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Large scale optimization of analog circuits with microwave applications

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LARGE SCALE OPTIMIZATION OF ANALOG CIRCUITS
WITH MICROWAVE APPLICATIONS
>7
By
Q I-JU N ZH ANG , B.Eng.
A T h esis
Subm itted to th e School o f G raduate S tu d ies
'
In P artial F u lfilm en t o f the R equirem ents
for the D egree
f
Doctor o f P hilosophy
*V*
&
' M cM aster L n fte r sity
Jutv 1987
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LARGE SCALE OPTLMIZATION OF ANALOG CIRCUITS
WITH MICROWAVE APPLICATIONS
k
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DOCTOR OF PH ILO SO PH Y (1987)
(E lectrical E ngineering)
Mc.MASTER U N IV ER SITY
H am ilton , O ntario
TITLE:
L a rg e S c a le O p tim iz a tio n o f A n a lo g C ir c u it s w ith
M icrow ave A pplications
AUTHOR:
Q i j u n Zhang, B.Eng. (E.E.)
(E ast C hina E n gin eerin g Institute)
SUPERVISOR:
J.W . B andler, Professor, D ep artm en t o f E lec tric a l and
C om puter E n gin eerin g
B.Sc. (E ng.), P h.D ., D.Sc. (Eng.) (U n iv ersity of London)
Q.I.C. (Im perial C ollege)
P .E ng. (P rovince o f Ontario)
C .E ng., F .I.E .E . (U nited Kingdom)
*
F ellow , I.E.E.E.
F ellow , R oyal.Society of Canada
NUMBER OF PAGES:
xvi. 236
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ABSTRACT
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T h is th e sis a d d resses its e lf to com puter orien ted techniques for large scale
o p tim ization o f a n a lo g c ir c u its.
N e w te c h n iq u e s for sim u la tio n and s e n s it iv it y
a n a ly s is a r e d e s c r ib e d a n d a r e u se d to im p r o v e th e p e rfo r m a n c e o f c ir c u it
optim ization .
A pow erful a u tom atic decom position t e c h n iq u e ^ developed d irectly
en a b lin g a norm al o p tim izer to solve large circuit problem s. Our theory is applied to .
the d esig n o f m icrow ave circu its, v
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T h e sta tu s o f large sca le circu it op tim ization and the state-of-th e-art of
s
m icrow ave CAD are review ed. The n ecessity o f circuit oriented optim ization tech ­
n iq ues is d em on strated by form ulating d esign , m odelling, d iagn osis a n d .tu n in g into
o p tim ization problem s.
A com p reh en sive trea tm en t of large ch an ge sen sitiv ity com p u lation for
lin earized circu its u sin g generalized H ouseholder form uius is presented. A technique
for circu it response updatin g via a m inim um order reduced systetn is developed. By
avoid in g r e -a n a lv sis o f the com plete circu it, our m ethod is responsible for efficien t
sim u la tio n o f large circu its w hen a su b set o f the circu it param eters is freq u en tly
perturbed.
A n e le g a n t theory for sim u lation and exact se n sitiv ity a n alysis o f branched
cascaded netw orks is described. Our approach ex p licitly ta k es the circuit-structure
into consid eration and does not d eteriorate as the overall network becom es large T he
p racticality o f the th eory is illu strated by e fficien t optim ization o f m icrow ave m u lti­
p lexers co n sistin g o f multi'-cavit^ filters distrib u ted along a w a vegu id e m an ifold .
E xam p les o f op tim izin g 1 2 - and 15-channel m u ltip lexers are provided
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A novel and gen eral au tom atic decom position tech n iq u e for large sc a le
op tim ization o f m icrow ave circu its is presented. The p artition in g approach proposed
by Kondoh for FET m od ellin g problem s is verified. T he a pplication o f our technique is
d em onstrated by the large sca le op tin yzation o f a 16-channel m u ltip lexer in volvin g
399 nonlinear fu n ctions and 240 variables.
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ACKNOWLEDGEMENTS .
T he a u th o r w ish es to ex p ress h is appreciation to Dr. J.W . B andler for his
en cou ra g em en t, con tinu ed a ssista n ce, expert guidance and supervision throughout
th e course o f th is work. H e a lso th an k s Drs. K.M. W ong and F. M irza, m em bers o f his
S u p erv iso ry C om m ittee, for th eir co n tin u in g in terest and useful su g g estio n s.
T h an k s are due to Dr. A .E . S alam a o f Cairo U n iv e rsity , G iza. Egypt, who
introduced the subject o f circu it d iagn osis to the author. The work on optim ization
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tech n iq u es for m odelling, d ia g n o sis an d tu n in g is m otiva^edjjy Dr. T, Ozawa o f Kyoto
U n iv e r sity , K yoto, Japan.
The author is gratefu l to Dr. K. M adsen o f the T echnical U n iversity of
D enm ark, Lyngby, D enm ark for d iscu ssio n s of optim ization problem s and for his
in p u ts into the au tom atic decom position tech n iq u e.' T echnical d iscu ssio n s with Dr.
M .H . C hen o f TRW , Redondo B each, CA, enhanced th e author's un d erstan d in g of
m u ltip lex er d ecom position properties.
It is the author's p lea su re to acknow ledge inspiring d iscu ssion s w ith his
co lle a g u e s, S.H . C hen, Dr. S. D aijavad, Dr. W. K ellcrm ann and M. R enault.
The
d ev elo p m en t o f the theory for branched cascaded netw ork a n a ly sis is the result of
clo se coop eration w ith Dr. S. D aijavad , now w ith th e U n iv e r sity o f C a lifo rn ia .
B erk eley , CA.
■
T he fin an cial a ssista n c e provided in part by th e N a tu r a l S c ien ce s and
E n g in e e r in g R esearch Council o f Canada through G ra n ts AT239 and G 1135, the
D epartm ent of E lectrical and C om puter E ngin eerin g through a T each in g A ssistants h ip , th e M in istry o f C o lle g e s and U n iv e r s itie s th ro u g h an O n tario G rad u ate
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
Scholarship, the M inistry o f Education o f the People's R epublic o f C hina through a
G overnm ent Scholarship, is g ra tefu lly acknow ledged.
T han ks go to tH? E n g in eerin g Word P rocessin g C enter for th eir patience
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and exp ert typ ing o f th is m anuscript. •
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TA BLE OF C O N T E N T S
PAGE
ABSTRA CT
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ACKNOWLEDGEMENTS
L IS T O F F I G U R E S
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xii
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L IS T O F T A B L E S
CHAPTER 1
CHAPTER 2
2.1
i
2 .2
xv
IN T R O D U C T IO N
1
, L A R G E S C A L E C IR C U IT O P T IM IZ A T IO N - REVIEW A N O B A S IC C O N C E P T S
. 7
Introduction
‘
7
-R eview o f Large Scale C ircuit O ptim ization
7
2.3
R eview o f M icrow ave CAD
10
2.4
F orm ulation o f C ircuit D esign as an O ptim ization Problem
2.4.1
T he C ircuit M odel
2.4/2
D esign S pecifications and Error Functions
2 .4 .3
T he Coding Schem e b etw een Indices o f Error *
F unctions, R esponses and Frequency Points
14
14
16
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2 .5
CH APTER 3
3.1
~
17
C oncluding R em arks
17
O P T IM IZ A T IO N T E C H N IQ U E S F O R M O D E L L IN G ,
D IA G N O S IS A N D T U N IN G
19
Introduction
19
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•X
T A B L E O F C O N T E N T S (continued)
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CHAPTER 3
v
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OPTIMIZATION TECHNIQUES FOR MODELLING,
DIAGNOSIS AND TUNING (continued)
19
3.2
~ C ircu it O rien ted O ptim ization T ech n iq u es
3.2.1
Introduction to M ath em atical P rogram m in g
3 .2 .2 L east pth O p tim ization
3 .2 .3 Q uadratic P rogram m ing
3 .2 .4 M INM AX ^nd M INBOX A pproaches in L inearization
3.2 .5 G radient and N on grad ien t A pproaches
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20
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3.3
G eneral F orm u lation o f D iagn osis a s O p tim ization Problem s
3.3.1 Introduction
3.3 .2 C on stra in t E quation
3.3 .3 T he C u rrent/V oltage Source S u b stitu tio n Model
3.3 .4
T he C om ponent C onqection M odel
’
3.3 .5 G en eral F orm ulation
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25
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27
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3.4
D ia g n o sis U s in g the L east-Squares M ethod
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3.5
D ia g n o sis U sin g the Q uadratic P rogram m in g M ethod "
33
3.6
D ia g n o sis U sin g th e Linear P rogram m in g M ethod
35
3.7
M odelling U sin g O ptim ization M ethods
36
3.7.1 B asic F orm ulation
36
3 7 .2 L im itation s o f the B asic F orm ulation ,
37
3.7
3
R eduction o f Model P aram eters and D ecom position
A pproaches
38
3 .7 .4 M ulti-C ircuit Approach
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3.8
T u n in g U sin g O p tim ization M ethods
3.8.1 P reproduction T u n in g
3 .8 .2 P ostproduction Tuning: Problem F orm u lation
3 .8 .3 Postproduction Tuning: F u n ction al A pproach
3 .8 .4 Postproduction T uning:*D eterm inistic Approach
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3.9
E xam p les
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3.9.1 DiajfnoSTS U sin g O ptim ization: An Illu stra tiv e
E xam p le
3 .9.2 D iagn osis o f a 2S Node C ircuit
3 .9 .3 G aA s FET M odelling: M ulti-C ircuit Approach
3 .9 .4 A H igh pass F ilter E xam ple for Postproduction
T u n in g
45
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50
53
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T A B L E O F C O N T E N T S (continued)
•.
CHAPTER 3
3.1 0
3.11
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O P T IM IZ A T IO N T E C H N IQ U E S F O R M O D E L L IN G ,
D IA G N O S I S A N D T U N IN G (continued)
D iscu ssio n s
.
3.10.1 U s e o f S e n sitiv ity Inform ation
3.1 0 .2 C onvergence and P ossib le D ifficu lties U sin g
O ptim ization T ech n iq u es
PACE
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58
•
63
C onclusion
64
L A R G E C H A N G E S E N S IT IV IT Y A N A L Y S IS O F
L IN E A R S Y S T E M S
„
§&,
4.1
Introduction
65
4 .2
A S et o f G eneralized H ouseholder F o « » » la s
4.2.1
G eneralized H ouseholder Form ulas
4 .2 .2
P rop erties o f G eneralized H ouseholder Form ulas
67
67
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4.3
C om putations o f O riginal and Adjoint Linear System
R esponses C orresponding to D ifferent N um bers of
Inputs and O utputs
4.3.1
D ifferent C a ses for C om puting R esponse C hanges
4.3 .2
D iscu ssio n s
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4 .4
Large C hange S e n sitiv ity A n a ly sis o f Linearized C ircuits
SO
4 .5
A N ew F orm ulation o f V, D and W for V ariables o f RCL
CHAPTER 4
Types
4 .6
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4 .7
82
E xam ples
4:6.1
A S y stem o f L inear E quations w ith R ectangular I)
4.6 .2
An E lectrical N etw ork w ith Its M inim um Order
Reduced S ystem A chieved
4.6.3
T he C ase of E xam ple 8.1.1 o f V lach and Singhal
(1983)
S3
S3
C onclusions
93
s
SS
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CHAPTER 5
5.1
E X A C T S IM U L A T IO N A N D S E N S IT IV IT Y A N A L Y S IS
O F M U L T IP L E X IN G N E T W O R K S
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Introduction
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94
94
T A B L E O F C O N T E N T S (continued) _
PAGE
CHAPTER 5
5.2
E X A C T S IM U L A T IO N A N D S E N S IT IV IT Y A N A L Y S IS
O F M U L T IP L E X IN G N E T W O R K S (continued)
94
> Branched C ascaded N etw ork s
5.2.1 P relim in a ry D escription o f th e N etw ork
5 .2 .2 R eduction o f Ju n ctio n s to 2-port R ep resen tation s
5.2 .3 C ascaded A n a ly sis
5 .2 .4 T h ev en in E q u ivalen t C ircuits and B asic R esponses
5 .2 .5
R esponses o f In terest
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100
100
103
107
A lgorithm for C a lcu lation o f T h even in E q u ivalen ts and
T h eir S e n s itiv itie s
107
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A 4-B ranch C ascaded N etw ork E xam ple
112
5.5
C om puter-A ided D esign o f M icrow ave M ultiplexers
5.5.1 — A n a ly sis o f S p ecific M u ltip lexer Structures
5.5 .2
O ptim ization o f a 12-C hannel 12 G Hz M ultiplexer
112
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129
C oncluding R em arks
131
A N A U T O M A T IC D E C O M P O S IT IO N A P P R O A C H
T O O P T IM IZ A T IO N O F L A R G E M IC R O W A V E
SY STEM S
134
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5.6
CHAPTER^
6.1
6.2
j
Introduction
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134
The D ecom position Approach for C ircuit O ptim ization
Problem s
6.2.1
C ircuit O p tim ization Problem s
6.2 .2
G rouping o f V a ria b les and F u n ction s U sin g
S e n sitiv ity Inform ation
6.2.3 D ecom position D ictionary
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137
138
142
r
6.3
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6.4
P ractical E xam p les o f D ecom position D ictionary
6.3.1 D ecom position D ictionary for FET D evice M odels
6 3.2 D ecom position D ictionary o f a 16-C hannel
" M u ltiplexer
143
143
A utom atic D eterm in ation o f Suboptim ization Problem s
6.4.1 T h eo retica l D escription
6.4 .2 An E xam p le for D eciding on a Subproblem and
C andidate P riority
153
153
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T A B L E O F C O N T E N T S (continued)
PAGE
CHAPTER 6
A N A U T O M A T IC D E C O M P O S IT IO N A P P R O A C H
T O O P T IM IZ A T IO N O F L A R G E M IC R O W A V E
S Y S T E M S (continued)
134
. A n A utom atic D ecom position A lgorithm for C ircuit
O p tim ization
156
6 .6
Large Scale O ptim ization o f M ultiplexers
159
6.7
C oncluding R em arks
171
6.5
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CHAPTER 7
C O N C L U S IO N S
A P P E N D IX A
S O M E D E F IN IT IO N S IN G R A P H T H E O R Y
.1 7 2
177
A P P E N D IX B
B R IE F D E S C R IP T IO N O F T H E C O M P U T E R P R O G R A M
F O R S IM U L A T IO N , S E N S IT IV IT Y A N A L Y S IS A N D
O P T IM IZ A T IO N O F B R A N C H E D C A S C A D E D
NETW ORKS
178
A P P E N D IX C
B R IE F D E S C R IP T IO N O F T H E P R O G R A M F O R
M U L T IP L E X E R O P T IM IZ A T IO N U SIN G
A U T O M A T IC D E C O M P O S IT IO N
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REFERENCES
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A U T H O R IN D E X
226
S U B J E C T IN D E X
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LIST OF FIGURES
PAGE
FIG URE
2.1
A g en eral rep resen tation o f m u lti-in p u t m u lti-ou tp u t a n alog system .
3.1
An eq u iv a le n t rep resen tation o f c h a n g e s in elem e n t values.
3.2
A p a ssiv e r e sistiv e circu it a s an ex a m p le for d ia g n o sis u sin g
o p tim ization techniques.
46
E q u iv a len t cu rren t sources r ep resen tin g th e effect o f ch a n g es in Gt,
i = 1 , 2 ,..., 5, for the c ir c u ito f F ig. 3.2.
^
4S
3.4
A r e sistiv e m esh netw ork (28 nodes);
52
3.5
E q u iv a len t circu it o f carrier-m ounted FET (D evice model B1824-2QC).
54
3.6
S m ith C hart d isp lay o f sca tterin g param eters S n , S™, S 12 and S 2 1
for th e carrier-m oun ted FET, before and a fter ad ju stm en ts on
param eters.
.
55
3.7
T h e high p ass notch filter circuit.
57
3.3a
T he resp on ses for the tu n in g o f th e h ig h p a ss notch filter u sin g
functional tunin g.
•
.
6Q '
T h e resp on ses for the tu n in g o f th e h ig h p a ss notch filter u sin g
d eterm in istic tuning.
61
4.1
An arbitrary 10 node netw ork w ith 7 variab le param eters.
71
4.2a
T h e orig in a l lin ea r sy stem and its so lu tio n s.
34
4.2b
M atrices V, W, P\- and vector R H S .
S5
4.2c
R esu lts corresponding to the first ch an ge o f variable param eters
represen ted by D
.
36
4.2d
R esu lts corresponding to the second ch an ge o f variSble param eters.
S7
4.3
T op ologitol rela tio n s for the circu it o f F ig. 4.1.
39
4.4
T h e sim p le circu it from V lach and S in gh al (1983).
92
3.3
3.3b
15
*
xii
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28
\
^— »
L IST OF F IG U R E S (continued)
FIGURE
PAGE
«
5.1
T he branched cascad ed netw ork under consideration.
97
5.2
D eta il o f th e kth se c tio n o f a branched cascaded circu it sh ow in g
referen ce p lan es a lo n g th e branch.
99
5.2f
5 .4
A 3 -p o rtju n ctio n in w hich ports 1 and 2 are considered alon g a
m ain ca scad e and port 3 represents a channel or branch o f the
. m ain cascade.
101
T h ev en in and N orton e q u iv a le n tsjit reference p lan es i and j, where
referen ce p la n e i is tow ards the source w.r.t. referen ce plane j.
104
Illu stra tio n o f an a rb itrary 4-branch cascaded circu it w ith short
circu it term in a tio n o f th e m ain cascade.
113
5 .6
E q u iv a len t circu it o f a con tiguous band m u ltip lexer.
124
5.7
Com m on port retu rn lo ss and chaiinel.output port insertion loss
resp on ses o f the 12-chan nel m ultiplexer before optim ization
130
C om m on port return lo ss and channel output port insertion loss
resp on ses o f the 12-channel m ultiplexer w ith optim ized spacings.
input-ou tp u t transform er ratios, cavity resonances and coupling
param eters.
132
A fic titio u s ex a m p le sh ow in g only the strong inter-connections
b etw een v a ria b les and function groups.
141
5.5
5.8 '
6.1
a
6.2
.
.
.
A FET eq u iv a le n t circuit.
6.3
R eturn and in sertio n toss responses of the 5-channel m ultiplexer
for each su boptim ization .
160
R eturn and in sertion toss responses o f the 16-channel m ultiplexer
before op tim ization .
167
R eturn and in sertion lo ss responses o f the 16-channel m ultiplexer
after 10 su b op tim ization s.
168
R eturn and in sertion lo ss responses o f the 16-channel, m ultiplexer
at th e ov era ll solu tion.
169
Block d iagram o f the com puter program for sim u la tio n , sen sitiv ity
a n a ly sis and op tim iza tio n o f branched cascaded netw orks.
179
6.4
6.5
6.6
B .l
'
'
xut
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144
•
LIST O F FIG U R E S (continued)
FIGURE
B.2
B.3
B.4
C. 1
C.2
PAGE
M ain program and th e com puter output for the sim u la tio n and
se n sitiv ity a n a ly sis o f a 2-port elem en t.
180
M ain program , block d a ta and th e com puter ou tp u t for th e sim u la tio n ,
se n sitiv ity com putation o f th e 4-branch cascaded netw ork.
183
M ain program , block da ta and th e com puter output for op tim ization
o f a 6th order m u ltica v ity filter.
190
B lock diagram o f th e com puter program for op tim ization of
m u ltip lexers u sin g au to m atic decom position.
203
C om puter output for th e op tim ization o f the 5-ch an n el m u ltip lexer
u sin g au tom atic decom p osition.
205
*
xiv
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-4
LIST OF TABLES
TABLE
3.1
3 .2
PAGE
R e su lts o f d iag n o sis u sin g op tim ization techniques for the circu it o f
F ig. 3 .2 , c a se 1.
r
R e su lts o f d ia g n o sis u sin g optim ization techniques for the circu it of
F ig . 3 .2 , ca se 2.
49
51
3 .3
R esu lts for th e G aA s FET exam ple.
56
3 .4
E lem en t v a lu e s for th e h ig h p ass filter o f Fig. 3.7.
59
4.1
O perational count for th e gen eralized H ouseholder form u ld ?
72
4 .2
F o rm u la s for th e com putation o f la rge ch an ges w hen A - 1 is involved
and w hen rj > ro.
76
Major com pu tation al effort for ca lcu la tin g A (B T A - 'C ) by form ulas
in T ab le 4.2 w here ri > rj.
73
E xp ressio n s appropriate for com putations for se n sitiv itie s w.r.t.
com p on en ts o f m atrix A w hen A - 1 is involved.
79
4.5
P a ra m eter ch a n g es for the circu it o f Fig. 4.1.
91
5.1
V ariou s frequency res’p on ses and th eir se n sitiv itie s exp ressed in
term s o f b asic voltage response and reflection coefficient.
4 .3
4 .4
5.2
5.3
5.4
5.5
5.6
'
108
N u m erica l v a lu es of th e resp on ses for the 4-branch cascaded netw ork
o f F ig. 5.5.
.
114
S e n s itiv itie s o f branch output vo lta g es w .r.t. variable p aram eters for
th e circu it o f F ig. 5.5.
115
S e n s itiv itie s o fT h e v e n in eq u iv a len t voltage sources w.r.t. variable
p aram eters for the circu it of Fig. 5.5.
.
116
S e n s itiv itie s o fT h e v e n in eq u iv a le n t im pedances w .r.t. variable
p a ram eters for the circu it o f Fig. 5.5.
117
S e n s itiv itie s o f in sertion loss w .r.t. variable param eters for the circu it
o f F ig. 5.5.
118
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LIST O F T A B L E S (continued)
PAGE
TABLE
5.7
S e n sitiv itie s o f branch port return lo ss w .r.t. variab le param eters
for the circu it o f Fig. 5.5.
,119
S e n sitiv itie s o f com m on port return lo ss w .r.t. variable param eters
for th e circu it o f Fig. 5.5.
120
S e n s itiv itie s o f various responses w.r.t, a n gu lar frequency w for
th e circu it o f F ig. 5.5.
121
5.10
G ain slop e and group delay for th e cir c u it o f F ig. 5.5.
122
5.11
E x a m p le o f tran sm issio n m atrices for su b n etw ork s in the
m u ltip le x e r o f F ig. 5.6.
125
5 .12
F irst-order se n s itiv itie s o f th e-tran sm ission m atrices in T able 5.11.
127
6.1
P a ra m eter v a lu e s for <J>° for th e FET cir cu it m odel.
6.2
T h e C m a trix for th e FET model.
147
6.3
N orm alized decom position d iction ary D for the FET m odel-
149
6.4
D ecom position dictionary for a 16-channel m u ltip lexer, w here
v a ria b les are cou plin g param eters and tran sform er ratios.
151
5 .8
5 .9
, 146
*
6.5
6.6
C.
D ecom position dictionary, for a 16-channel m u ltip lex er, w here
v a ria b les are thofdistances from the sh ort circuit.
152
C om parison o f\6 -c h a n n e l m u ltip lexer op tim ization w ith and
w ith o u t decom position.
170
1
In terp retation o f variable indices in F ig. C .2 for the 5-channel
m u ltip lexer.
214
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1
INTRODUCTION
C ir c u it o r ie n te d o p tim iza tio n te c h n iq u e s h ave been in s t r u m e n t a l in'
ad van cin g the sta te-o f-th e-a rt in com puter-aided circu it design. D uring the past two
d e c a d e s o f a c tiv e r e se a r c h an d d ev e lo p m e n t by b oth n u m e r ic a l a n a ly s t s and
e n g in eerin g p r o fe s s io n a ls , o p tim iz a tio n te c h n iq u e s h a v e g a in ed p o p u la rity and
appreciation by circu it d esig n ers. The power o f th ese techniques is further enhanced
due to the a sto n ish in g progress m ade in tbe com puter industry. W ith th eir high speed
and v a st cap acity in d ata processing, com puters a r e - now b e in g used to o p tim ize
c ircu its w ith accu ra cies and so p h istication th a t w ere only dream s of the previously
unaided d esig n ers.
In terestin g ly en ou gh , th e am bition o f electrical en gin eers grow s as fast as
their com p utin g cap ab ility . A s e r io is advance they have made is the increased size
and com p lexity o f today's a n a lo g sy stem s.
On the o th e r ‘hand, requirem ents on the
accuracy and p ractica lity o f d esig n and m od ellin g m ethods becom e more stringent,'
w hich in turn n e c e ssita te s the u se o f sop h isticated techniques such as m ulti-circuit
m o d e llin g an d y ie ld m a x im iz a tio n .
L a rg e s c a le p r o b le m s b eco m e a c r itic a l
consequence due to the increase in both the size o f circu its and the com plexity ot
design m ethods. T he solu tion o f large circu it problem s ch a llen g es researchers even
equipped w ith up-to-date c o m p u te r s.' C onsiderable efforts by softw are en g in eers are
in ev ita b le bdfore a com plicated sy stem can be designed.
The im m ed iate difficulty w ith large scale problem s is due to th e lim itation
of com puter hardw are. P roh ib itive CPC tim es and storage requirem ent often m ake
1
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♦ ordinary CAD softw are balk a t large problem s. .A freq u en t frustration w ith large
sca le optim ization' i s j a s e d
m um . O ther difficult!
,
likelihood o f stopping at a n undesired local m in i­
:ia lly in prototype and postproduction tu n in g, are d u e to
hum an in a b ility to cope with
in v o lv in g la rg e n u m b e rs o f in d ep en d en t
ta n eeo
varia b les to be adjusted sim u lta
o u sly to m eet a sp ecified resp on se pattern over a
w ide frequency ran ge (B an d ler and Z hang 1987c).
F un dam ental to a ll circu it op tim ization procedures is an efficien t circu it
sim u lator. A v ita l m ech an ism for a pow erful grad ien t based op tim izer1is a circu it
se n sitiv ity an alyzer. T h e p relim in ary step tow ards large sc a le optim ization is the
d evelop m en t o f e le g a n t sim u la tio n and se n sitiv ity a n a ly sis tech n iq u es.
L astly and
m o st im p o r ta n tly , th e m a th e m a tic a l o p tim iz e r i t s e l f m u st be m ade c a p a b le o f
h an d lin g th e large num bers o f v a ria b les and functions.
C onsider th e effort o f so lv in g N lin ear eq u ation s w ith N unkow ns a s an
exam p le. A s N in crea ses, th e sto rage req u irem en t in crea ses q u ad ratically and the
com putational effort in crea ses cu b ically.
The trouble often asso cia ted w ith m uch
e x istin g softw are is th a T so m e sim p le but redundant op eration s, a lm o st triv ia l for
o rd in a ry p ro b lem s, b eco m e u n b e a r a b le for la r g e s c a le p r o b le m s. • T h r e e su c h
situ a tio n s are w orth serio u s con sid eration .
One s itu a tio n occurs w h en a program , d esign ed to be powerful for a comp lete circu it sim u la tio n , is used very rep etitiv ely . E xam p les can be found in se n s iti­
v ity approxim ation, d esig n , tu n in g and y ield op tim ization w here only a few e le m e n ts
or on ly a subnetw ork are frequ en tly adjusted.
The straigh tforw ard approach is to
sim p ly repeat th e en tire circu it sim u la tio n each tim e even w hen a m ajority o f circu it
su b n e tw o r k s rem a in u n p ertu rb ed
For la rg e c ir c u its, th is b eco m es e x tr e m e ly
in efficien t.
<
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J
3
T h e second situ a tio n is w h en a general circu it a n a ly sis m ethod is applied to
a sp ecia lly stru ctured netw ork.
L
O b v io u sly , a ce r ta in a m o u n t o f o p e r a tio n s and
4
sto ra g e w ill becom e redundant. F or exam p le, large m icrow ave circu its b elonging to
th e category o f branched cascaded stru ctu res should not be treated u sin g the general
n o d a l a n a ly s is m e th o d a n d th e a d jo in t n e tw o r k a p p r o a c h , u n l e s s s p e c ia l
m a th em a tica l tools, e .g ., th e sp a rse m atrix technique is used.
T h e third situ a tio n can be observed ^directly in an op tim ization procedure.
A g en era l optim izer ta k e s a ll g iv e n fu n ction s and variab les into consideration. For
large s c a le o p tim ization problem s, th is is not alw ays n ecessary. For exam p le, the^O
often e x is t w eak in tercon n ection s b etw een certain variables and functions that could
nJ .
be decoupled durin g in itia l o p tim ization stages.
C
T he realiza tio n o f the a b o v e facts h a s prom pted in v e s tig a tio n s into a
num ber o f approaches to im p rove la rg e s c a le CAD te ch n iq u es.
W& can ex p lo it
p ossib le properties o f th e p a rticu lar circu it, e.g ., physical or topological properties.
W e can u se advanced m a th em a tica l tools.
W e can rearrange C A D so ftw a re into
su ita b le form ats for sp ecia l com p u ters, e.g ., the vector processors.
From th e sim u la tio n a n d a n a ly s is p o in ts o f v ie w , la r g e sc a le circu it
problem s have been fa irly treated in th e literatu re. But from the optim ization point'
of v iew , only sp arse and loo sely rela ted m a teria ls are availab le.
T h is th e sis a tte m p ts to offer a form al treatm en t to the problem o f large
sca le o p tim iz a tio n o f-a n a lo g c ir c u its.
W e propose new a p p r o a ch e s to im prove
*
sim u la tio n , s e n sitiv ity e v a lu a tio n an d o p tim iz a tio n , w h ich are a ll e s s e n tia l for
e x e c u tin g a circu it op tim ization. T he approaches include a com p reh en sive treatm ent
o f large chan ge s e n sitiv ity com p utation for repeated c ir c u it a n a ly s is , an e le g a n t
m ethod for sim u la tio n and s e n s itiv ity a n a ly sis o f branched cascaded netw orks anc^an
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au tom atic d ecom position tech n iq ue for circu it optim zation.
T he th e sis is based on
frequency dom ain eq u iv a len t circu it m odels. T he au tom atic decom position technique
d irectly en a b les a m athem atical o p tim izer to h an d le large circu it p rob lem s.
The
technique is used together w ith our branched cascaded a n a ly sis m ethod to produce the
optim al solu tion o f a 16-channel m icrow aye m u ltip lexer in v o lv in g 240 v a riab les and
399 n on lin ear fun ction s, rep resen tin g the state-of-th e-art in cir tu it op tim ization .
•
i
T he n ex t two chapters serv e as gen eral review and introduction to som e
im portant a sp e c ts o f contem porary circu it orien ted optim ization tech n iq u es, namely,*
op tim ization tech n iq u es for d esig n , m od ellin g, d ia g n o sis and tuning. In C hapter 2, w e
review th e state-of-th e-art in large sc a le circu it op tim ization and in m icrow ave CAD.
T he d esig n cen terin g problem is form ulated as a m in im ax o p tim ization problem .
C h a p ter 3 o ffers a r e v ie w o f o p tim iz a tio n te c h n iq u e s fo r m o d e llin g ,
d ia g n o si^ a n d tu n in g (MDT) o f e le c tr ica l circu its.
A general form u latiorrof circu it
d ia g n o sis a s an optim ization problem ,is introduced.
It *is follow ed by a d e ta ile d
in v e s tig a tio n in to th r e e sp e c ific fo r m u la tio n c a se s.
O p tim iza tio n m eth o d s for
m od ellin g and tu n in g are presented and com pared w ith those for d ia g n o sis
C h a p ter 4 p r e se n ts an e ffic ie n t app roach to la rg e c h a n g e s e n s it iv it y
a n a ly s is in lin e a r sy s te m s .
T h e ap p roach is basecT'upon a s e t o f g e n e r a liz e d
H ou seh older form ulas. E fficient sch em es for com p u tin g large change s e n s itiv itie s of
a lin e a r sy ste m w ith different num bers o f inputs and ou tp u ts are d eveloped.
The
co n cep t o f re sp o n se u p d a tin g v ia s o lv in g a m in im u m order red u ced s y s te m is
in trod u cetl^ A sy ste m a tic approach to form u latin g a m inim um order reduced sy stem
for lin ea r circu its is devised.
In C h a p ter 5, w e^ d escrib e a n o v e l a p p ro a ch to th e s im u la t io n a n d
se n sitiv ity a n a ly s is o f branched cascaded netw orks.
Form ulas are d erived for such
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5
resp o n ses a s in p u t or output reflection coefficien t, com m on port and branch output
/
‘
port retu rn lo ss, in sertio n loss, gain slope and group delay. E xact s e n s itiv itie s w.r.t.
a ll v a ria b les o f in terest, in clu d in g frequency, are evalu ated . An ex p licit algorith m is
4
provided d escrib in g th e d e ta ils o f the com putational a sp e c ts o f our th e o ry .
*
Our
Approach is u sed in the op tim al design o f m icrow ave m u ltip lexers co n sistin g o f m ultiv
s
ca v ity filte r s d istrib u ted a lo n g a w avegu id e m anifold.
In C hapter 6, w e describe a powerful and general decom position technique
for op tim ization o fla r g e m icrow ave sy stem s. U sin g se n sitiv ity inform ation, variab les
and functions are sy ste m a tic a lly grouped follow ing the construction o f a decom postion
dictionary. T he o v era ll problem is a u tom atically separated into a seq u en ce of subo p tim ization s.
T he p a rtitio n in g approach proposed by Kondoh for FET m odelling
♦
problem s is verified. T he technique is su ccessfu lly tested on large sca le optim ization
o f m icrow ave m u ltip lex ers in v o lv in g 16 ch a n n els, 399 nonlinear functions and 240
variables.
W e conclude in C hapter 7 w ith som e su g g estio n s for further research.
\
T h e a u t h o r c o n t r ib u t e d s u b s t a n t i a l l y to th ’e f o llo w in g o r ig in a l
d ev elo p m en ts presented in th is thesis:
(1)
T h e use o f g en eralized H ouseholder form ulas for large change sen sitiv ity
a n a ly s is , a n d 'a com p reh ensive treatm en t to the efTicient com putation of
response ch a n g es o f lin ear sy stem s w ith different num bers o f inputs and
outputs.
(2)
A sy ste m a tic schem e for d irect form ulation o f a m inim um order reduced
sy ste m for lin ea r circu it response updating.
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(3)
-
A sim p le a lg eb ra ic trea tm en t to branched cascaded netw ork s a n d a s e t o f
form ulas for c a lc u la tin g variou s responses and th eir s e n s itiv itie s for such
netw orks.
(4)
A n alg o rith m for sy ste m a tic sim u la tio n and exact se n sitiv ity a n a ly sis o fbranched cascaded netw orks.
(5)
"
'A theory and a n alg o rith m for au tom atic d ec o m p o sitio n in ta r g e sc a le
m icrow ave optim ization-problem s.
(6)
A th eoretica l d escrip tion o f m u ltip lexer decom position p roperties and th eir
u se in op tim al d e sig n o f practical m u ltiplexers.
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*\
LARGE SCALE CIRCUIT OPTIMIZATION CONCEPTS
2.1
REVI EW AND BASIC
V
'
IN TRO DUCTIO N
^ T h e use o f op tim ization tech n iq u es in circu it design has been advocated for
over 20 yea rs. T h ese tech n iq u es are now w id ely appreciated as e sse n tia l CAD tools in
the d esig n o f a n a lo g circuits. In the ca se o f large sca le circuit d esig n , h ow ever, direct
* ’
u se o f op tim ization is only sp arsely reported in the literatu re.
In th e sam e period, the u se o f com puters in m icrow ave circu it design also
received serio u s atten tio n .
In 1969, the IEEE TRA NSAC TIO NS O N M ICROWAVE
TH EO RY A N D T E C H N IQ U E S S p e cia l Issu e on C o m p u ter-O rien ted M idrow ave
P ractices sum m arized the early d evelop m en ts in this area.
S in ce then, exten siv e
research h as been perform ed resu ltin g in various su ccessfu l CAD te c h n iq u e s for
a n a ly sis, m od ellin g and d esign.
S op h isticated m icrow ave CAD so ftw a r e is being
m arketed and used.
In th is ch a p te r , w e rev ie w th e sta te -o f-th e -a r t in targe sc a le c ir c u it
o p tim ization and in m icrow ave CAD. T he basic •m athem atical form ulation o f circuit
d esig n as an op tim ization problem is introduced.
2.2
REVIEW OF LARGE SCALE CIRCUIT OPTIMIZATION
r
T em es and C alahan (1967) are am ong the ea rliest to form ally advocate the
u se o f ite r a tiv e optim ization in circuit, d esign .
W aren, Lasdon and Suchm an i^ 9 6 7 )
illu strated th a t optim ization m ethods can be applied to a wide range of engineering-.
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problem s. T h e.in trod u ction o f adjoint netw ork m ethods for c ir cu it s e n sitiv ity com ­
putation by D irector and R ohrer (1969) facilitated t h e u s e .o f pow erful grad ien t based
optim ization m ethods in circu it d esign . Bandler (1969, 1973) sy ste m a tic a lly treated
the form ulation o f error fu nction s, the le a st pth objective, n o n lin e a r c o n str a in ts,
optim ization m ethods and circu it s e n sitiv ity a n a ly sis. F urther a c tiv itie s in th is area
are summarize^! by D irector (1971), C haralam bous (1974), B andler and Rizk (1979),
B ravton, H achtel and S a n g io v a n n i-V in cen telli (1981). T he recen t review paper of
B andler and Chen (1987) addressed a variety o f circu it op tim ization tech n iq u es for
r e a listic d esig n and m od ellin g problem s. D ennis Jr. (1984) gave electrical en gin eers a
u ser’s gu ide to n on lin ear op tim ization algorith m s.
c
T he ev o lu tio n o f very large sca le in tegrated circu its (VLSI) prom pted the
research in so lv in g lar^e sca le problem s. T he im m ed iate work is to sim u la te and to
an a ly ze su ch circuits.
O ne major effort in th is area has been th e develop m en t of
netw ork p artition in g and variou s tea rin g m eth o d s'(e.g ., S a n g io v a n n i-V in c e n te lli.
Chen and C hua 1977). A nother effort is to use vector com puters (e.g., C alahan and
A m es 1979; Y am am oto and T a k ah ash i 1985) and parallel processors (e.g.,.H u an g and
W ing 1979).
Sparse m atrix tech n iq u es have been often involved (e.g., H uang and
W ing 1979). More d e ta ils o f large scale sim u la tio n are a v a ila b le in S'ew ton (19S1),
H achtel and S a n g io v a n n i-V in cen telli 1 1981) and Pederson (1984).
O ther w orks done for large sca le circu its are th e com putation o f p oles and
zeros (W ehrhahn R. 1979) and y ield estim a tio n (D ow ns, C ook an d R ogers 19S4).
L arge sca le n etw orks is also considered from the graph theory point o f view , e.g..
B oesch (1976).
Sp arse m atrix tech n iq u es and decom posittt^f'techniques are two powerful
m ath em a tica l tools for so lv in g large scale problem s. Both tech n iq u es take ad vantage
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
o f th e fa ct th a t a s a problem b e co m e s la r g e , m ore w eak in te r a c tio n s b etw een
subproblem s are lik e ly to occur.
T he sp arse m atrix tech n iq u es are used to avoid
u n n ecessa ry sto ra g e and com putation o f zero com ponents o f a m atrix. W ork th th is
a rea can be found in books o f Reid (1 9 7 1 ) an d D u ff (1981).
T h e. d eco m p o sitio n
tech n iq u es a re g en era lly used to so lve decoupled subproblem s sep arately and then to
solv e th e overall problem u sin g th e know lege o f subproblem solu tion s. An ex cellen t
su rv ey o f d ecom position for large scale problem s is provided by H im m elblau (1973).
L arge s c a le m ath em a tical program m ing becam e an im p o rta n t topic for
op eration s resea rch ers in th e I960'?. A major inspiration in th is fie Id-has been the
discovery o f th e decom position p rinciple by D an tzig and W olfe (1960). T he work of
G eoffrion (1970) and Lasdon (1970) sum m arized th e major p ion eerin g a c tiv itie s in
th is field.
More recen t rev iew s are a v a ila b le in H aim es (1982) and Luna (1984).
M any people h a v e used decom position approaches, e.g.. Bunch and”K aufm an (1981),
Shapiro and W hite (1982), Borison, M orris and Oren (198-4) and M andakovic and
Souder (1985). O thers used sp arse m atrix tech n iq u es (M urtagh and Saunders 1978.
1
C olem an 1984), m atrix sp littin g (O'Leary 1981), recursive' quadratic program m ing
(B ig g s and L aughton 1977) and dual optim ization m ethods (T em plem an 1979)
A lthou gh the sim u la tio n and a n a ly sis o f large scale an alog circu its have
r
been fairly treated in the lite r a tu r e , th e d irect optim ization of such c ircu its how ever,
is a m uch open subject. B andler, C hen, D aijavad, K ellerm an n , R enault and Zhqng
(1986) su ccessfu lly optim ized a 16-channel m icrow ave m u ltip lexer in volvin g as many
as 240 n on lin ear d esig n variables. The first form al attem pt to large sca le m icrow ave
optim ization is m ade by B andler and Zhang (1987a) who developed an a u to m a tic
d ecom position techn iqu e for d evice m odelling and large circu it design.
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2.3
REVIEW O F MICROWAVE CAD
The ea rly sta g e s o f com puter-oriented m icrow ave practices can be repre­
sented by the com m en ts o f G etsin g er (1969).
A s he pointed out, in th e late I960*s,
w ithin m icrow aves, th e electro m a g n etic field a n a ly sts V e r e th e group m o st fu lly
converted to th e com puter. In m icrow ave circu its, old d esig n m ethods w ere adapted to
th e com puter and new d esig n approaches w ere d evised .
C om puter program s w ere
used to do such th in g s a s a n a ly z e a r b itra r y m icr o w a v e c ir c u its, d e sig n filte r s ,
tra n sisto r am p lifiers and oth er com ponents.
B an dler (1974) e d ite d th e seco n d S p e c ia l Issu e o f C om p u ter-O rien ted '’
M icrow ave P ra ctices o f th e IEEE TR A N SA C T IO N S ON M ICROW AVE T H E O R Y
A N D T E C H N IQ U E S. A w ide range o f opinions h eld by contributors to the field as
w ell as u sers w ere revealed from the panel d iscu ssio n on the sta tu s o f com p u teroriented m icrow ave p ractices (C erm ak, G etsin ger, L eake, Vander V orst and Varon
1974). T he gap b etw een num erical tech n iq u es and real 'en g in eerin g problem s w as
brought into serio u s consideration.
R igorously d erived m ethods for d esign rather
than only for a n a ly sis w ere in creasin gly used (B andler 1974).
W ithout a ttem p tin g to trace a ll h istorical d e ta ils o f m icrow ave CAD, here
we “devote m ore a tten tio n to th e current state-of-th e-art in th is area.
A w id e range o f m icrow ave circu its can be covered a s candidates for th e tool
o f CAD. A s sta ted by HofTman (1984), e sse n tia lly it does not m atter very m uch if the
circuit is p a ssiv e or activ e, lin ea r or n on lin ear, w h eth er it is a\ sin g le com ponent or a
su bassem bly
A lso, th e frequency o f operation o f th e c ir c u it is in g e n e r a l n ot a
sig n ifica n t param eter.
A tr e a tm e n t o f th e su b ject a r e a is a lso co m p iled in to a
textbook by G upta, G ary and Chadha (19S1).
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
11
T he contem porary and recognized industry-standard in m icrow ave CAD
can be rep resented by th e com m ercially availab le softw are SU PER -C O M PA C T (1986)
and T O U C H STO N E (1985).
SU PER -C O M PA C T p ro v id es th e n e c e ssa r y tool for
choosin g a d esig n topology and a c tiv e elem en ts, sy n th e siz in g a m p lifie r m a tch in g
circu its, and ca n optim ize com p lete circu its h a vin g a s m any as four extern al ports.
C ir c u it e le n ie n ts include lum ped, m icrostrip and strip lin e, filter, thin film , coupler
and a ctiv e d evices. SU PE R -C O M PA C T can run under m ainfram e operating system s
su ch a s V A X /V M S, H P -U X (U N IX ) and IBM/CMS. It can also be run sim u ltan eou sly
b y mdife th a n o n e u se r , i.e ., a s a tim e sh a r in g so ftw a r e.
On th e o th er hand,
T O U C H S T O N E (19 8 5 ) is th e m o st ad van ced so ftw a re for R F /m icrow ave C A D ,
r u n n in g on personal com p u ters su ch as IBM PC-XT, AT and IBM -com patibles. With
a com plete ele m e n t and m ea su rem en t catalogu e, T O U C H ST O N E can be treated as a
laboratory in stru m en t to s e t up a m icrow ave c ir c u it and to perform sim u la tio n ,
op tim iza tio n and tun ing.
O ther com m ercially a v a ila b le m icrow ave CAD softw are
a lso e x ist, e.g., CIAO (1985) for circu it a n a ly sis and o p tim iz a tio n and CADEC+(1987) for-com puter aided d esig n o f electronic circu its, both ru n n in g on d esk top
com puters. M IDAS (1987) is a m icrow ave/R F CAD program incorporating a N etw ork «
D escrip tive L anguage, w hich a llo w s the use of algeb raic ex p ressio n s to d efin e any
v a lu e s for netw ork a n a ly sis. A llen an d M edley Jr. (1980) developed a set o f network
a n a ly sis program s for m icrow ave circu it design u sin g program m able calculators.
Today's research o f m icrow ave CAD c o n tin u e s to be a c tiv e in a broad
su bject area. T opics o f major in terest include m odelling o f activ e and passive m icro­
w a v e c o m p o n e n ts ( e .g ., S a lm e r 1987; B a n d le r , C h en and D a ija v a d 1 9 8 6 b ).
ch ara cteriza tio n and m o d ellin g o f tran sm ission stru ctu res and d isco n tin u ities (e.g..
P ram an ick and B h artia 1986; K oster and J a n sen -1986), lin ear and nonlinear a n a ly sis
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12
o f d ev ices and circu its (e.g.. R izzoli and Lipparini 1985: G ilm ore 1986; C urtice 1987),
large sca le num erical sim u la tio n and d esig n o f d evices and c ir cu its (e .g ., R izzoli,
F erlito and N eri 1986; B a n d ler an d Z h a n g 1987a) ar.d o p tim iz a tio n te c h n iq u e s
applicable to m icrow ave C A D (e.g ., B andler, K ellerm an n and M adsen 1985; Bandler
and C hen 1987).
In addition to th e popular frequency dom ain, fixed topology and
eq u iv a len t circu it'm od el d e sc r ip tio n o f m icro w a v e c ir c u its , m eth o d s u s in g tim e
dom ain (Sobhy and H osny 1981), ch an gab le topology (D ow son 1985), and physical
d evice m odels (Snowden 1986) are also developed. B esid es th e sc a tter in g p aram eter
approach used in softw are su ch a s SU PER -C O M PA C T (1986) and T O U C H ST O N E
(1985), th e w ave a n a ly sis approach h as been adopted in C AD p rogram s for n o ise
a n a ly sis o f interconnected m u ltiport netw orks (K an aglek ar, M cIntosh and B ryan t
1987). In Europe, e x te n siv e research in m icrow ave CAD is cu rren tly underw ay as
evid enced by the survey paper o f Gardiol (1986) and by th e IEE PR O C EED IN G S-H
Special Issue on C om puter-A ided D esign o f M icrow ave C ircu its ed ited by P en g ellv
*
(1986).
In d ealin g w ith problem s o f large num erical size , R izzoli, F erlito and N eri
(1986) exp loited the hardw are cap a b ility of vector processors (supercom puters) such
a s the C ray X-MP. They p resented a possible approach for vectorization o f m icrow ave
C A D program s.
As opposed to th e con ven tion al wav o f p rocessin g a circu it via a
seq u en ce o f sin g le frequency a n a ly sis, th ey perform a sin g le m u ltifreq u en cv a n a ly sis
o f th e circuit.
Common com pu tational op eration s o f th e c ir c u it a t d iffe r e n t fr e ­
q u en cies are exploited.
w ere obtained.
A ccording to th eir report, speed up factors o f the order o f 50
H ow ever, the m em ory requirem ents o f th is m ethod are in creased
sig n ifica n tly .
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B an d ler and Z hang (1987a) treated large sca le m icrow ave C AD problem s
by ex p lo itin g advanced m a th e m a tica l to o ls.
T h eir approach u se d an a u to m a tic
t
decom position sch em e. Large sca le op tim ization problem s w ere solved u sin g ordinary
m ain fram e com p u ters w ith m em ory lim ita tio n s w ith in reasonable com puter tim e.
T h eir m ethod is effectiv e on m icrow ave circu its havin g decom position properties.
O p tim iz a tio n m e th o d s a r e n ow c o n sid e r e d im p o r ta n t t o o ls in th e
m icrow ave C A D com m unity. 'T h e su rv ey papers by Bandler 1 1969), C haralam bous
(1 9 7 4 ) a n d B a n d le r an d C h en (1 9 8 7 ) su m m a r ize d m a th e m a tic a l p r o g r a m m in g
m ethods for so lv in g m icrow ave circu it d e sig n problem s. O ptim ization m ethods were
u sed in in teg ra ted d esig n cen terin g , toleran cin g and tu n in g (B andler, Liu a n d Trom p
1976b), in d ev ice m o d ellin g (B and ler, C hen and Daijavad 1986b) and postproduction
j
^
tu n in g (B an d ler and S a la m a 1985b) o f m icrow ave devices. A s a gen eral C AD tool,
o p tim ization tech n iq u es h ave a lso be£n used for d iagn osis (e.g., B andler and Zhang
1987b) and y ield m axim ization (e.g., H ocevar, L ightner and Trick 1984) o f electrical
circu its. M ath em atical p rogram m ing tech n iq u es involved ranging from th e random
op tim ization m eth od (T O U C H S T O N E 1985) w hich d oes not u se a n y d e r iv a tiv e
in fo r m a tio n , to v a r io u s g r a d ie n t m eth o d s u sin g e ith e r a p p ro x im a ted g r a d ie n t
(B andler, C hen, D aijavad and M adsen 1986), or exact first-orc^r d er iv a tiv es I Bandler.
K ellerm an n and M adsen 1985), or exact second-order d eriv a tiv es tlob ost and Zaki
1982). "Particularly, the m in im ax (H ald and M adsen 1981) and th e f i (H a ld and
M adsen 1985) op tim ization a lg rith m s developed by Hald and M adsen o f the T echnical
U n iv e r s ity o f D en m a rk h a v e b een very p ra c tica l for m o d e llin g an d d e s ig n o f
m icrow ave circu its (K ellerm an n 1986; D aijavad 1986).
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
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14
2.4
F O R M U L A T IO N O F C IR C U IT
D E S IG N A S A N
O P T IM IZ A T IO N
PROBLEM
B a sica lly , a d esig n problem is to find a s e t o f d esign ab le param eter va lu es
which let the circu it resp ose or perform ance o p tim a lly m eet som e g iven sp ecification s
(B andler and Rizk 1979). In th is section , the circu it d esig n problem is form ulated. .
2.4.1
T he C ircuit M odel
In com puter-aided d esig n , a circu it is u su a lly described by a m a th em atical
model. Let
4> — (<Pl <t>2---<J>nIT
represent th e d esig n param eters.
(2.1)
T he circu it resp on ses Fi<, k = 1, 2, ..., np, are
functions o f p a ram eters <J>and o f other in dependent v a ria b les »p, i.e.,
— Fi<(<}>, ip).
(2.2)
Fig. 2.1 d ep icts a g en era l circu it w ith m u lti-in p u ts and m u lti-outputs. T he resp on se
functions
F^ arc ev a lu a ted or m easured at output, p orts
voltage, cu rren t, in sertion loss,
and can
r e p r e se n t, e .g .,
return loss, group d elay and S p a r a m eter s.
Tw o
responses, e .g ., Fj and F^ are d istin gu ish ed eith e r by tw o d ifferen t output ports or by
two differen t typ es o f resp o n ses at the sam e port or by a m ixtu re o f both. T he circu it
topology is u su a lly fixed. T h e design param eters <}> can be accessed eith e r d irectly
(physical param eters), e .g .. len gth of.a w avegu id e, or indirectly (m odel param eters),
e.g., coupling p a ram eters o f a cavity filter. T he in d ep en d en t v a riab les ip rep resen t,
e.g., frequency, tim e, tem pretu re, etc. T he functions F ]<(<{>, ip) are assu m ed con tin u ou s
• in the ra n ges o f <{» and ip o f in terest. Perform ance sp ecifica tio n s are u su a lly fu n ction s
o f ip only (B and ler and Rizk 1979).
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INPUT
PORTS 1
OUTPUT
PORTS
LARGE
ANALOG
SYSTEM .
(a)
Fig. 2.1
(b )
A g e n e r a l r e p r e se n ta tio n o f m u lti-in p u t an d m u lti-o u tp u t a n a lo g
sy stem . F^, k = 1 ,2
np are resp on ses b ein g m easured, m onitored or
used as ou tp u ts subject to d esig n sp ecific a tio n s.
D ifferen t ty p e s o f
resp on ses (e.g., vo lta g e, current, return loss, insertion loss, S param ­
eters) m ay e x ist at the sam e output port, (a) system rep resen ta tio n
(b) resp on ses corresponding to each output port.
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16
/
2 .4 .2
D esign S p ecifica tio n s and Error F unctions
In th is th e sis, w e w ill tr ea t c ir c u it p ro b le m s m a in ly in th e freq u en cy
- - -
r
dom ain. In such a caseT"the in dep en dent variab les qi are com m only su b stitu ted by a
frequency param eter co. - Let S t'k(») and Suc(w) rep resen t th e upper and the low er
sp ecifica tio n s for the resp onse Fk(<}>, to), resp ectiv ely , k = 1 ,2 ,..., np. Let ca;, t = 1 ,2 ,
..., nu be a se t o f frequency p oints sam p led in th e frequency range o f in terest. In an
optim al d esig n problem , the ob jective function u su a lly in v o lv e s a se t o f n onlinear
error fun ction s fj(<|>), j = 1,2
m.
T y p ic a lly , th e error fu n c tio n s r e p r e se n t th e
w eigh ted differences b etw een circu it responses and g iven sp ecifica tio n s in th e form
^
a);) — Sufc(tOf)),
-
a);) - SLk(<i);)) ,
k € { 1 ,2 ,..., nF} ,
f € { l,2
n j,'
(2.3a)
(2.3b)
(2 .3 c ),
(2.3d)
w h ere w^k and w^k a r e ^ n o n -n e g a tiv e-) w e ig h tin g fa cto rs for u p p er and lo w er
sp ecification s, resp ectiv ely .
Let J be an index set d efin ed as
J = { 1 ,2 ,..., m } .
(2.4)
M(<<J>) = m ax fj(<t>).
j€ J
(2.5)
Let
T hen th e sig n o f Mf{<}>) in d icates w h eth er the sp ecification s are sa tisfied or violated.
As d escribed by B and ler and Rizk (1979), if
> 0 the sp ecification s are v iolated ,
M(<4>) s = 0 the sp ecification s are ju s t m et,
> ,< 0
th e sp ecification s are sa tisfied .
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17
/
ft
An optim ization approach to the circu it d esign problem is to find
such
th a t Mf<<J>) is m inim ized. T h is corresponds to the effort to m eet (w hen sp ecification s
are v io la ted ) or to exceed (w hen sp ecification s are satisfied ) d esig n sp ecification s as
m uch a s possib le. M a th em atically, th is is a m inim ax problem w here the m axim um of ■
a ll error fu nction s is m inim ized. A practical solver to th e m inim ax problem is the 2sta g e a lg o rith m developed by H ald and M adsen (1981).
t
2 .4 .3
T he C oding S ch em e B etw een Indices o f Error Functions. R esponses and
F requency P oin ts
'
T here e x ists a coding sch em e rep resen tin g the one-to-one correspondence
l / b etw een th e index o f fj and the in d ices o f the pair (F k, u>f) for the error functions of
b oth (2 .3 a ) an d (2.3b ).
W e d e fin e w e ig h t in g fa cto r m a tr ic e s W jj (for upper
sp ecification ) and W[_ (for low er sp ecification ). Both m atrices are np by nu
O th com ponent o f
The (k.
W^- and W L-a r e th e w e ig h tin g fa cto rs w ^ u i f ) and w ^ c o f ) ,
resp ectiv ely , wuk(oj^) or w ^ c o f ) is zero if no upper or low er specification is imposed
on F k(<J>, u f). T he coding sch em e r ela tin g the index of fj to the indices o f nonzeros in
W l' and W l are constructed by sy ste m a tic a lly scan n in g through Wjj and then W L.
resp ectiv ely (B andler and Zhang 1987c).
2.5
C O N C L U D IN G REM ARKS
In th is chapter, we provided a review o f large sca le circu it optim ization and
o f m icrow ave CAD. F airly treated in the litera tu re are the sim u lation and se n sitiv ity
a n a ly sis o fta r g e sca le circuits. T he op tim ization o f su ch circu its rem ain s a much open
subject.
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•i
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^
18
A lso in th is chapter, circu it d esign has been form u lated a s a m in im a x
op tim ization problem . Such a form ulation w ill be used for circu it d esig n problem s
throughout the th esis.
\
V
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X
3
i
>
OPTIMIZATION fECHNIQUES FOR MODELLING, DIAGNOSIS AND
»
*
TUNING
3.1
INTR O D U C TIO N
T h is chapter d ea ls w ith the ap p lication o f o p tim iz a tio n te c h n iq u e s for
✓
m o d ellin g, d ia g n o sis and tu n in g (MDT) o f electrical circuits. A conventional in ter­
p retation o f such tech n iq u es for m odelling and d iagn osis is the d e te r m in a tio n o f
Appropriate netw ork p aram eters lead in g to the b e st m a tch hot w een circuit responses
and m easured data. W hen the m ea su rem en ts are in sufficient to evalu ate all network
e le m e n ts, th e m o st lik ely fa u lts m ay be located. O th erw ise, if the m easurem ents are
♦
su fficien t, p aram eter id en tifica tio n is in itia ted , resu ltin g in a circu it mode! whose
perform ance b est fits the m easu rem en t data in the p resence o f u n ce rta in ties and
noise. Gfiosely related is the tu n in g problem w hich has been approached m ostly from
the optim iziftion point o f view . E x istin g softw are for m ath em atical program m ing can
be read ily exp loited in th is case.
T h is chapter is based on the work o f B andler and Zhang (1987b).
The
.1
p resen ta tio n is tu to ria l, b u tJ1 S » ig n ed to be h elp fu l for a sta te -o f-th e -a rt u n d e r ­
stan d in g.
We first review circu it oriented op tim ization m ethods with em p h a sis on
a s p e c ts im p o rta n t to MDT
A g e n e r a l fo rm u la tio n o f cir c u it d ia g n o s is a s an
optim ization problem is introduced.
It is follow ed by a detailed in vestigation into
th ree specific form ulation ca ses. O ptim ization m ethods for m odelling and tu n in g are
p resen ted and com pared w ilh th ose for d iagn osis. Illu stra tiv e exam ples are provided.
v j
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3.2,
CIRCUIT O R IEN TED OPTIMIZATION T E C H N IQ U E S
O ptim ization m ethods h a v e played an im portant role in com puter-aided
d esig n o f circu its ah& sy ste m s (see, e.g ., Bandler and Rizk 1979; B rayton, H achtel and
/
S a n g io v a h n i-V in c e n te lli 1981; B a n d ler and C hen 19871.
T y p ic a l c ir c u it d e sig n
objectives are to sa tisfy o f to exceed d esign sp ecification s a s m uch as possible. The
MDT problem s, how ever, are u su a lly orien ted eith er tow ards (response) d ata fittin g
or tow ards "param eter fitting" or a com bination o f both. T he "param eter fitting" can
be interp reted as forcing p ara m eters to approach a d esired pattern. Such a pattern is
constructed to best represent;
I)
' a n e s t im a t io n o f th e p a r a m e te r s , e .g ., r e s u l t s fr o m a d e lib e r a t e
perturbation to th e circu it (for more m easu rem en t inform ation), a projected
target param eter point for tuning;
2) /
an a ssu m p tion o f th e c ir c u it p h ilo so p h y , e .g ., ty p e o f fa u lts , w h e th e r
catastrophic or soft;
3)
a criterion for o p tim a lity , e .g ., th e o b jectiv e for m in im u m p a ra m eter
*
adjustm ent in tuning.
3.2.1
Introduction to M athem atical Program m ing
An o p tim ization problem cam be stated as
(3.1a)
m inim ize U(<{>)
<t>
subject to con strain ts
g(<J>)2:0
(3.1b)
and
h(<{>) = 0 ,
w here <J> i tcf>i
...
•
<t>niT. g £ Igi g-j ■• • gnclT, and h i [h! h2 . . .
(-3.1c)
hnJT.
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21
W hen U , g a n d h are a ll lin ea r functions of<}>, (3.1) is a lin ear program m ing
problem (LP), read ily solvable by th e sim p le x m ethod (se e L u en b erger 1934), a
c la ssic a l a p p ro a ch b ein g cu rren tly ch allen ged by Karm arkar's algorithm (K arm arkar
1984).
V '
'
^
To h an d le th e n on lin ear program m ing problem (N L P), i.e., the nonlinear
ca se o f (3.1), a variety o f m ethods h ave been developed. T he unconstrained N L P can
be so lv e d by c o n ju g a te d ir e c tio n m e th o d s a n d q u a s i-N e w to n m e th o d s.
The
con stra in ed N L P can be h a n d led u sin g , e .g ., p en a lty and b arrier m eth o d s, and
a u g m en ted L agrangian m ethods.
A sy ste m a tic trea tm en t to (3.1) can be found in m any text b ook s, c g..
L uen berger (1 9 8 4 ),, A com p reh en sive exam in ation o f optim ization from the circuit
d e sig n point o f view is provided in e .g ., T em es and C alahan (1937); Bandler
1 1973).
C h a r a la m b o u s (1 9 7 4 ); D ir e c to r (1 9 7 1 ); B r a y to n , H a c h te l and S a n g io v a n n iV in c e n te lli (19S1); B andler and C hen (1987).
In th is section, we h ig h lig h t th ose
a sp ects o f op tim ization which are relev a n t to MDT.
3 .2 .2
L east pth O ptim ization
A f r e q u e n t ly e n ^ u r i t e r e d
o b je c tiv e
U(t{>) is th e
p th
norm
of
ft<J>) = [fi(<J>) f2(<i» . . . fm(<l»IT. i.e..
/
\ LP
L-{#) = ( 2 j n < ! » | p )
•
I= 1
(3.2)
pal.
T h e la r g e r th e v a lu e o f p, t h e m o r e e m p h a s is is b e in g p u t on
m ax{|fi|, |f2[ , . . . , |fml}. At the so lu tio n , large (sm all) p typically produces m any f,;'s
w hich a re equal to m axflfjl. |f2|......... |fm|} (equal to zero).
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T he p = 1 ca se o f (3.2) corresponds to the f 1 norm o p tim ization , so lv a b le by
th e tw o-stage a lgorith m o f H ald and M adsen fl9 8 5 ). T he alg o rith m com b in es a firstorder m ethod th a t a p p roxim ates th e solu tion by su cc essiv e lin ea r program m ing, w ith
a q u asi-N ew to n m ethod th a t u se s approxim ate second-order inform ation to Solve the
sy ste m o f n o n lin ear eq u a tio n s a r is in g from the n ecessa ry first-order con d ition s at a
*
*
solution .
T he p = 2 ca se o f (3.2) (lea st-sq u ares or
w ide publicity.
approxim ation) is a problem o f
B oth first-order and second-order m ethods h ave b een d e r iv e d for,
gen era l n o n lin e a r ^ problem s (se e M arquardt 1963 and D ennis Jr. 1977). For certain
lin ea r
problem s, a closed form solu tion is o b ta in a b le by in v o k in g g e n e r a liz e d
m atrix in version (Rao and M itra 1971; N ash ed 1976).
T he objective function defined in (3.2) is used to p en alize th e m odulus o f f;.
•„ '
<$>
To p en alize th e valu e o ff,, w e u se the gen eralized le a st pth function (B an d ler and Rizk
1979)
u « |» = |
M .f N
fA' ^
(f (<{>)/MJq ) q if M = 0
'
r,
0
> .*
>
<3.3)
ifM r = 0-
w here
- m ax
f .(<J>)
i€ J
J
-
- { 1 ,2
(3.4)
m}
and
if Mf > 0 , th e n , K = {i| f. £ 0. i € -J}
if Mf < 0 , th en , K = J
‘
and q = p
(3.5)
.
an d q = T p .
A
;
In th e case o f M(- > 0 (M f < 0), the larger, th e-v a lu e o f p,„thp m ore nearly
would w e exp ect the m axim um fm inum um ) |fj to be em p h asized . T h e r e fo r e ,
the
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23
. m in im iza tio n o f (3.3) corresponds to the effort'to m eet (%vhen Mf > 0) or to exceed
V'
(w hen Mf < 0) a d esig n sp ecification a s m uch a s possible.
A s p—>►*, th e g en eralized le a st pth op tim ization approaches the m inim ax
op tim ization , th e la tte r b ein g e ffe c tiv e ly so lv e d by th e com bined LP and q u a siN ew ton m ethod o f H ald and M adsen (1981). T he algorith m is a tw o-stage one sim ila r
to th e
optim izat^pn a lg o rith m o f H ald and M adsen (1985). In itially, S tage I is used
an d a t each p oin t, f is approxim ated by lin ea r functions u sin g first-order inform ation.
In S ta g e 2, th e q u a si-N ew to n iteration is used to so lv e a se t o f n onlinear equations
th a t n e c e ssa r ily hold a t a local m inim um .
U su a lly , S tage 1 is used to obtain fast
co n v ergen ce to th e neighbourhood o f the solu tion .
S tage 2 is used to obtain super-
lin ea r fin al co n vergen ce, b u t.several sw itc h es b etw een the two sta g e s may. take place.
✓
T h e t w o - s t a g e a lg o r it h m s for £ i an d m in im a x o p t im iz a t io n s a r e
c o m p u ta tio n a lly practical and h ave been im p lem en ted by Bandler, K ellerm ann and
M adsen (1 9 8 5 ,1 9 8 7 ).
3 .2.3
__
Q uadratic P rogram m ing
In a q uadratic program m ing problem (Q P), the objective function is defined
as
U(<t>) = A + s T <}> +
w here A is a-scalar,
j^
s
^<J>T H<J>,-
'
'
(3 6 :
is a n-vector, and H is a n X n m atrix,
T he Q P problem s arise both in th eir own righ t and as subproblem s w ithin
gerieral n o n lin ea r optim ization m ethods. T y p ically, a QP problem is to m inim ize the
function o f (6) subject to linear eq u a lity and/or in eq u ality con strain ts. Such a problem
can be solved , e .g ., u s ^ ^ t h e itera tiv e m ethods described by Gil! and M urray (1977)
and Grill, M urray, S au nd ers and W right (1984). T h e linear inequality co n strain ts are
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treated u sin g -th e a c tiv e -se t m ethods in w hich a prediction o f th e se t o f co n strain ts
th a t are a c tiv e at th e so lu tio n is m aintained. T h is prediction is called th e w orking set
and is updated by ad d in g or d e le tin g c o n s tr a in ts a s the' ite r a tio n s proceed .
By-
treatin g the w ork in g s e t a s eq u a lity c o n s t r a in ^ th e con strain ed QP problem is tra n s­
formed in to an u n con strained one.
T he problem is r ela tiv e ly e a sy to so lv e i f the
origin al H is p o sitiv e d efin ite (G ill and M urray 1977).
For u n con stra in ed QP problem s, w ith H a s p o sitiv e d efin ite, th e m inim um
can be u n iq u ely located in a fin ite num ber o f ste p s, u sin g, e .g ., N ew ton's m ethod and
th e conjugate g ra d ien t m ethod.
3.2 .4
M INM AX and M 1NBOX A pproaches in L in earization
L in earization is often used in so lv in g n on lin ear program m ing problem s.
H achtel. S cott and Zug (1980) d escribed the M IN M A X and M IN B O X a p p ro a c h es
i
w here the ran ge o f th e v a lid ity o f a lin ea r approxim ation is sp ecified in the v a r ia b le '
dom ain and th e function dom ain, resp ectively. U sed in n on lin ear m inim ax o p tim iza­
tion, the M INM AX approach resem b les the con ven tion al way o f locatin g the m inim ax
point o f lin earized fu nction s subject to a prescribed "box co n stra in t" on <{>.
T he
M INBO X approach, on the oth er hand, eith er produces a sm a lle st step A<j> w hich
<•
a ch iev es u ser-sp ecified lev els o f im p ro v em en t in f, or s t a te s th a t th e le v e ls are
in feasib le.
’
>
3.2.5
G radient and N on grad ien t A pproaches •
^
*
T h e e m p lo y m e n t o f e x a c t g r a d ie n t in fo r m a tio n <JU/<3<{> s ig n if ic a n t ly
im proves th e e ffe c tiv en ess o f an o p tim ization algorith m .
T he w ell-know n a d join t
netw ork m ethod develop ed by D irector and R ohrer (1969a, 1969b) rem ain s a powerful
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25
tool for s e n sitiv ity calcu lation . A n eq u ivalen t, b u t pure a lg eb ra ic approach has also
been stu d ied (B ranin Jr. 1973; B a n d ler and Z hang 1986).
For sp ecia t ty p e s o f
n etw ork s, e.g., branched cascad ed netw orks, m ore effective m ethods can be derived
,
—
(B an d ler, D aijavad and Z hang 1986).
N ot in freq u en tly , th e gradient is d ifficult or even im p ossib le to o b ta in .
A pproxim ate g rad ien t m ethods ha v e been developed, in addition to the direct search
m ethods w hich do not depend e x p lic itly on evalu ation or estim a tio n o f gradients. The
#
th e o r e tic a l b a ck g ro u n d is th e B royden form u la (B royd en 19651, w h ich u tiliz e s
function v a lu es to im prove th e gradient estim ation a s'th e optim ization proceeds. T his
feature h a s been im p lem en ted in nonlinear
and m in im ax optim ization packages
j
(B an d ler, Chen, D aijavad and M adsen 1986).
J
.
3.3
G E N E R A L F O R M U L A T IO N OF D IA G N O S IS A S O P T IM IZ A T IO N
PROBLEM S
3.3.1
Introduction
T h e a n a lo g d ia g n o s is tech n iq u e s a re d e s c r ib e d h e r e u s in g a s in g t e
frequency m easurem ent.
sim p licity .
Such a description offers both conceptual and notati<>nul
P articu lar m a th em atical m an ipulations req u ired for m u lti-freq u en cy
c a se s are illu sti .ed w h en ever necessary.
Suppose from the circu it under te st (CUT), w e obtain a se t o f m easurem ents
represen ted by a n p -vectof FM. T^e corresponding resp on ses as functions o f circuit
p aram eters $ ^
[<{>1
$■> ... cpnP" are given by F ^ F(<J>, gj). For sin g le frequency cases,
F = F(<{>) is used for notation al convenience.
.
.
A n om in al d e sig n o f th e c ir c u it is
/
/
ch aracterized by <J>° and F°.
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f
W hen th e m ea su rem en ts are in su fficien t to identify all param eters, e.g.,
w hen np*< n, th e equation
F M = F(<J>°+,A<j>) .
(3 <)
is an u nderdeterm ined one. A n o p tim ization technique can he used to find the m ost
lik e ly A<J>, am on g a n 'in fin ite num ber o f so lu tio n s to (3.7).
sta ted a s
Such a problem can be
t
• - ize
■ iL(Aq>)
vaai
v m inim
A<J>
(3.8a)
a t . h (F M, A<J>) £ F(<}>0 + A $ ) - FM = 0 ,
(3 8b)
w h ere U is an in creasin g fu n ction o f jA<pi|, i = 1, 2 , . . . , n.
A co n v en ien t approach to so lv in g (3.8) is to u se p en a lty m eth od s.
For
ex a m p le, a le a st pth form ulation is
m inim ize {■ V ' w
w IA$.|
|a a Ip
p+
j. V
y
i - 1
. 1
* . [• F
- .($- °n + A<*>)- -■
P
= 1
P^)
,
(3 ‘9)
/
tw h ere w;, i = I , 2 , ..., n an d P;, i = 1 , 2 , - . . . , np are a p p rop riate w e ig h tin g fa c to r s
(B and ler, K ellerm an n and M adsen 19S7).
3 .3 .2
C on strain t E quation
Suppose the N -node circu it is characterized by its nodal equ ation
Y V = I
(3.10)
w here Y, V and I are th e nodal a d m itta n c e m atrix, v o lta g e v ecto r and cu rren t
ex cita tio n vector, resp ectiv ely .
W e a ssu m e, for convenience, th at th e m easu rab le
resp on ses o f the C U T, n am ely F, can be represented by lin ear com b in ation s o f nodal
K
«
v o lta g es u sin g a N
X
np m atrix C such th at
t
f
= ctv .
■ (3.11)
T h us,
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27
-
t
i
F = F ( $ ) = C T[Y(<i>)rl I.
(3.12:
T o sim p lify th e n on lin ear op tim ization o f (3.81 and (3.9), researchers have
em p loyed tw o effectiv e form ulations tran sform in g the constraint equation into linear
form s by in tro d u cin g in term ed iate p aram eters. T h ese form ulations arc the current/*
voltage sou rce su b stitu tio n model and th e com ponent connection model. T he former
m odel w ill be u se d throu ghou t th is chapter. A comprehensive* treatm ent to th e latter
can be found in R ansom and S a ek s (1973), and in DeCarlo and Sack s (1981).
3 .3 .3
T h e C u rren t/V oltage Source S u b stitu tio n Model
T
T h e c u r r e n t/v o lta g e so u rc e s u b s titu tio n m odel w as used by B an d ler,
B iern ack i and S a la m a (1981), B andler, B iernacki, Salam a and Starzyk (1982), and
B andler and S a la m a (1985a) for fa u lt d ia gn osis. In such a m odel, ch an ges in elem en t
v a lu es a re e q u iv a le n tly characterized by cu rren t or voltage sources. Fig. 3.1 show s
e q u iv a le n t rep resen ta tio n s for som e typ ical e lem en ts in linear circuits. W ithout loss
o f g e n e r a lity , w e a ssu m e th a t th e c h a n g es are represented by current sources only.
L et Alb he a n -vector c o n ta in in g su ch so u r c e s corresp o n d in g to th e n v a ria b le
e le m e n ts, and Q be a N 'X n incidence m atrix re la tin g the n b ra n c h e s co n ta in in g
v a ria b les to th e N -n od es o f th e circu it.
By in vok in g the superposition theorem , we
m ay w rite
Y lV ^ A V = - Q A I b .
13 I3)
w here A V is the d ev ia tio n o f a ctu al nodal v o lta g e s from their nom inal values. A lso,
A F ^ F — F° = C T AV = - C T [Y (4 > °)rl Q A Ib
(3 141
D e n o te .
A' = —CT lY « t > ° ) r 1 Q .
(315)
T hen w e h a v e th e con stra in t equation in lin ea r form as
^ j b = pM _ pO
(3.16)
or, in real form , a s
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28
Y +AY
V
o-
o-
o
VCCS
|Y !
v
I
►
—
(a+A a) Vk
A l b=AY V b
-o I
I
o — ►(XA I
V
V
o-
o-
W
\
M>
a
AI
=AaVk
s A jS Ik
cccs
b
VCVS
b
b
(r+Ar) I .
^ \A V
ccvs
Fig. 3.1
=ArI
An eq u iv a len t rep resen tation o f ch a n g es in elem en t values.
K
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29
•)
(3.17)
A x —h ,
w here
Re(A')
- lr a ( A ')
Im (A')
Re(A')
(3.18)
A =
b = [R e(FM - F°)T
tm (F M - F°)T]T
(3.19)
I m (A lW .
(3.20)
and
x = [Re(AIb)T
To com pute A<J> from x , w e sim u la te th e netw ork w ith a ll com ponents held
a t n o m in a l v a lu e s a n d w ith a d d itio n a l c u r r e n t e x c it a t io n s AI;b = x, + j x I + n,
i = 1 ,2 , . . . , n conn ected a cro ss co rr esp o n d in g c o m p o n en ts.
A fter m e a su rin g or
c a lc u la tin g branch v o lta g es V f , i = 1 , 2 , . . . , n, the com ponent change is evalu ated as
(3.21)
(jco)~ , i = 1 , 2, . . . . n
A<{>.i =
»
V
w here a = ai, w h o se v a lu e can be 0, 1 or — 1 d e p e n d in g upon w h eth er th e ith
com ponent is r e sistiv e , cap a ctiv e or inductive.
For m u lti-fr e q u e n c y d ia g n o sis, w e u se A<J> as o p tim iza tio n v a r ia b le s
d irectly. A , b 3nd x a re redefined accordingly. For exam p le,
i T rv
/ a.0
A. = —C [Y(<f>,a>.)]
1 yv
Q cfiag
1,-
, Q2 , . b ,
<j .)
/
,
. ,a n , . b ,
’
(j (o.) “ Vt>(co.), . . . . (jcaJ
i=1.2
' (3.22)
F M(Wl) - F °(« j)
A1
A’ =
,
b' =
(3.23)
F M(o>2) - F°(caJ
A2
Refb*)
Ret A")
A =
Im(AT
.
b =
(3.24)
Imlb")
and
*
r x = A<}>
-
(3.25)
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w h ere we have assu m ed th a t/fw o frequency points are taken. T he branch v o ltages
Vkb(o>i) k = 1 ,2 ,
, n are in /tia lly assu m ed . A n itera tiv e procedure updates Vkb(u;)
L
&
and a t th e sa m e tim e co m p u te? th e ch a n g es in <{>.
I f the nodal eq u a tio n o f (3.10) is replaced by a hybrid eq u ation , a m ore
g en eral form o f (3.17) ca n be sim ila r ly deduced w h e re both c u r r e n t and v o lta g e
so u rces e x ist for an eq u iv a len t rep resen tation o f A<}>.
3 .3 .4
T he C om ponent C onn ection Model
T he com ponent con n ection m odel w as used by R ansom and S a ek s (1973,
F
L 1I L 12
V
u
o
u'
r
1975). W e a ssu m e th a t th e sy ste m topology is described by a m atrix relation
(3.26)
H ere, u ' and v are the com ponent input and output variab les, resp ectiv ely , related by
v = Z u' ,
(3.27)
w here Z is the com ponent p aram eter m atrix. The u and F in (3.26) are th e sy stem
in pu t and output varia b les related u sin g the sy stem m atrix T as
/"T = r u.
(3.2S)
By introducing in term ed ia te v a ria b les R , we have lin ear relation
r ~ Lot R L 12 ,
(3.29)
R = (1 - Z L u ) - l Z.
(3.30)
w here R is related to Z, u iin g
It h a s been show n (S^e R ansom and S a ek s 1973) th at for sm all ch an ges in Z.
AZ =» A R .
"V
(3.31)
%
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A s su ch , R can be used in stea d o f Z for .optim ization.
F inal r e s u lts o f Z can be
com puted u sin g eith er th e ex a c t (i.e ., deduced form (3.30)) or th e approxim ate (i.e.,
deduced from (3.31)) rela tio n b etw een R and Z.
3.3 .5
G en eral.F orm ulation
T h e in term ed ia te v a ria b les x defined in (3.20) exh ib it a sim ila r pattern to
th e p a r a m e te r s A<J> s in c e a n e q u iv a le n t so u r c e c u r r e n t A Ib in c r e a s e s a s th e
corresp on d in g A<{> in crea ses. A lso, AIb = 0 if and only if A<{> = 0. Now, w c can solve
the op tim iza tio n problem w ith x as v a riab les and use the solution to find A<J>. A
sim p le y e t rea son ab le ob jective function is the least pth function o f x.
A general
form ulation o f d ia g n o sis a s an op tim ization problem is
... jx.
. . ip V 'P
m inim ize U (x ) «= (( _v w.
i=1
(3.32a)
(3.32b)
s.t. A x — b = 0 ,
w here w;, i = 1 , 2 , . . . , 2n are w eig h tin g factors and the constraint (3.32b) is derived
from (3.17-3.20). For th e m ulti-frequency case, (3.22)-(3.25) can be used to define A,
b, and x for the con stra in t equ ation (3.32b). In this case, the objective function U is
the w eig h ted le a st pth function o f x;, i = 1, 2, . .. n. A fter solvin g (3 32), A<J> can be
i
found u sin g (3.21) or (3.25).
3.4
(
DIA G N O SIS U SIN G TH E LEA ST-SQ U A R ES METHOD
T h e d ia g n o sis tech n iq u e u sin g le a st squares optim ization^was su ggested by
R ansom and S a ek s (1973). It is based on the assu m p tion that th e catastrophic faults
have been elim in a ted and the circu it failu re is due to com ponents d r iftin g out o f
toleran ce (as from a ge, tem p erature ch a n g es, etc.).
N ,.
A
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
'Vs'J
32
T he optim ization problem can be sta ted as
a
(3.33a)
t
m inim ize L (x )= x W x
x
»
(3 33b)
s.t. A x - b < ; 0 ,
w here th e co n stra in t eq uation (3.33b) is dCTined co n sisten tly w ith (3.32b).
W is a
d iagon al m atrix co n ta in in g w eig h tin g factors w^, i = 1 , 2 ' . . . ,2 n . A n ap propriate
*
choice o f the w e ig h tin g s can be such th at th e U o f (33a) approxim ates
1
,____
aof
i= l
under th e a ssu m p tio n th at A<J>S, i = 1 , 2 , . . . , n are quite sm all.
B an d lcr and S a la m a 1985a), for 1 S i f i n ,
1
k
*»
w ( = - (Re[(ju) ‘ V®])— ,
For exam p le (se e
„
*
(3.34)
w
t+ n
1 --------= -(I m [(jw ) 1 V “D — .
i
2o
T he so lu tio n o f the fo problem is«directly obtained u sin g g en eralized m atrix
in version (see Rao and M itra 1971), e.g.
(3.35)
x = W ' 1 A T(A W " 1 A T) _ 1 b .
Such a tech n iq u e using a co m p o n en tco n n ectio n model has been p resented
in R ansom and S a ek s (1973). -T he v a ria b les x co n sist o f elem e n ts o f the m atrix AR.
T h e op tim ization problem is to m in im ize th e fo norm of AR subject to
AT = L 2 1 A R Lj2 ,
^
(3.36)
w h ere AT is the d ifferen ce betw een the m easu red v a lu e s and the nom inal va lu es o f I \
T h e so lu tio n is the gen eralized in verse o f th e m a trix in (3. 36).
T h e co m p o n en t
co n n ection m odel is effective here sin ce A R = AZ under th e a ssu m p tio n th a t no
p ara m eters have sig n ific a n t d ev ia tio n from nom inal.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
3.5
D IA G N O SIS U S IN G TH E QUADRATIC PROGRAMM ING METHOD
T he q uadratic program m ing tech n iq u e for d ia g n o sis w as su g g e ste d by
M errill (1973).
H e consid ered su ch a cla ss o f situ a tio n s w here a sy stem becom es
in operative due to th e fa ilu re o f one or a few com ponents. He pointed out th at because
th e in divid ual sy ste m com p onents are gen erally high ly reliab le and w ell m ain tain ed ,
a d ia g n o sis th a t im p lic a te s m any com ponents a s h avin g failed is probably not correct.
T herefore, contrary to th e
€2
op tim ization technique, the m ain assum ption here is
r
th a t th e d ifferen ce b etw een th e a ctu al and the nom inal va^Ujfes for a few elem en ts,
w hich correspond to th e fau lty e le m e n ts, is m uch greater thafFthat for the rem ain in g
e le m e n ts th a t are nonfaulty.
4
T he o p tim iza tio n problem ^ an be described as
2n
».
\
m inim ize L (x) a >
»
X
__
x/ i i ' C
w. V |x.| 1- b
<3.3 4 a)
i = 1
s.t. A x - b = 0 ,
(3.37b)
w here the c o n stra in t equation (3.37b) is defined co n sisten tly w ith (3 .3 2 b ). T h e 5
under the radical p rev en ts th e d erivative o f the objective function from becom ing
unbounded.
To so lv e (3.37) e ffic ie n tly , M errill put th e c o n str a in t (3.37b) into the
objective function in a q uadratic form as a p en alty term , applied uniform w eig h tin g s,
wj = 1, i = 1. 2 , . . . . 2n and transform ed the problem into
. . .
-In
m inim ize L(y) — .
------ \ 7
,1 _ , “
uiT ,T
. ,
V y. .. r 5 + - P ( A y — b l ( A y — b)
•*
!
(3.38a)
O
1= 1
\
w here A = [ A
s.t. y £ 0 ,
V
—A ] and y is a 4 n v e c to r related to x via
*
(3.38b)
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
34
y . = x. a n d y_
M
J 2n +
i
y. = 0
i
_
. = 0
ifx . 2: 0
i
1
a n d y„
= —x.
2n + 1
i.
i = 1 .2
ifx . < 0
i
(3.39)
2n.
A lso, x can be ca lcu lated from y u sin g
v —V
V
x i - y i “ y 2n+i ’
i —10
On
i-l,2,...,2n.
(3.40)
_
F u rth erm ore, th e sq u a r e root p ortion o f U (y ) is lin e a r iz e d a t y - = yj,
r e su ltin g in
(3.41)
U.(y) = A + s Ty H— y T H y ,
j
2
w here
S = - [(yj + 5 ) " 172
2
(y i + 8)" 1/2
...
( y ^ + 5 ) _1/21T - p A b
(3.42)
and
ata
- ata
- at a
ata
(3.43)
H = P
T h e sca la r A is also a function o f P, 5, yj and b, but as its v a lu e is irrelev a n t to the
m in im iza tio n o f L'j(y), it w ill n ev er a c tu a lly have to be calcu lated . •
A s M errill ind icated , th e u se o f variab les v, in stead o f x, can e lim in a te the
\
d ifficu lty o f d eriv a tiv e d isco n tin u ity o f U at xt = 0. T he q u a si-lin ea r iza tio n o f U t am
(3 .3 S a ) to Uj o f (3 .4 1 ) le a d s to th e n a tu ra l a p p lic a tio n o f p o w e r fu l q u a d r a tic
program m in g m ethods (see G ill and M urray 197T).
*■
(3.3S) is solved itera tiv ely by the follow in g steps.
T he o p tim iza tio n problem o f
Step I
j = 0, yJ = 0.
Step 2
Com pute s a s a function o f vJ u sin g (3.42).
Step 3
M inim ize Uj(y) o f (41), su bject to y a 0 u sin g the quadratic program m ing
m ethod. T he so lu tio n is d efin ed as y i - 1.
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
35
S tep 4
If U(yj + 1)
L'(yj), th e n c a lc u la te x u sin g (3.40) an d stop; o th e r w ise ,
j « - j +■1 and go to S tep 2.
3.6
,
D IA G N O SIS U SIN G TH E LIN EAR PROGRAM MING METHOD
B and ler, B iernacki and S alam a (1981), and Bandler, B iernacki. S a la m a
/
I
and S tarzyk (1982) proposed the d ia g n o sis technique u sing the f t norm optim ization.
T he m ain a ssu m p tio n is sim ila r to th a t for the quadratic program m ing approach.
H ow ever, in stea d o f so lv in g a sequ ence o f quadratic optim ization problem s, a linear
p rogram m ing problem is form ulated,'taking ad van tage o f the nature o f the f j norm as
w ell a s th e lin e a r ity o f th e co n stra in t equation. A solution to suqh^a problem tends to
sa tisfy th e co n stra in t w ith m in im um num ber o f param eters different from zero
This
is co n siste n t w ith the a ssu m p tion th a t a few e lem en ts are actu ally faulty.
T he op tim ization problem can be expressed as
. . . ize t
m inim
L- ,(x)v i- V' w. \ I I
—
i 1 i1
x
i=l
s.t. A x — b = 0 ,
(3.44a)
A
(3.44b)
w here th e co n str a in t eq uation (3.44b) is defined co n sisten tly w ith (3.32b).
Such a problem can be solved d irectly u sing f j op tim ization algorith m s,
e.g ., H ald and M adsen (1985).
It can also be handled by u sin g a r eg u la r lin ea r
program m ing so lv e r in a sim ila r m anner to that o f Barrodale and Roberts (1978). Lot
y be d efin ed by (3.39).
T he problem of (3.44) is transform ed into a sta n d a rd LP
problem as
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
36
. ■ ■ fT, . a ,
.m inim ize U (y ) = [w 1 w 0 ...
a t . [A
,
w ^ ly
(3.45a)
(3.45b>
—A ]y = b
(3.45c)
y ^ 0 .
A t th e solu tion o f (3.45), x can be ca lcu la ted from (3.40).
3.7
M ODELLING U SIN G OPTIM IZATION M ETH O DS
In a m o d ellin g problem , it is req u ired to find p a r a m e te r v a lu e s o f an
%
eq u iv a len t device m odel to b est fit m easu rem en t data. A s H ach tel, S cott and Zug
(1980) have described, th e problem is o f a type th at is freq u en tly en cou n tered by
product a ssu ran ce en g in eers. T n ese en g in eers are faced w ith the fact th a t th e circu its
w h ich co m e o ff the product lin e d iffer from th e c ir c u its d esig n e d w ith c ir c u it
sim ulation^profecam s.
C on sequ ently, th ey n eed d ev ice m o d els wh'.ch a g r e e w ith
on-chip m ea su rem en ts in order to e s tim a te th e s t a t is t ic s o f th e o n -ch ip c ir c u it
perform ance.
3.7.1
B asic Form ulation
Let f = f(<t>) be a m -vector co n ta in in g the w e ig h te d d iffe r e n c e b e tw een
ca lcu la ted response F(«J>, to) and m easured data F^(o)) in the form o f
w.Cw.KF (4>,co.) - F M(w.)).
i j
i J j
i
j
i € {1,2
n_h
.
j € {1. 2 ........n } .
w
(346)
Due to m ea su rem en t errors and nonideal effects, f = 0 m ay not be possible. T herefore,
the m od ellin g problem can be sta ted as
-
m inim ize U (<}>)
$
(3.47a)
.
s.t. <i>L ^ <t> ^ 4>l .
(3.47b)
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
37
w here U is art in crea sin g function o f ]fi(<J>)|, i = 1 , 2 , . . . , m. <t>L and <J>l- arc low er and
upper bounds, r esp ectiv ely for <f>.
—
•
A rea son ab le objective functiorr-C(<{>) can be the le a st pth function o f f(<j>) in
1th e form o f (3.2).
W ith a sm a ll valu e o f p, the o b jectiv e fu n ctio n te n d s .to accom m od ate
m ea su rem en ts w h ich m ay contain a ccid en tal large errors. Large va lu es o f p uroduce
sa tisfa cto ry r e su lts w hen a ll m easu rem en t errors and nonidcal effe cts arc sm a ll.
S u ccessfu lly im p lem en ted a lg o rith m s have used p =_1 (e.g.; Bandler, K ellerm an n and
M adsen 1987), p = 2 (e.g. Kondoh 1986), and p = <='(e.g., H achtel, S co tt an d Zug
1980). •
3 .7 .2
L im ita tio n s o f the B asic F orm ulation
. T he b asic form ulation o f m od ellin g problem ’s is a tr a d itio n a l approach
w h ich is a lm o st en tirely directed a t a c h ie v in g th e best' p o ssib le m atch b etw een
m easured and calcu la ted responses. T h is approach has seriou s shortcom ings in two
fr e q u e n tly e n c o u n te r e d c a s e s .
T h e fir st c a s e is w h e n t he e q u i v a l e n t c i r c ui t
p ara m eters are not unique w .r.t. the r esp o n ses se le c te d and the secon d is w hen
nonideal effects are not m odelled ad eq u ately,-th e latter cau sin g an im perfect m atch,
’ even if sm a ll m ea su rem en t errors are allow ed for. In both cases, a fam ily o f solu tion s
for circu it m odel param eters m ay e x ist w hich produce a reasonable a n c^ im ij^ r m atch
b etw een m easured and sim u la ted responses (B andler, C hen and D aijavad, 1986b)
Such problem s becom e more difficult to handle w ith a large num ber o f v a r ia b le s
t
w here a d irect op tim ization is hop eless u n le ss started w ith accurate e stim a te s o f most
circu it e le m e n t v a lu es from independent m easu rem en ts or calcu lation s (T siron is and
M eierer 19S2; C urtice and C am isa 1934).
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
I
<
38
*
■
'
E ffo rts in a lle v ia t in g t h o s e /d if f ic u lt ie s h a v e b e e n m a d e in s e v e r a l
d irectio n s. S tr a ig h tfo r w a r d a p p ro a ch es in c lu d e s e e k in g a d d itio n a l in d e p e n d e n t
m easu rem ents and/or p red eterm in in g som e variab les. S in ce both a ctio n s reduce the
f freedom o f v a ria b les, they can be e ffe c tiv ely a p p lied i f a fu r th e r e x p lo ita tio n o f
physical properties o f a g iv e n d ev ice is perm itted.
H o w ev er, w h en faced w ith a
prescribed s e t o f p ossib le m ea su rem en ts and variab les, w e can proceed to gen eral
approaches such a s d ecom position and m u lti-circu it m easu rin g.
3.7 .3
R eduction o f Model P ara m eters and D ecom position A pproaches
R eduction o f m odel p a ram eters m ayJje^jjossible by full in v e stig a tio n q f
■physical p roperties o f the d ev ice to be m odelled. Such an approach w as d em on strated
by C urtice and C am isa (1984) in a FET m od ellin g problem . U sin g dc and zero bias
m easu rem en ts, th ey reduced-the num ber o f variab les from 16 to 8. The final r e su lts o f
th e m odelling w a s reported to be accurate and unique.
In laboratory ex p erim en ts, a’ repeated trial and error procedure m ay be
n ecessary.' R eduction of m odel v a r ia b le s can be a c h ie v e d by e x p lo itin g th e lab
exp erien ce w ith sa m p le d evices.
In sen sitiv e va ria b les should be r e m o v e d \ f ”m itia l
sta g e s o f an o p tim ization process.
V ariab les ten d in g to reach th e upper or low er
bounds du ring th e o p tim ization can be fixed in an appropriate m anner (H a ch tel, S cott
and Zug 1980).
T siro n is and M eierer (1982) and Kondoh (1986) have su g g ested to decom ­
pose-the o verall optim ization problem o f (3.47) into a seq u en ce o f su b op tim ization s.
T hey illu stra ted su ccessfu l FET m o d ellin g by properly d efin in g and ordering su b sets
o f p aram eters and resp onses.
In sen sitiv ely related p aram eters and resp on ses are
separated into .different subproblem s. A se r ie s o f su b op tim ization s can provide a good
4
R e p ro d u c e d with perm ission of the copyright owner. Furth er reproduction prohibited without permission.
■starting p oin t for th e o v era ll optim ization.
It a lso /im proves mod&l accu racy and
reduces th e p o ssib ility o f stop p in g a t an undesired local m in im u m .
A n au tom atic d eco m p o sitio n approach for d e v ic e m o d e llin g and large
circu it d e sig n w as d eveloped by B andler and Zhang (1987a).
U sing' th is approach,
su b op tim ization s for FE T m od ellin g problem s have been form ulated autom atically
u sin g com puterized s e n s itiv ity a n a ly sis o f the d evice.
T he re su lts w ere co n sisten t
w ith th ose o f Kondoh (1986).
3.7 .4
M ulti-C ircuit A pproach
T h is approach w as proposed by Bandler, C hen and D aijavad (1986b), The
*
fi-n o r m objective fu nction w a s used. Suppose th at after ta k in g m easu rem en ts on a
d evice a t a num ber o f frequency points, we m ake pn ea sy -to -a ch iev e physical adjust­
m en t, su ch that one or a few com p onents of <J> are changed in a d om inant fashion and .
the. rest rem ain co n sta n t or ch a n g e sligh tjy.
C onsider the follow ing optim ization
problem
2
m inim ize
^ 2
k
m
n
^
1^1 + ^ 13 |<t>1 — <J>*1
k = l >= I
j= l J J
J
(3.48)
w ith superscrip t k id en tify in g the origin al netw ork model (k = 1) or the model after
p h ysical adju stm ent (k = 2). Pj rep resen ts ah appropriate w e ig h tin g factor and m* is
an index w hose valu e depends on k, i.e ., a different num ber o f freq u en cies may be used
for th e o rig in a l and the perturbed m odel. 4*1 and <}>2 are vectors con tain in g circuit
p ara m eters of the orig in a l and perturbed netw orks, resp ectively.
* By ad din g the second seg m en t to the objective function, wc: tak e advantage
o f the kn ow ledge th a t only one or a few com ponents o f <{> shopld ch an ge dom inantly by
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
40
pertu rb in g a physical com ponent o f th e d evice. T herefore, w e p en a lize th e objective
function for any chan ge in <f>. H ow ever, by clev e rly s e le c tin g th e
norm , we still
allow for one or a few .large c h a n g es in 4>.
T h e co n fid e n c e in th e v a lid ity o f th e e q u iv a le n t c ir c u it p a r a m e te r s
in crea ses^ fltt) an op tim ization u sin g th e ob jective fu n c tio n o f (3 .4 8 ) r e s u lts in a
rea so n a b W n a x ch betw een ca lcu la ted and m easu red resp on ses for both c ircu its 1 and
*
2 (original and perturbed) and 2) th e e x a m in a tio n o f the so lu tio n re v e a ls ch a n g es from
<j>i to 4>2 w h ich a re co n sisten t w ith th e p h y sica l adjustm ent, i.e., o n ly th e expected
com p on en ts h ave changed sig n ifica n tly . W e can build upon our confidence even m ore
by exp a n d in g th e technique to m ore a d ju stm en ts, i.e., form u latin g th e op tim ization
problem a s
nc mk
m inim ize
<J>'
^
V
k = 1i = l
if^t +
\
n
^
^
0^ 1<{>? — <$>^1 ,
(3.49)
k=2
)
w here n<- circu its and th eir co rresp o n d in g s e t s o f r e sp o n se s, m e a s u r e m e n ts and
p a ram eters are considered and the first circu it is the referen ce m odel b efore a n y
a*
p hysical adju stm en t. <}>' con tain s a ll <}>k, k = 1. 2 , . . . , nc.
3.8
T U N IN G U SIN G OPTIM IZATION M ETHODS
Postproduction tu n in g is often e sse n tia l in the m an u factu rin g o f electrica l
circu its. T o lera n ces on the circu it com p on en ts, p arasitic effects and u n ce rta in tie s in
th e circu it m odel cau se d ev ia tio n s in th e m anufactured c ir c u it p e rfo r m a n c e, and
v io la tio n o f the d esig n sp ecifica tio n s m ay resu lt. T herefore, postproduction tu n in g is
in clu d ed in .th e fin a l sta g e s o f th e p ro d u ctio n p ro cess to rea d ju st th e n e tw o r k
perform ance in an effort to m eet th e sp ecification s.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
41
C om puter-aided d esig n ers h ave approached th e tu n in g problem in tw o
w ays, ea ch em p h a sizin g one d istin ct facet. B efore production, a t the tim e o f d esig n in g
a circu it, one ca n consider tu n in g a s an in teg ra l part o f the design process (B andler,
Liu an d T rom p 1976; P olak and S a n g io v a n n i-V in cen telli 1979), the objective b ein g to
r ela x 'the to lera n ces on the circu it com ponents and com pensate for th e u n ce rta in ties '
in th e m odel p aram eters. T he in tegral d esig n problem is form ulated and solved u sin g
o p tim iza tio n su ch th a t th e e sse n tia l dem and o f production cost reduction is op tim ally
m et. T h e so lu tio n o f the d esig n problem provides the m anufacturer w ith the allow ed
d esig n to lera n ces and th e tunable param eters.
In th e final production sta g e s, the m anufactured circuit is u su ally tested to
check w h eth er or not it m eets d esig n sp ecification s. T u n in g is often needed. H ere. it
is required to im p lem en t necessary ch an ges in the tunable param eters to adjust the
m anufactured c ircu it to sa tisfy the d esig n requirem ents (B andler and S alam a 1981).
3.8.1
P reproduction T u n in g
Suppose c = [ei co . . . cnF and t = [ti to . . . tn]T are vectors con tain in g
t o le r a n c e s a h d m a x im u m t u n in g a m o u n ts , r e s p e c t iv e ly , for th e p a r a m e te r
= [ $ i <f>2 • - - <t>nlT-
A n o n lin e a r p r o g r a m m in g p ro b lem in t e g r a t in g d e s ig n
cen terin g , to lera n cin g and tu n in g can be sta ted as:
m inim ize U(<J>°, e,t)
(3.50a)
- <J>°, c , t
s.t.
<J> — 4>° + E [ i + T p ( R t
fora ll u ,
u (R
som e p ,
p
and
(3.50b)
p( R ,
<
P
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
42
w here E and T a re n X n d ia g o n a l m a tr ic e s c o n ta in in g e*. i = 1 , 2
n an d t ,p
i = 1 , 2,..., n, resp ectiv ely , and
n * [px p2 ...
?2 -
P - [pl
pnr
• (3-5 1 ^
PnlT •
•
(3.52)
A lso, Re is a co n stra in t region in w hich a ll responses sa tisfy th eir sp ecification s. .R^ is
a region in w hich Jp;| £ 1, i = 1, 2
n. Rp is d efin ed a s the region {p[ — 1 ^ p ; s 1,
i = 1 , 2 ..........n} for tw o w a y t u n i n g a n d , {p| 0 s
{p| — 1 £ pi 5 0, i = 1 , 2
pj
1, i = 1, 2, ..., n} or
n} for on e w ay tuning. T he ob jective function can be an
in creasin g function o f jt;/({>i0| and a d ecreasin g function o f |ei/<}>i0[, resp ectiv ely .
A
d etailed trea tm en t o f th e preproduction tu n in g w as presen ted in B andler, Liu and
Trom p (1976) and in P olak and S a n g io v a n n i-V in cen telli (1979).
3.8.2
Postproduction Tuning: Problem F orm ulation
Prior to postproduction tu n in g , the m anufactured circu it is ch aracterized by
the a ctu al param eter v a lu e s g iv en by
<J>a =
E pa .
(3.53)
S u p p o se, for c o n v e n ie n c e , th a t th e p rep rod u ction s t a g e r e s u lt e d in t, > 0 for
i = 1, 2 ,. .*., nt and t; = 0 for i = nt + 1 , . . . , n. T h erefore, the tu n ab le p aram eters are
4>„ i = 1, 2 , . . . , n t. A s e t o f circu it perform ance fu n ction s given by
F(<|>. oj) = F(<J>° + E pa r T p, u)
(3-54)
are u su a lly m onitored d u rin g th e tu n in g process. T he d esired v a lu e s for F. denoted as
F 4 can be eith er an op tim al response or a d esign sp ecification . D efine f = f(<}>) a s a mvector w hose e le m e n ts are in the form o f
w
U
(co)(F (<b,o>.) — S .. (to))
j
iT
j
Li
j
(3.55)
—w
Li
Uo.)(F (<{>, to ) — S, (to))
j
tT
j
u
j
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
43
w here i € { 1 ,2
tip}, j€ { 1 ,2 ,„ „ nu j and <{> ■ 4>0 + E p ° + T p. S u i and S u are upper
and low er sp ecifica tio n s, resp ectiv ely,
w^i and w^i are w eig h tin g factors and are
n on n egative. If it is requ ired to m atch Fi(<J>, w) w ith its desired valu e F td(u), one can
e ith e r u se (3.55) by se ttin g
S Ui(ca) = S u (o)) = F d((a)
<3 -561
or define e le m e n ts o f f a s
w.( ca.) | F.(«J>, u.) — Fd(cj.) | .
(3 57)
i j
i
j
‘ J
T he postproduction tu n in g can be form ulated a s the optim ization problem
m inim ize U(p)
(3.58a).
P'
s.t. |p .[ £ 1 ,
j = 1, 2 ........nt .
<3.58b)
w here p' is a nt — vector co n ta in in g the first nt e le m e n ts in p. T he objective function
can be a le a st pth or a gen eralized lea st pth function o f fT<J>0+ E p a + T p), i.e., in the
form s o f (3.2) and (3.3), resp ectiv ely .
3 .8 .3
P ostproduction Tuning: F unctional Approach
F u n ction al tu n in g is a traditional approach. T he tunable param eters are
T
se q u e n tia lly adjusted u n til the circu it sp ecification s are m et.
H ere, the netw ork
e lem en ts are g e n e r a lly a ssu m ed unknown.
Let J be a m X nt Jacobian m atrix w hose (i, j)th elem en t is defined by
af.
J
i = 1 ,2
u
ap.
m
af
— t ,
aq>. j
and
j = 1 ,2
^
(3 53)
n.
The le a st sq u a res op tim ization o f (3.58), n a m e ly , ta k in g L‘ = fT f, w as
proposed by A ntreich , G leissn er and M uller (1975) and A dam s and M anaktala (1975).
The so lu tio n is given by
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
44
/
A p ' = —( J T J ) - 1 J T f(<{>0 + E | i a + T p ) .
(3-60)
T he m in im a x o p tim iz a tio n -o f (3 .5 8 ), n a m e ly , ta k in g U = m a x f;, w a s
ap proxim ated by B andler and S a la m a (1981, 1985b) w ho so lv ed th e follow in g lin ea r
program m ing problem
m inim ize z
\
(3.61a)
Ap\ z
n
s.t. r(<J>°+ E p a +
p ^ ^ A p .S p y .,
T p )+
t
T
j= i
J ip . S z
1J j
(3.61b)
i = l , 2 , . , , , m , j = l ,2 ....,n t .
p is in itia lly s e t to 0. A fter each so lu tio n o f (3.60) or (3.61), p is updated u sin g A p'. As
proposed by B an dler and S a la m a , sim u la te d s e n s itiv itie s and B royden form ula can be
used for ob ta in in g and u p d atin g J .
3 .8 ^
Postproduction T uning: D eterm in istic Approach
In c o n tr a st to th e fu n c tio n a l tu n in g a p p roach , d e t e r m in is t ic tu n in g
req u ires th a t a ll circu it p a ra m eters 4> and p ossib le p arasitic param eters £ (or its
effects) can be eith er m easured or identified.
By u tiliz in g th is in fo r m a tio n , th e
op tim iza tio n o f (3.58) becom es faster.
A seq u en tia l tu n in g a lg o rith m has been introduced by Lopresti (1977). Let
f be th e m -vector defin ed in (3.55) or (3.57). In itia lly , w e s e t p = 0 an d d efin e
f iA
V
t
/£ .
t
-T
ia S
(3.62)
S
w hich represen ts th e d evia tio n o f f from f(4>0) due to p a ra sitic effects and toleran ces in
u n tu n a b le param eters. In th e kth itera tio n , w e have
*
k = l , 2 ......................................13.63)
\
R e p ro d u c e d with perm ission of the copyright owner. F urther reproduction prohibited without permission.
45
x
B y d efin in g U o f (3.58) a s a q uad ratic function o f flu + l and ad d in g a term p en alizin g
Ihrge c h a n g e s in A p ', we obtain an op tim al control problem , i.e ., finding A p' su ch th at
( nnt +1
+ 1\\TT
U = ( f l
J Bf
n 1
V
*-1
.^
+ T
,* .
(3 .6 4 )
p.(Ap.r
j=1
is m inim ized subject to (3.63). B o f (3.64) is a p ositive sem id efin ite m atrix and Pj > 0,
j = 1 , 2 , . . . , n t. A closed form solu tion can be obtained in a form as
J
A pk = Yk * * ,
'
.
« .6 5 )
w here Yk is a m -vector calcu lated u sin g R iccatti equation (see Lopresti 1977).
Instead o f u sin g first-order se n sitiv ity inform ation J w hich becom es invalid
w hen com p on en ts o f A p ’ are not sm all en ou gh , A lajajian, Trick and E l-M asrv (1980)
h ave su g g e ste d a large ch an ge s e n sitiv ity m ethod for d e te r m in is tic tu n in g . The
r e su ltin g eq u ation is
Ap'
[JL -f(4»°)l
= —f
“),
(3.66)
w here J*- is th e large chan ge se n sitiv ity m atrix o f f w.r.t. p' and c is an unknown
variable.
3.9
E X A M PLES
In th is section , w e first p r ese n t the application o f optim ization techniques
for circu it d ia g n o sis through a sim p le illu s tr a tiv e ex a m p le fo llo w ed by se le c te d
problem s o f practical in terest for d ia g n o sis, m od ellin g and tuning.
3.9.1
D iagn osis U sin g O ptim ization: An Illu strative E xam ple
C onsider the p a ssiv e r e sistiv e netw ork o f Fig. 3.2.
N o m in a l v a lu e s for
e le m e n ts G;, i = 1 , 2 , . . . , 5 are equal to 1. E ach elem en t has ± 5 fo tolerance. The
m easu rab le responses are nodal v oltages, i.e.. F = (Vj V->
c a u sin g th e C o f
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
46
to
C3
-C Z D -
&c
c
3
05
05
©
Tf
Cfl
a
C5
CO
o
CM
*
cx
.£
©
©
c
a
3
©
o
©
G
>
o:
03
G
o:
o
3
C
c
*G
.2
wC
«
> .2
w
—
Ti
C/3 r»i
cn
S -3
< o
e
<N
C3
si
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
47
(3.11) to be a 3 X 3 id en tity m atrix. A lso for the exam p le. N = 3. np = 3 and n = 5.
T he incidence m atrix is g iv en by
I
Q =
1 0
0
0
- 1
1
0
0
0
1
-
0
(3.67)
0
1
1
T he v ariab le p a r a m e te r s a r e d e fin ed as <J> = [G i
G? G 3
adm ittance m atrix a t n om in al point 4 >° = [I
1
1
1
2 - 1
Y(<J>°) =
-I
G 4 G slT.
T he nodal
lF is
0
(3.6S)
3 - 1
0 - 1
2
For such a cir c u it, a ll q u a n titie s are re a l.
T h e re fo r e, the c o n str a in t
equ ations as w ell a s the related d efin ition s (3 .1 7 M 3 .2 0 ) becom es
(3.69)
A x = b ,
where
A = - C T [Y(<J>°)]1_1 Q
- 1
T
5
3
2
I
1
0
_2
4
2
2
1
-1
2
b = F M - F° = [V M - Y°
1 l
1
-3
5
• 0,T
V M - V°
2
(3.70)
V M - V°
2
3
(3.71)
3
I
and
(
X = [AI*
I
AI?
2
AlJ3
AlJ4
AI
b,T
5
(3.72)
'
w here AI,b, i = 1 , 2 .......... 5 are th e e q u iv a le n t c u rr en t so u r ces r e p r e se n tin g AG,,
i = 1 ,2 ......... 5 show n in Fig. 3.3. T he nom inal resp on ses F° = [V ^ V_>° Y j'^ T car.be
calcu lated as F° = [5/S 2/S
1 /SIT
C ase 1: Her^dTwe a ssu m e that no e le m e n ts have m uch greater d eviation from nom inal
than otheJ^s.
T ab le 3.1 sh o w s th e r e s u lts o f d ia g n o s is u s in g th e £ \, (•> and th e
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
Kig. 3.3
CD »
O CO
CM <►
0
<N
Equivalent current sources representing the effect of changes in G,-(
i = 1,2, ...,5 for the circuit of Eig. 3.2. A ljb =: AGj Vjb, where V/' is the
voltage across the ilh elem en t.
48
_a in
<
II
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
49
/
. -
CUT
'
T A B L E 3.1
R E SU L T S OF D IAG NO SIS U SIN G OPTIMIZATION TE C H N IQ U E S
FO R TH E.C IRC UIT OF FIG. 3.2, CASE 1 -
M easu rem en t
• VM
: A ctual
£Gi/Gi°%
i = l,2 ,...,5
D etected A G t/G ^ ^, i = 1.2 ..... 5
So
Q uadr
Program .
fi
Optim.
4.36
14.02
13.23
18.0
18.07
1.80
* 3.14
9.0
7.61
. 0 .0 0
0 .0 0
30.0
"33.04
0 .0 0
25.0
27.93
- 3 .8 5
0 .0 0
0 .0 0
0 .0 0
- 1 5 .0 7
- 1 5 .0 7
7.92
7,92
.
Optim .
•# 1
4.4
.5730
.2 3 2 6 '
..1186
#2-
.6437
-
.2241
2 .0
-2 .3 8
1 2 .0
- 1 1 .4 1
.1307
2 0 .0
23.73
4.35
4.35
15.0
1 8 .5 8
0 .0 0
0 ,0 0
3.0
- 0 .4 2
0 .0 0
0 .0 0
.0
- 2 .4 4
- 3 .1 2
- 3 .1 2
- 3 .4
r.96
0.60
0.60
1 0 .0
17.69
18.23
18.28
- 7 .0
-0 .5 0
0 .0 0
0 00
-S
-
"
4.00
512S
.62 6 6
.24 1 2
,
•
6 .0
.1145
#3
'
0
J?
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
.
quadratic program m ing m ethod.
It is dem onstrated that th e lea st squares m ethod
g iv e s a m ore reasonab le so lu tio n , w h ile th e oth er tw o m eth o d s h a v e m is ta k e n ly
d etected, e .g ., G4 a s n onfaulty w h ile th is e le m e n t actu ally changed 30 % for C U T # 1 .
H ow ever, the C2 optim ization m ethod m ay a lso fa il to give,correct resu lts, see C U T # 3
w here th e G2 and the G 5 are not d etected a s o u t o f tolerance.
C ase 2: In th is case, we a ssu m e th a t on ly a few e lem e n ts are fau lty h a v in g m uch
grea ter d eviation from nom inal than th e rest el.e'ment th a t a re w ith in th e sp ecified
to le r a n c e o f - ± 5 % .
T a b le 3 .2 sh o w s th e r e s u lts o f d ia g n o s is u s in g th e th r e e
op tim ization tech n iq u es presented. It can be se e n that both the f 1 and the quadratic
tech n iq u es give m uch sharper r e su lts than the
£0
technique. In m any ca ses, both
£1
and th e quadratic optim ization produce the sa m e solution. In som e c a se s, as show n
for C U T # 2 and C U T # 3 in T ab le 3.2, on e m ethod y ield s a better solu tion than the
other.
*t
For the quadratic p rogram m ing technique, we have used S = 10
6
and
P = 1 0 10. The QPSOL Fortran p a r a g e for q uadratic program m ing (G ill e t at 19S-)
w as u tilized to perform Step 3 in S ection 3.5 w ith a lim it on the num ber o f iteration s
for each quadratic program m ing as
3 .9 .2
3
.
D iagn osis o f a 28 N ode C ircuit
K cllerm an n (1986) ex p erim en ted w ith the n onlincar-optim ization problem
of (3.9) w ith p =
1.
on a 2S-node circu it show n in Fig. 3.4
The nom inal v a lu es o f the
ele m e n ts G, = 1.0 and toleran ces c, - ± 0 .0 5 , 1 = 1 ,2 ...... 52.
A ll ou trid e nodes are
assu m ed to be a ccessib le for m easu rem en ts. T he actual circu it in clu d es four fau lts
w here e le m e n ts G 4 1 , G 4 4 , G^s and G 4 S have —50Fc d eviation from nom inal. A ll oth er
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 3.2
R E SU L T S OF DIA G N O SIS U SIN G OPTIM IZATION T E C H N IQ U E S
FOR T H E CIRCUIT OF FIG. 3.2, CASE 2
CUT
M easu rem en t
VM
A ctual
AGi/Gi°%
i = 1.2..... 5
D etected AGl7 Gl°% ,i = l .2 ..... 5
ti
Quadr.
Program.
Optim.
#1
#2
.5000
0 .0
16.98
0 .0 0
0 .0 0
.3333
2 0 0 .0
149.06
2 0 0 .0 0
•2 0 0 . 0 0
.1667
.
0.0
- 8 .4 9
0 .0 0
0 .0 0
0 .0
- 3 3 .9 6
0 .0 0
0 .0 0
0 .0
- 3 3 .9 6
0 .0 0
0 .0 0
.5933 -
2 .0
1.95
1.77
5 77
.2207
6 .0
6.08
6.36
0 .0 0
- 3 .0 '
9.68
0 .0 0
0 .0 0
300.0
238.72
288.35
235,S9
3.0
- 1 2 .7 8
0 .0 0
- 13.51
.2683
2 0 0 .0
63.71
199.62
199 04
.1304
40.0
304.67
40.73
41.87
.0660
- 3 .0
93.19
- 0 .0 0
0 00
4.5
378.57
0 .0 0
2 45
2 .0
367.12
- 2 .3 9
0 .0 0
.1755
#3
fi
O p tim
•
✓
o-
.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
52
27
GiJ
28
rh
Fig. 3.4
A r e sistiv e m esh netw ork (2S nodes).
R e p ro d u c e d with perm ission of the copyright owner. F urther reproduction prohibited without permission.
e le m e n t v a lu es are w ithin th eir tolerances. T he d iagn osis w as perform ed su ccessfu lly
w ith on ly one excitation . R e su ltin g d eviation s for G 4 1 , G 4 4 , G 4 5 and G4 8 are -4 6 % ,
—54%, —45% and —53%, resp ectiv ely. D eviation s for other e le m en ts are m ostly zerb
ex cep t for a few sm a ll nonzero v a lu es.
3 .9 .3
G aA s FET M odelling: M ulti-C ircuit Approach
~~
T h is exam ple is due to B andler, Chen and D aijavad (T§8 6 b). T hey used the
eq u iv a le n t circu it at norm al o p eratin g b ias (including the carrier), as illu strated in
F ig. 3.5, and created a rtificia l m easu rem en ts u sin g TO U C H STO N E (1985). Two se ts
*
o fS -p a r a m e te r (scatterin g) m ea su rem en ts w ere created; one se t u sin g the param eters
9
reported by C urtice and C a m isa (1984) (operating bias Vds = S O V, VKS = - 2.0 V and
Ids = 128.0 mA) and th e other by ch an gin g th e valu es o f C i, C 2 , L,. and Ld_to sim u late
th e effect o f ta k in g differen t referen ce pltfties for the carriers. Both se ts of data are
show n in Fig. 3.6, w here th e S -p aram eters o f the two circu its are plotted on a Sm ith
C hart. A lth ough the m axim um num ber o f possible variab les, nam ely 32 (16 for each
circu it), w ere a llo w ed for in the o p tim ization , the in trin sic p aram eters were found to
be the sa m e betw een the tw o circu its, and as expected, C j, C^,
circu it
1
and Lj changed from
to 2. T able 3.3 su m m a rizes th e param eter va lu es Qbtained. The problem
involved 128 nonlinear fu nctions (real and im aginary parts o f 4 S-p aram eters, at
8
freq u en cies, for two circu its), 16 lin ear functions and 32 variables.
3 .9 .4
1
r
A H igh p ass F ilter E xam ple for Postproduction T uning
The h igh p ass notch filte r circuit show n in Fig. 3.7 was used by Bandler and
S a la m a (1981) to dem on strate postproduction tu n in g algorith m s. T he circuit exam ple
#
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
54
•£.<g d
<r
vwv—
Fig
3.5
<3*
Equivalent circuit of carrier-mounted FET (Device model B^824-20C).
*5*—
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
55
\
□
sn
+
S22
*
S21
X
S12
fl:
4 .0 0 0 0 0
fZ:
1 8 .0 0 0 0
m a g -3
r-
0
J
-3
>
Fig. 3 .6
S m ith C hart d isplay o f sca tterin g p aram eters S n , S ? ’, S i 2 and S o ] , for
th e carrier-m oun ted FET, before and after adjustm ents on p aram eters.
P oin ts a and b m ark the high frequency end o f original and perturbed
netw ork respon ses, respectively.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
56
TABLE 3.3
R E SU L T S FOR TH E G aA s FET EXAM PLE
P aram eter
.
O riginal C ircuit
Perturbed C ircuit
C‘
C2
<pF)
0.0440
0 .0 2 0 0
*
(pF)
0.0389
0 .0 2 0 0
*
Cdg
(pF)
0.0416
0.0416
^gS
(pF)
0.6869
0.6869
Cds
(pF)
0.1900
0.1900
Ct
(pF)
0 .0 1 0 0
0 .0 1 0 0
^g
(Q) '
0.5490
.0 .5 4 9 0
Rd
(0 )
1.3670
1.3670
R,
(Q)
1.0480
1.0486
R»
(Q>
1.0842
1.0842
0.3761
0.3763
0.3158
0.1500*
0.2515
0.1499*
G d-> (kQ)
*
Lg
(nH)
Ld
(nH)»
Ls
(nH)
0.0105
0.0105
Km
(S)
0.0423
0.0423
I
(ps)
7.4035
7*4035
•
-
* sign ifica n t ch a n g e in param eter value
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
57
'VW
/W
Q■
v
Q +
VW
AAA/
I
/77
r b
F ig. 3.7
/77
T he h igh p ass notch filter circuit.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
58
w a s o r ig in a lly em p lo y ed by A la ja jia n (1 9 7 9 ).
R3, R s, R 6 a n d R7 a r e tu n a b le
p aram eters. T he nom inal and a ctu a l e le m e n t v a lu es are g iv en in T a b le 3.4.
To use the fu n ction al tu n in g app roach o f (3 .6 1 ), B a n d le r an d S a la m a
defined f] a s th e a b solu te v a lu e o f V out from its n o m in a l, i.e ., u s in g (3 .5 7 ) w ith
F(<}>. u ) =
<o) and Fd(&>) = Vout(<£0, to). T w enty freq u en cies on th e in te r v a l
T h e lim it s in (3 .6 1 b ) a re PL-} = —P y = ^-02.
4 1 0 -5 0 5 Hz w ere u sed .
A fte r 11
itera tio n s, th e tuned resp o n ses v ery c lo se ly approached th e n om in al resp o n ses, as
s h o w n in F ig . 3 .8 ( a ) .
A f t e r t u n i n g , th e v a lu e s fo r t u n a b le p a r a m e t e r s
(R 3 R5 Rg R71 = [2 0 1 .9 5 2 2 .1 1 5
13.061 0.973],
T h e d e t e r m in is t ic a p p r o a c h o f ( 3 .6 2 ) - ( 3 .6 5 ) w a s p e r f o r m e d , w it h
F = [F j F 3 ... F 5 F , w here th e Fi are c o efficien ts in the tran sfer function o f the filter
T = (s2 + F js + F3) “
1
( F 3 s 2 + F 4 S + F 5 ).
B o f ( 3 .6 4 )
w as ta k en
a s d ia g .
{4, 0.0 4 , 4, 1012, 0.0625} and Pj = 0 .0 0 1 . The responses a sso cia ted w ith th e tu n in g
is sh o w n in F ig . 3 .8 (b ).
A fte r t u n in g , th e v a lu e s for t u n a b le p a r a m e t e r s
[R 3 Rs R^ Rt ] are • [1 8 4 .4 8 7 2.241
3 .1 0
13.747 0.9993].
DISCUSSION'S
C lose lin k s and s im ila r itie s e x is t b etw e en o p tim iz a tio n te c h n iq u e s for
m od ellin g , d ia g n o sis and tu n in g .
In t h is s e c tio n , r e le v a n t com m on a s p e c ts are
d iscu ssed .
3.10.1
U se o f S e n sitiv ity Inform ation
S e n sitiv ity M atrix
Suppose f(<J>) is defined by (3.46) for m od ellin g and d ia g n o sis and by (3.55)
or (3.57) for design and tu n in g .
Let <J>° be the d esign nom inal.
D efin e th e n X m
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\
59
4
TABLE 3.4
N om ih al V a lu e -
A ctual V alue
(kQ)
13.260
13.260
0 .0
(kQ)
93.0
93.0
0 .0
3
(kQ)
214.0
192.6
R*
(kQ)
2 .0
Rs
(kQ)
2 .0
Re
(kQ)'
12.467
Rt
(kQ)
1 0 .0 0
Cr
(pF)
0 .0 1
0.00973
1
E L E M E N T V A L U E S FOR THE H IG H PASS FILTER OF FIG. 3.7
C2
(pF)
0 .0 1
0.00965
- 3 .3 5
r
A
1 0 0 0 0 .0
'
P ercen tage D eviation
-
2 .0
1 0 .0
0 .0
-
1 0 .0
1 1 .2 2 1
-
1 0 .0
9 .0 0 r
-
1 0 .0
1 .8
1 0 0 0 0 .0
b
-o
E lem en t
0 .0
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60
20
nominal r e s p o n s e
be fo re tuni ng
dB
-10
gain
-2 0
voltage
after
-30
tuning
-40
-50
-60
300
400
500
frequency
Fig. 3.8(a)
600
700
800
H5
T he resp on ses for the tu n in g o f th e h igh p ass notch Filter u sin g fu n ction al
tuning.
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61
2 0
r
nomi nal
before
after
response
tu ni ng
t u n i ng
gain
-2 0
voltage
dB
-1 0
-30
-4 0
-50
300
400
500 ' 600
frequency
F ig. 3.S(b)
700
800
Hz
T h e r e sp o n se s for th e t u n in g o f th e h ig h p a s s n otch f ilt e r u s in g
d eterm in istic tun ing.
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62
s e n sitiv ity m atrix a s
''
S(«t>) ft d i a g f t 0} ^ —
oq>
d ia g { « * ° )} - 1 .
( 3 '731
«J>* is said to be a regular point (see S a ek s, S a n g io v a n n i-V in c en te lli and V isvan ath an
1981) o f S(<|>) if th ere e x ists an open neighbourhood o f
ran k .
in w hich S ( $ ) h as co n sta n t
P a r a m e te r id e n tific a tio n (or m o d e llin g ) is u s u a lly p er fo r m e d w ith th e
a ssu m p tio n th at the actu al p aram eter <J>“ is a t a regu lar point and Rank[S(<t>a)l = n.
O th erw ise, if Rank[S(<J>u)l < n, i.e., th e m ea su rem en t is n o t'su ffic ie n t, w e sh o u ld
eith er u se th e d ia g n o sis tech n iq u e introduced in S ection s 3.3 to 3.6, or se e k p ossib le
a d d ition al m easu rem en ts by c r e a tin g an y or a com bination o f 1 ) m ore a cc essib le nodes
■
3V.______________________________
_____
for e x c ita tio n and/or m e a su r e m e n t, 2) m ore frequeTTcy"'pbihts7 3) o th e r ty p e s o f
resp o n ses (e.g ., voltage and cu rren t), 4) a d d ition al circu its ob tain ed by p erturbing a
few p aram eters in th e C U T .
R esea rch h as b een perform ed on th e s e le c tio n o f
excitation^ and m easu rem en t ports and freq u en cies (B a n d ler.a n d S a la m a 1985a) as
w ell as th e m ulti-type response and m u lti-circu it concepts, e.g. (B andler, C hen and
Daijavad 19S6b).
In tu n in g problem s, it is desired th a t the su b m atrix co n ta in in g the first n t
rows o f S (a ssu m in g th a t on ly the first nt ele m e n ts in [<J>i
■ <{>nlT are tunable)
has a rank w hich should be a s high a s the rank o f S. Such rank com parison im p licates
the d egree o f difficu lty to a ch iev e th e d esired response by tu n in g <£,, i =
on ly.
1, 2
, . . . . nt
P
■ By checkin g the S m atrix, p o ssib le d ec o m p o sitio n can be c a r r ie d out,
se q u e n tia lly op tim izin g su b sets o f r e sp o n se : vs. v a r ia b le s w h ich are s e n s it iv e ly
I
rela ted (B a n k e r , C hen. D aijavad, K ellerm an n , R en au lt and Z h an g 1986).
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
63
LaW e C h an ge S e n s itiv ity
.
T he em b ed d in g o fla r g e ch an ge se n sitiv ity ca lcu la tio n s in an op tim ization
procedure, w h ere on ly a sm a ll su b set o f circu it param eters are updated each iteration ,
can g rea tly in crea se th e efficien cy. T he application o f H ouseholder’s form ula in fault
d ia g n o sis w a s reported by T em es (1977), Joh n son Jr. (1979) and Chen an d S a ek s
(1979). S u ch ap p lication can reduce re-evalu ation o f F(<{>) fro^n the order o f n to r, r be­
in g a rank m ea su re o f th e su b circu it to be updated, r is less ‘than or equal to the
«
num ber o f param eters in the su b circu it (H aley and C urrent 1985: B andler and Zhang
*
1986).
3 .1 0 .2
C on vergen ce and P o ssib le D ifficu lties U sin g O ptim izatij^^fC chniqucs
For problem s u sin g th e
and m in im ax op tim ization m ethod o f H ald and
M adsen (1 9 8 1 , 1985), su p erlin ea r or quadratic convergence are guaranteed. T he con ­
vergen ce for M errill’s quadratic approach w as reported to be about 2 or 3 iterations.
For a decom posed problem , seq u en tia l op tim ization may d iverge if the subproblem s
are not w ell d efin ed or not reason ab ly ordered. T herefore, it m ay be d esira b le to have
the sy ste m le s s d eco m p o sed a s th e so lu tio n is b e in g approach ed .
U s u a lly , an
o p tim iz a tio n c o n v e r g e s o n ly to a local m in im u m u n le ss th e o b je ctiv e and th e
c o n stra in ts sa tisfy certa in conditions. Global optim ization m ethods are being studied
(Groch, V id ig a l and D irector 1985).
Poor or u n accep tab le resu lts :n com puter-aided circu it op tim ization are felt
to be m ost lik e ly du e to bad preparation o f the problem , a lack of un d erstan d in g o f the .
hazards th a t can be en coun tered and the wrong, choice o f algorith m (B andler 1973).
Com pared w ith o th er tech n iq u es for m odelling, d ia g n o sis and tu n in g (if i^ p lic a b le ),
o p tim ization tech n iq u es often require more com puter tim e and storage. The choice o f
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4
64
s t a r t in g p o in t is o fte n a d e m a n d in g t a s k for s a t is f a c t o r y s o lu t io n a n d fa s t
convergence.
3.11
C O N C L U SIO N
W e h a v e p r e se n te d b a s ic p r in c ip le s o f o p tim iz a tio n t e c h n iq u e s for
m odellin g, d ia g n o sis and tun ing. E m p h a sis is cen tered on the problem form u lation
arid related properties, rath er than m a th e m a tic a l s o p h istic a tio n o f o p tim iz a tio n
procedures and d eta iled circu it a sp ects o f MDT.
F urther research can be directed
a
tow ard e ffe c tiv e m o d e llin g ’t e c h n iq u e s to im p r o v e th e v a lid it y o f id e n t if ie d
v»
*
p aram eters. T he u se and o rgan ization o f decom position needs further in v estig a tio n .
T h e d e sir e d o u tco m e is a n a u to m a tic p roced u re ca p a b le o f id e n t ify in g c ir c u it
■>
p a ram eters and m ak in g d ecision s o f p h y sica l ad ju stm en ts upon m on itored resp on ses
and id en tified p aram eters.
J
e
&
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4
L A R G E C H A N G E S E N S IT IV IT Y A N A L Y S IS O F L IN E A R S Y S T E M S
4.1
IN T R O D U C TIO N
In com puter aid ed circu it d esig n , it is often required to calcu late netw ork
resp on ses a fter a certa in s e t of p aram eters are changed. T h is problem , referred to as
large ch a n g e s e n sitiv ity problem , becom es esp ecially im portant when the network is
large and/or w h en a large num ber o f repeated circu it a n a ly sis is needed.
W ithout
sq lv in g the en tire netw ork eq uations for every se t o f param eter ch an ges, one can
update the n etw ork resp on ses m ore efficien tly by u sin g large ch a n g e s e n s itiv ity
a n a ly sis m ethods. T h is approach has been studied.by m any people. F idler 11976) and
S in g h a l, V lach and B ryant (1973), considered sin g le and m ultiple param eter ch an ges,
resp ectiv ely , and d eveloped m ethods to calcu late th e response function as a m ifltilin ea r form in. v ariab le param eters^ A n oth er m ethod is to fo rm u la te a reduced
sy ste m , w hose so lu tio n s are then used to update the resp on ses T his m ethod has been
treated from d ifferen t a n g le s, e.g ., the current source su b stitu tion approach o f Leung
and Spence (1975), the adjoint n etw ork approach o f T cm es and Cho (1 9 7 3 ). the
H ouseholder form ula approach (Leung and Spence 1975; Hajj 1981), the sca tterin g
m atrix approach o f H aley (1980) and the m atrix partition in g approach o f Vlach and
S in gh al (19S3). Hajj (19S1) derived and sum m arized a se t of algorith m s w here finite,
.(
in fin ite and zero p aram eter ch a n ges are a ll p erm itted and sp a r sity is e x p lo ite d
R auscher and Epprecht (1974) (also, see Gupta-, Gary and Chadha 19S1) used the
concept o f large ch ange se n sitiv itie s to update w ave variab les in an alyzin g perturbed
m icrow ave netw ork s. A recent overview of th is area is given by H alev and Current
■
/
65
^
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66
(1985) who presented general approaches en com p assin g m ost o f the previous m ethods.
A su rvey o f updating m atrix, in v erse form ulas w as given by H enderson and S earle
(1981).
As alread y noticed, large change a n a ly sis a lgorith m s w ill lose efficien cy
w hen too m any p aram eters are changed.
T h is is m ain ly b ecause th e a lg o r ith m s
involve the solu tion o f a reduced sy stem o f order n, th e num ber o f variables. H ow ever,
S
c a se s e x ist w here th is sy ste m is larger than needed. A lso, in ,a M onte-Carlo a n a ly sis
or in an optim ization procedure, it is possible th a t som e variables change slig h tly
w h ile others*change su b sta n tia lly .
In th is ca se, the sm a ll param eter chan ges m ay
4
*
cau se ill-conditionifig in a n on -iterative m ethod (e.g., L eung and S pence 1975). and
the large param eter ch a n g es m av affect th e con vergen ce rate in an itera tiv e m ethod
(Hajj 1981).
I
In th is chajSjpr, we p resen t a set o f gen eralized H ou seh old er form ulas w hich
is capable o f h an d lin g com plicated cases encountered in practice.
d eterm in in g a m inim um reduced sy stem is in v estigated .
T he problem qS
D ifferent asp ects o f th e
basic set o f form ulas are d iscu ssed in term s o f d u ality property and operational count.
A p p lica tio n s to g e n e r a l lin e a r s y s te m s are c o n sid e re d for o r ig in a l and a d jo in t
resp on ses w ith sin g le and m u ltip le input and output situ a tio n s. A lso ,.a s a sp ecial
case, a series o f first-order se n sitiv ity exp ression s are obtained w ith ou t reference to
T elleg en 's theorem .
N u m erical exam p les are g iven for a general sy stem o f lin ear
eq u a tio n s and for two electrica l circuits.
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4 .2
A SET OF G EN ER A LIZED H O USEH O LD ER FO RM ULAS
4.2.1
G en eralized H ouseholder F orm ulas .
•
'
Let the lin ea r sy ste m be ch aracterized by a N X N m atrix A . Suppose the
p a ra m eters $ o f th e sy ste m are ch an ged by A<p. The sy ste m m atrix A w ill then be
affected by AA. W e can exp ress
A A ^ V D W 1,
(4.1)
w here V , D and W are N X r i, ri X r 2 and N X r 2 m atrices, respectively. For a network
exam p le, D can be a n X n d iagon al m atrix con tain in g variab les and V and W are
N X n m atrices con tain in g + 1 and -1 (V lach and S in gh al 1983).
T he efTect o f A<J> in th e response m atrix A -1 is defined a s
A (A _I) 4 (A + A A ) " 1 - A " 1 .
(4.2)
For th e calcu la tio n o f A (A ‘ l ), co m m on ly su g g e ste d is th e H o u se h o ld e r form ula
(H ou seh older 1957), w hich can be represented by
A (A “l ) = - A
-1
V(D-1
WT A -1 V r l WT A * 1 .
N otice th at in order to obtain A(A
(4.3)
), one needs to deal w ith a separate
lin ea r sy stem ch aracterized by m atrix (D ~ 1 -t-WT A~l V).
T h is is u su a lly ca lled a
reduced sy stem and its size is u su a lly sm a ller than the original system .
In (4.3), D is required to be a square and non-singular' m atrix. Even if this
can be sa tisfie d , ill-co n d itio n in g m ay still happen w hen D is inverted. In fact, cases
e x is t w h ere D is sim p ly not in v e r tib le and a d d itio n a l m e a s u r e s su c h a s th e
p a rtitio n in g procedures developed by Hajj (1981), Vlach and S in gh al <1983) m ust he
applied. A nother form ula by H ouseholder (1953) is
A (A _1) = - A
-1
V D (D -r DW t A '
1
V D r 1 DW T A " 1 .
( 4 .4 )
T his form ula avoids a c tu a lly perform ing the inversion of D. But it still has
th e sam e lim itation a s that o f (4.3).
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68
i
A cco rd in g to th e fo r m u la tio n o f D , w e refer to (4 .3 ) a s S q u a r e w ith
Inversion F orm u la (SIF) and (4.4) a s S quare.w ithout Inversion F orm u la (SF).
To a lle v ia te th e lim ita tio n s, d ifferen t variation s o f th e H ou seh old er m atrix
• inversion form u las h a v e been derived (H enderson and S earle 1981).
B an d ler and
Zhang (1986) considered two im portant va ria tio n s and applied th em to lin ea r netw ork
se n sitiv ity a n a ly sis. T h e two va ria tion s are
A (A _1) = - A '
1
VD( 1 + WT A " 1 V D r1WT A '
(4.5)
1
and
A( A -1) = - A
' 1
V(1 + DW X A -1 V ) - 1 DW t A _1
.
T h ese two form ulas perm it D to be sin g u la r or even rectan gu lar.
(4.6)
Thus,
more freedom can be exploited u sin g d ifferen t form ulations o f D and ill-co n d itio n in g
can be avoided.
T h e reduced s y ste m s in (4 .5 ) a n d (4 .6 ) a re th e o r d e r o f ro a n d r !,
r esp ectiv ely , w here n is th e num ber o f row s o f D and rj is the num ber o f co lu m n s o f D.
T herefore, (4.5) m ay be preferred if rj > r2 , o th erw ise (4.6) should be used.
t
It is
reasonable to refer to (4.5) as V ertical R ectan gu lar F orm u la (V R F) an d (4 .6 ) as
H orizontal R ectangular Form ula (H R F), resp ectively, reflectin g the form o f D. Other
v ariation s o f the H ouseholder form ula also e x ist (H enderson and S e a rle I9S1), but the
reduced sy ste m s are as large as th e o rigin al svstem .
T h e ca se o f a rectangular D m ay occur, e.g ., when we co n stru ct a m inim um
order reduced sy stem in volvin g variab les th at are activ e e le m en t p a ra m eters, and
w hen large ch an ge a lgorith m s are applied to algeb raic linear sy ste m s o th e r than
electrical n etw ork s (B andler and Z hang 19S6). In th ose ca ses, th e recta n g u la r D may
be used in VRF and H RF w ithout m odification lea v in g V and W free o f v a lu e s
H ence. V and W need to be preprocessed only once.
)
i>
'
\
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<
69
- It should be noted th a t m ath em atically, the Square Form ulas arc special
cases o f th e R ectan gu lar ones. C om putationally, th e la tter have good sta b ility .
T h e v a rio u s form s o f H ouseholder form ulas p resen ted a b ove w ere a lso
studied by T y la v sk y and S oh ie (1986). T hey a ttem p ted a generalized'rep resentation
'i
and co n n ected th e H o u se h o ld e r form u las w ith o th er m eth od s o f s o ly in g lin e a r
equ ations, e .g ., th e low er-diagonal-upper (LD U ) decom position method.
■v
4.2 .2
^
P rop erties o f G eneralized H ouseholder Form ulas
D uality Property
T he HRF and th e VRF can be considered as dual to each other. If w e apply
the follow in g in terch a n g es
^
A -A t ,
!
14.7)
D **D t
v
(4.8)
and
V **W ,
(4.9)
then the tw o form u las, i.e. (4.5) and (4.6), are com p letely interchanged.
T h is d u a lity property can be em ployed to save our an alytical effort by half.
U n less o th erw ise sta te d , w e w ill focus on th e V er tica l F orm u la in th e e n su in g
sections. R esu lts for the H orizontal ones can be'^amilarly obtained.
T he M inim um O rder o f th e Reduced Svstcm
U sin g th e sca tterin g theory approach, H a ley and C urrent (19851 have
found th a t th e order o f th e reduced system can
be as low a s rank (AA).
U sin g our
approach o f on ly sim p le m atrix m an ipulations one can also verify that
m in
»
(V .D .W )
r; =
min
r0 = ran k (A A ). .
(4.10)
(V .D .W )
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70
T h is eq u ation y ie ld s th e conclusion th at, for e v a lu a tin g large change effects
in v o lv in g H ousehold er form u las, th e m inim um order o f th e reduced sy ste m is eq u al to
the rank o f AA (B an d ler and Zhang 1986).
C onsider th e circu it o f F ig. 4 . 1 in w hich 7 p aram eters are changed from
th eir nom inal v a lu es. By th e con ven tion al m ethods, e.g. V lach and S in gh al (1983),
th e reduced sy ste m is 7 X 7 .
H ow ever, the rank o f th e nodal ad m ittan ce d ev iation
m atrix is 4. T hus, an ev en sm a lle r sy stem o f size 4 X 4 is su fficie n t for th is problem .
O perational Count
C onsider th e com p utation o f A(A-1 ). Suppose ri + r2 < N and the m atrix A
h as already been LU factorized. U su a lly , V , D and W are form ulated-such th a t D
con tain s v ariab les and V an d W indicate the p o sitio n s o f the varia b les and are con ­
sta n t. Preparatory ca lcu la tio n s in v olvin g V and W a re perform ed only once for each
*
s e t o f variab les. T able 4 . 1 g iv e s operational counts (num ber o f op eration s, i.e., m u lti­
p lication s or d ivision s) for th e s e t o f generalized H ou seh old er form ulas. A s show n in
th e table, the com p utational sta b ility o f the HRF and th e VRF is achieved at the cost
o f one m ore m atrix m u ltip lica tio n , a s com pared w ith th e SIF.
Itf should be noticed
th a t these operation cou n ts are for arbitrary algeb raic lin ea r eq u ation s. W hen lin ea r
circu its are concerned, th e operational count is reduced a s d iscu ssed in Section 4.3.
4.3
. C O M PU T A T IO N S O F O RIG INAL A N D A D J O IN T L IN E A R SY ST E M
R E S P O N S E S C O R R E S P O N D IN G TO D IF F E R E N T N U M B E R S O F
IN P U T S A N D O U T P U T S
In th is sectio n , we ex a m in e the com p u tation s o f large ch an ge se n s itiv itie s
in different input and outp ut cases. T he VRF is ap p lied . A ll r e su lts o f Forward and
Backw ard S u b stitu tio n s (FBS) in v o lvin g A are calcu lated in the preparatory step and.
4
f
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
R eproduced
with perm ission
of the cop yrigh t o w n e r.
<5 +
i o a (T)
F urthe r reprod uctio n
■o —
prohibited
r
w ithout p e r m i s s i o n .
Fig. 4.1 '
' An arbitrary 10 node network w ith 7 variable param eters. A ll elem en t
valu es are assum ed as 1 V ariables <}>i, <J>2 . ■■■• 4*7 arc conductances o f the
/
associated com ponents.
-72
TABLE 4.1
O PERATIO NAL C O U N T FOR THE G ENERALIZED
H O U SE H O L D E R FORM ULAS
C a ses
Square w ith
Inversion
Form ula
(SIF) /
V ertical
R ectan gu lar
F orm ula
(VRF)
H orizontal
R ectangular
Form ula
(HRF)
-
cP
Cp
C 2
c.
cp
cp .
Square w ithout
Inversion
F orm ula
(SF) •
C ase I
rl *
r2
preparatory
ca lcu la tio n
ca lcu la tio n for
ea ch se t o f
p aram eter
ch a n g es
C ase
2
ri =
r2
’
~ r
preparatory
ca lcu la tio n
c a lcu la tio n for
each se t o f
p aram eter
ch a n g es
cp
2Ca+ Cb
3C a + C8
3C a —CB
cp
5C A -t-Cg
C p = N"(rj + r 2) + Nr jr.,
C j = r t(2 r jr 0 + r ^ + r.,N -f N 't . C., = r 2 (2 r 1 r 2 + r 2 2 -t- r rN + N 2)
C A = r3, C R = rN lr + N)
I
■if
'
■
R e p r o d u c e d w ith p e r m i s s i o n o f t h e c o p y r i g h t o w n e r . F u r t h e r r e p r o d u c t i o n p r o h i b i t e d w i t h o u t p e r m i s s i o n .
are represen ted by P and p for the'original sy stem (coefficien t m atrix A)
q for th e adjoint sy stem (coefficien t -matrix A T), To d istin gu ish th ese solutifi
d ifferen t R .H .S., w e use th e characters, sim ila r to the R .H .S., a s su b scr ip t.
For
exam p le, P y is th e solu tion o f
APV = V
(4.11)
AT qb = b .
(4.12)
and qb is th e solu tion o f
4.3.1
D ifferen t C a ses for C om puting R esponse C h an ges
.....
C ase 1: R esponse M atrix A - 1
A (A -1 ) = - P v D S Q w T ,
w here S is th e in v erse o f (1 +
(4.13)
WT P y D ) .
—
/
J
C ase 2: S v stem R esponses for a S in g le E xcitation V ector c
Suppose the'resp onse vector corresponding to ex citation c is x = (x; x;> ...
*NlT. :-e„
Ax = c
.
'
W e have
(4.14)
«
Ax = A(A_1 c)
;
= - Py Ds .
(4 15)
w here s is th e solu tion o f
(I + WT P v D)s = WT x .
•^
'
(4 16)
/-
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74
r
C ase 3: A djoint R esponses for a S in g le E xcitation V ector b
Suppose the adjoint response vector corresponding to ex citation b is y = [yi
y s - y s ] 7 . i.e..
AT y = b .
(4.17)
W e have
A y T = A ( b T A _1)
_
s -t
S
0t
(4.18)
W ’
w here s ' i s the solu tion o f
(1 + Q w T V D )T s ' = D T V T q b .
(4.19)
C ase 4: R esponse of S in g le-In p u t and S in gle-O u tp u t (SISO) S ystem
If we use vector b to se le c t the d esired output from response vector x , then
A (bT x) = A (bT A “ 1 c)
= -b 7 Ds
.
.
(4 '20)
w here s is defined in (4.16) and b [ eq u als P y Tb and is obtained in the preparatory
step.
.
C ase 5: R esponses o f M ulti-Input and M ulti-O utput (MIMO) S v stem
Suppose C is a N
vectors and B is an N
X
X n ’
m atrix w hose colum ns rep resen t different ex citation
m ’ m atrix w hose colum ns sele ct the d esired ou tp u t m ea su re­
m ents. Theft th e n '-in p u t m '-o u tp u t case can be exp ressed , form ally, by BT A '
T hus
1
C.
“v
A (B r A '
1
C) = - B T A -
1
V D (I + Wt A ' ! V D ) _1 WT A '
1
C
(4.21)
We notice th a t th e term B TA ‘ l V can be com puted e ith e r as B T P v or Q bTV
with a d ifference o f operation al count as X - In - m'). T herefore, com paring rj and m ’,
we can ca lcu la te (B T A * 1 V) as
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75
A
BT P v , ifr t £ m '
bta
(4.22a)
-1 v =
V ,
ifr > m'.
I
(4.22b)
, WT P
C = I
I Q^VC
ifr,' > n'
(4.23a)
O
S im ila rly ,
WT a
-1
.
if r , £ n'.
(4.23b)
A lso, a t le a st one o f (4.22a) and (4.23b) should be used in order to yield either P\- °r
Q w w hich is required in ca lc u la tin g
(1 + WTA_1 VD) = (1 + QWT VD)
= (1 + W t P v D ) .
(4.24)
Hence', according to the v a lu e s o f r j, r2 , m ’ and n ’, we can ch oose appropriate
form ulations. For ex a m p le, w hen m' < n' and m ’ < r>, we use
A (BT A - 1 C) = - S T Q WT C ,
(4.25)
w here S is the so lu tio n to
(
(I
Q w T V D )t S = (Q b T V D )t .
(
14 26)
T h is approach r e q u ir e s in' — ro F B S in th e ad join t sy ste m for Q 3 and Q w as
✓
preparatory ca lcu la tio n s, one LU factorization and m' F B S in the reduced system of
(4.26).
^
E xp ression s for D ifferent C ases o f Large C hange E valuation
f
In Table 4.2, we sum m arize the various c a s e s o f th e ab ove d iscu ssio n .
D ifferent situ a tio n s o f the MIMO case arc d istin g u ish ed so that the num ber of F B S in
the N
X
N sy stem eq u a ls the m inim um of m ’ + r^, n' •~'ri arid ri - r-> and the num ber
o f F B S in the reduced sy stem eq u a ls the m inim um of r i, r_>, m' and n ’, as show n in
.
*
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76
\
TABLE 4.2
FO R M U LA S FOR T H E C O M PU TA TIO N OF LARGE C H A N G E S
W H E N A ' 1 IS IN V O LV ED A N D W H EN ^ a r 2
Iden tification
F orm ula
D efinition o f S or s
J
Hj S = lo r H2 S =
A(bT A " 1)
- st O
“ wT
V ®
A(A -
- Pv D s
H 2 s = WT L c
c)
A (bT A
~ 1
c)
A (Bt A
" 1
C) .
(b T P v ) D s
H„ s = WT p
(1)
- S t (Q w t C)
(2 )
-(B T P v )D S
- - H o S = WT'P C
(3)
- ( Q bt V ) D S
H t S = Q wt C
(4)
- ( B t P v )D S ( Q vvt C)
Hj S = I or H 2 S =
(5)
- ( Q bT V ) D S ( Q w t C)
h
/ s = d t(v t q b i
'
w here Hj =
^
(1
1
t-*
1
1
= D T( ^ q b)
(/)
II
+
• - p v d s q wt
A ( A " 1)
X
'
*
+ Q WT V D) , Ho = (1 + WT P V D)
T able 4.3 can be used a s a gu id e to s e le c t am ong (1) to (5) by the m inim um FBS
, cril-Crion.
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T ab le 4.3. T h is m in im um F B S criterion can be u sed a s a guide to selec t appropriate
ex p ressio n s for th e calcu la tio n o f A(BT A '
1
C).
W hen th e n um b er o f F B S e x c e e d s th e order o f th e s y s te m , a m a trix
in version m ay be d irectly performed.
4 .3 .2
D iscu ssio n s
C om putational C ost C onsideration
In S ectio n 4 .2 .2 , the operational count h a s been discussed for a geneVallin ea r sy ste m o f eq u ation s. H ow ever, w hen an electric circu it is concerned, the cost is
m uch le ss.
W e consider the SISO netw ork a s an exam ple.
Suppose th e reduced
sy ste m is o f order r. In th e preparatory step , w e ca lcu la te P y whose operational count
is rN*2 and P Tv b> WT P v an<^
add itions.
x w h ich a re sim p ly e le m e n t s e le c tio n s and
T h en, for each se t o f param eter ch a n g es, we fo rm u la te and so lv e the
reduced sy stem by at w orst 4r 3 / 3 - r / 3 + r- operations.
The operational count for
\
updating*the output is r for the SIF and r + r ’ for the H RF and the V RF.
.-
f
S pecial Case: First-O rder S e n sitiv ity
A s a sp e c ia l c a se o f Iftrge c h a n g e s e n s it iv it y a n a ty s is . sm a ll c h a n g e
s e n s itiv ity com p u tation s can be deduced from our large change form ulas w ith o u t
re fe r en ce to T e lle g e n 's th eorem . T a b le 4 .4 g iv e s e x a m p le s o f su ch fir st-o r d e r
s e n s itiv itie s w .r.t. com ponents o f a m atrix. T h ese resu lts are obtained by putting A<p
into the denom in ator o f large change form ulas and then lettin g the param eter change
A(p approach zero. The form ulas in T able 4.4 are co n sisten t with the ex istin g ones
derived u sin g other approaches, e.g., B andler (1973).
.
J
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78
TABLE 4.3
MAJOR CO M PU TA TIO N A L EFFORT FOR C A LC U LA TIN G A(BT A '
BY FO R M U L A S IN TABLE 4 .2 W H ERE r.
•
%
*
C ategory
N o .o fL U
F actorization s
No. o f FBS
V
_
C orresponding
C ase in
T able 4 .2
(IM S )
(1 )
The N X N S y stem
R epresented
By A
'
.
y*
The ro x r 2 S ystem
R epresented
■By H i or H 2
X2
.
m'
(2 )
n'~+ r p
n'
(3 |
m ’ + r0
n'
(4)
rl +
r2
(5)
C)
I
1
m' +
1
r2
m ’ + r.,
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\
79
TABLE 4.4
E X P R E SSIO N S A PPRO PRIATE FOR CO M PUTATIO NS FOR SENSITIVITIES
W .R .T. C O M PO N EN T S OF MATRIX A W H EN A “ 1 IS INVO LVED
S en sitiv ity E xpression
Id en tification
(a) G eneral
(b) w hen A = A T-and i * j
dA~1
aA
—(P ut P ujT - P UJ P UIT> •
IJ
a(bTA - l c)
aA
a(BTA ~ l C)
aA
-< lb p c
-B T p
q TC
^ U l M U]
^ P b Pc + Pc Pb >
- B T(p
p T + ^pU J p T)C
* UJ
ij
a[BTA ~ 1C]flt
dA
Uj (Uj )
,
(^
T
-<lbPc
is a u n it N-v ector'con tain in g
1
at the ith (jth) row and zeros everyw here
t
else.
+
- ( P b P ,T - p ; p bT) ++
"where t*' tv is th e (£,k) th elem en t of m atrix *.
+t
w here b is the fth colum n o f B and c is the kth colum n of C Both b and c are
h
u sed a s the R.H.S. o f th e sy stem in volvin g A for original solu tion s p b, pc and
adjoint solu tion q h.
*
*
*
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80
4.4
LA R G E C H A N G E S E N S IT IV IT Y
A N A L Y S IS t)F
v ''
. •
L IN E A R IZ E D
CIRCUITS
In th is sectio n , we illu str a te how to form ulate the large change se n sitiv ity
*
problem o f an electrica l c ir c u it in to th e a lg e b r a ic r e p r e se n ta tio n s p r e se n te d in
S ection s 4.2 and 4.3.
H ere, w e in trod u ce th e c o n v e n tio n a l fo r m u la tio n .
A new
form ulation is presented in the n ext section.
Suppose a lin ear circu it is represented by
A x = b,
(4.27)
w here A i s a N x N m atrix ch ara cterizin g th e n etw ork, b is a N -vector-rep resen tin g
the ex cita tio n and x is a N -vector co n ta in in g sy ste m responses.
equation (4.27) is the nodal equ ation s o f the lin ear circuit.
W hen sy stem param eters <t>i, <{>2 .
A sim p le form o f
^
—■
<t>n are changed, c a u sin g the ch a n g e of
A by A A , resp on se c h a n g e s can be c a lc u la te d by large ch a n g e fo rm u la s.
T he
com m only used m ethod is to express AA a s a trip le product as (Hajj 19 8 1)
A A = V D WT
(4.28)
or u sin g p aram eter m atrix decom position o f A A as (H aley 1980; H aley and C urrent
1985)
iA = V
—
v ^
1
I
w T, r S n .
(4 '29)
1
1= 1
The response ch an ges Ax are then calcu lated u sin g the H ouseholder form ula or its
0
various eq u iv a len ts.
T h ese ca lcu la tion s in v o lv e the solution o f a reduced sy ste m
w hose size is d eterm in ed from- th e form ulation o f V, D and W. .We focus on this
form ulation. S u b seq u en t ca lcu la tio n s lead in g to A x can be perform ed according .to the
«
- •
p resen tation in S ection 4.3.
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U sin g th e w ell-esta b lish ed m ethods (e.g., Hajj 1981; Vlach and S in g h a l
•1983; H aley and C u rren t 1985), one can g en erate a n X n reduced sy ste m for un
arbitrary lin e a r netw ork by choosing
-
'
D = diag.{A^1, ^ 2,...,A 4.n},
\
V = [v t v
(430)
..vj-
-
<4 3 l >
f
and
W = (Wj w 2 ... w j
w here vj and wq i =
1 ........
,
n are N -veators c o n ta in in g
(432)
± 1
and
0
.
i=
1
, .... n
represent th e v a lu e o f variable i, being o f the tvpe th at en ter the tableau or m odified
nodal eq u ation s in th e form v, <pi WiT (Hajj 1981)
J
•—'*
It can be se e n th a t th is form ulation g iv e s each variab le an equal treatm en t
%
and no consid eration regardin g topological rela tio n s o f th ese variables is taken into
account.
f.
In a c a se -w h e r e the num ber o f p ertu rb ed v a r ia b le s is not very sm a ll
corfipared to the order o f the orig in a l system , th e efficiency o f large change algorith m s
is greatly d egen erated . Such a case occurred in E xam ple
8 .1 .1
o f Vlach and S in gh al
(1983) w here a 3 X 3 sy ste m had to be solved in order to update the response of a 2 x 2
.sy ste m , m erely b ecau se 3 varia b les exist,
-
A 'further tjsduction o f the reduced sy stem is m ade possible by the discovery
that the order o f such a sy ste m can be as low as the rank o f the original sy stem
d eviation m a trix (H aley and Current 1985).
T his m a n ifests its e lf as a m in im u m
sy stem (B andler and Z hang 1986). Such a m inim um sy stem can be achieved by a
thorough ex p lo ita tio n o f th e topological relation s am ong variables.
4
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4.5
'
D
A N EW FO RM ULATIO N OF V ,
TY PE S
A N D W FOR V A R IA BLES O F RCL
■\
*
A new form ulation o f V , D and W w as d eveloped to ach iev e th e m in im u m
order reduced
sy s te m for la r g e c h a n g e s e n s it iv it y co m p u ta tio n .
It w a s b r ie fly
introduced by B andler and Z hang (1986). A sy ste m a tic d escription is g iv en here.
Let the netw ork topology be represented b‘y graph G and th e edge s e t o f G
be represented by E, resp ectiv ely .
Let E' be a su b set o f E su ch th a t an ed ge in E
corresponding to a v ariab le is cla ssifie d in E ' . The induced subgraph o f G on ed ge set
%
E' is denoted a s G ' . S ep arate G' into blocks G\ , G?
U
G b \ b ^ 1, such th at G' = G\
Go' U ... Gb' and Gi’H G}' is e ith e r null or em pty c o n ta in in g only a cu t-v ertex o f G‘
for all i j = l,2 ,...,b and i.« j.
R e le v a n t te r m in o lo g ie s u sed here a r e d e fin e d in
A ppendix A.
j.
•
For RCL type v a ria b les in a linear netw ork, V , D and W can be form ulated
u sin g nodal rela tio n s in stead o f th e conventional branch r ela tio n s so a s to ach iev e a
m inim u m order reduced sy stem .
V, D and W are N X r , r X r and N X r m atrices,
resp ectiv ely and a re decom posed such th at
^
V - = [ ^ i V 2 ... v b] ,
(4.33)
W = [W , w
(4 . 3 4 )
2
. . . w bl
and
D - d ia g { D t , D o , , D b} ,
(4.35)
w here D; is the nodal a d m ittan ce m atrix o f G f u sin g the A ^fas p aram eter and V, and
»
W, a rc.in cid en ce m atrices o f G,' in d icatin g vertex location s o f G;’ a s seen from G.
Suppose G i* has m vertices, m Sr 2.
Dj is (m —I ) X ( /n — 1) sin c e one vertex can be
con sidered as "ground” and is ta k en as a referen ce v e r te x .
V; a n d W t a r e both
N X ( m - l) w here each colu m n vector corresponds to a n on -reference v ertex o f G;'. If
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th is non-reference v ertex and th e reference vertex o f G y appear in G as the kth and f S .
f th v ertices, resp ectiv ely , the corresponding colum ns o f V\ and W; are equal to u^ — u j
or Ufc, if th e fth v ertex corresponds-to th e ground,
b ein g a unit N -vector w ith I in
its kth position and zeros every w h ere else. For m ath em atical sim p licity , a vertex in
G y is taken
a referen ce v ertex if it corresponds to the ground o f th e overall circuit.
4 .6
EX A M PLES
4.6.1
A Sy stem o f L inear E q u ation s W ith R ectangular D
C onsider a 10 X 10 sy ste m o f linear equation:? w ith coefficien t m atrix as A.
Suppose th e in tersection ele m e n ts o f rows
2
, 5, 9 and colum ns 3 and
chan ged. W e form ulate V, D and W such th a t
W = [U 3 U 6 1 "
are constantly
1
V = [ U2 U5 ug I ,
j
6
‘
(4.36)
'
„
and
(4.37)
*
.'
^23
^26
(4.38)
D =
w here u t , i = 2 ,3 ,5 ,6 ,9 , is a u n it 10-vector con tain in g
1
in th e ith row and zeros
v
ev ery w h ere else. In th is w ay, no a d d ition al effort is involved w hen applying the VRF
and HRF.
If we use Vhe Square F orm ulas, elem en tary tra n sfo r m a tio n s m u st be
em ployed in order to obtkin a square m atrix D.
N u m erical so lu tio n s a s w elt as in term ed iate resu lts are show n in F ig 4.2
-
f
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
VECTOR ( B)
HATRIX: . [ * ]
1.0
5.0
5.0
1.0
S. O
2.0
1.0
1.0
7-0
- 2. 0
35.0
2.0
3.0
3.0
7.0
0.0
4.0
3^ 0
6.0
8.0
3-0 '
32.0
3.0
0.0
2.0
4.0
2.0
6.0
4.0
4.0
9.0
7.0
16.0
6.0
1.0
2.0
5.0
2.0
3.0
3.0
7.0
3.0
5.0
51.0
8.0
1.0
2.0
2.0
4.0
4.0
6.0
8.0
4.0
8.0
- 42.0
4.0
I'.O
6.0
7.0
3.0
5.0
7.0
3.0
5.0
3.0
19.0
.7.0
0.0
6.0
5.0
9.0
470' 8.0
9.0
2.0
■9. 0
34.0
;2.0
-0.0
4 JO
2.0
2.0
5.0
3.0
5.0
4.0
3
71.0
3.0
2.0
0.0
1.0
S.O
3.0
4.0
2.0
3.0
1.0
36.0
4.0
2.0
4.0
4.0
6.0
2.0
9.0
6.0
1.0
7.0
6?.C
m
SOLUTION BEFORE ANT CHANGE VECTOR [ X)
-8.8921.7
"
39.80097
■\
-3.00067
2.31014
-5.40S44
48.42778
-12.11626
-3.61726
-32.93004
16.99799
Fig. 4.2(a)
T he origin al lin ea r sy ste m and its so lu tio n s. <pA is a 1 0 X 1 0 m a trix
co n ta in in g p aram eters o f th e system , b is th e-excitation vector, x is the
solu tion vector.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
85
MATRIX [ V)
(
MATRIX [ V]
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
0.0
0 :0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
■o.o
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
\
MATRIX [PV
-.03684
-.30799
{
.19072
.00936
-.26526^
.04838
.00454
.09406
-.04645
-.23600
.02949
.02608
-.09579
.12002
-.48948
-.43199
.13012
.18919
.22984
.01487
.27060
.13754
. 36 321
.32717
.02238
-.27658
-.20670
.09789
.15865 '
«
.00846
‘ VECTOR [ RHS]
-3.00067
48.42778
)
F ig. 4.2(b)
M atrices V , W, P v and vector R H S , w here P v is the solu tion o f A P v =
V a n d R H S = W Tx .
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
86
«
(1ATRIX [ D]
2.00000
3.00000
4.00000 '
S . 00(500
2.00000
3.00000
-s
.
j
KATRIX [ H]
1.06802
.00797
-2.44666
-2.23801
VECTOR [ S ^
^
-2.66983
.
-18.72004
SOLUTION AFTER. THE F I R S T LARGE CHANGE
VECTOR [ X]
8.15496
m
-3.82546
-2.6698
-18.72004
24.40133
27.88727
22.14824
-27.58607
Fig. 4.2(c)
R e s u lts co r r e sp o n d in g to th e fir st c h a n g e o f v a r ia b le p a r a m e te r s
rep resen ted by D. H rep resen ts (1 + WT A - 1 V D ) and s is the so lu tio n o f
-*the reduced sy ste m H s = W Tx.
J
1
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
v a r ia b l e s
[vj’
and
chanc e
[U ]
a c a in
r e m a in
*
c a u s in g
ch anc e
op
[D].
un c h a n g e d .
MATRIX [DJ
6.00000
7.00000
5 . 00000
A .00000
3.00000
4.00000
KATRIX [H]
1 .0 2160
-.22657
-4.70646
-3.633B3
VECTOR [ S I
-4.57768
-7.39775
SOLUTION AFTER THE SECOND LARCE CHANCE
VECTOR [X]
-2.20615
3.56798
-4.57788
4
-12.47901
-3.16642
-7.39775
15.58395
25.41279
12.05534
-20.00992
Fig. 4.2(d)
R esu lts corresponding to the second change o f variable param eters,
and s are sim ila rly defined to those in Fig. 4.2(cl.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
88
4 .6 .2
An E lectrical N etw ork w ith Its M inim um Order Reduced S y ste m A ch ieved
T h e 10-node circu it o f F ig. 4.1 is solved u sin g th e gen era lized H ouseholder
form ulas w ith sim u lta n eo u s c h a n g e s o f ^ v a r ia b le com ponents. T opological relation s
sh ow in g th e netw ork graph G, th e induced subgraph o f G on edge se t E' and th e blocks
*
are g iv en in Fig. 4.3. G is divid ed into G t' and G J . T he m inim um order o f the
reduced sy stem is 4\ w hich is a ch iev ed by form u latin g V , D and W as
V =
tV i
= [U3 - U9
u 4 —u 9
u 8 —U9
0
0
0
0
0
0
0
0
'l
0
fr
0
0
1
0
1
0 >
0
0
0
0
0
0
0
0
0
0
•o
0
1
0
-1 #
-1
0
W
u —u
4
5
0
(4.39)
-I
-1
0
0
0
(4.40)
= V
and '
D.
D =
Aif> t + A<£„ + A<{>^
—A4 >^
-A $4
A(j>.4 + A«J>5 -i- A4>6
-A 4>2
“ ^ s
0
0
—A4>,,
0
0
^<t>2 ^
(4.41)
0
0
A 4),
Ci
N otice th at nodes 9 a n d 5 jy iv e been taken as referen ces for G i' and G V .resp ective!
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
89
to
\
(a)
9
(b)
g;
g;
(C)
Fig. 4.3
T opological rela tio n s for th e circu it o f Fig. 4.1.
induceci'subgraph G' and. (c) B locks.G i' and GV
(a) Graph G, (b) Fdge
*
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
90
T he ch a n g es o f variab les range from 0.00001 to 90. Zero ch a n g es are also
included a s show n in T able 4.5. T h ese sim u lta n eo u s sm a ll, large-and zero ch an ges are
handled directly by the VRF. f o r the two ex trem e c a se s o f A<b, the SIF can handle
»
A<}>—►<» w h ile the VRF and H RF accom m odate A({>—*•().
In a M onte-Carlo a n a ly sis,
netw ork op tim ization, id en tification and tu n in g, variou s unpredictable p a ttern s o f
A<{>—*0 in m u ltiparam eter ch a n g es m ay be possible w h ile A<$>—
is often lim ited by,
e.g., tolerances and tu n in g ra n g es or by step size co n strain ts. For 100 se ts o f variable
ch an ges o f $ i to <$>7 , the o p eration al c o u n t for ou r m eth od u sin g S IF ,*V R F . the
con vention al m ethod and the d irect m ethod are in the order of 11230, 1^030, 24930
and 43430. resp ectiv ely ’.
4.6 .3
The C ase o f E xam ple
8
.1.1 o f V lach and S in gh al (1983)
C onsider the circu it o f Fig. 4.4 w here G i, G-> and G3 are a ll v a r ia b le s.
E vidently, the "reduced" sy stem is of-order
2
u sin g the nodal based approach' w hich
giv es
A Gj
AG.,
-A G ,,
D = AA =
-a g 2
(4.42)
^ g 2+ a g 3
and
1
0
0
1
V = W = 1 =
C om pared w ith th e branch based m ethod w hich y ield s a 3 x 3 sy stem , th e operational
count is reduced frpm 23 to 16 for each se t o f valu es o f AGS, i — 1,2,3.
A lthough for th is circu it, one w ould rath er so lv e th e o r ig in a l n etw ork
e q u a tio n s,th a n use large chan ge form ulas, such a variab le structure can e x is t in a
large sy stem a s a subnetw ork w here an efficien t large ch an ge algorithm is ex tre m ely
im portant.
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91
TABLE 4.5
PARAM ETER C H A N G E S FOR THE CIRCUIT OF FIG. 4H
V ariable
^
T h e F irst
C hange
(I/O)
A $,
84.0
A <}>2
a
$ 3
**6
A<1>7
* '
T he Third
Changp
(I/O)
0 .0 0 0 0 1
0 .2
0.5
0 .0 0 1
0
0 .0 0 0 0 1
0 .1 2
3.0
0 .0 2
A<*>5
T he Second
C hange
(I/O)
•
40.
50.
0 .0 0 0 0 2
45. ■
0
0.00003
0 .0 2
90.
'
15
-2.
0 .1
\
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92
-o +
Vout
Fig. 4.4
T he sim p le circu it from V lach and S in g h a l (1983).
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
93
4.7
C O N C L U SIO N S
‘
1
.
.
W e have presented a m u ltip aram eter large change s e n s itiv ity a n a ly s is
approach for a general sy ste m in v o lv in g so lu tion s of lin ear equations.
Particular
a tten tio n h as been d evoted to the form ulation and order o f the reduced sy stem , which
in turn a ffects the sta b ility and efficien cy o f the sy stem response evalu ation .
The
m a th em a tica l essen ce o f th e g en eralized H ouseholder form ulas also provides basic
lin k s w ith o th er approaches, in d icatin g their theoretical eq u ivalen ce.
H ow ever, our
exten d ed form u las accom m odate more ca ses o f various form ulations o f the reduced
sy stem w h ich th e trad itional m ethods can n ot handle d irectly.
For a general circuit
w ith arbitrary d istrib u tion o f variabl^ com ponents, proper form ulations o f V , D and
W can be used to ensu re th e large ch an ge calcu lation to be performed via a m inim um
order reduced system . T hus, under certa in circum stances, large change algorithm s
are still fea sib le even if m any sy stem p aram eters are changed. Our work was recently
referred to by H aley and P ham (1987) as one o f the d istin ct, useful contributions to the
a n a ly sis o f m odified sy stem s. It is eftvisaged th at a general form ulation o f V , D and
W, togeth er w ith th e se t o f H ouseholder fo rm u la s/ca n be em bedded into the different
itera tiv e and n o n -iterative m ethods o f Hajj (1981) to y ield various powerful design
procedures.
>
-O
*
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J
<
5
E X A C T S IM U L A T IO N A N D S E N S IT IV IT Y A N A L Y S IS O F M U L T IP L E X IN G
NETW ORKS
V
5.1
IN TR O D U C TIO N
M any circu its can be ca teg o rized or reduced to the c la ss q f b ranched
cascaded netw orks. T he sim u la tio n and se n sitiv ity ev alu ation of such netw orks can
h e directly perform ed u sin g g en eral so ftw a re *vhich s o lv e s nodal e q u a tio n s and
adjoint netw orks.
H ow ever, w hen the circuit becom es large, the general m ethods
often d eterio ra te ra p id ly .
On th e oth er h an d , th e cascad ed stru c tu re (w ith o u t
branches) has been treated u sin g 2-port tran sm ission m atrices (e.g. Bandlcr. Rizk and
A bdel-M alek 1978; Iobost and Zaki 19S2)
Such treatm en t has been very efficien t
esp ecia lly for large cascaded circuits.
B andler, D aijavad and Zhang (1985, 1986) developed a novel and eleg a n t
approach to the sim u la tio n and se n sitiv ity a n a ly sis o f branched cascaded circu its
,
T h ey ex p licitly took th e circu it structure into consideration. The forward and reverse
a n a ly sis m ethod o f B andler, Rizk and A bdel-M alek (1978) was extended to general
branched cascaded netow rks. Our theory perm its a n efficien t and fast an alytical and
num erical in v e stig a tio n o f responses and se n sitiv itie s o f all functions of interest w .r.t
an y variable p aram eter, in clu d in g frequency.
T h even in eq u ivalen t circu its at any
referen ce plane and th eir se n sitiv itie s are a lso exp ressed a n a ly tica lly and calculated
sy stem a tica lly .
T h u s, respon ses such as com m on port return^to^s. branch o u tp u t
return loss, in sertion or tran sdu cer loss, gain slope and group delay can be handled
.e x a c t ly and efficien tly.
1
i
A ll a n a ly se s arc perform ed in the original circu it and no
94
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95
adjoint netw orks are needed in s e n s itiv ity co m p u ta tio n .
M ore im p o r ta n tly , th e
m ethod does not d eterio ra te for large circu its sin ce p ossib le redundant sto rage and
com putational req u irem en ts are elim in ated by ex p licit e x p lo ita tio n o f th e c ir c u it
structure.
*
“
In th is approach, each basic com ponent o f the stru ctu re is eith er a 2-port
m odel or a ^3-port ju n ctio n and can contain variables or be co n sta n t. T he fundam ental
requ irem en t for th e approach is th a t the tran sm ission m atrix d escription o f a ll b asic
com ponents and th eir d e r iv a tiv e s, if they co n ta in v a r ia b le s, a re provided.
T h is
inform ation is u tilized in a s y s te m a tic and e ffic ie n t sc h e m e w h ich le a d s to th e
ev a lu a tio n o f various resp on ses o f the netw ork at all ports o f in terest.
M icrow ave m u ltip le x e r s c o n s is tin g o f m u lti-c o u p le d c a v ity filte r s are
stru ctu rally branched cascaded. The d esign o f co n tigu ou s band m u ltip lexers w as a
problem o f sig n ifica n t th eo retica l in terest for sev era l y ea r s (A tia 1974; C hen, A ssal
and M ahle 1976), how ever, th e m anufacturing o f such stru c tu re s w ith m ore th an 5
ch a n n els did not appear to be feasib le.
R ecently, the subject h as tu rn ed in to an
ijAportant d evelopm ent area in m icrow ave en g in eerin g practice due to reports by
lea d in g m a n u fa ctu rers o f s u c c e s s fu l production o f
12
c h a n n e l c o n tig u o u s band
m u ltip lex ers for sa tellite- a p p lication s (Tong et al. 1982; C hen 1983; E gri, W illia m s
and A tia 1983; H olm e 1984). T he em ploym ent of o p tim ization tech n iq u es to d e te r ­
m ine th e best m u ltip lex er param eters has been an in d isp en sab le part o f th e d esign
procedures reported.
T h e use o f a powerful gradient-based m in im ax o p tim ization
tech n iq u e has reduced the C PU tim e required in th e d esig n procedure sig n ifica n tly
(B an d ler, K ellerm an n and M adsen 1985; Bandler, D aijavad and Z hang 1986).
-
T he im p lem en ta tio n o f a gradient based o p tim ization tech n iq u e in m u lti­
plexer d esig n requires, a s a v ita l step , a robust and efficien t algorith m for sim u lation
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and s e n sitiv ity a n a ly sis. T h is is ach ieved by applying our branched cascaded a n a ly sis
technique.
T h is chap ter is organized in the follow ing way. W e first describe the basic
cascaded a n a ly s is app roach a s a p p lied to a -g e n e r a l b ran ch ed ca sca d ed circu it.
F orm ulas for T h ev en in e q u iv a le n ts, reflection coefficien ts and branch output voltages
a s w ell a s th eir first- and second-order, s e n s it iv it ie s w .r.t. d e sig n v a r ia b le s and
frequency a t any referen ce p lan e are developed.
p resen ted to illu str a te our theory.
A 4-branch, ca sca d ed cir c u it is
W e consider m u ltip lexers co n sistin g o f m u lti­
c a v ity filte r s d istrib u ted alo n g a w aveguide m anifold.
T ran sm ission m atrices and
s e n sitiv ity ex p ressio n s for typ ical com ponents in a m u ltip lexer, which are required by
our approach, are tab u lated . T h e optim ization o f a 12-channel 12 GHz m u ltip lexer is
described.
5 .2
/
B R A N C H E D C A SC A D E D NETW O RK S
T he category o f a g e n e r a l-c la ss o f netw orks, n am ely, branched cascaded
stru ctu res, can be depicted as in Fig.' 5.1.
For such stru ctu res, we develop a novel
procedure to ca lcu la te the reflection coefficien ts at th e com m on port and branch
o u tp ut ports as well' a s branch ou tput voltages.
order d eriv a tiv es are e v a lu a te d
S im u lta n eo u sly , first- and second-
T h e approach is b a sed on th e co m p u ta tio n o f
T h e v e n in sou rce and im p e d a n c e e q u iv a le n ts and th e ir first- and seco jy i-o rd er
s e n sitiv itie s w.r.t. d esig n p a ram eters and frequency at the ports o f in terest
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R eproduced
with perm ission
section N
section
section k
1
CD
of the cop yrig ht o w n e r.
branch output
—1 -j
p o rts
bronch
(cha nn el)
subsection
|bronch input
(s p a c in g )
i— ■ p o rts
F urther rep ro du ctio n
I
i
t -
—o
—o
I •
r
1
I
2N
prohibited
&2 H
2 N*2
w ithout p e r m i s s i o n .
source
port
2 N«I
T
I
I
I
I
2N
—o
—o
o—
o—
T
2k
2H-1
r
i
i
to
T
12 k-
I
I
i
i
2H-1
2M1
2k
2k -1
1”
reference
planes
m oin cascade
term ination
common
port
r'i|» 5 . 1
,
The brunched cascaded nelw orlm inder consideration. The ju n ction s ore
arbitrarily denned 3-port junctions. B ranches or ch an n els are rep re­
sented in reduced cascade forms. Adjacent ju n ction s are separated by a j
su b se ctio n . P rin cip al co n cep ts o f r e fe r e n c e p la n e s , tr a n s m is s io n ‘
m atrices and typical ports are illu strated
>
98
5.2.1
^
P relim in a ry D escription o f the N etw ork
M odels o f B asic C om ponents
I
A lth o u g h th e basic com ponents o f a branched cascaded circu it are 2-port
►
e le m e n ts or 3-p o rt ju n c tio n s , in te r n a lly th e y ca n be co m p lica ted su b n e tw o r k s
characterized by a d m ittan ce, im pedance or hybrid m atrices. An exam p le o f such a
subnetw ork is th e m ulti-coupled ca vity filter described by an im pedance m atrix and
j
co n ta in in g m any d e sig n variables. A s a p rereq u isite step towards u sin g our theory,
th e tr a n s m is s i^ i m atrix for each
2
-port e le m e n t sh o u ld be d ed u ced e ith e r by a
red u ctio n p ro ced u re or by d ir e c t m e a su r e m e n ts.
A lso, if v a r ia b le s e x is t in a
subnetw ork, th e d e r iv a tiv e o f th e co rresp o n d in g tr a n sm issio n m a trix sh o u ld be
provided. For th e 3-port ju n ctio n s, how ever, a 3-port description in the form o f an
arbitrary hybrid m atrix, is su fficient.
R eference P la n es
C onsider th e branched cascaded netw ork o f Fig. 5.1. w hich c o n sists o f N
sectio n s. A typ ical sectio n , e .g ., the kth one, has a ju n ction , n(k) cascaded ele m e n ts of
branch k and a su b section alo n g the m ain cascade, as show n in Fig. 5.2. A ll reference
p lan es in th e en tire netw ork are defined uniform ly and num b ered c o n se c u tiv e ly
b e g in n in g from the m ain cascade term in ation , w hich is designated reference plane
1.
T h e source port is a t reference plane 2N + 2. T h e term ination o f the kth branch is
called referen ce plan e t(k) and the branch m ain cascade connection (branch input
port).is referen ce p la n eo (k ), k = 1 ,2 ,...,N . w here
t(l) = 2 N + 3
'
o (k) = i(k ) + n ( k ) ,
k = 1 ,2 ,..., N
.
t<k) = c (k — 1) -t- 1 , k = 2 , 3 , . . . , X .
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
(5 . 1 )
tr a n s m is s io n
matrix
re feren ce
plane
+ V k■
----------- Q ------O -----------Rk
H
At
^
- ( >—
<
-----------O - - - O
r + j -1
e le m e n t
j
i
cr- 1
elem ent
n(k)
p
A -----
2
k
2k
2k*1
2k-1
2k
Fig. 5.2
D e ta il o f th e kth se c tio n o f a b ran ch ed c a s c a ded c ir c u it s h o w i n
referen ce p la n es a lo n g the branch w here t = 't(k) and o = o(k).
(
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100
5 .2 .2
R eduction o f J u n ctio n s to 2-port R epresentations
B andler, Rizk and A bdel-M alek (1978) introduced the concept o f forward
0
an d r e v e r se a n a ly s is for ca sca d ed n etw o rk s.
co n sid eration to a cascade o f
2
'
To sim p lify th e str u c tu r e u n d er
-ports for w hich the forward and reverse a n a ly sis is
ap p licab le, th e 3-port junctions are reduced to 2-port rep resen tation s.
C onsider th e 3-port ju n ction show n in F ig. 5.3. ^To carry the a n a ly s is
through th e ju n ction alo n g th e m ain cascade, we term in ate port 3, e.g., by calcu latin g
the eq u iv a le n t a d m ittan ce seen a t th is port given by Y 3 = (—l3 )/V 3 and represent tho
tra n sm issio n m atrix b etw een ports 1 and 2 by A, T h e a n a ly sis can also be carried
th r o u g h th e ju n c tio n in to th e b ran ch by te r m in a tin g port
2
, e .g ., c a lc u la t in g
Y 2 = (—Io)/V 2 and d en o tin g the tra n sm issio n m atrix betw een ports 1 and 3 by D.
A s an exam p le, suppose th e 3-port junction is characterized by a hybri^T'"”’- '
m atrix H such th a t
,
*
[ \\
It
I3 ]T = H [ V
i3
2
v
^
%
3 it ,
(5 2)
w here H = [hjj]3 X3 . T hen A = [a,j]2 x2 can be found from
a.1 j = ( — D *- 1 [ h1 j.. — h i 3 h3 ] /(Y 3 + h 33 ) 1 .
15
3)
For variou s form s of hybrid m atrices H , the 2-port representation A or I) is
** ev a lu a ted in a sim ila r m anner u sin g e le m en ts of^H and the e q u iv a len t term ination at
p o r t3 o r '2 .
5 .2 .3
C ascaded A n a ly sis
, H a v in g red u ced th e ju n c tio n s to
2
-port r e p r e se n ta tio n s, the netw ork
stru ctu re betw een an y two referen ce planes is transform ed to a sim p le cascade o f twoports. A ssu m in g th a t the tra n sm issio n m atrices for all 2-ports are given , we define
th e eq u iv a len t tr a n sm issio n m atrix betw een reference p lan es i and j by
^
j
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
f
101
junction
A 3-port junction in w hich ports 1 and 2 arc considered alo n g a m ain
ca scad e and port 3 rep resen ts a ch an n el or branch o f the m ain cascade.
*
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
w here
«
•
A ..
»J
„
a
pu -
C ..
R
B
, q
u
-
u
D
'J
(5.5)
>J
In a forward (reverse) a n a ly sis, Q*j is com puted by in itia lizin g row vectors
u tT and uoT (colum n vectors Uj and uo) at referen ce p la n e i(j) and s u c c e s s iv e ly
p rem u ltip ly in g (p ostm u ltip lying) each tr a n sm issio n m a trix by th e r e s u ltin g row
(colum n) vector u n til reference plane j(i) is reached, u j and uo are u n it vectors g iv en
by (I
0]T a n d [0
11'1', resp ectively.
k.
Let $ be a generic notation th a t can be u sed to r e p r e se n t any d e sig n
variable in th e netw ork. 'S e n sitiv itie s o f Qij w.r.t. any variab le $ located betw een
reference p la n es i and j are evalu ated as
oQ
-2 1 =
0$
w h ere
1$
j
V
±
,Qf j
f f. I
-
<5 6)
1J
is a n in d ex s e t w h o se e le m e n ts id e n tify th e t r a n s m is s io n m a tr ic e s
co n ta in in g
and
is th e resu lt of a forward or rev erse a n a ly s is b etw een
reference p la n es i and j w ith th e fth m atrix-replaced by its d erivative w.r.t. 4 > Secondorder s e n s itiv itie s can be d erived in a sim ila r m anner as
° “ iJ _
\~
0
y iJ
[ iT‘i'
1°
(5^l5w ?77 k~
w here
and
are index se ts, not n ecessarily disjoint, id en tifyin g those m atrices
w hich are functions o f $ and to. A lso, we d efine 6 se n sitiv ity o f
a s if $ and
6 u>)
as the second-order
e x ist only in the fth and kth m atrices, respectively.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
5.2.4
' T h ev en in E q u iv a len t C ircu its and B asic R esponses
To ca lcu la te the input reflection coefficien t a t th e lcom m on port, the output
reflection co efficien ts a t th e b ra n ch o u tp u t p orts, a s w e ll a s th e b ran ch o u tp u t
v oltages and th eir se n s itiv itie s in a unified m anner, w e em ploy T h ev en in e q u iv a le n ts
at the ports o f in terest e v a lu a ted by th e m ethod o f forward and r e v e r se a n a ly s is
(B andler, Rizk and A bdel-M alek 1978). D en otin g the T h even in eq u iv a le n t vo lta g es
and im pedances at reference p la n e s i and j by Vgi, Zg‘, Vgi and Zgi, we have
v*
,.i
S
s = ------------'------
.. + z* a
a
ij
S
(5.8)
ij
and
Zg -
B i r ZS D l}
(5.9)
'w h ere reference plane i is located tow ards the source w.r.t. j, a s show n in Fig. 5.4. T he
s e n s itiv itie s are obtained as
/
,
(VM - ['(A . .) + Z‘ (C .). * (Z‘ ) C. 1 \ %
bo
ij $
S
ij q>
. Set
ij
S
<V ) = --------------------------- :----------------------*
Ai) + Z s‘ Cij
(5
_
im
(o.iu)
and
- zs
1 Z“](Q
.)
S
n j$
+ (Z‘ ) ID. - Zj, C .)
s -t>
1
(ZJA =
■
1J
s
ij
(5.115
A . + Z‘ C .
Ij
S . 1J
w here subscript <f>d enotes 3/o<f>.
If th e r e fle c tio n c o e f f ic ie n t a t th e k th b r a n c h o u tp u t p ort a n d its
s e n s itiv itie s are to be calcu lated , th en 1519) and (5.11) are sp ecialized to
Z^
s
* 1
-
D
—
A
(5.12)
and
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104
rtf*f«nce
r»»«r*ne*
z
(a)
- 4 -
CD-
vj
Fig. 5.4
i
T h ev en in and N orton eq u iv a len ts a t reference p la n es i and j, w here
referen ce plane i is tow ards the source w .r.t. referen ce p la n e j. <u)
referen ce plane j is in the m ain cascade, (b) reference plane j is in a
branch.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
w here A « A 2 n + 2 ,i + i . & m B 2 N + 2 ,c + l an<* * ” c(k). T h is is sim p ly due to th e fact
th a t there is no im pedance to the left o f reference plane 2 N + 2, i.e., Z$ 2 ^
'+ 2
= o. The
corresponding output reflection coefficien t is defined as
r*C + 1 f ^ k
S
~ *T.
kj
-
w
(5.14)
w here R l 11 is th e load resista n ce a t the kth ch an n el output. C lea rly , (5.13) is utilized ,
in the ev a lu a tio n o f (p1^ a s
.
k
=
‘
(pl*
(5.15)
*r+<4
B ranch o u tp u t v o lta g e is a ls o c o m p u te d by u t i l i z i n g th e T h e v e n in
eq u iv a le n t voltage source and im pedance at the branch o u tp u t port.
A t th e kth
{
th a n n e l w e have
*
Rk
^
Vk =
A<
® 16)
+ 2s +1)
’
I a ssu m in g a norm alized ex cita tio n at the source port. T h is can be e a sily exp lain ed by
n oticin g th a t
is ev a lu a ted u sin g a voltage d ivid er once V s 1 *
(5.S) and ta k in g into account th at Vg'-N+S
=
1
1
is known.
U sin g
and Zs2N + 2 = 0. w e h a v e V gi + l
= 1 1A. A lso
i-t- U
(A)
(Vk)
= -
r£
<z;
s>
vk
■&
(5.17)
The second-order s e n s itiv ity oV \*k w .r.t. <p and to, i.e .. t32 Vk/(d<pdcu)
«.
♦
r
obtain ed via ev a lu a tio n o f d- Z sl + 1/(<3(p 8 (a).
S u b s titu tin g co for cp in (5 .1 3 ) and
d iffe r e n tia tin g w .r.t. cp, g iv es
,r » l + U
o
J
4*0
(B)
vw
- Z‘ ~ l (A) " I (A) (Z‘ + 1) - <Z‘ + l ) (A)
o
^ -------<$>” —
5
L>
«♦>
.
"
.
(5
18)
k U. LOJ
\
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106
w here double sub scrip t $to d en otes 62 /(3<J> du>).
*
§
__
N ow , rep lacin g <J>by w in (5.171 and d ifferen tiatin g w .r.t. <{>, w e have
(V k) (Vk)
= ------♦------ «
(V k}
o* y*
Vk
-t A (A) — A A
--------& ____ * -J± +
^
I
A2
(Z‘ * V (Rk + Z l+ 1) - (Z‘ +1) (Z l+1)
5
<fcu> L b
o
u £> 9
cr£
(5 19)
+ z^+ Lr
N o rto n e q u iv a le n t a d m itta n c e s a n d c u r r e n t s o u r c e s a r e c a lc u la te d
siiiR larly to th e T h ev en in eq u iv a len ts. D en otin g the Norton eq u iv a len t cu rren ts and
a d m itta n ces a t referen ce plan es i a n d j by
y
Y l ‘. ItJ and YLJ, we have
C. + YJ. D
‘i
L ij
: =
(5.20)
A . + YJ, B . .
ij
I ij
and
(5.21)
I. L = IL S s 0 A lso,
[ —Y ‘L
1
! (Q i J )<p
(Y lV =
+ (YJ ) (D
Y
A
ij
1J
-Y !
L
B )
I J
(5.22)
+ YJ, B
L ij
•As special c a ses of (5.20), the eq u iv a len t ad m ittan ces Yj and Y-j required in
th e reduction o fju n ctio n s to 2 -port rep resen tation s are calculated as
yk
3
ycnk.i
L
'oik>. Uk>
ckk),
k = 1 .2 .: ... N
(5.23)
U k)
and, fo r a sh o rt-circu it m ain cascade term in ation .
Yk = Ya =
2
L
2k.1
B 2k , 1
k = 1 .2
N .
(5.24)
T he com m on port reflection co efficien t is also com puted u sin g the Norton
e q u iv a le n t (at the source reference plane) as
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107
0
.
2 R S ^ 2 N + 2.l
— rr.
p = i -
(5.25)
-
2N + 2.I
Its se n sitiv ity is g iv en b y '
(B) D - C D ) B
( p \=
2
(5.26)
R,
*
S
.
. B2
w h e r e B ■ B 2 N + 2 .I anc* ^ m ^ 2 N + 2 .t-
5.2.5
R esp onses o f In terest
Suppose th e frequency resp on ses o f in te re st include return loss, in sertio n
loss, tran sdu cer loss, gain slope and group d elay for each individual branch and the
return lo ss a t th e com m on port. T able 5.1 provides eq u ation s for c a lc u la tin g th e
responses and th e ir s e n s itiv itie s w .r.t. d e s ig n p a r a m e te r
It is c le a r th a t th e
ev a lu a tio n o f reflection co efficien ts a t th e com m on port and branch output ports (p°
and pk), branch ou tp u t v o lta g es (V*) and th e first- and second-order s e n s itiv itie s
ap°
ap k
avk
avk
a2 v k
Aj)
a<t>
a4>
aw
a<}>aw
a s described in S ectio n 5 .2 .4 , is su fficien t to com pute all responses and s e n s itiv itie s
tabulated.
5.3
ALGORITHM FOR C A L C U L A T IO N OF T H E V E N IN E Q U IV A L E N T S
A N D T H E IR SEN SIT IV IT IES
T he follow in g algorith m can be used to obtain T h even in e q u iv a le n ts a t
output ports and th eir s e n s itiv itie s w.r.t. any variable. T he algorith m a ssu m e s th a t
the tr a n sm issio n m atrices for a ll
2
-port e le m e n ts and th e hybrid m a tr ic e s for a ll
ju n ctio n s, a s w ell a s th eir se n sitiv itie s are given . T he reverse a n a ly sis a lo n g th e m ain
cascade is in itia liz e d by u;> for a short circu it term in ation or, m for an open c ir cu it
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108
T A B L E 5.1
V A R IO U S F R E Q U E N C Y R ESPO N SES A N D TH E IR SEN SITIV ITIES
E X P R E SS E D IN TERM S OF BASIC VOLTAGE RESPO NSE
A N D REFLECTION C O EFFIC IEN T
R esponse
Type
E xpression for
S e n sitiv ity w.r.t.
Form ula
return loss
(com m on port
or ch ann el
ou tp ut port)
- 2 0
cRe
lo g jp l
4 | V k|2 R
transducer lo ss *
*® ^
|Vkl (Rs +
in sertion lo ss *
( Vk).
cRe
r £)
(V k)
cRe
—2 0 log 10
( V k)
( V k)
cRe
g a in s lo p e '
irk
cRe
•Jxj
t
( Vk) ( V k)
^
( V k)2
( V k)
( Vk)
— Im
group d elay r
i
—Im
(JkaI
( Vk) ( V k)
<0
(Vkr
20
C
in
10
+ b etw een com m on port and channel k output port
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109
term in ation. C orrespondingly, the r esu ltin g a n a ly sis is rep resen ted by q vectors (as
in the algorithm ) or p vectors.
Step 1
For k =
S tep s
Step 1.1
1,
1 .1
C alcu late
2 ,.... N , se t o and t to o(k) and t(k ), re sp e ctiv ely , and ex ecu te
to 1.7.
+ i by reverse a n a ly sis f o r i = i +
Qoj by forward a n a ly sis for j = o, o - l ........t +
C om m ent
1,
t + 2, ...,o . C alcu late
1.
C ascaded a n a ly s is is perform ed on th e k th b ran ch .
T h e r e v e r se
(forward) a n a ly s is sta r ts from the branch ou tp u t (input) port and is
carried to th e branch input (output) port.
Step
1 .2
P<R — Qo.t + i A t e !. .
C a lcu la te Y 3 k u sin g (5.23).
C om m ent
The e q u iv a le n t a d m ittan ce o f the-kth branch, look in g from the branch
input port, is com puted. *This a d m itta n ce is u tilize d in th e 2-p ort
rep resen tation o f th e kth junction. ' .
*'
Step 1.3
C alculate dp^/dcf) u s in g (5.6) and dY 3 k/d<$> from (5.22) for a ll th e
variable <}>’s in the kth branch.
C om m ent
S e n s itiv itie s o f th e branch eq u iv a len t ad m itta n ce w .r.t. all varia b les
«*
in the branch are calculated. In e v a lu a tin g dY 3 k/ 3 4 >, w e u se a sp ecial
ca se for (5.22) w hich corresponds to Y3 k g iv en in (5.23).
Step
1 .4
C alcu late A^k u sin g (5.3). C alculate 3 A 2 k(°<{> for a ll th e variable $ ’s in
the kth ju n ction and the kth branch.
C om m ent
T he 2-port rep resen tation o f t ^ kth ju n ctio n , w h en ter m in a tin g its
port 3, is com puted. T he s e n s itiv itie s o f th e r e su ltin g tra n sm issio n
m atrix are rea d ily obtained.
Step 1.5
C alculate.q^k.i by reverse a n a ly sis and Y>k from (5?24).
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110
‘ «
0
\
C om m ent
The eq u iv a len t a d m itta n ce a t port 2 o f th e k th ju n c tio n , looking
tow ards th e m ain cascade term in ation , is calcu lated after a reverse
a n a ly sis from reference plane
- Step
1 .6
1
to reference plane
2
k.
‘ C a lc u la te dqjk.i/dcp u sin g (5.6) and dY 2 k/d<p u sin g (5 .2 2 )-ifo r a ll
v aria b les <p in section k \ k' < k and in the kth spacing.
C om m ent
T he s e n s itiv itie s o f the eq u iv a len t ad m ittan ce Y 2 k, w .r.t. all variables
g eo m etrica lly located to the right o f ju n ction k, are com p u ted .
In
e v a lu a tin g dY^Vdtp, we use a special case for (5.22) which corresponds
to Y 2 k, g iv en in (5.24).
S tep 1.7
‘ C alculate. D^k u sin g th e m ethod described in S ection 5.2.2. C alculate
dDjit/dcp for all the v ariab le <p’s in th e kth ju n ction and sp acin g, as welt
as in a ll k' sectio n s, k' < k.
C om m ent
T he 2-port rep resen tation o f junction k w hen te r m in a tin g its port 2, is
com puted.
T he s e n sitiv itie s o f the r e s u ltin g tr a n s m is s io n m atrix
w .r.t. a ll varia b les, included in or located to the right o f the junction,
are com puted.
S te p 2
t
C a lc u la te q 2 N-*-2 .i by e x te n d in g th e r e v e r s e a n a ly s is a lr e a d y
perform ed up to referen ce plane 2N in Step 1.5, to referen ce plane
2N + 2. N ote th a t A o.n has been evalu ated in Step 1.4.
C alcu late dq2N + 2 ,i/<)<P u sin g dq 2 N,i/d<p (ip b e lo n g s to th e se t o f all
v ariab les to the right o f section N and the N th spacing), which has
been ev a lu a ted in Step 1.6 and dA>\,7d<p <(p belongs to the se t of all
v a r ia b le s in th e N th branch and N th ju n c tio n ), w h ich h as been
evalu a ted in Step 1.4.
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111
/
C om m ent
T he reverse a n a ly sis from th e m ain cascade term in a tio n is carried
back to the source port.
calcu lated.
T he corresp on d in g s e n s it iv it ie s are a lso
T h ese r e su lts a r e u sed to c a lc u la te th e com m on port
reflection co efficien t and its se n sitiv itie s w .r.t. a ll v a r ia b le s in th e
1
en tire network.
Step 3
For k = Nr, N '-l, .... 1, se t o and t to o(k) and i(k ), r e sp ec tiv e ly and
ex ecu te Steps 3.1 to 3.3.
Step 3.1
C alcu la te Q2^ + 2 . 2 k + 1 W forward a n alysis.
C om m ent
T h e forward a n a ly sis is carried a lo n g th e m a in c a sc a d e from the
^
Stop 3.2
source port to th e input port o f ju n ction k.
Q2N +2. i + l
C alcu la te
Q2N +2.2k+1 Djk Qo.t + l
3 Q2 N + 2.
t + i^<{) u sin g (5.6) for a ll th e v a riab le <|>'s in the
e n tire m u ltip lexer.
C om m ent
A cascaded a n a ly sis from th e source port is carried through th e kth
ju n ction into the kth branch. T he se n sitiv itie s w .r.t. all v a ria b les are
com puted.
S tep 3 3
C alcu la te Vs i + * and
3Vgt-+
u sin g (5.8) and (5 .9 ).
A lso , c a lc u la te
and <3Zsl + 1/t)<j> u sin g (5.10) and (5.11) for a ll v a r ia b le s <J> in
th e en tire netw ork.
C om m ent
T h even in eq u iv a len ts and th eir se n sitiv itie s are com puted for th e kth
branch output port.
T h e theory and the a lg o rith m have been im p lem en ted in to a co m p u te r
program for sim u la tio n and se n sitiv ity a n a ly sis o f branched cascad ed netw orks. T h e^
■
-
*
num ber o f b ranches and th e num bers o f branch elem en ts are user d efin ed .
E xact
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112
se n s itiv ity a n a ly sis can be perform ed w .r.t. any variab le, including frequency. A b r ie f
description o f the com puter program is av a ila b le in A ppendix B.
5.4
. A 4-B R A N C H CA SC A D ED NETW ORK EXAM PLE
T h is exa m p le w as presented by B andler, D aijavad and Zhang (1985a) to „
i
illu str a te th e b asic concepts for sim u la tio n and s e n s it iv it y a n a ly s is o f b ran ch ed
cascaded netw orks.
T he circu it d iagram is show n in Fig. 5.5.
b ran ch es, i.e., N = 4.
T h e c ir c u it h as 4
T he num bers o f cascaded e lem e n ts for the 4 b r a n ch es a re
n (l) = 3, n(2) = 4, n(3) = 3 and n(4) = 2. A ccording to Eq. (5.1), the reference p lan es
for branch te r m in a tio n s are i ( l ) — 11, t(2) = 15, t(3 ) = 20 and t(4) = 24
'
The
*
A
reference p la n e s for b ra n ch -m a in c a sc a d e c o n n e c tio n s a re o (l) = 14. o(2) = 19.
o(3) = 2 3 a n d o ( l) = 26.
T he com puter program described in A ppendix B w as used R ecalcu late all
respon ses o f in te r e st and th eir s e n s it iv it ie s .
T a b le 5 .2 lis ts v a lu e s o f com p u ted
resp on ses (e.g., output vo lta g e, T h even in eq u iv a len t voltage and im pedance, in sertion
and return loss) for each branch. T he com m on port (i.e. a t referen ce plane 9) return
/
loss “is a lso ev alu ated .
T ab les 5 .3 - 5 .S provide s e n s it iv it ie s for each re sp o n se in
T able 5.2. T h ese s e n s itiv itie s arc ev alu ated w .r.t. circu it variab les <$>,, i = 1 ,2 , ...
8
T able 5.9 sh o w s se n sitiv itie s o f the circu it resp on ses w.r.t. frequency w. G ain slope
and group d ela y resp on ses are listed in T a b le 5.10.
5 .5
C O M PU TER -A ID ED DESIGN OF-MICROW AVE M U LTIPLEXERS
5.5.1
A n a ly sis o f Specific M u ltiplexer Stru ctu res
W h ile th e approach developed in S ection 5.2 is gen eral, as a special ca se,
the d esign o f m u ltip lex ers c o n sistin g o f coupled cavity filters d istrib u te d a lo n g a
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
113
------------C
O
^
n
m
co
----------------
fs.
------------
C5
Fig- 5
5
C
O
Illustration of an arbitrary 4-branch cascaded circuit with short-circuit
termination of the main cascade. I^ossy elements as well as tran sm ission
lines are included.
------------C
M
o
/
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T A B L E 5 .2 I
N U M E R IC A L V A L U E S OF TH E R ESPO N SES FOR THE 4-BRANCH
C A SC A D E D NETW ORK OF FIG. 5.5
1
T ype o f
R esponse
B ranch l f
Branch 2
Branch 3
Branch 4
o utput
voltage
0 .0 3 6 2 4
-jO. 07487
-0 .0 7 5 9 5
- j 0.06875
0.05983
-jO.04039
-15.00361
+ j 1.16405
T h ev en in
eq u iv a le n t
voltage*
0 .0 3 0 0 8
-jO.07785
0.03529
-jO.30176
0.03193
-jO.08172
-1 5 .6 5 3 4 6
-j'2.31876
T h ev en in
e q u iv a len t
im pedance*
0.00003
-jO. 08225
0.72129
+ j2 .41490
0 .00004
-jO.69080
0.02515
-fjO.23408
55.57S 92
53.76940
56.81050
10.42942
0 .00 0 5 5
1.72670
0.00052
'o. 41430
in sertion
lo ss (dB)
branch port
return loss
(dB)
com m on port return lo ss = 0 .4 1 2 4 3 dB
B ranch
1
is th e fu rth est from th e co'tnmon port.
T h ev en in e q u iv a le n ts for each branch are evalu ated at the reference plane ju st
before th e load corresponding to that branch.
J
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
TABLE 5.3
SENSITIVITIES OF BRANCH OUTPUT VOLTAGES W.R.T.
VARIABLE PARAMETERS FOR THE CIRCUIT OF FIG. 5.5
V ariab le
(p e r b m )
<t>4
B ranch
1
B ranch 2
Branch 3
B ranch 4
-0 .0 9 8 8 8
+ J 0 .19690
. 0 .0 1 6 0 2
-FjO. 0 1904
-0 .1 2 1 5 2
+ J0.07920
-0 .0 6 1 4 8
-FjO. 4 3 3 8 2
-0 .0 2 1 7 8
-Fj0.03689
0 .0 0 0 0 8
+ j 0 .00013
-0 .0 0 0 8 3
-FjO.00037
-0 .0 0 0 8 1
+ j 0 .0 0 2 6 3
0.41S40
-j 1.02730-
0 .4 2 3 4 0
-FjO.49683
-3 .1 7 4 6 1
-Fj2.09775
-1 .5 4 0 7 4
+ j 11.39034
-0 .0 0 0 1 5 .
•FjO.00018
0.02421
-FjO.02442
-0 .0 0 1 5 2
■ -Fj0.0007S
-0 .0 0 1 2 3
-FjO.00500
-0 .8 4 5 8 3
■ -j 1.25308
(per Gm)
0.42131
- j l . 0 1718
-1 .0 5 6 4 7
-jO.85004
0.75952
-jO. 57964
- 1 .3 2 1 6 1
+ j l0 .3 0 7 S l
0 .00216
-FjO.00231
0 .0 0 3 4 7
-j'0.00175
0.00061
-FjO.00267
0.16241
-FjO.04932
0.03997
-jO. 13157
- 0 .1 4 1 6 8
-jO. 09279
0.08734
-j0.0S 130
-1 2 .4 2 4 3 1
-FjO. 17372
♦k
+
o
o
o
o
o
o
0 .0 0 0 0 0
o
o
o
o
o
o
-FjO.OOOOO
o
o
o
o
o
o
+
,o
o
o
o
o
o
(per Gm)
t.
/
/
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
116
. TABLE 5.4
SENSITIVITIES OF THEVENIN EQUIVALENT VOLTAGE SOURCES
W.R.T. VARIABLE PARAMETERS FOR THE CIRCUIT OF FIG. 5^5
s
Branch 3
Branch 4
- 0 .0 8 2 1 7
+J0.20531
-0 .0 1 8 3 7
+ J0.07139
-0 .0 6 6 6 4
+ j 0 .16342
-0 .2 0 5 9 9
-jO. 03020
-0 .0 1 5 5 6
+ J 0 .0 4023 •
-0 .0 0 0 1 9
+ j0 .0 0 0 4 3
-0 .0 0 0 5 8
+ j0 .0 0 0 9 5
-0 .0 0 1 2 5
-jO. 00039
-1 .7 2 1 0 6
+ j4.29S 05
-5 .4 0 8 6 7
-jO.75803
-0 .0 0 0 9 7
+ j0 .0 0 l8 3
-0 .0 0 2 3 7
-jO. 00060
- 0 .0 0 0 1 4
+J0 . 0 0 0 2 0
-0 .0 1 4 2 7
+ j 0 .08775
+
o
o
o
o
o
o
(per Gm)
.
.
-0 .4 6 2 1 1
+j_l. 1S227
0 .3 3 9 6 4
-j 1.05 097
0.24151
-J4.04905
$7
0 .0 0 2 3 5
-i-jO.0 0213
-rj0.00537
0 .00246
+ j 0.00225
0.15861
-r-jO.03550
$
»H
0 .0 2 9 1 6
-JO. 13486
-0 .0 1 9 8 6
-j0 .5 0 lS 6
0.03119
-jO. 14165
-1 3 .3 5 1 9 9
‘ -j 1.80362
(perG m )
0 .0 1 0 2 1
0.36033
-j 1.10264 ’
o
b
o
o
o
o
-0 .4 7 0 0 1
+ j l . 87577
+
0 .3 3 5 7 2
-j 1.06095
o
b
o
o
o
o
Branch 2
o
b
o
o
o
o
--{per fem)
1
+
o o
b b
o o
o o
o o
o o
V ariable
B ranch
-4 .8 9 4 S 0
-jO.65144
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 5.5
SE N SITIV IT IES OF T H E V E N IN E Q U IV A L E N T IM PE D A N C E S W .R.T.
VARIABLE P A R A M E T E R S.fO R T H E CIRCUIT OF FIG. 5.5
to*
B ranch 1
V ariable
B ranch 4
*i
-0 .0 0 0 1 7
+ j 0 .00707
■0.00019
+ J0.00077
- -0 .0 0 0 1 6
+ j 0 .00450
0 .0 0 0 3 8
+ j 0 .0 3072
<t>.
-0 :0 0 0 0 3
+ j 0 .0 4250
O
©
o
o
o
©
B ranch 3
+
©
o
o
o
o
o
K
' Branch 2 _
- 0 .0 0 0 0 1
+ j 0 .00003
-0 .0 0 0 0 3
+ j 0 .00019
0.00085
+ j 0 .02364
0.00501
+ j0 .0 2 0 0 6
-0 .0 0 3 3 4
+ j 0 .11797
0 .0 1 4 9 8
+ j 0 .80592
0 .0 0 0 0 0
0.06160
+ j0 .1 1 2 2 2
- 0 .0 0 0 0 1
+ j 0 .00005
0 .0 0 0 0 0
-0 .0 0 1 2 9
+ j3 0 .9 3 S 4 S
(per Gm)
+ j0 .0 0 0 0 0
‘
0 00000
(Jbm )
+ j0
00000
+JO.OOOOO
(per Gm)
0.00066
+ j 0 .02605
0.17452
-i-jO.29796
4\.
0 .0 0 0 0 0
0 .0 0 0 0 0
+ j0 .0 0 0 0 0
~ j0 .0 0 0 0 1
0.00005
+ j0 .0 0 0 0 0
0.00058.'
-jo,:ooo2v&f
*
-0 .0 0 0 0 3
+ j 0 .00036
0 .0 0 0 0 0
-
+ j0
00000
&
0.00051
+ j 0 .02879
0 01S60
-r J0.72S54
0 .0 0 0 0 0
10.00052
-j-j0.00350
4-jO.OOOOO' ' *
, V* 0 .00005
+ j0 .0 0 0 0 0
0.042S3
-jO.05844
2
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
•
118
T A B L E 8.6
S E N S I T IV I T I E S O F IN S E R T I O N LO SS W.R.T.
V A R IA S L E P A R A M E T E R S FO R T H E C I R C U IT O F PIG . 5.5
V ariab le
+i
B ranch 1
Branch 2
2 3 .0 0 5 6 8
2 .09034
4 .45823
Branch 3
Branch 4
17.45152
-0 .0 5 4 7 5
0.01270
0.10816
-0 .0 0 0 5 9
-1 1 5 .5 9 0 9 6
54.88169
457.33905
0 .0 2 4 1 2
2 .9 1 1 4 4
0.20388
-0 .0 0 0 9 3
0 .0 0 0 0 0
0 .0 0 0 0 0
0 .0 0 0 0 0
0 .0 0 0 0 0
-1 1 4 .7 7 1 6 8
-1 1 4 .7 7 1 6 8
-1 1 4 .7 7 1 6 8
-1 .2 2 0 7 4
0 .1 1 8 5 9
0 .11859
0.11859
0.09126
-1 4 .1 3 4 6 6
-1 4 .1 8 4 6 6
-1 4 .1 8 4 6 6
-7 .1 5 7 4 0
-
-1 .3 9 5 1 7 •
\ .(perfcrn)
(per Gm)
(p erG m ) .
<t>T
C'
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119
T A B L E 5.7
' S E N S I T I V I T IE S O F B R A N C H P O R T R E T U R N L O S S W .R .T.
V A R IA B L E P A R A M E T E R S F O R T H E C IR C U I T O F FIG . 5.5
. ^
4>,
(per Gm)
-
Branch 2
B ranch 3
Branch 4
-0 .0 0 2 9 2
-0 .0 0 0 5 0
-0 .0 0 1 8 3
0 .00055
-0 .0 0 0 5 7
0 .0 0 0 0 0
-0 .0 0 0 0 6
-0 .0 0 0 5 0
0.01 4 6 5
-0 .0 1 2 7 8
-0 .0 3 9 1 7
0.09851
0 .0 0 0 0 0
-0 .0 0 0 7 1
-0 .0 0 0 0 8
-0 .0 0 0 5 9
0 .0 0 0 0 0
0 .0 0 0 0 0
0 .0 0 0 0 0
0 .01133
0.02123
0 .00598
0.17232
- 0 .0 0 0 0 1
- 0 .0 0 0 0 2
- 0 .0 0 0 0 1
-0 .0 0 9 1 3
0 .0 0 0 8 4
0.00155
0 .00063
0.71641
B ranch 1
V ariab le .
.
0 .0 0 0 0 0
(per Gm)
(per
6
4>7
m)
t
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
120
T A B L E 5.8
S E N S I T IV I T I E S O F C O M M O N P O R T R E T U R N LOSS W .R.T.
V A R IA B L E P A R A M E T E R S F O R T H E C IR C U IT O F FIG. 5.5
V ariable
S en sitiv ity
0.00533
-
0.00004
<{>3 (per
Gm)
0.13797
0.00003
**
<}>} (per Gm)
0 .0 0 0 0 0
<J>g(perG m )
0.12286
*7
**
•
*
4
-0 .0 0 9 0 9
0.71310
\
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T A B L E 5.9
SE N SIT IV IT IE S OF V A R IO U S R E SPO N SE S W.R.T. v
A N G U L A R FR E Q U E N C Y u FOR T H E CIRCUIT OF FIG. 5.5
1
Branch 2
Branch 3
B ranch 4
output
voltage
- 0 .1 7 7 7 8
+ j0 .3 3 1 2 0
0 .03944
+ j0 ,0 8 7 9 1
-0 .4 4 1 0 0
+ J0.26906
2 .3 9 0 6 8
+ j7 .3 6 2 3 5
T hevenin
eq u iv a len t
voltage
- 0 .1 4 8 8 9
+ j0 .3 4 6 6 6
-0 .1 0 8 8 0
+ j 0 .09567
-0 .2 4 3 8 4
+ j 0 .59055
0 .1 5 8 8 8
•fjO.51010
T h even in
eq u iv a len t
im pedance
- 0 .0 0 0 2 8
+ j 0 .02219
0.73081
+ j l . 3 2535
.-0.00051
+ j0 .2 8 l0 1
" -0 .0 0 1 3 8
+ j0 .5 0 6 2 4
branch port
return loss
-0 .0 0 4 8 4
-0 .0 0 5 9 0
-0 .1 1 5 9 7
Type o f
R espon se
B ranch
-0 .0 0 1 5 3 .
se n sitiv ity of'com m on port return loss = -0 .1 0 4 6 0
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122
T A B L E 5.10
G AIN SLO PE A N D G R O U P DELAY FOR TH E CIRCUIT OF FIG. 5.5
T ype o f
R esponse
g ain slope
(dB/Hz)
group delay
(s)
B ran ch
B ranch 2
1
2 4 6.411
47.006
0 .1 8 8 9 2
0.37785
Branch 3
3 90.162
0.32S62
Branch 4
6.579
0.50006
♦
V
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123
w aveguid e m anifold is con sidered here in more d eta il. C on tigu ou s or non-contiguous
band m u ltip lexers are trea ted in a sim ila r m anner. F ig. 5 .6 , w h ich is a sp ecial case for
th e structure in Fig. 5.1, illu str a te s a typical circu it e q u iv a le n t for a m u ltip lexer. A
branch co n sists o f a cou p led -cavity filter, togeth er w ith input-output transform ers,
and an im pedance in verter.
A su b section is th e w a v egu id e sectio n sep a ra tin g two
• adjacent filters and th e j'unction is the e q u iv a len t c ir c u it m od el for th e p h y sic a l
ju n ctio n betw een ch a n n el filte r s an d th e m an ifold .
T h e m a in ca sca d e is sh o rt-
circuited and th e resp on ses o f in te r est are com m on port return loss, channel output
f]
return loss, in sertion or tran sdu cer loss, gain slope and group d elay betw een com m on
port ^ }d ch ann el output ports.
To apply th e g en era l m ethod o f S ection 5 .2 , th e su b n e tw o r k s, n a m e lv ,
ch an n el filters, w avegu id e sp a cin g s and ju n ction s sh ou ld b e represented by
tra n sm issio n m atrices.
2
-port
R ecently, a com p reh en sive s e t o f form u las for reduction o f
m u lti-cavity filters to tw o-port eq u iv a len ts w hich a lso provides se n s itiv itie s w .r.t.
variab les o f in terest in the filte r stru ctu re, has been p resen ted by B andler, C hen and
D aijavad (1986a). T he form ulas ev a lu a te sh ort-circu it a d m itta n ce param eters and
th eir se n sitiv itie s w .r.t. a ll co u p lin gs a s w ell a s frequency for th e u n term in ated filter
m odel.. E valuation of tr a n sm issio n m atrices from sh ort-circu it a d m ittan ce m atrices is
straight-forw ard.
In T able 5.11, the tra n sm issio n m atrices for th e in d ivid u al com ponents o f
the m u ltip lexer stru ctu re show n in Fig. 5.6, have b een liste d .
T h e s e r ie s 3-port
ju n c tio n s are reduced to 2-port e q u iv a le n ts u sin g the m ethod described in S ection 5.2.
T ab le 5.12 lists th e s e n s itiv itie s o f tran sm ission m atrices in T a b le 5.11 w .r.t. relevan t
p a ra m eters and frequency.
>
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
R eproduced
with p erm issio n
output
lo ads
of the copy righ t o w n e r.
output transformers
Q„
Q
„.
pnnr| 2
prnrj
rrm
2
LuuJ
pnnq ni
rn m ni
m ulti-coupled
c a v ity
input
filte r s
transformers
LuuJ nN
fTTT)
F u rth er reprod uction
impedance inverters
A /W —
shor I-circuit
termination
JL
10
/
I"
prohibited
no n-ideal
series
w ithout p e r m i s s i o n .
Fig. 5.6
K>
junction
w aveguide
sp acin g
K quivalent circuit o f a contiguous band m ultiplexer. Kach channel has a
m ulti coupled ca v ity filter w ith input an d ^ u lp u t transform ers as w ell as
an im pedance inverter. The m ain cascade is a w aveguide m anifold w ith
a short circuit term ination, branches are connected to the m ain cascade
through nonidcal series junctions.
V
125
* TABLE 5.11
E X A M PL E S OF T R A N SM ISSIO N M ATRICES FOR SU B N E T W O R K S
IN TH E M U LTIPLEX ER OF FIG. 5.6
T ran sm ission M atrix
Subnetw ork
E xp ression
N o ta tio n
/
output transform er
n2:l
n2
0
0
—
1
,
A
1
n2
•to
m ulti-coup led
ca v ity filter*
J
_ <I n
1
A
_1
ql - P l %
"Pi
-0
*
input
transform er
l:n i
V
—
0
se r ie s ju n ctio n
term in a ted a t port 3
by Y3.lY = Yc + Y 3)
s e r ie s ju n ctio n
term in a ted at port 2
by Y2, ( Y = Y U + Y 2)
o
’
A
ni
- I f
Y
nt
i1
Y + Ya
Y
2Y Y + Y2
a
a
1
A
Y+Y
Y ^ YC
Y(Y + Y H Y Y
a
c
a c
a
D
1
Y+ Y
a
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
'
TABLE 5.11 (continued)
T ra n sm issio n M atrix
Subnetw ork
E xpression
V
N otation
cos 8
j ZQsin 8
iS i" 9
» s6
w avegu id e spacing**
Zo
.
A
■
Pi(qi) is th e ith e le m e n t o f vector p (q ) w h ich is th e so lu tio n o f Z p =
(Zq = u n), w here Z = j ( s l + M) + r l and s = ( u>o/A m Koj/ uo for a nth
order filte r w ith cou p lin g m atrix M centered a t u>o and havin g a bandw idth
p a ram eter A o and a uniform ca v ity d issip a tio n p a ra m eter r. 1 is a n X n
id en tity m atrix.
X
a w a v eg u id e se c tio n has a ch a racteristic im pedance Zq and 8 = p f , p =
w h ere i is th e se c tio n len g th and \ s is the guide w avelen gth .
2
n/.\j.t
V?
A
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
TABLE 5.12
FIRST-ORDER SE N SIT IV IT IE S OF T H E T R A N SM ISSIO N
M ATRICES IN TABLE 5.11
Subnetw ork
Id en tification
output
transform er
aA
S e n sitiv ity o f th e T ra n sm issio n M atrix
1
1
0
3n2
m ulti-coupled
ca v itv B iter
n;
.
A
8
aM
0
jc
t
2 <l
ub
J P A + V b 1A +
0
a Mb
jc
Pl^b + VaPb-^^b^Ph^J PaPb
q Tq
aa
)
a Cl)
input
transform er
p , q Tq t < i n p Tp -
2
q 1 p T‘i
pt p
1
- n“
aA
an.
se r ie s jun ction
term in a ted at
port 3 ”
0
0
aA
Y-
(Y 3>*K I
aA
—
0
<*>
tY I K , + (Y ) K„
, <$> € J
c 4> 1
a
2
oA
do)
(Y
7
3
-r Y ) K - (Y ) K ,
.c g 1
ti u> -
----------------------------
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128
TABLES. 12 (continued)
Subn etw ork
Identification
se r ie s
ju n ctio n
term in ated
a t port 2 m
S e n sitiv ity o f th e T ran sm ission M atrix
dD
<Y 2>»L 1
3D
— , <{) € J
(Y a )9 (L.I + L0)
+ (Yc )9 L.J
J
3$
3D
(Y + Y ) L, + (Y ) L
a co
2
3 a)
1
a u)
2
+ (Y ) L ,
c u
—sin9
w a v eg u id e
sp a cin g
3A
j Z cosQ
jcosQ
3£.
3
—sinO
L Zo
—sin9
3A
«P)
3<a
2
c —
j ZQcosO
jc o s 8
—sin 0
ifa * b
I ifa = b
Y
I
a
v-2
\
v\
Y
*++ L . = - —;
Y ;»Y C
1
YA
1
0
2(Y A+ Y )
1
1
T ——
•4 - Y
0
Y c+ •Y
•
0
I
•
1
I
3
——
Y
I
0
Y+Y
0
a
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
,
5.5 .2
129
O ptim ization o f a 12-C hannei 12 G H z M ultiplexer
A w id e r a n g e o f p o ssib le m u ltip le x e r o p tim iz a tio n p r o b le m s c a n be
form ulated and solv ed by appropriately d efin in g sp ecification s on va rio u s frequency
%
1
resp on ses o f interest.' T he se n sitiv itie s a re used in conjunction w ith th e grad ien tbased m in im ax a lg o rith m o f H ald and M adsen (1981) to en su re th e fa ste st p ossib le
solution s.
A s an ex a m p le, we h ave used our sim u la tio n and s e n sitiv ity form u las to
optim ize a 12-channel,
12
GHz m ultiplexer,, w ith ou t dum m y ch a n n els.
W avegu id e
spacings, input and outpu t transform er ratios, ca v ity resonant freq u en cies as w ell as
in terca v ity cou p lin gs are used a s o p tim iz a tio n v a r ia b le s. T h e o p tim iz a tio n w a s
executed by K ellerm an n (1986) and D aijavad (1986) and described in B an d ler, C h en ,
D aijavad and K ellerm ann (1984) and in B andler, D aijavad and Z hang (1986).
T he problem is described as follow s.
T here are tw elve
cav itv filters m ounted on the w a vegu id e m anifold.
6
th-order m ulti-
A n o p tim iz a tio n on a s in g ly
term in ated filte r w a s perform ed to o b ta in th e sta r tin g v a lu e s for th e n on -zero
cou p lin gs Mjo, M 2 3 , M 3 4 , M3 6 . M4 5 , M5 6 and th e sam e v a lu es w ere a ssu m ed for all
filters. T he model for the nonideal ju n ction s, i.e., th e eq u iv a len t a d m itta n ce s Y a and
Yc o f Fig. 5.6, w hich have also been a ssu m ed in th e tran sm ission m atrix d escrip tion
*
o f ju n ctio n s as ap p earing in T able 5 .1 1 . are c o n siste n t w ith the m odels su g g e ste d by
C hen, A ssa l and M ahle (1976).
F ig. 5.7 sh ow s th e com m on port retu rn lo s s an d
ch ann el in sertion loss responses a t the sta r tin g poin t for the o p tim ization o f the w h ole
structure.
T he sp ecific optim ization problem con sid ered in th is e x am p le w as to sa tisfy
a low er s p e c ific a tio n o f 20 dB on th e com m on port retu rn lo ss o v e r th e e n tir e
frequency band o f in te r e st for the m u ltip lexer.
From T ab le 5.1 it is cle a r th a t th e
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Oj
. ^
,®
°
UJ
a:
1i ° ^i U .
I'ig 5.7
;o
Common port return loss and channel output port insertion
responses of the 1 2 -channel multiplexer before optim ization. .
lo ss
130
o
o
o
LO
o
o
I EG ’ SSCT N O I id S S -; CNS NEGiSE
✓
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
131
/
e v a lu a tio n o f com m on port retu rn lo ss an d its s e n s it iv it ie s w .r.t. th e g e n e r ic
o p tim iza tio n v a r ia b le $ is str a ig h t-fo r w a r d o n c e th e c o m m o n p o rt r e f le c tio n
coefficien t p° and its s e n s itiv itie s
a re know n. W e w ill d escrib e the' p articu lar
v aria b les con sid ered in th is ex a m p le later. R eca llin g eq u ation s (5.25) and (5.26) and
* th e d efinition o f q;j in (5 .5 ),'p° and (p°)^> are ev a lu a ted from q 2 N + 2 . i and its s e n s i­
tiv itie s.
F in a lly , by referrin g to the a lg o rith m and sp ecifically Step 2 in th is ca se,
Q2 N + 2 . 1 . ^Q2 N + 2 . a r e calcu lated .
The optim ization in v o lv ed 60 v a ria b les, n am ely, 12 s e c tio n le n g th s , 14
variab les for each o f ch a n n els
1
and
12
(all
6
possible in tercavity co u p lin gs,
6
cavity
resonan t frequencies, input and output transform er ratios) and 4 va ria b les for each o f
c h a n n els
2
, 8 , 9, 10, and 11 (in p u t and ou tp u t transform er ratios, reso n a n t frequency
o f th e first cavity and cou p lin g M jj). T he total C PU tim e on th e Cyber 170/815 sy stem
w as about ten m inu tes. T he r e su lts-o f th e fir a l optim ization are sho^yn in Fig. 5.8.
E q u i-rip p le retu rn lo ss r e sp o n se s a t is f y in g th e r e q u ir e m e n t s o v e r th e e n t ir e
com m unication band h a s been ach ieved.
5.6
•
C O N C L U D IN G REM ARKS
We Have presented a new approach to sim u lation and se n sitiv ity a n a ly sis o f
branched cascaded netw orks. By u tiliz in g our form ulas o f T h even in eq u iv a le n ts and
their, s e n s itiv itie s w .r.t. netw ork p a ram eters a s w ell a s frequency, variou s frequency
respon ses and their s e n s itiv itie s a t arb itra rily ch osen referen ce p lan es a re ev a lu a ted .
T he m ethod presented-has been u tilized in the. optim al d esign o f a sta te o f th e art 12
ch ann el co n tigu ou s band m u ltip lexer. A ttra ctiv e and fast co m p u ter'resu lts obtained
u sin g a gradient-based op tim ization tech n iq u e ju stify our treatm en t o f s e n s itiv ity
ev a lu a tio n as an in tegral part o f th e a n a ly sis. A ll the se n sitiv ity form u las p resen ted
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I
133
«
in th is ch a p ter can be v e r ifie d in d e p e n d e n tly .
A ctu a l im p le m e n ta tio n o f o u r
approach, how ever, requires on ly an un d erstan d in g o f the d e fin itio n s o f the resp on ses,
form ulas for w hich are a v a ila b le 'in T a b le 5 .1 .. For m ore th e o r e tic a lly o r ie n te d
researchers or en g in eers, our m ethod o f d ea lin g w ith the s e n s itiv itie s (Section 5.2) is
straigh tforw ard and sh o u ld be a p p lic a b le to a lm o st a n y co m p le x lin e a r c ir c u it
stru ctu re in the frequency d om ain:
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»>
6
AN AUTOMATIC DECOMPOSITION APPROACH TO OPTIMIZATION OF
LARGE MICROWAVE SYSTEMS
*
INTRO DUCTIO N
6 .1
X
_
'V
A se r io u s ch a llen g e to research ers in m icrow ave CAD areas is due to the
siz e o f p ractical m icrow ave sy stem s.
E x istin g CAD techniques, m ature enough to
h an d le sy s te m s o f ordinary size, gen erally b ilk a t large circuits. T he reasons for their
fa ilu re in clu d e prohibitive com puter sto rage and C PU tim es required.
A frequent
fru stration w ith large scale o p tim ization is th e increased likelihood o f stop p in g a t an
u n d esired local optim um
O ther d ifficu lties, esp ecially in prototype and production
tu n in g , a re d ue to hum an in a b ility to cope w ith problem s in volvin g large num bers o f
in d ep en d en t variab les to be adjusted sim u lta n eo u sly to m eet a specified resp o n se
p attern over a w ide frequency range.
R ecen tly , FET m od ellin g (Kondoh 1986) and m anifold m u ltip lexer design
{B andler, C h en , D aijavad, K ellerm an n , R enault and Z hang 1986) p rob lem s w ere
so lv ed u s in g appropriate decom position schem es.
The optim ization problem s were
c le v e r ly trea ted by sy ste m a tic a lly or repeatedly selectin g and adjusting various sm all
se ts o f p a ra m eters and resp on ses u n til the sy stem becom es acceptably operational.
T h e su c c e ss o f th e s e efforts m otivated u s to pursue the gen eralization and autom ation
o f d ecom p osition approaches for m icrow ave optim ization problem s.
T h e concept o f decom p osition has been a traditional m ath em atically based
*
v eh ic le for ap proaching large sc a le problem s. H im m elblau (1973) h a s an ex cellen t
/
134
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135
'co llectio n o f su rv ey s from th e a rea s o f m a th em atics, e n g in e e r in g , e c o n o m ic s an d
m anagem ent scien ces.
In th is, chapter, w e w ill b e in te r e ste d in su ch a s p e c ts o f
d ecom position th a t are b en eficial to circu it op tim ization problem s.
^ -
-
D ecom position m ethods used in 'm ath em atical program m ing th eory u su a lly
assum e
A
cen ta in .stru ctu res for the objective function an d constraints.
T h eoretical
in v estig a tio n s h a v e been perform ed for lin e a r program m ing, nonlinear p rogram m ing
and m inim ax o p tim iza tio n s (e.g., GeofTrion 1970; Lasdon 1970; H im m elb lau 1973;
Luna 1984). ’.
In circu its and sy ste m s, diakoptic a n a ly sis, gen eralized hybrid a n a ly s is
(e.g. C hua and C hen 1976) and netw ork tea rin g m ethods (e.g., Wu 1976; T on g and
Chen 1986; A sa i, U rano and T a n ak a 1986) h ave been developed. Im portant to th ose
m ethods-are circu it rela tio n s, esp ecia lly topological relation s.
In addition to b ein g
used for circu it a n a ly sis, th^jdecom position tech n iq u es have b een u sed in d e sig n
(H im m elblau 1973) and fa u lt d ia g n o sis (S alam a, S tarzyk and B andler 1984).
D ecom position h a s also been an a ctiv e subject in electrical pow er sy ste m s
since such problem s e a s ily r esu lt in thousands o f va ria b les and eq u ation s. E xam p les
Jcan be found in optim al power flow (T alukdar, G iras and K alyan 1983; C o n ta x is,
jD elkis and K orres 1986), sta te e stim a tio n (Lo and M ahm oud 1986) a n d r e a l an d
r e a c tiv e ^ o w e r op tim ization problem s (B illin to n an d S haehdeva 1973). T h e decom ­
p o sitio n p a tte r n s in v o lv e d a r e o b ta in e d u s in g b o th p h y s ic a l a n d a n a ly t ic a l
%
in v estig a tio n s o f th e syste'm s.
M icro w a v e e n g in e e r s h a v e th e ir ow n sp e c ia l d iffic u ltie s .
T horough
lab o ra to ry e x p e r im e n ta tio n h a s to be p erform ed b efore u sin g c e r ta in fu n c tio n
s tr u c tu r e s a ssu m e d in m a th e m a tic a l p ro g ra m m in g th eory.
T h ey do n o t ta k e
ad van tage o f topological a n a ly sis often exp loited in th e a reas o f circu its an d sy ste m s
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136
sin ce m icro w a v e d e v ic e m o d els a r e o rien te d m ore to p h y sic a l th an to p o lo g ica l
a n a ly s is .
U n lik e pow er s y s te m s , m ost m ic r o w a v e r e s p o n s e s are. m uch m o re
com plicated and h ig h ly n onlinear.
It is often difficu lt for m icrow ave en g in eers to
a n a ly tic a lly in d icate p ossib le decom position patterns.
The sta te-o f-th e-a rt in large-scale op tim ization o f m icrow ave circu its is still
d ev ice dependent and based on h eu ristic judgem ent
Very recen tly, B a n d ler and
Z h ang (1 9 8 7 a ) m ade a fir s t a tte m p t to d ev elo p £ g e n e r a l and a b stra ct th e o r y
d escrib in g a decom p osition approach to m icrow ave circu it optim ization not requiring
particular ph ysical or top o lo g ica l know ledge o f the system .
In th is ch ap ter, w e p resen t the novel tec h n iq u e o f B an d ler and Z h an g
(1987a, 1987c) for th e o p tim ization o f large m icrow ave sy stem s.
U sin g se n sitiv ity
inform ation obtained from a su ita b le M onte-C arlo a n a ly s is , w e e x tr a c t p o ssib le
decom position p roperties w h ich could oth erw ise be deduced only through a p h ysical
and topological in v estig a tio n . The overall problem is a u tom atically separated into a
sequ en ce o f subproblem s, each b ein g characterized by the optim ization of a su b set o f
' circu it functions w .r.t. v a ria b les which are se n sitiv e to th e selected responses.
Our
s u g g e s te d te c h n iq u e h a s b een s u c c e ssfu lly te ste d on m ic r o w a v e m u ltip le x e r s
in v o lv in g up to 16 c h a n n els and 240 variables.
In Section 6.2, w e describe the basic concepts o f decom position for circu it
optim ization problem s.
U sin g th ese concepts, the p artition in g approach for FET
m o d ellin g problem s su g g ested by Kondoh (1986) is verified and the decom position
property o f m u ltip le x e r s is e x p la in e d , as p rese n ted in S e ctio n 6.3.
S ectio n 6.4
illu str a te s the au tom atic d eterm in ation o f suboptim ization problem s. An autom ated
decom position a lgorith m for large sca le m icrow ave o p tim iz a tio n is p r esen ted in
S ection 6:5. In S ection
6 .6
, th e m ethod is applied to th e optim ization o f m icrow ave
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137
I
m u ltip lexers.
In terestin g r e su lts d em on stratin g th e w h ole procedure o f autom ated
decom position for a 5-chamnel m u ltip lex er are depicted in illu str a tiv e graphs. The
resu lts o f op tim izin g a 16-channel m u ltip lexer u sin g our approach are provided.
6.2
TH E DECO M POSITIO N APPR O A C H FOR C IR C U IT O PTIM IZA TIO N
PROBLEM S
6.2.1
C ircuit O ptim ization P roblem s
Let
<}> = [ $ i <fc>
<M T
(6.1)
rep resen t the sy stem p aram eters. T he circu it responses, denoted a s Fk(<^, to), k = 1, 2,
..., np, are functions o f v a ria b les <J> and frequency <o. In an op tim ization problem for
c ircu it d e sig n , th e o b je c tiv e fu n c tio n u su a lly in v o lv e s a s e t o f n o n lin e a r error
fu n ction s
j = 1,2, ..., m. T y p ica lly , th e error functions rep resen t the w eighted
d ifferen ces betw een circu it resp o n ses and given sp ecification s in th e form defined in
(2.3).
Suppose sets
1
and J are defined as
r
I i{ 1 .2 ,...,n } .
'
J = { 1 ,2 ,..., m}.
( 6 .2 )
(6.3)
T he overall op tim ization problem , e .g ., a m inim ax op tim ization , is
m in im ize
<t>i, i € I
m ax
fj(<J>).
(6.4)-
j€J
In a decom position approach, one attem p ts to reach th e overall solution by
so lv in g a sequence of subproblem s. A typical subproblem is ch aracterized by
m in im ize
4>i, i€ I s
m ax
j€ J s
fj(<£).
( 6 .5 )
w here Is and J s are su b sets o f I and J , respectively.
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138
T he basic id ea for decom position is to decouple a variab le 4>i from a function
fj if th e in tera ctio n betw een them is w eak. A subproblem con tain s on ly the sen sitiv ely
related v a ria b les and functions. A proper arrangem ent o f the seq u en ce o f different
su bp roblem s to be solv ed is oflen im portant to ensure convergence and efficiency.
•6.2.2
G rouping o f V ariables and F u n ction s U sin g S e n sitiv ity Inform ation
.
T h e e s s e n tia l ta sk for th e a u to m a tic d eco m p o sitio n te ch n iq u e is th e
a u to m a tic d ecision on I3 and J 3, and th e au tom atic seq u en tial arran gem en t o f various
subproblem s. T h is is accom plished through an appropriate decom position dictionary
to be introduced ft^the en su in g text.
S e n sitiv ity A n a ly sis
'
■»
W e perform se n sitiv ity a n a ly sis a t a set o f random ly chosen p oin ts 4*^.
t — 1, 2 ,.... A m easure o f the interaction betw een 4>i and fj is defined as
/ af ($') <t>° ,2
S. I V f —
.
,J
Tr
^°4>
(6 .6 )
ij'
w here 4>i° and fj° are used for scalin g. A ll the S,j. i =
1,
2. ... n and j =
1,
2. .... m.
co n stitu te a nxm se n sitiv ity m atrix S. It is reasonable to conclude th at <(>, and fj can be
decoupled if Sjj is very sm all.
G rouping o f V ariables and F unctions
T he exam in ation o f various interaction patterns betw een 4 ),. i € 1. and fJ( j e
J, r e su lts in the breakdow n o f a ll variab les <J> into p g r o u p /id e n tifie d by index sets Ij.
N
Io
Ip, and a ll functions f into q groups id en tified by s^ts J j , J -j,.... J^. We have
I —
I|
U Io U ...
\
(6 . 4 )
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139
and
J = J t U Jo U ... U J , .
( 6 .8 )
T h e p a r t it io n in g o f <J> o r f c a n b e a c h ie v e d e i t h e r m a n u a l l y or
a u tom atically. T he m anual procedure corresponds to the m anual d eterm in a tio n o f
variab le groups and function groups u sin g a priori know ledge.
Such k n ow led ge is
ty p ic a lly o b ta in e d th rou gh e x te n s iv e la b o ra to ry e x p e r im e n t an d a n e x c e lle n t
u n d erstand ing o f the p articu lar d evice. T he au to m a tic procedure corresponds to the
com puterized p a rtitio n in g of <|> or f based upon th e se n sitiv ity m atrix S . T h e p arti­
tio n in g o f <}> and f ca n be perform ed
1
) b oth m a n u a lly ,
2
) m a n u a lly for
4>
an d
a u to m a tica lly for f, 3) au to m a tica lly for <J>and m a n u a lly for f, 4) both a u to m a tica lly .
A s an ex a m p le for m anual p a r titio n in g o f f, w e c o n sid e r a ><’-ch a n n el
m u ltip lexer.
T he com m on port return loss and c h a n n e l in se r tio n lo ss r e sp o n se s
a ssociated w ith th e sa m e ch ann el can be grouped together sin ce th eir b eh avior is
sim ila rly affected by variables <{>. T herefore, w e have N groups o f fu n ction s, i.e.,
<?= N. J f co n ta in s in d ices o f error fu n ction s related to channel f, t — 1 ,2 ,...,N .
J
A Procedure for A utom atic P a rtitio n in g o f V ariab les d>
S u p p o se th e fu n ction grou p s h a v e b een d ete rm in ed , i.e ., J h a s b ee n
decom posed into J f, £ = 1 , 2 , . . . . q. W e defin e a n xq m atrix C w hose (i, £)th com ponent
is
c
whehe w tJ is a w e ig h tin g factor
it
=
v
(W.
—
ij
S .) ,
ij
(6.9)
A very sm a ll v a lu e o f an entry in the C m atrix, say,
C t{, im p lie s t h a t th e ith v a r ia b le an d th e £ th fu n c tio n g r o u p a r e w e a k ly
*
ft
interconnected.
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<*.
Let Cave rep resen t th e average valu e o f a ll com ponents in th e C m atrix.
For a g iv en factor A, A £ 0, th e m atrix is m ade sp arse such that
is se t to zero if it is
le ss than ACave. By m ak in g C sparse, in se n sitiv e variab les are elim in ated and w eak
interaction s b etw een varia b les and function groups are decoupled.
Tw o v a ria b les <p; and (pj belong to the sam e group if they in teract only w ith
the sam e groups o f fu n ction s, i.e., if the ith and the jth rows o f C have the sam e
zero/nonzero pattern-. A thorough com puterized ch eck in g o f the C m atrix resu lts in
the au tom atic d eterm in a tio n o f index se ts Ik, k =
1 ,'2
p.
t
An Illu stra tiv e E xam p le o f M atrix C
C onsider th e fictitio u s relation s b etw een variab les and fu n ction groups
show n in F ig.
6 . 1 (a).
T h e functions f have been arranged into 5 groups T he C m atrix
(already m ade sparse) is
r
22
.
100
.
0
.
100
.
0
.
100
0
.
0
32. ^
0
.
0
.
0
.
0
.
0
.
.
0
.
0
.
0
.
0
.
83.
100
.
0
.
.
0
.
0
.
0
.
100
.
0
.
0
.
100.
86
.
0
.
0
.
0
.
100
.
0
.
0
.
0
.
55.
0
.
100
.
0
.
>
}
78.
0
.
.
100.
0
.
A s seen from Fig. 6.1(a), (po and
0
$3
. .
'u
both afTect only the 2nd function group.
In the C m atrix, row s 2 and 3 both have only one nonzero located at the 2nd colum n.
T herefore, varia b les cp? and
$3
are grouped together. S im ila rly , variab les <p4 and <pn
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141
*5
Jf J,
J'J,
*7
(a)
itl,
u i.
K l,
i*J.
\ wl
) t«u
uu
iti-j
(b)
F ig. 6.1
A fictitio u s exa m p le sh ow in g on ly th e stron g in tercon n ection s b e tw ^ n
v a ria b les and fu n ction groups, ta) sy ste m con figu ration corresponding to
m atrix C- (bl sy ste m configuration corresponding to m atrix D.
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142
b elon g to th e sa m e group. T h e resu ltin g index se ts for variable groups are Ij = {9}, I2
= {2 ,3 }, I3 = {7}, I4 = {5}, I5 = {4. 6 }, Ig = {1} and I7 = {8 }. T h e index s e ts have been
ordered such th a t the kth variable group correlates w ith no m ore function groups than
th e (k + l) t h variable group does, k = 1 ,2
6
. Such an arran gem en t is m ade to keep
su b seq u en t description sim ple.
6 .2 .3
D ecom position D ictionary
">■
♦
•
To m an ipu late d irectly w ith groups o f variab les and groups o f functions, we
con stru ct a px q d ictionary decom position m atrix D . D efine th e (k, f)th com ponent o f D
as
5
d
kf
= y
—
it:
y
( w .s .)
ij
ij
j€ J
■
IV.
i6Ik
If Dkf is zero, variables in the kth group are decoupled from functions in th e fth group.
O th e r w ise if D^j *
0
, w e s a y th a t <£i, i€ Ik,
fj. j ^ J f, a re co r re la ted .
The
decom position dictionary g iv e s a clea r picture o f the correlation p attern s betw een
g rou p s o f v a r ia b le s and fu n c tio n s, f a c i/ita tin g th e a u to m a tic d e te r m in a tio n o f
^'
su b o p tim iza tio n pro b lem s.
T h e id eal dIic tio n a r y is a d ia g o n a l m a trix w h ere a
V-
subproblem sim ply corresponds to a d iagonal elem en t. In th is ca se, only one variable
group and one function group is in volved in a subproblem . If a d iagon al dictionary
can be obtained w ithou t a r tific ia lly m ak in g C sp arse (i.e., u sin g sp arse factor X = 0),
th e n the sy ste m is com p letely decom posable. For a com pletely decom posable system ,
d ifferen t subproblem s can be calcu la ted in p a rallel. D eta ils o f decom posability can be
found in C ourtois (1977).
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143
C onsider the previous ex a m p le w ith th e r e su ltin g C m a tr ix d e fin e d in
( 6 .10). A ccording to th e index sets Ik. k = 1, 2 7 , th e decom position d ictio n a ry D
can be ob tain ed from C by adding rows
2
and 3 , and-adding rows 4 and
6
, resp ectiv ely .
T h e rela tio n s betw een groups o f v a riab les and fu n ction s are show n in F ig.
6
. 1(b). T he
resu ltin g dictionary is
100
.
0
.
0
.
0
.
0
.
0
.
.
200
.
0
.
0
.
0
.
0
.
0
.
1 .0 0
.
0
.
0
.
0
.
0
,
.
0
.
100
.
0
.
0
20
0
0
.
180.
.
100.
30.
.
70.
lob.
180.
o.
.
0
.
50..
0
.
0
rw here each entry has been rounded to m u ltip les o f
( 6 . 12 )
10
.
6.3
PRACTICAL EX A M PLES OF DECOM POSITION DICTIONARY
6.3.1
D ecom position D ictionary for FET D evice M odels
Through ex ten siv e ex p erim en t on practical FET d evices, K ondoh (1986)
sum m arized
8
suboptim ization problem s w hich can be repeatedly solved to y ie ld a
FET model w ith im proved accuracy.
T he eq u iv a le n t circu it is sh ow n in F ig . 6 .2 .
U sin g th e theory described in the p r e v io u s s e c tio n , B a n d ler an d Z h a n g (1 9 8 7 a )
p resented a decom position d ictionary for su ch d ev ices. T he 13 variab le p a ra m eters
w ere a u to m a tic a lly p a r titio n e d in to
a g reem en t w ith th e
8
8
grou p s.
T h e ir r e s u lt
w a s in c o m p le te
subproblem s o f Kondoh (1986).
H ere we describe the ex p erim en t reported by B andler an d Z hang (1987a).
4
A s e t o f true p aram eter v a lu es listed in Kondoh (1986) is used as a referen ce p oin t <J>°,
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144
'
A
FKT equivalent circuit
= TJ
co co
o>
A
Cn
Fig
6 2
CO
C>
O)
/T
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145
as show n in T able 6.1.
T he en tire frequency range o f m ea su rem en t is 1.5 G Hz to
26.5 GHz. W e perform se n sitiv ity a n a ly sis a t 10 r a n tb m ly ch o sen p a ra m ete r points in
the 10% neighbourhood of<f>°. T he function fj used in (6 . 6 ) is defined a s th e w eigh ted
difference b etw een the calcu la ted and th e m easu red v a lu es o f th e m odulus or th e
phase o f a particu lar S param eter.
F unctions asso cia ted w ith the sam e S param eter
are grouped together. T able 6.2 sh ow s the C m atrix o f (6.9) before b ein g m ade sp arse,
in d icatin g stron g as w ell a s w eak in terconnection p a ttern s b etw een each individual
param eter and differen t group^ o f functions. T able 6.3 provides a n ex a m p le o f the
decom position d iction ary calculated and norm alized from the C m atrix o f T ab le
6 .2
.
Table 6.3 yield s S subproblem s w hich agree w ith and fu rth er verify th e decom position
schem e proposed by Kondoh (1986).
W hen th e diction ary is m ade sp arse, certain
en tries, w hose valu es are only slig h tly le ss than the d om in an t o n es, are also set to
zero. T herefore, a s m entioned by K ondoh, rep eated cy c lin g and careful ord erin g o f the
8
su b o p tim iz a tio n s a re n e c e ssa r y .
T he fe a s ib ilit y o f c o m p u te r iz e d a u to m a tic
decom position is d em on strated by this'exam ple.
6.3.2
DecC&position D ictionary o f a 16-C hannel M u ltip lexer
M u ltip lex ers-co n sistin g o f m u lticavity filte rs d istr ib u te d a lo n g a w a v e ­
guide m anifold w as introduced in C hapter 5. It h as been observed th at p aram eters
associated w ith a particular ch an n el o f the m u ltip lex er stru ctu re have a stron g effect
on respon ses corresponding to th a t ch an n el and a w eak effect on resp on ses related to
✓
other ch an n els.
The theoretical description o f th is phenom enon w as p resented in
Bandler, C hen, D aijavad, K e lle r m a n n , R en a u lt and Z h an g (1 9 8 6 ).
A p rototyp e
decom position d ictionary w as con structed from m an u al p artition in g o f varia b les and
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,
146
TABLE 6.1
'
i
PARAM ETER V A L U E S FOR <J>0 FOR T H E FET CIRCUIT MODEL
________________ =_____________:________ ___ _______
i
.
P aram eter
U n it
V alue for 4>i
m
m S.
2
t
Ps
3.0
3
Cgs
pF
0.25
4
Cds
pF
0.08
5
Cdg
pF
0.025
1
8
6
50.0
Ohm
4.0
4.0
7
Rs
Ohm
8
Rd
Ohm
9
Rds
Ohm
10
Rt
Ohm
0 .2
pH
60.0
PH
25.0
11
12
Ld
13
Ls
*
3.0
250.
pH
— «*- - .........................- •
■
•
15
0
.x
•
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.
147
T A B L E 6.2
TH E C MATRIX FOR TH E FET MODEL
(a)
F U N C T IO N G R O U PS IN V O LV IN G T H E ENTIRE F R E Q U E N C Y BA N D
(1.5 GHZ TO 26.5 GHZ) •
-
F unction Groups
V ariab les
S u E ntire
Freq. Band
S 2 1 E n tire
Freq. Band
S j 2 E ntire
Freq. Band
S 2 2 E n tire
.* Freq. Band
&
18.55
1 0 0 .0 0
87.55
63.33
1 0 0 .0 0
8 9 .7 4
67.98
62.25
Cds
4.88
6 7 .7 4
45.73
1 0 0 .0 0
CdR
4.24
4 8 .88
1 0 0 .0 0
S I .27
35.53
3 7 .14
1 0 0 .0 0
5.88
17.44
9 7 .6 8
70.51
1 0 0 .0 0
c
R*
Rds
*
Each row o f the table h a a je e n scaled.
,
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148
TABLE 6.2 (continued)
T H E C MATRIX FO R TH E FET MODEL
(b)
F U N C T IO N G R O U PS IN V O LV IN G O NLY TH E U P PE R H A LF FR E Q U E N C Y
B A N D (14.0 GHZ TO 26.5 GHZ)
Function Groups
t
V aria b les
S n U pper
Freq. Band
t
■ '
S 2 1 U pper
Freq. Band
S t 2 Upper
Freq. Band
S 2 2 U pper
Freq. Band
31.91
1 0 0 .0 0
36.61
59.31
i*
1 0 0 .0 0
50.67
24.87
29 89
Rd
34.65
74.31
85.85
1 0 0 .0 0
Ri
1 0 0 .0 0
65.63
88.43
39.53
lk
1 0 0 .0 0
87.85
57.16
37.44
Ld
9.99
97.88
61.78
1 0 0 .0 0
l>3
62.94
31.31
1 0 0 .0 0
21 99
Each row o f th e tab le has been scaled.
#
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149
T A B L E 6.3
NORM ALIZED DECOM POSITION DICTIO NAR Y D FOR TH E FET M ODEL
%
(a)
C O R R ESPO N D IN G TO T H E SE N SIT IV IT Y A N A L Y SIS OF T A B L E 6.2(a)
F unction Groups
V ariable
Groups
$ l i E n tire
Freq. Band
S 2 1 E ntire
F req. Band
S i 2 E ntire
Freq. Band
.]
S 2 2 E n tire
Freq. Band
Rds, Cds
0 .0 0
0 .0 0
Cgs
1 .0 0
0 .0 0
0 .0 0
0 .0 0
1 ,0 0
0 .0 0
0 .0 0
1 .0 0
0 .0 0
0 .0 0
' CdK’ R,
8m
(b)
0 .0 0
j
1 .0 0
0 .0 0
C O R R E SPO N D IN G TO T H E SEN SITIV IT Y A N A L Y SIS OF T A BLE 6.2(b)
F unction G roups
V ariable
• Groups
Rd. Ld
S u U pper
Preq. Band
•
S 21 U pper
Freq. Band
S 12 U pper
Freq. Band
S 2 2 U pper
Freq. Band
0 .0 0
0 .0 0
0 .0 0
1 .0 0
R k. Ri. L«
1 .0 0
0 .0 0
0 .0 0
0 .0 0
Ls
0 .0 0
0 .0 0
1 .0 0
0 .0 0
t
0 .0 0
1 .0 0
0 .0 0
0 .0 0
*>
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150
fu n c tio n s.
'
T he fo llo w in g d e sc r ip tio n is b ased upon th e ir e x p e r im e n t and th e
S
s e n sitiv ity m atrices th ey used.
C onsider a 16-chan nel, 12 GHz con tigu ou s band m u ltip lexer whose eq u i­
v a le n t circu it fo llo w s F ig . 5 .6 .
A ll ch a n n e l filte r s a r e
6
th order.
W e perform
s e n sitiv ity a n a ly sis a t 30 p oin ts selected random ly w ith in 40% region of the optim al
point o f th e m u ltiplexer. S ix te e n function groups are com posed, each corresponding to
a p a rticu lar channel o f th e m u ltip lexer. T he kth function group con sists of com m onport return loss fu n ction s ca lcu la ted at 7 frequency p oin ts, 3 o f w hich are in th e
p’a ssb and o f chan nel k, and th e rem a in in g points a re in th e stopband o f channel k.
In th e fir st exp erim en t, w e have 16 groups o f v ariab les. T he kth variable
group inclu d es a ll cou p lin g paranyeters as w ell a s input and output transform er ratios
o f th e kth ch ann el filter.
T h e decom position diction ary is sh ow n in T able 6.4.
As
perform ed by B and ler et al. (19861, th is dictionary m atrix w as norm alized such that
each ele m e n t of th e m atrix is divided by the a v erage v alu e o f th e corresponding row
before n orm alization. The m atrix is then m ade sp arse by rounding o ff all en tries to
in teg ers. In the second exp erim en t, 16 variable groups w ere used, each group co n ­
ta in in g only one variab le. T he variab le in the kth group is th e d istan ce o f the kth
ch an n el filter from th e sh ort circu it m ain cascade term in ation . T he corresponding
d iction ary is show n in T able 6.5.
T he dictionary is n orm alized and m ade sp a r se
sim ila r ly to that for T able 6.4.
T hese d ictio n a ries provided a theoretical background for the large scale
m in im a x op tim ization o f th e 16-channel m u ltip lexer reported for the first tim e by
B andler, Chen, D aijavad, K ellerm ann, R enault and Z hang (1986).
T heir approach
w as a m anual m anip ulation o f th e decom position properties discovered. The band
form o f T able 6.4 and the near band form o f T able 6.5 correspond to the phenom enon
"S
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151
T A B L E 6.4
DECOM POSITION DICTIONARY F O R A 16-C H A N N EL M U LTIPLEX ER , W HERE
V A R IA B L ES ARE C O U PL IN G PA R A M ETER S A N D T R A N SFO R M E R RATIOS
channels (responses)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
8
8
8
0
1
8
6
0
0
0
0
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
0
0
0
1
0
0
0
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7
5
0
0
0
0
0
0
0
0
0
0
0
7
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
3
3
8
0
0
0
0
5
0
0
0
0
0
0
0
-
0
0
0
0
0
0
0
0
0
0
5
7
5
0
0
0
0
0
0
0
0
0
3
7
5
4
5
7
?•
6
3
0
0
0
0
0
0
0
0
0
0
.0
0
0
0
0
•
0
0
0
0
0
0
8
4
0
0
0
0
0
O’
0
0
0
0
0
0
6
1
0
0
5
5
5
0
0
5
7
5
0
3
7
6
*
o
>
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0
0
0
0
0
4
10
152
T A B L E 6.5
D ECO M PO SITION DICTIONARY FOR A 16-C H A N N EL M ULTIPLEXER, W HERE
V A R IA B L E S ARE T H E D IST A N C E S FROM THE SHORT CIRCUIT
channels (responses)
variables in channel
1
2
2
3
4
5
8
8
8
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
6
0
0
0
2
6
7
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1
.
1
0
0
0
0
0
0
0
0
0
0
0
0
0
3
4
0
0
0
0
o.
£
tr
0
0
0
0
0
4
0
0
0
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
0
0
„
6
7
8
9
0
0
0
0
3
7
4
0
0
0
0
0
2
6
0
0
0
0
0
0
2
0
0
0
0.
0
0
0
1
3
0
0
0
0
0
0
0
0
0
5
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
4
4
.
0
0
0
0
0
0
10
11
12
13
14
15
16
o'
0
0
1
1
1
1
1
2
1
0
0
2
1
0
0
2
1
0
0
0
0
0
0
0
0
2
1
0
0
0
0
1
0
3
3
5
5
3
3
5
5
5
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
1
6
4
7
0
1
1
1
1
2
2
4
7
1
0
0
0
0
0
-
3
3
3
0
0
0
0
O'
0
0
0
0
0
5
6
5
2
0
0
1
0
0
7
r
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153
th a t v ariab les in a ch an n el m ainly affect resp on ses a t the sam e ch an n el and adjacent
ch ann els. The nonzeros in the far off-diagonal region s in T able 6.5 in d icate the effect
betw een non-adjacent ch a n n els due to ad ju stm en ts on the distance o f a channel filter
from the m ain cascade.
i.
6
4
AUTOM ATIC DETERMINATION* OF SUBO PTIM IZATIO N PROBLEM S
6.4.1
T heoretical D escription
T he R eference F un ction Group
v
_
U su a lly , thejdecom position dictionary is not diagonal. A suboptim ization
often in v o lv es sev era l function groups and s e v e r a l v a r ia b le groups.
A m on g th e
function groups involved , th ere is a key group w hich w e call the referen ce group
Such a group ty p ica lly con tain s th e w orst error function. -T he referen ce fu n ctio n
group is used to in itia te a subproblem as described in the su bsequent text.
C andidate G roups o f V ariables
Suppose th e index se t J ; in d ica tes th e r eferen ce fu n ctio n group.
The
rundidate groups o f variab les to be used for the suboptim ization are th ose w hich affect
fj.j
6
J /.
In th e d eco m p o sitio n d ic tio n a r y , th e f t h colu m n a s s o c ia te s w ith th e
reference function group. Rows h avin g a nonzero in the fth colum n, are can d id ate
rows, each corresponding to a candidate v a r ia b le group.
T ak e F ig.
6
. lib ) a s an
exam ple. Suppose th a t the function group a ssociated w ith index set Jo is the referen ce
group, i.e., t —2. T he cand idate groups o f v a riab les are Io, Ig and I7 sin ce they corre­
late w ith the referen ce function group. C orrespondingly, in the D m atrix o f (6.12),
rows 2,
6
and 7 are candidateTOws sin ce th ey a ll have a nonzero in the 2nd golum n.
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154
D eterm ination o f a Suboptim ization Problem
A n au tom atic procedure for the d eterm in ation of P and J s for th e su b
optim zation o f 16.5) h a s been developed. Suppose
in d icates the reference function
group. For a selected candidate variable group, e.g ., the one corresponding to set 1^,
the index s e t J s in d ica tes the union o f a ll fu n ctio n groups w hich c o rre la te w ith
variable group k. I3 id en tifies v a ria b les in the kth group, a s w ell as all other variables
w hich correlate w ith functions only w ith in fj, j
co rrelatin g w ith any a ctiv e functions in fj.j
f > O.SMf
6
6
J s. A lso, P excludes variab les not
J s. A function f is said to be active if
w hen
Mf > 0
w hen
Mf <
*
(6.13)
f > 1.25M f
0
,
w here
M, ^
r
r
m ax
f .
j * .r
•
j
tfi
141
Priority o f C andidate Groups of V ariab les
It can be seen that a ^ a ir o f (Is. -P) a sso cia te w ith a pair o f (Ik,
For a
selected referen ce function group, each can d id ate v a r ia b le group lead s to a sub-
t
problem.' T he seq u en ce o f subproblem s used to pen alize fj, j € Jf, are determ ined by
the priority o f all resu ltin g candidates.
S in ce each candidate d eterm in es the function set .J' for a suboptim ization,
the priority o f the candidate is based upon the pattern o f error functions it wilfoiiTect.
i.e. pattern s o f fj, j € J s. F irstly , the few er the num ber o f function groups in J \ the
higher the priority. Secondly, the w orse the overall error functions in
the higher
the priority. The overall error functions in J s are ranked by the generalized lea st pth
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155
t
'
‘function (GLP) (B andler and Rizk 1979) a s
\lf ( Y
(fj (<{»)/Mf )q)1/q'
if Mf a:
0
(6.15)
GLP =
if Mf =
0
0
,
w here Mf w as defined in (6.14) and
ifM f. > 0 , then K = {j | f a 0 , j € J s}
and q = p
ifM f < 0 , t h e n K = J s
and q = —p .
(6.16)
T yp ically, we choose p = 2.
The priority o f can d id ate variable groups can be sim ila rly d eterm in ed in
th e decom position dictionary.
T h e few er the num ber o f non zeros th a t e x is t in a
can d id ate row, the h igh er the priority. For-two candidate row s co n tain in g an equal
num ber o f nonzeros, a high er priority is g iven to the can d id ate h a v in g a larger v alu e
in its generalized lea st pth function.
6 .4 .2
An Exam ple for D eciding on a Subproblem and C andidate P riority
For the exam p le o f F ig 6.1, suppose th at the m axim um error fu n c tio n s
w ith in each o f the 5 function groups are [3.S 4.
I. - 1 . 2.1. Suppose th at we choose
the w orst group, i.e., group 2, a s th e reference function group. A ccording to our pre­
v io u s d iscu ssion s, th e candidate variable groups are !•>, 1$ and I7 . Io has the h ig h est
priority since it a ffects few er (i.e. o n ly one) function groups than 1^ or
(7
does (Ig and I7
both affect three function groups). To rank the priority betw een can d id ates Ig and I7 ,
we com pare the overall error fun ctions th ey w ill affect.
T he fu n ction s affected by
✓
v a ria b les in Itj(or [7 ) are fj, j € J S = J i U J 2 U J 3 (or J s = Jo u J 3 U J 4 ). !<; has a high er
priority than I7 sin ce the o verall error functions in J i U J 2 U J 3 arfe w orse than th a t in
Jo tj J 3 U J 4 .
i
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. )
N
C orrespondingly, in th e decom position dictionary o f ( 6 .12), rows 2,
6
and 7
are cand id ates. Row 2 h as th e h ig h est priority since it con tain s few er nonzeros than
others. Row
6
h as th e second h ig h e st priority sin ce its GLP v a lu e is obviously larger
than the GLP valu e for row 7.
To form ulate a su b op tim ization problem , i.e., to decide I4 and J \ we choose
a pair o f (Ik, If), e.g., candidate v ariab le group Ig and reference function group J>. The
index s e t J s = J j U J ? U J 3 . T h e variab le index se t Is includes Ig (in d icatin g the
can d id ate variab le group), a s w ell a s Ij, I-? and I3 (in d icatin g a ll o th er v a r ia b le s
«
.
J *
affectin g fu n ction s only w ith in J s). Further, I3 can be excluded from Is sin ce variab les
in I3 do not a ffect a ctiv e fun ction s in.Js . T herefore, we have Is ■= Ig U It U !■>.
6.5
A N A U TO M A TIC D E C O M P O SIT IO N A LG O R ITH M FO R C IR C U IT
’
OPTIMIZATION
An autom atic d eco m p o sitio n a lg o r ith m for o p tim iz a tio n o f m icrow ave
sy stem s h a s been developed and im plem ented.
The algorithm can decide when to
update the se n sitiv ity m atrix and the decom position dictionary. T he form ulation and
the seq u en ce o f suboptim ization problem s are d ynam ically determ in ed
The degree o f
d ecom position is reduced a s the sy stem con verges to its overall solution. As a special
c a se, if a ll v a r ia b le s in te r a c t w ith a ll fu n ctio n s, our approach so lv e s o n ly on e
subproblem , th is being identical to the origin al overall optim ization.
S tep
1
In itialize sparse factor \ .
C alcu late the se n sitiv ity m atrix S and the
decom position d iction ary D. C alculate f.
C om m ent
•s.
T he in itia l s e n s itiv ity m atrix can be obtained from a su itab le MonteC arlo se n sitiv ity a n a ly sis perform ed off-line. A ll error fu n ctio n s are
ca lcu lated in th is step.
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Step 2
D efine ( such th a t
✓
fworst =
m ax fj = m ax fj.
j€ J
C om m ent
T he fth fu nction group co n ta in s the w orst response.
Such a function
group w ill be freq uently ch osen a s th e referen ce group to be penalized.
Step 3
For th e g iv e n t , d eterm in e the seq u en ce o f candidate rows in D.
th e ca n d id a tes in d ecrea sin g priority. S e t k = 0.
C om m ent
Rank
.
T h e fth fu n ctio n group is th e r e fe r en ce group to be p e n a liz ed . A ll
v ariab le groups co rrelatin g w ith th e fth function group are considered as
cand idates.
Stop 4
aJ f k = 0 then se t k to the row index o f th e first candidate, o th erw ise se t k
to the row index o f the n ext candidate. If such a candidate does not ex ist
then go to Step
C om m ent
8
.
T he can d id ate groups o f variab les are se q u e n tia lly selected . Each entry
into th is step r e su lts in a selectio n o f a can d id ate w ith a low er priority
than the cu rren t one.
Step 5
D efine Is and .P /u sin g the current k, (.
If I*1 and J s are id en tical w ith
th eir previou s valu es then go to S tep 4.
S o lv e th e su b o p tim iz a tio n
problem
,
m inim ize
m ax
to. i€Is
j€Js
fj(4 >) .
(
T erm in ate th e op tim ization if
m ax
j£ .P
C om m ent
\
fworst ■
A subproblem is form ulated and solvefa in th is step.
By ch eck in g the
functions not covered in th e present su b op tim ization , any s ig n ific a n t
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
158
d eterioration in the o v erall objective function is prevented. T he factor X'
can be, e .g ., 1 . 2 .
S tep
6
C om m ent
If Is = I a n d J s = J th en stop.
T h e p rogram t e r m in a t e s fo llo w in g th e c o m p le tio n o f an o v e r a ll
o p tim ization w hich is considered a s the la st subproblem .
S tep 7
C alcu late f. C a lcu la te
>'
fworst — fnax ^ .
jSJ
Go to Step 5.
C om m ent
*
An o v era ll sim u la tio n is performed.
By goin g to Step 5, the cu rren t
referen ce function group can be c o n tin u o u sly p en alized in th e n ex t
subproblem even if th is group does not include the w orst error functions
S tep
8
If
'
m ax fj
j€ J a
then go to Step 2.
<
m ax
~ jU
*
fj
If X =» 0 then stop oth erw ise, update S , reduce X.
update diction ary D and go to Step 3.
Comment:
W hen th e selectio n o f a candidate fa ils, a new seq u en ce o f candidates w ill
be defined by go in g to S tep s 2 or 3. By reducing the sparse factor X, the
d egree o f d ecom p osition is reduced a s th e o v e r a ll so lu tio n is b ein g
approached.
The reference fu n ction group w ill be read ju sted if th e
e x is t in g on e d o e s not co n ta in th e m a x im u m erro r fu n c tio n .
For
com p letely decom posable problem s, the term in a tin g conditions in Step
w ill not be sa tisfied and th e program w ill e x it from Step S.
T
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
8
6 .6
LARGE SCALE OPTIM IZATION OF M U LTIPLEXERS
The au tom atic decom position technique w as tested on th e o p tim ization o f
I
,
microwavfe m u ltip le x e r s u sed in s a t e llit e co m m u n ic a tio n s.
S p e c ific a tio n s w ere
im posed on the com m on port retu rn lo ss an d in d iv id u a l c h a n n e l in s e r tio n lo ss .
functions.
Each sub op tim ization w as solved u sin g a recen t m in im ax a lg o rith m o f
B andler, K ellerm an n and M adsen (1985).
U n til our recen t paper on ^ nultiplexers
(B andler, Chen, D aijavad, K ellerm a n n , R enault and Z hang 1'986; B an d ler and Zhang
1987a), the reported d esig n and m an u factu rin g o f th ese d evices w ere lim ited to 12
ch a n n els (e.g., E gri, W illia m s and A tia 1983; T ong and S m ith 1984; H olm e 1984;
yjp
C hen 1985; B andler, D aijavad and Z hang 1986).
A con tigu ou s band 5-ch annel m u ltip lex er w as s p e c ific a lly o p tim ize d to
illu s tr a te th e nov^l process'
a u to m a tic d e c o m p o sitio n , a s sh o w n in F ig . 6*3.
t
F unctions a ssociated w ith the .sam e channel are grouped together. V a riab les for each
channel include
12
coupling param eters, input and ou tp u t transform er ra tio s (n i and
no) and the distan ce m easu re from the ch an n el filter to the short circuit m ain cascade
term ination. The overall problem involved 75 variat>Ies dnd 124 n o n lin ear fu n c tio n sAs the param eters approached th eir solu tion , w eak in teraction s betw een v a riab les
and fu n ctio n s w ere a ls o c o n s id e r e d .
#
optim ization.
T h e fin a l su b p r o b le m w a s .th e o v e r a ll
"
^
We also tested our approach on a 16-channel m u ltip lexer in v o lv in g 240
"
0
variables arfd.399 n on lin ear functions. Th^ responses a t the sta r tin g point is show n in
■
f i g . 6.4. O n l^ O su b op tim ization s were perform ed before reach in g th e response o f
"Fig. 6.5. Ttfen a full optim ization is activated resu ltin g in all resp on ses sa tisfy in g
*
.
'*
their specifications as show n in Fig.
6 .6
. A com parison betw een the op tim al d esign
^ t h and w ithout decom p osition is provided in T able
6 .6
. W hen used to ob tain a good
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
Fig. 6.3
R eturn and in sertio n loss responses o f the 5-ehannel m u ltip lexer for each
sub op tim ization . T h e 20 dB sp ecification lin e indicated w hich channcl(s)
is to be op tim ized in th e n ext subproblem . T h e variab les to be selected
are in dicated in th e graph, e.g., 35 rep resen tin g coupling M 3 5 , d repre­
se n tin g the d ista n ce o f th e corresponding ch an n el filter from the short
c ircu it m ain ca scad e term ination. T he previously optim ized ch an n els
are h ig h lig h ted by th ick response cunres. (al responses at the startin g
point. (b M k ) resp o n ses for each suboptim ization, (f) responses a t the
fin al solu tion .
'
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
*0
90
r e t u r n
and
i n s e r t i o n
loss
<o b i
161
43
FREQUENCY
[MHZ)
90
93
r e t u r n
and
i n s e r t i o n
loss
(o b i
Ca)
30
^RECUCNCY
(MHZ)
(b)
i
rig.
6.3
(continued)
1
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
13 33 34
U 4S H
n l n3 0
lru/ bi mt t m
Z
o
H
29
01
H
0
< »
z
I
3
K
UJ
c
90 u—
114*0
17000
•
K 'P E T Q U E N C V
{MW’ )
Cc )
12
33
44
56
33
34
45
56
23
36
5S
nl
11 t 2 22
23 33 34
36 44 45
56 n l n2
J
M
ID
cn
M
o
; »
rr
ID
c
p 'S E C U E N C v
(M HZ)
(d)
Fig.
6.3
(continued)
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
163
12 2 2 23
3 3 34 3 6
4 4 4 4 36
ffl
Q
nl n2 d
t i i i n n i m n t ni t t t mi t t
M
K
rr
w
W
z
M 90
Q
2< 55
~
LI
c
12040
F f lE O U E N :Y
(M H Z )
(e)
9-
11
23
36
55
12
33
44
56
11
23
36
55
22
34
4S
66
nl n2 d
12
33
44
56
22
34
43
66
n l n2 d
a
U
in
90
90
U—
UUO
F B E C U E N C Y
[M HE)
(O
Fig.
6.3
(continued)
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
164
(OB)
12 23 33
33 34U
90
l l t t t t l l i t
f\ t t
30
. RETURN
AND
IN SERTIO N
LOSS
44 45 5*
o1 n2 0
t l 12 23
23 33 34
35 44 45
55 55 n l
n2 <2
FREQUENCY
(MHZ)
(g)
£
(OBI
0
13
AND
20
33
RE TURN
IN SERTIO N
LOSS
10
43
1.111.
30 L—
1IMO
FREQUENCY
(M H Z ]
(h)
F ig.
6.3
(continued)
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
165
11 1 2 2 2
( DO)
2 2 33 34
38 4 4 4$
56 n2
IS
20
RETURN
AND
INSERTION
LOBS
10
40
43 -
12000
f r e q u e n c y
(m
h z
:
(i) '
12
33
44
m
22
34
45
n2
13 -
2C
RETURN
AMD
INSERTION
LOSS
(OBI
11
23
36
56
tlM O
FREGUENCY £MhC3
(j)
Fig* 6 . 3
(conti nue d')
•
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
166
I t 12 22
(OBI
23 33 34
U « 4 4S
SSU K
36 44 4S
55 56 (6
a.
nl
n2 d
RETURN
AMD
INSERTION
LOSS
M ri2
>>fi)>>
23
30
39
40
FREQ U EN CY
r s iH Z )
INSERTION
33 -
RETURN
X
AMO
LOSS
( OBI .
(k)
3C
$
(O
Fig.
6. 1.
fcontinued)
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
Figc-6.4
Return and insertion loss responses of the 16-channel multiplexer before
optim ization.
Itf7
©
©
(sc)
1/
ITS
sson
o
<u
in
(V
©
m
N Q ix e a sN i
©
\n
cn v
Ul
N e n iiu
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
168
R e p ro d u c e d with perm ission of the copyright owner. Furth er reproduction prohibited without permission.
l-'ig 6 6
Return ;ind insertion loss responses of the 16-channel multiplexer at the
overall solution. All design specifications are satisfied.
169
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
170
%
T A B L E 6.6
COMPARISON' OF 16-C H A N N E L M ULTIPLEXER OPTIMIZATION
WITH A N D W ITHOUT DECOM POSITION
Purpose
of
O p tim ization +
to provide a
good s ta r tin g
p oin t for
fu rth er op ti­
m ization
to obtain a
near optim um
so lu tio n
R eduction in
O bjective
F unction
C riteria for
Com parison
from
13.46
C PU tim e *
99
250
w orking space
n eeded1.
2.197
483,036
num ber of
su b optim izations
10
-
from
13.46
C P U tim e *
651
553
73.97,2
483.036
*
_
w orking space
needed 1
num ber of
suboptim izations
51'
C PU tim e *
1045
12S9
w orking space
needed 1
4 83,036
4S3.036
from .
13.46
to
—0.09
4-
W ithout
Decomp.
to
2.4
to
0.32
to obtain
op tim u m
so lu tio n
W ith
Decom p\ -
\i u m b e r o f
su b optim izations
11
-
—•
d ifferen t sp arse factors .\ h ave been used to control the degree of decom position
•for th e th ree differen t purposes.
1
•
second s on the F P S-264 m ainfram e.
*
o f m ach ine m em ory u n its (one u n it per real numberj required by the mini max
o p tim ization package (B an dler, K ellerm ann and M adsen 1985)
-
1
\
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
171
s ta r tin g p o in t for su b se q u e n t o p tim iz a tio n , th e d ec o m p o sitio n a p p roach o ffe r s
* considerable reductions in both C PU tim e and storage. T he fe a sib ility o f ob ta in in g a
near optim um for la rg e p ro b lem s u sin g co m p u te rs w ith m em ory lim ita tio n s is
observed from th e table. H ow ever, w hen c lose to th e d esired solu tion , the siz e s o f the
subproblem s m ay approach those o f the overall problem . In th is ca se , the perform ance
o f optim ization does not d iffer sig n ifica n tly w ith or w ith ou t decom position, u n le ss the
or.jjinal problem is a lm o st com p letely decom posable.
i
6.7
C O N C L U D IN G REM ARKS
We have presented an autom ated decom position approach for op tim ization
o f large m icrow ave, sy s t e m s.
The approach is g en era lly applicable to the optim al
d esign o f large a n alog circu its. Com pared w ith the e x istin g decom position m ethods,
the novelty o f our approach lies in its g en era lity in term s o f device independency and
its autom ation. A d van tages o f the approach are I) a very sig n ific a n t sa v in g o f CPU
tim e and/or com puter sto ra g e and 2) e fficie n t d e co m p o sitio n by a u to m a tio n .
By
p artition ing th e o verall problem into sm a ller on es, the approach prom ises to'provide a
b asis for com p u ter-assisted tuning. It con trib u tes p o sitiv ely tow ards future general
com puter softw are for large-scale o p tim ization o f m icrow ave sv stem s.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
CT'*'""
7
C O N C L U S IO N S
T h is th e sis ad dressed th ree im portant phases in large sca le optim ization of
«
' +
an a lo g circu its, n am ely , sim u la tio n , se n sitiv ity a n a ly sis and opt+fnization.
N ovel
approaches have been described for large ch a n g e s e n s itiv ity a n a ly s is , branched
cascaded netw ork a n a ly sis, and au tom atic decom position o f circu it optim ization. The
u se o f n iicro w a v e c ir c u it e x a m p le s d e m o n stra te d th e p r a c tic a lity o f o u r new
approaches.
/
•
/
Our approaches offer im m ed iate reductions in com puter storage and CPU
tim e s req uired, e n a b lin g an e n g in e e r to o p tim iz e a la rg e c ir c u it w ith e x is tin g
s
.
\
com puters. The large ch a n g e se n sitiv ity a n a ly sis m ethod is used to perform repeated
circu it sim u lation . W hen used to solve adjoint sy stem s, the m ethod is also applicable
I’
'
to repeated se n sitiv ity ev a lu a tio n . T he branched cascaded a n a ly sis m ethod is a clever
a lte r n a tiv e to gen eral sim u la tio n and se n sitiv ity a n a ly sis m ethods such as the nodal
equation approach and the adjoint netw ork approach.
T he use of-our m ethods in
sim u la tio n and s e n sitiv ity ev a lu a tio n sig n ifica n tly speeds up a circu it optim ization
procedure. The au tom atic d ecom p osition technique directly p artition s the large scale
problem into sm all ones m anageab le by a m ath em atical program m ing softw are
In addition to th eir com putational efficien cy, our novel approaches also
provide better in sig h t into the various effects betw een variab les and responses, as^
V
discussed in C hapters 4, 5 and
6
, respectively
L ogically, the th e sis intends to fill th e gap betw een the svrale of cir c u it
design problem s and the practical lim it of a v a ila b le com puters.
,
i
'
,
;
• .s
,
It is interesting^.* O
'
>■'
172
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
-
\
173
notice th a t th is gap rem ains w idely open even though th e pow er o f com p u ters has
increased d ram atically over th e p a st tw o decades. It is w orth w h ile to m en tion th at
th e m ethods described in th e th e sis are d irected tow ards large sca le problem s. In case
o f sm a ll circu it problem s, th e efficien cy o f our m ethods m ay be le ss th an desired . T he
im p lem en tation of th ese m ethods is g e n e r a lly m uch m ore com plicated th an th a t o f
con ven tion al m ethods.
T he ap plication o f our m ethods is possible in m any situ a tio n s.
R epeated
sim u la tio n and grad ient ev a lu a tio n are often e s s e n tia l in o p tim iz a tio n p rob lem s
a r isin g from m odelling, d esig n , y ie ld m a x im ization etc. A large num ber o f repeated
circu it sim u la tio n s also e x ist in co n stru ctin g a fau lt d iction ary for c ircu it d iagn osis
(B and ler and Salam a 1985a) and in con stru ctin g th e d a tab ase for sta tistic a l d esign
u sin g the param etric sa m p lin g m ethod (S in gh al and P in el 1981). A ll th e se situ a tio n s
provide background for u se o f the large ch an ge se n sitiv ity a n a ly sis m ethod.
T he
efficien cy o f th e m ethod in crea ses if th e num ber o f p ertu rb ed v a r ia b le s is s m a ll
w h ereas the overall problem is large.
The branched c a sca d ed a n a ly s is m e th o d 'is d ir e c tly a p p lic a b le to an y
circu its stru ctu rally branched cascaded, or any circu its reducible to such a structure.
We have illu strated how -to ex ten d th e forw ard and r e v e r se a n a ly s is m eth od o f
B andler, Rizk and A bdel-M alek (1978) to branched cascaded stru ctu res. T h e critical
step for su ch -a n exten sio n isT h e reduction o f 3-port ju n ctio n s into su ita b le 2-port
rep resen tation s. U sin g th is idea, it is a lso natural to exten d the m ethod to cascaded
n etw ork s havin g m u ltip le le v e ls o f b ranches, form u latin g a p ossib le d ir e c tio n for
further research.
T h e a p p lic a tio n o f a u to m a tic d e c o m p o sitio n h a s b e e n .d e m o n str a te d
through a FET m odellin g problem and th e m inim ax d esig n o f a 16-channel m u lti­
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
plexer.
G en era l ap p lication s in m in im ax op tim ization problem s o r ig in a tin g from
d esig n and tu n in g can be en visaged .
decom position into
problem s.
It is also w orthw hile to em bed th e autom atic
and le a st sq u ares op tim ization used in large scale m odelling
T h e efficien cy o f our m ethod in crea ses a s th e cir c u it b eco m es h ig h ly
decom posable.
T he sp a rse m atrix technique is an oth er im portant to o ljo y so lv in g large
s c a le problem s.
T he essen ce o f our b ran ch ed casca d ed a n a ly s is m eth od can be
c o n s id e r e d a s e x p lic it ly ta k in g th e t o p o lo g ic a l s p a r s ity o f th e c ir c u it in to
con sid eration . T h e sparse m atrix tech n iq u e can be directly used to an alyze branched
cascaded n etw ork s by^solving th e nodal eq u ation s for the original and the adjoint
n etw orks. A s a further research effort, it is w orthw hile to com pare the efficiency o f
th e sp arse m a trix approach w ith th e branched cascaded a n a ly sis approach
On the
o th er hand,' th e au tom atic decom position tech n iq u e is to sy stem a tica lly exp loit the
sp arse pattern o f the Jacobian m atrix obtained from d ifferen tiatin g error functions
w .r.t. circu it varia b les. It is profitable to u se appropriate ideas from the sparse m atrix
tech niqu e to im prove the effect o f a u to m a tic d eco m p o sitio n , e .g .. th e se q u e n tia l
a rra n g em en t o f subproblem s
W e h a v e not co n sid ered a n o th e r a lte r n a tiv e o f tr e a t in g la r g e s c a le
p roblem s, i.e., ex p lo itin g special com puters such a s vector processors (C alahan and
A m es 1979; Y am am oto and T a k a h ash i 1985; R izzoli, F erlito and N'eri 1988) and
p arallel processors (H u an g and W ing 1979; Jacob, N ew ton and Pederson 1988)
For
exam p le, circu it a n a ly sis at d ifferent freq u en cies, sim u lation o f different circu its in a
m u lti-circu it approach (B andler, C hen and Daijavad 1986b), circu it sim u la tio n a t
differen t param eter points in a s ta tistic a l d esig n , and different su b optim izations in a
h ig h ly decom posable problem , are a ll su ita b le situ a tio n s for vector and/or parallel
V
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
175
processin g
R esearch in th is area is cu rren tly a c tiv e in th e circu its and s y s te m s
com m unity.
T he fundam ental m echanisnrw for ex e c u tin g a circu it op tim ization a re a
circu it sim u la to r, a s e n sitiv ity a n a lyzer and a m a th em atical o p tim iz er.
G en er a l
efforts tow ards large sca le circu it o p tim ization can be considered a s th e em b ed d in g o f
com m on a p p ro a ch es su ch a s d e c o m p o sitio n , s p a r s e m a tr ix m a n ip u la tio n an d
vectorization into the th ree fundam ental m ech an ism s. T he autom atic decom position
a lgorith m described in th e t h e s is h as b een d e sig n e d to o p era te e x t e r n a lly to a
m ath em atical optim izer.
It is en v isa g ed th at fu tu re optim ization o f la rg e a n a lo g
circuits can be perform ed by op tim izers h a v in g in tern a l cap ab ility o f decom position,
sparse m a trix m a n ip u lation or vectorization .
A n u m b er o f o th e r p r o b le m s a r e a ls o w o rth fu r th e r r e s e a r c h and
developm ent.
i
(a)
___
W e h ave considered th eo retical and com p u tation al aspects o f large ch an ge
s e n sitiv ity e v a lu a tio n s w ith r ela tiv ely sim p le algeb raic and e lec tric circu it
exam p les. The application in practical circu it d esign problem s should be
\
fully tested.
For in sta n c e , in a q u a d r a tic a p p ro x im a tio n to a c ir c u it
. >•
resp onse, one ne'eds to solve a large s e t o f lin e a r equations. T h e repeated
solution o f th e lin ear equations' is .necessary if th e quadratic approxim ation
is to be updated w ith rep lacem en ts o f sa m p lin g param eter p oin ts.
0
Large
ch a n g e se n sitiv ity form ulas in th is algeb raic ca se can be used to m in im ize
the effort of so lv in g the updated lin ear equations.
lb)
T he au tom atic d ecom position theory h a s been tested through a u to m a tic
p a rtitio n in g o f v ariab les and m anual p a rtitio n in g o f functions.
I
F urther
•
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176
'
•
■
"
V
-
•
research is needed to ex am in e autom atic p a r titio n in g o f fu n ctio n s and
com p letely a u to m a ted grouping o f both functions and variables.
■*
<c)
W hen w e tested the a u to m atic decom position algorith m on m u ltip lexers,
we Have
s
a ssu m ed th a t the, decom position dictionary tak es "a band m atrix
( '
form .
T h is a ss u m p tio n m a y not be tr u e fo r g e n e r a l c ir c u it s .
The
a rra n g em en t for d ifferen t subproblem s needs £o be fu rth er ‘tested w ith
d ecom p osition d ictio n a ries h avin g form s other th a n the band m atrix form.
•
(d)
v•
+
«
.The m u ltip le x e r problem has been used to dem onstrate the practical use o f
th e b r a n c h e d c a s c a d e d a n a ly s is an d th e a u to m a tic d e c o m p o s itio n
tech n iq u es p resen ted in th e thesis.- D uring our exp erim en ts w ith such a
device, it w as d iscovered th at as the num ber o f ch an n els in creases, strong
v
\
in teraction s e x is t not only betw een adjacent ch a n n els, but also b etw een
V c e r ta in n o n -a d ja c e n t c h a n n e ls , e .g ., th o s e a b o u t 7 c h a n n e ls a p a r t.
A b n o rm a lities in the response curve, particularly, sharp kinks, are lik ely
to occur for larg e m u ltip lex ers.
Such a p h en om en on has p la gu ed our
exp erim en t w ith m u ltip lex ers having m ore th a n ^
c h a n n e ls.
H ow to
control th e occurrence o f such ab n orm alities, is still unknown.
F u rth er
6
c>
research is n e e d e d
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
A P P E N D IX A
S O M E D E F I N I T IO N S IN G R A P H T H E O R Y
Let G = (V, B) d enote a graph w here V and E are the vertex set and the
ed ge se t, resp ectively. Let V' and E ' be su b sets o f V and E, resp ectively
D efin ition
1:
' G ’ = (V", E') is an edge-induced subgraph o f G if every vertex in V'
is the end vertex o f som e edge in
D efin ition 2 :
_ A vertex v is a cu t vertex o f a connected graph G if and only if there
e x ist tw o v e r tic e s u and w d istin ct from v such that v is on every u^w
path.
D efin ition 3 :
A block o f a sep arab le gfap h G is a m axim al nonseparable subgraph
o f G.
The books by Sw am y and T h u lasiram an (1981) and by Chen (1976) can be
referred to for the r elev a n t d efin itio n s.
i
177
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
•
-
APPENDIX B
*
BRIEF DESCRIPTION OFTHE COMPUTER PROGRAM FOR
SIMULATION. SENSITIVITY ANALYSIS AND OPTIMIZATION OF'
BRANCHED CASCADED NETWORKS
%
.
.
.
)
A com pu ter program h a s b een d evelop ed im p le m e n tin g th e branched
cascaded a n a ly sis m ethod described in C h ap ter 5. ’ T he program can be u sed to
perform sim u la tio n and s e n s itiv ity a n a ly sis for a general branched cascaded circuit.
L im ited op tim iza tio n ca p a b ility is also a v a ilab le. C ircuit elem en ts arc eith er 2-port
su b n etw o rk s or 3-port ju n ctio n s. A ca talogu e o f som e frequently used elem en ts are
coded. T h e option of user-defined e le m en ts.is also available.
*
• T h ere are three e n tr ie s to th e program . The first entry is used to perform
s im u la tio n and sen sitivift- a n a ly sis at the elem en t level.
It is designed^to analyze
c ircu it su b n etw ork s in d iv id u a lly or to help ch ecking the correctness o f user-.defined
e le m e n ts. T he second en try is used to perform sim u lation and/or se n sitiv ity a n a ly sis
o f a g en era l branched cascaded netw ork. The third entry is used to perform design
j
o p tim iza tio n o f branched cascaded circuits.
T he program is w ritten in Fortran-77. The block diagram o f the program is
show n in F ig. B. I. H ere w e briefly d escribe each o f the blocks.
M AIN1 is a m ain program defined by the user. It is used to execu te the
*\/
program through Entry 1. In Fig. B .2., a list of thp m ain program und an illu stra tiv e
**
«
se ssio n o f ex ecu tio n is provided. In th e ■‘execu tion , the elem en t tested is a sim p le 2port co n ta in in g on ly a seriesly .co n n ected resistor.
178
-
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
NAINi
. MAIN3
MAIN2
BLOCK DATA
BRH4
BRH5
BRH3
MMLC
BRHi .
USER DEFINED ELEMENTS
Fig. B. 1
•
•
(o p tio n a l}
B lock d iagram o f the com puter program for sim u lation , se n sitiv ity a n a ly sis
and op tim ization o f branched cascaded networks.
\
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
c
-C
SIMULATION AND SEN SI T I V I TY S a NALYSIS OF 2 - AND 3-PORT
ELEMENTS '
.
“" a .
c
c
C-*~
PROGRAM ENT1
CALL TESTfH
STOP
END
ELEMENT TYPE - ?
Input : I
NO. OF VARIABLES
( NX ) = ?
Input : 1
PARAMETER' VALUE ( 1 - 3 - ) = ?
Input : -r, 1, 1
CODINGS FOR PARAMETERS ( 1 - 3 ) =
I n p u t : 1, 1 , T
WHICH PARAMETERS ARE VARIABLES (
Input : I
TYPE
2 -FOR 2-PORT ELEMENT.
3
FOR 3-PORT ELEMENT.
Input : 2 ‘
FREQUENCY = ?
Input : 2
CHAIN MATRIX
Fig. B.2
( A.B.C '.D-)
1 INDICES
) =?
FOR ELEMENT-TYPE
A
1.00000
0 .0 0 0 0 0
C
0 .0 0 0 0 0
0 .0 0 0 0 0
B
1 .0 0 0 0 0
0 .0 0 0 0 0
D
1 .0 0 0 0 0
0 .0 0 0 0 0
M ain program and the com puter output for the sim u lation and se n sitiv ity
a n a ly sis o f a 2 -port elem en t.
v
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
PERTURBATION CHECK , OF S E N S I T I V I T Y .
VAR
S ENSITIVITY
I
i
1
1 o
1 o
I •
'0.0000.0
0
A
0 , 10E-05
0
C
0.10E -05
0.00
0.00000
0.00000
0
B
0.10E -05
0.00
0.00000
0.00000
0 •
D
0.,10E-Q5
0.00
0.00000
0.00000
1
A
0.10E -05
0.00
0/00000
0 . 0 0 0 0 0 ■■
1
C
0..10E-05 _
0.00
0.00000
o.ooooo-
1
B
0.10E+01
0.00
1.00000
0.00000
1
D
0.10E -05
0.00
0 . ’0 0 0 0 0
VARIABLE
i
^blFF(X)
I
ABCD^ ABS(A,‘ , D )
-------------------------
0
REF.
ANOTHER ELEMENT ?
Input : 2
FORTRAN STOP
*T0
Y/ N
02 *00000 " V
%
*”
0.00000
FREQUENCY.
_____ .
Fig.
B.2
1 / 2
(con tin u ed )
1
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
182
/
M A IN 2 is a m ain program defined by th e user. It is used to execu te the
program through E ntry 2. In F ig. B .3, is a lis t o f the m ain program for the 4-branch
cascad ed netw ork o f F ig. 5.5. T h is netw ork has been described in Section 5.4, The
output o f an in tera ctiv e sessio n is a lso ii^ lu d ed in Fig. B.3.
r
M A IN 3 is a m ain program defined by the user
program throu gh E ntry 3.
It is used to execu te the
F ig. B .4 g iv e s an e x a m p le o f th e m ain program for
o p tim izin g a m u ltica v ity filter, a d evice considered as a branched cascaded network
J
w ith 1 branch. Such a filter h as been described in T able 5.11. The filter is 6 th order
w ith cen ter frequency a s 4 GHz and b a n d w id th as 40 MHz.
/
T h e o u tp u t o f the
'
*
op tim iza tio n is a lso included in the figure.
i . BLOCK DATA is a Fortran data block defined by the user. In th is block,
i
th e user is required to define th e netw ork stru ctu re, the elem en ts, involved, th e source
and th e lo a d s, the v a ria b les, and the optim ization specifications. T he follow ing is a
b r ie f d escrip tion o f th e argum ents .in th is block.
' (al
■ S tructure o f the network: N is the total num ber of branches.
NIR is the
/
total num ber of reference p lanes.
branch k.
(b)
NK(k) is the num ber o f e le m e n ts, in
.
C ircu it ^elements: ITYPlj) is th e index for th e tjhpe o f th e jth e le m e n t.
R D A T (ij) or R D A T B (ij) or R D A T C lij) c o n ta in s the v a lu e o f the ith
p a ra m eterJ p the jth elem en t. 'A n e le m e n t m ay have up to 3. 17 and 10
p a ram eters for RDAT, RDATB and RDATC, resp ectiv ely .
ID A T (ij) or
IDATB(i jl'o r IDATCti j ) is an index for the ith param eter in the jth elem en t
(e.g., w hether in d uctive, r e sistiv e or capacitive) and corresponds to RDAT
■ or RDATB or RDATC, resp ectively.
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
183
C '
C
C
C .
'
SIMULATION AND. SENSITIVITY ANALYSIS OF BRANCHED .
CASCADED NETUORKS.
PROGRAM ENT2
CALL BRANCH'
STOP
END
’
'
BLOCK DATA
IMPLICIT REAL*8 ( A- H. Or- Z)
:
>>>>>>>
ADJUST .
PARAMETER ( N = 4 , . N I R = 2 6 , N X = 8 , N I N T = 3 )
'
DO NOT ALTER :
PARAMETER ( MN=6 , MNIR=27-, MNX=36, MNFR=91)
PARAMETER ( NDATB=17 , N I R B = 6 , NDATC=10, N I R C = 6 , MNFLT=6)
PARAMETER' ( MNINT=10)
CHARACTER*10 I N F L , OUTFL, I F I L E 1 , I F I L E 2 , O F I L E I , 0 F I L E 2 , FLNAM
COMPLEX*16 V S . R S . R L
COMMON / B L 0 K 1 /
ITYP( MNIR) ,NK( MN)
COMMON / B L O K 2 /
RDAT(3 , MNIR) , IDAT( 3 , MNIR)
COMMON / B L O K 3 / •■ IXX< 2 , MNX)
COMMON / B L 0 K 4 /
V S . R S . R L ( M N ) , ISO
COMMON / B L 0 K 5 /
0MG1.OMG2.NOMG.MODOMG
COMMON / B L 0 K 7 /
0 M G A ( 2 . M N I N T) . N P O ( MN I N T ) . I S P E C ( M N I N T ) ,
+
SPEC( 2 , MNINT)
'•
COMMON , / B L K l /
VLIGHT
COMMON / B L K 2 /
I N F L , O U T F L , I F I L E 1 , I F I L E 2 , OFILEI . 0 F I L E 2
COMMON / B L K 3 /
R DATB( NDATB. NIRB) , IDATB( 2 , NDATB, N I R B) ,
+
RDATC (NDA.TC , NIRC ) , IDATC ( 2 , NDATC , NIRC )
COMMON / B L K 3 3 /
OMG(MNFR)
COMMON / B L K 3 6 /
S P E ( 2 , MNFR) , ISPE(MNFR)
COMMON / B L K 3 7 /
N T , N I R T , NXT,NINTT
DATA NT, NIRT. 'NXT,'NINTT/N. N I R , NX, N I N T/
:
>>>>>>>
SET DATA :
'
“
‘
DATA ( N K ( I ) , 1 = 1 , N ) / 3 , 4 , 3 , 2 /
DATA V S . R S , ( R L ( I ) , 1 = 1 , N ) . I S O / I O O . , 5 * 1 . . 2 /
F ig B.3
M ain program b iock'data and the com puter output for the sim u la tio n and
s e n sitiv ity computuUixn'ut’the 4-branch cascaded network.
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
,»
X
'
•
S '
-DATA ( I T Y P ( I ) , I = 1 , N I R ) / .
+
1 0 5 ,7 ,1 0 5 ,2 ,1 0 5 ,1 ,1 0 5 .5 ,4 1 ,0 . 4 2 ,3 ,1 0 5 ,0 , 4 2 ,1 0 5 .3 ,2 ,0 ,
+ ,
4 2 ,1 0 5 ,1 ,0 ,
4 2 ,1 0 5 ,0 /
DATA ( ( I X X ( J , I ) , J = a , 2 ) , I = l , N X ) /
+•
1 2 , 1 , 1 2 , 2 , 3 , 1 , 1 8 , 1 , 2 1 , 1 , 7 , 1 . 8 , 3 , 25*,2/
DATA ( { I D A T ( I , J ) , I = 1 , 3 ) , J = 1 , N I R ) /
+
. 3 * 0 , 2 , 2 , 3 , 3 * 0 , 3*5,' 3 * 0 , 3 * 1 , 3 * 0 , 3 , 3 , 2 , 3 * 1 , 3 * 0 ,
'
+
•
3*1, 3 , 2 , 0 , 3*0, 3*0,
3*1, 3*0, 1 , 1 , 1 , 3*2, 3*0,
+
3*1, 0 ,0 ',0 ,. 3*3, 3*0,
3*1, 3 * 0 , - 0 , 0 , 0 /
DATA' ( <.RDAT( I , J > , 1 = 1 , 3 ) , J = 1 , N I R ) /
+
.fcal.,0., 1 - , 1 . , 2 . , ’ .06,1.,0., 0 . .0 . .0 . . .1.1..0., 0..0..0.,
+
. 0 5 , 1 . , 0 . , -1. , 1 . , 2 . ,
1 . , 1 . , 1 . , 0 . , 0 . 70 . , 1 . , 1 . , 1'. , 1 . , 2 . , 2 . ,
+
. 1,1.,0., 0.,0.,0.,l.,l.,l.,.0 6 ,l.,0 .,
.1,10.,0.,3.,3..3.,
+
0.,0.,0.,
1 . . . 1 . . 1 . , . 0 5 , 1 - , 0 . , 2. , 2 . . 2 . . 0 . . 0 . . 0 . , 1 . . 1 . . 1 . ,
+
.1,1.,0., 0.,0.,0./
_
v
DATA OMG1,OMG2,NOMG,MODOMG/6.2 8 3 1 8 5 3 , 6 . 2 8 3 1 8 5 3 , 1 , 2 /
DATA V L I G H T / . 3 /
DATA I N F L / ' S Y S S I N P U T ' /
DATA OU TF L/ ' S YS jSOUTPUT' /
END '
OUTPUT FILE NAME ?
I n p u t : SYSSOUTPUT
' f t * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
EXACT SIMULATION AND . SE N SI TI V I T Y ANALYSIS OF* MULTIPLEXING NETUORKS
rr>-'
NUMBER OF BRANCHES
( N
)
4
NUMBER OF VARIABLES
( NX )
8
(1)
>>
1.
.2.
SIMULATION,
OR
SIMULATION & SENSITIVITY
.
Input : 2 ■
(2) >>
S EN S I T IV I T Y U . R . T . :
1.
[ X] ,
OR
,
2.
[ X] & FREQUENCY.
OR
' *
x
3.
FREQUENCY.
'
Input : 2
( 4 ) >> CHANGE FREQUENCY?
Y/ N
...1 /2
( PREVIOUSLY, FREQ=
1.000000000000000 •
)
Input : 2
Fig.
B.3
(continued)
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
SIMULATION RESULTS
*
VARIABLES :
1.000
2.000
0.060
3.000
0.050
0.050
2.000
1.000
i
FREQUENCY :
'
1.00000
BRANCH VOLTAGES
:
0.03624
tO .07487
THEVENIN VOLTAGES
THEVENIN IMPEDANCES
0.00003
-0.08225
-0.07595
-0.0 68 75
0.05983
-0.0 4 0 3 9
-15.00361
1.16405
0.03529
-0.30176
0.03193
-0.08172
. -15.65346
-2.31 8 76
-0.72129
2.41490
0.00004
-0.69 08 0
0.02515
0.23408
53.76940
56.81050
10.42942
:
0.03008
-0 .07 78 5
INSERTION LOSS
,
:
:
55.57892
1
RETURN LOSS
:
0.00055
1.72670
COMMON PORT RETURN LOSS
0.00052
0.41430
:
0.41243
F ig.
B.3
(con tin u ed )
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without perm ission.
(6)
>> PRINT
PATA :
0
1. 2, 3,
4
Input : 0
.CONTINUE FOR S EN SI T I V I TY ?"
Input : 1
*
SENSITIVITIES V.R..T.
NONE. OR
■ BRIEF , ; , ,
Y/N
VARIABLES
*
____
DETAIL.
1 / 2
*
if
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
S E N S I T I V I T I E S OF BRANCH VOLTAGES
:
-0.09888
0.19690
0.01602
0.01904
-0.12152
0.07920
-0.06148
0.43382
-0.02178
0.03689
0.00008
0.00013
-0.00083
0.00037
-0.00081
0.00263
0.41840
-1.02730
0.42340
0.49683
-3.17461
2.09775
-1.54074
11.39034
-O.'OOO IS
0.00018.
0.02421
0.02442.
-0.00152
0.00078
-0.00123
0.00500
0.00000
0.00000
0.00000
0.00000
-0.84583
-1 .25308’
0.00000
0.00000
0.42131
-1.01718
-1.0-564 7
-0.85004
0.75952
-0.57964
-1.32161
10.30781
0.'00216
0.00231
0.00347
-0.0 0 1 7 5
0.00061
0.00267
0.16241
0.04932
0.03997
-0 .13 15 7
-0.14168
-0.09279
0.08734
-0.08130
-12.42431
0.173.72
Fig.
B.3
(continued)
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
S E N S I T I V I T I E S OF INSERTION LOSS
23.00568'
2.09034
’ 17.45152
4.45823
0.01270
0.10816
-0 .0 0 0 5 9
-115.5 90 96
54.88169
457.83905 .
-1.39517
0.02412
2.91144
0.20388
-0 .00093
0.00000
0.00000
. 0.00000
0.00000
-1 14 .77 16 8
-1 14 .77 16 8
-114.77 16 8
-1 .2 2 0 7 4
0.11859
0.11859
0.11859
0.09126
-14.18466
-14.18466
-14.18466
-7 .1 5 7 4 0
-0.00183
0.00055
S E N S I T I V I T I E S OF RETURN LOSS
\
■
:
-0 .05475'
:
-0.00292
-0,00050
-0.0 0 0 5 7
0.00000
0.01465
-0 .0 1 2 7 8
-0.03917
0.09851
0-. 0 0 0 0 0
-0.00071
-0.0 0 0 0 8
-0.00059
0.00000
0.00000
0.00000
0.00000
0.01133
0.02123
. 0.00598
0.17232
-0.0 0 0 0 1
-0.00002
-0.0 0 0 0 1
-0.00913
• 0.00084
0.00155
0.00063
0.71641
-0 .0 0 0 0 6 ■
S E N S I T I V I T I E S OF COMMON PORT RETURN LOSS
-0.00050
:
0.00533
0.00004
0.13797
0.00008
F ig.
B.3
(continued)
i
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
0
. 0
0
0
0
0
0.12286,
-0 .00 90 9
0.71310
( 7 ) >> CHECK S E N S I T I Vi E
W ?
Input : 2
CONTINUE S EN SI T I V I TY V . R . T .
Input : 1
1 / 2
Y/ N
. . 1/ 2 .
OMEG ? Y/N
a***********************************
*
S E N S I T I V I T I E S V . R- .T.
FREQUENCY
*
*
W
S E N S I T I V I T I E S OF BRANCH VOLTAGES
-0.17778
0.33120
0.03944
0.08791
S E N S I T I V I T I E S OF RETURN LOSS
-0.00484 .
-0 .0 0 1 5 3
:
-0.44100
0.26906
2.39068
7.36235
-0.00590
-0.11597
:
S E N S I T IV I T Y OF COMMON PORT RETURN LOSS
*
:
-0.10 46 0
GAIN SLOPE :
3 9 . 2 1 7 50
7.48130
6*2.09628
1 .04703
0.37785'
0.32862
0.50006
GROUP DELAY :
„
■
0.18892
Fig.
B.3
(continued)
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
\
189
( 8 ) >> CHECK S EN SI T I V I TY ’
Y/N
................
1 / 2
Input : 2
■ ( 9 ) >>
1 . SIMULATION AND S ENSI T I VI TY CONTINUED,
2.
EXIT
Input : 2
^
FORTRAN STOP
-
Fig.
B.3
'
*
OR
•
•
(con tin u ed )
i
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
190
C— —---- —--------------- ---------------------------------------------------- --------C
C *
OPTIMIZATION OF MULTIPLEXING NETWORKS
C
*'
C----------------------:---------------------------------------------- —
PROGRAM ENT3
IMPLICIT REAL*8 < A - H , 0 - Z )
> > > > > > > .ADJUST :
PARAMETER ( N X = 6 , L = 6 , N F R = 6 0 )
)
NX, L, a n d NFR a r e t h e n u m b e r , o f v a r i a b l e s " , t h e n um b e r o f
l i n e a r c o n s t r a i n t s a n d t h e n um b e r o f s a m p l e f r e q u e n c i e s ,
r e s p e c t i v e l y , for the c i r c u i t o p tim iz a tio n .
*
*
C
DO NOT .ALTER :
PARAMETER ( I U= 2* NF R * NX+ 5*NX* NX+ 5*NF R+1 0* NX+ 4*L)
DIMENSION C ( L , N X ) , B ( L ) , U ( I W ) , X ( N X )
K
* * * * * * * * * * * * * * *
*
>>>>>>>
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
' ADJUSf :
*
-
DATA
DATA
DATA
DATA
DATA
C ,B /^ 2*0./ '
LEQ.lPRf, I C H . K E Q S / 0 , 1 0 , 6 . , 3 / '
E P S .D X /l.E -6 ,l.E -2 /
M AX F / 4 0 /
MODE, MODX/2 , 0 /
»
*
*
*
*
*
_
,
j
C , B', LEQ, I P R , ICH, KEQS, E P S , DX an d MAXF a r e d e f i n e d c o n s i s t e n t l y
. w i t h t h e MMLC p a c k a g e .
%
MODE = 1 o r 2 ‘ f o r
Ll o r m in i m a x o p t i m i z a t i o n .
If
MODX = 0 , t h e i n i t i a l v a l u e s o f v a r i a b l e s a r e d e f i n e d i n
th e b lo c k d a t a , o t h e r w i s e th e y a r e d e f i n e d in a f i l e .
DO 20 1 = 1 , L
2 0 C ( I . I )'= 1 .
C (6,6)= -l.
‘
^
C
+
CALL M U L O P ( N X , N F R , L , L E Q , B , C , L , X , D X , E P S , M A X F , K E Q ? . U , I U . I C H . IPR,
MODE,MODX)
STOP
END
Fig. B.4
>
Main program , block data and the com puter output for optim ization o f a
order m u ltica v ity fijrer.
6
r
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
th
BLOCK DATA
IMPLICIT REAL* 8
(A-H.O-Z)
c * * * * , . * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
>>>>>>>
ADJUST :
PARAMETER ( N=gJL, N I R = 8 , N X =6 , N I N T = 3 , N FR = 6 0 )
c **************************************************************
C
DO NOT ALTER :
PARAMETER ( MN=6, M N I R =2 7 , MNX=36, MNFR=91)
PARAMETER ( NDATB=17 , NI R B = 6 , N DA T C = 1 0 , N I R C = 6 , M N F LT = 6 )
- PARAMETER ( MNINT=10 )
C H A R A C T E R S I N F L , OUTFL, I F I L E 1 , I F I L E 2 , O F I L E 1 , O F I L E 2 r FLNAM
* COMPLEX*16 V S , R S , R L
COMMON / B L O K l /
ITYP( MNIR) ,NK( MN)
• COMMON / B L 0 K 2 / ' R D A T ( 3 , M N I R ) , I D A T ( 3 , M N I R )
COMMON /BLOK.3/
IXX(2,MNX)
COMMON / 8 L O K 4 /
VS,RS,RL(MN)'lSO
COMMON / BLOKS/
OMGl, OMG2, NOMG, MODOMG .
COMMON / B L O K ? /
OMGA( 2, MNINT) ,NPO( MNINT) , ISPEC(MNINT) ' f
■+
SPEC( 2 , MNINT)
~ \ COMMON / B L K 1 /
.VLIGHT
COMMON / B L K 2 /
I N F L , O U T F L , I F I L E 1 , I F I L E 2 , O F I L E 1 , OFILE2
. COMMON / B L K 3 /
R DATB( NDATB. NI RB) , ID A TB( 2 , NDATB, N I R B ) ,
+
RDATC( NDATC, NIRC) , IDATC( 2 , NDATC, N I R C )
COMMON / B L K 3 3 /
OMG(MNFR)
COMMON / B L K 3 6 /
S P E ( 2 . M N F R ) , ISPE(MNFR)
COMMON / B L K 3 7 /
N T , NI R T , NX T, NI N T T
DATA N T , N I R T , N X T , N I N T T / N , N I R , N X , N I N T /
*,* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
C
^
^
*
>>>>>>>
SET DATA :
V ■
DATA (NK( I ) ,“1 = 1 , N) / 3 /
DATA V S . R S , ( R L ( I ) , 1 = 1 , N ) , I S O / 1 . , 1 . , J . , 2 /
•DATA ( I T Y P ( I ) , I = 1 , N I R ) /
+
. 5 ,2 ,4 1 ,0 ,
42 ,4 0 2 ,3 0 1 ,p/
DATA ( ( I X X ( I , J ) , J = 1 / N X ) , 1 = 1 , 2 ) /
+
6 * 6 , 4 , 6 , 7 , 8 , 9 , 1 0 / ..
DATA < ( I D A T ( I , J ) , I = 1 , 3 ) , J = 1 , N I R ) /
+
15*0, 6 , 7 , 0 , 6 * 0 /
DATA ( ( R D A T ( I , J ) , I = 1 , 3 ) , J = 1 , N I R ) /
+
3*0, 3*0, 3 * 1 ., 3*0, 3 * 1 ., 4 0 0 0 . , 4 0 . , 0 . , 6 * 0 . /
DATA ( RDATB( 1 , 1 ) , 1 = 1 , 7 ) /
+
■ .9 7 9 7 9 6 , .9 7 9 7 9 6 , . 8 1 0 1 , .4 8 9 4 , .8 4 5 0 . .1197,' - . 4 0 1 0 /
DATA ( R D A T B ( I , 2 ) , 1 = 1 , 8 ) / 0 . , 0 . ,
0 ..-1 .,
0 .,-!.,
0 .,0 ./
DATA ( ( I D A T B ( J , 1 , 2 ) , J = 1 , 2 ) , I = 1 , 8 ) / 1 6 * 0 /
DATA ( IDATB( 1 , 1 , 1 ) , IDATB( 2 , 1 , 1 " ) , 1 = 1 , 7 ) /
+
4*0, 1 ,2 ,
2,3,
3,4,
1,6,
2,5/
DATA V L I G H T / 1 1 8 0 2 . 8 5 /
DATA OMG1,OMG2.NOMG,MODOMG/3940.. 4 0 6 0 . , 1 2 1 , 21
Fig.
B.4
(continued)
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
I
192
DAiTA (OMGA(1 , 1 ) , 0 M G A ( 2 , I ) , 1 = 1 , N I N T ) /
3950 .,3 9 7 0 ., 3970 .,3 9 7 6 ., 3980 .,4 0 0 1 ./
DATA ( N P O ( I ) , 1 = 1 , N I N T ) / 2 1 , 1 7 , 2 2 /
DATA ( I S P E C ( I ) , 1 = 1 , N I N T ) / 0 , 0 , 0 /
DATA * ( S P E C ( 1 , I ) , S P E C ( 2 , I ) , I = 1 , N I N T ) / •.
+
- 1 0 0 ., .9993,;- - 1 0 0 . . . 9 9 9 3 ,
1 0 0 .,.1 /
DATA TNFL,OUTFL, I F I L E 1 , 0 F I L E 1 / ' S Y S S I N P U T ' , ' S Y S $ O U T P U T ' ,
+
' FLXXX' , ' FLXXC' /
. ..
DATA O F I L E 2 / ' F L R S P C ' /
END
+
«
"S
********************************************************************
COMPUTER AIDED DESIGN OF MULTIPLEXING NETWORKS
*
t
it
* NETWORK DESCRIPTION *
it
it
* * * * * * * * * * * * * * * * * * * * * * *
■J
*
S
NUMBER OF SECTIONS ( N ) .............................................
*■
NUMBER OF BRANCH ELEMENTS IN "SECTION
it i t * * it i t i r i t it it i r i t
it
*
I
’
l
•
. -
.
it it i t it it i t i t i t u it
it
* DESIGN OPTIMIZATION *
*
*********
METHOD
it
itJ t
*•
************
MINIMAX OPTIMIZATION
Fig.
B.4
(continued)
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
3
/
t
193
SPECIFICATION
.
FREQUENCY
INTERV.
-
# OF
PT.
LOWER
UPPER
21
17
22
3950.
3970.'
3980.
‘3 9 7 0 .
3976.
4001.
OUTPUT
RESP.
PORT
TYPE
WEIGHT
SPEC.
VALUE
I
1
2
3
REF. COEF
REF. COEF
REF. -COEF
OPTIMIZATION CONTROL DATA
__________________________
.
COM.
COM.
COM.
PT LOWER
PT LOWER
PT UPPER
100.0
100.0
100.0
0.999
0<999
0.100
v
/
NUMBER OF VARIABLES (NX)
. 1.
I
............................
.
6V
N.
NUMBER OF FUNCTIONS (M)
.
.
.X'
. ...............................
_ ........................ .
TOTAL NUMBER OF LINEAR CONSTRAINTS ( L ) .............................................
NUMBER OF EQUALITY CONSTRAINTS (LEQ)
STEP LENGTH (DX)
.
60
' 6
...............................................................
0
...................................................................................................i . 0 0 O E - 0 2
ACCURACY ( E P S ) .................................................................' . .....................................
MAX NUMBER OF FUNCTION EVALUATIONS
NUMBER OF SUCCESSIVE ITERATIONS
(MAXF) * -
(KEQS)
.
.
.
.
.
.
1.000E -06
.
.
.
40
.........................................................
WORKING SPACE ( I W ) ..............................................................................
3
1284
PRINTOUT CONTROL ( I P R ) ‘ ....................................................................................................................10
Fig.
B.4
(con tin u ed )
A
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
STARTING POINT -- VARIABLES AND FUNCTIONS
VARIABLE
IDENTIFICATION
PARAMETER
*
1
2
3
VALUE
OF ELEMENT
4
6
9 . 7 9 7 9 6 0 0 0 0 E -0 1
8 . 1Q 10Q 0000E -01
4 . 8 9 4 0 0 0 0 0 0 E -0 1
4
8
6
8.450000000E'3^I
5
9
10
6
1 .1 9 7 0 0 0 0 0 0 E - 0 1
- 4 .0 1 0 0 0 0 0 0 0 E - 0 1
6
6
6
6
7
6
FUNCTION
IDENTIFICATION
FREQUENCY
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
.
3950.00
3951.00
3952.00
3953.00
3954.00
3955.00
3956.00
3957.00
3958.00
3959.00
3960.00
3961.00
3962.00
3963.00
3964.00
3965.00
3966.00
3967.00
3968.00
3969.00
3970.00
3970.00
3970.38
3970.75
3971.13
VALUE
RESPONSE
OUTPUT • SPEC.
PORT
TYPE
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM..
COM.
COM.
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
'COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF'
COEF
COEF
COEF
COEF
COEF
F ig.
B.4
PT
PT
PT
PT
PT
PT
PT
PT
PT ’
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
WEIGHT
LOWER
LOWER
LOWER
LOWER
LOWERS
LOWER
LOWER
LOWER
LOWER
LOWER
LOWER
LOWER
LOWER
LOWER-.
LOWER
LOWER
LOWER
LOWER
LOWER
LOWER
LOWER
LOWER
LOWER
LOWER
LOWER
100.0
100.0
100.0
100.0
100.0
100.0
100*0
100,0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
-3.8468E -02
- 3 . 7153E-02
-3.5855E -02
- 3 . 4594E-02
- 3 . 3399E-02
- 3 . 2305E-02
- 3 . 1354E-02
-3.0598E -02
-3.0101E -02
-2.9939E -02
-3.0198E -02
-3.0981E -02
- 3 . 2396E-02
-3.4558E -02
- 3 . 7572E-02
-4.1514E -02
- 4 . 6391E-02
-5.2082E -02
-5.8244E -02
- 6 . 4 1 79E-02
- 6 .8 6 7 IE-02
-6.8671E -02
-6.9625E -02
- 6 . 9998E-02
-6.9661E -02
(continued)
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
195
\
26
27
28
29
30
31
32
33
34
35
36*
37
38
39
40
41
42
43
44
45
46
47
48
49
50 51
52
53
54
55
56
57
58
59
60
3971.50
REF.
3 9 7 1 . P7 ' REF.
-'3 9 7 2 .2 5
REF.
3972.63
REF.
3973.00
REF.
3973.38
REF.
3973.75
REF.
3974.13
REF.*
3974.50
REF.
3974.88
REF.
3975.25
REF.
3975.63
REF.
• 3976.00
REF.
3980.00
REF.
3981.00
REF.
3982.00
REF.
3963.00
REF.
3984.00
REF.
3985.00
REF.
3986.80
REF.
3987.00
REF.
3988.00
REF.
3989.00
REF.
3990.00
REF.
3991.00
REF.
3992.00
REF.
3993.00
REF.
3994.00
REF.
3995.00
REF.
3996.00
REF.
3 9 9 7 . 0 0 ' REF.
3998.00
REF.
3999.00
REF.
4000.00
REF.
4001.00
REF.
COEF
COM.
COEF
COM.
COEF
COM'.
COEF
COM.
COEF
COM.
COEF
COM.
COEF
COM.
COEF
COM.
COEF
COM.
COEF ■COM.
COEF
COM.
COEE COM.
COEF
COM.
COEF
COM.
COEF
COM.
COEF
COM.
COEF
COM.
COEF
COM.
COEF
COM.
COEF
COM.
COEF
COM.
COM.
COEF
COEF
COM.
COEF
COM.
COM.
COEF
COEF ■ COM.
COEF ' COM.
COM.
COEF
COEF
COM.
COM.
COEF
COM.
COEF
COEF
COM.
COEF
COM.
COEF
COM.
COEF
COM.
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT*PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
100.0
LOWER
100.0*'
LOWER
100.0
LOVER
LOWER
100.0
LOWER
100.0
LOWER
100.0
LOWER
100.0
LOWER
100.0
100.0
LOWER
LOWER
100.0.
100.0
LOWER
LOWER
100.0
LOWER
100.0
UPPER
100.0
UPPER
100.0
UPPER
100.0
UPPER
100.0
100.0
UPPER
UPPER
100.0 .
100.0
UPPER
UPPER
100.0
100.0
UPPER
UPPER
100.0
100.0
UPPER
UPPER
100.0
UPPER
100.0
UPPER
100.0
100.0
UPPER
UPPER ’ 1 0 0 . 0
UPPER
100.0
UPPER
100.0
UPPER . 1 0 0 . 0
100.0
UPPER
UPPER
100.0
100.0
UPPER
-6.8480E -02
-6.6333E -02
-6.3124E -02
-5.8812E -02
- 5 ; 3455E-02
- 4 . 7263E-02
- 4 . 0677E-02
- 3 . 4463E-02
-2.9 8 0 2 E -0 2
-2.8 3 3 1 E -0 2
- 3 . 1982E-02
-4.2 2 6 0 E -0 2
-5.8 0 7 4 E -0 2
2.5537E+00
- 3 . 2249E+00
- 2 . 2767E+00
-6.7620E +00
-8.0003E +00
- 3 . 7464E+00
-9.8285E -01
3.0138E-Q1
3.0779E -01
-7.0967E -01
- 2 . 4 9 5 5 E+ 0 G
-4.8093E +00
- 7 . 4 3 1 1E+0Q
- 9 . 8356E+00
- 7 . 1 6 1 IE- hOO
-4.6872E +00
-:2. 5274E+00
—7 . 6 8 1 1 E—01
5 . 2872E-01
1 . 3220E+00
1 . 5889E+00
■ 1 . 3222E+00
)
VALUE OF OBJECTIVE FUNCTION
Fig.
. . . . .
B.4
...................................
2.55371E+00
(continued)
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
196
AT SOLUTION ----- VARIABLES AND FUNCTIONS
VARIABLE
IDENTIFICATION
- PARAMETER
1
2
3
4
5
6
b
OF ELEMENT -
4
6
7
8
9
10
.
VALUE
6
6
6
6 .
6
6
9.898047042E-01
8 . 159142169E-01
5 . 1 0 5 9 3 3 0 1 3E-01
8.235514350E-01
9 . 23650732SE-02
- 3 . 5S7522286E-01
•
FUNCTION
IDENTIFICATION
FREQUENCY
RESPONSE
OUTPUT
PORT
VALUE
SPEC.
TYPE
WEIGHT
V
1
2
'3
4
5
6
7
8
9
10
11
12
13
14 .
15
16
17
18
19
20
21
22
23
24
is
3950.00
3951.00
. 3952.00
3953.00
3954,00
3955.00
3956.00
3957.00
3958.00
3959.00
3960.00
3961.00
3962.00
■ 3963.00
3964.00
3965.00
3966.00
3967.00
3968.00
' 3969.00
3970.00
3970.00
3970.38
3970.75
3971.13
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
_ REF.
' ref.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
REF.
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
COEF
F ig.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.
COM.'
COM.
COM.
-COM.
COM.
COM.
B-4
PT
PT
PT
PT
PT
PT
PT
PT
PT
‘PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
PT
LOUER
LOWER
LOUER
LOUER
LOUER
LOUER
LOWER
LOUER
LOUER
LOWER
LQUER
LOUER
LOUER
LOUER
LOUER
LOUER
LOUER
LOUER
LOUER
LOUER
_ LOUER
*“ LOUER
LOUER
LOUER
LOUER'
100.0
-5.3588E -02
lDO. O
- 5 . 308SE-02
100.0
-5.2624E -02
100.0
-5.2219E -02
100.0
- 5 . 1888E-02
100.0
-5.1655E -02
100.0
- 5 . 1544E-02
100.0
- 5 . 1586E-02
100.0
- 5 . 1 - 8 15E-02
100.0
-5.2268E -02
- 5 . 2985E-02
100.0
100.0
- 5 . 4004E-02
100.0
-5.5360E -02
100.0
- 5 . 70 7 4 E - 0 2
-5.9143E -02
100.0
100.0
- 6 . 1 521E-02-6.4090E -02
100.0
100.0
-6.6629E .-02
100.0
- 6 . 8766E-02
100.0
- 6 . 9.940E-02
100.0
-6.9413E -02
1 0 0 . 0 ' - 6 . 9 4 1 3E-02
100.0
- 6 . 8 6 1 4E-02
100.0
-6.7432E -02
- 6 . 5851E-02
100/0
(continued)
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
197
26
27
28.
' 29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
- 45
46 „
47
*48
49
50
51
52
53
54
55
56
57
58
59
60
«»
-
3971.50
3971.87
3972.25
3972.63
3973.00
3973.38
3973.75
3974.13
3974.50
3974.88’
3975.25
3975.63
3976.00
3980.00
3981.00
3982.00
3983.00
3984.00
3985.00
3986.00
3987.00
3988.00
3989.00
3990.00
3991.00
3992.00
3993.00
3994.00
3995.00
3996.00
3997.00
3998.00
3999.00
4000-00
4001.00
100.0
COM. PT
REF. COEF
LOWER
100.0
COM. PT • LOUER
REF. COEF
REF. COEF
com. ’ p t '
LOWER
100.0
REF. COEF
LOWER " 1 0 0 . 0
COM. PT
REF. . COEF
COM. PT
LOWER' ' 1 0 0 . 0
100.0
REF. COEF
COM. PT
LOWER
LOUER
COM. PT
REF; COEF
100,0
LOWER
100.0
REF! COEF
COM. PT
100.0
REF. COEF
COM. PT
LOWER
100.0
REF. COEF . COM. PT
LOWER
REF', COEF
LOWER
100.0
COM. PT
100:0
COM. PT
REF! COEF
LOWER
100.0
REF. .COEF
COM. PT
LOWER
REF. COEF
COM.' PT
100.0
UPPER
REF. COEF
COM. PT
UPPER
100.0
REF, COEF
COM. PT
UPPER
100.0
UPPER
100.0
REF. COEF
COM. PT
100.0
REF. COEF
COM. PT
UPPER
REF. COEF
COM. PT
UPPER
100.0
COM. PT
REF. COEF
100.0
UPPER'
REF. COEF
COM. PT
100.0
UPPER
REF. COEF
COM. PT
UPPER
100.0
REF. COEF
COM. PT
100.0
UPPER
REF. COEF ‘ COM. PT
100.0
UPPER
REF. COEF
COM. PT
UPPER
100.0
REF. COEF
COM. PT
100.0
yppER
REF. COEF
COM. PT
UPPER
100.0
REF. -COEF
COM. .PT
100.0
UPPER
REF. COEF
COM. PT
100.0
UPPER
COM. PT
REF. COE5
UPPER
100.0
REF. COEF
COM. PT
UPPER '■ 1 0 0 . 0
REF. COEF
COM. PT
UPPER
100.0
REF. COEF
COM. PT
100.0
UPPER
REF. COEF
COM. PT
100.0
UPPER
REF. COEF
COM. PT
UPPER
100.0
VALUE OF OBJECTIVE FUNCTION'
-6.38 80 E -0 2
- 6 . 1564E-02
-5 .9 0 0 5 E -0 2
-5.63 84 E -0 2
-5.3979E -02
-5.2192E -02
—5 . 1 5 4 4 E - 0 2
- 5 . 2638E-02
- 5 . 5985E-02
- 6 . 1575E-02
- 6 . 7844E-02
-6.9343E -02
-5.15 44 E -0 2
-5.1544E -02
-1.7255E + 00
-5.1544E -02
-4.0692E + 00
- 9 . 3586E+00
-5.9621E +Q 0
- 2 . 5942E+00
-6 - 6540E-01
-5 .1 5 4 4 E -0 2
- 5 . 3600E-01
-1.8782E + 00
-3.8407E + 00
-6.1989E + 00
-8.7470E + 00
-8.6<#9E +00
- 6 - 2926E+00
-4.1653E +00
-2.4159E +00
- 1 . 1 1 78E+0Q
- 3 . 2G37E-01
. - 5 . 1544E-02
- 3 . 2024E-01
-5.15445E -02
OPTIMIZATION CONCLUDING DATA
t
TYPE OF SOLUTION ( I F A l l )
.
.
.
0
NUMBER OF FUNCTION EVALUATIONS
F ig.
B.4
16
(con tin u ed )
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
*
198
NUMBER OF SHIFTS TO S T A G E - 2 " ............................................
STEP LENGTH ( DX) -
.
.
.
.
V EXECUTION- TIME <IN SECONDS)
.
.
...
2.
..................................................................... 4 . 0 3 0 E - 0 9
..............................................
54.200
FORTRAN STOP
Fig.
B.4
(continued)
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
.
199
(c)
Source and loads: V S co n ta in s the v a lu e o f th e v o lta g e e x c ita tio n .
RS
co n ta in s th e valu e o f source im pedance. RL(k) co n ta in s the load im pedance
Zr
o f the kth branch. ISO is
1
(or 2) if th e m ain cascad e term in ation is open (or
short) circuited.
(d)
V ariables: N X is th e num ber o f variab les. T h e jth variab le is id en tified as
th e 1X ^(2 j)th param eter in th e IXX(1 j ) t h ele m e n t.
<eF
O ptim ization: N IN T is th e num ber o f frequency su b in tervals. W ithin each
su b in terv a l, a uniform d e sig n sp ecification is im p osed .
N P O (k ) is th e
num ber o f frequency p oints in su b in terval k. O M G A (l.k) and OM GA(2,k)
contain the low er and the upper freq u en cies for su b in terv a l k. S P E C (l.k )
and SPE C (2,k) co n ta in the w eig h tin g and th e sp ecification for su b in terv a l
k. ISPEC(k) eq u a ls 0, i or —i if the sp ecification SPE C (2,k) is im posed on
the com m on-port reflection coefficien t, th e ith branch reflection coefficien t
or in sertio n lo ss, resp ectiv ely .
optim ization.
S P E (iJ) and ISPEtj) are reserved for
O M G l, OM G2, NGMG are th e low er frequency, th e upper
frequency and the num ber o f frequency p oin ts used for a com p lete circu it
sim u la tio n (o b ta in in g a ll c ir c u it r e sp o n se s) a t th e o p tim u m so lu tio n .
MODOMG in d icates th e mode o f such a sim u la tio n and is u su a lly se t to
(fi
2
.
Files: IN FL and O U T FL are ch aracter str in g s co n ta in in g th e in p u t and
output file nam es, resp ectively. 1FILE1, O F IL E l, IFILE2 and O FIL E 2 are
a lso ch aracter str in g s r e se r v e ^ fo r file nam es for com plicated u se o f the
program .
(g)
• C onstant: VLIGHT is the velocity o f ligh t.
A
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
200
In F ig. B.3,_the d a ta block for the 4 branch cascaded netw ork o f Fig. 5.5 is
provided. In F ig . B .4, the d a ta block for optim ization o f th e ntq.lticavitv filter exam ple
.
x?
is listed .
B R H 1 co n ta in s a s e t o f sub rou tin es used to perform various forward and
rev erse a n a ly sis in th e o v e r a ll circu it. T h is block incorporates^the branched cascaded
a n a ly sis tech n iq u e d e sc r ib e d in C h ap ter 5, fo r m u la tin g th e h ea rt o f th e e n tir e
program . It e x is ts in a lib rary form.
B R H 2 c o n ta in s a s e t o f su b r o u tin e s u sed to perform sim u la tio n an d
s e n s itiv ity ca lcu la tio n a t the e le m e n t level. A catalogu e o f standard 2-port and 3-port
ele m e n ts are coded here. B R H 2 e x ists in a library form.
B R H 3 co n ta in s a s e t o f sub rou tin es u sed to in itia liz e various arrays_for
sim u la tio n and s e n s itiv ity a n a ly sis o f the overall Circuit. It e x is ts in a library form
BR H 4 con ta in s a s e t o f sub rou tin es w hich provide in teractive access to the
program for sim u la tio n and s e n s itiv ity a n a ly sis both at the elem en t level and at the
o v era ll circu it le v e l. It e x ists in a library form.
\
8 R H 5 co n ta in s a s e t o f su b rou tin es used to perform circuit optim ization,
sp ecifica lly , to form ulate th e op tim ization problem , to call the m inim ax optim izer and
to prin t both th e in itia l d ata and the optim al solu tion s. It e x ists in a library form
U SE R D E F IN E D E L E M E N T S is an optional block which con tain s a set of
su b rou tin es w ritten by th e u ser to d efine h is or her own 2-port or 3-port elem en ts.
U sers are resp on sib le th e m se lv e s to rep resen t th eir 3-port e le m e n ts in twQoort forms
u sin g the m ethod o f Section 5.2.2.
All e lem en ts should be defined in tKe-form o f
tra n sm issio n m atrices.
MMLC is a m in itn ax optim izer.
based on th eir
2
It w as developed by HakKand M adsen
-sta g e a lg o rith m for n o n lin ear m in im a x o p tim iz a tio n (H ald and
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
201
M adsen 198^). It is a v a ila b le a s a standard softw are (B andler and Zuberek 1983).
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX C
BRIEF DESCRIPTION OFTHE PROGRAM FOR MULTIPLEXER
OPTIMIZATION USING AUTOMATIC DECOMPOSITION
.
k
A com puter program h a s been d ev elo p ed for m in h ^ ax o p tim iz a tio n o f
m ic r o w a v e m u ltip le x e r s u s in g a u to m a tic d eco m p o sitio n .
a lg o rith m h a v e b een described in C hapter
>
* *
6
T h e th e o r y a n d th e
. T he program is^written to be com patible
w ith th e M X SO S2 (1984) package developed by O ptim ization S ystem s A sso cia tes Inc.
U se r s are req uired o n ly to define param eter's for the m u ltiplexer before e x ec u tin g the
program . A ll control param eters for d ecom position are prompted in tera ctiv ely
T h e program is w ritten in Fortran-77. T he block diagram of the program is
show n in F ig. C. 1 . H ere we b riefly d escribe each o f the blocks.
M A IN is the m ain program used to open necessary files and to in itia lize
n ecessary p a ra m eters. U sers are not required to alter this part.
S E T M U X is a su b r o u tin e in w h ich u se r s are r'cquired to d e fin e a ll
n ecessa ry p a ra m eters and codes for th e m u ltip le x e r d evice.
T h is su b r o u tin e is
c o m p letely c o n siste n t w ith the m ain program o f the M XSOS2 package. T he M XSOS2
user's m an ual can be referred to for all d etailed definitions of a r g u m e n ts in th is
su b rou tin e.
PARAIO is a subroutine in w hich u sers are required to define in itia l valu es
or d efa u lt v a lu e s o f a ll variab les o f the m u ltip lexer. This subroutine is com p letely
c o n siste n t w ith th e m ain program o f M XSOS2 package. The M XSOS2 user's m anual
can be referred to for a ll necessary d efin itio n s.o f argu m en ts ia th is subroutine.
202
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
203
• SETMUX
MAIN
PARAIQ
DECOMP
SIMFF
AUTOMX
SUBTUN
IBMOPT
MMLC
FDF
MXS0S2
■t
Fig. C .l
Block diagram o f th e com puter program for optim ization o f m u ltip lex ers
u sin g a u to m a tic decom position.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
' 204
.
V
^
_
AU TO M X is th e m ajor su b ro u tin e for s e q u e n tia l a r r a n g e m e n t o f th e
a u to m a tic decom position stra teg y .
DECOM P co n ta in s a s e t o f subroutines for M onte-C arlo sen sitiv ity a n a ly sis
ahd for the con stru ction o f a decom position dictionary based upon a specified sparse
factor.
SIM FF co n ta in s a s e t o f subroutines for ca lcu la tin g n e c e ssa r y fu n ction
v a lu e s .
It is u se d to c h e c k th e p a tte r n o f a ll er ro r fu n c tio n s a fte r e v e r y
su boptim ization or du rin g a su b op tim ization.
S U B T U N co n ta in s h. s e t o f subroutines for c h o o sin g th e m ost su ita b le
su b op tim ization problem to so lv e. T he selection procedure is based upon the pattern ■.
1
o f a ll error fu n ction s and th e cu rren t decom position dictionary.
(
IBM OPT is th e sh orten ed version o f the m ain program o f t£ic M XSO S2
package. It m akes appropriate arra n gem en t for c a llin g the m in im ax optim izer.
MMLC is a m in im a x optim izer.
It w as developed by H ald and M adsen
b ased on th eir 2 -sta g e a lg o rith m for n onlinear m in im a x o p tim iz a tio n (H ald and
M adsen 1981). It is a v a ila b le as a standard softw are (B andler and Zuberek 1983).
FDI^ co n ta in s a s e t o f subroutines for c a lcu la tin g th e selected su b set o f
fun ction s and th eir s e n s itiv itie s w .r.t. the selected su b set o f variables. The d istan ces
o f ch an n el filters from th e m ain cascade term ination are converted to w a v eg u id e
sp a cin gs here.
y
M XSOS2 is a"Computer package for sim u la tio n , se n sitiv ity a n a ly sis and
op tim ization o f m icrow ave m u ltip lexers. It was developed by O ptim ization S ystem s
A sso cia tes Inc.
Fig. C .2 g iv e s th e com puter output o f op tim izin g the 5-channel m u ltip lexer
d escribed in S ection
/
6 .6
.
In th is ou tp u t, a S U B -D E S IG N m ea n s so lv in g a su b -
-
R e p r o d u c e d w ith p e r m i s s i o n o f t h e c o p y r i g h t o w n e r . F u r t h e r r e p r o d u c t i o n p r o h i b i t e d w i t h o u t p e r m i s s i o n .
205
-
•
OPTIMAL DESIGN AND TUNING USING DECOMPOSITION
A MULTIPLEXER EXAMPLE :
5
6
15
75
NUMBER OF CHANNELS
ORDER OF FILTERS
NUMBER OF VAR. PER CHANNEt
TOTAL NUMBER OF VARIABLES
AUTOMATED DESIGN
SELECT :
4
SEND
1.
2. '
3.
4.
OPTIMIZATION OUTPUT TO
A SEPARATE
FILE
SCREEN OUTPUT
OUTPUT FILE
NOT NEEDED
•
1
SELECT :
-
0 . PRINT BRIEFLY
1 . PRINT IN DETAIL
2 . SAVE X FOR SUB-DESIGN
0
DERIVATIVE VERIFICATION REQUIRED ?
( Y /
N
MAXOPT = ?
{ MA X # OF SUB-DESIGNS )
.
11
MCHK = ?
( SUGGEST :
5 <= MCHK <= 15 )
N )
6
MINIMUM # OF VAR PER SUB-DESIGN =?
(E. G.
5 )
5
<
•*. WANT TO READ: CM, PN1 , P N 2 ’, WGL FOR EACH CHANNEL ?
N
WORST OBJ COMPARED BY RATIO
( E.G,
.99, 1 ., 1.01
RATIO = ?
1.3000000
NEXT WORST DBJ COMPARED BY
( E*.G,
.4 )
RATIO = ? ,
•
•0.4000000
Fig. C .2
)
C om puter output for th e o p tim ization o f the 5-channel m u ltip lexer u sin g
au tom atic decom position.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
v
206
THRESHOJLD LAMD FOR LEVEL UPDATE ( E . G .
LAMD = ?
0 . 2000000 -
•
2.E-1
’
)
■*.
NO.
OF POINTS FOR EACH MONTE-CARLO ANALYS I S . = ?
3
I N I T I A L SPARSE FACTOR = ?
( E . G . . 6 , MUST < 1 )
0.6000000
(.
SELECT :
I N I T I A L I S I N G DECOMPOSITION DICTIONARY ( D) USING
1.
OLD RESULTS ( IN FltE-MONTE )
2.
NEW MONTE-CARLO ANALYSIS ( TO BE PERFORMED r
»
2
CHANNEL #
PARAMETER VALUES
' 1
0.00000
0.42471
0.76313
0.59395
0.00000
-0.39967
0.00000
0.00000 ^ 0.83646
2
0.00000
0.42471
0.76313
0.59395
-0.39967
0.00000
0:00000
0.42471
0.76313
o.'ooooo
0.53514
0.83371
1.04618
0.00000
0.00000
0.64385
0.00000
0.00000
0.83646
0.53514
0.83371
1.04618 ■
0.00000
0.00000
1.39143
0.59395
-0 .39 96 7
0.00000
0.00000
0.00000
0.83646
0.53514
0.83371
1-.04618
0.00000
0.00000
1.94280
0.59395
-0.39967
0.00000
0.00000
0.00000
0.83646
0.53514
0.83371.
1.04618
0.59395
-0.39967
0.00000
0.53514
0.83371
1.04618
I
3
4
-
*
5
0.42471
0.76313
0.00000
0.42471
0.76313
0
. 0
0
0
0
0
0 . 0 0 0 0 0
0.83646
.
0.00000
0.00000
2.59800
0 . 0 0 0 0 0
0 . 0 0 0 0 0
3.25709
>
A. SUB-DESIGN I S CHARACTERIZED BY
(1)
RESPONSES OF CHANNELS
N 1 ----- N2 .
(2)
NX V A R . ' S AS A SUBSET OF ALL VAR . '
s
.
RESULTS OF A SUB-DESIGN ARE GIVEN BY
IFALL MAXF
0
OBJ
OBJ(
1-- ------
5 )
3.037
0.898
16.025
15.117
Fig.
(continued)
C .2
5.076
16.025
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
" 207
V
N1 = 0 E X IT
H I .< 0 N E W D
N l , ' N 2 , NX = ?
- 1 0
0
NO.
OF M ONTE-CARLO P O IN T S
. 3
SPARSE FACTOR = ?
( E . G .
0.6000000 '
?
GRP
VAR . ' S
1
2
3
*4
5
* 6
Nl
N2
1
2
3
4
5
4
1
2
3
4
5
5
NX
9
9 '
9
9'
8
1
0
0
.01
t
17
32
47
62
73
OR >1 T O E X I T )
4
19
34
‘4 9
64
GROUP
6
21
36
51
66
16 .0 25 15 .117
Nl = 0 EXIT
Nl < 0 NEWD .
N1,N2,NX = ?
5
5
8 .
8 VAR. INDICES = ?
62
64 66
67 69 71 ‘ 74.
1
12 15 .143
SUBDESIGN it
1 IMPROVE
Nl = 0 EXIT
Nl •< 0 NEW D
Nl,N2,NX = ?
* 4
5 18
18 VAR. INDICES = ?
47 49 51 52 .54
64 66 67
69
71
1
8
IN
9
24
39
54
69
7
22
37
52
67
11
26
41
56
71
3.037
13
28
43
58
74
0 .8 9 8
75
1 5 . U 3 3.028
1. 936
14
29
44
59
75
15
30
45
60
5."076 16-025
'
9.605
9.605
0. 88+000
56 * 58
73
74
15 .187
59
75
60
62
15 .187 2 .5 9 5
SUBDESIGN it
2 IMPROVE-0.44-001
Nl = O' EXIT
Nl < 0 NEW D
Nl.N2.NX = ?
1
1
9
9 VAR. INDICES = ?
2
4
6
7
9 11
'13 14 15
1
15
7. 674
7 .6 7 4 7.6 7 4
Fig-
C.2
*
I
' ■
6.378
6.433
6.433
6.409 ^ 6 . 4 3 7
6.437
(con tin u ed )
i
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
208
'
/
A
\
SUBDESIGN #
Nl = 0 EXIT
Nl < 0 NEW D
N1,N2,NX = ?
- 1 0
NO.
3 IMPROVE 0 . 7 5 + 0 0 1
0
OF MONTE-CARLO POINTS ?
3
,
SPARSE FACTOR = ?
0.3000000
(E.G.
Nl
N2
NX
1-
1
1
11
2
2
2
14 .
3
3
. 3.
15
4
4
4
14
5
5
5
11
6
7
8
1
2
3
2
3
5
2
1
1
GRP
-
OR >1 TO EXIT)
VAR. >'S IN GROUP
T
0
.01
0
Nl = 0 EXIT
.
Nl < 0 NEW 0
N l , N 2 , NX = ?
1
2
27
27 VAR. INDICES = ?
1
4
3
5
2
•13
14
15
17
18
27
24
26
28
25
—
1
15
SUBDESIGN-#
4 IMPROVE
Nl = 0 EXIT
Nl < 0 NEW D
Nl,N2,NX = ?
3
5
41
VAR. . INDICES = ?
35
32
33
34
31
41
42
43
44
45
52
53
54
55
51
67
64
65
66
63
75
1
14
17.
27
31
41
46
56
62 .
. 75
13*.
16
73
2
3
4
5
6
7
8
9
11
18
28
32
42
47
58
63
19
29
.33
43
48
59
64
20
30
34
44
49
6‘0
65
21
22
23
24
,25
26
35
45
50
36 1 37
38
39
^0
51 / 52
53
54
55
66
67
69
71
74'
.
68
%
15
•
0
7.674
7.674
7 .674
-
6 .409
6. 437
6.437
7 - 8
9
11
6
19
20
22
23
21
29
30
0 . 121“ 0 . 1 1 8 ; 6 . 3 5 4
6. 489
6.409
6.489
0.12+001
• -
36
46
56
68
Fig.
37 ' 38
47
48
58
59
69
71
C.2
39
49
60
73
40
50
62
74
(continued)
>
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
I
209
1
5
4.028
SUBDESIGN #
5 IMPROVE
Nl = 0 EXIT
i
Nl < 0 NEU D
N l , N2. NX = ?
2
3
30
30 VAR. INDICES = ?
17
19
16
18
20
27
29
26
28
30
37
39
36
38
40
1
•5
♦
1.167
4.028
3.801
3.801
3.808
24
34
44
3 .519
25 ■
35
45
-0 .4 7 6
-0 .9 2 2
4.206
4.292
0.25+001
22
21
32
31
41
42
4. 292
23
33
43
SUBDESIGN ft
6 IMPROVE-O. 2 6 + 0 0 0
Nl = 0 EXIT
Nl < 0 NEW D
N1.N2.NX = ?
3
.5
26
2 6 VAR. INDICES = ? .
4 6 47
48 4 9
50
51 5 2
53
56 58
59 6 0
62 .6 3
64
65
68 69
71 73
74
75
1
31
3.322
SUBDESIGN ft
7
Nl = 0 EXIT
Nl < 0 NEU D
N l , N 2 , NX = ?
3
5
41
VAR. . INDICES
31
32
33
41
42
43
51
52
53
64
63
65
75
-
.
\
54 55
6 6 ' 67
0.975
0.664
0.668
0.668.
0.163
0.156
0.169
0 - 159
0.152
0.. 1 7 6
IMPROVE 0 . 9 7 + 0 0 0
>
1
34
35
44
45
54 * 5 5
67
66
1
37
47
58
69
36
46
56
68
5
39
49
60
73
38
48
59
71
3. 735
3 .735
40
50
62
74
2.729
V
SUBDESIGN ft
8 IMPROVE-O.4 1 + 0 0 0
Nl = 0 EXIT i
Nl < 0 NEU D
N l , N 2 , NX = ?
1
1
11
11 VAR. INDICES = ?
1
2
3
4
5
6
7
8
9
11
"I 4
1
8
2.726
Fig.
C .2
2.248
2.726
(continued)
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
210-
SUBDESIGN »
9 IMPROVE 0.10+ 001
Nl = 0 EXIT
Nl < 0 NEW D
Nl,N2,N X = ?
1
2 27
2 7 VAR. INDICES =
1 2
3
4
7
5
6
'8
9
13
14 15 17
18
19
2 0 ' 21
22
29
24
2 5 2 6 27
28
30
9
1
0.657
0.333
SUBDESIGN #
10 IMPROVE 0 . 2 1 + 0 0 1
Nl = 0 EXIT
N l < 0 NEU D
N l , N 2 , NX = ?
-1
5
11
NO. OF MONTE-CARLO POINTS
3
SPARSE FACTOR = ?
(E.G.
.01
1.0000000E -023
IRP
1
2
3
Nl
N2
NX
1
2
1
4
5
5
2
1
72
s VAR. ' S
10
70
1.
12
22
33
43
53
63
74
*
-
11
23
0.323
0.323
0.298
0.657
OR >1 TO EXIT)
IN GROUP
25
2
13
23
34
44
64
75
o
o
0.657
3
14
‘ 24
35
45
55
65
0.333
4
15
26
36
46
56
66
5
16
27
37
47
57
67
0 .323
6
17
28
38
48
58
68
' 7
18
29
"39
49
59
69
0 . 3 23
8
19.
30
40
50
60
71
9 ' 11
20
21
32
31
41
42'
51
52
61
62
72
73
0 . 298
0.6:
Nl = 0 EXIT
Nl < 0 NEU D
Nl , N 2 , N X = ?
1
5
73
73 VAR. INDICES = ?
Fig.
C.2
(continued)
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
211
2
13.
23
■\2
34
33
44
43
54
53
63*- 6 4
74
73
1
12
4
15
26
36
46
56
66
3
14
24
35
45
55
65
75
3
SUBDESIGN »
NO.
11
5
6
17
16
2 7 ’ 28
37
38
47
48
58
57
67
68
7
.
29
39
49
59
69
1t
6 4 /^ - 0 . , 0 2 5
9
8
20
19
3 0 " "31
41
40
50
51
60
61
71
70
11
21
32
42
52
62
72
-0 .025
-0
.695
IMPROVE 0 . 6 8 + 0 0 0
OF SUB-DESIGNS PERFORMED
11
SUGGEST TO PERFORM A SIMULATION AT THE SOLUTION
DETAILS OF OPTIMIZATIONS SAVED IN FILE
URITE:
Y
:
MXDOPT
CM.PNl , PN2.UGL* FOR EACH CHANNEL ?
V
WRITING CHANNEL DATA:
FLXX
ENTER LOCAL FILE NAME
RESULTS OF MONTE-CARLO ANALYSIS SAVED IN FILE
CHANNEL #
•
*
\\
\
\
• \
PARAMETER VALUES
:
MONTE
-0.04718
0.42314
0.62737
0.70930
-0 .37 63 7
-0.007j79
-0.10560
-0.02669
1.05925
0.55167
0.79935
0.91543
-0.04397
0.00000
0.73072
2
0.08258
0.40834
0.73380
0.61762
-0.43245
0.00847
-0.02519
-0.02801
0.90316
0T53718
0.83029
1.06973
-0 .02 07 8
0.00612
1.33536
3
0.11558
0.41570
0.74930'
0.62033
-0 .41449
0.00624
-0.02077
-0.02229
0.92125
0.53476
0.82319
1.07112
-0.01795
0.00430
1.95865
4
0.28426
0.39010
.0.74938
0.66443
-0.48177
0.00187
-0.00799
-0.01631
0.99691
0.54219
0.86340
1.10682
-0 .0 0 9 5 7
0.00991
2.56751
1
.
{
Fig.
C.2
(con tin u ed )
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
0 .0 8 0 1 4
0 .4 0 5 4 5
0 .7 1 1 0 3
- 0 .6 4 7 .0 8
-0 .4 5 2 0 3
-0 .0 2 8 6 0
TOTAL TIME SPENT (SECONDS)
0 .0 3 9 8 4
0 .0 0 3 3 0
0 .7 9 8 3 7
0 .5 5 2 8 2 ^
0 .8 4 8 9 6 *
1 .0 6 2 5 7
0 .0 1 4 1 5
- 0 .0 2 8 8 4
3 .0 8 7 7 1
179.2657
v.
»
F ig .
C .2 ( c o n t i n u e d )
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
■optimization problem . D uring a suboptim ization,.M C H K is the num ber o f iteration s
required per occurrence o f an o v erall functiotf th e ck .
T he m axim u m n u m b er o f ■
. iteration s for one su boptim ization is au to m a tica lly set to M Cf£K*M CHK.
T he te r m ,
OBJ d en otes th e objective function o f th e overall op tim ization or o f a suboptim ization
a s t h c t a s e m ay be. T he first ratio to com pare objective fu n ction s is used to com pare
th e overall objective function before (old OBJ) and a fter (new OBJ) each sub-design.
T he com parison r e su lts in the rejection or acceptance o f th e su b -d esign d ep en d in g
upon w h eth er the d iv isio n o f th e new OBJ by th e old OBJ is greater or le ss th an th e '
specified ratio.
T he second ratio is u s u a lly s e t to 0 .4 an d is u sed to ch eck the.
deterioration o f the o v era ll error fu n ction s du rin g a suboptim ization. T he "threshold
LAMD for level update" is u su a lly se t to 0.2.
A large valu e o f LAMD leads to the
t
quick and prem ature term in ation o f su b op tim ization s. A M onte-Carlo a n a ly sis w ill
■
.
£ -
be activated if the d ecom position diction ary is to be updated. E ach occurrence o f such
an updating ca u ses a reduction in the sp arse factor w hich is used in con stru ctin g t h e
■
■ decom position dictionary.
* -
«
'4
The- in itia l sp arse factor is sp ecified by the u ser
>
‘ su gg ested valu e for th is factor is about
0 .6
.
s i
The
*
IF A L L ^ indicates th e ty p e o f a su b \ -V*
o p tim iz a tio n 'so lu tio n , b e in g c o n s is te n t w ith th e IFA LL in th e MMLC p a c k a g e (Bandl'er and Z uberek 1983).
M A X F g iv e s th e n u m b er o f ite r a tio n s a c t u a lly
perform ed in a su b op tim ization. OBJ(k) indicate the objective function for the su b set
o f functions a sso cia ted w ith chan nel k, k = 1 ,2 , ...,5 .
In Table C .l, th e ind ices o f v a riab les appeared in F ig. C 2 are interpreted
into specific varia b les for the 5-channel m ultiplexer.
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 C .l
INTERPRETATION' OF VARIABLE IN D IC ES IN FIG. C . 2
FOR T H E 5-C H A N N EL-M U LTIPLEX ER
C hannel
Standard N otation for V ariables +
N u m b e r ---------------------------------------------------------——------------------M u M 1 2 M 2 2 M 2 3 M 3 3 M 3 4 M3 6 M4 4 M4 5 M 5 5 Msg Mb6
ni
no
d
1
1
2
3
4
5
6
7
8
9
10
11
12
13
1,4 15
2
16
17
18
19
20
21
22
23
24
25
26
27
28
29 30
3
31
32
33
34
35
36
37
38
39
40
41
42
43
44 45
4
46
47
48
49
50
51
52
53
54
55
56
57
58
59 60
1<*7
5
61
62
63
64
65
67
66
68
69
70
71
72
73
w here Mjj is th e c a v ity resonance or the coupling param eter, i, j € {1 , 2 , . . . .
n i and n 2 are th e in pu t and th e output transform er ratios,
75
74
6
}.
d is the distance o f a
ch an n el filter from the sh ort circu ited m ain cascade term ination.
•a
K
c.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
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4
A U T H O R IN D E X
H .L. A bdel-M alek
9 4 ,1 0 0 ,1 0 3 ,1 7 3
R.L. A dam s
43
C .J. A lajajian
r
/
J.Z . A llen
45, 58
11
I
W .G. A m es
8 ,1 7 4
R. A n treich
_
43
^
H. A sa i
135
F.
A ssa l
95, 129
A.E. Atia-
95, 159
J.W . Bandler
2, S, 9, 10. 11. 12. 13. 14. 16. 17, 1 9 .2 0 .2 1 .2 2 ,2 3 ,2 5 .
2 6 . 2 7 , 3 1 , 3 5 , 3 7 . 3 9 . 4 1 , 4 2 , 4 4 , 5 3 , 6 2 , 6 3 . 6Sl 70, 77,
8 1 , 8 2 . 9 4 , 9 5 , 1 0 0 . 1 0 3 . 1 1 2 , 123, 129, 134. 135. 136.
S.
143,145,150,155,159.170.173,174.201.204.213
I. B arrodale
35
P. B h artia
11
R.M . B iernacki
2 7 ,3 5
M.C. B iggs
9
T
.
R. B illin to n
135
F.T. Boesch
S
A .B . B orison
9
F.H . B ranin, Jr.
R.K . Bravton
- 25
8 .2 0 ,2 1
226
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
227
C.G. B.royden
25
P.R. B ryant
65
W .E. Bryant
12
J.R . Bunch
9
D.A. C alahan
7 .8 .2 1 .1 7 4
R.L. C am isa
3 7 v 3 8 ,5 3
I. A. Cerm ak
10
*
- R. Chadha
1 0 ,6 5
C. C haralam bous
8 ,1 3 ,2 1
H.S.M . Chen
63
■L.K. Chen
8
, 135
M.H. Chen
9 5 ,1 2 9 ,1 5 9
S.H . Chen
8 , 9 . 1 1 ,1 2 ,1 3 ,2 0 .
;
2 1 ,2 5 .3 7 ,3 9 .5 3 .6 2 ; 1 2 3 ,1 2 9 ,
1 3 4 .1 4 5 .1 5 0 .1 5 9 .1 7 4
W .K Chen
1 3 5 ,1 7 7
K M .C ho
65
L.O. Chua
8 ,1 3 5
T.F, C olem an
9
G.C. C ontaxis
135
' A S. Cook
' , ■
S
P.J. C ourtois
142
K.W. C urrent
6 3 . 6 5 .6 9 .SO .SI
W.R. C urtice
1 2 ,3 7 .3 8 ,5 3
S. Daijavad
9 , 1 1 , 1 3 .2 5 ,3 7 .3 9 .5 3 ,6 2 . 9 4 ,9 5 ,,1 1 2 , 123,129^ 134,
1 4 5 .1 5 0 .1 5 9 .1 7 4
G.B. D antzig
-
*
')
L
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
”
R. DeCarlo
C.
'2 7
O elkis
.135
J.E . D enn is, Jr.
8
S.W . D irector
8 ,2 1 ,2 4 .6 3
T. Downs
'
8
M. Dow son
"9
R .G .E g ri
9 5 ,1 5 9
E.I.-El-Nlasry
45
G.
65
E pprecht
M. F erlito
1 2 ,1 7 4
1
J.K . F id ler
F.E. C ardiol
65
•
12
R. Gary
1 0 .6 5
A.M . G eoffrion
9, 135
v f)} . G etsin ger
10
P.E. G ill
2 3 ,2 4 ,3 4 ,5 0
R. G ilm ore
12
T.C. G iras
135
E.
'
.
12
LS. D uff
*
,22
G leissn er
* 4 3
A. Groch
63
K.C. G upta
1 0 ,6 5
G.D. H ach tel
S, 20, 2 1 .2 4 , 3 6 ,3 7 . 38
Y .Y. H a im es
9
I.N .H a jj
'.
6 5 .6 6 ,6 7 ,8 0 ,8 1 ,9 3
4
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
229
J .H a ld
'
1 3 ,^ 2 2 ,2 3 ,3 5 ,6 3 ,1 2 9 ,2 0 0
S.B . H a ley
6 3 , 6 5 ,6 9 ,8 0 .8 1 ,9 3
H.V. H en d erson
6 6 ,6 8
D.M. H im m elb la u
9 ,1 3 4 ,1 3 5
D.E. H ocevar
13
G.R. H offm an
10
S.C. H olm e
9 5 ,1 6 9
E.A. H osny
12
A .S. H ou seh older -
67
J.W . H u an g
8 ,1 7 4
K.W. Iobst
1 3 ,9 4 .
-G-K. Jacob
174
R .H .J a n se n
11
A.T. J o h n son , J r
63
V.K. K alyan
135
N.G. K an ugiek ar
12
N. K arm arkar
21
L. K aufm an
9
* ^
>
Nl
*
/ ^ ~ 9 , 1 2 ,1 3 , 23, 26, 3 7 ,5 0 . 62, 95, 129, 134, 145, 150;
W. K ellerm a n n
1 5 9 ,1 7 0
H . KorVic
Joh
'
.f 37. 3 8 ,3 9 ,1 3 4 ,1 3 6 .1 4 3 , 145
G3. Korre:
K orres
135
N .H .L . K oster
11
L.S. Lasdon
7 ,9 ,1 3 5
M.A. L aughton
9
B.W. L eake
10
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
K.H. L eung
M.R. L igh tn er
A . Lipparini
P.C. Liu
K .L. Lo
* '
"P.V. Lopresti
*
D.G. L uenberger
H .P.L . L una ‘
K. M adsen
C. M ahle
Y.M . M ahm oud
V .K . M an ak tala
T. M andakovic
D.W. M arquardt
R.E. M cIntosh
M.W. M edley, Jr.
R. M eierer
H.M. M errill
S.K . M itra
■P.A. Morris
G. M uller
W. ^lurray
B.A. M urtagh
M.Z. N ashed
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
231
A. N eri
.
A.R. N ew ton
.
1 2 ,1 7 4
8 .1 7 4
>
D P. O’Leary
r,
9
.y >
S.S. Oren
9
D.O. Pederson
8 .1 7 4
.S. P engelly
12
B. V. Pham
93
J.F . P in el
173
E. Po ak
4 1 ,4 2
P. Pram anick
11
M.N. Ransom
2 7 ,3 0 ,3 1 ,3 2
C.R. Rao
oo
C. R auscher
65
J .K . Reid
9
M. R enault
9 ,6 2 ,1 3 4 ,1 4 5 , 1 5 0 ,1 5 9
M.R.M. Rizk
8 .1 4 ,1 6 ,2 0 ,
V. Rizzoli
'
-A
F.D .K . Roberts
V
12, 174
'
22, 94, 100, 103, 1 5 5 ,1 7 3
"
35
G. .Rogers
8
R.A. Rohrer
8
S.S.
135
Sachdeva
/
, 24
R .S a e k s
A
A.E. S alam a
2 7 ,3 0 ,3 1 ,3 2 ,6 2 ,6 3
G.
11
1 3 , 2 7 ,3 5 ,4 1 ,4 4 ,5 3 ,6 2 .1 3 5 ,1 7 3
S a lm er
A. S a n g io v a n n i-V in cen telli
M.A. Saunders
8
, 2 0 ,2 1 ,4 1 , 42, 62 •
9
*
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*
232
T .R . S cott
- 2 4 ,3 6 .3 7 . 38
S.R . S e a r le ..
v
,
6 6 ,6 8
J .F . Shapiro
9
K. S in g h a l
6 5 ,6 7 ,7 0 ,8 1 ,9 0 ,1 7 3
D.
159
S m ith
C.M. Snow den
.
12
M.I. Sobhy'
12
G.R.L. S oh ie
69
W .E. Souder
9
R. Sp ence
6 5 ,6 6
J .A . S tarzyk
27, 3 5 ,1 3 5
D.F. Suchfnan
7
M .N .S. Sw am y
177
S. T ak ah ash i
8
S .N .T a lu k d a r
v
, 174
135
\
%
M .T a n a k a
135
G.C. T em es
" 7; 2 1 ,6 3 ,6 5
A .B . T em p lem an
“
9
K. T h u la sira m a n
177
M .D .T on g
135
R. T ong
"
9 5 ,1 5 9
T .N . T rick
1 3 ,4 5
H .T ro m p
1 3 ,4 1 ,4 2
G .T sir o n is
D.J. T v la v sk v
4
3 7 ,3 8
69
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233
M. Urano
135
A .S. V ander V o rst-
10
D. V aron
10
L.M. V id igal
*
-
63
°
V. V isv a n a th a n
62
J . V lach
6 5 ,6 7 ,7 0 ,8 1 .9 0
■
.
.
1
A .D .W a r e n
7
v
E. W ehrhahn R.
8
f
*
D.E. W h ite
9
A:E. W illia m s
9 5 ,1 5 9
O. W in g
8 ,1 7 4
P. W olfe
9
F.F. W u
F.
.
Y am am oto
135
8 ,1 7 4
K .A .Z a k i
1 3 ,9 4
Q J. Z hang
2. 9, 1 2 ,1 3 ,1 7 ,1 9 ,2 5 , 39, 62, 63,
68
, 70. 81. 82, 94,
r 9 5 ,1 1 2 ,1 2 9 ,1 3 4 ,1 3 6 ,1 4 3 , 145, 150, 159
W.M. Zuberek
2 0 1 ,2 0 4 ,2 1 3
R.P. Zug
24. 36, 3 7 , 38
/ "
\
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 E X
Adjoint,
netw ork, 8 ,2 4 ,1 7 4
s y ste m , 7 0 ,1 7 2
B royden form ula, 25
C A D E C + ,W
C andidate,
row, 1 5 3 ,1 5 6
v a ria b le group, 153-158
ciAO.ii
>
r
C oupling,".
m atrix , 126 •
p ara m eters, 1 3 2 ,1 5 1 ,1 5 9
D ecom p osab ility, 142
D ecom position d iction ary, 1 4 2 -1 5 2 ,1 7 6
E q u iv a len ts,
N orton, 106
T h ev en in , 103-107,
111
t
, 112, 114, 116, 1 1 7 ,1 2 1 ,
T
Error fu n ction s, S, 1 6 ,1 7 , 1 3 7 ,1 5 4 ,1 5 5
■''FBS, 70, 57, 6 7 , 7S
FET,
m odel, 1 4 3 ,1 4 5
m o d ellin g, 5 ,3 8 ,3 9 ,5 3
Forward and rev erse a n a ly sis, 9 4 ,1 0 0 , 173
F u n ction al tu n in g , 43, 58
G ain slop e, 1 0 7 ,1 0 8 ,1 2 2 ,1 2 3
%
G eneralized,
H ouseholder form ulas, 4 ,5 , 6 6 -7 0 ,7 2 ,9 3
le a s t pth function, 22, 4 3 ,1 5 4 , 155
m atrix in version , 22, 32
234
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
235
Group delay, 14, 107, 1 0 8 ,1 2 2 ,1 2 3
H ouseholder form ula, 4, 5 ,6 5 - 7 0 ,7 2 ,8 0 ,9 3
H RF, 6 8 ,7 2 ,7 7 , 83
Im pedance in verter, 123
t*
Input-output transform er, 1 2 3 ,1 2 9 ,1 5 9
Insertion loss, 1 4 .1 0 7 ,1 0 8 ,1 1 8 ,1 2 3 , 1 2 9 ,1 5 9
Jacobian, 4 3 ,1 7 4
f j op tim ization, 1 3 ,2 2 ,2 5 , 35, 4 7 ,4 9 ,5 1 ,1 9 9
t± optim ization, 33, 47, 49, 51
Least pth,
function, 31, 37, 43
op tim ization , 2 1
L inear program m ing, 2 1 ,3 5 ,1 3 5
MDT, 4 , 1 9 , 2 0 , 2 1 , 6 4 •
M IDAS, 11
M INBO X, 24
M inim ax o p tim ization , 4 ,1 3 ,1 8 , 23, 2 5 ,1 3 7 , 170
M inim um order reduced sy ste m , 4, 5,
68
, 69, 8 2 ,9 3
M INM AX, 24
MMLC, 2 0 0 ,2 0 4 ,2 1 3
Model,
com ponent con n ection , 3 0 ,3 2
cu rren t/voltage source su b stitu tio n , 27
FET device, 143
M u lticavitv fitter. 5, 95, 96. 1 2 5 ,1 2 7 ,1 8 2 , 200
*
ft
M ulti-circu it approach, 3 9 ,5 3 , 174
M u ltiplexer.
,
d ecom p osition d iction ary, 1 4 5 ,1 5 0 -1 5 2 ,1 7 6
op tim ization , 5 .1 2 9 ,1 3 6 , 137, 159
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
s^ 6
*
stru ctu re,
112
M X S 0 S 2 ,202-204
N o n lin ea r program m ing, 2 1 ,1 3 5
P ostproduction tu n in g , 42-44, 53
Preproduction tu n in g , 41
P riority o f can d id ates, 154-157
R ank,
■
o f s e n s itiv ity m a trix , 62
of sy ste m d ev ia tio n m atrix , 69, 81
R eference,
f
function group, 153 -1 5 8
p lane, 9 8 ,1 0 0 ,1 0 3 ,1 1 2
-
R eflection co efficien t, 1 0 3 ,1 0 5 ,1 0 7 ,1 9 9
R eturn loss, 1 4 .1 0 7 ,1 0 8 ,1 1 9 ,1 2 0 ,1 2 3 ,1 2 9 ,1 5 0 ,1 5 9
S -p a ra m eters, 14, 5 3 ,1 4 5
^
S e n s itiv ity m atrix, 6 2 ,1 3 8
*
S pecifica tio n s, 1 6 ,4 2 , 1 2 9 ,1 3 7 ,1 5 9
'
Suboptim ization, 136, 1 5 3 ,1 5 4 , 156, 157, 159, 170
SU PER -C O M PA C T, 1 1 ,1 2
T O U C H ST O N E ,
11
,1 2 ,5 3
T ran sm issio n m atrix , 9 5 ,9 8 ,1 0 0 ,1 0 7 ,1 2 3 ,1 2 5 ,1 2 9
V R F,
68
. 7 0 .7 2 , 7 7 ,8 3
W aveguid e m anifold, 5 ,1 2 3 ,1 2 9
W eig h tin g factor, 1 6 ,1 7 , 26, 31, 32, 39, 43L, 139
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
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