ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR Progress Report 1 PART I. SOME-STUDIES OF TRANSISTOR MEASUREMENT TECHNIQUES AT HIGH FREQUENCIES P. M. Shaler,' Jr. PART II. GAS-TUBE STUDY W. ~Po Brown Approved by: N. Wo Spencer Project Supervisor ERI Project 2269 DEPARTMENT OF THE ARMY DIAMOND ORDNANCE FUZE LABORATORIES CONTRACT NOo DAI-49-186-502-ORD(P) -194 WASHINGTON, Do C. February 1958

The University of Michigan * Engineering Research Institute PERSONNEL EMPLOYED DURING THE PERIOD OF THE REPORT P. M. Shaler, Jr. Project Engineer Three-quarter time, student R. G. DeLosh Student Technician Part time W. P. Brown Student Engineer Part time N. W. Spencer Supervisor Part time W. G. Dow Consultant ii

The University of Michigan * Engineering Research Institute TABLE OF CONTENTS Page LIST OF FIGURES iv PART I ABSTRACT 1 OBJECTIVE 1 1. INTRODUCTION 2 2. CHARACTERISTICS OF LOW-FREQUENCY TRANSISTORS 2 3. SMALL SIGNAL A-C EQUIVALENT CIRCUIT PARAMETERS 5 3.1 The Equivalence of the T and T Representation 4 3.2 A Method of Determining Y and Z Parameters If the H Parameters Are Known 4. LOW-FREQUENCY CIRCUIT REPRESENTATION OF THE TRANSISTOR 6 4.1 High-Frequency Circuit Representation of the Transistor 7 5. NONLINEAR OPERATION AND OSCILLATORS 10 6. FORMULATIONS OF INPUT AND OUTPUT IMPEDANCE 11 7. OUTPUT IMPEDANCE OF THE GROUNDED BASE AND GROUNDED EMITTER CONFIGURATIONS 14 8. FURTHER DEVELOPMENT OF THE OUTPUT IMPEDANCE OF THE GROUNDED BASE CONFIGURATION 20 9. THE GROUNDED COLLECTOR CONFIGURATION 24 10. FUTURE PLANS 24 11. REFERENCES 25 PART II GAS-TUBE STUDY 26 Cold-Cathode Breakdown Study 26 iii

The University of Michigan * Engineering Research Institute LIST OF FIGURES No. Page 4.1 Low-frequency equivalent circuit of the grounded base configuration. 6 4.2 Admittance forms as treated by Middlebrooko 7 4.3 Equivalent circuit of the grounded base configuration including rb", rb', and Cc. 8 4.4 High-frequency equivalent circuit for the grounded base configuration. 8 4.5 High-frequency equivalent circuit for the grounded emitter configuration. 9 4.6 High-frequency equivalent circuit for the grounded collector configuration. 10 5.1 A transistor oscillator circuit. 11 6.1 The general four-terminal active network. 11 7.1 Circuit for measurement of Zol and Z02~ 14 7.2 Curves of |Zoll /G1 and IZo21 /G2 versus frequency for a type 2N128 transistor. 15 7.3 Circuit for measurement of Zo$ or Z02* 16 7.4 Curves of IZo21 /_2 and IZo31 /_3 versus frequency for a type 2N128 transistor. 16 7.5 Vector plot of a versus radian frequency. 17 7.6 Vector plot enabling the representation of (l-a). 17 I__________________________________ iv

The University of Michigan * Engineering Research Institute LIST OF FIGURES (Concluded) No. Page 7.7 Enlarged view of Fig. 7.6 showing the conditions for w = w.. 18 8.la Representation of the transistor grounded base output impedance. 20 8.lb Equivalent representation as seen by RX meter. 20 8.2 Curves of Rp/rbv and Cp/Cc versus c/2O. 22 9.1 Representation of the grounded collector input impedance. 24 -— v —--------

The University of Michigan * Engineering Research Institute PART I ABSTRACT Studies related to measurements of the external properties of transistors have led to techniques for the predictions of current gain 3-db cutoff frequencies, and of internal properties associated with a suggested equivalent circuit. OBJECTIVE The purpose of this investigation is to find methods for determining the internal properties of transistors designed for operation in the VHF and UHF frequency ranges through measurement of their external properties, 1

The University of Michigan * Engineering Research Institute 1o INTRODUCTION This progress report refers to the beginning phase of a new task under Contract No. DAI-49-186-502-ORD(P)-194, with the purpose of investigating instrumentation requirements and devising suitable instrumentation techniques for the experimental study and evaluation of very-high-frequency semiconductor devices. A particular objective is to evaluate very-high-frequency transistors in terms of measurements of their externally measurable parameters as represented, for example, by an equivalent circuito Work has been done which will first enable the measurement and understanding of transistors whose limitations become apparent at frequencies much lower than those in the VHF range. Following this, investigations will be confined largely to frequencies above 100 mcs. Thus this report summarizes these efforts, which include associated analytical review of low-frequency equivalent circuits, H parameters, some experimentation with oscillators at higher frequencies, and finally measurements of some input and output impedance under various conditions, 2o CHARACTERISTICS OF LOW-FREQUENCY TRANSISTORS To gain familiarity with the external characteristics of low-frequency transistors, an experiment was carried out involving the measurement of impedance and of gain versus frequency. Using a grounded base configuration, a relationship between a and frequency was obtained which verified that the gain cutoff curve had a slope of 6 db/octave (at about 300 kcs). Then RinGB (the input resistance of a grounded base amplifier) and its dependence on Ie were measured. Also, by noting resonance effects, measurements were made of collector capacitance Cc, as a function of collector voltage VCB, which verified that an alloy junction transistor has an effective capacitance inversely proportional to the square root of VC. Using a grounded emitter configuration, a curve was prepared of the grounded emitter current gain 3 versus frequency for a transistor, and the 3-db gain drop-off was observed to occur so that: fp n (l-a)fa. (2.1) The input resistance of a grounded emitter amplifier RinGE was then measured to verify that: in Rin GB (2.2) GE (1-a) ------------------------— 2 —------------

The University of Michigan * Engineering Research Institute 3. SMALL SIGNAL A-C EQUIVALENT CIRCUIT PARAMETERS Recognizing the need for a mathematical model to represent the transistor, reference was made to a paper' concerning the general active a-c equivalent circuit. This paper contains a discussion which deals with loop- and nodal-derive equivalent circuits, and terminates with tables of transformation formulas useful for converting any loop-derived equivalent circuit into any T configuration, or any nodal-derived equivalent circuit into any 3t configuration. The equations of interest concerning T equivalent configurations have the form: V1 = Z1 I1 + Z12 I2; (351) V2 = Z21 Il + Z22 12 (3.2) The quantities Z11, Z22, Z12, and Z21 are open-circuit impedances; for example, Z11 is equal to the ratio V1/I1 with the output open-circuited, i.eo, I2 = 0Zll = L (for I2 = 0), (3 3) I, Z12 = I (for I1 = 0),(.4) I2 tZ1 = -V2 (for I2 = O), and (5.5) II Z22 = 2 (for I, = O). (3.6) I2 If one knows the four open-circuit parameters Z1, Z12 Z21, and Z22 for a grounded base transistor configuration, one can use the transformation tables to find a new set of open-circuit parameters for either a grounded emitter or grounded collector configuration. The equations concerning ET equivalent configurations are: II = Y11 V1 + Y12 V2 and (5o7) 12 = Y21 V1 + Y22 V2. (3.8) The quantities Y11, Y12, Y21, and Y22 are short-circuit admittances: 5 5

The University of Michigan * Engineering Research Institute Y11 = I1 (for V2 = 0), (39) V1 I1 Y12 = (for V1 = 0), (3.10) V2 Y21 =. (for V2 = 0), and (3.11) V1 12 Y22 = (for V1 = 0). (3.12) V2 As before, once the short-circuit admittance parameters are known for one of the transistor configurations, transformations will allow the new parameters to be obtained for either of the remaining configurations. 351 THE EQUIVALENCE OF THE T AND Tc REPRESENTATION The equivalence between the T and -A representations of a particular transistor configuration can be shown as followso If the open-circuit parameters for a T configuration are known, the short-circuit parameters for an equivalent it of the same active circuit are: _xi = -, z22,, Y11 = Z22 (313) Z11 Z22 - Z12 Z21 Y21 = -21 (3.14) Z11 Z22 - Z12 Z21 Y12 = and (3.15) Zll Z22 - 212 221 Y22 = 1 (3.16) Zll Z22 - Z12 Z21 If the short-circuit parameters for a it configuration are known, the opencircuit parameters for an equivalent T of the same active circuit are: Zl Y22, (3.17) YI Y22 - Y12 Y21 ________________________4 -----------

The University of Michigan * Engineering Research Institute z21 = Y (3.18) Y11 Y22 - Y12 Y21 Z1 = 2, - Y - -, and (3519) Yll Y22 - Y12 Y21 Z22 =.1L --- - * (3.20) Y11 Y22 - Y12 Y21 352 A METHOD OF DETERMINING Y AND Z PARAMETERS IF THE H PARAMETERS ARE KNOWN Because of ease of measurement, H parameters are often used to represent the "black box" active circuit. The equations of interest are: V1 = hlIl + h12V2, (3.21) I2 = h2111 + h22V2, (3522) hll = VI (for V2 = 0), (3.23) II V1 h12 = (for I, = 0), (3.24) V2 h21 = -I (for V2 = 0), and (3.25) I1 h22 = (for Ii = 0) (3.26) V2 If the H parameters for an active circuit configuration are known, the opencircuit parameters for an equivalent T of the same configuration are: Z11 = h1 - h 2h2 (3.27) Z12 = 12, (3.28) h22 Z = 21-and (3.29) h22 Z22 = 1 (35.530) h22 5

The University of Michigan * Engineering Research Institute If the short-circuit parameters for an equivalent T of the same configuration are preferred: Y1 = 1, (3.31) hll hi2 Y12 h (3.32) hll Y21 h21 and (3335) hll h12h21 Y22 = h 22 h (334) 4. LOW-FREQUENCY CIRCUIT REPRESENTATION OF THE TRANSISTOR A low-frequency equivalent circuit is shown in Fig. 4.1. It has been assumed that the frequency of operation is low enough so that the effect of the capacitive reactance due to Cc, the collector capacitance, can be neglected. re rc arcIe E Ve V a B Fig. 4.1. Low-frequency equivalent circuit of the grounded base configuration. re is the effective emitter resistance;O KT (mv) qIe(ma)' rc is the effective collector resistance, rb is the effective low-frequency base resistance, and a is the short-circuit current gain of the grounded base configuration. The equations concerning this configuration are: Ve = ZllIe + Zl2IC and (4.1) Vc = Z2ile + Z22Ic. (4.2) 6

The University of Michigan * Engineering Research Institute The open-circuit parameters are: Zll = re + rb (4.3) Z12 = rb (4.4) Z21 = rb + arc, and (4.5) Z22 = r + r (4.6) If H parameters have been measured for a particular transistor in the grounded base configuration and it is desired to obtain representation using the simplified low-frequency equivalent circuit, the transformations are as follows: hi2 (h21 + 1) re = h -' (4.7) h22 b = h2 (4.8) rb =2h 1 - hi2 ^j 1 and r-h12, and (4.9) r = h22 h22 = - hi2 + h21 > - -2 — h2 h. (4.10) 1 - h12 4.1 HIGH-FREQUENCY CIRCUIT REPRESENTATION OF THE TRANSISTOR The low-frequency representation applies only when it is known that inherent reactances can be neglected and that the phase shift of the collector current with respect to emitter current is negligible. 2 Middlebrook accounts for high-frequency effects in a treatment involving admittance parameters. He has derived formulas for obtaining the four admittances after six basic measurements have been made. The representation of these parameters is in the form shown in Fig. 4.2. This elegant treatment is Y1o- I Y o2! I l Y Y 0 — - 4.. dmt0 —-i-^ --.- - --.o. Fig. 4.2. Admittance forms as treated by Middlebrook. 7

The University of Michigan * Engineering Research Institute more complete than the one by Shea3 but is also more complicated. The principal high-frequency circuit effects as described by Shea are those of the collector capacitance, the base spreading resistance, and of the complex value of a. Until it is necessary to do otherwise, the representation as described by Shea will be referred to in this report. The general form is shown in Fig. 4.3. re r aZ cIe Eo>r C 0" I 0 Fig. 4.3. Equivalent circuit of the grounded base configuration including rb", rb' and Cc. rb' is the high-frequency base spreading resistance, rb" is an apparent feed-back resistance due to the variation of the effective width of the base layer with collector voltage, Cc is the capacitance of the space change or depletion layer at the collector junction, and ao = 1 + j(/a7), where ao is the "zero frequency" current gain and ca is the radian frequency at which the magnitude of a drops to 0.707 ao. At higher frequencies, when the reactance of Cc is much smaller than rc, the effects of rb" can usually be neglected. By including only the most important frequency effects, practical circuits for the three transistor configurations can thus be obtained. aZcte re rc aZc C AV' Co\+ EE C 0e —-— I —------ c VE Vc B Figo 4.4. High-frequency equivalent circuit for the grounded base configuration. 8

The University of Michigan * Engineering Research Institute Summarizing, the grounded base case is as shown in Fig. 4.4. Where z = 1 —^ —.. (4.11) 1 + jc rcCc the open-circuit parameters are: zll = re + rb (4.12) Z12 = rb, (4.13) Z21 = rb' + Zc, and (4.14) Z22 = rb + Zc (4.15) rb rc/l+r azcrb B +- -- C Ib';.4-Ic 4 4I VB re VC E Figo 4.5. High-frequency equivalent circuit for the grounded emitter configuration. Similarly, the grounded emitter case is shown in Fig. 4.5. The shortcircuit grounded emitter current gain is the ratio of Ic to Ib: p - -— a *'(4.16) 1 -oc The open-circuit parameters are: Zll = rbt + re (4.17) Z12 = re, (4.18) Z21 = re - OZc, and (4.19) Za2 = re + c (4.20) 1+ 97

The University of Michigan * Engineering Research Institute rb re Bo - W- - A- -_oE B o V V E Ib- rc/l+ ICc(l)=) + Vb Ve aZclb C Figo 4.6. High-frequency equivalent circuit for the grounded collector configuration. Finally, the grounded collector is shown in Fig. 4,6. The open-circuit parameters are: Zll = rb + Zc, (4.21) Z2 =, (4.22) 1 + Z21 = Zc, and (4.23) Zc Z22 = re +. (4.24) 5. NONLINEAR OPERATION AND OSCILLATORS For most purposes it is desirable to obtain a wave form which is reasonably sinusoidal; thus a brief review of the effects of inherent oscillator nonlinearity is desirable. Oscillations start in a region in which the emitterbase diode is conducting, and the collector-base diode is nonconducting. As the amplitude increases, one or the other of the two diodes will experience a reversal of polarity, thus limiting the amplitude of oscillations. If the collector becomes positive (for a PNP transistor), the collector-base diode will then become conducting. In general, this will short-circuit the high impedance or resonant part of the circuit. If the emitter, on the other hand, goes negative, the emitter-base diode becomes nonconducting, generally causing a voltage surge across the inductance in the circuit. Compensation for these effects can be obtained by inserting resistance in series with the collector and in parallel with the emitter to base junction. A basic circuit useful for experimentation is shown in Fig. 5.1. 10

The University of Michigan * Engineering Research Institute C2 ^CIL Fig. 5.1. A transistor oscillator circuit. The frequency of oscillation can be determined from: 1 / C1 + C2 ~ L KC1C2 + CiCc + C2C) ) where C2 = C1 1 - (5.2) Using a radio receiver to detect the generated signal, and varying the frequency of oscillations by controlling L, C1, and C2Y fmax (the frequency at which the available power gain is no longer > 1 can be determined, For example, frequencies between 56 and 70 mcs were measured for type 2N128 transistors, between 70 and 79 mcs for type 2N129 transistors, and a maximum frequency of approximately 400 mcs for a type GA53233 diffused base transistor. 6. FORMULATIONS OF INPUT AND OUTPUT IMPEDANCE The general four-terminal active network allowing the derivation of input and output impedance is shown in Fig. 61l. Zg Vg( ZL Vi _= __V,' 1_V_2_ Fig. 6.1. The general four-terminal active network. In general, the input and output impedances can be written as follows: ------------------ ~~ ~ ~~11

The University of Michigan * Engineering Research Institute Zi = Z -122 1 and (6.1) Z22 + ZL Zo = 22 -1221 (6.2) Z2 1 Z1 + Zg Applying the above relationships to the grounded base configuration, and referring to Fig. 4.3 and Eqs. (4.11) through (4.15): ZiGB ='r(- + ZL+ and (6 3) Fzc (1-c) + ZL1 ZiGB = re + rb' Lrb' + ZL + an Z1 =arb' Z rb' (re + Zg) ZGB = re + rb + Zg + re + rbt + Zg (6.4) At lower frequencies where the reactance of Zc is much greater than ZL + rb' and Zc(1-a) is much greater than ZL, ZiGB can be simplified: ZiGB ^ re + rb (1-a), Zc > (rb' + ZL), Zc(l-a)> ZI (6.5) If Zg is short-circuited and if the parallel combination of rb' and re is negligibly small, which is generally the case, Zo can be simplified: ZGB c [1Z r + r Zg = 0. (6.6) OGB c e bJ If, on the other hand, Zg is made very large in comparison to rb', which in turn is much greater than re, Zo can be simplified another way: ZOGB %* Zc + rb' Zg > rb > re (6.7) Note here that ZoGB is shown in two forms depending upon the magnitude of Zg. In like manner the relationships for the grounded emitter configuration can be obtained: Zc + ZL ^GE * r' + re and (6-8) iGE = rb + re Lre + Zc(la) + ZL (6.8) ZGE Zc(1-) + re Lr' Zg re 12 —-— b- + Zg + ----- (6.9) 12

-The University of Michigan * Engineering Research Institute If the load impedance is so small that the reactive part of Zc(l-c) is much greater than (ZL + re) in the range of the frequencies used, Zi can be simplified: 1 ZiGE ~ rb' + re (1-) = rb + re (+), (6.10) Zc(l-a) > (ZL + re) To study the variation of ZoGE with frequency, neglect (rb' + Zg) in comparison with CaZc in Eq. (6.9) and consider: re ZOGE b (1 + Ka - a) Z, K Zg + rb' + re (6.11) The order of magnitude of re is about one-tenth of rb!, and if Zg is ten times as great as rb', the value of K is about 0.01. When the magnitude of (1-a) is much greater than 0.01, this value of K can be neglected and Eqo (6.11) yields: 1 ZGE Zc(l1-) = Zc (1+Z) Zg > rb' > re (6.12) Finally, the relationships for the grounded collector configuration can be obtained: Zi = rbl + Zc e re + ZL +, and (6.13) GC b c [Je + Z + zc/l+p zo re + (6.14) ZoGC = re + + L + Zg+ Zc (6.14) If a resistive load is used so that RL is much greater than re, the form of Zi becomes: RL Z ZRL (1+P) ZiGC rb + Zc rbL' + > /L (r + Z) (6.15) RL+Z, P RL (1+ZZ The form of Eq. (6.15) represents a resistance rb' in series with the parallel combination of RL(1+P) and Zc. If the frequency of operation is such that Zc is much greater than (rb' + Zg), Z0 can be simplified: 13

The University of Michigan * Engineering Research Institute ZGC r rb + Zg (6.16) ZOGC re + 7. OUTPUT IMPEDANCE OF THE GROUNDED BASE AND GROUNDED EMITTER CONFIGURATIONS The -grounded emitter and grounded base configurations are of interest considering output impedance measurements because of the inherent dependency of Zo upon Zc. These impedances will have a greater magnitude than those of the grounded collector configuration, which are primarily dependent upon Zg (Z being much smaller than Zc). Also, the magnitude of ZoGB and ZoGE can be expected to decrease with an increase in the frequency of measurement because of the shunt capacitive reactance of Zc, whereas the magnitude of ZoGC can be expected to increase with frequency due to the decrease in the magnitude of (1+f) which appears in the denominator in the expression for ZoGC. From the grounded base configuration, information can be obtained making use of Eqs. (6.6) and (6.7): Zol X Zc + rb' Zg > rb' > re (7.1) Z02 ZC r + rb ) Zg ~ * (7.2) To this end, measurements were made using the circuit of Fig. 7.1. 2N128 3.3K RX METER 15K.013.01= 5.2.v;'1.3 v Fig. 7.1. Circuit for measurement of Zol and Zo02 Zol was measured with the switch open and Z02 with the switch closed. Curves showing the resulting magnitude and phase angle of the impedance versus frequency are shown in Fig. 7.2. 14

The University of Michigan * Engineering Research Institute lOOK - - --—. —- - 80K — 1 _ t_ -- — 800 60K - - l- 2;-1 6 60~ u 0o 4z N 01l 40 K —-- — 400 ~ Q. 20K —' -- 20~ IZo21 0 Q I 11111 00.1.5 1 10 100 FREQUENCY (MC) Fig. 7.2. Curves of IZo0l| /i and IZo21 / 2 versus frequency for a type 2N128 transistor. Observing the curves of Fig. 7.2, one notes that the phase shift of ZOl is approximately 90~ at lower frequencies and then decreases at higher frequencies. Zc is primarily capacitive because the frequencies of operation are much higher than the frequency associated with one race time constant (at which the phase shift is 45~). At higher frequencies, the reactance of Cc approaches the magnitude of the series resistance rbg, and thus the decrease in phase shift would be expected. A more thorough treatment of the effect of rb' on ZoGB at high frequencies will be found in Section 8 of this report. One will notice also that the phase shift of Z02 is always less than that of Zolo This is due to the complex a. After the curves of the output impedance of the grounded emitter configuration have been discussed, the reason for this relationship will become clearer, From the grounded emitter configuration, information can be obtained making use of Eq. (6.12): Zo3 Zc(l-a) = Zc (l+), Zg > rb' (7-3) Measurements were made using the circuit of Fig. 7.3. In this regard, note that if the switch in Fig. 7.3 is closed, the circuit is identical with that shown in Fig. 7.1 also with the switch closed, and the conditions for meas urement of Z02 are met. Zo3 was determined with the switch in the circuit shown in Fig. 7.3 open. Curves showing the magnitude and phase angle of impedance versus frequency are shown in Figo 7~4 for both Zo3 and Z02, using a lower scale than in Fig. 7.2 to allow further comparison. 15

The University of Michigan * Engineering Research Institute 2N128 3.3K RX METER *5.2v, _ 15K1. 1.3v Fig. 73.5 Circuit for measurement of 7A03 or Z,O,. IOK - 900 8K E --- _ 0 —- 80o~6K- i L l6. sLJ'I — ]111i CD"'.1.5 I 10 100 FREQUENCY (MC) Figo 7040 Curves of [Zof | j /~ and 1z031 /G3 versus frequency for a type 2N128 tran sistor. The comparison of AQ_1 and AZ03 is of most interest because the difference between them is mainly due to the effects of the complex quantity (l -)o At requencies ower than Eq (7) reces to Zoi ^. ZC o (TA~ A01 A00 ( ) ol I 1111 7t 7 1- tttT- ~ mra 0 0o Comparing Eqs. (7o5) and (704),.Z3 = o(l) (105) where Fig Curves of I Zo and Z0 3 vers Compa = ng /s2 andd ( Z(o 3) =11 Zo(l-a) ~(7) o2-f = IlaZOI /g,an The absolute magnitude of Zo3 a Zh0en is: 16

The University of Michigan * Engineering Research Institute IZo31 =!ZolI x Il-oa, (7.6) and the phase angle of Zo3 is: 93 = G1 + 0(1 a) ~ (7~7) It is desirable at this point to study the nature of the quantity (1-a) as related to the complex form of a: aa - + J( T * (7.8) If a vector plot of a versus frequency is made using polar coordinates, the form will be as shown in Fig. 7.5.4 0 ao GU=00 =O \ 450 \ o //INCREASING \ i/ ~ FREQUENCY -w = Fig. 7.5. Vector plot of a versus radian frequency. A translation can be performed which will allow the vector (1-a) to be represented also as in Fig. 7.6. A value of ao =.9 was chosen for convenience. The vector (1-a) originates at the origin and terminates at some point on the left-hand semicircle. awa (~-a), \ \ //INCREASING FREQUENCY wua Fig. 7.6. Vector plot enabling the representation of (l-O). In general, the grounded emitter current gain is: 0 a |= - a bo -.. (7-9) At frequencies lower than that at which D decreases to.707 times its zero-fre1 —-------------------- 1 Y --------------------— ~~~~~~~~~~~~

The University of Michigan * Engineering Research Institute quency value, it is valid to say: a0 P " 1_. * (7.10) From Eq. (7.10) it is seen that the frequency fp (at which B decreases to.707 bo) will occur at the point where (1-a) has increased in magnitude to \2(1-ao). The radian frequency wd is shown in Fig. 7.6 and an enlarged view is shown in Fig. 7.7. / / I-a I-o ao Fig. 7.7. Enlarged view of Fig. 7.6 showing the conditions for X = cd. From Eq. (7.8), when w = Cd, the magnitude of 5G, which is the phase angle of a, is: ~a = arc tan _. (7-11) Wa Also from Fig. 7.7, for magnitudes of ao close to unity ~0 W arc tan 1o (7.12) ao then 1-ao a. C — P = - (7-13) a b By comparing the curves of Zol and Z03, an indication of the magnitude and phase of (1-a) is obtainable at any frequency in the range of measurement. From Eqs. (7.6) and (7.7)' IZo31 l-a| = z03J and (7.14) 1-a) ~= ~3 - ~i. (7.15) 18

The University of Michigan * Engineering Research Institute The frequency at which the grounded emitter current gain B drops 3 db should correspond to the point at which (1-~) = 45~ ~ f6 can thus be determined and if bo is known, it is possible to predict fa as from Eqo (7.13): fa = bofp (7.16) Using the curves of Zol and Zo3 in Figs. 7.2 and 7.4, (l_-a) = 45~ at f5 = 2 mcs. The magnitude bo calculated from the low-frequency characteristics was 40; thus the predicted fa would be 2 x 40 = 80 mcs. This frequency is high for a type 2N128 transistor, but using the same technique on another of the same type yielded fp = 1.7 mcs, bo = 38 for a predicted fa = 65 mcso These results are in closer agreement with nominal values. It is clear in the above cases that fp must be known very accurately to expect a good approximation of fa. The same accuracy can be required of bo as proves to be the case when measurements were made of the type GA53233 diffused base transistor. A test yielded fo = 65 mcs and a value of (1-a) = 1/6 /4~ at 4 mcs. The phase shift at this frequency is negligible and so this value of (1-a) is essentially 1-ao = l/l+bo; thus the magnitude of (l+bo) is 6 and bo = 5o The predicted fa for this transistor is then fa = 5 x 65 = 325 mcs. In this case, an error in the measurement of bo will have a considerably larger effect than an error in the measurement of fp in the prediction of fa. Moreover, it was noted that the value of bo as calculated from the absolute impedance ratio (bo e 5) did not coincide with the value as calculated from the static characteristics (bo'Q8)o The dynamic value as computed from the impedance ratio is undoubtedly a better approximation but some uncertainty does remain. The reason for including the curve of Z02 is that the difference between it and the curve of Z03 is due mainly to the relative magnitudes of re and rb' It can be seen, for example, that if rb' were much greater than re in Eqo (7.2), Z02 would approximately equal Zc(l-a), which is the relationship for Z03. Since the difference between Z02 and Z03 can be computed from measurements, the value of rb' will be calculated as a function of Zol, Z02, Z03, and re Using Eqs. (7.1), (7.2), and (7.53) and restricting the applicability to lower frequencies where the phase shift of (1-~) is negligible, i.eo, where the impedances Zol, Z02, and Z03 have a common phase shift: Z03 dividing Z03 by Zo1, 1 - a =; (7.17) Zoi a rb2 020 dividing Z02 by Zoi, 1- a rb (7.18) re + rb 2(7 Solving Eqs. (7.17) and (7.18) simultaneously for rb, Zol - 202 rb = re Z02 - 03 (719) 19

The University of Michigan * Engineering Research,, Institute In a test it must be assured that the emitter current is constant for all three impedances; then re 25 7/Ie(ma) for germanium transistors. Values calculated for rb' were 192 ohms for a type 2N128 transistor and 7.2 ohms for a type GA55235. A value for the same type 2N128 transistor calculated another way, to be discussed in the following section, was 190 ohmso Further analysis in the use of measurements of Zol, Z02, and Zo3 should prove more conclusively the validity of what has been presented in this section, and will examine frequencies above fp to determine how well the theoretical relationships for a and (1-a) predict the actual circumstances. 8. FURTHER DEVELOPMENT OF THE OUTPUT IMPEDANCE OF THE GROUNDED BASE CONFIGURATION Further study of the output impedance of the grounded base configuration has led to methods of obtaining Cc and rblo Under the conditions of large input impedance, from Eq. (6.7): Zo 0 Zc + rb. (8.1) Assuming constant values for Cc, rc, and rb' the circuit representation and the equivalent as seen by the RX meter are shown in Figs. 8.la and 8.lb, Cc o —-- Wr-/ —] o- ---- rc Zo - rb Zo - Rp Cp Fig. 8.la. Representation of the tran- Fig. 8.lb. Equivalent representasistor grounded base output impedance. tion as seen by RX meter. Because the magnitude of rc is ordinarily much greater than XCc at the frequencies of interest, the rb'Cc time constant will be of primary concern in the development below, The derived relationships between the transistor parameters and the parallel components in the equivalent representation are: 20

The University of Michigan * Engineering Research Institute (r rbl 2 + Trc2(U)/) 1 Rp = (r+rb) r o (8.2) r caL rb Cc rc+rbb +- (/0 W)2 c rb rc 2Cc Cp = A*~(8~3) p (rc+rb)2 + rc2(c/Uo)2 (85) Under the circumstances that r >> rb the parallel components can be simplified as follows: Rp X r c ~ 1 + (W/ o)2 O (84) 1+. — (c/w)' = rbC rb Cp -, Cc + (/7o)2 (8.5) At very low frequencies where rc/rb8 (c/Co)2 << 1, the value of rc can be measured directly as equal to Rp. This condition is not as easily met as it might seem because of the factor rc/rb' For example, typical values of rc and rb' are 106 and 102 ohms, respectively, and thus a typical ratio is rc/rb' = 104. Then in order to measure rc directly, << ^10~ 4 oo Therefore, in the example, the radian frequency of measurement must be 103 times the radian frequency associated with the rb'Cc time constant to expect a possible accuracy of 1% in the direct measurement of rc. The restrictions are not nearly as severe if direct measurement of Cc is desired. From Eqo (8.5) it can be seen that if C = 0.1 0o' the assumption that Cc is equal to Cp will be correct within 1%o For other ratios of rc/rbu the relationships are not the same as for the typical example; however, information concerning these ratios can be easily obtained if various curves are drawn for the normalized quantity Rp/rbv versus (/Uo for expected ratios of rc/rb' From Eq. (8.4): Rp = /rcb \ 1 + (e/mo)2 1 ~(/O2 Rp brc (/o)= r )2 (8.6) rb \b 1 + (/W0 )2 +(/lo) rb irc From Eqo (8.5): Cp _ 1 Cc =1+ (n/o)2 (8.7) 21

The University of Michigan * Engineering Research Institute Curves of Rp/rb8 (for four different ratios of rc/rb ) and of Cp/Cc versus 0/c0o are shown in Fig. 8.o2 ioooo lr000 lli <~~- "^ *I'^.l 00I 1000 rc, 105 -n 100 zCp/Ctg0 w/ w0Tlil,01 Rp/1rb E 2, / =: 1, (88) p/01 = 1, = 1 (89) W/Co Fig. 8o2o Curves of Rp/rb i and Cp/Cc versus /ao ~ It can be seen from Eqso (806) and (8.7) or from Fig. 8.2 that for C/cD = | Rp/rb 2, W/Wo = 1 i (8.8) CP/Cc = 1/2, CD/Co = 1 o (8.9) These conditions allow determination of rbI and cDo Once the low-frequency value of Cc is known, the frequency of measurement can be increased until Cp' 1/2 Cc; and at this point, the radian frequency is equal to: 1 0 rb Cc (8.10) Since Cc is known, rb' can be obtained directly: rb = oCc (8.11) Alternatively, since resistance measurements have been made at this critical frequency: rb - (8.12) 22

The University of Michigan * Engineering Research Institute rb' can thus be calculated from two interdependent relationships, one value obtained serving as a check upon the other. Impedance measurements were made using several different types of transistors and the values for 1/0o, rb' and Cc are tabulated in Figo 830 l/o = 1'C C= RbP rbl rbc Cc rb rb Transistor _oCc 2 Ipaisec lifd ohms ohms 2N129 1225 5 245 260 2N128 935 4.9 190 200 2N393 500 4.2 119 120 2N300 295 3.2 92 90 GA55323 <100 5 -- Fig. 8.3. Table of rb'Cc, Cc, and rb' for various transistors. Measurements were not obtainable for the 2N393 and 2N5300 above 250 mcs, but there was enough information to estimate the values shown because the slope of the Cp-versus-frequency curve allowed extrapolation to the frequency of interest. The maximum frequency of measurement was 250 mcs which corresponds to the minimum certain rb'Cc time of 636 pjjsec if the Boonton RX meter is used. If a method were devised to measure Cc alone at frequencies higher than 250 mcs, a good indication of the critical wo and thus rb' should still be obtainable. Two effects that have not been discussed concerning the measurement of output impedance of the grounded base configuration are: (1) the resistance Rp reading starts at some finite value and then increases with an increase in the frequency of measurement to higher than the "infinity" mark (as indicated on the dial of the RX meter), and (2) the capacitance Cp reading increases slightly to a maximum value with frequency just before the expected slope towards the 1/2 Cc value takes place. The increase in effective resistance is evidently due to regenerative feedback through the collector emitter capacitance developed across the high input impedance, and occurs at frequencies much lower than the critical frequency: f = 1 20 rb Cc The increase in capacitance is evidently due to the internal characteristics of the transistor itself. 25

The University of Michigan * Engineering Research Institute 9. THE GROUNDED COLLECTOR CONFIGURATION Measurements have not yet been made and the capabilities not thoroughly studied but first indications are that both input and output impedance of the grounded collector configuration should yield valuable information. From Eq. (6.15): ZcRL(1+P) Zi rb' + Zc RL (1+)' RL > re (9-1) The synthesis of (9.1) appears in Fig. 9.1: rb Z a R(IItp) frc TCc Fig. 9.1. Representation of the grounded collector input impedance. From Eq. (6.16), the output impedance for a resistive input impedance is: Zo re + rb g. (9o2) 1 + Here is a relationship which allows an increasing value of ZO as the quantity (1+f) decreases with frequency. Thus the accuracy of measurement should be good at high frequencies and the phase shift involved should be due principally to P. 10. FUTURE PLANS It is planned to continue work on the system of impedance measurement techniques. The output impedance of the grounded emitter configuration will be analyzed at frequencies above fs, and the capabilities of the grounded collector configurations will be studied. 24

The University of Michigan * Engineering Research Institute 11. REFERENCES 1. Giacoletto, Lo Jo, "Terminology and Equations for Linear Active Four Terminal Networks Including Transistors," RCA Review, 14, 28-46 (March, 1953). 2. Middlebrook, R. D., An Introduction to Junction Transistor Theory (John Wiley and Sons, Inc., New York, 1957), pp. 237-255. 3. Shea, R. F., Principles of Transistor Circuits (John Wiley and Sons, Inc., New York, 1953), Chs. 9-10. 4. Ibid., p. 205. 25

The University of Michigan * Engineering Research Institute PART II GAS-TUBE STUDY Cold-Cathode Breakdown Study The present objective of the gas-tube study is to obtain maximum familiarity with the effects of the various gaseous conduction parameters, including the geometry, on the volt-ampere behavior of Townsend current flow at very small currents. Interest will eventually be centered about the transient change from Townsend-current to glow discharge or arc behavior. The integral equations describing the volt-ampere and transient behavior must take into account the field distortion produced by space charge and the effect of this distortion upon the various condition parameters such as the first and second Townsend coefficients and the ionic mobilityo The initial phase of the work has consisted of a critical study of the origin of the general integral equation governing Townsend current phenomena under operating conditions for which space charge causes important field distortion. The equation was studied assuming a one-dimensional system which may be either planar or cylindrical. After obtaining an appropriate form for this equation, an extensive study of the literature was made to permit estimation of a reasonable functional dependence of the gaseous conduction parameters a, r, and ionic mobilityo It is well known that although adequate empirical equations for these parameters may be obtained for any restricted range of electric field and gas pressure, it is not legitimate to assign any particular functional dependence over an extended range. For example, although mobility is constant at low fields, it becomes dependent upon the electric field at high fields. Because of this variability of the various parameters with the field, the solution of the integral equation requires numerical or analog computer techniques. Analog computer techniques seem appropriate because of the speed and flexibility they offer for studies in which the objective is to gain maximum familiarity with gross behavior. Analog solutions are currently being obtained and will be presented and discussed in a subsequent technical reporto 26