Technical Report ECOM-01870-17 November 1967 COMPUTATIONAL UTILITY OF NONLINEAR TRANSISTOR MODELS C. E. L. Technical Memorandum No. 98 Contract No. DA 28-043-AMC-01870(E) DA Project No. 1 PO 21101 A042. 01. 02 Prepared by Alan B. Macnee COOLEY ELECTRONICS LABORATORY Department of Electrical Engineering The University of Michigan Ann Arbor, Michigan for U. S. Army Electronics Command, Fort Monmouth, N. J. DISTRIBUTION STATEMENT This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of CG, U. S. Army Electronics Command, Fort Monmouth, N. J. Attn: AMSEL-WL- S.

ABSTRACT Three nonlinear models for junction transistors are shown to be equivalent, physically. Their differences lie in the state variables employed. The incremental behavior of these models is compared with that of the Hybrid-n and high-frequency tee. The computational utility of the voltage- controlled and the charged- controlled models are compared for a switching circuit example. If a single model is used to describe the transistor, all models are "strongly" nonlinear in one or more operating regions. In any single operating region, however, an appropriate model choice leads to "almost linear" equations. iii

TABLE OF CONTENTS Page ABSTRACT iii LIST OF ILLUSTRATIONS v LIST OF TABLES vi 1. INTRODUCTION 1 1. 1 Calahan's Voltage-Controlled Model 1 2. THE NET-1 MODEL 7 3. THE CHARGE-CONTROLLED MODEL 10 4. CHOICE OF A LARGE SIGNAL MODEL 21 4. 1 Voltage-Controlled Model in Saturation Region 23 4. 2 Charge-Controlled Model in Saturation Region 24 4. 3 The Linear and Cut-Off Regions 29 4. 3.1 Using Voltage-Controlled Model 29 4. 3.2 Using Charge-Controlled Model 31 4. 3. 3 Combined Voltage and Charge Control 32 5. SUMMARY 37 REFERENCES 38 DISTRIBUTION LIST 39 iv

LIST OF ILLUSTRATIONS Figure Title Page 1 Large signal, voltage-controlled transistor model and defining equations 2 2 (a) Incremental voltage-controlled model; (b) Simplified model for "normal" operation; and (c) Simplified model converted to hybrid-7r form 6 3 Large signal, transistor model used in NET-1 analysis program 8 4 Charge-control, large signal equivalent circuit 12 5 (a) Incremental current-controlled model and (b) Simplified model for "normal" operation region kCI 1- N aI 0o o 6 T and hybrid-vT equivalent circuit including basewidth modulation effects 18 7 Switching transistor storage time test circuit 22 8 Calculated current response of saturated transistor (See Fig. 7) 36

LIST OF TABLES Table Title Page I Numerical integration of Eqs. 66a and 67a 25 II Comparison of linear charge-controlled and nonlinear voltage-controlled equation solutions 29 vi

1. INTRODUCTION This memorandum demonstrates the equivalence of three large signal transistor models: the model used in the NET- 1 circuit analysis program, a model used by D. A. Calahan in his computer circuit analysis course, and D. Koehler's charge-control model (Refs. 1, 2, 3). These models differ primarily in the choice made for the controlling, state variables. Despite their equivalence, the differing variable choices can significantly influence the computational utility of these models in particular circumstances. The "optimum" variable choice depends upon the transistor's operating condition (saturation, linear or cut-off regions) and upon the external circuit. In general computer-aided analysis programs, it may prove impractical to change transistor models in the course of the solution calculations, but such a step is practical and worthwhile in hand calculated solutions. 1.1 Calahan's Voltage-Controlled Model This model together with the defining nonlinear equations are given in Fig. 1. Equations 1. 1 through 1. 4 represent first-order effects and will be essential to any large signal analysis. Equations 1. 1 and 1. 2 give the exponential behavior of an ideal semiconductor junction, and Eqs. 1. 3 and 1. 4 describe nonlinear capacitances which account for the stored charge associated with each junction. The voltage dependence of the transition capacitances CTE and CTC and of the forward and reverse alphas, indicated by Eqs. 1. 5 through 1. 8, are often second-order effects. The voltage dependence of the width of the junction transistion (space- charge) regions is responsible for most of these nonlinear effects. For purposes of comparison with other models it is convenient to write out the voltage dependence of the emitter and collector currents.

aR iR aF iF C E iC iF T E CTC + VF - - VR + Rbb' B RiR = lCs (eXVR - (1.1) if = I (eS XvF_ 1) (1.2) IES Xv CDE ES (e F 1) (1.3) CDC -cRVR (e R (1.V4) TC = CTC(vR) (1. 5) CTE = CTE(VF) (1.6) aF = aF(VF vR) (1.7) R = aR(VF, vR) (1.8) Fig. 1. Large signal, voltage-controlled transistor model and defining equations These are iE ES(e +F ) dt [(TE +CDE)VF aR T eCS( -I =IES (e 1) ICS XR TES- eVF dvF a T (CTEF) (1)

and e) -ICS vR d dvR iC = e _1 + dt (CTC+CDC)vR aF'ES(e F1) = ICS R 1) a IEs )+C dt d + t- (CTcVR) (2) These equations and Fig. 1 are for the PNP case. For an NPN transistor ICS, IES, VR, VF, should all be multiplied by minus one. The equations are first order, nonlinear, ordinary differential equations in the junction voltages vF and vR (vEB and vcB). This model is therefore convenient for node voltage or state-variable formulations of network equilibrium equations. Bulk resistances can easily be included in the emitter and collector leads when they are significant. It is of interest to examine the incremental behavior predicted by Eqs. 1 and 2, even though one plans to use them for large signal calculations, since numerical integration techniques effectively reduce the nonlinear model to a sequence of linear models. Writing each variable E IE + e iC = IC + ic VF VEB +Ve R = VCB + c as a quiescent value plus an increment, Eqs. 1 and 2 give IE= I ES(e B a 1) CB 1 ) (3) C = FIE(e s eCB ),(4)'Ic ~ig(OxV -l~'IC ~3

ie = IEs e + e + oF TE+ VEB dvT dt - aR CS e B)vc (5) and i= (XlCS e (C )+(+C + a TC aFIES e ve ic Cs c WR dt T CB dv'R dt F (6) Equations 3 and 4 are the familiar Ebers-Moll equations, relating the quiescent currents to the quiescent voltages (Ref. 4). The incremental Eqs. 5 and 6 are conveniently rewritten as dv 1i e (7) e r ve + (Cde + Cte) dt mr Vc'(7) and dv c r c + (Cdc tc) dt gmfe;(8) where rEB' (9) e AVEB IES e de wF r Cte (CTE + VEB dvF E (11) BEB A 1 r = (12) c XVcB klCs e dc r' (13)

Ctc (cTC + VCB dvR (14) VcB AaR (15) mr r (5) and a gm= re (16) The incremental circuit implied by Eqs.. 7 and 8 is given in Fig. 2(a). In the saturation region, this complete model may be required. The relationship between this model and the more common "normal" operation incremental models is illustrated in Figs. 2(b) and (c). If the emitter-base junction is forward biased by at least one-tenth of a volt and the collectorbase junction reverse biased by at least the same amount, Eqs. 3 and 4 reduce to XVEB XVEB E IES e + aR ICS IES e (17) and IC -F IE - ICS(- aFaR). (18) In this region gmr' Cdc' and 1/rc all approach zero, and the model reduces to Fig. 2(b) where now r = (19a) e XE gmf = XIEaF (19b) and Cc = Ctc (20a) e Cde + Cte (20b) If we change the input-controlling variable to is is seen to be the usual highfrequency T-model without the "excess-phase shift" correction (Ref. 5). On the other hand,

if the model is reoriented, one can easily convert it to the hybrid-7r form shown in Fig. 2(c).* + V r gmf Ve i Cde + Cte i dc Ctc e Cc + c rrb gmrvc i V + (a) Fig.e v v rbb, e ]Cc (b)!C c r C Fig. 2(a) Incremental voltage-controlled model; (b) Simplified model for "normal" operation; and (c) Simplified model converted to hybrid-w form * Here the voltage-controlled current source has been replaced by two equivalent sources, and one of these is converted to the negative conductance of gmf mhos in parallel with re. 6~~~~~~~~

2. THE NET- 1 MODEL This model and the appropriate defining equations are given in Fig. 3. In comparing this with the voltage-controlled model in Fig. 1 several points can be noted. The transition capacities in Fig. 3 are assumed to have the specific voltage dependence indicated; the NET- 1 user selects three parameters for each capacity. The diffusion capacities in Fig. 3 are incremental values, and they are written as functions of the internal current components iCF and iEF. The choice of incremental values means that the current through the emitter diffusion capacity in Fig. 3 is dvF de Cde dt whereas in Fig. 1 the same current is d ide = dt (CTE VF) The NET-1 model includes ohmic bulk resistance in each lead (REE, RBB, RCc) plus ohmic leakage resistance across each junction (RE and Rc). This model also introduces two emission constants ME and MC to allow for junction voltages and incremental resistances which are greater than predicted by the ideal e;v factor. The equivalence between the NET- I model and the voltage-controlled model is established by writing the emitter and collector currents in Fig. 3 as a function of vF and vR for the case of RE and RC infinite. Kirchhoff's current law at the emitter node gives dvF d iE i + Cde dt+ ~ (Cte VF). (21) Now, substituting Eqs. 3. 3 and 3. 4 into Eqs. 3. 1 and 3. 5; and then using Eq. 3. 1 to eliminate ii in Eq. 21 gives

/ F Xv\ __ IEO M E -- R ICO e C ) FR 1 XvF' EO ME )C dld + E w F (1-aFaR) e dt dt (CteVF) (22) i1 = ~ VVF 1' EF -E lF [ -j (3. c) i2 = iCF- FiEF (3. 2) C l Iaa. ] (3. 4) Cde = MXF FF + EQ] (3 5) Cdc MCR [iCF + 1-RaF] (3.6) E~C Ctc Cte D1/(VzE- VF) (3 7) a F aFvF) (3.19) = aR(R) EF (3.10) 8CF = -I- a e ~(3.4) Cde = MWF EF + 1-aF (3. 5) C + VR)"[ (3.6) a F a F(v F) (3.9) Fig. 3. Large signa~l, transistor model used in NET-1 analysis program

Similarly one finds for the collector current AXVR \ X VF 1 aFEO ME -! c 1-a a 1-a a XvR (Mc ico eC dwc d (23) (MC R(1-aF-R-) e dt +dt (Ctc VR) If we let ME = MC = 1, and write IES ('EQ (24) CES (l-aFaR) I a a (25) Equations 22 and 23 are identical with Eqs. 1 and 2. As presented in Fig. 3, the NET-1 model is neither voltage- or currentcontrolled. The current sources i1 and i2 are controlled by two current variables iEF and iCF (which are proportional to the normal and inverse minority-carrier charge densities in the base region) as are the diffusion-capacities; but the transition capacities and the current gains are written as functions of the voltages. If Eqs. 3. 3 and 3. 4 are used to eliminate the voltages vF and vR in Eqs. 3. 7 through 3. 10, then one is left with a model which is completely current- controlled. Since the controlling currents are proportional to stored charge, this is also the charge-controlled model.

3. THE CHARGE- CONTROLLED MODEL Koehler's charge-controlled equivalent circuit and defining equations are shown in Fig. 4. In this circuit the relationships equivalent to Eqs. 3. 5 and 3. 6 or Eqs. 1. 3 and 1. 4 have been replaced by stores SN and SI. These new circuit elements are defined by two relations: (1) the voltage across the store is always zero and (2) the current flow through the store, out of the straight-line side, is the rate of change of the stored charge [qN(t) and qI(t) in this circuit]. The normal and inverse alphas aN and aI can be considered constants or expressed as functions of qN and qI. 0 The currents iE and iC are obtained by applying Kirchhoff's current law at the emitter and collector nodes. dqN qN qI d iEadq = -aiq +2- (26) IE d+ TEN I T- + dt (CTEVF)' (26) T EN Io TCI and dqI q1 qN d i- N- +.(Cv (27) C dt TCI No TEN Comparing these equations with Eqs. 22 and 23 or Eqs. 1 and 2 for the previous models, one is struck by their simplicity and "lack" of nonlinearity. If the alphas are assumed constant, the only nonlinear terms which affect the system dynamics are due to the currents flowing to the transition capacities CTE and C In the saturation region vF and vR can be expected to be relatively constant (they will change only 26 millivolts for a 2. 3:1 change in qI or qN); hence these nonlinear current terms often will be negligible. As qN and qI pass through zero, however, these currents become the dominant ones. The beauty of Koehler's charge-controlled model is the simplicity and linearity of Eqs. 26 and 27 in the saturation region. Its equivalence with the voltagecontrolled model of Fig. 1 is easily demonstrated by using Eqs. 4. 1 and 4. 2 to eliminate qN and qI in Eqs. 26 and 27. Solving Eqs. 4. 1 and 4. 2 for qN and qI gives 10_

qN(t)'EoN (e F_ 1 (28) qN~) =1- aN6ai/ ql(t) ='CoC (eXVR ) (29) O I Differentiating Eq. 28 and then substituting into Eq. 26 one obtains iE x E e. - aI aCn -n qNF / IT('E = hIES EN dt + IES dt (CTEVF), (30) and similarly for Eq. 27.R dv (lcsrclF de e C ICS T e R -I a IES + (C (31) If TEN - (32) Al T = - (33) a N a F (34) 0 aI a R a (35) 0 these equations are identical with Eqs. 1 and 2. Further, comparing Eqs. 28 and 29 with Eqs. 1. I and 1. 2 one sees F qN =ENiF = co (36) A i R qI= TCIiR cR (37) 11~~

qI(t) Em rI TN B Collector Junction I Junction + t) + CTE CE(F) E =S....- c C( N TCl e VC(t) qN qN(t) qN(t) boEN Cl Emitter I EN B 1Collector Junction I I Junction Base Region qN(t) (1- aN aI vF(t) = in + (4.1) F TEN IEO vR(t) = in (4.2) CTC = CTC(VR) (4.3) CTE = CTE(VF) (4.4) Fig. 4. Charge-control, large signal equivalent circuit Making the usual split of the dependent variables into quiescent and incremental values, Eqs. 26 and 27 can be split into two sets of equations QN QI I E a = (38) E TEN Io I T

and - -n n a - C dv- (40) e dt TEN T te dt dq. q n dv 1 -aC+ C (4) C dt T -C NO TCN te dt (41) Similarly expanding Eqs. 4. 1 and 4. 2 in power series and retaining only two terms gives QN(1-a N a0) I VEB X n EN IE ] (42) QI(1- EQN J) and V e =e T (44) V rc (45) dt + TCI where and A 1 Tin c=X I+... (47) VE B = TENo T~ =-a a~ En CIC N Ic 0 0 1 n

Equations 38 and 39 emphasize the linear relationship between the stored charges and the external currents while Eqs. 42 and 43 exhibit the usual logarithmic dependence of the voltages upon these charges. Using Eqs. 42 and 43 to eliminate QN/ TEN and QI/ TCI in Eqs. 38 and 39 yields the familiar Ebers-Moll Eqs. 3 and 4. If we let A qn T (48) EN qi 12 - q (49) 2CI Equations 40, 41, 44 and 45 can be combined into 1 EN e (50) i =- + C a i (50) e r Ve+ + Cte) dt 1 2 e e e and i + (T +i- aN i. (51) C C O Comparing these equations with Eqs. 7 and 8, taking cognizance of Eqs. 32 through 35, and Eqs. 10 and 13, one sees that they correspond exactly with the exception of the "crosscoupling terms" which are voltage controlled in Eqs. 7 and 8 and current controlled in Eqs. 50 and 51. The incremental model implied by Eqs. 50 and 51 is given in Fig. 5(a). In the special case of "normal" operation this reduces to the high-frequency "T" circuit shown in Fig. 5(b). Here, as in Fig. 2, the diffusion and transition incremental capacities have been combined into single elements C and C. he i C Comparing the normal operation, incremental models of Figs. 2(c) and 5(b)

with commonly used incremental models one observes two omissions: (1) base width modulation effects [rb,c and rce in Fig. 2(c) and rc and the L vc generator in Fig. 5(b)], and (2) an excess-phase shift factor. It should be recognized, however, that the large re Cc re e Irbbt e - C (a) e 1 e c (b) Fig. 5(a) Incremental current-controlled model and:b) Simplified model for "normal" operation region Q TCI l-aNo Io signal models will include the major base width modulation effects, if the voltage dependence of the alphas is included. For example, when this is done in the charge-controlled model, the incremental Eqs. 40, 41, 44, and 45 become dqn qn qi dye I v (40a)

dqi q dv Q (aN aN \ i = r c N I v) v), (41a) e TI Ne T EN tAcdt N dvRcc avF eo \ oa re-re (1.aN a1) 0 vc (44a) q. CN-c atT~ and v = r + -1 v. (45a) C C T CT X1-aN a avF e The new terms in Eqs. 40a and 41a can be represented in the model of Fig. 5(a) by slight changes in the values of re, rc, aN and a1I, and they are usually negligible when the junction in question is forward biased. In the normal operating region, however, Eq. 41(a) reduces to qn dvc 1N v (41b) c No TEN + C dt TEN avc c 0 EN EN C which can be written tc -iN i + C + (52) 0 c where cM1 ( aN (53) c QN aaN N TEN avR E avR This is the usual collector resistance of the low frequency T-model; it is typically in the megohm range and varies inversely as the quiescent emitter current IE (Ref. 6). The added terms in Eqs. 44(a) and 45(a) can be represented by voltage-controlled * Recall Eqs. 26, 27, 4. 1 and 4. 2 are all written for a PNP device and therefore anis negative. avR

voltage sources in series with r and r in Fig. 5(a). These generators will have neglie c gible effect when the junctions are reverse biased, but their influence can be observed at the forward-biased junctions. In the normal operating region, for example, Eq. 44(a) is usually written ve = r i1 + vc (54) where ~Mec X(1~ OaN a ) (55) X(l -a N aI) avR O O To the extent that the diffusion processes, in the base region, are linear tLec, rC', r and aN are related by the equation o r a' aN ec c o (56)* e 1 - N 2 0 Figure 6(a) gives the "normal" T incremental equivalent circuit when these space-charge-layer widening effects are included. If the alpha voltage dependences are included in the voltage controlled circuit, the complete hybrid-7r circuit shown in Fig. 6(b) results. Equivalently the hybrid-7r model parameters can be derived directly from the Tmodel of Fig. 6(a) (Refs 7, 8). The relations are r e rb, T?~ (57) N r rce (58) ec N - tec N gm e r c rN (59) g r r e e *This useful relationship is pointed out in the Mullard Reference Manual of Transistor Circuits, 1st edition, 1960, pp. 82-83; but it does not appear to be very generally recognized.

rb,C =-a )r rC (+aN (60) ec N c o tc1 -(N' r Cbfc = Cc (61) and Cbe = C. (62) e c v e + rrb C rbb' Cb- c +, v + e c (a) E rbWc B rbb' bc C modulation effects 0 C e b'_ gm c (b) Fig. 6. T and hybrid-? equivalent circuits including base-width modulation effects The one effect all of these large signal models ignore is the excess-phase shift factor. At the incremental model level, Fig. 6(a), this effect may be approximately included by multiplying aN by a pure phase shift factor.* m is a real constant which usually is in the range 0.25 to 1.0.

ms WF aN -aN e (63) o o When this is done and the T-model is converted to the hybrid-r, only the values of gmand Cb'e are modified; ms a N - rg - F (59a) gm r e and Cb'e C(1+ maN ). (62a) 0 The excess-phase shift correction given in Eq. 63 represents a pure time delay of m/wF = m TEN seconds. This effect could be introduced into the large signal models by delaying the transfer current generators. For example, in Fig. 1, the current-controlled current generator would be aRiR(t) - aRiR (t -R m R (64) aF iF(t) - aF i (t mF and in Fig. 4 ql(t) ql (t - mRTCI) 1o CI o TCF (65) N qN (t) N(t- mFTEN) aN0 aN T No tEN No EN One can reach some general conclusions about all three large signal models. First, they are essentially equivalent, differing only in the choice of controlling, or state, variables. It will be shown in the next section that this may not be a trivial difference in large signal calculations. Secondly, if fixed values are assumed for the normal and inverse alphas, all models neglect base-width modulation and excess-phase shift effects. When the voltage

dependence of the alphas is introduced, the predominant base-width modulation effects are reproduced. * The excess-phase shift effects could be included by adding suitable time delays in the appropriate transfer current generators which represent the collected currents. To the writer's knowledge workers in. the area of large signal transistor circuits have not added the excess-phase, time delays. If the analysis program employed is a time domain program, this should be relatively easy to do, and it would appear that the effects might well be significant for fast switching circuits and high-frequency high power amplifiers or oscillators. *This is the normal capability in the NET- 1 model, for example. 20

4. CHOICE OF A LARGE SIGNAL MODEL The relative merits of various incremental models of junction transistors have been discussed at great length in the literature* (Refs. 7 and 9). The two most popular models are the hybrid-pi and the high-frequency tee. Despite the protestations of various writers concerning the fundamental nature of one model or the other, there can be a complete equivalence between their two models. This equivalence is given in the previous section in Fig. 6 plus Eqs. 57 through 62(a). Each model, in its greatest generality, contains eight parameters which are related rather directly to physical processes in the transistor. At the higher frequencies two parameters Ice and rc', in Fig. 6(a) and rb,c and rce in Fig. 6(b) can be neglected, leaving only six parameters. For those applications which are insensitive to the excess-phase shift, only five parameters are left in either model; and in each case one of these is predictable without measurement (re and gm). For large signal analyses, as has been pointed out, two models have been used which correspond closely to these small signal cases. The large signal version of the tee model is the charge- controlled tee model, and the large signal version of the hybrid- fr is the voltage-controlled model. These two large signal models are equivalent, and one should expect them to yield the same results if the nonlinear differential equations are solved exactly. In practice, however, such equations are usually integrated numerically. Such a solution involves breaking down the nonlinear equations into a sequence of linear equations. The allowable "step" size in such an approximate solution can vary markedly between the two "equivalent" models. This difference can reflect directly into cost per computation run and hence can be of considerable engineering importance. To illustrate these general remarks a specific example will be used. The *R. L. Pritchard, loc. cit., gives an excellent summary as of 1956; see also Elementary Circuit Properties of Transistors, SEEC, Vol. 3, Wiley, 1964, pp. 212-217.

circuit in Fig. 7 is one which is commonly specified for measurement of the storage time of a switching transistor. Assuming the circuit has reached equilibrium in the saturated condition before t = 0, the problem is to calculate the storage and turn-off transients. For this example, the transistor parameters are taken to be aF = 0.98 = aN o aR = 0.50 = aI o Ebb -volts E11 -10 volts 500 R K 0. 9 1 fd. R B + inv (t) E 0.. oF WR = 10 rad/sec = CTC -TE ICs =IE = 10 amps ICS = ES For simplicity both the alphas and the transition capacities are taken as constants. 22

4. 1 Voltage-Controlled Model in Saturation Region Using the voltage-controlled model of Fig. 1, with vR and vF as the dependent state variables, the equilibrium equations are obtained by writing Kirchhoff's current law at the collector and the emitter: ICS XXvR dv Xv Xv v+E c VR CTC +WR dt - 1 +aFIES e 1F + F CC (66) and /(CE IES F)v dv -IF) + a vR V- vF- ECC + -- 1 + TE WF dt ERCSRC vF + Eth ~~R.(67) th Here Eth is the Thevenin voltage seen looking into the external circuit from the baseemitter terminals of the transistor, and Rth is the Thevenin resistance at the same point. In the steady state, prior to t=O, Eth = -11 volts and Rth = 1000 ohms. When v. in(t) jumps to +10 volts, the Thevenin voltage is 10 minus the initial voltage on the coupling capacitor; and the Thevenin resistance is 500 ohms. These equations have been integrated numerically for the particular case presented in Fig. 7.* The initial equilibrium values were found by a Newton-Raphson solution of the equations with the derivatives set to zero and with Eth = -11 volts anc Rth = 1000 ohms. Taking X = 40 volts, the initial values are vF(O0) = 373 mv vR(O-) = 363 mv vC (0-) = -5. 687 volts Therefore for t _> 0 *This problem was assigned by Prof. D. A. Calahan as part of his course in computer analysis of electrical networks; the writer used solutions obtained by N. E. Abbott and R. J. Talsky. 23

Eth = +4. 313 volts, and Rth = 50052. (The change in v is negligible for the times of interest.) Writing c1 vF(t) and vR(t) as these initial values plus changes vf(t) and vr(t), Eqs. 66 and 67 reduce to 40v 40v dvr -20.06 e + 30.01e ef Vf vr 9. 99 dt = 40V -9(66a) (2 + 804 e x 10-9 and 40v 40v dvf -30.07e + 10.03e V+ -3v 10. 618 dt /40v (67a) 2 + 1228 e f x 10-9 where now vf(O+) = vr(0+) = 0. Table I summarizes the results of some numerical integrations of Eqs. 66(a) and 67(a) for the first two nanoseconds of the solution. The first three pairs of columns give the results of Euler integration with time-step sizes of 1, 0. 5, and 0. 2 nanoseconds. The last pair of columns give the results of a Runge-Kutta integration with a 0. 1 nanosecond time step, and this is taken as the "exact" solution. Since the exponential terms decrease 9. 5 percent every time vf or vr change by 2. 5 millivolts, it is clearly desirable to choose a step size small enough to limit Avr and Avf to this value or less. Here a uniform At of 0. 1 nanosecond would probably be adequate for most purposes. A more sophisticated integration —such as the Runge-Kutta method — allows much larger step sizes; in this example, a step as large as 1 nanosecond did not seriously change the Runge-Kutta result. 4. 2 Charge Controlled Model in Saturation Region Consider now the same example using the charge controlled model of Fig. 4. Again, the appropriate equilibrium equations can be obtained by applying Kirchhoff's current law at the collector and the emitter terminals. 24

Euler Integration Runge-Kutta At= 1 At= 0.5 At=0.2 At= 0.1 t in v v Nanoseconds r f r f r f rf 0 0 0 0 0 0 0 0 0 0.1 0.2 - 0.010 - 3.158 - 0.432 - 3.154 0.3 0.4 - 0.90 - 6.083 - 1. 558 - 6. 128 0.5 - 0.0249 - 7.895 - 2.325 - 7.561 0.6 -2.395 - 8.947 - 3.210 - 8.966 0.7 0.8 -4.371 -11.567 - 5.300 -11.712 0.9 1.0 - 0.0497 -15.78 - 5.080 -14. 173 -6.827 -14. 111 - 7.759 -14.401 1. 1 1.2 1.3 1.4 1. 5 -12.160 -20. 066 -15.557 -21.077 1. 6 1.7 1.8 1.9 2.0 -18.49 -23.86 -26.256 -28.033 All voltages are in millivolts Table I. Numerical integration of Eqs. 66(a) and 67(a) X(qI + iCS/WR) dt 1 + -E'q+~ q N + E(68)S 25

and ( 1 C dq+ C TE dqN X(qN + ES/W F) dt E ES ECC + n1 n ES co q + a,q F FqN +R RqI R RB (69) In writing Eqs. 68 and 69, Eqs. 32 through 35, 24 and 25 have been used to make the comparison with Eqs. 66 and 67 evident. On the face of things, one might conclude that there is little to choose between Eqs. 68, 69, or Eqs. 66 and 67; both are rather "messy" pairs of nonlinear differential equations. Notice, however, that the importance of the nonlinear terms depends on the operating range. For example, the only nonlinear term on the righthand side of Eq. 68 is simply the collector-emitter voltage divided by RC. Clearly, this term will be negligible in the saturation region (qI> 0). On the right side of Eq. 69 an additional nonlinear term is just vEB/ RB, and we know that even a 100:1 change in qN(t) will only require a 100 millivolt change in vBE. The same exponential law which produced such rapid changes in the coefficients of Eqs. 66 and 67 with voltage, is working for us! The only other nonlinearities in Eqs. 68 and 69 are in the coefficients of dqI/dt and dqN/dt. These nonlinear effects will be negligible provided CTC ICS qI(t) > T C (70) X R and CTE IES qN(t) > > TE - (71) F For the problem at hand CTC/X = CTE/X = 5 x 10- coulombs while ICS/WR and IES/WF are both 1017 coulombs. Further, the initial values of the stored charges are 26

qN(0-) = S( e 1 = 3. 07 x 10-11 coulombs and qi(0-) -i R e 1 = 2.01 x 10-11 coulombs Conditions in Eqs. 70 and 71 imply that the storage (or diffusion- capacity) current is much greater than the transition capacity current. In the present example the stored charges can drop to one percent of their initial values before the transition capacitors represent five percent of the total. In summary, while the voltage- controlled Eqs.66 and 67 are very nonlinear in the saturation region, the charge-controlled equations, are almost linear! The fact that the storage time can be calculated closely, by solving linear differential equations was pointed out orginally by J. L. Moll (Ref. 10). In the present example, writing the stored charges as their initial values plus changes qn(t) and qi(t) and substituting the numerical values reduces Eqs. 68 and 69 to /3.07 x 10 11 + qn -q + 0. 98 q + 2.5 x 10-14n 11 dqi n 2.01 x 10 + qi 2. 49 x 10-3 -9 1+ x 10 - qi 2.01x 10 and /3.07x1011 +qn 14(30710617 -qn+O. 5qi- 1.928x10 -12.5x10O i4n - 5x10- n(3.07x106+1017 qn) dqn 1. 629 x 10- + AL, qn 1 + 1'ef 1 3. 07 x 10- l/(69a) 27

Linear approximations to these equations are 9 dqi 10 - = -qi + 0. 98 qn (68b) and 9 dq n 10 dt - qn + 0 5qi 2. 005 x 10 (69b) Solving these linear equations with the specified initial value one finds qi(t) = (-3.849 + 4. 674 e.t - 0.825 e t x 11 coulombs (72) and -0 3t - 17t 11 qn(t) = (-3. 927 + 3. 338 e + 0. 589 e )x 10 coulombs (73) for Iqi(t)l > 0. 0201 x 10-i coulombs and t in nanoseconds. Table II compares these solutions with the results of the Runge-Kutta Integration of Eqs. 66(a) and 67(a). The voltages are related to the charges by Eqs. 4. 1 and 4.2 which for the present problem reduce to + 3.07 ] vf = 25Tn (1 + 101%) and yr = 25fn 1 + 2.01 In calculating Table II a slide rule was used to evaluate the necessary exponentials and logarithms. Clearly, the agreement between the numerical integration of the nonlinear equations of the voltage- controlled formulation and the linear approximation to the chargecontrolled equations is excellent over the saturation region of operation. For this example, the transistor remains saturated for 3. 1 nanoseconds. 28

Runge- Kutta Linear Approximation Eqs. 68(b) and 69(b) Solutions Integration of Eqs. 66(a) and 67(a) Charge in Picacoulombs Voltage in mvy Voltage in mv Time in Nanoseconds i(t) | (t) v Vf vr f 0 0 0 0 0 0 0 0.2 - 0.3443 - 3.639 - 0.434 - 3.153 - 0.432 - 3.154 0o 4 - 1.213 - 6.677 - 1.556 - 6.13 - 1.558 - 6. 128 0,~6 - 2.418 - 9.260 - 3.200 - 8.91 - 3.210 - 8.966 0.8 - 3.849 -11.503 - 5.315 -11.76 - 5.300 -11.712 10 - 5.362 -13.460 - 7.760 -14.42 - 7.759 -14.40 1.2 - 6.957 -15.218 -10.620 -17.11 -10.591 -17.06 1.4 - 8.546 -16.794 -13.83 -19.77 -13.80 -19.73 1o 6 -10.101 -18. 220 -17.46 -22.50 -17.43 -22.43 1. 8 -11.646 -19.547 -21.65 -25.35 -21.55 -25. 19 Table 11. Comparison of linear charge-controlled and nonlinear voltage-controlled equation solutions 4. 3 The Linear and Cut-Off Regions 4. 3. 1 Using Voltage- Controlled Model. The previous sections have compared the voltage- controlled and the charge- controlled transistor models for calculations in the saturation region. In this region the charge-controlled equations are "almost" linear whereas the voltage-controlled equations are "strongly" nonlinear. It is illuminating to continue the comparison of the two models into the normal, or linear operating region. In this region: vR and ql are less than zero, but vR and qN are still positive. The voltage-controlled Eqs. 66(a) and 67(a) and the charge- controlled Eqs. 68(a) and 69(a) are written in terms of incremental voltages and charges. At the "edge" of the linear region vr(t) = -0. 363 volts and 29

qi(t) = -2.01 x 10 coulombs The Runge-Kutta integration of Eqs. 66(a) and 67(a) gives for the time to reach this point (the storage time) T = 3.23 nanosec, and at this time Vf(3. 20) = 0. 0492 volts and therefore qn(3.23) = -2. 64 x 1011 coulombs As the operating point moves into the linear region, v (t) is going to become more negative, and therefore the terms in Eqs. 66a and 67a which depend exponentially on v r(t) are negligible. These equations then reduce to 40vf dvr 30.01 e + - v - 9.99 r f r (67b) 2 x 10-9 (67b) and 40v dvf -30. 07 e e + vr - 3vf + 0. 618 2 + 1228 e x 10These are still "strongly" nonlinear in vf, but no difficulty is to be expected if the numerical integrations, begun in the saturation region, are continued into the linear region. The edge of the "cut-off" region is defined by the point vf(t) = -0. 373 volts This solution of Eqs. 66(a) and 66(b) reaches this point at t = 4. 69 monoseconds, and vr(4. 69) = -4.63 volts 30

This is less than half way to the cut-off equilibrium value of - 10 volts! This brings out a point, which does not seem to have been recognized in the literature; there can be a substantial difference between the time required to reach "cut-off, " as defined by both junctions being reverse biased, and the time needed to bring the collector current down to 10 percent of its saturation value. For times greater than 4. 69 nanoseconds the remaining exponential terms in Eqs. 66(b) and 67(b) become negligible, and the voltage- controlled equations reduce to the pair of linear equations dvr vf -vr- 9. 99 dt - -9 (66c) 2xO10 and dvf Vf - 3vf + 0. 618 dt 109 (67c) 2 x 10 These equations are linear because of the assumption, for this example, that the transition capacities CTC and CTE are constants. The expected voltage dependence of these capacities would make the equations nonlinear. Continuing the integration of Eqs. 66(c) and 67(c) one finds that the collector current reaches - 1 milliampere at t = 10. 7 nanoseconds. Therefore the turn-off time for the collector current is 10. 7 - 3. 2 = 7. 5 nanoseconds while the transistor junction cut-off time is only 4. 69 - 3. 23 = 1. 46 nanoseconds! 4. 3. 2 Using Charge-Controlled Model. Turning to the charge-controlled Eqs. 68(a) and 69(a) as qi(t) - -2.01 x 1011 coulombs one would expect that the linear approximation Eqs. 68(b) and 69(b) would no longer be valid, since two of the neglected logarithmic terms in Eqs. 68(a) and 69(a) approach infinity! Nevertheless, solving Eq. 72 for the time required for qi(t) to reach -2. 01 x 1011 (1 - 0. 498 x 10-6) coulombs (this corresponds to vr = 0.o 363 volts or vR = 0) gives 31

Ts = 3.16 nanoseconds and at this time, Eq. 73 gives qn(3. 16) = -2. 61 x 10 4 coulombs. These values agree very closely with those obtained from the numerical solution of Eqs. 66(a) and 67(a). Thus, the linear approximations, Eqs. 68(b) and 69(b), are seen to be useful right up to the very edge of the linear region. Certainly it is unreasonable to expect Eqs. 68(b) and 69(b) to continue to be accurate in the linear region. If one chooses to continue the solutions with the charges as state variables, the resulting nonlinear equations must be integrated numerically. 4. 3. 3 Combined-Voltage and Charge-Control. An alternative, at this point, is to go back and make a new selection of variables. Returning to Fig. 4, as qi(t) becomes negligible, Kirchhoff's current law at the collector node of the charge controlled model gives (see Eq. 27) aN d C = TEN q(t) dt (CTC VR) and at the emitter node dqN qN d (75) -ic- iB = dt + + dt (CTE F) EN In the circuit of Fig. 7 -vR + VF + EC C iC R (76) C and Eth + vF iB =Rt ~(77) As long as the transistor is in the normal, or forward operating region, 0. 324 volts < vF < 0. Over most of this region vF will lie withing 50 millivolts of 32

0. 324 volts. This suggests neglecting the change of vF in this region. When this is done, Eqs. 74 through 77 reduce to d VR aN ECC + 0. 324 dt (CTC R)+ff NO C (78) c EN c and qN +. q (t) - ECC - 0. 324 -Et - 0. 324 dt + q N(t) = + (79) EN c c th For the present example, CTC is a constant, and these equations are linear in the variables vR(t) and qN(t). Introducing numerical values gives dvR + 0.5 x 109 vR - 0. 49 x 1021= 48384x10 (78a) dt' N 102 and dqN 9 3 2 dt' + 10 qN - 10 vR = 4.02 x 10; (79a) with initial conditions VR(0+) = 0, and qN(0+) = 3.07 - 2. 61 = 0. 46 x 10 11 coulombs.* The solutions of this pair of equations are t' 3t 12 t t' 10 qN(t') = -463.7 + 467. 559 e + 0.741 e 2 coulombs (80) and t' 3t vR(t') = -464.1 + 464.441 e - 0. 341 e volts (81) where t' is expressed in nanoseconds. Equations 78 and 79, and hence their solutions, apply as long as qN(t') > 0. Using Eqo 80 one finds that qN(t') reaches zero at t' = 1. 27 nanoseconds, and vR(1.27) = -3.65volts Here, the time origin has been shifted to the instant at which the transistor enters the "linear" region. This is 3. 23 nanoseconds after the application of the base voltage step. 33

This "cut-off time" is 15 percent shorter than the value calculated by the voltage-controlled Eqs. 66 and 67, and the voltage drop is about 27 percent less. It is clear that over this traversal of the transistor's normal operating region the exponential, exp (- ), in Eqs. 80 and 81 is well approximated by the first two terms in its power series expansion so that 3 t 12 10 qN(t') F 3. 859- 3. 117t' +0.711e e (82) and vR(t) -3. 096t' + 0. 341 (1- e ) (83) as long as 0 < t' < 1. 5. These solutions predict an almost linear time dependence for both qN(t) and vR(t). This essentially linear time behavior is observed in the voltage- controlled equation solutions. Beyond t = 3. 23 + 1. 27 = 4. 5 nanoseconds, the assumption of a fixed vF is no longer valid and as a result, Eq. 4. 1 must be used to eliminate vF in Eq. 75. The result is a nonlinear differential equation in qN(t)o However, the voltage-controlled Eqs. 66 and 67, in the cut-off region became linear dv 2 x 10 dt"+ vR vF = -10 (66d) and 9 dv 2 x 10 dt + 3vF - vR = 1. 687 (67d) The solutions of these equations with the initial conditions vR(O) = -3. 65 and vF(O) = 0. 324 are vR(t") = -14. 313 + 10. 582 e -0 29 29t + 0. 081 e-1 7071t volts, (84) and 34

vF(t) = -4. 313 + 4. 4489 e-0u 2929t + 0. 1881 e- 7 t volts (85) The collector- emitter voltage is the difference of these two junction voltages vCE(t") = -10 + 6. 133 e 2929t _- 0. 107 e 7071t" volts(86) In all three of these solutions t" is the time in nanoseconds measured from the instant at which vR(t) = 0. Equation 86 is a direct indication of ic(t) since vCE = ECC - iC RC. Equation 86 predicts the time required for ic to reach -1. 0 ma, t" = 6. 19 nanoseconds, which corresponds to a total time t = 4. 5 + 6. 19 = 10. 69 nanoseconds from the time of application of the base voltage pulse. Figure 8 summarizes the numerical example presented here. The solid curves are plots of the numerical integration of the voltage-controlled nonlinear equations, and the crosses represent the values obtained with the three linear sets of equations. 35

8 x 6 iB Linear ~a~~~x X 2 Saturation -[ 14 Cut-off 2 I -4 1 2=~~~~~ 0 2 4 6 8 1 IK Time in Nanoseconds -2 Nonlinear voltage-controlled model -6- x x —x Linear models 1. Charge controlled in saturation region -8 Kx 2. Charge and voltage controlled in 1/ 3. Voltage controlled in cut-off region x -10 K Fig. 8 Calculated current response of saturated transistor (see Fig. 7) 36

5. SUMMARY The example of Sec. 4 brings into focus some of the situations which may be encountered when a particularly large signal transistor model is used to predict circuit behavior over a wide range of operating conditions. This example carries the transistor through all three operating regions: saturation, normal (or linear), and cut-off. If one wishes to describe the transistor by a single model, then either the voltage-controlled or the charge-controlled models lead to "strongly" nonlinear differential equations over two of the three operating regions. In the saturation region the voltage-controlled formulation leads to strongly nonlinear equations while the charge-controlled equations are "almost" linear. In the cut-off region the voltage-controlled equations are almost linear, and the charge-controlled equations are strongly nonlinear. Finally in the normal operating region both the charge- controlled and the voltage- controlled models yield strongly nonlinear differential equations. In this region, however, one can obtain a set of almost linear equations by choosing as variables one charge and one voltage (qN and vR). In computer analysis programs one may prefer a single model for all regions of operation. This makes the analysis of nonlinear differential equations inevitable. If one is willing to change models, whenever a transistor transitions from cut-off-to-normal or normal-to-saturation, for example; then three linear models may be accurate enough for many calculations. The linear models are particularly attractive when hand calculations are involved and in giving the circuit designer a "feel" for the circuit behavior. 37

REFERENCES 1. Electronic Design, Vol. 15, No. 12, June 7, 1967, pp. 57-59 or A. F. Mahnberg, F. Q. Cornwell, and F. N. Hofer, NET-1 Network Analysis Program, 7090/94 Version, LA3119, Los Almos Scientific Laboratory, Los Almos, New Mexico, 1964. 2. D. A. Calahan, Notes on Computer Analysis of Circuits, Winter-Term 1967, University of Michigan, Ann Arbor, Michigan. 3. Dankwark Koehler, "The Charge-Control Concept in the Form of Equivalent Circuits,...,"B.S. T.J., Vol. XLVI, No. 3, March 1967, pp. 523-576. 4. J. J. Ebers and J. L. Moll, "Large Signal Behavior of Junction Transistors, " Proc. IRE, XLII, December 1954, pp. 1761-1772. 5. M. V. Joyce and K. K. Clarke, Transistor Circuit Analysis, Addison-Wesley Pub. Co., (96), pp. 231-233. 6. J. M. Early, "Effects of Space-Charge-Layer Widening in Junction Transistors, " Proc. IRE, Vol. 40, November 1952, pp.- 1401-1405. 7. R. L. Pritchard, "Electric-Network Representations of Transistors - A Survey, " IRE Trans. Circuit Theory, Vol. CT-3, pp. 5-21. 8. Mullard Reference Manual of Transistor Circuits, 1st Edition, 1960, pp. 83-84. 9. Elementary Circuit Properties of Transistors, SEEC, Vol. 3, Wiley, 1964, pp. 212-217. 10. J. L. Moll, "Large-Signal Transient Response of Junction Transistors, " Proc. IRE, Vol. 42, December 1954, pp. 1773-1783. 38

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UNCLASSIFIED Security Classification DOCUMENT CONTROL DATA R&D (Security claaeification of title, body of abetract and Indexing annotation must be entered when the overall report ia clasailied) 1. ORIGINATIN G ACTIVIY (Corporate author) Za. REPORT SECURITY C LASSIFICATION Cooley Electronics Laboratory UNCLASSIFIED The University of Michigan 2b. GCouP Ann Arbor. Michgan. 3. REPORT TITLE COMPUTATIONAL UTILITY OF NONLINEAR TRANSISTOR MODELS 4. DESCRIPTIVE NOTES (Typo of report and Inclulve date.) 7695-17-T (TM 98) - November 1967 S. AUTHOR(S) (Last name, firt name, Initial) Macnee, Alan B. 6. REPO RT DATE i.. TOTAL NO. OF PAGES 7b. NO. OF REPS November 1967 10 Oa. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) DA 28-043-AM-01870(E) b. PROJECT NO. C. E. L. TM 98 PO 21101 A042. 01. 02 6C. 9 6. AT1 AgR R 1. T LERRP REPOT NO(S) (Any other numbere that may be asslgned dlt report d. ECOM- 01870- 17-T 10. AVA ABSLlTY/LIMITA*ION NOTICLS This document is subject to special export controls an each transmittal to foreign governments or foreign nationals may be made only with prior approval of CG U.S. Army Electronics Command, Fort Monmouth, N.J. 07703 Attn: AMS EL-WL-S. 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY U. S. Army Electronics Command Electronic Warfare Lab., AMSEL-WLFort Monmouth, N. J. 07703 13. ABSTRACT Three nonlinear models for function transistors are shown to be equivalent, physically. Their differences lie in the state variables employed. The incremental behavior of these models is compared with that of the Hybrid-, r and high frequency tee. The computational utility of the voltage- controlled and the chargedcontrolled models are compared for a switching circuit example. If a single model is used to describe the transistor, all models are "strongly" nonlinear in one or more operating regions. In any single operating region, however, an appropriate model choice leads to "almost linear" equations. DD 1 JAN 64 1473 UNCLASSIFIED Security Classification

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