ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR Progress Report No. 4 COMPUTER COMPONENTS DEVELOPMENT K... E.. Monroe Harvey L. Garner Project Supervisor Project 2452 NATIONAL SECURITY AGENCY SIGNAL CORPS PROCUREMENT OFFICE CONTRACT NO. DA-49-170-sc-1791 WASHINGTON, D. C. March 1957

The University of Michigan * Engineering Research Institute TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS iii ABSTRACT v OBJECTIVE v 1. TEEORETICAL ANALYSIS OF PULSE AMPLIFIERS 1 Qualitative Discussion of a Pentode, Tetrode, or Cascode Triode Pulse Amplifier. 1 Qualitative Discussion of a Triode Pulse Amplifier 4 Linear Circuit Analysis of Possible Amplifier Configurations Graphical Circuit Analysis 10 2. EXPERIM4ENTAL RESULTS AND CIRCUITS 24 437A Cascode Circuit 24 436A Circuit 26 3. CONCLUSIONS 28 APPENDIX —TRIODE INPUT CAPACITANCE 31 ii

The University of Michigan ~ Engineering Research Institute LIST OF ILLUSTRATIONS Table Page I. Results of the Graphical Analysis Example 20 II. Variation of Tube Currents with Load Current for Cascode 437A's 24 III. Variation of Tube Currents with Load Current for the 436A 28 Figure 1. Pentode operating path. 2 2. Pentode waveforms. 3 3. Triode operating path. 4 4. Triode waveforms. 5 5. Triode test circuit. 7 6. Waveforms for the triode circuit. 8 7. Pentode-triode pulse amplifier. 9 8. Variation of number of gate drives with frequency. 11 9. Approximate transformer equivalent circuit. 12 10. Approximate plate load presented by the transformer and its load. 12 11. Approximation of a waveform by steps. 13 12. The problem arising from a step grid voltage. 14 13. Cascode circuit used in the graphical analysis example. 16 14. Input signal 17 15. Plate characteristics for graphical analysis. 17 16. Results of the graphical analysis example. 18 '__ __ ~iii

The University of Michigan ~ Engineering Research Institute Figure Page 17. Experimental waveforms. 19 18. Cascode 437A dynamic flip-flop for 5-me operation. 25 19. Waveforms for the 437A flip-flop. 26 20. 436A dynamic flip-flop for 5-mc operation. 27 21. Waveforms for 436A flip-flop. 29 iv

The University of Michigan ~ Engineering Research Institute ABSTRACT This report is an analysis of a pulse amplifier of the type employed in the SEAC computer-gating package. There are two ways of investigating the performance of a given circuit. The first of these is a theoretical analysis of the circuit, and the second is experimentation. The theoretical work, unless grossly inapt assumptions are made, will give the investigator at least an order-of-magnitude feeling for values of circuit parameters and frequencies of operation. In this report linear circuit analysis and a graphical method of analysis are used, and the results are compared with experimental results. Agreement between the theoretical and experimental results is reasonable on the basis of the assumptions made. OBJECTIVE The objects of this phase of the computer-components development program are to determine the pulse amplifier-circuit parameters which limit the pulse-repetition rate, to develop theoretical methods of analysis and to use these methods to predict the theoretical limit imposed by presently available components and to experimentally determine the upper limit using available components. v

The University of Michigan * Engineering Research Institute 1o THEORETICAL ANALYSIS OF PULSE AMPLIFIERS 1.1 QUALITATIVE DISCUSSION OF A PENTODE, TETRODE, OR CASCODE TRIODE PULSE AMPLIFIER If the amplifying component is a vacuum tube, then a pulse transformer is required to match the tube to the lower impedance of the gating structure. The tube and transformer then comprise a pulse amplifier. Operation of such a pulse amplifier is best understood by consideration of three different phases of the pulse: the rise, the flat top or duration, and the fall or decay. 1.1.1 The rise.-Initially the pulse amplifier has on its grid a voltage egl, which keeps the tube in a low-conduction state. The plate voltage is the supply voltage since there is negligible d-c drop in the transformer. This quiescent point corresponds to point A of Fig. 1. When a pulse is applied by the gating structure, the grid is suddenly raised to the positive value eg2. Since there is plate-circuit shunt capacitance, the plate voltage cannot change instantly; hence, the plate characteristics are traversed to point B. The tube is now drawing much more current than previously, the capacitance begins to charge, and the plate voltage begins to fall. The plate characteristics are traversed along the eg2 curve towards zero plate volts and the voltage across the transformer increases. This process continues until the current in the transformer secondary is in equilibrium with the plate current, At the equilibrium point, the capacitance is charged (draws zero current at this instant) and no further increase in transformer voltage will occur. This point is where the line of the load resistance, referred to the primary, intersects the eg2 curve, and it marks the end of the rise-time response of the pulse amplifier. This point is marked C on the figure, and it occurs where the tube is "bottomed." Bottomed operation is desirable since the plate voltage in the bottomed condition changes very little as the tube agesl Hence, the amplitude of the output pulse should remain near the standard value over the useful life of the tube. 11,12 The duration. —The rise time is so short that the transformer primary inductance does not have time to begin drawing a great deal of current. However, oncethe capacitance is charged, 1. CC. Harris, "The 'Hard-Bottoming' Technique in Nuclear Instrumentation, " Institute of Radio Engineers, Professional Group on Nuclear Instrumentation, Vol. NS-3, No. 2, March 1956. 1

The University of Michigan ~ Engineering Research Institute z B E F PLATE VOLTAGE Fig. 1. Pentode operating path. the transformer has an almost constant voltage impressed and magnetizing current will begin to build up comparatively rapidly, The point of operation on the plate characteristics will move away from point C back up the eg2 curve toward the knee. This means that the plate voltage is ins creasing, the transformer voltage is falling, and the output pulse is drooping. The magnitude of the droop will depend upon the primary inductance, the pulse duration. and the plate resistance of the tube in the bottomed condition. If the transformer inductance could be made large, then the magnetizing current and output pulse droop could be kept small. However, as will be seen in the next section, the amount of transformer inductance that may be used at a given repetition rate is fixed by factors other than magnetizing current. 1.1.3 The falle-At the end of the pulse, the increased magnetD D izing current has increased the total tube current up to point D on the curves. At this time. the grid is again switched suddenly to eg, and again, because of plate shunt capacitance, the voltage cannot change instantly and the plate characteristics are traversed to point E. At this time we have essentially a parallel RLC circuit with an initial charge on the capacitance and with an initial current in the inductance. The R is the load resistance referred to the primary side of the transformer. As the capacitor discharges, the operating point travels along eg to A, at which point a critical damping resistance is switched in E F inductance then decays and produces a back voltage out to point F and fcurrenll back to poin to A before the next pulse comes alongy Rough sketches of important waveforms during a pulse period are shown in Fig. 20 In t speeding up'a SEC-type circuitryl an important factor is plateacircuit shunt capaclta gce, If the transformer is to recover after one pulse before the next one comes along, the resonant frequency of the transformer primary inductance and the platecircuit shunt capacitance must be equal to or greater than the desired repetition rate i 2 Rinsta Dentonly W Brown, and Wh latmer Compuarter Comsonen Detvetlopment, Report No, 2, January 1957, 2

The University of Michigan ~ Engineering Research Institute LLJ B C D 0. TI ME A E F z B w D X C Fig 2 Pentode waveforms TIME OI AB,B A TIME ILl D)E ao Fig~ 2, Pentode waveforms. tube and stray capacitances so that the magnetizing current and pulsetolerated., If the inductance is made lower than is necessary to allow sufficiently fast recovery, and/or if the load resistance is made too small, the magnetizing current will carry the path of operation back around the knee' of the curve where a very small further increase in supply voltage and the load current and voltage to fall to zero before the grid pulse is over, That is, the amplifier will not "hold up" a pulse of the width being amplified~ This condition indicates that the limit to which the circuit may be loaded has been exceeded. The same circuit may be loaded more heavily if the grid is 5

The University of Michigan ~ Engineering Research Institute driven more positive so that the knee to which the magnetizing current must creep is higher. The extent to which this may be carried is limited by either the current or power-dissipation ratings of the tube employed. Usually the plate dissipation is low since high plate current occurs only at low plate voltage and vice versa in this type of amplifier. Consequently, the limit is set by screen dissipation or cathode current, according to which rating is reached first. 1.2 QUALITATIVE DISCUSSION OF A TRIODE PULSE AMPLIFIER From the considerations similar to the preceding, we see that the path of operation on triode plate characteristics is ABCDEFA, as shown in Fig. 3. The waveforms are sketched in Fig. 4. These waveforms are similar to the pentode case, except that the plate current increases IF PLATE VOLTAGE. Fig. 3. Triode operating path. during the transformer backswing where the plate voltage rises above the supply voltage. This additional current may become a significant part of the average plate current, which is undesirable because, in the case of a triode, the tube rating which is likely to be exceeded first is that of average cathode current. The current increase may be eliminated by using a large quiescent bias or possibly by feedback. The first of these would require a large grid swing to drive the tube into heavy conduction, and this is undesirable. Hence, the solution appears to be feedback which would bias the grid more negative as the plate swings positive. This feedback can presumably be derived from the regular package output which is swinging negative at the required time. Further discussion of a triode amplifier is given in section 1.3.1. - - - - ~~~~~4

The University of Michigan ~ Engineering Research Institute L BC D > TIME E,F, A z B D LLU r F F TIME 0 F~-J ~ ~ ~ 0 J TIME (DG ~, C Fig. 4. Triode waveforms. 1.3 LINEAR CIRCUIT ANALYSIS OF POSSIBLE AMPLIFIER CONFIGURATIONS 1.3.1 General discussion.-It is the purpose of this section to develop formulae to be used to estimate the number of gate drives that may be expected from several amplifier configurations. In Appendix B of Computer Components Development, Report No. 2, it is shown that the maximum load current from a tube-transformer pulse amplifier is given by Lox l140 a2 C f E - - fEC () 2 Fig 4 Tioe avfoms

The University of Michigan ~ Engineering Research Institute where: AI = pulse-plate current Cp = primary shunt capacitance Es = secondary voltage Cs = secondary shunt capacitance f = frequency of operation If we assume: (1) that a linear analysis is valid; i.ee that I gm eg, where gm is the tube transconductance and eg is the grid voltage change, (2) that one volt of noise clipping is necessary and there is a drop of 2 volts in the diode logic so that Es = eg + 3, and (3) that secondary capacitance may be neglected, then Eq. (1) may be rewritten as I~max = 111723 gm2 eg2 (2) (2) nLmax - 140I2 Cpf (eg + 3) The current required per gate is calculated by considering the circuit of Fig. 5. A current I1 is required to charge and discharge the input capacitance of the pulse amplifier; hence, a current of 2IL is required in the "and" pull-up resistor. From noise considerations it is necessary to have a current I2 flowing in clamp diode D1. Hence, the input gate current, Igate is Igate = 2I1 + I (3) The current I1 is given by I - Cg e I = OT = lof Cgeg, (4) 0.1 T where T = 1/f = pulse repetition period. The number of gates that may be driven by a gating package is then 23 gm2 eg2 ILmax 140r2 Cpf (eg + 3) Igate 20 f Cg eg + I (5) Eq. (5) applies only as long as none of the ratings of the tube employed are exceeded. 6

The University of Michigan ~ Engineering Research Institute + 50v 15:3 +50 -1.0 -0.5 0 211 3.9K 51-2.0 - te50 Figs 5o Triode test circuit. Within the ratings of the tube employed, Eq. (5) applies to several amplifier configurations provided the proper input and output capacitances and gm's are used, Capacitances for several /cO'nfigurations are as follows: (a) If a pentode or tetrode tube is used, the input capacitance may be taken as the capacitance from the control grid to the cathode and to the screen grid plus stray wiring capacitance. The output capacitance is the capacitance from the plate to the cathode and to the screen and suppressor grids plus 'transformer and stray capacitances. These input and output capacitances, except for the stray components, are ordinarily published by tube manufacturers. (b) In a triode the Miller Effect capacitance becomes the dominating part of the input capacitance, This capacitance has a transient and a steady-state component. However, as shown in the appendix of this report, the steady-state value is reached in such a short time that the transient may be neglected for operation at frequencies equal to or less than 10 mcps. The input capacitance is then effectively given by Cg =cgk + (l-A)Cpg ~ (6) 7

h......e lniversity of Michigan Engineering Research Institute:Some expev:r imentationll has been -erformted w i. bL the t.I5"A triode to -i...lus-trate the Mi]..r Eftfect. the circu: it used was the same as that shown i.n Fig, 5. ReIsu]l.ts of the experimenx are lshown i Fig. 6. Fi: g. 66(a) s-howrs tile grr:i.d waveform;:i,lt w ie-th the fpl_.ate su-S}pply turned off.::ln this case th e ceap;eitanee1lc at the f ->i.d..i s ju.:st t he g rida.to-est:Lhot e (a) O..1 [tsec/mtiajo7r.i divisi on (b) 0.1. pst:ec/major d vi.vi...on 2.O v/lmajor di.v-i..;i. -on 2.0 v/major d.ivi.s -ion Fig, 6. Wavefortms; for the t-riode circui.t;, capacixtance tand the;str-ay capacitance, The gate - pl].-.up aIcad pu 3.l.-lidown currents are l.arge enough to charge and discharge the t gr:i.d capacitance. [In Fig. 6(b) the p]l.ate suppl.y Yvoltage i.s turn ed on, the tube has gain, and the Mi.ler capaci-tancel, which is now pre.sent,:is large enough -to require a comrparat;ive ly l.ong 'time to charge and dics-; charge I.resumtiably it i.s possible Lto i ncrease -gate curren-ts to the po:int wlhere they woul.,dc be able tLo charge and di.scharge even thle largteJ ' MI].- I-r. capa.:ci-tance i.n a suff:ic.ientl.y short ti. mte. Hlowevert such..arge gate cu rrents would. have adverse e-fects upon the transien-ts in -the gating diodes and woul.d require a grea-ter amount o:f- noise el.iApping. Al.-so, a given triode would be able to drive only a very sm-all numtber of such h;ij.gh-current gat- -.es. It shoul.d be possibble to drive the i.npu-t cra,)pacitaance of -the triode by a rather2 low...power pentode as in F]ig. F, whose small. input capacitance cou].d be driven by comiparatively low-ecurrent gat-;e. he backswi.ng of the pentode 's tre;)-ans'former woult7 d then tend -to kecpj -_the t.riode biiased off during its tra nsf.-Cormer' s )ekswckswi.ng, -thereby reducing the tr:iode,average plate current. t-owever~ if 'the ri se, time of the pulise is not to deteriorate too in this cascade pul.s e ampi]ifier, -then 'the rise time of the peyntode sta[fge and of the -triode stage must be the same as for a si^ngle-state ampl.ifi:er. Since rise -times of cascaated stages, with rise ti:mes l, and T2, add.in -the square roo-t accord.ing to the formula:> ). (..7) the rise..-t;i.me rOequi. rements on the individual. stags,- bcome more str:i.ngentt. In fact, each st-age must now rise I:n onlJy O.'(707 of -the time that a s inge3...stage puls.e ampli.elrt wouttl.d e al.lo'Ltecd.

The University of Michigan ~ Engineering Research Institute +'Eg2 +Ebb -E1 +Ebb OUTPUT TO GATES INPUT -E2 FROM GATES Fig. 7. Pentode-triode pulse amplifier. (c) If two triodes are used in the cascode configuration, the input and output capacitances will be given by Cp = Cpg + 1 - Cpk, (8) gm RL RL Cg = Cgk+ (2 + r )Cgp (9) as has been pointed out in the Phase I report, 1.3,2 Predicted gate.. drives for particular tubes and configurations. 15.3.2..1 436A Tetrode. The current 12 of Eq. (3) has been determined experimentally to be about 4 ma. Allowing 5 pF for grid-circuit stray capacitance and a voltage gain of 40, the input capacitance for the 436A is calculated to be 23 pF. The output capacitance with the transformer in place was calculated from the undamped transformer ringing frequency to be about 10 pF. The transconductance is 30 millimhos. Hence, 9

The University of Michigan ~ Engineering Research Institute Eq. (5) may be rewritten as (10) N 23 (30)2 (106) eg 1 140t2 f (eg + 3) 10"11 20(23)(10-') f eg + 4 x l-3 This equation is plotted as a function of frequency with grid swing as a parameter in Fig. 8(a). 1,.32.2 Cascode 437A. Again assuming a voltage gain of 40, the input capacitance of 437A's in cascode is calculated to be 27.5 pF. The output capacitance is about 10 pF and the transconductance is 45 millimhos, so that Eq. (5) is evaluated to be 23 (45)2(10-6 ) eg2 1 () N - 140m2 f (eg + 3) 10-11 20 f (27.5)(10-12) eg + 4(103). and is plotted in Fig. 8(b)d 1. 4 GRAPHICAL CIRCUIT ANALYSIS Linear circuit analysis is useful in analyzing this type of pulse amplifier in an approximate manner. It tells one the tube parameters which are important and gives one the ability to estimate with very little calculation the number of gate drives that may be expected from a particular amplifier configuration. If a more precise description of the pulse waveform is desired, one can resort to a graphical analysis of the pulse amplifier. 1.4.1 General discussion —If the grid signal waveform is assumed to be known, then from the plate characteristics of the tube and a known RLC load, the output-voltage waveform may be plotted. From the same set of calculations, the magnetizing current, the capacitive current, the load current, and the total tube current and voltage may be plotted. If the tube has a screen grid, and if the screen-grid characteristics are available, then a plot of screen-grid current may also be obtained by reading from the screen current, point by point, for the previously calculated pairs of values of plate voltage and control-grid voltage. The transformer and its load may be approximated by the circuit of Fig. 9. The leakage inductance is small3 and may be 3. R. Denton, W. Brown, and W. Kilmer, Computer Components Development, Report No. 2, p. 12, Table I, January 1957. 10

The University of Michigan * Engineering Research Institute 100 90 z 80 (I) w 70 > (a)SINGLE 436A TETRODE 0 60 LJ < 50 LL o 40 rn 30' m *f~ e g=4v z 20 10 Iv 5 6 7 8 9 10 FREQUENCY - MC 100 90 (b) CASCODE 437A's z 80 w 70 60 LJ. < 50 o40 3V W 30 20 10 20 0 -5 6 7 8 9 10 FREQUENCY-MC Fig. 8. Variation of number of gate drives with frequency. 11

The University of Michigan ~ Engineering Research Institute. Cp Cp^ Lp n2RL 2 ~ = Transformer leakage inductance P = Transformer' primary inductance p = Primary shunt capacitance Cs = Secondary shunt capacitance RL = Secondary load resistance n = Step-down turns ratio Fig. 9. Approximate transformer equivalent circuit. neglected without serious error. In this case, the plate load for the tube reduces to a parallel RLC circuit as in Fig. 10o C m Lt Rib WHERE:C=Cp+CsxnR =n2 RL Fig. 10. Approximate plate load presented by the transformer and its load. 12

The University of Michigan * Engineering Research Institute The essence of the analysis is that the grid waveform may be approximated by true step functions occurring at regular intervals of time as in Fig. 11. The approximation becomes more accurate as the number of steps is increased and the time between steps is decreased. The approximation becomes exact in the limit as the number of steps tends to infinity and the time between steps tends to zero. eg t F AT Fig. 11. Approximation of a waveform by steps. The steps of voltage at the grid will result in steps of current through the tube. The tube current-step increase may flow through any of the three paths: R, L, or C. However, a true step increase will flow through C at the initial instant of application, and, if sufficiently short times, At-'are used, then it is approximately true that the capacitor current will not change appreciably during one time interval, Hence, the voltage across the capacitor will increase during the first interval, and during the second interval-this voltage will begin to force current through the resistive load according to Ohm's Law and will begin to change the current in the inductance according to Lenz's Law. Note that an error has been introduced: actually some resistive current will flow as soon as the capacitor has acquired the most minute charge, and also the inductive current will begin to change. However- if the time interval is taken quite small, then-the error will be small. Before the pulse is applied, the grid is at some steady potential, there is some steady flow of plate current, and all this current flows through the primary of the transformer, which is a d-c short circuit. Hence, initially, there is no output voltage ep and 13

The University of Michigan ~ Engineering Research Institute there is no current in either the capacitor or the resistor. Then at t = 0+, the grid voltage is suddenly increased and causes a sudden increase in the plate current of ibl. The problem to be solved then is as illustrated in Fig. 12. ITL Li'I R O. I IRI 0 Fig. 12. The problem arising from a step grid voltage. For times t << RC, the currents may be shown to be approximately iR = i t (1 it3 (13) L t (14) ic1 = ib (1 t, (15) provided that the circuit is critically damped. Presumably the increase in plate current ib is not known. However, it is known that the output voltage is given by ep - Ebb - eb, (16) and it is also known that the output voltage must be given 'by c - dct i R = L i (17) If we now apply the approximation that the capacitive current is a constant for short times, we have ep ct for t << 18) epI = icz and at the end of the first interval we have ic At Ebb - ebt (19) 14

The University of Michigan ~ Engineering Research Institute where t = At- (just before the second grid step), and eb = the plate voltage at t = At- and is unknown. But ic is given by I i i -20 = ib iR L. (20) Therefore, (ib iRiL) C Ebb - eb (21) This last equation may be rewritten in the form of a load line, drawn on the plate characteristics, and eb and ib read off. Given eb and ib, the currents ic 9 i L and ir ma be calculated. This procedure may (e generalized to the following recursion formulae: E -= Ebb + (irj- + )Lj1) At/C - epj (22) ebj = E (At/C)ibj (23) ePj = Ebb - eb (24) ij = ep/R (25) iL. = iLj + ep (At/L) (26) Cj = ibj iRj iL (27) From these equations, known initial conditions, given grid waveform, and the plate characteristics of the tube used, a complete solution may be obtained. 1.4,2 A specific example of graphical analysis.-=The discussion of the preceeding section and the method of applying the formulae to a particular situation can probably be clarified by consideration of a particular example. Consider the cascode 437A circuit of Fig. 13 with input grid signal as in Fig. 14 and plate characteristics as in Fig. 15. Assume a transformer primary inductance of 200 Why, a shunt capacitance of 10 pF, and a load resistance referred to the primary of 2k (this value of resistance slightly overdamps the circuit). The plate supply voltage will be taken as 200 volts and the upper grid supply as 75 volts. Assuming that there has been no input pulse for a sufficiently long time for all transients to have died out, all the plate current will flow through the inductance, and there will be no output voltage and no 15

The University of Michigan * Engineering Research Institute resistive or capacitive component of current, The grid voltage will be -1,O volt. Evaluating Eq. (22) for these conditions gives Eo = 203.2 volts and substituting into Eq. (23) gives ebo = 203.2 - 200 ibo ~ (28) This load line is the j = 0 line of Fig. 15, and at the intersection with the eg =.loO-volt curve, one can read ebo and ibo to be 200 volts and 16 ma, respectively. From Eq. (24) through (27), respectively, one calculates ePo = O, iro = 09 iLo = 16 ma, and ico = 0. This concludes the first iteration, E92 Ebb c L R: I WI ' Oep 437A eg9 0! { )437A Fig. 13. Cascode circuit used in the graphical analysis example. C = tube interelectrode capacitance, plus stray capacitance = 10 pF; L = 150 ~hy; R =2Ko 16

The University of Michigan * Engineering Research Institute + 100 A o 0 [Lw90S+/ > / 2/ 8 9298e ibo 6 7-0.5 TI M E'- M/z S EC 13:: Fig. 14. Pnput signal. 120 e- =+,O volt 800 w 90 70 60 t =2 At 0.0 5 0j a 40 4,. t =,A t - 0.5 30 -j=: 2 0 t0 -1.0 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 PLATE VOLTS Fig. 15. Plate characteristics for graphical analysis. 17 —

The University of Michigan * Engineering Research Institute I10 90 80 70 60 50 40 0 --I 0 20 60 80 00g 160 180 200 220 240 260 20graphical -40 - 50 - 60 -70 - 80 - 90 -I 00 18

....... le niversity of Michigan ltnginteeling Researclh Institute FIor thIe second:i. ttt on, j.. t At.e. tp. see, c and the grid vol.laGe a.s i.o vol.. as in Fig. t..t[.: quat:ion (. ) Ji agaai. eval.tiuated to be 203.2 volts, (29).'therefo-"re' ti:t, (23.):i.s' aga:i.n,b::0.. 2. 20 0:j.. (30) F'ro:m the in.tersect;i.on thi.s j:::: ].o lad. ine tz:iith the e 0. O.5-volt curve, e,, andc i, are reads 1.96 voal.-ts aand 36 'na, respe c-tivel.y. From Eq. (2:) 'through (2') O (.ul.C.tated, re:peet. eC. v-.... 4 volts, ir. 2 ma. ]-.6 0) ma, and 1J.. 1.8 i33ima I. T.e% second Jteratin..t -'l.o.Jn.i. ow ended. Fturther I. teraatl.(:on.s may be xerfoFrmed s:i.l mil.ari.y w:ith the resul.ts be:ing. as shown in -tlabl -e I...he: resu.tl s 1 re presented graph:i. ca. l.y i.n Fig. 16. Xp menttal. resul.-ts using " the same vol-tages and circui t.~paraxr<e-t~ers are preo sent9-ted for comparison.ln.th1 photogralph of' opla v..ateoa vo.ta, e in Fig.. 7. -(a) Gr-i-d wavcfo-rrti (b3) -.)l.-tate wavef'oi-i I. vol.t;/major d.:i.v:i. sion 100 volts/major div:.l.on 1].00 mpsec/m;ajor div:i sion 1.00 mp.:;:ec/miajor div:i.sion F i. g. 13 (. Exper: rI. me~:: ln' -Vta]. wavefo:)orms. The graphica. c-. method of ana.ys;is y^ie. y3ds.re.sulTts whi.ch agree qulte wel.. with experiment-tresuts. t owev,.t.:i. a very 'ti-me.,con. suming rmeth od and., there'foree d.oes.not lend i.ts(l. -to an i.lvestigation onf the eff ects< of' vary:iXnf.g c.i:rlcul. t:arrame-tersi. f' c acul.l.ions ar. e perform...ed by handl. '.herefore the probltem of p.rogrammn g i an aut;omat i.e cotipu-ter to pe;rtform 'the gra-ph:ical. anal..ys:i.s rouiti. ne i:.s presently being con:;idered, Such a program canl be modified. easil.y, or even automaLical..ly, so tha-t c:i.rcuit param-eters; can be varie.(dd at; wil] l.. iurht; rtore, 'the;speed of a computer a.s comt'pa-red wi'th lhand ctmpu't;ation mlakes feas ibl e the a naly is cf <a c ircui.'t when subljected. to a tr.ain of't pulses;e rather than jtust can i.solated p:ul.Se, as considered here. Any di:fferences between -the:f'irst and Nth pul.ses of the train can be obser-ved as can 'the bui..ld-uz p of the magne tL.zi.zig currenwt 'to its steady -s'.tate waveform. The l.oad cur rent can be varied 'to dedtermi.ne 'the ulti.mnate ntumber of gate drives;. Average powers and curren'ts c1an be cotpu'ted for eatch )set of):i.mpresse3.d vol.';age and l.oawd condit:i.onsa:3r:i. efy, th:is 1.-9

The University of Michigan * Engineering Research Institute TABLE I RESULTS OF THE GRAPHICAL ANALYSIS EXAMPLE egj -eb i ep iR iL icj E t 0 -1 200 16 0 0 16.00oo 0 203.2 0 1 -0.5 196 36 4 2 16.05 18 203.2 2 2 0.0 188 58 12 6 16.21 35.8 199.6 4 3 +0.5 178 79 22 11 16.50 51.5 192.44 6 4 +1,0 162 109 38 19 16,90 73.1 183.5 8 5 +1.0 149 106 51 25.5 17.58 63 169.2 10 6 +1,0 138 104 62 31 18.4 54.6 157.6 12 7 +1.o 128 100 72 36 19,36 44,6 147.7 14 8 +1.0 120 97 80 40 20.42 36 139 16 9 +1o0 114 93 86 43 21,56 29,4 132.1 18 10 +10 o 108 90 92 46 22.8 21.2 126.9 20 11 +1.0 105 86 95 47.5 24.06 14.4 121.8 22 12 +1.0 102 83 98 49 25.36 8.6 119o 3 24 13 +1.0 100 80 100 50 26.69 3.3 116.9 26 14 +1 o0 99'.5 79 100.5 50.25 28.03 0.72 115.3 28 15 +1o0 98.6 77.5 101.4 50o7 29038 - 2.58 114.2 30 16 +1,0 99 78 101 50.5 30,72 - 3,22 114.6 32 17 +1,0 99,5 79 100.5 50.25 32,06 - 3.31 115o24 34 18 +1,0 100 80 100 50 33,3 - 3.39 116 36 19 +1. 0 100o6 80,7 99.4 49.7 34 71 - 3.7 116.7 38 20 +1.0 101,1 81.2 98.9 49,5 36,03 - 4,33 11705 40 21 +1,0 101o8 82.1 98,~2 49.1 37~34 - 4.34 118.2 42 22 +1.0 102,4 83 97 6 48.8 38.64 - 4.44 119.1 44 23 +1,0 103 84 97 48.5 39.94 - 4.44 119.9 46 24 +1,0 103.8 84.7 96,2 48.1 41,32 - 4.72 120,67 48 25 +1,0 104.5 85.3 95.5 47.8 42,59 - 5.09 121,7 50 26 +1,0 105.3 86.1 94,7 47,35 43o85 - 5,1 122,6 52 27 +1.0 106 87 94 47 45.10 - 5.1 123,54 54 28 +1O 0 106.8 87.9 93.2 46,6 46.34 - 5e04 124o42 56 29 t 10 107.7 88.6 92,3 46.15 47.57 - 5.12 125.4 58 30 +1.0 108,4 89.2 91.6 45.8 48,79 - 5.39 126.4 60 31 +1,0 1091ol 90 90o9 45.45 50.00 - 5.45 127o32 62 32 +1.0.110 90,4 90 45 51.20 5p.80 128.2 64 33 +1.0 111 91.2 89 44,5 52.38 - 5568 129.24 66 34 +1,0 112 92 88 44 53.55 - 5,55 130o38 68 20

The University of Michigan * Engineering Research Institute TABLE I (Continued) j e eb ib ep iR. iC. EJ t g' i -.P I L.. j j J j j J J J 35 +1o 0 112,9 92.9 87o.1 43.55 54.71 - 5.36 131.51 70 36 +1.0 113o8 93.4 86.2 43.1 55.86 - 5.56 132.55 72 37 +1.o 0 114o6 94 85.4 42. 7 57.00 - 570 133.59 74 38 +1.0 115 7 94,7 84,3 42.15 58.12 - 5.57 134.54 76 39 +1.0 116.7 95.1 83.3 41,65 59.23 - 5.78 135o8 78 40 +1.0 117.7 95.9 82.3 41.15 60o.31 - 5.56 1369 80 41 +1.0 118.5 96.3 81.5 40,75 61,40 5 585 138.o0 82 42 +1,0 119.7 97 80.3 40o.15 62.47 - 5.62 138.9 84 43 +1.0 120,6 97.6 79 4 39o7 63o53 - 5.63 140,o2 86 44 +1.o0 121,6 98 78.4 39.2 64.57 - 5.77 141.25 88 45 +1,0 122,7 98.6 77o3 38.65 65.6o 565 142,35 90 46 +1 o,0 123.7 99 76.o3 38 15 66.61 - 5.76 143 35 92 47 +05 1291ol 77 70.9 35.45 67.55 -26 144.65 94 48 0o0 138o3 56 61.7 30.85 68,37 -43 22 149.7 96 49 -0.5 151 35 49 24,5 69.02.V58,52 158014 98 50 -1 0 166.6 151 33.4 16o 7 69.46.71 06 169o 7 100 51 -1,0 179.8 15o1 20,2 10.1 70o33 -65~33 182.83 102 52 -1oO 192.7 1504 7.3 3065 70.43 58~68 195.9 104 53 -1,0 204.4 15.7 - 4.4 - 2o2 70.37 -52.47 207.5 106 54 -1.0 214.0 15.9 -14.0 - 7,0 70,28 -47.38 217.0 108 55 -1.0 223.5 16.0 -23.5 -11,75 69 96 -42,21 226.7 110 56 -1,0 231,8 16o0 -31.8 -15.9 69.54 -37.64 235.1 112 57 -1.0 239o3 16.0 -39~3 -19o65 69,02 -33.37 242.5 114 58 -1,0 246 16,0 -46 -23 68o41 -29.41 249.2 116 59 -1 0 252 16.1 -52 -26 67 72 -25.72 255.1 118 60 -1.0 257 16.2 -57 -28.5 66.96 -22.26 260,3 120 61 -1.0 261,3 16 2 -61.3 -30 65 66.14 -19. 29 264 7 122 62 -1.0 265.1 16o 3 -65 1 -3255 65.27 -16.42 268.4 124 63 -1.0 268.1 16,3 -68.1 -34 o05 6436 -14,01 271.6 126 64 -1.0 270.9 16,4 -70.9 -35.45 63.42 -11.57 274.2 128 65 -1.0 272o1 16.5 -72.1 -36.05 62,46 - 9.91 275..5 130 66 -1.0 274.0 16.6 -74 -37 61.48 - 7.88 277.3 132 67 -1,0 275.7 16.6 -75.7 -37 85 60o47 - o 02 278.9 134 68 -1.0 276.9 16.6 -76.9 -38.45 59.45 - 4,4 280.2 136 69 -1o0 277.9 16 o -77.9 -38o95 58.44 - 2.79 281 o1 138 70 -1,0 278.o7 16.7 -78o7 -39. 35 57.39 - 1o34 281.9 140 71 -1.0 2790 16.7 -790. 39 o 5 56 34 - 0,14 282 3 142 72 -lo,0 279,1 1607 -79o1 -39055 55.29 + o096 28204 144 73 -1.0 278.8 16o7 -78.8 -39.4 54.24 + 1o86 282.1 146 21

The University of Michigan * Engineering Research Institute TABLE I (Continued) J gj e e bi. ep iR Lj iC Ej t 74 -1.0 278.5 16.7 -78,5 -39,25 53.20 + 2.75 281.8 148 75 -1.0 278.0 16o7 -78o0 -39 52.16 + 3054 281,3 150 76 -1.0 277.3 16.7 -77. 3 -38,65 5113 + 4~22 280.6 152 77 -1o0 276.5 16,6 -76.5 -38o25 50o11 4~74 279,8 154 78 -1,0 275~7 16,6 -75o7 -37.85 49,o10 5.35 278,9 156 79 -1,0 274.8 16.6 -74o8 -37.4 48.11 5.89 277.95 158 80 -1.0 273.7 16.6 -737 -3685 47o13 6632 276094 160 81 -1,0 272.5 16,5 -72.5 -36.25 46,17 6.58 275.8 162 82 -1.0 271.2 16,5 -71.2 -35o6 45.22 6.88 274,5 164 83 -1.0 270.0 16.5 -7000 -35 44.28 7,22 273.1 166 84 -1o0 268,6 16.4 -68.6 -34.3 43537 7.33 271.9 168 85 -1.0 267.2 1603 -672 -336 42,48 7042 270,4 170 86 -1,0 26509 16.3 -65o9 -32.95 41,61 7.64 269 172 87 -1,0 264.3 16.3 -64.3 -32.15 40,75 7.70 267.6 174 88 -1o0 262.8 1602 -62.8 -31.4 39.94 7066 266 176 89 -1 0 261,1 16o2 -611 -30.55 39.13 7,62 264.5 178 90 -1o0 259.7 1602 -5907 -29385 38934 7,71 262,8 180 91 -1o0 258.0 162 -58,0 -29,0 37~57 7.63 261.4 182 92 -1.0 256.4 16,1 -56,4 -28,2 36,82 7o48 259 7 184 93 -1,0 255 16.1 55 -27.5 56.09 7051 258,1 186 94 -1o0 25304 16.1 534 -267 35o38 7.42 256,7 188 95 -1,0 252 16.1 -52 -260 34.69 7.41 2551 190 96 -1.0 250,4 16.0 -50,4 -25,2 34.02 7.18 253.7 192 97 -10 249.0 16,0 -49 -24,5 33037 7413 252.2 194 98 -1 0 247,6 160 -47.6 -23o8 32a74 7,06 250, 8 196 99 -1.0 246 16,0 -46 -23 32,13 6 87 249.4 198 100 -1o.0 244,4 16o0 -444 -22o2 31o54 6.66 247,8 200 101 -1,0 243 16.o -43 -21.5 30097 6 o53 246,3 202 102 -1.0 241,7 16o0 -4107 -20.85 30 42 6,43 244,9 204 103 -1 0 240,4 16,0 -40.4 -20.2 29 88 6,22 243.6 206 104 -1.0 239,2 16.o -39,2 -19.6 29736 6,24 242,4 208 105 -1.0 2380 16,o 0 -38 -19 28086 6,14 241,2 210 106 -1,0 236,9 16,0 -36.9 -18.45 28,37 6.08 240,0 212 107 -1.0 235 9 16.0 -3559 -17 95 27.90 6o05 238,9 214 108 -o0 234.8 16o0 -34o8 -17,4 27o44 5.96 23709 216 109 -lo0 23308 16 o -3308 -16.9 26.99 5.91 236,8 218 110 -1,0 232.6 16o0 -32o6 -16,3 26.56 5.74 235o8 220 111 -1o0 231 4 16.o -.314 -15.7 26.14 5.56 234.6 222 112 -1 0 230,3 16o0 -30.3 -15015 25074 5.41 23304 224 22

The University of Michigan * Engineering Research Institute TABLE I (Concluded) e eb. 'b e 'L.. t 113 -1,0 229.5 16eo -29.5 -14.75 25.35 5.40 232.4 226 114 -1.0 228.4 16.0 -28.4 -14,2 24.98 5.22 231.6 228 115 -1.0 227.4 16.0 -27.4 -13.7 24.62 5.08 230.6 230 116 -1,0 226.4 16.0 -26.4 -13o2 24.27 4.93 229.6 232 117 -1.0 225.3 16.0 -25.3 -12.65 23.94 4.71 228.6 234 118 -1.0 224.4 16 o -24~4 -12.2 23.62 4.18 227.5 236 119 -1.0 223.9 16.0 -23.9 -11.95 23.30 4.65 226.8 238 120 -1.0 223.0 16.0 -23 -11,5 23.00 4.50 226.2 240 121 -1o0 222.1 16.0 -22.1 -11.05 22.71 4034 225,3 242 122 -1.0 221.3 15.9 -21.3 -10.65 22.43 4,12 224.4 244 123 -1.0 220.5 15.9 -20.5 -10.25 22.16 3.99 223.7 246 124 -1.0 219.8 15.9 9.8 - 9.9 21.90 4.o 222.9 248 125 -1.0 218.9 1 -18.9 - 9.45 21,66 3.69 222.2 250 126 -1.0 218.1 15.9 -18.1 - 9005 21.42 3.53 221.3 252 127 -1.0 217.5 15.9 -17,5 - 8.75 21019 3.46 220.6 254 128 -1.0 216.9 15.9 -16.9 - 8.45 20.97 3.38 220.0 256 129 -1.0 216.4 15.8 -16.4 - 8.2 20.75 3.35 219.4 258 130 -1.0 215.7 15.8 -1507 - 7.85 20.54 3.11 218.9 260 131 -1.0 215.0 1508 -15.0 - 7.50 20.34 2.96 218.2 262 132 -1.0 214.3 15.8 -14.3 - 7.15 20.15 2.80 217.6 264 133 -1.0 213.8 15.7 -13.8 - 6o 9 1997 2.73 216.9 266 134 -1.0 213.3 15.7 -13.3 - 6.69 19.79 2.56 216.4 268 135 -1.0 212.7 15.7 -12.7 - 6.35 19.62 243 215,9 270 136 -1.0 212.3 15o7 -12.3 - 6.15 19.46 2.39 215.4 272 137 -1.0 212.0 15.7 -12,0 - 6.0 19.30 2.40 215.0 274 138 -1.0 211.4 15.7 -11.4 - 5.7 19.15 2.25 214.6 276 139 -1.0 211.0 1507 -11.0 5-.5 19.01 2.19 214.1 278 140 -1.0 210.5 15,7 -10.5 - 525 18.87 2,08 213.7 280 141 -1.0 210.0 15.7 -10o.0 5.0 18,74 1.96 213.2 282 142 -1.0 209.5 15.7 - 9.5 - 4.75 18.61 1.84 212.7 284 143 -1.0 209.1 15.7 - 9.1 - 4.55 18o49 1.86 212.3 286 144 -1.0 208.7 15.7 - 8.7 - 4.35 18.37 1.72 211.9 288 145 -1.0 208.3 15.7 - 8.3 - 4.15 18o26 1 o59 211.5 290 146 -1.0 207.9 15.7 - 7.9 - 3.95 18.15 1.50 211.1 292 147 -1.0 207.5 15.7 - 7,5 - 3,75 18.05 1.40 210o7 294 148 207.2 - 7.5 - A20o 1 7 360 17.9 1.3 210,4 296 149 -1.0 207.0 157 - 700 - 3.50 17,86 1.34 210.2 298 150 -1.0 206.7 15.7 - 6.7 335 17.77 1.28 209.9 300 23

The University of Michigan ~ Engineering Research Institute analysis can be used to investigate pulse amplifiers in considerable detail and with an accuracy which is consistent with the accuracy of an average family of plate-characteristic curves. 2. EXPERIMENTAL RESULTS AND CIRCUITS As a check on the validity of our results, an experimental investigation was made of the circuits which had been analyzed theoretically. Since no generator was available which was capable of producing the desired signal input, it was decided that the pulse amplifier should be made to furnish its own input by means of the dynamic flip-flop circuit. Consequently, dynamic flip-flops using the 436A tetrode and cascode 437A triodes have been built and tested. Both of these circuits were designed for 5 mcps operation; the transformers used were chosen to have a primary inductance which allowed for a sufficiently rapid recovery and with a turns ratio which gave the secondary voltage which was necessary to drive the delay lines. Diagrams for the 437A and the 436A circuits are shown in Figs. 18 and 20, respectively. 2.1 437A CASCODE CIRCUIT This circuit was capable of supplying 200-ma load current before the flip-flop stopped working. The plate and upper-grid supply voltages are the maximum that may be used to remain within the rated 45-ma average cathode current of the tube. Table II gives the plate, upper-grid, and total cathode currents as a function of the load current. The transformer secondary should always be heavily loaded to keep the grid dissipation and cathode current low, Western Electric recommends that the grid dissipation be kept below 20 milliwatts, and this is done in this circuit at all loads. However, the average cathode-current rating is exceeded unless the secondary is loaded. TABLE II VARIATION OF TUBE CURRENTS WITH LOAD CURRENT FOR CASCODE 437A'S IL (ma) 0 25 50 75 100 125 150 Ip (ma) 24 26 29 31 33 35 36 Igrid(ma) 30 26 24 20 17 14 11 Icathode (ma) 54 52 53 51 50 49 47 The 200-ma load current represents about sixteen 12-ma gates that this circuit is capable of driving, 24

REGENERATIVE BROADENING LOOP + 65 +150 19:3 (B) (C) 47 II *1~ I I (H I 5K/.., WVT~~. ~~~1100-OHM LINE TERMI (A) -2.0 INATION CLOCK 3.3 K -0.5 37 (G) 437A w __ $31.3 K 5 START - PULSE S~s K (F)' I NAI ' PL 51 3.3K (E)5 ~STARTl~ II I I I I 0~~~~-50v D) + 40 0 0 TN0 -2.0 -1.,0 -L40 DELAY LINE (,3 FEETOF RG 65/U) Fig. 18. Cascode 437A dynamic flip-flop for 5-ic operation.

Tlhe linivecrsity of Michigan Intgitneerinlg Resea'hil Institute 'yp:.ical. w.- veforms for this. ( c ircui-. aL(. ' f;lo)\wra ill aF'i {. 1.9 Po l a, (A) 10 v/cm a i W I 1 25 ma Po I.........:' (.'.''.B):***:~' 10'::*':...v/cm.. 50..W%%t~gM............^ - WWW W::.::..................o:i.nt..(1)3) 5. ().V/Cm...g..........i...nt.).......... v/cm....... i............V.........) V L...."....1 ' (.. m a -:gWiiFlX iI'~1oin-t (}') 1.0 v/cm * "i '1:4: 205 ma' Ej Poi~nt (1) 2 v/cr|-1ma C)(A 0 t )v(56A-) ^.X.*R<.Sgni^-n C. ri: f.r.c.. 1.,c^ ' cuiit he.ci oflc~. opiert i: the i. te 1.)t. ( reat eXC id 25 et (c): -- i e6

5 K 5TYPECTP309's (C) CLOCK (A) 3.3 K (A)~ 210 IL OHMS 100 OHMS +75 +125 0 TO -50v 5K~~~~~~~~~~~~~~~~~~~~= START PULSE 20:5 TYPE CTP 309's 20:21 IL (D) 51 OHMS I 1100 -~ ~~~~~~~~~~3.3 K 1OHMS = I, 3 HM -o.5 5~~~~~~~ ~ K. (E) 9 (E) L~I, — IL _ 436A -2.5 DELAY (F)(G) LINE 3.3 K IOK LINE $.$10K -2.5 66DIODES ARE TYPE HD2182 -1.5 -40 +40 Fig. 20. 436A dynamic flip-flop for 5-mc operation.

The University of Michigan * Engineering Research Institute 350-ma or about twenty-nine 12-ma gates. Again the transformer secondary should always be loaded in order to keep the screen dissipation below the rated value as indicated in Table III. Waveforms for this circuit are given in Fig. 21. TABLE III VARIATION OF TUBE CURRENTS WITH LOAD CURRENT FOR THE 436A IL (ma) 0 25 50 75 100 125 150 175 200 Ip (ma) 15 17 18 19 20 21 22 23 24 Is (ma) 37 35 34 32 31 30 29 28 26 Ps (watts) 2~77 2.62 2~55 2.4 2032 2.25 2.17 2,10 1.95 3. CONCLUSIONS Either the 436A or the cascode 437A circuits work fairly well at 5 mcps and provide enough gate drives to be useful computer packages. Neither of these tubes was able to produce as many gate drives as the theory predicted, This is partially because the gate drives were predicted from a linear circuit analysis, and this type of analysis cannot hope to accurately predict the behavior of this non linear pulse amplifier but can only give more or less qualitative information about the desirable properties of the components to be used, However, the main reason why these circuits are not actually able to produce the number of gate drives predicted is that they are both limited by power or current considerations before the theoretical number of gate drives can be reached. Thus, a more realistic analysis should include the tube limitations on cathode current and power dissipation, The theoretical number of gate drives minus the experimental number indicates roughly the increase in gate drives that we would obtain if the tube dissipation could be increased with the other tube parameters held constant. Further testing of these 5-mcps circuits will be conducted with a view to determining how they may be interconnected. If it turns out that the interconnection problem can be solved satisfactorily at 5 mcps, then the frequency will be pushed somewhat higher. Preliminary tests with essentially the same circuits, but with different lengths of delay line and with different (lower inductance) transformers, indicate 28

- -.g ~.~ ~ ~ ~,, ~ '- ~ - -—,' ct-5 0 -~-. "-~. 0 0 0 C~ CP 0 0 ~ -~~~~-P > 4 o cE ~w o:~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~: _Z — ~ ~ 5-.0 A' 00 ONa -- __ __ 0 4-aJa'? <~ ~ ~ ~ ~.. 4. 4. 4.) 4.) 4 e ~~ 2 p 2 2 2 2 2 0.sC. — W.' C(S t**tt1*,t E*, (S ~ ~ ~ ~ ~ ~ ~ 3O(3\ 3 *

The University of Michigan * Engineering Research Institute that, if the interconnection problem is not too severe, it should be possible to obtain approximately 10 gate drives from the 436A circuit at 7 mcps. In view of the fact that these pulse amplifiers are limited by current or power ratings, tubes with high power ratings are desirable. If tubes were available which would deliver a larger plate-current swing than the tubes used here with the same grid-voltage swing, that is, with higher gm's, then it seems possible to design an amplifier of this type to operate at higher frequencies. However, considering the careful manufacturing processes that were necessary to produce the tubes used here3, it seems unlikely that significantly better tubes will be developed in the near future4 3. Ford, G.T., "The 404A-A Broadband Amplifier Tube," Bell Labs Record, 27, p.59, 1949. 4 Beck, A.H.W., "Thermionic Valves-Their Theory and Design," Cambridge University Press, 1953, p.291. 30

The University of Michigan ~ Engineering Research Institute APPENDIX TRIODE INPUT CAPACITANCE The grid is driven by a gating structure which may be approximated by a constant-current step source. The circuit to be considered is then as shown in Fig. 1-A, and the small-signal equivalent of this circuit is shown in Fig. 2-A. Applying; Norton's theorem to the voltage generator and letting R = (rp P~9/(rp + RL) gives the final circuit of Fig. 3-A. C3 RL C2 CFig. 1-A C2 CI RL Fig. 2-A 31

The University of Michigan ~ Engineering Research Institute NODE I C2 NODE 2 1C gm' g I 1 eQ REFERENCE NODEt Fig. 3-A From this last circuit, one may write by inspection I1 = (eo + eg) pC2 I2 = eg PC1 13 = eoPC3 I4 = eo R For Node 1 I = + I2 = (eo + eg) pC2 + eg PCL For Node 2 gm eg = I1 + I + 14 = (eo + eg) pC2 + eopC3 + ~ Rewriting: p (C1 + C2) eg + pC2eo = I (gm - pC2) eg - (pC2 + pC3 + R) eo = P(C1 + C2) pC2 e] I gm - PC2 - (Cl + Cs) - e = oi We wish to know the input capacitance, and since 1 r de eg = I dt or I(t) Cin dt Cin We then wish to find (deg/dt)(l/I(t). Hence, solve for peg for a stepfunction input I/p: peg = p (C2 + C3) + R I p -(C(1 + C2) [P(C2 + C3) + ] + C2(gm - PC2) 32

The University of Michigan ~ Engineering Research Institute 1 + pR (C2 + C3) R(ClC2 + C1C3 + C2C3) PCp1 + (1 + gmR) C2 R(ClC2 + C2C3 + CC3C C2 + C3 CIC2 + CiC3 + C2C3 B = 1 and = 1 C1 + (1 ~gmR) C2 R(C1C2 + ClC3 + C2C3) R(C1C2 + C1C3 + C2C3) I -, Epe_ g = A B I p +< p(p+Xa) I de _cXt B and I = Ae- 0 +- e(1 -0t ) I dt (C2 + C3) e.t (1 - e-t) CIC2 + C1C3 + C2C3 + C + (1 + gmR) C2 _ 1 r 0C2+C3 1 C7 C1 + ( + gmR) c2 + LClC2 + ClC3 + C2C3 C + (+gmR) C2 1 Cin Cin C0+(l+gmR)C2 C0C2+C1C3+C20C3 C2+C3 time The input capacitance has reached 95% of its final value when t = 3/a or when t = 3 R(C1C0+ C+C 0 + C+ 2C3) C0 + (1 + gmR) C2 33

The University of Michigan ~ Engineering Research Institute For the 437A triode with R = lk, Cj = 11 pF, C2 = 4 pF, and C3 = 1 pF, tc is calculated to be 0.89 x lO- sec. That is, the input capacitance has reached 95% of C1 + (1 + gmR) C2 in a time of the order of 1 misec for the 437A. This means that for 10 -mcps (or less) operation, the input capacitance may be considered to be just Cin = C1 + (1 + gm R) C2. 34