Modelling procedures for interactions between thermal and electrical device parameters

Sol&State

Electronrcs

Pergamon

MODELLING BETWEEN

Press 1967. Vol. 10, pp. 737-744.

PROCEDURES

THERMAL

AND

Prmted m Great Bntam

FOR INTERACTIONS ELECTRICAL

DEVICE

PARAMETERS* H. POTASH Department

of Engmeermg,

Umversity

of Cahfornia,

Los Angeles,

U S A.

and W. W. HAPP National Aeronautics

and Space Admimstration,

Cambridge,

Massachusetts,

U S A.

(Received 19 May 1966; zn reusedform 4 January 1967)

Abstract-Flowgraph models are developed to describe relationships between thermal and electrical parameters of devices and associated circuits. Flowgraph techmques permit a systematic mvestlgatlon of a wide range of device phenomena such as thermal feedback, thermal runaway and temperature induced drift. Figures of merit are derived for thermal sensitivity, and for thermal stability and are applied to temperature dependent parameter varlatrons m transistors Design criteria for compensation of thermal effects by associated circuits are evaluated by flowgraph techniques and flowgraph models serve to Justify design hmltatlons and design approximations based on thermal dependence.

R&urn&-Des modeles mathematlques a plans graphlques ont et& developpes pour d&me les relations entre les paramttres thermlques et Blectriques.des dlsposmfs et circuits associes. Les techniques a plans graphlques permettent un examen systematlque d’une gamme &endue de phenomenes des drsposmfs tels que la reaction thermlque, la fmte thermlque et l’apport de temperature mduit Des facteurs de mente sont d&iv& pour la senslbllne thermlque et la stablhte thermique et 11s sont apphquds au varrations des param&tres de transistors dependants de la temperature. Des critbres de construction pour la compensation d’effets thermlques par des circuits associes sont evalues par des techmques a plans graphlques et des modeles a plans graphiques servent B justifier les hmltes et approximations de construction basees sur la dependance thermique

Zusammenfassung-Flussdiagramme zur Darstellung der Bezlehungen zwlschen thermlschen und elektrischen Parametem von Bauelementen und zugehorlgen Schaltungen werden entwickelt. Die Zelchnung von Flussdlagrammen ermoglicht eme systematlsche Untersuchung solcher Phanomene wle thermische Ruckkopplung, thernusche Instabihtat, Veranderung der Kenngrossen mn der Temperatur. Die thermlsche Empfmdhchkelt und Knterien fur die therrmsche Stab&it werden abgeleltet und auf das Temperaturverhalten von Transistoren angewendet. Aus Flussdiagrammen werden Richtlmlen fur die Kompensatlon von Temperatureffekten mltemander gekoppelter Bauelemente gewonnen. Flussdiagramme zeigen zudem die temperaturbedmgten Grenxen mtegrierter Schaltungen.

* This work supported

by the National Aeronautics

and Space 737

Admmistratlon

NAS 12-16.

738

H.

POTASH

and

INTRODUCTION THE RELATIONSHIPS between component parameters and thermal behavior is investigated with the following objectives. (a) To develop a physical and analytical explanation of the interaction between thermal and electrical effects. (b) To determine criterra which describe effectively thermal instabihty caused by positive feedback between the thermal circuit and the electrical circuit. This instabihty is often referred to as thermal runaway. (c) To assess quantltatlvely the sensitivity of cncmt parameters to temperature variations and develop a set of judicious approximations valid over suitably specified temperature ranges. Although many techmcal pubhcations reveal extensive work on thermal sensrtivity, the approaches proposed fall short of the objectives stated above m several aspects; specifically:(l-lO’

W

W.

HAPP

(b) Approximations cannot be justified but the designer suspects the linear assumptions to be valid. The lmear approximation is useful as a trial solutron subject to subsequent verification. (c) Approximations are valid but range of validity must be specified. (d) Small signal analysis applies, all approxrmations are valid. The appropriate stated exphcltly.

assumptions

should

always be

OPERATION OF THE p-n DIODE Analysis of d.c. bias, stability and temperature sensitrvlty of transistor circuits mvolves the small signal transistor parameters as well as temperature dependent ‘steady-state’ (or ‘operating point’ or ‘large signal’) parameters. Temperature sensmvrty of these parameters depends on semiconductor properties which are usually provrded as empulcal design data or derived from solid state physics.

(u) The hmitatrons under which the approximations are valid are not known or not stated.(203*6*8) (b) The concept of thermal feedback and the parameters used to describe it are defined such that ambiguity is possible and frequently misapplication results.(2.3,4.6,7,8) (c) The circuit sensitivity is calculated before the thermal runaway and the temperature dependence of the transistor 1s evaluated. This approach entails making approximations without ascertainmg their validity.(3*4*6*8*g) This investigation arms to overcome these shortcommgs by employmg recently developed techmques of flowgraph analysrs.(1° - w Flowgraphs are mathematical models establishing the mterrelatron between variables of systems with large numbers of components. By establishing a model for both the thermal circurt and the electncal circuit and by defining clearly the mteractions, It is possible to describe the system accurately, then to make a well-defined set of approximations, and finally, to advance a design procedure based upon the approximate model. Four distinct sets of assumptions are mvolved. (a) Approximations cannot be used smce the transistor is operating in the nonlmear region, thus the design must resort to experimental data.

FIG 1 Regions of operation of blased diode F-Forward R-Reverse

btased base to emitter diode blased collector to base diode

First m importance is the leakage current I,,, m the back% ard biased diode and the voltage drop VF m a forward biased diode (Fig. 1). Both parameters are temperature dependent. In sihcon transistors an increase m temperature of 10°C causes I,,, to double. As indicated in the mtroduction, small signal analysis techniques are employed here for thermally dependent variations of the ‘steady state’ (or ‘operating point’ or ‘large signal’) current. A notation, such as H,, = AV,,/AI,, will be used

THERMAL

AND

ELECTRICAL

FIG 2 Defimtlon of currents m transistor. to denote thermally dependent two-port ‘large signal’ parameters. Capital letter H parameters are used m order to drfferentiate ‘operating point’ from ‘small signal’ parameters such ash parameters. The H parameters can be obtained from the large signal curve noting that for AVnE, AI,, etc. the entire linear regron changes, whrle the h parameters are obtained from the small srgnal scope,

DEVICE

PARAMETERS

Consider the steady state current flowmg m a normally operated forward biased transistor, Frg. 2. The base to emitter diode is forward biased and the collector diode 1s reversed biased. Only under these condrtions will I,,, flow between collector and base. An increase m I,,o decreases the base current and mcreases the collector current. Thus can be verified with the ard of Fig. 2 from the The temperature-induced defimtron of Icso. and V, are mcorporated in the changes of I,,, common base equivalent circuit (Frg. 3) and in the correspondmg flowgraph model Frg. 4. COMMON EMITTER CONFIGURATION The sum of the currents entering and leaving the closed volume A m Fig. 2 are described by the

FIG 3 Common base cmxnt model. COMMON BASE CONFIGURATION Dejnitzon. I,,, is dejned as the current jlowing between the base and collector ;f the emrtte-r is disconnected. Thus : 1, = - HFJE+ICBO and I* = (1 +HFBYE-ICBO.

FIG 5 Common emrtter flowgraph

VF

Hi33

:: VEB ’

.IE

HIS

. VCE ’ Hoe

5

-

HFB

*c

ko FIG 4. Common base flowgraph model,

739

FIG. 6 Path mversion for Fig. 5

H. POTASH

740

and

Aowgraph of Fig. 5. Path inversion gives a flowgraph appropriate for common emitter configuration (Fig. 6). Smce (1 f HFE) is usually about SO, the effect of I,,0 is much more significant m the common base configuration.

FIG 7. Common

emitter

W. W. HAPP and

IcBo = ICBO*2(T-W*~

where IcBo* is the initial value at T = T,. To a first-order approximation ICBO x IcBo* J,AT where J, = In Z/TT yielding a transmittance m Fig. 9(a), which does vary greatly with temperature. The temperature will reach a steady state value only when all the heat generated (P,) by the transistor is dissipated by the heat sink (PJ, as Fig. 9(b) shows. In turn the heat generated in the transistor is -IICEVCE, modelled suitably in Fig. 9(c). r*CBO

circuit model

JT

hT *

. A’crio (4

%

l

l

*cl30 FIG

8. Common

emitter

pd

C Al-*

flowgraph model

cd)

The current ICBo flowing across the emitter junction will be amplified as if it were a regular base current. The amphfied current H,,I, will and HFEICBO be HPEICao. In addition of I,,, gives the total current flowing from emitter to collector. Addmg HRE, HIE: and HOE gives the temperature dependent common emitter circuit model (Fig. 7) and the Aowgraph model (Fig. 8). THERMAL CONSTRAINTS For changes m temperature up to 75°C above the specified operating temperature of the transistor, ICBo increases by a factor of two for an increase in temperature T, = 10°C in silicon transistors and of T, = 7°C m germanrum transistors. Thus AI --=-CR0 T a CBO

FIG.

9. FunctIonal

relations for thermal dependence.

The heat dissipated by the heat sink is proportional to the temperature difference (AT) between the transistor and the heat sink and to the heat smk constant, C defined as the heat flow per unit temperature. Combining the flowgraphs m Fig. 9 gives the relation between IcBo and T (Fig. 10). Combining the flowgraphs in Figs. 8 and 10 yields Fig. 11 for calculating temperature biasing dependence and thermal runaway; thus* I,

=

I,H,E*

+Ic,o*(l

f HFE+) -I- ~&JOE*

1- VCBOIIF)

where I, is the critical thermal feedback current

THERMAL

AND

ELECTRICAL

xc00

GO JT

.

I

AT FIG.

10.

Relations

I/C

P,j

between transistor temperature

currents

and 10

I&ma)

6

‘0

Texas FIG 11. Temperature

Ruttaway limit It is advantageous loon

dependence

for configuratlon.

to define the thermal feedback *T=+d

JTvCE,Q,(l ~/IF(Q)

c

At the thermal runaway

Icno*

*

M IF and

JY%E ~J~FcR,

M -

c

8

IO

FIG. 12 Transistor charactenstlcs. Instruments 2N33.5 (Redrawn from SENSITIVE

Ref

1)

PARAMETERS

IBHFE*

+~C,O*(l

C(Q) =

+HFE*)

+ VCEHOE*

l-F,

Comparison with I, calculated from Ftg. 5 yields the temperature dependent parameters

+HFE*)

=

4 6 VcE (VOLTS)

TEMPERATURE

I

point of Q point when

2

The variation of transistor components depends on the I, at the Q point. The value of IF(o) is about one-half of the value of IFcR)

I

l F(Q) at the operating

741

PARAMETERS

represents an upper limit, which increases as the thermal circuit improves. Therefore, IFtR) IS a figure of merit of the cooling design. The response of the temperature feedback, has a time constant in the 0.1-10 set range, it is slow relative to the small signal variations in electronic circuits, therefore, it is Justified to neglect the variations of I, with respect to the small signal variation of the input.

%BO

--I

DEVICE

t1 + HF.E*).

I, is minimum when V,, equals V,,(max). Smce Vcx(max) usually equals the circuit d.c. supply voltage (E,,), the crmcal runaway current I FcR)1s calculated for VcE = ECE. Runaway ~111 occur as IcBo* approaches IFcR). Hence, a large value of IFcR) is desirable and

H FE* I-F,

and

H OE* -. l-F,

The forward amphfication of the temperature dependent H,, differs from HFE* by a factor of (1 - F,)-l. This increase causes an upward shift in the transistor charactenstrcs, particularly m the characteristics of V,, vs. I, at constant I,. Similarly, changes by the same factor (1 - FT) -I are observed for HOE. The changes in H,, and Ho, are illustrated by the transrstor characteristic curves in Fig. 12.

742

H.

POTASH

and W

TEMPERATURJI SENSITMTY

In the desrgn of a circuit, the possible occurrence of thermal runaway must first be examined, only then can thermal sensitivity be considered. The thermal sensitivity parameters S,, S, and S, specify the changes m the large signal currents and voltages with respect to the temperature dependent parameters.(2) &=---,

1,

S, =:-

I CBO

and

V

S, = 2.

I CEO

CBO

The supply voltage Vl is assumed to be constant. Srmilarly voltage stability factors at constant I,,, can be formulated.

. FIG. 13.

I

I

I

.

Transrstor model for sensltrvrty calculatrons DESIGN EXAMPLES

,4 temperature feedback factor F, of 0.2 or less 1s considered adequate m design. If further the amplified base current, IBHFE, exceeds sigmficantly the ohmic current, PeEHoE in the collector port, the direct contrrbution of T/e, to I,, can be neglected. Hence, I, = IBN,,+lCao(l +H,,). It 1s only under these restrictrons that thus formula 1s justified, although rt 1s frequently used irrespective of the thermal crrcurt and frequently leads to erroneous results. Although the drrect contrrbution of V,, to I, can be often neglected, the effect of V,, through the thermal loop must remain as a factor of F,. The transistor model used m sensrtrvrty calculatrons is given m Fig. 13. (a)

F,

< 0.2.

(b)

V,,Ho,

(cl

H,,

<

H = +.

IBHFE-

*

l---1’,

The above criteria for temperature dependent transistor parameters will now be rllustrated by examples. The temperature dependent transistor model makes rt possible to define a hnear graph for the

W.

HAPP

crrcurt using accepted flowgraph techniques.(14-21) All voltage sources are represented as branches (heavy line), all current sources are represented as links (dashed hne) and the branches form a connected but not closed structure (tree). A tree IS achieved since passive elements of the cn-curt can be assigned to branches or links. The flowgraph 1s constructed from the equrvalent crrcuit by assigning two nodes to each parameter, a voltage node and a current node. The nodes are mterconnected by defining : a current nodes for the branches as function of (&rent nodes for the links (b) voltage nodes for the iinks as functron of voltage nodes for the branches, (c) the current node as function of the voltage node and impedance for passive parameters that were chosen to be links, (d) the voltage node as function of the current node and resistance for passive parameters that were chosen to be branches and (e) active parameters m terms of condmons and controlled variables. Results are summarized m Table 1. CONCLUSION

Flowgraph models provide a useful techmque to descrrbe the interactions between thermal and electrrcal device parameters under temperature variation. Devrce performance 1s analyzed by first estabhshing a flowgraph model which is systematically constructed on the basis of physical phenomena and of system constraints. Thermal and electrical performance characteristics of the device result from this model. The model is then exammed to determine the presence of feedback and similar relatronships between thermal and electrical parameters. Of particular mterest are conditions for stability of the system. The flowgraph model 1s utrhzed to determine other devrce propertres, for example. (a) assumptions necessary to permrt approxrmatrons in performance charactenstrcs, (b) procedure for evaluatmg temperature sensitivity of parameters and (c) figures-of-merrt for stabihty of a system against thermal ‘runaway’. Flowgraph techniques thereby provide a vahd as well as practical desrgn approach.

THERMAL

AND

ELECTRICAL

Table 1. Sensitivity

DEVICE

PARAMETERS

criteria for stabilizing circuits

Schemau CWCUK

Flowgraph

REFERENCES 1. L. P HUNTER, Handbook of Semaconductor Electronics, McGraw-Hill, New York (1962). 2. J. F PIERCE, Tranststor Czrcuit Theory and Design, Merrill, Ohio (1963). 3. S S. HAKIM, Junctron Transrstor-Carcult Analysts, Iliffe, London (1962).

4. R. H MATTSON, Busts Junctzon Devzces and Czrcutts, Wllev. New York I19631 5. M. SA&, Soled Stale Thiory, McGraw-H& New York (1963). 6. R F. SHEA, Prtncaples of Tranststor Ctrcutts, Wiley, New York (1955). 7 L B ARGUIMBAN, Vacuum Tubes Carcutt and Tranststors, Merrtll, Ohlo (1956)

744

H.

POTASH

and W.

8 R 9. 10 11

12 13.

14

F. SHEA, Transsstor Ctrcuit Engineering, Wiley, New York (1957). M. V. JOYCE and K. K CLARK, Transzstor Czrcuit Analyszs, Addrson Wesley, London (1961). J. L. BIXROLIGHSand W. W. HAPP, IEEE Trans. Aerospace 2, 1127 (1964). J. B. COMPTON and W. W. HAPP, IEEE natn Conv. Rec. 11,445 (1963) P KAUFMANN and J. J. KLEIN, Semiconductor Products, pp. 37-40, October, 1959 W. W. HAPP, Apphcatlon of Flowgraph Techniques to the Solutron of Rehabrhty Problems m (M. F. GOLDBERG and J VOCCARO, Eds ) Phystcs of Fazlure m EZectronzcs, Washmgton, US. Dept of Commerce, Office of Tech&al Servrces -AD4341329. DD 375-423 (1964). D. E: M&Y and W.. W. ‘HAPP, IEEE Xrans.

PG-ANE-11,248

(1964)

15 J. B. COMPTON and W. W. HAPP, IEEE Trans. Aerospace 2,259 (1964) 16 J. B. COMPTON and W. W HAPP, IEEE Trans. Aerospace 3,94 and 372 (1965) 17. W. W. HAPP, IEEE Trans. Aerospace 3. 252 (1966) 18. E R. ROBBI&, D. E. MOODY and w. W ‘HAPS, IEEE Trans. Aerospace 3. 370 (1966). 19. J. STAUDHAMMER and W. w. H~PP, khtv. elekt. fjbertr. 20, 329 (1966).

W.

HAPP

20. A. A. B. PRITZKER and W W. HAPP, J. tnd. Engng 17.267 (1966). 21. W. W. tiApp,’ IEEE Trans. Educatton 9, 69 (1966). 22. R M. CARPENTERand W. W. HAPP, Proc Allerton Conf systems and ctrcust Theory 4, 454-464 (1966). 23. R. M. CARPENTER and W. W. HAPP, Electronrcs, December (1966). 24. G E. WHITEHOUSEand A. A. B PRITZKER, J. ind Engng 17 (6), 293 (1966). 25 W F. ROMBALSKIand R M CARPENTER, Desensitizing of Microcncutts to Variattons m Temperature and Production Spread, Phase IV, Determtnation of the Effect of Productron Spread of Parameters, Proc 2nd NASA Mzcroelectron. Conf , September (1966). 26. W. W. HAPP, E R ROBBINS and D. E. MOODY, IEEE Trans. Aerospace 2, 370 (1966). 27. M J. DEVANEY, G W. ZOBRIST and W. W. HAPP, Proc 10th Midwest Symp. Circuit Theory, Purdue Untverszty, May (1967) 28. H POTASH and W. W. HAPP, Proc 10th Mzduest Symp. Czrcuzt Theory, Purdue Untversity, May (1967). 29. C S. LORENS, Flowgraphs, McGraw-Hrll, New York (1964).