sourcewithoutthe signal source being isolated fromhe nonlinear element.Thus, he average impedance of the nonlinear nductorcan be controlledwith a small amount of signal powerindependent of pumpcurrentandvoltage,and poweramplification sobt8ained. It is further evident hat, hepower gain is proportional to he bandwidt,h,which isinversely proportional to the amount f isolation. Becausethis type of isolation depends upon frequency separation,the power gain cannot exceed unity when the signalfrequency is equal o hepump frequency. Thus, heupper limit to the power gain as a function of frequencyappears t,o be of the form

=Constant . -.S Ps (1%for the case when an inductcr is used to obtain isolation.This result is readily vcriiied experimentally, but is dif-ficult to derive analytically.It has been shown t,hat thc addi tion of one element tothe basic circuit of Fig. 2 makesamplification possiblebecause of the isolation it provides between the controlandpump sources. Anothermethod of isolationexists;namely, the use of balanced ircuits,,he implest o fwhich is the familiar series-connected saturable inductorshown in Fig. 4. Detailed analysis of th is circuit is givenin Johannessen, ' I 6 ] and it is shown th at tjhc upper limitto thepower gain is

a relation similar to that of (12).In he balancedcircuit, solation is obtainedby fre-

quency eparation. A detailed discussion of frequent;\*separationproperties of balanced ircuits is given byHelgesson. I Because of th e nonlinear elements presentin the circuit of Fig. 4, voltage and current components ffrequencies + n w S +mu, for n, rn = 0, 1, 2, . . ., are

components areeparat'ed such that components offrequencies n odd and 'na even flow in t,he control circuit,and n even and n odd flow in the out'put ircuit.

Thus, we have demonstrat,ed th at t he following condi-tion is necessary to provide power amplification:

Condition 2: Some form of isolation, depending on thetype of nonlinear element, must exist bet,ween the signasource and the pump source.Comnmdy used forms of isolation are frequency selectivenet,n-orks, balancedircuits? and nonlinear resistors(diodes). Conditions 1 and 2 form a set' of necessary andsufficient conditions for power amplification in nonlinearreactance amplifiers.

GAIN-BANDWIDTH LIMITATIOXThe input impedance of the series-connect,ed saturable

inductor circuit, of Fig. 4 is given by tmhexpression21=jd, (14

n-hereL =-E,.20 ,

7 i

To achieve the maximumgain-ba,ndwidt'hproduct, theratio of the current s is maximized subject t'o the int,egraconst8ra.int. mposed by he inductance given in (15)ITsing m a.pproach similar to that employed by Bode,[llthe integralconst,raint is given by

.i\-here Yz1 s the short-circuitransfer admittance parmneter. Since l Y 2 1 1 z =[ I L , / E 3 ! ? ,e have from (16) th at

Tbe power gain expression is given by.zn,! ~ j 2K,, =_ _ - . (I S)R, I s

I-sing (17) and (18), the gain-bandwidth product becomesK,,. B =4f. (19

If a simple resistor is used in series with the controwinding, i n accordance with Fig. 5, again-bandwidthproduc,t o f S f P / ' ~s obtained. Thus, the use of the opt imum~nut~chingetwork increases t,he gain-bandwidth product11)- a factor of T / 2 .

UNILATERALAINA\.\lason's nilateral gain quantity U has meaning only

for circuits in which feedback is possible. For the circuit ofFig. 4 the input modes different' from the outputmode soth at feedback is impossible. The use of a resistive (diodedo.it-nconverter athe utuu t, however,onverts theenerated.Because of th e balanced configurat,ion, these

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JOHANNESSEN ET bL.: NONLIXE-IR REACTANCE MIPLIFIERS 379BANDPASSRESPONSE

Fig . 5. Low-pass andbandpass frequency response chara,ct.eristic.

output mode to t'he input' mode and feedback is possible.From R'Iason[21 wehave

1x12 - 2112i =4(R11R22 - R12R21)' (2 0

From Johannessenr51 we obta inx11 =Rs +R,(1 -x2 2 =R, +R,(1 +e-iwT)x12 =&(- l +e - j w T ) xP l = --&,(I +e - jwT )where R, is the nmgnetizing impedance of the satura.bleinductor. Use of (21) in (20) givesu = Rm2

[RsRp+ +R,) +(Rs- R p ) R m ~ 0 ~ ~ I T ] '(22)

For practica,l circuitsR, >>R, and R,; Rs =R, ( 2 3 )

giving

Since R, is nverselyproportional t o the loss in thenonlinear nductor, (24) sta tes hat he unilateralgainapproaches nfinitywhen the nonlinear nductor lossesapproach zero. Xot e hat ' he circuit just described isa well-known magnetic amplifier.

P A R A 4 M E T R I CAMPLIFIERSIn he foregoing section, amplifiers in which all fre-

quency components were allowed to exist were considered.The parametric amplifier is a limited class of these am-plifiers inwhichonlya few (usually hree) requencycomponents arepermitted. In a paractical circuit. thiscondition cannot be satisfied, so th at an analysis based 011th e existence of exactly hree frequencies is validonlyfor narrowband parametric amplifiers. Helgesson haspointed out the contradict,ion in attempting to establish abroadband theory of parametric amplificat'ion based on a3-frequency analysis. The problem is of part,icular sig-

nificance in a single varactorparametric amplifier. I nthis amplifier filters are used t o provide the dual functionof frequencyiltering and impedancematching. Toobtain a broadband amplifier the bandwidth of thesefilters must be extended, but this extension decreases thefiltering proper ty which, in urn , decreases the gain.From the discussion given in the foregoing section it wasseen that these filters provide the isolation between signaland pump sources required for power amplification. Thus ,elimination of these ilters also eliminates power gain.Helge~son[~1~[~1howed that his contradiction anbeovercome by using balanced circuits n which amplificationisobtainedwith no estrictionon the frequency com-ponents. Thus, impedancematching, or broadbanding,canbeperformedndependently of signal and powersource isolation.

Inan analysis that limits thenumber of frequencycomponents, the impedances to bematched becomefrequencyependent.; andhe matchingroblem isexceedingly difficult to solve. For the noninvertingup-converters the upper limit is given bg7 the Manley-Rowepower relat'ion

J s

this relation being very similar to (13).I < U [ ~ ~erived the following upper bound for ower gainFor the negative mpedance (or reflection) amplifier,

This expression is a maximum when C,/C,, is maximized.The maximum value of this ratio s obtained from physicaconsiderations. Since

C ( t ) =CO+2C1cos ut, and C( t ) 2 0we get

Substituting this value in(26) gives

For wIS0=wp/2

For w p / w o close to uni ty thi s expression is also close tounity.Thus, we againind that he gain-bandwidthproduct is related t o wp /Wh .

The noninvertingpconverter amplifier frequencyresponse is limited by the curve j,/f, in Fig. 5, which isbasically a low-pass response. If the lower sidebandcomponents re ermitted to flow, then a bandpassresponse can be obtained as shown in Fig. 5 . The gain-

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bandwidt,h product is still limited to j , (or slightly higherif we use opt8imum matching etmorlm) ; he gain improve-mentobtained over the passband is due to he tradeoff between low-frequency response and bandpass esponseso as to maintain a constmt gain-bandwidth product.

However, if the usc of :t resistive down-converter and afeedbackembeddingnetwork is permitt,ed, any desiredgain at frequencies below f , can be obtained. The mainproblems with such an implementation re racticalones such as availability of coupling networl