Technical Report ECOM-0138- 19-T HIGH- FREQUENCY TRANSISTOR MODELING FOR CIRCUIT DESIGN C. E. L. Technical Report No. 205 Contract No. DAAB07-68-C-0138 DA Project No. 1 HO 62102 A042 01 02 Prepared by A. B. Macnee R. J. Talsky COOLEY ELECTRONICS LABORATORY Department of Electrical Engineering The University of Michigan Ann Arbor, Michigan for U. S. Army Electronics Command, Fort Monmouth, N. J.

ABSTRACT It has been found that hybrid-pi or high-frequency T are inadequate high-frequency models for certain transistor types even though the models are supplemented by reasonable extrinsic elements. The hybrid-pi can be modified to model these transistors by replacing the r C7f circuit by an RC ladder. Using a computer optimization program an optimal, N-lump model is generated. For the 2N918 transistor a two-lump model extends the frequency range of the hybrid-pi model to fT/ 2. Typical circuit examples show most of the improvement in model performance can be obtained with a two-lump model. 111

FOREWORD This work was supported in part by the United States Army Electronics Command, Fort Monmouth, N. J., under contract DA28-043-AMC01870(E). The report is based on a report submitted by R. J. Talsky in partial fulfillment of the requirements for the E. E. Degree, University of Michigan, 1969.

TABLE OF CONTENTS Page ABSTRACT iii FOREWORD iv LIST OF ILLUSTRATIONS vi LIST OF TABLES viii 1. Introduction 1 2. Admittance Loci 5 3. Multilump Models 12 4. Typical Circuit Applications 28 5. Conclusions 35 REFERENCES 38 APPENDIX A 39 APPENDIX B 44 DISTRIBUTION LIST 55

LIST OF ILLUSTRATIONS Figure Title Page 1 Popular high-frequency, incremental models of a bipolar transistor; (a) high-frequency T and (b) high-frequencv hvbrid-oi and the parameter interrelations 2 2 Measured short circuit admittance loci for a 2N918 at Vce =4V 6 3 Effect of interlead capacitances on the short circuit input admittance locus of 2N918 at Vce = 4V and I 2 mA 11 c 4 One-, two- and N-lump models of distributed base charge 13 5 Flow chart of transistor model optimization program 16 6 (a) N-lump extension of hybrid-pi model plus extrinsic capacitances; (b) reduced model for Yie and Yfe calculations 18 7 -Measured and calculated y and yre values for a 2N918 at 4 V and 2 ~meA 20 8 (a) Computer optimized models for Yie and yfe (b) A complete 2-lump model for 2N918 at V = 4V, Ic= 2 mA (ohms, picafarad, nanohenrys, ana mhos) 23 9 Percentage error magnitude versus frequency for computer optimized 1-, 2-, and 3-lump models of a 2N918 25 10 Measured and calculated admittance loci for 2N918 at Ic = 2mA, Vce = 4V; frequency is in megahertz 27 11 Typical circuits for comparing transistor models; (a) emitter-follower (b) 200 MHz tuned amplifier, (ohms, picofarads, nanohenrys) 29 vi

LIST OF ILLUSTRATIONS (Cont.) Figure Title Page 12 Calculated emitter follower step responses using 1-, 2-, and 3-lump 2N918 models 31 13 Calculated frequency response of 200 MHz amplifier using 1-, 2-, and 3-lump 2N918 models 34 14 Plots of measured y parameters for 2N918: (a) Yie (b) Yre (c) Yfe (d) y 39 vii

LIST OF TABLES Table Title Page I Model comparison in emitter follower circuit 30 II Model comparison in tuned amplifier 33 III Frequencies of susceptance maxima for typical transistors 36 IV Short circuit admittance parameters of a 2N918 transistor measured on G. R. Transfer function and Immittance Bridge 41 V Short circuit current gain of a 2N918 transistor measured on G. R. Transfer-Function Bridge 42 VI Short circuit admittance parameters of a 2N918 measured with RX meter and a Transistor Test 43 Jig * * *

1. Introduction Analysis and syntnesls ot circuits contammg bipolar transistors require suitable device models. For this purpose, the most popular high-frequency, incremental models are the hybrid-pi and the high-frequency T (Ref. 1, 2). Under the usual high-frequency assumptions these two models are equivalent as indicated in Fig. 1. The popularity of these models stems from their simplicity and the relatively good correspondence between elements of the models and the basic physical processes in the device. In particular, the capacities Ce and C each have two components: a barrier, or depletion layer, capacitance and a base charge, or diffusion, capacitance. In modeling the dynamics of the charge stored in the base by a single lumped capacitance, one neglects the distributed nature of the base charge. The excess phase factor exp(-) multiplying the controlled source in each model is an attempt to include the first-order effects of the distributed nature of the base change. It also should be noted that these are models of the intrinsic devices; header capacities and lead inductances must be added to describe most packaged devices accurately. For the circuit designer, determination of the model parameters for a particular transistor type can be a significant problem. Device manufacturers usually do not provide enough data to allow

ns raile e I reOQ - reC b (Ce c C! A1 C uT b V g eL Cgr = Ce(I+n) rT=v(/+I)re (b) CAr =C m re Fig. 1. Popular high-frequency, incremental models of a bipolar transistor; (a) hi(h-frequency T and (b) high-frequency hybrid-pi and the parameter interrelations

calculation of all the parameters of the models in Fig. 1. The values for rx, C and n are particularly hard to find unless rather complete high-frequency data, such as a set of short-circuit admittance parameters, are provided. A perusal of manufacturer's data sheets reveals that extensive high-frequency data are available for a very limited number of devices. In 1966, Sidney Chao reported on the y-parameters of a collection of twenty transistor types produced by seven different manufacturers. That report seems to include most of the generally available data as of that date (Ref. 3). In connection with wideband amplifier and oscillator studies, the writers had occasion to measure the short circuit admittance parameters of a small, planar, high-frequency transistor, the 2N918 (Appendix A). Analysis of this measured data revealed that using the circuits of Fig. 1, it was not possible to model this transistor satisfactorily over a frequency range greater than 0 to 0. 1 fT' even though extrinsic elements were added. The observed differences between the hybrid-pi model predictions and the measured data are most easily observed in plots of the short-circuit admittance loci as indicated in the second section of this paper. These differences suggest that it is the "single-lump" representation of the distributed base charge which largely is responsible for the model's failure. In Section 3, two- and three-lump approximations to the distributed

base charge are added to the basic hybrid-pi, and a computer optimization program is used to select the parameters of this extended model. Finally, in Section 4 the responses of some typical circuits using the hybrid-pi and the extended hybrid-pi are compared. It is our conclusion that it is worthwhile to extend the hybrid-pi to a "two-lump" approximation, which requires the addition of one R and one C to the model. This extended model fits the 2N918 measured y-parameter data to within 10 percent up to about fT/2, and can produce significant changes in the predicted response of typical circuits. The model can be extended further, but this does not seem to be worthwhile for the 2N918.

2. Admittance Loci The short circuitadmittance parameters of 2N918 transistors were measured over the frequency range 1 to 900 MHz for collectoremitter voltages from 1 to 8 V and collector currents from 0. 5 to 8 mA. Some of the lowest frequency measurements were made with a Hewlett-Packard Vector voltmeter, but the bulk of the data was taken with a General Radio Transfer- Function and Immittance Bridge (50900 Mhz range) and with a Boonton RX meter (10-200 MHz). In general there is good correspondence between the RX- meter and GR Bridge data over their common frequency range (50-200 MHz). These data are also qualitatively in good agreement with typical data published by manufacturers of this transistor type (Ref. 4). The inadequacy of the simple hybrid-pi model for this transistor is most easily seen if one examines the Yie and yfe admittance loci with frequency as a parameter. Figure 2 plots these loci using the data in the appendix for VCE = 4 V and Ic = 0. 5, 2. 0 and 8. 0 mA. It will be noted that the frequency at which bie(w) reaches a maximum is much higher than the frequency at which bfe(W) is a minimum for each quiescent current. This is in conflict with the predictions of the usual hybrid-pi model. The short circuit input and forward transfer admittances of the hybrid-pi model in Fig. l(b) are

bie in 900 IC 0.5 ma mmhos 15 15 700 900 0""0"2.0 ma 500 MHz 8.0 m. 100 0 8 5 10 15 20 25 gie in mmhos bfe in mmhos 20 40 60 80 100 0,,, I I I I I 0 MHz gfe In mmhos 500 0.5ma2.0 ma -200 50 100 MHz -40 - 60t 100 = 0 8 ma 0 50 MHz Fig. 2. Measured short circuit admittance loci for a 2N918 at Vce 4V c e

1 + sr C 7 T Yie r + r + sr r C x V x r T and ns WO! P3e -sr C (2) fe r + r + srrC (2) x 7r x 7 T where C C - C T v U Both of these admittances have the same pole at the transverse cutoff frequency r +r "b rrC (3) For s = jw, this pole dominates the frequency dependence of the susceptances r C b() T w bie(w) (4) (rX+ rf) 1+ WOb and

gr r2 CT cos(nw) + (r + r) [wr 1C + sin(no)] - bfe(w) = 2 2 (5) (r +rd ) (1+ ) Wb The value of wb is normally an order of magnitude smaller than wa, so that in the vicinity of wb the cosine and sine terms in (5) can be replaced by the first term in their respective power series. When this is done, Eq. 5 reduces to gmrr 2 CT + (r + r ) (r C + )L b() mx (6) fe (r +r)2 1+ Wb which has the same frequency dependence as bie(w). Equations (4) and (6) show that the hybrid-pi model predicts that the magnitude of bie and bfe will both reach a maximum at the same frequency, wb. Further examination of Eqs. 1 and 2 shows that as long as the excess phase exponential is approximated by the first two terms in its series, both admittance loci are semicircles. Returning to the measured loci for the 2N918 it is evident that one can infer very different values for wb depending upon whether one uses the input or the transfer admittance locus. The frequencies of the various susceptance maximum are tabulated as follows:

Frequency of Susceptance Frequency maxima ratio bie bfe I = 0. 5 mA >900 MHz 500 MHz >1.8 I 2. 0 mA 750 MHz 150 MHz 5 I = 8.0 mA 500 MHz 50 10 Since Fig. 1 and Eqs. 1 and 2 are for the intrinsic transistor, one is led naturally to ask if the observed differences between the model predictions and the measured data can be explained by extrinsic elements. It is our conclusion that this is not the case. The principal extrinsic elements are the bulk resistances, the lead inductances, and the interlead capacities. Taken alone, the interlead capacities simply add linear susceptance terms to Eqs. 4 and 6. These can lead to susceptance maximum at differing frequencies, but nothing like the observed frequency ratios can be explained with reasonable capacity values. For example, subtracting the linear susceptance of 2 pF from the Yie data at 8 mA only shifts the bie maximum frequency to about 450 MHz. Lead inductances can also cause some changes in the frequency of the susceptance maxima, but they tend to move the input susceptances down toward negative values at higher frequencies. There is

little evidence of this in the 2N918 data for the collector current and frequency ranges investigated. The 2N918 is mounted in a standard TO- 18 package. Measurements on such a package with the transistor disconnected give interlead capacities of the order of 0. 6 pF between each pair of leads. Figure 3 replots the yie(jw) locus for I = 2. 0 mA, and shows the ie C result of subtracting linear susceptances corresponding to 0. 6, 1. 2 and 1. 8 pF. It will be noted that even for the 1. 8 pF case, which reduces the magnitude of the susceptance maximum by a factor of 2. 2, the frequency of the maximum has only been reduced about 100 MHz to 650 MHz which is still 4. 3 times the frequency of the transfer susceptance maximum. The other interesting feature of these loci is their shape. Rather than being semicircles, as the hybrid-pi would predict, they look more like two semicircles joined by a tangent line. This observation naturally led to the consideration of the extension of the hybridpi to a two- or N-lump model.

bie- (CBE+CBC) in mmhos 15 700 CBE + CBC =0 900 MHz ~~I0-~500 xx /6 pf 1l10 2 1.2pf I00 50 0 5 10 15 20 25 g;, in mmhos Fig. 3. Effect of interlead capacitances on the short circuit input admittance locus of 2N918 at V = 4V and I = 2 mA ce c

3. Multilump Models In the hybrid-pi model the dynamics of the minority charge diffusing across the base region of the transistor are modeled by the parallel r C circuit. The element C also includes a component representing the barrier capacitance across the emitter-base space charge region, but for the usual emitter currents this term is dominated by the "diffusion" capacity. Physically, we know the diffusing charge is distributed across the base region, but both the hybrid-pi and the high-frequency T equivalent circuits treat this charge as though it were lumped at a single point. A distributed RC circuit could model the distributed charge exactly, but it would be awkward to use in circuit calculations. As an alternative it is natural to consider replacing the one-lump approximation to the distributed charge by a multilump approximation as shown in Fig. 4. It is easily seen that introducing these RC ladder networks can give rise to differences between the driving-point and the transfer susceptance of the kind observed in the 2N918 measurements. The driving-point admittance has alternating simple zeros and poles along the negative real axis of the s-plane; whereas, if the voltage controlling the g -generator is taken from the end of the line, the transfer admittance is an all-pole function. By taking the gm-generator control voltage at other points along the line one can introduce a 12

ao~Ltzq aseq palnqT.lsip Jo slapouw dunIl-N pu3 -omx'-auo'7' (D) NZ I. 00 (q) (D) 1%10 A xI 1 t~ + ~ 1 I~....8 ^/k/~'-A/ 9,,. o~~(o

further degree of flexibility into the zeros of the transfer function. Initially Bode plots of the Yie and Yfe data were used to select the poles and zeros for a two-section ladder of the form shown in Fig. 4(b). With plots of the log magnitude and angle versus log frequency it is not difficult to make reasonable estimates of suitable asymptote break frequencies. Having the poles and zeros, it is a straightforward matter to synthesize the RC ladder. After several examples had been done'"by hand, " a digital computer program that selects the component values to minimize a suitable error measure was written. The error function selected is c data c model ERR = W iof Yie Yie (2 c dcatai freq. c data c model P Yfe Yfe + (W2 (7) c data Yfe where W1 and W2 are weight constants, and P is a positive integer specified by the user. The quantities Appendix B gives tile FORTRAN IV listing of this program.

Yie Yie- S(CBE+ CBC) (8) and Yfe Yfe Yfe CBC (9) are the short-circuit input and forward transfer admittances of the intrinsic transistor, i. e., corrected for the interlead capacities CBE and CBC. The transfer and input admittances of the ladder network are evaluated efficiently by a simple cumulent algorithm (Ref. 5). The derivatives are estimated by incrementing element values in the simplest fashion. A flow chart of the optimization program is given in Fig. 5. The program subtracts the admittances of the interlead capacities to obtain Yie and Yfe of the intrinsic transistor. A FletcherPowell minimization routine is then used to minimize Eq. 7. The minimization routine is one given by Calahan (Ref. 6). A similar routine is available as a part of the IBM Scientific Subroutine Programs package (Ref. 7). When the minimization routine cannot reduce the error more than a specified percentage, the program adds the interlead capacities to the optimized model y-parameters and prints out the y-parameters and errors as a function of frequency as well as the "optimized" model values.

READ: Wl, W2, P, N, CBC AND CBE yie AND ife DATA RI, Cl,.. R2N+l Gm, AND LB CALCULATE CYie AND Cyfe CALCULATE ERR, AERR..., AERR ARI AL6 ERRLAST - ERR > EPSI YES NO l~CALL MIN 1PRINT ELEMENT (Fletcher- Powell) VALUES, y's AND INCREMENT ERROR COMPONENTS FRI' 5''LB i Fig. 5. Flow chart of transistor model optimization program

The complete N-lump transistor model is shown in Fig, 6(a). Our program optimizes R1, C2,..., LB to match the measured Yie and Yfe data. The program could be extended to optimize simultaneously on all four y-parameters, but initial experience has suggested that this is often unnecessary because the output and reverse admittances are relatively independent of the stored-base charge. When the collector-emitter terminals of the model are shorted, CCE is removed from the circuit, C is in parallel with C2, and CBC is in parallel with CBE as shown in Fig. 6(b). The optimization program takes account of the specified CBC and CBE, but the C2 it calculates includes C. The y and y data for the transistor are used to select CCE and C;at high frequencies re -' -sCBC (10) and C C2 -oe s(CCE + CBC + C +2) (11) Aoe frequenC +cies 2 At low frequencies

CBC 8 C2 C R BE CCE (a) E ILB R3 I + +ie VCT 2 02N v mYfVI CBE+ CBC R2N+I iCYle - I (b) Fig. 6. (a) N-lump extension of hybrid-pi model plus extrinsic capacitances; (b) reduced model for Yie and yfe calculations

R3 +'. + R2N+ 1 Yre - CBC R+ C R 3+... +R(12) 1 - 1'' + R2N+ 1 and gmR1 R s C C +BC +C [+R Yoe E C + C R + +... + R5 (13) In principle, having found the ladder element values R1, C2,..., R2N+ 1 and Gm, Eqs. 10 and 11 can be used to determine C C and C. As a practical matter, real data may not be BC' CE' precise enough at low frequencies to show a significant low frequency asymptote shift. As an illustration, consider the 2N918 at IC = 2 mA, VCE = 4V. The y-parameter measured for one unit are tabulated in the appendix. The input and transfer-admittance loci for this transistor already have been presented in Fig. 2. The Yoe and yre parameters, which are predominantly susceptive, are plotted in Fig. 7. Log-log coordinates are used so that the asymptotic susceptances are straight lines with a slope of +1. Looking at the data one cannot discern a significant difference between the low- and high-frequency susceptance asymptotes. The -bre data can be fitted very well by a single line corresponding to a capacity of 0. 68 pF, and the b data are well matched by a 1.63 pF capacity. On the basis of these data we can conclude for this transistor that C is I-L

20 100 90 80 70 60 _ c MEASURED POINTS 50 - - MODEL CALCULATION 40 - if oe O 4030 25 20 - o/ /.bre 15 0 0.9 0.8 0.70.6 -oe 0.5 0 0.4 0.3- 0 025 0 -9re 0.2- 0 0 0.15 0.1 10 15 20 25 30 40 50 70 100 150 200 300 400 600 1000 Frequency in MHz Fig. 7. Measured and calculated Y and y values for a 2N918 at 4 V and 2 mS re

less than 0. 068 pF (10 percent of the measured b re), and for circuit design purposes it can be neglected. For the conductance data, gre was found to be too small for reliable measurement with the setup used. The measurement of g is also difficult for this small transistor, and a number of additional measurements were made in an attempt to improve the data. Particularly at the lower frequencies, data spreads of up to ~30 percent are evident. Above 500 MHz, however, the data becomes reasonably smooth, and goe seems to approach an asymptotic slope of +2. At low frequencies, the data "appear" to approach a constant value between 0. 15 and 0.2 millimhos. The susceptance data are well-modeled by the circuit of Fig. 6(a), by making CBC 0. 68 pF and CCE 0. 95 pF. Adding a shunt resistor across CCE with a value of 6000 ohms matches the low-frequency value of goe. The square-law frequency asymptote suggests a fixed resistance in series with the output capacity. For series RC circuit jwC Y (14) series RC 1 + jw RC which reduces to 22 w C R+jw C (15)

22 as long as w R C << 1. In Fig. 7, g at 900 MHz is 1. 04 millimhos so that the oe series resistance should contribute 1. 04 - 0. 167 = 0. 873 millimhos. Taking C as 1. 63 pF one finds the needed series resistance is 10. 4 ohms. Figure 8(a) gives the one-, two-, and three-lump models optimized by the computer program to match the Yie and Yfe data given in the appendix for the 2N918 at 4 V and 2 mA. Before carrying out the optimization the measured data (real and imaginary parts of the four y's) were plotted and "smoothed" by eye. The data fed into the optimization program were read from these smoothed curves at 15 frequencies spread relatively uniformly on a logarithmic frequency scale from 2 to 450 MHz. With the weighting factors all taken as unity the mean square errors for these three models are 0. 793, 0. 107, and 0. 0435. The improvement achieved by going from a oneto a two-lump model is significant; for circuit design purposes the additional improvement possible with three or more lumps seems less worthwhile for this transistor. The models in Fig. 8(a) are for the input and forward characteristics only. Adding on CBC, CCE, CBE, and the output resistance to any of these circuits gives a complete model. This is illustrated in Fig. 8(b) for the two-lump case. * That is, 2, 3, 4, 5, 7, 10, 15, 20, 45,..., 450.

23 120 1430 Z6 < F~v ),.0618 v 6.56 3.81 218 2.4i 5,48; <V.t.067v 12.5 3.72 227 25.5 0.9,,1300.0675 v 2.43 0.992 4.87 0.68 6.56 381 218 11 + 6000 2.47 5.48T 67 0.95 (b) Fig. 8. (a) Computer optimized models for Yie and Yfe (b) A complete 2-lump model for 2N918 at Vce = 4V, Ic=2mA (ohms, picafarad, nanohenrys, and mhos)

24 The nature of the model approximations is seen by examining the individual terms in Eq. 7. One hundred times these errors are plotted versus frequency in Fig. 9. For the one-lump (hybrid-pi) approximation, the Yie error is less than 10 percent up to 95 MHz, but the Yfe error exceeds 10 percent above 28 MHz. The large difference in these two frequencies is in keeping with our observations in Section 1 of the inadequacy of the simple hybrid-pi for this transistor. With the two-lump model, the Yie error is less than 10 percent up to 440 MHz, and the Yfe error reaches 10 percent at 325 MHz. The addition of just two elements allows a 3:1 improvement in the useable frequency range for the model and gives a much more balanced error. The addition of two more elements to make a three-lump approximation gives a further improvement particularly in Yfe above 50 MHz. Now the error magnitude never exceeds 7 percent over the full frequency range of 2 to 450 MHz. The solid lines in Fig. 7 are calculated for the two-lump model of Fig. 8(b). The errors in boe, bre' and g are less than 10 percent out to 900 MHz. The calculated gre is larger than our measurements indicated. (Our best estimate is that g is about -0. 2 millimhos at 900 MHz. ) For most circuit calculations this discrepency will not be significant since Yre is dominated by

25 15 0 "5 i/ -oI LUMP 2 LUMP o, /I 1,_ C~iQ/ / ~L5</ \, 0 10 100 1000 Frequency in MHz Fig. 9. Percentage error magnitude versus frequency for computer optimized 1-, 2-, and 3 —lump models of a 2N918

26 b re. If the refinement is desired, one can reduce the magnitude of re gre by splitting CBC into two portions, one as shown, and a second directly between the base and collector external terminals as indicated by the dotted lines in Fig. 8(b). The Yie and Yfe admittance loci are plotted in Fig. 10 for the two-lump model of Fig. 8(b). The circled points are the input data for the computer optimization program.

27 bie in mmhos 500 10 300 200/ o SMOOTHED MEASURED DATA 5 150 2 LUMP MODEL 100 — 3 LUMP MODEL 70 45 30 5 10 15 iie in mmhos tbfe in mmhos i mmhos -10 0 10 20 30 40 50 60 10 0 15 0o -10 20 0 30 0 -20 045 450 O 200 150 10 70 -30 -40 Fig. 10. Measured and calculated admittance loci for 2N918 at Ic = 2mA, Vce = 4V; frequency is in megahertz

4. Typical Circuit Applications The utility of one transistor model over another depends on the proposed application of the model. Since all models considered here are identical at low frequencies, we would expect that in lowpass circuits the differences in performance would be less than in bandpass circuits, particularly as the center frequencies exceed fT/ 10 (about 90 MHz for the 2N918 example). Two simple transistor amplifier circuits are shown in Fig. 11, an emitter-follower and a simple, single-tuned amplifier. The emitter-follower with a 1000 ohm resistive source and a parallel RC load of 100 ohms and 30 pF is a low-pass circuit with a -3dB bandwidth of the order of 100 MHz. In the second circuit the output inductors transform the 50-ohm load to match yoe at 225 MHz, and the input simply cancels the input susceptance at the same frequency. The overall frequency response is essentially that of the output singletuned circuit. Both of the circuits in Fig. 11 were analyzed using a linear circuit analysis program that calculated the poles and zeros of the specified input-output pairs as well as frequency and time responses. Table I summarizes the calculated poles and zeros, the frequency response, and the step response characteristics of the emitter-follower circuit. The step responses are plotted in Fig. 12. There are clear 28

29 1000 1+ + 100:30 VL (a) 334 50 [ * ~ ~+ +S 4 i 15e6 p 3940 50 VL'S 15.6 S (b) Fig. 11. Typical circuits for comparing transistor models; (a) emitter-follower (b) 200 MHz tuned amplifier, (ohms, picofarads, nanohenrys)

30 Calculated Transistor Model Used [Fig. 8(a)] Characteristics hybrid-pi 2- lump 3- lump Poles (Gigarad) -109. 3 - 98. 41 - 99. 15 - 14.07 -1. 787tj17. 02 -51. 56 -.3319~j. 3994 -,3078+j. 4334 -1. 231~j 12. 36 -2. 603 -. 3063~j. 4225 -2. 305 Zeros (Gigarad) -6. 928j 32. 89 -11. 03~j28. 48 -51. 59 -8. 533 -23. 09 9. 854~j22. 05 - 1. 276~j4. 608 - 18. 05 -2. 265~j4. 011 Voltage Gain -2. 187 dB -2. 179 dB - 2. 162 dB Bandwidth (-3dB) 90 MHz 95. 5 MHz 93. 5 MHz Gain Peak -2. 067 dB at - 1. 813 dB - 1. 836 dB 31.6 MHz at 42. 2 MHz at 42. 2 MHz Rise Time in 3.78 3.54 3. 58 ns (10- 90%) Delay Time 2. 64 2. 80 2. 80 Overshoot 7.5% 10.4% 10. 2% Table I. Model comparison in emitter follower circuit

1.0 2- AND 3-LUMP - MODELS 0.8 HYBRID-PI 0.6 U) a:.4 HYBRID-PI 0.4 0.4 HYBDI 2-AND 3- LUMP cn |. MODELS 0.2 0 2 4 6 8 10 Time in nsec Fig. 12. Calculated emitter follower step responses using 1-, 2-, and 3-lump 2N918 models

32 differences between the responses calculated with the hybrid-pi and those calculated using either the two- or three-lump model; the larger models predict more overshoot and delay and shorter risetimes. These differences are all small, and those between the twoand three-lump models are certainly negligible in this circuit. Table II summarizes calculated performance of the tuned amplifier. As in the emitter-follower example, there is a significant change when the hybrid-pi is replaced by the two- or three-lump model but the difference between the performance with the last two is much smaller. The three, calculated frequency response curves are given in Fig. 13. In this example, the small difference between the two- and three-lump models is attributable in part to the circuit design, which makes the relatively unchanging output admittance the most important frequency determining element. In some bandpass designs we have observed significant differences between the predictions of the two larger models.

33 Calculated Transistor Model Used [Fig. 8(a)] Characteristic hybrid-pi 2-lump 3-lump Poles (gigarad) -68. 5 - 35.3 -51.0 -3.27 -10.9 -5.77 -1.14 -0. 933 -0. 830 -0. 0116 -0. 0116 -0. 0116 -. 0792j 1. 23 -. 0981j 1. 23 -0. 978~jl. 24 -3. 44~j4. 93 -2. 36~j4. 80 -42. 5 Zeros (gigarad) -10.6 -0. 993~j 10. 3 -51. 6 9.41 6.19 -1.95~j 8.39 0. -7.62 6.21 0. 0. -5.02 0. 0. Maximum T. P. G. 20. 0 dB 17. 8 dB 17. 4 dB Center Frequency 195 MHz 195 MHz 197 MHz Bandwidth (-3dB) 25. 5 MHz 31. 6 MHz 31 MHz Table II. Model comparison in tuned amplifier

34 20 HYBRID-PI 18 16 14 C 2-LUMP &12 3-LUMP 6C 8 6 150 200 250 Frequency in MHz Fig. 13. Calculated frequency response of 200 MHz amplifier using 1-, 2-, and 3-lump 2N918 models

5. Conclusions At the outset it was pointed out that the hybrid-pi circuit plus extrinsic elements cannot accurately model certain small transistors over wide frequency ranges. The suitability of the hybrid-pi model for a given transistor is established easily with the Yie and Yfe admittance loci. The hybrid model can be expected to be satisfactory if the susceptance maxima of these two admittances occur at about the same frequency. We have examined the admittance data compiled by Sidney C. Chao for 16 transistor types (Ref. 3). For 14 of these we plotted Chao's data and observed or estimated the frequency of the susceptance maxima. * Table III summarizes the results of these observations. If the ratio of the frequencies at which the susceptance maxima occurs is 1. 5 or less, it seems probable that a one-lump or hybrid-pi model will be satisfactory. When the ratio exceeds 1. 5, it is expected that a multilump model will be required to match the transistor characteristics closely over a wide frequency range. On this basis, five of the transistors listed in Table III would require a multilump model: the 2N918, 2N2219, 2N2708, 2N2929, and 2N3663. It is not obvious to the writers, from published information, what makes these five types electrically so different from the other members of the table. The two transistors omitted were both experimental units. 35

36 volt m.Frequency of suscep-.s Quiescent point Frequency VCE in volt I in mA tance maxma ti VCE inMHz Yie Yfe 2N918 4 2. 0 750 150 5. 0 2N2219 20 20.0 150 <50 >3. 0 2N2415 6 2. 0 300 300 1. 0 2N2708 15 2. 0 600 50 12. 0 2N2857 6 1. 5 500 300 1. 7 2N2865 10 4. 0 400 300 1. 3 2N2929 - 10 - 10. 0 100 40 2. 5 2N3282 -10 - 3.0 300 200 1. 5 2N3553 28 25.0 29 33 0, 88 2N3570 10 5. 0 400 400 1.0 2N3662 10 5. 0 450 120 3. 7 2N3688 10 4. 0 150 100 1.5 2N3783 - 10 - 3. 0 200 200 1. 0 2N3866 15 10. 0 130 150 0. 87 Table III. Frequencies of susceptance maxima for typical transistors From the circuit designer's viewpoint, the why of the need for a distributed, stored-charge model is less important than the fact that such a model may be needed if certain transistor types are to be employed. If the transistor manufacturer furnishes typical data that establishes the y-parameters as a function of frequency, it is a simple matter to determine whether a multilump model should be considered.

37 If not, the measurement of Yie and Yfe versus frequency at a single quiescent point should be sufficient to establish the frequency ratio of Table III. Finally, if one wishes to play probabilities, our investigation of Chao's data suggests that there is at least a 60 percent probability that the simple hybrid-pi will be adequate. A two-lump extension of the hybrid-pi model does not represent a great increase in circuit complexity, particularly when computer analysis programs are used. This is particularly true since in the frequency ranges where these effects are noticeable, some of the extrinsic transistor parameters are almost certain to be important. The multilump models can match the transistor characteristics over the complete frequency range 0 to about fT. When only a limited frequency interval is of interest, a simple hybrid-pi model can be optimized for a particular frequency range, such as 0. 4 fT to 0. 5 fT' even though a multilump model might be required for the full range, 0 to 0. 5 fT. In this connection, it should be noted that an approximate "excess phase" can be added easily to the usual hybridpi model by paralleling the gm-generator with a second current generator flowing from emitter to collector and controlled by the current in thle C capacity (Ref. 8).

RE FERENCES 1. R. L. Pritchard, Electrical Characteristics of Transistors, McGraw-Hill, 1967, Chapters 5 and 6; contains a very complete bibliography. 2. P. E. Gray, et al., Physical Electronics and Circuit Models of Transistors, SEEC, Vol. 2, John Wiley & Sons, 1969, Chapter 8. 3. Sidney C. Chao, Application of Computers to RF Circuit Design, Final Report on Contract DA28-043 AMC-01347(E), ECOM-01347-F. June 1966. 4. Fairchild Semiconductor Transistor and Diode Data Catalog 1970, pp. 2-66. 5. J. L. Herrero and G. Willoner, Synthesis of Filters, Prentice-Hall, 1966, Chapter 1. 6. D. A. Calahan, Computer-aided Network Design, Preliminary Edition, McGraw-Hill, 1968, pp. 289-293. 7. IBM System/ 360 Scientific Subroutine Package, FMFP and FMCG, H20-0205-2, pp. 202-206. 8. Semiconductor Circuits, Engineering Summer Conference Course 7007, University of Michigan, Ann Arbor, June 1-5, 1970, Chapter 12. 38

30 60 U) ie o 25 Y E: V2 VCE 4 V 50 re I 2 mA VCE 4 V 20 - IC = 2 mA oG.R. Data Re[Yie. o 15;'ARX Meter Data Data A RX Meter Data / co~, HP Vector Volt$ 10- 20 - meter Data Im [Yi.], 5- g/ 10- k [Yfe 5 10 50 100 500 1000 5 10 50 10 0 oo 1000 Frequency (MHz) -10 Frequency (MHz) (a) (c) Io 6 Yre 12 Y yoe.~~ ~VCE=4V U 4 CEo ~ 10 VCE =4V e'::O IC = 2mA I = 2mA E 4 -- o G. R. Bridge Data 8 - G.R. Data 0 3 6a RX Meter Data r 6 2 1 2 F| o~ ~_L-:~! o I..R., eoe 10 50 100 500 1000 10 50 100 500 1000 Frequency (MIHz) Frequency (MHz) (b) (d) Fig. 14. Plots of measured y parameters for 2N918: (a) Yie (b) Yre (c) Yfe (d) Yoe

Frequency VCE = 4V VCE = 4V VCE = 4 VCE = 4V VCE = 4V VCE = 1 VCE4V VCE = 2V VCE = 8V (MHz) IC =.5mA I IC = 2 mA IC = 2 mA IC = mA IC I = 8 mA ICIC = 2mA A IC = 2mA 50 0.6 +j 1.6 0.6 +j 1.6 1.2 +j 2.2 2.0 +i 3.0 3.4 + 3.4 1.4 +j 2.6 1.4 +j 2.6 1.4 +j 2.6 70 0.4 +j 2.2 1.0 +j 2.4 1.6 +j 3.0 2.6 +j 3.6 4.0 +j 4.4 1.6 +j 3.2 1.6 + j 3.2 1.4 +j 2.8 100 0.8 +j 2.8 1.4 +j 3.2 2.2 +j 3.6 3.4 + j 4.4 5.0 + j 6.8 2.4 +j 4.0 2.4 +j 3.8 2.0 +j 3.4 (a) Yie in nillimhos 200 2.0 +j 5.0 2.8 + j 5.6 3.8 +j 6.0 5.4 + j 6.4 7.8 + j 8.0 4.2 +j 6.6 4.0 + j 6.2 3.4 + j 5.8 500 4.4 + j 9.8 5.8 + j10. 2 7.8 + j11.O 11.2 + j1.4 16.0 + jlO.8 8.8 +j12.2 8.4 + j10. 6 7.2 + jlO. 6 700 8.4 +j13.8 10.6 + j14.2 13.8 + j4.0 18.2 + j12.4 23.0 + jll1.0 15.6 + j15.6 14.8 + j14.6 12.6 + j15.8 900 12.6 + j16.0 15.8 + j15.6 19.8 + j13.4 23.4+j 8.7 25.5 + j 2.7 22.1 + j14.7 21.3 + j14.4 18.6+ j13.5 Frequency VCE =4V VCE 4V VCE = 4V VCE = 4V VCE = 4V VCE = 1V VCE = 2V VCE = 8V (MHz) IC =.5 mA IC =1 nA IC = 2mA IC = 4mA IC 8 mA IC = 2mA IC = 2 mA IC = 2mA 50 jO0.6 j 0.6 j 0.6 j 0.6 j 0.6 j 0.6 j 0.6 j 0.6 70 j 0.8 j 0.8 j 0.8 j O.8 j 0.8 j 0.8 i 0.8 j 0.8 100 0.2+j 1.2 0.2+j 1.2 0.2+j 1.2 0.2+j 1.2 0.2+j 1.2 0.2+j 1.6 0. 2j+ 1.4 0. 2+j 1.2 (b) yoe in millimbos 200 0.2 +j 2.6 0.2 + j 2.6 0.2 +j 2.6 0.2 +j 2.6 0.2 +j 2.6 0.2 +j 3.0 0.2 +j 3.0 0. 2+j 2.4 500 0.4 +j 6.0 0.4 +j 6.0 0.4 +j 6.2 0.3 +j 6.2 0.8 +j 6.2 1.0 + j 7.2 0.8 +j 6.6 0.6 +j 5.8 700 0.6 + j 8.4 0.8 + j 8.4 1.0 +j 8.4 1.0 + j 8.4 1.2 +j 8.4 1.6 + j10.2 1.2 + j 9.0 0.8 + j 7.8 900 1.2 + j12.4 1.4 + j12.2 1.8 + j12. 2 2.0 + j12.0 2.2 + j1l.8 3.2 + j15.0 2.2 + j12.8 1.4 + jll.0

Frequency VCE=4V VCE =4V VCE = 4 V VCE4V VCE=4V VCE= V VCE 2V VCE = 8V (MHz) IC=.5mA IC= 1mA IC = 2mA IC =4mA IC = 8mA IC = 2mA |IC 2mA IC= 2mA 50 18.4 - j 3.8 31.2- j 8.7 52.0 - j20.4 73.8 - j40.8 93.0 - j67.0 52.0 - j21.6 52.8 - j20.8 51.0 - j20.4 70 17.4 - j 4.8 29.0 - j10.8 45.6 - j24.8 61.2 - j44.4 70.8 - j66.6 44.4 - j18.0 44.4 - j23.4 45.6 - j24.6 100 15.6 - j 6.4 25.2- j13.4 36.6- j26.1 45.2- j42.8 50.4 -j57.6 34.2- j25.2 35.4 - j25.8 35.4 - j25. 5 (c) yfe in millimhos 200 11.0 - j 8.8 16.0 - j16.0 20.6 - j26.2 24.0 - j37.2 25. 2- j49.2 20.0-j27.6 20.6- j26.8 20.2- j25.2 500 4.6-j10.8 5.6- 16.0 6.0-j22.0 3.6- j30.3 -1.5- j36.9 5.4 -j23.1 5. 6- j23.0 6. 4- j21. 6 700 1.8 - j10.0 1.8 - j14.4 - j20.6 -4.4 - j26.8 -11.6 - j29.4 1.4 - j22.4 0.2 - j21.6 -0.8 - j20.0 900 0.4 - j1O.4 -0.2 - j14.4 -5.0 - j20.4 12.4 - j23.4 -18.0 - j21.0 -8.0 - j22.4 -6.4 - j21.4 -3.2 - j19.4 Frequency VCE = 4V VCE = 4V VCE = 4V VCE = 4V VCE = 4V VCE = 1V VCE = 2V VCE = 8V |P (MHz)'IC =.5 mA IC = 1 mA IC = 2 mA IC = 4 mA IC = 8mA IC = 2mA IC = 2 mA IC = 2 mA 50 0 0 0 0 0 0 0 0 70 -j 0.2 -j 0.2 -j 0.2 -j 0.2 -j 0.2 -j 0.2 1 -j 0.2 -j 0.2 100 - j 0.4 -j 0.4 -j 0.4 - j 0.4 - j 0.4 -j 0.4 -j 0.4 - j 0.4 (d) yre in millimhos 200 - j 0.8 - 0.6 - j 0.6 - j 0.6 - j 0.6 - j 1.0 - j 0.8 - j 0.6 500 -j 2.0 -j 2.0 -j 2.0 -j 2.0 - j 2.0 -j 3.2 -j 2.4 -j 1.8 700 -0.2-j 3.2 -0.2-j 3.2 -0.2-j 3.2 -0.2 -j 3.2 -0.2-j 3.2 -0.4-j 4.4 -0.2- j 3.8 -j 2.4 900 -0.4 - j 4.0 -0.4 - j 4.0 -0.4 - j 4.0 -0.4 - j 3.8 -0.4 -j 3.8 -1.8 -j 5.8 -1.0 -j 4.6 -0.2 -j 3.2 Table IV. Short circuit admittance parameters of a 2N918 transistor measured on G. R. Transfer function and Immittance Bridge

Frequency VCE =4V VCE = 4V VCE = 4V VCE = 4V VCE = 4V VCE = 1V VCE = 2V VCE = 8V (MHz) IC -5 mA IC = mlA IC = 2mA Ic = 4mA IC = 8 mA IC = 2 mA IC = 2mA 50 2. 20 - j 9.40 3. 10 - j13. 50 3. 60 - j16. 60 3.60- j19.20 3.20 - j19.00 2. 80 - j14.80 3. 20 - j15. 60 3. 60 - j16.80 70 0. 30 - j 7. 25 0. 50 - jlO. 20 0. 50 - j13.40 0.40 - j14. 20 0. 20 - j13. 80 0.20 - j2.30 0.40 - j13.00 0.40 - j14.20 100 -0.1. 40 - j 5.15 -0.60 j07.20 -0.90-j 9.30 -1.20- jl10.40 -1.30 - jlO. 10 -0. 90 - j 8.40 -0.80 - j 8.90 -0.90 - j 9.70 2060 - j 470 -.78j 2.46 11 - j 3.45-144j 4.70-150j 4.60-1.45-j 3.85-1.-50 j 4.15-1.55j 4.40 500 -0.55 - j 0.82 -0.73-j 1.08 -0.88-j 1.33 -0.96 - j 1.52-0.93 - j 1.50-0. 90 - j 1.23-0.89 - j 1.29 -090 -j 1.38 700 -0.43 - j 0.38 -0.58 j- 0.58 -0.72- j 0.78 -0.81 - j 0.94 -0.80 - j 0.98 -0.75 - j 0.63 -0.74 - j 0.77 -0.72 - j 0.79 900 -0.40 j 0.32-0. 51 O.45-0.67 j 0611-0.78-j 0.70-0.77 - j 0.74-0. 62-j 0.56-0.67-j 0.59-0. 67-j 0.60 Table V. Short circuit current gain of a 2N918 transistor measured on G. R. Transfer- Function Bridge

Frequency |VE =4V VE = 4V VCE = 4V VCE = 4V VE =4V VCE I V VCE =2V CE - 8 V (MHz) C =.5mA = mA I = 2 mA IC= 4mA IC = 8 mA Ic = 2mA IC = 2nA Ic 2mA 2 0.21 + jO. 04 0.38+ jO.07 0.62 +jO.08 1.11 + jO.12 1.89 + jO.16 0.68 + jO.10 0.65+ jO. 10 0.57 + j.09 5 0.22 +jO.16 0.40 +jO.20 0.64 + j0.25 1.15 +-jO.36 1.96 + jO.59 0.68 + jO.27 0.66 +jO.26 0.58 + jO.25 10 0.24 +jO.30 0.43 +jO.39 0.67+ jO.47 1.19+ jO.64 2.13+ jO.90 0.74 +jO.52 0.70 +jO.50 0.59+ jO.44.25 0.26 + jO.71 0.49 + jO.91 0.83 + jl. 15 1.42 + jl.43 2.55 + jl.91 0.87 + jl.24 0.80 + j.1 6 0.67 + jl.04 50 0.33 +jl.41 0.64 + jl.72 1. 12 + j2.06 2.01 + j2. 50 3.39 + j2. 87 1.33 + j2.37 1.20 +j2.16 0..99 + j1.89 100 0.64 +j2.63 1.23+ j2.97 1.96 + j3.29 2.94 + j3.62 4.52+ j4.11 2.10 + j3.65 2. 00 + j3.41 1.72 + j3.13 200 1.77 +j4.77 2..58+j5.15 3.32+j5.76 4.60+j6.33 7.39 +j7.22 3.60+j6.43 3.31 +j5.86 3.06+j5.30 (a) Yie in millimhos Frequency Yoe Yib Yfe (MHz) (mmhos) (mmhos) (mmhos) 10 0.01 +j 0.14 59.26 - j 4.74 58.58 - j 5.35 25 0. 01 + j 0.33 58.00 - j1l. 24 57.16 - j12.72 50 0. 06 + j 0. 66 52. 57 - j19. 97 51.39 - j22.69 100 0. 12 + j 1.31 42. 53 - j24. 54 40.44 - j29.13 200 0.09 +j 2.44 25.11 - j21.33 21.70 - j29. 52 (b) Yoe' Yib' and calculated Yfe at VCE = 4V, IC = 2mA Table VI. Short circuit admittance parameters of a 2N918 measured with RX meter and a Transistor Test Jig

I C4*****TtiiS PROGPAI MAAY 6E USED IL V WE VELCTfIU JIFF ERENGCE FURi RU TH 2 C YII AND Y21 BFTWEE!N THE UPSI EF:'. P SPGNSE A ND rHE ACTUAL RESPONSE 3 C FCR THE LAPd2EV TY PE Til L U P2 I;A NSI Sk At D E L 4 C 5 C(3M4PL EX FL T It 2 0 F C T N2 ( 2 C ),StJ A Y / ( O. O)I.)/,AL2)/2L*, O O)/ 6 CUMPLEX CUMUL, DEN vb/, 2(3) /20* 09 G. *O Y 1,Y21 7 D I MEN S-1 CN AA ( 2O ), FkE- (2 D),G(2 C) 8 INTLGEP Pt ER.ANCF 9 JUPt.P=C 10 I T E' 11 C 12 C PE.AF, 0 NUfBER LE` ELEVENTS — N=NUM 3EP R OF LALCER ELEM'EENTS + 1 13 C P-P&tfP TO'WHICH ERROR-CRITERION IS KAISEU 14 C t2RAN'%lh=THF CURRENT GE NERP TOP'S CONTROLLING BRANCH NUMBER 15 REAu(5v2) N,8RANCHiP 16 2 FCMAIT( 312 17 NM-=N-1 1 8 NN =N-3 3 19 NT=N~-1 20 c 21 C READ ELEM ENT VALUES AA( 1I)... AA(N) t AA(N) IS GM, AA(N+1) IS LB W 22 C NET',CNRK IS FREENCY SCALED BY LO*"9 23 C NE TkORK IS MAGNITUDE SCALED BY 104*-3 24 REAU5,tL) (AA(J),vJ=19NT) 25 ~ FURMAT(6F124.0) 26 C READ DESIRED RESPONSE DATA 2 7 READ(5,3) NFREQ, (FREQ(J), FCTN 1 (J),rfCTN2(J), J=1,NFREQ) 28 3 FUPMAT(12/(5FIO.O)) q9. -t 1 C EAD(,i vCLF, C8c W Y I IY2 1 30 160 FORMAT (4FluG.O) 3 1 WR I IE (6,1 65) C 3 E v C BC i WY 1, Wv'Y 2 1 32 lofS FO RM AT (/ / l1OX i, 4HC, 8 E.=,,F 8.4, 1 G X, 4HC 3C, F 8 4/ 1 OX,, 5H W Yl I,F 8.409X, 33 1 m-)H WYY21 F8.4 ) 34 30 170 I=1,NFPEQ 3o 1 7 V CTIN ( Ih) =F CT N2 ( I ) +JAY *6.2 8 31 854 *EREF Q (( I ) *Ct%3 CRC

3.7 1O.O WRI TE(6b,4) {AA( J) J=1,NT) ":3;!' 4 FORt.ATI(//, OX,23H'HE MO E I.. ELEiEN T S ARF t//(. FLO.5)) 39 ALFSAV=. 000001 40 IF( ITER ) 19,2- t 2, 19 41 19 STOP 42 C 43 20 CALL MIN(ERR,G, AA, NT,.99 JU'iP,NT, ALFSAV,2. 1 ) 44 I J=I TER*JUMP 45 ERR=0. 46 IF(IJ)8, 7,8 47 7 WRIRTE(6,6) 48 6 FORMAT(///3X,4HFREQ 5X, 7HPE( I1) t,3X 7HI M(YII) 3X,7HRE(Y21), 3X, 49 17H IM(Y21 ) 3X16HERRY11,4X,6H EtRY21,/3Xt 5H( GiHZ),4X,6H(MMH),t4X? 50 26H(MMHO) t 4Xt oH( MMHO), 4X 6h ( PMHO),4X, 6H(MM H0 ),4X, 6H(4 M1 ) / ) 51 8 DO 9 J=INFRE', 52 S=JAY*FREQ( J)*6.2831854 53 DO 90 K=l,NN,2 54 -A(K)=AAAK c) 55, 90 A (K+1/) =;S AA K+i ) 56 A( )=A( 1 )+S*AA(NT) 57 A(NM) =AA ( M) 58.D OEN=C UMUL ( A,NM) 59 Y 1I=CUMUL (A( 2 ) t NM-1 )/DEN 60 Y21=AA(N) CUMUL (A(bRANCH+1 )tNMBRANCH)/DEN 61 ERRY1 I=CABS( FCTNI(J)-Yl)1 ) /CA S(FCTN 1J)) 62 RRYZ= CABS ( FCTN2 (J )-Y21 )/CABS( FC rN 2( J ) ) 63 IF( IJ) 911,9 64 11 Y 1=Y11+S*(CBC+CBE) 65 Y21=Y21-S*C'C 66 WR.I TEi(6,tI10) FREQ(J), Y,YY21,ERY1 I, ERRY21 67. 10 FORMAT (7F10.4) 6-8 I F(J-NFREQ) 9,53,9 69 53 WRITE(6,55) 70 55 FORMAT(//,w27HTHE MEAN SQUARE ERRORS ARE /) 71 9 ERR=ERR+ ( WiY. 11*ERR Y1 )**P+('Y 2 1*ERRY 21 )**P 72-__ C

73 WP RI T E&(6, 2'3) ER P. 74 29 FOPRMATF ( 15 7) 75 I F ( JUMP )O2100,p 2- 4 76 C 77 C CALCULATE THE GRADIENTS 78 24 L 00 14 J=1,NT 79 AASAV AA(J)*0C001 80 AA(J)=AA(J)4-AASAV 81 ERRI=CO. 82 00 17 K=l vNFREO 83 S=JAY*FREQ(K)*6.2831854 84 D0 99 9 =1,Nt,2 85 A(I)=AA(T) 86 99 A(I+1)=S*AA(1+1) 87 A ( I ) =A( 1) +S* AA ( NT) 88 A(NM)=AA(lNM) 89 DEN=CUMUL(AtiNM) 90 YII=CUMUL (A(2 )NM- /DEN 9 ~ V'2 1 Y~1=AA(N ) *CUMUL (A ( B3RANC14-1)+ I AN-BRANCH) /DEN 92 17 ERR I'ERR I + (WY1 I *C ALS ( F CT I (K ) -Y 1 1 ) / C ALIS (FCTN l(K) ** 93 1+ (WYV2 1*'CABS ( FCTN2 ( K) -Y21 ) / CA3S (FCTN2 (K) **P 94 IF (ABS(ERR1-ERR).GEe.000 i. CJP.ABS(AA(J)).GE..901) GO TO 300 95 G(J)=1.CE-10 96 GO TO 301 97 300 G(J)=(ERR1-ERR)fAASAV 98 301 CONTINUE 99 AA(J)=AA(J)-AASAV 100 WRITE (6,92010) G(J),AA(J) 101 200 FURMAT(lCX1CXF5.75XFlO.5) 102 14 CONTINUE 103 c 104 IlfE=ITEP+i 105 G6 TO 20 1C6 ENO 4D OF FILE

I C >Pi t.X FU CTIG CI J UL (, r h ) 2 C 3 Cef-IPL. LX (20,C, OLDC Nt WC 4 [ =: 5 C-i. 6'JLDC=1. 7 6 1-I+1 8 N CE WCC-, CB ( I I +LOL)C 9 OLDC=(. 10 C =NEv C I11 IF{I-*J)6, 111l 12 11 CUMUL=C 13 EN!) 14 C 15 C FND OF F ILE $CC'PY FPOCM *SOUPCE* T[) -M IN $LIST -MIN 1 SUBRCJTIl NE MIN r( F,GX, ERP CR,, JMP NI TERt ALFSV,ALFi'UL,NPRK NT ) 2 )I.rIENSI jN f 2C ),X(20),GSAVE( 2i ),S ( 20)1 SIG(20),tG(20)), H (20, 20 3 1 A ( 2t,2i ).,( 2 &,? ),C (20,20 ) XSAVtif 2, ) 4 Cc-'t**1TTttIS S UBROCUTIt.E MINtIMZES A FtiNCTION F 5 C USING FLETCHER-POWELL WITH A SEARCH TO FINlD MIN 6 C X(J) IS VARIA{,LEG(J) IS GRADItN1-,F IS FJNCT IJN 7 C Ju,~P=. AT STrART AND FINISH OF ITFRATION 8 C JUPI ri.i IRES CALCUJLAT iCN OF FJNCTIUN ANI) GA KAIt'NT 9 IF (JU t Fi) 2C,' 1,2C 10 1 KEY=l 11 JltjMp=1l 12 1 2 R E T UR 13 20 Gd) TC ( L I,17, 3 6, 38,3'.,38, 1), KEY 14 1 3 FSAVF=F 15 1 IT f " I'T[r. P

lb 00 14 J=lN 17 XSAVE(J)=X(J) 18 14 GSAVECJ)=G(J) 19 34 1 =0, 20 IMAX=l1 21 KEV=2 22 FI=FSAVE 23 F2=FSAVE 24 X1=O. 25 X2=0. 26 IF(JUMP)5bt58i5b 27 56 DO 12 J=1,PN 28 DC 13 K=1,N 29 13 H-1(JgK)=0,. 30 12 H(JtJ)=1. 31 NGRAD-=0 32 jUMp=-1 33 58 CALL MATMUL(HNtNG,1,St,-1.20,1 20 ) o 34 CALL 14A TMUL ( S, 1I N,GGtidEE 1, t 1,*,r I I 1 p,1) 35 IF(DEM))5,56,556 36 5 ALF=ALFSAV 37 A LF SA V=O01. 38 GO TO 15 39 C HAS MINIMUM BEEN BCUNCED 40 17 IF(F)44,44,31 41 44 ALFSAV=ALFSAV-ALF 42 ALF=ALF/ALFMUL**4 43 GU TO 1L 44 31. IF((F-F 2)IF-.G00061)2b,28,3 45 C DECREASE SEARCH STEP SIZE IF CN FIRST ITERATIUN 46 3 GU0 TO (4030),IMAX 47 4 ALFSAV=ALI FSAV/ ( ALFMUL )**2 48 GO TO 5 49 C MINIMUM NGT BOUNtEOWtCUILE STEP SIZE 50 28 F1-=F2 51 F 2=F

52 x1tX2 53 X2=ALFSAV 54 IMAX=2 55 ALF ALFM(U)L*ALF 56 GC TO 1. 57 C MINIMUM HAS BEEN BOUNDED,MAKE N SEARCH ITERATIONS 58 30 TEMP=X2 59 X2=ALFSAV 60 ALFSAV=TEMP 61 TEMP=F2 62 F2=F 63 F=TE(fP 64 I(F NP I I NTfl3) 8 03 36,80 65 80 WRITE(6,r250) 66 250 FCRMAT(lHC) 67 36 CALL QUAD(X1,X2,ALFSAVIITER KEY,FIFF F2) 68 IF(KEY-4)27, 37,99 69 37 IFINPRINT)39t38i39 70 39 WRITE(6i4C) ii 40 FCP.MAT(IHF) 72 38 CALL FI 3MIN ( X1 tX2 vALFSAV, ITER 9KEYvFF I F2) 73 IF(KEY-7)27936, 36 74 99 JUmP=1 75 GO TO 27 76 1.5 ALFSAV=ALF SAV+ALF 77 27 DO 21 J=1,N 78 21 X(J) =XSAVE( J) +ALF SAV*S(A) 79 IF (ALFSAV.LE..000000I.) GC TO 25 80 IF(JUMP-1 )26 t25,26 81 25 IF(NPR INT)8I,26,81 82 81 WRITE(6,24) 83 24 FORMAT(IHG) 84 KEY=7 85 26 RETURN 86 C EXIT IF F HAS DECREASED. INSIGNIFICANTLY U7 19 I F-( ABS ( F/ FS AVE)-E-F.RROR) 1 1 10,0 10

8 8 1 J UP=C 89 RETURN 90 C FORM H MATRIX VIA FLETCEEP-PO]WELL 91 11 D0 7 J~l N 92 SIG(JAL ) FSAV*S( J) 93 7 DG(J)=G(J [-GSAVE(J) 94 CALL MATMUL(SI;,1t,NDG, 1,rEMl.,,, t 1 ) 95 CALL MIATMUL(SIGvNPSI G,,NA,0EA, lt,1,20) 96 CALL MATML L(DGLn\,Ht,N,,1., 1,~, z) 97 CALL MATMUL(B,1,tND fSG,1,E t l,2O1, 1 ) 98 CALL MATMUL.(DG," N,1, 8 f,N,1C, 1.,,2 uu,20?) 99 LALL MA'TMUL(HN,t\,C,N,BdE:I,2C,20,i2 0) 100 00 8 J=ltN 101 30 8 K=1,N 102 8 H(JK)=H(JK)+A(JK) -E3J,9)( 103 JuMP=-1 104 NGPAD=1 105 GO TO 18 106 END END OF FILE $COPY FROM *.SOURCE* TO -QUAD'$ LST -QUAD I C 2 SUBROUTINE QUJAD(XAXCXMINITERKEY1,FFAFC) 3 C 4 IF(F121,21, 30 5 30 KEY=KEY1-1 6 GO TO( 10C,3nCd 1C0v1C,1C,1CC1 )KEY 7 100 XAS=XA**2 8 XCS=XC**2 9 XI3XMIN 10 XBS=XB**2 1L FB=F 12 J=0

13 2.1 KEYI1=KEYVI1 14 2 lF(KEYl-4)4v23v23 15 2 3 XMIN=XB 16 ITER=ITER-J 17 F=FB 18 RETURN 19 4 XD=.5*((XBS-XCS)*FA+(XCS-XAS)*Fi3(XAS-X3 S)*FC) 20 XD=XD/I ( XB-XC)*FA+(XC-XA)*FB+(XA-Xb)*FC) 21 IF(XL-XC)90,90 21 22 9C IF(XA-XD)91,9l,2l 23 91 IF(ITER)89i,21,39 24 89 (US=XD**2 25 XMIN=XD 26 RETURN 27 300 FU=F 28 22 IF (FD-FF) 3, 3,19 01 29 19 IF(J)21,21,218 8 30 18 ITER=O 31 KEY1=KEY1+2 32 GC TGO21 33 3 IF(XD-X3)'99,8 34 8 FA=FB 35 XA=XB 36 XAS=XBS 37 GO TO 1C 38 9 FC=FB 39 XC-XB 4C XCSoXBS 41 10 FB=Ft) 42 XB=XD 43 XBSZXDS 44 J=J+1 45 [F(J-ITER)2,2112 46 END EN4D OF FILE

$COPY FROM *SOUFCE* TO -MATMUL $LIST -MATMUL 1 C 2 SUBR OUlIT I NE MATIAU L(At r,\,tvBt r', LLr C, DIV, NPUW1,jNk0UW _,N R W3) 3 C 4 OIM ENS ICN A (1 ),8(H 1) 9C(1) 5 D)O 2 L=lLL 6 DO 2 J=1,N 7 JJ=J+NROW3* (L-1) 8 C(JihO. 9 DO 1 K=1,M IC JK=J+(K-1) *NRGWI. 11 KL=K+(1-1)*oNPCW2 12 1 C(JJ)=C ( JJ) +A(JK)*B( KL) 13 2 C(JJ)=C(JJ)/ DIV 14 RETURN 15 END END OF FILE $CUPY' FROM *SOURCE* TO -FIgMIN $LI'ST -FIBMIN 1 C 2 SUBROUTINE FIBMIN(X1,X2,XMIN,.ITERKEY1,FF1,F2) 3 C 4 C*****THIS SUBRCUTINE MINIMIZES t FUNCTION F BY A FIBONACCI SEARCH 5 C XMIN IS THE VALUE RETURNED AS A MIN, N IS THE NUMBER 6 C OF ITEkATIONS MADE, ALF1 IS TIE WIJTH OF THE ORIGINAL INTERVAL 7 C F IS THE VALUJE Of FUNICUICN AT XMIN 8 C XA AND XP ARE SEARCH POINTS AND FA AND FB ARE FUNCTION VALUES 9 C ORIGINALLY X PIN IS THE GREATER L IMIT OF THE BOUNDED INTERVAL 10 KEY f=KEYl-3 11 GO TO(10092000,0C),KEYD 12 100 IF(ITEP)23,23,22 13 23 KEY1=KEVl+3

14 RETUR.N 15 22 I Y1= 16 1 Y2=1 17 KEYI=KEY 1+1 18 2 J=0 19 C CALCULATE APPROPRIATE F:IBCNACCI NUMBERS 20 30 D00 3 I= 11ITER 2.1 ITEMP=IY2 22 I[Y2=IY2+IY 23 3 IYI=ITEMP 24 YI=IY1 25 Y2= IY2 26 C DETERMINE FIRST SEARCH POINT 27 1 XA= X1+( X2-X 1) *Y 1/(YI+Y2) 28 GO TO 4CGO 29 C DETERMINE NEXT SEARCF POINIT AND VALUE OF FUNCTION 30 300 IF(KEY) 14, 14.13 c 31 13 FA=F 32 GO TO 6 33 14 FB= F 34 6 J=J+l 35 c DETERMINE MINIMAL SEARCH PCINT 36 8 IF( FA-FB)4,4+,5 37 4 X2=XB 38 F2=FB 39 KEY = 1 40 XB=XA 41 FB=FA 42 C DETERMINE NEXT SEARCF POINT AND VALUE OF FUNCTION 43 XA=XI+X2-XB 44 400 XMI N= XA 45 21 IF(J- IT ER)22,9,9 46 20 RE TURN 47 200 FA=F 48 12 KEY1=KEY I+1 49. GO TO 500

50 5 X 1=XA 51 F1=FA 52 XA=XB 53 FA=FB 54 500 KEY=-1 55 XB= X2-XA+X 1 56 XMIN=XB 57 GO To 21 58 9 KEY =KEY1+l.59 C RETURN APPROPRIATE VALUE FUr- XMIN 60 ITER=O 61 IF(KEY)lOt 10 11 62 10 XMIN=XA 63 F=FA 64 RETURN 65 11 XMIN= X 66 F=FB 67 RETURN 68 END END OF FILE

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60 DISTRIBUTION LIST (Cont.) No. of Copies Headquarters U. S. Army Combat Developments Command Attn: CDCLN- EL Fort Belvoir, Virginia 22060 USAECOM Liaison Officer MIT, Bldg. 26, Rm. 131 77 Massachusetts Avenue Cambridge, Mass. 02139 Commanding General U. S. Army Electronics Command Fort Monmouth, New Jersey 07703 Attn: AMSEL- EW 1 AMSEL- PP AMSEL- 10- T AMSEL-GG-DD 1 AMSEL- RD- LNJ 1 AMSEL-XL- D 1 AMSEL-NL-D 1 AMSEL-VJ6D 1 AMSE L- KL- D 1 AMSEL- HL- CT- D 3 AMSEL-BL-D 1 AMSEL-WL-S 3 AMSEL-WL-S (office of record) 1 AMSEL-SC 1 Dr. T. W. Butler, Jr., Director 1 Cooley Electronics Laboratory The University of Michigan Ann Arbor, Michigan 48105 Cooley Electronics Laboratory 16 The University of Michigan Ann Arbor, Michigan 48105

Securitvy Classification DOCUMENT CONTROL DATA. R & D!Security classification of title, body of abstract and indexing annotation must be entered when the overall report is classified) 1. ORIGINA TING ACTIVITY (Corporate author) 2a. REPORT SECURITY CLASSIFICATION Cooley Electronics Laboratory Unclassified University of Michigan 2b. GROUP Ann Arbor, Michigan 48105 3 REPORT TITLE High- Frequency Transistor Modeling for Circuit Design 4. DESCRIPTIVE NOTES (Type of report and.inclusive dates) Technical Report No. 205, May 1971 5. AU THOR (S (First name, middle initial, last name) A. B. Macnee and R. J. Talsky 6. REPORT DATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS May 1971 72 8 8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) DAABO07-68-C-0138 b. PROJECT NO.TR05 1 HO 21101 A04 01 02 c. 9b. OTHER REPORT NO(S) (Any other numbers that may be assigned this report) d. | ECOM-0138-19-T 014820-19-T 10. DISTRIBUTION STATEMENT t 1. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTI.VITY U. S. Army Electronics Command Fort Monmouth, New Jersey 07703 __________________ Attn' AMSEL-WL-S. 13. ABSTRACT It has been found that hybrid-pi or high-frequency T are inadequate high-frequency models for certain transistor types even though the models are supplemented by reasonable extrinsic elements. The hybrid-pi can be modified to model these transistors by replacing the r f C. circuit by an RC ladder. Using a computer optimization program an optimal, N-lump model is generated. For the 2N918 transistor a two-lump model extends the frequency range of the hybrid-pi model to fT/ 2. Typical circuit examples show most of the improvement in model performance can be obtained with a two- lump model. DD FOR 1473 (PAGE 1) S/N 01 01 -807-6811...Security Classification A-31408 1-31408

Seuritv Cla ssification 14 LINK A LINK B LINK C KEY WORDS ROLE WT ROLE' WT ROLE WT High-frequency transistor modeling Hybrid-pi models High-frequency T Computer optimization program S/N 0101 807-2 Security Classification A-31409 I 118762

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