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Advances in yield-driven design of microwave circuits

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a d v a n c e s in y ie l d
- d r iv e n d e s ig n o f m ic r o w a v e c ir c u it s
By
J I A N S O N G , B. E ng ., M. E ng.
(C h o n g q in g U n iv e rsity )
A Thesis
S u b m itte d to the School o f G r a d u a t e Stu d ie s
in P artial F u lf ilm e n t o f th e R e q u ir e m e n t s
f o r th e D egree
D o ctor o f P h ilosophy
M c M a s te r U n iv e r s ity
A p ril 1991
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A D V A N C E S IN Y I E L D -D R I V E N D E S IG N
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M c M A S T E R U N I V E R S IT Y
H a m ilto n , O n ta rio
D O C T O R O F PH IL O S O P H Y ( 1991)
(E le c tric al E n g in e e rin g )
TITLE:
A d v a n c e s in Y ie ld - D r i v e n D esign o f M ic ro w a v e C ircuits
AUTHOR:
J ia n Song
B .E ng .(E .E .), M .E n g .(E .E -), ( C h o n g q in g U n iv e rsity )
SUPERVISOR:
J. W. B a n d le r
P ro fe ssor, D e p a r tm e n t o f E le c tric al a n d C o m p u te r E n g in e e rin g
B.Sc.(Eng.), Ph.D ., D.Sc.(Eng.) ( U n iv e r s ity o f London)
D .l.C. (Im perial C ollege)
P .E ng. (P rov in c e o f O n ta rio )
C .E n g . F .l.E .E . ( U n i t e d K in g d o m )
F ellow , I.E.E.E.
Fellow , R oyal Society o f C a n a d a
NUMBER OF PAGES:
x iii, 163
n
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ABSTRACT
T h is
thesis a d d re s se s
itse lf to c o m p u t e r - a i d e d
y ield -d riv en
desig n o f
m ic ro w a v e c ir c u its u sin g im p le m e n ta b le , e f f i c i e n t a p p r o x i m a t i o n a n d o p tim iz a tio n
te c h n iq u e s.
Basic co n ce p ts o f y ie l d - d r iv e n d e s ig n a r e i d e n t i f i e d . A n u m b e r o f a p p ro a c h e s
to statistical desig n a r e re v ie w e d . T h e i r f e a tu r e s a n d lim ita tio n s a r e discussed. T h e
r e c e n t ge n e ra liz ed £p c e n te r i n g a p p r o a c h a n d o n e - s i d e d t x o p ti m i z a tio n a lg o r ith m a re
a d d resse d .
A hig hly e f f i c i e n t q u a d r a tic a p p r o x im a ti o n , s p e c ia lly a p p lic a b le to statistical
d e s ig n , is p re se n te d . A set o f v e ry sim p le a n d e a s y - t o - i m p l e m e n t fo rm u la s is d e r iv e d .
T h is a p p r o x im a tio n te c h n iq u e is also a p p lie d to g r a d i e n t f u n c t i o n s o f c ir c u i t responses
to p r o v id e h ig h e r a c c u ra c y .
A c o m b in e d a p p r o a c h to a tt a c k larg e sc a le p r o b l e m is p r e s e n te d , w h ic h
e x p lo re s th e most p o w e r f u l c a p a b ilitie s o f h a r d w a r e a n d s o f tw a r e a v a ila b le to us,
n a m e ly , the s u p e r c o m p u t e r ,
e f f ic ie n t q u a d r a t i c
sim u la tio n , a n d s t a t e - o f - t h e - a r t o p tim iz a tio n .
m o d e l in g ,
f a s t a n d d e d ic a te d
Y i e l d - d r i v e n d e s ig n te c h n iq u e s a re
e x te n d e d to deal w ith tu n a b le cir c u its by c o n s id e r in g t u n i n g to le ra n c e s. A 5 - c h a n n e l
w a v e g u id e m u ltip le x e r is c o n s id e r e d as a n e x a m p le b o th f o r th e c o m b in e d a p p r o a c h
a n d f o r the tr e a tm e n t o f tu n a b le circuits.
Y ie ld -d r iv e n
d es ig n
of
n o n lin e a r
m ic r o w a v e
c ir c u it s
w ith
statistically
c h a ra c te r iz e d devices is c o n s id e re d . R e le v a n t c o n c e p ts a r e in tr o d u c e d . T h e e f f i c ie n t
In te g ra te d G r a d i e n t A p p r o x im a t io n T e c h n i q u e ( / C A T ) is p r e s e n te d in th e sta tistic a l
iii
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d e sig n e n v ir o n m e n t, w h ic h a voids th e p r o h ib itiv e c o m p u ta tio n a l b u r d e n r e s u ltin g
f r o m th e tra d itio n a l p e r tu r b a tio n sc h e m e .
A novel a p p r o a c h , called Feasible A d j o i n t S en sitiv ity T e c h n iq u e (.FAST), is
d e r iv e d to c alcu late se nsitivities o f n o n lin e a r c ir c u its t h a t a re sim u la te d in th e
h a rm o n ic b alance e n v iro n m e n t. By ta k in g a d v a n ta g e o f th e c o m p u ta tio n a l e f f ic ie n c y
o f a d jo in t analysis a n d th e im p le m e n ta tio n a l sim p lic ity o f th e p e r tu r b a tio n t e c h n iq u e ,
F A S T is responsible f o r g rea t sa ving s o f c o m p u ta tio n a l e f f o r t r e q u ir e d f o r y ie ld d r iv e n d esign o f n o n lin e a r circuits.
iv
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ACKNOW LEDGEM ENTS
T h e a u t h o r w ish e s to ex p re ss his s in c e r e a p p r e c ia tio n to Dr. J.W . H andler f o r
his e n c o u r a g e m e n t , e x p e r t g u id a n c e a n d s u p e r v is io n th ro u g h o u t th e c o u rse o f this
w o rk .
He also th a n k s Dr. J.F. M a c G r e g o r a n d D r. N.K.. S in ha, m e m b e rs o f his
S u p e r v is o r y C o m m it te e , f o r th e i r valuable su gg estions a n d c o n tin u in g interest.
T h e a u t h o r a p p r e c ia te s the o p p o r t u n it y g iv e n to h im by Dr. Bandlcr to be
in v o lv e d
in
in d u s tr ia lly
re le v a n t
p r o je c ts
and
a c a d e m ic
r es e a rc h
activitie s.
C o o p e r a tio n b e tw e e n Dr. B and lcr a n d so m e le ad in g in d u s tria l e stab lis h m e n ts has
alw ays b e en a s ig n if ic a n t m o tiv a tio n f o r th e w o r k o n y i e l d - d r i v e n de sig n.
T h e a u t h o r o f f e r s his d e e p g r a titu d e to Dr. R .M . B iernacki o f M cM aster
U n i v e r s ity a n d O p tim iz a tio n S ystem s A ssociates In c. f o r his e x p e r t a d v ic e , th ro u g h
f r e q u e n t a n d v e ry b e n e f ic ia l discussions.
It is th e a u t h o r ’s p le a s u r e to h a v e b e e n a sso c ia te d w i t h Dr. S.H. C h e n a n d Dr.
Q .J. Z h a n g .
T h e c o n tin u o u s a n d i n - d e p t h in te ra c tio n w ith th e m has led to the
d e v e l o p m e n t a n d i m p r o v e m e n t o f new ideas. S o m e o f t h e i r e x c ellen t w o rk has been
v e ry h e lp fu l in th e a u t h o r ’s research. T h e a u t h o r has also b e n e f itte d fr o m in sp irin g
d iscu ssions w ith his colleagues S. Ye a n d Q . Cai.
T h e assistan ce
o f M .L . R e n a u lt a n d G .R . S im p s o n o f th e S im u la tio n
O p tim iz a tio n System s R e s e a rc h L aboratory' (S O S R L ) in p r e p a r in g p ro g ra m s a n d
p ro v id in g c o m p u t e r a n d w o rd p ro cessing fa c ilitie s is a c k n o w le d g e d .
T h e fin a n c ia l assistance p ro v id e d by th e N a tu ra l Sciences a n d E n g in e e rin g
Research
C o u n c il
of
Canada
th r o u g h
G ran ts
OGP0007239,
STR0040923,
v
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E Q P0043573, a n d the D e p a r tm e n t o f E lectrical a n d C o m p u t e r E n g in e e r in g th ro u g h
a T e a c h in g A ssistsntsh ip is g r a te fu lly a c k n o w le d g e d .
F inally, th an ks a r e d u e to m y fa m ily f o r th e ir c o n tin u o u s e n c o u r a g e m e n t,
especially m y w ife , Sh uxi Li, f o r s h o w in g he r u n d e r s ta n d i n g , s u p p o r t and to le ra n c e
w hile she w orks to w ard s h e r o w n Ph.D. deg re e .
VI
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TA B LE O F C O N TEN TS
PA G E
ABSTRACT
,ii
ACKNOWLEDGMENTS
v
L IST OF F IG U R E S
xi
L IST O F T A B L E S
CHAPTER 1
CHAPTER 2
xiii
INTRODUCTION
I
REVIEW OF STATISTICAL DESIGN APPROACHES AND
O PT IM IZ A T IO N T E C H N IQ U E S
7
2.1
I n tro d u c tio n
7
2.2
N o ta tio n an d F o r m u la tio n s
2.2.1
C ir c u it P a ra m e te rs , Design V ariables
a n d T h e i r Sta tistic s
2.2.2 C irc u it S im u la tio n
2.2.3 Design S p e c ific a tio n s a n d E r r o r F u n c tio n s
2.2.4 Yield E stim a tio n
2.2.5 Y ie ld - D riv e n C ir c u it D esign
9
9
i1
12
15
16
2.3
R e v ie w o f A p p r o a c h e s to S tatistical Design
2.3.1 T h e Sim plicial A p p r o x im a t io n T e c h n iq u e
2.3.2 T h e C e n t e r - o f - G r a v i t y M e th o d
2.3.3 U p d a t e d A p p r o x im a t io n s a n d C u ts
2.3.4 Stochastic A p p r o x im a tio n
2.3.5 P a r a m e '. k‘ Sa m p li n g
2.3.6 Sensiti
.* F ig u r e f o r Y ield I m p r o v e m e n t
2.3.7 SimuiuiCa A n n e a lin g O p tim iz a tio n
i8
19
20
22
23
24
25
27
2.4
T h e G e n e r a liz e d t p C e n t e r in g A p p r o a c h
2.4.1 F o rm u la tio n o f Y ie ld O p tim iz a tio n
2.4.2 Im p ie m e n ta tio n a i A sp e cts
28
28
30
vii
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T A B LE O F C O N TEN T S(continued)
PA G E
CHAPTER 2
REVIEW OF STATISTICAL DESIGN APPROACHES AND
OPTIM IZATION TECHNIQUES
(c o n tin u e d )
2.5
2.6
CHAPTER 3
v
31
31
33
C o n c lu d in g R e m a rk s
36
EFFICIENTiQUADRATIC APPROXIMATION
FOR STATISTICAL DESIGN
39
3.1
In tro d u c tio n
39
3.2
T h e M ax im a lly Flat Q u a d r a tic A p p r o x im a t io n
40
3.3
A p p r o a c h U s in g a F ix e d P a tte n o f Base Points
3.3.1
D e riv a tio n a n d A lg o r ith m o f th e A p p r o a c h
3.3.2
C o m p u ta tio n a l E f f ic ie n c y
42
42
46
3.4
Q u a d r a t ic A p p r o x im a tio n o f C ir c u it R e sp onses
3.4.1
I m p le m e n ta tio n
3.4.2
D esign o f a 1 1 - E le m e n t L o w - P a s s F ilte r
48
48
48
3.5
G r a d i e n t Q u a d r a tic A p p r o x im a tio n S c h e m e
3.5.1
Q u a d r a t i c A p p r o x i m a tio n to R e sp on ses
a n d G r a d ie n ts
3.5.2
A 1 3 -E le m e n t L o w - P a s s F ilt e r
3.3.3
A T w o - S ta g e G a A s M M I C F e e d b a c k A m p l i f i e r
52
52
55
57
C o n c lu d in g R e m a r k s
63
3.6
CHAPTER 4
O n e - S id e d
M a th e m a tic a l P r o g r a m m in g
2.5.1
F o rm u la tio n o f the P ro b le m
2.5.2
A lg o r ith m s f o r the O n e - S id e d
O p tim iz a tio n
SUPERCOM PUTER-AIDED STATISTICAL DESIGN O F
A LARGE SCALE C IR C U IT
A 5-CHANNEL
MICROWAVE MULTIPLEXER
65
4.1
In tr o d u c tio n
65
4.2
Y ie ld - D r iv e n D esign f o r N o n - M a s s iv e P r o d u c tio n
-
67
viii
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T A B LE O F CO N T ENTS(co nti nu ed )
"
^
o
CHAPTER 4
CHAPTER 5
-
PAGE
S U P E R C O M P U T E R - A I D E D S T A T I S T I C A L D E S IG N O F
A LARGE SCALE C IR C U IT
A 5-C H A N N E L
M IC R O W A V E M U L T I P L E X E R
( c o n tin u e d )
4.3
Y ie ld - D r iv e n D e sig n o f T u n a b le C ir c u its
4.3.1
T u n i n g ar.d Its T o le ra n c e s
4.3.2
Y i e ld - D r iv e n D esig n w ith T u n i n g T o le r a n c e s
67
67
72
4 .4
U tiliz in g th e V e c t o r P ip e lin e S u p e r c o m p u te r
4.4.1 Basics a b o u t the V e c to r P ro cessor
4.4.2 S u p e r c o m p u t e r - A i d e d Y ield O p tim iz a tio n
4.4.3 T h e C r a y X - M P / 2 2 E n v i r o n m e n t
74
74
75
76
4.5
Yield O p tim iz a tio n o f a 5 - C h a n n e l M u ltip le x e r
4.5.1
D esign V a ria b le s a n d S p e c ific a tio n s
4.5.2
D esign P ro c e d u re a n d R e sults
76
79
81
4.6
C o n c lu d in g R e m a r k s
85
N O N L IN E A R C I R C U I T Y IE L D O P T I M I Z A T I O N
W IT H G R A D IE N T A P P R O X IM A T IO N S
87
5.1
I n tro d u c tio n
87
5.2
T h e H a r m o n ic Balance S im u la tio n T e c h n i q u e
5.2.1 T h e N o n li n e a r M o d e l a n d Its
T i m e - D o m a i n S im u la tio n
5.2.2 L in e a r S u b n e tw o r k S im u la tio n
5.2.3 H a r m o n ic B alance E q u a tio n
5.2.4 R e sp o n se C a lc u la tio n
89
92
95
99
101
S p e cific a tio n s a n d E rr o r s f o r N o n lin e a r
C ir c u it Y ield O p t im iz a tio n
102
E f f ic ie n t G r a d i e n t A p p r o x im a tio n fo r
Y ie l d - D r iv e n D esign
5.4.1 IGAT fo r N o m in a l D esign
5.4.3 IGAT f o r Yield O p tim iz a tio n
103
104
105
N o n lin e a r F E T Statistical M odels a n d S tatistical
O u tc o m e G e n e r a ti o n
106
5.3
5.4
5.5
ix
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TABLE O F C O N TEN TS(continued)
PA G E
CHAPTER 5
CHAPTER 6
N O N L IN E A R C I R C U I T Y IE L D O P T I M I Z A T I O N
W IT H G R A D IE N T A P P R O X I M A T I O N S
( c on tinu ed )
5.6
Yield O p tim iz a tio n o f a F r e q u e n c y ^ D o u b le r
5.6.1
D e sc rip tio n o f the Design
5.6.2 Design P ro c e d u r e
5.6.3 Results a n d D iscussions
107
107
112
5.7
C o n c lu d in g R e m a r k s
118
FAST G R A D IEN T BASED N O N LIN EA R C IR C U IT
S T A T IS T IC A L D E S IG N
119
6.1
In tro d u c tio n
119
6.2
Feasible A d j o i n t S e n s itiv ity T e c h n iq u e
6.2.1 G e n e r ic F o r m u la s o f A d j o in t
S e n sitiv ity A n aly sis
6.2.2 F A S T fo r N o m in a l D esign
6.2.3 F A S T fo r Y ie ld O p tim iz a tio n
121
C o m p a riso n s o f V a rio u s A p p ro a c h e s
6.3.1 I m p le m e n ta tio n a l C o m p a riso n s ot PAST,
IGAT, E A S T a n d F A S T
6.3.2 N u m e r ic a l C o m p a riso n s o f P A S T ,
E A S T a n d FAST.
128
6.4
Yield O p tim iz a tio n o f a F r e q u e n c y D o u b le r
130
6.5
C o n c lu d in g R e m a r k s
132
6.3
CHAPTER 7
C O N C LU SIO N S
i 21
122
127
128
130
137
REFERENCES
143
A U T H O R INDEX
153
S U B J E C T INDEX
159
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LIST O F F IG U R E S
F IG U R E
2.1
PA G E
U p p e r a n d low er sp e c ific a tio n s f o r a n a m p lif ie r :p_ be d e sig n ed
to o p e r a te o v e r a sp e c ifie d t e m p e r a tu r e range.
14
2.2
Illu stratio n o f the sim p licial a p p r o x im a ti o n a p p r o a c h .
21
2.3
F lo w c h a rt o f yie ld o p tim iz a tio n .
32
3.1
T h e L C lo w - p a s s filte r.
49
3.2
C ir c u i t sc h e m a tic o f th e L C 1 3 - e l e m e n t filte r.
56
3.3
N o r m a liz e d G a A s M E S F E T m o de l.
3.4
A tw o - s ta g e a m p lif ie r .
60
4.1
Illu stra tio n o f yield o p tim iz a tio n w ith the fa b r ic a tio n to le ra n c e
region R f only.
71
Illu stra tio n o f yield o p tim iz a tio n w ith the c o m b in e d s pread
region R fitjt -
73
4.3
E q u iv a le n t c ir c u it o f a 5 - c h a n n e l c o n tig u o u s b a n d m u ltip le x e r.
77
4.4
R e t u r n a n d in se rtio n loss o f the 5 - c h a n n e l m u ltip le x e r at the
m in im a x solution.
80
R e t u r n loss e n v e lo p e o f 3000 5 - c h a n n e l m u ltip le x e r c ir c u it
o u tc o m e s a f t e r yield o p tim iz a tio n .
83
R e t u r n loss e n v e lo p e o f s a tis f a c to r y 5 - c h a n n e l m u ltip le x e rs
a m o n g 300 0 c i r c u i t ou tco m e s a f t e r y ie ld o p tim iz a tio n .
84
5.1
S c h e m a tic r e p re se n ta tio n o f the o n e - F E T c ir c u it.
90
5.2
D e c o m p o sitio n o f the c ir c u it in Fig. 5.1.
91
5.3
T h e n o n lin e a r in trin sic F E T m odel.
93
5.4
C ir c u it d ia g r a m o f the F E T m ic r o w a v e f r e q u e n c y d o u b le r.
111
5.5
H isto g ra m o ' “ c o n v e rsio n gain s o f the f r e q u e n c y d o u b le r at
th e s ta r tin g point.
114
4.2
4.5
4.6
r:
59
xi
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5.6
5.7
5.8
6.1
6.2
H isto g ra m o f c o n v e rsio n gains o f th e f r e q u e n c y d o u b l e r a t
th e s o lu tio n o f yield o p tim iz a tio n usin g IG AT.
115
H isto gram o f sp e c tra l p u ritie s o f the f r e q u e n c y d o u b l e r a t
th e s ta r tin g -p o in t.
116
H isto g ra m o f s p e c tra l p u ritie s o f the f r e q u e n c y d o u b l e r a t
the so lu tio n o f yield
o p tim iz a tio n
u sin g IGAT.
117
H isto g ra m o f c o n v e rsio n g ains o f th e f r e q u e n c y d o u b le r a t
the so lu tio n o f yield
o p tim iz a tio n
u sin g F A ST .
133
H isto g ra m o f s p e c tra l purities o f th e f r e q u e n c y d o u b l e r a t
the so lu tio n o f yield
o p tim iz a tio n
using F A ST .
134
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LIST O F TA BLES
TABLE
3.1
PAGE
C o m p a riso n o f sta tistic a l d e sig n o f a lo w -p a ss filte r w ith a n d
w ith o u t q u a d r a tic a p p r o x im a tio n s .
51
Y ie ld -o p tim iz a tio n o f th e L C 1 3 - e le m e n t f ilte r w ith a n d
w ith o u t q u a d r a tic a p p r o x im a tio n s .
58
3.3
P a ra m e te r values a n d to le ra n c e s f o r th e M M IC a m p lif ie r .
61
3.4
Yield o p tim iz a tio n o f th e M M I C a m p lif ie r w ith a n d w ith o u t
q u a d r a tic a p p r o x im a tio n s .
62
Statistical d esig n o f u 5 - c h a n n e l m u ltip le x e r using q u a d r a tic
a p p ro x im a tio n s .
82
3.2
4.1
5.1
A ssu m e d statistical d is tr ib u tio n s f o r th e F E T p a ra m e te rs .
108
5.2
F E T m odel p a r a m e t e r c o rre la tio n s.
109
5.3
Yield o p tim iz a tio n o f the f r e q u e n c y d o u b le r using IGAT.
113
6.1
Yield o p tim iz a tio n o f th e f r e q u e n c y d o u b l e r using FAST.
131
xiii
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xiv
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INTRODUCTION
F o r m o re th a n tw o d e c a d e s, w e h av e b een w itnesses to the r a p id p rogress o f
•c irc u it c o m p u t e r - a i d e d d e sig n (C A D ), f r o m a set o f a b s tr a c t m a th e m a tic fo rm u la s
u n d e rsto o d b y o n ly a h a lf do z e n u n iv e r s ity p ro fe sso rs to a n in d isp e n s a b le a n d
e v e ry d a y tool used b y v irtu a lly all c ir c u i t d e sig n e rs. Progress on s u c h a scale is the
result o f tw o m a jo r d r iv in g forces, n am ely , a c tiv e re se arc h a n d d e v e lo p m e n t o f
nu m e rica l c ir c u it analysis a n d o p tim iz a tio n , a n d th e d r a s tic e v o lu tio n o f c o m p u te r
h a rd w a r e a n d so ftw a re . C A D te c h n iq u e s c o n tin u e to th riv e , p e n e tr a tin g all c ircu it
design areas w ith r e v o lu tio n a r y ideas.
T h e m o n o lith ic m ic r o w a v e in te g ra te d c i r c u i t ( M M I C ) is p a r t o f a d e ve lop in g
techno lo gy . T h e m a jo r d i f f e r e n c e b e tw e e n M M I C a n d its p re v io u s g e n e r a tio n , the
h y b r id m ic ro w a v e in te g r a te d c ir c u it (M IC ), is th e follow in g: M M IC s allow various
a c tive a n d passive c ir c u it ele m e n ts , s u c h as tra n sm issio n lines, resistors, cap ac ito rs,
in d u c to rs, f i e l d - e f f e c t tra n s isto rs ( F E T s ) a n d d io d e s o f m a n y ty pes to be inte gra te d
on a single c h ip , p e r f o r m in g c e r ta in fu n c tio n s , su c h as a m p l if ic a tio n , m ix in g , f ilte rin g
a n d p h a s e - s h if ti n g .
T h e M I C tec h n o lo g y , h o w e v e r , is to m o u n t active c ir c u it
ele m e n ts a n d o th e r c o m p o n e n ts , f o r e x a m p le , c h ip c a p a c ito r s , on a d ie le c tric su b stra te
with p re v io u s ly f a b r ic a te d tra n sm issio n lines a n d o th e r e le m e n ts. In M IC e n g in e e rin g ,
tu n in g is possible by c h a n g in g the physical d im e n s io n s o f e le m e n ts
o r e v en by
su b s titu tin g o n e F E T f o r a n o th e r . But f o r M M IC s, th e r e is no o p p o r t u n i t y fo r d e v ic e
re p la c e m e n t a n d v e r y lim ite d scope f o r c ir c u it tu n in g .
I
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T h e tra d itio n a l " p e r f o r m a n c e - d r iv e n " d e sig n is c o n s id e r e d a d e q u a te e n o u g h
fo r M IC s. T h is is because o f th e a b ilit y to re p la c e d e v ic e s a n d to t u n e c irc u its to
c o u n te ra c t to lerance e f f e c ts .
T h e fina l p r o d u c tio n yield a f t e r tu n in g can be
dra m a tic a lly increased fro m th e y ield b e fo re tu n in g . H o w e v e r, th e tu n in g p r o c e d u r e
based on an in d iv id ua l c irc u it is v e r y e x p e n siv e a n d tim e c o n s u m in g . It m eans th a t
a t the e x p e n se o f h ig h e r costs, th e f in a l yield c an b e im p r o v e d b y tu n in g .
Because t h e r e is essentially n o t u n in g a llo w e d in M M I C p r o d u c tio n , im po sin g
p e r f o r m a n c e - d r i v e n d esign c a n lead to very low yie ld e v e n f o r q u it e m o d e st circuitsa n d , c o n se q u e n tly , to p ro h ib itiv e costs. A d i f f e r e n t d e s ig n a p p r o a c h , ca lle d "y ie ld driven " design o r statistical d e sig n , is n ecessary f o r M M I C . Y ie l d - d r i v e n design takes
u n c e rta in tie s,
su c h
as
process
to le ra n c e s,
e n v ir o n m e n ta l
f lu c tu a tio n s ,
m o del
in a c c u ra cy , etc., into a c c o u n t to m a x im iz e m a n u f a c tu r in g yield.
C o m p a re d w ith d igital c ir c u it s im u la tio n , m ic r o w a v e c i r c u i t s im u la tio n is fa r
m o re involved a n d c o m p le x because o f the analo g n a tu r e o f s u c h c ir c u its . M ic ro w a v e
o r an a lo g c irc u its can h ave m a n y d i f f e r e n t types o f e le m e n ts a n d m ost o f th e m usually
have c o n tin u o u s value d is tr ib u tio n s . C i r c u it responses a r e also c o n tin u o u s fu n c tio n s
w h ic h d e m a n d c o n sid e ra b le c o m p u ta tio n a l e f f o r t to solve . S im u la tio n o f n o n lin e a r
c ir c u it responses is e v e n m o r e c o m p u ta tio n a lly in te n s iv e sin c e it re q u ir e s ite ra tio n .
In y ie ld - d r iv e n d e sig n , m u ltip le sets o f c irc u its , s o m e tim e s , a q u ite la r g e n u m b e r o f
th e m , need be sim u la te d .
O p tim iz a tio n te c h n iq u e s a re o f te n u tiliz e d to a u t o m a t e y i e l d - d r i v e n design.
Because they a re iterative in n a tu r e , m ost o p tim iz a tio n a lg o r ith m s r e q u ir e m a n y
o b je c tiv e f u n c tio n e v a lu a tio n s, ea c h o f w h ic h is th e re su lt o f m a n y c i r c u i t sim ulatio ns.
G r a d ie n t- b a s e d
o p tim iz a tio n
te c h n iq u e s d e m o n s tr a te ,
in g e n e r a l,
f a r s u p e r io r
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3
p e r f o r m a n c e to d ir e c t ( n o n - g r a d ie n t) m e th o d s .
O n th e o t h e r h a n d , g ra d ie n t
c a lc u la tio n involves c ir c u it se n sitiv ity analysis.
A s a re su lt o f m a n y o p tim iz a tio n ite r a tio n s , m u ltip le c ir c u it s im u la tio n s an d
se n sitiv ity analyses, a n d possible ite ra tiv e c ir c u it sim u la tio n p r o c e d u r e s , th e collective
c o m p u ta tio n a l e f f o r t c re a te s special d if f i c u l tie s in te rm s o f p r o h ib it iv e c o m p u ta tio n a l
costs a n d le n g th y d esig n cycles. T o m a te ria liz e t h e y i e l d - d r i v e n d e s ig n m eth o d o lo g y ,
th e re is still a c o m p e llin g n e e d to f u r t h e r r e s e a rc h a n d d e v e lo p c o m p u ta tio n a lly
feasible te c h n iq u e s to such an e x te n t th a t y i e l d - d r i v e n d e s ig n is no lo n g e r m erely an
o p tio n , b u t a n in d isp e n sa b le a n d r o b u s t tool f o r c ir c u it d e sig n e rs d e a lin g with
pra c tic a l circu its.
T h is thesis is in te n d e d to s u m m a r iz e n e w rese a rc h re su lts o f c o m p u t e r - a i d e d
y i e ld - d r iv e n d e sig n f o r m ic ro w a v e c ir c u its . We p ro p o se n e w a p p r o a c h e s to push yield
o p tim iz a tio n te c h n iq u e s fo r w a r d to m e e t the ch a lle n g e s.
T h e thesis consists o f 7
ch a p te rs.
In C h a p te r 2, we id e n tify c o n c e p ts a n d no ta tio n in v o lv e d in y ie ld -d r iv e n
design. T h e n , w e rev iew som e o f th e m ost r e p re s e n ta tiv e a p p r o a c h e s d e v e lo p e d in the
past a n d th e m ost re c e n t a p p ro a c h e s. O u r e m p h a s is is laid o n th e a p p r o a c h by B andlcr
a n d C h e n (1989) since it will b e used a n d f u r t h e r d e v e lo p e d in s u c c e e d in g c h a p te rs .
T h e g en e ra l fo r m u la tio n o f this a p p r o a c h , w h ic h c o n v e r ts th e p ro b le m o f y i e ld - d r iv e n
d esign to an t y o p tim iz a tio n p r o b le m , is g iv e n . We c o n s id e r th e r e le v a n t co n c e p ts and
d e f in iti o n s used in th e o n e - s id e d
o p tim iz a tio n . A tw o - s ta g e a lg o r ith m to solve the
o n e - s i d e d £j o p tim iz a tio n p ro b le m is d e s c r ib e d .
In C h a p t e r 3, we p r e s e n t a hig h ly e f f i c i e n t q u a d r a tic a p p r o x im a tio n te c h n iq u e .
T h e ne w a p p r o a c h takes a d v a n ta g e o f th e m ax im a lly fla t i n te r p o la tio n a n d o f a fix e d
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p a tte r n o f base p o in ts, th u s su b s ta n tia lly re d u c in g co m p u ta tio n a l e f f o r t a n d r e q u ir e d
s to ra g e. A set o f e x tr e m e ly sim p le f o r m u la s to ca lc u la te m o d e l c o e f f ic ie n ts is d e r i v e d .
M o re o v e r, this a p p r o a c h is e x t e n d e d to a p p r o x im a te g r a d ie n ts o f c ir c u it re s p o n s e
fu n c tio n s. T h e ele ga n ce o f this a p p r o a c h is its conciseness a n d a p p lic a b ility . H ig h
e f f ic ie n c y a n d fe a sib ility T o r y i e l d - d r i v e n design a re d e m o n s tr a te d by th e results o f
y ie l d - d r iv e n c irc u it d esign.
C h a p te r 4 deals w ith sta tistic a l desig n o f tu n a b le c ir c u its w ith tu n in g
to lerances. T h e issue o f statistical d e s ig n o f large scale c ir c u its is also a d d re s s e d . We.
p ro po se a c o m b in e d a p p r o a c h
in te g ra tin g the use o f s u p e r c o m p u te r , e f f i c i e n t
q u a d r a ti c m odeling , and d e d ic a te d s im u la to r.
M o d e rn s u p e r c o m p u te r s have f o u n d
v a lu a b le a p p lic a tio n s in m ic ro w a v e c i r c u i t C A D w ith a ttr a c tiv e p e r f o r m a n c e - t o - c o s t
ratios. O u r s o f tw a r e , w h ic h c a rrie s o u t statistical d esig n o f m ic r o w a v e m u ltip le x e r s ,
has b e e n d e v e lo p e d fo r th e s u p e r c o m p u t e r . T h e c o m p u ta tio n a l results o f a 5 - c h a n n e l
m u ltip le x e r design p e r f o r m a n c e on th e C r a y X - M P are r e p o r te d in th e c h a p te r .
C h a p te r 5 o f f e rs the fir s t c o m p r e h e n s iv e d e m o n s tr a tio n o f y i e l d - d r i v e n d e sig n
o f m ic ro w a v e c irc u its w ith sta tistic a lly c h a r a c te r iz e d d e v ic e s.
E f f i c i e n t h a r m o n ic
balance s im u la tio n is e x p lo re d . A p o w e r f u l g r a d ie n t a p p r o x im a tio n te c h n iq u e , called
IC AT , is in tro d u c e d to av o id e x t r e m e l y e x p e n s iv e c o m p u ta tio n a l e f f o r t re q u ir e d b y
the tra d itio n a l p e r tu r b a tio n m e th o d . L a rg e -sig n a l F E T p a r a m e te r sta tistic s a r e fu lly
f a c ilita te d . E x te n siv e n u m e ric a l e x p e r i m e n t d ir e c te d at y i e l d - d r i v e n d e sig n o f a F E T
f r e q u e n c y d o u b le r v e rifie s o u r a p p r o a c h .
C h a p te r 6 pre se n ts a h i g h - s p e e d g r a d ie n t ca lc u la tio n t e c h n iq u e f o r m ic ro w a v e
n o n lin e a r c irc u its o p e ra tin g w it h in th e h a r m o n ic b alance s im u la tio n e n v i r o n m e n t . T h e
te c h n iq u e , called F A S T , is i m p le m e n ta tio n a lly feasib le f o r tr u ly g e n e r a l - p u r p o s e
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5
n o n lin e a r c ir c u it C A D , c o m b in in g the e f f ic ie n c y a n d a c c u r a c y o f the a d jo in t
se n sitiv ity analysis w ith the sim p lic ity o f th e p e r t u r b a ti o n m e th o d .
T h ro u g h
n u m e ric a l e x p e r im e n ts , th e su b sta n tia l a d v a n ta g e s o f F A S T o v e r o th e r g r a d ie n t
c a lc u la tio n a p p r o a c h e s h a v e b e e n o b se rv e d .
We c o n c lu d e in C h a p t e r 7 w ith som e su gg estio ns f o r f u t u r e research.
The
au th o r
has
c o n tr ib u te d
su b sta n tia lly
to
the
fo llow ing
orig in al
d e v e lo p m e n ts p r e s e n te d in this thesis:
(1)
A n e f f i c i e n t q u a d r a tic a p p r o x im a tio n t e c h n iq u e f o r statistical design.
(2)
E x te n s io n o f y ie ld - d r i v e n design to tu n a b le c irc u its w ith tu n in g tolerances.
(3)
A c o m b in e d a p p r o a c h using s u p e rc o m p u te rs , re s p o n s e a p p r o x im a tio n , a n d
d e d ic a te d s im u la to r, to statistical d esign o f large scale circuits.
(4)
A first in te g ra te d tr e a tm e n t o f y ie ld - d r iv e n d esign f o r n o n lin e a r m icrow av e
c ir c u its in th e h a r m o n ic balance e n v ir o n m e n t.
(5)
A g r a d ie n t a p p r o x im a tio n a p p r o a c h su ita b le fo r statistical desig n.
(6)
A n ew , e f f i c i e n t , a n d e a s y - t o - i m p l e m e n t s e n sitiv ity analysis te c h n iq u e o f
n o n lin e a r m ic ro w a v e c ir c u its a n d its a p p lic a tio n in statistical design.
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
REVIEW OF STATISTICAL DESIGN APPROACHES AND OPTIM IZATION
TECHNIQUES
l\
2.1
IN T R O D U C T IO N
T h e use o f o p tim iz a tio n te c h n iq u e s c a n be tra c e d b a c k to the very first
d e v e lo p m e n t o f c ir c u it C A D . Since th e n , d u e to d esign o f m o re a n d m o re c o m p le x
c irc u its a n d th e e n o r m o u s c o m p u ta tio n a l p o w e r a v aila b le , th e in c re a sin g d e m a n d s
have m a d e o p tim iz a tio n te c h n iq u e s w id e ly a d o p te d as p r i m a r y tools in c ir c u it C A D
pro gram s.
T h e tra d itio n a l c ir c u it d esign is th e s o - c a lle d n o m in a l d esig n th a t focuses on
the i m p r o v e m e n t o f in d iv id u a l c irc u it responses o f interest. In re a lity , h o w e v e r, it is
im p ossib le to m a n u f a c tu r e a c ir c u it w ith e x a c t d e sig n e d p a r a m e te rs fo r th e follow ing
reasons: th e e x is te n c e o f u n c e rta in tie s a n d to lerances i n h e r e n t in the m a n u f a c tu r in g
p rocess, the in a c c u ra c y o f m a th e m a tic a l m odels to a p p r o x im a t e the real physical
b e h a v io u r
of
c ir c u it
e le m e n ts,
etc.
R e c o g n itio n
su c h
flu c tu a tio n s
in
th e
m a n u f a c tu r in g e n v i r o n m e n t a n d d esig n process leads to th e a p p ro a c h o f y i e ld - d r i v e n
c ir c u it de sig n . T h e n a tu r e o f y ie ld - d r iv e n d e sig n s ug gests th a t it c a n be c o n v e rte d to
a n o p tim iz a tio n p ro b le m , w h e re th e o b je c tiv e is no lo n g e r to im p r o v e in d iv id u a l
c ir c u it p e r f o r m a n c e , b ut to increase th e e stim a te d m a n u f a c t u r i n g yield. Y ie ld - d r iv e n
d e sig n has b een g iven m u c h a tte n tio n f o r m o re th a n tw o d ec a d e s. M an y a p p ro a c h e s
have been d e r iv e d . T h e r e is a w id e range o f lite ra tu re . A special issue o f th e I E E E
T ra n s a c tio n s on C o m p u t e r - A i d e d Design on S tatistical D esign o f VLSI C irc u its E d ite d
7
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by S tro jw a s a n d S a n g io v a n n i-V in c e n te lli (1986) w as p u b lis h e d .
T h e r e h a v e b een
m a n y C A D b o o k s a n d d e d ic a te d c h a p te rs o n this s u b je c t ( S tro jw a s 1987, S o in .a n d
S p e n c e 1988, L a d b r o o k e 1989, a n d V e n d e lin , Pavio a n d R o h d e 1990).
F irst, in this c h a p te r , we in tro d u c e n o ta tio n f o r c i r c u i t p a r a m e te r s , d esign
v ariab les, a n d th e ir statistics.
a d d re sse d .
C ir c u it sim u la tio n a n d re s p o n s e c a lc u la tio n a re
R e la tio n s b e tw e e n response fu n c tio n s, d esig n sp e c if ic a tio n s , as well as
e r r o r f u n c tio n s a re discussed.
T h e con cep ts o f c ir c u i t o u tc o m e s , n o m in a l values,
tolerances, a n d m a n u f a c t u r in g yVild a re id e n tifie d . T h e n , th e fo r m a l d e s c rip tio n o f
y i e ld - d r iv e n d esign is pre se n te d . V ario us p ro b le m s a risin g f r o m sta tistic a l d e sig n a re
also d e s c rib e d .
A . r e v ie w o f a n u m b e r o f a p p r o a c h e s to y i e l d - d r i v e n d e s ig n is given .
We
d e s c r ib e in so m e d e ta il several m e th o d s w h ic h r e p r e s e n t m a n y y ears rese a rc h on this
s u b je c t.
'
Fin ally , w e p r e s e n t th e gene ra liz ed £p c e n te r in g a p p r o a c h (B a n d le r a n d C h e n
1989) a n d e la b o ra te o n a special case: th e £j a p p r o a c h . We c o n c e n tr a t e in d e ta il on th e
o n e - s id e d £j o p tim iz a tio n p ro b le m a n d its im p le m e n ta tio n a l a sp e c ts because it will be
used in o u r a p p r o a c h to y ie ld - d r iv e n design. T h e b asic c o n c e p ts , re le v a n t d e f in itio n s ,
a n d m a th e m a tic a l fo rm u la tio n o f the
o p tim iz a tio n p r o b le m a re p re s e n te d . A t w o -
stage a lg o rith m c o m b in in g th e tru st G a u s s - N e w t o n m e th o d a n d th e q u a s i- N e w t o n
m e th o d is o u tlin e d .
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2.2
N O T A T IO N A N D F O R M U L A T IO N S
In o r d e r to d e s c r ib e th e fo r m u la tio n f o r y i e ld - d r iv e n d e sig n , wc will give
re le v a n t d e f in itio n s a n d in tr o d u c e u sefu l n o ta tio n .
2.2.1
C ir c u it P a r a m e te r s , D esign V a ria b le s a n d T h e i r S tatistics
T h e c ir c u i t c o n s id e r e d w ill h a v e a fi x e d to p o lo g y a n d k n o w n c o m p o n e n t ty p e s,
th a t is, the s t r u c t u r e o f th e c i r c u i t is not a n o b j e c t o f d e s ig n a n d th e c o m p o n e n ts
in v o lv e d have th e p ro p e r m a th e m a tic a l m odels to a p p r o x i m a te th e ir b e h a v io u r.
A-'
g iv e n set o f p a r a m e te r values will d e te r m in e th e p e r f o r m a n c e o f th e c irc u it. T h e
p a r a m e t e r values may b e tra d itio n a l dis c re te e le m e n ts , s u c h as resistors a n d c a p a c ito rs ,
d e v ic e p a r a m e te rs su ch as c o e f f i c ie n ts o f c h a r a c te r is tic e q u a tio n s o f a F E T m o del,
g e o m e tric a l d im e n s io n s su c h as th e w id th o f a m ic ro strip lin e a n d th e gate len gth o f
a F E T , as well as m a n u f a c t u r in g p a r a m e te rs s u c h as th e p e r m e a b ility a n d c o n d u c tiv ity
o f the m aterial. O th e r c o n tro llin g o p e r a tio n a l a n d e n v ir o n m e n ta l fa c to rs, such as bias
voltages a n d th e e x c ita tio n level, can also be in c lu d e d . We use
<f>A
(2 . 1)
'N
to d e n o te this s e t o f values w h e r e s u b s c rip t N is th e total n u m b e r o f p a ra m e te rs .
R elatio n sh ip s b e tw e e n these p a r a m e te rs can be v e r y c o m p lic a te d .
h ie ra rc h ic a l s tr u c tu r e a m o n g th e p a r a m e te rs m a y exist.
(1) A
F o r e x a m p le , the actual
proc e ss p a r a m e te rs a r e d e f i n e d , h o w e v e r, the c ir c u it is s im u la te d using e q u iv a le n t
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
c irc u it m odels.
A m odel is n ee d e d to c o n v e r t process level p a r a m e te rs to the
e q u iv a le n t c ir c u it p a ra m e te rs . (2) Som e o f p a r a m e te rs m ay b e fi x e d o r d is c re te fo r
c e r ta in reasons.
(3) Som e o f th e p a ra m e te rs m a y be c o rr e la te d b e c au se th e y a re
co n tro lle d by a g r o u p o f lo w er level p a ra m e te rs . F o r instance, a n u m b e r o f e q u iv a le n t
c ir c u it p a ra m e te rs are c o ntrolle d by a g r o u p o f c o m m o n process p a r a m e te rs .
In a p ra c tic a l d esig n process, o n ly so m e o f th e 4 a r e c o n s id e r e d as designable.
H o w e ve r, we shall assum e th a t ^ are all d e sig n variables unless o th e r w i s e s ta te d . It
is also ph ysically m e a n in g fu l to a ssum e th a t d e s ig n variables a r e i n d e p e n d e n t.
In the classical design p ro b le m we a re in te re s te d in f in d in g o n e sing le p o in t
(c irc u it) in the d esig n variable space w h ic h satisfies the design sp e c ific a tio n s . T h is
kind o f so lu tio n is im p ractical f r o m th e m a n u f a c tu r in g p o in t o f v ie w s in c e th e r e is a
n u m b e r o f fa c to rs w hich in flu e n c e s th e p e r f o r m a n c e o f a m a n u f a c t u r e d d esign .
A m o n g these fa c to rs are: (a) m a n u f a c tu r i n g to le ra n c e s, (b) m o d e l u n c e rta in tie s , (c)
parasitic e f f e c ts , a n d (d) e n v ir o n m e n t flu c tu a tio n s.
D ue to the fo re g o in g u n c e r ta in tie s , th e p a ra m e te rs o f a c tu a lly m a n u f a c t u r e d
circ u its a r e s p re a d o v e r a reg io n.
In the follow in g, we use
4> to d e n o t e a
m a n u f a c t u r e d c ir c u it. Sup po se the p r o b a b ility d e n sity fu n c tio n ( p d f ) o f 4 is
In a d d iti o n to the g e n e ric f u n c tio n f o r m , tw o sets o f p a r a m e te rs a r e n e e d e d to
d e sc rib e the p ro b a b ility d e n s ity f u n c tio n .
T h e firs t set
c o n ta in s th e n o m in a l
values, also called the no m inal d esig n o r n o m in a l c ir c u it in c ir c u it d e sig n . T h e sec o n d
set o f p a ra m e te rs can be to le ra n c e e x tr e m e s f o r u n if o rm d is tr ib u tio n , s t a n d a r d
d e v ia tio n s associated w ith n o rm a l d is tr ib u tio n s , a n d c o rre la tio n c o e f f ic ie n ts f o r j o i n t
d istrib u tio n s. We use
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
II
e&
to d e n o te this set o f a d d itio n a l p a ra m e te rs ,
since
m ore
p a ra m e te rs
d is trib u tio n s .
are
needed
to
c m a y have a h ig h e r d im e n s io n th a n 4>
d e s c rib e a sy m m e tric a l
and
c o rre la te d
N o w , th e a u g m e n te d f o r m o f the p ro b a b ility d e n sity fu n c tio n o f ^
beco m es
/>(*. A
«)■
(2.2)
T h e to le ra n c e region is d e te r m i n e d b y the p ro b a b ility d e n sity f u n c tio n an d c ir c u it
o u tc o m e s a re alw ays fall in to it
<f>6 Re(<fP, «),
(2.3)
w h e re Rc{<f>'-) is the tolera n c e region . In th e fo llo w in g , w e m ay use p(4>), p($, ^°)
o r p{4>,
e) to d e n o te th e p ro b a b ility d e n s ity f u n c tio n a c c o rd in g to d i f f e r e n t
c irc u m s ta n c es .
2.2.2
C ir c u it Sim u la tio n
C irc u it respo nse c a lc u la tio n u su a lly involves a t w o - s ta g e process.
First, the
c ir c u it is s im u la te d by solving a set o f c ir c u it e q u a tio n s
/ < * , « = 0,
(2.4)
w h e re z , th e so lution o f the e q u a tio n s , is a v e c to r usually con sisting o f node voltages,
b r a n c h voltages, b r a n c h c u r r e n ts , etc.
In g e n e ra l, solv in g (2.4) involves s y ste m a tic
a p p r o a c h e s to c ir c u it an a ly sis ( C h u a a n d Lin 1975, Viach a n d Singhal 1983). T h is set
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o f c irc u it e q u a tio n s c a n b e lin e ar, n o n lin e a r, d if f e r e n t i a l - i n t e g r a l , o r m ix e d . T h e
w
c irc u it e q u a tio n s can also be based on nodal, loop, ta b u la r , m ix e d ana lysis, etc.
T h e n , responses o f d esig n in te re st a re ca lc u la ted f r o m the s im u la tio n results
o f (2.4). T h e responses a r e d e n o te d by
F tf)
f 2(4>)
(2.5)
F{4>) &
h 0 )
w h e re M is the n u m b e r o f re spo nse s c o n sid e re d . In a d d itio n to ^ , F(4>) is a fu n c tio n
o f z , w hic h is in te n tio n a lly o m it te d f ro m F(4) to a v o id c o m p le x ity .
C o n sid e r a voltage a m p lif ie r c irc u it. T h e respo nse o f in te re s t is th e voltage
gain Gv, F irst, a set o f nodal e q u a tio n s can b e set up
Y (4 )V = / ,
w h ere
(2.6)
is th e n odal a d m itta n c e m atrix w h ic h is a s s u m e d to e x ist, V th e nodal
voltage v ector a n d I th e n o da l c u r r e n t e x c ita tio n v ec to r -.We n eed to solve
P(V, 4>) =Y(<f>)V - / = 0 .
(2.7)
T h e n , Gv is ca lc u la ted f r o m th e voltage a t the o u t p u t n o d e a n d th e i n p u t voltage.
2.2.3
Design S p e c ific a tio n s a n d E r r o r F u n c tio n s
In c ir c u it d e sig n , sp e c if ic a tio n s a r e use d to d e sc rib e th e d e s ir e d p e r f o r m a n c e .
S p e c ific a tio n s a r e usually g iv e n in a set o f d is c re te values. T h e s e values a r e f u n c tio n s
o f in d e p e n d e n t p a r a m e te rs , su c h as f r e q u e n c y , tim e , te m p e r a tu r e , i n p u t p o w e r level,
etc. (B and ler a n d R izk 1979, B a n d le r, B ie rn a c k i, C h e n , S ong, Y e a n d Z h a n g 1990 a n d
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
\\
13
\\
1991a). We use i> to d e n o t e t h e in d e p e n d e n t p a ram e te rs . S p e c ific a tio n s, d e n o te d b y
S(i}>), a r e a r ra n g e d as th e v e c to r
r?i» )
s 2(.i>)
S(i>) &
(2.S)
s L(i>)
w h e re L is th e total n u m b e r o f sp e c if ic a tio n s im posed. F o r s im p l ic i ty , if we o m it ^
in (2.8), then
(2.9)
S p e c ific a tio n s ca n be o f th e fo llo w in g forms: (1) an u p p e r s p e c ific a tio n w hich
r e q u ire s th e respo nse be b elow it, (2) a lo w e r sp e c ific a tio n w hic h r e q u ire s the response
be a b o v e it, a n d (3) u p p e r a n d lo w e r s p e c ific a tio n s w hic h r e q u ir e th e sam e respo nse
be b e tw e e n th e two.
T o d is tin g u is h u p p e r a n d low er s p e c if ic a tio n s , w e in tro d u c e
s u b s c r ip ts u a n d / , th a t is, S uj a n d S{j. F o r e x a m p le , a n a m p l i f i e r is to be d esign ed
to m e e t a set o f p r e d e t e r m i n e d f r e q u e n c y s p e c ific a tio n s a n d , a t th e sam e tim e, to
e x h ib it sta b le responses in a p a r tic u l a r te m p e r a tu r e region ( B a n d le r a n d Rizk, 1979).
A ty p ic a l g ra p h ic a l p r e s e n ta tio n o f su c h a case is sho w n in F ig 2.1. T w o in d e p e n d e n t
p a r a m e te rs , te m p e r a tu r e a n d f r e q u e n c y , a r e involved to d e f i n e th e spe c ifica tio n s.
Fo r the case o f u p p e r o r lo w e r sp e c ific a tio n s , w e d e f in e th e e r r o r f u n c tio n s
as
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frequency
Fig. 2.1
U p p e r a n d low er sp e c ific a tio n s fo r a n a m p lifie r to be d e sig n e d to o p e ra te
o v e r a sp e c ifie d te m p e ra tu re ran g e (B a n d le r a n d R iz k 1979).
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
15
(2.10)
I < m< M ,
1 < / < L,
or
et}<4>) & wij(Sij ~
( 2 . 11 )
1 < m< M ,
1 < j < L,
w h ere wui a n d *v/;- a re n o n n e g a tiv e w e ig h tin g fa c to rs . S u b sc rip ts m a n d i (o r j ) m ay
be d if f e r e n t, if u p p e r a n d lo w e r s p e c ific a tio n s a re im p o sed o n th e sa m e response. In
su ch a c ase, S tli a n d S tj
* j ) a n d F/n($ ) a re th e c o rre sp o n d in g s p e c ific a tio n s a n d
resp o n se. H o w e v e r, m ju s t in d ic a te s its p o sitio n in (2.5) a n d is n o t n e ce ssa rily equal
to e ith e r i o r j . A p o s itiv e (n o n p o sitiv e ) e rro r f u n c tio n im p lies th a t th e c o rre sp o n d in g
sp e c ific a tio n is v io la te d (s a tis fie d ).
T o u n ify in d ic e s o f e r r o r fu n c tio n s , w e a sse m b le all e rro r fu n c tio n s re su ltin g
fro m ( 2 . 10) an d ( 2 . 11) in a v e c to r
<?i( 0
e 2(tf)
(2. 12)
c{4>) =
eL(4>)
T h e a c c e p ta b le re g io n Ra , w ith re sp e c t to a se t o f g iv en sp e c ific a tio n s , is d e fin e d by
Ra & {$ \ e(<f>) < 0) .
2.2.4
(2.13)
Y ield E stim a tio n
T h e re e x ist tw o c a te g o rie s o f a p p ro a c h e s to y ield e stim a tio n , n am e ly , the
g e o m e tric a l (o r d e te rm in is tic ) m e th o d a n d th e M o n te C arlo (o r sta tistic a l) m eth o d .
T h e g e o m e tric a l m e th o d c o n sid e rs a c o n tin u o u s sp re a d o f p a ra m e te r values.
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
T h e ty p ic a l fo rm u la to e v a lu a te th e y ie ld fo r th e g e o m e tric a l m e th o d is
OO
A4>\ c) = J / ( « / < * , *°, c)d<f>,
(2.14)
-OO
w h e re fu n c tio n
1(4) is th e a c c e p ta n c e in d e x d e fin e d by th e fo llo w in g
f: 1
i f e(4) < 0
(2.15)
0
and
o th e rw ise ,
p{4>, 4°-. c) is th e p a ra m e te r p ro b a b ility d e n s ity fu n c tio n .
T h e M o n te C a rlo m eth o d u tiliz e s a n u m b e r o f sa m p le p o in ts g e n e ra te d
a c c o rd in g to th e g iv e n p ro b a b ility fu n c tio n o f c ir c u it p a ra m e te rs.
4 - 4 ° ^
(2.16)
4 € Re(4>, «), / = I, 2 , . . . K,
w h ere K is th e total n u m b e r o f sa m p le p o in ts. T h e se sa m p le p o in ts re p re s e n t c irc u it
o u tc o m e s in th e m a n u fa c tu rin g p ro cess.
statistical outcomes.
We w ill re f e r th e se sam p le p o in ts as
F o r th e sta tistic a l m e th o d , th e m a n u fa c tu rin g yield ca n b e
e stim a te d by th e fo llo w in g
i K
(2. 17)
K M
2.2.5
Y ie ld -D riv e n C irc u it D esign
A n im p o rta n t p ro b le m in y ie ld -d riv e n d e sig n is d esig n c e n te rin g (B a n d le r an d
A b d e t-M a lc k 1978, B an d ler a n d K e lle rm a n n 1983). T h e p u rp o se o f d e sig n c e n te rin g
is to e n h a n c e yield b y o p tim iz in g o n ly nom inal p a ra m e te rs 4° a n d k e ep in g th e
to le ra n c e s fix e d .
T h is is a sp e c ia l case o f sta tistic a l d e sig n .
T h e fix e d to le ra n c e
p ro b le m has its p ra c tic a lly m e a n in g fu l a p p lic a tio n w h en th e m a n u fa c tu rin g p ro c e d u re
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
a lw a y s e x h ib its lim ite d p re c is io n as a c o n sta n t b a c k g ro u n d a n d m o d el in a c c u ra cy is
k n o w n . T h e e s tim a te d y ie ld is d ire c tly o r in d ire c tly u sed as th e d esig n o b je c tiv e . T h e
g e n e ra l fo rm u la tio n to so lv e su c h a p ro b le m c a n be s ta te d as
OO
m ax-
(2.18)
f o r g e o m e tric a l a p p ro a c h e s , o r
m a x ' Y(4>°)
(2.19)
f o r M o n te C a rlo a p p ro a c h e s.
F o r so m e o th e r cases, it is po ssib le to in flu e n c e to le ra n c e s b y a d ju s tin g th e
m a n u f a c tu r in g p ro c e ss. P re su m a b ly , th e sm a lle r th e to le ra n c e s a ro u n d a v alid n o m in al
d e s ig n a re , th e h ig h e r th e y ie ld is.
H o w e v e r, tig h te n in g to le ra n c e s w ill re su lt in
in c re a s e d m a n u fa c tu rin g co st.
A n o th e r ty p e o f p ro b le m in v o lv es th e d esig n o f
to le ra n c e s.
is k n o w n as o p tim a l
S u ch
a p ro b le m
to le ra n c in g
(B an d le r an d
A b d e l-M a le k 1978), o p tim a l to le ra n c e assig n m e n t (K a ra fin 1971, 1974), o r th e
v a ria b le to le ra n c e p ro b le m (B a n d le r a n d K e lle rm a n n 1983).
In su ch cases, th e
p a ra m e te rs in <
j>°, a lo n g w ith th o se in £ , a re c o n sid e re d as o p tim iz a tio n variab les.
T h e o b je c tiv e f u n c tio n c o n sists o f costs w h ic h re fle c t assig n e d to lera n c es. A yield
f ig u r e s h o u ld also b e a tta c h e d as one m o re d esig n c o n s tra in t to e n su re th e d e sire d
y ie ld . T h e ty p ic a l fo rm u la tio n is
m in im iz e C (^°, e ) ,
s u b je c t to
(2.20)
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w h e re YL is a y ie ld s p e c ific a tio n . W hen a 1 0 0 -p e rc e n t y ie ld is r e q u ir e d , th is p ro b le m
b eco m es th e w o rs t case d e sig n p ro b le m a n d th e fo rm u la tio n c o rre sp o n d in g to ( 2 . 20)
is
m in im iz e C (^°, e ) ,
(2 -21 )
s u b je c t to
e tf ) < 0 , V * e R e,
w h e re R e is th e to le ra n c e re g io n .
2.3
R E V IE W O F A P P R O A C H E S T O S T A T IS T IC A L D E S IG N
T h e s ta tis tic a l d e sig n a p p ro a c h w as o rig in a te d in th e e a rly 70’s. A m o n g o th e rs ,
K a r a f in (1971 a n d 1974), B u tle r (1 9 7 1 ), P in el a n d R o b e rts (1972), E lia s (1 975),
B a n d le r (1 9 7 2 , 1973 a n d 1974), B a n d le r, L iu a n d T ro m p (1976) a re p io n e e rs m a k in g
fu n d a m e n ta l c o n trib u tio n s to th e re se a rc h . T h e n , m an y o th e rs , D ire c to r a n d H a c h te l
(1 9 7 4 , s im p lic ia l a p p ro x im a tio n ), S oin a n d S p en ce (1978, th e c e n te r o f g ra v ity
m e th o d ), B a n d le r a n d A b d e l-M a le k (1 978, u p d a te d a p p ro x im a tio n s a n d c u ts ), P o la k
a n d S a n g io v a n n i-V in c e n te lli (1979, o u te r a p p ro x im a tio n ), T a h im a n d S p en ce (1 979,
th e ra d ia l e x p lo ra tio n a p p ro a c h ), A n tre ic h a n d K o b litz (1982, d e sig n c e n te rin g b y
y ie ld p re d ic tio n ) , S ty b lin s k i a n d R u sz c z y n sk i (1 983, sto c h a stic a p p ro x im a tio n ),
S in g h a l a n d P in e l (1 9 8 1 , p a ra m e tric sa m p lin g ), B a n d le r a n d C h e n (1 988, g e n e ra liz e d
t p c e n te rin g ), B ie rn a c k i a n d S ty b lin sk i (1 9 8 6 , d y n a m ic c o n stra in ts a p p ro x im a tio n ),
S ev erso n a n d S im p k in s (1 9 8 7 , w o rst case m eth o d s v ia H a d a m a rd a n a ly sis), P u rv ia n c e
a n d M e e h a n (1 9 8 8 , s e n s itiv ity f ig u re f o r y ie ld im p ro v e m e n t), V a i, P ra sa d a n d
M e sk o o b
(1 9 9 0 ,
y ie ld o p tim iz a tio n
th ro u g h
sim u la te d
a n n e a lin g )
have
m ade
su b s ta n tia l f u r th e r c o n trib u tio n s . In th is se c tio n , w e d e sc rib e se v e ra l re p re s e n ta tiv e
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
19
a p p ro a c h e s.
2.3.1
;■ \
T h e S im p lic ia l A p p ro x im a tio n T e c h n iq u e
We e x a m in e th e s im p lic ia l a p p ro x im a tio n te c h n iq u e o f D ire c to r an d H ach tel
(1 9 7 7 ), a n d B ra y to n , D ire c to r a n d H a ch te l (1980). T h e ir m e th o d is a g e o m e tric based
o n e w h ich uses th e s im p le r e x p re ssio n s to a p p ro x im a te th e a c c e p ta b le region.
F o r a g iv e n a c c e p ta b le re g io n Ra, th e b o u n d a ry o f it is d e fin e d as
dRa = {
| et{<f>) < 0 , V /, a n d e/tf) = 0 , 3 j \ /, j e {1,2 ,..., £.11.
(2.22)
G iv e n a se t o f m (> n * 1) p o in ts o n b o u n d a ry dRa, a g ro u p o f ( n - l) - d im e n s io n a l
sim p iices c a n b e o b ta in e d . T h e s im p lic ia l a p p ro x im a tio n to Ra , d e n o te d by Ra , is a
p o ly h e d ro n w hose faces a re ( n - 1)-d im e n s io n a l sim p iices. A p ro c e d u re was su g g ested
by D ire c to r a n d H a c h te l (19 7 7 ) to o b ta in a co nvex hull w hich fo rm s such an
a p p ro x im a tio n to Ra. S p e c ific a lly , th e sim p lic ia l a p p ro x im a tio n is d e fin e d by
Ra = I 4> I vj<t> < bh i = 1, 2 , ..., Nf 1,
(2.23)
w h e re th e i;,- a r e o u tw a rd p o in tin g no rm als to th e b o u n d in g h y p e rp la n e s d e fin e d by
n p o in ts o n
dRa, th e 6; a r e th e d ista n c e s b e tw e e n th ese h y p e rp la n e s in th e
a p p ro x im a tio n a n d th e o r ig in , a n d Nf is th e to tal n u m b e r o f h y p e rp la n e s.
W hen
o b ta in in g a sim p lic ia l a p p ro x im a tio n , w e ca n fin d an e s tim a te o f th e new nom inal
p o in t by d e te rm in in g th e c e n te r o f th e larg est h y p e rs p h e re in sc rib e d inside o f th e
p o ly h e d ro n . A lin e a r p ro g ra m m in g a p p ro a c h is re c o m m e n d e d to f in d th e c e n te r o f
the larg est h y p e rsp h e re .
T o im p ro v e th e s im p lic ia l a p p ro x im a tio n , m ore a n d m ore p o in ts on the
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
20
b o u n d a ry sh o u ld be fo u n d to e x p a n d th e p o ly h e d ro n . E ach o f th e s e p o in ts is fo u n d
a lo n g th e o u tw a rd n o rm a l d ire c tio n o f th e larg e st fa ce e x te n d in g fro m th e m id p o in t
o
th e sam e face o f th e c u r r e n t p o ly h e d ro n . T h e n o m in a l p o in t is also u p d a te d w h e n a
new a p p ro x im a tio n is av ailab le.
A sim p le illu stra tio n o f th e sim p lic ia l a p p ro x im a tio n is g iv e n in Fig. 2.2.
Fu rth e r d e v e lo p m e n t based o n th e sim p lic ia l a p p ro x im a tio n a p p ro a c h h as b een
m ad e to in c lu d e a r b itr a r y sta tistic a l d is trib u tio n s b y B ra y to n , D ire c to r, an d H a c h te l
(1980), a n d to allow th e use o f m u ltip le c rite rio n o p tim iz a tio n fo r y ie ld by L ig h tn e ran d D ire c to r (1981). T h e sim p licial a p p ro x im a tio n a p p ro a c h is b ased o n an essen tial
assu m p tio n o f a c o n v ex a c c e p ta b le re g io n .
T h is a ssu m p tio n lim its its a p p lic a tio n
becau se d e te rm in in g w h e th e r o r n o t a g iv e n p ro b le m has a c o n v e x a c c e p ta b le re g io n
is v e ry d if f ic u lt by itse lf.
2.3.2
T h e C e n te r - o f - G r a v ity M e th o d
Soin a n d S pen ce (1980) p ro p o se d to use M o n te C a rlo a n a ly sis to c o n s titu te a
ran d o m sa m p lin g , o r sta tis tic a l e x p lo ra tio n , o f th e to le ra n c e re g io n . F o llo w in g M o n te
C arlo a n aly sis, w h ic h id e n tifie s e a c h c ir c u it o u tc o m e as ’pass’ o r ’f a il’ a c c o rd in g ly , th e
c e n te rs o f g ra v ity o f b o th th e pass a n d fa il o u tc o m e s a re d e te rm in e d . T h e c e n te r f o r
p ass(fail) o u tc o m e s, d e n o te d by Gp(Gj) , is sim p ly th e a rith m e tic m ean o f values o f th e
co o rd in a te s o f p a ss(fa il) c irc u it o u tco m e s.
T h e n , a n ew n o m in al p o in t alo n g th e
d ire c tio n fro m Gy to Gp is fo u n d so th a t th is p o in t has a h ig h e r yield th a n the p re v io u s
n o m in al p o in t. T h e p ro p o se d re la tio n sh ip b e tw e e n the o ld a n d n ew n o m in al p o in ts is
C . = fe u + x &p - 9 T h e w ay to c a lc u la te th e ste p size,
<2 -2 4 >
A, is v e ry d e c isiv e in d e te rm in in g th e
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
21
acceptable region.
(R a )
^
simplicial approximation
F ig. 2.2
Illu stra tio n o f th e s im p lic ia l a p p ro x im a tio n a p p ro a c h (D ire c to r an d H achtel
1977).
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
c o m p u ta tio n a l e f f o r t n e e d e d f o r the e n tire process.
A ru le fo r th e ch o ice o f A is
d e riv e d fo r a v ery re s tric tiv e case w h e re i 00% y ie ld can be a c h ie v e d w ith the g ;en
to le ra n c e s a n d th e n o m in al p o in t w ith 100% yield m u st b e o n th e str a ig h t line b e tw e e n
G p a n d G f. F or m o re g e n e ra l cases, a m ore p rec ise ru le to ch o o se A is d if f ic u lt to
o b ta in . T h e re fo re , sev eral A sh o u ld be c alc u lated a n d o n ly th e o n e w ith h ig h est y ield
is se le cte d .
M an y ite ra tio n s m a y be n ecessary b e fo re th e e stim a te d yield does n o t
in crease. T w o o u tc o m e sa m p lin g sc h e m es a re d e v elo p e d to a c h ie v e a high c o n fid e n c e
level at an a c c e p ta b ly low c o m p u ta tio n a l cost.
T h e serio u s p ro b le m s w ith th is a p p ro a c h are: (1) th e u n c le a r re la tio n sh ip
b e tw e e n th e tw o g ra v ity c e n te rs a n d y ie ld , c o n se q u e n tly , the u n c e rta in ty to fin d th e
o p tim a l n o m in al p o in t a lo n g th e s tra ig h t line b etw e en tw o c e n te rs; ( 2 ) lack o f th e
a u to m a tic p ro c e d u re to f in d th e ste p size.
2.3.3
U p d a te d A p p ro x im a tio n s an d C uts
In th is m e th o d , p ro p o se d b y B a n d le r a n d A b d e l-M a le k (1 9 7 8 ,1 9 8 0 ), a lo w e r-
o rd e r m u ltid im e n sio n a l p o ly n o m ia l a p p ro x im a tio n is m ad e to th e a c c e p ta b le reg io n .
F irs t, th e c irc u it is s im u la te d a t so m e p o in ts, c alled b ase p o in ts, w h ic h a re se le c te d
a c c o rd in g to a d e riv e d sc h e m e to p re se rv e o n e -d im e n sio n a l c o n v e x ity /c o n c a v ity o f
th e c irc u it resp o n se fu n c tio n . T h e n , th e c o e ffic ie n ts o f th e a p p ro x im a tio n fu n c tio n ,
a q u a d ra tic p o ly n o m ial in th e ir c ase, can be d e te rm in e d b y so lv in g a set o f lin e a r
e q u a tio n s based on th e s im u la te d resp o n se values a n d th e c o o rd in a te s o f th e base
p o in ts. T o p re v e n t p o ssib le loss o f a c c u ra c y a t so m e p o in ts c o n s id e re d as v e ry c ritic a l
in w o rst case d e s ig n , m u ltip le reg io n s a ro u n d
th e se p o in ts a re su g g ested
be
a p p ro x im a te d in d iv id u a lly . It is also n ece ssa ry to u p d a te th e a p p ro x im a tio n fu n c tio n s
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
23
as th e n o m in a l p o in t m oves.
A n a n a ly tic a l a p p ro a c h to th e e v a lu a tio n o f yield a n d y ield se n sitiv itie s is
p ro p o se d b ased on c o m p u ta tio n o f th e h y p e rv o lu m e o f th e in te rse c tio n o f th e n o n a c c e p ta b le reg io n a n d th e to le ra n c e re g io n . T h e h y p e rv o lu m e fo rm u la is d e riv e d fro m
lin e a r c u ts o f th e to le ra n c e re g io n .
T h e lin e a r c u ts are fu n c tio n s o f th e n o n lin e a r
c o n s tra in ts d e f in in g th e b o u n d a ry o f th e a c c e p ta b le region. T h e sc h e m e to c o n stru c t
th e lin e a r c u t fro m th e lin e a r o r q u a d ra tic c o n stra in ts, w h ich a p p ro x im a te the o rig in a l
e x p e n siv e n o n lin e a r c o n s tra in ts , is g iv e n . S e n sitiv itie s o f th e yield e stim a te a re also
a v a ila b le to be used in a g r a d ie n t-b a s e d o p tim iz a tio n a lg o rith m .
2.3.4
S to c h a stic A p p ro x im a tio n
S ty b lin sk i a n d R u sz c z y n sk i (1 9 8 3 ) fo u n d an an alo g y b e tw e e n sta tistic a l desig n
c e n te rin g a n d th e p ro b le m o f fin d in g th e m axim um o f th e re g re ssio n fu n c tio n
m ax- W " ) = ^ ( 4 > ) p (0 ) d 0
=EUQ)}
(2.25)
w h e re 0 is an n -d im e n s io n a l ra n d o m v a ria b le w ith zero e x p e c ta tio n , th e o u tc o m e
<j> = <j>° - 0 , a n d p(9) is th e p ro b a b ility d e n sity fu n c tio n o f 0 .
It ise v id e n t th a t
(2 .2 5 ) c a n be o b ta in e d by re d e fin in g th e o rig in o f th e p a ra m e te r sp ace.
T h e sto c h a stic a p p ro x im a tio n a p p ro a c h , th e n , is used to so lv e th e p ro b lem in
(2.2 5 ). T h e fo llo w in g ite ra tio n s a re used
,o
y
dk= 0
w h e re k is th e c u r r e n t ite ra tio n ,
,o
1
,
- h + Tkdk
" P k ) d { k - 1)
+ PlA tt>
(2.26)
0 ^
Pic <
an e s tim a te o f th e g ra d ie n t o f
1
W ° ) , a n d r* —* 0
an d pk -* 0 a re n o n -n e g a tiv e c o e ffic ie n ts . T h e s ig n ific a n t a d v a n ta g e o f th is a p p ro a c h
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
24
is th a t th e g ra d ie n t o f y ie ld , £*., c a n b e e stim a te d b ased o n o n ly o n e o u tc o m e p o in t.
F o rm u la s to g e n e ra te rk a n d pk a re g iv e n .
F ro m th e e x a m p le g iv e n by th e a u th o rs , a d ra m a tic y ie ld in c re a se is o b se rv e d
d u r in g th e f ir s t se v e ra l ite ra tio n s o f th e o p tim iz a tio n pro cess. It is su sp e c te d th a t the
s e le c te d s ta rtin g n o m in al p o in t b u t o f th e a c c e p ta b le re g io n m ig h t c o n trib u te to th e
fa s t in itia l c o n v e rg e n c e . T h e a lg o rith m e x h ib its slow c o n v e rg e n c e w h en close to the
so lu tio n .
L ik e m an y sto ch astic a lg o rith m s , th is m eth o d m ay s u f f e r fro m v ery slow"
c o n v e rg e n c e rate w hen a p p ro a c h in g th e so lu tio n . A n o th e r lim ita tio n o f th is m eth o d
is th a t th e g ra d ie n t e v a lu a tio n re s tric te d to d iff e re n tia b le fu n c tio n s .
2.3.5
P a ra m e tric S am pling
T h e key p rin c ip le o f th e a p p ro a c h d u e to S in g h a la n d P in e l (1981) is to re p la c e
th e o rig in a l p ro b a b ility d e n s ity fu n c tio n b y so m e o th e r d e n s ity fu n c tio n
y(4>°) = j
/(*)
P i t ,* 0 )
K t , 4>°)
(2.27)
w h e re /i(& ^ ° ) is a cho sen sa m p lin g d e n s ity . A($, $ °) c a n b e a r b itr a r y e x c e p t fo r th e
re q u ire m e n t th a t h{<j>, # ° ) £ 0 w h e n e v e r /(^)p (& _ $ °) £ 0 - T h e c o rre s p o n d in g M o n te
C a rlo v ersio n o f (2.27) is
K
(2.28)
K4f )
1=1
w h e re <f>' a re sam p le p o in ts g e n e ra te d fro m th e sa m p lin g d e n s ity h {$) ra th e r th a n the
c o m p o n e n t d e n sity p ( ^ ') . T h e w e ig h t fa c to rs
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
co m p e n sa te fo r th e use o f a d if f e r e n t d e n sity fu n c tio n .
T h e M o n te C a rlo analy sis is c a rrie d
o u t on ce o n ly u sing a sa m p lin g
d is trib u tio n h(<f>) . T h e sa m p le d s ta tistic a l o u tc o m e s a n d th e c irc u it resp o n ses a re
\ '
"T1
sto re d in a d a ta b a se . T h e a d v a n ta g e o f th e p a ra m e tric sa m p lin g a p p ro a c h is th a t no
new c irc u it sim u la tio n s a re re q u ire d d u rin g th e o p tim iz a tio n if th e to le ra n c e reg io n
does n o t m ove o u t o f th e te r r ito r y o f th e d a ta b a se . H o w ev e r, w hen th e o p tim iz a tio n ,
leads som e o u tc o m e s o u t o f th e d a ta b a se , a d y n a m ic u p d a tin g sch em e m ay be
n ecessary to e n la rg e th e d a ta b a se .
2.3.6
S e n sitiv ity F ig u re fo r Y ield Im p ro v e m e n t
P u rv ia n c e a n d M eeh an (1988) in tro d u c e d a s e n sitiv ity fig u re fo r use in
g ra d ie n t o p tim iz a tio n . T h e m ain h y p o th e sis o f th e ir a p p ro a c h is th a t use o f th is new
s e n s itiv ity fig u re in a g ra d ie n t-b a s e d o p tim iz a tio n p ro cess w ill re su lt in a c ir c u it
desig n w ith im p ro v e d y ield .
S u p p o se th a t th e c irc u it p e rfo rm a n c e o f in te re s t is F(<j>). T h e s e n sitiv ity to a
p a ra m e te r 4>j is c a lc u la te d as (G u p ta , G a rg a n d C h a d h a 1981)
<2'29)
T h e resp on se fu n c tio n is f ir s t n o rm a lize d w ith re sp e c t to th e n o m in a l p o in t a n d the
to le ra n c e e x tre m e s su c h th at th e n o rm alized n o m in a l p o in t is a t th e o rig in a n d th e
no rm alized to le ra n c e e x tre m e s a re -1 a n d 1. T h e n the fu n c tio n is e x p a n d e d as a
p o ly n o m ial fu n c tio n a t th e n o m in a l p o in t
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
" 26
F tf) ~ gQ>)
**
-
N
- -
N
& i'o + 52 aA' + 52 a«°)M ' + 52
/-I
/./-l
i.j.k-1
(2.30)
a t<*ja kfa4j4,k
+
w h ere fa sta n d s fo r th e n o rm alized v a ria b le . D e fin e ~g(fa) as th e a v e ra g e value o f g(fa)
w ith re sp e c t to all th e p a ra m e te rs e x c e p t fa, th a t is
i
gifa) = [g(hp(fa*---,fa-i,fa+i,---,4>N)d4>i'''dfa-idfa.l “ 'd $ N ,
(2.31)
-i
.‘j
w h ere
p(<f>i-.----,fa-i->fa,\
*s l he p ro b a b ility d e n s ity
fu n c tio n o f all the
n o rm alized p a ra m e te rs e x c e p t fa. N o tic e th a t a n a ssu m p tio n m a d e is th a t it is possible
to se p a ra te fa fro m th e p ro b a b ility d e n s ity fu n c tio n o f alt p a ra m e te rs . T h e d e riv a tiv e
o f th e av e ra g e p e rfo rm a n c e fu n c tio n value is g iv e n by
-
dfa
OO
\ ^ ^ - p ( f a , - . . , f a . 1,fa ,i^--> (PN)d f a m" d h - i d ^ i — d ~4>N (2.32)
-oo dfa
T h e d e riv a tiv e s
Zsifa)
.
. ,
w
;— , t = 1, 2, . . . , N,
dfa
a re used as th e g ra d ie n t re q u ire d by th e o p tim iz e r to in flu e n c e th e o p tim iz a tio n
so lu tio n .
T h e p ro b lem s w ith th is a p p ro a c h a re (1) th e a ssu m p tio n on th e se p a ra b le
p ro b a b ility d e n sity fu n c tio n m ig h t n o t be a p p lic a b le to m a n y p ra c tic a l p ro b le m s, an d
( 2)
th e u n c le a r c o n n e c tio n b e tw e e n th e so lu tio n fo u n d by th e p ro p o se d se n sitiv ity
fig u re a n d th e n o m in al p o in t f o r o p tim a l y ield.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
27
j;
2.3.7
S im u la te d A n n e a lin g O p tim iz a tio n
S im u la te d a n n e a lin g is a te c h n iq u e fo r th e so lu tio n o f d if f ic u lt c o m b in a to ria l
o p tim iz a tio n p ro b le m s ( K ir k p a tr ic k , G e la tt a n d V ecchi 19S3). It has b ee n e x te n siv e ly
used f o r c ir c u it g e o m e tric a l p a rtitio n in g , c o m p o n e n t p la c e m e n t, a n d c irc u it w irin g
(Je p s e n a n d G e la tt 1983, C a s o tto , R o m e o , a n d S a n g io v a n n i-V in c e n te lli 1987). V ery
re c e n tly , th is m e th o d w as a p p lie d to m o d ellin g o f m icro w av e se m ic o n d u c to r d e v ice s
(V a i, P ra sa d , L i, a n d K a i 1989). V a i, P rasad , a n d M eskoob (1990) also used sim u late d
a n n e a lin g o p tim iz a tio n f o r m ic ro w a v e c irc u it y ie ld -d riv e n d esig n .
S im u la te d a n n e a lin g
b elo n g s to th e ran d o m o p tim iz a tio n ca te g o ry .
It
c o n d itio n a lly a c c e p ts h ig h in te rm e d ia te values o f the o b je c tiv e fu n c tio n to allow
p ro b a b ilis tic h ill-c lim b in g .
F o llo w in g the m ech an ism o f th e a n n e a lin g p rocess, a
c o n tro llin g
p a ra m e te r c a lle d
p s e u d o -te m p e ra tu re
p ro c e d u re .
P s e u d o -te m p e ra tu re is se t re la tiv e ly high a t th e in itia l stag e, th en is
d e c re a se d a r tific ia lly a n d slo w ly .
is used
in
th e o p tim iz a tio n
I f th e p re se n t ite ra tiv e so lu tio n d ecrea ses the
o b je c tiv e fu n c tio n v a lu e , th e s o lu tio n is a c c e p te d as in c o n v e n tio n a l o p tim iz a tio n
m e th o d s. If an in te rm e d ia te so lu tio n in creases th e o b je c tiv e fu n c tio n v a lu e, th e n th e
a c c e p ta n c e is c o n d itio n a l o n th e re s u lt o f a ra n d o m e x p e rim e n t su ch th a t p ro b a b ility
o f th e a c c e p ta n c e o b e y s a B o ltz m a n n d istrib u tio n
-a A
e
‘k
(2.33)
w h ere A(/j. is th e d if f e r e n c e b e tw e e n tw o o b je c tiv e fu n c tio n v alu e s, Tk is the c u rre n t
te m p e ra tu re , a n d o is a w e ig h tin g fa c to r. F or th e sam e a m o u n t o f o b je c tiv e fu n c tio n
in c re m e n t, it is m o re lik e ly to b e a c c e p te d at a h ig h e r te m p e ra tu re th a n a t a low er one.
T h is o p tim iz a tio n p ro c e d u re is a p p lie d to a d is trib u te d a m p lifie r c irc u it (V ai,
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
28
P ra sa d , a n d M esk o o b 1990). T h e o b je c tiv e fu n c tio n f o r y ie ld o p tim iz a tio n is d e fin e d
u sing th e p re d ic te d yield ra te .
( U n fo rtu n a te ly , th e fo rm u la tio n o f th e o b je c tiv e
fu n c tio n fo r y ield o p tim iz a tio n is n o t m ade a v a ila b le in th e ir p a p e r). S olu tio n s w ith
im p ro v ed yield rates a re a c c e p te d a u to m a tic a lly w h ile th e a c c e p ta n c e o f th o se w ith
it.
d e c re a sed yield rates is g o v e rn e d b y th e B o ltzm an n d is trib u tio n .
A in te re stin g
a d v a n ta g e o f th e ir a p p ro a c h is th a t th e y allo w th e n u m b e r o f stag es in th e d is trib u te d
a m p lifie r as v ariab le.
Sim u lated an n e a l ing o p tim iz a tio n re q u ire s e x tre m e ly little c o m p u ta tio n a l effo rt-'
itse lf a n d is v e ry easy to im p le m e n t.
T h e m ost d is tin c t a d v a n ta g e o f sim u la te d
a n n e a lin g is its a b ility to reach th e global o p tim a l so lu tio n w ith o u t re q u irin g a good
in itia l
s ta rtin g p o in t.
H o w e v e r, it is c o m m o n ly
a d m itte d th a t y ie ld o p tim iz a tio n
sh o u ld s ta r t w ith a n o m in al d e sig n w h ich u su ally is a re a so n a b ly go o d p o in t. T h e
m a jo r d ra w b a c k o f sim u la te d an n e a lin g o p tim iz a tio n is th e v e ry slow co n v e rg e n c e ra te
w hen th e te m p e ra tu re is low . V ery h igh c o m p u ta tio n a l co sts d u e to re p e a te d c irc u it
sim u la tio n m ay need be re q u ire d .
2.4
T H E G E N E R A L IZ E D l p C E N T E R IN G A P P R O A C H
2.4.1
F o rm u la tio n o f Y ield O p tim iz a tio n
It is alw ay s o u r d e s ire to c o n v e rt th e p ro b le m o f y ield o p tim iz a tio n to a w ell
b eh a v e d
m a th e m a tic a l
p ro g ra m m in g
o p tim iz a tio n te c h n iq u e s can be a p p lie d .
p ro b le m
so
th a t
m o d e rn
m a th e m a tic a l
B an d le r a n d C h e n (1 9 8 9 ) p ro p o se d th e
g e n e ra liz ed £p c e n te rin g a p p ro a c h .
In th e fo llo w in g th e d e sig n v ariab les a re th e n o m in a l v a lu es <f>°. K. sta tistic a l
o u tco m es a re g e n e ra te d fro m th e g iv en p ro b a b ility d e n s ity fu n c tio n . A lth o u g h o n ly
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
29
th e o u tco m es 4 a p p e a r in th e e r r o r fu n c tio n s, th e y d e p e n d o n
becau se th e 4 a re
re la te d to $ ° .
A f te r th e e r r o r v e c to r 4 f o r th e o u tco m e 4 has b ee n a ssem b led as
(2.34)
et (4 )
w h e re L is th e to tal n u m b e r o f e rro rs c o n sid e re d , the fo rm u la tio n o f the o b je c tiv e
fu n c tio n fo r o p tim iz a tio n c a n fo llo w th e p ro c e d u re d e s c rib e d in B an d ler an d C h en
(1988). F irst, we c re a te th e g e n e ra liz e d l p fu n c tio n v‘ fro m d ,
(2.35)
E {-ej{4))-p
j -1
p, i f J ( 4 ) = 0
w h ere
A 4 ) = ij\e j(4 ) > 0 ).
(2.36)
T h e n w e d e fin e th e o n e - s id e d t x o b je c tiv e fu n c tio n f o r y ield o p tim iz a tio n as
w °) =
le t
.
(2.37)
w h e re
I = { /tv ' > 0 )
(2.38)
a n d dj a re p o sitiv e m u ltip lie rs . If th e a,- w ere ch o sen as
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
th e n fu n c tio n U(<f>°) w ould b e c o m e th e ex ac t n u m b e r o f u n a c c e p ta b le c irc u its, th a t
is,
„ f ,o ,
n u m b e r o f u n a c c e p ta b le c irc u its
U(<p ) = -------------------------^ -------------------------
(2.40J
a n d th e yield w ould be
YU?) = 1 - W p - .
K.
(2.41)
T h e m ech an ism o f th e o n e - s id e d £2 fu n c tio n n a tu ra lly im ita te s th e relatio n
b e tw e e n th e yield an d u n a c c e p ta b le o r a c c e p ta b le o u tc o m e s.
m a x im iz in g yield Y is c o n v e rte d to o n e o f m in im iz in g
m in im ize
N o w , the task o f
T h a t is
.
(2.42)
We use (2.39) to assig n m u ltip lie rs a,- a t the s ta rtin g p o in t a n d fix th em d u rin g
th e o p tim iz a tio n process. T h e n
is no lo n g er th e c o u n t o f u n a c c e p ta b le ou tco m es
d u rin g o p tim iz a tio n , b u t a c o n tin u o u s a p p ro x im a te fu n c tio n to it.
2.4 .2
Im p le m e n ta tio n a l A sp ects
S u p p o se th e value o f p in (2 .3 5) is ch o sen as 1. T h e o b je c tiv e fu n c tio n fo r the
o n e -s id e d
o p tim iz a tio n b eco m es
tw°> = E E
(2.43)
/e/ j ^ W )
w h e re a,-, /
and
o p tim iz a tio n a re e ft ') .
a re d e fin e d as b e fo re .
In (2 .4 3 ), e rro r fu n c tio n s fo r
In (2 .3 7 ), e r r o r fu n c tio n s f o r o p tim iz a tio n are v‘.
fu n c tio n s e j$ ) a r e d if f e r e n tia b le , b u t the fu n c tio n s
The
v* o f (2 .3 5 ) mo.y n o t be.
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31
H o w e v e r, (2 .4 3 ) has m o re e r r o r fu n c tio n s th a n (2 .3 7 ) i f m o re th a n o n e sp e c ific a tio n
is im p o sed .
c
In th e fo llo w in g c h a p te rs , w e use b o th (2.37) a n d (2 .4 3 ) in th e d if f e r e n t yield
o p tim iz a tio n p ro b lem s.
T o d istin g u ish th e m , w e re f e r to (2.37) a n d (2.43) as
Im p le m e n ta tio n I a n d Im p le m e n ta tio n II o f th e o n e -s id e d £j c e n te rin g ap p ro a ch ,
re sp e c tiv e ly .
S ev eral re o p tim iz a tio n s w ith u p d a te d ct; m ay b e a p p lie d to f u r th e r increase
y ield . E ach c a n use a d if f e r e n t n u m b e r o f sta tistic a l o u tc o m e s o r a d iffe r e n t set o f
o u tco m es.
T o su m m a riz e th e d iscu ssio n in th is s e c tio n , all s te p s in v o lv e d in o u r yield
o p tim iz a tio n a re sh o w n in Fig. 2.3.
2.5
O N E -S ID E D €x M A T H E M A T IC A L P R O G R A M M IN G
A h ig h ly e f f ic ie n t o n e -s id e d
o p tim iz a tio n a lg o rith m (B a n d le r, C h en a n d
M ad sen 1988) is u sed to so lv e (2.37) o r (2.43). T h e a lg o rith m is based on a tw o -sta g e
m e th o d c o m b in in g a f ir s t- o r d e r m e th o d , th e tru s t reg io n G a u s s -N e w to n m e th o d , w ith
a s e c o n d - o r d e r m e th o d , th e q u a s i-N e w to n m e th o d .
S w itc h in g b e tw e e n th e tw o
m eth o d s is a u to m a tic a lly m a d e to e n s u re g lo b al c o n v e rg e n c e o f th e co m b in ed
a lg o rith m .
2.5.1
F o rm u la tio n o f th e P roblem
T h e o p tim iz a tio n p ro b le m to be c o n sid e re d has th e fo llo w in g m ath em a tica l
fo rm u la tio n . Let
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32
determine design specifications,
design variables, statistics o f parameters
and th e starting n o m inal circuit
generate statistical outcomes <j>S
initialization
outcom e generation
solve circuit equations P(Z‘ <3>') = 0
circuit simulation &
sensitivity analysis
calculate reponses Fj (<$>') and VI*.
update <J>
calculate errors ej(<t>1) and V ej (<}>*),
error function &
gradient calculation
adjust <p°
one-sided £ j
optimization
test convergence
next iteration
local minimum o f U(<J>°) reached
F ig. 2.3
F lo w c h a rt o f y ield o p tim iz a tio n
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33
/i(* )
/ 2(x)
A x) =
(2.44)
fjx )
be a s e t o f n o n lin e a r, c o n tin u o u sly d iffe re n tia b le fu n c tio n s. T h e v e cto r
x
is th e set o f p a ra m e te rs to be o p tim iz ed .
(2.45)
We d e fin e an in d e x set w hich co n ta in s
in d ices o f all fu n c tio n s w ith p o sitiv e values
J(x) = ( j | f}{x) > 0 1.
T h e n th e o n e -s id e d
(2.46)
o p tim iz a tio n p ro b le m ca n be sta te d as
m in im iz e M x )
(2.47)
S u b s titu tin g e ith e r (2.37) o r (2.43) in to (2.42) g iv es a o n e -s id e d l x p ro b le m as d e fin e d
in (2.47).
2.5.2
A lg o rith m s fo r th e O n e -S id e d i l P roblem
T h e tru s t reg io n G a u s s -N e w to n an d Q u a s i-N e w to n m eth o d s have th e ir ow n
a d v a n ta g e s. T h e tru st reg io n m e th o d is su p p o sed to w o rk w ell a t the b e g in n in g stage
o f o p tim iz a tio n , b u t to c o n v e rg e v e ry slow ly w h en close to a s in g u la r so lu tio n . T h e
q u a s i-N e w to n m e th o d has a fa s t ra te o f c o n v e rg e n c e n e a r a s o lu tio n b u t is not reliab le
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
34
fro m a bad s ta rtin g p o in t. By c o m b in in g th em to g e th e r, w e h o p e to fu lly e x p lo re th e ir
ad v an tag es a n d to e ffe c tiv e ly av o id th e ir sh o rtc o m in g s (H a ld an d M adsen 1981, 1985).
S ev eral sw itc h e s b etw een th e tw o m e th o d s m ay ta k e p lac e a n d th e sw itc h in g c rite ria
e n s u re g lo b al co n v e rg e n c e o f th e c o m b in e d a lg o rith m .
A T ru s t R eg io n G a u ss-N e w to n M eth od (M e th o d f)
A t th e A'th ite ra tio n w ith th e p re se n t so lu tio n x k , a local b o u n d Ak is ch o se n .
T h e fo llo w in g su b p ro b le m is to be solved
m i n i m i z e ^ .!>•
!>.y /■=l
s u b je c t to
(2.48)
j = 1, 2, . . . , w ,
* fjUk) * / / ( x i f h ,
y, i
A* > //;,
w h ere
Ak > -Itj,
o,
/ «= 1 , 2
N,
is th e g ra d ie n t o f fj. A sta n d a rd lin e a r p ro g ra m m in g a lg o rith m c an be used
to so lv e th is p ro b le m f o r hk . I f hk red u ces th e o b je c tiv e fu n c tio n o f th e o rig in al
p ro b le m , i.e., if U(xk + hk) < U(xk), th en x k + hk is a c c e p te d as a n im p ro v e d so lu tio n
to (2 .4 8 ) a n d th e n e x t ite ra te sta rts. O th e rw ise , th is ite ra tio n is c o n sid e re d as a fa ilu re ,
an d th e p re v io u s so lu tio n is k ep t.
T h e local b o u n d Ak sh o u ld be a d ju s te d in e v e ry ite ra tio n .
i\
T h e rule f o r
a d ju s tin g Ak is based on w h e th e r o r n ot th e lin e a riz e d su b p ro b le m (2.48) is a good
a p p ro x im a tio n to th e o rig in al n o n lin e ar p ro b le m in the p re se n t tru s t reg io n d e fin e d
by th e c u r r e n t local b o u n d Ak . A d e ta ile d d e s c rip tio n o f th e a lg o rith m to controlA ^
is g iv en by B an d lcr, K e lle rm a n n a n d M ad sen (1987).
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
35
'-V ,
A Q u a s i-N e w to n M e th o d (M e th o d i n
T h is m e th o d so lv es th e o p tim a lity c o n d itio n o f th e o n e - s id e d
p ro b lem . T h e
o p tim a lity c o n d itio n is a se t o f e q u a tio n s
E /j(x) ♦ £«,•//<*) =0,
o
w h e re
(2.49)
a n d J 0 a re d e fin e d as
j . A { j I fj{x) > 0 }
(2.50)
J Q * U \ fj(x ) = 0 )>
(2.51)
and
re s p e c tiv e ly , a n d th e m u ltip lie rs m ust sa tisfy
1>
> 0,
j e
J 0-
(2.52)
T h e se o p tim a lity e q u a tio n s re s u lt fro m a p p ly in g th e K u h n - T u c k e r c o n d itio n s to the
o n e -s id e d t x p ro b le m .
T h e q u a s i-N e w to n m e th o d is used to so lv e (2 .4 9 ). S e c o n d -o rd e r d e riv a tiv e
in fo rm a tio n is re q u ire d .
A m o d ifie d B FG S fo rm u la (P o w e ll 1978, and B an d ler,
K e lle rm a n n a n d M ad sen 1987) can b e a d o p te d to g e n e ra te a n d u p d a te th e H essian
m a trix .
A C o m b in e d 2 -S ta a e A lg o rith m
Based o n th e th e o ry o f H ald a n d M ad se n (1981 a n d 1985), the a lg o rith m
c o m b in e s th e tru s t re g io n G a u s s-N e w to n m e th o d (M e th o d I) w ith the q u a si-N e w to n
m e th o d (M e th o d II).
A t th e b e g in n in g , th e tru st reg io n is used. A d d itio n a l to the c o m p u ta tio n o f
th e tru s t re g io n m e th o d , th e p re p a ra tio n f o r M e th o d II is also m a d e, w hich in clu d es
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
e stim a tio n o f a c tiv e sets J 0 in (2.51), estim a tio n o f th e m u ltip lie rs S; in (2.52), an d
u p d a tin g o f th e a p p ro x im a te Ja c o b ia n using a m o d ifie d B FG S fo rm u la (P ow ell 1978,
also B a n d le r, K e lle rm a n n a n d M adsen 1987).
A sw itc h fro m M e th o d I to M e th o d 11 is m ad e i f th e fo llo w in g co n d itio n s a re
met:
( 1)
T h e estim ated a c tiv e se t J 0 has b ee n u n c h an g e d o v e r a p re d e te rm in e d n u m b e r
v
(2)
o f c o n secu tiv e ite ra tio n s.
T h e estim a te d m u ltip lie rs c o rre sp o n d in g to J 0 s a tisfy (2.52).
If M eth o d II is u n su c c e ssfu l, th en the sa fe m e th o d , M e th o d I, sh o u ld be used
ag a in . A sw itch fro m M eth o d II back to M ethod I is m ade if o n e o f th e fo llow ing
c o n d itio n s is:m et:
(1)
T h e c o n te n ts o f e ith e r J0 o r J^ need b e u p d a te d b e c a u se a fu n c tio n n o t
in c lu d e d in J Q has beco m e zero o r c h a n g e d sig n .
(2)
A t least on e m u ltip lie r has v io late d th e c o n s tra in t.
(3)
A q u a si-N e w to n step fa ils to d e c re a se th e re sid u a l o f th e o p tim a lity equations:
0.999 \\Rk ||
(2.53)
S ev eral sw itch es b etw een th e tw o m eth o d s m ay ta k e p la c e u n til co n v erg en ce
is re a c h e d . T h is tw o -sta g e a lg o rith m w ill be used in o u r y ield o p tim iz a tio n in th e
fo llo w in g c h a p te rs.
2.6
C O N C L U D IN G R E M A R K S
In th is c h a p te r, w e h av e c o n s id e re d th e y ie ld -d riv e n c ir c u it d e sig n p ro b lem .
T h e re le v a n t d e fin itio n s a n d n o ta tio n s h av e b e en in tro d u c e d . T h e fo rm u la tio n o f the
s ta tistic a l d esig n p ro b le m has b een p re se n te d . A n u m b e r o f e x is tin g ap p ro a c h e s to
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
37
s ta tis tic a l d e sig n h av e been rev iew ed in c e rta in d e ta il. T h e ir in d iv id u a l ad v a n ta g e s
a n d s h o rtc o m in g s h av e b een a d d re sse d .
In o r d e r to in tro d u c e th e new re su lts p re se n te d in th e u p co m in g c h a p te rs , wc
h a v e e m p h a s is e d on th e g e n e ra liz e d £p c e n te r in g a p p ro a c h .
D u e to its im p o rta n t
p ro p e rtie s , a sp e c ia l case o f th is a p p ro a c h , n a m e ly , th e o n e -s id e d £j a p p ro a c h , has
b e e n d isc u sse d . T w o p o ssib le im p le m e n ta tio n s o f the o n e -s id e d
m e th o d h av e been
id e n tifie d . A tw o -s ta g e a lg o rith m c o m b in in g th e tru s t G a u s s -N e w to n m eth o d an d the
q u a s i-N e w to n m e th o d has b e e n o u tlin e d .
R e p ro d u c e d with perm ission of th e copyright owner. Fu rth er reproduction prohibited without perm ission
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
E F F IC IE N T Q U A D R A T IC A P P R O X IM A T IO N FO R S T A T IS T IC A L D ESIG N
3.1
IN T R O D U C T IO N
ij
In o r d e r to m a k e e x istin g sta tistic a l c irc u it design m eth o d s m ore p ra c tic a lly
u sa b le , m a n y a p p ro a c h e s h av e b e e n d e v ise d to re d u c e v ery co stly c o m p u ta tio n a l e f fo rt
by a p p ro x im a tin g a c c e p ta b le re g io n s o r c irc u it responses. Q u a d ra tic a p p ro x im a tio n
has p ro v e n s u ita b le a n d su c c e ssfu l (B a n d le r an d A b d e l-M a le k 1978, A b d e l-M a le k an d
.y''
B a n d le r 1980, H o c e v a r, L ig h tn e r a n d T ric k 1983, a n d B iern ack i a n d S ly b lin sk i 1986).
H o w e v e r, th e d e te rm in a tio n o f a q u a d ra tic m odel itse lf fo r a p ro b lem w ith a large
n u m b e r o f v a ria b le s m a y be too e x p e n siv e .
For a c irc u it w ith 50 e le m e n ts, th e n u m b e r o f c o e ffic ie n ts in the q u a d ra tic
m o d el is 1326. T h e c a lc u la tio n o f th e c o e ffic ie n ts in a tra d itio n a l m a n n e r involves
1326 c ir c u it s im u la tio n s a n d s o lv in g a lin e a r sy stem o f 1326 e q u a tio n s. Besides all th e
c o e f f ic ie n ts , th e m a trix o f th e lin e a r sy stem re q u ire s sto ra g e o f a 1326 by 1326 a rra y .
D e te rm in in g a q u a d r a tic a p p ro x im a tio n to th e resp o n se o f s u c h a c irc u it c re a te s q u ite
a la rg e p ro b le m in te rm s o f c o m p u te r tim e a n d s to ra g e , a lth o u g h th e c irc u it itself m ay
b e o f a m o d e ra te scale. T h e r e f o r e , f o r la rg e scale p ro b le m s th e tra d itio n a l a p p ro a c h e s
th a t a im to o b ta in u n iq u e q u a d r a tic m odels do n ot e ffe c tiv e ly re d u c e co m p u tatio n a l
costs.
B ie rn a c k i a n d S ty b lin sk i (19 86) in tro d u c e d th e c o n c e p t o f th e m axim ally fla t
in te rp o la tio n a n d p re s e n te d an u p d a tin g a lg o rith m . T h e m ost s ig n ific a n t p ro p e rty o f
th e ir a p p ro a c h is th a t th e m e th o d allow s the n u m b e r o f a c tu a l c irc u it sim u la tio n s
39
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
40
re q u ire d fo r a n a c c u ra te m o d el to b e m u ch less th a n th a t n e e d e d f o r a fu ll u n iq u e
q u a d ra tic a p p ro x im a tio n . H o w e v e r, th e co m p u ta tio n a l re q u ire m e n t o f th e m e th o d ,
e sp e c ia lly sto ra g e sp a c e , is still h ig h .
In th is c h a p te r we su b sta n tia lly e n h a n c e th e m a x im a lly f la t q u a d ra tic
in te rp o la tio n . O u r a p p ro a c h m akes use o f a fix e d p a tte rn o f p o in ts a t w h ich s im u ­
latio n is p e rfo rm e d , re su ltin g in v e ry low co m p u ta tio n al re q u ire m e n ts f o r b o th C P U
tim e a n d sto ra g e (B ie rn a c k i, B a n d le r, Song a n d Z h an g 1989). T h e b asic c o n c e p t is
re v ie w e d in S ectio n 3.2. O u r new a p p ro a c h is d e sc rib e d in S ec tio n 3 .3 . C om parisonso f th e e ffic ie n c y o f o u r a p p ro a c h a n d th a t o f th e o rig in al m a x im ally f la t q u a d ra tic
a p p ro x im a tio n a re g iv e n in th e sam e sectio n . In S ection 3.4, th e p ro p o se d q u a d ra tic
a p p ro x im a tio n te c h n iq u e is a p p lie d to m odel c irc u it resp o n se fu n c tio n s . A lo w -p a ss
filte r se rv e s as an e x a m p le to d e m o n stra te th is im p le m e n ta tio n . In S ectio n 3 .5, w e
u tiliz e th e q u a d ra tic a p p ro x im a tio n to m odel n o t o n ly c irc u it p e rfo rm a n c e fu n c tio n s,
b u t also th e ir g ra d ie n ts (B a n d le r, B ie rn a c k i, C h e n , Song, Ye a n d Z h a n g 1991c). O u r
g r a d ie n t-b a s e d o p tim iz a tio n p ro c e d u re , th e o n e -s id e d l x c e n te rin g a p p ro a c h (B a n d le r
an d C h e n 1989), re q u ire s g ra d ie n t in fo rm a tio n .
H ig h e r g r a d ie n t a c c u ra c y w ill
im p ro v e th e o v e ra ll p e rfo rm a n c e o f th e o p tim iz a tio n process.
F in a lly , S ectio n 3.6 c o n ta in s th e co nclusions.
3.2
T H E M A X IM A L L Y F L A T Q U A D R A T IC A P P R O X IM A T IO N
A q u a d ra tic m odel in p o ly n o m ia l fo rm to be u sed to a p p ro x im a te a g iv e n
fu n c tio n / ( x ) , x = [x j x 2 . . .
can be w ritte n as
it
q(x) = a0 * T , a i( xi ~ ri) +
i- l
n
£ a ij( xi ~ riK*j i j -!,/< /
(3.1)
R e p ro d u c e d with perm ission of the copyright owner. Fu rther reproduction prohibited without permission.
cs
41
w h e re r = [/-j r 2 . . . rn ]r is a k n o w n re fe re n c e p o in t. T h e fo rm o f th e q u a d ra tic
f u n c tio n u sed is sim ila r to th a t o f B ie rn a c k i a n d S ty b lin sk i (1986). H o w e v e r, q(x) is
d e f in e d h e re w .r.t. th e re fe re n c e p o in t r ra th e r th a n w .r.t. th e o rig in . N o te th a t th e
s u b s c rip t n o ta tio n is su c h th a t e a c h c o e ffic ie n t c a n b e easily id e n tifie d w ith its
c o rre s p o n d in g x te rm , e .g .,
is th e c o e f fic ie n t o f (Xj - r;)(Xj - r; ) . D e te rm in in g a
q u a d ra tic m o d e l is e q u iv a le n t to d e te rm in in g all its c o e ffic ie n ts , w h ic h a re now
u n k n o w n s in (3.1).
S u p p o se th a t m (m > n + 1) e v a lu a tio n s o f / ( * ) a re p e rfo rm e d a t som e p o in ts
y , / = 1, 2 , . . ., m . T h e se p o in ts a re c a lle d the b ase p o in ts. U sin g th e values o f
/ ( j 0 , w e se t u p a sy stem o f lin e a r eq u a tio n s
G i l Q \2
a
' / r
G21 Q22
V
fz
w h e re a a n d v a re a rra n g e d to h a v e th e fo llo w in g orders:
a = [ a0
ax
.
.
.
^ ]r
(3.3)
and
v = [ an
a 22 .
,
.
a12
a 13
.
re s p e c tiv e ly .
T h e v e c to rs f j a n d f x a re o f d im e n sio n s
re s p e c tiv e ly .
T h e y c o n ta in f u n c tio n
values /(■*').
.
.
]
,
(« + 1) a n d m
T h e m a trix £?,y, /, j =
( 3 .4 )
- (« + 1) ,
I, 2, is
d e te r m in e d b y th e c o o rd in a te s o f th e base p o in ts a n d o f r .
S im ila rly to th e a p p ro a c h d u e to B ie rn a c k i a n d S ty b lin sk i (1986), th e re d u c e d
s y ste m w ith v a ria b le s v is o b ta in e d as
C v = e,
(3.5)
w h e re
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
jj
42
C = 0 22 -
021Q u Q 12
(3.6)
and
e = / 2 - 621611 A *
m < f o —..
\\
£
(3.7)
^ , th e a b o v e sy stem is u n d e r-d e te rm in e d .
'l
W hen th e le a s t-s q u a re s c o n s tra in t is a p p lie d to v, th e u n iq u e so lu tio n to (3.5)
c a n b e fo u n d as
v =
Cr(CCrr 1e
(3.8)
a n d a is re a d ily o b ta in e d as
a = On 1/1 - 6 i i 012 v -
(3.9)
T h e n v , c a lle d th e m in im a l E u c lid e a n n o rm so lu tio n o f (3 .5 ), a n d a g iv e th e
m a x im a lly f la t q u a d ra tic in te rp o la tio n in th e fo rm o f (3.1) to f ( x ) . T h e te rm o f th e
m a x im a lly f la t q u a d ra tic a p p ro x im a tio n com es fro m th e m e c h a n ism o f th e le a s tsq u a re s c o n s tr a in t th a t fo rc e s th e s e c o n d - o rd e r d e riv a tiv e s to b e as sm all as po ssib le.
3.3
A P P R O A C H U S IN G A F IX E D P A T T E R N O F B A SE P O IN T S
3.3.1
D e riv a tio n a n d A lg o rith m o f th e A p p ro a c h
In th e o rig in a l sc h e m e o f B ie rn a c k i a n d S ty b lin sk i (1986) all base p o in ts a re
ra n d o m ly se le c te d . T h is ty p e o f sele c tio n allow s c e rta in fre e d o m . H o w e v e r, se v e ra l
larg e m a tric e s h a v e to be s to re d a n d m a n ip u la te d . F o r in s ta n c e , m a tr ix C in (3.6)
n e e d s a n a rra y w ith d im e n sio n
— M +- ) [ m^ —
?il .
M e a n w h ile , so m e fa ir ly
in v o lv e d c a lc u la tio n s , su c h as m a trix in v e rs io n , o r e q u iv a le n t c a lc u la tio n s sh o w n in
(3 .6 )—(3 .9 ), a re re q u ire d . E v e n a c ir c u it o f a re a so n a b le size m ay d e m a n d la rg e sto ra g e
sp a c e a n d C P U tim e . H e re , w e sh all p ro p o se a n ew a p p ro a c h w h ic h is b a se d o n a
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
43
fix e d p a tte rn o f base p o in ts. T h e re g u la rity o f th e p a tte rn w ill g re a tly re d u c e sto rag e
a n d sim p lify th e c a lc u la tio n o f c o e ffic ie n ts .
In o u r a p p ro a c h , o n ly m ( n +• 1 < m < 2n + 1) base p o in ts a re used. ~ rhc
r e fe re n c e p o in t r is se le c te d as th e f ir s t base p o in t jc1. T h e n e x t it base p o in ts arc
d e te rm in e d by p e rtu rb in g o n e v a ria b le at a tim e a ro u n d r , i.e..
[o . . . 0
0-t
0 . . . o f,
(3.10)
/ = I , 2, . . ., « ,
w h e re 0; is a p re d e te rm in e d p e r tu r b a tio n . It can be sh o w n th a t th e firs t ( n * 1 ) base
p o in ts lead to v ery sim p le fo rm s o f m a tric e s
1
1
a n d Q u.Q iz- T h e y are
0
1
X
0
0
h
0
0
(3.11)
Q'n =
0
o
_L
1
o
Tn
K
and
0
0
0
1
01
0
0
I
0
I
o X
Q n Q -iz ~
0
0
0
[
0
0
0
I
0
0
0
I
0
0
0n
I
(3.12)
B ecause o f th is sim p le p a tte rn th e y need n o t be sto re d in m a trix fo rm .
A fte r th e f irs t n * I b ase p o in ts, th e re m a in in g m - (« + 1 ) p o in ts fo llo w to
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
44
p ro v id e th e s e c o n d - c r d e r in fo rm a tio n o n th e fu n c tio n . S im ila rly to th e base p o in ts
d e fin e d in (3 .1 0 ), c o n se c u tiv e base p o in ts a re se lec ted b y also p e rtu rb in g one v a ria b le
a t a tim e.
F or sim p lic ity , th e se b ase p o in ts a re d e te rm in e d b y co n se c u tiv e ly
p e rtu rb in g th e v ariab les in r , th a t is
= r *
0
[o . . ; o -y; o . . . o f ,
(3-13)
/' = 1, 2 , . .
k,
w h ere 7/ is a n o th e r p e r tu r b a tio n o f r , w h ich m u st n o t e q u al P,, a n d
k = m - (« + 1).
U n d e r this a rra n g e m e n t m a tric e s Q 2i an d Q22 have re g u la r s tru c tu re s . S u b stitu tin g
Q2l a n d 0 22 in to (3 .6 ), th e m a trix C takes a co n cise a n a ly tic a l fo rm
o
~ Pihi
o
o
o
- Pihj
o
(3.14)
o
o
o
h k - Pkhk 0
an d th e v e c to r e c a n be e x p re sse d by
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
45
i
2i
Pi
2i
Pi
12
2!
~/?2
'K
o
o
e - f2 ~
i
2
Pi
fx
2
p.-
(3.15)
0
0
I
2
2l
—
Pk
Pk
o
S u b s titu tin g (3.14) a n d (3.15) in to (3.8), th e c o e ffic ie n ts a re d e te rm in e d by
/ ( x " t l w ) - / ( x 1) _
l i - Pi
/ l x M ) - / ( * 1)
1i
Pi
( = 1 , 2
Qji = 0 ,
(3.16a)
k ,
n,
r = A: 1
(3.16b)
and
fl/y = 0 ,
i * j,
i, j = 1 ,2
n.
(3.16c)
T h e c o e ffic ie n ts fl0 a n d a- a re easily o b ta in e d as
o0 = / ( x 1)
(3.17a)
and
[ / (xM > - / ( x 1)) - 0 faj7 ,
Pi
( = 1 , 2 ..................it.
(3.17b)
T h e m ax im ally fla t q u a d ra tic in te rp o la tio n , u sing a fix e d p a tte rn o f base
p o in ts d e fin e d h e re , has an in te re stin g p ro p e rty .
A ll the c o e f fic ie n ts o f the m ixed
te rm s, fly fo r i *■ y , a re c o n v e n ie n tly fo rc e d to b e zero becau se no re la te d in fo rm a tio n
ca n be e x tra c te d fro m th e fix e d p a tte rn .
A ny o f th e ai;'s in (3 .1 6 a ), i< k , can be
R e p ro d u c e d with perm ission of the copyright owner. Fu rth er reproduction prohibited without permission.
n o n z e ro b ecau se d o u b le p e rtu rb a tio n s a re m ad e alo n g a s tra ig h t lin e p a ra lle l to th e ith
ax is. I f a th ird p e rtu rb a tio n is m ade alo n g th e sam e lin e , it can b e sh o w n th a t th e C
m atrix w ill not have fu ll ro w ra n k , a n d , th e re fo re , th e th ird p e rtu rb a tio n d oes not
p ro v id e a n y e x tra u sefu l in fo rm a tio n fo r the q u a d ra tic m odel.
It sh o u ld be n o te d ,
h o w ev er, th at th e fa c t that th e m ix ed te rm c o e ffic ie n ts b ecom e zero is a c o n se q u en c e
o f th e m ax im ally fla t in te rp o la tio n .
In a situ a tio n w h ere th e m ix ed te rm s a re im p o rta n t, th is a p p ro a c h can easily
be m o d ifie d by in tro d u c in g a n a p p ro p ria te tra n sfo rm a tio n o f v a ria b le s. In su c h a case
the p e rtu rb a tio n s c a n be c a rrie d o u t alo n g lines not n ecessarily p a ra lle l to th e ax es.
T h e p ro p o sed fix e d p a tte rn o f base p o in ts can th u s be g e n e ra liz e d w h ile p re se rv in g
the m ain a d v a n ta g e s o f o u r a p p ro a c h .
T h e o re tic a lly sp e a k in g , th e e ffic ie n c y an d sim p lic ity o f o u r m odels a re
a c h ie v e d at th e ex p e n se o f so m e m odel a c c u ra c y . It sh o u ld be stre sse d , h o w e v e r, th a t
even so -c a lle d ex a c t c irc u it sim u la tio n c a rrie s c e rta in a p p ro x im a tio n s o f ac tu a l
p h y sical b e h a v io u r.
T h e re fo re , o u r a p p ro a c h p ro v id e s a n e x c e lle n t m o d elin g
te c h n iq u e fo r m an y p ra c tic a l p ro b lem s, esp ecially w h en a v e ry h ig h a c c u ra c y is n o t
really n ecessary . T h e m e th o d is su ita b le fo r a m ax im u m n u m b e r o f base p o in ts o f
2n * 1. It tak es a d v a n ta g e o f th e c o n c e p t o f m ax im ally fla t q u a d ra tic in te rp o la tio n an d
th u s a n y n u m b e r o f base p o in ts b e tw e e n /t + 1 an d 2« + I can be used.
3.3.2
C o m p u ta tio n a l E ffic ie n c y
A d y n a m ic u p d a tin g sch em e w as p ro p o se d by B ie rn ac k i a n d S ty b lin sk i (1986),
w h ich allow s th e e x istin g m odel to be re v ise d w hen a new base p o in t is a d d e d .
H o w ev er, th is sim p le u p d a tin g m ay n o t b e su ita b le if so m e o f th e base p o in ts a re f a r
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
47
fro m th e re g io n o f in te re s t. T o m a in ta in a c c u ra c y , it m ay. b e d e sira b le to d isre g a rd
s u c h p o in ts. O u r m e th o d c a n b e u sed to re b u ild th e m odel v e ry e ffic ie n tly w h e n e v e r
it is n e e d e d . H o w e v e r, it c a n a lso b e u sed w ith in th e c o n c e p t o f d y n a m ic u p d a tin g
p ro v id e d th a t th e b ase p o in ts a re se le c te d in th e a fo re m e n tio n e d m a n n e r a n d th e ir
n u m b e r d o es n o t e x c e e d 2n + 1.
In th is sectio n w e c o m p a re co m p u ta tio n a l e ffic ie n c y o f o u r a p p ro a c h w ith th a t
o f th e o rig in a l m e th o d o f B ie rn a c k i a n d S ty b lin sk i (1986). T o u n ify th e c o m p a riso n
w e assu m e th a t e x a c tly 2n +1 b a se p o in ts a re u sed to b u ild th e m odel. In o u r ap p ro a c h
th e r e q u ir e d sto ra g e is re d u c e d to a m in im u m .
f u n c tio n v a lu e s a r e to b e sto re d .
O n ly 2 n p e rtu rb a tio n s a n d 2» + 1
N o m a trix m a n ip u la tio n s a re n e e d e d .
A ll
c a lc u la tio n s a re s im p lifie d to (3 .1 6) a n d (3 .1 7 ). T h e o p e ra tio n a l c o u n t to c a lc u la te all
c o e ffic ie n ts u sin g th is p a tte r n c a n b e m e re ly 4/z. In th is n ew a o o ro a c h . th e sto rag e
re q u ire m e n t a n d c o m p u ta tio n a l c o u n t v a ry lin e a rly w ith th e n u m b e r o f v a ria b les. F o r
th e o rig in a l m e th o d o f B ie rn a c k i a n d S ty b lin sk i (1 986), a t least, all base p o in ts,
m a tric e s
Q11 a n d C a re s to re d in th re e a rra y s w ith d im e n sio n s n x ( 2 n + l ) ,
( n + I ) x ( n + l) and
b x b x ( b + l ) / 2 , re sp e c tiv e ly , a n d th e c o m p u ta tio n a l c o u n t is
0 ( n 4). F o r th e o rig in a l a p p ro a c h , th e sto ra g e re q u ire m e n t a n d c o m p u ta tio n a l co u n t
v a ry c u b ic a llv a n d o u a rtic lv . re sp e c tiv e ly , w ith th e n u m b e r o f v ariab les. F o r a c irc u it
w ith 50 e le m e n ts a n d m c h o se n as 101, w e n e e d sto rag e c o n sistin g o f tw o a rra y s o f 101
a n d 100, a n d c o m p u ta tio n a l e f f o r t o f 200 m u ltip lic a tio n s.
T h e o rig in a l a p p ro a c h
w o u ld re q u ir e sto ra g e c o n sistin g o f th re e a rra y s o f 50 b y 101, 51 b y 51 a n d 50 by
1275, re s p e c tiv e ly , a n d in c o m p a ra b le c o m p u ta tio n a l e ff o r t.
S ta tistic a l d e sig n o r d e sig n c e n te rin g in v o lv es a v e ry larg e n u m b e r o f c irc u it
s im u la tio n s . I t is e sse n tia l to re d u c e th e C P U tim e d u e to s im u la tio n w h ic h u sually
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
ta k e s a la rg e p o rtio n o f o v e ra ll d e sig n tim e . In su c h d e sig n , th e tre n d o f th e c ir c u it
re sp o n se h y p e rs u rfa c e is m u c h m o re im p o rta n t th a n th e a c c u ra c y o f th e in d iv id u a l
c ir c u it re sp o n se s. T h e re fo re , o u r m a x im ally f la t q u a d ra tic a p p ro x im a tio n is c a p a b le
o f a c c e le ra tin g th e d e sig n p ro cess w ith o u t lo o sin g m u c h a c c u ra c y in th e v ic in ity o f a
n o m in a l c irc u it. T h e a p p ro x im a te m odel is e v a lu a te d a t all d e sig n o u tco m e s th a t a re
sa m p le d a c c o rd in g to th e c h a ra c te riz e d sta tistic a l p ro p e rtie s o f th e v a ria b le s, s u c h as
m e a n s, v a ria n c e s a n d d e p e n d e n c ie s .
3.4
Q U A D R A T IC A P P R O X IM A T IO N O F C IR C U IT R E SPO N SE S
3.4.1
Im p le m e n ta tio n
T h e q u a d r a tic a p p ro x im a tio n te c h n iq u e h as b e en im p le m e n te d w ith in th e
f ra m e w o rk o f a c ir c u it d e sig n p ro g ra m w h ich u ses a g e n e ra l-p u rp o s e sim u la to r a n d
im p le m e n ta tio n I o f th e g e n e ra liz e d
c e n te rin g a p p ro a c h o u tlin e d in th e p re c e d in g
s e c tio n . T h e in d iv id u a l c irc u it re sp o n ses a re a p p ro x im a te d by q u a d ra tic fu n c tio n s .
T h e r e fe re n c e p o in t is d e fin e d as th e n o m in a l p o in t.
q u a d r a tic m o d els is b u ilt.
A t each ite ra tio n , a se t o f
T h e m o d els a re e v a lu a te d f o r all o u tc o m e s, i.e , th e
s ta tis tic a lly sa m p le d c irc u its . T h e o b je c tiv e fu n c tio n is c a lc u la te d fro m th e re s u ltin g
a p p ro x im a te e r r o r fu n c tio n s.
3.4.2
D e sig n o f a 11- E le m e n t L o w -P a ss F ilte r
A lo w -p a s s la d d e r f ilte r w ith 11 e le m e n ts u sed b y S in g h al a n d P in e l 1981, a n d
W eh rh ah n a n d S p e n c e 1984, sh o w n in F ig .3 .1 , w as u se d in th is e x a m p le . T h e u p p e r
s p e c ific a tio n w as 0 .32dB in th e fre q u e n c y ra n g e fro m 0 .0 2 H z to 1H z, a n d th e lo w e r
s p e c ific a tio n 52dB a t 1.3H z, o n th e in se rtio n loss. T h e fre q u e n c y sam p le p o in ts are
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
49
4
Xx
X2
rm n
r m
T *7
: > X3
X4
X5
r r~r\
rr~m
m n
-X 9
‘-10
‘•11
—o o-
Fig. 3.1
The LC low-pass filter.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
50
{0.02, 0 .0 4 ,..., 1, 1.3H z). A re la tiv e to le ra n c e o f 1.5% w as a ssu m ed f o r all e lem en ts.
O u tc o m e s w ere u n ifo rm ly d is trib u te d b e tw e e n to le ra n c e e x tre m e s. T h e sta rtin g p o in t
w as th e re s u lt o f a sy n th e sis p ro c e d u re (S inghal a n d P in el 1981).
R e su lts a n d c o m p a riso n s a r e g iv en in T a b le 3.1.
g e n e ra liz e d
f.x c e n te rin g a p p ro a c h w as used.
Im p le m e n ta tio n I o f th e
T w o d esig n s w ith in
th e sam e
o p tim iz a tio n e n v iro n m e n t w e re c a rrie d o u t. T h e o n ly d if fe re n c e b e tw e e n th e tw o
a p p ro a c h e s w as th e w ay o f c a lc u la tin g th e c irc u it resp o n ses.
A c tu a l c irc u it
s im u la tio n s w ere u sed in th e f ir s t d e sig n a n d o u r a p p ro x im a tio n m e th o d w as u tiliz e d
//
in th e se c o n d o n e. E ach d e sig n c o n siste d o f tw o su ccessiv e c e n te rin g p ro cesses sh o w n
as p h a se 1 a n d p h a se 2 in T a b le 3.1. T w o phases o f d e sig n w ith a c tu a l sim u la tio n s
to o k a p p ro x im a te ly 158.6 m in u te s o n th e V A X 8600 a n d re q u ire d 79200 c irc u it
s im u la tio n s. T h e fin a l y ie ld w as 63.7% . T h e v e ry larg e n u m b e r o f c ir c u it sim u la tio n s
w as d u e to re p e a te d c irc u it sim u la tio n o f 200 o u tc o m es, g ra d ie n t c a lc u la tio n s b ased
on th e p e rtu rb a tio n a p p ro a c h , a n d m a n y o p tim iz a tio n ite ra tio n s. T w o p h a se s o f d esig n
w ith o u r a p p ro x im a tio n m e th o d u se d o n ly 6.4 m in u te s a n d 1357 a c tu a l sim u la tio n s.
T h e fin a l y ie ld w as 79.7% . In a ll cases th e yield v alu es w e re e stim a te d fro m 1000
M o n te C a rlo sam p les u sin g e x a c t c ir c u it sim u la tio n s. F o r th e p u rp o se o f c la rity , th e
C P U tim e n e e d e d f o r th is y ie ld e s tim a tio n is n o t in c lu d e d in th e a fo re m e n tio n e d C P U
tim e s. In te re s tin g ly , f o r th is e x a m p le , o u r m e th o d n o t o n ly p re se n te d g re a tly re d u c e d
c o m p u ta tio n a l e f f o r t as c o m p a re d w ith th e a c tu a l s im u la tio n a p p ro a c h , b u t also
re a c h e d a h ig h e r fin a l y ie ld . It su g g ests th a t a c c u ra te , tim e -c o n s u m in g e x a c t c ir c u it
s im u la tio n d o es n o t n e c e ssa rily re su lt in a b e tte r fin a l y ie ld .
W hile n o t illu s tra te d in th e ta b le , w e h av e u sed th is e x a m p le to c o m p a re th e
e ff ic ie n c y o f o u r m e th o d w .r.t. th e o rig in a l m a x im ally f la t a p p ro a c h (B ie rn a c k i a n d
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
51
T A B L E 3.1
■'
C O M P A R IS O N O F S T A T IS T IC A L D ESIG N O F A L O W -P A S S
F IL T E R W IT H A N D W IT H O U T Q U A D R A T IC A P P R O X IM A T IO N S
V
C om ponent
N o m in al
D esign
E x act S im ulation
Q u a d ra tic A p p ro x im a tio n
Phase 1
x1
Phase 2
x2
P hase 1
x3
Phase 2
x4
0.22510
0.24940
0.25230
0.24940
0.22510
0.21490
0.36360
0.37610
0.37610
0.36360
0.21490
0.22572
0.24903
0.25269
0.24908
0.22568
0.21589
0.36313
0.37625
0.37633
0.36313
0.21587
0.22512
0.24944
0.25276
0.24882
0.22594
0.21658
0.36275
0.37698
0.37561
0.36305
0.21674
0 .22266
0.25045
0.25268
0.25028
0.22335
0 .22163
0.36291
0.37938
0.37156
0.36226
0.22168
0.21669
0.25131
0.25083
0.24067
Yield*
54.0%
61.7%
63.7%
70.2%
79.7%
Yield™
54.0%
74.0%
84.5%
xs
x1
x2
x3
x4
xs
x6
x7
xg
xg
x 10
xn
0.22120
0.23347
0.37008
0.37217
0.38529
0.37232
0.21893
200
200
200
200
x°
X1
x°
x3
48000
31200
529
828
9
7
10
19
C P U (V A X 8600)
96.3m in.*
62.4m in.*
2 .5 m in .
3.9m in.
C P U (M ic ro V A X )
4 8 1 m in .
312m in.
1 2 .3m in.
I9.5m in.*
N u m b e r o f O u tc o m e s
U sed f o r O p tim iz a tio n
S ta rtin g P o in t
N u m b e r o f S im u la tio n s
N u m b e r o f Ite ra tio n s
C P U tim es d o n o t in c lu d e y ie ld e stim a tio n based o n a c tu a l s im u la tio n .
*
T h e y ie ld is e s tim a te d u sin g e x a c t sim u la tio n a n d 1000 o u tc o m e s.
™
T h e y ie ld is e s tim a te d u sin g q u a d ra tic a p p ro x im a tio n a n d 200 o u tc o m e s used
in d esig n .
*
T h e C P U tim e is a p p ro x im a te ly given by a ssu m in g th a t th e sp e e d ra tio o f
V A X 8600 to M ic ro V A X is 5.
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
S ty b lin sk i 1986).
U sing th e sam e base p o in ts a n d e m p lo y in g th e sam e sc h e m e o f
re b u ild in g th e m o d els a t e ach ite ra tio n , th e o rig in a l a p p ro a c h re q u ire d a p p ro x im a te ly
5.3 a n d 8.3 m in u te s f o r th e sam e tw o p h ases th a t o u r m e th o d to o k 2.5 a n d 3.9 m in u te s
to fin ish . It sh o u ld b e noted th a t in b o th cases the C P U tim e n e e d ed to b u ild a n d /o r
to e v a lu a te th e m odel c o n stitu te s o n ly a fra c tio n o f th e o v era ll tim e , th u s th e
re m a in in g p o rtio n s a rc c o m m o n to th e tw o a p p ro a c h e s.
3.5
G R A D IE N T Q U A D R A T IC A P P R O X IM A T IO N S C H E M E
In th e p re v io u s p re s e n ta tio n , w e have a p p lie d q u a d ra tic a p p ro x im a tio n to
c irc u it resp o n se fu n c tio n s.
In th is se c tio n , we u tiliz e a n e ffic ie n t q u a d ra tic
a p p ro x im a tio n sch em e to re p la c e th e e x p e n siv e re p e a te d c irc u it sim u la tio n s a n d
g ra d ie n t e v a lu a tio n s, in o rd e r to sp eed u p th e process. .The n o v e lty o f th is u tiliz a tio n
is th a t not o nly
c irc u it p e rfo rm a n c e
fu n c tio n s , b u t also
th e ir g ra d ie n ts .a r e
a p p ro x im a te d . In a g ra d ie n t-b a s e d o p tim iz a tio n p ro c e d u re , su c h as th e o n e -s id e d l x
c e n te rin g a p p ro a c h (B a n d le r a n d C h e n 1989), g ra d ie n t in fo rm a tio n is c ritic a l in
d e te rm in in g th e d ire c tio n fo r o p tim iz a tio n ite ra tio n s to fo llo w .
H ig h e r g ra d ie n t
a c c u ra cy w ill im p ro v e th e o v e ra ll p e rfo rm a n c e o f th e o p tim iz a tio n p rocess.
3.5.1
Q u a d ra tic A p p ro x im a tio n s to R e sp o n se s and G ra d ie n ts
In m ost a p p ro x im a tio n a p p ro a c h e s fo r sta tistic a l d e sig n , su ch as B ie rn a ck i a n d
S ty b lin sk i (1986) a n d B ie rn a c k i, B a n d le r, Song a n d Z h a n g (1989), o n ly c ir c u it
resp o n ses a rc m o d eled b y q u a d ra tic fu n c tio n s .
T h e g ra d ie n ts o f th e resp o n ses a re
e ith e r not used o r th e ir a p p ro x im a te values a re c a lc u la te d by d if f e r e n tia tin g the
q u a d ra tic a p p ro x im a te resp o n ses.
T o f u r th e r im p ro v e th e p e rfo rm a n c e o f th e
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
v/'
'53
g r a d ie n t-b a s e d y ie ld - d r iv e n o p tim iz a tio n , m o re a c c u ra te g ra d ie n ts a re p re fe ra b le .
C o n s id e r a re sp o n se f u n c tio n w ith n v a riab le s. T h e g ra d ie n t o f th e resp o n se
V
is a v e c to r o f fu n c tio n s o f th e s a m e n v aria b les, each o f th e fu n c tio n s bein g the p a rtia l
d e riv a tiv e o f th e re sp o n se w .r.t. o n e designable v a ria b le .
In yield o p tim iz a tio n w e
>'7
ty p ic a lly d e a l w ith th re e ty p e s o f v a ria b le s, n am ely , nDS d esig n a b le v a riab le s x DS w ith
s ta tistic s, nD d e sig n a b le v a ria b le s x D w ith o u t sta tistic s, an d ns n o n -d e sig n a b lc
v a ria b le s x s w h ic h a re s u b je c t to sta tistic al v a ria tio n s.
S up p o se th a t th e re a re A:
resp o n ses, R;, i = 1, 2 , . . . , k . T h e g ra d ie n ts o f th e resp o n ses w ith re sp ec t to the
r
CD
T
r
1
d e sig n a b le v a ria b le s a re
Vi?,- »
(3.18)
d XDS
w h e re x
d x D
sta n d s fo r th e n o m in a l v alu es an d th e d im e n sio n o f the g ra d ie n t v e c to r is
( nDS *nD).
F o r y ie ld -d riv e n d e s ig n , c ir c u it responses an d th e ir g ra d ie n ts have to be
e v a lu a te d a t a n u m b e r o f s ta tis tic a l o u tco m es. E ach sta tistic a l o u tco m e is g e n e ra te d
in a 0 1ds + ns )-d im e n s io n a l sp a c e a c c o rd in g to a k n o w n sta tistic a l d is trib u tio n an d can
be e x p re sse d as
XDS
xs
1D S
A xD S
AXc
(3.19)
w h ere bccDS a n d A xs a re o u tc o m e s p e c ific d e v ia te s fro m the n o m in a l values. B ecause
o f a large n u m b e r o f s ta tis tic a l o u tc o m e s n ee d ed fo r a m e a n in g fu l yield e stim a te th e
m ain sa v in g o f th e c o m p u ta tio n a l e f f o r t is a c h ie v e d b y b u ild in g th e m o d els in the
(nDS
+,,s) - d im e n s io n a l s p a c e o f th e sta tistic a l v aria b les (3.19). In o th e r w o rd s, we
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
0
V
54
c o n s id e r (3 .1 9 ) as th e v e c to r x in (3 .1 ), o r
x =
XDS
(3.20)
L o c a lity o f sta tistic a l s p re a d s assu res a good level o f m o d el a c c u ra c y . T h e
m odels a re b u ilt fo r th e c u r r e n t (o p tim iz a tio n sp e c ific ) n o m in a l p o in t a n d u tiliz e d fo r
as m an y sta tistic a l o u tc o m e s as d e s ire d . In a d d itio n to th e resp o n se fu n c tio n s , e a ch
e n try to th e g ra d ie n t v e c to rs c a n b e a p p ro x im a te d b y a se p a ra te q u a d ra tic fu n c tio n in
a s im ila r m a n n e r as th e re sp o n se fu n c tio n s a re .
F o r k re sp o n se s, th u s , th e re a re
to tally fcx( 1 +>il)S *nD) fu n c tio n s to be a p p ro x im a te d , i.e.,
(i
(3-21)
It sh o u ld be p o in te d o u t th a t, if th e a d jo in t te c h n iq u e is u sed , th e g ra d ie n t can
be a v a ila b le at a low a d d itio n a l c o st to th e c irc u it s im u la tio n , a n d c a n b e re tu rn e d
fro m th e s im u la to r re g a rd le ss o f w h e th e r it is u tiliz e d o r n o t. T h e r e f o r e , th e p ro p o sed
m e th o d can n o t only u tiliz e in fo rm a tio n th a t w ould o th e rw ise b e lo st, b u t also allow s
fo r re d u c tio n o f th e m o d el d im e n s io n a lity by itD, as is c le a rly seen fro m (3.21).
A g e n e ra l-p u rp o s e c ir c u it d e sig n p ro g ra m , called M c C A E , is u n d e r c o n tin u o u s
d e v e lo p m e n t in the S im u la tio n O p tim iz a tio n System s R e se a rc h L a b o ra to ry (SO SR L )
o f M c M a ste r U n iv e rsity . T h is p ro g ra m uses p ro p rie ta ry m o d u le s o f O p tim iz a tio n
R e p ro d u c e d with perm ission of the copyright owner. Fu rth er reproduction prohibited without permission.
55
S y stem s A sso ciates Inc.
In th is p ro g ra m , th e g ra d ie n t in fo rm a tio n h as b een m ade
a v a ila b le . Im p le m e n ta tio n II o f th e g e n e ra liz e d
c e n te rin g a p p ro a c h is u sed . We
a p p ly o u r q u a d ra tic a p p ro x im a tio n te c h n iq u e to g ra d ie n t
fu n c tio n s as w ell.
+ ns) + 1 b a se p o in ts, d e fin e d b y (3.10) a n d (3 .13), a re u se d . A n in te rfa c e has
b e e n d e v e lo p e d f o r th e re sp o n se a n d g ra d ie n t a p p ro x im a tio n m o d u le w h ic h is very
fle x ib le in d e a lin g w ith d if f e r e n t ty p e s o f v a ria b le s in v o lv e d in y ie ld - d r iv e n design.
‘r
A d d itio n a l,to re sp o n se fu n c tio n s , th e ir g ra d ie n t fu n c tio n s /ir e also ta k e n as c a n d id a te s
f:
ii
to b e a p p ro x im a te d . T h e sam e se t o f 2 x (n DS +ns ) + \ base p o in ts, d e fin e d b y (3.10)
a n d (3 .1 3 ), is u sed b o th f o r re sp o n se fu n c tio n s a n d fo r th e ir g ra d ie n t fu n c tio n s.
A g a in , th e r e fe re n c e p o in t is d e fin e d as th e n o m in a l p o in t. A t e a ch ite ra tio n , a se t o f
q u a d r a tic m o d els f o r b o th resp o n ses a n d g ra d ie n ts is b u ilt. T h e m o d els are e v a lu ated
f o r a ll o u tc o m e s. T h e re s u ltin g q u a d ra tic m odel fo r th e g ra d ie n t is m o re a c c u ra te than
th e o n e th a t c o u ld b e o b ta in e d b y d if f e r e n tia tin g th e q u a d ra tic m odel o f the response,
b e c a u se th e p a rtia l s e c o n d - o r d e r in fo rm a tio n is in c o rp o ra te d in the m odel.
3.5.2
A 1 3 -E le m e n t L o w -P a ss F ilte r
T h e lo w -p a ss f ilte r (W e h rh a h n a n d S pence 1984) sh o w n in Fig. 3.2 is
c o n s id e re d . T h e c irc u it m u st m eet th e sp e c ific a tio n s: in se rtio n loss less th an 0.4dB at
th e a n g u la r fre q u e n c ie s
{0.25, 0.27, 0.2 9 , 0 .3 1 , 0 .3 3 , 0 .6 7 , 0.69, 0 .7 1 , 0 .7 3 , 0.75, 0.90,
0 .9 0 5 , 0 .9 1 , 0.9 2 , 0 .9 3 , 0 .9 7 8 , 0 .981, 0 .984, 0.9 8 6 , 0.9 8 8 , 1),
a n d g r e a te r th a n 49dB a t
(1 .0 4 5 6 9 , 1.056, 1.059, 1.063, 1.067, 1.071, 1.115).
T h e r e a re 13 d e sig n v a ria b le s.
A n o rm a l d is trib u tio n w ith 0.5% sta n d a rd
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
56
xi2
X
r r r \
r r r \
O— W ' t f
zz X 4
F ig . 3.2.
± X 7
±zx
z=x
C ir c u it s c h e m a tic o f th e L C 1 3 -e le m e n t f ilte r (W e h rh a h n a n d S p en ce 1984).
R e p ro d u c e d with permission of the copyright owner. F urther reproduction prohibited without perm ission.
57
d e v ia tio n is a ssu m ed f o r a ll v a ria b les.
T h e s ta rtin g p o in t is th e o p tim a l m in im ax
s o lu tio n , w h ic h h as a n e stim a te d y ie ld o f 33.4% . T o illu stra te th e e ffic ie n c y o f the
n e w q u a d ra tic a p p ro x im a tio n a p p ro a c h , w e solve th e p ro b le m u sin g b o th a p p ro x im a te
s im u la tio n s fro m th e q u a d ra tic m o d e fa n d e x a c t sim u la tio n s. T h e fin a l yields f o r b o th
a p p ro a c h e s a re 75.6 a n d 80.7% . C o m p u ta tio n a l d e ta ils a re given in T ab le 3.2. C P U
tim e s f o r th e tw o d e sig n s w e re 7 a n d 30 m in u te s, re sp e c tiv e ly .
3.5.3
A T w o -S ta g e G aA s M M IC F e e d b a c k A m p lifie r
We c o n s id e r a tw o -s ta g e 2 - 6 G H z G aA s M M IC fe e d b a c k a m p lifie r (V e n d e lin ,
P a v io a n d R o h d e 1990).
T h e s p e c ific a tio n s a re a sm a ll-sig n a l g a in o f 8d B ± ld B ,
V SW R a t th e in p u t p o rt o f less th a n 2 , a n d VSW R a t th e o u tp u t p o rt less th a n 2.2. T h e
c ir c u it a n d th e e q u iv a le n t c ir c u it m o d el f o r th e F E T a re show n in F igs. 3.3 a n d 3.4.
It is in te n d e d to m a n u fa c tu re h ig h -v o lu m e , h ig h -y ie ld , a n d , c o n se q u e n tly , lo w -c o st
m ic ro w a v e a m p lifie rs . T h e size o f th e IC has a s tro n g e f f e c t on th e cost. T h e re fo re ,
w e c o n sid e r th e m ean v a lu e s o f m o st c a p a c ito rs as fix e d to k eep th e size o f the c h ip
re a so n a b le . T h e m e a n v alu e o f th e g a te w id th is fix e d b ec au se o f th e assu m ed F E T
p ro c e ss, b u t a 3% s ta n d a rd d e v ia tio n is allo w ed . We assu m e th a t th e bias c irc u it is
w ell d e sig n e d su c h th a t c h a n g e s o f bias re sisto r v a lu es d o n o t stro n g ly in flu e n c e the
R F re sp o n se a ro u n d th e o p e ra tin g p o in t. T h e r e f o re , no to le ra n c e s a re assigned to the
re s isto rs.
v a ria b le s .
T w o fe e d b a c k re sisto rs a n d a fo rw a rd c a p a c ito r a re ch o sen as d esign
A s ta n d a rd d e v ia tio n o f 2% is assu m ed f o r th e d e sig n v a ria b le s.
For
n o m in a l v alu es a n d s ta n d a rd d e v ia tio n s o f o th e r e le m e n ts, see T a b le 3.3. C o rre la tio n s
a m o n g e le m e n ts a re n o t c o n s id e re d in this c irc u it.
T h e f ir s t s te p in th e e n tir e o p tim iz a tio n p ro c e d u re is to fin d a m in im ax
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
58
T A B L E 3.2
' ^
Y IE L D O P T IM IZ A T IO N O F T H E L C 1 3 -E L E M E N T F IL T E R
W ITH A N D W IT H O U T Q U A D R A T IC A P P R O X IM A T IO N S
Solution^
S o lu tio n ^
0.2088
0.03594
0 .1822
0 .2340
0.2424
0.08776
0.1333
0.3549
0.06477
0.1674
0.1422
0.1140
0.1433
0.2145
0.03642
0.1800
0.2347
0.2426
0.08702
0.1290
0.3535
0.06496
0.1625
0.1435
0.1414
0.2205
0.03929
0.1775
0.2266
0.2556
0 .08426
0.1234
0.3551
0.06481
0.1561
0.1498
0.1098
0.1303
33.4%
75.6%
80.7%
N um ber of
S im u latio n s
3780
18200
C PU *
7 m in .
3 0 m in .
P a ra m e te r
xi
X2
x3
x4
x5
X6
x7
x8
x9
X10
X11
x12
X13
Y ield
E stim a te
In itia l
0.1120
t T h e so lu tio n a f te r o n e p hase o f y ie ld o p tim iz a tio n
w ith q u a d r a tic a p p ro x im a tio n s.
T h e so lu tio n a f te r o n e p h ase o f y ie ld o p tim iz a tio n
w ith e x a c t sim u la tio n s a n d n u m e ric a l g ra d ie n ts.
O n th e Sun S P A R C sta tio n 1.
C o m m en ts: N o rm a l d is trib u tio n o f a = 0.5% is
a ssu m e d f o r a ll p a ra m e te rs . 100 o u tc o m es a re
u sed in th e o p tim iz a tio n .
1000 o u tc o m e s a re u sed in th e y ie ld e stim a tio n .
P a ra m e te rs a re r ca led d o w n b y th e f a c to r 2jt, e .g .,
th e a c tu a l e le m e n t v alu e o f x 1 is 2irx0.2088.
R e p ro d u c e d with perm ission of the copyright owner. Fu rther reproduction prohibited without perm ission
59
0.65/Z
0.14Z
0.85/Z
A A A ,-
drain
1.1Z == V
65/Z
0.65/Z
source
Fig. 3.
N o rm a liz e d G aA s M E S F E T m odel (V e n d c lin , Pavio a n d R o h d e , 1990). Z
is th e g a te w id th in m illim e te rs. g m = 0 .1 7 Z , an d t - 2.5ps. A ll resisto rs arc
in o h m s. A ll c a p a c ito rs a re in p ic o fa ra d s.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
Ri
C5
RF output
C 3
-AAA/— j |—
—w v
input
— o
r9
oR8
W V
R6
R4
-L C-
C i3 —
R 12
R 11
ig. 3.4
A tw o -s ta g e a m p lifie r (Y e n d e lin , Pavio an d R o h d e , 1990).
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
61
T A B L E 3.3
PA RA M ETER VALUES AND TOLERANCES
F O R T H E M M IC A M P L IF IE R
E le m e n t
P a ra m e te r
M ean
V alue
Z (/itn)
R 4(fl)
C s(p F )
300
400
R e(« )
C 7(p F )
R 8(n )
R » (« )
C 10(p F )
R u (n )
20
10
R « (n )
C 13(p F )
4
3%
V/
145
2200
4
6000
500
10
S tan d a rd
D e v ia tio n
0%
2%
— 2%
2%
2%
0%
2%
0%
. 2%
2%
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
T A B L E 3.4
Y IE L D O P T IM IZ A T IO N O F T H E M M IC
A M P L IF IE R W IT H A N D W IT H O U T
Q U A D R A T IC A P P R O X IM A T IO N S
P a ra m e te r
In itia l
Solution*
Solution**
Ri
201.02
R2
c3
504.82
5.3501
207.63
627.94
2.7742
207.73
630.53
2.7563
32.1%
77.8%
77.3%
N um ber of
S im u la tio n s
1380
6400
C PU *
9 m in .
3 9 m in .
Y ield
E s tim a te
* T h e so lu tio n a f te r o n e p h ase o f y ie ld o p tim iz a tio n
w ith q u a d ra tic a p p ro x im a tio n s.
it
f t T h e so lu tio n a f te r o n e p h ase o f y ie ld o p tim iz a tio n
w ith e x a c t sim u la tio n s a n d n u m e ric a l g ra d ie n ts.
O n th e S u n S P A R C sta tio n 1.
C o m m en ts: 100 o u tc o m e s a re used in th e o p tim iz a tio n .
1000 o u tc o m e s a re u sed in th e y ie ld e stim a tio n .
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
63
s o lu tio n as th e s ta rtin g p o in t f o r y ie ld -d riv e n d e sig n . T h e m in im a x so lu tio n is fo u n d
a n d liste d in T a b le 3.4. T h e y ie ld e stim a te a t this p o in t is 32.1% . T w o y ie ld -d riv e n
o p tim iz a tio n
p ro cesses a re c a rrie d
o u t w ith a n d
w ith o u t th e
new q u a d ra tic
a p p ro x im a tio n to b o th th e resp o n ses a n d th e ir g ra d ie n ts. T w o so lu tio n s a n d th e fin a l
y ie ld a re g iv e n in T a b le 3.4. T h e a c tu a l yields b ased o n a M o n te C arlo an aly sis o f
1000 o u tc o m e s a re 77.8% a n d 77.3% , resp e ctiv ely .
T h e C P U tim es a re 9 a n d 39
m in u te s.
3 .6
C O N C L U D IN G R E M A R K S
-
In th is c h a p te r w e h av e p re se n te d a h ig h ly e f f ic ie n t q u a d ra tic a p p ro x im a tio n
te c h n iq u e . T h e n e w a p p ro a c h takes a d v a n ta g e o f th e m ax im ally fla t in te rp o la tio n a n d
o f a fix e d p a tte r n o f base p o in ts , th u s su b sta n tia lly re d u c in g th e c o m p u ta tio n a l e f f o r t
a n d re q u ire d sto ra g e .
A se t o f e x tre m e ly sim p le fo rm u la s to c alc u la te m odel
c o e ffic ie n ts has b e e n d e riv e d . T h e eleg an ce o f th is a p p ro a c h is its co n ciseness a n d
a p p lic a b ility . T h e v e ry s tro n g im p a c t o f o u r a p p ro a c h o n th e fe a sib ility o f sta tistic a l
d e sig n o f la rg e r c irc u its sh o u ld n o t be u n d e re stim a te d .
T h is a p p ro x im a tio n a p p ro a c h has been a p p lie d e ith e r to c irc u it resp o n se
fu n c tio n s o n ly , o r to re sp o n se s a n d th e ir g ra d ie n ts sim u ltan e o u sly . F o r the firs t case,
th e re su lts o f a s ta tis tic a l d e sig n e x a m p le has p ro v e n v e ry e ffic ie n t. F o r the second
c a se , it is fo u n d th a t o u r a p p ro a c h ca n be esp e c ia lly su ita b le f o r g ra d ie n t-b a s e d y ie ld d riv e n d e sig n , m a k in g th e m o d e l m o re a c c u ra te a n d ro b u st. A s ta n d a rd test p ro b lem
a n d a n M M IC a m p lif ie r d e sig n illu stra te th e m e rits o f th is im p le m e n ta tio n .
It sh o u ld also b e
n o te d th a t o u r a p p ro a c h is s u ita b le fo r a large v a rie ty o f
a p p lic a tio n s w h e re a la rg e n u m b e r o f ex p e n siv e sim u la tio n s is in v o lv ed .
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4
S U P E R C O M P U T E R -A ID E D S T A T IS T IC A L D E S IG N O F A L A R G E S C A L E
C IR C U IT
A 5 -C H A N N E L M IC R O W A V E M U L T IP L E X E R
0
4.1
'if
IN T R O D U C T IO N
T h e a d v a n ta g e o f m o d e rn tech n o lo g y has been c re a tin g in crea sin g ly c o m p le x
c irc u its . T h e d e m a n d f o r re lia b le d esig n w h ich leads to s h o rte r d e v e lo p m e n t tim e ,
re q u ire s m o re a n d m o re a c c u ra te , a t the sam e ..time, m ore a n d m ore c o m p u ta tio n a lly
in v o lv e d m o d els fo r c ir c u it ele m e n ts an d devices.
S tatistic al d e sig n o f large scale
c irc u its p re se n ts a g re a t c h a lle n g e .
D esign o f m ic ro w a v e m u ltip le x e rs is a large scale p ro b le m . C o n tig u o u s -b a n d
s;
m u ltip le x e rs c o n sistin g o f m u lti-c a v ity filte rs d is trib u te d a lo n g a w a v eg u id e m a n ifo ld
a re u sed in sa te llite c o m m u n ic a tio n s. T h e p ro b lem o f o p tim a l d esig n a n d m a n u fa c tu re
o f su c h c irc u its has been o f s ig n ific a n t, p ra c tic a l in te re st (A tia 1974; C h e n , Assa! an d
M ah le 1976, C h en 1983 a n d 1985) fo r a n u m b e r o f y ears. T h e re has been sy ste m a tic
re se a rc h
to
p ro v id e
v e ry
c o m p re h e n siv e sim u la tio n , s e n sitiv ity
a n a ly sis, an d
o p tim iz a tio n d e sig n tools f o r th ese c irc u its (B an d ler. K .ellerm ann a n d M adsen 1985
a n d 1987, B a n d le r, C h e n , D a ija v a d a n d K elle rm a n n 1984 a n d 1988, B a n d ler, D aijav ad
a n d Z h a n g 1985 a n d 1986, a n d B an d ler a n d Z h a n g 1987).
B a n d le r, B ie rn a c k i, C h e n , R e n au lt,. Song a n d Z h a n g (1988) d e v elo p e d a
c o m b in e d a p p ro a c h to larg e scale c irc u it sta tistic a l d e sig n . T h e y a tta c k e d th is d iff ic u lt
p ro b le m by: ( 1) th e use o f s u p e rc o m p u te rs, ( 2 ) e f f ic ie n t a p p ro x im a tio n to c irc u it
re sp o n se s, an d (3) th e u se o f fa st, d e d ic a te d sim u la tio n te c h n iq u e s.
65
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66
In th e m a n u fa c tu re o f e le ctric al c irc u its, tu n in g c an b e a n /e s s e n tia l a n d
e ffe c tiv e p a rt o f th e p ro d u c tio n process to im p ro v e c ir c u it p e rfo rm a n c e a n d to
in crease th e fin a l m a n u fa c tu rin g y ield . T h ro u g h a d ju s tm e n t o f c ir c u it c o m p o n e n ts .
to m eet d e sig n s p e c ific a tio n s , tu n in g co u n te ra c ts c o lle c tiv e e ffe c ts cau sed by th e
to le ra n c e s in h e re n t in the p ro d u c tio n process, m odel in a c c u ra c y a n d o th e r fa c to rs
ig n o red in th e d e sig n process. In g e n e ra l, tu n in g re q u ire s sim u lta n e o u s a d ju s tm e n t o f
se v e ra l e le m e n ts to re a c h a sa tisfa c to ry result.
E v en w h e n th e n u m b e r o f tu n a b le
e le m e n ts is m o d e ra te , th e e f f o r f jn a d e to fin d th e rig h t c o m b in a tio n o f ele m en ts a n d
a m o u n ts to be a d ju s te d is e n o rm o u s. T h is leads to th e m ost a p p a re n t p ro b le m w ith th e
tu n in g process: th e en o rm o u s a m o u n t o f tim e re q u ire d .
In th e tu n in g p ro ce ss, th e
a m o u n t o f tu n in g is also s u b je c t to im precisions o f a d ju s tm e n ts a n d u n c e rta in tie s o f
c irc u it m odels. T h is c h a p te r also addresses th e issue o f th e a p p lic a tio n o f sta tistic a l
d esig n to tu n a b le c irc u its w ith to leran ces asso ciated w ith b o th th e fa b ric a tin g p ro cess
p rio r to tu n in g an d tu n in g p ro cess itself.
T h is c h a p te r is o rg a n iz e d in th e fo llo w in g w ay . We f ir s t p ro v id e a p h y sica l
m ean in g o f a p p ly in g y ie ld -d riv e n d esign p rin c ip le s to d e sig n o f sm all p ro d u c tio n
volum es. T h e n sp ecial c o n sid e ra tio n is giv en to th e a p p lic a tio n o f y ie ld - d riv e n d esig n
to tu n a b le c irc u its . We discu ss th e use o f th e s u p e rc o m p u te r to d e a l w ith large scale
c irc u its. F in a lly , a 5 c h a n n e l m u ltip le x e r w ith 75 d e sig n a n d to le ra n c e d v ariab le s, 124
c o n s tra in ts a n d up to 200 sta tistic a lly p e rtu rb e d o u tc o m e s, is u se d to d e m o n stra te th e
fe a sib ility o f y ield o p tim iz a tio n o f large scale p ro b le m s u sin g o u r co m b in e d stra te g y .
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
X i:
>)
n.
-■
..
67
4 .2
Y IE L D -D R IV E N D E S IG N F O R N O N -M A S S IV E P R O D U C T IO N
C o n v e n tio n a l y ie ld -d riv e n d e sig n is in te n d e d to in c re ase y ield o f m ass
p ro d u c tio n o f e le c tro n ic c irc u its . In so m e cases, h o w e v e r, a re la tiv e ly sm all n u m b e r
o f c irc u its a re re q u e s te d b y c u sto m ers.
T h e n th e te rm y ield its e lf is " m f1 o n g e r
m a th e m a tic a lly (m o re p re c ise ly , sta tistic a lly ) ju s tifia b le b ecau se o f th e sm all n u m b e r
o f o u tco m es. U n d e r s u c h c irc u m sta n c e s, th e m a jo r c o n c e rn o f the desig n is n ot the
y ield , b u t th e p ro b a b ility o f p ro d u c in g sa tisfa c to ry c irc u its a t th e low est cost.
T o le ra n c es, in e v ita b ly in h e re n t in th e m a n u fa c tu rin g p ro cess, a re still a c ritic a l fa c to r
'v'V
a ffe c tin g th e fin al o u tc o m e s. T h e logical re su lt o f a p p ly in g th e y ie ld -d riv e n d e sig n
m eth o d o lo g y to c ir c u it p ro d u c tio n on a sm all scale is th e im p ro v e d p ro b a b ility o f
o b ta in in g c irc u its m e e tin g s p e c ific a tio n s.
T h e fo rm al m a th e m a tic a l ju s tific a tio n is based on th e o rig in a l d e fin itio n o f
yield
Y(^°) = P ro b a b ility o f ( 4>‘ € R a K
w h ere Ra is th e a c c e p ta b le re g io n an d fix e d to le ran c es g iv e n by c
(4.1)
a re a ssu m ed .
T h e n , th e u tiliz a tio n o f o rd in a ry y ie ld -d riv e n d e sig n te c h n iq u e s can in crease th e
p ro b a b ility o f m a n u fa c tu re d c irc u it o u tc o m e s to m eet d e sig n sp e c ific a tio n s.
4.3
Y IE L D -D R IV E N D E S IG N O F T U N A B L E C IR C U IT S
4.3.1
T u n in g a n d Its T o le ra n c e s
T o a p p ly th e a v a ila b le y ie ld -d riv e n d esign a p p ro a c h to tu n a b le c irc u its , a
n u m b e r o f asp ects sh o u ld be id e n tifie d . In th e fo llo w in g , to le ra n c e s in h e re n t in th e
c irc u it fa b ric a tio n p ro cess b e fo re tu n in g a re re fe rr e d to as fa b ric a tio n to le ra n c e s, a n d
to le ra n c e s in th e tu n in g p ro c e ss as to tu n in g to le ra n c e s. H e re , we fo cu s o u r a tte n tio n
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
on th e case o f fix e d fa b ric a tio n to lera n c es, tu n in g re g io n , a n d a ssu m e d tu n in g
to le ra n c e s. We assu m e th a t th e fa b ric a tio n to leran ces a re s y m m e tric a l a n d th a t tu n in g
is tw o -w a y a n d sy m m e tric a l. L et
c
=[ei
«2 - • - £ „ ] r ,
'
= [ ‘i
h
(4.2)
■ ■ ■ ‘n } T ,
(4-3)
’ ' * S
^
an d
= [ <, , S
f
d e n o te th e fa b ric a tio n to le ra n c e e x tre m e s, m axim al tu n in g a m o u n ts , an d tu n in g
to le ra n c e e x tre m e s, re sp e c tiv e ly .
F o r those e lem en ts th a t c a n n o t b e tu n e d , th e
c o rre sp o n d in g tu n in g a m o u n ts in (4.3) a re zero. T h e n , th e a sso c ia te d tu n in g to le ra n c e s
in (4 .4 ) a re also zero . We w ill assu m e th a t th e p a ra m e te rs c a n b e v a rie d c o n tin u o u sly
an d in d e p e n d e n tly .
4>° is used to d e n o te th e n o m in al d e sig n .
T h e k th tu n a b le
o u tc o m e <
f>k m ay be d e s c rib e d by
4>k =
- A**,
(4.5)
A <j>k = A<f>k + t k + A t k,
k
k
w h e re A <
f> n eed s th e fo llo w in g e x p la n a tio n : A $£ re p re s e n ts d e v ia tio n s d u e to th e
ac tu a l m odel u n c e rta in tie s a n d fa b ric a tio n to leran c es, g e n e ra te d fro m
A *£ = E f i ,
(4.6)
where
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
69
f 1
Ml
m2
f 2
e
"&
-! < f t < 1. / = 1, 2
M &
•
N:
•
£w_
I d e n o te s p o ssib le p o s tp ro d u c tio n tu n in g a d ju stm e n ts, ta k in g fo rm o f
r
= tp ,
(4.7)
w h e re
11
Pi
P2
T&
-1 < Pi < 1, i = 1 ,2 .......... N\
,
P 4
•
lN
and A t
Pn
re p re s e n ts a c tu a l tu n in g to le ra n c e s, d e fin e d by
At
= E ,o ,
a &
*
*
(4-8)
w h e re
-
£h
■
S
&
*
,
-1 < a i < 1,
/ =1,2
N.
°N
In th e se fo rm u la s , P; a n d o,- a re ra n d o m n u m b e rs, g e n e ra te d a c c o rd in g to th e ir ow n
s ta tis tic s , Pi is th e re la tiv e tu n in g a m o u n t. F o r id eally a c c u ra te tu n in g ( A t * = 0 ) we
can o b ta in A 4> = 0 p ro v id e d th a t th e tu n in g ra n g es are larg e e n o u g h to a c co m m o d a te
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
70
th e sp re a d s d u e to th e fa b ric a tio n to le ra n c e s. T h e n , fo r re a listic (im p re c ise ) tu n in g
a n d s u f f ic ie n t tu n in g re g io n , w e o b ta in A^* - A /* . O u tc o m e s o f n o n tu n a b le c irc u its
can be c o n sid e re d as a sp e c ia l case o f (4 .5 ), w h e re
= A ^ .
If th e yield o p tim iz a tio n o n ly tak es m a n u fa c tu rin g to leran ces in to a c c o u n t,
we m ay fa c e a v ery p o o r y ield fig u re b ecau se m an y fa b ric a te d c irc u its c a n n o t be
tu n e d in to th e a c c e p ta b le reg io n d u e to e x is tin g to le ra n c e s.
C o n sid e r a tw o -
a n d e 2 a re fa b ric a tio n to le ra n c e s. If
d im e n sio n a l case sh o w n in F ig . 4 .1 , w h e re
o n ly th e fa b ric a tio n to le ra n c e s a re 'c o n s id e re d in yield o p tim iz a tio n , d esig n te n d s to
'If
)
m ove th e fa b ric a tio n to le ra n c e region', R , to o v e rla p s th e a c c e p ta b le re g io n , R , as
I
!f
£
m uch as p o ssib le. T h e n w e m ay h av e a so lu tio n 4>°. A t th is p o in t, c irc u it o u tc o m e s
fa b ric a te d can s p re a d o v e r th e a re a Rt . A ssum e th a t tu n in g on th e sec o n d e le m e n t is
a v a ila b le a n d th a t th e m ax im al tu n in g a m o u n t is t2. T h e n , tu n in g o n th e sec o n d
p a ra m e te r trie s to b rin g o u tc o m e s in to th e a re a R( ,(^°, e ) , d e fin e d by th e n o m in al
p o in t an d
(4-9)
N
w h ere
f; A max(0,£j - /;) .
(4.10)
B ecause o f tu n in g to le ra n c e s, fin a l o u tc o m e s ( a f te r tu n in g ) d o n ot all fa ll in to
Ri r In ste a d , th ey s p re a d o v e r th e re g io n
(4 -H )
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
f<
tu n a b le
71
nontunable
*1
Fig. 4.1
Illu stra tio n o f yield o p tim iz a tio n w ith th e fa b ric a tio n to le ra n c e reg io n R t
only. R a is th e a c c e p ta b le reg io n , R £ t the sp re a d reg io n w ith ex a ct
tu n in g , a n d R t. t t is th e actu al sp rea d reg io n a f te r to lc ra n c e d tu n in g .
Y ield a f te r tu n in g £is ( R an R { t t () / R tit t£.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
72
w h e re
C
=
£
+C,
S om e o f c irc u its m ay fall o u tsid e o f Ra d u e to th e tu n in g to le ra n c e s, in d ic a tin g th a t
th o se o u tc o m e s v io late the d e sig n sp e c ific a tio n s. F o r in sta n c e , 4*1 sh o w n in Fig. 4.1
is su ch a c irc u it.
4.3.2
Y ie ld -D riv e n D esign w ith T u n in g T o le ra n c e s
It is n a tu ra l to e x te n d y ie ld -d r iv e n desig n te c h n iq u e s to tu n a b le c irc u its.
C o n sid e r
4>k = * ° ♦ A**,
(4.12)
A / = EQ ,
w h e re
*1
ei
T2
«2
E 4
. 0 &
-
, -1 < &, < 1, / = 1, 2 , . . . , Af(4.13)
•
*N
If th e c o m b in e d o u tc o m e sp re a d re g io n , Rc , , W ° ,T ) , is tre a te d in th e sam e w ay as
th e n o rm al to le ra n c e re g io n , th e n y ie ld -d riv e n d esig n w ill g iv e a so lu tio n su c h th a t
tu n a b ility is fu lly u tilized to can cel an y d e v ia tio n cau sed b y fa b ric a tio n to leran ces
a n d , m e a n w h ile , tu n in g to le ra n c e s a re also a c c o m m o d a te d . Fig. 4.2 p re s e n ts su c h a
fo rm u la tio n
w h ere tu n in g a n d tu n in g
to le ra n ce s a re c o n sid e re d .
U n lik e th e
fo rm u la tio n o f Fig. 4 .1 , th is p ro c e d u re te n d s to m ove th e n o m in a l p o in t su c h th a t the
c o m b in e d reg io n Re , , ( ,
e ) o v e rlap s th e a c c e p ta b le reg io n Ra os m u c h as p o ssib le,
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
tunable
73
e ,t
non tun able
F ig. 4.2
Illu stra tio n o f y ield o p tim iz a tio n w ith the c o m b in e d sp re a d reg io n R f 11 .
Y ield a f te r tu n in g is h ig h e r th a n th a t in F ig . 4.1-
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
g iv in g a h ig h e r y ield a fte i-tu n in g /w ith th e sam e tu n a b le re g io n a n d tu n in g to le ra n ce s.
N ow , 4
is w ell in th e a c c e p ta b le reg io n .
'> \
T h e d esig n o f tu n a b le c irc u its u sin g (4.12) has tw o goals. O n e is to d riv e up
th e p ro b a b ility o f o b ta in in g c irc u its th a t e x h ib it good in itia l re sp o n se s fo r th e tu n in g
p rocess. T h e o th e r is to in crease th e p o ssib ility o f c irc u it o u tc o m e s s a tis fy in g s p e c i­
fic a tio n s a f te r tu n in g e a s y - to - tu n e ele m e n ts. F o r th e la tte r, w e assig n la rg e r tu n in g
to leran ces to m o re d if f i c u l t- t o - t u n e e le m e n ts in a n e f f o r t to a v o id p a in s ta k in g , and
ted io u s tu n in g . T h is s h o u ld have a n im p a c t o n th e o v e ra ll e f f o r t o f tu n in g .
4.4
U T IL IZ IN G T H E V E C T O R P IP E L IN E S U P E R C O M P U T E R
4.4.1
Basics a b o u t th e V ecto r P ro cesso r
T h e larg est a v a ila b le c o m p u te rs, n am ely su p e rc o m p u te rs , h a v e im p ressiv e
n u m b e r-c ru n c h in g a n d sto ra g e c a p a b ilitie s. O n e ty p ic al ty p e o f s u p e rc o m p u te r is the
v ecto r p ip e lin e c o m p u te r.
O n e o f th e m ost fu n d a m e n ta l m ech an ism s th a t m ak es a v e c to r p ip e lin e
s u p e rc o m p u te r v ery p o w e rfu l is its m a ch in e a rc h ite c tu re s , w h ic h in c lu d e v e c to r
in s tru c tio n a n d p ip e lin in g .
F ro m th e u se r’s p o in t o f v ie w , th e p ro g ra m sh o u ld be
re o rg a n iz e d su ch th a t it b est fits th e m a c h in e a rc h ite c tu re . H e re , w e c o n c e n tra te o n ly
on th is ty p e o f re o rg a n iz a tio n called p ro g ra m v e c to riz a tio n .
P ro g ra m v e c to riz a tio n is th e m o st im p o rta n t a c tio n to be ta k e n to fu lly ta k e
a d v a n ta g e o f a v e c to r p ip e lin e s u p e rc o m p u te r. T h e v e c to riz a tio n is c a rrie d o u t b y the
c o m p ile r. W hat th e c o m p ile r a c tu a lly v e cto rize is d o -lo o p s. S p e c ific a lly , it v e cto rizes
in n e rm o st d o -lo o p s , so in a n y se c tio n o f co d e w ith d o -lo o p s n e ste d m o re th a n o n e
level d e e p , th e c o m p ile r w ill a tte m p t to v ec to riz e o n ly th e m ost d e e p ly - n e s te d loop.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
75
In c ir c u it s im u la tio n a n d o p tim iz a tio n p ro g ra m s, a s u b sta n tia l a m o u n t o f w o rk is d o n e
b y m a n y v e c to r a n d m a trix m a n ip u la tio n s, re su ltin g in m an y po ssib ly v c c to riz a b le d o lo o p s.
S u c c e ssfu l v e c to riz a tio n o f th ese p ro g ra m s w ill get d ra m a tic p e rfo rm a n c e
im p ro v e m e n t.
T h e fo llo w in g ste p s s h o u ld b e taken to m ak e a p ro g ram b e tte r v e c to riz e d .
O fte n , w e e n c o u n te r a p a ir o f d o -lo o p s nested f o r w h ich the o u te r loop has a m u ch
g re a te r ra n g e th a n th e in n e r loop. We can re a rra n g e th e loops by e x c h a n g in g th e m to
g a in e f fic ie n c y . K e e p th e in n e r loop as sim p le as p o ssible. M any e x trin s ic fu n c tio n
c a lls, I /O s ta te m e n ts , G O T O a n d IF sta te m e n ts, a n d in d e x cross re fe re n c e s in h ib it
v e c to riz a tio n . C e rta in p re p ro c e ssin g m a y be h e lp fu l to re o rg a n ize th e co d e s tru c tu r e
to a c h ie v e b e tte r v e c to riz a tio n re su lts a t th e cost o f m ore m em o ry w h ich u su ally is not
a big p ro b le m w ith s u p e rc o m p u te rs .
4.2.2
S u p e rc o m p u te r-A id e d Y ie ld O p tim iz a tio n
V e c to r p ip e lin e s u p e rc o m p u te rs have b e e n used b y m an y re se a rc h ers in the
c irc u it C A D
fie ld (C a la h a n
Y am am o to a n d T a k a h a s h i
1979 a n d
1985).
1980, V la d im ire sc u and
P ed erso n
1982,
S pecially , e f f o r t has been m ade to e x p lo re
s u p e rc o m p u te rs fo r m ic ro w a v e c irc u it C A D (R iz z o li, F e r lito a n d N e ri 1986, R izzo li,
C e c c h e tti a n d L ip p a rin i 1986, R izzo li a n d N e ri 1988, B an d le r, B ie rn a c k i, C h e n ,
R e n a u lt, Song a n d Z h a n g 1988, R iz zo li e t al 1991).
F ro m th e C A D v ie w p o in t, y ie ld -d riv e n d esig n p ro b le m s c o n c e rn in g re a listic
m ic ro w a v e c irc u its a re o f v e ry la rg e scale. W ith e x istin g desig n m e th o d s, th e m ost
p o w e rfu l a v a ila b le c o m p u ta tio n a l tools, n a m e ly , su p e rc o m p u te rs, c an p ro v id e us w ith
an e f f e c tiv e m ean s to d e a l w ith p ra c tic a l d esig n p ro b le m s.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
4.4.3
T h e C ra y X - M P /2 2 E n v iro n m e n t '
\\Y ie Id -d riv e n d e sig n has b een c a rrie d o u t o n th e C R A Y X - M P /2 2 in th e
O n ta rio C e n te r f o r L a rg e S cale C o m putation^ lo cate d a t th e U n iv e rs ity o f T o ro n to .
O'
In clu d ed in th e m a in fra m e a re tw o c e n tra l p ro ce ssin g u n its (C P U s), e a c h c o n tro lle d
by a 9.5 nan o seco n d (n s) c lo c k , a n d 2 M illio n 6 4 - b it w o rd s (16M B ) o f sh a re d c e n tra l
m em ory. T h e C ra y O p e ra tin g S ystem (C O S) e ffic ie n tly allo c a te s sy ste m re so u rc e s a n d
co n tro ls th e e x e c u tio n o f u se r jo b s o n o n e o r b o th o f th e C P U s.
A t th e tim e th e
sy stem ran u n d e r V ersio n 1.15 o f (C O S). T h e F O R T R A N c o m p ile r u sed is C ra y ’s
F O R T R A N -7 7 (C F T 7 7 ).
T h is c o m p ile r p e rfo rm s o p tim iz a tio n s s u c h as a u to m a tic
v e c to riz a tio n a n d .in s tru c tio n re s e q u e n c in g in o rd e r to m ost e f fe c tiv e ly u tiliz e th e X M P h ard w are.
4.5.
Y IE L D O P T IM IZ A T IO N O F A 5 -C H A N N E L M U L T IP L E X E R
T h is p ro b le m is a 5 - c h a n n e l I2 G H z c o n tig u o u s b a n d m ic ro w a v e m u ltip le x e r
c o n sistin g o f m u lti- c a v ity filte rs d is trib u te d alo n g a w a v e g u id e m a n ifo ld (B a n d le r,
C h e n , D a ija v a d a n d K e lle rm a n n 1984). F ig. 4 .3 illu stra te s th e e q u iv a le n t c irc u it o f
th e m u ltip le x e r.
T u n in g is esse n tia l a n d e x p e n siv e fo r m u ltip le x e rs to sa tis fy th e
u ltim a te sp e c ific a tio n s . T h e g o al o f th is d e sig n is to p ro v id e s u c h a w e ll- c e n te re d
n o m in al d e sig n th a t th e tu n in g p ro cess ca n be g re a tly e asied .
G e n e ra l m u ltip le x e r o p tim a l n o m in al d e sig n p ro c e d u re s u sin g p o w e rfu l
g ra d ie n t-b a s e d
m in im a x
and
a lg o rith m s
K e lle rm a n n a n d M adsen (1985 a n d 1987).
have
b een d e s c rib e d
by
B a n d le r,
T h e c ir c u it s im u la tio n a n d s e n s itiv ity
an aly sis asp ect o f th e p ro b le m to g e th e r w ith a n u m b e r o f e x a m p le s o f m u ltip le x e r
o p tim iz a tio n hav e b een p re s e n te d by B a n d le r, C h e n , D a ija v a d a n d IC ellerm an n (1 984).
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
V
O
77
^ o
<N
<S
o">
cn o
^ <N
u-i
HH
Fig. 4.3
E q u iv a le n t c irc u it o f a 5 -c h a n n e l c o n tig u o u s b an d m u ltip lex e r.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
A novel a p p ro a c h to th e e x a c t s im u la tio n a n d s e n sitiv ity a n aly sis o f m u ltip le x in g
'7''
• '■
n e tw o rk s has been d e riv e d b y B a n d ler, D a ijav a d an d Z h a n g (1 9 8 6 ), based o n the
g e n e ra l b ra n c h e d c a sc a d e d n e tw o rk s tr u c tu r e (B a n d ler, D a ijav ad a n d Z h an g 1985).
A n a u to m a tic d e c o m p o s itio n a p p ro a c h to o p tim iz a tio n o f larg e sc a le p ro b lem s was
a p p lie d to a 1 6 -c h a n n e l m u ltip le x e r d e sig n (B a n d ler a n d Z h a n g 1987). R e c e n tly , a 5 c h a n n c l m u ltip le x e r has a lso been u sed to d e m o n stra te a g ra d ie n t a p p ro x im a tio n
sc h e m e used in g r a d ie n t- b a s e d o p tim iz a tio n (B a n d le r, C h e n , D a ija v a d a n d M adsen
1988).
A
m u ltip le x e r d e sig n sy stem
has been d ev elo p ed
in
th e S im u la tio n
O p tim iz a tio n S ystem s R e se a rc h L a b o ra to ry , M c M a ste r U n iv e rsity .
O u r y ield o p tim iz a tio n uses th e g en e ralized l x c e n te rin g a lg o rith m d e sc rib e d
in C h a p te r 2.
E ven w ith th e p o w e r o f th e s u p e rc o m p u te r, su ch d e sig n u sin g e x a c t
s im u la tio n w o u ld ta k e a v e ry long tim e to c o m p lete . T h e cost w o u ld be e x tre m e ly
h ig h . T o sp eed up th e d e sig n p ro cess, w e use th e q u a d ra tic a p p ro x im a tio n sc h e m e ,
p re s e n te d in C h a p te r 3 , to m o d el th e m u ltip le x e r responses.
T h e re fo re , o n ly the
s im u la tio n p o rtio n o f th e m u ltip le x e r d e sig n sy stem is to b e u se d .
T h e g ra d ie n t
in fo rm a tio n w ill be p ro v id e d u sin g th e q u a d ra tic m odels o f th e resp o n se fu n c tio n s .
T h e p ro g ra m is im p le m e n te d o n th e C R A Y X - M P /2 2 . T h e m u ltip le x e r d e sig n
sy stem is ta ilo re d to o b ta in a d e d ic a te d sim u la to r to p e rfo rm fa s t c ir c u it sim u la tio n .
T h e n th is p a rt is c o n n e c te d to the sta tistic a l d esig n d riv e r , w ith an in te rfa c e o f
q u a d ra tic a p p ro x im a tio n .
T o fu lly e x p lo re th e v e c to r p ip e lin e s u p e rc o m p u te r, th e p ro g ra m , in c lu d in g
s im u la tio n , o p tim iz a tio n a n d s ta tis tic a l o u tc o m e g e n e ra tio n , is c a re fu lly re o rg a n iz e d
by re s tru c tu r in g th e p ro g ra m , re o rd e rin g so m e DO loops, a n d re d ire c tin g in p u t an d
o u tp u t flo w s.
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without perm ission.
4.5.1
D esign V a ria b le s a n d S p e c ific a tio n s
A s sh o w n in Fig. 4. 3 , e ach b ra n c h u n it consists o f a n im p ed a n ce in v e rte r Is,
an in p u t tra n s fo rm e r w ith th e tra n s fo rm ratio n ^ , a m u lti-c o u p le d c a v ity f ilte r Z ;,
an o u tp u t tra n s fo rm e r w ith th e tra n s fo rm ratio n2l, an d a n o u tp u t load w ith v o ltag e
V ;.
T h e m a in cascad e is a w a v e g u id e m a n ifo ld w ith sp a cin g /. b etw e en tw o
b ra n c h e s. B ran ch es a re c o n n e c te d to th e m ain cascad e th ro u g h se rie s ju n c tio n s . T h e
se rie s ju n c tio n s a re assu m ed n o n - id e a l.
E ach o f the m u lti-c o u p le d c a v ity filte rs is
d e te rm in e d by 12 in d e p e n d e n t c o u p lin g c o e ffic ie n ts .
In o rd e r to ta k e th e a p p r o p ria te to lera n ce s into a c c o u n t, sp e c ific a tio n s w e re
ch o sen to be lOdB f o r th e c o m m o n p o r t re tu rn loss an d fo r th e in d iv id u a l c h a n n e l
sto p b a n d in s e rtio n losses, re s u ltin g in 124 n o n lin e a r c o n stra in t fu n c tio n s.
D esign
v a ria b le s in c lu d e d 60 c o u p lin g s, 10 in p u t a n d o u tp u t tra n s fo rm e r ratio s a n d 5
w a v e g u id e sp acin g s. T o le ra n c e s o f 5% w e re assu m ed fo r th e sp a c in g s, an d to leran c es
o f 0.5% f o r th e re m a in in g v a ria b les.
c o n v e n tio n a l
m in im a x
n o m in a l
T h e sta rtin g p o in t w as th e so lu tio n o f th e
d e sig n
w .r.t.
c o rre s p o n d in g resp o n ses a re sh o w n in Fig. 4.4.
sp e c ific a tio n s
of
20dB .
The
T h e e stim a te d yield w .r.t. th e
s p e c ific a tio n s o f IQdB a t th is p o in t w as 75%.
C a r e f u l a n d tim e -c o n s u m in g tu n in g is e sse n tial fo r m u ltip le x e rs.
A ny
a d ju s tm e n t in v o lv in g p h y sic a l d is c o n n e c tio n o f th e s tru c tu re is p a rtic u la ry ex p e n siv e .
F o r th e sa k e o f illu stra tio n w e fo c u s a tte n tio n in th is ex am p le on the w av e g u id e sp a c ­
ings as b e in g e x p e n siv e a d ju s tm e n ts to m a k e. T h e re fo re , th e la rg e r tu n in g to le ra n c e s
are a ssu m e d fo r th e sp acin g s. We c o n sid e r to tal to le ra n ce s o f 5% fo r th e sp acin g s, a n d
0.5% fo r th e c o u p lin g s a n d th e tra n s fo rm e r ra tio s.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
\\
80
0
5
return and insertion loss (dB)
10
15
20
25
30
35
40
45
501--------- 1------.—
11880
11920
11960
12000
12040
12080
12120
frequency (MHz)
Fig. 4.4
R e tu rn a n d in se rtio n loss o f th e 5 -c h a n n e l m u ltip le x e r a t the m in im ax
so lu tio n .
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
4 .5 .2
D esign P ro c e d u re a n d R e su lts
T h e p ro cess c o n sists o f 4 p h a se s a s sh o w n in T a b le 4.1. A t th e b e g in n in g o f
e a c h p h a se , a se t o f q u a d ra tic m o d els c o rre s p o n d in g to 124 resp o n ses is c o n stru c te d .
T h e se m o d els, th e n , a re used f o r a ll o u tc o m e s in th e e n tir e p h ase. N o tice th a t th is is
|r
d if f e r e n t fro m th e cases in th e la st c h a p te r w h e re q u a d ra tic m odels a re re b u ilt fo r
e a c h o p tim iz a tio n ite ra tio n . T o b u ild q u a d ra tic m o d e ls, 151 sim u la tio n s a re p e rfo rm e d
a t th e base p o in ts f o r ea c h p h a se , le a d in g to a to ta l 604 e x a c t sim u la tio n s fo r th e e n tire
d e sig n p ro cess.
M o re a n d m o re o u tc o m e s, fro m 50 to 2 0 0 , a re used in fo u r
c o n se c u tiv e d e sig n p h ases. A to ta l o f 20 o p tim iz a tio n ite ra tio n s a re in v o lv e d . W ithout
q u a d ra tic m o d elin g , 2500 e x a c t sim u la tio n s a n d se n s itiv ity analy ses w o u ld have b een
re q u ire d ( if th e sam e n u m b e r o f o p tim iz a tio n ite ra tio n s a r e necessary).
F o u r phases to o k to ta lly 69.5 seco n d s o n th e C R A Y X - M P /2 2 to reach a 90%
e stim a te d y ield . T h e e stim a te d y ie ld is g ra d u a lly in c re a se d in th e firs t th re e ph ases
a n d b ecom es a lm o st s te a d y in th e la st p h ase.
T o s tu d y th e c o lle c tiv e p e rfo rm a n c e o f
th e c ir c u it, th e r e tu rn losses o f all 3000 o u tc o m e s a n d o f th e s a tis fa c to ry o u tco m es a re
c o n ta in e d w ith in th e e n v e lo p e s in F ig s. 4 .5 a n d 4 .6 , re sp e c tiv e ly . A v e ry sm all p o rtio n
a b o v e th e s p e c ific a tio n in F ig . 4.5 re fle c ts th e p ro b a b ility o f n e a r 10% to p ro d u c e a
b ad c irc u it.
T h is a p p ro a c h allow s us to h a n d le th is la rg e o p tim iz a tio n p ro b le m (w ith 75
to le ra n c e d v a ria b le s, 124 c o n s tra in ts a n d up to 2 00 sta tis tic a lly p e rtu rb e d c irc u its ) in
a c c e p ta b le C P U tim e.
R e p ro d u c e d with perm ission of the copyright owner. Fu rther reproduction prohibited without permission.
82
TA B L E 4.1
--v>
S T A T IS T IC A L D E S IG N O F A 5 -C H A N N E L M U L T IP L E X E R
U S IN G Q U A D R A T IC A P P R O X IM A T IO N
P hase 1
Phase 2
S o lu tio n
o f P h ase 2
Phase 4
S ta rtin g P o in t
o f th e Phase
N o m in al
D esign
In itia l Y ield
75.0%
81.0%
84.3%
90.0%
In itia l Y ield
56.3%
69.0%
69.3%
92.0%
N u m b e r o f O u tco m es
U sed fo r O p tim iz a tio n
50
100
150
200
N u m b e r o f Ite ra tio n s
4
6
6
4
F in al Yield*
81.0%
84.3%
90.0%
90.3%
F in al Yield**
77.3%
77.3%
91.3%
94.0%
C P U T im e
(C R A Y X -M P /2 2 )
16.5s
17.6s
17.8s
!7.6s
*
**
S o lu tio n
o f Phase 1
P hase 3
S o lution
o f Phase 3
Y ield e stim a te d using a c tu a l sim u la tio n .
Y ield e stim a te d using q u a d ra tic a p p ro x im a tio n .
C P U tim es d o not in c lu d e y ield e stim atio n s based o n a c tu a l sim u la tio n .
A ll y ield s a re e stim a te d using 300 sam p les.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
(:■
return loss (dB)
83
20
11880
11920
11960
12000
12040
12080
12120
frequency (MHz)
F ig . 4.5
R e tu rn loss e n v e lo p e o f 3 0 00 5 -c h a n n e l m u ltip le x e r c ir c u it ou tco m es a f te r
y ield o p tim iz a tio n .
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
84
0
5
return loss (dB)
10
15
20
25
301
fflfl iH
11880
11920
11960
12000
12040
12080
12120
frequency (MHz)
Fig. 4.6
R e tu rn loss e n v e lo p e o f s a tis fa c to ry 5 -c h a n n e l m u ltip le x e rs am ong 3000
c ir c u it o u tco m es a f t e r y ield o p tim iz a tio n .
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
4.6
C O N C L U D IN G R E M A R K S
In th is c h a p te r, w e h a v e sh o w n th a t y ie ld -d riv e n d e sig n te c h n iq u e s c a n w ell
be a p p lie d to n o n -m a ssiv e p ro d u c tio n a n d tu n a b le c irc u its .
We have id e n tifie d a
n u m b e r o f asp e c ts related to sta tistic a l d e sig n o f tu n a b le c irc u its w ith tu n in g
to le ra n c e s. Som e d iscu ssio n a d d re sse s th e use o f o n e ty p e o f s u p e rc o m p u te r, n am ely ,
th e v e c to r p ip e lin e s u p e rc o m p u te r.
T o co p e w ith s ta tis tic a l d e sig n o f larg e scale
c irc u its , s u p e rc o m p u te rs a n d v e ry e f fic ie n t a p p ro x im a tio n te c h n iq u e s have been used
to c o o p e ra te w ith a p o w e rfu l y ie ld - d riv e n a p p ro a c h . T o d e m o n stra te the fe a sib ility
o f s u c h y ie ld o p tim iz a tio n o f la rg e scale m ic ro w a v e c irc u its , w e have su c c e ssfu lly
a c c o m p lish e d th e y ield o p tim iz a tio n o f a 5 c h a n n e l m u ltip le x e r w ith 75 d esig n a n d
to le ra n c e d v a ria b le s, 124 c o n s tra in ts a n d up to 200 sta tistic a lly p e rtu rb e d c irc u its . N o
m ic ro w a v e c irc u it d esig n o p tim iz a tio n o f th is ty p e a n d o n th is scale has e v e r been
re p o rte d .
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A
v*
/
5
NONLINEAR CIRCUIT YIELD OPTIM IZATION WITH GRADIENT
APPROXIMATIONS
5.1
IN T R O D U C T IO N
M a n y e n g in e e rin g a p p lic a tio n s re q u ire th e use o f n o n lin e a r c irc u its , such as
sw itc h e s , o sc illa to rs, a n d m ix e rs. T h e ra p id d e v e lo p m e n t o f m o n o lith ic m icrow ave
in te g ra te d c irc u its (M M IG s) has m a d e possible th e in te g ra tio n o f th e se n o n lin e a r
c irc u its w ith in o n e d ie . T h is d e m a n d s e ff ic ie n t a n d e ffe c tiv e C A D tools to design
n o n lin e a r c irc u its . N ot o n ly m u st th e tools p ro d u c e v alid n o m in a l d e sig n s, b ut th ey
m u st e n s u re d e sig n m a n u fa c tu ra b ility a n d s a tis fa c to ry y ield . F o r M M IC te ch n o lo g y ,
in
p a r tic u la r ,
y ie ld - d r iv e n ,
c o s t- e f f e c tiv e
d esig n
is
vital
to
co m m ercial
c o m p e titiv e n e ss.
R iz z o li, L ip p a rin i a n d M a ra z z i (1983) a tte m p te d to a c c o m p lish in g nom inal
d e sig n o f n o n lin e a r c irc u its . T h e ir a p p ro a c h c o m b in e s th e h a rm o n ic balance (H B )
s im u la tio n a n d d e sig n in to o n e o p tim iz a tio n loop b y c o n s id e rin g d e sig n v ariab les a n d
HB s ta te v a ria b le as o p tim iz a tio n v a ria b le s sim u lta n e o u sly . T h e c ir c u it d o es n o t have
to be c o m p le te ly so lv e d by H B d u rin g o p tim iz a tio n u n til th e fin a l so lu tio n is reach ed .
H o w e v e r, th is a p p ro a c h is n o t c o m p a tib le w ith m ost y ie ld -d riv e n a p p ro a c h e s since
y ie ld - d r iv e n d e sig n re q u ire s a c tu a l c ir c u it resp o n ses to c a lc u la te y ield as the o b je c tiv e .
T o m eet th is new a n d v ery d if f ic u lt c h a lle n g e , a n u m b e r o f im p o rta n t p o in ts m ust be
in c o rp o ra te d in y ie ld -d riv e n d e sig n o f n o n lin e a r c ircu its:
(1)
A n e f f ic ie n t HB s im u la tio n sc h em e.
87
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
(2)
H ig h -s p e e d g ra d ie n t c a lc u la tio n i f g r a d ie n t- b a s e d o p tim iz a tio n is used.
(3)
W e ll-m o d eled n o n lin e a r passive a n d a c tiv e d e v ic e s , su c h as F E T s, d io d e s, a n d
tra n sm issio n lin es, a n d valid sta tistic a l m o d els f o r c irc u it c o m p o n e n ts,
I;
e sp e c ia lly fo r a c tiv e d ev ic e s.
(4)
A w e ll-s tru c tu re d ov/er all d e sig n p ro cess c o n s o lid a tin g th e a b o v e po in ts.
T h is c h a p te r is o rg a n iz e d as follow s.
We s ta r t b y re v ie w in g a n e f fic ie n t
sim u la tio n m e th o d s u ita b le f o r m ic ro w av e n o n lin e a r c ir c u its , n a m e ly , th e HB m eth o d
(H B ). In stead o f w o rk in g o n v e ry a b s tra c t an d in tric a te fo rm u la tio n s f o r th e g e n era l
HB a p p ro a c h , w e use a sim p le c irc u it, a c n e - F E T c ir c u it, to illu stra te h o w th e HB
m eth o d w o rk s. T h e HB m e th o d is im p le m e n te d w ith e x a c t Ja c o b ia n m a tric e s fo r fast
c o n v e rg e n c e a n d im p ro v e d ro b u stn ess. N ex t w e in tro d u c e s p e c ific a tio n s a n d e rro rs
fo r n o n lin e a r c irc u it yield o p tim iz a tio n , an d fo rm u la te th e y ie ld -d r iv e n d esig n
p ro b lem f o r n o n lin e a r c irc u its .
We o f f e r a n a p p ro a c h to e f f ic ie n t y ie ld -d riv e n
o p tim iz a tio n o f n o n lin e a r m ic ro w a v e c irc u its w ith s ta tis tic a lly c h a ra c te riz e d d ev ices
(B a n d le r, Z h a n g , Song a n d B iern ack i 1989). T h is a p p ro a c h u tilize s a p o w e rfu l an d
ro b u st o n e -s id e d £, o p tim iz a tio n a lg o rith m f o r d e sig n c e n te rin g (B a n d le r a n d C h en
1988). T h e e ffe c tiv e g ra d ie n t a p p ro x im a tio n te c h n iq u e (B a n d le r, C h e n , D a ija v a d a n d
M adsen 1988) is a d o p te d .
p ro v id e d .
An
e x te n sio n o f th is a p p ro a c h f o r sta tistic a l desig n is
In d e p e n d e n t a n d / o r c o rre la te d
n o rm a l
d is trib u tio n s
and
u n ifo rm
d is trib u tio n s d e s c rib in g la rg e -s ig n a l F E T m odel p a ra m e te rs a n d passiv e e lem en ts a re
fu lly a c c o m m o d a ted .
T h e yield o p tim iz a tio n o f a m icro w av e fre q u e n c y d o u b le r w ith a la rg e -sig n a l
s ta tistic a lly sim u la te d F E T m o d el is su c c e ssfu lly c a rrie d o u t. T h e p e rfo rm a n c e y ield
is in creased fro m 40% to 70% . We b e liev e th a t th is is th e f ir s t d e m o n s tra tio n o f yield
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
89
o
o p tim iz a tio n o f n o n lin e a r c irc u its o p e ra tin g u n d e r la rg e -sig n a l s te a d y - s ta te p e rio d ic
o r a lm o st p e rio d ic c o n d itio n s.
5.2
T H E H A R M O N IC B A L A N C E S IM U L A T IO N T E C H N IQ U E
A q u ic k re v ie w o f th e re c e n tly d e v e lo p e d HB m e th o d is w o rth w h ile because
it is a key p a r t o f o u r y ie ld - d r iv e n n o n lin e a r c irc u it d esig n a p p ro a c h .
The
fo rm u la tio n w ill also h e lp us d e riv e th e s e n sitiv ity analysis in th e n e x t c h a p te r. T h e
HB m e th o d is a h y b rid tim e - a n d fre q u e n c y -d o m a in ap p ro a c h w h ic h allow s all th e
a d v a n ta g e s o f tim e -d o m a in n o n lin e a r d e v ice m o dels, such as F E T s a n d d io d es, an d
th e p o w e r o f s te a d y -s ta te f re q u e n c y -d o m a in tec h n iq u e s f o r lu m p ed a n d d istrib u te d
c irc u it e le m e n ts, su ch as m ic ro s trip lines and strip lin e s. A v a rie ty o f a tte m p ts has
b een m ade to im p ro v e th e e ffic ie n c y an d v e rsatility o f H B b y K u n d e rt and
S a n g iv a n n i-V in c e n te lli(1 9 8 6 ), R iz z o li, L ip p a r in ia n d M arazzi (1 9 8 3 ), C u rtic e (1987),
a n d G ilm o re (1 9 8 6 ), a n d m an y o th e rs.
T h e key c o n c e p t in th e HB m e th o d is to d ecom pose th e c ir c u it into tw o parts,
lin e a r an d n o n lin e a r su b n e tw o rk s . T h e r e fo re , tim e - and fr e q u e n c y -d o m a in analyses
c a n be p e rfo rm e d on th e n o n lin e a r a n d lin ear su b n e tw o rk s, re sp e c tiv e ly . T o illu stra te
h o w th e HB m e th o d c a n be u se d to s im u la te a n o n lin e a r c irc u it, w e c o n sid e r a typical
o n e - F E T c irc u it sh o w n in F ig . 5.1. T h e c irc u it is d eco m p o sed in to th e lin ea r p a rt,
N l , a n d th e n o n lin e a r p a rt, N n l , as sh o w n in F ig. 5.2. We assu m e th a t the c irc u it is
e x c ite d by a p e rio d ic s tim u lu s w ith fu n d a m e n ta l fre q u e n c y w.
R e p ro d u c e d with perm ission of the copyright owner. Fu rther reproduction prohibited without permission.
90
FET
input
Fig. 5.1
input
output
m atching
m atching
gate
drain
bias
bias
output
BD
S c h e m a tic re p re s e n ta tio n o f th e o n e - F E T c irc u it.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
91
nonlinear
intrin sic
FET
Fig. 5.2
D eco m p o sitio n o f th e c ir c u it in F ig. 5.1. N L a n d N nl sta n d fo r th e lin e a r
a n d n o n lin e a r su b n e tw o rk s.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
92
5.2.1
T h e N o n lin e a r M o d e l a n d Its T im e -D o m a in S im u la tio n
T h e F E T u sed is a m o d ifie d M a te rk a a n d K a c p rz a k m odel (M a te rk a a n d
K a c p rz a k 1985, S u p e r-C o m p a c t 1987) sh o w n in F ig . 5 .3. T h e n o n lin e a ritie s in h e re n t
s'in th e in trin sic p a rt a re d e s c rib e d by
to = n v G0 - ^ to(OKi + *^-7—-)>
‘D SS
(E * K f G)
^ ( " o ‘to) - hiss
.
1 vp 0
ta n h
S l vD
+
‘G = l G d * * V ( a G vG )
" ' 1,
to = //«»exP[ato(>to - ‘’I -
^1
= * io (l
^
= 0,
“ ^e
if
vg
kbc)]'
)>
>
k r vg
1,
C x = c 10(l - ^ V o )-1/ 2,
C j = C jqi/ s " +
if ^ i Vg ^ 0.8,
and
j
C> = C>»[1 ^ Cp = CpfffS ,
- tto)](-1/2),
if K ffvj - *to) -
w h ere i,1, t'c , a n d vD a re c o n tro llin g v o ltages as in d ic a te d in F ig . 5.3, th e re m a in in g
c o e ffic ie n ts b ein g m odel p a ra m e te rs. N o tice th a t such d ec o m p o sitio n allow s tw o o f
th e c o n tro llin g v o ltag es, ^ a n d vp, to a p p e a r a t th e in te rfa c in g p o rt.
S u ppose th a t th e c o n tro llin g voltages can be e x p ressed as
//
>'*(0 = £
h -0
VkW ^ hwt.
k = I , C , D,
(5.1)
w h ere H is th e h ig h e st h a rm o n ic c o n sid e re d an d V^(h) a re c o m p le x c o e ffic ie n ts o f the
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
93
n on lin ear in trin sic FE T
«»
F ig. 5.3
T h e n o n lin e a r in trin s ic F E T m odel.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
94
F o u rie r series. T h e n w e d e f in e
v*\)
k = 1, C , D.
,
Vk =
(5-2)
Vk { H )
T h e c u rre n ts fro m c o n tro lle d so u rc e s, th e c u rre n t th ro u g h th e n o n lin e a r re sisto r, a n d
th e c h arg es on th e c a p a c ito rs a re fu n c tio n s o f V1, VG an d V d , th a t is,
ik{VG , VD ), k = D, C7, B
■
v ,
W
Vg) =
v g *v d
)
- vg (Y g )]
5 T O ---------
)> k = \ , F
T lie c a lc u la tio n o f th e c h a rg e s o n th e tw o n o n lin e a r c a p a c ito rs in v o lv e s in te g ra tio n .
T h e fre q u e n c y -d o m a in e x p re ssio n s o f these c u rre n ts a n d c h a rg e s c a n b e o b ta in e d by
th e F o u rie r tra n s fo rm a tio n . T h e y a re o f th e fo rm s
lk( V ) =<7[ik( V ) l
k = D ,G , B, R lt
(5.3)
Qk{V) - r M V ) } ,
k = I, F ,
w h ere .9" sta n d s f o r F o u rie r tra n s fo rm , Ik a n d Qk a r e v e c to rs c o n sistin g o f th e
co m p lex c o e ffic ie n ts o f th e c o rre s p o n d in g F o u rie r serieses, a n d V is d e fin e d b y th e
fo llo w in g
(5.4)
D
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
5.2.2
L in e a r S u b c irc u it S im u la tio n
B a n d le r, Y e a n d Song (1990) p ro p o se d a n e ff ic ie n t a p p ro a c h to c o n stru c t th e
m u ltip o rt m a trix to a c c e le ra te H B a n a ly sis. H ere w e use th is a p p ro a c h to d e al w ith
th e c ir c u it in Fig. 5.1.
C o n s id e r th e lin e a r s u b n e tw o rk in F ig . 5.2.
T h e r e a re fiv e p o rts, a p o rt
c o n n e c tin g w ith th e in trin s ic F E T , a n o u tp u t v o lta g e p o rt, a n in p u t e x c ita tio n p o rt,
a n d tw o D C bias p o rts. A t a g iv e n fre q u e n c y , th e no d al e q u a tio n ca n be w ritte n as
(5.5)
0
V,
0
w h e re I',, , k = g , d , s, a re n o d e v o ltag e s a t th e th re e te rm in a ls o f the in te rfa c in g
p o rt, a n d V,,., i = 1, 2 , . . . , b, a re in te rn a l n ode voltages in th e lin e a r su b n e tw o rk .
F irs t, w e ex p ress Y in th e fo rm
Y = [ y„t y,xd y„, yi yo ybg vao y,h ■ • • yab ]
(5-6)
w h e re e ach e n tr y is a c o lu m n v e c to r. By a d d in g a n d , a t th e sam e tim e, s u b tra c tin g
y„ V„ a n d yn V,. to a n d fro m th e rig h t h a n d sid e o f (5 .5 ), w e c an re w rite it as
f t
A t
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
,1
"
9%
1
\
1
96
"
ln t
^
^
v,
h
Vo
lo
v
m >
(5
=
Vug
lBG
'
l BD
0
0
J
w h ere
/ '
= [ y„t
y nj
t V n ^ y , i s + y n)
J7
yo
vbg
vbd
y ,h
• • y Hh\
( 5 -8 )
N o tic e th a t
We hav e
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
Second, w e express Y ' in its row vector form
y[
y 'd
y».
y{
y'o
Y' =
y/:o
(5-10)
y/m
y /
yttj
w h e re each e n try is a ro w v e c to r. I f w e a d d th e f irs t a n d se co n d row s to th e th ird row
in (5 .9 ), th e n , w e have
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
because, from K.CL,
f»g + !"d +
N o w , we su p p ress in te rn a l n o d e s, fro m
°to
a n d th e n o d e /ty fro m (5.11).
T h e n , th e fo llo w in g eq u a tio n c a n be o b ta in e d :
"
J!
yi
f'l
yH
yf
Vt
u
y&
vo
lO
V BG
l BG
VBD
?BD
ym
yuc
w h ere th e c o e ffic ie n t m atrix is w ritte n in th e row v e c to r fo rm .
(5.12)
N o tic e th a t at th e
o u tp u t p o rt Iq = 0 . H ence, VQ c a n be su p p re sse d so th a t o n ly V ly VD, V{, ^ g ^ a n d Vbd
are le ft in tlic re su lta n t eq u a tio n s
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
99
>1
"
VD
rut
Vi
l "d
=
h
Vb g
h o
Vbd _
Ib d
(5.13)
E q u a tio n (5.13) can be used a t d if f e r e n t h a rm o n ic s. F o r g iv e n V 1 a n d VD at
a c e rta in fre q u e n c y hu>, 0 < h < M, a n d th e sig n al in p u t (w h e n h = 1) o r D C biases
(w h e n h - 0 ) , fro m (5.13), th e c u rre n ts g oing in to th e lin e a r p a rt fro m th e in te rfa c in g
p o rt w ith n o n lin e a r p a rt can be c o m p u te d as
!„k(h),
0 </i<//,
k = g, d ,
(5.14)
w h ere th e h a rm o n ic ind ex h has b e e n in c lu d e d to in d ic a te th e h arm o n ic c o n sid e re d .
We use /„
to d e n o te th e co m p lex v e c to r c o n ta in in g all h a rm o n ic s, th a t is,
V°>
'„ / 2 )
,
I,lk ”
5.2.3
k = g , d.
(5.15)
H a rm o n ic B alance E q u atio n
HB s ta rts by a ssu m in g a n in itia l guess o f V, d e fin e d in (5 .4 ), th a t is
V =
Vg
Vd
w h e re Vk, k ~ 1, (7, D , a re g iv e n in (5 .2 ).
T h e c u r r e n t su m m a tio n s a t th e th re e
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
100
nd , a n d «# , a re
n o d es,
- I J Y ) * jK Qt iV ) + W
.\
4i
=
h,sy) *hw w Qpin
-
«
-
«■
(5.16)
'^ k
( 5 . 17)
an d
*1
=V
1
K> - W
)
- ; n < 2 c< ^ ),
(5-18)
an d Qc a re fro m (5 .3 ), /„ t , k = g, d, a re d e fin e d in
w h ere /w, / a , / 0 , //* ,
(5.1 5 ), and
n
= w diagiO , 1, 2
____ H).
(5.19)
In th ese e q u a tio n s , V is in c lu d e d to sy m b o liz e th o se te rm s w h ich a re fu n c tio n s o f V.
We o b ta in th e HB e q u a tio n b y asse m b lin g , fro m (5.16) to (5 .1 8 ),
FH' V )
F{V) =
Fn jV
>
(5.20)
K C L re q u ire s
F{V) = 0 ,
(5.21)
p ro v id e d th a t th e V ? re c o rre c tly assu m ed .
T h e e q u a tio n in (5.21) is in co m p le x fo rm .
We c a n re w rite it in to a s e t o f
a lg e b ra ic re a l v alue e q u a tio n s by s p littin g th e real a n d im a g in a ry p a rts,
F(K)=0,
(5.22)
w h ere the b a r sta n d s fo r a real v e c to r re su ltin g fro m th e c o m p le x c o u n te rp a rt, i.e.,
R e(K ) '
lm(K)
" R HF) '
,
F =
_
1m(F)
_
T h e HB m eth o d s ta rts by a ssu m in g an in itia l se t V . T h e n , a N e w to n -lik e
R e p ro d u c e d with perm ission of the copyright owner. Fu rth er reproduction prohibited without permission.
101
m e th o d is used
v ne„
=
(5.23)
Void - W o t * ) ' 1n v o u )
w h e re J(.Vold) is th e J a c o b ia n a n d e v a lu a te d a t VoldT h e Ja c o b ia n c a n b e c o n s tru c te d e ith e r n u m e ric a lly o r a n a ly tic a lly . In o rd e r
to o b ta in fa s te r c o n v e rg e n c e a n d h ig h e r a c c u ra c y , th e a n a ly tic a l Ja c o b ia n a p p ro a c h is
a d o p te d in o u r yield o p tim iz a tio n .
5.2.4
R esp o n se C a lc u la tio n
C o n sid e r th e o u tp u t v o ltag e as th e d e sire d re sp o n se.
A lin e a r e q u a tio n in
(5.12) can be used to c a lc u la te th e o u tp u t
(5.24)
If we w rite o u t th e ro w v e c to r y $ a n d n o tic e th a t I q = 0 , th en w e have
(5.25)
It is th en possible to fin d f/ o{h),
h = 0 , 1, 2 , . . ., / / , a f te r so lv in g th e HU e q u a tio n .
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
5.3
S P E C IF IC A T IO N S A N D E R R O R S F O R N O N L IN E A R C IR C U I T Y IE L D
O P T IM IZ A T IO N
C o n sid e r a n o n lin e a r m icro w av e c irc u it o p e ra tin g u n d e r la rg e -s ig n a l s te a d y -
sta te p e rio d ic c o n d itio n s. R esp o n se fu n c tio n s f o r s u c h a c ir c u it m ay in v o lv e D C a n d
h a rm o n ic c o m p o n e n ts o f th e o u tp u t sig n al. U n lik e lin e a r c ir c u its , th e re f o re , d e sig n
s p e c ific a tio n s c a n be im p o sed a t D C and se v eral h a rm o n ic s. T h e j t h sp e c ific a tio n c a n
be d e n o te d by
S UJ<h ),
(5 .2 6 )
Sh{h)
(5.27)
if it is an u p p e r sp e c ific a tio n , o r
in th e case o f lo w er sp e c ific a tio n s , w h ere
h =
(5.28)
H
is th e
h a rm o n ic in d e x v e c to r, 0 a n d H re p re se n t D C a n d th e h ig h e st h a rm o n ic s,
re sp e c tiv e ly .
A s p e c ific c ir c u it resp onse m ay in v o lv e a ll o r som e
o f th e (H + 1)
sp e c tra l c o m p o n e n ts.
We still use 4> to d e n o te c irc u it o u tco m e s, n a m e ly ,
4t = / ♦ A * \ / = 1 , 2
K.
(5.29)
T h e response o f each o u tc o m e , d e n o te d by
R 0 , V \ h)
(5.30)
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
103
is c a lc u la te d a f te r so lv in g th e H B e q u atio n s
'V
F t f W 1) - 0 ,
(5.31)
w h e re V ' c o m p rise s th e s p lit real a n d im a g in ary p a rts o f th e s ta te v a ria b le s in th e 11B
e q u a tio n fo r th e o u tc o m e 4>‘ . T h e c o rre sp o n d in g e r ro r f u n c tio n is d e fin e d as
\\
R 0 , V \ h) ^ S uj(h)
(5.32)
S , f h ) - /?■(£ V \ h).
(5.33)
o r as
We assem b le all e rro rs f o r th e o u tc o m e 4' in to one v e c to r d . I f all e n trie s o f this
v e c to r a re n o n p o sitiv e , th e o u tc o m e <t>' rep resen ts an a c c e p ta b le c itc u ii.
F o llo w in g th e g e n e ra liz e d t x c e n te rin g a p p ro a c h d e s c rib e d in C h a p te r 2 , the
o b je c tiv e fu n c tio n fo r th e o n e -s id e d £j o p tim iz atio n is
m in im iz e
w °) = E
E
W0
(5.34)
)
ie! j e W )
w h e re efrfi) a re e le m e n ts in e{$), a n d
) , / , a n d a,- a re d e f in e d in (2.36), (2.38)
a n d (2.3 9 ), re sp e c tiv e ly .
5.4
E F F IC IE N T G R A D I E N T A P P R O X IM A T IO N F O R Y IE L D -D R IV E N
D E SIG N
It is a lw a y s a d i f f ic u l t p ro b le m to o b ta in th e g ra d ie n ts o f c irc u it resp o n se
fu n c tio n s w hen p o w e rfu l g ra d ie n t- b a s e d o p tim iz a tio n is u sed .
F o r y ie ld -d riv e n
d e sig n o f n o n lin e a r c irc u its , th e c o m p u ta tio n a l e f f o r t fo r e v a lu a tin g g ra d ie n ts m ay be
p ro h ib itiv e sin c e m a n y n o n lin e a r c irc u it outco m es a re in v o lv e d . H ow to o b ta in the
g ra d ie n t in fo rm a tio n e f f ic ie n tly is o f ex trem e im p o rta n c e . M u c h resea rc h has been
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without perm ission.
d e v o te d to d e v ise h ig h -sp e e d g ra d ie n t a p p ro x im a tio n m e th o d s w h en a n a ly tic a l
g ra d ie n ts a re n o n e x iste n t o r v e ry d if f ic u lt to get. A s im p le a n d tra d itio n a l w ay to
o b ta in n u m e ric a l g ra d ie n ts is th e p e r tu rb a tio n a p p ro x im a te s e n s itiv ity te c h n iq u e
(/M S T )(R iz z o Ii, L ip p a rin i a n d M arazzi 1983). B andler, C h e n , D a ija v a d a n d M adsen
(1988) p ro p o sed an in te g ra te d g ra d ie n t ap p ro x im a tio n te c h n iq u e ( IG AT ) f o r c irc u it
d esig n .
1
In th e fo llo w in g se c tio n w e w ill b rie fly review IG AT in th e sin g le fu n c tio n
case, th e n e x te n d IGAT to yield o p tim iz a tio n .
5.4.1
IGAT fo r N o m in al D esign
Since th e a p p lic a tio n o f IGAT is n o t re stric te d to c ir c u it re sp o n se fu n c tio n s ,
let us use / ( $ )
to d en o te a g e n e ric fu n c tio n .
A p p ro x im a tin g D e riv a tiv e s by PAST
T h e f ir s t- o r d e r d e riv a tiv e o f f{<f>) w .r.t th e fcth v a ria b le ca n b e e stim a te d by
df(<f>)
dh
w h ere 4> +
f(4> + a ^ . ) - f(4>)
A4>k
(5.35)
d e n o te s th e p e rtu rb a tio n o f th e ifcth v a ria b le , A ^ . is th e p e rtu rb a tio n
len g th an d uk is a c o lu m n v e c to r w hich has I in th e /cth p o s itio n a n d z ero s else w h e re .
A n a p p ro x im a tio n to th e g ra d ie n t, V / ( $ ) , c a n be o b ta in e d b y p e r tu r b in g all v a riab le s
one at a tim e.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
i'\
105
V
A p p ro x im a tin g D e riv a tiv e s b v IG A T
T o s ta r t th e p ro c e ss, P A S T is u sed as in B a n d ler, C h e n , D a ija v a d a n d M ad sen
(1988) to c a lc u la te th e a p p ro x im a te g ra d ie n t.
T h e B ro y d e n u p d a te g e n e rates the n ew a p p ro x im a te g ra d ie n t fro m
th e
p re v io u s g ra d ie n t, n a m e ly ,
f(A
^
~ fi& o ld ) ~ ( V f( $ o ld ) ) T
*f{4>„ew)
= X
VJ /{ (( A<
f>old)\ * ------------------------------------------
A<f>,(5.36)
w h ere $old a n d <
j>new a re tw o d if f e r e n t p o in ts an d A<f> = $new- 4>old. If 4>0id and
a re ite ra te s o f o p tim iz a tio n , /(<f>0id) a n d / ( ^ „ Ch.) need to be e v a lu a te d an y w ay . T h u s ,
th e u p d a te d g ra d ie n t c a n be o b ta in e d w ith o u t a d d itio n a l fu n c tio n e v a lu a tio n s (c irc u it
sim u la tio n s).
T o o v e rc o m e a p a rtic u la r d e fic ie n c y o f th e B ro y d e n u p d a te , a f te r a few
u p d a te s, a sp e c ia l ite ra tio n o f P ow ell g en e rates a sp ec ial s te p
to g u a ra n te e stric tly
lin e a rly in d e p e n d e n t d ire c tio n s. A f te r a n u m b e r o f o p tim iz a tio n ite ra tio n s, w e m ay
also a p p ly P A ST to m a in ta in th e a c c u ra c y o f th e a p p ro x im a te g ra d ie n ts at a d e sira b le
level.
5.4.2
IG A T f o r Y ield O p tim iz a tio n
N otice th a t — ~t— and —
dA
ar e d iffe re n t. T h e latter is o f interest, becau se ^0
84
is c o n s id e re d v a ria b le in o u r y ie ld - d riv e n d esig n .
T h u s , alt p e rtu rb a tio n s a re m ade to
in th e in itia liz a tio n a n d re in itia liz a tio n
step s u sin g P A S T (B a n d le r, C h e n , D a ija v a d a n d M ad sen 1988). W hen 4>° is p e rtu rb e d
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
lo <£° ♦ A <f%uky d e n o te d b y f t ^
fo r s h o rt, o u tco m es sh o u ld b e re g e n e ra te d from<£°
in o rd e r to g e t p e rtu rb e d c ir c u it resp o n ses. T h e se o u tc o m e s a re d e n o te d b y f t . p m N o tic e th e v e ry sa m e p ro c e d u re to g e n e ra te <f>
‘ sh o u ld b e re p e a te d to g e n e ra te S p o i ­
lt m ean s th a t th e in itia l s ta tu s o f th e ra n d o m n u m b e r g e n e ra tio n , u su a lly , a s e t o f
in itia l valu es c a lle d se e d s, is sto re d .
F ro m B a n d le r, C h e n , D a ija v ad a n d M ad sen {1 9 8 8 ), th e a p p ro x im a te d e riv a tiv e
o f th e resp o n se f ( 4 > ) is d e fin e d as
d /if)
^
f^k.pcrt)
d<j>l
~
^
21)
A<t>l
W hen th e B ro y d en u p d a te o r the sp ecial ite ra tio n o f P ow ell a re u se d , AtfP is
a n d $ ICW as g e n e ra te d e ith e r b y th e o p tim iz a tio n o r b y th e sp e c ia l
c o m p u te d fro m
ite ra tio n .
O u tc o m e s <
f>old a n d C w
a re o u tco m es g e n e ra te d fro m f^id a n d
re sp e c tiv e ly . T h e g ra d ie n t o f th e resp o n se
v / < o
-
w .r.t.
c a n be u p d a te d as
( 5. 38 )
(A *°)r A *°
In c irc u it s im u la tio n , th e re a re u su ally sev eral resp o n se levels in v o lv e d .
S u p p o se th e re sp o n se o f in te re s t, o n w h ich th e d e sig n s p e c ific a tio n is im p o se d , is the
p o w er g ain . In th e c ir c u it sim u la tio n , th e p o w e r gain is c a lc u la te d fro m th e o u tp u t
p o w e r w h ic h , in tu r n , is c a lc u la te d fro m th e o u tp u t vo ltag e.
T h is im p lies th re e
d if f e r e n t re sp o n se levels. IGAT c an be a p p lie d a t a n y resp o n se lev el.
5.5
N O N L IN E A R F E T S T A T IS T IC A L M O D E L S A N D S T A T IS T IC A L
O U T C O M E G E N E R A T IO N
P u rv ia n c e , C riss a n d M o n te ith (1988) tre a te d th e sta tistic a l c h a ra c te riz a tio n o f
R e p ro d u c e d with perm ission of the copyright owner. Fu rther reproduction prohibited without permission.
s m a ll-sig n a l F E T m odels.
O u r p ro p o se d y ield o p tim iz a tio n re q u ire s sta tistic a lly
I
d e sc rib e d la rg e -sig n a l F E T m o d els. O u r la rg e -sig n a l F E T sta tistic a l m odel in clu d es
y.
V *.
"
an in trin s ic la rg e -sig n a l F E T m odel m o d ifie d fro m th e M a te rk a a n d K a cp rz ak m odel
-X'v
(M a te rk a a n d K a c p rz a k 1985, M ic ro w av e H a rm o n ic a 1987), sta tistic a l d is trib u tio n s
a n d c o rre la tio n s o f p a ra m e te rs . T h e m u ltid im e n sio n a l n o rm al d is trib u tio n is assu m ed
fo r all F E T in trin sic a n d e x trin s ic p a ra m e te rs. T h e m ean s a n d s ta n d a rd d e v ia tio n s a re
listed in T a b le 5.1. T h e c o rre la tio n s b etw ee n p ara m e te rs a re assum ed a c co rd in g to th e
resu lts p u b lish e d by P u rv ia n c e , C riss a n d M o n te ith (1988).
C e rta in m o d ific a tio n s
hav e been m ade to m ake th e c o rre la tio n s fo r the la rg e -sig n a l F E T m odel to be
c o n siste n t w ith those fo r th e s m a ll-sig n a l F E T m odel d e a lt w ith in P u rv ian c e, C riss
a n d M o n te ith (1988). T h e c o r r e c tio n c o e ffic ie n ts a re g iv e n in T a b le 5.2.
We use a ran d o m n u m b e r g e n e ra to r cap ab le o f g e n e ra tin g sta tistic a l o u tco m es
fro m th e in d e p e n d e n t a n d m u ltid im e n sio n a l c o rre la te d n o rm a l d is trib u tio n s a n d fro m
u n ifo rm d istrib u tio n s .
P a ra m e te rs o f th e n o n lin e a r la rg e -sig n a l m odels h av e c e rta in physical lim its.
A n o rm al d is trib u tio n ra n d o m g e n e ra to r m ay g en era te o u tc o m e s fa r b ey o n d these
lim its. T h e se o u tco m es m a y c a u se serio u s p ro b le m s e ith e r in HB sim u la tio n , su ch as
v e ry slow c o n v e rg e n c e a n d d iv e rg e n c e , o r in o p tim iz a tio n . Such outcom es m ust be
c a re fu lly d e te c te d a n d e lim in a te d .
5.6
Y IE L D O P T IM IZ A T IO N O F A F R E Q U E N C Y D O U B L E R
5.6.1
D escrip tio n o f th e D esign
A F E T fre q u e n c y d o u b le r sh o w n in F ig. 5.4 is c o n s id e re d .
It c o n sists o f a
c o m m o n -s o u rc e F E T w ith a lu m p e d in p u t m a tc h in g n e tw o rk a n d a m ic ro strip o u tp u t
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
108
T A B L E 5.1
A S S U M E D S T A T IS T IC A L D IS T R IB U T IO N S
FOR T H E FE T PA RA M ETERS
v;
FET
P a ra m e te r
N o m in al
V alue
S ta n d a rd
D e v iatio n
<%)
L q ( o H)
R D (fi)
L s (n H )
R S(D)
0.16
2.153
0.07
1.144
440
1.15
5
3
5
5
14
3
4.5
5
0.65
0.65
0.65
^DE^
C ds ( p F)
0.12
^D S S ^)
6.0x1 O' 2
-1 .9 0 6
-1 5 x l0 ’2
V po(V)
E
1.8
FET
P a ra m e te r
N o m in a l
V alue
S ta n d a rd
D ev ia tio n
(%)
S/
Kg
T<pS)
0 .6 7 6 X 1 0 '1
0.65
0.65
7.0
6
Ss
1. 666x 1O' 3
0.65
3
3
3
3
“G
F.; 0(D)
C 10(P F )
C F0(p F )
1.1
0 .7 1 3 x l0 " 5
38.46
-0 .7 1 3 x l0 * 5
-3 8 .4 6
3.5
0.42
0.02
8
4.16
6.64
T h e fo llo w in g p a ra m e te rs a re c o n s id e re d as d e te rm in istic : K.E = 0 .0 , K R = 1. 111, K j
= 1.282, C 1S = 0. 0, a n d K P = 1.282.
F or d e fin itio n s o f th e F E T p a ra m e te rs , see S ection 5.2.1.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
109
T A B L E 5.2
F E T M O D E L P A R A M E T E R C O R R E L A T IO N S
Rs
Rs
Ls
r de
Cos
Sjtl
r
r in
C gs
C cd
1.00
-0.16
0.11
-0.22
-0.20
0.15
0.06
0.15
0.25
0.04
-0.16
1.00
-0.28
0.02
0.06
-0.09
-0.16
0.12
-0.24
0.26
: Ls
0.11
-0.28
1.00
0.11
-0.26
0.53
0.41
-0.52
0.78
-0.12
r de
r
C ds
-0.22 -0.20
0.02
0.06
0.11
-0.26
1.00 -0.44
-0.44
1.00
0.03 -0.13
0.04 -0.14
-0.54
0.23
0.02 -0.24
- 0 .1 4 -0.04
0.15
-0.09
0.53
0.03
-0.13
1.00
-0.08
-0.26
0.78
0.38
0.06
-0.16
0.4!
0.04
-0.14
-0.08
1.00
-0.19
0.27
-0.46
r in
C qs
9 gd
0.15
0.12
-0.52
-0.54
0.23
-0.26
-0.19
1.00
-0.35
0.05
0.25
-0.24
0.78
0.02
-0.24
0.78
0.27
-0.35
1.00
0.15
0.04
0.26
-0.12
-0.14
-0.04
0.38
-0.46
0.05
0.15
1.00
C e rta in m o d ific a tio n s have b e e n m ade to a d ju s t th ese sm a ll-sig n a l p a ra m e te r
c o rre la tio n s (P u rv ia n c e , C riss a n d M o n te ith 1988) to b e c o n siste n t w ith th e la rg e sig n al F E T m o d el.
•
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110
m a tc h in g a n d f ilte r se c tio n . T h e o p tim iz a tio n v a ria b le s in c lu d e th e in p u t in d u c ta n c e
L j a n d th e m ic ro s trip le n g th s l x a n d / 2. T w o bias v o lta g es V BG a n d V BD a n d th e
d riv in g p o w e r lev el PIN a re also c o n sid e re d as o p tim iz a tio n v aria b le s.
The
fu n d a m e n ta l fre q u e n c y is 5 G H z. R esp o n ses o f in te re st a re th e c o n v e rsio n g a in a n d
sp e c tra l p u rity , d e fin e d by
p o w e r o f th e se c o n d h a rm o n ic a t th e o u tp u t p o rt
c o n v e rsio n g ain = 10 lo g — - ------ ^----------------— -------------— :--------------p o w e r o f th e fu n d a m e n ta l f re q u e n c y a t th e in p u t p o rt
and
,
,n .
p o w e r o f th e se co n d h a rm o n ic a t th e o u tp u t p o rt
sp ectral p u rity = 10 log— ---------------- -— .— .----:---------- ----------------------to tal p o w er o f all o th e r h a rm o n ic s a t th e o u tp u t p o rt
resp ectiv ely . T h e sp e c ific a tio n s f o r th e co n v e rsio n g ain a n d s p e c tra l p u r ity a re 2.5dB
a n d 20dB , re sp e c tiv e ly . T h e y a re b o th lo w er sp e c ific a tio n s.
T h e la rg e -sig n a l F E T sta tis tic a l m o d el in clu d es an in trin s ic la rg e -sig n a l F E T
m odel m o d ifie d fro m th e M a te r k a a n d K a c p rz a k m o d e l(M a te rk a a n d K a c p rz a k 1985),
statistical d is trib u tio n s a n d c o rre la tio n s o f p a ra m e te rs . T h e m u ltid im e n sio n a l n o rm al
d is trib u tio n is assu m ed fo r all F E T in trin s ic a n d e x trin sic p a ra m e te rs . T h e m ean s a n d
sta n d a rd d e v ia tio n s a re listed in T a b le 5.1. T h e c o rre la tio n c o e ffic ie n ts a re g iv e n in
T a b le 5.2. U n ifo rm d is trib u tio n s w ith fix e d to le ra n c e s o f 3% a re a ssu m ed f o r P IN,
V BG, V BD, L j, / j a n d / 2. T h e n o m in a l values fo r n o n o p tim iz a b le v ariab les a re : L 2 =
15nl 1, L3 = 15nH , C j = 2 0 p F , C 2 = 2 0 p F , Wj = 0 . 1 x l 0 r 3m , w 2 = 0 .6 3 5 x l0 ~ 3m , R^OAD
= 50H, R tNPUT = 5 0 n , R bg = lOfl, a n d R BD = 10n. F in a lly , th e u n ifo rm d is trib u tio n s
w ith fix e d to le ra n c e s o f 5% a re assu m ed fox L 2, L 3, C lt C 2, Wj a n d w 2. T h e ra n d o m
n u m b e r g e n e ra to r used is c a p a b le o f g e n e ra tin g o u tc o m e s fro m th e in d e p e n d e n t a n d
m u ltid im e n sio n a l c o rre la te d n o rm al d is trib u tio n s a n d fro m u n ifo rm d is trib u tio n s .
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
Ill
Rd
C2
11
R lo a d
R input L i
I
W,
n r n
y v
w
DS
©
P
in
BG
BG
Fig. 5.4
T
C irc u it d ia g ra m o f th e F E T m icro w av e fre q u e n c y d o u b ler.
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112
5.6.2
D esign P ro ced u re
In o u r p ro g ra m , Im p le m e n ta tio n II o f th e g e n e ra liz ed t x c e n te rin g a p p ro a c h
o f (2 .4 3 ) is used.
In m o re d e ta il, th e e r r o r fu n c tio n s re su ltin g fro m th e sim u la te d
c o n v e rsio n g ain a n d sp e c tra l p u r ity a re c a lc u la te d , th e n th e se e r ro r fu n c tio n s w ith
th e ir m u ltip lie rs d e fin e d
in (2 .4 3) a n d (2.39) a re fe d
in to
th e o n e -sid e d ^
o p tim iz a tio n . / GAT c a lc u la te s a p p ro x im a te se n sitiv itie s o f th e c o n v e rsio n g ain a n d
s p e c tra l p u rity . T h e c o m p u te r used is th e M u ltiflo w T ra c e 1 4 /3 0 0 1.
T h e s ta rtin g p o in t f o r y ield o p tim iz a tio n is th e so lu tio n o f th e m in im ax
n o m in al d e sig n w .r.t. th e sam e s p e c ific a tio n s , using the sa m e six d e sig n variab les. A t
(his p o in t, th e estim ated y ield b ased on 500 o u tco m es is 39.6% .
We c o n d u c t a d esig n u sin g IGAT g ra d ie n t c a lc u la tio n . C o m p u ta tio n a l d e tails
a re g iv en in T a b le 5.3. T h e d e sig n has tw o c o n se cu tiv e p h ases, th a t is, th e s ta rtin g
p o in t fo r th e seco n d p h ase is th e so lu tio n o f th e firs t ph ase. T h e seco n d p hase is to
re o p tim iz e th e firs t so lu tio n w ith u p d a te d a ;.
5.6.3
R e su lts a n d D iscussions
U sin g IGAT , th e f irs t p h a se rea c h e s 71% y ield a n d th e sec o n d phase c o n firm s
th a t th e s o lu tio n o f th e f irs t p h a se has b e e n o p tim iz e d in te rm s o f th e e stim a te d yield.
T h e tw o p h ases use 61 o p tim iz a tio n ite ra tio n s an d 184 fu n c tio n e v a lu atio n s.
Figs. 5.5, 5.6, 5.7 a n d 5.8 sh o w h isto g ram s o f th e co n v e rsio n g ain a n d th e
sp e c tra l p u r ity , re sp e c tiv e ly . 500 o u tc o m e s a re used to c a lc u la te b o th d istrib u tio n s.
1T h is m a c h in e has th e p e rfo rm a n c e o f 10.04M FL O PS in a te st o f so lv in g 100 lin e a r
e q u a tio n s. T h e ro u tin e s u sed a re th e s ta n d a rd U N P A C K ro u tin e s. F u ll p rec isio n (64
b its) a rith m e tic w as used. T h e V A X 11/7 8 0 c o m p u te r has th e p e rfo rm a n c e o f
0 . 14M F L O P S fo r th e sam e test.
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113
T A B L E 5.3
Y IE L D O P T IM IZ A T IO N O F T H E F R E Q U E N C Y D O U B L E R U S IN G IGAT
P a ra m e te r
S ta rtin g
P o in t
N o m in al
D esig n
S o lu tio n I
S o lu tio n 11
P in (W)
V BG(V )
V BD(V )
L j( n H )
/ x( m )
2.0x1 O '3
-1 .9
5.0
5.0
l.O xlO -3
5 .0 x l0 -3
2.49048x10 '3
-1 .7 0 3 2 9
6.50000
5.29066
1.77190x10 '3
5 .7 3 0 8 7 x l0 '3
1 .9 8 4 8 8 x I0 '3
-1.93468
6.50000
5.68905
1.73378x10"3
5 .7 5 0 1 l x l O '3
1.92366x10 '3
-1 .9 2 5 4 2
6.50000
5.633822
1.73740xlQ —3
5.74907x10 - 3
39.6%
71.0%
71.0%
23
38
4450
4750
18.6m in
19.1m in
/2(m )
Y ield
N o . o f O p tim iz a tio n
Ite ra tio n s
N o . o f C ir c u it
S im u la tio n s
C P U (M u ltiflo w
T ra c e 1 4 /3 0 0 )
T h e y ie ld is e s tim a te d fro m 500 o u tco m es.
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114
30:
FREQUENCY
25
20
15
10
5
^
0
1
2
3
4
5
6
C O N V E R S I O N GAIN t d B )
Fig. 5.5
H isto g ram o f c o n v e rsio n gain s o f th e fre q u e n c y d o u b le r a t th e s ta rtin g
p o in t. 500 o u tc o m e s a re used.
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115
C O N V E R S I O N GAIN CdB)
Fig. 5.6
H isto g ra m o f c o n v e rs io n g a in s o f th e fre q u e n c y d o u b le r a t th e so lu tio n o f
y ield o p tim iz a tio n u sin g IGAT. 500 o u tco m es a re used.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
7626
li
16
18
20
22
24
26
28
30
32
S P E C T R A L P U R I T Y CdB)
F ig. 5.7
H isto g ram o f sp e c tra l p u ritie s o f th e fre q u e n c y d o u b le r a t th e sta rtin g
p o in t. 500 o u tco m es a re used.
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117
30
FREQUENCY
25
20
15
10
5
0
U
16
18
20
22
24
26
28
30
32
S P E C T R A L PURITY ( d B )
F ig. 5.8
H isto g ram o f sp e c tra l p u ritie s o f th e fre q u e n c y d o u b le r at th e so lu tio n o f
yield o p tim iz a tio n u sin g IGAT. 500 o u tc o m es a re used.
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118
Fig. 5.5 is th e c o n v e rsio n g ain d is trib u tio n b e fo re y ield o p tim iz a tio n . T h e h isto g ram
in Fig. 5.6 is b ased on th e fin a l so lu tio n . F ig . 5.7 is th e s p e c tra l p u r ity d istrib u tio n
b e fo re yield o p tim iz a tio n . T h e h isto g ra m in Fig. 5.8 is b a se d on th e fin a l so lu tio n .
( i
T h e im p ro v e m e n t in sp ectral p u r ity is v e ry c le a rly illu stra te d b y th e h isto g ra m s in Fig.
5.8.
B efore yield o p tim iz a tio n , th e c e n te r o f th e d is trib u tio n is clo se to th e desig n
s p e c ific a tio n o f 20dB , in d ic a tin g th a t m an y o u tco m es a re u n a c c e p ta b le . A f te r yield
o p tim iz a tio n , th e c e n te r o f th e d is trib u tio n is s h ifte d ito th e r ig h t- h a n d sid e o f the
s p e c ific a tio n . M o st o u tco m es th e n sa tis fy th e sp e c ific a tio n .
5.7
C O N C L U D IN G R E M A R K S
We h a v e p re se n te d a c o m p re h e n siv e ap p ro a c h to y ie ld - d r iv e n design o f
n o n lin e a r m ic ro w a v e c irc u its o p e ra tin g w ith in th e HB sim u la tio n e n v iro n m e n t. T h is
h ad b een th e f ir s t c o n v in c in g d e m o n s tra tio n o f y ield o p tim iz a tio n o f sta tistic a lly
c h a ra c te riz e d n o n lin e a r m ic ro w a v e c irc u its . O u r success is d u e to th e so p h istic a te d
c o m b in a tio n o f th e fo llo w in g a d v a n c e d te ch n iq u e s: e f f ic ie n t H B s im u la tio n using
e x a c t Ja c o b ia n s , p o w e rfu l o n e -s id e d
d e sig n c e n te rin g , e ff e c tiv e a n d ro b u s t g ra d ie n t
a p p ro x im a tio n , a n d fle x ib ility o f s ta tis tic a l h a n d lin g to allo w d if f e r e n t k in d s o f
n o n lin e a r d e v ic e p a ra m e te r sta tistic s.
In o u r d e sig n e x a m p le , th e la rg e -s ig n a l F E T
sta tistic a l m o d el is fu lly fa c ilita te d . C o m p re h e n siv e n u m e ric a l e x p e rim e n ts d ire c te d
at y ie ld -d riv e n o p tim iz a tio n o f a F E T fre q u e n c y d o u b le r v e r ify o u r a p p ro a c h .
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F A S T G R A D IE N T BA SED N O N L IN E A R C IR C U IT S T A T IS T IC A L D E SIG N
6.1
IN T R O D U C T IO N
T h is c h a p te r ad d resses fast g ra d ie n t ca lc u la tio n fo r n o n lin e a r c irc u it y ie ld -
d riv e n d e s ig n .
S tatistical d e sig n o f p ra c tic a l n o n lin e a r m icro w a v e c irc u its is a
c h a lle n g e . O n e se rio u s in h e re n t d if f ic u lty is th e p ro h ib itiv e c o m p u ta tio n a l cost: m any
c irc u its h a v e to b e sim u la te d re p e a te d ly a n d e ac h c ir c u it sim u la tio n involves C P U
in te n s iv e ite ra tio n s to solve h a rm o n ic b a la n c e eq u atio n s. F u rth e rm o re , g ra d ie n t-b a s e d
o p tim iz a tio n
re q u ire s e f f o r t to e s tim a te th e g ra d ie n ts o f th e e rro r fu n c tio n s.
T h e r e f o r e , a n e ffe c tiv e an d e f f ic ie n t a p p ro a c h to g ra d ie n t ca lc u la tio n is o f the utm o st
im p o rta n c e .
T h e c o n v e n tio n a l P e rtu rb a tio n A p p ro x im a te S e n sitiv ity T e c h n iq u e (PAST) is
c o n c e p tu a lly sim p le . Since P A ST n eed s to p e rtu rb all v aria b le s o n e a t a tim e, th e
c o m p u ta tio n a l e f f o r t in v o lv ed gro w s in p ro p o rtio n to th e n u m b e r o f variab les.
R iz z o li, L ip p a rin i a n d M arazzi (1983) u se d th is m e th o d in th e ir s in g le -lo o p a p p ro a c h
to n o m in al c ir c u it d esig n . In yield o p tim iz a tio n , h o w e v e r, P A S T becom es e x tre m e ly
in e f f ic ie n t b e c a u se o f th e larg e n u m b e r o f c irc u it o u tco m e s to b e d e a lt w ith.
T h e E x a c t A d jo in t S e n s itiv ity T e c h n iq u e (EAST) has b ee n re c e n tly d e v e lo p ed
b y B a n d le r, Z h a n g a n d B iern ack i (1 9 8 8 a, 1988b) f o r th e h a rm o n ic b alan ce te c h n iq u e .
In c o n tra s t to PAST, E A ST involves so lv in g a se t o f lin e a r e q u a tio n s w hose c o e ffic ie n t
m a trix is a v a ila b le a f te r c irc u it sim u la tio n . T h e so lu tio n o f a sin g le a d jo in t sy stem is
s u f f ic ie n t f o r th e c a lc u la tio n o f s e n s itiv itie s w ith re sp e c t to all v ariables.
No
119
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
120
■>
p e rtu rb a tio n o r ite ra tiv e sim u la tio n s a re re q u ire d . E A S T e n jo y s h ig h c o m p u ta tio n a l
e ffic ie n c y , b u t is v e ry d if f ic u lt to im p lem en t.
We in tro d u c e a p o w e rfu l a p p ro a c h to g ra d ie n t c a lc u la tio n , th e f e a s ib le A d jo in t
S e n s itiv ity T e c h n iq u e (FAST), b y B an d ler, Z h a n g a n d B ie rn a c k i (1989) a n d B a n d le r,
Z h a n g , Song a n d B ie rn a c k i (1990). M o tiv ated by th e p o te n tia l im p a c t o f th e a d jo in t
se n sitiv ity a p p ro a c h o n g e n e ra l p u rp o se C A D
p ro g ra m s w e h a v e s tu d ie d
its
im p le m c n ta tio n a l a sp e c ts. F A ST co m b in es th e e ffic ie n c y a n d a c c u ra c y o f th e a d jo in t
'\
se n sitiv ity te c h n iq u e w ith th e sim p lic ity o f th e p e rtu rb a tio n te c h n iq u e .
F A S T is
d e m o n stra te d to b e an im p le m e n ta b le , h ig h -sp e e d g ra d ie n t c a lc u la tio n te c h n iq u e .
F A ST re ta in s m ost o f th e e ffic ie n c y and a c c u ra cy o f E A S T w h ile a c c o m m o d a tin g th e
sim p lic ity o f PAST. FAST, lin k in g s t a te - o f - th e - a r t o p tim iz a tio n a n d e f f ic ie n t
h a rm o n ic b a lan ce sim u la tio n , is th e k ey to m a k in g o u r a p p ro a c h to n o n lin e a r
m ic ro w a v e c irc u it d e sig n th e m ost p o w e rfu l a v a ila b le .
F irs t, we d e sc rib e a g e n e ra l fo rm u la tio n f o r th e a d jo in t s e n s itiv ity a n aly sis
te c h n iq u e . T h e n , th is fo rm u la tio n is ap p lie d to d e v e lo p F A S T fo r th e n o m in a l d e sig n
case.
F A S T is e x te n d e d to a fo rm a p p lic a b le to s ta tis tic a l d e sig n .
D e ta ile d
co m p a riso n s am o n g FAST, IGAT, PAST, a n d E A S T a re g iv e n .
F A ST is used in y ie ld o p tim iz a tio n o f a m ic ro w a v e fre q u e n c y d o u b le r. T h e
p e rfo rm a n c e yield is in creased fro m 40% to 70%. T h is e x a m p le is th e sam e c irc u it
used in th e last c h a p te r so th a t th e c o m p a riso n b e tw e e n IGAT a n d F A S T c a n be
d ra w n . F A S T e x h ib its s u p e rio r e ffic ie n c y .
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121
6 .2
F E A S IB L E A D J O IN T S E N S IT IV IT Y T E C H N IQ U E
6.2.1
G e n e ric F o rm u las f o r A d jo in t S e n sitiv ity A n aly sis
'*-7
.1
We d e riv e a s e t o f g e n e ric fo rm u la s to c a lc u la te se n sitiv itie s o f a resp o n se in
th e h a rm o n ic b a la n c e sim u la tio n e n v iro n m e n t.
T h e se n s itiv ity o f a response w ith
re sp e c t to o n e v a ria b le , h *
d R jtt, v , h)
( 6 . 1)
dh
sh o u ld be c o m p u te d a t th e so lu tio n o f th e h a rm o n ic b a la n c e e q u a tio n , th a t is, this
s e n s itiv ity c a lc u la tio n is p e rfo rm e d s u b je c t to th e c o n stra in ts o f
F tf, V ) = 0.
F irs t, (6.1) is fo u n d to be
dR}(<f>,V,h)
3 R , ( t ,V ,h )
3 R ^ V ,h )
dh
3V
( 6 -2 )
dh
T h e n , w e d if f e r e n tia te b o th sid es o f th e h a rm o n ic b a la n c e e q u a tio n w ith resp ect to h *
IT
d F ( h K :)„ .
dh
—
a m v )
dV
av
ah
= 0.
(6.3)
N o w , fro m (6 .3 ), w e can f in d th a t
dV
d<t>k
dF
dF
av
ah
(6.4)
_ _ T - i 9F
a h
'
w h e re J is th e Ja c o b ia n m a trix d e fin e d in (5.23).
By su b s titu tin g (6.4) in to it, (6.2) can b e re w ritte n as:
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122
a/tjit, V, h) t_
Wk
3Rj(t, v, h)
If’ ° =
d*k
dRjfa'V, h) i
_
d<Pk
dV
(6 ' 5)
N ow th e th re e te rm s on th e rig h t- h a n d s id e o f (6 .5 ) sh o u ld be a v a ila b le b e fo re
c o m p u tin g th e s e n sitiv ity . F o r d if f e r e n t ty p e s o f resp o n ses a n d v aria b le s, th o se te rm s
can b eco m e v ery in v o lv ed . (It is a h ead a ch e to im p le m e n t th is a p p ro a c h in a g e n e ra lp u rp o se p ro g ra m w h ile v a rio u s responses a n d u s e r- d e fin e d responses a re allo w ed .)
B an d ler, Z h a n g a n d B iern ack i (1 9 8 8 a, 1988b) h av e p ro p o se d E A S T to c a lc u la te
se n sitiv itie s. T h e s e n s itiv ity e x p re ssio n s f o r v a rio u s e le m e n ts hav e b e e n d e riv e d an d
listed (B a n d le r, Z h a n g a n d B ie rn a c k i 1988b).
6.2 .2
F A S T fo r N o m in al D esign
T o sim p lify th e d e riv a tio n a n d to e m b o d y th e p ro c e d u re , w e c o n sid e r th e
c irc u it in Fig. 5.1 in th e last c h a p te r. T h e resp o n se o f in te re s t is th e v o ltag e a t the
o u tp u t p o rt in th e c ir c u it sh o w n in F ig. 5.2. T h e o u tp u t v o lta g e a t a c e rta in h a rm o n ic
can be c a lc u la te d fro m (5.25). We m ay w a n t to in c lu d e V^ f i ) in to (5.25),
V x{h)
W
V (/;) = - [•V(Z 1^
°
yO,D^‘) yo.liV y'OfidJ1) yofiD^h) ]
y o ,o W
)
VD{h)
V /W
• ( 6 .6 )
vBo (*>
Vb d W
We d e fin e a v e c to r fo r th e o u tp u t v o ltag e s p e c tra ,
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123
Vo (0)
V0 ( D
Vo A
V0 ( H)
We also d e fin e a v e c to r c o n ta in in g th e in p u t a n d D C bias voltages
Vt {\)
Vs A
VbgW
^ d (O)
N o w , w e can relate th e o u tp u t v o ltag e a t all h a rm o n ic s to th e h a rm o n ic b alan ce
so lu tio n V , in p u t e x c ita tio n an d D C b ia s su p p lie s
V
Vs
(6-8)
w h ere
Vo,o 6 diag {j'o o (0) j>o0 ( 1) • - • y o to(H)} ,
J'otI(0)
0
0
^ (1 )
(6-9)
0
0 y fo H )
0 y 'o jP O
(6 . 10)
an d
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
124
o
y'ojidV) y'ofio^
4 /(0
0
0
o
^ o,s A “
o
( 6 . 1 1)
From (6.8), w e have
(6.12a)
Vo = y 0 , o [ Yo.F Yo . s ]
or
y0 = [a
b
(6 .1 2 b )
]
w h ere m a tric e s A a n d B a re a p p ro p ria te ly d e fin e d .
By s p littin g th e real a n d im a g in a ry p a rts o f (6 .1 2 ), th e re a l-v a lu e d v e rsio n can
be fo u n d as
Re(J^)
Vo A
Im(F0 )
R eW ) -Im (/4) Re(Z?) -Im (S )
Im (K )
lm(/4)
Re(Ky)
R eU )
Im (R)
R e(R )
M f^ )
N ow , we d e fin e th e fo llo w in g
Re(fO
[a
b
]
Re(/1) - I m U ) R e(R ) -Im (fi)
Im (P )
Im(/1)
R e (F 5)
R e(/1)
Im (B )
lie (R )
,
(6-13)
Im(Ks )
w h ere Vs d e n o te s th e s p lit real a n d im a g in a ry p a rts in th e sp e c tra o f e x c ita tio n
v o ltag es, a n d V is a c tu a lly th e s o lu tio n to th e h a rm o n ic b a la n c e e q u a tio n s (5.22).
Then,
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
125
VQ = [A B]
V
Ac
vs
V
(6.14)
7s
has a lin e a r tra n s fe r m a trix lin k in g th e o u tp u t v o lta g e w ith
Ys an d V.
T he
c o e ffic ie n ts o f C a re fu n c tio n s o f e le m e n t p a ra m e te rs in th e lin e a r su b n e tw o rk . V
a re d e p e n d e n t n o t o n ly u p o n e le m e n t p a ra m e te rs in b o th lin e a r an d n o n lin ea r
su b n e tw o rk s , b u t also th e e x c ita tio n a n d D C b iases. O b v io u sly , Ys is d e te rm in e d by
th e e x c ita tio n voltage a n d D C b ias vo ltag es. We m ay w a n t to c h a n g e Ys to im p ro v e
th e c irc u it p e rfo rm a n c e .
T o m ake th e fo rm u la tio n 'c o n c is e , w e o n ly c o n sid e r th e d e riv a tiv e o f
VR{h) = Re{Ko (/0 )
w ith re sp e c t to
T h e c o rre s p o n d in g row in (6.14) gives
~
■
V
, .it
I'o W = [aT bT
- C
T
V
(6.15)
Vs
Vs
F ro m th is e q u a tio n , th e a p p ro x im a te d e riv a tiv e o f VQ w .r.t. 4>k can be calc u lated as
AcT
r
av
+ o, r ------
(6.16)
by p e rtu rb in g $ to <f> + A ^k^kA.
L et V be th e a d jo in t voltag es o b ta in e d b y so lv in g th e lin e a r eq u ation
_ _A
J TV=a,
(6.17)
w h e re J is th e Ja c o b ia n o f F w ith re sp e c tiv e to V a t th e h a rm o n ic balan ce so lu tio n .
S u b s titu tin g (6.17) in to th e th ird term o n th e rig h t h an d sid e o f (6.16) gives the
— T
__
V JAY.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
126
From (6.4), we have
- J A V fs A F.
N o w , w e can ex p re ss
A l'£ « [ c 7 (<j> + A^Ufc) - cT ]
(6 . 18)
/ V
+ 6T [F5 (<£ ♦ A tfafc) - v s] - V A F .
T h e in c re m e n ta l te rm A F c a n be a p p ro x im a te d by
A F « F ( ^ + A^.Ujt. , F )
(6-19)
f o r a s m a l l A<^..
E q u a tio n (6 .1 8 ) is a sp ecial case o f (6.5).
C o n sid e rin g th e d if f e r e n t e le m e n ts, (6 .1 8 ) c a n be f u r th e r e x p re sse d as
V
\c T(<i> + A <pkiik ) - c7]
V F(<f> * A <(>k uk , V ) ,
Vs
if 4>ic € lin e a r s u b n e tw o rk ;
a f
;
bT[vs{<f> +
A <j>LMk ) - F j ] - K F ( f + A ^ a fc, F ) ,
(6.20)
if 4>k e so u rc e s;
Ar _________ _
V F(<j> + A<f>k uk ,V ),
if <
j>k € n o n lin e a r s u b n e tw o rk .
T h is fo rm u la is m u ch e a sie r to im p le m e n t th a n th e c o rre s p o n d in g fo rm u la fo r
E /157' (B a n d le r, Z h a n g a n d B ie rn a ck i 1988b).
T h e fu n c tio n F{4> + &4kuk *V) is
e v a lu a te d b y p e rtu rb a tio n . T h e e f f o r t fo r so lv in g th e lin e a r e q u a tio n s (6 .1 7 ) is sm all
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
127
sin ce th e L U . fa c to rs o f th e Ja c o b ia n m a trix a r e a lre a d y a v a ila b le fro m th e fin a l
r
_
_
;!
h a rm o n ic b alan ce ite ra tio n . T h e te rm s V a n d Vs a re also a v a ila b le fro m th e h arm o n ic
+ A^u^.) an d b{<f> + A<j>k uj.) ca n be
b alan ce sim u la tio n . T h e p e rtu rb e d v e c to rs
easily c a lc u la te d sin c e th e y in v o lv e th e lin e a r su b n e tw o rk o n ly . F in a lly , th e p e rtu rb e d
e x c ita tio n s Vs(4> + A $ kuk) c a n b e e ffo rtle ssly o b ta in e d . It is c le a r th a t th e ca lc u la tio n
o f all th e te rm s in (6 .1 8 ) o r (6.20) c a n be re a d ily im p le m e n te d .
F in a lly , th e a p p ro x im a te s e n s itiv ity o f o u tp u t voltage V0 (Ji) w ith resp e c tiv e
to <
i>k c a n be c o m p u te d as
avi(h)
AF > )
ah
6.2.2
F A S T f o r Y ield O p tim iz a tio n
F o r th e /th o u tc o m e , th e fo llo w in g term s a re d e fin e d , c o rre sp o n d in g to th e ir
c o u n te rp a rts in th e n o m in al c irc u it case,
V o (A
c iA
b (A
c iA
V (A
V 0 ).
(6.22)
T h e h a rm o n ic b a la n c e e q u a tio n a n d a d jo in t sy stem a re
F t f , V{4)) = 0
(6.23)
and
[ 7 V )
] V tf)
= a { 4 ').
(6.24)
S im ila r to ICAT p e rtu rb a tio n s in yield o p tim iz a tio n , p e rtu rb a tio n s used in
F A ST a re also m ad e to th e n o m in al values <t>°. 4 a n d <f>k
a re o u tc o m e s g e n e ra te d
fro m th e u n p e rtu rb e d an d p e rtu rb e d nom inal v alu es <f>° a n d $ktpcrn resp e c tiv e ly . T h e
in c re m e n t o f th e o u tp u t v o ltag e o f th e /th o u tc o m e d u e to th e p e rtu rb a tio n is
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
128
c a lc u la ted by
I.
(6.25)
*V )
A
k.pm) - r & h \ - y
t
.
-
w h ere
(6.26)
A
.(
V(4>‘) is th e h a rm o n ic b alan ce so lu tio n o f th e /th o u tc o m e , a n d
is th e a d jo in t
!i
sy stem so lu tio n c o rre sp o n d in g to (6.17),
F o r d if f e r e n t types o f e le m e n ts, a fo rm u la sim ila r to (6.20) can be d e riv e d .
A lth o u g h w e p ro v id e d fo rm u la s to c a lc u la te th e s e n sitiv ity o f th e o u tp u t
vo ltag e, th e p rin c ip le s b e h in d o u r a p p ro a c h a re a p p lic a b le to o th e r fo rm s o f resp o n ses
o f in te re st. S in ce m an y o th e r re sp o n ses a re fu n c tio n s o f th e o u tp u t v o ltag e,
th e ir
s e n s itiv itie s c a n be c a lc u la te d fro m th e se n sitiv itie s o f th e o u tp u t vo ltag e. In n o n lin e a r
m ic ro w a v e c ir c u it d e sig n , th e p o w e r p e rfo rm a n c e is u su a lly o f m a jo r in te re s t. T h e
s e n sitiv ity in fo rm a tio n o f th e o u tp u t v o ltag e c a n be tra n sla te d in to th a t o f o u tp u t
p o w e r th ro u g h th e c h a in ru le .
6.3
C O M P A R IS O N S O F V A R IO U S A P P R O A C H E S
6.3.1
Im p lc m e n ta tio n a l C o m p a riso n s o f PAST, IGAT, E A S T a n d F A S T
P A ST a n d IGAT do n o t n e e d a n y m o d ific a tio n o f th e c irc u it sim u la to r.
P A ST is a w id ely used a p p ro a c h , b e c a u se it is very easy to im p le m e n t.
H o w e v e r, c o m p u ta tio n a l costs m a y be p ro h ib itiv e .
S u p p o se th e re a re 10 d e sig n
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
129
v ariab les in th e n o n lin e a r c irc u it. U sing P A ST to c a lc u la te th e g ra d ie n t, o n e n eed s
to p e rtu rb all d e sig n v a ria b le s an d to solve th e e n tir e n o n lin e a r c ir c u it f o r e ac h
p e rtu rb a tio n , i.e ., 10 tim es. T h e b est possible situ a tio n f o r th is a p p ro a c h is th a t all 10
s im u la tio n s use th e sa m e J a c o b ia n a n d all c o n v e rg e in o n e ite ra tio n . T h is a p p lie s to
nom inal c irc u it d e sig n . F o r y ie ld o p tim iz a tio n , a larg e n u m b e r o f sta tistic a l o u tco m es
m ay m ake P A ST p ro h ib itiv e .
T h e d is tin c t a d v a n ta g e o f IGAT o v e r P A S T is th a t IGAT o n ly re q u ire s th e
c irc u it resp o n se fu n c tio n on ce to u p d a te th e p re v io u sly c a lc u la te d g ra d ie n t fo r m ost
o p tim iz a tio n ite ra tio n s . IGAT e n jo y s th e sim p lic ity o f th e p e rtu rb a tio n m e th o d so th a t
yield o p tim iz a tio n c a n b e c a rrie d o u t w ith o u t m o d ify in g th e c irc u it sim u la to r to
c a lc u la te e x a c t d e riv a tiv e s . IGAT is v ery d e sira b le w h e n th e c irc u it sim u la to r c a n n o t
be m o d ifie d .
Both E A S T a n d F A ST re q u ire m o d ific a tio n to th e c irc u it sim u la to r.
T h e g e n e ric e x a c t a d jo in t se n sitiv ity te c h n iq u e (B a n d le r, Z h a n g an d B iern ack i
1988a, 1988b) is a c c e p te d by all c irc u it th e o re tic ia n s as th e m ost p o w e rfu l tool.
H o w ev er, to im p le m e n t it, we h av e to keep tra c k o f all a r b itr a r y lo catio n s o f v aria b le s
a n d to c o m p u te b ra n c h v oltages at all these lo cations. M ic ro w a v e s o ftw a re e n g in e e rs
h av e, to d a te , fo u n d th e se o b stacles in su rm o u n ta b le .
U si.ig F A ST , w e also need to p e rtu rb all v a ria b le s.
F o r a c irc u it w ith 10
d e sig n v a ria b le s, in ste a d o f c o m p le te ly solving 10 n o n lin e a r c ir c u its , w e o n ly e v a lu a te
10 re sid u a ls in th e fo rm o f (6.19) a n d calc u late th e p e rtu rb e d lin e a r su b n e tw o rk . T h e
so lu tio n o f a d jo in t e q u a tio n (6.17) can be a c c o m p lish e d by u sin g fo rw a rd a n d
b a c k w a rd s u b stitu tio n s . In FAST, w e co m p le te ly e lim in a te th e need to tra c k v a ria b le
lo catio n s.
We o n ly n e e d to id e n tify th e o u tp u t p o rt, w h ich is th e sim p le st ste p in
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
130
a d jo in t s e n sitiv ity th e o ry .
6.3 .2
N u m e ric a l C o m p a riso n o f PAST, E A S T a n d F A S T
We use a M E S F E T m ix e r (C a m a c h o -P e n a lo sa a n d A itc h is o n 1987, B a n d ler,
Z h a n g a n d B iern ack i 1988a, 1988b) to in v e stig a te th e a c c u ra c y a n d a c tu a l tim e
e ffic ie n c y o f FA ST (B a n d le r, Z h an g a n d B ie rn a c k i 1989). S e n sitiv itie s o f th e m ix e r
c o n v e rsio n gain w .r.t. 26 v a ria b le s w ere c a lc u la te d by th e FAST, E A S T and P A S T
a p p ro a c h e s, re sp e c tiv e ly . T h e v a ria b les in c lu d e a ll p a ra m e te rs in th e lin e a r as w ell as
in th e n o n lin e a r p a r t, DC b ia s, LO p o w e r, IF , L O a n d R F te rm in a tio n s . T h e re su lts
sh o w th a t th e F A ST se n s itiv itie s a re a lm o st id e n tic a l to th e e x a c t se n sitiv itie s, w h e rea s
th e se n sitiv itie s c o m p u te d b y P A S T a r e ty p ic a lly 1 to 2 p e rc e n t d if f e r e n t fro m th e ir
e x a c t values. T h is fa c t rev eals th a t F A S T p ro m ises to b e m u c h m o re relia b le th a n
PAST. T h e C P U tim e c o m p a ris o n show s th a t F A S T is 3 tim e s slo w e r th a n E A ST b u t
23 tim e fa ste r th a n P A S T f o r o n e c o m p le te s e n s itiv ity an a ly sis o f th e m ix er c irc u it.
T h e c o m p a riso n b e tw e e n IGAT a n d F A S T w ill b e giv en in th e fo llo w in g
se c tio n by a sta tistic a l d e s ig n e x a m p le .
6.4
YI E L D O P T IM IZ A T IO N O F A F R E Q U E N C Y D O U B L E R
T h e sam e F E T fre q u e n c y d o u b le r sh o w n in Fig. 5.1 in C h a p te r 5 is c o n sid e re d .
We im p le m e n t F A S T a n d c o n d u c t a d e sig n u sin g F A S T g ra d ie n t c a lc u la tio n in th e
sa m e e n v iro n m e n t as IGAT w as im p le m e n te d .
C o m p u ta tio n a l d e ta ils a re g iv e n in
T a b le 6.1. T h e d esig n has tw o c o n se c u tiv e p h a ses. T h e f ir s t p h ase uses 19 f u n c tio n
e v a lu a tio n s a n d g ra d ie n t c a lc u la tio n s to g iv e 70.6% y ield . T h e se c o n d p hase s lig h tly
in creases th e e stim a te d y ield to 71%, v e rify in g th e so lu tio n o f th e f ir s t p h ase. T h e
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
•'\
131
T A B L E 6.1
Y IE L D O P T IM IZ A T IO N O F T H E F R E Q U E N C Y D O U B L E R U S IN G F A S T
P a ra m e te r
S ta rtin g
P o in t
N o m in a l
D esig n
S o lu tio n I
P IN(W)
V BG(V )
V BD(V )
L j( n H )
/j( m )
/ 2(m )
2 .0 x 1 0*3
-1 -9
5.0
5.0
1.0x1 O’3
5.0x1 O’3
2 .49 0 4 8 x 1 0 '3
-1 .7 0 3 2 9
6.50000
5.29066
1.77190x1 O '3
5 .7 3 0 8 7 x l0 _s
2 .0 2 3 l 3 x l 0 '3
-1 .9 3 9 3 0
6.50000
5.71547
1 .7 3 5 3 1 x l0 '3
5.74965x1 O '3
I.9 4 4 4 4 x l0 " 3
-1 .9 2 9 2 7
6.50000
5.63312
1.74046x10 3
5.74956x10“ 3
39.6%
70.6%
71.0%
19
29
950
1450
7 .9 m in
12.1m in
Y ie ld
N o . o f O p tim iz a tio n
Ite ra tio n s
N o . o f C ir c u it S im u la tio n s
a n d F A S T A n aly ses
C P U (M u ltiflo w
T ra c e 1 4 /3 0 0 )
S o lu tio n II
T h e y ie ld is e stim a te d fro m 500 o u tco m e s.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
e ff ic ie n c y o f F A S T is w ell d e m o n stra te d . T o re a c h th e sa m e y ie ld le v e l, th e C P U
tim e u sed b y th e F A S T a p p ro a c h is m u ch less th a n th a t u sed b y th e IG A T a p p ro a c h .
F ig s. 6.1 a n d 6.2 sh o w h isto g ram s o f th e c o n v e rsio n g a in s a n d th e s p e c tra l
p u ritie s , re s p e c tiv e ly , a t th e so lu tio n o f y ie ld o p tim iz a tio n . 500 o u tc o m e s a re u se d to
c a lc u la te b o th d is trib u tio n s . T h e h isto e ra m s o f F ig . 6.1 a n d F ig . 6 .2 a re v e ry sim ila r
to th o se in F ig . 5.6 a n d F ig . 5 .8 , re sp e c tiv e ly .
T h e c e n te r c f th e s p e c tra l p u rity
h isto g ra m has b e e n m o v e d w ell a b o v e th e sp e c ific a tio n .
6.5
C O N C L U D IN G R E M A R K S
T h is c h a p te r has p re s e n te d th e c o m p re h e n siv e fo rm u la tio n o f g ra d ie n t
c a lc u la tio n o f F A ST .
C o m b in in g p e rtu rb a tio n s , a n d a d jo in t a n a ly sis te c h n iq u e s,
F A S T has s ig n ific a n tly im p ro v e d c o m p u ta tio n a l e ffic ie n c y as c o m p a re d w ith m ost
e x istin g m e th o d s. T h e
s ig n ific a n t a d v a n ta g e s F A S T o v er P A S T a n d IG AT a re its
u n m a tc h e d sp e e d s a n d a c c u ra c y , a n d o v e r E A S T a re its im p le m e n ta tio n a l sim p lic ity .
IG AT is a d e sira b le c h o ic e w h en th e c irc u it s im u la to r c a n n o t b e m o d ifie d . F A S T is
p a rtic u la rly su ita b le fo r im p le m e n ta tio n in g e n e ra l p u rp o se m ic ro w a v e C A D so ftw a re .
N u m e ric a l e x p e rim e n ts d ire c te d a t y ie ld - d riv e n o p tim iz a tio n o f a F E T
f re q u e n c y d o u b le r v e rify th e n e w g ra d ie n t c a lc u la tio n a p p ro a c h .
T h e s u b s ta n tia l
c o m p u ta tio n a l a d v a n ta g e o f IGAT a n d F A S T h a v e b e e n o b se rv e d .
S in ce m o st m ic ro w a v e c ir c u it C A D p ack ag es c u rre n tly u se d in in d u s try u tiliz e
th e tra d itio n a l p e r tu r b a tio n a p p ro a c h to e v a lu a te g ra d ie n ts , d e sig n o p tim iz a tio n
is
s e v e re ly lim ite d b y p o o r e ffic ie n c y . B ecause o f its s u p e rio r p ro p e rtie s , th e g ra d ie n t
c a lc u la tio n te c h n iq u e b ased on a d jo in t s e n s itiv ity an aly sis has a ttra c te d m u c h a tte n tio n
f ro m th e m ic ro w a v e C A D c o m m u n ity (G ilrn o re a n d S te e r 1991). T h e a u th o r stro n g ly
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
751899
0
1
2
3
*
5
S
C O N V E R S I O N GAIN ( d B )
F ig . 6.1
H isto g ra m o f c o n v e rsio n g a in s o f the fre q u e n c y d o u b le r a t th e so lu tio n o f
yield o p tim iz a tio n u sing F A ST . 500 o u tco m es a re used.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
3125
14
16
18
20
22
24
26
28
.
30
32
S P E C T R A L PURITY ( d B )
Fig. 6.2
H isto g ram o f sp e c tra l p u ritie s o f th e fre q u e n c y d o u b le r a t th e so lu tio n o f
yield o p tim iz a tio n u sin g FA ST. 500 o u tco m es a re used.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
b eliev es th a t F A ST w ill b e c o m e a n e x p e d ie n t tool to m e e t th e p ressin g n e ed f o r
e f f ic ie n t m ic ro w a v e n o n lin e a r c irc u it d esig n . O u r success sh o u ld s tro n g ly m o tiv a te
o th e r d e v e lo p m e n ts in n o n lin e a r m icro w av e c irc u it C A D , su c h as y ie ld -d riv e n d e sig n
u sing phy sicas based d e v ic e m odels a n d statistica l m o d ellin g o f m icro w a v e d e v ic e s fo r
la rg e -sig n a l a p p lic a tio n s.
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‘K.
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7
C O N C L U S IO N S
T h is thesis h as a d d re sse d y ie ld - d riv e n d esig n o f m icro w av e c irc u its u sin g
e f f ic ie n t c o m p u te r - a id e d d e sig n te c h n iq u e s. N ovel a p p ro a c h e s to a p p ro x im a tio n s to
b o th c ir c u it resp o n ses a n d g ra d ie n ts h av e b ee n d e sc rib e d . E m p h asis has b e en p u t on
th e fe a s ib ility o f th e se a p p ro a c h e s. T h is is e sse n tia l b o th in d e a lin g w ith p ra c tic a l
larg e p ro b le m s a n d a llo w in g y ie ld -d r iv e n d e sig n to e x ist iti g e n e ra l-p u rp o s e C A D
tools. O u r a p p ro a c h e s o f f e r g re a t re d u c tio n s in co m p u ta tio n a l re q u ire m e n ts, in c lu d in g
sto ra g e a n d C P U tim e .
M e a n w h ile , it is easy to im p le m e n t th e m into a new C A D
p ro g ra m s o r to in te g ra te th e m w ith e x is tin g p ro g ra m s.
T h e e f f ic ie n t q u a d ra tic m o d e lin g p re se n te d in C h a p te r 3 o ffe rs su b sta n tia l
sav in g s o f C P U tim e a n d s to ra g e th ro u g h a v o id in g re p e a te d e x p e n siv e sim u la tio n s a n d
g ra d ie n t e v a lu a tio n s. T h is te c h n iq u e c a n be a p p lie d to a b ro ad ra n g e o f a p p lic a tio n s
w h e re m an y c o m p u ta tio n a lly in te n siv e sim u la tio n s a re re q u ire d , su ch as sta tistic a l
m o d elin g , M o n te C a rlo s im u la tio n , ite ra tiv e n u m e ric a l c a lc u la tio n in v o lv in g very
c o m p lic a te d fu n c tio n e v a lu a tio n s, e tc . F o r e x a m p le , it has b een su g g ested b y R izzoli
e t al. (1991) th a t c o m p le x c o m p o n e n ts th a t can o n ly be c o m p u te d by e le c tro m a g n e tic
m e th o d s a re a p p ro x im a te d b y o u r q u a d r a tic a p p ro x im a tio n .
A c tu a l tu n in g is a lw a y s a sso ciate d w ith to le ra n c e s. In C h a p te r 4 , y ie ld -d r iv e n
d e sig n o f su c h c irc u its is d isc u sse d . By ta k in g tu n in g to leran c es in to c o n s id e ra tio n ,
o u r y ie ld - d r iv e n d e s ig n w ill ease th e tu n in g p ro cess a n d th e d e sig n resu lts p re se n t
b e tte r yield s a f te r tu n in g .
F o r v ery larg e sca le c irc u its , y ie ld -d riv e n d e sig n usin g
137
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138
g e n e ra l-p u rp o s e C A D p ro g ra m s o n a v a ila b le c o m p u te r p la tfo rm s m ay b e c o m e
u n m an ag eab le. O u r c o m b in e d a p p ro a c h p re se n te d in C h a p te r 4 p ro v id e s a n e ff e c tiv e
m eans to m eet th is c h a lle n g e . D ue to o u r w o rk , th e a p p lic a tio n o f s u p e rc o m p u te rs to
y ie ld -d riv e n d esig n task has g e n e ra te d in te re sts in m ic ro w a v e C A D a re a (R iz z o li e t
a l. 1991)
O u r a p p ro a c h e s to g ra d ie n t c a lc u la tio n in th e h a rm o n ic b a la n c e e n v iro n m e n t,
p re se n te d in C h a p te rs 5 a n d 6 , a re d ire c tly a p p lic a b le to y ie ld - d riv e n d esig n u sin g
o p tim iz a tio n te c h n iq u e s. T h e y have sh o w n v ery p ro m isin g fe a tu re s in c lu d in g h ig h
c o m p u ta tio n a l e ffic ie n c y a n d im p le m e n ta tio n a l fe a sib ility . We h av e illu stra te d h ow
these m eth o d s can b e im p le m e n te d to s u it d if f e r e n t situ a tio n s. T h is w o rk has b e e n
h ig h ly re g a rd e d by o th e r re se a rc h ers. P u rv ia n c e a n d M e e h a n (1 9 9 1 ) sta te d th a t th e
s te p fro m lin ear c irc u its to n o n lin e a r c irc u its in y ie ld o p tim iz a tio n w as f ir s t p re s e n te d
to th e m icro w av e C A D c o m m u n ity by us. T h e se m eth o d s h a v e fo u n d th e ir w ay in to
g e n e ra l-p u rp o s e C A D so ftw a re . We b e lie v e th a t th e y w ill p la y a n im p o rta n t ro le in
th e new g e n e ra tio n m ic ro w a v e c irc u it C A D s o ftw a re . R e lia b le a n d e f f ic ie n t y ie ld d riv e n te c h n iq u e s w ill also fa c ilita te a n d b ro a d e n th e n ew C A D re se a rc h .
C o m p re h e n siv e te stin g e x a m p le s h a v e b e e n p re se n te d in th is th e sis to e x a m in e
o u r new a p p ro a c h e s. T h e m e rits o f th ese a p p ro a c h e s h av e b e e n w ell d e m o n s tra te d .
T h e a u th o r has fe lt, d u rin g th e c o u rse o f th is w o rk , th a t th e issue o f h ow to
im p ro v e m a n u fa c tu ra b ility a n d yield has b e en re a liz e d to be o f e x tre m e im p o rta n c e
by m ore a n d m o re e n g in e e rs in th e M M 1C fie ld .
D e v e lo p m e n t o f re lia b le a n d
e f f ic ie n t y ie ld -d riv e n d e sig n tools has lagged b e h in d th e d e m a n d . In o r d e r to re d u c e
th is g ap , th e fo llo w in g p ro b le m s c o u ld be c o n sid e re d in f u tu r e re sea rch a n d
d e v e lo p m e n t.
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139
(a )
In th e p ro p o sed g ra d ie n t q u a d ra tic a p p ro x im a tio n sc h e m e o f C h a p te r 3 , a
c irc u it resp o n se an d its g ra d ie n t a re tre a te d as se p a ra te d fu n c tio n s . H o w ev er,
;r
th e y a re a ctu ally re la te d .
T h e re fo re , it is su g g e ste d th a t th e g ra d ie n t
in fo rm a tio n , to g e th e r w ith re sp o n se in fo rm a tio n , sh o u ld be u tilized to b u ild
th e resp o n se m odel. T h e n , c o n siste n c y b e tw e e n th e q u a d r a tic m odels o f th e
resp o n se a n d g ra d ie n t a n d h ig h e r a c c u ra c y o f th e resp o n se m odel arc logical
co n seq u en ces. A p ro b lem is th a t th e re s u lta n t fo rm u la s m a y n ot be as sim p le
as those in C h a p te r 3.
(b )
S ta tistic a l d esig n o f larg e scale c irc u its is still o n e o f th e m ost c h a llen g in g tasks
in C A D .
A p p ro x im a tio n te c h n iq u e s w ill c o n tin u e to p la y an e x tre m ely
im p o rta n t ro le in th e c o m in g g e n e ra tio n o f C A D . T h e y c an be ap p lied at the
sy stem response level, su b sy ste m lev el, a n d /o r d e v ic e an d elem en t level.
B esides som e
tra d itio n a l
a p p ro x im a tio n
te c h n iq u e s, su c h
as q u a d ra tic
a p p ro x im a tio n o f Low a n d D ire c to r (1 9 8 9 ) a n d sp lin e a p p ro x im a tio n o f B arby,
V lach a n d Singhai (1988), th e re sp o n se s u rfa c e m e th o d b ased on e x p erim en ta l
desig n th e o ry (B ox, H u n te r a n d H u n te r 1978) has b e used to g u id e sta tistica l
d esig n o f V LSI d e v ic e s (A lv a re z , A b d i, Y o u n g , W eed a n d H erald 1988,
M cD o n ald , M a in i, S p a n g le r a n d W eed 1989, a n d A o k i, M a su d a , S h im ad a, a n d
Sato 1987). A p p ly in g th is m e th o d to m icro w av e c irc u its a n d devices w ould
tre m e n d o u sly im p ro v e th e e f fic ie n c y o f larg e scale n e tw o rk y ie ld -d riv e n
desig n .
(c )
As we k n o w , th e u ltim a te goal o f d e sig n is to d ir e c t th e m a n u fa c tu rin g
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140
•' ..1
p ro cess. It is m u ch m o re m e a n in g fu l to use a c tu a l g e o m e tric sizes a n d p ro c ess
c o n tro llin g p a ra m e te rs as d e sig n v a ria b le s. P hysics b ase d m odels o f passiv e
a n d a c tiv e d ev ices a r e n e cessary to lin k th e d e sig n p ro c e d u re a n d th e
/s
m a n u fa c tu rin g p ro cess (Y o sh ii, T o m iz a w a a n d Y o k o y am a 1983, S now den a n d
L o re t 1987, K h a tib z a d e h a n d T re w 1988, a n d C u rtic e 1989). R e search in to
u sin g p h y sics based d e v ic e m o d els in d e sig n has b ee n u n d e rta k e n b y B an d le r,
Z h a n g a n d C ai (1990). S im u la tio n o f a c tiv e d e v ic e s, su ch as G aA s F E T s, is
still th e c o m p u ta tio n a l b o ttle n e c k .
In th e case o f y ie ld - d riv e n d e sig n w ith
p h y sics based m o d els, c o m p u ta tio n a l e f f o r t w ill b e ev en m o re p ro h ib itiv e .
A p p ro x im a tio n to d e v ic e level s im u la tio n sh o u ld b e d e v ise d . T h e tab le lo o k u p
a p p ro a c h (B u rn s, N e w to n a n d P e d e rso n 1983, J a in , A g n e w a n d N ak h la 1983)
a n d th e m a c ro m o d e lin g a p p ro a c h ( T u r c h e tti an d M a se tti 1983, C asin o v i a n d
S a n g io v a n n i-V in c e n ie lli 1991) c a n b e v e ry good c a n d id a te s to be a p p lie d in
m icro w av e c irc u it d e s ig n .
It is b e tte r to c o m p le te th e a p p ro x im a tio n
c a lc u la tio n in th e p re p ro c e s s in g p h a se p r io r to a c tu a l d e sig n o p tim iz a tio n .
(d )
A n o th e r u rg e n t p ro b le m e m e rg in g is s ta tis tic a l m o d e lin g o f d if f e r e n t ty p e s o f
ele m e n ts a n d d e v ic e s.
R e lia b le sta tis tic a l m odels a re key to d e te rm in in g
a c c u ra c y o f yield e s tim a te s a n d , c o n s e q u e n tly , o f y ie ld -d riv e n d esig n . T h e
m o re d if f ic u lt a sp e c t w ill be th e s ta tis tic a l m o d elin g o f d e v ices using p h y sic s
b ased m odels. R e se a rc h re p o rte d is m a in ly fo cu se d o n d ig ita l c irc u it d e v ic e s
(C o x , Y ang, M a h a n t-S h e tti a n d C h a tte r je e 1985, H e rr a n d B arnes 1986, a n d
Y u, K a n g , H a jj a n d T r ic k 1987). O n e o f th e f ir s t re se a rc h resu lts re la te d to
th e s ta tistic a l m o d e lin g o f G a A s M E S F E T s is g iv e n b y B a n d ler, B ie rn a c k i,
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141
C h e n , S o n g , Y e a n d Z h an g (1991 b).
F u r th e r re se a rc h is d e fin ite ly n e ed ed to
p ro d u c e e f f ic ie n t, co n sisten t a n d u n iq u e so lu tio n s.
(e )
P arallel p ro cessin g a n d d is trib u te d c o m p u ta tio n w ill p ro v id e us w ith ev en
g re a te r c o m p u ta tio n a l p o w er. C a lc u la tio n s re q u ire d in y ie ld -d riv e n d esign
h av e a n in h e re n t p arallel n a tu re w h ic h is h ig h ly su ita b le fo r parallel
pro cessin g (R izzo li et al. 1991).
R esea rch in to th is parallelism sh o u ld be
c o n d u c te d to e ffic ie n tly e x p lo re a v a ila b le n e tw o rk e d c o m p u te rs.
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without perm ission.
142
O
'1
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A .R . T h o r b jo r n s e n a n d S.W. D ire c to r (1973), " C o m p u te r-a id e d to le ra n c e assig n m e n t
f o r lin e a r c irc u its w ith c o rre la te d e lem en ts", IEEE Trans. Circuit Theory, vol. C T -2 0 ,
p p . 5 1 8 -5 2 3 .
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d e v ic e s u sin g sim u la te d a n n e a lin g o p tim iz a tio n ", IEEE Trans. Electron Devices, vol.
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M . V a i, S. P ra sa d a n d B, M esk o o b (1 9 9 0 ), "C o m p u te r-a id e d desig n o f m o n o lith ic
d is tr ib u te d a m p lifie rs w ith y ie ld o p tim iz a tio n ", IEEE Int. Microwave Sym p. Dig.,
(D a lla s, T X ) , p p . 3 4 7 -3 5 0 .
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Linear and Nonlinear Techniques. N ew Y ork: W iley, 1990.
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m e th o d s", Proc. IEEE Int. Sym p. Circuits Syst. (M o n tre a l, C a n a d a ), pp. 1424-1438.
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s u b m ic r o m e te r - g a te Si a n d G a A s M E S F E T ’s u sin g tw o -d im e n sio n a l p a rtic le
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vol. C A D - 6 , p p . 1013-1 0 2 2 .
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
AUTHOR INDEX
H .L . A b d e l-M a le k
1 6 - 1 8 ,2 2 ,3 9
B.L. A b d i
139
D. A gn ew
140
'6'
A .R . A lv arez
139
K..J. A n tre ic h
18
Y. A o k i
139
F. Assal
65
A .E . A tia
65
J.W . B andler
3, 8 , 12-14, 16-18, 22, 28, 2 9 , 31, 34, 35, 39, 4 0 , 52,
88, 95, 104-106, 119, 120. 122, 127, 129, 130, 140
J.A. B arby
139
J.J. B arnes
140
R .M . B iern ack i
12, 18,
3 9 -4 2 , 46, 47, 50, 52, 6 5 , 75, 88 , 119, 120,
122, 127, 129, 130, 140
G . Box
139
R.K.. B rayton
19, 20
J.L . B urns
140
E.M . B u tler
18
Q . C ai
140
D .A . C alah an
75
G . C asinovi
140
A. C aso tto
27
153
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
154
C. C e c c h e tti
75
R. C h a d h a
25
P; C h a tte rjc c
140
S .ll. C hen
3, 8 , 12, 18, 2 8 , 29, 31, 4 0 , 52, 6 5 , 75, 7 6 , 78, 88,
10 4 -106, 140
M .H . C h en
65
L.O . C h u a
11
P. C ox
140
D. Ci iss :
106-108
W. R . C u rtic e
89, 140
S. D aijav ad
65, 76, 78, 88, 104-106
S.W. D ire c to r
18, 19, 21, 20, 139
M. F erlilo
75
R. G a rg
25
R. G ilm o re
89, 153
C . G e la tt
26, 27
K..C. G u p ta
25
G .D . H ach tcl
18, 19, 21, 20
I.N. H a jj
140
J. H ald
33, 35
II. R. H er did
139
N. H err
140
D .ll. H o ccv ar
39
W. H u n te r
139
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
155
J . H u n te r
139
N .K . J a in
140
D. Je p se n
- 27
T . K a c p rz a k
92, 106, 1 10
F. K ai
27
S.M . K an g
140
B.J. K a ra fin
17, 18
W. K e tle rm a n n
16, 17, 3 4 , 35, 65, 76
M .A . K h a tib z a d e h
140
S. K irk p a tric k
26
R .K . K o b titz
18
K .S. K u n d e rt
89
P.H . L a d b ro o k e
8
N .C . Li
27
M .R . L ig h tn e r
20, 39
P -M . L in
11
A . L ip p a rin i
75, 87, 89, 1C4, 119
P.C. L iu
18
D. L o re t
140
K . M adsen
31, 3 3 -3 5 , 6 5 , 76, 78, 88 ,
S.S. M a h a n t-S h e tti
140
C . M ah le
65
R. K a in i
139
E. M arazzi
87, 89, 104, 119
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
156
_H. M asuda
139
A . M alerk a
9 2 , 106, 110
J . M cD onald
139
M .D . M cch an
18, 25
H. M eskoob
18, 27
D. N lom cith
106-108
M fs. N ak h la
140
A . N eri
75
A .R . N ew ton
S, 31, 3 3 -3 7 , 101, 140
A .M . Pavio
S, 57, 5S
D .O . P ederson
75, 140
J .F . Pinel
18, 2 4 , 4 8 , 50
1Z. Polak
IS
M .J.D . Powell
35, 105, 106
S. Prasad
18, 27
J. P u rv ian ce
18, 25, 106-108
M .L . R e n a u lt
6 5 , 75
V. R izzoli
75, 87, 89, 104, 119, 137, 138, 141
K..A. R o b e rts
18
U .L . R o h d e
8 , 57, 58
F. R om eo
27
A . R uszczy n sk i
18, 23
A . S a n g io v a n n i-V in c e n te lli
8 , 18, 2 7 , 140
S. Sato
139
<
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
F . S ev erso n
18
S. S h im a d a
139
S. S im p k in s
18
K . S in g h a l
11, 18, 2 4 ,
C .M . S n o w d e n
140
R .S. S oin
8, 18, 20
J. Song
12, 4 0 , 52,
48, 50, 139
65, 75, 88, 9 5 , 120, 140
r, 139
L . S p a n g le r
R . Spence
8, 18, 2 0 , 48, 55, 56
M . S te e r
132,
M .A . S ty b lin sk i
'
18, 23, 39, 41, 42, 4 6 , 4 7 , 52
A .J . S tro jw a s
8
K .S. T a h im
18
S. T a k a h a s h i
75
M . T o m iz a w a
139
R .J. T re w
140
T .N . T r ic k
39, 140
H.
18
T ro m p
M . V ai
18, 27
M .P . V e c c h i
26
G .D . V e n d e lin
8, 5 7 , 58
J. V lach
1 1 ,1 3 9
A . V la d im ire sc u
75
H .D . W eed
139
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
E .W e h r h a h n
48, 55, 56
F . Y am am o to
75
P. Y an g
140
S. Y e
1 2 , 4 0 ,9 5 ,1 4 0
K . Y o k o y am a
139
A . Y o sh ii
139
D .L . Y o u n g
139
T . Yu
140
Q .J . Z h a n g
1 2 , 4 0 ,5 2 , 6 5 , 7 5 , 7 8 , 8 8 , 119, 120, 122, 127, 129, 130,
140
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
S U B J E C T IN D EX
A d jo in : sy ste m , 119, 127
A c c e p ta n c e in d e x , 16
A p p ro x im a tio n ,
d y n a m ic c o n s tra in ts , 18
g ra d ie n t, 55, 78, 88 , 103, 118
q u a d ra tic , 3, 39, 7 8 , 82
o u te r, 18
s im p lic ia l, 18, 20
s to c h a stic , 18, 23
Base p o in ts, 22, 4 1 , 5 5 , 6 3 , 81
B ro y d en u p d a te , 105, 106
C e n te r o f g ra v ity m e th o d , 18
C e n te rin g ,
d e sig n , 16, 2 3 , 4 7 , SS
g e n e ra liz ed £p, 8 , 18, 28
o n e -s id e d l l% 31, 40, 52, 48, 55
C irc u its ,
an a lo g , 2
d ig ita l, 2
h y b rid m ic ro w a v e in te g ra te d (M IC ), 1
m ic ro w a v e , 2 , 75, 102, 120, 138
n o n lin e a r, 2, 87, 102, 119, 129, 135
m o n o lith ic m ic ro w a v e in te g ra te d (M M IC ), 1, 87, 138
C irc u it a n a ly sis, 1, 11
C ir c u it e q u a tio n s , 11
C irc u it re sp o n se , 4, I I , 102,
C o m p u te r-a id e d d e sig n (C A D ), I, 7, 75, 120, 137
C o n s tra in ts, 17, 36, 79
159
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Design,
n o m in a l, 7, 10, 68, 79, 87, 104, 112, 120
p e rfo rm a n c e -d riv e n , 2
s ta tis tic a l, 2 , 119
y ie ld - d r iv e n , 2, 7, 16, 27, 5 3 , 66, 87, 103, 135
D esignable v a ria b le , 53
D istrib u tio n , 2
c o rre la te d , 11
jo in t, 10
n o rm a l, 10, 58, 107, 110
u n ifo rm , 10, 110
E q u iv alen t c irc u it m o d el, 48, 57
E rro r fu n c tio n s, 12, 1 5 ,2 8 , 103, 119
F easib le a d jo in t se n sitiv ity te c h n iq u e ( / v i s r ) , - i 19, 112, 127, 135
F e a sib ility , 4, 66, 8 5 , 137
F E T m o d el, 9 , 88, 107
in trin s ic , 93
la rg e -sig n a l (n o n lin e a r), 9 2 , 107, 110
s m a ll-s ig n a l, 107, 109
sta tis tic a l, 106, 110
' G a u s s -N e w to n m e th o d , 34
G e n e ra liz e d i p fu n c tio n , 29
M adm and a n a ly sis, 18
H a rm o n ic b alan ce (H B ), 4, 87, 120
H ie ra rc h ica l s tru c tu re , 9
Im p le m e n ta tio n 1, 31, 50
Im p le m e n ta tio n 11, 3 1 ,5 5 , 112
In te g ra te d g ra d ie n t a p p ro x im a tio n te c h n iq u e ( IG AT ), 104, 112, 129, 135
J a c o b ia n , 35, 101, 121, 125
I v u h n -T u c k e r c o n d itio n s , 35
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16 i
M a n u fa c tu ra b ility , 8 7 , 138
M a th e m a tic a l p ro g ra m m in g , 28, 31
lin e a r, 19, 34
M ax im ally fla t in te rp o la tio n , 3, 39
M in im al E u c lid e a n n o rm , 42
M o d ifie d B FG S fo rm u la , 35
M o n te C a rlo a n a ly sis, 15
M u ltip le x e r, 4 , 65, 7 6 , 80, 85
M u ltip lie rs , 29, 35, 112
N o m in al v a lu e s, 10,
O p tim a l to le ra n c e a s s ig n m e n t, 17
O p tim a l to le ra n c in g , 17
O b je c tiv e f u n c tio n , 2 , 17, 27, 48, 103
O p tim iz a tio n
g r a d ie n t-b a s e d , 2, 2 3 , 4 0 , 78
o n e -s id e d £1, 3, 8 , 30, 88 , 103, 112
O u tc o m e s, 11, 16
sta tis tic a l, 16
P a ra m e tric sa m p lin g , 24
P e rtu rb a tio n , 4 3 , 104
P e rtu rb a tio n a p p ro x im a te s e n s itiv ity te c h n iq u e (P /l-S r), 104
P ro b a b ility d e n sity f u n c tio n , 10
Q u a d ra tic m o d el, 39 ,
Q u a s i-N e w to n m e th o d , 34
R a d ia l e x p lo ra tio n a p p ro a c h ,
18
R a n d o m n u m b e r g e n e ra to r, 107
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R efe re n c e p o in t, 41
//
Region,
'
a c c e p ta b le , 15
c o m b in e d , 72
to le ra n c e , 11
tu n a b le , 74
R esponse fu n c tio n s , 12, 101, 121
S am ple p o in ts, 16, 24
S e n sitiv ity , 25, 12!
S en sitiv ity an aly sis, 6 5 , 120
S im u lated a n n e a lin g , 26
S im u latio n ,
c irc u it, 11
fre q u e n c y -d o m a in , 89, 94
tim e -d o m a in , 89, 92
S p e c ific a tio n s, 12, 13, 103
lo w er, 13, 102
u p p e r, 13, 102
S ta n d a rd d e v ia tio n , 56, 107
S te a d y -s ta te , 88, 102
S u p e rc o m p u te r, 65, 74
T o le ra n c es, 10
fa b ric a tio n , 67
fix e d , 16
tu n in g , 16
T o le ra n c e a ssig n m e n t, 17
T o le ra n c e re g io n , 11
T u n a b le c ir c u it, 67
T u n ab le re g io n , 74
T u n in g to le ra n c e s, 67
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163
T w o -s ta g e a lg o rith m , 3
U p d a te d a p p ro x im a tio n s a n d c u ts , 22
V e c to r p ip e lin e c o m p u te r, 14
V e c to riz a tio n , 74
W eig h tin g fa c to rs, 15
Y ield ,
e stim a tio n , 15
p re d ic tio n , 18
p ro d u c tio n , 2
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