Payne: Distributed Amplifier Theory

substituting r for s (r = s for Fig. 2A) in the expressionfor the capacity of system IV for small s, we obtain anexpression identical with the capacity for system I underan average-power limitation for small r. It follows thatthe capacity and optimum statistics are the same forsystem I as for system IV under a peak-power limitationwhen s is small.The same considerations applied to system V witlh

small s lead to the same capacity for system I, but dif-ferent optimum statistics are obtained. It can be shownby an analysis of system I in the same way we haveanalyzed the other systems at small signal-to-noiseratios that, to a first approximation, the rate of trans-

mission of information is

R= (A2- A/012)W/2 U2. (H-1)

This rate is greatest when the amplitude of the trans-mitted signal is constant and equal to B, and the dis-tribution of its phase has no Fourier first-harmoniccomponent. Such is the case for the phase distributionof Fig. 2A; it is also the case for the amplitude distribu-tion of Fig. 2E if the latter is regarded as having aconstant amplitude but phase 0 or wx, each with prob-ability . In this approximation, the rate of transmissionof information is not affected by the difference betweenthese distributions.

Distributed Amplifier Theory*DELMAR V. PAYNEt, SENIOR MEMBER, IRE

Summary-The mathematical theory of the operation of the plateline of a distributed amplifier is developed using matrix algebra as atool. The effects of a finite number of current generators placed atregular intervals on what would otherwise be an ordinary lumped-constant transmission line are determined. The theory predicts themanner in which the output of a distributed amplifier is changedwith propagation constant of the grid and plate lines, termination ofthe grid and plate lines, number of tubes, and grid driving voltage.Some of the deductions made were tested using a six tube distributedamplifier; and, the experimental results were found to be in generalagreement with predicted results.

INTRODUCTION

fl HE DISTRIBUTED AMPLIFIER, invented byW. S. Percival, was an attempt to increase thegain-bandwidth product of an amplifier by sepa-

rating the individual tube interelectrode capacitanceswhile adding their transconductances." 2 Considerablework has been done in the United States on the applica-tions and improvement of operation of this type of am-plifier.3-5 The objective of increased gain-bandwidthproduct was attained by separating individual tubes bya portion of a lumped-constant transmission line as in-

* Decimal classification: R363.4XR132. Original manuscript re-ceived by the Institute, August 26, 1951; revised manuscript received,November 10, 1952. Extract from a report submitted in partial ful-fillment of requirements for the M.S. degree at the Graduate Schoolof Kansas State College.

t Bendix Aviation Corporation, Research Laboratories, Detroit,Mich.

' William S. Percival, British Patent 460,562.2 H. A. Wheeler, "Wide-band Amplifiers for Television," PROC.

I.R.E., vol. 27, pp. 429-438; July, 1939.3 E. L. Gintzon, W. R. Hewlett, J. W. Jasberg, and J. D. Noe,

"Distributed Amplification," PROC. I.R.E., vol. 36, pp. 956-969;August, 1948.

4 W. W. Horton, J. W. Jasberg and J. D. Noe, "Distributed Am-plifiers: Practical Considerations and Experimental Results," PROC.I.R.E., vol. 38, pp. 748-753; July, 1950.

'f H. G. Rudenberg and F. Kennedy, "300-mc Traveling-waveChain Amplifier," Electronics, vol. 22, pp. 106-109; December, 1949.

dicated in Fig. 1. An examination of Fig. 1 shows that,for Class A operation, the current and voltage equationsfor the grid line may be found by the usual methods ofanalysis of lumped-constant transmission lines; but, theplate line, since it contains the tubes as active elements,may not be analyzed by these methods. This paper isconcerned with the analysis of the plate line.

Z T Zr

T Zr0

Fig. 1-Schematic of a distributed amplifier.

ANALYSIS OF THE PLATE LINE

In order to simplify the mathematics and to approxi-mate existing conditions over the useful frequency range,a number of assumptions were made.

1. All elements other than tubes are assumed to belinear and passive.

2. There is no magnetic coupling between inductiveelements.

3. The lines, both plate and grid, are assumed to becomposed of identical sections.

4. The tubes are capable of being represented by cur-rent generators.

5. The currents and voltages may be expressed bysine and cosine functions of time.

6

1953 759

PROCEEDINGS OF THE I.R.E.

Zs Zr

Fig. 2-Block diagram of the plate line of a distributed amplifiershowing assumed directions of currents and voltages.

Consider the periodic structure shown schematicallyin Fig. 2. The construction of each identical three-termi-nal pair is shown in more detail in Fig. 3. The equationsof the kth three-terminal pair are of the form

Ek-1 all a12 0] Ekl'1

Ik-I = a21 a22 0 Ik-li

Ek-1 1 0 0 E

Ik--11 1 -ik0k0

Et- -bil b12 ° - Ek

Ilk1 = b2l b22 ° IkL§kiJ Lo0 :12 1

If the proper substitutions are made in these equationsand the indicated multiplications are performed, a 3 X3matrix of the form

Ek-1 Cll C12 C131k1[ Ek 1

Ik-1 C21 C22 C23ik Ik (1)

L IJ Loot ILiis obtained. Equation (1) is not particularly useful sincethe 3 X3 matrix has elements containing ik. Equation(I) may be rearranged, after introducing ik - 1, to obtain

-Ek_l -Cll C12 C13 - Ek -

Ik-1 C21 C22 C23 Ik

Ek.1 - L Cii C12 Cii E ~(2)

where will be of the form er. F is complex and is recog-nized to be the propagation constant of the grid line.

In order to investigate the plate line without obscur-ing the results by the details, divide the 3 X3 matrix of(2) into submatrices as indicated. Call

Cll C121D =

C21 C22

Xk =

F C131LC23

=Ek[:1

l ITk-1J

Equation (2) may now be written

[X]k-1 = [ -F] [Xk]

Fig. 3-Block diagram of the kth section of the plate line of a dis-tributed amplifier which is composed of n identical sections.

where 0 = [O 0]. The matrix equation for any three-ter-minal pair of the plate line is identical to (3) except forsubscripts, so that the process of relating the kth voltageand currents to the k-nth voltage and current centersaround the nth power of the matrix

ED F

LO tz(4)

It may be proven by induction that the form of the nthpower of the matrix is

-D F- n

[Dn Dn-'F+D-2Fr+.;. +DF¢n-2+Fn ](

D is the 2 X 2 transformation matrix relating input andoutput voltages and current when there is no currentik, as in the grid line. It is observed that Dn relates

Xo to X",Dn-'F + Dn-2Fr + .. . + DFtn-2 + F1n-l

relates XO to in and ;n relates io to in (io is a fictitiousquantity).

Propagation constant is related to elements of D by6

ez, e-7 = 1/2(cnl + C22) ± 1/2 [(c1l + C22) 2 *.±. 4] 1/2. (6)

y is associated with the right traveling components ofvoltage and current; and, -'y is associated with the lefttraveling components of voltage and current. Let

En= A1 + A2, In = B1 + B2, in = C1 + C2. (7)

where the subscript 1 designates the right travelingcomponents and the subscript 2 designates the lefttraveling components.7 Represent the iterative imped-ances of the plate line at some point k by Zo1 and Z02;also, represent the iterative impedances of the line atthe point of entry of some current ik by Zol' and Z02'.Let

A1B1 = -,

zo1

Cl =

Z01' + Z02'

A2B2 =

-Z02

C2--Z0zol Z02,

(8)

6 Leon Brillouin, Wave Propagation in Periodic Structures, Mc-Graw-Hill Book Company, Inc., New York, N. Y., 1946, p. 211.

(3) 7 Ernst A. Guillemin, Communication Networks, Vol. II. JohnWiley and Sons, Inc., New York, N. Y., 1935, p. 165.

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Payne: Distributed Amplifier Theory

F i,n gives a voltage E'=cl13 i,, and a current I'= c2si,.The impedance relation between E' and I' is

E' c13

STI C23

Split E' and I' into right and left traveling components

inC2,3ZO2' ±n2Z1I' = C23C1 + C23C2 = 763Z + C2Zl

Zo11 + Z02' Z01' + Z02'

E' = C23CI - + C23C2 (-23 C23

inCl3Z02' inC13ZOiZolI+ Z02' ZOI' + Z02'

Substitute (7) and (8) into the equation

LXo][D F] Xn]

C13$nZo2'to get Eo = Aleny + A2e-ny + C1,3+ZO2'

Z01' + Z02'C13inZOZ'

Zo1' + Z02'c23inZO2

lo =Blenty + B2e7-yt + tol + Z021+O + 2.

+C23inZOl'

Zo1' +I Z02'where

= e'(n-1) + ve(n-2)y + ±.. + n-l

= e-(n-1) + re-(n-2) + . . . + n-1

In order to solve for A1 and A2, make use of (8) andboundary conditions Eo+ IoZ8 =0, E -InZr =0.

-i,(Z8c23 + C13)A1 =

(Z8 + Z01) (Zol0' + Z02')

- il(Z8c23 + C13)A2 =

(Zs + Z01) (ZOl' + Z02')

whereZs - Z02 Zr - Z01

Rs = - and Rr=Z8 + Z02 Zr + Z0I

The output voltage, En, of the plate line may be foundin terms of the signal impressed on the grid line, eo, bysubstituting (11) into the first of (7) and letting i,n =gmen= gmeotn

where gm is the transconductance of the tubes and is as-sumed to be constant. The conditions usually attemptedin the construction of a distributed amplifier areZ01 = Z02 = ZO, ZOl' = Z02' = Zo', = ey, and R8 =0.

CONCLUSIONS

Several things may be concluded from an inspectionof (12).

(a) A plot of gain versus frequency will fluctuate(9) about some average in a nearly periodic manner.

(b) MakingZ8C23 - C13 1Z8C23 + C13

or making the real part of the propagation constant ofthe plate line greater than zero will reduce the magni-tude of the variations of gain with frequency change.

(c) Increasing the number of tubes should make theratio of maximum to minimum gain in the usable fre-quency spectrum approach one provided that the realpart of the plate line propagation constant is not zero.

(103 (d) The gain of the amplifier should be doubled bymaking the reflection factor, Rr, at the receiving end ofthe plate line be equal to one.

(e) In order that the average gain of the amplifier beconstant and high, the real part of the propagation con-stant of the grid line should be as nearly zero .s possible.An amplifier was constructed using six tubes and con-

stant-k sections for the grid and plate lines to test theconclusions listed as (a) and (d). m-derived half-sections

the were used to terminate the lines in their characteristicimpedance. A typical section of the amplifier is shown

(ZsC23 C13)[Z02'l + (Z + C13) Zoi12j ZOI

(eny - R,R,e-ny)

[z02'(l± 8 o1%] RRZ02

(e "l - RsRre-n'y)

in Fig. 4. Computation reveals that the elements of the3 X 3 matrix of (1) are in this case

Cii = 1 + 2YZ

C21 = 1

C31 = 0

C12 = YZ2 + Z C13 = -2Z

C22 = 1 + 2YZ

C32 = 0

C23 = - 1

C33 = 1.

(11)

[z02'1 + (Zc23 -C13) z01'%] (Zol + R,Z02)-gmeoU-(Z8c23 + C13) (Z8C23 + C13)

En =(-(Z, + Z01) (ZOl' + Z021) (e,,y- R*Rre-"'y)

(12)

1953 761

PROCEEDINGS OF THE I.R.E.

It was assumed that Z01'=Z02'=ZO', R,=O, ¢=e",and y=jf. The absolute value of the gain was com-puted from the equation

The first experiment was designed to show the pres-ence of the cosine terms and the e-5i multiplier. HereRrO= . The results are shown in Fig. 5. It is seen that thetheory has been borne out reasonably well.

G| =gm(ZO + 1Z) 3 (ZO + cz)IGI= 3+ (Cosf.3+ Cos 303+cos 5f)e-5i# 1 +1?..2 ~~(ZO +IV)

Fig. 4-Typical section of a six section experimental amplifier.

2.0 ___ _ __ _ A-EXPERIMENTAL GAINB-CALCULATED GAINZs, Zr Zo

1.2- -

I -IT-o 10 20 30 40 50

FREQUENCY-MEGACYCLES60 70

Fig. 5-Predicted and experimental gain characteristics of a six tubedistributed amplifier with the plate and grid lines terminated intheir characteristic impedances.

The second experiment attempted to show the pres-ence of the 1+Rr term in the numerator of (13). Rr wasmade as large as possible consistent with the measuringtechnique used. The results are shown graphically inFig. 6. The difference between the calculated and ex-perimental gain curves may be explained on the basisof R85 0 at all frequencies.

ACKNOWLEDGMENT

The author would like to thank Professor J. E. Wolfefor checking the electrical concepts of the theory and forhis advice on the construction of the experimental am-plifier. Dr. P. M. Young is to be thanked for his reviewof the matrix algebra used; and Professors Joe. E. Wardand W. R. Ford are to be thanked for their help andadvice on the construction and testing of the experi-mental amplifier.

Dr. A. C. Hall and Mr. M. T. Gannon of BendixAviation Corporation have been helpful in their criti-cisms of the paper and getting it into a form suitable forpublication.

FREQUENCY- MEGACYCLES

Fig. 6-Predicted and experimental gain characteristics of a six tubedistributed amplifier with the output end of the plate line open-circuited and the other end as well as both ends of the grid lineterminated in the characteristic impedance of the lines.

C\:)5

(13)

z

(9

762 June

(