DA Analysis

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C A R R I E R D I F F U S I O N IN BASE R E G I O N OF T R A N S I S T O R 1297The value of a is

00 W[ (nrrL)2 (PL)2 J1El UnsechL 1+ B + I i +jWTa=-----------------------------------

For y=B this reduces to00 n[1 - (-1)n coshpJ sech:[ + ( n ~ Y + ( 4 Y+ jWTr

a= 4rr2 L -------------------------------------------[(nrr)2+ p2J2= lThis is the current amplification factor for uniform current density at the emitter and transverse field.The authors would like to acknowledge suggestions from Dr. J. S. Saby and Dr. P. Weiss. The work wassupported by the U. S. Air Force Air Research and Development Command, Army Signal Corps, and NavyBureau of Ships under contract AF 33 (600)-17793.

J O U R N A L OF A P P L I E D P H Y S I C S V O L U M E 2 5. N U M B E R 10 O C T O B E R . 1954

Analyses of Basic Dielectric Amplifier CircuitsSHou-HsIEN CHOWBurroughs Corporation Research Center, Philadelphia, Pennsylvania

(Received December 16, 1953)Dielectric amplifiers with high-input impedance and low-output impedance have high power gain and slowresponse. Analyses of two basic types of dielectric amplifier circuits, i.e., parallel and series, are given here.

Both steady-state and transient responses are studied.I. INTRODUCTION

DIELECTRIC amplifiers offer a solution to theproblem of obtaining high-power amplification ofsignals from a source with a high internal resistance. Inthis paper mathematical analyses of two basic types ofdielectric amplifiers, i.e., parallel-connected and seriesconnected circuits, are given. The basis of a mathematical analysis of a nonlinear circuit is the faithfulrepresentation of the characteristics of the nonlinearelement in the circuit. For a nonlinear capacitance thevoltage V across the capacitance is not a single-valuedfunction of the charge Q accumulated because of thepresence of hysteresis. To make a mathematical analysispossible, the Q- V characteristics will be represented bya single-valued function. This representation is a good

one when the hysteresis loop is narrow.A power series can faithfully represent a single-valuedQ- V characteristic to any degree of accuracy, depending on how many terms are being used in the series.A typical Q- V characteristic of a nonlinear capacitanceis shown in Fig. 1. In the analysis of the parallel-connected circuit, this characteristic will be represented byQ(V)=k1V- k 3V3+k6V5-k7V7+ . . . ; (1)

and in the analysis of the series-connected circuit, thecharacteristic will be represented byV(Q) = kx'Q+k3Q3+k6Q6+ . . . . (2)

In Eq. (1) many terms are necessary to represent aQ- V characteristic, whereas in Eq. (2) two or threeterms are sufficient.

II. BASIC DmLECTRIC AMPLIFmR CIRCUITSThe basic mechanism of dielectric amplifiers dependson some degree of isolation of the input and output

Q

v

FIG. 1. Q- V characteristic of a nonlinear capacitor.

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1298 SHOU -HS IEN CHOWcircuits. This can be achieved by a push-pull or balancedconnection. Figures (2) and (3) show the two basic typesof push-pull connections. Figure (2) shows a parallelconnected circuit! in which two nonlinear capacitancesCI and CII in parallel are connected to the ac carriersource. Figure (3) shows a series-connected circuit, inwhich CI and Cu in series are connected to the ac carriersource. In both circuits, Vo is the input voltage, Ro theinput resistance, Vm sinwt the ac carrier voltage, and Rzthe load resistance.

III. PARALLEL-CONNECTED AMPLIFIERIn the circuit shown in Fig. 2, C1=C2=C are twolinear capacitances which serve to balance the circuit.The capacitance C must be a high capacitance so that itsreactance is very low in the ac circuit. When this is true,there are negligible ac voltages across C1 and C2 andtherefore the ac current through Ro is negligible. Thecircuit equations are

(3a)

where

and qI, qu are related to VI, Vu by the Q- V characteristic (1). On combining Eqs. (1), (4), (6), and (7) it isfound that

Vm f [dQ(V) I ]- coswt= vdt+2R z-- vw dV V=Vo/200 1 1 [d2n+!Q(V) I ] }2Rz L v2n+!n=! (2n+l)! dV2n+l V=VO/2

wheredQ(V)1Cv=--dV V=VO/2'Vm sinwt= izRz+VI+ VI,Vm sinwt=izRz+vu+ V2, (3b) and

where VI and VII are voltages across the nonlinearcapacitors CI and CII, respectively, and VI and V2 arevoltages across the linear capacitors C1 and C2, respectively, with - V 1=V 2=Vo/2. Therefore,VoVI= t '+ - , (4a)2Vo

Vu=V-- , (4b)2where v is a periodic function to be determined. Onsubstituting Eqs. (4), Eqs. (3) both become

Vm sinwt = izRz+ v, (5)or(6)

IIVo

FIG. 2. Parallel-connected dielectric amplifier.1 A. M. Vincent, Electronics 24, 84 (1951).

~ W = L ~ B0 { 1 [d2n+!Q(V) I ] }n=! (2n+l)! dV2n+l V=VO/2 .Equation (8) can be solved for V by using the Laplacetransformation. The method has been used by Pipes insolving a similar equation of this type2 and discussed bythe author.3 Take the Laplace transform of Eq. (8):

where p is the Laplace variable, L the Laplace operator,L(v)=fJ(p)=v, and A is a constant. On rearranging Eq.(9) it is found thatVmm p2v= - - - - - - - - -w (p2+W2) (p+m)

Am 1 [ m]+--- 1 - - { L [ ~ ( v ) J } ,p+m Cv p+m

(10)where

1m=--.2RzCv

Take the inverse transform of Eq. (10). Since only thesteady-state response is of interest here, the transientpart of the inverse transform will be neglected. By theuse of the Faltung theorem of the Laplace transforma-

2 L. A. Pipes, J. App!. Phys. 23.1 625 (1952).3 S. H. Chow, J. App!. Phys. 2:', 216 (1954).



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BASIC D I E L E C T R I C A M P L I F I E R C I R C U I T S 1299

FIG. 3. Series-connected dielectric amplifier.tion4 the inverse transform of Eq. (10) is

Vmm 1v (m sinwt-w coswt)---(v)(m2+w2) Cv+ ~ t cmU-u).[v(u)]du.CvJo (11)

The integral equation Eq. (11) can he solved hy themethod of successive approximations. By taking thefirst approximation of 'lJ as

the second approximation is1'11(2) = v(l) - --[v(l)]Cv mIt- E-m( t -u ) ['11(1) (u) ]du,

Cv 0(13)

etc. Although (v) occurring in Eq. (8) is an infiniteseries, a finite number of terms will be sufficient. Thenumber of terms necessary depends on the accuracydesired, and the amplitude of V 0 and V.,.. When Vo andVm are small only a few terms are necessary. The loadcurrent i l is 1i l= - (V m sinwt-v).Rl

(14)On carrying out the details of the suggested procedure,the steady-state solution can be found to any desireddegree of accuracy.The response time of the parallel-connected dielectricamplifier depends mainly on the time constant RoC /2 , or

the time necessary to charge the linear capacitors inseries. The nonlinear capacitances, on account of theirlow values compared with C, affect the response time but4 See, for instance, L. A. Pipes, Applied Mathematics for Engineers and Physicist (McGraw-Hill Book Company, Inc., NewYork, 1946), Chap. XXI.

little. The circuit is inherently a slow-response circuit.In order to avoid interactions between input and outputcircuits, either Ro or C has to be high, and therefore thetime constant is long.It is desirable that the load resistance be low. Thereason can be visualized in the following manner. Whenthe infinite series (v) is neglected in Eq. (8), the circuitequation is simply that of a resistance Rl and a linearcapacitance 2C v in series. When V0 is zero, 2C v is a highcapacitance whose value will be denoted by C,,; andwhen Vo is high, 2C v is a low capacitance whose valuewill be denoted by C/. For a high-gain amplifier, thedifference in the power delivered to Rl when thecapacitance is high and when the capacitance is low isgiven by

t:.P-Rl[ Em2 Em2 1 15)-2 (RI2+_1) (RZ2+_1 ) ,W2C,,2 W2CZ2

which should be high. By neglecting the second term inEq. (15) and differentiating the resulting equation withrespect to R l , it is found that t:.P is a maximum whenR 1=1/WCh. Therefore, Rl should be a low resistance.When a shorter response time is desired, either Ro or Cshould be low. In this case, the current through Ro willbe appreciable. The interaction between input andoutput circuits will then reduce the effectiveness of theinput voltage in controlling the value of the nonlinearcapacitances. This interaction may be called "inherentnegative feedback."

IV. SERIESCONNECTED AMPLIFmRFigure 3 shows a series-connected dielectric amplifier.

In this circuit Ro should be high to avoid interactionbetween input and output circuits. A step input voltageVo will charge the nonlinear capacitors C1 and CI l to Voand - Yo, respectively. When a steady state sets in thecurrent through Ro is insignificant and will be neglected.Therefore, the steady-state circuit reduces to that

,,1..

2R

FIG. 4. Steady-state circuit of a series-connecteddielectric amplifier.

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1300 SHOU-HS IEN CHOWshown in Fig. 4. The circuit equations read

V m sinwt= vI+VU+iIRI,V m sinwt= 2Ri2+i IR I,

i l=i 1+ i 2,

(16a)(16b)(16c)

where VI, VII are the voltages across Gr and Gn , respectively, and are related to the charges on Gr and Gnby the Q- V characteristic (2). Since the same accurrent flows through G1 and Gn , the charges on Gr andGIl are qI=ql+QO,qn=ql-Qo,

(17a)(17b)

where Qo is given by Vo on the Q - V characteristic (2),and ql= f i1dt is a periodic function to be determined.On combining Eqs. (2), (16), and (17) it is found that

2R 2RRI dql [dV(Q) I ]--Vm s i nw t= - - - - -+2 -- ql2R+R I 2R+R I dt dQ Q=Qo

00 { 1 [d2nHV(Q) I ] }2E q ~ Hn=1 (2n+ 1)! dQ2n+l Q=Qo ' (18)or dqlVm' sinwt= R l l-+2Sql+2i1 (ql) , (19)dtwhere 2R

Vm '= - - - V m ,2R+R I2RRIR l1=--- ,2R+R I

S= dV(Q) IdQ Q=Qo'

Equation (19) can be transformed into an integralequation by the same procedure used in obtaining Eqs.(9), (10), and (11). The resultant integral equation isVm'ql em' sinwt-w coswtJR ll (m'2+w2)

2 II- C m'(I-uli1[ql(U)]du,Rl l 0

(20)

where m' = 2S R H By taking the first approximation ofVm'ql( l l= em' sinwt-w coswt],R 11 (m'2+w2) (21)

the second approximation is

etc. Although i1(ql) has been expressed as an infiniteseries, one or two terms will prove to be sufficient. FromEq. (16) the load current i l is

Vm 2R dqli l =- - - s i nw t+ - - - - - . (23)2R+R I 2R+R I dtOn carrying out the details of the suggested procedure,the steady-state solution can be found to any desireddegree of accuracy.

The transient response of the series circuit may bestudied with the ac voltage short-circuited. This devicehas been used by the author in the t ransient analysis ofa parallel-connected magnetic amplifier. Such a simplification proves to be necessary when the Laplacetransformation is used in solving nonlinear equations. I fthe ac voltage is included in the analysis, the resultanttransient is the transient excited when both the inputvoltage Vo and the ac voltage Vm sinwt are suddenlyapplied to the circuit at the instant t=O. Such is not thecase in practice, since the ac voltage is always present.Furthermore, neglecting the ac voltage introduces onlya small error. The reasoning here is the same as thatstated in reference 3. The simplified transient circuitthen becomes that shown in Fig. 5, where G1(= Gn ) isthe nonlinear capacitance. The charging of the nonlinearcapacitance when a step input voltage Vo is applied is

R (A)R.

~ N \ / ' - - - - - l l l t - - - - ' - - . . . L . . !2Flo V.

FIG. 5. Transient circuit of a series-connecteddielectric amplifier.



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BASIC DIELECTR IC AMPLIF IER CIRCUITS 1301governed by the following equation: Vt

dqtVo= (2Ro+R)i t+vt= (2Ro+R)-+v(qt) , (24)dtwhere v(qt) is given by Eq. (2). Equation (24) may betransformed into an integral equation by the sameprocedure used in obtaining Eqs. (9), (10), and (11).The resultant integral equation isqt- Vo +[Q._ Vo ]e-m"t- (2Ro+ R)m" ' ( 2R o+ R)m"

1 ft2Ro+R 0 e-m"(t-u). [kaql(u)+k5'qt6(U)+ . . . ]du, (25)

where m"=k//(2Ro+R) and Qi is the initial dc chargeon the nonlinear capacitor. By taking the first approximation of qt as

Vo [ VO]1 )_ + Q._ e-m"tqt - (2Ro+R)m" ' ( 2R o+R)m" , (26)

etc. The charging of a nonlinear capacitor by a dcvoltage can be visualized in the following manner.Initially the nonlinear capacitor has been charged to adc potential Vi, corresponding to a dc charge Q . Thedifferential capacitance of Cr is1C i=- - - - -

d : ~ Q ) Q = Q i(28)

When the nonlinear capacitor is then charged to Vo,corresponding to a dc charge Qo, the differentialcapacitance is 1Co=-----

d : ~ Q ) I Q = Q O(29)

v.

~ - - - - - - - - - - - - - - - - - - - - - - tFIG. 6. Charging of a nonlinear capacitor by a dc voltage.

I f the transient response curves of two linear series RCcircuits with a resistance (2Ro+ R) and one with acapacitance C i and the other with a capacitance Co areplotted, the transient voltage wave form of the nonlinear circuit must lie between the two curves. Figure 6shows a case when Q;=O.In the series-connected amplifier the input resistanceshould be high and the load resistance should be low forthe reasons mentioned in the analysis of the parallelconnected amplifier. I f the input resistance is not high,the interaction between input and output circuits willcause an "inherent negative feedback" as in the parallelconnected circuit.

v. CONCLUSIONDielectric amplifiers, with high input resistance andlow load resistance, are inherently devices with highpower gain and slow response. In the present paper, thesteady-state response of both parallel- and series-connected dielectric amplifiers has been studied. By a seriesof successive approximations, the steady-state solutioncan be carried out to a high degree of accuracy. Thetransient response has also been studied. The entireanalysis assumes a single-valued relation connecting Qand V of the nonlinear capacitors. Interactions betweeninput and output circuits, which are negligible in highgain, slow-response dielectric amplifier circuits, havebeen neglected. Certain of the engineering aspects ofdielectric amplifiers may be found in reference 1 and willnot be repeated here.


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