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Thermal effects on RF/microwave circuit and DC bias network design

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THERMAL EFFECTS ON RF / MICROWAVE CIRCUIT
AND DC BIAS NETWORK DESIGN
By
AHMED EJAZ NADEEM
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1998
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T his dissertation is dedicated to m y parents.
M ay G od rest the departed soul o f my late father, Justice M uftakhiruddin, and bless my
dearest m other w ith health and happiness, rew ard h er abundantly for her good deeds and
give her m any m ore years to be w ith us, am en.
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ACKNOWLEDGEMENTS
I w ould like to acknow ledge and thank the G overnm ent o f Pakistan, M inistry of
Science & T echnology for aw arding m e the scholarship for studies leading to my Ph.D. I
also gratefu lly acknow ledge Pakistan A ir Force for granting me leave o f absence for the
duration to com p lete m y work, w ithout w h ich this w ork would not have been possible.
I w ould like to express my p rofound gratitude to Professor W illiam R. E isenstadt.
w ho served as a chairm an o f my supervisory com m ittee. His sincere guidance, continuous
encou rag em en t, constructive criticism , invaluable technical advice, and unceasing and
unconditional su p p o rt in my academ ic and personal affairs made this w ork possible. I also
extend m y d eepest appreciation to him for being very friendly, polite, understanding, and
m aking h im se lf available during all w orking days to discuss anything I w anted to, and also
for sp en d in g his precious time at hom e review ing and editing this m anuscript. I m ust
adm it that it has been a pleasure to have been his student.
I am also indebted to Professors G ijs B osm an, M ark E. Law, Khai D. T. N go, and
O scar D. C risalle w ho very kindly agreed to serve on m y Ph.D. com m ittee. I appreciate
their interest in my w o rk and their valuable suggestions and com m ents from the stage o f
research proposal to its realization. I w ould like to specially thank P rofessor G ijs B osm an
for his invaluable and m uch needed en couragem ent and advise to continue w orking on my
degree w hen m y fath e r’s sudden and u n tim ely dem ise shook my self beyond description. I
w ould also like to appreciate P rofessor K hai D. T. Ngo for his ever increasing generosity,
iii
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support, and accom m odation during my graduate program . I w ould also like to express my
thanks to Professors M ark E. Law and O scar D. C risalle for alw ays being very helpful and
accom m odating.
I also gratefully acknow ledge the contributions o f m y friend. Dr. D avid E. Bockelman o f M otorola R adio Products G roup A pplied R esearch. His valuable assistance and
suggestions greatly helped me obtain reliable scattering param eter d ata from m easure­
ments. W ithout his support this w ork would not have been possible.
I w ould also like to express m y special thanks to Scott D. Langford o f H arris S em i­
conductor, for providing m e both the m aterial and o pportunity to use their facility to per­
form m easurem ents on DC bias circuits.
I w ould like to gratefully acknow ledge the valuable help and assistance o f K eith J.
Ram bo, Jam es P. G oetten, and Ali A nghaie tow ards the use o f com puter resources. T heir
ever increasing generosity and alw ays very prom pt response is greatly appreciated.
Special thanks are given to my friends. P rofessors Saeed Khan and A nw er S.
Ahmed and th eir fam ilies for being very nice to me, m y w ife, and my son. T heir friendship
and hospitality m ade ou r social life in G ainesville, Florida, very pleasant and unforgetta­
ble. I w ould also like to thank the entire M uslim co m m unity in G ainesville, F lorida for
their friendliness and for m aking us feel com fortable during ou r stay here.
I am also grateful to m y brothers, A hm ed Pervaiz M uftakhir, Dr. A hm ed Javaid,
and Dr. A hm ed Shahzad and their w ives, w ho, despite being far away, kept in touch regu­
larly and provided help and encouragem ent throughout the graduate program . I w ould also
like to thank m y in-law s for their support and interest in m y education.
O f all w ho supported and provided assistance, none w as as critical and valuable as
iv
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my d ear w ife, N u v era A rjum and. I w ould like to express m y deepest gratitude and love to
her, for sharing not only the m om ents o f happiness and joy, but also being w ith me in dif­
ficult tim es o f m y life, w hen I needed her the most. H er patience, understanding, en co u r­
agem ent, help and com m itm ent served as the guiding light w hich led the w ay to this
conclusion. I w o u ld also like to acknow ledge my son, A hm ed S haharyar E jaz, w ho pro­
vided m e m uch n eed ed relief and com fort through his innocent and unforgettable conver­
sations, activities an d love.
Finally, I w o u ld like to express my profound respects and thanks to my d ear par­
ents for their en d less love and support. U nfortunately, my father. Justice M uftakhiruddin
did not live long en o u g h to w itness the realization o f his dream s fo r me. He w as my m en­
tor, m y guide, and m y best friend. W ithout his help, and sacrifices o f my bo th parents, it
w ould not have been possible to pursue my graduate studies. H ence, this dissertation is
dedicated to my parents.
v
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TABLE OF CONTENTS
A C K N O W LED G EM EN TS
............................................................................................................... iii
ABSTRACT
ix
C H A PT E R S
1
IN T R O D U C TIO N
................................................................................................................. 1
2
R E V IE W O F L IT E R A T U R E ...................................................................................................7
Scattering P aram e ters.................................................................................................................9
S cattering Param eter M easu rem en ts.....................................................................................12
C om pact M odels For B ipolar Ju n ctio n T r a n s is to r ..........................................................19
T ransm ission L in es................................................................................................................... 21
M icrostrip L i n e s ..........................................................................................................25
C oplanar W av eg u id es................................................................................................ 27
A m bient T em perature Effects on D C B ias N etw o rk s.................................................... 28
Self-H eating Effects on C irc u its ........................................................................................... 29
Self-H eating Effects on L arge-S ignal C ir c u its ..................................................30
Self-H eating Effects on S m all-S ig n al C ir c u its ..................................................30
3
SM A U L-SIG N A L S -PA R A M E T E R S A N D B IA S S E N S IT IV IT Y ..........................32
S -P aram eters and Sm all-Signal H ig h -F req u en cy M odel F or B J T ............................ 34
Experim ental V erification o f C lo sed -F o rm E x p ressio n s for S -P a ra m e te rs
40
C om parison o f M easured, C alculated, and S im u lated Sm all-S ignal
S -P a ra m e te rs .......................................................................................................................... 54
Bias Sensitivity o f Sm all-Signal S -P aram eters.................................................................59
.............................................................................................................................65
Sum m ary
4
A N A L Y T IC A L M O D E L L IN G O F P A C K A G E D T R A N S IS T O R S A N D
PR ED IC T IO N O F T H E R M A L EFFE C T S O N S M A L L -S IG N A L
S -P A R A M E T E R S .................................................................................................................69
T herm al M easurem ents o f S m all-S ignal S - P a r a m e te r s ............................................... 70
Selection o f A ctive D e v ic e s .................................................................................... 72
Selection o f L am inate M aterial............................................................................... 73
Selection o f T ransm ission M ed ia and C o n n e c to r s ...........................................74
D esign o f T ransm ission L i n e s ............................................................................... 75
L ayout o f B o a r d ..........................................................................................................77
C haracterization o f B o a rd ......................................................................................... 77
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S-Param eter M easurem ents o f P ackaged T ran sisto rs.................................................... 82
S-Param eter P rediction fo r Packaged D evices Using C losed-Form
Expressions D erived from Intrinsic D evice M odel for B ipolar T ran sisto rs . . . 89
D erivation o f Im proved C lo sed -F o rm E xpressions for S-Param eters
o f B ipolar T ransistors U sing M odified G um m el-Poon M o d e l...............................92
T herm al Effects on S m all-S ig n al S -P a ra m e te rs ............................................................. 99
Sum m ary
.......................................................................................................................... 104
5
SEN SITIV ITY A N A LY SIS O F S M A L L -SIG N A L S-P A R A M E T E R S .................108
Sensitivity: C oncept, Im portance, and D efin itio n ......................................................... 109
Sensitivity A nalysis o f S m all-S ignal S -P aram eters o f B J T s .................................... I l l
Sum m ary
.......................................................................................................................... 135
6
A D ESIG N A N A L Y SIS O F A M B IE N T T E M PER A TU R E E FFE C T S
ON C O M M O N M IC R O W A V E D C BIA S C IR C U IT S ......................................... 137
Sim plified M odeling o f B ip o lar Junction T ransistor for DC B ias A n aly sis. . . . 139
Therm al Effects on T em p eratu re S ensitive Independent V a ria b le s ........................ 142
Tem perature C oefficient o f S aturation C u r r e n t..............................................142
Tem perature C oefficient o f F orw ard C urrent G ain......................................... 145
Tem perature C oefficient o f B ase E m itter V o lta g e ......................................... 148
E xam ple Passive and A ctive D C B ias N etw orks: Analysis and V erification . . 150
Passive B ias N e tw o r k ............................................................................................. 152
Active Bias N e tw o rk ................................................................................................159
Sim ple C urrent S ource (C urrent M i r r o r ) ......................................................... 166
D C B ias Design O p tim iz a tio n ........................................................................................... 168
O ptim ization o f Passive Bias N e tw o rk .............................................................. 170
O ptim ization o f A ctive B ias N e tw o r k .............................................................. 175
Sum m ary
.......................................................................................................................... 177
7
C O N C LU SIO N S A N D F U T U R E W O R K .......................................................................180
D esign-O riented T h erm al A n a ly s is ............................................................................... 184
Therm al E ffects on S ta b ility ................................................................................. 185
Therm al E ffects on Im pedance M atching N e tw o rk s .................................... 185
Therm al E ffects on U nilateral Figure o f M erit................................................ 185
Self-H eating Effects on D C B ias N etw orks used in R F/M icrow ave C ircu its . . 186
O ptim ization o f DC B ias D esigns for A m bient and Self-H eating E f f e c ts
186
A PPE N D IC ES
A
C LO SED -FO R M E X P R E S S IO N S F O R SM A LL-SIG N A L
S -P A R A M E T E R S ...............................................................................................................187
B
BIAS D E PE N D E N C E O F S M A L L -S IG N A L D EV ICE M O D EL
P A R A M E T E R S ................................................................................................................... 199
C
EX PER IM EN TA L V E R IF IC A T IO N O F C LO SED -FO R M E X P R E S S IO N S
vii
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FOR S M A L L -S IG N A L S C A TTER IN G P A R A M E T E R S ................................... 202
D
A N A LY TIC A L E X PR ESSIO N S FO R PA SSIV E DC B IA S C IR C U IT S ........... 214
E
A N A LY TIC A L E X PR ESSIO N S FO R A C TIV E DC BIA S C IR C U IT S ..............217
R EFER EN C E LIST
.................................................................................................................. 220
B IO G R A PH IC A L S K E T C H ...........................................................................................................232
viii
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A bstract o f D issertation P resented to the G raduate School
o f the U niversity o f F lorida in Partial Fulfillm ent o f the
R equirem ents for the D egree o f D octor o f Philosophy
T H E R M A L EFFEC TS O N R F / M ICROW AVE CIR C U IT
A N D DC BIA S N E T W O R K D ESIG N
By
A hm ed E jaz N adeem
M ay 1998
C hairm an: W illiam R. E isenstadt
M ajor D epartm ent: Electrical and C om puter E ngineering
The developm ents m ade in the low er frequency electronic circuits also influenced
the grow th o f the m icrow ave circuit design. Today, m icrow ave planar circuits have found
applications in the fields o f com m unications, electronic w arfare, radars, and w eapon sys­
tem s. A m ong many environm ental variables, tem perature is the m ost obvious one that nei­
ther rem ains the same in different fields o f ap p licatio n s, nor rem ains constant in any one
system . Virtually, every transistor param eter is d irectly or indirectly affected by tem pera­
ture because o f therm al dependence o f the physical properties o f the materials used to fab­
ricate sem iconductor devices.
A design-oriented analysis o f therm al effects on R F/M icrow ave circuits has been
presented. A n insightful relationship betw een sm all-signal scattering-(s-) param eters and
circuit m odel for bipolar ju n ctio n transistor has been developed. C losed-form expressions
have been verified against on-w afer m easurem ents from 45 M H z to 20.045 G H z. Bias and
ix
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therm al sensitivity o f s-p aram eters has been presented. L im itatio n s o f intrinsic m odel for
bipolar transistor have been pointed o u t and the m odel has b een m odified appropriately to
account for excess phase and to im prove the predictions o f m agnitude o f s-param eters.
Utility o f w ork to pred ict s-p aram eter for packaged devices at different tem peratures has
been shown.
Sensitivity an aly sis o f sm all-signal s-param eters o f b ip o lar ju n ctio n transistors
w ith respect to the d ev ice m odel param eters has been p erfo rm ed . T he device model
param eters w h ich play d o m in an t role in causing variations in s-param eters have been
identified. T he link b etw een the changes in m odel p aram eters caused by eith er variations
in bias conditions o r tem p eratu re and their co rresp o n d in g effects on sm all-signal sparam eters has b een estab lish ed .
A sim plified m odel has been developed to estim ate the effects o f am bient
tem perature on bias cu rren t in com m only used bias n etw orks. A m ethodology has been
presented to d eterm in e key contributions o f circuit elem en ts in bias current variations with
tem perature. Insightful an aly sis o f dc bias netw orks has been presented and results have
been experim entally verified. An optim ization strategy h as been developed w hereby
selection o f suitable tech n o lo g y for passive circuit elem en t(s) im proved the design while
the basic circuit to p o lo g y rem ained unaltered. N eed to revisit the standard RF/M icrow ave
design procedures to acco u n t for therm al effects and to fo rm u late m ethodologies to design
therm ally stable circu its has been show n.
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C H A PT E R 1
IN TR O D U C TIO N
The transistor, since its invention at the Bell Telephone L aboratories in 1948, not
on ly has brought a revolution in electro n ics industry, but has also found its way into varied
applications in com m ercial, in dustrial, and m ilitary fields. D uring the first decade of sem i­
co n d u cto r devices ( 1950s), m ost o f th e research and engineering w as directed towards ger­
m anium sem iconductors, but w ith the developm ent o f planar tech nology in 1958, the
silicon devices overtook the g erm an iu m devices [Gro67], T he p la n ar silicon technology
show ed a new frontier o f integrating b oth active and passive dev ices on the same silicon
substrate. The num ber o f co m p o n en ts accom m odated on a single silicon integrated circuit
(IC ) has exhibited phenom enal grow th, virtually doubling every y ear [M 0 0 8 O] and inte­
gration o f several m illion tran sisto rs has already been reported. H ow ever, prior to 1965,
m ostly all m icrowave equipm ent utilized waveguides, coaxial lines, o r striplines. The
technological advancem ents m ade in the low er frequency electro n ic circuits also influ­
enced the grow th o f the m icrow ave c irc u it design. Today m icrow ave circuits can be classi­
fied into three categories: m icrow ave discrete circuits (M D C ’s), m icrow ave monolithic
integrated circuits (M M IC ’s), and m icrow ave integrated circuits (M IC ’s) [Lia87],
M icrow ave planar circuits have found applications in th e fields o f com m unica­
tions, electronic w arfare, radar, an d w eapon systems. The co m m u n icatio n applications
include satellites, space, long d istan ce telephone, m arine, cellu lar telephone, mobile, air­
craft, personal, data, w ireless, L A N , vehicles, and others. W hile rad ar has applications for
1
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aircraft, w eather, collision avoidance, im aging, air defence, ships, police, intrusion detec­
tion. sm art w eapons and so on [Cha94]. A m o n g st m any different environm ental variables,
tem perature is the m ost obvious one that n e ith e r rem ains same in different fields o f m icro­
w ave applications, no r remains constant d u rin g the use o f any m icrow ave circuits in any
one application.
Today, bipolar transistor is one o f th e m ost w idely used sem iconductor devices for
m icrow ave applications. It is the dom inant active device in microwave circu it design from
U H F (3 00-3000 M H z) to S-band (2-4 G H z). W ith the im provem ents in p lan ar technology,
the upper frequency limit for bipolar ju n c tio n transistors (BJTs) is contin u o u sly being
extended, and devices with 30 G H z f j an d 4 0 G H z fmax are now available [A il97]. The
m ajority o f bipolar transistors o f current in terest are made by silicon p lan ar technology.
Silicon (Si) due to its natural abundance, b e tte r m echanical properties, and superior natu­
ral dielectric is far and away the m ajor sem ico n d u cto r material in high sp eed electronics
[Sze90]. The Si bipolar transistor is inexpensive, durable, integratable, an d offers much
higher transconductance than com peting field-effect devices. It has m oderate noise figure
in radio frequency (RF) application, and 1/ f noise characteristics are 10-20 dB superior to
G aA s M ESFETs. F or these reasons, it d o m in ates in the lower m icrow ave bands [Bah 8 8 ].
A lm ost every transistor p aram eter is d irectly or indirectly affected by tem perature
because o f the therm al dependence o f the p h y sical properties o f the m aterials used to fab­
ricate sem ico n d u cto r devices [M ul77, S ze8 1 , Fre85, Sah91]. T herefore, the electrical
properties (currents and potentials) o f a d ev ice are also tem perature dependent. T he RF/
M icrow ave design param eters such as g ain an d noise figure are w ell know n to be strong
functions o f the collector current in tran sisto r am plifiers involving B JTs [G on84,V en90].
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3
W hile the co llecto r current itself is quite sensitive to tem perature variations. T he operating
tem perature o f a device is determ ined by the am b ien t tem perature and the pow er dissi­
pated in the device (self-heating effect) [Str59, Lia93J. In o rd er to design a therm ally sta­
ble R F/m icrow ave circuit, the am bient tem perature and self-heating effects need to be
considered both individually and com bined.
T his w ork presents a design-oriented analysis o f therm al effects on R F/M icrow ave
circuits. Since scattering- (s-) param eters are used extensively in R F/M icrow ave circuit
design, so know ledge o f therm al dependence o f s-param eters is considered essential. It has
been show n that since the nature o f s-p aram eter m easurem ents is external to the netw ork,
therefore, it tends to lim it o r com pletely obscure the insight to the device b ehavior w hich
in fact causes the variations in s-param eters. Sim ilarly, co m p u ter sim ulations, w hen used
to predict s-param eters at different bias conditions, tend to m ask the device behavior. To
gain an insight into the netw ork itself, a relatio n sh ip has been developed betw een sm allsignal s-param eters and equivalent circu it m odel for bip o lar junction transistors. The
closed-form expression derived for sm all-signal s-param eters have been extensively veri­
fied against on-w afer m easurem ents, an d excellen t agreem ent has been achieved for a
broad frequency range o f 45 M Hz to 20.045 G H z. T he s-param eters calcu lated using
closed-form expressions derived durin g this w ork using intrinsic hybrid- 7t m odel have
been com pared w ith m easured data and M D S sim ulations. T he results have show n that sparam eters calculated using closed-form expressions m atched m easured data m ore closely
than s-param eters obtained through M D S sim ulations. B ased upon the insight gained
through the closed-form expressions, a qualitative description o f bias sensitivity o f sm allsignal s-param eter has been provided. It has been show n through m easurem ents that both
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4
m agnitude and phase o f all four s-param eters for B JT in com m o n -em itter configuration
are sensitive to bias co n d itio n s.
A step-by-step p ro ced u re to measure sm all-signal s-param eters for packaged
devices at different tem p eratu res has been show n. T h e lim itations o f intrinsic device
m odel at higher freq u en cies have been pointed out. It has been show n that m agnitudes o f
S | i and S 22 and phase an g les o f S (2 and S 21 are overestim ated, w here as the m agnitudes o f
S 12 and S2) and phase an g les o f S! 1 and S 22 are u nderestim ated w hen only intrinsic device
m odel is used. T he m o d el has been modified by adding ex trin sic base to collector capaci­
tance and im proved clo sed -fo rm expressions have been derived. It has been shown that the
im proved clo sed -fo rm ex p ressio n s not only account for n eed ed excess phase but they also
im prove the m agnitude p redictions. The approach used in this w ork has been com pared
w ith prior w ork and a b rie f sum m ary has been com piled. T he closed-form expressions
derived during this w o rk have been used to predict tem perature effects on sm all-signal sparam eters. T he an aly tical predictions have been extensively verified for three different
packaged bip o lar ju n c tio n transistors for a w ide range o f tem peratures from -35°C to
+85°C . E xcellent ag ree m en t betw een the m easurem ents and analytical predictions has
been shown for b oth m ag n itu d e and phase o f sm all-signal s-param eters.
To identify the so u rces inside the active device w hich cause variations in sm all-sig­
nal s-param eters due to ch an g es in tem perature, a system atic sensitivity analysis has been
presented. The device m o d el param eters which play a do m in an t role and cause significant
changes in m agnitude an d phase o f sm all-signal s-param eters have been identified.
H aving show n th at sm all-signal s-param eters are sensitive to bias current, bias
voltage, and tem p eratu re, it has been established that, to realize a therm ally stable R F/
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5
M icrow ave circuit, it is necessary to m aintain a constant bias cu rren t despite variations in
tem perature. A sim plified G u m m el-P o o n based large-signal m odel has been developed to
estim ate changes in bias cu rren t due to variations in am b ien t tem perature. Several popular
DC bias circuits have been an aly zed using the technique dev elo p ed during this work. A na­
lytical results have been verified against therm al m easurem ents o f D C bias circuits. This
new analytical approach p ro v id es a valuable insight into th e design an d key contributions
can be determ ined for each d esig n . B ased upon the insight gained during this work, an
optim ization strategy has been developed, w hereby the d esig n can be improved signifi­
cantly by selection o f the ap p ro p riate and suitable technology for the passive circuit ele­
m ents w hile the circuit topology rem ains unaltered.
T he dissertation has b een organized in seven chapters. A b rie f overview o f the
advancem ents in electronics, its im pacts on m icrowave circu it design, and the motivation
for this w ork is given in the cu rren t chapter. C hapter 2 co n tain s a literature review o f the
relevant topics. This chapter p ro v id es theoretical b ackground for quick reference to key
concepts m entioned in the later chapters. Derivations and proofs involving m athem atical
m anipulations have not been in clu d ed , in the interest o f sim p licity and to provide a more
physical than m athem atical review o f the topics. However, sufficient references have been
provided.
In C hapter 3, the relatio n sh ip betw een sm all-signal s-param eters and high fre­
quency equivalent circuit m odel fo r bipolar junction transistors, verification o f closedform expressions for s-param eters against on-w afer m easurem ents, com parison o f m ea­
sured, calculated and sim ulated s-param eters, and bias sensitivity o f the s-param eters has
been presented.
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6
C hapter 4 presents sm all-signal s-param eter m easurem ents o f packaged bipolar
junctio n transistors done at different am bient tem peratures, com parison betw een m easured
data and s-param eters calculated using closed-form expressions derived in C hapter 3. a
derivation o f im proved closed-form expressions for sm all-signal s-param eters o f bipolar
junctio n transistors. A lso included in this chapter is therm al sensitivity o f s-param eters
based upon the m easured data.
C hapter 5 presents the sensitivity analysis o f sm all-signal s-param eters o f bipolar
junctio n transistors w ith respect to the device m odel param eters. T he device model
param eters w hich play dom inant role in causing variations in s-param eters have been
identified. A lso included in this ch ap ter is a com parison o f sensitivities o f sm all-signal sparam eters w ith respect to each device model param eter. A system atic approach is
presented to establish the link betw een the changes in m odel param eters caused by either
variations in bias conditions o r tem perature and their co rresponding effects on smallsignal s-param eters.
In C hapter 6 , an im proved G um m el-Poon based analysis o f the am bient
tem perature effects on co llecto r currents o f bias designs com m only used in RF/M icrow ave
circuits and a design optim ization strategy has been presented. T he analysis o f popular DC
bias netw orks was perform ed using three different npn transistors and in all cases the
m easurem ent data, sim ulation results and the analytical predictions w ere found to be in
excellent agreem ent. A lso, the optim ization strategy w orked eq u ally w ell for all three
transistors, thus adding validity to the approach presented in this chapter.
Finally, in C h ap ter 7, the w ork presented in this dissertation has been sum m arized
follow ed by proposals for future research.
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CHAPTER 2
R E V IE W O F L IT E R A T U R E
A m plification and detection o f received signal are the fundam ental processes o f any
com m unication system . For am plification o f R F/M icrow ave signals both vacuum tubes
and solid state devices are used. V acuum tubes such as klystrons and travelling-w ave tubes
(TW T) are used for high pow er ap plications [H ar47, Pie50, Lia80, G an 8 1 ], w hereas solid
state devices are more suitable for low noise and m edium power levels up to a couple o f
hundred w atts [Bah 8 8 ]. W hen co m p ared w ith the vacuum tubes, solid state devices
require low voltages to operate and they are relatively com pact, lightw eight, efficient, and
extrem ely reliable. T hese distinguishing features m ake them particularly useful for spe­
cialized applications such as space, m ilitary, and portable w ireless, w here both w eight and
size are o f param ount im portance.
A m plifiers using solid state devices can be classified into two categories: (1) reflec­
tion-type o r negative resistance am plifiers and (2) transm ission type am plifiers. The first
category o f am plifiers m ainly em ploys tw o-term inal solid state d ev ices—diodes, such as
varactor, tunnel, and G unn to m ention a few. H ow ever, negative resistan ce circuits, for
exam ple, oscillators are also realized w ith three term inal devices—transistors. Design,
analysis and characterization o f these ty p es o f am plifiers are d iscussed extensively in the
literature [Pen62, Bla64, C ha64, S te67, W at69, S te71, Eas73, H ad73, Ven82, G on84,
B ah 8 8 , C ha94, Kai94]. T he transm ission type am plifiers use three term in al devices—tran­
sistors, w hich could be bipolar ju n c tio n tran sisto rs (B JT s), m etal field effect transistors
7
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8
(M E S F E T s), high electron m obility transistors (H EM Ts), o r hetro ju n ctio n bip o lar transis­
tors (H B Ts) [Coo71, H a81, Ven82, G on84, B ah 8 8 , C ha94, K ai94]. T h e fundam ental dif­
ference in the basic configuration o f these tw o types o f am plifiers is th at reflection type
am plifiers use a circulator to isolate the input from the output, w h ereas in transm ission
type am plifiers, sim pler input an d output m atching netw orks are required. A block dia­
gram o f a typical transistor am plifier is show n in Fig. 2-1. T he am plifier consists o f an
input-m atching netw ork, a transistor, and an output-m atching netw ork. T he am plifier is
connected to the source and load through transm ission lines having characteristic im ped­
ance o f ZQ.
T ransistor
Source
Input
M atching
N etw ork
O utput
M atching
N etw ork
F ig .2 -1. B lock diagram o f a typical transistor amplifier.
T his chapter provides theoretical background for quick referen ce to key concepts
m entioned in the later chapters.T he topics review ed here include scattering-param eters
and their m easurem ents, com pact m odels for the bipolar ju n ctio n transistor, transm ission
lines, am bient tem perature effects on DC bias netw orks, and self-h eatin g effects on cir­
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9
cuits. D erivations and proofs involving m athem atical m anipulations have not been
included in this chapter, in the interest o f sim plicity and to provide a m ore physical than
m athem atical review o f the topics. S ufficient references and leads have been provided.
Scattering Param eters
S cattering-param eters are being used extensively in R F/M icrow ave circuit design
because they are m ore suitable for high frequency m easurem ent than o ther sm all-signal
param eters. In addition, they characterize circuits at a transm ission im pedance (50 ohms)
that is typical o f m icrow ave signal lines. O th er tw o-port param eters, for exam ple, hybrid
(h-), adm ittance (y-), and im pedance (z-) characterize two-port circuits by relating total
voltages and total currents at each o f the tw o ports. Also, they require both input and out­
put ports o f the netw ork to be open and sh o rt circuited, in order to m easure the param eters.
As the frequency goes higher and higher, th ese necessary conditions for h, y, and z param ­
eter m easurem ents becom e very difficult to achieve. First, equipm ent is not available to
m easure total voltages and currents at the netw ork ports at high frequencies. Second, per­
fect short and open circuits—necessary co n d itio n s to measure h, y, and z param eters—are
very difficult to achieve over a band o f frequencies because m agnetically induced induc­
tances in short term inations and electrical field fringing in open circuit term inations make
the m easurem ents exceedingly difficult. T h ird , active devices, such as transistors are not
usually stable w hen term inated w ith open circu its and often result in oscillations [Fit78].
T he o nly and obvious choice in su ch situations is to use s-param eters. S -param e­
ters are basically a m eans o f characterizing n-port networks and their dom ain o f applica­
tions is not lim ited to only two o r three port networks. S-param eters have num erous
advantages over other conventional netw o rk param eters at radio frequencies (RF) and also
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10
at microwave frequencies. S-param eters are m easured w ith resistive term inations (usually
50 ohm s) w hich m inim ize the chance o f oscillations in active devices. These param eters
are defined in term s o f quantities w hich are m easurable at h igh frequencies i.e. pow er
waves. The application o f signal flow graph techniques on s-param eters helps to analyze
com plex netw orks. T h e origin o f s-param eters can be traced b ack to the transm ission line
theory and pow er param eters. S-param eters are related to incident and reflected power,
w hich do not change in m agnitude along lossless lines. Fig. 2-2 show s a pictorial view o f a
tw o-port netw ork an d its s-param eters.
Two-Port
N etw ork
Fig.2-2. Tw o-port netw o rk and its s-param eters.
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11
The fam iliar m easu rem en t based definition o f s-param eters is as under:
S
11
( 2 - 1)
= —
“i
=
5
12
0
( 2- 2 )
= -!
aj = 0
(2-3)
5 21 " a [
=
0
(2-4)
22
aj = 0
w here a,- and bt are norm alized co m p lex voltage waves incident on and reflected from the
ith port o f the netw ork. From th e definitions, it can be seen th at S j i and S 22 represent the
input and output reflection coefficients, respectively. S n is m easured by term inating the
output port with m atched load (i.e. Z l = Z q ) , this sets a 2 eq u al to zero. Sim ilarly S 22 is
m easured by setting input p o w er equal to zero and m atching the port im pedance with Z q
(i.e. Z S=Z0 ). S2i represents the forw ard transm ission gain w ith the output port term inated
by a m atched load (i.e. Z l = Z q ), and S 12 m eans the reverse transm ission gain w ith the
input port term inated by a m atch ed load (i.e. Z$=Zo).
M athem atical derivations and proofs for s-param eters fro m the transm issions line
theory have been dealt w ith ex ten siv ely in literature [C 0 I6 6 , D w o79, G up81, Ven82,
G on84, R od 8 6 , Ven90]. T he referen ced literature also has th e relationships betw een sparam eters and other netw ork p aram eters in tabulated form.
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12
Scattering P aram eter Measurements
B eing able to d irectly m easure the quantities o f im portance h as a direct bearing on
understanding and con tro llin g those quantities. A m ajor problem in the rapidly growing
m icrowave sem iconductor in d u stry today is achieving tim ely developm ent o f new m icro­
wave processes and devices. T h e advancem ent o f high speed integrated circuit processes
and devices is adversely effected by an inability to m easure high frequency effects and
m odel them accurately.
Before the developm ent o f m icrow ave w afer probing techniques, devices w ere first
packaged and fixtured in o rd er to m easure th eir perform ance. A fter the m easurem ents, the
bond wires, package, and fixture effects w ere subtracted to d eterm in e the device perfor­
m ance. This m ethod o f m easu rin g scattering param eters is still in p ractice w hen on-w afer
m easurem ents are neither p o ssib le n o r convenient. To perform such m easurem ents, the
packaged devices are m ounted on su itab ly designed boards. T hese boards are com prised
o f ( 1 ) pads com patible to th e fo o tp rin t o f th e package under test, ( 2 ) transm ission media
from the pads to the edges o f the board, and (3) RF/M icrow ave connectors. T he connec­
tors facilitate insertion o f b o ard in the m easurem ent path. A variety o f different types o f
transm ission lines can be u sed to convey the RF/M icrow ave signal fro m the connectors to
the pads and back such as m icro strip lines, striplines, co p lan ar w aveguides, and coplanar
striplines, etc. A lso, several d ifferen t types o f connectors are available to choose from , for
exam ple flange m ount, panel m ount, bulkhead P.C. m ount, and ed g e m ount. Fig. 2-3 (a)
show s a typical board layout fo r m easuring the packaged devices, w h ile Fig. 2-3 (b) shows
a typical high frequency m easu rem en t set up. The s-param eter test set show n in the figure
is a part o f the netw ork analyzer. T he m ax im u m frequency at w hich s-param eters o f pack­
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13
aged devices m ounted on boards can be m ade depends largely on the design and perfor­
m ance o f the boards. T he RF/M icrowave signal undergoes several discontinuities and gets
subjected to undesirable effects from the very start to the end, such as: ( I ) discontinuities
w ithin the connectors due to right angle o r other bends, ( 2 ) discontinuities associated with
m ode o f launching the signals on to the boards, (3) disco n tin u ities and losses w ithin the
transm ission m edia, and (4) discontinuities and losses d u rin g travel betw een transm ission
m edia and the pads and also to the device under test. T h ese losses and adverse effects can
be reduced through careful design, but certainly can not be elim inated com pletely. Despite
the above, fairly accurate and reliable m easurem ents have been m ade and reported up to
several G H z. T he w ork presented in the later chapters has been com pared against the sparam eters o f packaged transistors m easured upto 2 G H z through careful design o f the
boards and adherence to the correct m easurem ent techniques.
O n-w afer m icrow ave probing opened new avenues for im provem ent in high-speed
circuits. T he current state o f the art for perform ing high frequency m easurem ents is to use
coplanar probes in conjunction with netw ork analyzer and a calibration substrate. These
coplanar probes are designed to provide transm ission lines right up to the pads on the
w afer o f the device under test (DUT). These probes have been reported to function up to
120 G H z. The setup for on-w afer m easurem ent is sim ilar to that shown in Fig. 2-3 (b),
except the w afer is placed on the chuck and held in place firm ly by a vacuum . T he connec­
tions to the pads on the w afer are made through the co p lan ar probes.
C alibration o f m ost test equipm ent is thought o f as a procedure done once o r twice
a year to keep the instrum ent operating w ithin certain specifications. This is not the case in
R F/M icrow ave m easurem ents, which are extrem ely sensitive to the norm al w ear o f fix-
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14
T ransm ission Line
T ransistor
C onnector
G round Plane
PORT 1
PORT 2
S-PARAMETER TEST SET
DUT
Fig.2-3. Typical high frequency m easu rem en t setup, (a) board layout, (b) test setup.
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15
tures. probes, connectors, and cables. Thus, prio r to m aking m easurem ents o f either p a c k ­
aged or on-w afer devices, the netw ork analyzer needs to be calibrated. At m icrow ave
frequencies, system atic effects such as leakage, test port m ism atch, and frequency
response also effect the m easured data. The situation is not that detrim ental, because in a
stable m easurem ent en v ironm ent however, these effects are repeatable and can be m ea­
sured accurately by the netw ork analyzers. T his process o f m easuring the repeated sy stem ­
atic effects is called m easurem ent calibration. D uring calibration, a series o f know n
devices (standards) are inserted in the m easurem ent path. Since the response o f these stan ­
dards is know n, the system atic effects can be determ ined reliably. A fter characterization,
the system atic effects are m athem atically rem oved an d this is term ed as “error-correction.”
Several different approaches have been reported in th e literature to calibrate the netw ork
analyzers such as T h ru-S hort-D elay (TSD ), O pen-S hort-L oad (O SL ), and T hru-R eflectLine (TRL) [Fra75, E ng79]. A m ong them . O SL and T R L are the tw o popular calibration
techniques in use. The standards used in these calibration techniques are shown schem ati­
cally in Fig. 2-4. The open, short, and load standards show n in Fig. 2-4 are used for reflec­
tion calibration at both ports 1 and 2. In order to perform full tw o-port calibration using
O SL technique, an additional “T H R U ” standard is needed to correct the transm ission m ea­
surem ent (called “SO LT” calibration). Both full tw o-port and T R L tw o-port calibration
techniques use the sam e erro r m odel and the sam e accuracy enhancem ent m athem atics is
em ployed. T he full tw o -p o rt erro r model im plem ented in H P8510 netw ork analyzers, is
based on tw elve error-term m odel and provides full directivity, isolation, source m atch,
load match, and frequency response vector erro r correction for transm ission and reflection
m easurem ents o f tw o-port devices connected to s-param eter test sets [Fit78]. This m odel
is considered to provide the best m agnitude and phase m easurem ent accuracy for tw o-port
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16
o
o
O PE N
a
o —
o
a
o —
o
On
rO
k>
SHORT
TH R U
~
°
1
rP
—
c
F
Lb
R EFLE C T
R =50 ohm s
0
0 >
^
ZQ = 50 ohms
LO A D
(a)
o
^
o
>
LIN E
(b)
Fig.2-4. Schem atic representation o f com m only used calibration standards, (a) O SL
standards, (b) T R L standards.
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17
devices, but requires m easurem ent o f all four s-param eters o f the tw o-port device. The dif­
ference between the tw o calibration techniques is in the standards used and the m easure­
m ents made to quantify the erro r term s—T R L requires few er standards, hence fewer
m easurem ents are needed. U nder ideal conditions, these system atic effects would be com ­
pletely characterized and rem oved. In practice, the accuracy to w hich the standards are
know n establishes how well these system atic effects can be characterized. In fact, a wellestablished figure o f m erit for a calibrated system is the m agnitude o f the residual system ­
atic effects. The residual effects are the portion o f the u ncorrected system atic error that
rem ains because o f the im perfection in the calibration standards o r im proper m easurem ent
practices. As a result o f calibration, the reference plane is m oved up to the very tips o f the
coplanar probes o r to the edges o f the connectors depending upon w hich method o f m ea­
surem ent is being followed.
As stated earlier, at R F/M icrow ave frequencies it is often im possible to directly
m easure the s-param eters o f a device under test such as transistor. Instead, m easurem ents
are m ade at the reference plane o f the probe tips or cable connectors, w hich is removed
both physically and electrically from the D U T by an intervening test fixture or test struc­
ture. Normally, these test fixtures have high isolation betw een their input and output ports,
thus can be described electrically by two port networks. O nce the s-param eters o f the twoport networks w hich describe the fixture are known, the s-param eters o f the em bedded
DUT may be determ ined. T h is procedure is term ed as de-em bedding. A block diagram
and signal flow graph o f a typical m easurem ent setup is show n in Fig. 2-5. The fixtures
labeled A and C are represented as tw o-port netw orks in the signal flow graph. These net­
w orks are required to be characterized so as to rem ove their contributions from the m ea­
surem ents, yielding the s-param eters for the transistor only.
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18
N etw o rk A nalyzer
(a)
D U T em bedded in fixtures
a0
^2IA
11A
t>!
22A
a-
^21B
I IB
22B
^ 2 ic
tic
t>3
22C
(b)
Fig.2-5. T ypical m easurem ent set up. (a) B lock diagram , (b) S ignal flow graph.
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19
T he co ntinuous process o f m iniaturization o f the active devices not only requires
sophisticated and accurate instrum entation, but also req u ires an effective and accurate p ro ­
cedure for de-em b ed d in g the parasitics associated w ith the device during o n -w afer m ea­
surem ents. T hese parasitics, w hich m ay include the effects o f bonding pads, device
connections and substrate losses have quite significant effects on the m easured s-p aram e­
ters o f the device. Several correction approaches have been proposed by the researchers
and engineers [Lan84, W ij87, W il90, C ho91, K oo91, L ee91]. T hese correction p rocedures
o r de-em bedding techniques take into account the effects o f bonding pads, series parasitics
and also parallel parasitics due to interconnection betw een the pads and the devices. In
sum m ary, these procedures suggest the follow ing: 1) m easure s-param eters o f the D U T. 2)
m easure s-param eters o f the dum m y structure, 3) su b tract the parasitic effects using d iffer­
ent p aram eter m atrix m anipulations and the resulting d ata w ould be s-param eter d ata o f
the actual device.
T he w ork presented in this dissertation has b een com pared w ith the s-p aram eter
m easurem ent d ata o b tain ed using both packaged devices and on-w afer and an ex cellen t
agreem ent betw een the m easurem ent data and theory has been achieved.
Compact Models For Bipolar Junction Transistor
T he design and optim ization o f high frequency b ip o lar integrated circuits require
accurate and physical com pact m odels. A com pact m odel is defined by an eq uivalent cir­
cuit consisting o f lum ped elem ents like diodes, resistan ces, capacitances, and co n tro lled
sources. T he lum ped elem en t values and the asso ciated branch currents and voltages
should be as sim ple as possible functions o f operating po in t and tem perature.
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20
The Ebers-M oll m odel was developed in 1954 [Ebe54], and is one o f the earliest
attem pts to model the b eh av io r o f transistor. O riginal E bers-M oll m odel, also know n as
E M I model, is a sim ple n o n lin ear DC m odel. It is valid for all four regions o f operations.
T he charge storage effects in the device are not m odeled in E M I. W ith time, the basic
EM 1 has been im proved significantly by adding m ore elem ents and param eters to account
for parasitics and several first and second order effects. T he progression from E M I to
EM 2 and later to EM 3 has b een nicely sum m arized by G etreu [G et79]. The im provem ents
m ade in EM3 are m ainly in three areas: DC perform ance, ac perform ance, and am bient
tem perature effects. H ow ever, the effects such as basew idth m odulation, variation o f cur­
rent gain (P) w ith current (I), im proved charge storage, and variation o f parasitic em itter
and base resistances w ith am b ien t tem perature have been accounted for separately and the
m odel has been altered piece by piece.
In 1970, G um m el an d Poon provided a physical supplem ent to the Ebers-M oll
m odel [Gum70]. The G um m el-P oon m odel, also denoted as G P m odel, was based on the
one-dim ensional integral ch arg e control relation (IC C R ). A lthough, EM 3 and G P m odels
are basically equivalent, but the unified treatm ent o f different effects in G P model provides
a better appreciation o f the lim itations and assum ptions involved in the m odels [Get79].
Along w ith the technological advancem ents in the fields o f bip o lar transistors, the
researchers have also p ro p o sed several com pact m odels. T he E xtended G um m el-Poon
m odel [Kul85], M ost E x quisite T ransistor M odel (M E X T R A M ) [D eG 90], and M M SPIC E
[Jeo89] are am ong the m ost cited m odels today. T he E xtended G um m el-P oon, as im plied
from its name, is based on th e form ulation o f the original G um m el-P oon model and inher­
its m ost o f its param eters fro m the original m odel. Due to its easy im plem entation and
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21
fam iliarity to m ost circuit d esigners, the Extended G um m el-P oon m odel w ith slight varia­
tions has been w idely em ployed in a variety o f com m ercial circu it sim ulators like PSPICE
[Psp90], HSPICE [H sp92], an d M icrow ave Design System [H p96]. For the same reasons,
the large-signal DC and sm all-signal high frequency equivalent circuit m odels derived
from Extended G um m el-Poon m odel have been used in this w ork. H owever, the approach
follow ed during the entire w o rk is quite general rather than being m odel specific. T here­
fore, this work can be applied to new er com pact m odels as w ell.
Transmission Lines
In electrical and electronics technology, both voltage and current waves are widely
encountered and m onitored. T he oscilloscope, a popular m easurem ent equipm ent, pro­
vides a fam iliar m eans o f d isp lay in g voltage-tim e w aveform s. A lthough the display is
referred to as a w aveform , it ex hibits only one aspect o f a w ave, the am plitude-tim e varia­
tion. A nother equally im portant characteristic o f any wave is the w avelength, which is
show n by am plitude-distance curves. W hen the w avelength, a function o f frequency, is
m uch larger than the physical d im ensions o f the circuit, then am plitude-distance variation,
being small, can be ignored. T his happens at the low er frequencies. However, w hen the
w avelength o f the signal propagating through the system is sm all o r com parable to the cir­
cuit dim ensions—a situation at h ig h er frequencies, then am plitude-distance variations m ust
be taken into account. This is called distributed circuit analysis.
The term transm ission line has been defined in the sim plest form as, “a pair o f con­
ductors linking together tw o electrical system s, com ponents, o r devices” [Che85]. From
such a broad definition, any sy stem o f w ires can becom e a candidate for consideration as a
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22
transm ission line. However, to allow the use o f this term only for certain types o f intercon­
necting w ires, a com paratively restrictive definition is needed. In electronic system s, the
need for interconnection betw een two points th at are som e distance ap art from each other,
is often encountered. If the frequency o f signal conveyed from one point to an o th er is high
enough, then the distance betw een the points (length o f the interconnection) m ay becom e
an appreciable fraction o f the w avelength o f th e signal being propagated. It then becom es
essential to take into account the properties o f the interconnecting w ires, because they no
longer behave as short circuits. U nder these conditions, separation and physical dim en­
sions o f the interconnecting w ires and sep aratin g dielectric becom es im portant. A rigorous
and form al expression for transm ission line has been proposed by D worsky, such as, “any
structure that guides a propagating EM w ave from p o in t a to point b , o r a set o f boundary
conditions to M axw ell’s equations w hich describe wave propagation betw een two
points” [Dwo79].
Transm ission lines in R F/M icrow ave circu its are used (1) to carry signal from one
point to another and ( 2 ) as circuits elem ents for b oth passive and active circuits such as fil­
ters, im pedance transform ers, couplers, and am plifiers etc. A variety o f transm ission struc­
tures are used at RF/M icrow ave frequencies w hich include coaxial lines, w aveguides,
striplines, m icrostrip lines, coplanar w aveguides, co p lan ar strips, slot lines, coupled striplines, and coupled m icrostrip lines. C ro ss-sectio n al view s o f these transm ission structures
are show n in Fig. 2-6. The choice o f a suitable transm ission m edium depends upon electri­
cal and m echanical consideration in the d esign. E lectrical considerations include transm is­
sion line loss, dispersion, higher order m odes, range o f characteristic im pedance levels,
m axim um operating frequency, and suitability for im plem entation. M echanical trade-offs
include ease o f fabrication, tolerance, and reliability.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
23
/ / / / / / / / /
K W
C oaxial Line
C ircu lar W aveguide
V///A7ZflTOB
Stripline
I
/W
/V /^ I
R ectan g u lar W aveguide
W 7//////3OT73
C o u p led Stripline
7 ////////y /y /
///Aw zazzaj
M icrostrip Line
C o u p led M icrostrip Line
C oplanar W aveguide
C o p lan ar Strips
C oplanar W aveguide
w ith B ackside G rounded
S lot Line
Fig.2-6. T ransm ission lines com m only used in R F /M icro w av e passive and active circuits
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
24
T ransm ission m edia can be classified into tw o categories: ( 1) C onventional trans­
m ission lines and (2) P lanar transm ission lines. C oaxial lines. C ircular w aveguides, and
rectan g u lar w aveguides can be grouped together as conventional transm ission lines, w hile
m any o f the rem aining can be term ed as p lanar transm ission lines. C oaxial lines are co m ­
m only used for interconnections, signal transm ission, and m easurem ents. They provide
perfect shielding betw een the fields inside and outside o f th e line and have low loss w ith
very little dispersion. T he dom inant m ode o f propagation in these lines is transverse ele c ­
tric and m agnetic (TEM ) i.e. no axial electric and m agnetic field com ponents. T hese lines
can be m ade quite flexible and have no cut o ff frequency due to higher order modes. P res­
ently, these lines can be used for frequencies up to 50 G H z w ithout excessive attenuation.
R ectangular w aveguides are norm ally rigid m etal pipes used for guided wave propagation.
Precision m atching is req u ired to use rectangular w aveguides, w hich m akes them expen­
sive. N orm ally, they are o p erated in transverse electric 10 ( T E 10) mode and their operating
frequency is defined by the cu to ff frequency at the low er en d and the excitation o f a higher
ord er m ode at the upper end. D etailed review o f these conventional transm ission lines is
beyond the scope o f this chapter, however, they have been d ealt w ith extensively in litera­
ture covering im portant m icrow ave properties such as characteristic im pedance, phase
constant, attenuation, m odes o f propagation, etc. [M on48, R am 65, G up81, Ram 84, B ah 8 8 ,
C ha94, Kai94]
Today, m icrow ave p lan ar circuits have found applications in the fields o f co m m u ­
nications, electronic w arfare, radars, and sophisticated w eapon system s. For a transm is­
sion line to be suitable as a circuit elem ent in m icrow ave integrated circuits, one o f the
m ajor requirem ents is th at the structure should be planar. A p lanar geom etry as show n in
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
25
Fig. 2-6 for planar transm ission lines, im plies that the characteristics o f the elements can
be determ ined and controlled from the dim ensions in a single plane. A m ong many differ­
ent planar transm ission structures, m icrostrip lines and coplanar w aveguides are the most
popular structures, thus they are briefly review ed in the following paragraphs. The remain­
ing structures though less popular y et very useful have been described, characterized, and
analyzed in detail in literature, references for som e o f w hich are b eing provided here
[C oh54, C oh55a, Ven70, H ow 74, B ah78, W he78, G up79, G up81, B ah 8 8 , Cha94, Kai94],
The circuits designed and im plem ented using any one o f the planar transm ission lines or
com binations o f them have distinct advantages, such as light w eight, sm all size, improved
perform ance, better reliability and reproducibility, and much low er co st as com pared to
the circuits w ith conventional transm issions lines [How74, G up74, You74, Gup79].
M icrostrip Lines
Several different physical descriptions are associated with the term m icrostrip. For
instance, one describes m icrostrip as a stripline m ade o f a dielectric sheet w ith a shieldedplane conductor bonded on the b o tto m side and a pattern o f a stripline on the top side
[W he77], O ther definitions include, a tw o conductor transm ission line evolved conceptu­
ally from a tw o-w ire line [G up79], an open stripline consisting o f a thin p lan ar conductor
supported on a dielectric sheet w ith a grounded m etallic plane on the opposite side o f the
dielectric [Ful90], and also a m etal interconnect line placed on a dielectric substrate o f
finite w idth w ith a ground plane below the substrate [Eo93a],
There are three fundam ental m odes o f propagation in m icrostrip lines: (1) dielectric-quasi TE M mode, (2) skin-effect m ode, and (3) slow-wave m ode [Has71, Sek84,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
26
Y ua91, Eis92]. A t low frequencies, m icrostrips w ith substrate o f m oderate resistivity
ex h ib it the slow -w ave mode. As the frequency goes higher, and the p ro d u ct o f frequency
and substrate resistivity becom e large enough to produce a sm all dielectric loss angle, then
the fundam ental m ode closely resem bles the T E M m ode, provided that the w avelength is
m uch larger than the thickness o f substrate (dielectric). This situation is term ed as dielectric-quasi-T E M m ode o r quasi-TE M m ode. T he skin-effect m ode is observed w hen the
p rodu ct o f frequency and substrate conductivity is large enough to y ield a sm all depth of
penetration into substrate. In that situation, the dielectric acts like a lossy conductor and
m etallic ground at the bottom is replaced by an im perfect ground plane m ade o f silicon.
M icrostrip lines differ considerably from other transm ission structures, for exam ­
ple, w hen com pared w ith striplines, one obvious difference is that the m icrostrip is open to
air on its top. T he open air configuration has several advantages o v er o th er p lan ar struc­
tures, for exam ple, b etter interconnection features (convenience for use in M IC and active
device test structures), easier fabrication and possibility o f m inor trim m ing o r adjustm ents
w hen used as m atching or circuit elem ents. H owever, along w ith these advantages, the
open air structure o f m icrostrip has one m ajor disadvantages, w hich co m p licates both its
analysis and design. T he fact that the field is partly in air and partly in the dielectric, m od­
ifies the m ode o f propagation to a non-T E M hybrid m ode as com pared to a m uch desired
purely T E M m ode. D etailed analysis, design, m odeling, and characterizatio n o f m icrostrip
lines has been reported in literature [W he65, Sch69, G et73, H am 75, Edw 76, W he77,
B ah77a, B ah77b, G up79, Kue79, K ue80, E is92, Eo93a, Eo93b]. B esides the aforem en­
tioned, a relatively detailed and com prehensive review o f transm ission lines in general,
and m odes o f propagation, m ethods o f analysis, and m odeling o f m icro strip lines and its
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
27
discontinuities such as bends, T-junctions, gaps, etc. in particular along w ith num erous
references o f relevant literature have been reported in [Nad94],
C oplanar Waveguide
A coplanar w aveguide (C PW ) co n sists o f a strip o f thin m etallic film d ep o sited on
the surface o f a d ielectric slab w ith tw o g round electrodes running adjacent an d parallel to
the strip on the sam e surface as show n in Fig. 2-6 [Wen69]. B esides this basic configura­
tion for CPW , there are several variations o f this type o f transm ission m ed iu m such as
C P W w ith asym m etric ground. C P W w ith finite ground, and CPW w ith back -sid e ground.
C oplanar w aveguides (C PW s) offer several advantages over m icrostrips for m ono­
lithic o r hybrid m icrow ave integrated circu its [H ou76]. These advantages include ease o f
parallel and series connection o f both passive and active com ponents, and high circu it d en­
sity. D espite these advantages over m icrostrips the use o f CPW s has been less w idespread
than expected. A long w ith advantages o f having all the conductor on the sam e plane, there
are som e lim itations as well. The fact th a t the fields in CPWs are less confined com pared
to the m icrostrips m akes them very sensitive to th eir environm ental co n strain ts such as
upper shielding, lateral ground plane truncation, and line-to-line coupling. T he upper
shielding low er the im pedance o f the line, w hile truncation o f lateral g ro u n d planes
increases the im pedance, thereby in creasin g line-to-line coupling. C oplanar w aveguides
w ith back-side g ro u n d tend to reduce som e o f the unw anted effects associated w ith trunca­
tion o f lateral g round planes. B ack-side g ro u n d low ers the im pedance significantly and
thus counters som e o f the effects o f lateral g ro u n d truncation. B ack-side g ro u n d is often
introduced to im prove the m echanical stren g th and the power handling cap ab ility o f the
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28
line. A lso it allow s easy im plem entation o f m ixed coplanar-m icrostrip circuits. The d is­
persion o f C PW s w ithout back-side ground is sim ilar to that o f m icrostrip, however, w ith
back-side grounded the C PW s become rem arkably less dispersive com pared to m icros­
trips. A nother feature o f C PW s w ith back-side ground is that their sensitivity to the upper
shield gets reduced com pared w ith free-standing C P W s i.e. CPW s w ithout back-side
ground.
T he characteristic im pedance, effective dielectric constant, effect o f m etallic strip
and substrate thickness, losses, phase velocity, upper bound o f attenuation, parasitic
effects o f upper shielding, influence of finite lateral ground planes, and line-to-line co u ­
pling has been d ealt w ith extensively in literature [W en69, Dav73, Hou76, Kit76, H an84,
L eo 8 6 , G ho87].
Ambient Temperature Effects on DC Bias Networks
P rior w ork in this area can be divided into tw o categories: (1) different circuit
topologies proposed and im plem ented to achieve stable quiescent point, and ( 2 ) design
analysis o f DC bias netw orks used in low frequency and R F/M icrow ave circuits. The
form er category being a m acro approach tow ards the goal to design/realize stable DC bias
circuits, does not focus on am bient tem perature effects alone, rather takes a general view
and considers the factors as (a) initial tolerances in the param eter values o f both active and
passive elem ents, (b) tem perature variations o f circu it elem ents, (c) changes in supply
voltages, and (d) o th er effects as radiation and aging etc. [W id65, M ei69, R yn 8 6 ].
T he later category takes a micro approach and analyzes the DC bias circuits using
the equivalent circu it m odel for the active device. T he device model used is EM 1 and the
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29
tem perature sensitive param eters include only leakage cu rren t (ICBo ) ’ current gain ((3),
and base-em itter voltage (V BE). T he therm al stability o f the D C bias design is ascertained
by adding the changes in co llecto r current (Ic ) caused by (a) changes in I c b o (b) changes
in |3, and (c) changes in V BE due to tem perature [H P75, G o n 8 4 , R yn 8 6 , Ven90].
The applicability o f the aforem entioned analytical approach on the DC bias net­
w orks com m only used now adays in the R F/M icrow ave circu its is limited since (a) the
equivalent m odel E M I is obsolete, particularly the value fo r param eter Ic b o *s rarely
available for m odem devices, (b) the parasitic resistances (rb, re, rc), not considered in
EM 1 m odel, need to be included in the large signal m odel u sed for bias calculations, and
(c) the tem perature sensitive param eters are not lim ited to the ones considered in the prior
work. A com prehensive design-oriented analysis using G um m el-P oon based large-signal
equivalent circuit m odel has b een done during this w ork and included in C hapter 6 .
Self-Heating Effects on Circuits
The term self-heating can be defined in the sim plest form as, a phenom enon o f
tem perature rise in sem iconductor structure due to dissipation o f power. A nother relatively
specific definition has been proposed by H efner as, w hen the devices are heated signifi­
cantly by the pow er dissipation w ithin the device [H ef92], T h e significance o f self-heating
has becom e m ore evident as circu it and device integration level continue to im prove w ith
tim e. T he sm aller devices resu lt in increased current densities and restriction o f heat flow.
These tw o factors tend to increase the therm al im pedance an d raise the device operating
tem perature [D en89, Fox91, Zw e93]. A nother technological feature w hich results in
increase o f therm al im pedance is the use o f S i0 2 for isolation in m odem devices. Due to
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30
different therm al conductivities o f SiC>2 and Si, a barrier gets created for heat flow and this
causes an increase in th erm al im pedance [Fox93, Z w e93, Zw e97].
Self-Heating Effects on Large-Signal Circuits
Self-heating effects on large signal circuit behavior w ere studied by Fox [Fox93].
It has been suggested that circu its w hich depend on precise control o f B JT characteristics
are more sensitive to self-h eatin g . T he current m irror circuits w hich are frequently used in
active bias netw orks in R F /M icrow ave circuits w ould show increased m ism atch in the ref­
erence and output currents due to self-heating. T his undesirable effect is due to the differ­
ence in operating co n d itio n s o f the transistors induced by self-heating. It has also been
suggested that long therm al tim e constants can slow dow n the electrical response o f a cir­
cuit. W hile the errors cau sed by self-heating can be reduced through careful design, they
can not be com pletely rem oved.
Self-Heating Effects on Small-Signal Circuits
The early w ork on self-heating effects on sm all-signal perform ance o f BJT was
done by M ueller in 1964 [M ue64], The equivalent circuit m odel used in that w ork was
Ebers-M oll. L ater in 1993, Fox e t al. updated the w ork using G um m el-P oon m odel and
also investigated the co n d itio n s u n d er w hich self-heating can significantly effect the BJT
circuit behavior [Fox93]. It has been suggested that errors in output adm ittance and
reverse transadm ittance ca n be significant w ithout m uch pow er dissipation. O ther sm allsignal param eters are less effected unless pow er dissipation is significant. A lso, the volt­
age gain o f the BJT am plifiers has been reported to be reduced due to self-heating.
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31
The next chapter p resen ts the relationship betw een sm all-signal s-param eters and
high-frequency equivalent circ u it m odels for bipolar ju n c tio n transistors, verification o f
closed-form expressions for s-p aram eters against o n -w afer m easurem ents, com parison o f
m easured, calculated and sim u lated s-param eters, and bias sensitivity o f the sm all-signal
s-param eters.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER 3
S M A LL -SIG N A L S-PA RA M ETER S A N D BIAS S E N S IT IV IT Y
In the 1970s, with the advancem ents in the m easurem ent eq u ip m en t for RF/
M icrow ave frequency ranges, mainly the netw ork analyzers and on-chip probes, w hich
perform
the
s-param eter m easurem ents
readily
and reliably,
the
popularity
and
applications o f s-param eters increased trem endously. Now, RF and m icrow ave circuit
design m ethodologies are centered around the s-param eter data o f active and passive
devices. S-param eters which are used alm ost universally for the design o f sm all-signal
am plifiers, in conjunction with the noise param eters characterize the device com pletely in
term s o f its sm all-signal circuit perform ance.
Virtually, every transistor p aram eter is directly or indirectly
affected by
tem perature because o f therm al dependence o f the physical properties o f the m aterials
used to fabricate sem iconductor devices. T herefore, the electrical properties (currents and
voltages) o f a device are also tem perature dependent. This broad therm al dependence o f
device param eters im plies that s-param eters m ust also be tem perature dependent.
M icrow ave circuit gain is a strong function o f collector current in b ip o lar transistor
am plifiers. W hile the collector current itse lf is quite sensitive to tem perature. In order to
design therm ally stable RF/M icrowave circuits, the tem perature effects (b o th am bient and
self-heating) need to be considered. Since, the s-param eters are used extensively in design
w ork, so therm al dependence o f s-param eters is a logical starting point for such an
analysis. S -param eters o f an active or a passive device are defined and also m easured as
32
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
33
the ratios o f the incident and reflected p o w er w aves o f n-port netw ork as m entioned in
Chapter 2. These incident and reflected p o w er w aves are related to the external voltages
and currents at each port o f the network- T herefore, s-param eter characterization o f a
transistor is lim ited to the external voltages and currents at each o f the tw o ports o f the
network representing the device. M easurem ent data is the most reliable m eans to quantify
the dependence o f s-param eters on tem p eratu re sensitive quantities such as bias current,
etc., but the nature o f the m easurem ents is external to the netw ork. T hus direct
m easurem ents tend to lim it o r com pletely obscure the insight to the device behavior,
which in fact cau ses the variations in s-param eters with tem perature. C om puter
sim ulations; when used to quantify the sen sitiv ity o f s-param eters to the bias current by
predicting s-param eters at different bias co n d itio n s, tend to mask device therm al behavior.
To perform design-oriented analysis o f th erm al effects on s-param eters, it is necessary to
(a) gain an insight into the netw ork itself to identify the sources w hich cause variations in
s-param eters, and (b) describe them both q u antitatively and qualitatively. To realize these
objectives, s-param eters o f a transistor need to be linked w ith the device model
param eters. After extensive literature search , the author w ould like to report here that no
prior w ork has been published w hich (a) relates the s-param eters w ith tem perature, o r (b)
expresses s-param eters in term s o f sm all-sig n al equivalent circuit m odel. However,
transistor gain has been reported to be d ep en d en t on tem perature sensitive bias current
(collector current in B JTs), w hich im plies th a t S 21 is sensitive to tem perature. As regards
to the relationship betw een s-param eters an d device m odel param eters, the only references
found are in the context o f feedback effects on broadband am plifier design [Gon84,
Eis95]. The prior w ork neither w as focused o n therm al analysis nor intended to be used for
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34
therm al analysis. A lso, the prio r work by G onzalez [G on84] is based on overly sim plified
low frequency negative feedback model fo r BJTs. H ence, the w ork reported in this c h ap ter
is the original w ork.
This ch ap ter p resen ts the relationship betw een sm all-signal s-param eters and high
frequency equivalent circu it m odels for bipolar ju n c tio n transistors, verification o f clo sed form expressions for s-param eters against o n -w afer m easurem ents, co m p ariso n o f
m easured, calcu lated and sim ulated s-param eters, an d bias sensitivity o f the s-p aram eters.
S-Param eters and Small-Signal High Frequency Model For BJT
To derive the relationship betw een sm ail-signal s-param eters and the device m odel
param eters, the intrinsic high frequency hybrid- 7t m odel show n in Fig 3-1 has been used.
To develop the tw o -p o rt representation in term s o f the intrinsic device p aram eters, a
matrix analysis o f the high-frequency equivalent circu it in Fig. 3-1 is essential. To ex p ress
s-param eters in term s o f device m odel param eters, the indefinite adm ittance m atrix
approach was em p lo y ed [C o t6 1]. The preference for indefinite adm ittance m atrix over
other established tech n iq u es w as based on these properties; (a) it expresses the b eh av io r o f
the circuit relative to som e com m on, but rather in d istinct reference point, (b) any row o r
any colum n adds to zero —a step by step verification tool for rather tedious w ork involving
matrix m anipulation, (c) if o n e o f the nodes is m ade coincident w ith the reference n ode, a
matrix w hich d escrib es the behavior o f the resulting circuit can be form ed by sim ply
crossing out the row and co lu m n corresponding to the com m on node. The last p roperty is
very useful in situ atio n s w h ere one o f the term inals is grounded as in co m m o n -em itter o r
com m on base co n fig u ratio n s, and (d) in cases w here one o r m ore nodes in a c irc u it are
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35
bO
e
Fig. 3-1. Intrinsic high-frequency H ybrid- k m odel for b ip o lar transistor
in accessib le—a situation quite com m on in equivalent circu it m odels—then the effect o f
inaccessible node(s) can still be accounted for.
To proceed with th e s-param eter representation in term s o f device model
p aram eters, an equivalent circu it in term s o f adm ittances is draw n along with the current
sources co n nected to each node as show n in Fig. 3-2. A system o f nodal equations for the
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36
4¥I V2
%
yb
.V 6
yc
gm (V 2 - V3)
i6
©
i3
i5
com m on node
Fig. 3-2. Equivalent circuit for d eterm ining the indefinite adm ittance m atrix
voltages betw een nodes 1,2 ,3,4,5, and 6 and an external reference node indicated as a
com m on node are w ritten.
> i = y bV i - y bv 2
'2 = -•>'il / I + <3'fc + >V + V V2 - > ’n V V 5
'3
= -^ K + ^
V2 + ^ K + ye + y o + im ^ V3 - y e V4 - > ' o V 5
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3- n
(3-2)
(3-3)
37
(3-4)
(3-5)
(3-6)
T his results in a 6 x 6 m atrix since there are six nodes in the circuits. N odes 2, 3, and 5 are
com pletely inaccessible. These nodes are m athem atically rem oved from the above
equations. This is accom plished by elim in atin g the current source i2, solving for V2 in
term s o f the other node voltages, and substituting this result back into the rem aining five
nodal equations. The expression for V2 after setting i2 = 0 becom es
(3-7)
T his expression for V2 is substituted in Eq. (3-1), (3-3) and (3-5). A fter th ese equations are
updated they becom e independent o f V2. This procedure is repeated for o th er tw o nodes
i.e. 3 and 5 and eventually the resulting m atrix is a 3x3 adm ittance m atrix. T his 3x3 m atrix
does not have any term corresponding to inaccessible nodes, how ever, the effects have
been accounted for. Using the property o f the indefinite matrix m entioned earlier, node 4 is
elim inated for com m on em itter configuration, and the result is a 2 x 2 adm ittance matrix.
The elem ents o f the matrix or y-param eters are then norm alized w ith respect to 50 ohm s
and converted into s-param eters using the standard conversion tables available in texts
[C 0 I6 6 , D w o79, G up81, Ven82, G on84, R od 8 6 , Ven90], The entire seq u en ce o f derivation
o f sm all-signal s-param eters in term s o f sm all-signal high-frequency equivalent circuit
m odel param eters is shown in Fig. 3-3. T his conversion and sim plification o f the
expressions was done using M A PLE and hand analysis together. T h e detailed step-by-step
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
38
Sm all-Signal H ig h -F req u en cy M odel
Equivalent C ircu it in term s o f
A d m ittances
System o f N odal E q u atio n s and M atrix
M an ip u latio n s
Sim plifications an d C lo sed Form
E xpressions fo r [S]
Fig. 3-3. Pictorial representation o f derivation o f sm all-signal s-param eters in term s o f
sm all-signal high-frequency m odel.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
39
derivation o f closed-form ex p ressio n s for sm all-signal s-p aram eters for BJTs is given in
A ppendix A. T he resu ltin g sm all-signal s-param eters in term s o f sm all-signal highfrequency equivalent circu it m odel param eters are as follow s:
- C Z 20 + a r b - r c ) C - r ( B r o - r n ) ) Z 0 + D + E + F
xn
= ------------------------------------a 1 ^ ,---------------------------------
C
_
12
2Z (C r + A r r
° _£_____ °
D enom
2Z ( C r
-i
=
0
+ (A -g
e
D enom
)
(T._Q\
r ..) r _ r )
m V- K °
(3-10)
c
_ ~CZ2° - « r b - rc ) C - r^ B r o - rK) ) Z o + D + E + F
*22 ~
~Denom
w here
A = 1+ J r ^ ,
(3‘8)
5 = I +s r ^ ,
(
U
C = A ^ ( 1 + V o ) + 5 (A rQ+ r^ ) ,
D = (A((r + r ) r , + r r ) + r r , . ) r , r e ,
v vv c e ' o
e c'
e 11 it o° m
E = ( ( A 5 ( r + r ) + fir,, + A r _ ) r , + ( Ar + r ) ( 5 r + r ))r ,
vvv c
e’
p.
K b
c
p 'v e
K"
o
F = (A r +B r , , ) ( ( r + r ) r , + r r ) + r , r ( r + r ),and
v K
p ' vv c
eJ b
e cJ
p 7tv c
eJ
9
D enom = CZ~ + ( (r, + r + 2 r )C + (A r + r, )/■ + (A r_ + B rtl )r )Z
o
b
c
e ’
K o
p'Tt
p ’o ’ o
+D +E +F
The closed-form ex p ressio n s derived above estab lish a link betw een tw o-port sparam eters representation o f a B JT and its sm all-signal high-frequency equivalent circu it
model. These closed-form ex p ressio n s have been verified against on-w afer high-frequency
s-param eter m easurem ents o v e r a broad range o f frequencies in the following section.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
40
Experimental Verification of Closed-Form Expressions for S-Parameters
Before proceeding w ith experim ental verification o f closed-form expressions, it is
appropriate to briefly revisit the b ehavior o f sm all-signal m odel param eters with respect to
bias conditions and also to state the assum ptions w hich have been m ade during the course
o f calculating the sm all-signal s-param eters using the closed-form expressions derived in
the previous section. T h e sm all-signal model param eters are com puted autom atically
inside the circuit sim ulators such as SPIC E and H SPIC E by linearizing the static and large
signal model param eters for a transistor [M as93]. T he num erical values for the sm allsignal model param eters used du rin g this w ork have been com puted using HSPICE.
The sm all-signal transconductance (gm) is directly proportional to the bias current
(Ic ) and increases linearly w ith the increase in the bias current, w here as, the output
resistance looking into the co llecto r (r0) and the input resistance betw een base and em itter,
w hile looking into the base (rrt) are inversely proportional to Ic , as they decrease w ith the
increase in collector current. H ow ever, unlike gm, both rQ and rn are sensitive to VCE. The
capacitances cK and c^ rep resen t the em itter-base and collector-base ju n ctio n capacitances.
In forward active m ode, c K is the sum o f two parallel capacitances present in the em itterbase junction: a d ep letion-layer capacitance and a diffusion capacitance, w hile the reverse
biased collector-base ju n ctio n only exhibits a depletion capacitance and that is represented
by c^. Both cK and c^ are dep en d en t upon the transistor type, ju n ctio n areas, and the DC
bias point. The em itter-base cap acitan ce (cK) is a strong function o f the bias current and
increases with the increase in the co llecto r current, w here as the collector-base capacitance
(c^) is very weekly dependent upon Ic . For R F and m icrow ave transistors, crt can range
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
41
from a few tens o f fem to farads ( 1CT15F) to a few pico farads ( 1 0 ' 12F). T he collector-base
capacitance (c^) is small com pared to crt, b ut it plays a very significant role in determ ining
the effective input capacitance o f tran sisto r due to M iller m ultiplication [Sed87]. A
graphical representation showing variation o f these sm all-signal m odel param eters with Ic
and VCE has been included as A ppendix B o f this dissertation. T hree resistances rb, re, and
rc, w hich are integral part o f the intrinsic equivalent circuit m odel also deserve a mention
at this ju ncture. T hese resistance rb, re, and rc, are norm ally term ed as ohm ic resistances
and they represent the resistance betw een base region and base term inal, em itter region
and em itter term inal, and collector region and co llecto r term inal, respectively. In fact, rb
and rc vary w ith the variations in bias co n d itio n s, w hile re is generally considered to stay
constant. However, these resistances have been assum ed constant for this w ork and their
values have been taken from the m odel param eters for the transistor used.
The underlying motivation to use hybrid-Jt model to derive the closed-form
expressions for sm all-signal s-param eters is that all its constituents are sim ple resistors or
capacitors w hose frequency behavior is w ell know n except controlled source (gm vbe). The
hybrid-Jt m odel represents the bipolar ju n ctio n transistor over a w ide range o f frequencies,
from D C to higher frequencies, so long as the base transit tim e t B is m u ch sm aller than the
signal tim e period (T). At higher frequencies, the sm all-signal transconductance (gm)
becom e com plex and can be expressed as [Zie76].
—
= -------- 1-------8 m0
1+ U j-fA
(3-12)
v J or
W here g m represents the sm all-signal high-frequency transconductance, g m0 is the small-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
42
signal transconductance at D C ,/s ig n ifie s the frequency o f the app lied signal, a n d /a is the
frequency up to w hich gm is practically eq u al to g m0 w ith only a very sm all phase angle.
Beyond the frequency
w hich can be estim ated using the ex p ressio n given below, the
magnitude o f gm decreases slow ly w ith th e increasing frequency.
'«
* 2 jrtg
, 3 - 1 3 )
The high-frequency lim it for the applicability o f hybrid- 7i m o d el can be estim ated
by satisfying the follow ing requirem ent [Lev84]
t D
(£>tb
= 2k
—
«
2k
(3-14)
or
t n
Y « l
(3-15)
W here t B is the diffusion transit tim e th ro u g h the base region and ca n be expressed as
i
W~B
=
“
( 3 '
n
1 6 )
or
XB ~ re Cd i f f
W here re represents e m itter resistance an d
(3' l7)
denotes the b ase-em itter diffusion
capacitance. To estim ate the diffusion tran sit tim e through base t B e ith e r equation (3-16)
o r (3-17) can be used. T h e base-em itter diffu sio n capacitance can be estim ated as
Cd i f f = K v~t XF
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3~18)
43
W here K is a num erical factor (fo r uniform base K = 2/3), w h ich accounts for the fact that
not all excess m inority charge leaves the neutral region [Lev84]. As stated earlier, beyond
f a the sm all-signal transconductance becom es com plex and has both m agnitude and phase
angle. However, during this w ork a real value for sm all-signal transconductance calculated
using H SPIC E has been used. T his was considered sufficient because for all the active
devices used during this w ork, the frequency f a was fo u n d to be several orders o f
m agnitude higher than the frequency o f interest i.e. the frequency range o f m easurem ents
and analytical predictions o f sm all-signal s-param eters.
To experim entally verify the closed-form expressions for com m on-em itter (CE)
configuration o f BJTs derived earlier, on-w afer high-frequency s-param eter m easurem ents
w ere done. Two different devices were m easured using an A utom atic N etw ork A nalyzer
H P 85 IOC [Hp94] and co p lan ar probes over a broad range o f frequencies from 45 M H z to
20.045 G Hz. T he m easurem ents were done at room tem perature inside an RE screen room
w ith solid co p p er w alls all around to provide a high degree o f isolation from external
noise. The m easurem ent uncertainty o f netw ork analyzer for m agnitude and phase o f both
reflection (Sj j and S 22 ) and transm ission (S 2i and S i 2) coefficients is presented in Table 31 and Table 3-2 [H p84]. T he devices w hich w ere m easured shall be referred to as n p nl
and n p n l here and their G um m el-P oon m odel param eters are given in Table 3-3. P rior to
device m easurem ents, the netw ork analyzer was duly calib rated using TR L technique and
the full tw o-port erro r m odel im plem ented in H P8510C w as em ployed. B esides the
transistors on-w afer, dum m y test structures (w ithout the transistors) were also m easured
under the sam e eq uipm ent setting and calibration. The m easu rem en t data w as corrected
and the effects o f the surrounding test structure were su b tracted using y-param eter de-
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44
Table 3-1. M easu rem en t uncertainty fo r m ag n itu d e and phase o f Sj | and S 22
M easurem ent U ncertainty o f H P8510C
Reflection C oefficient
(Linear M agnitude)
At 18 G H z
At 8 G H z
M agnitude
(Linear)
Phase
(D egrees)
M agnitude
(Linear)
Phase
(D egrees)
0.0
0.0035
5.0
0.0040
5.0
0.1
0.0045
2.7
0.0050
2.8
0.2
0.006
1.7
0.0060
1.8
0.3
0.007
1.4
0.0075
1.5
0.4
0.008
1.3
0.0090
1.4
0.5
0.011
1.2
0.012
1.25
0.6
0.0125
1.15
0.0135
1.2
0.7
0.0145
1.15
0.0155
1.2
0.8
0.017
1.25
0.0175
1.3
0.9
0.019
1.3
0.02
1.35
1.0
0.022
1.4
0.023
1.5
Table 3-2. M easu rem en t uncertainty for m ag n itu d e and phase o f S 21 and S l2
F requency
M agnitude (dB )
Phase (D egrees)
8 GHz
0.06
0.40
18 G H z
0.08
0.5
em bedding technique [W ij87]. A fter de-em bedding, the corrected y-param eters o f the
transistors w ere converted back to s-param eters u sing the conversion tables given in the
num erous texts. A typical com parison o f m easu red (un-corrected) and de-em bedded
(corrected) sm all-signal s-param eters to x n p n l is show n in Fig. 3-4. Also show n in F ig.3-4
is the characteristic o f an o p en (dum m y) test stru ctu re m easured to de-em bed the effects o f
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
45
Table 3-3: G u m m el-P oon m odel param eters for bipolar ju n c tio n transistors m easured
npnl
npnl
npn2
IS
5.28E-17
7.92E-17
M JE
0 .197
0.197
BF
101
101
TF
9.22PS
9.22PS
NF
1.0
1.0
ITF
105E-3
157E-3
VAF
86.8
86.8
X TF
5.67
5.67
IKF
335E-3
503E-3
V TF
1.86
1.86
ISE
1.32E-15
1.98E-15
PTF
41.43
41.43
NE
2.0
2.0
TR
6 4 IPS
64 IPS
BR
11.6
11.6
CJC
141FF
212FF
VAR
4.125
4.125
V JC
0.641
0.641
IKR
1.20E-2
1.80E-2
M JC
0 .267
0.267
ISC
6.68E-15
1.00E-14
X CJC
1.0
1.0
NC
2.0
2.0
CJS
8 7 .5 F F
132FF
RB
5.86
4.1
VJS
0.5
0.5
IRB
357E-6
510E -6
MJS
0.171
0.171
RBM
201E-3
141E-3
EG
1.135
1.135
RE
431E-3
302E-3
X TI
4 .177
AM I
RC
7.80
4 .8 0
XTB
0.632
0.632
CJE
315FF
4 73F F
FC
0.961
0.961
VJE
0.817
0.817
Param eter
P aram eter
npn2
the test structure from the m easured sm all-signal s-param eters for the actual transistor. A s
clearly visible in Fig. 3-4, the test structure is sym m etrical as its s-param eter are identical
i.e. Si j = S 22, and S j2 = S 22 - Identical s-param eters (both in m agnitude and phase) for the
open test structure is a m easu re o f certainty and reliability o f the on-w afer m easurem ents.
The clean and sm ooth m easu rem en t data presented in this ch ap ter indicates the high
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
46
standard o f eq uipm ent m ain ten an ce and ad h eren ce to good m easurem ent practices.
M easurem ents w ere p erfo rm ed several tim es
to ensure repeatability—an essential
requirem ent for any ex p erim en tal w ork and m ore so for high-frequency on-w afer
m easurem ents. For the tw o d ifferen t npn tran sisto rs m easured on-w afer. sm all-signal sparam eters w ere calcu lated u sin g the clo sed -fo rm expressions derived in the previous
section. The p aram eter values fo r the expressions w ere obtained using H SPIC E. G um m elPoon model param eters w ere u sed in M O D E L statem ents (.M O D EL) and the transistor
w as biased in C E co n fig u ratio n for required VCE and Ic . To extract sm all-signal m odel
param eters, operating p o in t statem en t (.O P) w as included in netlist. A fter H SPICE
sim ulation at room tem p eratu re, the sm all-signal param eters were read from the operating
point inform ation in the o u tp u t file.
The com parison o f m easu red and calcu lated sm all-signal s-param eters for n p n l is
given in Fig.3-5 to Fig.3-8. M ag n itu d e and phase have been plotted separately to provide a
clear and m eaningful co m p ariso n and also to estim ate the difference betw een the
quantities depicted in the figures. As show n in the aforem entioned figures, the sm allsignal s-param eters o b tain ed from on-w afer m easurem ents and calculated by using the
closed-form expressions have excellen t agreem en t over the entire frequency range. The
m agnitudes and phase angles for all four s-p aram eters both m easured and calculated
except m agnitude o f S22 not o n ly follow the sam e trend through out the frequency range,
but they are very clo se to e a c h other. T he clo sed -fo rm expressions overestim ate the
m agnitudes o f S21 and S j 2 by approxim ately 1.0 an d 3.0 dBs respectively, w hile the phase
angles for Sl j and S2i are u n d erestim ated by ap p roxim ately 8 and 10 degrees respectively
w hen com pared to m easu rem en ts. T hese differences betw een the m easured and calculated
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
47
Fig. 3-4. C om parison o f m easured (b old), and de-em bedded or corrected (light) smallsignal s-param eters o f n p n l at V CE = IV, Ic = 3m A. A lso show n is the
characteristic o f open (dum m y) structure (dash). Note that sam e scale has been
used for plotting S j2 and S2i o f open structure.
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48
1
npnl
VCE = 3 V, IC = 1mA
q—o
ta—□
0
Calculated
Measured
03
3
u
1
-3
3
(a)
'5
CO
3
2
on
3
•4
8
0
12
16
20
Frequency (G H z)
npnl
VC£ = 3 V, IC = 1mA
q—o
Calculated
t3—q
Measured
-40
<U
O
-80
T3
(b)
-120
CO
-160
-200
0
4
8
12
16
20
Frequency (G H z)
Fig. 3-5. C o m parison o f m easured and calcu lated sm all-signal s-param eters, (a) S u
m agnitude, (b) S u phase.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
49
npnl
VCE = 3V , I C = 1mA
Calculated
Measured
S 2j magnitude (dB)
o—©
q—q
-10
-15
0
4
8
12
16
20
Frequency (G H z)
200
npnl
V CE = 3V , I C = 1mA
o —o
Calculated
q—q
S2| phase (degree)
150
Measured
100
50
0
4
8
12
16
20
Frequency (G H z)
Fig. 3-6. C om parison o f m easured and calcu lated sm all-signal s-param eters, (a) S2i
m agnitude, (b) S2J phase.
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50
-10
cq
-20
ts
0
•o
1
-30
eo
c3
ri
C/5
-40
npnl
V c e = 3V, IC = 1mA
-50
o—o
□—q
Calculated
Measured
-60
0
4
8
12
16
20
Frequency (G H z)
100
npnl
VCE = 3V, IC = Im A
o —o
Calculated
a—q
Measured
u<D
00
<D
T3
-20
-60
0
4
8
12
16
20
Frequency (G H z)
Fig. 3-7. C om parison o f m easured and calculated sm all-signal s-param eters, (a) S t 2
m agnitude, (b) Sjo phase.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
51
npnl
VCE = 3V, IC = 1mA
o—©
b— q
Calculated
Measured
CQ
-a
-10
-1 5
0
4
8
12
16
20
Frequency (G H z)
40
npnl
VC£ = 3 V, I C = 1mA
o—o
Calculated
□—q
Measured
-4 0
-80
0
4
8
12
16
20
Frequency (G H z)
Fig. 3-8. C om parison o f m easured and calculated sm all-signal s-param eters, (a) S 22
m agnitude, (b) S 22 phase.
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52
s-param eters are at the higher en d o f the frequency range. T he sm all differences observed
betw een the calculated and m easured s-param eters can be attrib u ted to several factors: (a)
inaccuracies during extraction o f device m odel param eters; (b) p ro cess variations during
fabrication; (c) assum ptions o f co n stan t rb and rc when they are in fact sensitive to bias
conditions; and (d) neglecting capacitance between external b ase and collector and the
substrate effects during the course o f derivation o f sm all-signal s-p aram eter expressions
(using only intrinsic high-frequency equivalent circuit m odel).
In order to gain an insight into the possible cause(s) for the difference in
m agnitudes of m easured and calcu lated output reflection coefficient (S 22 ). the data has
been plotted on im pedance (Z) S m ith chart and also on polar p lo t in Fig. 3-9. As show n on
Sm ith chart, at approxim ately 8 G H z the magnitude o f m easured S 22 starts to depart from
the calculated S 22, and then sh o rtly after it starts following alm o st a constant resistance
circle. This clockw ise rotation along a constant resistance circle is m ore pronounced for
the case when m easured S 22 for VCE = 3V and Ic = 3m A is p lo tted on the same Sm ith
chart as shown in Fig. 3-9 (a). In both bias conditions i.e. VCE = 3 V, Ic = 1mA and VCE =
3V, Ic = 3mA, the shift in m agnitude o f S 22 is from a h ig h er im pedance to a lower
im pedance and the plot is m oving tow ards the center o f Sm ith ch a rt w hich signifies 50
ohm s. W hen m easured S 22 for VCE = 3 V, Ic = 3mA is com pared w ith S 22 for V ce = 3 V, Ic
= 1mA, it is observed that not o nly the clockw ise rotation is m ore p ronounced but there is
a loss o f phase—a signature o f series inductance along w ith red u ced capacitive reactance.
H aving described briefly the beh av io r o f S 22 with the help o f Z S m ith ch art and polar plot,
it can be sum m arized that the departure o f m easured S 22 fro m the calculated S 22 in
m agnitude could be attributed to eith er (a) inductive behavior o f collector at higher
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
53
M easured
(VCE = 3V. IC =3mA)
(a)
M easured
<VCE = 3V’ i c = lm A)
8 GHz
Calculated
(VCE = 3V. IC = lmA)
90°
(b)
M easu red
<VCE = 3V- IC =3m A)
180°
0°
M easured
(Vc e = 3V. IC = 1mA)
8 GHz
Calculated
(VC £ = 3V, IC = 1mA)
-90°
Fig. 3-9. O utput reflection coefficient (S 22 ) o f npnl. (a) Im pedance (Z) sm ith chart
representation; (b) P olar plot.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
54
frequencies, o r (b) due to the substrate effects w hich becom e significant at higher
frequencies. It is w orth m entioning here that in addition to collector to substrate
capacitance, the substrate co ntributes b oth cap acitan ce and substrate resistance. C ollector
to substrate capacitance in series w ith su b strate resistance ap p ear in parallel w ith sm allsignal output resistance (rQ). A t h ig h er frequencies w hen th e substrate effect is greater,
then the parallel com bination o f large rD and relatively sm aller collector to substrate
capacitance plus a parallel com b in atio n o f substrate resistance and capacitance m ay result
into sm aller output im pedance. T he effects o f substrate have not been included in the
derivation o f closed-form ex p ressio n s d u e to the fact that neither substrate resistance is
part o f G um m el-Poon m odel used d u rin g this w ork, nor this data is provided by the
m anufacturers as part o f d evice m odel param eters. A lso, to estim ate substrate resistance,
detailed inform ation about the layout geom etry and dim ensions is necessary. Despite
neglecting these substrate effects, the deviation o f calculated S 22 from the m easured S 22 in
m agnitude is less than 1 dB upto 12 G H z and aro u n d 3 dB at 20 G H z. As regards to the
phase o f S 22 in both cases, there has b een an excellent agreem ent throughout the entire
frequency range. C losed-form expressions for sm all-signal s-param eters have been
extensively verified for both transistors n p n l an d npn2 under several different bias
conditions and the results have been p resented in A ppendix C.
Comparison of Measured, Calculated, and Simulated Small-Signal S-Parameters
B esides m easurem ents
using
au to m atic
netw ork
analyzers,
sm all-signal
s-
param eters can also be o btained th rough co m p u ter sim ulations using H SPIC E or M DS. In
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
55
these circuit sim ulators, a com plete set o f device m odel param eters for the bipolar junction
transistor is supplied by the user or accessed through the built in com ponent libraries for
active devices. T he sim ulator num erically com putes the sm all-signal m odel param eters by
linearizing the static and large-signal model param eters. T hese sm all-signal model
param eters are then used for com puting the sm all-signal s-param eters for the user
specified frequency range. In both sim ulation tools, com plete device m odel such as
G um m el-P oon in H SPIC E or G um m el-Poon or M E X T R A M in M DS is em ployed for
these s-param eter com putations. Parasitics elem ents o f the m odel such as capacitance
betw een external base and collector and collector to substrate capacitance are also
included. T his section presents a com parison o f sm all-signal s-param eters obtained from
(a) on-w afer m easurem ents, (b) evaluation o f closed-form expressions derived during this
work, and (c) com puter sim ulations using M DS. T he underlying m otivation for com paring
the sm all-signal s-param eters obtained from three distinct sources is (a) to determ ine the
relative rank o f s-param eters calculated using closed-form expressions, w hen com pared
w ith sim ulation against m easurem ents, (b) to identify the strengths and w eaknesses o f the
results obtained using closed-form expressions vis-a-vis com puter sim ulations, and (c) to
establish the utility, significance, and m erits o f the closed-form
expressions for
calculating/predicting sm all-signal s-param eters o f bipolar ju n ctio n transistor for any bias
condition.
A com parison o f m easured, calculated, and sim ulated sm all-signal s-param eters for
n p n l in com m on-em itter configuration at VCE = IV and Ic = 1mA is presented in Fig. 310. Input and output reflection coefficients ( S u and S22) have been plotted on Z Sm ith
charts, w hile p o lar plots have been used for the forw ard and reverse transm ission
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56
Fig. 3-10.C om parison o f m easured (light), calculated (bold), and sim ulated (dash) sm allsignal s-param eters for n p n l in co m m o n -em itter configuration at VCE = IV and
Ic = 1mA.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
57
coefficients (S2) an d S !2). As show n in Fig. 3-10. for S[
m easurem ents and clo sed-form
predictions are in excellent agreem ent both in m agnitude and phase for th e entire
frequency range, w here as the sim ulation greatly underestim ates the m agnitude, w hile the
phase agrees fairly w ell w ith m easurem ents. C om parison o f reverse transm ission
coefficient ( S p ) show s once again good agreem ent for m agnitude and an ex cellen t m atch
o f phase o f m easured and calculated data, w hile sim ulation greatly overestim ates
m agnitude and underestim ates phase. F o r S 2I, m easured and calculated data ag rees w ell
for the entire frequency range, w here as sim ulated S 21 has m uch d ifferen t phase.
S im ulated S 22 also exhibits a different tren d com pared to the m easured S 22, w here as
calculated S22 d isplays a deviation from the m easurem ent data only at higher frequencies,
w hile the phase m atches very w ell w ith m easurem ent. To facilitate separate co m p ariso n o f
m agnitudes and phase angles o f these three different sets o f sm all-signal s-param eters.
Fig. 3-1 la n d Fig. 3-12 show ing m agnitudes in dB s and phase angles in degrees have been
included. As show n in Fig. 3-11, sm all-signal s-param eters calculated from clo sed-form
expressions are in excellent agreem ent w ith the m easurem ents ex cep t for S 22 at higher
frequencies, w here as the sim ulation n ot only underestim ates the m agnitudes o f S p and
overestim ates S I2, but also follow s a different trend. For S 21 and S22 sim ulation
underestim ates th e m agnitudes for low er h alf and overestim ates for the upper h a lf o f the
frequency range. A s regards to the p h ase angles o f sm all-signal s-param eters com pared.
Fig. 3-12 show s th at the calculated d ata tracks the m easurem ents very w ell over the entire
range o f frequencies, w here as except fo r S p , the phase angles o f the o th e r three sparam eters i.e. S p , S21, and S22, the difference betw een m easurem ent and sim u lation is
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
58
-15
-4 .
-45
-12
-55
4 5 .0
MHz
FREQ
2 0 .0 4 5
MHz
GHz
FR EQ
2 0 .0 4 5
GHz
2 0 .0 4 5
G Hz
Magnitude (S12)in dB
Magni tude (S ji)in dB
15*
-15
-25
4 5 .0
MHz
FREQ
2 0 .0 4 5
Magnitude (S21)in dB
GHz
45.0
FR EQ
Magni tude (S22)in dB
Fig. 3 -1 1.C om parison o f m agnitude o f m easured (light), calculated (bold), an d sim ulated
(dash) sm all-signal s-param eters for np nl in com m on-em itter configuration at
Vce = IV and Ic = 1mA.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
59
-
180°
-
4 5 .0
MHz
FR EQ
2 0 .0 4 5
80 "
FREQ
GHz
Phase (Su) in degree
2 0 .0 4 5
GHz
Phase (S12)in degree
%•
■
4 5 .0
MHz
FR EQ
2 0 .0 4 5
Phase (S2|) in degree
GHz
FR EQ
2 0 .0 4 5
GHz
Phase (S22)in degree
Fig. 3 -l2 .C o m p ariso n o f phase o f m easured (light), calcu lated (bold), and sim ulated
(dash) sm all-signal s-param eters for n p n l in com m on-em itter configuration at
V c e = IV and Ic = ImA.
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60
fairly large. For S21, although the trend seem s to be sim ilar to the m easurem ent, but the
sim ulation greatly overestim ates the phase. The phase o f S (2 is predicted correctly for
only a very narrow frequency range and there after the variation in phase w ith frequency is
significantly different both in values and trend.
Bias Sensitivity of Small-Signal S-Param eters
It is w ell known that a stable BJT am plifier requires the tran sisto r to be biased in
the forw ard active region and the bias point to be kept constant despite variations in
tem perature and device param eters. The R F/M icrow ave design param eters such as gain
and noise figure are known to be strong functions o f the co llecto r cu rren t in transistor
am plifiers involving BJTs [Hp75, Ven90]. A s stated earlier, the sm all-signal device model
param eters such as gm, rn, rQ, rb, rc, c^, and c^ are sensitive to changes in bias conditions.
T hese sm all-signal model param eters have been shown to be p art o f the closed-form
expressions for sm all-signal s-param eters derived and experim entally verified in the
previous sections. Since sm all-signal s-param eters are functions o f these bias sensitive
device m odel param eters, they m ust be sensitive to changes in co llecto r current as well.
T his section is intended to qualitatively describes the bias sensitivity o f sm allsignal s-param eters. O n-w afer s-param eter m easurem ents o f transistors n p n l and n p n l at
different bias conditions have been used to ascertain the bias sensitivity. V ariations in both
m agnitude and phase o f s-param eters caused by changes in bias co n d itio n s have been
included here. C hanges in bias conditions have been divided into tw o categories: (a)
changes in collector current w hile VCE is held constant, and (b) ch an g es is VCE while
collector current is kept constant. Due to sm all sam ple size (only tw o devices), no attem pt
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
61
has been m ade to quantify the changes in e ith e r m agnitude or phase due to a som e percent
change in collector current. However, q u alitativ e description o f bias sen sitiv ity o f sm allsignal s-param eters alone is considered q u ite valuable, which is included in this section.
Variations in S | i w ith changes is co lle c to r current and VCE are sh o w n fo r n p n l in
Fig. 3-13. A t low er frequencies, the m ag n itu d e o f S u decreases w ith the increase in
collector current, w hile at higher frequencies, the effect is opposite. A lso, the increase in
collector current causes an increase in phase qu ite significantly. A t low er frequencies, the
capacitive reactance w hich is m ainly due to b ase-em itter junction cap acitan ce is dom inant,
but as the frequency increases, this reactan ce decreases and the resistive co m p o n en ts o f
input im pedance start to becom e pro m in en t an d this behavior is clearly v isib le as the Sj (
plot m oves clockw ise w ith the increase in frequency tow ards low er cap acitiv e reactance
w hen plotted on Z -sm ith chart. W ith the in crease in collector current, c K increases as
stated earlier and this results into low er capacitive reactance and increase in phase.
Increase in VCE causes an increase in co llecto r-b ase depletion cap acitan ce an d thereby
reduction in phase. Since the change in c^ is sm all w ith the increase in VCE, therefore, the
reduction in phase o f S ( j is also sm all. T he effects o f changes in co llecto r c u rre n t and VCE
on m agnitude and phase o f S J2 are show n in Fig. 3-14. Increase in both co lle c to r current
and VCE causes a decrease in m agnitude an d also phase o f S j 2- S ince b ip o la r junction
transistor is not a unilateral device so rev erse transm ission coefficient ( S 12) is also a
m easure o f isolation betw een the o u tp u t an d input. T he unilateral assu m p tio n m akes the
design o f an am plifier m uch sim pler, but su ch an assum ption is not p ractical due to
stability considerations. However, red u ctio n in m agnitude o f S J2 w ith th e increase in
collecto r current and V CE is a favorable effect.
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62
(c)
(a)
0
0
VCE = IV IC
VCE = 3 V IC
VCE = I V I C = ImA (Bold)
VCE = IV IC = 2mA (Light)
VCE = IV [C = 'mA (Dash)
4 5 . 0 MHz
FREQ
2 0 .0 4 5
GHz
M agnitude ( S n ) in dB
4 5 .0
MHz
FREQ
ImA (Bold)
ImA (Light)
2 0 .0 4 5
GHz
M agnitude (S ( t ) in dB
250
VCE = IV I C = ImA (Bold)
VCE = 3V IC = ImA (Lighl)
VCE = 1 V I C = ImA (Bold)
VCE = 1V IC = 2mA (Light)
VCE = IV I C = 3mA (Dash)
-250
4 5 .0
MHz
FREQ
Phase ( S 11) in degrees
(b)
2 0 .0 4 5
GHz
4 5 .0
MHZ
FREQ
2 0 .0 4 5
GHz
P hase (S j {) in degrees
(d)
Fig. 3-13.B ias sensitivity o f sm all-signal Sj j. (a) V ariation o f m agnitude w ith changes in
Ic ; (b) V ariation in phase w ith changes in 1^; (c) Variation in m agnitude with
ch anges in VCE; (d) Variation in phase w ith changes in VCE.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
63
(c)
(a)
VCE = I V I C = Im A(Bold)
VCE = IV 1C = 2mA (Light)
VCE = IV IC = 3mA (Dash)
VCE = IV IC = ImA (Bold)
VCE = 3 V I C = ImA(Lighi)
-55
-55 •
4 5 .0
FP.EQ
MHz
2 0 .0 4 5
0 MHz
GHz
FREQ
2 C .0 4 5
GHz
M agnitude ( S I2) in dB
M agnitude ( S j2) in dB
160
160
V C E = IV IC = Im AiB old)
VCE = 3V I C = 1mA (Light)
VCE = IV IC = I mA ( Bold)
VCE = IV IC = 2mA (UghO
VCE = IV IC = 3mA (Dash)
%#
-40
4 5 .0
MHz
FREQ
Phase ( S 12) in degrees
(b)
2 0 .0 4 5
GHz
0 MH:
FREQ
2 0 .0 4 5
OH;
Phase ( S j2) in degrees
(d)
Fig. 3-14.B ias sensitivity o f sm all-signal S i2. (a) V ariation o f m agnitude w ith changes in
Ic ; (b) V ariation in phase w ith changes in Ic ; (c) Variation in m agnitude w ith
changes in VCE; (d) V ariation in phase w ith changes in VCE.
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64
Forward transm ission coefficient S 21 is w ell know n to be sensitive to changes in
bias conditions. Variation in S 21 w ith changes in co llector cu rren t and VCE are shown in
Fig. 3-15. The m agnitude o f S 2j increases linearly w ith the increase in collector current
and also with the increase in V CE. H owever, the phase o f S 2J decreases w ith the increase
in eith er collector current o r V CE. V ariations in o utput reflection coefficient S22 w ith
changes in collector current an d V CE are shown in Fig.3-16. B oth m agnitude and phase o f
S22 decrease with the increase in either collector current o r VCE.
This behavior can be d escrib ed in term s o f real and im aginary parts o f the com plex
output im pedance o f B JT in com m on-em itter configuration. T he output reflection
coefficient S22 o f a transistor w hen plotted on Z -Sm ith chart, starts in m ost cases in the
capacitive reactance (low er) p o rtion and w ith the increase in frequency m oves tow ards
larger phase angles. As the p h ase angle spanned by the plot increases, the corresponding
capacitive reactance decreases. T he radial distance o f the plot d eterm ines the m agnitude o f
S22 and the angle co rresponding to the distance traversed by the plot determ ines the phase
o f S22. The radial distance o f S 22 plot is a function o f real part, and the angle spanned on
Sm ith chart is dependent upon the reactive com ponent o f the o u tp u t im pedance. Since the
real part o f transistor’s output im pedance is inversely proportional to its transconductance
(gm), therefore w ith the increase in co llecto r current, the real p art decreases. T he decrease
in real part of the output im p ed an ce results into reduction o f m agnitude o f S22. The
increase in VCE decreases the depletion capacitances, w hich in turn the increase the
overall capacitive reactance an d thus reduction in phase. T he variations in m agnitude and
phase o f sm all-signal s-param eters due to changes in co llector cu rren t and VCE have been
sum m arized in Table 3-4 and T able 3-5 respectively.
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65
Table 3-4. Effect o f C o llecto r C urrent on S m all-Signal S-Param eters
C o llecto r C urrent Increases
P aram eter
6 G H z to 20 G H z
1 G H z to 6 G H z
M agnitude
Phase
M agnitude
Phase
S ll
D ecreases
Increases
Increases
Increases
S12
D ecreases
D ecreases
D ecreases
Increases
S21
Increases
D ecreases
Increases
Increases
S22
D ecreases
Increases
D ecreases
Decreases
Table 3-5. E ffect o f V CE on S m all-S ignal S-Param eters
V CE Increases
Param eter
M agnitude
Phase
S ,i
Increases
D ecreases
S 12
D ecreases
D ecreases
S 21
Increases
D ecreases
S 22
D ecreases
D ecreases
Sum m ary
In this chapter, a relatio n sh ip betw een sm all-signal s-param eters and highfrequency equivalent circuit m odel fo r b ip o lar ju n c tio n transistor has been established.
C losed -fo rm expressions fo r tw o -p o rt s-p aram eters in a com m on-em itter configuration
have been derived using in trin sic h igh-frequency sm all-signal m odel for BJTs. The
closed -fo rm
expressions
have
b een
verified
against
on-w afer
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
high-frequency
66
(c)
(a)
6
6
VCE = IV IC = ImA (Bold)
VCE = 3V IC = ImA (Light)
V C E = 1V IC = 1mA (Bold)
V C E = IV IC = 2mA (Light)
V C E = IV IC = 3mA (Dash)
•14
-14
0 MHz
FREQ
2 0 .0 4 5
4 5 .0
GH2
MHz
FREQ
2 0 .0 4 5
GHz
M agnitude (S 22 ) in dB
M agnitude (S 22 ) in dB
20
20
V C E = IV I C = ImA (Bold)
VCE = 1V IC = 2mA (Light)
V CE = IV IC = 3mA (Dash)
VCE = IV IC = ImA (Bold)
VCE = 3V IC = ImA (Light)
-80
-80
FREQ
Phase (S 22 ) in deg rees
(b)
2 0 .0 4 5
GHz
4 5 .0
MHz
FREQ
2 0 .0 4 5
GHz
Phase (S 22 ) in degrees
(d)
Fig. 3-16.B ias sen sitiv ity o f sm all-signal S22 - (a) V ariation o f m agnitude w ith changes in
Ic ; (b) V ariation in phase w ith changes in I<s (c) Variation in m agnitude w ith
changes in VCe; (d) Variation in phase w ith changes in VCE.
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67
(c)
(a)
25
VCE = IV IC
VCE = 1V IC
VCE = IV C
0 MHz
FREQ
VCE = IV [ C = 1mA (Bold)
VCE = 3V IC = ImA (Light)
Im A (Bold)
2mA (Light)
3mA (Dash)
2 0 .0 4 5
GHz
M agnitude (S2 l) in dB
• 1 5 . 0 MHz
FREQ
M agnitude (S21) in dB
ISO
VCE = IV I C = ImA (Bold)
V CE = .IV IC = 1mA (Light)
VCE = 1 V I C = 1mA (Bold)
VCE = IV IC = 2mA (Light)
VCE = IV IC = 3mA (Dash)
-20
-20
4 5 .0
MHz
FREQ
P hase (S21) in degrees
2 0 . 0 4 5 GHz
45 .0
MHz
FREQ
2 0 .0 4 5
GHz
Phase (S 21) in degrees
(b)
(d)
Fig. 3-15.B ias sensitivity o f sm all-sig n al S 21. (a) Variation o f m agnitude w ith changes in
Ic ; (b) Variation in phase w ith changes in
(c) V ariation in m agnitude with
changes in VCE; (d) V ariation in phase with changes in VCE.
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68
m easurem ents from 45 M H z to 20.045 G H z. Excellent agreem ent betw een the sparam eters calcu lated using closed-form expressions and m easured has been shown over
the entire frequency range. A com parison betw een m easured, calculated and sim ulated
sm all-signal s-param eters has been presented. It has been show n that s-param eter
calculated
using
closed-form
expressions
derived
during
this
w ork
m atch
the
m easurem ents m ore closely com pared to s-param eters obtained through M DS sim ulation.
A lso included in this ch ap ter is a qualitative analysis and description o f bias-sensitivity o f
sm all-signal s-param eters. M easured data has been analyzed and results have been
tabulated.
The w o rk presented in C hapter 4 includes m easurem ents o f sm all-signal sparam eters o f pack ag ed transistors under different am bient tem peratures, verification o f sparam eter pred ictio n s at different tem peratures using closed-form expression presented in
this chapter ag ain st m easured data, and a com parison o f m easured, predicted, and
sim ulated s-p aram eters at different tem peratures. Also included in the next chapter is
therm al sensitivity o f sm all-signal s-param eters.
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CHAPTER 4
A N A L Y T IC A L M O D E L L IN G O F PAC KA GED T R A N SISTO R S A N D PR ED ICTIO N
O F T H E R M A L E F FE C T S O N SM A L L -SIG N A L S-PA R A M ETER S
It is w ell know n that operating tem perature strongly effects the behavior o f sem i­
co n d u cto r devices. T he operating tem perature o f a device is d eterm ined by the am bient
tem perature and the pow er dissipated inside the device (self-heating effect). As stated ear­
lier, due to th erm al dependence o f the physical properties o f the m aterials used to fabricate
the sem ico n d u cto r devices, alm ost every transistor param eter is e ith e r directly or indi­
rectly effected b y tem perature. T his bro ad therm al dependence m akes the currents and
voltages (both in sid e the device and at the term inals) also sensitive to tem perature.
W ith the advancem ents in m icrow ave planar technology, the u p p e r frequency limit
for b ip o lar ju n c tio n transistors is continuously being extended and th ese devices are find­
ing applications in a w ide variety o f fields such as com m unications, electronic w arfare,
radars, and so p h isticated w eapon system s. T he system s em ploying R F/M icrow ave circuits
operate on land, at sea, in air, and in space. B esides m any other environm ental variables,
tem perature is the m o st obvious w hich changes from application to application and even
w ithin the sam e system . Since, R F and m icrow ave circuit design m ethodologies are cen­
tered aro u n d the s-p aram eter data o f the active and passive devices u sed in the circuit,
therefore, it is very im portant to know the therm al behavior o f sm all-signal s-param eters.
In C h ap ter 3, it has been show n that sm all-signal s-param eters are sensitive to bias current.
T he bias cu rren t (Ic in B JTs) is itself know n to be quite sensitive to tem perature. In addi­
tion to the bias dep en d en ce, sm all-signal device m odel param eters also exhibit sensitivity
69
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70
to tem perature. This dependence o f sm all-signal device param eters (a) on tem perature and
(b) on bias current w hich itself is tem perature sensitive, presents an interesting situation.
This chapter presents sm all-signal s-param eter m easurem ents o f packaged bipolar
ju n c tio n transistors done at different am bient tem peratures. A com parison betw een m ea­
sured d ata and s-param eters calculated using closed-form expressions derived in C hapter
3, and a derivation o f im proved closed-form expressions for sm all-signal s-param eters o f
bip o lar ju n ctio n transistors is perform ed. T herm al sensitivity o f s-param eters based upon
the m easured data is included in this chapter. T h is chapter dem onstrates the ability to ana­
lytically m odel packaged transistors and boards and predict tem perature effects on sm allsignal s-param eters.
Thermal Measurements of Small-Signal S-Parameters
N orm ally, sm all-signal s-param eters o f bip o lar junction transistors are m easured at
room tem perature. T hese room tem perature m easurem ents are perform ed eith er on-wafer,
o r after having m ounted the packaged devices on suitably designed boards. F or on-w afer
s-param eter m easurem ents at tem peratures o th er than norm al room tem perature, usually,
therm al chucks are used. The tem perature o f these therm al chucks is held co n stant by an
external tem perature controller connected to the chuck. T he w afer is placed on the chuck
and after som e tim e the w afer tem perature equalizes w ith the chuck and then m easure­
m ents are m ade. For this w ork, packaged devices w ere m ounted on specially designed
boards and sm all-signal s-param eters w ere m easured. T he board with the device on it was
placed inside a tem perature controller (oven). S chem atic representation o f the m easure­
m ent set-up is shown in Fig. 4-1. T his m easurem ent set-up provided an opportunity to
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71
Tem perature C ontroller
t
& O
Bias-Tee
Bias-Tee
RF
s
OUT
Of
DC
DC
R F + DC
Pow er Supply
VBE
^
VCE
o
o
Fig. 4-1. S chem atic representation o f set-up for therm al m easurem ents o f sm all-signal
s-param eters o f packaged bipolar ju n ctio n transistors m ounted on boards
placed inside a controller.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
72
quantify the follow ing w ith sm all-signal s-param eters: (a) effect o f tem perature change
w hen bias conditions are held constant, (b) effects o f ch an g es in bias at different tem p era­
tures, (c) com posite effect w hen both tem perature an d b ias are changing. Study o f effects
m entioned in (b) and (c) is im portant because these are the conditions in w hich R F /M icrowave circuits are n o rm ally operated (varying tem perature, varying bias). The situation
w here the bias cu rren t rem ain s constant despite ch an g es in tem perature (a) is an ideal and
m ost desirable co n d itio n . T h is is useful to study sin ce it can show the reduction in varia­
tion in s-param eters w ith tem perature by im proving the DC bias design. The im prove­
m ents in overall ac perform ance with reduced variations in s-param eters due to
tem perature can be related to im proving the D C bias netw ork.
In order to p erfo rm reliable sm all-signal s-p aram eter m easurem ents o f packaged
bipolar ju n ction tran sisto rs at different tem peratures, follow ing factors were exam ined: (a)
selection o f device, (b) d esig n o f board, (c) transm ission m edia, and (d) type o f connectors
to em bed the board in th e m easurem ent path.
Selection of Active Devices
To choose the d ev ice for this experim ent the follow ing inform ation was required;
com ponent size and sh ap e, lead size and spacing, m o u n tin g , and therm al considerations,
reliable device m odel param eters, sufficient details o f the package used, and sm all-signal
s-param eter m easu rem en t d ata at room tem perature. T h e inform ation pertaining to the
device m odel p aram eters and the package m odel w as necessary to calculate s-param eters
at different tem peratures. C losed-form expressions fo r p red ictin g tem perature effects w ere
derived for this w ork. S m all-signal s-param eter d a ta p rovided in the data sheet has been
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
73
used to verify and validate th e m easurem ents perform ed during this research at room tem ­
perature against m easurem ents done independently by the m anufacturer. Sm all-signal sparam eter sim ulations w ere perfo rm ed on M DS using the device m odel param eters and
package model provided by th e m anufacturers in data sheets. T he sim ulation results were
com pared against published s-param eters for the sam e device to establish reliability o f
both device and package m odel param eters. Three surface-m ount npn transistors
MRJF927, M R F947, and M M B R 941, w ere selected after reasonable agreem ent was
observed betw een the p u b lished d ata and the data obtained from M DS sim ulations. The
package style for M R F927, and MRJF947 is SC -70 or SOT-323, w hile M M B R941 has
been packaged in SOT-23 [M ot97a, M ot97b].
Selection of Laminate M aterial
Selection o f suitable lam inate m aterial for the board is essential p rio r to laying out
the board. For applications above 500 M H z, the selection o f lam inates available to design­
ers reduces significantly. T h e m aterials com m only used for printed circu it board applica­
tions such as FR-2, C E M -1, C E M -3, FR -4, FR-5 are not suitable for high frequency
application due to (a) higher dielectric loss, and (b) greater sensitivity o f their dielectric
constants to frequency. A n o th er factor th at lim its their use, w hen surface m ount com po­
nents are em ployed, is their in co m p atib le therm al coefficients o f ex p an sio n (C TE s), w hich
strain the solder jo in ts and often result into solder jo in t cracking. For high frequency appli­
cations, where interconnections act as transm ission lines, these conventional lam inates
w ith higher dielectric constants and greater dissipation factors, effect b o th the speed and
integrity o f the signal p assin g th rough them . They exhibit greater propagation delays and
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74
m ore attenuation at higher frequencies--factors w hich are to be avoided [H in 8 8 , C 0 0 8 8 ,
S er89]. To choose a lam inate fo r high-frequency application such as the one presented in
this chapter, following lam inate selection criteria was form ulated: ( I ) low dielectric con­
stant, (2) low dielectric loss, (3) low tem perature coefficient o f dielectric constant, (4) sta­
ble dielectric constant over a b ro ad frequency range, (5) com patible therm al coefficient o f
expansion, (6 ) cost effectiveness, and (7) process boards using standard conventional cir­
cuit board techniques. A fter extensive search and evaluation, a double sided copper clad
glass reinforced hydrocarbon/ceram ic laminate from R 0 4 0 0 0 series w as selected for
designing the boards for this w ork. Lam inate material R 0 4 0 0 3 has a typical dielectric
constan t o f 3.38, therm al coefficient o f dielectric +40 ppm /°C , and therm al coefficient o f
expansion sim ilar to copper [R og97]. This lam inate is a low loss m aterial with stable
dielectric constant over environm ental conditions, thus allow ing repeatable designs o f
controlled im pedance transm ission lines. The tem perature coefficient o f R 0 4 0 0 3 is am ong
the low est, thus m aking it m o st suitable for tem perature sensitive applications. These
properties besides others m ake R 0 4 0 0 3 a superior m aterial for use at higher frequencies
w hen com pared with conventional base materials m entioned earlier. T he circuit boards
used during this work w ere do u b le sided copper clad w ith 60 ± 4 mil (1.52 ± 0.10 mm)
thick base m aterial (R 0 4 0 0 3 ). T he thickness o f electrodeposited copper on each side was
one ounce (35 pm ).
Selection of Transmission M edia and Connectors
H aving selected the b o ard m aterial, the next phase in the board design was selec­
tion o f transm ission m edia an d type o f connectors. C oplanar w aveguides with backside
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
75
gounded w ere chosen for this w ork. T h eir preference over m icrostrip lines was due to (a)
ease o f series connection—transistor (D U T ) and end connectors in this case, (b) both signal
and ground conductors on the sam e plan e—a requirem ent for com m on-em itter configura­
tion, (c) rem arkably reduced dispersion, and (d) im proved m echanical strength and pow er
handling capability due to back side ground. A s regards to the connectors, female SM A
end launch connectors with thick b road tab contact PE4543 w ere selected [Pas98]. W ith
these connectors, signal is launched on to the transm ission m edia directly.
Design of Transmission Lines
To design the coplanar w aveguide w ith a characteristic im pedance o f 50 ohm s,
several com binations o f signal co n d u cto r w idths (w) and spacing betw een signal and
ground conductor (s) were exam ined. A cross-sectional view o f coplanar waveguide w ith
backside grounded is shown in Fig. 4-2. T he criterion to choose the optim al design for 50
w : w idth o f signal conductor
s : spacing betw een signal &
ground conductors
h : height o f substrate material
Fig. 4-2. C ross-sectional view o f co p lan ar w aveguide with backside grounded.
ohm s transm ission line during this w ork w as based on G hione and N ald i’s work that in
coplanar w aveguides with backside ground, electrical param eters such as characteristic
im pedance (ZQ) and effective dielectric co n stan t (£,.) are sensitive to (a) shape ratio w /
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
76
(w +2s) and (b) substrate thickness ratio h /(w +2s) [G hi87]. T herefore, the com bination o f
w and s w hich offered m inim um reflection, m axim um return loss, o r voltage standing
w ave ratio (V SW R ) clo sest to unity was chosen as the optim al design. Return loss (RL) is
defined as the ratio o f th e pow er o f the incident w ave to the pow er o f the reflected wave
(R L = Pin / Pref). A co m p ariso n o f return loss for three different w and s com binations cho­
sen from m any co m p u ted from M DS sim ulations is show n in Fig. 4-3. For an ideal trans-
200
w =58 m il, s = 8 mil
x-x- w =76 m il. s=12 mil
■ ■ w =87 m il, s = l 6 mil
h = 60 mil
50
300 K H z
Frequency
3 GHz
Fig. 4-3. C om parison o f retu rn loss for different com binations o f signal conductor w idth
(w) and spacing betw een signal and ground co nductors (s). N ote all three com bi­
nations result into characteristic im pedance o f 50 o h m s at DC.
m ission m edium i.e. one th at is perfectly m atched, there should be no reflection i.e. Pref =
0 regardless o f the value o f P in. This results in an infinite return loss. This ideal situation
can be expressed as having zero reflection coefficient, o r V S W R equal to 1. As shown in
Fig. 4-3, the sim ulation fo r co p lan ar w aveguide w ith w =76 m il and s=12 mil exhibits the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
77
highest return loss. T h u s for this w ork, the coplanar w aveguides w ere designed and im ple­
m ented using the optim al shape ratio o f w = 76 mil and s = 12 mil.
Layout of Board
Layout o f the b o ard used for therm al m easurem ents o f sm all-signal s-param eters
o f bipolar ju n ctio n tran sisto rs is shown in Fig. 4-4. As m en tio n ed earlier the active devices
selected for m easurem ents w ere packaged in tw o different styles i.e. SOT-323 and SOT23. This required a sep arate board for each type o f tran sisto r due to dim ensional d iffer­
ences in pad footprints fo r tw o packages. To keep the b o ard dim ensions and layout etc. the
sam e for all three tran sisto rs, the dim ensional differences in their pad footprints needed to
be removed. T he pad lay o u t was changed in such a m an n er that only one footprint w as
needed to accom m odate the tw o different packages. N ew p ad layout is shown in Fig. 44(b). The extrem e cases for both packages i.e. m in im u m and m axim um dim ensions for
SOT-323 and SOT-23 w ere used as lim its for pad size an d separation from each other. T h e
m odified pad layout as sh o w n in Fig. 4-4(c), accom m odates both packages satisfactorily.
Characterization of Board
In order to ex tract reliable sm all-signal s-p aram eters for packaged devices from
m easurem ents w hen they are m ounted on boards, the effects o f the intervening test stru c ­
ture o r fixture betw een the devices and the cable co n n ecto rs need to be subtracted. T h is
process is term ed as de-em bedding. To characterize th e b o ard com pletely, test structure
w ithout an active device (D um m y) show n in Fig. 4-4 (a), “th ru ” structure shown in Fig. 45, and connectors w ere m easured. A s m entioned earlier, fem ale SM A connectors w ere
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
78
.................
04
0.012
IS'S^'^5ss2Si!????^5
0012
•r^ % ^ S 5 ao iS ^ ^ B
(a)
*F%-g*~3r* rzy^XTZrir
0012
0 489
0.040
• 0.040 -
0 129
(b)
0.056
scrr-323
(max. dim )
SOT-323
(min. dim.)
SOT-23
(mm. dim.)
SOT-23
(max. dim .)
LiL i
tm
(C)
Fig. 4-4. Layout o f a ty p ical board used for therm al m easurem ents o f sm all-signal sparam eters o f B JT s. (a) Top side o f board; (b) M odified pad layout for active
devices. Sym bols B, C , and E on pads signify base, collector, and em itter
respectively; (c) P lacem ent o f SOT-323 and SOT-23 packages on single pad
footprint. All dim en sio n s shown are in inches an d figures are draw n to scale.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Fig. 4 -5. L ayout o f thru-structure on board. N ote all dim ensions are in inches.
used in th is w ork which rendered the board non-insertable, therefore, fem ale to male
adapter w as used to em bed the board in the m easurem ent path. The ad ap ter w as also m ea­
sured to effectively subtract its effect from the m easurem ent data. T hese m easurem ents
w ere m ade under the sam e conditions inside the tem perature co n tro ller using H P8753C
(N etw ork A nalyzer) and with the sam e calibration set that was used for sm all-signal sp aram eter m easurem ents o f active device at different am bient tem peratures. T he boards
w ere ch aracterized at tem peratures from -35°C to +85°C, the sam e range o f tem perature
for w h ich devices were m easured. T h e transm ission coefficient and phase delay o f thrustructure m easured between 0.2 G H z and 2.2 G H z at 25°C is show n in Fig. 4-6. Also
show n in F ig. 4-6 is the m easured phase response o f the connector an d adaptor, an d a com ­
parison betw een the sim ulation results and the m easured data for the co p lan ar w aveguide
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
80
designed and im plem ented in this w ork. T h e m atch betw een the d e-em b ed d ed phase (m ea­
sured data) and sim ulation (Fig.4-6 (d)) o v er the en tire frequency range is excellent.
75
200
0.5
*\
—
•-
u.
DJO
t)
3
M easured Thm-Structure
A daptor
Connectors
^3
o
■o
3
C
00
3
S
(N
De-embedded Thru-Structure
JH
o
Q
ot:
rs
*-* S :i M agnitude
04)
Phase
-a *
CO
.
a
.
a
rt
a.
t- - V—*
*
a
c*i
c/o
-0 5
0.2 GHz
Frequency
2.2 GHz
uZJ
CJQ
3
y
Q
CO
CL -200
y.
-1 2 0
0 2 GHz
Frequency
2.2 GHz
(b)
(a)
—
S 2 1 Measured
S-«| Simulated
Frequency
*-* S :i M easured
— S->> Sim ulated
Frequency
Fig. 4-6. M agnitude and phase o f forw ard tran sm issio n coefficient (S 2 1)o f thru -structure.
(a) D e-em bedding o f S21 phase; (b) M ag n itu d e o f S2t and d e-em b ed d ed S 21
phase; (c) Com parison o f sim ulated and m easured S2j m agnitude; (d) C o m p ar­
ison o f sim ulated and m easured S 2j phase.
F rom the m easured s-param eter d ata for the coplanar w aveguide, its attenuation
and phase shift can be determ ined. T he atten u atio n is obtained from the m ag n itu d e o f S 21
and phase sh ift is equal to the phase o f S 21. A tten u atio n o f any circu it elem en t (coplanar
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81
w aveguide in this w ork) w hen represented as a tw o-port netw ork can be related to S21 as
A tte n u a tio n (in d B ) = 20 log
1
(4-1)
’21
The m agnitude and phase o f S21 o f a thru structure m easured at -35oC, +25oC , and +85oC
is shown in Fig. 4-7. A s shown in Fig. 4-7, the attenuation in the coplanar w aveguide
0.5
(a)
00
<N
00
+ 85 °C -
-0.5
0.2 G H z
F requency
2.2 GHz
(b)
a.
-120
0.2 G H z
F requency
2.2 GHz
Fig. 4-7. M easured S2i at -35°C , +25°C , and +85°C . (a) M agnitude; (b) Phase.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
82
increases w ith the increase in tem perature. A t -35°C , there is virtually no loss and the
transm ission line is alm ost lossless. H owever, w ith 120°C rise in tem perature i.e. at
+85°C , the co p lan ar w aveguide has a negligibly sm all loss of 0.1 dB at 2.2 G H z. T here­
fore, during this w ork the transm ission m edia w as treated as a lossless line. A s regards to
the phase, as show n in Fig. 4-7. the tem perature effects are also negligible.
S-Parameter Measurements of Packaged Transistors
Having found the perform ance o f board satisfactory, the surface m o u n t transistors
w ere mounted on the boards and m easurem ents were made using netw o rk analyzer
HP8753C from 0.2 G H z to 2.0 GHz. D uring this process as shown in Fig. 4 -1 , the boards
w ere placed inside the tem perature controller. As m entioned earlier, due to sm all package
dim ensions, it is often im practical to attach R F connectors to the actual p ack ag e term inals,
therefore transm ission lines (coplanar w aveguides in this work) are used to ex ten d connec­
tions from the package to R F connectors. S ince s-param eters are m easured as ratios o f
incident and reflected w aves at and bf- respectively, and both
and bj bein g travelling
waves (function o f position and time). T herefore it is necessary to specify th e position or
reference plane w here m easurem ents are m ade. In cases where transm ission lines (even if
lossless) are used as connections betw een the D U T and the m easuring equipm ent, the ref­
erence plane w here m easurem ents are m ade is not the sam e as that o f the D UT. However,
using transm ission line concepts, a relationship betw een the s-param eters at tw o different
reference planes can be established and s-param eters o f DUT can be ascertained confi­
dently from the d ata m easured at the reference planes away from the reference planes o f
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
83
the DUT. This is term ed as shifting the reference p lan es [H p72, G on84]. Pictorial rep re­
sentation o f shifting the reference planes is show n in Fig. 4-8, where DUT represent the
surface m ount packaged transistor and the transm ission lines on both sides represent
coplanar w aveguides, connectors, and adaptor used to ex ten d connection from the package
to netw ork analyzer. The phase angles <J>! and <j>2 show n in Fig. 4-8 represent the electrical
R eference Planes
(Packaged Transistor)
M easurem ent
P lan e -1
DUT
Plane-2
DUT
P lane-I
M easurem ent
Plane-2
T ransm ission L ines
Fig. 4-8. Pictorial representation o f shifting the reference planes.
lengths o f transm ission lines connected on both sides o f DUT. The m easurem ent p la n e s-1
and -2 signify the positions w here m easurem ents are m ade. The s-param eters m easured at
the m easurem ent p la n e s-1 and -2 can be related to the s-param eters o f D U T at D U T
p la n es-1 and -2 by the follow ing
/
5
= <P S <t>
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4-2)
84
w here S
and 5 represent the s-p aram eter m atrices at m easurem ent and D U T planes
respectively. A s shown in eq u atio n (4-2), the s-p aram eter m atrix at the m easurem ent
planes is the result o f pre-m u ltip licatio n and p o st-m ultiplication o f s-param eter m atrix o f
D U T by the diagonal m atrix <t>. T h e diagonal m atrix <t> represents the s-param eters for
lossless transm ission lines [C ha94].
-y < t>
0
(4-3)
—y<j>.
0
E xpanding equation (4-2) in term s o f s-p aram eter and diagonal m atrices, w e get
-j<$>
5 1 1 5 12
5 ' S'
21 22
0
0
5 11 5 12
—
e v<l
^21 ^ 2 2
> 9
0
0
(4-4)
-y'0o
e
after m atrix m ultiplication, the s-param eters at m easurem ent p la n e s-1 and -2 can be w rit­
ten as
f
/
s 1, 1. s 12
s'
21
S
-y'2*!
5 ji e
-y '(4 > ! +<i>2)
5 p e
(4-5)
/
—y'(4> | +<t>2 )
22
S9 j e
„
- J 2^
S 22 e
w here the m ultiplier e’^ rep resen ts the phase difference betw een the tw o reference planes.
/
A s show n in Fig. 4-8, for
^ w hich is a ratio o f bj ’ and a / ’, the incident w ave travels
through the length <|)| and p art o f it gets reflected, and then the reflected w ave travels
through the sam e length
again. T h u s as show n in eq u atio n (4-5), S n o r S 22 term s are
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
85
m ultiplied tw ice by the m u ltiplication factor. For forw ard transm ission coefficient, the
incident wave travels th rough the length o f transm ission line 0 j and then the transm itted
w ave travels through the length <jH- Similarly, for 5 p
, the m u ltip lier is the sum o f the
electrical lengths o f tran sm issio n lines connected on each side o f the DUT i.e.
From equation (4-5), w e can d eterm in e the s-param eters for the D U T using
,
5 i l S 12
^21 ^22
720.
Sl l ‘
S'
21
7(0. + 0 9)
e
, y'( , +<M
0
512e
(4-6)
/
S
22
y'209
e
To determ ine the sm all-sig n al s-param eters o f the packaged devices, the m easured
data was post processed u sin g M ATLAB and reference planes w ere shifted using the sparam eter data o f the d um m y structure shown in Fig. 4-4. To verify the m easurem ents and
to validate the assum ption to treat the coplanar w aveguides designed and im plem ented
during this work as lossless, the s-param eters for three packaged devices m easured at
+25°C were com pared w ith the s-param eter data provided by the m anufacturer in the data
sheets. As shown in Fig. 4 -9 to Fig. 4-11, the s-param eters for the packaged devices m ea­
sured during this w ork agree very w ell with the s-p aram eter d ata provided by the m anufac­
turer. A s shown in Fig. 4-9 to Fig. 4-11, for the entire frequency range o f m easurem ents
i.e. 0.2 G H z to 2.0 G H z, th e agreem en t is excellent b oth for m agnitude and phase o f all
four tw o-port s-param eters fo r all three packaged devices m easured during this work. The
sm all-signal s-param eters fo r th e packaged devices w ere m easured at several different
tem peratures betw een -35 °C and +85°C . The m easurem ent results, as w ould be presented
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
86
0.2 GHz
Frequency
2.0 GHz
0 2 GHz
Frequency
2.0 GHz
S12
sn
Frequency
Frequency
Fig. 4-9. Com parison betw een the m easured s-param eters for M R F927 (solid line) and
data provided by the m an u factu rer (x x) at VCE = 3 V and IC = 3mA.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
87
„ - —
-
------
.
15
V
I
© •
20
©
i 25
2.0 GHz
i
Frequency
1
11
0 2 GHz
Frequency
0 2 GHz
2.0 GHz
S12
S ll
.
— ---------- - - -
—
\\
.•
i5
4
0 2 GHz
S21
*3
2
1
*'
Frequency
0 2 GHz
2 0 GHz
Frequency
2 0 GHz
S22
Fig. 4-10. C om parison betw een the m easured s-param eters for M RF947 (solid line) and
data provided by the m an u factu rer (x x) at V C E = IV and IC = Im A .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
88
25
0 2 GHz
Frequency
2.0 GHz
S ll
15
10
05
0 2 GHz
Frequency
2 0 GHz
0 2 GHz
Frequency
2 0 GHz
S 12
.5
'4
0 2 GHz
S21
20
'3
2
I
*
Frequency
2.0 GHz
S22
Fig. 4-11. C om parison b etw een the m easured s-param eters for M M BR941 (solid line)
and d ata pro v id ed by the m anufacturer (x x) at V C E = IV and IC = 1mA.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
89
in the follow ing sections, have been used to verify s-param eter predictions at different
tem peratures using closed-form expressions derived during this w ork. A lso, a qualitative
study o f therm al effects on sm all-signal s-param eters has been do n e using the m easured
data.
S-Param eter Prediction for Packaged Devices Using CIosed-Form Expressions
Derived from Intrinsic Device Model for Bipolar Transistors
It has been shown in C hapter 3 that sm all-signal s-param eters for bipolar junction
transistors can be calculated reliably using the closed-form expressions derived during this
w ork. T he calculated s-param eters w ere com pared against on -w afer m easurem ents at sev­
eral d ifferent bias conditions and excellent agreem ent betw een the two was observed
betw een 45 M H z to 20.045 G H z. T his section is intended to present a com parison
betw een sm all-signal s-param eters for the packaged devices m easured and the s-param e­
ters calcu lated using closed-form expressions.
To calculate sm all-signal s-param eters for the packaged devices, both the device
m odel (G um m el-P oon) param eters and package model provided by the m anufacturer were
used. S m all-signal s-param eters for the transistor without package w ere calculated using
the procedure outlined in C hapter 3. T he calculated s-param eters w ere em bedded as a data
set inside the package m odel and s-param eters w ere com puted fo r the entire tw o-port net­
w ork using M D S. This arrangem ent is show n in Fig. 4-12. The com parison betw een m ea­
surem ents and sm all-signal s-param eters for packaged transistor M R F927 predicted using
closed -fo rm expressions in conjunction w ith M D S sim ulation is show n in Fig. 4-13. As
show n in Fig. 4-13, the prediction o f sm all-signal s-param eters for the packaged device is
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
90
CCB
LBL
LBB
••
^
•
•
’ DATASET
Fig. 4-12. Schem atic for M DS sim ulation show ing bipolar ju n c tio n transistor em bedded
as a tw o-port s-param eter data set inside package m odel. DATASET shown is
com prised o f sm all-signal s-param eters calculated using closed-form expres­
sions.
quite different from the m easurem ent data. T his difference betw een the m easurem ent data
and predictions is neither a surprise, nor in any way underm ines the validity and useful­
ness o f the closed-form expressions derived and experim entally verified in C hapter 3. To
gain an understanding o f the discrepancies betw een m easured and calculated data, it is
appropriate to investigate the differences betw een the tw o sets o f s-param eters. A closer
look at Fig. 4-13 show s that the m agnitudes o f S ] | and S 22 are overestim ated, while for
S j 2 and S2 j, the situation is quite the opposite i.e. their m agnitude predicted by the closedform expressions are less than m easured. As regards to the phases, for both S 1( and S 22,
the phase angle predicted by the closed-form expressions is less than the phase measured.
For S j2, and S 2j, again the difference betw een th e m easurem ents and predictions is oppo-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
91
M easured
C alculated
05
C alcu lated
M easured
0.2 GHz
Frequency
0.2 GHz
2.0 GHz
Frequency
2.0 GHz
S 12
S ll
M easured
M easured
C alculated
C alculated
0 2 GHz
S21
Frequency
2.0 GHz
0 2 GHz
Frequency
2 0 GHz
S22
Fig. 4-13. C om parison o f sm all-signal s-param eters m easured and calcu lated using
closed-form expressions derived in C h ap ter using intrinsic device m odel for
bipolar ju n c tio n transistor.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
92
site i.e. their phase angles are m ore than the m easured phase at lower frequencies and as
the frequency increases, the calculated phase fo r both
and S 21 is less com pared to the
m easured phase.
As stated ea rlie r in C hapter 3, the clo sed -fo rm expressions were derived using only
the intrinsic G um m el-P oon m odel for b ip o lar ju n c tio n transistors. Also, the co m parison
presented in C h ap ter 3 betw een the m easured and calcu lated s-param eters pertained to
devices on-w afer. W h ere as the com parison betw een the predicted and m easured data
show n in this section is for the packaged devices, w here not only the active device on-chip
inside the package needs to be m odeled properly, but the package also requires an accurate
and reliable characterization. To account for ex cess phase and to better predict the m agni­
tude o f sm all-signal s-param eters, the intrinsic G um m el-P oon model has been m odified by
adding an extrinsic elem ent. D erivation o f clo sed -fo rm expressions for sm all-signal sparam eters for bip o lar ju n ctio n transistors using the m odified G um m el-Poon m odel is pre­
sented in the follow ing section.
Derivation of Improved Closed-Form Expressions for S-Parameters of Bipolar T ran­
sistors Using Modified Gummel-Poon Model
It is w ell know n that at high frequencies the m easured phase shift o f bipolar ju n c ­
tion transistors often exceeds the phase shift predicted by the hybrid-7t or T m odels
[Coo71, H ar72, M ac72, W ei78, Gou82, C h a8 5 , W ij88, A nt88, M as93]. T hese m odels
have been form ulated by lum ped representations o f som e physically distributed resistive
and capacitive effects. This results into a lum ped sets o f poles and zeros w ithin the m odel.
Several different ap proaches have been p roposed in literature to overcom e this d iscrep ­
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
93
ancy and m odel the excess phase needed to better m atch the m easured data. These sug­
gested approaches include (a) lum p partitioning o f the device plus addition o f extrinsic
elem ents [G ho65, Lan72, M ac72, 01172, C ha85], (b) addition o f tim e delays at base and
collecto r and one-section R C ladder netw ork to m odel base-em itter junction [Gou82], and
(c) addition o f extrinsic base to collector capacitance [W ij88, A nt88]. The approaches
m entioned in (a) and (b) em ploy m odel param eter o p tim ization and also em pirical extrac­
tion o f ratio factors based upon m easured data to achieve better m atch between calculated
and m easured s-param eter.
To account for excess phase and to better predict the m agnitude of sm all-signal sparam eters, the intrinsic G um m el-P oon model has been m odified as shown in Fig. 4-14.
T he m odified m odel is sim ilar to the sm all-signal m odel im plem ented in SPICE, HSPICE,
o r M D S w ith the exceptions that (a) rb, rc, and rc have been assum ed constant, and (b) col­
lector to substrate capacitance and other parasitic cap acitan ces have not been included.
T he elem ent (Cbcx) added to the intrinsic model is called extrinsic base to collector capac­
itance and it is betw een external base and internal collector. W ith the addition o f C bcx, the
base-collector capacitance is m odeled as a distributed capacitance. The base-collector
ju n c tio n capacitance (C jc)h as been divided into tw o parts; (a) capacitance between inter­
nal base node and collector (C^); and (b) capacitance betw een external base and collector
(Cbcx)- These tw o capacitances (C ^ and C bcx) are related to total base-collector capaci­
tance C jc by a m odel p aram eter XCJC. M odel p aram eter X CJC, w hich is between 0 and 1,
specifies the ratio o f partition betw een
and C bcx i.e. C |i = XCJC C jc represents a frac­
tion o f base-collector capacitance betw een internal base an d collector, while Cbcx = (1x CJc) C jC is part o f b ase-collector capacitance betw een external base and collector. W hen
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
94
^bcx
O c
b O
mv be
e
Fig. 4-14. M odified G um m el-P oon m odel for derivation o f closed-form expression o f
sm all-signal s-param eters for bipolar junction transistors.
m odel param eter XCJC is equal to 1, the entire base-collector cap acitan ce is m odeled
betw een internal base and collector, w hile for non-zero values o f XCJC, C bcx represents a
fraction o f CJC betw een external base and collector. For the packaged d ev ices M RF927,
MRJF947, and M M BR941 m easured during this work, XCJC has been rep o rted 0.2, 0.5,
and 0.3 respectively by the m anufacturer in data sheets.
To derive the closed-form expression for sm all-signal s-param eters, sam e proce­
dure was followed as described in C hapter 3. A fter having expressed the equivalent circuit
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95
in term s o f adm ittance, a sy stem o f follow ing nodal equations w ere w ritten
*'l = <-v6 + -v f c « ) Vl ~ > b V2 ~>'bcx V5
‘2 = ~ ? b V \ + (>'A + -v lt + V
<3
=
-f.yn + gm) V 2 + ^
‘V
V
V
(4-7)
V
'5
<4"8>
+ y e + >'o+ S , ^ V3 - > ' e V4 - > o V5
‘4 = ~ y e V 3 + y e V4
‘5 = - • V . r V'l +( - -V)li
- <-vo + *m >V3 +( - V t +>(i
( 4
‘ 9
)
i4 ' l0)
+ •vo + >c )V5- - ' A * 4 - 11>
‘6 = ~ y c V 5 + y c V 6
<4 - 12)
These nodal equations are sam e as presented in C hapter 3 ex cep t equations (4-7) and (411) w hich include the n ecessary term s for C bcx. A fter m atrix m anipulations and sim plifi­
cations to m athem atically rem ove the inaccessible nodes m entioned
in C h ap ter 3. the
im proved closed-form expressions presented below were obtained.
S J I = - ((((D E ra+ A s CK + D) r^ + A (E ra + I )) rK + (D ra + A) r^ + A ra) Z 2 + (((((G
s
+ B) s CK + (G s
+ s Cbcx rc) gm) ra + G F + s Cbcx rc - I)
+ G (E r a + I))
rK + ((G s C yi + B) ra + G) ril + G r0) ZQ + ((((-H s
- B) E re - (rc s
+ B) s CK
rb ' ( rb rc Sm + H) s
- 1 - s Cbcx rc) rQ -(H F + B ) r e - rc - rb rc F)
+ ((-H re - rb
rc) E - H) r0 -H re - rb rc) rK + (((-H s
- B) re - B rb - rb rc s C^) ra -H re - rh rc) r ^
- ( H re + rb rc) ra) / Denom
(4-13)
5 /2 = 2 Z 0 ((((D Ere + s2 rb Cbcx Crt + D) ra + (A s CK + D) re)
+ (Ere + 1) Ar0 + A re)
rK + ((D re + s Cbcx rb) ra + A re) r^ + A r e rQ) / Denom
(4-14)
5 2 / = 2 Z a ((((D Ere + s2 rb Cbcx CK + D - gm) ra + (A s CK + D) re) r^ +(Ere +1) Ar0 +
A re) rn + ((D re + s Cbcx rb) rQ + A re) r[i + A re ra) / Denom
(4-15)
522 = - ((((D E r Q+ A s C K + D) r^ + A (Er0 + /) ) rK+ (D r0 + A ) r il + A rQ) Z 2+ ((((-(G s
Cp + B) s CK - (G s
+ s Cbcx rc) gm) r0 - G F - s Cbcx rc + 1) r^ - G (E rQ + I )) rK -
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96
((G s
B) ra + G) r ^ -G ra) ZQ + ((((-H s
- B) Ere - (rc s + B) s CK rb - (rb
rc gm + H) s C^ - I - s Cbcx rc) rQ-(H F + B) re - rc - rb rc F)
+ ((-H re - rb rc) E H) rQ -H re - rb rc) rn + (((-H s C ^ - B) re - B rb - rb rc s C^) rQ -H re - rb rc)
+ rb rc) ra) / D enom
(4-16)
Where A = / + s Cbcx rb, B = I + s Cbcx rc, D = A s C\L + s Cbcx, E = s C k + gm, F = s CK
+ s C^, G = A rc - rb, H = A rc + rb, J = H s
+ B, and
Denom = ((((D E ra + A s CK +D) r^ + A(E rn + I )) rK + (D ra + A)
+ A rQ) ZQ2 + ((((2
D E re + (J + 2 rb s Cbcx) s CK + (H gm + 2A) j C ^ + 5 Cbcx(2 + rc g j ) ra + (2(A s
Q + D)) re + H F + B)
+ (2 A(Ere + I ) + H E) ra + A (2 re + rc) + rb) rK + ((2D
re + J + 2 r b s Cbcx) ra + A ( 2 r e + rc) + rb)
+ (A(2 re + rc) + rb) ra) ZQ + (((J(Ere
+ 1) + (rc s Cp + B) rb s CK + rb rc g^ s C^) ra + ( H F + B) re + (1 + rb F) rc) r^ +
((H re + rb rc) E + rb + A rc) ra + H re + rb rc) rK + ((J re + (rc s
+ B) rb) r„ + H
re + rb rc> r\x + (H re + rb rc) r0)
A com parison betw een the m easured d ata and th e s-param eters calculated using
the im proved ex p ressio n s at +25°C is presented in Fig. 4 -1 5 . and Fig. 4-16.
Also shown
in the figures are the p redictions o f m agnitude and phase using the closed-form expres­
sions derived from in trin sic G um m el-Poon m odel. F rom Fig. 4-15 and Fig. 4-16 it can be
seen clearly that overall situation both in term s o f m agnitude and phase has im proved
greatly w ith the ad d itio n o f extrinsic base to co llecto r capacitan ce. T he difference in m ag­
nitude o f S ] | betw een m easured and predicted using im proved m odel has reduced from 4
dB to less than 2 dB . A lso the m atch appears to be im prove as the frequency increases.
The difference in S i2 m agnitude between m easured and calcu la ted using intrinsic m odel is
about 5 dB, and this h as reduced significantly and erro r b etw een m easured and im proved
model is alm ost n eg ligible. F or m ost of the frequency ran g e, the e rro r between the m ea­
sured and predicted S 21 is less than a dB, how ever, at 2.0 G H z the e rro r increases to about
1.5 dB. The basic m o d el greatly overestim ated the m agnitude o f S22- but the agreem ent
between m easured an d im proved model is fairly good. A s for the phase o f sm all-signal s-
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97
o
12
intrinsic Model
Measured S
Improved Model
Improved Model
Intrinsic Model
Measured
-20
-32
....
0.2 GHz
Frequency
0.2 GHz
2.0 GHz
Frequency
2.0 GHz
S12 M agnitude (dB)
S 1 1 M agnitude (dB)
o
Intrinsic Model
Intrinsic Model
Measured
Improved Model
Improved Model
Measured
0
-10
0.2 GHz
Frequency
S21 M agnitude (dB)
2.0 GHz
0.2 GHz
Frequency
2.0 GHz
S22 M agnitude (dB)
Fig. 4-15. Com parison o f m agnitude o f s-param eters o f M R F927 m easured and calcu­
lated using the im proved m odel at +25°C , VCE = 3V and IC = 3m A . N ote
intrinsic model represents th e s-param eters calculated using expressions derived
in C hapter 3.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
98
85
o
Intrinsic Model
| Intrinsic Model
Measured
Improved Model
Measured
Improved Model
-120
35
.
0.2 GHz
Frequency
2.0 GHz
S 11 Phase (degree)
0.2 GHz
Frequency
2.0 GHz
S 12 Phase (degree)
o
180
----------—■
—■
Measured
Improved Model
Intrinsic Model
Intrinsic Model
Measured
Improved Model
60
0.2 GHz
Frequency
S 2 1 Phase (degree)
2.0 GHz
0.2 GHz
Frequency
2.0 GHz
S 22 Phase (degree)
Fig. 4-16. C om parison o f phase o f s-p aram eters o f M RF927 m easured an d calculated
using the improved m odel at +25°C , V C E = 3V and IC = 3m A . N ote intrinsic
m odel represents the s-param eters calculated using expressions derived in
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99
param eters, the e rro r betw een m easured an d calculated is no m ore than 8 to 10 degree at
any point w here as the basic m odel greatly underestim ated the phase.
T he com parison betw een the m easu red and calculated sm all-signal s-param eters
for M R F927 at -35°C and +85°C is p resented in Fig. 4-17 to Fig. 4-20. It can be seen from
the figures that the agreem ent betw een th e m easured and calculated sm all-signal s-param ­
eters is excellent over the entire frequency range at the tem perature extrem es i.e. -35°C
and +85°C . It has been shown that the sm all-signal s-param eters predicted using im proved
closed-form expressions derived in this ch ap ter agree very w ell w ith th e m easured data.
Thus, the utility o f the closed-form ex p ressio n s to predict sm all-signal s-param eters reli­
ably has been dem onstrated.
The erro r betw een m easured and calcu lated sm all-signal s-param eters even after
addition o f extrinsic base to collector cap acitan ce can be attributed to (a) inaccuracies in
extraction o f device model param eters in general and for XCJC in particular, (b) errors in
package m odel characterization, (c) process variations, and (d) neglecting parasitic capac­
itances during the course o f derivation o f closed-form expressions for sm all-signal sparam eters by using a relatively sim ple m odel for high-frequency application.
Thermal Effects on Small-Signal S-Parameters
As m entioned earlier, alm ost every transistor param eter is directly o r indirectly
effected by tem perature and sm all-signal s-param eters being functions o f these param eters
are definitely sensitive to tem perature as w ell. It has been show n in C h ap ter 3 that both
m agnitude and phase o f s-param eters ch an g e w ith the change in bias co n d itio n s mainly
the bias cu rren t—w hich itself is quite sensitive to tem perature.
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100
12
o
Measured
Improved Mode!
Improved Model
Measured
-20
-32
0.2 GHz
Frequency
0.2 GHz
2.0 GHz
Frequency
2.0 GHz
S I 2 M agnitude (d B )
S 1 1 M agnitude (dB )
Improved Model
Improved Model
Measured
Measured
-10
.'TV
0.2 GHz
Frequency
S21 M agnitude (dB )
2.0 GHz
0.2 GHz
Frequency
2.0 GHz
S22 M agnitude (dB)
Fig. 4-17. C om parison o f m agnitude o f s-param eters o f M RF927 measured and c alcu ­
lated using th e im proved m odel at -35°C , V C E = 3V and IC = 3m A.
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101
90
o
Measured
Improved Model
Measured
Improved Model
-1 8 0
0
.
0.2 GHz
Frequency
2.0 GHz
S 11 Phase (degree)
0.2 GHz
Frequency
2.0 GHz
S 12 Phase (degree)
o
180
Improved Model
Measured
Measured
Improved Model
0
0.2 GHz
Frequency
S21 Phase (degree)
2.0 GHz
0.2 GHz
Frequency
2.0 GHz
S22 Phase (degree)
Fig. 4-18. C om parison o f phase o f s-param eters o f M R F927 m easured and calculated
using the im proved m odel at -35°C, V C E = 3V and IC = 3m A.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
102
-12
o
Improved Model
Measured
Improved Model
Measured
-20
-32
0.2 GHz
Frequency
0.2 GHz
2.0 GHz
2.0 GHz
Frequency
S 12 M agnitude (dB)
S I 1 M agnitude (dB)
Measured
Improved Model
Improved Model
Measured
-10
0.2 GHz
Frequency
S21 M agnitude (dB)
2.0 GHz
0.2 GHz
Frequency
2.0 GHz
S22 M agnitude (dB)
Fig. 4-19. C om parison o f m agnitude o f s-param eters o f M R F927 m easured and calcu­
lated using the im proved m odel at +85°C , V C E = 3V and IC = 3m A .
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103
Improved Model
Measured
Measured
Improved Model
0
0.2 GHz
Frequency
2.0 GHz
S 11 Phase (degree)
I
0.2 GHz
I
I
I
■
Frequency
2.0 GHz
S12 Phase (degree)
o
180"
Measured
Improved Model
Improved Model
Measured
-90
0.2 GHz
Frequency
S21 Phase (degree)
2.0 GHz
0.2 GHz
Frequency
2.0 GHz
S22 Phase (degree)
Fig. 4 -20. C om parison o f phase o f s-param eters o f M R F927 m easured and calcu lated
using the im proved model at +85°C . V C E = 3V and IC = 3mA.
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104
T his section is intended to qualitatively d escrib e therm al effects on sm all-sig n al sparam eters. T his study is based on s-param eter m easurem ents o f packaged dev ices m ea­
sured at different am b ien t tem peratures. As show n in Fig. 4-21 and Fig. 4 -2 2 , all four sparam eters ex hibit noticeable changes due to tem perature variations. T he m ag nitude o f
both input and o u tp u t reflection coefficients (S j i an d S 22 ) and reverse tran sm issio n coeffi­
cient S p increases w ith the increase in tem perature. W hile the m agnitude o f So 1 decreases
w ith the increase in tem perature. As regards to the phase angles o f sm all-signal s-param eters, for both input and output reflection coefficients ( S p and S 22 ). it d ecreases w ith the
increase in tem perature, w here as phase angle fo r S ]2 and S 21 increases w ith th e increase
in tem perature.
Sum m ary
In this chapter, a procedure to m easure sm all-signal s-param eters fo r packaged
b ipolar ju n ctio n transistors has been described. T he devices were m ounted on specially
designed boards and the m easurem ent were p erfo rm ed by placing the boards inside a tem ­
perature controller. D ifferent phases o f w ork have been described in detail and criteria for
selection o f active devices, base m aterial (lam inate) for the board, pad lay o u t, and tran s­
m ission m edia has been outlined. W ith careful d esig n and selection o f tran sm issio n lines,
an alm ost lossless an d therm ally stable co p lan ar w aveguide has been im plem en ted. T he
m easurem ents have been com pared w ith M D S sim ulations over a frequency range o f 0.2
G H z to 2.0 G H z an d excellent agreem ent has been presented. A detailed d escrip tio n o f
shifting the referen ce planes to extract sm all-sig n al s-param eters o f p ack ag ed devices
m ounted on boards has been given. The m easu red s-param eters have b een co m p ared
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105
-14
+ 25 C
+ 85 C
- 35 C
-44
2.0 GHz
0 2 GHz
S l l Magnitude
0 2 GHz
Frequency
S12 Magnitude
(dB)
20
2.0 GHz
(dB)
o
+ 85°C
- 35°C
+ 25°C
+ 85°C
-10
0 2 GHz
S21 Magnitude
Frequency
(dB]
2.0 GHz
0.2 GHz
Frequency
S22 Magnitude
2.0 GHz
(dB)
Fig. 4-21. C o m p ariso n o f magnitude o f s-param eters o f M R F927 at -35°C, +25°C , and
+ 8 5°C . M easurem ents were m ade at V C E = 3V and IC = 3mA.
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106
o
80
+ 25°C
- 35°C
- 35°C
-120--------------------- -i-------- -----------0 2 GHz
SllPhase
Frequency
30
0 2 GHz
2.0 GHz
S12 Phase
(degree)
L_l
Frequency
1 0 GHz
(degree)
0
165
- 35°C
yj
65
i
0 2 GHz
in
i
Frequency
S21 Phase (degree)
2.0 GHz
0.2 G Hz
S22 Phase
Frequency
2.0 GHz
(degree)
Fig. 4-22. C om parison o f phase o f s-param eters o f M R F 927 at -35°C, +25°C , and
+85°C . M easurem ents were made at V C E = 3V and IC = 3mA.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
107
against the data pro v id ed by the m anufacturer an d ex cellen t m atch has been achieved. It
has been shown that intrinsic G um m el-Poon m odel for bip o lar transistors does not esti­
mate m agnitude and p h ase o f s-param eters o f the dev ice correctly. A b rie f survey o f earlier
approaches to overcom e this problem and better m odel an d sim ulate the B JTs has been
included. T he m odel has been im proved by adding ex tern al base to co llecto r capacitance
and it has been show n th at im proved m odel not o n ly accounts for excess phase but it also
im proves greatly the p red ictio n o f m agnitudes for all fo u r sm all-signal s-param eters. It has
been shown that for dev ices having X ^ jc equal to unity, the im proved m odel w ould reduce
to the basic m odel an d the s-param eter ex p ressio n s d erived in C hapter 3 w ould be suffi­
cient for predicting s-p aram eter reliably. B ased upo n th erm al m easurem ents, a b rie f q u ali­
tative study o f therm al effects on m agnitude and p h ase o f s-param eters has been presented.
The w ork p resen ted in the next ch ap ter in clu d es the sensitivity analysis o f sm allsignal s-param eters o f b ip o lar junction tran sisto rs w ith respect to the device m odel
param eters. The device m odel param eters w hich play do m in an t role in causing variations
in s-param eters have b een identified. Also in clu d ed in the next chapter is a co m p arison o f
sensitivities o f sm all-signal s-param eters w ith resp ect to each device m odel param eter. A
system atic approach is presented to establish the link betw een the changes in m odel
param eters caused by eith er variations in bias co n d itio n s o r tem perature and their
corresponding effects on sm all-signal s-param eters.
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C H A PT E R 5
SEN SITIV ITY A N A LY SIS O F S M A L L -S IG N A L S-PARA M ETERS
With the advancem ents in p lan ar silicon technology, microwave planar circuits
have found applications in a w id e variety o f fields. A m o n g m any environm ental variables,
tem perature neither rem ains th e sam e in different fields o f applications, nor rem ains
constan t during the use o f any m icrow ave circuit in any one application. A s stated earlier,
s-param eters are used alm o st universally for the design o f sm all-signal am plifiers
em ployed in m icrowave system s. It has been show n in C hapters 3 and 4 that both the
m agnitude and phase o f sm all-sig n al s-param eters o f the bipolar ju n ction transistors
change with the variations in bias conditions an d tem perature. T hese ch anges in
m agnitude and phase o f sm all-sig n al s-param eters are due to variations in device m odel
param eters caused by changes in bias conditions and tem perature. For reliable d esigns, it
is essential to know in advance the effects o f v ariatio n s in device model param eters on
sm all-signal s-param eters and th e ir corresponding im plications on the perform ance o f the
microwave circuits. To acquire this inform ation, it is n ecessary to gain an insight into the
active device itself to identify the sources which cau se variations in s-param eters.
This chapter presents the sensitivity analysis o f sm all-signal s-param eters o f
bip o lar junction transistors w ith resp ect to the device m odel param eters. T he device m odel
param eters w hich play d o m in an t role in causing variations in s-param eters have been
identified. A lso included in th is ch a p te r is a co m p ariso n o f sensitivities o f sm all-signal sparam eters w ith respect to eac h device m odel param eter. A system atic approach is
108
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109
presented to establish the lin k betw een the changes in m odel param eters caused by either
variations in bias conditions o r tem perature and their co rresp o n d in g effects on sm allsignal s-param eters.
Sensitivity: Concept, Importance, and Definition
It is w ell know n that a change in the value o f a circuit co m p o n en t alters the circuit
perform ance in som e way. T he am ount by w hich a given change in a com ponent value
alters the circuit perform ance is a m easure o f the sensitivity o f th e circuit. The im portance
o f sensitivity analysis in electrical circuits can be described by considering a practical
situation in w hich the co m p o n en t values can be predicted to vary w ithin certain limits as a
function o f tim e o r environm ent. T his leads to the w orst case o r determ inistic problem ,
w here values o f circuit co m p o n en ts such as resistors and cap acito rs are known to change
w ith tim e and tem perature. In such a case, it is required to know in advance how the circuit
w ould perform as the co m p o n en t value changes. This in form ation pertaining to circuit
perform ance in advance can be used to either com pare alternative designs or to optim ize
the circuit.
Sensitivity in the sim p lest form has been defined as the partial derivative o f a
differentiable function F w ith respect to any p aram eter .r, [C hu75, Tem77, G up81,
D ob91]. M athem atically it ca n be expressed as:
D X ( F) =
7\F
(5-1)
This definition is convenient and useful for com puter applications, but it is not
independent o f the scale. T h is is also referred to as u n-norm alized sensitivity [Chu75,
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110
G up81]. A more form al and w idely u sed definition is the fractional change in function F
resulting due to a fractional ch an g e in any p aram eter .r, [Chu75, Fid78, G u p 8 l, D ob91].
T his can be expressed m athem atically as:
s /
xi
=
Hm
A-X^O ( A x ^ / i X ; )
(5-2)
T his is term ed as the increm ental sensitivity o f the function F to a change in param eter .r,-.
For a sufficiently sm all in crem en t in p aram eter .t(- (Ax,- approaching zero), the change in F
i.e. AF can be approxim ated as:
dF
AF = S— A r.
Bx i '
(5-3)
Substituting equation (5-3) into equation (5-2) results into more com m only used
expression for sensitivity.
SF
-n
= X
-1*L
F dx .i
T he expression in equation (5-4) is know n as the differential
w ith respect to param eter .r,-.
(5-4)
(
}
sensitivity o f the function F
T h is expression for sensitivity is more often term ed as the
norm alized sensitivity. B esides the above expressions for sensitivity, several other
equivalent expressions have also been reported in the literature [Chu75, Fid78, D ob91].
In RF/M icrow ave circu its, the function w hose sensitivity needs to be determ ined is
usually a com plex function, w h ich can be represented in the form
F = | F \eJi>
(5-5)
Taking natural logarithm o f b o th sides yields
I n F = ln |F | + y<j)
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(5-6)
Ill
D ifferentiating w ith respect to a p a ra m e te r .rf- results into
1 9F
1 8 |
. 9<(>
F a ^ = I f i a ^ |F |+ ^
< 5 ' 7 )
To obtain the norm alized sensitivity, w e need to m ultiply both sides o f equation (5-7) by
.r,. T he resulting com plex eq u atio n can be sp lit into two parts and b y equating real and
im aginary parts on both sides we g e t tw o equations as under:
(5-8)
< 5 ' 9 )
W here Re and Im represent the real and im aginary parts. The eq u atio n s (5-8) and (5-9)
provide the useful relationships to d eterm in e the sensitivities o f the m agnitude and phase
values o f the com plex function w ith resp ect to param eter .t,-. In the follow ing sections,
these relationships have been u sed to d eterm in e the sensitivity o f b oth m agnitude and
phase o f sm all-signal s-p aram eters w ith resp ect to device m odel param eters.
Sensitivity Analysis of Small-Signal S-Parameters of BJTs
To identify the sources w h ich cau se variations in both m agnitude and phase o f
sm all-signal s-param eters w ith c h an g e s in bias and tem perature, the sensitivity o f sparam eters w ith respect to d ev ice m odel param eters needs to be determ ined. As
m entio n ed earlier, these active d ev ice m odel param eters depend on b ias and tem perature
and exhibit noticeable ch an g es in th e ir values w ith the changes in bias and operating
tem perature. In C hapters 3 and 4 , the closed-form expressions fo r sm all-signal s-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
112
param eters for BJTs have been derived. T hese closed-form expressions relate the sparam eters w ith the sm all-signal active device m odel param eters. To gain an insight into
the active device itself, the sensitivity o f each sm all-signal s-param eter w ith respect to
sm all-signal model param eters sh o u ld be ascertained. The expressions to determ ine the
sensitivity o f m agnitude and phase o f sm all-signal s-param eter can be stated as:
(5-10)
(5-11)
W here Sij represents the sm all-signal s-param eters o f BJT and Pi denotes the sm all-signal
m odel param eters. For this w ork, six sm all-signal model param eters
C^, C bcx, rK, rQ,
and g m have been chosen w ith resp ect to w hich the sensitivity analysis o f s-param eters has
been perform ed. The dependence o f these m odel param eters on bias and tem perature is
w ell know n. The bias dependencies o f sm all-signal active device m odel param eters have
also been included in A ppendix B.
T he following procedure has been form ulated and im plem ented to determ ine the
sensitivity o f m agnitude and phase o f s-param eters w ith respect to m odel param eters: (1)
evaluate the closed-form expressions for the s-param eter(s) w hose sensitivity needs to be
determ ined at the desired frequency; (2) determ ine the partial derivative(s) o f the closedform expression(s) with respect to the m odel param eter(s) o f interest; (3) evaluate the
partial derivative(s) at the desired frequency; (4) evaluate the expressions in equations (510) and (5-11) to determ ine the sensitivity o f m agnitude and phase w ith respect to the
m odel param eter(s).
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113
To dem onstrate the sensitivity o f m agnitude and phase o f each sm all-signal sparam eter w ith respect to CK, C^, Cbcx, rK, rQ, and gm, the norm alized sensitivities have
been plotted in Fig. 5-1 to Fig. 5-4. T he active device used here as an exam ple is M RF927
and the values o f sm all-signal m odel param eters have been calculated at bias condition
VCC = 3V and IC = 3m A using H SPIC E . The sm all-signal s-param eters have been
calculated using the im proved clo sed -fo rm expressions presented in C h ap ter 4. The
norm alized sensitivity expressions p resen ted in equations (5-10) and ( 5 - 1 1) have been
evaluated for a frequency range o f 0.1 G H z to 5 G H z using M ATLAB. As show n in Fig. 51 (a), the sm all-signal param eters w ith respect to w hich the m agnitude o f S n is more
sensitive are C^,
C bcx, and gm. T h e resistive param eters r^ and r0 have virtually no
effect on the m agnitude o f Sj t . A m ong the dom inant sm all-signal m odel param eters. C bcx,
Cjp and gm effect the m agnitude o f S | ] m ore than the rest o f the m odel param eters. The
phase o f S (! at low er frequencies exhibits m ore sensitivity to
and C bcx. As the
frequency goes higher the sensitivity o f phase o f S [ j to sm all-signal m odel param eters
reduces significantly. The m agnitude o f S 12 is m ost sensitive to C bcx as show n in Fig. 5-2
(a), w hile the phase o f S )2 is m ore sensitive to CK. O ther sm all-signal m odel param eters
except rrt, rG also have noticeable effect on both m agnitude and phase o f S 12. Sim ilar to
S n , both m agnitude and phase o f S q show negligible sensitivity to rrt, r0. As shown in
Fig. 5-3 (a) and Fig. 5-3 (b) the m agnitude and phase o f S21 is m ost sensitive to gm, C^,
and C bcx, w hile the fractional change in both m agnitude and phase o f S 2] due to fractional
change in the resistive m odel p aram eters rrt and rQ is negligible. T he m agnitude and phase
o f S22 as presented in Fig. 5-4 (a) and Fig. 5-4 (b) show more sensitivity w ith respect to
C n, C^, C bcx, and g m com pared to the resistive sm all-signal m odel param eters.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
114
0.3
O
-o
1 0 .2
eo
rs
2
bcx
9m
o
>
0 .0
(a)
& -0.1
O
N
T3
1 -0 .2
-0.3
Frequency (G H z)
0.5
3
bcx
0.4
9m
o
I
0 .2
s
E
o
Z
0 .0
(b)
-0 .2
F req u en cy (G H z)
Fig. 5-1. N o rm alized sensitivity o f S j t w ith resp ect to sm all-signal m odel param eters Cjt
C^, C bcx, r ^ rQ, and gm. (a) S ensitivity o f m agnitude value o f S j [; (b) Sensitivity
o f Phase value o f S j j .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
115
1.0
o
~o
3
'c
BO
3
oi
0 .5
t/T
o
’bcx
’>
7
c
o
00 0 .0
-o
<u
(a)
9m
_N
*3
C
i_
o
Z
-0.5
F requency (G H z)
0 .4
o
’bcx
>
9m
(b)
o
Z
-0 .2
F requency (G H z)
Fig. 5-2. N orm alized sensitivity o f S (2 w ith respect to sm all-signal m odel param eters C n
C H, C(jCX, r rt, r0, and g m. (a) Sensitivity o f m agnitude value o f S ^ ; (b) S ensitivity
o f Phase value o f S )2.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
116
•o
3
c
so
3
0.5
ol
CO
o
0 .0
(a)
-J'.
c
o
CO
T3
u -0.5
"3
E
c
o
bcx
z
-1 .0
9m
1
0
2
3
Frequency (G H z)
4
5
0 .2
•bcx
9m
0 .0
>
I -0 .2
O
Z
-0.3
Frequency (G H z)
Fig. 5-3. N orm alized sensitivity o f S 21 with respect to sm all-signal model param eters C n
CH, Cbcx, r^, rQ, and gm. (a) Sensitivity o f m agnitude value o f S21; (b) Sensitivity
o f Phase value o f S 2i-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
117
0
"O
'c
CD
ca
2 0 .5
CM
CM
bcx
9m
CO
O
f * 0 .0
CO
c
0
CO
0 -0 .5
N
-1 .0
Frequency (GHz)
o
:!\r. o.o
C
1)
C/5
•o
oN
^ -0 .5
E
o
bcx
Z
-1 .0
Frequency (GHz)
Fig. 5-4. N orm alized sensitivity o f S 22 w ith respect to sm all-signal m odel param eters Cn
C^, C bcx, rn, rQ, and gm. (a) Sensitivity o f m agnitude value o f S 22 - (b) Sensitivity
o f Phase value o f S 22 -
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118
The norm alized sensitivities o f sm all-sig n al s-param eters with respect to sm allsignal m odel param eters o f M RF927 presented in Fig. 5 - 1 to Fig. 5-4 provide a valuable
insight into the active device. However, in o rd er to estab lish the trend o f the sensitivities o f
m agnitudes and phase angles o f sm all-signal s-param eters with respect to m odel
param eters, the sensitivity analysis for three d ifferen t discrete devices M R F927, M R F947,
and M M B R 941 has been perform ed. T he results have been presented in Fig. 5-5 to Fig. 510. T h e sensitivities o f m agnitudes an d p h ases o f all sm all-signal s-param eters w ith
respect to the sm all-signal model param eters have show n sim ilar trends for three devices
analyzed during this work. The m agnitudes o f S | j and S 22 show a positive sensitivity to
C n i.e. for a fractional increase in the value o f C n, the m agnitude o f S n an d S22 also
increases. T he sensitivity o f phase o f S lt is h ig h er at low er frequencies an d as the
frequency increases the S t ( phase becom es less sensitive to the changes in the value o f C K.
A fractional increase in the value o f CK causes a decrease in the m agnitudes o f S j2 and S 2J
thereby m aking the sensitivities o f m agnitudes o f S 12 and S 2t negative. The phase o f S J2
has a positive sensitivity to C^, w here as the p h ase o f both S21 and S22 ex h ib it negative
sensitivity to C n at low er frequencies, but as the frequency increases, the sensitivity o f
phase o f S2J becom es independent o f frequency w hile for S22, the phase value starts to
increase.
As show n in Fig. 5-6, a fractional in crease in
causes a decrease in the
m agnitudes o f all four sm all-signal s-param eters. A m ong the four m agnitudes, the least
sensitive to changes in
is
m agnitude. A s regards to the phase sensitivity o f the
sm all-signal s-param eters, S 22 phase is the m o st sensitive to the changes in C ^, w hile S 2[
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
119
ft
040
MMBR941
MRF947
MRF927
0.30
• - » MMBR941
• - « MRF947
MRF927
K
06
|
020
a o io
04
000
02
5 *010
-0 20
4
00
5
0
4
Frequency (GHzI
Frequency (GHz)
00
-0 1
MMBR94I
m-m MRF947
MRF927
MMBR94I
MRF947
MRF927
09
07
05
sc
Jr
-0 2
03
-0
3
-0 1
>»
=
-04
-03
-05
-05,
0
4
Frequency (GHz)
Frequency (GHz)
00
K
~ mmbr^tt
•—» MMBR941
MRF947
MRF927
»-■ MRF947
MRF927
* -02
3
SC
i
*0 4
>»
•06
0
*>
4
5
0
1
Frequency (GHz)
0.8
K
06
2
3
Frequency (GHz)
00
MMBR941
MRF947
MRF927
4
5
« -» MMBR94I
MRF947
MRF927
K
m
>J -02
-0.4
04
3
■X
s
>•*
0 2
.0.6
>.
00
-0.2
0
4
5
0
Frequency (GHz)
■>
4
5
Frequency (GHz)
Fig. 5-5. N o rm alized sensitivity o f sm all-signal s-param eters w ith respect to Cn for three
d ifferen t devices M R F927, M R F947, an d M M B R 941.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
120
000
04
MMBR94I
MRF947
MRF927
w
MMBR941
MRF947
MRF927
-002
-0 06
-0 1
0
-0 08
Frequency (GHz)
00
4
5
4
5
Frequency (GHz)
00
• - » MMBR94I
MRF947
MRF927
01
3
•02
5 -0 6
•si
-04
0
■»
-08
4
♦-* MMBR941
• - « MRF947
* - • MRF927
0
Frequency (GHz)
00
\
Frequency (GHz)
000
MMBR94I
MRF947
» - • MRF927
m
^
■01
-0 02
MMBR941
MRF947
MRF927
•0 04
5 -0 2
•0.06
•03
-0.4
•0 08
0
3
-0 10
4
5
0
0.0
♦-* MMBrtiuT
5
«-» MMBR94I
m-m MRF947
MRF927
09
• - « MRF947
MRF927
Z -0 1
4
Frequency (GHz)
Frequency (GHz)
0.8
S 07
■02
06
0.4
i*-0.3
=
03
■04
-0.5
0
3
4
5
00
0
Frequency (GHz)
4
5
Frequency (GHz)
Fig. 5-6. N orm alized sensitivity o f sm all-signal s-param eters w ith respect to
different devices M R F 927, M RF947, and M M B R941.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
for three
121
02
04
MMBR94I
MRF947
MRF927 .
MMBR94I
MRF947
MRF927
- 03
- 00
00
•0 1
Frequency (GHz)
Frequency (GHz)
• - » MMBR94I
MRF947
MRF927
- OS
• - * V1MBR941
MRF947
MRF927
X.
06
00
=« 00
0
■>
-02
4
5
Frequency (GHz)
Frequency (GHz)
00
000
• - » MMBR941
MRF947
MRF927
^
-0 05
« -* MMBR941
»-■ MRF947
MRF927
-0 1
i -0 1 0
-0 2
-0 15
•0 20
>• -03
■025
0
00
3
Frequency (GHz)
4
5
•0 30
Frequency (GHz)
• - » MMBR94I
MRF947
MRF927
-o 0.8
-02
• MMBR941
MRF947
MRF927
06
5 -04
04
0.2
0
4
Frequency (GHz)
5
0.0
0
4
5
Frequency (GHz)
Fig. 5-7. N orm alized sensitivity o f sm all-signal s-param eters w ith respect to Ct,cx for
three different devices M R F927, M R F947, and M M B R 941.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
122
o 10
008
008
•—» SIMBR941
» - • MRR47
* - • MRR27
♦—» MMBR94I
m
MRF947
MRF927
K
006
006
004
^ 0 04
s 0 02
002
S 000
000
-
0.02
0
4
5
4
5
4
5
Frequency (GHz)
Frequency (GHz)
000
006
MMBR941
MRF947
MRF927
004
•e
-0.05
i
rt
002
000
■0 15
• - » MN1BR941
M RR47
MRR27
•002
z
-0 04
0
•>
4
5
-0 20
Frequency (GHz)
Frequency (GHz)
0 10
008
0000
MMBR94I
* -« MRF947
MRF927
MMBR941
M R R47
M R R27
-0 005
0.06
•OOIO
ue‘004
■0 015
002
OOO
0
-0 0 2 0
T
4
5
0
Frequency (GHz)
Frequency (GHz)
0000
006
MMBR941
MRF947
MRF927
-OOIO
• - » MMBR94I
• - « M RR47
M RR27
00 4
■0020
3 0 02
=« -0 030
0.00
•002
-0 050
0
-0 04
3
Frequency (GHz)
4
5
0
Frequency (GHz)
Fig. 5-8. N orm alized sensitivity o f sm all-signal s-param eters w ith resp ect to rn for three
different devices M R F 927, M R F 947, and M M BR941.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
123
00010
0004
« -* MMBR941
• - « MRF947
* - • MRF927
00008
00006
•—» MMBR94I
MRF947
MRF927
0003
00002
00000
0 002
S -0 0002
-0 0004
0 001
-0.0006
-0 0008
-00010
0
0010
I
■>
3
Frequency (GHz)
0000
4
0
5
■»
4
5
4
5
4
5
Frequency (GHz)
OOIO
N1MBR941
MRF947
MRF927
MMBR941
MRF947
MRF927
0008
-
s
0 005
0006
2 0000
■S 0 004
>
0002
0000
-0 005
-OOIO
0
4
5
Frequency (GHz)
■»
Frequency (GHz)
00000
OOIO
• MMBR941
—■ MRF947
* - • MRF927
0008
0
t -0 0005
ri
0006
■r' 0004
-0 0010
« - » MMBR941
m-m MRF947
MRF927
0002
OOOO
0
-0 0015
■>
4
0
5
Frequency (GHz)
3
Frequency (GHz)
0 005
0025
•— MMBR941
• - « MRF947
MRF927
MMBR94I
MRF947
MRF927
0003
0020
0001
2 0 015
-0001
= 0010
S -0.003
it
y:
0005
0
-0.005
3
Frequency (GHz)
4
5
0
i
3
Frequency (GHz)
Fig. 5-9. N orm alized sensitivity o f sm all-signal s-param eters w ith respect to r0 for three
different devices M R F 927, M R F 947, and M M B R 941.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
124
02
♦ -* MMBR94I
MR PM 7
MRF927
3
« -* MMBR94I
MRF947
• - « MRF927
V
•0 1
00
-0 2
-04
0
-t
4
5
Frequency (GHz)
Frequency (GHz)
04
M.MBR94I
MRF947
MRF927
« - • MMBR94I
MRF947
• - « MRF927
03
02
•J
^ 00
|-0 2
= -0.3
y*. -0.3
-04
0
-0 4
■>
3
Frequency (GHz)
4
5
Frequency (GHz)
02
• MMBR941
MRF947
MRF927
MMBR941
MRF947
MRF927
so
09
s
-3
08
-00
= 06
•Jr.
05
0
-02
->
4
5
Frequency (GHz)
0
•>
00
*-* MMBR94I
» - • MRF947
MRF927
-02
4
5
Frequency (GHz)
MMBR94I
MRF947
MRF927
08
| 06
-04
04
•08
0
■>
3
Frequency (GHz)
4
5
00
0
■>
4
5
Frequency (GHz)
Fig. 5-10.N orm alized sensitivity o f sm all-signal s-param eters with respect to g m fo r three
different devices M R F927, M R F947, and M M B R 941.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
125
phase is the least sensitive. A fractional increase in C bcx causes an increase in m agnitudes
o f S 11 and S 12, w here as the m agnitudes o f S 2I and S22 exhibit a negative sensitivity to
C bcx. T he phase o f S j ( and S22 show an increase w ith the increase in C bcx, while for S 12
and S2l the sensitivity o f th eir phases to C bcx is negative.
As stated earlier, changes in both r^ and rQ do not cause significant changes in
m agnitude or phase o f any sm all-signal param eter. S im ilar trend has been observed for all
three devices as show n in Fig. 5-8 and Fig. 5-9. T he sensitivities o f sm all-signal sparam eters with respect to gm are show n in Fig. 5-10. A fractional increase in gm causes a
fractional decrease in both m agnitude and phase o f S j j and S 12, while S21 shows a
positive sensitive both for its m agnitude and phase. As regards to the effect o f an increase
in g m on S22, its m agnitude decreases, w hile its phase show s a positive sensitivity.
It has been show n that am ong the sm all-signal m odel param eters chosen during
this w ork with respect to w hich the sensitivity analysis has been perform ed for three
different devices, sm all-signal s-param eters are m ost sensitive to C^, C^, C bcx. and g m.
A lm ost all the sm all-signal s-param eters have exhibited sensitivity to these four m odel
param eters. To gain fu rth er insight into this m atter and to determ ine a relative rank am ong
CK, C^, C bcx, and gm, a com parison o f sensitivities o f sm all-signal s-param eters to these
model param eters is presented for M R F927 in Fig. 5-11 to Fig. 5-14. As shown in Fig. 511, the magnitudes o f S21 and S22 are the m ost sensitive to CK. At 5 G H z for a 10%
increase in the value o f C^, the m agnitudes o f Sj j , S 12, S2i, and S22 will change
approxim ately 2% , 0 % , -5% , and 6% respectively. A s regards to the phase, the least
sensitive among the
sm all-signal s-param eters
is Sj j ,
w hile S |2 and S22 show
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
126
S11m
S12m
S21m
S22m
u
0 .5
2? 0.0
"o
:1
x -0 .5
C
o
C/2
F requency (GHz)
S lip
S12p
S21p
S22p
V
0 .5
o
>
C -0 .5
(U
c/5
-
1.0
F requency (GHz)
Fig. 5-11 .C om parison o f sensitivities o f sm all-signal s-param eters w ith resp ect to
M RF927. (a) Sensitivity o f m agnitudes; (b) Sensitivity o f phase.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
for
127
0 .0 0
S11m
S12m
S21m
S22m
u -0 .0 5
cu -0.10
a -0.15
-
0.20
F requency (G Hz)
0 .4
S11p
S12p
S21p
S22p
0 .3
0.2
O
>
■j-.
I
0.0
-
0.1
F requency (G Hz)
Fig. 5-12.C om parison o f sensitivities o f sm all-signal s-param eters w ith respect to
M R F927. (a) Sensitivity o f m agnitudes; (b) Sensitivity o f phase.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
for
128
1.0
00
•- 0.5
5
o
TJ
3
S11m
S12m
S21m
S22m
I 0.0
o
>
-0.5
F requency (G H z)
0.8
s 0.6
S lip
S 12p
S 21p
S 22p
JS
cu
o
>, 0.2
>
-
0.2
F requency (G H z)
Fig. 5-14.C om parison o f sensitivities o f sm all-signal s-param eters w ith respect to g m for
M R F 927. (a) Sensitivity o f m agnitudes; (b) Sensitivity o f phase.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
129
0
S11m
S12m
S21m
S22m
-C
U
0.5
u
TD
(a)
0.0
3
•_
O
-0.5
c
u
on
-
1.0
0.9
0
?
3
Frequency (G H z)
• —•
■— ■
4— ♦
k — k
4
S11P
S12p
S 2 lp
S22p
0.7
•C
u
0.5
£
:3 0.3
ra,
(b)
o
>
e
1)
CO
-
0.1
-0.3
-0.5
4
5
Frequency (G H z)
Fig. 5-13.C om parison o f sensitivities o f sm all-signal s-param eters w ith respect to Ct,cx
for M R F927. (a) Sensitivity o f m agnitudes; (b) Sensitivity o f phase.
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130
approxim ately 3.5% an d -5% change at 5 G H z for 10% change in
sensitivity o f sm all-signal s-param eters to
respectively. The
p resen ted in Fig. 5-12 show s that both
m agnitude and phase o f S 22 are the m ost sensitive. T h e m agnitude o f S22 decreases w ith
the fractional increase in the value o f
w hile its p h ase yields a positive sensitivity to an
increase in C Jl.
It can be seen from Fig. 5-13 that a fractional increase in C bcx causes an increase in
m agnitudes o f S n and S J2, w here as the m agnitudes o f S2i and S22 exhibit a negative
sensitivity to Cbcx. T he phase o f S n and S22 show an increase w ith the increase in C bcx.
w hile for S i2 and S21 the sensitivity o f their phases to C bcx is negative. The sensitivity o f
sm all-signal s-param eters w ith respect to g m is p resen ted in Fig. 5-14. It can be seen that
S 21 and
S-n
are
m ore
sensitive
to changes
in
the
value
o f the
sm all-signal
transconductance than the o th er tw o s-param eters. B oth m agnitude and phase o f S21
increase w ith an increase in the value o f g m. For S 22. the m agnitude reduces significantly
due to in increase in g m, w hile its phase show s a positive sensitivity to g m. T he other
sm all-signal s-param eters S n and S 12 also show sen sitiv ity to gm but that is significantly
less com pared to S 2J and S 22.
The results o f the sensitivity analysis presen ted so far have been sum m arized in
tabulated form an d p resented in Table 5-1 and Table 5 -2. T he sensitivity o f sm all-signal sparam eters w ith respect to sm all-signal m odel param eters at I G H z are given in Table 5-1,
w hile the sensitivity o f sm all-signal s-param eters at 5 G H z has been sum m arized in Table
5-2. The percentage change in m agnitude and phase o f each sm all-signal s-param eter has
been calculated for 10% change in sm all-signal m odel p aram eters Crt, C^, C bcx, and g m.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
131
The characteristics o f sm all-signal s-param eters (m agnitude o r phase) w hich have show n a
change o f m ore than 2% due to 10% change in m odel param eters have been underlined to
facilitate a com parison o f sensitivities at a glance.
Table 5-1. Sensitivity o f Sm all-Signal S-Param eters to Sm all-Signal M odel Param eters
P ercentage C hange in Sm all-Signal S-Param eters o f M R F927 at 1 G H z
w ith 10% C hange in Sm all-S ignal M odel Param eters.
M odel
Param eter
S 12
S 11
S 2l
S 22
M ag.
Phase
M ag.
Phase
M ag.
Phase
M ag.
Phase
c*
-1.35
3.50
-2.68
-1.04
-3.19
-1.41
-0.76
-4.59
Cn
-0.57
0.56
-0.66
-0.73
-0.66
-0.31
-0.83
2.41
Cbcx
-0.54
3.07
6.44
-1.60
-1.64
-0.91
-2.18
6.72
a
Dm
-0.83
-0.48
-0.59
-1.00
6.90
-0.08
-1.88
6.52
Table 5-2. Sensitivity o f Sm all-Signal S-Param eters to Sm all-Signal M odel P aram eters
Percentage C hange in Sm all-Signal S -P aram eters o f M R F927 at 5 G H z
w ith 10% C hange in Sm all-S ignal M odel Param eters.
M odel
Param eter
S 12
s li
M ag.
Phase
M ag.
Phase
3.67
-4 .9 9 H
-1.57
5.66
-4.79
-1.20
-0.67
-1.13
-0.46
-1.68
2.44
1.28
5.70
-1.79
-2.38
-2.11
-5.32
7.38
-0.62
-2.11
-0.65
5.74
1.01
-6.78
3.90
M ag.
Phase
M ag.
Phase
c*
1.51
0.42
-0.23
cn
-0.67
-0.06
1.92
-2.59
^bcx
§m
S tt
S 21
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132
In order to identify the dom inant sm all-signal m odel param eters w hich cause
variations in sm all-signal s-param eters due to changes in bias o r tem perature, it is required
to (a) perform param etric sensitivity o f s-param eters w ith respect to m odel param eters,
and (b) observe variations in m odel param eters them selves cau sed by the changes in bias
or tem perature. T he sensitivity analysis presented earlier in this section has dem onstrated
that both m agnitudes and phases o f alm ost all the sm all-signal s-param eters show changes
due to changes in the values o f m ainly four sm all-signal m odel param eters C^,
C bcx,
and g m. A m ong these four m odel param eters, only CK and g m show significant changes in
their values due to changes in either bias o r tem perature. T he bias dependence o f sm allsignal m odel param eters has been given in A ppendix B. V ariations in CK, C^, C bcx, and gm
w ith bias and tem perature for M R F927 have been show n in Fig. 5-15. The variations in
Cjp
C bcx, and gm due to tem perature have been shown in Fig. 5-15 (a), (b), and (c),
w here as the effect o f change in bias (-15% to +15% ) on these m odel param eters has been
depicted in Fig. 5-15 (d), (e), and (f). Besides a ready reference for the discussion to
identify the dom inant m odel param eters, the m otivation to present variations in
and g m
side by side is to dem onstrate that the effect o f changes in tem peratures on the dom inant
param eters Cn and gm is opposite to that o f bias. The o th er tw o param eters
and C bcx
although w ith respect to w hich sm all-signal s-param eters show noticeable sensitivity, can
not be considered as candidates for being dom inant param eters since the variations in their
ow n values due to changes in tem perature and bias are negligible.
H aving identified the dom inant m odel param eters w hich cause changes in sm allsignal s-param eters w ith bias and tem perature, it is appropriate to revisit the bias and
therm al sensitivity o f sm all-signal
s-param eters
presented
in Chapters 3 and 4
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133
2.4
2.4
2.0
-35
f
-15
5
25
45
65
-15
85
-10
•5
0
5
Temperature (°C)
Variation in Bias Current (‘T)
(a)
(d)
0 25
025
0 20
020
0 .5
-
0 15
010
=
0 10
10
Cu
•C
1
Ck .
005
0 05
000
000
-35
-15
5
25
45
65
85
-15
-10
-5
0
5
Temperature (°C)
V ariation in Bias Current (^-1
(b)
(e)
10
15
10
15
016
0.16
-
15
014
014,
•U
010
0 10
0 08
008
-35
15
5
25
45
Temperature <"C)
(C)
65
85
-15
-1 0
■5
0
5
Variation in Bias Current (“Tl
(f)
Fig. 5 -1 5 .Variations in sm all-signal m odel param eters o f M R F 927 w ith changes in
tem perature and bias current. Tem perature d ependence is show n in (a), (b), and
(c), while (d), (e), and (f) depict the effect o f bias on sm all-signal model
param eters.
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134
respectively. A s show n in Fig. 3-13 (a), at low er frequencies, the m agnitude o f S I(
decreases w ith the increase in bias cu rren t. T he increase in current cau ses an increase in
the values o f CK and g m w hile the sen sitiv ity o f S ] f m agnitude is negative to both these
m odel param eters, thereby d ecreases w ith the increase in bias current. A t higher
frequencies, S t j m agnitude show s a positively increasing sensitivity to C rt, thus the net
increase in its m agnitude reduces as the frequency goes higher since the effect o f g m on
S j | m agnitude gets cancelled by the decrease caused by C K. As regards to the phase of
S j | , since it is m ost sensitive to CK, and show s positive sensitivity, therefore w ith the
increase in C^, the phase o f S n increases. However, at higher frequencies the sensitivity
o f S j j phase to m odel param eters b eco m es negligible, therefore the n et change in S j t
phase at higher frequencies becom es sm aller.
T he m agnitude o f S 12 yields a negative sensitivity to both C n and g m, therefore,
w ith the increase in both the m odel param eters, the m agnitude o f S j 2 decreases. A t lower
frequencies the phase o f S 12 show s a reduction while at higher frequencies it increases
w ith the increase in Cn since its sen sitiv ity to
changes from negative to positive. The
m agnitude o f S 2 j increases w ith the in crease in bias current as the n et fractional change in
its m agnitude cau sed by the d o m in an t m odel param eters is positive. As regards to its
phase the negative sensitivity to
a n d g m causes a decrease at low er frequencies. But as
the frequency increases, the trend starts to becom e opposite as show n in Fig. 3-15, Fig. 511, Fig. 5-14, and Table 5-1 and Table 5-2. The change in m agnitude and phase o f S 22 also
agrees very w ell w ith the trends o f th e ir sensitivity to
and g m.
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135
T herm al sensitivity o f sm all-signal s-p aram eters presented in C h ap ter 4 (Fig. 4-21
and Fig. 4-22) can also be explained w ith confidence w ith the help o f the sensitivity
analysis perform ed and results presented in th is section. As shown in Fig. 4-21. the
m agnitudes o f S | j, S 12, and S22 increase w ith the increase in tem perature. T h e increase in
tem perature causes a decrease in the values o f both CK and g m, while the sensitivities o f
the m agnitudes o f S ,„ S I2, and S22 as show n in Table 5 - 1 and Table 5-2 are negative,
thereby causing an increase in their m agnitudes. A s for the m agnitude o f S21, it is m ore
positively sensitive to gm com pared to its negative sensitivity to C^. T herefore, its
m agnitude decreases w ith the decrease in the value o f gm caused by the increase o f
tem perature. Sim ilarly, the sensitivity o f phase angles o f sm all-signal s-param eters to
tem perature variations can be explained w ith th e help o f sensitivity analysis presented
earlier in this section.
Sum m ary
T his ch ap ter has presented a system atic sensitivity analysis o f sm all-signal sparam eters to the sm all-signal m odel param eters. A formal procedure to perform
sensitivity analysis o f sm all-signal s-param eters has been form ulated and successfully
im plem ented. A com parison o f the norm alized sensitivities o f both m agnitude and phase
o f each sm all-signal s-param eter w ith respect to C^, C^, C bcx, rrt, rQ, and g m has been
presented to establish a relative rank o f sm all-signal m odel param eters in term s o f their
effect on sm all-signal s-param eters. It has been show n that the resistive m odel param eters
rrt and rQ have negligible effect on sm all-signal s-param eters although they them selves are
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136
quite sensitive to ch an g es in both bias current and tem perature. The tw o capacitive model
param eters
and C bcx though effect the sm all-signal s-param eters significantly, but their
ow n variations w ith bias current and tem perature are negligible. The other com ponents o f
the model rb, re, an d rc, since have been assum ed co n stan t during this w ork, so their effect
on
sm all-signal s-p aram eter was
neglected.
T hrough
this system atic process
of
elim ination, C n an d g m have been identified as the dom inant model param eters. By
com paring the results o f the sensitivity analysis perform ed on three devices, the typical
trend for the sensitivity o f sm all-signal s-param eters w ith respect to sm all-signal model
param eters has been established. W ith the help o f the results presented in this chapter, the
sensitivity o f sm all-signal s-param eters to changes in bias current and tem perature
presented in C hapters 3 and 4 has been explained. The sensitivity analysis presented in
this chapter has not only provided a valuable insight into the active device itself, b ut it has
also highlighted the significance o f the closed-form expressions for sm all-signal sparam eters derived in C hapters 3 and 4.
The w ork p resented in the next ch ap ter includes a design-oriented analysis o f
am bient tem perature effects on dc bias netw orks. T he analysis is based upon a sim plified
G um m el-Poon m odel developed during this w o rk and show n to be adequate for dc bias
design and analysis o f therm al effects. T he analytical results have been experim entally
verified for several different bias networks. C losed-form expressions have been presented
for popular dc bias netw orks used in RF / M icrow ave circuits. Also included in the next
chapter is an optim ization strategy to make the d esig n s therm ally stable.
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C H A PTER 6
A D ESIG N A NA LY SIS O F A M B IE N T TEM PER A TU R E E F F E C T S O N CO M M O N
M ICROW AVE DC BIAS C IR C U IT S
P rior to perfo rm in g the RF/M icrow ave circuit design, the D C biasing point needs
to be established. O ften, the factor w hich is paid the least attention in RF/M icrow ave
transistor circuit design is the DC bias network. Stable B JT am plifier designs require that
the transistor be biased in the forw ard-active region and the bias/quiescent point be kept
constant despite variations in tem perature. Since the cost p e r dB o f m icrowave gain or
noise figure is so high, the circu it designers can not afford to sacrifice the R F perform ance
by inattention to the D C bias netw ork design [H p75], T h e R F/M icrow ave design
param eters such as g ain and noise figure are well know n to be strong functions o f the
collector current in tran sisto r am plifiers involving BJTs [H p75, Ven90]. The percentage
change in both gain an d noise figure due to a sm all percentage change in the collector
current is significant en o u g h to ju stify stabilizing the bias current w ith reference to
tem perature variations. T h is m inim izes changes in R F p erform ance over a range o f
am bient conditions.
Bias sensitivity o f sm all-signal s-param eters w as presented in C hapter 3 and it w as
shown that both m agnitude and phase o f s-param eters are sensitive to changes in collector
current. For input reflection coefficient S u and forw ard transm ission coefficient S 21 , the
m agnitude increases w ith the increase in tem perature, w hile reverse transm ission
coefficient S 12 and o u tp u t reflection coefficient S22 d ecrease in m agnitude w ith the
137
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138
increase in collector current. T he phase for all four tw o-port sm all-signal s-param eters has
also exhibited sensitivity to changes in collector cu rren t. S-param eters are used
extensively in RF/M icrow ave circu it designs, therefore to realize a therm ally stable
desig n , it is necessary to m aintain a co n stan t collector cu rren t in the circuit.
N um erous text and jo u rn al articles have pointed o u t th at therm al behavior o f DC
bias netw orks can be determ ined by know ing the therm al stability o f the collector
currents. The techniques used at low frequencies such as em p lo y in g an em itter resistor to
introduce substantial degeneration at DC to achieve in sensitivity to pow er supply noise
and tem perature and a bypass cap acito r to reduce deg en eratio n at the frequency o f
am plification [Lin56], w hen em p lo y ed for R F/M icrow ave d esig n s, give rise to problem s
o f frequency instability and oscillations. The bypassed e m itter resistor also degrades the
noise perform ance o f R F/M icrow ave circuits [H p75, V en90]. M oreover, the bypass
cap a cito r is usually too large to be im plem ented in integrated circuits [W id65]. H ence a
carefu l study o f circuit elem ents and transistor m odel param eters that are tem perature
sensitive and influence the co llecto r current is essential tow ards realization of a therm ally
stable D C bias network.
As stated earlier, alm ost every transistor param eter is d irectly o r indirectly affected
by tem perature because o f the therm al dependence o f th e physical properties o f the
m aterials used to fabricate sem ico n d u cto r devices [M ul77, Sze81, Fre85, Sah91].
T herefore, the electrical properties (currents and p o ten tials) o f a device are also
tem perature dependent. T he o p eratin g tem perature o f a device is determ ined by the
am bien t tem perature and the pow er dissipated in the device (self-heating effect) [Lu92,
L ia93]. In order to design a th erm ally stable DC bias n etw ork, the am bient tem perature
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139
and self-heating effects need to be considered both individually and com bined. Several
researchers have studied self-heating effects in BJTs [Str59. Fuk76, M u n9l, Fox93,
Z w e93], but the am bient tem perature effects have been analyzed using the Ebers-M oll
m odel only in [Hp75] and cited by others [Gon84, Ven90].
This chapter presents an im proved G um m el-Poon based analysis o f the am bient
tem perature effects on co llecto r currents o f bias designs com m only used in RF/M icrowave
circuits and also a design optim ization strategy. The analysis o f po p u lar DC bias networks
w as perform ed using several different npn transistors and in all cases the measurem ent
data, sim ulation results and the analytical predictions w ere found to be in excellent
agreem ent. Also, the optim ization strategy w orked equally w ell for all transistors, thus
adding validity to the approach presented in this chapter.
Simplified Modeling of Bipolar Junction Transistor for DC Bias Analysis
The G um m el-Poon m odel for BJT [Gum70] has been sim plified for bias circuit
calculations for the BJT in the forw ard-active region. In the sim plified model shown in
Fig. 6-1 for both npn and pnp transistors, the base-em itter diode m odels the dom inant
com ponent o f the base current due to injection o f holes from the base into the emitter. The
controlled current source m odels the collector current in the forw ard-active mode. Also
show n in Fig.6-1 are q,, re, and rc, w hich model the ohm ic resistances from the active
regions to the base, em itter, an d collector term inals respectively. T hese resistances are
constant for the DC bias calcu latio n s during this work.
Generally, the co llecto r cu rren t (Ic ) can be expressed as a function o f saturation
current (/$), com m on-em itter forw ard current gain (|3y), b ase-em itter voltage (VBE),
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140
rb
AM— °
BE
■/V
B
rb
AM— °
EB
yv
EB
B
(b )
Fig. 6-1. Sim plified BJT M odel for D C bias analysis, (a) E quivalent circuit model for npn
transistor, (b) Equivalent circu it m odel for pnp transistor.
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141
therm al voltage (Vf), and the resistive elem en ts o f the circuit (/?,) as:
Ic = fUs -Py • VB£ • vt • Ri )
(6 -1 )
W here Is, Py VBE, Vt, and Rj are functions o f tem perature (T) [C am 69, G et7 9 , G ra84.
Sed87, H od88, G ha94]. Since 1^ is a co n tin u o u s function o f the tem p eratu re sensitive
variables w ith continuous partial derivatives, the total derivative o f Ic w ith respect to T
can be w ritten using chain rule for functions o f functions:
dl
dlr <//-
dlr
dT
dls d T
dpy-
d $ f dl r
dVR F
/ + _ _ ---------- .
d T dVB £
dT
dfr dV. d l r d R .
+ _ L — + — _ ------dV{ d T d R . d T
w here d lc = dIc /dT.dT and substituting d lc = AI^ and dT = AT yields AI c =
.-.AIc -
(d lf-
dl^
8/^, d $ j
d l£
d V gE
d l^
dls
dT
ap;
dVB E
dT
dV( d T
dT
dV{
d l^
jr
dR,
dT
dT
AT
'
AT
(6-3)
T he following procedure has b een form ulated and im plem ented to estim ate the
dependence o f collector current on tem p eratu re in the DC bias netw orks p resen ted in this
chapter: ( I ) replace the active device(s) in the D C bias netw ork w ith th e equivalent
circuit(s) in Fig. 6-1, (2) w rite an expression for collector current in term s o f the device
m odel param eters and other passive co m p o n en ts o f the network, (3) calcu late the partial
derivative o f the collector current w ith resp ect to the tem perature sensitive independent
variables using M ATLAB, M aple [M at92, M ap96], and hand analysis, an d (4) evaluate
equation (6-3) for any specified range o f tem p eratu re variation.
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142
Therm al Effects on Tem perature Sensitive Independent Variables
In DC bias designs em ploying b ip o la r ju n ctio n transistors, to estim ate the change
in collector current due to tem perature, the variations in tem perature sensitive independent
variables Is,
VBE’ Vr and Rj need to be know n. Literature search show ed that Is varies
approxim ately 15% /°C rise in tem perature [H od88], T he variation in / j due to tem perature
reported by H odges [Hod88] is based on th eoretical calculations from expression for Is
used in ideal diode equation. The changes in P^-are relatively sm all i.e. +0.3 to +0.7% /°C
[Cam 69, G ra84], The rise in transistor tem p eratu re results in a decrease in its VBE (-2m V /
°C) [C am 69, G ra84, Sed87, H od88]. T h e therm al voltage Vt being KT/q increases
approxim ately 86|iV /°C . For various types o f resistors used in bipolar integrated circuits
the tem perature coefficients range from ± 10 to 3000 ppm /°C [C am 69, G ra84, G ha94],
D uring this w ork, tem perature coefficients o f 1$, VBE’ and py have been determ ined
experim entally and the results are p resented in the follow ing sections.
Temperature Coefficient of Saturation C urrent
The saturation current for a b ip o lar ju n ctio n transistor is obtained from the
m easurem ent d ata for the collector cu rren t. M ainly there are tw o com m on m ethods o f
m easuring co llecto r current for the p u rp o se o f extracting Is , (a) m easurem ent o f 1^ at
single value o f V BE = V CE (single-point m easurem ent), and (b) m easurem ent o f 1^ as a
function o f V BE [G et79]. In both m ethods, th e value o f Is is obtained after post-processing
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143
the m easured data. In single-point m easurem ent m ethod, the extraction o f Is is straight
forward using the ex p ressio n for 1^ at low current levels [Get79]
(6-4)
where as in the o th er m ethod, first natural logarithm (In) o f m easured Ic as a function o f
VBE is plotted and then by extrapolating the plot to V BE=0. the value o f Is is obtained.
O btaining Is through extrapolation o f ln(Ic ) vs. V BE plot is m ore com m on and popular
com pared to sin g le-p o in t m easurem ent schem e. H owever, For this w ork, single-point
m easurem ents being sim p ler and faster were used to obtain Is at different tem peratures
and to determ ine its tem perature coefficient.
In single-point m easurem ent m ethod, Is is considered as a function o f VBE w hen
the transistor is in th e active region (base-em itter ju n ctio n forw ard biased) w hile zero bias
is applied across b ase-co llecto r junction w hen VCE equals VBE—the bias point at w hich
the m easured 1^ is used to obtain Is . The m easurem ent set up shown in Fig. 6-2 was used
to determ ine the tem perature coefficients o f Is, (3f, an d VBE■Due to the sensitivity o f base
current to VBE and ex p o n en tial dependence o f Ic on VBE, to m onitor VBE and VCE high
resolution digital v oltm eters (Fluke 8050A) with high m easurem ent accuracy w ere used.
Also, for m easuring the co llecto r current accurately, Keithley 480 Pico am pm eter w as
used. The bias at base and collector was supplied through precision pow er supply
H P6626A and m o nitored using the digital voltm eters show n in Fig. 6-2. This precaution
was taken due to sen sitiv ity o f base current to VBE, since an erro r o f lm V can cause a
significant change in base cu rren t and a corresponding change in the collector current.
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144
CE
BE
Precision
Pow er ~Zt
Supply
Precision
Pow er
S u p p ljr
Fig. 6-2. M easurem ent set up to determ ine tem perature dependence o f (a) saturation
current (Is ), (b) base em itter voltage (V BE), and (c) DC forw ard current gain (P).
being m easured to obtain the saturation cu rren t T he bypass capacitors shown in the set up
w ere used to remove the pow er supply fluctuations and noise, if any. The device used in
this w ork w as an npn transistor from C A 3096 N PN /PN P transistor array. The transistor
was m ounted on a square pad circuit board p laced inside tem perature controller (oven) and
m easurem ents were made from +15°C to + 3 5 °C . The base-em itter voltages at w hich
m easurem ents were perform ed w ere 0.60 V an d 0.65 V and they were kept low to ensure
that the transistor was operated in low current region only and junction tem perature was
not raised significantly due to excessive pow er d issipation (self-heating). The variations in
saturation current with tem perature obtained th ro u g h m easurem ent are show n in Fig. 6-3.
The saturation current m easured at room tem p eratu re (25°C) for C A 3096 is 9.826 E - 15 A,
w hich m atches very well w ith the value for Is in d ata sheet i.e. 10 E-15 A. The difference
o f 1.7% betw een the Is m easured during this w o rk and reported by the m anufacturer is
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
145
5e-14
4e-14
<
3e-14
u
c
o
3U.
3
3
2 e ‘ 14
1e-14
15
20
25
30
35
Tem perature (°C)
Fig. 6-3. Variation in satu ratio n current (Is ) w ith tem perature.
negligible. The tem perature coefficient o f Is determ ined experim entally during this work
is +16.3% /°C rise in tem perature and the sam e value has been used for analyses in the
follow ing sections.
Temperature Coefficient of Forw ard C urrent Gain
The DC forw ard co m m o n -em itter current gain (py) is the ratio o f the DC collector
current to the D C base cu rren t w hen the tran sisto r is in the forw ard active region i.e. w hen
the base-em itter ju n c tio n is forw ard biased and the base-collector ju n ctio n is reverse
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
146
biased. Since, Pyis know n to vary w ith th e collector current, therefore, it is appropriate to
know the value o f (3yat the bias co n d itio n under w hich the transistor is ex p ected to operate.
N orm ally, this inform ation is provided by the m anufacturer in d ata sheets as a plot o f Py vs.
Ic . In o rd er to determ ine the tem p eratu re coefficient o f Py free o f its d ependence on Ic ,
m easurem ents w ere m ade at three d ifferen t bias conditions (V c c = 3V & Ic = 1mA, Vc c
= 5V & Ic = 3m A , and V c c = 10V & Ic = 5m A ). For each set o f m easurem ents, the
collector current w as kept co n stan t by controlling the base em itter voltage w hile
tem perature was varied betw een -55°C to +100°C . As m entioned earlier, the m easurem ent
set up show n in Fig. 6-2 was used for this w ork w ith the exception that K eithley 480 Pico
am pm eter w as replaced w ith digital m ultim eter (Fluke 8050A ). T his w as necessary
because the co llecto r currents (3m A an d 5m A ) to be m easured ex ceeded the m axim um
m easurable current lim it o f Keithley 4 8 0 P ico am pmeter.
T he variations in Py w ith tem p eratu re m easured at three d ifferent bias conditions
are show n in Fig. 6-4. As show n in Fig. 6-4, the rate o f change o f DC forw ard current gain
w ith tem perature depends on the bias co n d itio n , mainly on the co llecto r current. Also, Py
does not exhibit a uniform rate o f ch an g e through out the tem perature cycle, instead, the
slope o f the g raph Py vs. T is g reater betw een -55°C to approxim ately +25°C (room
tem perature) than for tem peratures above +25°C . The variations in Py w ith tem perature at
different co llecto r currents have been su m m arized in Table 6-1. A s listed in Table 6-1, the
average rate o f change o f Py per d eg ree C elsius rise in tem perature is m ore at low er
collector currents than at higher co lle c to r currents. To determ ine the tem perature
coefficient o f py, it is appropriate to k n o w the collector current at w hich the device is
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
147
500
V C C = 3 V. IC = 1mA
C£L
I 400
V CC = 5V, IC = 3m A
o
C
C3
|
LL.
U
Q
300
VCC = 10V, IC = 5m A
200
-55
-35
-15
5
25
45
65
85
105
T em p eratu re (°C)
Fig. 6-4. V ariation in DC forward current gain (Pf) w ith tem perature.
expected to operate. The tem perature coefficient determ ined through m easurem ents
during this w ork is w ell within the range +0.3% to +0.7% reported in literature A lso, it
w ill be show n in later sections that change in co llecto r current due to change in Pydue to
tem perature is sm aller com pared to the ch an g es in co llecto r current due to o th er factors,
therefore it is acceptable to use an average value for the tem perature coefficient o f Py for
the entire tem perature cycle.
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148
Table 6 - 1. Variations in (3f w ith tem perature
Collector C urrent
(mA)
Apf/AT (%)
Average Apf/AT (% )
-55°C to + I0 0 °C
-5 5 °C to +25°C
+ 2 5 °C to +100°C
1.00
0.653
0.271
0.537
3.00
0.520
0.260
0.447
5.00
0.419
0.206
0.349
Temperature Coefficient of Base Emitter Voltage
In bipolar ju n c tio n transistors, the b ase-em itter voltage (VB£) decreases w ith the
increase in tem perature. To determ ine the tem perature dependence o f VBE' the set up
shown in Fig. 6-2 w as used. The collector current w as kept constant by controlling the
voltage across base em itter ju n c tio n w hile the tem perature was varied from -55°C to
+ 100°C. T he variations in VBE due to tem perature w ere recorded for three different bias
conditions as was d o n e to determ ine the tem perature coefficient o f (3^ The effect o f
tem perature on b ase-em itter voltage is shown in Fig. 6-5. As shown in Fig. 6-5, the
tem perature coefficient o f V BE decreases w ith the increase in collector current. For
analytical w ork p resen ted in th is chapter, the m easured values o f tem perature coefficients
o f VBE have been used.
For pnp transistors, th e variations in Py-and VBE exhibited sim ilar trends as for npn
transistors. For sim p licity o f analysis the sam e tem perature coefficients o f py-and VBE have
been used for npn an d pn p transistors.
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149
0.9
Base-Emilter Voltage (V)
V c c = 10V, Ic = 5m A ,A V BE/A T = -1.759 m V /°C
0.8
0.7
0.6
v c c = 3V, Ic = I m A , AVb e/A T = -1.891 m V /°C
v c c = 5V, Ic = 3mA, AVb e /A T = -1-837 m V /°C
0.5
-55
-35
-15
5
25
45
65
Tem perature (°C)
Fig. 6-5. Effect o f tem perature on b ase-em itter voltage (V BE).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
85
105
150
Example Passive and Active DC Bias Networks: Analysis and Verification
D uring this work four p o p u lar passive and three active D C bias designs shown in
Fig. 6-6 and Fig. 6-7 respectively, w ere stu d ied and optim ized for am bient tem perature
effects on therm al stability. T he netw orks show n in Fig. 6-6 (b), Fig. 6-7 (a) and Fig. 6-7
(c) are analyzed in detail in this ch ap ter in accordance w ith the procedure outlined earlier,
w hile for the rem aining netw orks (Fig. 6-6(a), Fig. 6-6(c), Fig. 6-6(d), and Fig. 6-7(b)),
analytical expressions are included in A ppendices D and E.
RB
r— V W
R1
RC
MV—o vcc
Wr
RB
— W r-£
RF/Microwave
Transistor
RC
VA— o VCC
RF/Microwave
Transistor
R2
Passive 1
Passive 2
(a)
(b)
R1
RF/Microwave
Transistor
RF/Microwave
Transistor
R2
Passive 4
Fig. 6-6. Passive DC bias netw orks co m m o n ly used in R F/M icrow ave circuits
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
151
o
R1
Active 1
vcc
R3
►
R2
RF/Microwave
Transistor
R4
(a)
o
R1
R3
►
R4
vcc
RF/Microwave
Transistor
R2
R5
R6
(b)
VCC
Active 3
RF/Microwave
Transistor
RF/Microwave
Transistor
(c)
Fig. 6-7. Active D C bias netw orks com m only used in RF/M icrow ave circuits
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
152
Passive Bias Network
To analyze the passive DC bias netw ork in Fig. 6-6(b), the equivalent circuit is
drawn in Fig. 6-8 using the sim plified m odel for npn transistor. T his bias netw ork w ould
Rl
W —»
BE
R2
RC
Vcc
Fig. 6-8. Equivalent circu it for DC bias netw ork show n in Fig. 6-6 (b)
be referred to as Passive N etw ork 2 in this work. T he analytical expression for the
collector current in term s o f the device model param eters and other passive com ponents o f
the netw ork is:
V BE/ V t
Al^l>('1 - e1
cp
BVoc
BE
- —
V
R0
cc 2
(6-5)
H----------- -
/
where
A = (flj + R2 +Rc)(re + rb +
+ (R i +RC)R2 '
C = (flj + R2 + Bc)rg + RcR2 ■F rom (6-5), it can be seen that
B = R { +R^ + Rc ,
depends on Is, VB£,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
and
Vt,
153
RB, Rj, R2, and R c . The partial derivatives o f 1^ w ith respect to these tem perature sensitive
variables are:
*(l-<
31 c
VB E / V t
cp
die.
( 6- 6 )
/
Ms\e
dv B E
B
c i f vt
V
dic
A/^l[ l1 -- e
VB E / V t
ais vbe
's
VBE/ V,
Cp
/
oR,
rb + R B + R l ) l S
cp
% ,
BE
C +
1- e
a/?,
{re +
V B E / V r'
Al cr I - e
S e
( 6-
10)
(6-
11 )
VBE/ V t '
■>
/
c-p /
B V DCr
V R~r
BE e
cc 2 e
J2
~
J2
f
a/C
(6-9)
c*f vt
a/
a/?,
')
cp
/
= ~~
3 /„
/V
BE
( 6- 8 )
ap r
a /c
(6-7)
C
rb +RB +Rl +RC)Is{l ~e
cp
/
^ R F ^ ^ t\
J
r RF' r
(
* / S( r * + * c V
"e
9
c “p
/
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
j
154
BE
dlc
W,
B V BE('re + RC)
cc
V
(
VB E / V t \
(‘r e + r b + R B + R 2 ) l s { ] ~ e
J
cp
VB E
R~ ( r + / ? - )
cc 2 e
C
(6-12)
(
VB E / V i
A /5 {ff + /?2 ( 1 _
7
c-p
/
B V B E (^r e + R 2 )
+ ----------
C"
f
Vc c R 2^r e + R 2 )
(6-13)
CT
T he DC bias design show n in Fig. 6-6(b) (Passive N etw ork 2) was im plem ented
for three bias conditions listed in Table 6-2 using discrete com ponents. The active device
used w as an npn transistor from C A 3096 N P N /P N P tran sisto r array, while for passive
circuit elem ents i.e. resistors, carbon film resistors
from M C F
1/4 W series o f
M U LTIC O M P w ere used. T hese resistors have a tem p eratu re coefficient o f 0 to -700 ppm /
°C. T he resistor values used both in m easurem ents and an aly sis are show n in Table 6-2.
Table 6-2. R esistor Values for Passive N etw o rk 2
Passive
Elem ents
B ias C o nditions
V qq = 3V &
—1mA
Vc c = 5V & Ic = 3mA
= 10V &
R1
6.8 KQ.
8 .2 1 0 2
7.5 102
R2
8.2 K Q
3.3 102
2 102
RC
1.3 KJ2
4 30 £2
910 £2
RB
68 KQ.
39 102
20 102
—5 mA
T he variations in Ic w ith tem perature m easured an d pred icted by the analysis are
show n in Fig. 6-9 to Fig. 6-11 for three different bias co n d itio n s. A lso show n in the above
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
155
1.3
M easurem ent
Collector Current (mA)
1.2
1.1
1.0
Sim ulation
0.9
0.8
A nalysis
0.7
-55
-35
-15
5
25
45
65
85
105
Tem perature (°C)
Fig. 6-9. C om parison o f m easured, analytical prediction, and H SP IC E sim ulation o f
variation in Ic o f Passive 2 netw ork w ith tem perature. T he bias conditions are
V c c = 3 V and Ic = 1mA.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
156
Collector Current (mA)
M easurem ent
Sim ulation
A nalysis
T em perature (°C)
Fig. 6-10. C om parison o f m easured, an alytical prediction, and H SP IC E sim ulation o f
variation in Ic o f Passive 2 n etw ork w ith tem perature. T he bias conditions are
V c c = 5V and Ic = 3m A .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
157
6.0
M easurem ent
V r r = 10V & Ir = 5mA
Collector Current (mA)
5.5
5.0
Sim ulation
4.5
A nalysis
4.0
-55
-35
-15
5
25
45
65
85
105
T em p eratu re (°C )
Fig. 6-11. C o m p ariso n o f m easured, analytical p red ictio n , and H SPICE sim ulation o f
variation in Ic o f Passive netw ork 2 w ith tem p eratu re. The bias conditions are
V c c = 10V an d Ic = 5m A .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
158
m entioned figures are the results o f HSPICE sim ulations. T he com parison presented in
Fig. 6-9 to Fig. 6-11 show s that the analytical results are in ex cellen t agreem ent w ith the
m easurem ents for the en tire range o f tem perature i.e. from -55°C to +100°C. A
sum m arized com parison o f am b ien t tem perature effects o n Passive N etw ork 2 m easured,
predicted analytically, and sim u lated using H SPICE is given in Table 6-3. The reference to
Table 6-3. C om parison o f M easured, A nalytical, and S im u lated R esults o f A m bient
T em p eratu re E ffects on Ic o f Passive N etw ork 2
P ercentage Variation in C o llecto r C u rren t w ith Tem perature
B ias C ondition
+100°C
-55°C
u
u
>
k:
Measured
3V
1mA
-27.07
-24.36
-19.81
+25.37
+23.13
+20.34
5V
3m A
-25.38
-20.71
-19.50
+22.25
+ 19.66
+ 19.15
10V
5m A
-18.05
-16.16
-13.03
+ 18.19
+ 15.30
+ 13.20
Analytical Simulated
Measured
Analytical Simulated
calculate the percentage variations has been the collector cu rren t at +25°C. It can be seen
from Table 6-3 that the results o b tain ed from the analytical procedure presented in this
chapter are very close to the m easured data. T he variations in Ic at the tem perature
extrem es i.e. -55 °C and + 100°C, predicted analytically are approxim ately 3% less than
the m easured data. W hereas, the difference betw een the m easu red and the sim ulated d ata
is from 3% to 1%. T he sim u latio n s also underestim ated the variations in collector current
w ith change in am bient tem p eratu re. Sim ilar trends w ere observed when m easured,
analytical predictions, and sim u latio n s for other D C bias netw orks show n in Fig. 6-6 w ere
com pared. T he analytical ex p ressio n s for the passive b ias circu it show n in Fig. 6-6 (a).
Fig. 6-6 (c), and Fig. 6-6 (d) are given in A ppendix.D.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
159
Active Bias Network
The equivalent circuit for the active DC bias netw ork in Fig 6-7 (a) is shown in Fig
6-12. The analytical expression for
n> 1
an d its partial derivatives w ith respect to the
rc
re 2
rb 2
l
EB
R2
BE
R4
R3
vw
~T~
Vcc
Fig. 6-12. Equivalent circuit for D C bias netw ork in Fig. 6-5(a)
tem perature sensitive device m odel param eters (both in npn and pnp) and circuit elem ents
are
f VE B \ / V t
C2
A / s r Ce
\
J
or
( VB
' B E 2T/ V' t
CDp
cp / 2
/I
J
R4 VE B 1
( r e l + R 3 ) V BE2
R.R,V
1 4 cc
(6-14)
CD
w here A = R4(R{R2 + (K, + R2Krb \ + re \ + R3» ' B = (R4 + r b 2 + r e 2 K r e \ +R3y
C = re2(re[ +/?3)+ /? 3/?4 , and D = R x +R2 .
a / C2
df
SI
( VE B l / V t
41[ e,
A
-lj
CD P
/I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(6-15)
160
f KESl/V r
51
3/C2
A
" ‘J
(6-16)
CDp2/!
3P/1
r ve b i / vi\
^ 7S l L e
a/C2
J *4
CDP/ ,V'f
dVEBl
(6-17)
C
a/C
2
(6-18)
CP/2
d lS2
Bl
f V* £ 2 / V '
81S 2{e
a iC2
aP/2
_ 1,)J
(6-19)
Cp2/2
( VBE2 / Vt)
3 /C2
BE2
AV
a/C2
dV~
d l c ~,
~
# t VBE2/ \
J
8
B E2 l S 2 \ e
+
+ /?3)/5l[e
V EB
)
,
\/V t
\
(
- 1 j A/Sl(e
VE B l / V t
\
-1J
+
R4 V cc
+^ ~
( 6 - 22 )
R \ R4 V cc
CD2
t VE B \ / V t
R 4 (^R \ + r b l + r e \ + R 3) I S l [ e
-
3/C 2 _
3/?2
3R 3
\
1J
f VE B \ / V t
AIS l [ f
~
CDP/1
*„
< * /!
^
1j
CD2p/l
V , . ( > |/v '- .)
“
,,
CP/2V ^
^ 7 i
_
( 6- 20 )
^
( VE B \ / V A
,
ESI 5l(e
C D P /iV5
RA^R 2 ^ rb \ + r e\
^ ~ =
re \ + R
^ f l ^ t
(
^
J
S l{e
R \ R4 Vcc (6
CD2
^ s 2( ^ s £ 2 / " ' - 0
+
C2 0 P/ |
~
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3 , C2
~ 'J
A,Sl(‘
3*4
3 S ll/
“ 'J
<r e l +
c 20|J^ i
CDR^fl
C|J/ 2
v BE, / v l
'
7
C“p/2
8 R 3IS2{ ‘
+
‘ 'J
‘'ffi, .''4 ''£ BI«3 .< '-,I+R 3»l'fi£2'i3
C +
~>
+
■*
C
C
+ —!—— — !—- —-——
CD
(6-25)
c 2d
T he D C bias design show n in Fig. 6-7(a) (Active N etw ork 1) w as im plem ented for
three bias co n d itio n s listed in Table 6-4 using discrete com ponents. T he resistor values
used in m easurem ents, analysis, and H SP IC E sim ulations are show n in Table 6-4.
Table 6-4 R esisto r Values for Active N etw ork I
Passive
E lem ents
Bias C onditions
VCc = 3V & Ic = 1mA
Vc c = 5V & Ic = 3mA
Vc c = 10V& Ic = 5mA
R1
5.1 K D
7.5 KD
33 KD
R2
2.4 K D
9.1 KD
24 KD
R3
6.2 K D
470 D
820 D
R4
1.3 K ft
2 KD
820 n
T he variations in 1^2 w ith tem perature m easured, p red icted by the analysis, and
H SP IC E sim ulation are show n in Fig. 6-13 to Fig. 6-15 for three different bias conditions.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
162
1.3
CC = 3 V & Ic = 1mA
Sim ulation
Collector Current (mA)
1.2
1.1
A nalysis
1.0
0 .9
M easurem ent
0.8
0 .7
-55
-35
-15
5
25
45
65
85
105
T em perature (°C)
Fig. 6-13. C om parison o f m easured, analytical prediction, and H S P IC E sim ulation of
variation in 1^ o f A ctive 1 netw o rk w ith tem perature. T he bias conditions are
Vc c = 3V and Ic = 1mA.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
163
4.0
3.8
Sim ulation
V r r = 5V & Ir = 3m A
Collector Current (mA)
3.6
3.4
3.2
3.0
A nalysis
2.8
M easurem ent
2.6
2.4
2.2
2.0
-55
-35
-15
25
45
65
85
105
T em perature (°C)
Fig. 6-14. C om parison o f m easured, analytical prediction, and H SPIC E sim ulation o f
variation in 1^ o f Active 1 netw ork w ith tem perature. The bias conditions are
V Cc = 5V and Ic = 3m A.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
164
6.0
c c = 1 0 V & Ic = 5 mA
Sim ulation
Collector Current (inA)
5.5
5.0
A nalysis
4.5
M easurem ent
4.0
-55
-35
-15
5
25
45
65
85
105
T em p eratu re (°C)
Fig. 6-15. C om parison o f m easured, analytical pred ictio n , an d H SPICE sim ulation o f
variation in Ic o f A ctive 1 netw ork w ith tem perature. The bias conditions are
Vc c = 10V an d Ic = 5mA.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
165
It can be seen that the analytical results are in excellent agreem ent w ith the m easurem ents
for the entire range o f tem perature. A sum m arized com parison o f am bient tem perature
effects on A ctive N etw ork I m easured, predicted analytically, and sim ulated using
H SPICE is given in Table 6-5. It can be seen from Table 6-5 th at the results obtained from
Table 6-5. C om parison o f M easured, A nalytical, and S im ulated R esults o f A m bient
T em perature Effects on IC2 o f A ctive N etw ork I
P ercentage Variation in C o llecto r C u rren t w ith Tem perature
Bias C ondition
-55°C
u
u
>
Measured
+105°C
Analytical Simulated
Measured
Analytical Simulated
3V
1mA
-17.95
-19.19
-18.93
+ 19.69
+ 19.19
+ 21.96
5V
3m A
-15.54
-16.93
-17.34
+ 17.56
+ 16.93
+20.08
10V
5m A
-10.39
-8.18
-12.53
+ 11.64
+8.18
+ 14.28
the analytical procedure presented in this chapter are very close to the m easured data. The
differences in percentage variations in IC2 at the tem perature extrem es i.e. -55 °C and
+105°C ,
p redicted
analytically
and
m easured,
sim ulated
using
HSPICE
are
<
approxim ately 3.5% . T he seem ingly large difference betw een H SP IC E sim ulation and
m easurem ent for V C C = 10V and IC = 5m A is prim arily due to the difference in co llecto r
current at +25°C . T he m easured co llecto r current is 4.81 m A , w hile the sim ulation yielded
the collector cu rren t o f 5.09 m A . T he sam e resistor values w ere used for m easurem ents,
analytical predictions, and H SP IC E sim ulations.
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166
Simple C urrent Source (C urrent M irror)
To estim ate the effects o f the am bient tem perature on current m irror circuit in Fig.
6-7 (c), the equivalent circuit is show n in Fig. 6-16. The analytical expressions for IQ and
Cl
BE1
■=r V c c
Fig. 6-16. Equivalent circuit for cu rren t m irror in Fig. 6-7 (c)
its partial derivatives with resp ect to the tem perature sensitive model param eters and
circuit elem ent R are
, v RF/ v .
's e
Vcc - ^ B E ~
.
- l ) l r b + re + 2R)
------------------0 7 ---------------------
l n = ------------------------------------- L-----------------(R + r g )
O
dlO
dFs ~
i C BE
t ~ l } rb + r e + 2R)
{R + re)$f
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(6-26)
(6-27)
167
^
f V RE/ V r
*Q
!s { e
-
^
b * r e + 2R)
(6-28)
(R + re) P27
aP/
f VBE/ V t
ls \ e
-
dl
1
* f V<
O
dV
\ rb + r e + 2R)
-
(6-29)
BE
* 0
( VRE/ V t\
IS V B E { e
j rb + re + 2 R ^
d V.
, vRF/v.
'k
v /v
3 /„
'O
dR
-2 /s ( / S £
(R + re ) $ f
(6-30)
( R + r e ) V>V~t
'-■ )
Vc c - VB E - -
,
b
e * 2R)
} r>-*r.
7
(6-31)
(R + r e r
As m entioned earlier, to im plem ent bo th the passive and the active bias networks
in this work, discrete com ponents w ere used. The necessary conditions for the simple
current source shown in Fig. 6 -7 (c) to operate are: (a) the transistors are connected in
current-m irror configuration, (b) the transistors are perfectly m atched, and (c) the bases
and em itters o f both the transistors are connected together, thus ensuring sam e VBE.
Im plem entation o f a sim ple cu rren t source, w hich utilizes transistors in a current-m irror
configuration and to satisfy the n ecessary conditions for the circu it to function as needed,
using discrete com ponents is alm ost im possible. In integrated circuits realization o f
m atched transistors having their bases and em itters connected to g eth er to achieve the same
V BE is routinely done by fabricating transistors on the sam e chip. However, to
dem onstrate the applicability o f the procedure developed during this w ork for estim ating
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
168
the am b ien t tem perature effects on the output current o f sim p le current source, the
analytical predictions were co m p ared against the results o f H S P IC E sim ulations. In the
H SPIC E sim ulation, the necessary co n d itio n s for a sim ple cu rren t source can be satisfied
easily, by specifying the sam e m odel param eters for both the tran sisto rs. The variations in
IQ w ith tem perature predicted by th e analysis and H SPICE sim u latio n are shown in Fig. 617. T he npn transistor used for H S P IC E sim ulation and an aly tical w ork is NTD1X2P5
from M otorola. From Fig. 6-17, it can be seen that the an aly tical result is in excellent
agreem ent w ith the sim ulation for the entire range o f tem perature. T h e percent change in
IQ estim ated analytically and by H S P IC E sim ulation is show n in Fig. 6 - 17(b). It is shown
that th e difference betw een the an aly tical and sim ulation results is less than 2% over the
entire range o f tem perature from -5 5 °C to +125°C.
DC Bias Design Optimization
The analysis presented in th is ch ap ter provides a valuable insight into the DC bias
designs. T he behavior o f each tem p eratu re sensitive device m o d el param eter and passive
circu it elem en t can be ascertain ed w ith confidence. Then, th e ir individual contributions
tow ards the resulting d rift/variation in collector current d u e to change in am bient
tem perature can be estim ated. T h is inform ation can not be o b tain ed through any circuit
sim u lato r o r analytical approach reported in literature. To select the com pensating
com ponents, the focus during this w o rk has been on passive circ u it elem ents. The active
elem ents leave m uch less o p p o rtu n ity for optim ization since the d esig n er m ust negotiate
w ith the process engineer to ch an g e active device param eters.
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169
1.10
1.0 5
Simulation
,0
1.00
0.95
Analysis
0.90
-70
-40
-10
20
50
80
110
140
Percent change in l0 (%)
Temperature (°C)
Simulation
Analysis
-10
-70
-40
-10
20
50
80
110
140
Temperature (°C)
Fig. 6-17. A m bient tem perature effects on cu rren t mirror, (a) Variation in ou tp u t current,
(b) Percent change in o u tp u t current.
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170
Optimization of Passive Bias Network
T he partial differentials o f Ic w ith resp ect to tem perature sensitive device m odel
param eters and passive circuit elem ents o f D C bias design in Fig 6-6(b) are show n in Fig
6-18 (a) and (b) respectively. It can be seen from Fig 6 - 18(b) that the partial differentials
o f Ic w ith respect to R ^, R b - R-i * and R 2 are positive, w hich im plies that the partial
derivatives o f Ic (rate o f change o f Ic ) w ith respect to these circuit elem ents are negative
w ith the increase in am bient tem perature sin ce th eir tem perature coefficients are negative.
The overall effect is an increase in c o llecto r current due to increase in am bient
tem perature. T hus to improve the bias netw ork and reduce the collector cu rren t drift, the
current contributions o f passive elem ents n eed to be decreasing w ith tem perature. This
im provem ent can be achieved by replacing carbon film resistors having negative
tem perature coefficient (used in initial design/analysis) for R c, RB, R j, and R 2 w ith m etal
film resistors o f M F12 series from M U L T IC O M P having positive tem perature coefficient
o f 50 ppm /°C [Far97]. T he im provem ent in perform ance o f DC bias netw o rk in Fig. 6-6(b)
is shown in F ig 6-19. As shown in Fig. 6-19, the perform ance has im proved significantly
and variation in Ic has decreased from +25.37% to +20.77% at 105°C an d from -27.07%
to -18.48% at -55°C . A lso shown in Fig. 6 -1 9 is that the analytical prediction o f variation
in Ic for the im proved design m atches very w ell w ith the m easured response for the entire
range o f tem perature from -55°C to +105°C . T h e passive bias network can be m ade m ore
stable w ith respect to am bient tem perature by replacing the passive circu it elem ents w ith
resistors having even higher positive tem perature coefficients such as base diffused or
epitaxial resistors having tem perature coefficients o f +2000 ppm/°C and + 3 0 0 0 ppm /°C
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Partial differentials of Ic w.r.t. device param eters
4e-3
BE
2e-3
-2 e-3
-4e-3
-55
-35
-15
5
45
25
65
85
105
65
85
105
T em p eratu re (°C)
Partial differentials of Ic w.r.t. passive elements
10e-5
Rc
Rb
5e-5
R2
-5e-5
-10e-5‘
-55
-35
-15
5
25
45
T em p eratu re (°C)
Fig. 6-18. Partial differentials o f IC w ith respect to device model param eters and passive
circuit elem en ts
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172
Measured response - Initial design
(using carbon film resistors)
Measured response - Improved design
(using Metal film resistors)
Analytical Prediction - Improved design
(using Metal film resistors)
<
E
c
a
Urn
3
u
!_.
u
J£
~3
U
0.8
0.6
-55
-35
-15
5
25
45
65
85
105
T em perature (°C )
Fig. 6-19. C om parison o f initial design and im proved response o f passive DC bias
network in Fig 6-6 (b). A lso show n is the analytical prediction o f the im proved
response.
respectively [Cam 69, G ra84, G ha94]. The im provem ents in perform ance predicted
analytically for the passive b ias netw ork shown in Fig. 6-6 (b) by replacing the passive
circu it elem ents are show n in Fig. 6-20. As shown in Fig. 6-20, by choosing suitable types
o f resistors having different tem p eratu re coefficients, even the passive bias design can be
m ade therm ally stable. T he initial design using carbon film resistors yielded a variation o f
+ 25.37% at +100°C, w hich w as reduced to 20.77% sim ply by replacing all carbon film
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173
1.3
Initial design - Carbon film resistors
Metal film resistors
Rc replaced with base-diffused resistor
Rc replaced with epitaxial resistor
Collector Current (mA)
1.2
1.1
1.0
0.9
O— €> Rq & R1 replaced with base-diffused resistor
A —A R2 replaced with base-diffused resistor
0.8
'Ti
v-r Rc and Rb replaced with epitaxial
<3
<3
and base-diffused resistors respectively
R2 and R1 replaced with base-diffused
and metal film resistors respectively
0.7
-55
-35
-15
5
25
45
65
85
105
T em perature (°C)
Fig. 6-20. A nalytical predictions o f the im provem ents in passive bias netw ork by
replacing the passive circu it elem ents w ith resistors o f higher positive
tem perature coefficients. Initial design and im proved design with metal film
resistors show n w ith filled circles and squares have b een experim entally
verified. T hese results have been included for ready reference. The
im provem ents in perform ance with unfilled sym bols are analytical predictions.
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174
resistors w ith m etal film resistors. A s show n in Fig. 6-18 (b), the rate o f change o f
collector current with respect to R2 is the highest com pared to o th er passive elements,
w hile
is the next m ost dom inant resisto r to cause significant changes in the collector
current. By replacing R 2 w ith the base diffused resistor, the variation in collector current
has been reduced from +25.37% to o nly +3.99% at 105°C, w here as replacem ent o f Rc
w ith base diffused resistor resulted in a reduction o f approxim ately 10% o f variation at
+ 105°C. A sum m ary o f variations in co llecto r current predicted analytically is presented
in Table 6-6. T he variations in co llecto r current with tem perature have been reduced to
less than 1% at +105°C by replacing R 2 and R.[ w ith base diffused and m etal film resistors
respectively. In the im proved designs, all the resistors are carbon film resistors except
those w hich have been replaced w ith o th e r types o f resistors and m entioned in Fig. 6-20
and Table 6-6.
Table 6-6. Analytical Predictions o f AIC o f Passive 2 N etw ork w ith Tem perature.
D escription o f D esign
= 3V & I^ = 1mA)
AIC at +105°C (%)
Initial design w ith carbon film resistors
+23.13
Im proved design with m etal film resistors
+ 18.15
Im proved design - RC replaced w ith base diffused resistor
+ 15.44
Im proved design - RC replaced w ith ep itax ial resistor
+6.635
Im proved design - RC & R l replaced w ith base diffused resistors
+4.150
Im proved design - R2 replaced w ith base diffused resistor
+3.990
Im proved design - RC & RB replaced w ith epitaxial and base dif­
fused resistors respectively
+ 1.060
Im proved design - R2 & R l replaced w ith base diffused an d m etal
film resistors respectively
+0.852
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175
Optimization of Active Bias Network
To gain an insight into the perform ance o f the active DC bias netw ork in Fig. 6-7
(a) and to optim ize the design for am b ien t tem perature effects, the partial differentials o f
Iq 2 w ith respect to passive circuit elem en ts are shown in Fig. 6-21. T he partied differentials
2
e
£
5
o
4e-4
R2
R3
R4
2e-4
i
CN
u
o
-2e-4
-o
*2
£
-4e-4
-55
-35
-15
5
25
45
65
85
105
T em perature (°C)
Fig. 6-21. Partial differentials o f 1^ w ith respect to passive circuit elem ents
o f Ic 2 w ith respect to R2 and R3 are positive, thus m aking them good candidates for being
chosen as the com pensating elem ent(s). T he initial design gets im proved significantly by
replacing the carbon film resistors fo r R2 and R3 only (in the initial design) w ith m etal film
resistors o f sam e value. The im proved perform ance o f active bias netw ork in Fig. 6-7 (a) is
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176
show n in Fig. 6-22. As shown in Fig. 6-22, th e variation in bias cu rren t ( I c i) has reduced
1.2
1.1
Measured response - Initial design
(using carbon film resistors)
Measured response - Improved design
(R2 & R3 replaced with metal film resistors)
Analytical Prediction - Improved design
(R2 & R3 replaced with metal film resistors)
c
<
>
;—L
i
3
u
u
jj
"o
U
5
25
45
105
T em p eratu re (°C)
Fig. 6-22. C om parison o f initial design and im p ro v ed response o f active D C bias netw ork
in Fig 6-7 (a). A lso shown is the an aly tic al prediction o f the im p ro v ed response.
significantly from -17.95% at -55°C to - 1 1.60% and from 19.69% to + 1 1 .9 1 % at +105°C .
T he active bias netw ork can be further im p ro v ed by replacing R 2 and R 3 w ith resistors o f
suitable tem perature coefficients as show n in Fig. 6-23. Figure 6-23 show s that by
replacing only R 2 w ith base diffused resistor, the change in co llecto r cu rren t at +105°C
has been reduced by over 11% i.e. from + 1 9 .6 9 % to 8.65% . By ch o o sin g ep itax ial resistor
for R 2, the design im proves even m ore and A I^ at +105°C becom es o n ly + 4.63% . A nother
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
177
com bination o f base diffused resistor for R 2 an d m etal film resistor for R 3 results in further
reduction o f variation in collector current w ith am b ien t tem perature i.e. only +2.00% at
+105°C . A sum m ary o f variations in co llecto r current predicted analytically is presented
in Table 6-6. T he variations in collector cu rren t w ith tem perature have been reduced to 2%
at +105°C by replacing R2 and R 3 w ith base diffused and m etal film resistors respectively.
In the im proved designs, all the resistors are carbon film resistors except those w hich have
been replaced w ith other types o f resistors an d m entioned in Fig. 6-20 and Table 6-7.
Table 6-7. A nalytical Predictions o f AIC o f A ctive N etw ork w ith Tem perature.
D escription o f D esign
(V CC = 3V & Ic = 1mA)
AIC at + 1 05°C (% )
Initial design w ith carbon film resistors
19.69
Im proved design - R2 & R3 replaced w ith m etal film resistors
11.91
Im proved design - R2 replaced w ith base diffused resisto r
8.65
Im proved design - R2 replaced w ith epitaxial resistor
4.63
Im proved design - R2 & R3 replaced w ith base diffused and
m etal film resistors respectively
2.00
Sum m ary
T h is chapter has presented a desig n -o rien ted analysis o f am bient tem perature
effects on D C bias networks. A G um m el-P oon b ased sim plified model for B JT (both npn,
pnp and diode connected npns) has been developed, used and verified for D C bias
calculations. Several bias circuits com m only used in RF/M icrow ave circuits have been
analyzed and com pared against m easurem ents and H SP IC E sim ulations. The analysis and
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
178
Initial design - Carbon film resistors
R2 & R3 replaced with metal film resistors
<
c
o
3
U
o
o
u
o
U
0.9
O— O R2 replaced with base-diffused resistor
A —A R2 replaced with epitaxial resistor
y r ? R2 and R3 replaced with base diffused
and metal film resistors respectively
0.8
-55
-35
-1 5
5
25
45
65
85
105
Tem perature (°C)
Fig. 6-23. A nalytical predictions o f the im provem ents in active bias netw ork by replacing
the passive circu it elem ents w ith resistors o f hig h er positive tem perature
coefficients. Initial desig n and im proved design w ith R2 and R3 replaced with
metal film resistors show n with filled circles and squares have been
experim entally verified. T hese results have been included for ready reference.
The im provem ents in perform ance w ith unfilled sym bols are analytical
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
179
m easurem ent results have shown excellent agreem ent over the entire tem perature range.
T he m easurem ents show ed that the variation in collector current due to am bient
tem perature is linear and the analysis presented in this chapter predicted the sim ilar trend.
It has been shown in this chapter that both passive and active DC bias netw orks can be
optim ized by selecting suitable technology for the com pensating elem ent(s) while
retaining the basic topology o f the design. A nalysis, experim ental verification, and
optim ization o f one passive and tw o active D C bias netw orks has been included in this
chapter. The analytical procedure presented in this chapter has been successfully
im plem ented for all the bias netw orks m entioned in this chapter during this w ork.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER 7
C O N C L U SIO N S A N D F U T U R E W O R K
This dissertation has presented d esig n -o rien ted analysis o f therm al effects on RF/
M icrow ave circuits. T he m otivation for this w o rk is the technological advancem ents being
m ade in the R F/M icrow ave circuit design. T h is phenom enal grow th has resulted in ever
increasing ap plications o f p lanar R F /M icrow ave circu its in the fields o f com m unications,
electronic w arfare, radar, and sophisticated w eap o n system s. The system s em ploying RF/
M icrow ave circu its operate on land, at sea, in air. an d in space. B esides m any other
environm ental variables, tem perature is th e m o st obvious one w hich changes from
application to application and even w ithin th e sam e system . D ue to therm al dependence of
the physical properties o f the m aterials u sed to fabricate the sem iconductor devices,
alm ost every tran sisto r param eter gets directly o r indirectly influenced by tem perature.
T herefore, the electrical properties (currents an d voltages) o f a device are also tem perature
dependent. T h is broad therm al d ependence im plies that s-param eters m ust also be
tem perature dependent. Since s-param eters are used extensively is R F/M icrow ave circuit
design, so th erm al dependence o f s-param eters is a logical starting p o in t for analysis o f
therm al effects on R F/M icrow ave circuits. A lso, in o rd er to design therm ally stable RF/
M icrow ave circu its, the tem perature effects (b o th am bient and self-heating) need to be
considered.
B ipolar ju n c tio n transistors w ere ch o sen fo r this work due to th e ir dom inance in
R F/M icrow ave circu its from U H F to S -band. A lso, w ith the im provem ents in planar
180
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
181
technology, the upper frequency lim it for bipolar transistors is being continuously
extended, thus offering an even g reater prom ise for increased applications in M icrowave
circuits at m uch higher frequencies than for w hich they are being used at present.
T he n eed to establish a link betw een sm all-signal s-param eters and equivalent
circuit m odel for active devices w as ju stified in C hapter 3. It w as show n that since sparam eters w ere defined as the ratios o f the incident and reflected pow er w aves o f n-port
networks, therefore neither m easurem ents nor com puter sim ulations to predict sparam eters co u ld provide the m uch needed insight into the device itse lf to identify the
sources w hich caused variations in s-param eters due to tem perature. A relationship
between sm all-signal s-param eters an d high-frequency equivalent circuit model for
bipolar ju n ctio n transistor was estab lish ed , since no prior w ork in this regard w as found.
C losed-form ex p ressio n s for tw o-port s-param eters in the co m m o n -em itter configuration
derived using the intrinsic high-frequency sm all-signal m odel fo r BJTs have been
experim entally
verified.
E xcellent agreem ent has been
show n
betw een
on-w afer
m easurem ents and s-param eters pred icted using closed-form expressions from 45 M Hz to
20.045 G H z. It w as shown that s-param eters predicted using clo sed -fo rm expressions
m atched the m easurem ents m ore clo sely com pared to s-param eters obtained through
MDS sim ulations.
S cattering-param eter m easurem ents for packaged active devices m ounted on
specially desig n ed boards w ere presen ted in C hapter 4. A step-by-step procedure to obtain
s-param eters fo r packaged devices reliab ly was presented and the results w ere com pared
with the m easurem ent data p rovided by the m anufacturer. It was show n th at intrinsic highfrequency m odel for B JTs did not estim ate m agnitude and phase o f s-param eters correctly.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
182
Im proved closed-form expressions w ere derived after m odifying the equivalent circuit
m odel. The com parison between the sm all-signal s-param eters predicted using the
im proved expressions and the m easured d ata show ed a significant im provem ent. It was
also show n that for BJTs having p aram eter X ^ jc equal to one, the im proved model
reduced to the intrinsic model and clo sed -fo rm expressions derived in C h ap ter 3 were
sufficient to p red ict s-param eters reliably.
Q ualitative study was presented in C h ap ter 3 to dem onstrate that b o th m agnitude
and phase o f sm all-signal s-param eters o f bipolar junction transistors change w ith
variations in bias. T he sm all-signal s-param eters m easured on-w afer at several different
collector currents w ere com pared and an insightful sum m ary was com piled in C hapter 3.
A fter having established the reliability o f sm all-signal s-param eter m easurem ents for
packaged transistors in Chapter 4, a com parison o f sm all-signal s-param eters m easured at
several different tem peratures was show n to describe qualitatively the tem perature effects.
It was show n that both m agnitude and phase o f sm all-signal s-param eters w ere effected by
the change in am bient tem perature.
To gain an insight into the active device itself to identify the sources w hich cause
variations in sm all-signal s-param eters d u e to changes in bias and tem perature, a
system atic sensitivity analysis o f sm all-signal s-param eters to the sm all-signal model
param eters w as presented in C hapter 5. A form al procedure form ulated during this
research to perfo rm sensitivity analysis o f sm all-signal s-param eters w as described and
successfully
im plem ented. A com parison
o f the norm alized sensitivities o f both
m agnitude an d phase o f each sm all-signal s-p aram eter w ith respect to C^, C^, C bcx, rrt, r0,
and gm was presented to establish a relative rank o f sm all-signal m odel param eters in
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
183
term s o f their effect on sm all-signal s-param eters. It was shown that the resistive m odel
param eters rrt and ru had negligible effect on sm all-signal s-param eters alth o u gh they
them selves w ere quite sensitive to changes in both bias current and tem perature. T he tw o
capacitive m odel param eters
and C bcx though effect the sm all-signal s-param eters
significantly, but their ow n variations w ith bias cu rren t and tem perature w ere negligible.
T h e other com ponents o f the m odel rb, re, an d rc, since had been assum ed co n stan t during
this w ork, so their effect on sm all-signal s-p aram eter w as neglected. T hrough a system atic
process o f elim ination, CK an d gm were identified as the dom inant model param eters. By
com paring the results o f the sensitivity analysis perform ed on three devices, a typical
trend for the sensitivity o f sm all-signal s-param eters w ith respect to sm all-signal m odel
param eters was established. W ith the help o f the results presented in C h ap ter 5, the
sensitivity o f sm all-signal s-param eters to changes in bias current and tem perature
presented in C hapters 3 and 4 w as explained. T he sensitivity analysis presented C h ap ter 5
has not only provided a valuable insight into the active device itself, but it also highlighted
the significance o f the closed-form expressions for sm all-signal s-param eters derived in
C hapters 3 and 4.
Bias sensitivity o f sm all-signal s-param eters presented in C hapter 3 show ed that
both m agnitude and phase o f s-param eters w ere sensitive to changes in co llecto r current.
W hile the collector current itself is quite sensitive to tem perature. T herefore, in order to
realize a therm ally stable R F/M icrow ave circuit, the need to m aintain a co n stan t collector
cu rren t in the circu it w as justified. A design analysis o f the am bient tem perature effects on
typical DC bias circuits used in R F/M icrow ave circu its was presented in C h ap ter 6. A
sim plified m odel for the BJTs based on the G um m el-P oon model w as developed and used
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184
to analyze the D C bias circuits in forw ard-active m ode. A step-by-step procedure w as
show n that p redicts the am bient tem perature effects on the operating point. The analytical
results w ere verified w ith m easurem ents and H SP IC E sim ulations for a w ide range o f
am bient tem p eratu re variations (-55° C to 125° C). Estim ations o f collector current w ith
tem perature sh o w ed excellent agreem ent w ith the m easurem ents and sim ulations over the
entire tem p eratu re range. Seven different DC bias circuits w ere analyzed. Key factors in
the tem p eratu re sensitivity o f each circuit w ere identified. Based upon the insight gained,
an optim izatio n strategy was developed, w hereby DC bias design could be im proved
significantly by selection o f the appropriate and suitable technology for the passive circu it
elem en ts w hile the circu it topology rem ained unaltered.
T h e orig in al w ork presented in this dissertation has established the necessary
foundations to undertake som e very interesting an d m uch needed research in the areas
such as th erm al effects on (a) stability o f b oth RF/M icrow ave transistors and am plifiers,
(b) im pedance m atching networks, and (c) unilateral figure o f m erit. Besides having
provided a basis to proceed w ith the proposed research, need for (a) self-heating effects on
D C bias netw orks, and (b) optim ization o f D C bias netw orks for both am bient tem perature
and self-h eatin g effects has also been highlighted and justified. A b rief description o f the
proposed future w o rk is as under:
Design-Oriented Therm al Analysis
A s an ap plication o f the results o f sensitivity analysis proposed above, it is fu rth er
suggested th at the stan d ard RF/M icrow ave circu it design procedure be revisited in view o f
the therm al effects on s-param eters. The areas o f proposed study include but not lim ited to
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185
stability, im pedance m atching netw orks, unilateral figure o f m erit, and gain circles etc.
Thermal Effects on Stability
One very im portant consideration in R F/M icrow ave am plifier design is its stability,
or its resistance to oscillate. The stability o f a tw o-port is determ ined using the sparam eters data. It is proposed that therm al effects on stability o f a tw o-port be ascertained
in view o f the variations in s-param eters due to tem perature.
Thermal Effects on Impedance Matching Networks
For an am plifier to deliver m axim um pow er to the load, o r to perform in a certain
manner, p ro p er term inations m ust be connected both at the input and also at the output
ports. These term inations are usually term ed as im pedance m atching networks. To
determ ine w hat im pedance is m atched at input or output port, know ledge o f s-param eter
data is necessary. Since s-param eters are functions o f therm ally sensitive device
param eters, therefore it is proposed that therm al effects on both the requirem ents and
design o f im pedance m atching netw orks be analyzed.
Thermal Effects on Unilateral Figure of Merit
U nilateral figure o f m erit is a m easure o f determ ining the erro r involved in
assum ing the device to be unilateral. This is done be setting S 12 equal to zero. The
motivation for settin g S 12 to zero, is the ease in RF/M icrow ave design in unilateral case.
U nilateral figure o f m erit is know n to be a function o f frequency. H owever, in view o f the
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186
therm al dependence o f s-param eters, it is proposed that therm al dependence o f unilateral
figure o f merit be studied so as to quantify the error involved both as a function o f
frequency and also tem perature.
Self-Heating Effects on DC Bias Networks used in RF/Microwave Circuits
It is well known th a t the operating tem perature o f a device is determ ined by the
am bient tem perature and th e self-heating o f the device. T he am bient tem perature effects
on D C bias netw orks have already been analyzed and presented in this dissertation. To
analyze the DC bias netw orks com pletely, self-heating effects need to be considered. T h e
significance o f self-heating in view o f technological advancem ents and its effects on largesignal circuit behavior have been discussed in C hapter 2. Future work in this area is
expected to result in interesting and valuable contributions.
Optimization of DC Bias Designs for Ambient and Self-Heating Effects
O ptim ization o f D C bias netw orks for am bient tem perature effects was p resented
in C hapter 5. It is pro p o sed that after com plete therm al characterization o f DC bias
netw ork for both am bient and self-heating effects, a unified strategy be developed to
optim ize the bias netw orks so as to m inim ize the changes in RF/M icrow ave perform ance
over a range o f am bient tem perature and pow er dissipation.
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A P P E N D IX A
C LO SED -FO R M E X P R E S S IO N S FO R SM A L L -SIG N A L S-PARA M ETERS
This appendix is intended to p resen t the derivation o f closed-form expressions for
sm all-signal s-param eters for bip o lar ju n c tio n transistors. T hese closed-form expressions
have been derived using the intrinsic high-frequency equivalent circuit m odel shown in
Fig. 3-1. T he entire sequence o f the derivation has already been given in C hapter 3.
To proceed with the s-p aram eter representation in term s o f device m odel
param eters, an equivalent circuit in term s o f adm ittances is draw n along w ith the current
sources connected to each node as show n in Fig. 3-2. T h e sam e equivalent circuit is again
show n in Fig A -l for ready reference. A system o f nodal equations for the voltages
betw een nodes 1,2,3,4,5, and 6 an d an external reference node indicated as a com m on
node are w ritten.
Il = Yb V , - Y b V 2
(A -l)
h = ' Yb v i + (Y b + Ypi + Yu) V 2 - Ypi V 3 - Yu V5
(A -2)
l3 = - ( Y p i + gm) V 2 + (Ypi + Ye + Y0 + g m) V 3 - Y e V4 - Y 0 V5
(A -3)
U = - Ye V 3 + Ye V4
(A -4)
15 = (Sm - Yu) V 2 - (Y0 + g m) V 3 + (Y u + Y0 + Y c) V5 - Y c V6
(A -5)
16 = - Y c V5 + Y c V6
(A -6)
187
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188
VI
V5
V2
,V6
gm (V2 - V3)
V3
V4
------- #
/
common node
Fig. A -j. Equivalent circuit for determ ining the indefinite adm ittance m atrix.
N odes 2, 3, and 5 are inaccessible, so we need to parcel o ut th eir effect on circuit
behavior to the other nodes. T his is done by sequentially setting th e cu rren t sources I2, 13,
and I5 to zero and substituting V 2, V 3, and V 5 in the nodal eq uations until we get expres­
sions independent o f V 2, V 3, and V 5. Set I2=0 and solve for V 2 in term s o f other node volt­
ages. T he expression for V 2 after setting I2 = 0 becom es
V 2_n ew = - (- Yb V , - Y pi V 3 - Yu V5) / (Y b + Y pi + Yu)
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(A-7)
189
Now substitute V 2_new for V 2 in Ij, I3, and I5. This w ould elim inate V 2 from the indefi­
nite m atrix. T he resulting equations independent o f V2 are
I |_ w „ _ V 2 = Y b V , + Y b (- Y b V ,
- Y pi V , - Yu V 5) / (Y b +
Y pi+ Yn )
(A -8 )
l 3 _ w ^ V , = ( Y pi + gm) (- Y „ V , - Y ^ V 3 - Y„ V s ) / (Y b + Y pi + Y„) + ( Y p, + Ye + Y0 +
(A - 9 )
V 3 - Ye V 4 - Yc V 5
I s - w o - v 2 = - (gm - Y„) <- Y „ V , - Ypi V 3 - YuV 5) / (Y „ + Ypi +
gm)
Yu)- (Y 0 + gm> V 3 + (Y„
+ Y0 + Y c ) V 3 - Y c V 6
(A -10)
This procedure is repeated for other two nodes 3 and 5 and eventually the follow ing nodal
equations independent o f V 2, V 3, and V5 are obtained.
Il_ w o _ .V i_ V 3 _ .V 5 = Y b (- Yp, Ye V 4 Yu + V j Yp, Y c Ye + Yc V | Yc Yu - Y„ Ye V 4 Y pi + Yc
V , Yu Y c - Yc V4 Y p, Y c - Y C YU Y c V 6 + V , Y p, Y0 Ye + V , Y p, Ye Y„ + Y0 V , gm
Yu - Ye V 4 Yb Yu - Ye V 4 g m Y0 - Y c V 6 Y p, Yc + V , Y pi Y c Y0 - Y c V 6 Y pi Y„ + V ,
Ypi Y„ Y c + V , Yu Y c Yb + V , Yu Y c g m - Yu Y c V 6 Yc - Y„ Y c V 6 g m) (Y pi Y b +
Ypi Yu + Ye Y b + Ye Yp; + Ye Yu + Y0 Y b + Y0 Y pi + Yb Yu + g m Y b + gm Yu) /
((Y p, Y b + Yp, Yu + Yc Y „ + Ye Y p, + Yc Y„ + Y0Y b + Yb Y pi + Y0 Yu + g m Y b + g m
Yu) (Y b Yp, Y„ + Y b Ye Y„ + Yb Y„ Yu + Yb gm Y„ + Y„ Yb Ypi + Yb Yb Ye + Yc Y„
Yp, + Yb Yc Ye + Yb Yc Y0 + Y„ Yc gm + Yb Yc Yu + Yc Ye Y pi + Ypi Ye Y„ + Yc
gm Yu + gm Yu Ye + Yc Ye Yu + Yc Yb Yu + Yb Ye Y pi + Yc Ypi Y„ + Yc Y„ Ypi))
(A-l 1)
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190
Io-W oJ V j .V ,- = - Yc (Y b V , Y0 Ypj - Yb Ypi V4 Y„ + Y b V , Y pl Yc + Yb V , Yc gm- Y„ V4
Yo Yu + Yb Yc V6 Y0 + Yb V , Y„ Y„ - Y„ Ypi V4 Yc + Yb V , Y„ g m - Yb V4 Y„ Yc
- Y b V 4 g m Yu - Y b V 4 Y pj Y„ - Y b V 4 g m Y c + Y b V , Y pi Y„ + Y c V 6 Y p, Y0 + Y c
V 6 Y pi Yu + Yu Y c V 6 Y„ + Yu Y c V 6 g m - Y pi V 4 Y c Y „ - V 4 Y pi Y c Y„ - V 4 Y „ Y c
Yu - V 4 g m Y c Yu) ( Y pj Y b + Y p i Y11 + Ye Y b + Y e Y pj + Yc Y„ + Yc Y b + Y0 Y p, +
Y0 Yu + g m Y b + g„, Yu) / ((Y pj Y b + Y p j Y„ + Yc Y „ + Ye Y pj + Ye Yu + Y0 Y b +
Y0 Y pj + Yc Yu + gm Y b + g m Yu) ( Y b Y pi Y„ + Y b Yc Yu + Y b Y„ Yu + Y b g m Yu +
Y „ Y0 Ypj + Y b Y0 Ye + Yc Y b Y pi + Y b Y c Ye + Y b Y c Y0 + Y b Y c gm + Y„ Yc Y„
+ Y c Ye Y„j + Y pj Ye Yu + Y c gltl Yu + g m Y „ Yc + Y c Ye Yu + Y c Y „ Yu + Y„ Yc Y pi
+ Y c Ypj Yu + Y c Y0 Ypj))
(A -12)
= - (Y b V , Y0 Yu + Y„ V , Y0 Ypi + Yb g„, Yc V4 + Yb V , Ypi Yu - V6 Yu Ye Yb
+ Y b Y0 Ye V4 - Y b V6 Yo Ye - V 6 Yu Y b Y pj - V 6 Yo Yb Ypj - V6 Y„ Y b Yu - Y b g m
V[ Yc + Yb V | Yu gm - V6 gm Yu Yb + Yb V , Ye Yu - V6 Y„ Yc Yu - V 6 Y0 Ye Y pi
- V6 gm Yu Ye + Yp, Ye V4 Yu + Y0 Yc V4 Ypi + Yc V4 Yc Y„ + Yc V4 gm Yu - V6 Yu
Yc Ypj) Yg / (Y b Ypj Yu + Yb Ye Yu + Y b Y„ Yu + Yb gm Ya + Y„ Yc Ypi + Y b Y0
Ye + Yc Yb Yp, + Yb Yg Ye + Yb Yc Y0 + Yb Y c g m + Y0 Ye Yu + Yc Ye Ypj + Ypj
YgYu + Y g g m Y„ + gIn Yu Yg + Y gY 0 Yu + Y gY o Yu + Y0 YgYpj + Yc Y pjY u + Yg
Y0 Ypj)
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(A -13)
191
N ow having w ritten nodal equations for nodes 1, 4, and 6. To get Y -param eters for
com m o n -em itter configuration, we need to ground node 4 (connecting the em itter term i­
nal). This can be achieved by crossing o u t the row and colu m n corresponding to node 4.
We can sim ply w rite Y-param eters for tw o ports (com m on-em itter) by neglecting the V 4
term s from the latest expressions for Ij and I6. T hus Y -param eters are as under:
Y11 = Y b (((Y c + Ye) Y pi + (Y C+ Ye) Yu) Y0 + ((Y C+ Y e) Yu + Y c Ye) Ypi + ((Ye + gm)
Y c + g m Ye) Yu) / (((Y c + Y pi + Yu + Ye) Y0 + (Y c + Yu) Ypi + (Ye + g m) Y c + (Ye
+ gm) Yu) Yb + ((Yc + Ye) Y ^ + (Y c + Ye) Yu) Y0 + ((Y c + Ye) Yu + Y c Ye) Y pi +
((Ye + g m) Y c + gm Ye) Yu)
(A -14)
Y 12 = Y b ((- Ypi - Yu) Y0 - Y ^ Yu + (- Ye - g m) Yu) YC/(((Y C + Yp, + Yu + Ye) Y0 + (Y c +
Yu)
+ (Ye + gm) Yc + (Ye + g m) Yu) Yb+ ((Yc + Ye) Y p, + (Yc + Ye) Yu) Y0 +
((Y c + Ye) Yu + Yc Ye) Y pi+ ((Ye + g m) Yc + g m Ye) Yu)
(A -15)
Y 2 i = - (( Y pi + Yu) Y0 + Y ^ Yu + (Yc + g m) Yu - gm Ye) Y b Y C/( ( ( Y C + Y p, + Yu + Ye) Y0
+ (Y c + Yu) Ypi + (Y e + g m) Y c + (Y e + g m) Y u) Y b+ (( Y c + Ye ) Y pi + ( Y c + Ye ) Yu)
Y0 + ((Y c + Ye) Yu + Y c Ye ) Y pi+ ((Y e + g m) Y c + g m Ye) Yu)
(A -16)
Y 2 2 = " (((* Yu - Ypj - Ye) Y0 - Ypj Yu + (- Ye - g m) Yu) Y b + (- Ye Y pi - Ye Yu) Y0- Y pi Ye
Yu - g m Yu Ye) Y c / (((Y c + Y pi + Yu + Yc) Y0 + (Y c + Yu) Y pi + (Ye + gm) Y c +
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192
(Ye + gm) Yu) Yb+ ((Yc + Ye) Y pj + (Yc + Ye) Yu) Y0 + ((Y c + Ye ) Yu + Yc Ye) Ypi+
((Yc + gm) Yc + gm Ye) Yu)
(A -17)
Before proceeding fu rth er the Y -param eter equations n eed to be m odified appropri­
ately to take into account the n o rm alizatio n with respect to th e characteristic impedance.
N orm alized Y-parameters are as under:
Y ,,_ n = Y n Z o
CA - 18)
Y12—n= Y12 Z0
(A -19)
Y 2l-n = Y 21 ZQ
(A-20)
Y 22-n=Y 22 Z0
(A-21)
To convert norm alized co m m o n em itter Y -param eters into com m on em itter Sparam eters, we use conversion tables given in m any texts. T h e expression for D2 in term s
o f norm alized tw o-port adm ittance param eters is
D 2=( 1+YI l_ n) ( l+ Y 22_nH Y l2-nY 21-n)
(A-22)
D 2 = (((((Z o + Z02 Yc)Y e + 2 Z 0 Yc + l ) Y 0+ ((Z 0 + Z02 Yc) Ye + 2 Z 0 Yc + I) Yu + Y c +
ZG Yc Ye) Ypi + (((Z 0 + Z 02 Y c) Ye + 2 Z0 Yc + I) Yu + (1 + Z 0 Yc) Ye + Yc) Y0+
(((2 Z 0 + ZQ2 g m) Yc + I + Z0 g m) Ye + gm + 2 Z0 Yc gm) Yu + Yc g m + Yc Ye) Yb
+ (((1 + Z0 Yc) Ye + Yc ) Y0 + ((1 + Z 0 Yc) Ye + Yc) Yu + Yc Ye) Y pi + ((1 + Z 0 Yc)
Ye + Yc) Yu Y0 + ( ( ( ! + Z Q g m) Yc + gm) Ye + Yc gm) Yu) / ((('Yu + Y0 + Yc) Ypi +
(Ye + Yu + Yc)Y 0 + (Ye + gm) Y u + Ycgm + Yc Ye) Yb + ((Ye + Yc) Y0 + (Ye + Yc)
Yu + Yc Ye) Ypi + (Ye + Y c) Yu Y0+ ((gm + Yc) Ye + Y c g m) Yu)
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(A -23)
193
Now the S -param eters can be found using norm alized Y-parameters and D 2
S ,1= ((1-Y 1i_ n)( l+ Y 22_ n)+(Y 12-0^2l_ n ) ) ^ 2
(A -24)
s , I = - (((((z 0 + Z 02 Yc) Ye - 1) Y0 + ((ZQ + ZQ2 Yc) Ye - 1) Yu + ZG Yc Ye - Y c) Y pi +
(((Z G + ZQ2 Yc) Ye -1 ) Yu + (- ZG Y c - 1) Ye - Yc) Y0+ ((Z0 gm + ZQ2 g m Yc - 1) Ye
- gm) Yu - Y c Ye - Yc g j Yb+ (((- Z Q Yc - 1) Ye - Yc) Y0 + ((- Z0 Yc - 1) Ye - Yc)
Yu - Y c Ye) Ypi+ ((- Z0 Yc - 1) Ye - Yc) Yu Y0 + (((- 1 - Z0 gm) Yc - gm) Ye - Y c gm)
Yu) / (((((Z 0 + Z 02 Yc) Ye + 2 ZQ Yc + I ) Y0+ ((ZQ+ Z 02 Yc) Ye + 2 Z Q Yc + I) Yu
+ Yc + ZQ Yc Ye) Ypi+ (((ZQ+ Z02 Yc) Ye + 2 Z0 Yc + 1) Yu + (1 + Z0 Yc ) Ye + Yc)
Y0 + ( ( ( 2 Z 0 + Z 02 gm) Yc + 1 + ZQ g m) Ye + g m + 2 ZQ Yc §m) Yu + Yc g m+ YC Ye)
Yb + (((1 + Z 0 Yc)Y e + Yc)Y 0 + ( ( l + Z 0 Yc) Ye + Yc)Y u + Yc Ye ) Y pi + ((l + Z G
Yc) Ye + Yc) Yu Y0 + (((1 + ZG gm) Yc + gm) Ye + Yc gm) Yu)
S i2 = -2 Y 12_ n/D 2
(A -25)
(A -26)
S 12 = 2 Z 0 Yc ((Yu + Y0) Ypi + Y0 Yu + (Ye + gm) Yu) Yb / (((((Z0 + ZQ2 Yc) Ye + 2 Z 0 Yc
+ l ) Y o + ((Z o + Z02 Yc) Y e + 2 Z o Yc + 1) Yu + Yc + Z0 Yc Ye) Ypi + (((Z 0 + ZG2
Yc) Ye + 2 ZQ Yc + 1) Yu + (1 + ZG Yc) Ye + Yc) Y0 + (((2 Z0 + Z D“ g m) Yc + I +
Z o g m)Y e + g m + 2 Z 0 Yc gm)Y u + Yc gm + Yc Ye) Yb + (((1 + Z 0 Y c) Ye + Yc) Y0
+ ( ( l + Z 0 Yc)Y e + Yc) Y u + Yc Ye) Ypj + ((1 + Z 0 Yc) Ye + Yc) Yu Y0 + (((1 + Z 0
gm)
Y C
+ gm) Ye + Yc gm) Yu)
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(A -27)
194
S2I=-2Y2In/D 2
(A-28)
5 21 = 2 Z0 Yc Y b ((Yu + Y0) Y pi + Y0 Yu + (Ye + g m) Yu - g m Ye) / (((((ZD + ZD2 Y c) Ye +
2 Z 0 YC+ 1 ) Y 0 + ((Z 0 + Z 02 Yc) Ye + 2 Z o Y0 + l ) Y u + Yc + Z0 Yc Ye) Y pi +
(((ZD + ZG2 Yc) Ye + 2 ZD Yc + 1) Yu + (1 + Z 0 Yc) Ye + Yc) Y0 + (((2 Z 0 + Z02
gm) Yc + l + Z Dg m) Ye + cm + - Zo Yc gm) Yu + Yc gm + Yc Ye) Yb + (((1 + Z0 Yc)
Ye + Yc)Y 0 + ( ( I + Z 0 Yc)Y e + Yc)Y u + Yc Ye) Ypi + ((1 + Z0 Yc) Ye + Yc) Yu Y0
+ ( ( ( ! + Z0 gm) Yc + g m) Ye + Yc gm) Yu)
S22= (( 1+Y i j_n)( )-Y22n )+ (Y |2riY2i_n))/D2
(A-29)
(A-30)
5 22 = - (((((Z02 Yc - Z0) Ye - 1) Y0 + ((ZG2 Yc - Z 0) Ye - 1) Yu - Z0 Yc Ye - Yc) Y pi
+
(((ZQ2 Yc - Z0) Ye - 1) Yu + (ZQ Yc - 1) Ye - Y c) Y0 + ((- 1 + Z02 gm Yc - Z Q g m) Ye
- g m) Y u - Y c gm - Y c Ye) Y b + (((Z0 YC- l ) Y e - Y c) Y 0 + ((Zo Yc - I) Ye - Yc) Yu
- Yc Yc) Ypi + ((Z Q Yc - 1) Ye - Yc) Yu Y0 + (((- I + Z0 gm) Yc - gm) Ye - Yc g m) Yu)
/ (((((Z0 + Z02 Yc) YC + 2 Z 0 YC+ I) Y0 + ((Z 0 + Z 02 Yc) Ye + 2 ZD Yc + 1) Yu +
Yc + Z0 Yc Ye) Y pi + (((Z Q + ZQ2 Yc) Ye + 2 Z Q Yc + 1) Yu + (1 + ZG Yc) Ye + Yc)
Y0 + (((2 ZQ + Zu2 g m) Y c + 1 + ZQ gm) Ye + g m + 2 Z 0 Yc g m) Yu + Yc g m + Yc
Ye) Yb + ( ( ( l + Z 0 Y c) Y e + Yc)Y 0 + ( ( l + Z 0 Yc) Ye + Yc) Yu + Yc Ye) Y pi + ((1 +
Z0 Yc) Ye + Yc) Yu Y0 + (((1 + ZQ gm) Yc + g m) Ye + Yc gm) Yu)
(A -3 1)
In the above expressions w e have been using the adm ittances shown in Fig. A -1.
T hese adm ittances in term s o f sm all-signal m odel param eters are as under
Yb = 1 / rb
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(A -32)
195
Ye = 1 / re
(A-33)
Yc = I / rc
(A -34)
Y0 = 1 / r0
(A -35)
Ypi = (1 + s rpj C pi) / rpi
(A -36)
Yu = (1 + s ru C u) / ru
(A -37)
Substituting the adm ittances into the expressions for S I 1, S12, S21, and S22, we g et the
scattering param eter expressions in term s o f the sm all-signal m odel param eters.
^1 i_new= (((■ 1 ■ s ru C u) r pi ro §m
s C u) ru - 1) Tpj - ru) Z 0
+ ((- s Cpj - s
s Cpj * s~ Cpj ru C u) rp; - I - s ru C u) rQ + ((- s C pj -
+ (((1 + s ru C u) rb rpj + (- 1 - s ru C u) rpj rc) r0 gm + (% 2 rb
Cpj ru C u) Tpj - 1 - s ru Cu) rc - ru - ru s rpj Cpj) rQ+ (% 1 + ru) rb + (((-
s Cpj - s C u) ru - 1) rpi- ru) rc + rpiru) ZQ + (((1 + s ruC u)
rpi rc+ (1 + s ruC u) rpj re)
rb + (1 + s ru C u) rpire rc + ru rerpi) rGg m + ((% 2 rc + % 2 re + % 1 + ru) rb + (% 2 re
+ (1 + s ru C u) r pj) rc + (ru + ru s rpi C pj) re + rpi ru) rQ + ((% 1 + ru) rc + (% 1 + ru) re)
rb + ((% 1 + ru) re + rpi r u) rc + ru re rpi) / (((1 + s ru C u) rpi rGg m + %2 r0 + % 1 +
ru) Z02+ (((1 + s ru C u) rb rpi + ( I + s ru C u) rpi rc + (2 + 2 s ru C u) rpi re) r0 g m +
(%2 rb + %2 rc + ((2 s C pi + 2 s2 Cpi ru C u) rpi + 2 + 2 s ru C u) re + ((s C pi + 2 s C u)
ru + 2) rpi + ru) rQ + (% l + ru) rb + (% I + ru) rc + (((2 s C pi + 2 s C u) ru + 2) rpi + 2
ru) te + rpi ru) Z Q + (((1 + s ru C u) rpj rc + ( I + s ru C u) rpj re) rb + (1 + s ru C u) rpj re
rc + ru re rpi)
r0
grn + (( % 2 rc + %2 re + % 1 + ru) rb + ( % 2 re + (1 + s ru C u) rpi) rc +
(ru + ru s rpi C pi) re + rpi ru) rQ + ((% 1 + ru) rc + (% 1 + ru) re) rb + ((% 1 + ru) re + rpi
ru) rc + ru re rpj)
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(A -38)
196
W here
%1 =
((s Cpi + s C u) ru + 1) rpj
(A-39)
%2 =
(s Cpj + s2 Cpj ru Cu) rpi + 1 + s ru C u
(A-40)
s I2_new = 2 Z0 ( ( I + s ru Cu) rpi re r0 gm + (%2 re + (1+ s ruC u) rpi) r0 +
(% 1 + ru) re) /(((1
+ s ru Cu) rpi rQ gm + %2 rQ + % 1 + ru) Z02+ (((1 + s ru Cu) rb rpi + (1 + s ru Cu) rpi
rc + (2 + 2 s ru Cu) rpi rc) r0 gm + (%2 rb + %2 rc + ((2 s C pi + 2 s2 Cpi ru Cu) rpi +
2 + 2 s ru Cu) re + ((s C pi + 2 s C u) ru + 2) rpi + ru) r0 + (% 1 + ru) rb + (% 1 + ru) rc
+ (((2 s Cpj + 2 s Cu) ru + 2) rpi + 2 ru) re + rpi ru) Z 0 + (((1 + s ru Cu) rpi rc + ( I + s
ru C u) rpi re) rb + ( I + s ru C u) rpi re rc + ru re rpi) rQ gm + ((%2 rc + %2 re + % 1 + ru)
rb + (%2 re + ( I + s ru C u) rpi) rc + (ru + ru s rpi C pi) re + rpi ru) r0 + ( ( % ! + ru) rc +
(% 1 + ru) re) rb + ((% 1 + ru) re + rpi ru) rc + ru re rpi)
(A -4 1)
W here
% l = ( ( s C ^ + s Cu) ru + 1) rpi
%2 —(s C pj + s- Cpj ru Cu) rpj + I + s ru C u
^2l_new
^ Zo ( ( ( 1 + s ru Cu) rpj re - rpj ru) r0 g m + ( %2 re + ( 1 + s ru Cu) rpj) rQ + {% I + ru)
re) / (( (I + s ru C u) rpi rQ g m + %2 r0 + % 1 + ru) Z 02 + (((1 + s ru C u) rb rpi + (1 + s
ru C„) rpi rc + (2 + 2 s ru C u) rpi re) r0 gm + (% 2 rb + %2 rc + ((2 s C pi + 2 s2 C pi ru
C„) rpi + 2 + 2 s ru C u) re + ((s C pi + 2 s Cu) ru + 2) rpi + ru) r0 + (% 1 + ru) rb + (% 1
+ ru) rc + (((2 s Cpj + 2 s C u) ru + 2) rpi + 2 ru) re + rpi r u) ZQ + (((1 + s ru Cu) rpi rc
+ ( l + s r u C u) rpj re) rb + ( l + s r u C u) rpi re rc + ru re rpi) r0 g m + ((% 2 rc + %2 re
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
197
+% 1 + ru) rb + (%2 re + (1 + s ru Cu) rpi) rc + (ru + ru s rpi Cpi) re + rpi ru) r0 + ((% I
+ ru) rc + {% 1 + ru) rc) rb + ((% 1 + ru) re + rpi ru) rc + ru re rpi)
(A-42)
Where
% 1 = ((s Cpi + s Cu) ru + 1) rpi
%2 —(s C pi + s
Cpi ru Cu) rpi + 1 + s ru C u
^22_new — (((” ^ ” s l"u C|j) '■'pi *o §m
^ C pj ”
C pj ^"u C u) *"pi “ ^
^ ^"u C u) *o
^ C pj ”
s Cy) ru - 1) rpi - ru) ZQ + (((■ 1 _ s ru Cu) rpj rb + (1 + s ru C u) rpj rc) r0 gm +(((- s
Cpi ” ^ Cp; ru Cu) rpi - 1 - s ru Cu) rb + %2 rc + ru + ru s rpj Cpj) rQ+ (((- s Cpj - s Cu)
ru - D r pi- ru) rb + (% I + ru) rc - rpi ru) Z0 + (((I + s ru Cu) rpi rc + (1 + s ru Cu) rpi
re) rb + (1 + s ru Cu) rpi re rc + ru re rpi) r0 g m + ((% 2 rc + %2 re + % I + ru) rb + (%2
re + (1 + s ru Cu) rpi) rc + (ru + ru s rpi Cpi) re + rpi ru) ru + ((% 1 + ru) rc + (% 1 + ru)
re) rb + ((% 1 + ru) re + rpi ru) rc + ru re rpi) / (((1 + s ru C u) rpi r0 g m + %2 r0 + % 1
^u) Zo“
(((^
^ ru C u) rb r pi •+■(1 + s ru C u) rpj rc + (2 + 2 s ru C u) rpj re) rQ gm +
(% 2 rb + %2 rc + ((2 s C pi + 2 s2 C pi ru C u) rpi + 2 + 2 s ru C u) re + ((s C pi + 2 s Cu)
ru + 2) rPi + ru) ro + (% 1 + ru) rb + (% 1 + ru) rc + (((2 s Cpi + 2 s C u) ru + 2) rpi + 2
ru) re
rpi ru) ZQ + (((1 + s ru C u) rpj rc + (1 + s ru C u) rpj re) rb + (1 + s ru C u) rpj re
rc + rQre rpi)
r0
gm + ((% 2 rc + % 2 re + % 1 + ru) rb + ( % 2 re + (1 + s ru C u) rpi) rc +
(ru + ru s rPi C pi) re + rpi ru) rD + ((% 1 + ru) rc + (% 1 + ru) re) rb + ((% 1 + ru) re + rpi
'"u) rc
*"u re rpi)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A -4 3 )
198
W here
% 1 = ((s C pi + s C u) ru + 1) rpi
%2 = (s
+ s2 Cpj ru C u) rpi + 1 + s ru C u
These expressions for sm all-signal s-param eters can be further sim plified using
both M A PLE and hand analysis. The sim plified closed-form expressions are as u nder
S , j= (-C Z 02+((rb-rc) C -ru (B r0-rpi)) Z 0+D +E+F) / D enom
(A-44)
S 12=(2 Z q (C re+A rpi rQ)) / D enom
(A-45)
S2 i =(2 Z q (C re+(A -gm ru) rpi rQ)) / D enom
(A-46)
S22= (-C Z02-((rb-rc) C -ru (B r0-rpi)) Z 0+D +E+F) / D enom
(A-47)
W here A = l+ s ru C u, B = l + s rpi C pi, C = A rpi ( l + g m r0)+B (A r0+ru).
D =(A ((rc+re) rb+re rc)+re ru) rpi rQ g m,
E=((A B (rc+ re)+B ru+A rpi) rb+(A rc+ ru) (B re+ rpi)) r0,
F=(A rpi+B ru) ((rc+re) rb+re rc)+ru rpi (rc+re),
and D enom =C ZQ2+((rb+rc+2 re) C +(A r0+ ru) rpi+(A rpi+B ru) rQ) Z 0+D +E+F.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A P P E N D IX B
BIA S D E PE N D E N C E O F S M A L L -S IG N A L D E V IC E M O D EL PA R A M E T E R S
T his appendix presents the variation in sm all-sig n al equivalent circuit m odel
param eters for B JT w ith bias. The param eters fo r w h ich bias dependence has been
included here are rQ, rrt, C^, C^, and Cbcx.
70
M RF927 (V C E =
M RF927 (V C E =
M RF947 (V C E =
M RF947 (V C E =
M M BR941(VCE =
M M B R 94I(V C E =
60
50
IV )
3 V)
IV )
3V )
IV )
3 V)
xz
o
12
C 30.
20
2.0
2 .5
3 .0
Fig. B - 1. V ariations in sm all-signal o u tp u t resistan ce (rQ) w ith changes in VCE an d 1^.
199
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
200
M RF927 (V CE =
M R F927 (V CE =
m
M R F947 (V C E =
M RF947 (V C E =
o -o M M BR941(VCE =
M M BR94K V CE =
—
1.5
IV )
3V )
IV )
3V)
IV )
3V )
2 .0
Ic (m A )
3 .0
2 .5
2.0
UL.
Q.
U
o-o M R F927 (V CE =
— M R F927 (V C E =
m
M R F947 (V C E =
M RF947 (V CE =
o -o M M B R 94l(V C E =
M M B R94K V C E =
0 .5
2.0
2 .5
Ic (m A )
Fig. B-2. Variations in rn an d C^ w ith changes in V Cg an d Ic-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
IV )
3 V)
IV )
3V)
IV )
3V)
3 .0
201
0.4
1
1
—
M R F927 (V CE = IV)
M R F927 (V CE = 3V)
s - a M R F947 (V CE = IV )
m M R F947 (V CE = 3V) .
o -o M M BR941(VCE = IV)
♦ -» M M B R 94K V C E = 3V)
0 .3
u.
a.
0.2
■<
U
[3------------------ EJ -
0.1 <F
T
LJ
r
L1
r
—
----------- F 2------------------ F]
=5F
=0
£
o.
%
1.0
1 .5
2 .0
2 .5
3 .0
Ic (mA)
0 .4
M R F927 (V C E
M R F927 (V CE
b- b M R F947 (V C E
» - • M R F947 (V CE
o~o M M B R 9 4 (V C E
M M B R 94K V C E
0 .3
=
=
=
=
=
=
IV )
3V)
IV)
3V)
IV )
3 V)
U.
Q.
U
0.1
o.y
1.0
1 .5
Ic (mA)
2 .0
2 .5
Fig. B-3. V ariations in sm all-signal collector-base capacitance (C |i) and capacitance
betw een ex tern al base and collector (Cbcx) w ith changes in V CE and Ic .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3 .0
A PPE N D IX C
EX PERIM EN TA L V E R IFIC A T IO N O F C L O S E D -F O R M EX PR ESSIO N S FOR
S M A L L -SIG N A L SCA TTERIN G P A R A M ETE R S
This appendix presents a com parison o f sm all-signal s-param eters obtained from
on-w afer high-frequency m easurem ents and by evaluating closed-form expressions
derived in C hapter 3. T he s-param eters com pared pertain to b ipolar junction transistors
referred to as npnl and n p n l in C hapter 3. The input and o u tp u t reflection coefficients (S j [
and S 22 ) have been plotted on im pedance sm ith chart, w hile the forw ard and reverse trans­
m ission coefficients (S 2i and S ^ ) have been presented on p olar plots. Fig. C -l to Fig. C-5
show the com parison betw een the m easured data and the calculated sm all-signal s-param ­
eters for active device n p n l , w hile the com parison for n p n l is presented in Fig. C -6 to Fig.
C - 11 o f this appendix.
202
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
203
45.0
MHz
FREQ
20.045
S ll
MHz
FREQ
MHz
FREQ
20.045
GHz
Full scale = 5.0
20.045
GHz
Full scale = 0 25
S12
45.0
S21
45.0
GHz
4 5.0
MHz
FREQ
20.045
GHz
S22
Fig. C - l . C om parison o f calcu la ted (bold) and m easured (light) sm all-sig n al s-param eters for n p n l in co m m o n -em itter configuration from 45 M H z to 20.045 G H z at
VC E = I V and Ic = 1mA.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
204
45.0
MHz
F REQ
20.045
MHz
FREQ
45.0
MHz
FREQ
20.045
45.0
GHz
sca*e ~ 15
20.045
GHz
Full scale = 0 25
S12
S ll
22]_
45.0
GHz
MHz
FR E Q
20.045
GHz
S22
Fig. C-2. C om parison o f calculated (bold) and m easured (light) sm all-signal s-parameters for npnl in co m m o n -em itter configuration from 45 M H z to 20.045 GHz at
VCE = IV and Ic = 2m A .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
205
45.0
MHz
FREQ
20.04 5
S ll
MKz
FREQ
MHz
FREQ
20.04 5
Full scal e = 15
GHz
20.045
GHz
Full scal e = 0 25
S12
45.0
S21
45.0
GHz
45.0
MHz
FREQ
20.045
GHz
S22
Fig. C -3. C om parison o f calculated (bold) an d m easured (light) sm all-signal s-param e­
ters for npnl in com m o n -em itter configuration from 45 M H z to 20.045 GHz at
VCE = IV and Ic = 3m A.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
206
45.0
MHz
FREQ
20.045
GHz
S ll
MHz
FREQ
MHz
FREQ
20.045
45.0
GHz
Full scale = 15
20.045
GHz
Full scale = 0 25
S12
45 . 0
S21
45.0
MHz
FREQ
20.045
GHz
S22
Fig. C-4. C o m parison o f calculated (bold) an d m easured (light) sm all-signal s-p aram e­
ters for n p n l in com m on-em itter co n figuration from 45 M H z to 2 0.045 G H z at
V c e = 3V and 1^ = 2mA.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
207
O
45.0
MHz
FREQ
20.045
MHz
FREQ
45.0
MHz
FREQ
20.045
4 5.0
GHz
Full scale = 15
20.345
GHz
Full scale = 0 25
S12
S ll
S21
45.0
GHz
MHz
FREQ
20.04 5
GHz
S22
Fig. C -5. C o m p ariso n o f calculated (bold) an d m easured (light) sm all-signal s-param eters for n p n l in com m on-em itter configuration from 45 M H z to 20.045 G H z at
V c e = 3V and Ic = 2mA.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
208
45.0
SII
MHz
FREQ
20.045
GHz
45.0
MHZ
FREQ
S12
20.045
GHz
Full Scale = 0 25
Fig. C-6. C om parison o f calcu lated (bold) and m easured (light) sm all-signal s-param e­
ters for npn2 in co m m o n -em itter configuration from 45 M H z to 20.045 G H z at
V ce = IV and Ic = 1mA.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4 5 .0
MHz
FREQ
2 0 .0 4 5
4 5 .0
GHz
MHZ
FREQ
20.0 4 5
GHz
Full Scale = 0 25
S12
S ll
4 5 .0
MHz
FR EQ
2 0 .0 4 5
4 5 .0
GHz
MHz
FREQ
20.0 4 5
GHz
Full Scale = 15.0
S21
S22
Fig. C-7. C om parison o f calculated (bold) and m easured (light) sm all-signal s-param eters for npn2 in com m on-em itter configuration fro m 45 M Hz to 20.045 G H z at
V c e = IV and Ic = 2m A.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
210
: iJ
4 5 .0
MH=
FREQ
2 0 .0 4 5
FREQ
SHs
2 0 .3 4 5
GHz
Full Scale = 0 25
S 12
S ll
y
20 .0 4 5
GHz
FP.EQ
2 0 .0 4 5
GHz
Full Scale = 15.0
S21
S22
Fig. C-8. C om parison o f calculated (bold) and m easured (light) sm all-signal s-param e­
ters for npn2 in co m m o n -em itter configuration from 4 5 M H z to 20.045 G H z at
V Ce = IV and Ic = 3m A .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
211
FP.EQ
2 0 .0 4 5
4 5 .0
GHz
MHz
FREQ
2 0 .0 4 5
3Hz
Full Scale = 5 0
S21
S22
Fig. C-9. C om parison o f calcu lated (bold) and m easured (light) sm all-sig n al s-param e­
ters for npn2 in co m m o n -em itter configuration from 45 M H z to 20.045 G H z at
V CE = 3 V and Ic = 1mA.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4 5 .0
KH=
FREQ
2 0 .0 4 5
45 .0
GH=
MH=
FREQ
2 0 .0 4 5
OH 2
Full Scale = 0.25
Sll
S 12
Fig. C -10. C om parison o f calculated (bold) and m easured (light) sm all-signal s-param eters for npn2 in com m on-em itter configuration from 45 M H z to 20.045 GHz at
V c e = 3V and Ic = 2mA.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
213
Full Scale = 0 25
S ll
S 12
Fig. C -l 1. C o m p ariso n o f calculated (bold) and m easured (light) sm all-signal s-param eters for npn2 in com m on-em itter configuration from 45 M H z to 20.045 G H z at
VCe = 3V and Ic = 3mA.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX D
ANALYTICAL EXPRESSIONS FOR PASSIVE DC BIAS CIRCUITS
T he purpose o f this appendix is to p ro v id e the expressions for the collector
currents and th eir partial derivatives w ith respect to tem perature sensitive device m odel
param eters and passive circu it elem ents for the passive dc bias netw orks show n in Fig 54(a),(c), and (d).
T he analytical expressions for dc bias n etw ork show n in Fig 3-2(a) are:
I
(D -l)
C ~
W here A = R nC + r b. + r e + R BD and B = R C
r + r .
e
(D-2)
AIs e
dv
BE
apf
vt
(D-3)
b
(D-4)
(D-5)
BVf V~t
dRg
ZTP/
B
214
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(D - 6 )
215
(
VBE/ V t\
dic
ls [ l e
dRB ~
B$f
J
(D-7)
T he analytical expressions for dc bias netw o rk show n in Fig 5-4(c) are:
,,
v
(
BE
/v
Vf
A/S e
V oir
~ 1
l r = ----- ^ ----- =
c
BVf
V
*■*
L - B E +J ^ L 2
re
B
(D - 8 )
W here A = (rg +rb + R^)R\ + (re + ei, ) R'7 and B = re(R~>+ R \)-
^C_
dfs
J VBE/ V < A
T ;
B$jr
lO
3/c
A ls e
VRF/ V ,
,
w jvt
ai/^ =
(D -9)
( D - 10)
r
V nr. / v .
a/ c
A ,s { ' BE
ap/
(D-ll)
flp2/
VB E / V t
b! c
AIs VBEe
(D -12)
5
dV
dVt
BVf V~t
/ V flF/ V ,
3/c
(re + r/,+/?2)/5(e
^ =
W }
/ V RF/ V ,
{re + rb + R \ )ls [ e
sir2
—
b
b
BP/
~0
\
, V RF/ V ,
+
\
“ 'J
\
‘ ‘J
Are l s { e
V RF/ V ,
Are l s { e
+ ----H
(D -13)
?"~
7 i f
r
VccreR2
\
" ‘J
VccreR2
v,
------ ----- r-= + -^(D-i4)
f i" p /
B
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
216
The analytical expressions fo rd c bias network show n in Fig 5-4 (d) are:
M s[ \-e
VBE/ V t
Bp
V D[~
BE
V
r -i- f i r
e
E
/
+
R~.
cc 1
B
(D -15)
where A = (Bj + R-) ) ( r g + rb + R^) + R^R0 , and B = ( R j + R^ ) ( r g + R e ) .
a/c
a
(
VB E / V t \
[l-e
J
d/s
dlc
dV
(D-16)
Bfijr
AIs e
VBE/ V t
(D-17)
W f Vt
BE
r e + RL
VBE / V t
d I C _ A I S V BE e
(D-18)
r 1
dVt
B$fV
di_c
A l ^| 1 - e
ap
VBE/ V r
(D-19)
BP
f
dIc
M
7
r + r^
e
VSE/ V ,
re
* ri -W^Tr)
Bp
s{ ' -
f
V
Rn
cc 2
B (B , + R 2 )
CC
dlC
M
Bp
r.
d RE
( r e + RE ) $ f
*2
V
_____________________
BE
cc 2
+ R C)2
(-r e + RE }
]
R j + R-y
B
I
Vbe/vnJ
1-
(D-20)
B{-re JrRE )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(D-21)
(D-22)
APPENDIX E
ANALYTICAL EXPRESSIONS FOR ACTIVE BIAS CIRCUIT
The purpose o f this ap pendix is to provide the expressions for the collector current
and its partial derivatives w ith re sp e c t to tem perature sensitive device model param eters
and passive circuit elem ents for the active dc bias netw ork show n in Fig 5-5(b).
f VE B \ / V t
v ccRs * i
„
D
A/4
e
\
-'J
f VBE1/ V t
s/4
e
\
- lJ
£
VE B \ K5 ~ L V BE2
£P
P
_______________________________________ 1 1 _________________ Li
/
C2 “
D
fE-n
1
w here A = ((rbl + rg] + /?3)(fl, + R 2) + R l R2 )R5 , B = i re2 + rb2 + R6 + R5)('re\ +/? 3 ) ’
C = rg l + R 3 , D = (re2 + R6 )(rel +/?3 ) + /?3^ 5 , and E = /?, + « 2 .
J
VM l / V i
31 a
4"
3 , SI - ~
dIC 2
DEf f l
<E -2 >
[ e EBl
'-i)
AI S \ l e
ap/ i
A
"'J
(E-3)
D£P 2/1
VE B \ / V t
3i C2
- R e ~-
S\e
5-
(E-4)
d y EB\
D
( VBE2/ V t
*C2 =_ r
3 / S2
4
\
(E-5)
°P /2
217
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
218
»C2
ap
(E-6)
/2
(372
D
VB E 2 / V t
-r , - R eX
3
a/ C 2
BIS2e
P/I^r
a y BE2
(E-7)
D
VE B \ / V t
S 1 EB \ e
VB E 2 / V t
S2
+ -------
BE2e
v 7 fy~2
EV} frl
BIC 2
dV.
(E-8)
D
VE B l / V t
( r M + r e \ + * 3 + * 2 ^ * 5 IS \ { (
-1
)
( VE B \ / V t
A f Si [ e
EP
a/ c 2
a/?.
£2P
71
7 1
D
^* 5
£
-0]
^*5*1
"
£“
(E-9)
D
V CD, / V .
Vr.n , / V .
- 1
e
a/ C2
dR.
£ p71
_ v
£2p
7i
D
Vc c R5*1
(E-10)
D
VE B \ / V t
a/ C2
a/?,
-V £ £ 2
*5,S\{ e
f VBE2/ V t
\
J
('r e2 + rb2 + * 6 +
5^ S 2 [ e
72
Vl
D
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 0
219
W *J
£P
(
-1
\
VB E 2 /V t
^2 ( e
J
72
/I
( re2 + ^ 6 + ^ 5 )
D*
(V
RcR,
££_£_L_V'
r
-CV
EBr 5
BE2
E
■>
(E-ll)
(re2 + / ? 6 + /?5 )
D~
VB E 2 / V t
V
a /c o
3*7
i
£
-Vn
-
AIs i [e
££2
£p
72
/I
=
1
D
VE B l / V :
Vc c R 5*1
— T A - VEBXR5 - C V B E 2
A /5 l ( e
“ £P
- R-
'/I
72
(E-12)
D‘
a /C2
m : ~
~
( VBE2/ V t A
C /52'7e
-')
~
Dp
f-
f
V'
~ £~ E
-
A l S V7e
V E B \ R 5 ' C V BE2
V£B' /V ' nJ * , 52 Ir 1 VB E 2 / V t - 1w
£P
C
/I
72
D‘
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B IO G R A PH IC A L SK E T C H
A hm ed E jaz N adeem w as b om on D ecem ber 21, 1961, at Quetta, Pakistan. In
1976, he secured 4th position in the entire province in S econdary Schools Exam ination
and he was aw arded a M erit Scholarship for higher secondary education. In 1978, he
received a P resid en t’s M erit Scholarship for studies lead in g to a B achelor o f Engineering
degree for securing the first position in the board o f Interm ediate and Secondary E duca­
tion. He received his B achelor o f Engineering in A vionics from N.E.D. University o f
Engineering & Technology, K arachi, Pakistan in January, 1983.
A fter graduation from Pakistan Air Force, C ollege o f A eronautical Engineering, he
w as aw arded the P resid en t’s C om m ission in the Pakistan A ir Force and since he has been
w orking as a professional engineer. In 1986, he was selected for a specialist training w ith
the U nited States A ir Force. He com pleted the advanced electronics m aintenance calibra­
tion training at Low ery A ir Force B ase, Denver, C olorado, as an honor graduate.
In 1990, after a nationw ide com petition, the G overnm ent o f Pakistan, M inistry o f
Science & Technology aw arded him a scholarship fo r post-graduate studies leading to
Ph.D. in electrical engineering. A fter receiving his M .S.E.E . from the University o f F lor­
ida in 1994, he started w orking for a doctorate in electrical engineering. His current
research interests are high-frequency interconnect characterization, modeling, sim ulation,
and m icrow ave integrated circuit design.
232
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I certify that I have read this study and that in my opinion it conform s to acceptable
standards o f scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree o f D octor o f Philosophy.
W illiam R. E isenstadt, C hairm an
A ssociate P ro fesso r o f Electrical and
C om puter E ngineering
I certify that I have read this study and that in my o p inion it conform s to acceptable
standards o f scholarly p resentation and is fully adequate, in scope and quality, as a
dissertation for the degree o f D octor o f Philosophy.
/O
__________
G ijs Bosnutfi
Professor o f E lectrical and C om puter
E ngineering
I certify that I have read this study and that in my o pinion it conform s to acceptable
standards o f scholarly p resentation and is fully adequate, in scope and quality, as a
dissertation for the degree o f D octor o f Philosophy.
, f
jf/J l
---------------------M ark E. Law
Professor o f E lectrical and C om puter
E ngineering
I certify that I have read this study and that in my opinion it conform s to acceptable
standards o f scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree o f D octor o f Philosophy.
[ (A a u n
, j _________________
Khai D. T. N go
A ssociate P ro fesso r o f Electrical and
C om puter E ngineering
I certify that I have read this study and that in my opinion it conform s to acceptable
standards o f scholarly presen tatio n and is fully adequate, in scope and quality, as a
dissertation for the degree o f D octor o f Philosophy.
O scar D. C risalle
A ssociate P ro fesso r o f C hem ical
Engineering
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T h is dissertation was su bm itted to the G raduate F aculty o f the C ollege o f
E ngin eerin g and to the G raduate School and was accepted as partial fulfillm ent
o f the requirem ents for the d eg ree o f D octor o f Philosophy.
M ay 1998
W infred M. P hillips
D ean, C ollege o f Engineering
K aren A. H o lb r o o ^
D ean, G rad u ate Scnool
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