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A "IIGhI FREOUENCY POI{ER TRA$IS'ISTOR IIIODEL
A TTIESIS SUEMITTED TO T.HE
UI{IVERSIT{ OF AUCIUAND
FOR THE DEGREE OF
DoCTOR 0F PHII0S0P}Y
by
BRIAN EGAN
Departnent of Electrical Engineerlng
iluly 1976
'f.\t
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ABSTRACT
The development of a 'largesignal power transistor model applicable at
radio frequencies is described. This model which has its basis in the classical
largesignal models is valid for cutoff and active region operation but does
not include saturation operation. The model is intended primarily fep use jn
broadband linear radio frequency amplifier apolications and is useful up to
frequencies of the order of l/.5 ft.
The model is described by two first order nonlinear differential equations
and a number of algebraic equations. Equation coefficients are determined
from measurements made on the devices under study. Two methods are described
for the solution of the model equations. The first and principal nethod is an
iterative one requiring computer assistance whilst the seconcl is analytical
and depends upon piecewise linearisation of the device transfer characteristic.
This analytical method whilst in some respects inaclequate, €.9. distortion
level predictions, is easy to implement and desp.ite its limitations affords
useful ins'ight into output power capability and frequency limitations of
speci fi c devi ces.
The model contains all transistor nonlinearities and parasitic elements
of significance and an important feature is the inclusion of device temperature
as a model variable, resulting in good accuracy over a wide range of operating
conditions. A simplified input impedance representation is evolved and it is
demonstrated that input impedance npasurements provide a useful window on modei
structure and aid in the evaluation of parameter values.
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ACKNOltILE DGEMENTS
The author wishes to thank most sincerely his supervisor
Professor A.G. Bogle, Head of the Department of Electrica] Engineering, ftlr
his encouragement and advice and for providing the opportunity for the research
described in this thesis to be undertaken.
The author also wishes to thank the following:
l4r.H.J. Purchas of the staff of Al^lA (A'sia) N.Z. Ltd for many helpfu'l
discussions and for the supply of semiconductor devices.
Professor I,R. Kauffman of the University of Arizona for helpful
discussions and for assistance in the procurement of semiconductor device
data.
Professor F.T. Haddock of the University of l4ichigan for providing
study faci li ties during the author's study 'leave in 197I2.
The Electrica'l Engineering Department Technical Officer, l4r A.Thorley
and his technical staff for their assistance
The staff of the Engineering Library.
Miss J.l,l. Bradshaw for typing this thesis.
The University Grants Committee and the Auckland University Research
Grants Crnrnittee for financial support.
Finally the many people, fellow members of staff, past and present
students, and others who have contributed in some measure to the content
' of this thesis.
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iii
ABSTRACT
ACKNOIdLEDGEI.IENTS
,LIST OF SYI{BOLS
CHAPTER I
CHAPJER 2
g z.CI
$ ?.1
g 2.2
5 2.3
g 2.,4
g 2,5
C0l'ITENTS
A GENERAL INTRODUCTIO}I
A REVIEhJ OF
CTASSICAL LARGESIGNAL I4ODELS
Introductlon
Classical LargeSignal Flodel s
5 2.:l.l
g 2.1.2
The Base Region
The Steady State EbersMoll l4odel
g 2.2.1
l{oll [.lodel
,g 2.2.2
The First Order Junation lriodel
Conurnon Emitter. Configuration of the Ebers
Frequency Dependence in the Ebers1,loll
l,lodeil
Frequency Dependenee of the Incremental
' Comnon Emitter Cument Gain
The Beaufoy and Sparkes Charge Control l.{odel
$ 2.3.1
S 2.3.2
The Second Charge. Contrsl Equation
g 2.3.3 Alternative Form of the Charge Oontrol
Equatio,ns
g 2.3.4
The Active Region Equivalent Circuit
The Linvill Lumped ,Element ffiodel
Addi tr'onal ltechani sns
5 2.5.1
The Emitter Crowding Effect
g 2.5.2
Current upon Collector Vo:ltage
S 2.2.3
The First Charge Control Equation
Dependence of Current Gatn and Gollector
THE
7
I
9
l0
lt
t4
l8
2l
22
22
24
26
2.6
27
2g
29
Page 6
lv
5 2.5.3
5 2.5.4
References
Para.sitic Resistance
Junction Transition Capacitances
30
30
30
CHAPTER 3
5 3.0
5 3.1
s 3.2
5 3.3
5 3.4
CHAPTER 4
5 4.0
5 4.1
s 4.2
s 4.3
References
SPECIFIC DEVICE MODELS
Introduction
The Hami son llodel
The Bailey I'tode'l
The Gurnnel Model
The Hi I bers l'loclel
5 3.4.1
References
34
35
4l
42
45
47
48
Sunmary
DERIVATION AI{D AI{ALYSIS OF A I.ODIFIED CHARGE
COI'ITROL I,IODEL
I ntroducti on
A New Model
5 4.1.1 Analysis of the Basic ltodel
9 4,1.2
An Approximate Analytical Solution
$ 4.1.3
Model Calculations
Base Resistance
g 4.?,1
I'leasurement of Base Resistance
Frequency Dependent irtodel parameters
5 4.3.1
Emitter Lead Inductance
5 1.3.2 Junction Transition Capacitances
5 4.3.3
BaseEmitter Junction Capacitance
5 4.3.4
CollectorBase Junction Capacitance
g 4,3.5
Base Charge Storage parameter
THE TEST IIODULES
Introducti on
Test t'{odule Circuit Characteristics and Design
Design Philosophy
Broadband Toroidal Transmission Line Transformers
5 5.3.1 Analysis of the Input Transformer
Confi guration
50
50
52
53
57
59
62
63
64
65
65
67
71
72
CHAPTER 5
5 5.0
E 5.1
5 5.2
5 5.3
75
75
77
82
83
Page 7
5 5.4
g 5.3.2
The l4easured Performance of the Test Modules
g 5.4.1
Effect on Amptifier Linearity of
Input Hatching
References
Input Impedance l4easurement
Temperature
5 6.3. I
s 6. 3.2
s 6.3.3
TRANSISTCR DISTORTIOi.I STUDI ES
I ntroducti on
Distortion Perfornance of the piecewise Linearisecl
Devi ce i'lodel
90
92
101
l0r
104
105
109
ll0
112
115
llB
ll9
t?0
120
123
126
't26
128
128
IJJ
135
136
CHAPTER 6
5 6.0
$ 6.1
s 6.2
THERML PROPERTIES OF POI,IER TMNSISTORS
Introduction
Therma'l Stabi I ity
5 6.1.1
Determiriation of rhermar and Erectrical
Resi stances
5 6.1.2
Experimental Results for the 2N5642
Transi stor
5 6.1.3
Experimental Results for the 2N503g
Trans i s tor
Thermal Dependence of Current Gain
g 6.2.1
Consideration of Avalanche
Mul ti pl ication
g 6.2.2
Cument Gain Dependence of the ZNS64?
Transi stor
E 6.2.3
Current Gain Dependence of the 2N503g
Trans i s tor
Dependence of Co1lector Current
Temperature Dependence of 15
Evaluation of IS(T)
IS(T) Characteristics of the ZilS64Z
Transi s tor
5 6.3.4
IS(T) Characteristics of the 2N503g
Transi stor
Summary
References
5 6.3
g 6.4
CHAPTER 7
5 7.0
5 7.1
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vi
7.1.1
Spectra'l Analysis of the Truncated
Cosinusoid
Cornparison of Linearised Model
Distortion predictions with Test
Module Measurements
'
7 .l.Z
g 72 Nurnerical computation of collector curre4t hraveforms
and Spectral Composition
s 7,2r
consideration of the 2t.b64z rransistor
9 7.2.2 consideration of the 2N503g Transistor
5 7. 3
Sunimary
References
l38
140
I4t
142
153
155
155
CHAPTER B
5 8.0
s 8.1
5 8.2
5 8.3
5 8.4
5 8.5
CHAPTER 9
s 9.0
5 9.1
E 9.2
VERIFICATION OF THE 2N5642 MODEL
I ntroducti on
Model Data
Determination of Junction Temperature
Comparison of l4easured and predicted Module
Performance
Study of Test I'lodule Input Impedance
5 8,4.1 A Linear Impedance l4oclel
5 8.4.2
Input Impedance Hode.l Analysis
g 8.4.3 Verification of the Impedance Model
Summary
VERIFICATION OF THE 2N5038 MODEL
I ntroducti on
Model Parameters and Circuit Data
5 9.1 .l
l4odel Data Summary
5 9.1.2 Input Signal Level Dependence of Device
Temperature
t4odel Veriffcation  Low Frequency Model
g 9.2.1
High Frequency Model I
g 9.2.2
High Frequency Model z
S 9.2.3
Summary
E 9.3 Input Impedance Calculations
5 9.3. I
liodel Deri vati on
g 9.3.2
Conrputation of yrn.,
g 9.3.3
Computation of yin2
159
160
161
163
167
167
171
175
182

184
184
t85
l86
186
l86
190
202
204
204
207
207
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vii
9 9,3.4 Veriftcation of ZNFO38 Input Impedance
llodel
References
CIIAPTER IIO EONCLUS.ION
APPE!{DIX I
GEI'IERAL s0LuTI0N 0F EQuATIOr.r (4.10)
APPENDIX 2
FOUR,IER ANALYSIS OF THE COLLECTOR CU,RR,EI,IT }IAVEFORI,I
OF TI{E PIECEI{XS:E LINEARISED TRAI,ISrcTOR }IODEL FOR
. CLASS B CONDITIONS
ilg
APPENDIX 3
I4EASIJREI'IENT OF TI{E II''ITERI'IAL AND OVERLAP COI4PONENTS
OF COLLECTOR.SASE JUI{CTiON CAPACITANCE III A SI\IALL
SIGNAL TRANSISTOR
APPENDIX 4
A4.t
TEST HODULE ADMITTANCE AND IIIIPEDANCE MEASUREFIEI,ITS 230
l4easured values of collectortocollecto Load
Admr_ttances
44.2 lhasured varues of zNE64t rest lbdrle lnp.ut
Inpedanee Referred to the TransistorBase Terminals A3l
A43 Measuned values of zNSO3g rest thdule Inprrrt
Impedance Referred t0 the, TransistorBase Tenninals 233
230
208
2r0
2li3
216
225
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viil
LIST .OF SY!{BOLs
A
all, a12r a.l, ae'
b
Gt C'
%
Gj
GfE
C16, Co, C
Dil
Dp, Dn
E, Ex
Es' Eg(T)
fT
fp
%
%
h
IB' IC, IE
lB, iC' iE
Cuffent carrying cross.sectional area of transistor
Coefficients, in equations (2.9) and (2.10)
Reciprocal p:f ttrermal voltage
constants
Base diffusion capacitance
dunctlon capacitance
Emltterbaso j:unction capacitance
Collectorbase junction capacitances
Nth order" distsrtfon leve,,l
Hole and electron diffusion constants
Electric fteld strength
Ener(U bandgap
Transition fnequency
F cutoff frequency
LFrgesi'gnal fundamental frequency transconductance
Smallsignal transconduetance
lteration interval
'
DC components.of basen collector and emitter cur"rents
Instantaneous values o base, collector and emitter
currents
.
IES, ICS
trn
tp
Junction reverse saturatlon currents
. Electrsn current
Hole current
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.ix
Is
I0" Il , IA ., .
Itdp, Jn
K
L
Lp
M
f
tn
NA' ltlD
NC, NV
h, F
ni
n*' Pno (Po,)
F
nrfu
q
R,, t
=$IEs
Modified Bessel Functirons
llole and electron curr€nt densit,ies
Boltanann's Oon:stant
$elf tnductance
Hole lrcombinatJon length
fii:ller Avalanehe Multiplication Factor
Avalanche rnultlpli;cat'ion factor in equation (b.S,Z)
dunction emission fac;torr, also excess phase factor
' Aceqpton and donsn atom concentrations
Density o,f state constants (S 6.2)
Electro,n and hoile denslties
ilntrfnsic electron densrity
Electron and ho'le equillbrium densities
Excess hole densitJ
Total base charge
ileetron charge
Resistance
RB' tn
Rc'h
RL' RI,CC
S
s
T
TurO, To
t
V
YA.
Bbse paras:itic resistances
0ollector and ernitter parasitic resistances
toad resistance
Storance
eompl:ex freouency vaniabl,e
Absolute temperature
Anrbient ard reference temperatures
Time
.
Voltage
l, Ft,
Ehr"ly ttoltage
r' r.
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x
vBB' vcc, vEE
Clreuit supplty voltages
Transistor breakdown voltages
Junction vol tarEe
Veloci ty
Instdn esus potentials at transistor tennlnals and
circuit nodes. A second subscript denotes the
reference pofnt lf other than the circuit cormon
rail
Base wtdth
Emitter w'idth
Fo.nrard a,nd reyerse cofimonbase current gains
tow f..requengy conmonbase cunent gains
Base transport factor
Fqryard and reverse comrnon emitter current gains
Emrltter injection efficiency
Electron and hsle mobilit,ies
Resisti:vi ty
Therma'l resistance Junctiol to case
Thermal resistance case to air
Ba.se ninority camier lifetime parameter (S 2.3.I1,
Base region transit time
Oliarge control model parameters
Hole rccombinatisn I ifetirne
Junctfon barrier potential
Angular frequency
Angular half power freguencies (EbersMoll modet)
Angular transition frequency
vcgo, vcro
vj
vc' vBE
v
vBt
l,l, WB
l.lE
"r' h
qFo' ho
t' to
FF' h
T
h'h
p
oJo
oaa
"B
"b
'rF' rBF
tp
uo
trl
oft sR
h
Page 13
xl
N0TE: synbols listed ane those which largety aro no,t defJned in the body
of the text' Insofa,r as possible the symbols adopted are those
cons'idered to be in most cormon usage in semicond,ucton device
modelllng. In eertain instances it has been considered appropriate
to retain s,ymbol definitisns used ln particular source referencgs
even when this results in multiple usdg6 of a particular symbol.
In these ea$es specific definitions have been inelluded at the
approprfate points in the text.