ENGINEERING RESEARCH INSTITUTE TE UNIVERSITY OF MICHIGAN ANN ARBOR Scientific Report SOME STUDIES OF THE DISCHARGE INITIATIONS IN A COLD-CATHODE GAS TETRODE No Wo Spencer Ro Co Kiene Wo Go Dow Project 2269 DEPARTMENT OF THE ARMY DIAMOND ORDNANCE FUZE LABORATORIES WASHINGTON, Do Co CONTRACT NOo DAI-R49-186-502-ORD-(P)-1994 August 1957

The University of Michigan * Engineering Research Institute TABLE OF CON'TENTS Page LIST OF FIGURES iii ABSTRACT vi OBJECTIVE vi 1o INTRODUCTION AND ACKNOWLEDGMENTS 1 2o THE EXPERIMENTAL CIRC-rIT 1 35 SIMPLIFICATION OF THE EXPERIMENTAL CIRCUIT 5 4o NONLINEAR DIFFERENTIAL EQUATION DESCRIBING CURRENT IN EQUIVALENT CIRCUIT 7 5o BREAKDOWN DATA FOR TEST CIRCUIT FOR e(t) = 8t 60 CORRELATION OF EXPERIMENTAL AND LINEARIZED, EQUIVALENT CIRCUIT DATA 10 7. CORRELATION OF EXPERIMENTAL AND NONLINEAR EQUIVBALENT CIRCUIT DATA 16 71l Variation of the Nonlinear I:dductance as L = Ca/ig 18 7~2 Effect of Changing Initial, and, Filial Current Values in Equivalent Circuit 21 7 5 Investigation of Other Nonlinnear lF ements 21 80 INVESTIGATION OF OSCILLATTONS BY GROUP FIRING DATA 25 90 GRID-CATHODE RESONANCE EFYECT.AS A FUJNCTION OF THE INITIAL OPERATING CURRENT io 55 10. GRID-CATHODE RESONANC'E DUEr TO SHUNT:APACITY Cgc 38 11 CONCLUSIONS AND SUMMARY 42 APPENDICES 43 Appendix I. Analytical Solution of a Series Circuit Employing a Nonlinear Inductance and, a Negative Resistance 44 Appendix IIo Graphical Method of Solving a Nonlinear Differen ial Equati on 47 Appendix IIIo Axnalog Computer Technology 52 Appendix IVo Grid-to-Cathode Static Characteristics 54 ii

The University of Michigan * Engineering Research Institute LIST OF FIGURES Noo Page 2ol Experimental, circuit employing cold-cathode gas tetrode. 2 3ol Equivalent circuit after elimination of anode and second, grid circuitso 4 352 Equivalent circuit including an inductance to simulate the dynamic time lag characteristic of the grid-cathode gapo 4 3~3 Typical grid-cathode static characteristicso 5 354 Equivalent circuit with assumed dynamic-volt-ampere characteristic for grid-cathode gapo 6 535 Voltage source equivalent circuit of Fig, 354o 6 306 Final equivalent circuit simulating the dynamic characteristics of the grid-cathode circuit of the type QF-391 tubeo 7 501 Experimental setup for obtaining dynamic characteristic of test circuito 9 502 Comparison of experimental data from tube firings and linear equivalent circuit results. 11 60l Circuit involving linear inductance and negative resistance0 13 602 Normalized. computer setup for solution of current in Eqo 4,lo 13 603 Curves lshowing effects of different values of X on current variation in linear circuit. 15 6o4 Curves showing effects of different values of (L) with constant (k)o 15 70l Simplified circuit involving nonlinear inductance and negative resistance. 16 702 Normalized computer setup for solution of current in Eq 7.2o 17 7o5 Comparison of current variation of linear and nonlinear circuits 19 --------------------------- ~ ~ ~ ~ ~ l l —-----------

The University of Michigan * Engineering Research Institute LIST OF FIGURES (Continued) No. Page 7.4 Comparison of linear and nonlinear equivalent circuit results. 20 7.5 Results of varying initial current (i0) in nonlinear equivalent circuit of Fig. 7.1. 22 7.6 Results of variation of a in equivalent circuit. 25 7,7 Results of variation in final current (i0) in the equivalent circuit. 24 7.8 Comparison of various nonlinear circuit results. 26 8.1 Group firing data for QF-591 tube. 28 8.2 Graph showing percent of firings occurring during various groups. 50 8.5 Current variation of grid circuit assuming linear components. 50 8o4 \ vs t curve. 51 9.1 Setup for investigating grid-cathode resonance., 34 9.2 Resonance curve of test circuit. 55 9.5 Typical resonance curves for operating currents il and iC. 6 9~4 Resonant frequency of grid circuit as a function of operating current. 57 10.1 Setup for investigating grid resonance. 59 10.2 Grid-to-cathode dynamic circuit. 59 1005 Resonance curves of QF-591 No. 7295 Lot 8N1 for various operating currents. 40 10.4 Grid-cathode resonant frequency vs operating current, 41 -----------------------— iv —------------

The University of Michigan * Engineering Research Institute LIST OF FIGURES (Concluded) No Page A-I-1 Circuit containing nonlinear inductance and negative resistance. 45 A-I-2 Current vs time for various values of ao 45 A-II-1 Figure shows method of constructing phase-plane ploto 50 A-II-2 Phase-plane plot of Eqo 12o3 (' = 10)o 50 A-III-1 Analog computer technologyo 53 A-IV-1 Grid-cathode static-characteristic setupo 55 A-IV-2 Comparison of static characteristics of QF-391 No' 7295 Lot 8Nlo 56 _ _ _ _ _ —-----------------— Y —-----------------------

The University of Michigan * Engineering Research Institute ABSTRACT Studies related to the verification of a suggested equivalent circuit for the grid-cathode discharge in a subminiature cold-cathode gas tetrode have led to modification of the equivalent circuit, and to consideration of the firing time delays as basically a statistical matter. Further investigation is necessary before these conclusions can be firmly established. OBJECTIVE The purpose of this investigation was to obtain a better understanding of the fundamental reasons for delays occurring in the initiations of a low-level gas discharge. vi

The University of Michigan * Engineering Research Institute 1 INTRODUCTION ANFD ACKNOWLEDGMENTS The Electrical Engineering Department of The University of Michigan has been conducting a study, primarily analytical, of certain properties of a subminiature, cold-cathode gas tetrode known as QF-391L Generally speaking, the investigations have been concerned with. a particular circuit configuration using the tube which enables illustration of the readily observable property of a delay between application of a ramp function input current and breakdown of the control grid-cathode gap. This research has been carried out for, and'under sponsorship of, the Diamond Ordnance Fuze Laboratories~ The QF-391 was developed by the Raytheon Electric Corporation under contract with the Diamond Ordnance Fuze Laboratories, Several members of the staffs of the Electrical Engineering Department and the Diamond Ordnance Fuze Laboratorie- have contributed to this studyo As is natural. in a program to which, different individual s contribute at different times, it is not aLways possible to give thte proper credit for various aspects of the worko It isL eispecially difficult in this. case where some portions of the work have a bearing upon clas;ified applications, and, where some appropriate references are classified and thus camnot be noted. The contributors from the Diamond. Ordnance e Fze Laboratories include To Mo Liimatainen (scientific officer), Martin Reddon, Alfred Ward and Jay Lathrop; from The University of Michiga.n, a,side from those noted on the title page of this report, Willia Kerr, Ho Go Hedges (now at Michigan State U:niversity), and Jules Needle (now at Northw estern University)o The bulk of the investigation d.iscu-ssed in this report wras carried out by Ro Co Kiene, now at Mirmeapolis:-Honr.eywell, who was at the time a graduate student. MIr Kiene likewise prepared much of the rough draft of this report 2o THE E.XPERIMEINAL CIRCUIT In connection with certain classified work, it was observed that appreciable delays occur betwee:n the application of test signals to some coldcathode gas-tetrode grid-cathode gaps and the breakdo-wn of the gas in the gapo One of the circuits that illustrates the effect quite readily is shown in Figo 20o1 In this example, a current source is employed to initiate the flow of ------------------------— 1 —------------

The University of Michigan * Engineering Research Institute R3 C XR R2 SCOPE _y T~ ~ T c+: FIG. 2.1 EXPERIMENTAL CIRCUIT EMPLOYING COLD-CATHODE GAS TETRODE current in the g-c gap which operates, initially at least, in the Townsend region. The capacitance Cg is that associated with the current generator and test setup wiring. The plate and grid circuit elements and the battery are arranged to bias these tube elements appropriately, and to enable observation of breakdown by providing an easily observable magnitude of plate current. In operation, a current input signal is initiated which, after a delay, fires the g-c gap which in turn causes plate-cathode conduction and an indicating signal, for observation across R5. This circuit was chosen for study. The general objective was to (a) obtain an understanding of the factors governing the observed delays, and (b) establish the validity of an equivalent circuit that had been suggested to simulate the observed effects. A current-type input signal is not always the most convenient experimentally, thus one of the first steps taken in this work was to arrange an equivalent voltage source. In this regard replace (Fig. 2.1) the current source i(t) and the shunt capacitance with a voltage generator e(t) and a'series capacitance Cg by Norton's theorem. In substantiation, from Fig. 2.1 we can write: i(t) = Cg dt + il(t), (2.1)

The University of Michigan * Engineering Research Institute elit) = l-)t J i(t)t - (t)dt and (2.2) Cg -g e(t) = el(t) + 1- i(t)dt (23) Equation 235 states that we can have equivalent voltage source e(t) operating in series with the capacitor Cg provided the voltage e(t) = - i(t)dt (2,4) C.An assmnption should be noted at this point. In changing from a current source to a voltage source equivalent circuilt, the complete shmunt capacitance Cg became a series capacitance This step is something of an approximation. In an experimental setup of the test circuit, the capacitance Cg will consist of all capacitance to be found in parallel with the current generator including stray and tube capacitanceso The stray and. tube capacitances will of course always be present and. can never be completel.y eliminated from the shunt branch and placed in a series circuit arrangement, Tlse capacitances, although in a sense neglected here ha.,ve an effect, considered in Section 10, on the tube operation O 35 S$IMPLTFICATION OF THE EXPEIRIM:ENTAL CLRCJLIT in this section, the equivalent circuit of the experimental circuit (Fig. 5.1) will be considered for possible simplificationso It can be demonstrated that the time interval between breakdown of the grid-cathode gap and anode-cathode gap is relatively small compared to the time interval'between the application of the input siglnal and the grid-cathode breakdown. Accordingly, this study is confined. to the time lag mechanism of the grid-cathode gap of the QF-391 gas tetrode Thus the equivalent circuit may be reduced to that shown in Fig. 35l, where the second grid and anode circuits of the tube are omitted. As a parallel, during bench testing, the anode and second grid-circuit parameters are maintained constant Thus it is assumed t:hat any contributions that these circuits might make to alter the time response of the grid-cathode circuit can be considered constantO Typical static characteristics of the grid-cathode gap are shown in Figo 353. The operating point of the gap is usual:ly chosen at some initial value of current such as i: if illustrating a typical breakdown pointo *Grid-to-Cathode Static Characteristics - Appendix IVo 5

- The University of Michigan * Engineering Research Institute i (t) INPUT SIGNAL I | l g | - GRID-CATHODE 9 _|_ } GAP OF QF 391 R I I 1 cg TEb I DYNAMIC CHARACTERISTICS OF GRID-CATHODE GAP FIG. 3.1 EQUIVALENT CIRCUIT AFTER ELIMINATION OF ANODE AND SECOND GRID CIRCUITS L- -i i(t) g <, >9 v DYNAMIC VOLT-AMPERE 5RJ^ — \fCg ao'a \ ^CHARACTERISTIC OF i = 9 i R GRID-CATHODE GAP E FIG. 3.2 EQUIVALENT CIRCUIT INCLUDING AN INDUCTANCE TO SIMULATE THE DYNAMIC TIME LAG CHARACTERISTIC OF THE GRID-CATHODE GAP -4

The University of Michigan * Engineering Research Institute V g - ACTUAL STATIC CHARACTERISTIC -— ASSUMED DYNAMIC CHARACTERISTIC FOR ANALYTICAL STUDIES O Vd _ > Vi Vg= Vd -P ig 0 ifci jo gi GRID-CATHODE STATIC CURRENT FIG. 3.3 TYPICAL GRID- CATHODE STATIC CHARACTERISTICS When the imput signal is applied to the grid circuit, the increasing current through the grid-cathode gap does not follow the static characteristics shown, but instead a dynamic characteristic (Fig. 3.2) which depends on the tube constants and the ionization mechanism of the gap. The current buildup across a gaseous gap operating in the Townsend region of the static characteristics can presumably be explained by a dynamic volt-ampere characteristic and in part simulated by an inductance whose magnitude varies inversely with the current. The parameter is defined as L = a (3.1) where a is a constant dependent upon gas parameters. The dynamic resistance of the grid-cathode gap is a function of the current through the gap. However, as a first approximation, this resistance is considered to have a constant magnitude p corresponding to the negative slope of the static characteristic, since operation of the tube is confined largely to this region in all experimental investigations. Then, assuming p as the constant dynamic resistance of the gap, we obtain a relationship from the linear portion of the volt-ampere curve of Fig. 3.,3 that is, for i > io, Vg = Vd - pig. (3.2) Thus considering both the negative resistance and special inductance, the grid circuit of the test circuit has the form shown in Fig. 5.34, retaining, for the moment, the current generator form. (Figure 35.5 illustrates the voltage source equivalent circuit.) 5

The University of Michigan * Engineering Research Institute AMPERE CHARACTERISTIC FOR GRID-CATHODE GAP a il(t) e(t) L = (R g I iCg RR V ertI__i______ I_ _ FIG. 3.5 VOLTAGE SOURCE EQUIVALENT CIRCUIT OF FIG. 3.4 -6 —--

The University of Michigan * Engineering Research Institute The resistance of the parallel combination of R', circuit leakage resistance, and Rg, grid circuit resistance, as compared with the rest of the circuit, is very high; thus these branches can be neglected, The battery Vd can also be ignored, since its effect is only that of setting the initial condition io in the tube. The elimination of these parameters has no appreciable effect on the dynamic response of the equivalent circuit; however, it must be remembered that the inductance L has an initial value corresponding to the initial condition i = ioo The resulting simplified equivalent circuit employed to simulate the dynamic properties of the test circuit is shown in Fig. 5.6. Through analysis of this circuit, the time interval required for the current to increase from io to if will be determined. For this purpose a relationship between this time interval as a function of e(t) and the differential equation describing the current ig in this circuit will be developed. a ~~~~C L.. Cg ig'9 iO INITIAL CURRENT CONDITION -P e (t) FIG. 3.6 FINAL EQUIVALENT CIRCUIT SIMULATING THE DYNAMIC CHARACTERISTICS OF THE GRID-CATHODE CIRCUIT OF THE TYPE QF-391 TUBE 4, NONLINEAR DIFFERENTIAL EQUATION DESCRIBING CURRENT IN EQUIVALENT CIRCUIT Having determined the simplified equivalent circuit for simulating the dynamic characteristics of the fuze circuit, we proceed to solve for the current ig as a function of the various parameters. Due to the variable inductance in the equivalent circuit, the differential equation describing the current is nonlinear: (i-) -P ig + C Jig dt = e(t) (41) N

The University of Michigan * Engineering Research Institute The initial condition of this equation at time t = 0 is that ig = o i (4.2) Differentiating Eq. 4ol to remove the integral, we obtain a secondorder nonlinear differential. equations.. +- -ig (4~3) d, ^i- ig 7 r di -=g d e(t) d( t g d.)_ d.t Cg dt Modest attempts to solve this equation analytically were not fruitful; however, through use of a graphical approach, the phase-plane method, a solution was determinable It is feasible to obtain a solution through use of an analog computer; thus, during the course of the investigation described in this report, a solution was obtained in that manner alsoo The analog computer setup of Eqo 4l1 is given in Section 7~ 5o BREA(KDOWN DATA FOR TE5i" CIRC't lR T FOR e(t) = Xt The dynamic behavior of the test circuit was investigated experimentally by applying a particular voltage signa:l and f:iring the tubeo Thie voltage signal employed, had the form e(t) Xt, (ol) and was used because of its relative simpl.icityo The experimental test- circuit arrangement usaed. to obtain the dynamic firing characteristics as a function of i. is shown in Figo 501 The battery X and the resistance RI- capacitance CL com'bination generated the voltage e(t) = Xto Since the tube fired after relatively short time intervals (in the 0-50millisecond range), a simple RBCS1 combination could be used. to generate the xt signalo The voltage appearing across the capacitance (Cl) has a linear slope of,1/RC1, at t = O0 Also, for small time inte:rvals below the RC2j time constant, the slope is nearly the sameo Therefore, by making the time constant RICz equal to,, say, econd, the voltage rise acros the condenser can be considered just Xt for short time intervalso'Use sugge sted by:DOFL personnel 8

The University of Michigan * Engineering Research Institute RL Eb Rg2 RI Cg Rg, EgI C2 R Eg, R2 io = INITIAL CURRENT TRIGGER SWEEP TO VERTICAL OF SCOPE AMPLIFIER OF SCOPE CIRCUIT VALUES: TUBE - QF-391 NO. 4092 LOT 8NI X - VARIABLE VOLTAGE FOR ADJUSTING VALUE OF LINEAR RATE OF RISE VOLTAGE APPLIED TO TUBE. R, = I X 106 OHM Eb - 180 VOLTS C, = I X 106 FARAD E = 155 VOLTS Cg = 10 x 112 FARAD Rg2 I X 10 OHM C2 =.05 X 10 FARAD Rgl I X 10'OHM R2 = 50 X 103 OHM i0 = 5.4 X 109 OHM RL: 6.8 X O6 OHM (INITIAL CURRENT) FIG. 5.1 EXPERIMENTAL SETUP FOR OBTAINING DYNAMIC CHARACTERISTIC OF TEST CIRCUIT The capacitance Cg represents the total input capacitance as shown in Fig. 3o6, and has a value of 10 [ii farad.o The various values of the other parameters are given in Figo 5olo Operation of the circuit is as follows, Egl is adjusted until an initial current io (steady) flows through the grid-cathode gap of the QF-391, When the switch is thrown, voltage X is applied to the RIC0 combination and, for the small time interval of interest, the voltage across C1 is 2t, a linear rate of rise signalO At the same time the switch is thrown, a signal is applied to an oscilloscope to initiate the time-calibrated horizontal sweepo When the tube fires, evidence of the fact appears on the face of the scope as a vertical displacemento From the horizontal distance traveled by the beam up to the time of firing, the time delay between the applied signal and firing can ----------------------— 9 —----------

The University of Michigan * Engineering Research Institute be determined. Typical results are plotted in Fig, 5~2 wnere computed curves (to be described) are included for comparisono The points plotted represent the results of experimental data for one tube at a particular value of operating or initial current ioo It was noticed in obtaining these data that the firing time was not a unique function of a particular value of k, as might be expected, but rather a statistical functiono Thus the points plotted on Figo 5.2 represent an average firing time for twenty firings at each different value of Xo The statistical nature of firings makes it difficult to obtain a unique curve to represent the dynamic characteristic of the fuze circuito The curve shown is a function of the tube's characteristics and of the chosen boundary conditionso By trying many tubes, most of the results indicate the general shape of the curve showin for initial values1 of operating current near io = 10"9 amp and firing currents of if = 10~8 amp and greater. The only region which shows appreciable divergence is for low values of X near the "forbidden region " defined as a range of values of \'below which the tube ceases to fireO For the particular tube shown, the lowest value of X aet'which the tube fired was Xf = 1o5 x 102 volts/seco The time delay between application of this af and firing was T = 70 millisecondso In. general, for the operating regions of interest between 10-9 amp to 1.08 amp the forbidden region varies from k = loO x 102 volts/sec to A - l03 volts/sec with the forbidden region firing times ranging from 80 milliseconds to about 10 milliseconds, respectively, The curve shown in Fig. 5.2 should not be considered a unique representation of the dynamic characteristics of the circuit, butt ilnstead an average picture showing the general character of operationo For the initial stages of our investigations with the equivalent circuit, this experimental curve was used as a basis of comparison for the equivalent circuit results. This enabled us to obtain a rough value for the nonlinear inductance to be used. in the equivalent circuito The dashed curve of Fig. 5~2 represents the type of dynamic data which it was desired to simulate by the equivalent circuit of Fig. 3560 60 CORRELATION OF EXPERIMENTAL.AND1 LIEARBIZED EQUIVALENT CIRCUIT DATA After the experimental data shown on Fig 5~2 were obtained, the equivalent circuit response was investigated in hope that a correlation between the experimental and analytical data could be foundo However, prior to this, a similar linear circuit with a constant inductance was solved in anticipation of t"boundingt the experimental resultso This "bounding~ property of the linear circuit can be explained as followso 1.0

35 I' —--- TUBE CEASES TO FIRE AT X 1.5 X 102 VOLTS/sec. F 5. FIG. 5.2 AND T= 70 MILLISEC.. 30 - COMPARISON OF EXPERIMENTAL DATA FROM TUBE C': 30r l \ = b.I I I\ - 3 _ |_l_ \ _ |_|_ |||FIRINGS AND LINEAR EQUIVALENT CIRCUIT RESULTS 25 n L 07h. -------------- -- -- L —1 —- -- _ — -— c —--- TIME REQUIRED FOR TUBE TO FIRE - o \ s _- | _I__ \ I I ______________T______ - TIME FOR CURRENT TO RISE FROM |: 0 20 -... IO-9\^ 109 AMPS TO IO-AMPS IN LINEAR 3 \n CIRCUIT. a0E / \ TUBE FIRINGS =: U — i 0 w L= 106h. w. 3~ ----- - "- -~~~~-z 10-h." —- - 0 I02 103 104 LINEAR RATE OF RISE SIGNAL X -VOLTS/ sec.

The University of Michigan * Engineering Research Institute In the nonlinear equivalent circuit, th.e range of values over which the nonlinear inductance varied (determined by the current) was unknowno A rough value for this inductance can be computed readily, but for the particular tube used, this value was inadequateo In testing various tubes it was found that the range of current variation across t:he grid-cathode gap was from about i = 109 amp to the breakdown current of approximately if = 10- amp. Therefore, the proposed nonlinear inductance function L = O/ig used in the equivalent circuit would vary by a factor of 10 in this current rangeo This a, ssumes that the current ig increases continuously from io to if and no discontinuities of the function L = al/i are encountered., Thus the dynamic characteristics of the nonlinear circuit could presumably'be considered "bounded& -by the characteristics of two linear circuits using the maximum and minimum values of the nonlinear element L = C/ig instead of the varying inductance. By this method the experimental curve shown in Fig. 5.2 can'be t.reated as "boun.lded'by the linear equivalent circuit s resl.ults a:s how'ni r:'o this wyt r tihe ran.ge of vaui. es in which the nonlinear element L = C/i had to'vary was established. The linear circuit which wals. usEed is showl inl Figo 6c r. cc by the analog computer setup (Fig. 6.,2) used to simulate the circuit As noted, the inductance did not vary with current ig as in the nonlinear case, but was maintained constant The equation of the circuit is L dig,P L p ig + J- ig dt = Xt, (6ol) where ig i. at time (t) -0. (6.2) The solution of this linear differential equation is i - ) j — + -- e - sin (t)t- I + io lf e in (ft+ |. (6.3) = L o PoP where O=,P 2L I- a'p2 pDo = _ + p2 o -1.2

The University of Michigan * Engineering Research Institute L CONSTANT C (e) ^-R i FIG. 6.1 CIRCUIT INVOLVING LINEAR INDUCTANCE AND NEGATIVE RESISTANCE RECORDER I- I —- -io — Ri 0~ lL R i i < -.~Jidt FIG. 6.2 NORMALIZED COMPUTER SETUP FOR SOLUTION OF CURRENT IN EQUATION 4.1 15 -

The University of Michigan * Engineering Research Institute The curves shown in Fig 5 2 are for the following values of the linear circuit s parameterso X = 102 to 104 volts/sec L = 105, 106, and 107 henries C = 10 x 10-12 farads g p = -108 ohms The solid curves of Figo 5~2 represent the time it takes the current ig to increase from an initial value of io = 10-9 amp to a firing current if = 10-8 amp for various values of \o The results obtained from the linear circuit indicate that the experimental data are bounded for L = 107 to 105 henries. For values of X greater than 103 volts/sec, the experimental results correspond very closely to L = 105 henrieso For lower values of L, close to the forbidden region, there is a greater variation between the experimental and analytical data. To discuss the correlation between the experimental curve shown in Figo 5o2 and the curves describing the linear equivalent circuit, three curves are drawn in Fig. 605 to show the effect of varying L and.o As can be inferred from Eq, 6o53 the current ig is oscillatory and exponentially increasing in nature. With X sufficiently high, the current in the equivalent circuit rises to a much greater value than if = 10-8 amp during the first peak or oscillation, This can be considered as corresponding to the grid-cathode current reaching the breakdown current ig before the first peak or oscillation occurs. With X at some lower value such as X2, the current just reaches 10-8 amp during the first peako This corresponds to the boundary of the forbidden region, that is, the lowest value of X needed for the tube to fire on the first oscillation, When X is reduced to less than?2, the current no longer reaches 10-8 amp during the first peak, but can exceed 10-8 amp during some later cycleo Thus, when X = X3, \ can be associated with values which lie in the forbidden region, k2 representing the edge of the forbidden regiono Also to be noted from the equations and curves is the fact that the time to reach 10 8 amp decreases with increasing X, which is likewise observed experimentallyo The curves shown in Fig. 6o4 are for constant X and variable inductance L. As the value of L is increased, the resonant frequency of the circuit decreases, consequently resulting in a greater time required for the current to reach if = 10-8 Similarly, as can be seen from the curves of Fig. 5o2, when L is increased, the time required for the current to reach 10-8 amp, for the same constant value of X, also increaseso From the bounding of the experimental data by the linear equivalent circuit data, it can be seen that the nonlinear function L = a/ig should probably vary from 105 to 104 henries for high values of Ao For lower values of X, -14

The University of Michigan * Engineering Research Institute L: 106 h. X, IC S v/sec. X2 - 4 x IC2 v/sec. X = 2 x 102 v/sec. XI F I ^ ^ i0 - 10 AMPS 1 Z t-4.5 t10 t TIME (MILLISEC. - TIME TO REACH \\ 10-8 AMPS. FIG. 6.3 CURVES SHOWING EFFECTS OF DIFFERENT VALUES OF X ON CURRENJT VARIATION IN LINEAR CIRCUIT.,L 0 lO'h. X = l03 v/sec..L-L 106 h.j X 1 03 v/sec. / \ \f I o AAMP, s _r - __\_ _ _____ _ ____ i -- _ __ _ C.. /)/'1 \ \ TIME (MILLISEC.) I ^ I_ t2\ \ t 1.25 MILLISEC \ \ t 4.5 MILLISEC FIG. 6.4 CURVES SHOWING EFFECTS OF DIFFERENT VALUES OF (L) WITH CONSTANT ION IN LINEAR CIRCUIT. — L:0 - h. —: 10 15 v/sec.

The University of Michigan * Engineering Research Institute some other range of greater L seems to be necessary. To investigate this point further, we now examine the nonlinear circuit of Fig. 3.6. 7o CORRELATION OF EXPERIMENTAL AND NONLINEAR EQUIVALENT CIRCUIT DATA In the previous section it was shown that the experimental data of the test circuit could be "bounded" by analytical results obtained from the linear circuit of the form in Fig. 6,1. From these results, the range of values over which the nonlinear parameter L = f(ig) would have to vary to approximate the experimental data is determined, However, the exact variation of this function is unknown. In this section the initially proposed nonlinear function L = a/ig is studied with regard to its effect on the dynamic characteristics of the circuit shown in Fig. 7.10 The dynamic characteristics of the nonlinear circuit are then compared with the experimental and linear circuit data. The effects of several other types of nonlinear variations of the inductance parameter are also considered, and the results of employing these nonlinear inductances are compared with the experimental results for possible better correlation, C ~~L f(i) e = Xt < -R FIG. 7.1 SIMPLIFIED CIRCUIT INVOLVING NON-LINEAR INDUCTANCE AND NEGATIVE RESISTANCE To solve the nonlinear equation describing the current variation in the equivalent circuit, an analog computer is used. Since the equivalent circult is considered with various nonlinear inductances, Eq. 4.1 is put in the following general form: 16

The University of Michigan * Engineering Research Institute f(ig) di - P i dt = t (71) dt ( i g C+ g gt =.t ( The corresponding computer equation is: dig (p _i - 1C ig dt + t) (7 2) dt-~ f(ig) c After normalizing this equation, the computer setup for the solution of ig as a function of time was arranged as shown in Figc 7~2~ Appendix III briefly explains the operation of this setupc RECORDER + PHOTO-FORMER - Ri - C |idt + Xt I -- f (i) C / J F(i) i+io SERVO- PHOTOMULTIPLIER FORMER FIG. 7.2 NORMALIZED COMPUTER SETUP FOR SOLUTION OF CURRENT IN EQUATION 7.2. -------------------- 1,,

The University of Michigan * Engineering Research Institute 71l VARIATION OF THE NONLINEAR INDUCTANCE AS L = a/ig In Section 3 it was shown that the nonlinear element simulating the time lag of current buildup across the grid-cathode gap could have the form L = /ig Having determined a range of variation of L (Section 6), an approximate value for the constant a can'be calculatedo Thus from the results obtained using a constant L, it is seen that, to approximate the test circuit's characteristics for low values of X, the nonlinear element would, have to vary from about 105 to 106 henrieso This means that since the corresponding current range is from io = 10-9 amp to if = 10-8 amp, the coefficient of the dig/dt term would vary as f(ig) = to 10to = o 106 to 5 henries o 10-9 10"8 10-9 10-8 The characteristics of the nonlinear circuit for two values of a are shown in Fig~ 7~4 in comparison with experimental and linear circuit data to illustrate the boundary effect of the assumed linear circuit arrangement, Figure 7o3 shows the current variation of the linear circuit and the nonlinear circuit where L = a/igo The curves are drawn for values of X for which the current ig just reaches the firing current if during the first peako For lower values of X, the current variation ig will not reach the value if on the first peak but on some later peak, and, as previously defined, these values of \ lie in the forbidden regiono It is of interest to note that the current ig varies monotonically from io to if without discontinuity, and of particular interest that this representation shows clearly how the element of chance is a factor in the operation of this tube. By observing the current variation of the nonlinear circuit in comparison with the linear case, it can be seen that the bounding effect concept of the linear case is not valid for times greater than that to the first peak, for the reason that the inductance seems to become infinite as the c-urrent passes through zero. This aspect of the problem has not been pursued other than to note the aboveo Actually it may prove upon investigation that the apparent passing through zero of the current is erroneous, resulting from assumptions in establishing the computer setup, and that in fact the current only approaches zero, as suggested by the flattening of the dashed curve of Figo 7~35 The presence of the second appearing peak is real and is observed in the experimental setup, as will be considered latero The time required for the current ig to reach if for the nonlinear and linear cases as a function of X is shown in Figo 7o4o As an illustration of the relationship between FigSo 7~3 and 7o4, the boundary value of X is X = 5o0 x 102 volts/sec and X = 4o0 x 102 volts/sec for the nonlinear L = OoOl/ig 1.8

— _ ___- _ —___ / cr ~ - - NON-LINEAR CIRCUIT L= —h. - W / -8 / I' 0-^ ~ ~_ _ if: 10 AMPS / z0 oa =.01 X,5.0 X 10 v/sec. I \o I I I I 1', — 1 1 1 1 1 1 1 1 1 1 CI | 0 / \ FIG. 7.3 COMPARISON F OF LINEAR AN NON — LINEAR CIRCUIT L= CONSTANT 4o 4_{_/ __ = 4.0 X 102 v/sec. - 4 / \..... -9 o /: 0 w -2 4 8 12 16 20 2428 32 36- 4044 4852 56 60_______ N I I I I3 -J ___TIME- MILLISECONDS ~~~~~~~~~~~~~~~~~~~~0 -6~~~~0 ~~~~~~-10c~~~~~~~~~~~~~~~~~" FIG. 7.3 COMPARISON OF CURRENT VARIATION OF LINEAR AND NON-LINEAR CIRCUITS

3,5 " —---....... - - - - -................... \.... - -- -- -\ - ---------- -- __ _ _ ~F IG. 7.4 6 wo 30\.__ — COMPARISON OF LINEAR AND NON-LINEAR -J, ---------— \ _______ __ __ _ - EQUIVALENT CIRCUIT RESULTS I2~~~ \ \ L: 107 h. 0D Q -----------— TEST DAgv ^TA - _ —-- -- -- — X- L_: -0 INDUCTANCE VARIES FROM - - S:25....'o I 0 I \ \ TS hATA "x. 0I r)../ \INDUCTANCE VARIES FROM I __________- -'_'g_ 107 h. TO1h. _ 6 M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0 ____ ___ \ 0. n- 20 ~ \ I% mL= INDUCTANCE VARIES FROM ^,o ---------— v _^ _ _ —— ^ ^ ^ —--— _ _ _ _, _0_ N 106h. TO 105h., la,. 15 w aO \ a~~~~~~~~~~~~~~~~=.01-' 0 1o~~~~~~~~~~~ \ z 0:u 106h. %~ M M o 0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~5 LL ~ ~ ~ ~ z1 0 i02 10 - v/ sec.

The University of Michigan * Engineering Research Institute and linear (L = 106 henries) examples, respectivelyo The time for the current to rise from io to if is shown to be t = 16 milliseconds and t = 10 milliseconds for the nonlinear and. linear cases, respectivelyo These points are shown to'be the end points of the L = 106 and a = Oo01 curves shown in Figo 7o4o 7.2 EFFECT OF CHANGING INITIAL AND FINAL CUREENT VALUES IN EQUIVALENT CIRCUIT From the nonlinear data of Figo 7o4 it can be seen that the experimental circuit operation is relatively poorly approximated by the equivalent circuito Experimentally, the operating point io of the tube, on the gridcathode static characteristic, 11has an important effect on the time it takes for the tube to fire with an applied linear rate of rise signai0 With the operating point io near the breakdown current if of othe tube, instability occur as is to be expected, but shorter firing timer also result, and, conve:rsely, with the operating point near or on the positive slope region of the static characteristics, stability is increased at the sacrifice of longer firing timeso Since the values of a, io, and if used in Fig. 7o4 do not approximate the circuit s performance very well, the effects of changing the initial current and also the effects of extending the range of the firing current if were considered, Figure 7~5 shows the results of a change in'the initial current io, Also drawn on this graph is the experimental c ircuit characteris.tic obtained from the setup shown in Figo 5ol with the corresponding operating conditions. As expected, shorter time intervals re.sult as the initia: current approa hes the critical value. Sharper carves als.o retsult from higher values of initial current~ The "breakpoint" shown on t;he circuit characteristic is,; the point at which the time of firing ceases to occur after an exact time intervals For lower values of A the experimental curve represent.s a statistical average of the firings, as noted earlier, In Figo 706 the effect of changing a to a lower value is shown for two values of initial currento Finally, in Figo 7o7, the effect of changing the terminal current in the equivalent circuit is shown. Variations of these factors do not improve the correlation between analytical and experimental data at low ko 7o3 INVESTIGATION OF OTHER NONLINEAR ELEME.NTS The adoption of the parameter L = O/ig appears valid only for values of X greater than 5 x 102 volts/seco For lower'there is a large deviation between experimental and equivalent circuit resultso Since variation of the parameters a, io, and if in the equivalent circuit do not appear to improve the fit to any appreciable degree, it was decided to alter the nonlinear element in some manner in the hope that the experimental data would be simulated to a closer degree for the lower X regionso Take, for example, the nonlinear case where f(ig) = /igZo 21.

The University of Michigan * Engineering Research Institute 10 o -J\ 8 L i = X 10 AMPS BREAKPOINT OF TEST DATA 7 IL (I) LO o 5 ——.-,,o --— TEST DATA FROM FIG. 5.1-QF —91 No. 4092 LOT 8NI L-9 FG. iSo:A INT0 AMPS 0, EQUIVALENT CIRCUIT) io I 169 M IZ~ i = 2 X r ~10'9AMPS o -3 —---------— i0 =3X 10- AMPS LL. 9 LIw -— D —--- io = 5 X 10 AMPS BREAKPOINT OF TEST 1 -- - o 2 3 4 5 6 7 8 9 10 LINEAR RATE OF RISE SIGNAL (X) - v/sec. X 102 FIG. 7.5 RESULTS OF VARYING INITIAL CURRENT (iO) IN NON-LINEAR EQUIVALENT CIRCUIT OF FIG. 7.1 --------------- ~~ ~~22 --------------

The University of Michigan * Engineering Research Institute ~~8 —\ ~^~~~~~ TEST DATA -\ \ w 5 X I0 TO 5 X I0 AMPS;a-=.00066 X, —-v/sec. X 102 1) 3 —-— \- \ —23 w 0 z -o - - 5 X 169 TO 5 X Id AMPS a =.001 5 X 10 TO 5 X 10 AMPS; a =.00066 BREAK POINT ^^ \ 2 3 4 5 6 7 8 9 10 X- v/sec. X 102 FIG. 7.6 RESULTS OF VARIATION OF a IN EQUIVALENT CIRCUIT ----------------------------- 2 5

The University of Michigan * Engineering Research Institute 10 TEST DATA 0 7. w -J\ -9 -8 ------ 5 X 10 TO 5 X 10 AMPS; a =.001 U) W 5 —-- -9 -8 ----- 5 X 10 TO 2 X 10 AMPS; a -.001 >5 -9 -8 - ------ 5 X 10 TO I X 10 AMPS; a =.001 w _ _ _ _ IL w U) 24 r 1 2 3 or \ 102 o \ \ \ EQ CIRU ---------------------- 4 -----------

The University of Michigan * Engineering Research Institute Returning to the linear results of Figo 5o1, it can be seen that the circuit characteristic values can be bounded to a closer degree for L = 107 henries and L = 105 henries. Therefore, the nonlinear element should vary from 107 to 105 henries in the lower regions of x for a closer approximation to the experimental resultso This means that the f(ig) coefficient in Eq. 71o should vary in the following manners _ 10-11 f(ig) = P_ = ---------- = io7 to 105 henries g i2 10-18 to 10-16 to henrie The result of using this function to simulate the time-lag mechanism of the tube is shown in Fig. 708. As can be seen, this nonlinear f'unction seems to approximate the experimental data better in the low X region.o Since the correlation is still. not as good as one might d.esire, a third nonlinear function can be tried. Thus consider the nonlinear case where f(ig) = l/ig3 In this case we let the function f(ig) vary from 108 to 105 henries. -~= 10-19 f(i ) - = - LO = 108 to 105 henries g 3ig 10-27 to 10-24 10 henrie The results of employing this nonlinear function are shown to correlate a little closer for the low X regions. Therefore it seems that, as the inverse power of ig was increased in the f(ig) coefficient, the equivalent circuit data better approximate the experimental resultso Returning to the case employing the first power of ig, the apparent poor simulation at low X region is likely due to the fact that operation at low \ is not according to our conceptiono The computer results of the equivalent circuit suggest that there is another effect present in test circuit operation asLide from the statistical aspect common to breakdown phenomena Thus it may be possible to augment the explanation by considering firing at some "later oscillation" than the first peak for lower values of Xo To substantiate thiL, the problem was considered in this respect as described in the next section. 8. INVESTIGATION OF OSCILLATIONS BY GROUP FIRING DATA In this section the experimental X-vs-t curves are considered in detail for evidence of oscillations occurring across the grid-cathode gap before breakdown for the low values of Xo The data in this section can be obtained. by use of the setup shown in Figo 5olo 25

(1) -1 —__- _____ __ _ - tFIG. 7.8 0 I 32 [ \30 ___________-1____,__ _ COMPARISON OF VARIOUS NON-LINEAR 25 \ - I ------- -- - L 10h^~.- -- - - CIRCUIT RESULTSD 1 3) o 20 —- -- 033 104 0 \.TESTDATATEST DATA - - m o z -1 7 - _\_ \ I. _....LI... I_0 h. 102 10 X ofi-:sec.

The University of Michigan * Engineering Research Institute Prior to this point, the experimental firing data of the circuit were considered a continuous X-vs-t curve of the type shown on Fig. 5o2o For the higher values of X, it was noted that the tube consistently fired at the same time for each particular value of Xo In that region the equivalent circuit is valid with respect to predicting the firing time interval for the circuito However, as A is decreased, it is noted that the firing times are no longer a unique function of \ but appear somewhat random in natureo Therefore, the continuous curve for values after the breakpoint of the gap represented an average picture of a wide scattering of points for each Xo It is also in this region that the equivalent circuit data deviate from the experimental data to a large degreeo This deviation between the two sets of data can be explained as followso The wide scattering of firing time- for low values of \ is attributed to the occurrence of current oscillations. acros the grid.-cathode gap before breakdowno To substantiate the concept, many tube firirngs were conducted to obtain a quantity of data which would enable detection of the possibility of the tube firing after more than a particular time interval. Figure 8ol. shows the results of firing a- tube 100 times at a value of A = 500 volts/sec and an initial. operating current of io = 29,4 x lO09 ampo At this value of X there appears to be a 50-50 chance that the tube will fire during the second group or oscillation of grid cu' rrent i go As can be seen from the graph, there is a distinct interval of time occurring between'the groupsO Therefore, if these groups are regarded a- representative of a particular oscillation of grid current, then the approximate 5-millisecond time interval between groups represents a period of oscillationo Returning to the data presented in Fig. 7t53 as representative of the current oscillation in the nonlinear equivalent circuit, it i'S seen that the oscillations are not equally spaced in. regard, to time as in the linear equivalent circuit o Therefore, it may be that the apparent inductance is not a nonlinear element depending on the instantaneous current ig, but perhaps rather a constant which is dependent on the initial current ioO'This possibility will be considered later in the report From the data of Figo 8.o, the' frequency of oscillation is about 200 cpSo These data represent a group firing for one value of \ and ioo Other values of A were tried with the same operacting current io'with the following results o (1) When relatively high values of X were applied X > 103 volts/sec), more than 95% of the firings occurred during the first groupo This indicated that the firing times were occurring sometime during the rise of current on the first peak of oscillationo As \ was increased, the time interval between t = 0 and firing was shown to decrease o This meant that the firing current was reacetd nearer the beginning of the oscillation than beforeO 27

,______________34 34 |,"t - -FIG. 8.1 GROUP FIRING DATA FOR OF-391 TUBE 321 A l t {; | t'313 LOT AC5 = 30 X 500 v/sec. CIRCUIT AS SHOWN IN FIG. 2.1 X = 500 v/sec. 28e l -----------— i Igo= 2.4 x I0' AMPS.,_______2 6 _ _ _ _ _ _ _ _ _ _ _ _ ___________ y _ | ( O P E R A T I N G C U R R E N T ),/^: | { | / | 6 | Rg, 3.1 x 10' ) a< 24 —---------------------------------- ---- R g2 I1.I x 1 0.a,~22 __________ ____ _ _^ — _____~ Cn: IOn/L f'22. Eb = 180 v. 20 ---— __ - NUMBER OF FIRINGS = 100 19 // 9 -REPRESENTS PROBABLE TIME O3 18 I — /// —-------- I ~ AT WHICH A PARTICULAR E" y//7^T //, OSCILLATION OCCURS 1 16 14 10 6^'/ - _ _ 2 2 __ 1.0 2.6 7.8 13.0 17.0 TIME OF FIRING - MILLISECONDS

The University of Michigan * Engineering Research Institute (2) As k was decreased to about X = 800 volts/sec, the percentage of firings occurring during the first group dropped to 71. The percentage of firings in the second group increased slightly to 16. This shows that the probability of the tube firing on the second oscillation was increased as X was decreased (3) At X = 500 volts/sec, the percentage of firings occurring during the first group dropped to 21, while the second group increased to 44. Also at this low value of X, evidence of a third and possibly a fourth group appearedo This further indicated that at very low values of X, near the forbidden region, a number of current oscillations might occur before the grid-cathode breakdown current is reached. This apparently explains. the sharp deviation bet'ween the analytical. and experimental data at the low values of X, for the analytical data were taken on the basis of the tube firing dauriing the first oscill.ation forall Xo Therefore, if the analytical data from the equivalen;t circuit are to approximate the experimental data, the presence of the later oscillations. must be considered (4) As X was decreased further,'the percentage of firings occurring in group one decreased rapidly, while the percentage of firings in groups three and four increasedo Finally, at low X (X < 350 volts,/sec), the tube ceased to fire O (5) As the previous values of X were varied and the grid.-cathode cnaracteristics remained constant, it wa,. noted, that the interval between all groups remained constant. From the preceding information it is seen that' the co:ntinuous type of X-vs-t curve cannot be considered valid for representing accurately the time at which the tube may fire as a function of.o In order tha tthe oscillations can be represented by the X-vs-t curves and thus give an acceptable representation of the circuit s dynamlic performance, a curve of the form shown in Figo 8,2 should be drawno This type of presentat;ion gives. complete information regarding oscillations, from which we can construct a more accurate t.ype of,X-vs-t curve. Then assuming that the current oscillation acrosos the grid-cathode gap results from a constant inductance paramester as indicatted by the exaperimeental results, -the X-vs-t curves should be drawn in the follo.'ing maIner: POINT (1) Assume firing a tube a number of times at a relatively high -value of X. Of these firings, the greater percentage occur during the first group or oscillation, as indicated by the heavily cross-hatched bar at ~4 as sh own by Fig. 8,2, The time for firing of this predominant group is plotted as point (1) on the X-vs-t graph shown in Figo 804 where Xc = k4 of Figo 8,2. Then assuming that the grid- athode cirrent ean be represented a sine wave of exponentially increasing amplitude, due presumably to the.linear induc29

The University of Michigan * Engineering Research Institute | |43RD OSCILLATION OR GROUP z 3 ~T~LLA ~| l 2 2ND OSCILLATION t,L / p 1IST OSCILLATION I ^ s //I I Xfr XI VARIOUS GROUPS / Xc2 FIG. 8.3/ / V -XA / BREAKDOWN GRID CURRENT 0/ TIME FIG. 8.3 CURRENT VARIATION OF GRID CIRCUIT ASSUMING LINEAR COMPONENTS 30

The University of Michigan Engineering Research Institute tfr 0t t I~ Et4 Xfr Xa b Xc FIG. 8.4 VERSUS t CURVE tance parameter, we can explain point (1) in the following manner. At t = 0 the grid-cathode current is at the bias value of io. After the application of the %t signal to the circuit, this current rises to a value depending on X and the components in the circuit. Since the value Xc = %4 is large enough to cause the current to increase to the value of grid-cathode breakdown current if, during the first oscillation the tube fires at some time to as indicated in Figo 8o3o However, as Fig. 8.2 shows firing does not always occur during the first cycleo Thus one must consider that the curve labeled Xc, Figo 8.3, represents the most likely situation under the A = X4 conditions. This seems a reasonable conclusion, for the curves drawn in Figo 8.3 are only approximate representations of a largely statistical process. POINTS (2) AND (3) LL I I > %2, groups one and two become almost equal in regard to percentage of firings. Therefore, to describe this point on the graph, we define this value of X as equal to some Xbo By knowing the times at which groups %2 and 3 appear, we locate the points (2) and (5) on the X-vs-t curve. -.. II-.'. Xtr, Xc Xb Xc FIG. 8.4 X VERSUS t CURVE tance parameter, we can explain point (1) in the following manner. At t = we find that this is just the value the grid-cathode current is at the bias value of io. After the application of the it signal to the circuit, this current rises to a value depending on t and the components in the circuit~ Since the value kc = k4 is large enough to caus 8~3~ However, as Fig. 8.2 shows, firing does not always occur during the first cycle. Thus one must consider that the curve labeled Xc, Fig~ 8.3, represents the most likely situation under the X = X4 conditions. This seems a reasonable conclusion, for the curves drawn in Fig. 8.3 are only approximate representations of a largely statistical process. POINTS (2) AID (3) Therefore, to describe this point on the graph, we define this value of X as equal to some kbo By knowing the times at which groups X2 and X3 appear, we locate the points (2) and (3) on the X-vs-t curve. Returning to the linear circuit assumption, we can describe these points in the following manner. At X = Xb we find that this is just the value 51

The University of Michigan * Engineering Research Institute lation. However, since it is so close that the current might not reach if for some reason on the first oscillation t = t2 it would reach if sometime later on the second oscillation t = t3o Therefore, since the tube fired 50% of the time on the first oscillation and 50% of the time on the second oscillation, we can represent these values of X = Xb by points (2) and (3), POINTS (3) AND (4) In this range the percentage of firings during the second oscillation *will prevail. The time at which this group occ'urred would be represented by time from t3 to t4o POINTS (4) AND (5) These points will represent the value of X = Xk where 50% of the firings occur on the second, oscillation and. 50% on the third osc illation. oThese points are explained as points (2) and (3)o X1 < Xa < <2 POINT (6) At this value of X, the grid-cathode gap ceas es: to'break dovrn, Va.,ues of. less than Xfr define the "forbidden: region." It is interesting to note that by ass:uming linear operat-ion, Fig. 803 can be constructed from Fig. 8o4, Then, from the curre:nt wave fo:rm of Fig. 803, the values of inductance, resistance, and capacitance neeeded. in a linear circuit to produce this wave form can be calculated Therefore, by obtaining the experimental data in the form of Fig, 802, andd then co:nst:ructing Figo 804, a linear circuit could be determined to dup:iicate the experimental.:reswtst, It is observed that the intervals occurring bettwreen group firings are constant. However, from the nonlinear equivalent circuit dairta sho'on in Figo 75, the period between current peaks is not con+Kst:ant for the varioutsU values of Xo Moreover, the time between two successive oscillations is quite large with respect to the experimental observed effectso Therefore,'the linear equivalent circuit's oscillations seemed to correlate with the experimental group firing data much more closely than the nonlinear circuit where L = C/igo To investigate this point further, additional data were obtained to illustrate the dependence upon the initial current io as rwell as Xo.It was noted that the interval occurring between two successive groups decreases as the initial current io is increased, indicating that a relationship exists between the time of current buildup across the gap and. the initial operating cur32

The University of Michigan * Engineering Research Institute rent. Furthermore, since the period of oscillation, as indicated by the group intervals, changes with io, there exists a definite relationship between the initial current and the frequency of oscillatory current'buildup in the gap. This suggests that the inductance element used to simulate the time lag of current buildup through the gap has a constant value dependent primarily on the magnitude of the initial operating currento It was thought that since the time required to obtain the group firing data was excessive, some other means of investigating the oscillation phenomena would prove beneficial, From the standpoint of tube life, an experimental setup which would require fewer firings would also be desirable. Thus, in view of the relationship which seemed apparent between the period of oscillation and the operating current io, we were led to try to investigate these oscillations by observing some resonance phenomena~ 9o GRID-CATHODE RESONANCE EFFECT AS A FUNCTION OF THE INITIAL OPERATING CURRENT io In the preceding section, it was noted that the time interval between group firings is a function of the initial grid-cathode current io, denoting that the frequency of oscillation of the grid-cathode current i is a function of io. Then since the time interval between any of the groups for a particular X seems constant, this dependence further suggests that the inductance parameter has a. constant value whose magnitude is a function of the initial current io, rather than a nonlinear inductance whose value varies with the instantaneous values of grid-cathode current ig To investigate this dependence, an experimental procedure was adopted using an alternating input signal for the driving voltage instead of the Xtsignal. The resonant frequency of oscillation for various operating currents io could then be determined directly by noting at what input frequency the tube becomes most sensitive. The setup shown in Figo 9ol was used, and the initial operating conditions were adjusted equivalent to those of Fig 8ol.o The rms alternating voltage ea was then applied to the grid circuit as shown, and the frequency increaed from some low value until the tube fired, the firing determined by a signal appear ing across Rxo Further increase in frequency resulted in continuous firing of the tube, but finally, at a still higher frequency, the tube ceased firingc The alternating voltage ea was then decreased and the procedure repeated. The result of this type of experimentation for the same tube and operating conditions of Fig. 81l is shown in Figo 9~20 A distinct resonance occurs at about 200 cps, which is the frequency of oscillation calculated from Figo 81 for this particular value of ioo Thus, the resonant frequency of the grid circuit found by group firings and by applying an alternating voltage to the grid circuit correspondo Figures 9o3 and 9o4 illustrate the dependence of the apparent resonant frequency on the initial currento 33

RL m QF-391 Cg 10^+ /<^^t Rg2 c R gR 2 R- I Qx KE1THLEY \ n ELECTROMETER AUDIO- SCOPE E b OSCILLATOR SCOPE m Rx FIG. 9.1 SETUP FOR INVESTIGATING GRID-CATHODE RESONANCE:3 Vi (A <T> FIG. 9.1 SETUP FOR INVESTIGATING GRID-CATHODE RESONANCE ^~~~~~~C

0.7 0.6 = 2 R AREGION OF CONTINUOUS FIRING 0.5 z - 0.4 0 LL ______ = 2.4 x 1"~(OPERATING CURRENT) ~ R 2 E RESONANCE OF GRID CIRCUIT 20 50 100 200 300 400 600 800 100 0~ ~ ~ ~ o _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _RA I N 1UREN..0. 501020 0 0 00 80 10 ~~~~~~~~~~FEQU=ENC = cp VLS t:'

The University of Michigan * Engineering Research Institute z S REGION OF o. 9. FIRING / o |e R w II II CURRENTS i AND i z I II I I i fT fl f2 fR f3 f4 SIGNAL FREQUENCY FIG. 9.3 TYPICAL RESONANCE CURVES FOR OPERATING CURRENTS iI AND i2

FIG. 9.4 7 ___l _- 4 RESONANT FREQENCY OF GRID CIRCUIT - l-l_________ ____ AS A FUNCTION OF OPERATING CURRENT ______ (CRTICAL CURRENT) 6 TUBE - QF-391 313 LOT AC5 ___ _________ Eb 180 VOLTS <...5.,...._ z Li. "D' 3 z w 0 100 200 300 400 500 RESONANT FREQUENCY- cps

The University of Michigan * Engineering Research Institute 10o GRID-CATHODE RESONANCE DUE TO SHUNT CAPACITY Cgc The interesting resonant effect discussed in the preceding section is clearly representative of the total equivalent circuit as illustrated in Fig. 9.2, To obtain more information concerning the grid-cathode resonance of the tube, as contrasted with tube plus circuit, a revised setup eliminating the capacitor Cg was used, as shown in Fig. 10ol, and the corresponding (Fig. 10.2) equivalent circuit, With this setup, the resonance effect seems to be due only to the grid-to-cathode capacitance, 1.5 mmfd, the stray capacitance, and the inductance effect of the tube. Since the resistance shunting the Keithley electrometer and the 109 ohms are in series with the audio oscillator, the sensitivity of the system is not as high. However, our interest lies in the resonance phenomena of the gap and the sensitivityo Eg was adjusted until the grid-cathode gap was operating at some initial current io, a value chosen to keep the tube operating on the negative slope of its static characteristic (see Appendix IV for characteristic)o Resonance data were then obtained following the procedure of Section 9. Figure 10.3 shows three somewhat irregular resonance curves obtained for different values of operating currento Curves at other values of current were also taken, but for the sake of clarity only three are shown here, It is noted -that there appear two distinct resonance points for large values of operating current io, and that the frequency is much higher than that found from the previous'setup (Figo 9.2). Circuitwise, the only change is substitution of the series-connected Keithley for the Cg = 10 mm capacitor, and addition of the 109-ohm resistor Insufficient time prevented justification of the magnitude of the frequency change and the reason for a double-peaked curveo It is clear that the effective capacitance in the circuit was reduced by the above circuit change which is consistent with the frequency increase, However, it is not apparent why the capacitance change should be the factor of 100 required for a factor of 10 frequency increase. The presence of the 109-ohm resistors seems to have the effect of isolating the Keithley input capacitance (n 6 mm) from the grid, for the capacitance reactance is a factor of 100 less than the series resistance. Leaving this point unresolved, the assumption is made that the only effective capacitance in the grid circuit is the Cgc = 1o5 mmfd, and that the resonance effect observed involved the gap inductance o Figure 1004 illustrates the dependence of frequency upon initial 38

The University of Michigan * Engineering Research Institute 6.8 x 106, 9 109 F 391 10 I0 0 eL CX AUDIO KEITHLEY 108 TO SCOPE OSCILLATOR ELECTROMETERS 10 ov Eg Et' 180 V. FIG. 10.1 SETUP FOR INVESTIGATING GRID RESONANCE 109 o19 r"' —S L=f(io) / INPUT CAPACITY OF- | KEITHLEY ELECTRO- __ METERS C E6 /_Lf CI. FIG. 10.2 GRID-TO-CATHODE DYNAMIC CIRCUIT 359

3.5...... I i.I io 140 x 10 AMPSAMPS i = 20x I0 AMPS =3 vr 2.5 u_ I I I I \ f I' o+ > LL w2.O w LJ Z (1 0 > 1.5 1 * (I) FIG. 10.3 1.0-::3 RESONANCE CURVES OF QF-391 NO. 7295 LOT 8NI - FOR VARIOUS OPERATING CURRENTS - - PARALLEL RESONANCE Cgc: 1.5//af 0.5- - C 0' 100 200 400 600 800 1000 2000 4000 6000 8000 10,000 RESONANT FREQUENCY -cps

FIG. 10.4 GRID-CATHODE RESONANT FREQUENCY |. VS. OPERATING CURRENT (TUBE -OF-391 NO. 7295 LOT 8NI) = 3000,, 2800 — __, us Cgc = 1.5/zft I / 0. 2600. I C 0 2400 z u 2200. uL_ 2000Z 800.. f= Olio + 800.. 1600... 1400 - --—...... 1200... 6 8 I 0 12 14 16 18 20 -9 OPERATING CURRENT - (i0 X 10 AMPS)

The University of Michigan * Engineering Research Institute current From the curve: fr = 1011 io + 800 o (lOol) On the basis of the assumption made above, in regard to Cgc, a relationship can be written: 1 fr = 1011 io + 800 = r2J-g (10o2) 11o CONCLUSIONS AND SUMMARY As a result of this research work and the preparation of this report, several conclusions seem clear, although additional investigations should be carried out to establish firmly their validity and to "fill in" some missing details o lo The delays observed in firing of a small gap under the influence of an applied signal, such as a ramp-function voltage, can be reasonably accurately simulated withan equivalent circuit which employs capacitance, inductance, and negative resistance The nature of the inductance is not completely clear, but present indications are that it should be essentially constant, and of a magnitude determined by the initial Townsend current flowing in the gaso 2o The desired observations (employing an equivalent circuit) can best be made using an analog computer, which facilitates simulation of the negative resistance o Normally encountered minor variations of the computer elements seem to contribute to simulation of the statistical "firing effect"' which is observed with the test circuit employing the tubeo 3o The nature of the current buildup in the gap is oscillatory, and although satisfactory prediction of the frequency has not been realized, it appears to be predicted by the static capacitances of the circuit and the inductance mentioned in (1) above 4o Static characteristics of subminiature tubes employing currents in the millimicroampere range can be readily and satisfactorily determined by the techniques employed in the course of this research program. 42

APPENDICES

The University of Michigan * Engineering Research Institute APPENDIX I ANALYTICAL SOLUTION OF A SERIES CIRCUIT EMPLOYING A NONLINEAR INDUCTANCE AND A NEGATIVE RESISTANCE It is noted from investigating the nonlinear equation shown in Section 4 that a similar circuit omitting the capacitance can be solved analytically. The following is a solution of the circuit shown in Figo A-I-lo The circuit equation is di Ri E (A-I-l) Rearranging Eqo A-I-1, we obtain: di i2 = i (A-I-2) dt a a ~i r\~dt Ai i(Rdi f o dt c(A-I-3) 0 i(Ri+E) - Jo: - i 1 io t 1 &n I_ n- 1 Q 1~ _ 1 (A-I-4) E Ri+E E Rio +E a Et R +E i e ( = E (A-I-5 ) Finally Eqo A-I-5 can be solved for i. Et io ea i = t (A-I-6) 1 - R io(e- O 1) E Using the values of the parameters as 44

The University of Michigan *Engineering Research Institute io. L a R, i oE o- t'O FIG.A'-I CIRCUIT CONTAINING NON-LINEAR INDUCTANCE AND NEGATIVE RESISTANCE / / / / -7 / a0 / ~Q= /:~~/ / 3, 01 I.-o. // -3 10 sec. (FOR a =. ) So lX 105 sec.(FOR a =.001) t ---- G.A —------------ R 45

The University of Michigan * Engineering Research Institute L = a/i, a =.1, o01, o001 R = -108 Q E = 100 volts, and io = 10-9 amp (initial condition at t = 0)o From the denominator of Eq. A-I-6, it can be seen that i goes to infinity when Et R io (e' - 1) = 1 (A-I-7) E For the values of the parameters given in the circuit, the time for the current to reach infinity is t = 0.0691 sec o (A-I-8) The results of Eqo A-I-6 are plotted on Fig. A-I-2. 46

The University of Michigan * Engineering Research Institute APPENDIX II GRAPHICAL METHOD OF SOLVING A NONLINEAR DIFFERENTIAL EQUATION In Section 4 the equation which described the current variation in. the equivalent circuit is nonlinear, and a solution was not determined by ordinary analytical meanso However, it was found that it could be solved by the following graphical technique, which is referred to as the phase-plane methodo To illustrate it, we will consider the equation describing the test equivalent circuit Et R io (e- - 1) = 1 0 (A-II-1) We now let a = 0001, R = -108, C = 1011 farads, i = 10-9 I, t = 10-3 T, and A = 103 volts/sec. Substituting these values, Eq. A-II-1 becomes 1 d- -_ 10- + 10 /IdT = T o(A-II-2) I dT To solve Eq. A-II-2 by the phase-plane method, we first differentiate with respect to T to eliminate the integral term and the T term on the right-hand side of the equation. d101 + 10 I = 1 (A-I-3) I dT2 12\d T/ lI3dT Any nth-order differential equation can be written as n separate first-order differential equationso For example: 47

The University of Michigan * Engineering Research Institute dny d3y dy dy n C o3 0 + 2 o +d + oy = 0. (A-II-4) dxn dX;3 dx2+ Let v =: dx dv z = _ dx dz w = -, and n = 2 etco d2 dy Ol Ua2 2 y = d (A-II-5) IV2 Co a d + Clv + CoQy = O (A-I-6) dv a O Tudx E.cbe e-rsne by th tw is e (A-I I-7 ) - — v -- y = o ( Thus Eq, A-II-5 can be represented by the two first-order differential equations (A-II-6 and A-II-8)o Therefore, returning to Eqo A-II-3 and letting V: d, (A-II-8) dT we get 1 dV lO- V + 1 = 1.(A-II-9) I dT IThen since dV dV/dT dI dV dI dl/dT ~'dT =- d - (A1- ) we can now substitute Eqo A-II-10 into Eqo A-II-9 and eliminate the independent variable (T)o 48

The University of Michigan * Engineering Research Institute V dV V2 VV V -- 10o- V + 101 = 1 (A-II-11) Rearranging Eq. A-II-11, we have dV V -1 10-1 I2 I d = 10I V + (A-I Equation A-II-12 is a differential equation involving only the current and the rate of change of current, independent of time, When the initial values of V and I are known from the initial conditions of the original equation, the dV/dI term can be computed. The value of this term (slope) is then plotted (see Fig. A-II-1) on a graph of V vs I. Through some point IV1 which this line intersects, another value of dV/dI again is computed from the new values of il and V,. This process is continued until a so-called phase-plane curv of V vs I is finally drawn, The accuracy of this plot is determined by the interval chosen between successive points of calculation; the smaller this interval, the greater the accuracy. After establishing a phase-plane plot of the solution of Eqo A-II-12, the next step is to find the value of T for each value of current I and rate of change of current V. From Eqo A-II-8 we get T = J dI. (A-II-13) 0 V Therefore, the time interval from Io to I is represented by the area under the 1/V-vs-I curve between these two values of I. By finding enough values of time for different values of current, we finally obtain the current-vstime curve which is the solution to Eqo A-II-30 Since we are interested only in finding the time it takes for the current to increase from its initial value to some arbitrary value, it is not necessary to find a number of points on the current-vs-time curve. Therefore, in summary, the step-by-step procedure in solving a nonlinear differential equation by this method is: (1) Normalize the given equation so that the phase-plane plot lies on the graph paper, This is done by the same method used to find the computer solutio (2) Eliminate all integrals by differentiation, 49 -

The University of Michigan * Engineering Research Institute V dI dT dV PLOT OF =F (I, V) dI V2 VI Io II I2 CURRENT (I) FIG.A-II-I FIGURE SHOWS METHOD OF CONSTRUCTING PHASE-PLANE PLOT. V I=10 4 20!-V -VS I CURVE 3 15 (V) VS(I) PHASE-PLANE CURVE (X=103 VOLTS/SEC.) 2 - 10 5 I I - 5 I O Io1. /-T V. dIO 1.4 SEC. 00.-o I ^^770 0 2 4 6 8 I=10 CURRENT -(I) FIG. A-I1-2PHASE -PLANE PLOT OF EQUATION 12.3. ( X = 103 ) 50

The University of Michigan * Engineering Research Institute (3) If the given equation is nth order, then represent it by n firstorder differential equationso (4) Finally, eliminate the independent variable by appropriate substitutions o (5) From the equation obtained in step 4, construct a phase-plane ploto (6) After the velocity-vs-distance or (dI/dT = V)-vs-I curve of step 5 is drawn, the (1/V)-vs-I curve is constructed (7) The value of the independent variable for each value of dependent va iable is then found by calculating the area under the (1/V)-vs-I curve of step 6. A phase-plane plot of Eqo A-II-3 is shown in Figo A-II-2 for \ = 103 volts/sec. Also, the time for the current I to increase from the normalized values of Io = 1 to I = 10 is found, corresponding to a current change of io = 109 amp to i = 108 amp of the original equation (A-II-l)o 51

The University of Michigan * Engineering Research Institute APPENDIX III ANALOG COMPUTER TECHNOLOGY Figure 7.2 in Section 7 represented an analog computer setup used to simulate Eq. 7~.2 The following is a brief description of the setup. Functions of various symbols (see Fig, A-III-I): (1) Integrater: The output voltage is equal to tthe time integral of the input voltage, (2) Adder: The output voltage is equal to the sum of the input voltages multiplied by some constant usually greater than 1 (5) Potentiometer: Used when voltage is to be multiplied by a constant less than 1. (4) Servo-Multiplier: An electromechanical device which is capable of multiplying two variables, (5) Photo-Former: Consists of a cathode-ray tube with a short persistence screen, an outline (opaq-ue on one side of the line) of the desired function in front of the screen, and a photo-multiplier tube in front of the screen. The cathode-ray spot in normal operation is positioned along the x axis by a voltage representi;ng the input or forcing function; the. output of the photo-multiplier tube is connec.ted through high-gain amplifiers to the vertical deflection plates of the cathode-ray tube, and so dr:ives the spot to the line separating the opaque and clear portions of the outline, i.e,, the function shape. The voltage required. to hold the spot on this line is a direct measure of the y value of the function at that point. In simulating Eqo 7.2 by computer, the dependent and independent variables are first normalized into computer units. The computer setup is then connected as shown in Fig. 7.2 with the appropriate initial conditionso The Xt voltage is then applied to the setup and the voltage representing the variable (i) is recorded. Figure 7.3 shows a typical representation of the variable from a computer solution. 52

The University of Michigan *Engineering Research Institute INTEGRATOR: eI- e-o- eo - e, dt ADDER: el -n eo 2 eOUT, -K? MULTIPLIER: el e2 e e0 e 100 PHOTO- FORMER e, f(e,) e e (e)' -1- ~t;^'^ -- ^eo eo =fe,) FIG. A-III-I ANALOG COMPUTER TECHNOLOGY 553

The University of Michigan * Engineering Research Institute APPENDIX IV GRID-TO-CATHODE STATIC CHARACTERISTICS In Fig. 353 of Section 3 a sketch of the static characteristics of the grid-to-cathode gap of a typical QF-391 gas tetrode is shown. These static characteristics were obtained with the setup shown in Fig. A-IV-lo The values of resistance used in the second grid circuit will. affect the static characteristics of the first grid., but for the purposes of this study, was held constant. To obtain the static characteristic of the first grid of a tube, we proceeded in the following manner (Fig. A-IV-I)o Eg was increased until about 1 x 10-9 amp were indicated by the Keithley electrometer (io). Then E2 and E1 were adjusted until zero current flowed through this shunt circuito The value of io was thus the true grid current, and the E2 + El was the grid-to-cathode voltage. We then proceeded increasing Eg and io, keeping ix equal to zero. In this way the grid-cathode static characteristic was obtained. The maximum error in reading io was estimated to be ~.01. x 109 ampo The minimum current which could be detected by ix was ~ i.012 amp. Therefore, the maximum error in determining eg was ixR = 1012? x 1010 =.01 volt. This accuracy was reasonable for our purposes. All runs were made in a shielded. box using Keithley battery-poowered electrometers and Victoreen Hi-Meg resistorso The fact that consistent, reproducible results were obtained lent confidence that the resulting curves were meaningful. A typical plot is shown (Fig. A-IV-2) where DOFL data (for the same tube) are shown for comparison. The agreement is remarkably good, considering that the curves were taken in different laboratories independently. -----------------— 5 4 —----------

6.8 x 1O6 OF 331 ---— VW- j C IO9^ i 91 ^-^ 10 tO. < e r^~~~0 ~ 9^ 0 I 08di e'O 9~~~~~~~e 10'~^ < (ix Cf9 w'/Eg -— Eb 180 VOLTS Vn9 E 2 E %J1 mn I -~~~~~~~~~~~~~~~~~~~3 Er 5 <^ cu FIG.A-I-I GRID-CATHODE STATIC-CHARACTERISTIC SETUP io - INDICATED BY KEITHLEY ELECTROMETER i- INDICATED BY KEITHLEY ELECTROMETER E2-0-150 VOLTS E,-0-4 VOLTS V- VOLTMETER

92.2 FIG. A-IV-2 COMPARISON OF STATIC CHARACTERISTICS 92 -—. —- --- --- --— 1 — 1 I OF OF-391 NO. 7295 LOT 8NI 92.0 _ / --- --- --- --—.- O DOFL'S DATA " —- _ /'< 0I 91.9, I N'I ~Y ~T ---- ---- ---- ---- B..S OAJ -/ ^ ^ ^^ —-- ---- ---- ---- ---- ---- ----- *) w 9 I I 0 2 4 i6 10 12 14 16 s 2 2 24 261 0 2 3 9 1 i - - ^II \ P IA CE