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818 ROBERT S . ELL IOTTmust be satisfied. For all practical purposes, there isnot much that can be done in the way of design to improve Qo and V over the values presently attainable,so one may argue that

(19)where K has been optimized about as well as can beexpected.For a given temperature UAv has a minimum valueestablished by the noise level. Thus, essentially, thefrequency bound is set by the current density of thebeam. Only by achieving higher beam densities wouldit seem reasonable to expect any significant increasein the frequencies obtainable.Referring to Fig. 2, we observe that with an ac beamcurrent density of 1 ampere!cm2, one would not expect to generate in a magnetron a wavelength shorterthan about 30 microns. The assumptions made inderiving this bound were generous enough so that itis perhaps off by one magnitude or more. Should thisprove to be the case, it would suggest that experimenters have just about reached the frequency limitwith present techniques. I t would suggest further

that there are only two favorable lines of attack. Onewould be to concentrate on developing denser beams.The other would be to abandon Group I oscillatorsand investigate some of the more recent schemes forproducing coherent oscillations which do not requirea resonant energy extractor, such as the travelingwave tub4 and devices in which a bunched beamradia tes directly.Is, 6

ACKNOWLEDGMENTThe author is indebted to Dr. Lloyd T. De Vorewho first suggested this problem, and to ProfessorsW. L. Everitt, H. von Foerster, E. C. Jordan, C. Nash,and N. Wax for their generous counsel.t There is some reason to believe that the foregoing analysiscan be extended to traveling wave tubes by changing to a movingcoordinate system. In a sense, the traveling wave tube has adistributed resonant energy-extractor for which a Qo and V canbe computed. Due to its distributed characteristics, it offers more

promise as a coherent generator of short wavelengths than GroupI oscillators, The author hopes to consider this problem in a laterpaper.15 P. D. Coleman, "Generation of Millimeter Waves," Ph,D,thesis, Massachusetts Institute of Technology, May, 1951.16 H. Motz, J. App!. Phys. 22, 527 (1951).

JOURNAL OF APPL I ED PHYS IC S VOLUME 23 , NUMBER 8 AUGUST . 1952

A Mathematical Analysis of a D i e l e c t ~ i c AmplifierLOUIS A. PIPESDepartment of Engineering, University of California, Los Angdes, California

(Received April 1, 1952)This paper presents a mathematical analysis of the fundamental circuit on which the operation of dielectric amplifiers depends, The analysis is based on the assumption that the effective hysteresis curve of theamplifier's dielectric material may be represented by a hyperbolic sine function. The case of resistive loadis analyzed and expressions for the steady-state input and output currents are calculated. From a consideration of the transient response of the amplifier, an estimate of its time constant is obtained.

I. INTRODUCTIONA GREAT deal of the important progress made inall phases of electrical engineering for the pastthirty years is due to the successful use of circuitrybased on the vacuum tube. Despite the excellent performance of the equipment based on this type of circuitry, its use is limited because the tube is a verydelicate device. The tube itself has a relatively shortlife and is sensitive to overloads, shocks, and vibrations.This limits the reliability of this type of electronicequipment to that of the vacuum tube and makes itsuse, particularly in military operations, hazardous. Thislack of complete reliability has encouraged the study ofthe theory and possible application of tubeless devices.I t appears that the most promising competitors of thevacuum tube today are the transistor, the resistance orcrystal amplifier, and the magnetic and dielectric ampli-

fier. The transistor and the magnetic amplifier havebeen covered quite extensively in recent literature.! 2The dielectric amplifier is a relatively new device

from the standpoint of applications.s I t has the advantages of ruggedness, compactness, efficiency, reliability,and high gain. I t s less expensive than tube or magneticamplifiers and is particularly adaptable to high impedance control circuits. I t has the disadvantages of gaindrift, and is often noisy as a result of molecular disturbances sometimes found in its dielectric. Dielectricamplifiers seem particularly well-suited for use in regulators, relays, limiters, servo systems, phase shifters,modulators, and multivibrators.

1 R. L. Wallace, Bell System Tech, J. 30, 381 (1951).2 J. G. Miles, "Bibliography of Magnet ic Amplifier Devices andthe Saturable Reactor Art," Am. Inst. Elec. Engrs. TechnicalPaper 51-388. (This is a comprehensive bibliography concerningMagnetic Amplifiers and their practical application to the middleof the year 1951.)3 A, M. Vincent, Electronics 24, Part II.84 (1951).

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M AT H E M A T I C:A L A N A L Y S IS 0 FAD I E LEe T R IC A !VI P L I F IE R 81 9I t is difficult to credit anyone individual with theinvention of the dielectric amplifier. However, the discovery of the phenomenon that certain dielectrics haven:)lllinear characteristics naturally suggested their usein amplifiers. As early as 1912, Debye4 discovered thatwhen atoms having different electron affinities com

bined, electric dipoles were formed. The nonlinearity ofthe magnetization curves of materials containing iron isusually termed a "ferro-magnetic effect." This effect isknown to depend on the parallel orientation of electronspins of the magnetic dipoles present in the magneticmaterial. By analogy to the magnetic case, nonlinearityof the dielectric constant, caused by the alignment ofelectric dipoles of atomic ions in certain nonmagneticcrystalline structures, has been termed the "ferro-electriceffect," although there is no iron involved.Many substances that exhibit the ferro-electric effecthave been discovered. Rochelle salt was found to be anonlinear dielectric by Valasek5 in 1921. The ferroelectric properties of the dihydrogen phosphates andaresenates were studied in 1935 by Busch and Scherrer.6Wainer and Salomon7 discovered the unusual dielectricproperties of barium titanate in 1942. The ferro-electricproperties of the t itanates and some of their applicationshave been studied by A. von Hippel and his associatesat the Massachusetts Institute of Technology.S,9The barium titanate compositions that are frequentlyused today in dielectric amplifiers are synthetic crystalline materials which have a maximum dielectric constantin the range 1500 to 10,000. The dielectric constant ofbarium titanate changes greatly with temperature variations. The temperature at which the dielectric constantis a maximum is called the Curie point. I t is possibleto shift the location of the Curie point of barium titanateby the addition of strontium titanate. The greatest gainof dielectric amplifiers is obtained when they are operated at the Curie point of their dielectric material.Usually they are designed to be operated at a temperature just above the Curie point. It has been foundpossible to compensate for the change in gain due totemperature variations by using temperature controldevices, or a combination of two dielectrics, one operating above its Curie point and the other one below it.The dielectric constants of some materials such asbarium-Iead-zirconate remain approximately constantfor a considerable temperature variation. When thesematerials are used, temperature control is not a seriousconsidera tion.

4 P. Debye, Polare Molekeln (S. Hirzel, Leipzig, Germany,1929).J. Valasek, Phy. Rev. 17,475 (1921).6 G. Busch, Phys. Acta. 11, 269 (1938).7 E. Wainer, Trans. Am. E!ectrochem. Soc. 89, 138 (1946).8 A. von Hippe!, ONR Research Reviews 24, No.3 (1951).9 A. von Hippe!, Dielectric Relaxation Phenomena in Liquids andSolids (Division of Solid State Physics; Am. Phys. Soc. Pittsburgh, March 8-10, 1951).

E f L - - - - - - - - - - 1 r = ~E.. .sIN (uN}

FIG. 1. Fundamental circuit.II. THE BASIC PRINCIPLES OF DIELECTRICAMPLIFIER OPERATION

The fundamental principles underlying the operationof a dielectric amplifier may be illustrated by means ofFig. 1. The elementary dielectric amplifier of Fig. 1can be compared to a circuit in which the variations ofthe grid potential of a vacuum tube control a relativelylarge amount of power in the plate circuit. This elementary dielectric amplifier consists of a capacitor inseries with a source of harmonic electromotive forceand a constant potential source. The amplitude of thecurrent that flows in the circuit is controlled by varyingthe impedance of the capacitor. The variation of theimpedance of the capacitor is provided by changing thedegree of saturation of its dielectric by varying themagnitude of the direct potential Eo..The dielectric constant of a nonlinear dielectric material decreases with its degree of saturation. The capacitance, and, therefore, the admittance of a nonlinearcapacitor is diminished with saturation. Therefore thecurrent in the circuit may be reduced by saturatingthe dielectric of the capacitor.The Saturation Curve of the Nonlinear Capacitor

In order to study the behavior of the circuit of Fig. 1mathematically, it is necessary to represent the saturation curve V(q), expressing the potential drop acrossthe plates of the capacitor when it carries a charge q,by an analytical expression. A typical saturation curveof a nonlinear capacitor has the general form given byFig. 2.The typical saturation curve V(Q) may be represented by a variety of analytical expressions. A veryuseful one that can be adjusted to fit the usual experi-

\IfQJ

1 ~FIG. 2. Saturation curve of the capacitor.



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820 LOUIS A. PIPESmental V(Q) curve closely is the hyperbolic sine curve Therefore,

SoV(q)=- sinh(aq). (2.1)aIn this expression, So is the initial elastance of the

capacitor defined bySo= ( dV )dg q=O (2.2)

This is the reciprocal of the initial capacitance de-fined by( d

q )Co= -dV q=O (2.3)The constant a is a measure of the nonlinearity of

q = ~ sinh-1( aE ) .a So (2.8)I f (2.8) is differentiated with respect to t, the followingexpression for the current i is obtained:

i(t) = (a 2E2+S02)-idEjdt. (2.9)I f it is assumed that the control potential, Eo, is muchgreater than the maximum value of the applied harmonic potential of the circuit, so that

(2.10)then the expression (2.9) may be written in the form,

i(t) = A0 cos(wt), (2.11)the saturation curve of the capacitor. The slope of the whereV(q) curve is given by (2.12)dV-=So cosh(aq) = V'.dq (2.4)Therefore the curve V(q) has constant slope and is

E."t{i} ::0 /ljml,-IifI_U/l_11_f?\- I.i- ec;.,,sIN[Wt,

FIG. 3. Dielectric amplifier.therefore a straight line only when a=O. I f (2.4) isdivided by (2.1), the result may be written in the form

V'a= - tanh(aq).V (2.S)For large values of q, tanh(aq) = 1. Therefore (2.S) maybe written in the form

(2.6)Equation (2.6) may be used to obtain an estimate fora from the empirical saturation curve of the nonlinearcapacitor.In order to accentuate the most important featuresof the elementary dielectric amplifier of Fig. 1 and tokeep the mathematical complexity of the analysis to aminimum, let the resistance of the circuit be neglected.The equation of the circuit may be obtained byequating the applied potential of the circuit to thepotential drop across the plates of the capacitor V(q)in the form

SoV(q)=- sinh(aq) = Eo+Em sin(wt)=E(t). (2.7)a

A measure of the gain of the dielectric amplifier is(2.13)

A small increase !!.Eo of the control potential isaccompanied by a change of !!.Ao in the amplitude ofthe current given by(2.14)

Since the gain of the amplifier }J.o is negative, a slightincrease of the control potential corresponds to a decreasein the amplitude of the current in the amplifier.III. A PRACTICAL DIELECTRIC AMPLIFIERCIRCUIT NOTATION

Having outlined the basic principles involved in theoperation of dielectric amplifiers, a basic circuit offundamental importance in the design of dielectric amplifiers will now be studied. This circuit is illustratedby Fig. 3.The more elaborate dielectric amplifier circuits suggested for practical applications are essentially variations and combinations of this basic circuit. Becauseof its technical importance a mathematical analysis ofits operation will now be presented. The followingnotation will be used:qo(l) = the circulating charge in the input circuit.q(t) = the circulating charge in the output circuit.io(t)=qo=the input current.i(t)=q=the output current.So= the initial charge of the nonlinear capacitor.Co= the initial capacitance of the nonlinearcapacitor.Ro=the resistance of the input circuit.R= the resistance of the output circuit.Eo= the control or input potential.



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MATHEMATICAL ANALYSIS OF A DIELECTRIC AMPL IF IER 821Em sin(wt) = the potential applied to the output circuit.Q(t) = (qo-q) = the charge of the capacitor.V(Q) = the potential between the plates of the nonlinear capacitor.V(Q) = (So/a) sinh(aQ)=an analytical expressionfor the saturation curve of the nonlinear

capacitor.a= a constant determined from the empiricalsaturation curve of the nonlinear capacitor.R. = RoR/(R o+ R) = the effective resistance ofthe circuit.

b= 1/R eCo=a constant.To= 1/b=ReCo= the initial time constant of thecircuit.Qo(t) = the first approximation to the charge of thecapacitor.A = EoCoR.a/Ro= a constant.B=Ema/Rb2+W2!= a constant.I n(X) = the modified Bessel function of the firstkind of order n and argument x.

The General Equations of the Dielectric AmplifierI f Kirchoff's electromotive force law is applied to thetwo loops of the circuit of Fig. 3, the following equationsare obtained: Roio+ V(Q) = Eo, (3.1)

R i- V(Q) = Em sin(wt), (3.2)whereQ= (qo-q) = j t(io_i)dt. (3.3)

oV(Q) is the potential drop across the plates of the nonlinear capacitor, and qo and q are the circulating chargesin the two loops of the dielectric amplifier circuit.Equations (3.1) and (3.2) may be written in thefollowing form:and i o+V(Q)/Ro=Eo/Ro

i- V(Q)/R=Em sin(wt)/R.I f (3.5) is subtracted from (3.4), the result is

(3.4)(3.5)

( i O - i ) + ( ~ + ~ ) V ( Q ) = E o / R o - E m sin(wt)/R (3.6)Ro Ror

where. V(Q) Eo EmQ+--=--- sin(wt),R. Ro R

Re=RoR/(Ro+R).

(3.7)

(3.8)Equation (3.7) is a nonlinear differential equation forthe determination of Q(t). When Qhas been found bysolving (3.7), then the input current io may be determined by means of (3.1) in the form

io(t) = [Eo- V(Q)J/Ro. (3.9)Similarly, as a consequence of (3.2), the output current

i is given byi(t) = [Em sin(wt)+ V(Q)J/R.

In order to effect the solution of (3.7) it is necessaryto represent the saturation curve V(Q) by an analytical expression. In this analysis, the expression givenby (2.1) will be assumed. With this assumption, Eq.(3.7) takes the form

So Eo Em dDQ+ - s inh (aQ)=- - - sin(wt) , D=- R ~ Ro R &I f the term Qo/CoR. is added to both members of (3.11),the result is

Eo Em b(D+b)Q( t )= - - - sin(wt)+-[aQ-sinh(aQ)].Ro R aI f it is assumed that the circuit is inert at t= 0, thenQ(O) =0 and the differential Eq. (3.12) may be writtenas a nonlinear integral equation having the followingform :10,11

b tQ(t) = Qo(t)+-j e-b(t-u){ aQ(u) -sinh[aQ(u)J}du,a 0where Qo(t) is the solution of the linear differentialequation

(D+b)Qo(t) = Eo/Ro-Em sin(wt)/R,with Qo(O) = O.The nonlinear integral Eq. (3.13) is a special caseof an integral equation studied by Lalesco. 12 It s solution may be effected by setting up the following sequenceof functions:

b tQ1(t) = Qo(t)+-f e-b(t-u)a 0X (aQo(u)-sinh[aQo(u)J}du,

b f tQ2(t)=QO(t)+- e-b(t-u)a 0 (3.15)X{aQ1(u)-sinh[aQ1(u)]}du,

10 L. A. Pipes, "Operational Methods in Nonlinear Mechanics,"Report 51-to (Department of Engineering, University of California, Los Angeles, 1951).11 L. A. Pipes, Applied Mathematics for Engineers and Physicists(McGraw-Hill Book Company, Inc., New York, 1946), ChapterXXI.12 Vito Volterra, Ler;ons sur les Equations Integrales (GauthierVillars, Paris, France, 1913), p. 90.



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822 LOUIS A. P IPESThe limit of the sequence of functions, Qo(t) , Ql(t),

. . . , Qk(t), defined by (3.15), is the solution of (3.13).That is, limit Qk(t) = Q(t).

100(3.16)

The function Qo(t) is the solution of (3.14) withQo(O) =0; it is given by

(1- e-bt ) Em [e- bt sin(wt-)]Qo(t)=Eo - -w - - -+ (3.17)Rob R W+w2) WW+W2)tan =w/b.

This is the variation of the charge on the plates of thecapacitor if it had a cons tant capacitance Co. I t is thefirst approximation to the solution of the integral Eq.(3.13). An investigation of the convergence of thesequence (3.15) indicates that it converges rapidly, particularly for small values of a.IV. THE THEORY OF THE FIRST APPROXIMATION

The response of the dielectric amplifier based uponthe first approximation to the solution of the integralEq. (3.13) may be obtained by substituting Qo(t) intothe Eqs. (3.9) and (3.10) for the input and outputcurrents of the amplifier and thus obtainingio(t) = l/Ro{Eo- S osinh[aQo(t)J/a} (4.1)and

1 { So }i(t) = R Em sin(wt)+-;; sinh[aQo(t)] . (4.2)A. The Effective Time Constant of the AmplifierI t can be seen from the form of the transient solutionfor Qo(t) given by (4.12) that the quantity

1 R.To--=R.Co= -b So (4.3)is a measure of the rapidity with which transient disturbances in the dielectric amplifier circuit attenuate.So is the initial elastance of the nonlinear capacitordefined bywhere 50= (dV d Q ) Q ~ o ,V(Q) = So sinh(aQ)/a

(4.4)(4.5)

is the saturation curve of the nonlinear capacitor. Abetter estimate of the response time of the dielectricamplifier may be obtained if the incremental elastanceof the nonlinear capacitor is used instead of So. Theincremental elastance of the nonlinear capacitor is defined by the equation,

Si= (dV /dQ)Q=Q=So cosh(aQ). (4.6)Then if Q. is the mean operating value of Q when deviations from the steady-state operation are initiated, aneffective value of the time constant of the amplifier

T cmay be defined by the expressionRe

(dV I dQ)Q=Qe So cosh(aQ.) (4.7)For any case other than Qe=O, we have Te


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M A T H EM A TIC A LAN A L Y SIS 0 FAD I E LE e T RIC AM P L I F IE R 823The following identities are established in works onBessel functions: 13

00cosh[B cos(wt)]=Io(B)+2L:I 2n(B) cos(2nwt) (4.16)n= 1

and'"sinh[B cos(wt)]= 2L: J(2n+l) (B) cos[(2n+ l)wt], (4.17)

11=0

where J ,,(x) is the modified Bessel function of the firstkind of order n and argument x. These functions havebeen extensively tabulated. 14 By the use of the identities (4.15), (4.16), and (4.17), the expressions (4.13)and (4.14) for the steady-state input and output currents may be written in the following form:

Eo So [io . ( t )=-- -s inh(A) Io(B)Ro aRo00 ] 2So+2:LJ(2n)(B) cos(2nwt) - - co sh (A)

n= 1 aRo

andEm So [i 8( t )= - s in(wt)+- sinh(A) Io(B)R aR

'" ] 2So+2L:J(2n)(B) cos(2nwt) +-cosh(A)n= 1 aR

I t is thus seen that the steady-state output and inputcurrents contain even and odd harmonics of the fundamental applied potential. I f greater accuracy than thatgiven by the first approximation is desired, the secondterm, Ql(t), of the sequence (3.1S) may be calculatedand substituted into the Eqs. (3.9) and (3.10).

V. AN ALTERNATIVE METHODI f the control potential is much greater than theamplitude of the harmonic potential so that the condition

EoE", (5.1)is satisfied, then the following alternative method maybe used to determine the amplitude of the outputcurrent.

13 N. W. McLachlan, Besfel Functions for Engineers (OxfordUniversity Press, London, England, 1934).14 E. Jahnke and F. Emde', Tables of Functions (Dover Publications, New York, 1943), pp. 232-233.

Equation (3.7) may be written in the formSo Eo Em .- s i nh ( aQ)= - - - sin(wt)-Q.aRe Ro R (5.2)

An approximate steady-state solution of this equationmay be obtained by taking Q o be given by Q08 andwriting it in the following form:

So Eo Em .- s inh(aQ)=---- sin(wt)-Q08' (5.3)aRe Ro RThe value of Q08 is given by (4.11); it is

Eo Re Em cos(wt)Qo.=- -+ - . (5.4)So Ro R (b2+W2)tIf this is differentiated and substituted into (5.3), thefollowing equation is obtained:

So Eo- sinh(aQ)=-+P sin(wt)=I(t);aRe Ro (5.5)where

Em[ W ]P=R" W+w 2)! 1 . (5.6)I f (5.5) is solved for Q the result is

Q = ~ sinh-{ablj- (5.7)By differentiating (5.7) with respect to t one obtainsSince Q= (a

2J2+b2)-!dJldt.J(t) = (Eol R o)+ P sin(wt),

dJldt=wP cos(wt),

(5.8)(5.9)

(5.10)and hence substituting (5.10) into (5.8) we obtain

. wP cos(wt)Q .(a2J2+b2)! (5.11)Now if (3.1) and (3.2) are added together the followingresul t is obtained:

ioRo+iR=Eo+Em sin(wt). (5.12)From the definition of Q given by (3.3), we have

(5.13)I f Eqs. (5.12) and (3.13) are solved simultaneously forio and i, the results are

Eo+Em sin(wt)+RQio(t) (R+Ro) (5.14)



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824 LOUIS A. PIPESand Eo+Em sin(wt)-RoQi(t) (R+Ro) (5.15)Hence by eliminating Qby means of (5.11), the follow-ing expression for the steady-state output current isobtained:

Eo 1i,(t) +-(R+Ro) (R+Ro)

[ wP COS(wt)]X Em sin(wt)-Ro .(a2[2+b 2)tNow if EoEm,

(5.16)

(5.17)since the term P sin(wt) in (5.9) can then be neglected.Equation (5.16) can be written in the form

where

and

Eoi,(t) +Ao sin(wt-cp),(R+Ro)Ao (Em

2+Ko2)!(R+Ro) ,

Kot ancp=-Em

(5.18)

(5.19)

(5.20)As can be seen from (5.19) and (5.20), the effect ofincreasing the control potential Eo is to decrease Koand hence to decrease the amplitude of the harmonicload current Ao. The effective gain of the amplifiermay be defined by the expression

(5.21)

I f the relative harmonic content of the output currentis desired, it may be obtained by substituting the complete expression for let) as given by (5.9) into (5.16)and obtainingi,(t) Eo + 1 {Em sin(wt)

(R+Ro) (R+Ro)RowP cos(wt) } (5.22)[a 2(E/Ro+P sinwt)2+b2Ji

The binomial expansion of the radical term in (5.22)yields the required harmonic terms.The method of analysis discussed in this section is ageneralization of the one presented in Sec. I I generalized to take into account the effects of resistance. I tgives good quantitative results particularly when thecontrol potential Eo has a value much larger than themaximum harmonic potential Em.

VI. CONCLUSIONA mathematical method for determining the responseof a fundamental dielectric amplifier circuit has been de-veloped. Although the analysis is based on the assumption that the saturation curve of the nonlinear capacitorof the amplifier has the form V(q) = (So/a) sinh(aq), itcan easily be extended to cases where other analyticalexpressions are given for the saturation curve V(q).The solution of the fundamental nonlinear differentialEq. (3.7) may be effected either by an iteration process

similar to that of (3.15) or by a method such as thatdiscussed in Sec. V for whatever analytical expressionmay be used to represent the saturation curve of thecapacitor. Once this solution is obtained the amplification constant, time constant, and steady-state responseof the amplifier may be readily computed.

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