Active RC networks utilizing the voltage follower
by James Llafet Hogin
A thesis submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY in Electrical Engineering
Montana State University
© Copyright by James Llafet Hogin (1966)
Abstract:
The (RCVF) is defined as an active network whose passive portion consists of a three-port grounded
RC structure and whose active portion consists of the three-terminal grounded voltage follower. The
necessary and sufficient conditions on a voltage transfer function for its realization by the (RCVF) are
developed. Synthesis methods for the passive portion of the (RCVF) are reviewed, developed, and
discussed. A digital computer program is presented which demonstrates the feasibility of optimizing
(RCVF) networks by the method of steepest descent. A discussion and analysis of practical voltage
followers and the practical considerations involved with their use is presented. In addition, a useful
three stage voltage follower is developed which has very high input impedance and a gain very close to
unity.
Illustrative examples and experimental verification are given. It is concluded that the (RCVF) is a
highly useful and practical network. A C T I V E RC NETWORKS U T I L I Z I N G T H E VOLTAGE F O L L O W E R
by
JAMES LLAFET H O G I N
A thesis submitted to the Graduate Fac u l t y
in partial fulfillment
of the requirements for the degree
of
D O CTOR OF PHILOSOPHY
in
Electrical Engine e r i n g
Approved:
H e a d „ Ma j o r Department
D e a n 9 Graduate Division
MON T A N A S T A T E UNI V E R S I T Y
B o z e m a n 9 Mon t a n a
M a r c h 9 1966
ACKNOWLEDGMENTS
The author p a r t i c u l a r l y wishes to thank Prof e s s o r B 0 J c
Bennett for his encouragement and helpful suggestions.
is also grateful for the f inancial support provi d e d b y a
Douglas Aircraft C o m p a n y scholarship.
iii
He
TABLE OE CONTENTS
Page
Vita
'
Acknowle d g m e n t s
ii
iii
Table of Contents
iv
List of Tables
vi
List of Figures
vii
List of Symbols
x
Abstract
I,
xii
Int r o duction
I
A0
The Active Ne t w o r k
2
B0
The Voltage .Follower .
4-
C0
Comparison of Active and Passive Transfer
F u n c t i o n Synthesis
II0
5
Statement of Pro b l e m
11
IIIo N e c e s s a r y and Sufficient Conditions for the
R e a l i z a t i o n of the Voltage Transfer F u n c t i o n T^
A0
IVo
Surplus F a ctor U t i l i z a t i o n
21
(RCVF) R e a l i z a t i o n Techniques
A0
26
A P ictorial R e p r e s e n t a t i o n of the Passive
Synthesis P r o b l e m
B0
13
,
R e v i e w of P r e v i o u s l y Proposed Configurations
iv
26
$0
Page
Co
A R e a l i z a t i o n Technique f o r the Biquadratic
Transfer F u n c t i o n
B0
Jl
Ladder Ne t w o r k R e a l i z a t i o n for Low-pass and
High-pass T r a nsfer F unctions
E0
Fo
Vo
VIo
J6
A N etwork Co n f i g u r a t i o n f o r Fourth-'order
Band-pass T r a n s f e r Functions
47
Examples
51
Net w o r k O p t i m i z a t i o n
59
A,
Opt i m i z a t i o n Parameters
59
B0
Computer P r o g r a m D e s c r i p t i o n
67
Co
Example N etwork
85
P r a c t i c a l Considerations
90
A0
The Non-ideal Voltage F o l l o w e r
90
B0
Voltage F o l l o w e r Circuits
91
Co
Experimental Results
96
VIIo Con c l u s i o n and Rec o m m e n d a t i o n s
107
Ao
Con c l u s i o n
107
B0
Recommendations
107
I,
111
Appendix
A p p e n d i x 11,
O p t i m i z a t i o n C o m puter Program L i s t i n g
Opti m i z a t i o n C o m p u t e r Program Input and
Output Data for Example Problem
121
A p p e n d i x IIIo N umerical Computations
125
Literature Consulted
IJO
v
LIST OF TABLES
F r e q u e n c y - a t t e n u a t i o n t a b u lation of experimental
and t h eoretical values for network of Figure 6.8.
F r e q u e n c y - a t t e n u a t i o n t a b u lation of experimental
and t h eoretical values f o r n etwork of Figure 6.9»
F r e q u e n c y - a t t e n u a t i o n t a b u lation of experimental
and theoretical valu e s for.n e t w o r k of Figure 6.10.
LIST OB’ FIGURES
1.1
The active RC n e t w o r k u t i l i z i n g the v oltage f o l l o w ­
er.
2
. / I , = # s _ +2.8280+4 ^ ^ h
out
in
s ^ +1.4 1 4 s +1
less net w o r k terminated h y r e s i s t o r s .
1.2
R e a l i z a t i o n of E
1.3
R e a l i z a t i o n of E ^ / E ^
==0.2
a loss-
s +2.8288+4
s^+1.414s+l
P a n t e l l 1s method.
1.4
Simple r e a l i z a t i o n of E
1.5
R e a l i z a t i o n of
. / I . = - +2?828s+4 ^
out
in
s2 +lo4l4s+1
2
X Sg + 2 °'
8'
2 8s+4 w i t h the
s +1.414s+l
(RCYP)
3.1
The
"A" and
3.2
Interco n n e c t i o n of "A" and
3.3
The
"A" and
"B" two-port unbalanced n e t w o r k s .
"B" networks.
"B" n e t works connected to the Voltage
follower.
4.1
A p o s sible current distribution.
4.2
■Current d i s t r i b u t i o n of passive p o r t i o n of (RCYP)
2
apS +a^s+a
n e t w o r k to realize T = --- g-------- .
P 2 3 +Pi8+P 0
4.3
Net w o r k co n f i g u r a t i o n w i t h input current d i s t r i b u ­
t i o n of Figure 4.2.
4.4
Current d i s t r i b u t i o n for case of a-, / a s r n /r
4.5
Current d i s t r i b u t i o n for case of r ^ / r Q s a^/a^
4.6
The basic ladder network.
J-
O
J-
O
.
.
4 .7
The basic decomposed ladd e r network.
4.8
Decomposed ladder n e t w o r k wit h shunt arm capacitors
and series arm resistors.
4=9
Current d i s t r i b u t i o n for bas i c . l a d d e r ne t w o r k r e a l ­
ization of y - Q .
4.10
P ole-zero d i s tributions for
to y ^ ^ \
4.11
Three basic b u i l d i n g bloc k s for the f o u r t h-order
band-pass t r a nsfer f u n c t i o n realization.
4.12
Current d i s t r i b u t i o n for network c o r responding to
t r a nsfer f u n c t i o n of Eq.. (4.23)•
2
4.13
4.14
Net w o r k r e a l i z a t i o n for T =
v
%
.
■ s^ +1»4 1 4 s +1
Net w o r k r e a l i z a t i o n for T ^ = I / sZ*"+2,6132s^+3<>4143s^
+2.6132s+l.
4.15
Net w o r k r e a l i z a t i o n f o r T^=2.172s^/s^+1.414s^+3s^
+1.4!4s + 1 0
'3.1.
P l o w chart f o r n e t w o r k optimization c o m puter p r o ­
gram.
5.2
5 =3
Config u r a t i o n of example network.
Optimized n e t w o r k for realiz a t i o n of T^l/s^"
+2»6132s 5+3.4143s 2+2.6132s +1.
6=1
Current d i s t r i b u t i o n di a g r a m for n o n u n i t y voltage
follower gai n k.
6.2
Voltage f o l l o w e r
6.3
Altered network.
of Refe r e n c e 32.
viii
6<>4-
S y s t e m d i a g r a m f o r n e t works of Figures 6.2 and 6.3.
6,5
Three stage voltage f o l lower network.
606
6.7
Current d i s t r i b u t i o n f o r twin-T network.
p
Net w o r k for r e a l i z a t i o n of T ^ = O .ls/s +.ls+1.
6.8
(HCVF) for r e a l i z a t i o n of second-order band-pass
t r a nsfer f u n c t i o n d e n o r m a l i zed to 30Q 9000 ohms and
150 cps.
6.9
(RCVF) for r e a l i z a t i o n of fourth-order low-pass
t r a nsfer f u n c t i o n d enormalized to I m e g o h m and 85
cps.
6.10
(RCVF) for r e a l i z a t i o n of fourth-order band-pass
t r a nsfer f u n c t i o n d enormalized to 500»000 ohms and
39»8 cps.
ix
LIST OF SYMBOLS
EC
A b b r e v i a t i o n for the t e r m "resistance-capacitance".,
EOVF
A b b r e v i a t i o n for the active network composed of
resistors,
capacitors,
and a single grounded v o l t ­
age f o l l o w e r o
E
I
i
I
y Ij
Voltage p a r a m e t e r o
Current p a r a m e t e r c
Short-circuit admittance p a r a m e t e r s „
Voltage t r a n s f e r function.
Complex f r e q u e n c y v a r i a b l e „
N
nume r a t o r po l y n o m i a l of y ^
for i=j.
Negative of
n u m e r a t o r po l y n o m i a l for i / j.
D(G)
D e nominator p o l y n o m i a l of y^^.
A(G)
Nume r a t o r p o l y n o m i a l of T^,
P(8)
Denomi n a t o r p o l y n o m i a l of T^,
B( G )
Polynomial equal to P ( s ) - A ( s ) ,
E(G)
A u g m e n t i n g po l y n o m i a l such that P(s)+E(s) h a s - n e g a ­
tive real zeros.
ai
P1
ri
ei
Coefficients of A(s).
Coefficients of P(s) „
Coefficients of E.(s ) .■
Coefficients of E(s).
Coefficients of D ( s ).
Admittance parameter,
x
Impedance parameter=
R e s i stance parameter=
Capacitance parameter*
7H
6±
Zeros of admittance*
Poles of admittance*
Opt i m i z a t i o n parameter.
S e n s i t i v i t y of T w i t h respect to k*
Polynomial root s e n s i t i v i t y (p = any p o l y nomial
r o o t )*
Coefficients of T^.
Capacitance d i s t r i b u t i o n parameter.
Et
Resistance d i s t r i b u t i o n parameter.
Capacitance summ a t i o n parameter.
yR
Capacitance ratio parameter.
Resistance ratio parameter.
Opt i m i z a t i o n p a r a m e t e r weig h t i n g functions.
2(8)
Sec o n d - o r d e r po l y n o m i a l with zeros lying on unit
circle in the RHP*
Angle
zeros of F(s) mak e w i t h the p o s i t i v e real
axis*
B(s)
Au g m e n t i n g po l y n o m i a l for F ( s ) .
Zeros of B ( s ) »
A
Coefficients of B ( s )*
F u n c t i o n of the coefficients of B ( s ) .
xi
AB S T R A C T
The (RCVB) is defined as an active n e t w o r k whose p a s ­
sive p o r t i o n consists of a three-port grounded RC structure
and whose active p o r t i o n consists of the three-terminal
grounded voltage follower*
The n e c e s s a r y and sufficient
conditions on a voltage t r a n s f e r function for its r e a l i z a ­
tio n "by the (RCVF) are developed*
Synthesis met h o d s for the
p a s s i v e p o r t i o n of the (RCVF) are r e v i e w e d $ developed, and
discussed*
A digital c o m puter p r o g r a m is pres e n t e d which
demonstrates the f e a s i b i l i t y of optimizing (RCVF) networks
b y the me thod of steepest descent*
A dis c u s s i o n and a n a l y ­
sis of p r a c t i c a l v oltage f ollowers and the p r a c t i c a l c o n ­
siderations involved w i t h th e i r use is presented*
In a d d i ­
tion, a u s e f u l three stage vol t a g e follower is developed
w h i c h has v e r y h i g h input impedance and a gain v e r y close to
unity*
Illustrative examples and experimental v e r i f i c a t i o n
are given*
It is concluded that the (RCVF) is a h i g h l y u s e ­
ful and p r actical network*
xii
I.
INTRODUCTION
A c t i v e RC networks are networks containing only r e s i s ­
tors g capacitors and active e l e m e n t s »
These networks ax-e
attractive at the lower frequencies where m a g netic elements
are large,
expensive and d ifficult or impossible to realize.
In addition, the advent of mic r o c i r c u i t technology, where a
stable inductance p a r a m e t e r is difficult to attain, has r e ­
sulted i n an increased interest in this type of network.
The need for a p o w e r supp l y f o r the active elements is often
u nimport a n t if, as f r e q u e n t l y happens, the n e t w o r k is a part
of a system for w h i c h p o w e r is required regardless of the
network.
The purpose of this study is to develop synthesis p r o ­
cedures concerned w i t h a p a r t i c u l a r type of active RC n e t ­
work:
an active RC n e t w o r k whose active element consists of
a single ideal voltage follower.
The ideal vol t a g e follower
is defined as a 2-port net w o r k whose input admittance and
output impedance are zero and whose ratio of output to in ­
put volt a g e is unity.
The cathode,
emitter,
or source f o l ­
lower circuit m a y be considered a practical approx i m a t i o n to
the ideal voltage follower.
In addit i o n to de v e lo p i n g synthesis procedures, this
study is concerned w i t h the opti m i z a t i o n of the synthesized
™2™
networks b y means of the method of steepest descent as ap ­
plied b y the digital C o m p u t e r 0
F i n a l l y 9 the study, is c o n ­
cerned wit h the p r a c t i c a l d e m o n s t r a t i o n of the synthesis
methods and w i t h the p r o b l e m of obtaining a vol t a g e f o l l o w ­
er w h i c h clo s e l y a p p r o x i m a t e s .the I d e a l 0
A«
THE ACTIVE HETWORK
The active RC net w o r k c o n f i g u r a t i o n of c o n c e r n in this
r e s e a r c h is shown in Figure I 0I 0
The passive p o r t i o n of.
the n e t wo r k consists of a 3-port grounded RC structure.
The
active p o r t i o n consists of the 3™terminal grounded voltage
f o l l o w e r defined above.
F o r convenience, this type of ac ­
tive n e t w o r k will be referred to as the
( R CVF)0
I
A n important c o n s i d e r a t i o n in the d e s i g n of active n e t ­
works is the sensi t i v i t y of the desired net w o r k function to
variations of the active element p o r t i o n of the active n e t ­
work.
This co n s i d e r a t i o n is in addition to that of the s e n ­
siti v i t y associated w i t h va r i a t i o n s of the p a s s i v e n etwork
elements.
It should be pointed out, however, that from a
pra c t i c a l po i n t of view, the g a i n of the voltage follower
proposed as the active element in this study c a n be made
more stable t h a n alternate types of active devices which r e ­
quire a gai n other t h a n unity.
This observation is based
RG
P A S S I V E NETWORK
I
I
VM
P A S S I V E POR T I O N
OF NETWORK
Figure 1.1.
ACTI V E PORTION
OF NETWORK
The active RC network u t i l i z i n g the voltage follower
(RCVF)
-4on the fact that the voltage f o l l o w e r is c o n c e r n e d ■wit h the
direct c o m p arison of input vol t a g e and output voltage r a ther
t h a n wit h their indirect comparison.
Thus 9 a direct c o m p a r ­
is o n of active networks on the basis of the magn i t u d e of the
sensit i v i t y f u n c t i o n associated w i t h the active element could
be misleading.
It is more informative to specify network
f u n c t i o n variations caused b y expected variations in the a c t ­
ive device.
In conclusion, the active element p r o p o s e d for
this stu d y m a y be m ade h i g h l y stable and at least p a r t i a l l y
compensate for the fact that the networks are quite sensi­
tive to changes in the active element.
A n additional r e a s o n f o r considering the voltage f o l l o w ­
er as an active device is that, in its elementary f o r m s , it
is an extremely simple and w i d e l y used device.
Bo
THE VOLTAGE TRANSFER FUNCTION
The development of the v o l t a g e transfer f u n c t i o n for
the
(RCVF) proceeds as follows.
Referring to the passive
p o r t i o n of the net w o r k shown in Figure 1.1 and observing the
conventions shown, the first of the short-circuit admittance
equations is
F or the p a r t i c u l a r c o n f i g u r a t i o n shown in Figure 1.1,
I^=O
-5“
and E^=E^o
E q u a t i o n (1.1). t h e n "becomes
yI1E1+712E2 +yl3E1=0
(1-2)
and the voltage t r a n s f e r f u n c t i o n T
ip _E ont_E l._
m a y "be solved as
-y 12
V Eiii E2 yi:L+y15
(1.3)
Since the poles of the short circuit admittance p a r a ­
meters .are identical
(provided c o m m o n factors have not "been
cancelled), we m a y c o n v e n i e n t l y define
for i = j
(1.4)
?ij=<
" N i,1
I D, .
for i/j
where the N 0s' and D are polynomials,
and write the voltage
transfer f u n ction as
,m
V
E
,
N 12
.„ O U t
.
■
n
II-n I3
„ A(s)
"■ F < s ; .
(1.5)
This m a n n e r of d e f ining the y^j allows us to consider all
of the coefficients of the
as h a v i n g the same sign
(which we a r bitrarily choose positive).
Go
COM P A R I S O N OE A C T I V E A N D P A S S I V E T R A N S F E R FUNCTION.
SYNTHESIS
The m a i n difference b e t w e e n active and pas s i v e transfer
f u n c t i o n synthesis is t h a t p a s s i v e synthesis is g enerally
-6concerned w i t h the direct r e a l i z a t i o n of a g i v e n transfer
quan t i t y "by means of one of the transfer i m m i t t a n c e s „
Ac­
tive synthesis is g e n e r a l l y concerned w ith the indirect r e ­
aliz a t i o n of the same t r a n s f e r q u a ntity b y means of the sum
or difference of two t r a nsfer immittances =,
The active ele­
ments of the active n e t w or k serve as sum or difference t a k ­
ing d e v i c e s o
As an e x a m p l e 9 let us c o n sider a given t r a n s f e r func- ,
t i o n Tv and examine some of the ways this f u n c t i o n m a y be
realized w i t h passive circuitry.
T h e n it will be shown h o w
this same f u n c t i o n m a y be r e a lized w i t h the '(EClfB1) active
network.
Con sider the t r a n s f e r f u n c t i o n T
= K
where K is an a rbitrary posit i v e constant.
s + 2 o8 2 8 s »4
s ^ +I.. 414s +1
This function
m a y be realized b y a passive n e t w o r k since its poles lie in
the left-half of the com p l e x f r e q u e n c y plane
(LHP).
The
f u n c t i o n is a m i n i m u m phase f u n c t i o n since its zeros also
lie in the LHP.
G uillemin (25) shows that a m i n i m u m phase
tran s f e r f u n c t i o n cannot be realized b y a lossless network
terminated in a single resistor.
He shows $ h o w e v e r 9 a r e a l ­
i z a tion co n s i s t i n g of a lossless network terminated b y more
th a n one resistor.
A p p l y i n g this method to the present case
results in the n e t w o r k of Figure 1.2.
(See A p p e n d i x III-A
-7O530 H
177 H
707 H
707 H
R e a l i z a t i o n of E
,/I.
=j4(s^+2o828s+4/s^+lo414s +
I) w i t h a lossless ne t w o r k terminated b y resistors®
106 H
yTlfTv
Pigure 1.3<>
R e a l i z a t i o n of E outZ E ^ n = O ®2 (s2 +2®8 2 8 s + 4 / s 2 +
l®414s+l) b y P a n t e l l 1s method ®
-8f or n ume r i c a l c o m p u t a t i o n s ) ,
The chief disadvantage of this
realization, besides the large n u m b e r of elements,
is that
the input and output do not share a common ground.
Other p o s sibilities for the realization, of T^ include
its re a l i z a t i o n as a driving po i n t impedance or admittance
(T^ is p o s itive real), its r e a l i z a t i o n in the f o r m of a c o n ­
stant resistance ladder (29), or b y P a n t e l l 1S meth o d
T
v
(41).
has b e e n realized w i t h the r esultant net w o r k shown in
Figure 1.5
(see A p p e n d i x III-B for numerical computations).
Note that P a n t e l l 1s method has the advantage of a driving
and t erm i n a t i n g resistor.
Finally,
about t e r m i n a t i n g resistors,
if one does not care
etc. and a simple n e t w o r k is
sought w h i c h realizes Tv , the c o n figuration of Figure 1.4
is among the simplest available.
This n etwork was not syn­
thesized b y any rigorous method, but b y the applic a t i o n of
heuristic means.
To realize Tv b y means of the
( R CVF), we f o l l o w m e t h ­
ods to be pres e n t e d later in this thesis to develop the n e t ­
w o r k shown in Figure I . 5.
Note that the n u m b e r of passive
elements required iq less t h a n the number required in the
m a j o r i t y of thq passive networks presented above.
Also note
that the same n u m b e r of reactive and only one additional r e ­
sistive element
is required b y the
(RCVF) r e a l i z a t i o n t h a n
"by the simplest of the p a s s i v e networks
-10-
4.248H
Figure lo4„
Simple r e a l i z a t i o n of
a
'out
s £+2^828 s +4 o
‘in
s 2+1.414s +1°
4F.
547F,
I «64F c
.1390
----AA/'---__
^
<
:.646Q
<
VO L T A G E
FOLLOWER
(RCVF).
jOUt
-O
O-
Figure l.$o
-q
R e a l i z a t i o n of
3^+2.828 8 +4 wit h the
s^+1.414s+l
II.
S TATEMENT OE PRO B L E M
A t t e n t i o n will n o w Be given to a statement of the p r o b ­
lem for this research.
The basic p r o b l e m of d e v e loping s y n ­
thesis procedures for the
(RCYE) is b r o k e n into the f o l l o w ­
ing five' parts
I.
Determine the n e c e s s a r y and sufficient conditions
on a voltage t r a n s f e r f u n c t i o n T =A(s)/P(s) such
that it m a y be realized in the configu r a t i o n of the
(RCYE) as shown in Figure I.I.
2p
G-iyen the desired vol t a g e transfer f u n c t i o n T ^ = A ( s )/
P(s) whose n u m e r a t o r and denominator polynomials
meet the n e c e s s a r y and sufficient conditions s pec­
ified above, determine the c o r responding network
functions of the p a s s i v e RC po r t i o n of the
3P
( R CVE).
Realize the determined n e t w o r k functions in the
for m of a f o u r-terminal p assive RC n e t w o r k such
that w h e n the n e t w o r k is used in c o n j u n c t i o n w i t h
the voltage f o l l o w e r as in Figure 1.1, the overall
n e t w o r k p ossesses the v o l t a g e transfer function T^=
A(8)/P(s).
4-»
Take advantage of the digital computer topological
methods of circuit analysis p r e s e n t l y available
and the method of steepest descent in order to i m ­
prove and optimize the realized networks.
5P
Demonstrate the p r a c t i c a l i t y of the above items b y
-12W i l d i n g and tes t i n g some example n e t w o r k s „
Item 2 above is complicated b y the fact that if there
i s 'one w a y t h e n in general there are an infinite n u mber of
ways that Ty m a y be b r o k e n up into the ne t w o r k functions of
a p a s siv e EG f o u r-terminal network.
Thus, we are faced wit h
the p o s s i b i l i t y of s electing the one set of n e t w o r k f u n c ­
tions w h i c h optimizes some attribute or combi n a t i o n of a t ­
tributes of the network.
S u c h attributes include m i n i m i z i n g
the spread of element values, m i n i m i z i n g the sensitivity of
to va r i ations in circuit values,
etc.
A significant p o r ­
t i o n of this r e s e a r c h is devoted to this opt i m i z a t i o n p r o ­
blem
III.
N E C E S S A R Y A N B S U R E I CIENT CONDITIONS E O R T H E R E A L ­
I Z A T I O N OE T H E V O L T A G E T R A N S F E R F U N C T I O N Tyo
Considering Tv as the r a t i o .of polynomials A(s) to
P (s ) $ this section will "be concerned with the e s tablish­
ment •of the n e c e s s a r y and sufficient conditions on A(s) and.
P(s) such that a desired stable voltage t r a n s f e r f u n c t i o n Tv
m a y be realized with the
(RCVE) configuration.
Since we
are onl y interested in stable t r a n s f e r functions, it follows
i m m e d i a t e l y that the zeros of P(s) are confined to the lefthal f of the com p l e x f r e q u e n c y pl a n e wit h the p o s sible excep­
t i o n of a single zero at the origin.
Next we call on the E i a l k o w c ondition (29) which states
F or an u n b a lanced two-port w i t h no mutual inductance
the n u m e r a t o r coefficients of -Ypgs ^ l l 9 an^ Y g 2 are
all n o n n e g a t i v e , the coefficients of -y^g "being no
greater t h a n the c o r r e s p o n d i n g ones in y ^
or
J22°
This c o n d i t i o n can obvi o u s l y b e extended to the threeport unb a l a n c e d p assive RC n e t w o r k of Figure 1.1 w ith the
res u l t i n g statement t h a t :
F o r an unbalanced RC three-port with no m u t u a l i n d u c t ­
ance the n u m e r a t o r coefficients of - y ^ s
“Y i g 9' an^ Y n
are all nonnegative, the coefficients of - y ^
be i n g no
gre a t e r t h a n the c o r r e s p o n d i n g coefficients in (y-^+y-^)
-14This latter cond i t i o n follows f r o m the fact that if we f orm
a two-port n e t w o r k f rom the three-port p assive n e t w o r k of
f igure I 0I b y joining terminals 2 and 3 together, t hen the
short-circuit t r a n s f e r admittance of the two-port n etwork
is the sum of the short-circuit admittance p a r a meters
y^^)
of the three-port network*
Cy^+
The second p art of the
f i a l k o w c o n d i t i o n for u n b a lanced two-ports m a y t h e n be r e ­
stated as "the coefficients
b e i n g no greater t h a n
the corres p o n d i n g ones in y ^ " »
If y-^ is added to each
side of the inequality, the p r e v i o u s l y stated cond i t i o n is
verified*
A n o t h e r w a y of e x p r essing this cond i t i o n is that
the coefficients of
of equation (1*5) must be
greater t h a n or equal to the corresp o n d i n g coefficients of
ELo*
Since the n u m e r a t o r A(s)
of the transfer f u n ction T^
is the n u m e r a t o r of the t r a n s f e r admittance of a passive
RG unbalanced t w o - p o r t , it follows that the zeros of the
tran s f e r f u n c t i o n m a y appear anywhere except on the positive
real axis*
The f ollowing n e c e s s a r y c o n d i t i o n for the voltage trans
fer f u n c t i o n of the
(RCVR) m a y n o w be s t a t e d :
The voltage t r a n s f e r f u n c t i o n of an unb a l a n c e d R O
three-port n e t w o r k used in c o njunction w i t h a voltage
-15fol l o w e r n e t w o r k as shown in I1Igure 1.1 must be a. ratio
of nonne g a t i v e coefficient polynomials in s with the
coefficients of the n u m e r a t o r polynomial b e i n g no
greater t h a n the c o r r e s p o n d i n g coefficients in the d e ­
nom i n a t o r polynomial.
In addition, f o r stable t r a n s ­
fer functions, the zeros of the denominator polynomial
must be located in the L H P w i t h the p o s sible exception
of a single zero at the origin=
The zeros of the n u ­
m e r a t o r po l y n o m i a l m a y appear anywhere except on the
pos i t i v e real axis.
A synthesis proc e d u r e will n o w be given wh i c h shows
that a voltage t r a n s f e r f u n c t i o n fulfilling the above n e c ­
essary c ondition m a y always be realized.
T h u s , the stated
c onditio n on a t r a nsfer f u n c t i o n will be shown to be suffi­
cient as well as n e c e s s a r y for its realization.
Consider the two-port un b a l a n c e d EC networks,
"Bfl, shown in Figure $.I.
"A" and
A s sume that the short-circuit a d ­
mittance parameters of bot h n e t works have the identical d e ­
nomi n a t o r p o l y nomial .D(s) and that their n u m e r a t o r p o l y n o ­
mials are defined as i n equation (1.4) with the addition of
the subscript A or B to denote net w o r k "A" or “B" r e s p e c t i v e ­
ly.
If B volts are placed on the input terminals of each
n e t w o r k and the output terminals g r o u n d e d , t h e n f rom the
—16—
N
"12A
IlA
N11B
N12B
11A~ 12A
Figure $.1.
The
"A" and "B" two-port unbalanced RC n e t w o r k s „
Figure $.2.
I n t e rconnection of "A" and "B" networks
-17short-clrcuit admittance equations we k n o w that currents, are
established as shown in Figure 3*1°
Now if the
"A" and riB tl
networks are interconnected as shown in Figure 3 * 2 ? the same
currents will f l o w since the conditions p r e v i o u s l y imposed
still apply*
(terminal
nal
The int e r c o n n e c t i o n forms a n e w three-port
(I) to ground, termi n a l
(2) to g r o u n d , and t e r m i ­
(3) to g r o u n d ) unb a l an c e d RC net w o r k whose short-circuit
admittance parameters are seen to be
jIl"
n IIA+nIIB
„
D
= y I l A +yIlB
.-~N 12A
J12"
y I 3~
I
'='y12A
(3*1)
N1 2 A +N1 2B"N 11A n IIB
; ■*■■ r ■ D
If this n e t w o r k is used for the p assive p o r t i o n of the
.(RCVF), the overall voltage t r a n s f e r function will be
,12A
“y12
y l l +y13
n =
v
- • '12A
N 1 2 A +N12B
N 1 1 A +N1 1 B +N1 2 A +N12B"N 1 1 A _ N 11B
= A(s)
(3.2)
■
This equation is intere s t i n g in that its element .poly­
nomials are polynomials wh i c h are the numerators of RC
transfer admittances and not numerators of RC driving point
V
admittanceso
The e q u ation suggests the f ollowing synthesis
/
procedur e for a t r a n s f e r f u n c t i o n which meets the p r e v i o u s l y
-18stated n e c e s s a r y conditions.
1.
Ghoose a p o l y n o m i a l D whose zeros are restricted
to the negat i v e real a x i s .
2.
Synthesize an E G
"A" ne t w o r k whose t r a n s f e r admit­
tance is
_ -A(s)
D(s)
J12A~
This m a y always be done provided the coefficients
of A(s) are posit i v e
(the n e c e s s a r y cond i t i o n g u a r ­
antees that t h e y are) and that the order of D and
the magnitudes of its coefficients are chosen s u f ­
f i c i e n t l y large
5»
Synthesize an EG
(see S e c t i o n 15-3 of reference 25)®
"B" n e t w o r k whose t r a nsfer a d m i t ­
tance is:
y 12B
_ -(P(s)-A(s))
Zttsl
A g a i n the coefficients of
(P(s)-A(s)) are g u aran­
teed to be p o s itive b y the nece s s a r y c o n d i t i o n ^
and the
nB rt n e t w o r k m a y always be realized provided
the order of D and the magnitudes of its c o effi­
cients are c h osen s u f f i ciently large.
4-o
A f t e r b e i n g synthesized, the
"A" and
"B" networks
are connected as i n .Figure J.2 and joined to a . v o l t ­
age follower as shown in Figure 1.1.
This new
overall c o n n e c t i o n is sh o w n in Figure 3®3®
-19-
Figure 3»5«
followero
"A" and
l1B ri n e t works connected to the voltage
-20The voltage t r a n s f e r f u n c t i o n of the active c o n f i g u r a ­
t i o n will t h e n he T y= :
Thus, the p r e v i o u s l y quoted
con d i t i o n is sufficient as well as necessary.
The synthesis proc e du r e outlined above will,
in general,
p r o duce configurations with an excessive n u m b e r of e l e m e n t s .
F o r example, consider the band pass voltage t r a nsfer f u n c ­
tion
,Ools
-Ly.-' "7r^T ..
v
s^+.ls+l
(3o3)
If we applied the suggested synthesis p r o c e d u r e , we could
separate the f u n c t i o n as
y 12A=
-Ools
-Di
and
•(s2 +l)
y12B=
(3.4)
y 1 2B m a y ^3e realized in the f o r m of a twin-T RC network
w h i c h requires 3 capacitors and 3.resistors for a total of
6 elements.
y]_2A co"t:i^^ t h e n be realized w i t h 2 elements.
It is known, however, that it is possible to realize the
transfer f u n c t i o n w ith fewer elements
(reference 14) tha n 8.
Thus the suggested synthesis procedure, although p o w e r f u l ,
in general requires an excessive numb e r of elements.
For
this reason, it is desirable to examine other methods of
synthesis.
The next chapter will be concerned w i t h this
problem.
/
-21A. • SUBPLUS F A C T O R U T I L I Z A T I O N
Before going on to other synthesis methods 9 it is a p ­
propriat e to examine the p o s s i b i l i t y of u t i l i z i n g surplus
factors to convert a t r a n s f e r funct i o n T^ w h i c h apparently
does not meet the n e c e s s a r y and sufficient conditions of
this chapter into a t r a n sf e r f u n c t i o n which does meet these
conditions, o
It is well knoTim in circuit t h e o r y that there are
cases w h e n it is p o s sible to m u l t i p l y a po l y n o m i a l with
negative coefficients b y a p o l y n o m i a l with L H P zeros such
that the resultant p o l y n o m i a l has nonnegative coefficients.
T h u s , if we have a T
whose poles are restricted to the LHP
and its n u m e r a t o r p o l y nomial contains some negative coef f i ­
cients, there still exists a p o s s i b i l i t y that surplus f a c ­
tors mig h t be utilized so that T ^ meets the prescribed n e c ­
essary and sufficient conditions.
In the literature, the overall pro b l e m is b r o k e n d o w n
into the fundamental p r o b l e m of fin d i n g a p o l y n o m i a l B(s)
w i t h L H P zeros such that B(s)<F(s) has n o nnegative coeffi­
cients where F ( s ) =s^-(2 c o s 0 ) s + l
and
„
This is
a valid simpl i f i c a t i o n of the p r o b l e m since factors of the
fo r m of F(s) m h y t>e separated out of any po l y n o m i a l with
-
negative coefficients
22-
(provided an appropriate f r e q u e n c y
t r a n s f o r m a t i o n is m a d e ) „
These factors m a y t h e n be ind i ­
v i d u a l l y operated u p o n to pr o d u c e nonnegative coefficient
polynomials.
One appro a c h has b e e n to sho w that a po l y n o m i a l of the
type B(s)=(s+l)n could pr o d u c e the desired coefficients and
to r e a s o n t h e n that for s u f f i c i e n t l y small and distinct
n
the surplus fact o r B(s)= Tf ( s + l + 6\ ) will a p p roach the
i=l
1
same result (28,40).
%
The smaller the value of (f) , the larger the vhlue of n
required to produce h o nnegative coefficients.
es zero, the required n approaches o o ,
of
As 0
a p p r o a c h­
The optimum choice
f or B(s) = (s+y3 )n will be considered so that for a
given n, 0
m a y take on the smallest value p o s sible tionsis-
tent wi t h the n o nnegative coefficient r e s t r i c t i o n on
. #(s)B(s).
odd is
It. is s h o w n that the optimum choice of
JJq^ = I , but that for n eve n
The corres p o n d i n g values for 0
c o a "1"
JJ for n
or'
m i n are cbs”"^
■n
and
^
.Consider
B(s) = (s 0})n =bo sn +b 1 sn '
+b
“•s+b
n
(3.5)
-23where
-K = —
(g= 0 sl s2 ,»o o o ?n)
j
(3 »6 )
The p r o d u c t of B(s) and F(s) m a y t h e n he w r i t t e n as:
Y)-f-"]
o
B(s)F(s)= ‘Y,
( h . T + b . ^ - 2 G o s 0 b - ) s n + 1 ™3
(3-7)
j=-l
3-13+1
where
Thus, it is seen that the f o l l o w i n g set of inequalities m ust
he satisfied in order that B(s)F(s) have n o nnegative c o e f f i ­
cients
hj-i+ho-t
Gj= ■l
J
-b ^ ' - 2 cos,
(j=0,l,...,n)
■ (3-8)
J
Substi t u t i n g the values from Eq.
S 4- —
3
(3-6) into
(3-8)
-JLf-
/](n-a+l)
(j+1)
(5.9)
The basic p r o b l e m m a y n o w be equated to that of choosing a
n onnegat i v e value of J3 such that the minimum of the set
(Gq , G ^ , p « --,Gn ) is maximized-
Call this value of G,
G
m i n max'
As seep from Eqeontinous functions of
(3-9), f o r arbitrary h, the G^ are
J3 w h i c h possess n o m ore t h a n one
m i n i m u m and no m a x i m a in the range ©-=
j!] <co»
In addition,
if g ( y O ) is defined as m i n (Gq 9G 1 , -- - - ,Gn ) , t h e n g =0 for
^3=0 and co ■-
J3 corresponds to
The values of J3 for
T h u s , the optimum value of
inter s e c t i o n of two of the G curvee»
w h i c h Go and G . intersect m a y be computed b y me a n s of
U
■
an
equation (3°90 as
P
C'i)3—© jI j o o o o jn 51^ 3°)
- : - (3 .10 )
A n exami n a t i o n of this equation shows that f o r a given 3%
the value of i =* 3 w h i c h results in a mi n i m u m value for
i=g’4-l„
P is
Since S=G q for yf}=09 it follows that as yQ is in ­
creased fro m 0 , g corresponds successively to ©"os©i$ <>»»»^Gn ,
Substi t u t i n g (3 +1) for I i n e q u ation (3»10)„ the optimum
value of
P> is that element of the set
Ch4'3 R n - 3 + 1 J
(.3~O.5:00® »n—I)
w h i c h produedp the largest value of G. from E q e
U
(3 .11 )
( 3 . 9 ).
That is 9 what permi s s i b l e Value of 3 maximizes
(nr 3OC 3 +jT
(5.12)
Differen t i a t i n g wit h respect to 3 :
9G3
= (n-2i+l)(n+2)
372
37^
(n-g) ■(3 + 1 )7^Cn-3 +Ijpz^(g+
2)
(5.13)
n-l
wh i c h is seen to be p o s i t i v e „ negative, and zero for 3 < — g— »
5 and 3
n-l
respectively0
It follows that the opti
m u m value of y0 (yQ 0p^) and the corresponding value of m i n i mum(^)=cos""^
.nLa? ' is found b y substituting 3=2^1 for n
odd and 3 = 2^1..+
y2 f o r n even into E q s » (3.11) and ( 3 . 1 2 ) e
E o r the case of n even, it is found that the two possible
values of 3 result in identical values of Gm i n ma3C»
Thus,
-25-
\
/^opt ^
\ n odd
(3.14)
0 min=cos 1
/
and
\
/ n even
(3.15)
/
The above shows the opt i m u m surplus factor of the form
(s+yO)n .
In the pre s e n t case, we are not limited to surplus
factors wh i c h lie on the n e g ative real axis.
I H P complex
conjugate zeros are p e r m i s s i b l e and, in g e n e r a l , the y are
superior to negative real axis zeros.
The p r o b l e m of f i n d ­
ing thei r optimum l o c ation appears to be much, mor e difficult
ahd ho solut i o n is offered to this problem.
however,
is left as a p r o b l e m c h a l l e n g e . ;
The question,
■IV.
(HCVF) HEALIZATIOH TECHNIQUES
This sec t i o n is concerned wit h the synthesis of special
n e t w o r k configurations to he used as the pas s i v e p o r t i o n of
the
( H CV F ) .
These configurations are of interest since, in
general, t h e y require fewer elements than the c o r responding
networks realized b y the general synthesis p r o c e d u r e of
Chapter I I I e. S e c t i o n A of this chapter gives a pictorial
r e p r e s e n t a t i o n of the pas s i v e synthesis p r o b l e m which s i m p l i­
fies the v i e w i n g of the p r o b l e m and offers insight into the
sensitivity, problem.
S e c t i o n B reviews n e t w o r k c o n f i g u r a ­
tions pres e n t e d b y other authors.
This mater i a l is solely
concerned w i t h second-order t r a n s f e r functions.
Sec t i o n C
presents a n e w l y developed synthesis procedure for the b i ­
quadratic transfer function.
S e c t i o n B presents a ladder
n e t w o r k c o n f i g u r a t i o n for low-pass and high-pass transfer
functions.
S e c t i o n E presents a special n e t w o r k c o n f i g u r a ­
t i o n for f o u r t h-order band pass t r a nsfer functions.
Finally,
S e c t i o n F presents several n u m e r i c a l examples of the syn­
thesis procedures pres e n t e d in this chapter.
A.
A P I C T O R I A L R E P R E S E N T A T I O N OF T H E PASSIVE SYNTHESIS
PROBLEM
There are a n u mber of ways to state the requirements on
the pass i v e p o r t i o n of the
(ECVF).
A simple w a y of vie w i n g
-27t?he p r o b l e m and one w h i c h offers some insight into the s e n ­
sitiv i t y p r o b l e m is offered below*
Con sider a (RCVF) vol t a g e trans f e r f u n c t i o n Tv m e e t i n g
the n e c e s s a r y and sufficient conditions of Cha p t e r III*
F r o m equ ation (1*5)
rn _ A(s) _
:.A(s)
_ ■ %2
V - F t s T - R(a)+I(s) " Ir11- H 1 ,
■
(4.1)
where R(s) = 'P(s)-A(s) = rn sn + *.* * * * * * *»+r^s>rQ
has nonnegative coefficients since the coefficients of A(s)
are required to be less t h a n or equal to the c o r responding
coefficients of P ( s ) ,
■Since the p o lynomials
and
5 are the n u m e r ­
ators of short-circuit RC admittance p a r a m e t e r s 9 it is p o s s i ­
ble to associate the p o lynomials A ( s ) s P(s), and R(s) with
the currents f l o w i n g in a f o u r-terminal RC n e t w o r k whose
short-circuit admittance pa r a m e t e r s possess t h e s e nume r a t o r
polynomials*
This p o s sible current dist r i b u t i o n is shown in
Figure 4*1 and is onl y valid w h e n terminal
(I) is driven b y
a voltage source equal to the characteristic polynomial B(s)
of the network, and the other terminals are grounded*
Since —
is a d r i v i n g point admittance, the
zeros of its n u m e r a t o r and denom i n a t o r must be interlaced
-28-
Figure 4.1.
A p o s sible current distribution.
A ( S ) = S - S ^ a n s+a
P(s)+E(s)=
o
P 2 S +(p 1 +e 1 )s+pl
R(s)=r-s +r-, s+r
Figure 4.2.
Current d i s t r i b u t i o n of passive p o r t i o n of
• 2
2
(RCVF) net w o r k to realize T ^ = ( a 2 s +a 1s+a 0 )/(p2 s +p^s+p^).
-29™
along the negative real axis*
Thus E(s) m a y be thought of
as a pol y n o m i a l whose coefficients serve the sole purpose
of augmenting the co r r e s p o n d i n g coefficients of P(s) in
such a m a n n e r that the zeros of the resulting po l y n o m i a l are
shifted onto the negative real a x i s .
Since E(s) is also the
num e r a t o r po l y n o m i a l of the s e nsitivity f u n c t i o n associated
wi t h the g ain of the vol t a g e follower
(as will be shown
l a t e r ) „ it is desirable to choose E(s) so that its effect
on this s e nsitivity is minimized*
This p r o b l e m is d i s c u s s ­
ed fu l l y in Chapter V*
In conclusion, Figure 4-*I is a pictorial r e p r e s e n t a ­
ti o n of the f ollowing synthesis problem*
Synthesize a fo u r - t e r m i n a l HG n etwork wh o s e c haracter­
istic polynomial D poss e s s e s arbitrary negative real
axis zeros, suc h that w h e n terminal
(I) is driven b y
D volts and the other terminals are shorted to ground,
A(s) amps f l o w t h r o u g h t e r m i n a l
t h r o u g h terminal
min a l
(4)*
(2), E(s) amps flow
($), and R(s) amps f low thr o u g h t e r ­
Once this n e t w o r k is realized,
nected as shown in Fi gure 1*1 w i t h terminal
ing the ground terminal*
it is c o n ­
(4-) b e c o m ­
The resulting v oltage t r a n s ­
fer f u n c t i o n is then, of course, Tv = A ( s ) / P ( s ) »
This pict o r i a l r e p r e s e n t a t i o n as discussed above and shown
in Figure 4*1 will be used extens i v e l y in this chapter*
_ JO—
■B „
R E V I E W OF P R E V I O U S L Y P R O P O S E D O O H F IGURATIOES
Before p r e s e n t i n g n e w synthesis techniques, a revi e w
of existing techniques will he given,
S a l l e n and K e y (14)
p r e sent a catalog of 18 circuits which, w h e n used in c o n ­
junction w i t h an a mplifier of g a i n K q , produce second order
transfer functions of the general form
a 9Sv +a 1s+an
V
" —
-4 —
P 2 S ^ p 1 Stp 0
0 .2 )
A l t h o u g h the a mplifier gain K q of these circuits is neither
restricted to n o r required to he u n i t y in all cases, m any
of the cataloged circuits are usef u l at this va l u e of gain,
S a l l e n and Key, in fact, point out the advantage of using
this value.
W h e n u n i t y g a i n is permissible, the networks of
S a l l e n and K e y reduce to the
(RGVF) case.
wh i c h are permi s s i b l e are 10 in number,
Those networks
A study of them
shows that there is a wide choice of network element values
for the r e a l i z a t i o n of a specific n etwork f u n c t i o n b e longing
to a giv e n n e t w o r k f u n ction form.
It is i n t e r e s t i n g to note
that none of the networks of S a l l e n and K e y are valid for
the real i z a t i o n of the n e t w o r k f u n ction of the f o r m T^ =
A(S)ZP(S ) = U g S + a ^ s + a g / p g S + p ^ s + p Q in the (RCVF) conf i g u r a ­
tion,
It is also i n teresting to note that the wor k of
Bal a b a n i a n and Patel
(IJ) w h i c h is an extension of that of
S a l l e n and K e y is co m p l e t e l y inapplicable to the
(RGVF)
-31case in that the g ain of the active element is in the v i c i n ­
ity of u n i t y but that a gain of exactly u n i t y is i n admissi­
ble.
F o r the synthesis of the networks of S a l l e n and K e y 9
the r ead e r is referred to t h e i r report
C.
(14)„
A R E A L I Z A T I O N T E C H N I Q U E F O R T H E BIQUADRATIC TRANSFER
FUNCTION
Consider the b i quadratic t r a nsfer function
' 2
a 0s +a.-, s +3/-,
T
K - 2 - 1 —
0
(4.3)
P 28 +P 1 S +P 0
whose coefficients s atisfy the n e c e s s a r y and sufficient c o n ­
ditions of the
(RCVF )i0
Assume all coefficients are greater
t h a n zero and K is mad e s u f f i c i e n t l y small such that the r e ­
sulting n u m e r a t o r coefficients are less than the co r r e s p o n d ­
ing deno m i n a t o r c o e f f i c i e n t s »
.In order to rea l i z e T^ as the
voltage t r a nsfer f u n c t i o n of the
(RCYF )9 it is sufficient
to reali z e a four-terminal n e t w o r k whose current distr i b u ­
t i o n as defined in S e c t i o n IY A is shown in F i gure 4.2»
Since the c o m p l e x left-half plane zeros of a quadratic
pol y n o m i a l m a y be shifted onto the negative real axis b y i n ­
creasing the coefficient of the first order term,"it is p e r ­
missible that E(s) be no more complicated t h a n E ( s ) = e ^ s e-
-32It will n o w be shown that the biquadratic 3?
w i t h the
p r e v i o u s l y m entioned restrictions m a y be r e a lized with no
more t h a n three capacitors and f our resistors <>
The effect
of r e m o v i n g the restrictions on the coefficients will then
be examined and it wil l be shown that the r e q u i r e d number
of elements m a y o f t e n be r e d u c e d o
A ss u m e the characteristic polynomial,
D ( s ) , of the four
terminal p assive n e t w o r k to be realized is D(s)=d^s+l where
d ^ m i n C a 1Z a ^ r 1Zr0 ) and is greater t h a n zero since aQ , S 1 „
r 0 , r 1 are all greater t ha n zero,
BTq w consider the net w o r k c o n figuration s h o w n in Figure
4,3,
It is easily verified that this c o n figuration has an
input current d i s t r i b u t i on the same as that of Figure 4,2
and that the characteristic p o l y n o m i a l is of the form D(s)
=
d 1 s+l»
Thus, the dri v i n g po i n t current m a y be considered
Q as PgS +(P 1 ^e 1 )Sfp 0 as shown in Figure 4,3,
To calculate the current d i s t r ibution for the r e mainder
of the network, it is p o s s i b l e to p e r f o r m a continued f r a c ­
ti o n expansion of the dri v i n g p o int.admittance to obtain the
values of E 1 and G 1 as
E
1_
I
pO
0i"P1 +ei- <a1Po
-33-
d-, s +1
2
2
Ia =P 28 +(P 1 H-B1 )S^-P0
Ib =P 28 +(P 1 +e 1- d 1p 0 )s
Figure 4.3.
Network c o n f i g u r a t i o n with input current
distri b u t i o n of Figure 4.2.
oH :
c
u -------
T
1C^I -O
I
P^xj --
W -------+
aI
-—s+1
R1
aO
1
I
f
Ib
I - Cr0 aI 7 aO )s+r0
a
I
b =alSpa0
I =r0 s^
c
^ 2
I — d.^S
d
I
e - (rl-r 0 al/a 0 )s
I = B1 S
f
:: °>
1C
j
I
r3
T
4
f
4
R 1=(IZr0 ) Q
Rg=(IZa0 ) Q
(ai-a 0P 2Z C 1 )Z(a 0ri-r0 ai
^al^l“P 2a 0 ^ ^Cl aO el ^
C l = ^rp l +el“ alp 0 //a0 ^) F.
F<
T g Z C a 1Z aQ-RgZCi)
S 2Z C a 1Za 0 - P 2 Z C 1 )
Figure 4.4.
Current d i s t r i b u t i o n for case of
Ft
(S 1Za 0 )=(T 1Zr 0 ) o
-
34
™
The current flowing thr o u g h
is t h e n seen to he (d^QS+Pg,)
p
amperes and the current t h r o u g h
is p^s +G-^se The eontinued f r a c t i o n e x pansion is continued to obtain
p2
—
and
'
d-, - p2
E 0= —
%
1
The voltage
in Figure 4,3°
P2
S1 - C,
and the currents th r o u g h Gg and Eg are shown
It is easily s een that G^, O g , and Eg are
posi t i v e provided e^ is c h o s e n sufficiently l a r g e „
The elements E 1 , C 0 and E 0 are n o w divided in such a
m a n n e r that paths for the currents
rOaI
(— — — s+rD ) 9 (an s+an ) $
2
2
rOaI
0
(rgS ; ^ (a.gS^), (r^- ■a ;,'J~)s, and e^s are provided for the
case of Ca 1Za 0 )-Cr 1Zr 0 ) o
F o r the case of Cr 1Z ^ 0 )-Ca 1Za 0 )9
the elements are divided in s uch a manner that paths for the
a0r 2
2
2
aGr I
currents C^1S +rQ ) 9 C ™ — 8 +a0 ), C^gG ), (agB ), Ca1)s
0
and S1 S are provided.
V
The current distributions for these
two cases are shown in Figures 4-.4- and 4.5$ respectively.
It is r e a d i l y seen that the networks of Figures 4.4
and 4.5 m a y be put into the f orm of Figure 4.2.
Thus,
these two networks will realize in the
(BCVF) config u r a t i o n
ag s 2 +a1 s +a 0
any biquadratic voltage t r a n s f e r function T = — — g— 1
-----P2S
+P1 S +p0
to w i t h i n a constant m u l t i p l i e r K provided all coefficients
are greater t h a n zero.
-35-
O
'TT
I Bi
O
t
I,
1S = crI V rO ^ +aO
1I==r I s Ir O
S1-(Va0)Q
E2=(Vr0)Q
1C = a S s 2
Crl-ro V 0l)/CrOal-aprl>|ft
^a=r S s
crI0I-PsrO v c c IrOeI V
1C = caI=aOr Iz r O ^
1I = eIs
Figure 4.5»
0I = cP I +eI-r Ip O z r O 5 P
a 2/(r i/r o - p 2^ C l ^
r 2/ (^1Zr 0 - P 2Z C 1 )
Current d i s t r i b u t i o n for case of
F,
F,
( r r Q )=(a ^ / a0 )
-36The p r e c e d i n g was based on the fact that K be chosen
s u f f i c i e n t l y small that the n u m e r a t o r coefficients of T^
be less t h a n the co r r e s p o n d i n g denominator c o e f f i c i e n t s <,
Since m a x i m u m gai n is u s u a l l y desired, it is desirable to
increase K until at least one of the n umerator coefficients
is- equal to its c o r r e s p o nd i n g denominator coefficient.
In
other words, K is increased un t i l at least one of the' c o e f f i ­
cients of the p o l y n o m i a l R(s) is zero.
A s s u m i n g that o nly one of the coefficients of R.(s) is
reduced to zero, three pos s i b i l i t i e s will be e x a m i n e d ,
If
rg is z e r o , it is noted that the n u mber of r e q uired c a p a c i ­
tors is reduced b y one.
If
Tq is zero, the number* of re- .
quired resistors is reduced b y one,
Ior the ease of r-^=G,
the prop o s e d net w o r k c o n fi g u r a t i o n no longer h o l d s , and one
must either b a c k off on the value of K or attempt a different
n e t w o r k configuration.
The four remaining cases, w hen two
or three of the coefficients of R ( s ) simultaneously vani s h
as K is increased, are s imilarly analyzed.
D.
L A D D E R N E T W O R K R E A L I Z A T I O N E O R LOW-PASS' A N D HIGH-PASS
T R A N S F E R FUNCTIONS
A r e v i e w of a synthesis p r o c e d u r e proposed b y the p r e ­
sent author (16) for the synthesis of a rbitrary order low-
-37pass active filters in the
giveno
(EGVF) configu r a t i o n will "be
As an e xtension of this work, it will "be p r o v e n
that the 4 t h order low-pass t r a n s f e r function m a y always
"be realized "by this method.
In addition, the method will
"be extended to include h igh-pass filters.
Consider the "basic ladder n e t w o r k shown In Figure 4,6
and th e n consider each Y . (j=l, 2 ,,*,,n) decomposed into two
9
n
subnetworks whose admittances are Y. and Y . ' such that
9
it
U
n
U
Y.+Y,=Y.o
Eac h of the Y, ground terminals m a y be d i sconnectJ
J
J
J
ed fro m ground and soldered t o g e t h e r to form a n e w terminal
as shown in Figure 4,7*
,This altered net w o r k m a y be thought
of as a decomposed ladder network.
The n o t a t i o n has b e e n
changed in Figure 4,7 to agree w i t h the current dis t r i b u t i o n
n o t a t i o n used in previous sections.
It is s een that the p u r ­
pose of decomp o s i n g the admittances is to shunt the currents
B(s) and E(s) to separate terminals.
The success of the s y n ­
thesis proc e d u r e p resented in this section depends u p o n the
ability to p e r f o r m this separation,
added in series w i t h terminal
f r o m terminal
Eote that an impedance
(I) or an admittance placed
(2 ) to the refe r e n c e node has no effect on the
voltage transfer f u n c t i o n of the corresponding
(EGVF),
A synthesis p r o c e d u r e for low-pass t r a nsfer functions
in the for m of the decomposed ladder will n o w be given w i t h
-38-
Figure 4.6.
The basic ladder network.
P(s)+E(s)
Figure 4.7.
The basic decomposed ladder n e t w o r k
no guarantee that the r e su l t i n g p assive elements hav e non nega t i v e v a l u e s o
Experience has shown that all of the n e t ­
w o r k functions so f a r encountered of the low-pass form
(4.4)
V
® m sn+V l sIl“ 1+'
m a y be r e a l i z e d 0
— +PjS+1
P(s)
A p r o o f is offered that all functions of
this fo r m for the case n = 4 m a y be realized b y the decomposed
ladder n e t w o r k »
The p r o p o s e d synthesis p rocedure n o w follows
w i t h the shunt arms of Figures 4-»6 and 4<>7 assumed to be
single capacitors and the series arms assumed to be single
resistors as shown in Figure 4,8»
1,
Choose a.polynomial-E(s)
of degree less t h a n or
equal to n and a p o l y n o m i a l D(s) of degree
(n-1)
such that y 11='fp(sO+E(s)J /D(s) is a dri v i n g point
admittance conforming to the config u r a t i o n of
Figure 4,6»
E(O)-D(O)=Io
Magn i t u d e scale y 11 so that P(O)+
Since. P ( O ) = I 9 t h e n E ( O ) = O o , Eote that
the zeros of . [p(s)+E(s)j
and D(s) must be interlaced
along the negative r eal axis of the com p l e x f r e q u e n ­
cy plane,
2,
P e r f o r m a continued f r a c t i o n expansion on
Jp(s)+E(s)] /D(s) to determine the element values
for the c o n f i g u r a t i o n s h o w n in Figure 4.6»
step of the expansion,
At each,
label the denom i n a t o r p o l y n o ­
mia l of the ratio of polynomials used to determine
—
40
-
—
Figure 4.8,
Decomposed ladder network with shunt arm
capacitors and series arm resistors.
D ( S ) = K ^ (s +S1 ) (s +(5^) (s +66 )
1S=k I s (S +61 )(S +64 )(S +6 5 )
cs +62 x s +S5 )
Vb =K^ C s +5 5 )
Ic=k$ 8 (s+Sy)
Vc=^4
kj=CjEj
Id~k4 s
(j=l,2,),4)
Figure 4.9«
Current d i s t r i b u t i o n for basic ladd e r network
realiz a t i o n of
.
-
41
-
-
C , as D.*
d:
3o
u
Tl
To determine the 0. of Figure 4*8, the f o llowing
tJ
proc e d u r e is carried out*
A continued fraction
ft
expansion of E(s)/D(s)
is b e g u n to o b tain C ^ s *
It
Rote that O^s is E(oo)/D(oo)*
mined as
[e (s )-C^s D (s )J
Eg(Co)ZDg(Co) *
EgQs) is next d e t e r ­
and C^s is determined as
This pro c e s s is continued to determine
ft
the r e m a i n i n g C .*
If E(s) and D(s) are considered
Jt
as E 1 (s) and D 1 (s) respectively, t h e n C . s = E . (s)/
Jo 1 3.
V
D.
4-o
(s)*
o
Finally, the O . of F i gure 4.8 are d e t e r m i n e d as
I
„
J
CL=C,-C . *
u
d
u
The a p p endix of r eference 16 contains a p r o o f that the
synthesis p rocedure outlined above is valid*
The reference
does not con t a i n any guarantee that the element values so
realized are nonne g a t i v e *
It is shown b e l o w that the 4-th o r ­
der low-pass t r a n s f e r functions m a y always be realized w i t h
n onnegat i v e elements b y this m e t h o d »
Figure 4*9 shows the basic three-terminal passive RC,'
n e t w o r k w h i c h must be decomposed in order to realize a 4 t h
order low-pass t r a n s f e r f u n c t i o n b y the method outlined in
this section*
A s ■explained previously, this n e t w o r k has
b e e n synthesized fro m an RC d r i v i n g point admittance
-424
(p.+e^, )s"l'+(p,+e^)s:?+(pp+ep)s 2 +(p -1+en )s+l
J 1 1 = ----- ^
2.------- p ~ ^ ------ :----- ---- d 3 s +d2 s +di s+1
(4,5)
where the trailing coefficient of the numerator and d e n o m i ­
n a t o r po lynomials have b e e n n o r m alized to unity.
If the
current and voltage d i s t r i b u t i o n of figure 4,9 is calculated 9
it is seen to have the for m shown.
Since the object of a
continued fraction e x pansion of y-^ is to reduce the order
of the p o lynomials associated w i t h the current and voltage
distributions b y one at each step of the expansion, the d i s ­
tributions of figure 4,9 are valid.
The p ole-zero p a t t e r n of the continued f r a c t i o n ex-,
p e n s i o n of y-^ will n o w be r e v iewed to show t h a t 6 /
°
It is well k n o w n that the poles and zeros of y - ^ must be
interlaced along the n e g ative real axis.
Thus, y-^ is of
the form
(s+7^ ) (s+7^) (s+7^) (s +7|7 )
Jll= =E
(4.6)
( S ^ 1 ) (S4<54 ) (s+(5g)
£) < TJjz & 2± 7]
6 ^
where
7] <
bers.
The p ole-zero d i s t r i b u t i o n of y-^ is shown in figure
4’
.10.( a).
7] ^ are all p o s i t i v e real n u m ­
The first step of the continued f r a c t i o n expansion
is to remove the pole of y ^
at infinity. - The admittance
r emainin g after this removal is y-^^^ ^ r2SrI i ^ ^ " " ^ l s where
is the original y^^°
Since
is guaranteed positive,
-43-
^7
4$
%
^4
c Cb
;in
5
<
I
I
I
I
I
I
I
I
y.- - - - - - - -
(a)
^3
4
7?/
^--------- y ------- -------y ---I
I
I
i
i
i
I
i
i
I
i
I
I
I
ft — I
x
o-l
I
I
I
I
I
i
I
I
i
-------A ---------V ^
-— « --------
I ^
(b)
(c)
(a)
4—
o — I
4
—
JrU 5
©-
(e)
6
-
* ----
-e------ x "* '
—
e-
^
5
(f)
4 ---e-
JrI l 5
(s)
-----
Figure 4,10,
yg
P ole-zero distributions for y^l^
^ l l ^0
5
-44it is easily verified that the effect of the above pole r e ­
moval is to shift all of the zeros to the l eft except for
the most -negative zero w h i c h d i s a p p e a r s o
changed.
The poles are unfl)
The resu l t i n g pole - z e r o pa t t e r n for
is
shown in Figure 4,10
(b)„
The nex t step in the expansion
is to shift the most n e g ative pol e of
to infinity.
This is accomplished b y s u btracting a positive constant
from
/'-lb°
=rIi
The r e s u l t i n g admittance is y , , ^ ^ = ^—
}
— — ■
—
,
I
CTT -H i
y Il
The effect of this extraction is to leave the zeros u n ­
touched a n d .to shift all poles to the left,
is chosen so
as to shift the most negative pol e to infinity.
The r esult(o)
ing pole - z e r o p a t t e r n for
J
s^10im
Fi.gu.re 4,10 ( c ),
The p r e c e d i n g two steps are repeated until the order of the
admittance is reduced to zero.
The pole-zero patterns for
each of these subsequent steps is shown in Figure 4,10,
Notice that the poles of y ^ - ^ ^
for k even b e come the zeros
of the currents flowing thr o u g h the capacitors in the eprrent
d i s t r i b u t i o n dia g r a m of Figure 4,9,
A study of the regions
to w h i c h these poles are restricted leads to a subscript
n o t a t i o n for the ($'a such that 6
2 .^ ~~— — —
one exception to this ine q u a l i t y is that (5
t h a n 64.0
The
m a y be larger
This e x ception is of no consequence in the f o l l o w ­
ing development.
-45If the proposed synthesis proc e d u r e is to he s u c c e s s f u l 9
|p(s)+E(s)J
and D(s) must he c h o s e n to have p r o d u c e d currents
t h r ough the capacitors of E i gure 4o9 such that a set of IVs
m a y he c h o s e n to s atisfy the equality
h^s(s+(5^ ) ( s + 64 ) (s + S g )+h-2 s (s +62 )(s +6 ^ ) +h^ s (s
p Zt_s4 +p 5s 5 +p 2s 2 +p1 s
where O-h.-k.
J J
) +h ,8=
3y
(4.7)
( j = l 52 5394 )o
E q u a t i n g coefficients leads to
the foll o w i n g set of equalities
Ii1=P4
(4.8)
hi(6i+64^g)+h.2:=P3
(4.9)
Ii1 (6 264
(4 *10)
) +bg (6 2 +65 ) +Ii^ =P 2
Ii1 ( ( S ^ ^ )+h2 (62$^) + h ^ ^h4=P1
(4.11)
E r o m the pol y n o m i a l D ( s ) = d ^ s ^ + d 2s 2 +d 1s+l=K 1 (s+ 61 )(s+ 64 )(s+ 6 g)
we have
(4 .12 )
6164
+54<5q =^
^
(4.13)
(4.14)
If
(S 1 -^ 4 -KS6 ) is c h o s e n equal to P ^ Z p 49 the n a solution of
equations
(4.8) t h r o u g h (4.14) acquires the h's in terms of
the coefficients of P(s) and D(s) and the q u a n t i t y
h1=p4
h g =0
dI
h 3 -p2 =p4 d,
P4 2
h/4,=p
=Pnl ”
- d7 Z)-z(Po-P;i “V -—)
“ 6 5 (P2 -p 4 d 7 ' )
3
3
Sj<>
(4.15)
(4.16)
(4.1?)
(4.18)
-46Since the k's i n the n e c e s s a r y conditions O - h . - k .
J
J
(3= I 92-95 $4) m a y be made a r b i t r a r i l y large b y choosing the K
of E q u a t i o n (4o6) a r b i t r a r i l y large, the conditions a c t ually
reduQe to 0 - h.
(3= 1 ,2.,5»4)„
These conditions are reflected
U
in Equations
(4ol7) and
(4«18) as
p 2"-p 4 I ™ * “ 0
(4.19)
p4
r ,
p l~ d^- . - 6 5 (p 2-p 4 d ^ " )
(4.20)
Since gg sund
(6
g reater t h a n zero and the quantity
)=<a1/^39 m a y be mad e as small as desired b y
m a k i n g S 1 andS^ (for example) arbit r a r i l y small while h o l d i n g
(S1 - K^4S g ) = dg/d^ c o n s t a n t , the c ondition of E q u a t i o n (4.19)
m a y always be fulfilled.
L i k ewise
(S1S ^ S g ) =
m a y also be
made as small as desired b y m a k i n g S 1 and S ^ arbitrarily
small.
The q u a ntity (S1 -Sx) m a y be made as small as desired
b y choosing the zero of
[p(s)+E(s)j
ar b i t r a r i l y close to the
S 1 term.
c hosen as small as desired,
as small as desired.
nearest the origin ( ^ 1 )
Since
S 1 Ir t u r n m a y be
it follows that S ^ m a y be c h o s e n
T h u s , the c o ndition of E q u a t i o n (4.20)
m a y always be fulfilled, and it follows that
j]?(s)+.E(s)J
and D(s) m a y always be c h osen suc h that the t r a nsfer f u n c t i o n
T^ of Eq u a t i o n (4.4) wit h n = 4 m a y always be realized.
A means for r e a l i z i n g h i gh-pass functions of the form
-47-
V
_n— I
Prn~
v,S11+P
t
-"n-l"
'+— --+P1S+!
(4.21)
based on the methods of this s e c t i o n will n o w be given.
The
method is based on the foll o w i n g steps.
1.
P e r f o r m a high-pass to low-pass t r a n s f o r m a t i o n on
the T
of E q u a t i o n (4.21) to obtain a T^ of the
form of E q u a t i o n (4=4).
2.
Realize this t r ansformed T^ b y the methods of this
section.
5»
Divide the admittance of each of the elements of
the realized n e t w o r k b y s.
This is equivalent to
m u l t i p l y i n g the characteristic po l y n o m i a l b y s.
T h u s 5 the current d i s t r i b u t i o n of the ne t w o r k is
unchanged.
4.
N o w p e r f o r m a low-pass to h i g h ^pass t r a n s f o r m a t i o n
on the n e t w o r k to o b t a i n the high-pass r e a l i z a t i o n
of the original T^.
It is easily verified that the proposed high-pass synthesis
method is valid.
A N E T W O R K CONFIGU R A T I O N EOR E O U R T H - ORDER BAND-PASS
TRAN S F E R FUNCTIONS.
A p assive n e t w o r k c o n f i g u r a t i o n is n o w pres e n t e d which
m a y be used in c o n j u n c t i o n w i t h the
(ROVE) to realize a
—
48
-
—
fourth-o r d e r hand-pass', t r a nsfer f u n ction of the form
ap sd
,2 .
ZS-^tp2 S' +P1 S+ !
(4.22)
where the coefficients are assumed to meet the n e c e s s a r y c o n
ditions of Chapter I I I 0
W i t h r espect to active device s e n ­
sitivity and spread of element values, this n e t w o r k r e p r e ­
sents the best of a considerable n u mber of p o s s i b l e con f i g u ­
rations i n v e s t i g a t e d o
The three basic building, bl ocks of the proposed c o n f i g ­
u r a t i o n are shown in Figure -4-.11«,
placed in parallel,
These basic blocks are
and elements are disconnected from
ground and joined t o g ether to f orm terminals
the sought after three-port
(2 ) and
(EGVF) passive network.
terminals are p r o p e r l y marked in Figure 4.11.
(3 ) of
These
Since the,
three networks are to be paralleled, they m a y be assumed to
be d rive n b y the same v oltage source
(D(s)) for the purpose
of current dis t r i b u t i o n calculation.
Thus, it is seen that
the present co n f i g u r a t i o n was obtained b y an examination of
the current d i s t r i b u t i o n for v arious subnetworks.
A d i s c u s s i o n of the choice of network parameters to
realize a transfer f u n c t i o n of the form of E q u a t i o n (4.22)
will n o w be given.
First choose a third-order denominator
'a" net w o r k
"b" network
'c " network
1
vO
I
0 Ia =(7I l ^ i ) F.
E i b =i Q
E2b=((62-/72)/62)
Rla=(( 7I l A ) Z 7I f ) Q
Ia 2 =S 2 D,
V = ( V A - 7I2 )) R-
V ( ( 7Ii A ) A )
Vb = ^ 2~^ 2')I)2
=7Ii s d
d Is
Ib 2-sD2
Figure 4.11.
Three basic b u i l d i n g blocks for the fourth-order band-pass transfer
function realization.
—50p o l y nomi a l
D(S) = ( S ^ 1 ) (s+ 6 2 ) (s+ 6 j)
(4.25)
where 5 1 , (j2 ? 6 3 ? are un e q u a l p o s i t i v e real quantities.
f e r ring to Figure 4 , I l 9 D 1 is see n to be
Dg, is
(s^Ks+fij)-
(s+(52 ) (s+(5j) ■and
Notice that the current f l o w i n g through
the grounded c a pacitor of the
<52 (53 s 2 ,
Re­
"a" n etwork is sZf"+(S2 +63 )s^+
If (62 +(53 ) is c h o s e n equal to P 3Zp^ and the ad m i t ­
tance of the "a" n e t w o r k is magn i t u d e scaled "by the factor
P ^ 9 this current becomes p Z|_sZt"+p3s^+ 62
6 3p^s^»
The "a" n e t ­
w o r k is s e e n to take care of the realiz a t i o n of the two l e a d ­
ing coefficients of the po l y n o m i a l P ( s ),
Also notice that
the current flowing t h r o u g h the grounded r e s i s t o r of the "b"
n e t w o r k i s ^ s 2 + ^ (Sp+S 3 ) S-^ 2 S 1 S 3 -
If (Sp-I-S3 )ZS 1 6 3 is
c h o s e n equal to P 1 and the admittance level of the "b" net wor k is scaled b y IZTlp
S
j
S 3 , this
current becomes
(IzS 1 S 3 )G
+PpSHl and m a y be considered to realize the two trailing c o ­
efficients of P ( s ) ,
The
"c " net w o r k accomplishes the r e a l i z a t i o n of the Ty
nume r a t o r and d e nominator quadratic term,
ure 4,11
(c) are c h osen to interlace with the
an RG admittance function.
and 7^^ of F i g ­
S ’s to produce
The admittance T = s (s +T^3 ) (s +7^.)Z
(s+Sp)(s+S 2 )(s+S3 ) is t h e n r e a lized in the config u r a t i o n
shown,
A current d i s t r i b u t i o n c o mputation for the
"c" n e t ­
w o r k shows that the current t h r o u g h the final r e s i s t o r is
-51The admittance level of the ne t w o r k m a y t h e n be
scaled,
Since a quadratic t erm current. (
has
6163
already b e e n ' r e a l i z e d , the s caling factor is c h o s e n such
that the r e s u l t i n g current f l o w i n g through the final r e s i s ­
t or is
(p2 -
--- 6 2 6 ^
It is seen that this current
is the A(s) p o r t i o n of the r e a l i z a t i o n and is equal to ap.
Thus, the proposed n e t w o r k realized the desired band-pass
transfer f u n c t i o n to w i t h i n a constant multiplier*
It is
apparent that the success of the method depends on. the a b i l ­
i ty to choose the
-
C r
2
6 's such that the quantity Cp0- v— 7—
is positive.
0103
N o tice that no r e s t r i c t i o n has
b e e n p la c e d on the relative m a g n itudes of the
($'s.
The
element values of the "c" n e t w o r k are hot shown as they b e ­
come quite complicated in terms of the
(51s a n d
is far easier to w o r k w i t h n u m e r i c a l values.
7]’s »
The synthesis
method as presented is only valid w h e n ( 6 2 +63 ) ma N
sen equal to p ^ / p ^ and
to equal p^.
It
cho­
(5163 s i m u l t aneously chosen
It is obvious that there are cases where this
A a y not be done.
N o r those cases it is suggested that a
current d i s t r i b u t i o n dia gr a m applied to the subnetworks
might suggest alternative realizations.
N.
EXAMPLES
This section will present n umerical examples of the syn
-52thesis p r o c edures p r esented in the previous sections of Ghap- ter IVo
Each example p r o b l e m will be presented in the c u r ­
rent d i s t r i b u t i o n for m of Figure 4.1.
,F or the first example, c o n s i d e r the biquadratic t r a n s ­
fer func t i o n of S e c t i o n I eC e
's^+2,o.828s+4
(4*23)
s ,+Io 4 1 4 s +1
/
This tra nsfer f u n c t i o n will be realized b y the methods of
Section IVoC0
The current d i s t r i b u t i o n for this T
in Figure 4*12 for the case of
V
is shown
K=1
A- (maximum value allowable),
aI
rI
Since — — < — — 9 we take the n e t w o r k of Figure 4 » 4 and easily
”/
'r
0x
xr
O\
calculate the element v a l u e s o The resultant n e t w o r k is shown
in Figure 4,13 for the case of e-^ h a ving b e e n c h o s e n equal
to 3.o293.<>
Note that e^ could have b e e n chosen c o n s i derably
less since the only requi r e m e n t o n e ^
is that the polynomial
nected to the voltage follower is shown in Figu r e l»3o
The next example will demonstrate the low-pass ladder
n e t work r e a l i z a t i o n of Se c t i o n I V 0D 0
Consider the Butter-
wo r t h low-pass t r a nsfer f u n ction
I
(4.24)
s4 +2 o6131s 5+3 o4 1 4 2 s 2 +2»6131s +1
The first step of the proc e d u r e consists of c h o o s i n g a po l y -
-53-
A (s )= X s ^ + o? 07 s +1
P(s)+E(s)=s +
(lo4-14-+e-, )s +1
Figure 4.12.
Current d i s t r i b u t i o n for network corresponding
to transfer f u n c t i o n of E q . (4.23)«
Figure 4.13°
Network r e a l i z a t i o n for T =X s + 2 ° 8 2 8 s + 4 ^
v
s^+1„414s+l
-54nomial E(s) ©f degree less t h a n or equal to four and a p o l y ­
nomial D(s) of degree three such that y ^ ^ = jp(s)+E(s)]/D(s)
is a dri v i n g point admittance conforming to the c o n f i g u r a ­
t i o n of Figure 4.6.
and
The p o lynomials
E ( s ) = 1 2 . 9 7 1 0 s 5 +25. 1 9 1 9 s 2 +ll.1407s
(4.25)
D ( s ) = 2 . 6 5 5 s 5 +6.9391s 2 +5.2260s+l
(4.26)
fulfill these requirements.
A continued f r a c t i o n e x pansion of
jp(s)+E(s)J/D(s) is
performed to determine the element values for the c o n f i g u r a ­
tio n shown in Figure 4.6.
The D. polynomials are identified
U
d uring this p rocedure as sh o w n below.
C1 S=O .57 6 5 s
D 1 = 2 .6 5 5 s 5 + 6 .95 9 1 s 2 + 5 « 2 2 6 0 s + l / s ^ + 1 5 .5842s5 + 2 8 .6 0 6 2 s 2 +!5. 7559s +1
s 4 + 2 .6 1 5 2 s ^ + 1 .9 6 7 6 s 2 +
.5 7 6 5 s
E 1=O. 2047.
1 2 .9 7 1 0 s 5 + 2 6 .6 5 8 6 s 2 + 1 5 » 5 7 7 4 s + l / 2 . 6 5 5 s 3 + 6 .9 5 9 1 s 2 + 5»2260 s +1
2 . 6 5 5 s ^ + 5 » 4 5 2 9 s 2 + 2 .7 5 8 4 s + .2 0 4 7
CgS=S ..728s
D 2 =I.4 8 6 2 s 2 +2.4876s+.7955
/l2.97l0s 5+26.6586s 2 +15-5774s+l
12.9710 s ^ + 2 1 . 7 1 2 6 s 2 + 6.9405s
E 2 = O . 5017
4 . 9 2 6 0 s 2 +6 . 4 5 6 9 s +1 / 1 . 4 8 6 2 s 2 +2 .4 8 7 6 s + 0 . 7955
1 .4 8 6 2 s 2 + ! o9 4 2 0 S + Q .5017
-55C =s= 9 .027s
D 5 = O »54-56s+0.o4-956 /4 » 9 2 6 0 s 2 +6»4-369 s +1
4 .9 2 6 0 s 2 +4-»454-8s
E 5=O.2753
l»9821s+l |0»5456s+0.4-936
0.54568+0.2753
0^ 3 = 9.0843
D4=O »2183 /l.9821s+l
1 .9821s
I
E4=O.2183
/0o2183
0.2185
The C . m a y n o w he determined as
U
028=8.7288
D2=Io4862s2 +2.48 7 6 s + 0 o 7953 /12 .9 7 1 0 s 3 + 2 5 .1919s2+ll.1470s
12o9710s^+21.7126s2+ 6»9405s
0^ 8 = 6.5778
D 5= O.5456s+0.4936
/3 .4793s2 +4o2002s
5.4795s2 +5.1477s
C4s =4o828s
D4 =O.2183 /1I0525 S
1.05258
0
and O 0 determined as
T
The above calculations
-56restilt in the n e t w o r k shown in Figure 4» 14«
'The final example is concerned with a demons t r a t i o n of
the four t h-order band-pass t r a n s f e r function synthesis of
S e c tion I-V 0Ec
This example f u n c t i o n is
_________ sf______ ,
V
(4.27)
e-
sV
l o414s
V
S 2 +1
o4 1 4 s +1
The first step is to choose the zeros of D ( s )0
c h osen as (5-1= 2 042, S p = O 04 1 4 9 S % = I * 0 0 o
t y (p2 - g
r
These are
Note that the qu a n t i-
-6263P 4 ) is equal to 2 .172 ,
T h u s 9 the above
choice is p e rmissible and leads to a value of K = 2„172 and
D ( s ) = ( s + 2 04 2 ) ( s + 0414)(s+1.00)
The
(4.28)
T^tS of the n e t w o r k are c h o s e n as 7|]_=2. 6 6 7 5 4 3 s
T^j = O 06 0 0 5 and
»667»
These values lead to the network
r e a l i z a t i o n of Figure 4 .15 w i t h the terminals for the c o n ­
fig u r a t i o n of Figure 4.1 p r o p e r l y marked.
A(s)=l
P(s)+E(s)
3o414s +2c6132s
Figure 4,14.
Net w o r k r e alization for T
v s4 +2.6132s 5+3o4143s 2 +2.6132s+l'
©
3.85 F,
5850
,
W v ----1 (-
O
+ (-
1.71 F,
1.109 F 1
.550^
.05520K 14.232.
1720
Iigure
.15»
N etwork r e a l i z a t i o n for T >
2 a 72s 2
s4+ 1 . 4 1 4 s 3 +3s2 +l.4143-1-1*
V,
Z E T W O B K OPT I M I Z A T I O N
This chapter investigates the p o s s i b i l i t y of util i z i n g
the digital computer to optimize a given n e t w o r k conf i g u r a ­
tion=
A p a r t i c u l a r class of networks is selected for use as
an example of the o p t i m i z a t i o n process and a computer p r o ­
gram is developed w h i c h optimizes any low-pass or, b y a
suitable transformation,
any h igh-pass transfer function
realized b y the decomposed ladder method of S e c t i o n I V o D 0
S e c t i o n A of the pre s e n t chapter is devoted to a discussion
of the n e t w o r k p a r a meters to be o p t i m i z e d «
S e c t i o n B is
concerned wit h a d e s c r i p t i o n of the optimization computer
p r o g r a m and the results of the p r o g r a m on an example n e t ­
work,
The prime pur p o s e of S e c t i o n A is to define the op ­
t i m i z a t i o n p a r a m e t e r Q while the latter section will be c o n ­
cerned w i t h the m echanics involved in m i n i m i z i n g this p a r a ­
meter,
A0
OPTI M I Z A T I O N PABAMETEBS
The first p a r a m e t e r to be discussed is the sensitivity
parameter.
Bode first introduced the term "sensitivity" in
c o n n ect i o n with f e e d b a c k analysis
(30),
Since t h e n the t e r m
has b eco m e c o m monly defined as the logarithmic derivative of
T(s,k) w i t h respect to the l o g a r i t h m of k where T(s,k) is a
n e t w o r k f u n c t i o n of interest
(in the present case the voltage
-60tran s f e r f u n c t i o n T ) and k is a variable element,
Thus,
the sensitivity p a r a m e t e r is
qT
9 (I n T )
k . 9T
k - g(lhk7 “ T gk
(5.U
The above de f i n i t i o n of s e n s i t i v i t y m a y be looked u p o n as
r e p r e s e n t i n g the ratio of a pe r c e n t a g e changb in a given n e t ­
wo r k fun c t i o n to a pe r c e n t a g e change in a variable element
of the n e t w o r k function.
T h u s , ignoring phase a n g l e s 9 if the
n e t w o r k f u n c t i o n varies b y 0,01 db for a I db v a r i a t i o n in
k, we say the s e nsitivity is 0,01»
As an example of the use
of the s e nsitivity p a r a m e t e r in the present investigation,
consider that the g ain of the active device departs from
u n i t y and is represented b y the variable k.
This represents
the actual s ituation since in a n y practical case, the active
device will depart from the ideal b y some amount.
Ex p r e s s i on
(1 .5 ) w h i c h is the expression for Tv in terms of the a p p l i ­
cable admittance polyn o m i a l s must the n be altered to
T__=
, = MsI
v N 11- ^ 13
p(.s)
(5 .2 )
A p p l y i n g the d e f i n i t i o n of (5*1) to this e q u ation results in
5®v_
k '
klL
■15
(5.5)
N ll“k % 3
Since k is approx i m a t e l y u n i t y and
(B-^-kir^) is P ( s ) ,
E q u a t i o n (5*3) m a y be w r i t t e n as
SkV ~
PTsJ
(5*4)
-61Since the n e t w o r k r e a l i z a t i o n has no control of P ( s )9 this
equation shows that the only p o s s i b i l i t y of sensitivity
m i n i m i z a t i o n lies in the choice of
wh i c h is the E(s)
of the p i c t o r i a l r e p r e s e n t a t i o n of Figure 4.1.
A n alternate approach to sensitivity is to utilize the
pol e - z e r o s e nsitivity w h i c h follows w h e n the s e nsitivity of
T ( s $k) is obtained in terms of its poles and zeros
Let
TC
(31)°
(s+Zi ).
i=l
T ( s $k ) - g
n
(5.5)
TT (s+p.)
1
i=l
where the
p ^ ? and g are 9 in g e n e r a l $ functions of some
p a r a m e t e r k.
Taki n g the loga r i t h m of each side $ d i f f e r e n ­
t i a ting w i t h respect to k, and m u l t i p l y i n g both sides b y k
i
k
where
-I.
sI + .E 1
'9 k
s+zi
’
i»i
sEi
"k"
s +P 1
(5,6)
~sp 1 5 k f h
k
gk
are defined r e s p e c t i v e l y as the zero and pole sensitivity of
the n e t w o r k f u n ction T.
Still another type of s e nsitivity is "coefficient senn
sitivity" (18).
If T ( s ?k)=
t . (k)s 1 9 t hen coefficient
i=Q
sensit i v i t y is defined as
Y
-
4..
A t 3- A j
s?1=
62
-
A(Int1 )
-----—
Ak/k
■
d (Ink)
wh i c h is seen to be the ratio of a percentage change in a
gi v e n p o IynomiaJl coefficient to a percentage change in the
variable element k.
In the present investigation, we are interested in the
variable element k r e p r e s e n t i n g the values of the components
in the passive p o r t i o n of the n e t w o r k as well as the gain
of the active p o r t i o n of the
( E C O ’) .
Other net w o r k p a r a meters w h i c h will be of interest
are element d i s t r ibution (33 )» element sum, and element
ratioo
Capacitance and r e s i stance distribution are de ­
fined re s p e c t i v e l y as
Op=
v
and E b =
^
(O 1-H)Z
i=l
x
y)
f r (E 1- E )2
i=l
^
(5 .8 )
where H and \E are the average values of net w o r k capacitance
and resi stance respectively.
Capacitance sum C q .is the t o ­
tal capacity of the n etwork
n„
S 1
(5.9)
This qua n t i t y is of interest since it is p r o p o r t i o n a l to the
—63~
volume and cost of the resu l t a n t network.
Element ratio is defined as the ratio of m a x i m u m ele­
men t value to m i n i m u m element v a l u e »
T h u s 5 capacitance and
resistan c e ratio are defined r e s p e c t i v e l y as
max
R “ '0;
min
max
and
TT B
(5.10)
min
It is d e s i r a b l e 5 in p r a c t i c e 9 to minimize each of the
parameters listed a b o v e „
In g e n e r a l s h o w e v e r 5 it is im ­
possible to minimize t h e m simultaneously.
In order to r e ­
duce a m ultitude of o p t i m i z a t i o n parameters to a single op ­
tim i z a t i o n p a r a m e t e r to be mini m i z e d $ an overall o p t i m i z a ­
t i o n par a m e t e r w ill n o w be defined together w i t h its a s s o ­
ciated o p t i m ization problem.
The optimization para m e t e r Q
will be defined as
SJi
k
+ Zw1
i
Sgi
+ Z Zw0
i j 'j'Pi C j
8
.•^
k
S ai
S^i + T s
7
% %&i . 7
+ / .W
I ^ i
Z Z i 'I
i-j
+ ZZw.R j 5P i 8R i
• U
i. j
wOB0B+wHEEI)+Wc,^Os+wc,^Oji+Wjj^SH
where all quantities have p r e v i o u s l y b e e n defined
gai n of the v o l t a g e .follower)
Ed
(5.11)
(k is-the
except for the w " s which are
arbitr a r i l y assigned w e i g h t i n g factors.
These m a y be used
"-64"—
to upgra d e or downgrade the r e l a t i v e importance of the c o m ­
p onent n e t w o r k parameters.
The corresponding optimization
p r o b l e m is to m i n imize Q for a gi v e n set of weig h t i n g f a c ­
tors w subject to the u s u a l n e t w o r k constraints of n o n n e g a ­
tive element values.
A n important c o n s i d e r a t i o n in optimization problems is
that of the constraints.
It m a y be reasoned in the present
p r o b l e m that the constraints are, in effect, absent.
This
is true since the n e t w o r k element values are continous f u n c ­
tions', of :the
independent parameters.
The element values of
the initial n e t w o r k are all positive.
becomes negative,
zero.
If. an element value
it mus t have p a ssed through a value of
This means that either the value of
t h r o u g h infinity.
or
passed
As long as W r,
and W d have finite valitR
u e s , the c o n d i t i o n of n e g ative element value will either
never occur or else signify that too large a step size has
b e e n t a k e n in the o p t i m i z a t i o n process.
This latter c o n ­
d i t i o n is handled b y t a king the step over a g a i n with r e ­
duced step size.
Thus, in the present problem, the con­
straints are e f fectively absent.
A n e x amination of the opt i m i z a t i o n p a r a m e t e r Q shows
t h a t , once a n e t w o r k has b e e n synthesized, all of the c o m ­
ponents of Q m a y easily be evaluated wit h the p o s s i b l e ex-
_ g Ex­
c e p tion of the component s e n s i t i v i t y p a r a m e t e r s .
The eva l ­
u a t i o n of the component sensi t i v i t y parameters involves i n ­
creasing each element in t u r n "by one^percent,
appropriate t r a n s f e r admittances,
evaluating the
and deter m i n i n g the change
in the coefficients of the t r a n s f e r function T -
Fo r t u n a t e -
V
ly, there are digital computer routines available wh i c h e val­
uate the transfer admittances of a net w o r k fro m a d e s c r i p t i on
of its topology.
There are a n u m b e r of ways a solution to the o p t imiza­
tio n p r o b l e m m a y be attempted.
It should be p ointed out at'
the outset that the opti m i z a t i o n pr o b l e m is extremely c o m p l i ­
cated and that no exact s o l u t i o n is attempted.
One p o s s i ­
b i l i t y for an approximate s o l ution is to p e r f o r m a p u r e l y
r a n d o m choice of the independent parameters of the subject
n e t w o r k a large n u mber of times w i t h Q.being evaluated for
each of the resultant networks.
That n etwork w h i c h c o r r e ­
sponds to the lowest value of Q is. the n selected as the
final design.
A n o t h e r possibility,
and the one used in this i nvesti­
gation, is the method of steepest descent
(35)»
This m e thod
is often, compared in the literature to the p r edicament of
b e i n g lost in the woods.
Since civilization is u s u a l l y c o n ­
centrated at the lower elevations, one p o s s i b i l i t y for
—66 —
.,reaching shelter is to travel at all times in that dire c t i o n
w h i c h ma ximizes one's descent.
In optimization problems
this corresponds to c o m p u t i n g gradients at each step in the
developm e n t of the solution and taking a n e w step in the
dire c t i o n of steepest descent.
This process is continued
u n t i l no f urther significant improvement is possible.
The
method consists of seeking a local minimum' and it is seen
that other m i n i m a are i g n o r e d .
Since the compu t a t i o n of
gradients can r equire considerable computational effort <, a
n umber of s t e p s 'm a y be t a k e n in a given dire c t i o n in order
to f u l l y exploit a g i v e n gradient computation.
This p o s s i ­
b i l i t y is t a k e n advantage of in the present investigation.
A p r e l i m i n a r y step in the present o p t i m ization p r o b l e m
is the de t e r m i n a t i o n of a r e a s onable network t o p o l o g y and
initial element values for a g i v e n transfer f u n c t i o n T^.
This t o p o l o g y m a y either be selected from those presented in
this report or devised b y other mieans.
The experience and
ability of the n e t w o r k d e s i g n e r are utilized in this step.
A "reasonable" n e t w o r k is defined here as one w h i c h appears
;
to be amenable to optimization.
The net w o r k synthesized in
this step is called the initial network.
The computer p r o g r a m to be described in the next sec t i on
accepts the initial n e t w o r k and optimization p a r a m e t e r
w e i g h t i n g factors and, b y apply i n g the method of steepest
d e s c e n t , determines improved n e t w o r k s ,
Bo
COMP U T E R P R O G R A M DESCRIPTION.
T h e .c o m puter p r o g r a m for n e t w o r k optimization has b e e n
w r i t t e n in F O R T R A N for the IBM 1620
storage capability.
(60K) computer with dis k
As explained previously, the computer
p r o g r a m is not w r i t t e n for a general network but for a p a r ­
t i c ular class of n e t w o r k s .
This class of networks is the
low-pass decomposed ladder n e t w o r k of Section IVoD.
Since
m a n y of the subroutines of.the computer p r o g r a m m a y be used
if similar computer programs are wri t t e n for other classes
of networks,
and jsince the present computer p r o g r a m is valid
for a w i d e l y used class of n e t w o r k f u n c t i o n s , the FOR T R A N
statements for the computer p r o g r a m are printed in A p p endix
I,
In addition, the method and purpose of each subroutine
are explained in this section.
This should enable the read-'
er to recognize and adapt those routines w h i c h are of in t e r ­
est to him.
The network o p t i mization computer p r o g r a m consists of
16 subroutines and one main, program.
The m a i n pr o g r a m has
b e e n b r o k e n into five linked subprograms b ecause of the
limited core storage of the 1620 computer.
These five sub-
-68programs, which have b e e n named CHOKE, C H T W O , C H T R I , CHFOUR
and C H F I V E 9 are t o g ether considered as the m a i n program.
The f l o w chart for this m a i n p r o g r a m is shown in Fignre $.1.
The circuit analysis p o r t i o n of the pro g r a m is accomplished
in terms of topological formulas based on linear graph t h e o ­
ry.
Ho attempt is made here to present the basic theory or
to explain the terminology.
The interested read e r is r e ­
ferred to references $6 and 37 for an exposition of the t h e ­
ory and f a m i l i a r i z a t i o n w i t h the terminology.
Before d i s ­
c u s sing the f l o w c h a r t 9 each of the 16 subroutines will be
explained.
S U B R G U T IHE R D I H .
This subroutine is concerned w i t h reading
in input data b y means of the card reader and c o mputing some
constants which will be used later in the program.
The in­
put data consists of the f o l l o w i n g quantities.
HOD
Humber of .branches in a complete tree of the
network.
HEL P
Humb e r of n e t w o r k elements exclusive of the
auxi l i a r y elements.
HO
Order of the denomi n a t o r of T^.
HTT P
A r r a y for identifying the type of each ele­
ment- of the network.
+1 for a Capacitor, 0
for a resis t o r and -I for an inductor
(induc­
tors are not used in the present c a s e ) .
-69-
(
START
GALL R D I N
I N C REASE ELEMENTS OF IFR & ITO
_________ ARRAYS BY ONE__________
CALL T R G O M
A L P H ( I ) = A L B A R ( I ) ,1=1,NO
B E T A (I)= B T B A R ( I ) ,1=1,NOMT
CALL SYNTHZ
CALL E V LVTR
CALL SENSIT
CALL EVOB
VOB A R = V O B J
PRINT H E A D I N G F O R INITIAL NETWORK
CALL DMPOT
Figure 5»1»
program.
F l o w chart for net w o r k optimization computer
—yo—
Figure 5,1.
(Continued)
-71-
ST p =STBES
CAL L TKSTP
A L B A R ( I ) = A L P H ( I ) ,1=1,NO
B T B A R ( I ) = B E T A ( I ) ,1=1,NOMT
VOB A R = V O B J
SENSE
SWITCH
D E C S = ( (VORJ-VOBJ)/ V O R J )*10000
D E C S = A B S F (DECS)
PRINT OUTPUT H E A D I N G F O R FINAL N E T W O R K
CALL DEPOT
CALL EXIT
Figure 5« I
(Concluded)
-72-
IB
Identifying n u m b e r of an element connected b e ­
t w e e n the input and reference node of the s u b ­
ject network.
IJ
Identifying n u mber of an element connected b e ­
tw e e n the input and output node of the n e t ­
work.
KB
Identifying n u m b e r of an element connected b e ­
t w e e n the output and reference node of the n e t ­
work .
IBR
A r r a y for ident i f y i n g the node
"from" which
each element is connected.
ITO
A r r a y for i d entifying the node
"to" which each
element is connected.
SP
A r r a y for the coefficients of the denominator
(P(s)) of T .
SP(I) is the coefficient of the
s t erm raised to the
AIiBAR •
(I-I).power..
A r r a y for zeros of the initial
(P(s)+E(s)) p o ­
lynomial .
BTBAR
A r r a y for zeros of the initial D(s) polynomial,
WK
WX
A r r a y f o r the w e i g h t i n g factors wJgpi1
A r r a y for the w e i g h t i n g factors w.
WELH
A r r a y for the w e i g h t i n g factors w n
and
,1?Pi
.-R-j
WCD
»,
Set equal to the weighting factor W q
-73WED
WCS
Set equal to the w eighting factor W to .
-D
Set equal to the
w e ighting factor W r, .
WCR
Set equal to the
ESP
E u m h e r of coefficients in P(s)
ETT
. E u mher of branches in a 2-tree
w eighting f a ctor W n .
uE
WEE
Set equal to the w eighting factor wTO .
ltE
The subroutine computes the foll o w i n g q u a n t i t i e s „
EVB
(EO.+l) „
(EOD - I )0
Eumher of independent variables associated
wit h the n e t w o r k (2 x E Q - 2 )0
EOMI
(RO-I)i
EOM T
(NO-2).
EPAE
Eumber of core storage locations required for
the storage of one tree (5 elements m a y be
stored in one core storage l o c a t i o n ) „
S E B R O E T IEE T E C O M .
Determines and stores those 2-trees of
the n e t w o r k w h i c h are u t i l i z e d in the determ i n a t i o n of the
num e r a t o r p o l y nomial of y^g - E p j «
U pon co m p l e t i o n of this
s u b r o u t i n e , the applicable 2-trees are stored in the ITE
array with each m e m b e r of the a r r a y containing as m a n y as
five n e t w o r k element i d e n t i f i c a t i o n n u m b e r s »
The quantity
ETS is set equal to the n u m b e r of 2-trees stored in the ITR
array.
S E B E O E T IEE T E D E T .
This subroutine is called b y T R C O M for
- rJQ-
the purp o s e of d e t e r m i n i n g if a given set of ne t w o r k b r a n c h ­
es forms a network tree.
Given a set of BTOD integers
(LD
array) identi f y i n g a set of b r a nches of a n e t w o r k c o n s i s t ­
ing of N O D independent node p a i r s ? it is determined if the
set forms a tree of the n e t w o r k .
The subroutine returns the
integer variable N E L as 0 if the set forms a tree.
Other­
wise NEL- is set equal to K to denote that m e m b e r of the LD
array w h i c h represents the b r a n c h which first forms a c i r ­
cuit with those branches p r e v i o u s l y examined
(the branches
are examined b e g i n n i n g wit h b r a n c h number L D ( N O D ) ).
The
method of determ i n i n g whe t h e r or not a given set of b r a n c h ­
es constitutes a tree is the following.
1,
Take each b r a n c h b e g i n n i n g with the b r a n c h i d e n t i ­
fied b y LD(NOD) and pr o c e e d thr o u g h L D ( N O D - I )9
e t c o 9 to L D ( I ) .
As each b r a n c h is s e l e c t e d 9 d e t e r ­
mine if the nodes of this branch w h e n considered
w ith the nodes of those branches alr e a d y selected:
2.
a.
Adds no n e w nodes.
b.
Adds one n e w node.
c.
Adds two n e w nodes.
If two n e w nodes are added 9 the b r a n c h selected
forms a subgraph w h i c h is not connected to any
w h i c h m a y have alr e a d y b e e n c o n s i d e r e d .
3°
If one n e w node is a d d e d 9 the branch b e i n g c o n s i d ­
ered forms an e xtension to one of the subgraphs
-75p r e v i o u s l y g e n e r a t e d ».
No circuits are generated
in this case.,
4o
If no n e w nodes are generated, there are two p o s s i ­
bilities .
a0
Two uncon n e c t e d subgraphs hav e b e e n c o n ­
nected to f orm a single subgraph.
No cir- •
cults are generated in this case,
b«
A circuit has b e e n generated b y the a d d i ­
t i o n of this node in wh i c h case the q u a n ­
t i t y N E L is set equal to the subscript
of the m e m b e r of the LD a r r i y w h i c h is
c u r r e n t l y involved, and the examination of
branches is terminated since the set of
branches b e i n g examined cannot p o s s i b l y
form a tree.
5o
If all branches are examined without encountering
the c ondition of 4b, the set of branches c o nsti­
tutes a tree and the q u a n t i t y N E L is set equal to
zero o-
SUBROTJTINE T R G L .
of six categories.
This subroutine classifies a tree into one
O nly one of these categories is of in ­
terest in the present case, but since the toiitine is of use
for other c o n f i g u r a t i o n s , it is discussed here.
routine will accept a set of N O D integers
The s u b ­
(LD array) i d e n t i ­
-76fying the branches of a tree of a net w o r k and assign the
tree to one of six classifications b y setting the integer
variable ICLS equal to the c l a s s i fication n u m b e r of the
tree* . .Before d e f ining the clas s i f i c a t i o n n u m b e r s ? it is
n e c e s s a r y to define several t e r m s »
A tree w h i c h contains the b r a n c h I B 0
IHPTJT THEE
GUTPUT TREE
A tree w h i c h contains the b r a n c h K B 0
PLU S TRE E
A n input tree w h i c h forms a tree whe n
b r a n c h K B is substituted for IB but
does not f o r m a tree w h e n b r a n c h IJ is
substituted for I B 0
HIH U S TREE
A n input tree w h i c h forms a tree bot h
w h e n b r a n c h K B is substituted for IB
and again w h e n IJ is substituted for I B 0
The ICLS integers m a y n o w be d e f i n e d 0
ICLS =. I
.the tree does not fall into any of the c a t e ­
gories given b e l o w 0
=- 2 ? the tree is an input t r e e 0
= 3 » the
tree
is an output t r e e 0
= .4 ? the
tree
is b o t h an input
and output tree,
■= 5 ? the
tree
is b o t h an input
and minus tree,
= 6 , the
S U B R O U T IHE S E Q G E H 0
tree is. b o t h an input
and plus tree.
This subroutine is an aid in the generai
t i o n of all p o s sible combinations of H things t a k e n M at a
—
77
~
In .the pre s e n t c a s e 9 the objective is to generate all
time.
possible trees f r o m a given set of elements and n o d e s »
S p ecific a l l y , . t h e subroutine takes a set of integers,
LD(M) i LD(M-I) .......... . LD(2), LD(I)
where LD(I)
LD(2)
LD(M-I) ^ LD(M) and LD(I)Si N, and
generates a n e w set of integers $ L D 8(M), L D 8(M-I),
....... L D 8(2), L D 8(I)
subject to the same constraints given' above and the a d d i t i o n ­
al constraint that the quant i t y
Y
(LD8(J)-LD(J))Nj
J=I'
is posit i v e and the m i n i m u m value possible.
In other words,
the next highest in r a n k of N integers ta k e n M at a time has
n o w b e e n generated*
A n additional feature of this subroutine is the ability
to skip over combinations.
Suppose we have the sequence
LD(M) , LD(M-I) *...... . L d C k +1) , 'LD(K)', LD(E-I) >,, , .LD(I)
and, instead of generating the next ranking sequence, we d e ­
sire to skip to the next ran k i n g sequence after
LD(M) ,.'LD(M-I)
,LD (K+l)., LD(K) ,(N+2-K)
(-N-1) ,N»
This is accomplished b y the incl u s i o n of the integer variable
N E L in the input ar r a y of the subroutine where N E L is set
equal to K.
occurs.
If N E L is set less t h a n 2, normal sequencing
U p o n r e t u r n i n g f rom the subroutine,
if N E L is zero,
-78the sequence presented to the subroutine was the highest
r a n k i n g p o s sible and the g e n e r a t i o n of all p o s s i b l e c o m ­
bination s was p r e v i o u s l y completed,
For any other condi-
.tion, K will be u n changed if its initial value was great­
er th a n I or set to 2 if.its initial value was less than 2 ,
SUBROUTINE SYNT H Z ,
This subroutine performs the low-pass
synthesis p rocedure of S e c t i o n I Y ,B,
of P(s)
Given the coefficients
(SR array), the NO zeros of the pol y n o m i a l
+E(s)),' and the
(P(s)
(NO-2) zeros of the polynomial D(s), this .
subroutine computes the
($xNQ-2)
■ of the r esultant low-pass filter.
element values
(ELEM array)
The first NO element v a l ­
ues correspond to R^ thr o u g h R n res p e c t i v e l y (see Figure
ff
4,8),
The next two element values correspond to
respectively.
The r e m a i n i n g element values a l ternately
correspond to G= and G=
^
tl
and
U
'
(0= 3 )4 ,*,.,NO),
Subroutine SYNTHZ
u
calls on subroutines P O A D and P O H U L ,
These two subroutines
are explained below.
SUBROUTINE P O A D ,
The input a r r a y of this subroutine is
(POO, NOG, POD, NOD, R P Q , N P O ).
NOD coefficients
mial
The subroutine adds the
(POD array) of a (N0D-1) order p o l y n o ­
(POD(I) is the t r a i l i n g coefficient) to the NOG c o ­
efficients
(P0C array) of a (NOC-I) order polynomial.
The
coefficients of the r e s u l t i n g pol y n o m i a l are stored in the
-79“
E P O a r r a y and the q u a ntity EPO is set equal to the numb e r
of coefficients of the r e s u l t i n g polynomial.
STJBROTJTIEE POMUL.
The input a r r a y of this subroutine is
( A E ?I A E ?P © B 9E O L sP 0 G 9E 0 0 ?EERR).<>
The subroutine multiplies
an (IAE-I) order p o l y n o m i a l ? whose IAE coefficients are
stored in the A E a r r a y , b y a (EOD-I) order po l y n o m i a l whose
coefficients are stored in the P O D array.
The coefficients
of the r e s u l t i n g po l y n o m i a l are stored in the POO array and
the quan t i t y EOG is set equal to the numb e r of coefficients
of the r esulting polynomial.
S E B E O U T IEE E V L Y T E .
This subroutine evaluates the 2-tree
admittance products of the ETS 2-trees stored in the. ITE
array.
The admittance p roduct for 2-tree n u m b e r I is
stored at l o c ation V T E ( I ) .
S U B R O U T IEE S E E S I T .
This subroutine evaluates the se n s i ­
tivities of the coefficients of P(s) to changes in the
element values of the n e t w o r k synthesized b y subroutine
SYETHZ.
These sensitivities are stored in the S E L M array.
S E L M ( M 11J) gives the p e r c entage change in coefficient p^_^
of the po l y n o m i a l P(s) due to a
lj% increase in the element
value of the low-pass n e t w o r k whose value is stored at l o ­
c a t i o n ELEM(J)..
-80SUBROUTIETE E T O B 0
The sole pur p o s e of this subroutine is to
evaluate the o p t i mization p a r a m e t e r Q of E q u a t i o n (5°11 )0
This qua n t i t y is stored at l o c a t i o n T O B J 0
SUBROUTINE.-DMPOT o
This is an output subroutine which h a n ­
dles the data associated w i t h the initial n e t w o r k and the
final n e t w o r k as explained i n the previous section,
.subroutine tests sense switch I,
l
Hhe
If -sense switch I is off,
the output is printed on the on line p rinter only.
If on,
output is also punched on cards,
SUBROUTINE E T G R A D „
The independent variables of the p r e s ­
ent opti m ization p r o b l e m are the NO zeros of (P(s)+E(s))
and
(N"0-2) zeros of D ( s ) <>
variables is N T B = 2 x N 0 - 2 „
T h u s , the number of independent
The p u r p o s e of the present sub­
routine is to evaluate the N T B gradients associated w i t h the
o p timiza t i o n problem.
The cqmputed gradients are stored in
the GRAD a r r a y and are weighted in direct p r o p o r t i o n to the
magnitud e of the associated p o l y n o m i a l zero,
SUBROUTINE STCO M P .
This subroutine determines the size of
step to be t a k e n in the method of steepest descent.
The
step size is stored at l o c ation S T P 0
SUBROUTINE T E B T P »
This subroutine takes the step of value
■81S TP and computes the r esultant optimization p a r a m e t e r V 6BJ»
S U B R O U T I N E OOUSTR.
T h i s subroutine checks to see that a
given set of zeros for .(P(s)+E(s))
gi v e n set of zeros for D(s)
terlaced »
(ALPH array) and a •
(BETA array) are p r o p e r l y in ­
U p o n r e t u r n i n g f r o m the subroutine, the integer
variable N O O U is zero if the zeros are p r o p e r l y interlaced
and set to one if the y are not p r o p e r l y interlaced.
This completes the e x p l a n a t i o n of the 16 subroutines.
The m a i n p r o g r a m of Figure 5,1 w ill n o w be discussed.
First5
subroutine R D l U is. called to read the input dat a fro m cards.
Since the elements of the IFR and ITO arrays are to be used
later as subscripts and since these elements c o n t a i n the i n ­
teger O 9 each of the elements are incremented b y I,
Uext9
subroutine TROOP? is called to determine and store the .appli­
cable 2-trees of the input network.
The A L P H and BETA a r ­
rays are set equal to the A L B A R and BitBAR arrays r e s p e c t i v e ­
ly,
Subroutines S T U T H Z 9 E V L V T R , SEUSlT and E V O B are then
called in turn, and it is seen that at this po i n t the value
of the objective f u n ction of the ne t w o r k init i a l l y input
has b e e n ■computed along w i t h the element values of the n e t ­
work,
V O B A R is set equal to t h e current value of the ob ­
jective f u n c t i o n and an output h e a d i n g for the initial n e t ­
w o r k is printed.
Subroutine DEPOT is next called to print
—82—
out the computed parameters of the initial network.
The r e ­
m a i n i n g p o r t i o n of the m a i n p r o g r a m is concerned with the
improvement of the initial n e t w o r k "by the method of steep­
est d e s c e n t o
As each improvement to the net w o r k is accom­
plished, the value of the r e s u l t i n g objective function is
typed out if sense switch I is turned on.
The pr o g r a m is
terminated automat i c a l l y if the objective f u n c t i o n im p r o v e ­
ment obtainable for a given set of gradients is less t han
0 , 0 5 % of its previous value.
If it is desired to terminate
the p r o g r a m before this c o n d i t i o n is f u l f i l l e d , sense switch
3 is turned on.
U p o n termination,
subroutine DMPOT is c a l l ­
ed to print out the parameters of the final improved n e t ­
wor k and the program,is terminated.
The method of p r e p a r i n g the input data will n o w be ex­
plained,
R e f e r to the expla n a t i o n of subroutine RDIET for a
def i n i t i o n of the variables to be discussed below.
The
first input card is used to input the 7 integer variables
ETOD9 ETELP9 ETO9 I B 9 I J 9 KB and ETELM,
These are input as
7-two digit, numbers w h i c h o c cupy the first 14- columns of
the first input card.
ETTYP array,
Card n u m b e r 2 is used to input the
A m a x i m u m of 30-two digit numbers m a y occupy
the first 60 columns of this card.
The same format is used
to input the IPR ar r a y on card numb e r 3 and the ITO array on
card n um b e r 4-,
The first 50 columns of the next card are
-83used to input five F l O „2 numbers corresponding to the values
of W O D 9 E R D 9 E C S 9 W O E and W E E r e s p e c t i v e l y <>
N e x t 9 the E E L M
a r r a y is input b y p l a c i n g as m a n y as 8 F l O «2 values on a
card u n t i l the array is i n p u t .
This same format is used to
input the E K 9 E Y 9 S P 9 A L B A E and BT B A B arrays in turn,
0.
EXAMPLE NETWORK„
A n example is n o w p r esented to clarify the p r e p a r a t i o n
of the input data and to d e monstrate the accomplishments
wh i c h are possible w i t h the program=
The t r a n s f e r function
selected is the basic low-pass B u tterworth fourth-order
transfer f u n ction
T = ----------- ;
-- ---- ----------------V
ZL
Z
O
8^ + 2 .61328^+ 3 .4-14-3s^+2.6132s+!.
(5, 11 )
It is k n o w n that the a p p l i c a t i o n of the synthesis procedure
of S e c ti o n IV.D. will result in the 10 element network c o n ­
fi g u r a t i o n of Figure 5°2.
The elements must be placed in the
'position s shown in order that the computer p r o g r a m operate
properly.
Node and element identifying numbers are properly-
assigned a n d 9 since there are no elements connected across
the input and output terminals of the network, dummy (aux­
iliary)
elements are added to the network and assigned the
i d e n t i fi c a t i o n numbers 11 and 12.
This is done so that, the
quantities IB and K B will have values other t h a n zero.
An
-84-
DUMMY
IO-L
DUMMY
Figure 5»2*
5.677F.
C o n f i g u r a t io n of example n e t w o r k »
6 o 313F o -
7.324F.:
VO L T A G E
FOLLOWER
5.567Fo -p. 2.732F,
Figure 5»3»
rp _
v
.363F,
Optimized net w o r k for r e a l i z a t i o n of
____ »
s 4 + 2 o6 1 3 2 s 5 + 3 o4-14-3s 2 + 2«6132 s +1°
-85examinat i o n of Figure 5»2 enables the d e t e r m i n a t i o n of the
followin g input variables.
NOD=5
NELP=IO'
N0=4
IB=Il
IJ=5
KB= 12
NELM=IE
NTYP(T)=O
N T Y P (2)=0
N T Y P (3)=0
N T Y P (4)=0
NTYP(5)=1
NTYP(G)=I
N TYP(7)=1
NTYP(S)=I
NTYP(9)=1
NTYP(IO)=I
NTYP(Il)=O
NTYP(IE)=O
IFB(I)=I
I F E ( E ) =2
IPE(3)=3
IFE(4)=4
IFE(5)=1 .
I F E ( G ) =2
IFE(7)=3
IFE(S)=3'
IFE(9)=4
IFE(IO)=4 .
IFE(Il)=I
IFE(12)=5■
ITO( I ) =2
IT0(2)=3
I T O (3)=4
IT0(4)=0
ITO(5)=5
ITO(G)=O
IT0(7)=5
ITO(S)=O
ITO(9)=5
ITO(IO)=O
ITO(Il)=O
ITO(IE)=O
The wei g h t i n g functions were i nitially set to I with the
thought that these quantities could be altered after running
t h r o u g h the c o m puter p r o g r a m once in order to obtain a feel
for the magnitudes of the various •elements of the optimizet i o n parameter.
It was found desirable to increase WEB and
W E E to 100 and 10 respectively bec a u s e of the c h a r a c t e r i s t i c­
al l y low value of E D and E E in c o m p arison w i t h the other opti
m i z a t l o n p a r a m e t e r elements.
L i k e w i s e 9 the W K 1s and W T 1S
were increased to 10 and 50 respectively.
The rema i n i n g quantities w h i c h are to be input are set
as f o l l o w s :
-86-
SP(I)=I0 .
SP(2)=206132
SP(3)=3o4143
SP(4)=2o6132
SP(5)=10
ABBAR(I)=.0500
ALBAE(2)=o2500
ALBAR(3) = o3500
ALBAR(4)=4o0000
BTBAR (.1)= 01000
BTBAR(2) = 03000
These quantities are punched at their p r o p e r input card
locations and are shown on lines 3 through 19 on page 122
of A p p e n d i x I I 0
below.
1.
The results are interpreted and discussed
See pages 123 and 124 for the printout.
The results are printed out in two steps.
Pirst9
the parameters of the initial net w o r k are printed 9
and the n the parameters of the final network.
2.
The first item considered is the pole and zero l o ­
cations of tfte input admittance
It is shown
that the initial zero locations of 0 .059 0 .23 $
Oo 35 9, and 4.00 are shifted to the final zero lo ­
cations of 0.0644$ 0.4432$ I .04385 and 1.5467.
The poles are shifted f r o m 0.1000$ 0.3000$
2.2132 to 0 . 2 8 2 3 9 0.8960$
and 1.4348.
and
Note that
these quantities are the distances fro m the origin
to the l o c ation of the poles or zeros on the n e g a ­
tive real axis and not their actual location.
3.
The next item considered is the coefficients of the
input admittance
The results show that the
input admittance is shifted from 57 .143s^"+265.714s^
—87—
+155<>286s 2 + 2 7 <.1 0 7 s + 1 /1 5 oO61 s 5 +39<»358s 2 + 13 o7 8 5 s + 1 o
t o 2 1 „ 7 1 6 s Z|"+ 6 7 .2 7 7 s 5 +64.02 3 1 s 2 + 19 o3 9 9 s + 1 / 2 ,7 3 5 s 5
+7 o 1 9 9 s 2 +5 o 353s + 1 o
4».
N e x t 9 the sensitivities of the coefficients of
P(s) wit h respect to variations in the network
element values are printed o u t «
There are too
m a n y of these quantities to be listed individually
hereo
As an example $ note that the optimization
process changes the sensitivity of coefficient 3
wit h respect to element 10 from Q 0534 to 0 »309 =
5o
Note that the capacitance d i s t ribution p arameter
is reduced f rom 1160 to 33 °87 <> the r e s i stance d i s ­
t r i b u t i o n p a r a m e t e r f r o m 0*0861 to 0.00622, and the
-total capac i t y of the net w o r k is reduced from 89°39
to 2 7 o9 8 o
The capacitance and resistance ratios
are reduced f rom 308.74 to 20.18 and 4*337 to 1*381
respectively*
6.
The sensitivities of the coefficients of P(s) w i t h
respect to the voltage follower gain are next c o n ­
sidered*
Notice that the sensitivity of c o effi­
cient 2 to a one per c e n t vari a t i o n in the voltage
follower gain is decreased from 0 *207% for the
initial net w o r k to 0 *0926% for the final network*
Finally, the s e n s i t i v i t y of the coefficients of
P(s) w i t h respect to the voltage f o l lower input
—88—
conductance are c o n s i d e r e d o
Hotice that the sen-n
sitivity of c o e f f i c i e n t 3 to a voltage follower
w i t h an input c o nductance equal to one percent of
the r e c i procal of the total-resistance of the p a s 'sive p o r t i o n of the n e t w o r k is decreased from
0 .395 # to 0,0719#.
7.
At.'this point, the sensitivities and other ele-ments of the opt i m i z a t i o n para m e t e r have "been
.printed o u t . . The next item' to he considered is
the element values of the associated n e t w o r k s .
Notice that the 35*778 farad capacitor of the i n i ­
tial n e t w o r k has b e e n r educed to 6.312 farads in
the final network.
8.
The final item considered is the value of the o p ­
t i m i z a t i o n parameter.
Notice that it has b e e n d e ­
creased f rom an initial value of 1884.5 to 126 .58 .
This,
of c o u r s e , m a y be considered a worthwhile i m ­
p rovement .
The final optimized n e t w o r k is shown in -Figure 5»5»
Several observations and recommendations w i t h regard to the
optimi z a t i o n computer p r o g r a m w i l l n o w be made.
First of
all, it was observed that the optimization p a r a m e t e r c o n ­
verged to the s a m e .value for several starting points.
This
s u g g e s t s , a l t h o u g h it c e r t a i n l y does not p r o v e , that the
—89—
m i n i m u m obtained is a global m i n i m u m and not a local m i n i ­
mum.
A l t h o u g h the opti m i z a t i o n p a r a m e t e r converged to the
same value for each starting point, the rate of convergence
is g r e at l y influenced b y the starting point.
It is felt
that it would be advantageous to ca r r y out an initial search
technique w h i c h would r a n d o m l y generate a n u m b e r of start­
ing points.
■That n e t w o r k w h i c h results in the lowest value
of optim i z a t i o n p a r a m e t e r would t h e n be selected as the i n i ­
tial n e t w o r k of the o p t i m i z a t i o n program.
A f a ctor w h i c h was found to influence g r e a t l y the rate
of convergence is the w e i g h t i n g factors applied to the g r a ­
dients .
In the present case, it was found most advantageous
to weight the gradients b y the magnitude squared of their
associated independent parameters.
tions,
Z o r other c o n f i g u r a ­
it m a y be desirable to change this to a weig h t i n g of
the gradients b y the m agnitude or some other f u n c t i o n of its
associated independent parameter.
A n y desired change is
carried out b y altering statements 3 and 4 of subroutine
EZGEAD.
VI,
P R A C T I C A L C O N S I D ERATIONS
S e c t i o n A of this chapter will consider the effect of
the departure of the voltage f o l lower from the ideally
assumed conditions of zero input admittance and u n i t y v o l t ­
age gain.
S e c t i o n B will he concerned w i t h the design of
voltage follower networks and S e c t i o n C will p resent the
experimental results of some example n e t w o r k s .
A.
T H E NON-IDEAL- V O L T A G E F O L L O W E R
As s een in Eigure ! . I 9 if the voltage f o l lower p o s ­
sesses a finite input admittance Y ^ 9 this admittance m a y
he considered to appear across the input terminals of the
p a s sive RC network.
If Y
is assumed to consist of a con-
ductance Ga in p a r allel w ith a susceptance Ca , t hen Y a =Ga
+Ga s and the characteristic equation D(s) of the RG net w o r k
is unchanged.
T h u s 9 the q u a ntity N ^ - N ^
the quantity G D+C Ds:.
is increased h y
One w a y in which the effect of Y
m a y he t a k e n into account is the following:
I.
Synthesize a p assive n e t w o r k for the transfer
f u n ction T =A(s)/P(s) as t h ough Y & does not exist.
One of the results of this step is the d e t e r m i n a ­
tio n of the admittance scaling factor K and the
characteristic p o l y n o m i a l D(s). of the network.
2.
N o w repeat the synthesis procedure for the trans-
-91fer f u n c t i o n T = A ( s ) / P ' (s) where P 1(s)=P(s)
V
-G-a B(s)-C^sD(s) 0
If the K and the D(s) of the
n e w n e t w o r k are the same as those of the p r e v i o u s ­
ly synthesized network,
Ty=A(s)/P(s)
t hen the t r a n s f e r function
is o b t a i n e d .w h e n the second network
is used in c o n j u n c t i o n w i t h a voltage follower
whose input admittance is Y^=G^+C^s.
To take account of,a v oltage follower g ain k other t h a n
u n i t y reca l l that the v oltage trans f e r funct i o n T^ is from
E q u a t i o n (1.5)
12
(6 .1 )
N 11 kH13
This equation m a y be interpreted in the t e r m i n o l o g y of the
current dis t r i b u t i o n dia g r a m of Figure 4.1 As thou g h the
quantity kE(s) is subtracted fro m the current of terminal I
to determine P(s) instead of the quantity E(s).
An alter­
nate current di a g r a m r e p r e s e n t a t i o n is shown in Figure 6.1.
If the q u a ntity P'(s) is set equal to
( P ( s ) -(l-kE(s)) and
the usua l synthesis procedures are applied, t h e n the r e s u l t ­
ing passive network, w h e n used in conjunction w i t h a vol t a g e
follower whose gain is k, will p roduce a t r a n s f e r function
Ty=A(s)/P(s).
B.
V O L T A G E F O L L O W E R CIRCUITS
- 9 2 -
P' (s) + (l-k)E(s)
+kE(s)
Figure 6.1.
Current d i s t r i b u t i o n diagram for non- u n i t y
voltage follower gain.
+30 v
2N2222
Figure 6.2.
2N2222
Voltage follower of Reference $2
-93This s ection is concerned w ith the p r o b l e m of d e s i g n ­
ing p r ac t i c a l voltage f o l lower c i r c u i t s e
mentioned 9 the cathode,
As p r e v i o u s l y
emitter or source f o l lower circuit
offers a simple ap p r o x i m a t i o n to a voltage follower*
Ex­
cept for the less critical applications, however, these
circuits have b e e n found to depart too far from the ideal
to p e r f o r m satisfactorily.
It will be shown that voltage
followers which v e r y clo s e l y approximate the ideal m a y be
designed.
The first thought that occurs is that a two stage am ­
p l i f i e r u s i n g the field effect t r a n sistor as an input stage
would make an ideal network.
Analysis of such a network r e ­
veals, h o w e v e r , that while extr e m e l y hig h input impedances
are a t t a i n a b l e , a g a i n of the order of 0.98 is all that is
possible.
A r e v i e w of the literature does not immediately
reveal a simple s a t i sfactory two stage voltage follower.
The two stage voltage follower to be presented here, however,
is one which is adopted from the literature.
circuit
(32) is shown in Figure 6.1.
The original
This circuit has an
input impedance of ap p r o x i m a t e l y $ 0,000 ohms and a gain of
a p p r oxi m a t e l y 0.998.
The altered network shown in Figure
6.2 has a m uch h i g h e r input impedance and a g a i n m u c h closer
to unity.
This improvement is accomplished b y substituting
a field effect t r a n sistor for a conventional transistor.
-94—
Notice also that the c i r c u i t r y of the altered net w o r k is
g r e atly simplified and that the low frequency response
has b e e n extended d o w n to d-c„
operati o n of the two networks,
To gain insight into the
consider the system d i a ­
gra m of Figure 6„3 w h i c h is valid for b o t h c i r c u i t s e
and Ag r epresent the v oltage gains of the first and sec­
ond stages r e s p e c t i v e l y e
Note that the first stage of
each amplifier is operated in the common base c o n f i g u r a ­
t i o n but that the load impedance for the stage in Figure
6 ®I is low in co m p a r i s o n w i t h that for the corresponding
stage in Figure S e 2»
Since the voltage gai n of the c o m ­
m o n base config u r a t i o n is p r o p o r t i o n a l to load i m p e d a n c e 9
A ^ . for the first n e t w o r k is m u c h lower t han A^ for the
second network.
Straight forward analysis of Figure 6.3
show that
A =
®out
'
I
CD
ein
A 1A2
A ^ A g +l—Ag
zi
(6e2)
(6e3)
‘ 1 In
where
is the emitter-base impedance of the input stage.
Note that Ag is essentially the voltage g a i n ■of an emitter
follower and m a y be considered a p p r o x i m a t e l y 0.93=
Typical
values of A^ are 35 and 500 for the circuits of Figures 6.1
and 6.2 respectively.
quations
(6 .2 ) and
S u b s t i t u t i n g typical values into e-
(6 .3 ) r esults in A = O . 9985 and A ^ = 23 g300
IOM
2N4-04
6+
e
X - 1004
X / -
in
o—
10K>
_J=
'out
— =0
+10 v,
Figure
S„3 o
Altered network,
out
Figure 6.4
S y stem dia g r a m for networks of Figures 6.2 & 6.3°
-96—
ohms for Figure 6 d .
The corresp o n d i n g results for the
n e t w o r k of Figure 6.2 are A ~ 0.9999 and Ziri ^ $$0,000 ohms.
T h u s , improvement has b e e n accomplished b y a l l owing the
voltage g ain
to a p p roach the m a x i m u m attainable from a
conventional t r a n s i s t o r .■
A three stage Voltage follo w e r w h i c h follows n a t u r a l ­
ly fro m the net w o r k of Figure 6.2 is shown in Figure 6 .5 .
Actually, the net w o r k contains fou r t r a n s i s t o r s .
Since one
of the transistors only b ehaves as a current source to p r o ­
p e r l y bias the first stage, the net w o r k is classified as a
three stage network.
The result accomplished b y this n e t ­
wor k is to make Ag of Figure 6.3 approach muc h closer to
unity.
Notice that the low f r e q u e n c y response of this n e t ­
wo r k also extends to'd-c.
This net w o r k will be used in the
experimental v e r i f i c a t i o n of S e c t i o n C.
0.
EXPERIMENTAL R E S U L T S „
This section will present the experimental results of
'three transfer functions realized b y the methods of this
report.
The three transfer functions include a second-or­
der band-pass t r a nsfer function,
a fourth-order low-pass
tran s f e r f u n ction and a fou r t h - o r d e r band-pass transfer
function.
These transfer functions in normalized form are
-97-
-20 v«
15K<
2N3114
2N4-04
O+
X-1004
h
°— I
+o
'in
9 -20 Vc
'out
IOK
8.2E
6+10 v,
2N3114
IOK
6 -20 Vc
Figure 6.3«
Three stage v oltage follower net w o r k
-98respectivelj
T = ■ - 2 °-^ -.s + O o l s +1
T = “TT
v
(GA)
%
'
: p
'
s%2o6132s:)+3o4145s^+2e6132s+l
.
(6 o5)
2
V
v
-
%
----------------- A
s
—
:-------------------
sAlo414-s^+3s^+lo4-14s+l
C s - 6
)
The tran s f e r f u n c t i o n (6.4) corresponds to a band-pass f u n c ­
t i o n whose center frequency is I r a d i a n per second and whose
pass band is 0.1 r a d i a n p e r second.
E q u ation (6.5) is the
low-pass f o u r t h-order m a x i m a l l y flat
(Butterworth) charac-
-
teristic whose cutoff f r e q u e n c y is one r a dian p e r second.
Equa t i o n (6 .6 ) is the fou r t h - o r d e r m a x i m a l l y flat band-pass
transfer f u n c t i o n whose center freq u e n c y is one r a dian p e r
second and whose pass— band is one r a d i a n p e r second.
This
means that the u p p e r cutoff f r e q u e n c y is 1.619 radians p e r
second and the lower cutoff f r e q u e n c y is 0.619 radians p e r
second.
A n explicit synthesis p r o c e d u r e for the second-order
b a nd-pas s t r a nsfer f u n ction was not presented in Chapter IV.
One of the networks of S a l l e n and K e y m a y be utili z e d or %
more simply, the current d i s t r i b u t i o n d iagram for the f a m i l ­
-99iar twin-T EC net w o r k m a y be computed as shown in Eigure 6 *6 ,
r
If
0 9 kjj, and
are chos e n 0.8, 1 . 6 7 9 and I respectively,
a n e t w o r k w i t h p o s i t i v e elements results which can be c o n ­
verted to the form of Figure 6.7 for use in c o njunction w i t h
the
(ECYE) to realize the t r a n s f e r function of (6=4),
This
n e t w o r k is shown in Figure 6 .8 =
The fou r t h - o r d e r low-pass f u n c t i o n of E q u a t i o n (6=5) is
the same as that used in the c o m puter o p t i m ization example
of Chapter Y=
The n e t w o r k used in the present case is one
that was obtained f r o m the c o m puter pro g r a m b e fore the c o m ­
p u t e r p r o g r a m was co m p l e t e l y debugged.
The n e t w o r k is, not
an opti m u m one although it turns out to be v e r y n e a r l y op ­
timum.
The p r ocedure of IY E m a y be applied in the case of the
transfe r f u n ction of E q u a t i o n (6 .6 ).
A p p l i c a t i o n of this
p r ocedur e results in the n e t w o r k of Figure 6 .6 .
wor k realizes
This n e t ­
(6 .6 ) w i t h i n a constant multiplier.
The realized networks will be impedance and frequency
scaled to typical values.
The circuit for the second-order
band-pas s transfer f u n c t i o n was impedance scaled to $ 00,000
ohms and freq u e n c y scaled to 1$0 cycles p e r second.
sultant
(ECYF) co n f i g u r a t i o n is shown in Figure 6.7«
The r e ­
— 1 0 0 -
Figure 6 .6 .
Current d i st r i b u t i o n for tw i n - T network.
R(s)=s +1
Figure 6.7
Network for r e a l i z a t i o n of T.
0 .1s
s +.Is +1
-101-
<66t>6K
600K
,00443 F.
o-- W V -
VOL T A G E
180 K
F O L LOWER
ydut
.0296 F,
,0177 S’■6 0 K
Figure 6.8./
(ROVF) for r e a l i z a t i o n of second-order bandpass t r a nsfer f u n c t i o n denormalized to 300)000 ohms and
150 cps.
,0104 F.
- W
+
-
,0132 F.
,0104 F,
286E
-AZW-
m -
"VOLTAGE"
FOLLOWER
,012 F,
-4,440pF.
780pF,
"out.
Figure 6.9«
(EGVF) for r e a l i z a t i o n of fourth-order low-pass
trans f e r f u n c t i o n denormalized to I m e gohm and 89 cps.
- 1 0 2 -
The circuit for the f o u r t h-order low-pass transfer
func t i o n is impedance scaled to I m e g o h m and freq u e n c y
scaled to 85 cps»
This denormalized n etwork is shown in
Figure 6 .8 .
The circuit for the fou r t h - o r d e r hand-pass transfer
func t i o n is impedance scaled to 500,000 ohms and f requency
scaled to 5 9 »8 cps,
The denormalized network is shown in
Figure 6.9.
The networks of Figures 6.7, 6 .8 , and 6.9 were built
and tested us i n g the voltage f o l lower circuit of Figure
6 .4-0
A com p a r i s o n of theor e t i c a l and experimental results
for these networks are shown in Tables 6.1 thr o u g h 6.5°
- £ 0T114.3%
,1133 F.
2?. 6K'
0-- W v
Hf
,0307 Fe
if
292K
— W\A-
VOLTAGE
.01362 F,
FOLL O W E R
<415%
,00884 Fe
W W
,1359 %.
% 85 .6E
17 .35%
'out
.0862 F,
Figure 6.10.
(RCVF) for r e a l i z a t i o n of fourth-order band-pass transfer function
denormalized to 500^000 ohms and 39=8 cps.
104
T A B L E 6.1
FREQUEinICY-ATTENUAT IOM TAu UL a T IOiQ OF EXPERIMENTAL AiMO
THEORETICAL VALUES FOR NETWORK OF FIGURE 6. ti
FREQUENCY
(CPS)
17.0
25.0
38.0
55.0
75.0
96.0
109.0
119.1
125.1
130 =5
134.8
138.5
140.1
141.7
142.6
143.3
144.2
144.9
145.8
146.8
147.9
150.2
152.6
154.0.
154.9
155.8
156.6
157.5
158 =3
159.3
161.1
163.0
167.3
172.8 .
180.4
189.8
20 7.9
236.1
303.1
4u9.2
618.2
939.0
1476.0
DB
ATTENUATION
(EXPERIMENTAL)
40.00
36.00
32.00
28.00
24.00
20.00
.17.00
14.00
12.00
10.00
8.00
6.00
5.00
4.00
3.50
' 3.00
2.50
2.00
1.50
I .00
.50
0.00
.50
I .00
1.50
2.00
2.50
3.00
3.50
4.00
5.00
6.00
8.00
10.00
12.00
14.0
17.00
2 0.00
24.00
28.00
32.00
36.00
4 0.00
DB
ATTENUATION
(THEORETICAL I
38.80
35.31
31.35
27.46
23.54
19.35
16.35
13.55
11.56
9.44
7.46
5.50
4.57
3.61
3.06
2.63
2.10
1.69
1.21
.74
.33
0.00
.48
1.06
1.50
1.97
2.41 .
2.90
DB
ERROR
1.198
.680
.6 46
.531
.458
.649
•6 46
.446
.438
.551
.530
.497
.424
.387
.436
.361
.398
.300
.285
.259
.167
-.003
.015
- .0 6 2
-.002
.024
.089
.093
3.34
.152
3.89
.109
«163
4.83
5.76
7.62
7.57
I 1.69
13.72
16.54
19.49
23.63
27.47
31.77
35.70
39.77
.237
.3 77
.427
.303
.277
.452
.500
.311
.328
.223
.291
.229
105
TABLE 6.2
FREQUENCY-ATTENUATION TABULATION OF EXPERIMENTAL AND
THEORETICAL VALUES FOR NETWORK OF FIGURE 6. 9
FREQUENCY
(CPS)
50.0
65.7
71.8
75.9
79.5
82.4
84.9
87.2
89.7
93.8
97.5
105.0
113.0
120.0
128.0
140.0
153.0
172.0
195.0
222.0
267.0
DB
ATTENUATION
IEXPERIMENTAL)
DB
ATTENUAT ION
ITHEORETICAL)
‘ DB
ERROR
0.00
.50
a06
.52
1.00
1.47
-.062
-.021
1.00
1.50
2.00
2.50
3.00
3.50
4.00
5.00
6.00
8.00
10.00
12.00
14.00
17.00
20.00
24.00
28.00
32.00
3 6.00
2.00
2.50
2.99
— .001
.025
-.002
-.004
.009
3.47
.022
4.04
5.05
6.01
8.07
10.31
12.24
14.38
17.41
-.045
— .050
-.017
-.077
-.316
20.46
24.50 .
28.85
3 3.35
39.76
-.248
-.384
-.416
-.461
-.504
-.855
-1.356
-3.767
106
TABLE 6.3
FREQUENCY-ATTENUATION TABULATION OF EXPERIMENTAL AND
THEORETICAL-VALUES FOR NETWORK OF FIGURE 6.10
FREQUENCY
(CPS)
4.9
5.9
7.6
9.3
11.3
13.3
15.1
16.4
13.1
19.7
21.3
22.2
23.1 '
23.8
24.3
24.9
25.6 ■
26.5
27.5
29.0
39.8
50.8
54.6
56.9
59.6
61.5
63.0
65.0
66.6
69.8
72.7
79.1
86.1
94.1
102.5
118.0
136.0
166.0
206.0
255.0
319.0
401.0
--
DB
ATTENUATION
(EXPERIMENTAL)
DB
ATTENUATION
ITHEORETICAL I
DB
ERROR
35.50
32 .00
28.00
24.00
20.00
17.00
14.00
12.00
10.00
8.00
6.00
5.00
4.00
3..50
3.00
2.50
2.00
1.50
I.00
.50
0.00
.50
1.00
I .50
2.00
2.50
3.00
3.50
4.00
5.00
6.00
8.00
10.00
12.00
14.00
17.00
20.00
24.00
28.00
32.00
36.00
40.00
36.12
32.77
28.12
24.29
20.45
17.06
14.30
12.42
10.10
8.06
6.19
5.21
4.31
3.67
3.24
2.77
2.2 7
1.72
1.22
.68
0.00
.24
.68
1.08
1.68
2.17
2.59
3.19
3.68
4.71
5.65
7.67
9.71
11.81
13.77
16.87
19.83
23.79
27.90.
31.84
35.88
39.95
-.623
-.777
-.124
-.296
-.451
— a069
— .300
-.424
-.107
-.069
-.190
-.218
-.318
-.175
-.249
-.275
-.278
-.226
-.228
— aI87
0.000
.251
.315
.414
.316
.325
.404
I «308
.310
.284
.345
«326 ;
.282
.180
»220
■,164:
-
m
t m
. .041:
YIIo
A,
CONCLUSION AND RECOMMENDATIONS
CON C L U S I O N
This thesis has described and demonstrated the f e a s i ­
b i l i t y of u t i l i z i n g the grounded voltage f o l l o w e r as the
active p o r t i o n of an active RC network.
Chapter III es­
tablished the n e c e s s a r y and sufficient conditions for the
r e a l i z a t i o n of the voltage t r a n s f e r function of such a d e ­
vice
(the R C Y F ),
cedures for the
Chapter IY described some synthesis p r o ­
(RCYF) and Chapter V demonstrated that the
digital computer m a y be u t i lized to optimize the synthe­
sized networks.
Finally,
Chapter YI gave a demons t r a t i o n
of several transfer functions realized b y the
(RCYF) active
n e t w o r k wh i c h indicates the p r a c t i c a l value of the network
b y showing excellent agreement b e t w e e n predicted and actual
results.
Indeed, the results of b o t h the theoretical and
experimental investi g a t i o n lead one to conclude that the
(RCVF) is a h i g h l y u s eful and p r a c t i c a l network,
B.
RECOMMENDATIONS.
This study has resulted in a number of possibilities
for future wor k b e c o m i n g apparent.
Recommendations for
future w ork are listed below.
I.
The first,
and p r o b a b l y the most o b v i o u s , recom-
-108m e n d a t i o n that comes to mind is the further i n v e s ­
t i g a t i o n of synthesis t e c h n i q u e s „
Although Chap­
ter III showed that a t r a nsfer f u n ction which
meets specified n e c e s s a r y conditions m a y always
he realized 9 the synthesis of optimum networks is
another matter.
Thus $ the further i n v estigation
of networks for specific classes of t r a nsfer f u n c ­
tions is indicated.
A n area wh i c h was not touched on in this i n v e s t i g a ­
t i o n but appears inter e s t i n g is that of the p o s s i ­
b i l i t y of v a r y i n g t r a n s f e r functions b y switching.
The concept of the current dis t r i b u t i o n diagram
offers a great deal of insight into the problem.
A simple example of the possibilities for s w itch­
ing m a y be seen b y examining the n e t w o r k of Figure
6.7'.
This n e t w o r k is used in conju n c t i o n wit h the
voltage f o l l o w e r to realize the t r a n s f e r function
o
T^= a^s/pgS +a^s+pQ w h e r e a^= 0 »l and Pg=PQ=Io
Notice that a^ m a y be halved -m e r e l y b y splitting
the 2- ohm r e s istor c a r r y i n g the current 0 .1s into
two 4 -ohm resistors and switching one of them over
to the terminal associated w ith the current E(s).
The result of this o p e r a t i o n is to cut the b a n d ­
w i d t h of the band - p a s s f u n ction in half.
The above
example represents onl y one p o s s i b i l i t y of a n u m b er
-109w h i c h r e a d i l y come to m i n d „
3<,
The next r e c o m m e n d a t i o n concerns the o p t i mization
of networks b y means of the digital computer.
It
is obvious that n e t w o r k optimization is an ex­
t r e m e l y c o m p l e x p r o b l e m and that it has only b e e n
touched u p o n in this investigation.
t i m i z a t i o n method
O n l y one o p ­
(the method of steepest descent)'
has b e e p attempted on a single class of networks.
The opti m i z a t i o n att Ciipt was successful from the
standpoint that significant improvement was a c c o m ­
plished, but the p r o b l e m is important enough to d e ­
serve a great d eal,of further study.
It is s p e c i ­
f i c a l l y r e commended that the approach of continousIy equivalent networks
(20 ,53 ) be investigated.
This a p p r o a c h is not only concerned w i t h the o p t i ­
m i z a t i o n of a gi v e n n e t w o r k configu r a t i o n but wit h
the ge n e r a t i o n of n e w configurations.
4*
An o t h e r important area for further invest i g a t i o n is
that concerned w i t h p r a c t i c a l considerations.
Such
considerations would include temperature effects
and compensation, component selection, voltage
follower circuitry, the max i m u m impedance level of
a network, the m i n i m u m freq u e n c y a l l o w a b l e , e t c .
A n o t h e r important ite m u n d e r this h e a d i n g would be
the effect of joining two or more
(EGVT) networks
-IlOr-
togetihero
.These n e t works could be connected in
cascade or. some other desirable configuration.
5.
The final r e c o m m e n d a t i o n concerns the driving
point impedance of the
(ECVF)»
This thesis has
done n o t h i n g w i t h this p o s s i b l y important subject.
P r e l i m i n a r y i n v e s t i g a t i o n has indicated that
driving point f u n c t i o n synthesis u t i l i z i n g the
v oltage f o l lower m a y be more difficult t h a n t r a n s ­
fer f u n c t i o n synthesis.
It is felt that the p r o b ­
lem deserves a separate paper.
\
PLEASE NOTE:
This is not original
copy. Appendix contains
pages with extremely fine
print. Filmed as received.
University Microfilms, Inc.
APPENDIX I
O P T I M I Z A T I O N C O M PUTER P R O G R A M ' L I S T I N G
n
ZZJOS 5
ZZFOR 5
*LDISKCH0NE
D I M E N S I O N L D O O J , N T Y P t 3 J I , V T R ( S O O ) , S E L M I 1 1 , 3 0 ) , S P ( 1 1 ) » W E L M ( 1 1 ,30 )»
I E L E M t 3 0 ) , D T ( I l ) , W K ( I l ) , W Y t 1 1 ) , A L B A R ( I O ) , B T B A R t 1 0 ) , G R A D ( Z O ) , I F R (30)
2 , 1 T O ( 3 0 ) , I T R ( I O O O ) , F N C O ( I l ) , F D C O ( I l ) , B T ( I l ) ,A L P H ( 1 0 ) , B E T A ( 1 0 ) , F T C O
3( 11 )
C O M M O N N O D ,N TT »N E L P *N T S , N O »N S P ,L D »NT Y P »V T R »S E L M *S P , W E L M , W C D , W R D *W C
1 5 ' » W C R , W R R , E L E M , D T , W K ,W Y , N VB ,A L B A R , B T B A R ♦ V 0 3 A R , G R A D , G R M A X , N G R I , S T P ,
2 N 0 M I , N O M T , N C O N , IJ , I B ,K B , I F R , IT O , N E L M , N IT R , IT R , N F A R ,F N C 0 , F D C 0 ^ B T ,A L
3 P H , B E T A , F T C O , C M A X ,CM IN ,R M A X ,R M IN , C D ,R D , C S U M , V O B J
CALL RDIN
DO I 1=1,NELM
IFR(I)=IFR(I)+!
1 ITO(I)=ITO(I)+!
CALL TRCOM
D O 2 I = I ,NO
2 A L P H ( I ) = A L B A R t 11
DO 3 I=I,NOMT
3 BETA(I)=BTBARt I)
C A L L L IN K ( C H T W O )
END
IS
»WCR»WRR»ELEM,DT,WK »WY,NVB,ALBAR,8T3A R ,VOBAR,GRAD,GRMAX,NGRI»STP,
2N0MI,NOMT,NCON,IJ ,IB,KB »IF R ,ITO»N£LM,NITR,ITR,NFAR,FNCO,FDCO,BT,AL
3PH,BETA1FTCO,CMAX,CMIN,RMAX1RM IN,CD,RD,C5UM,V0BJ
I CALL DMPOT
CALL LINK(CHFOUR)
END
ZZJ06 5
ZZFOR 5
♦LDISKCHFOUR
D I M E N S I O N L D (30) , N T Y P t 3 0 ) ♦ V P R (500) , S E L M ( 1 1 ,30 I , S P ( I l ) , W E L M ( 1 1 , 3 0 ) ,
1 E L E M I 3 0 ) , D T t 1 1 ) , W K t 1 1 ) » W Y ( 1 1 ) , A L B A R ( I O ) , B T B A R ( I O ) , G R A D ( Z O ) »1 F R (30)
2 , 1 T O ( 3 0 ) , I T R ( I O O O ) , F N C O ( I l ) , F D C O I 1 1 ) , B T t 1 1 ) , A L P H ( I O ) , B E T A t 1 0 ) ,F T C O
3( 11 )
COMMON NO D , N T T, NE LP fN TS iNO , N S P f L D ,NTY P »VTR»SELM,SP»WELM»WCD*WRD»WC
IS »W C R , W R R » E L E M » D T , W K , W Y , N V B ,A L B A R ,B T B A R , V O B A R ,G R A D , G R M A X , N G R I , S T P ,
2 N 0 M I » N O M T , N C O N , IJ , I B , K B , !F R, I T O , N E L M , N I T R , I T R , N F A R , F N C O , F D C O f B T t A L
3 P H , B E T A ,F T C O f C M A X f C M I N ,R M A X ,RM I N ,C D ,R D , C S U M , V O B J
21 S T M U L = I ,
NY =O
N f =o
VORJ=VOBAR
CALL EVGRAD
CALL STCOMP
ZZJOB 5
ZZFOR 5
* L D ISKCHTWO
COMMON NOD,NTT,NELP,NTS,NO,NSP,LD,NTYP,VTR,SELM,SP,WELM,WCD,WRD,WC
IS»WCR»WRR*ELEM»DT,W K »W Y *NVB»ALBAR,8TBAR,VOBAR»GRAD,GRMAX,NGRI»STP*
2N0MI,NOMTtNCON,IJ,IB,KB ,'IFR,I TO,NELM,N ITR,ITR,NFAR.FNCO,FDCO1BT,AL
3P H ,BETA,FTCO♦CM A X ,CM IN,RMAX,RMIN,CD,R D »CSUM»VOBJ
CALL SYNTHZ
CALL EVLVTR
CALL SENSIT
CALL EVOB
VOBAR=VOBJ
PRINT I
I
FORMATtlHl,6H*******26X,15HINITIAL NETWORK,27X,6H**»***)
IF (SENSE SWITCH I) 17,18
17 PUNCH 19
19 FORMAT!6H******,26X,15HINITIAL NETWORK,26X,6H******)
10
CONTINUE
CALL LINKtCHTRI)
22
201
76
202
2
3
I
104
41
106
END
12
ZZJOB 5
ZZFOR 5
♦LDISKCHTRI
D I M E N S I O N L D ( 3 0 ) , N T Y P t 3 0 ) * V T R ( 5 0 0 ) , S E L M t1 1 , 3 0 ) , S P ( 1 1 ) , W E L M t 1 1 , 3 0 ) ,
lELEM( -3 0) ,D T (1 1) , WKt 1 1 ) , W Y ( 1 1 1, A L B A R t 1 0 ) , B T B A R t 1 0 ) , GRAD(ZO)', IFRt 30)
2 , ITOt3 0 ) ,I TR t1 0 0 0 ) , F N C O t 1 1 ) ,F D C O t 1 1 ) , B Tt1 1 ) ,A L P H t 1 0 ) ,B ET At1 0 ) ,FTCO
3( 11 )
CO M M O N N O D, NT T,N E L P , N T S , N O ,N SP ,L D, NT YP ,V TR ,S EL M, SP ,W EL M, WC D,WRDeWC
13
27
35
36
VLAS=VOBJ
STBAR=STP
STBES=O,
STP=STBAR*STMUL
CALL TKSTP
I F t S E N S E S W I T C H 2) 2 0 1 , 2 0 2
TYPE 76,VOBJ
FORMAT(2HTO»E17.8»/)
IF(NCON) 1,2,1
IF(VOBJ-VLAS) 3,1,1
NY=NY+!
STBES=StP
VLAS=VOBJ
STMUL=Z.*STMUL
I F (N F) 1 0 6 , 2 2 , 1 0 6
IF(NF-IO) 10 4, 10 4, 10 6
I F (N Y) 4 1 , 4 1 , 1 0 6
NF=NF+1 ST M U L = O . 5*STMUL
G O T O 22
STP=STBES
CALL TKSTP
D O 12 1 = 1 , NO
ALBARtn=ALPH(I)
D O 13 1 = 1 , N O M I
BTBARtn=BETA(I)
VOBAR=VOBJ
I F t S E N S E S W I T C H 3) 3 5 , 2 7
D E C S = ( tV O R U - V O B J )/ V O R J ) * 1 0 0 0 0 ,
D E C s = A B S F (D E C S )
IFtDECSr-S.) 3 5 , 3 5 , 2 1
P R I N T 36
FORMAT t7H1****** »27X.13HFINAL NETWORK,26X»6H******)
I F t S E N S E S W I T C H I) 3 0 1 , 3 0 2
112
D I M E N S I O N L D (.30) ,NTY Pt 30) , V T R ( S O O ) ,S E L M (1 1 , 3 0 ) , S P ( I l ) , W E L M (1 1 ,30 ) *
IEL EM(30),DT(Il),WKdll,WY(Il),ALBAR(IO),BTBAR(IO),GRAO(ZO),IFR(SO)
2 , I T 0 ( 3 0 ) * I T R ( 1 0 0 0 ) , F N C O I 1 1 ) , F D C O I 1 1 ) , B T ( 1 1 ) , A L P H l 1 0 ) , B E T A ( 1 0 ) ,FTCO
31 11 )
VOBJ=VOSAR
30 1 P U N C H 3 0 3
3 0 3 F O R M A T (6 H * * * * * * , 2 7 X , 1 3 H F I N A L N E T W O R K , 2 8 X , 6 H * * * * * * )
302 CO N T I N U E
CALL LINK(CHFIVE)
END
ZZJ06 5
ZZFOR 5
# L D IS K C H F IVE
D I M E N S I O N L D ( 3 D ) , N T Y P t 3 J ) , V T R ( S O O ) , S E L M t 1 1 , 3 0 ) , S P ( I l ) »W E L M ( 1 1 * 3 0 ) #
I E L E M ( 3 0 ) , D T ( I l ) , W K ( I l ) , W Y I 1 1 ) , A L B A R ( I O ) , B T B A R ( I O ) , G R A D ( 2 0 ) ,I F R (30)
2 , I T O ( 3 0 ) , I T R ( I C O O ) ,FNC.O(ll) , F D C O ( I l ) , B T ( I l ) , A L P H ( I O ) ,BET A! 10) , F T C O .
3( 1 1 )
C O M M O N N O D ,N T T , N E L P ,N T S , N O ,N S P , L D , N T Y P , V T R , S E L M ,S P ,W E L M , W C D » W R D » WC
I S , W C R , W R R , E L E M , D T ,W K , W Y , N V U ,A L B A R , U T 3 A R ,V 0 3 A R , G R A D , G R M A X , N G R I »STP,
2 N 0 M I , N O M T , N C O N , IJ , 13 , K S ,I F R ,-I T O , N E L M , N IT R , I T R , N F A R , F N C O , F D C O , B T ,AL
3 P H , B ET A ,F T C O , C M A X ,CM IN , R M A X ,R M IN ,C D , R D ,C S U M , V O B J
I CALL DMPOT
C A L L EXIT
END
COMMON NOD,NTT,NELP,NTS,NO,NSP,LD,NTYP *VTR,SELM,SP,WELM,WCD,WRD,WC
IS,WCR,WRRfELEM,DTiWK,W Y, NVBfALdAR,UTBAR,VOBAR,GRAD,GRMAXeNGR I,STP,
2NOMI ,NOMT,NCON,IJ,IB,KB,IFR,ITO,NELM,NITR,I TRfNFAR,FNCO,FDCO,BT,AL
3PH,BETA,FTCO,CMAXfCM IN,RMAX,R M INiCDfRD,CSUMtVOBJ
B E T A (N O M I)=S P (N O )
DO I I=IfNOMT
1 BETA(N OM I) =B ET A( NO MI )- BE TA t I)
I=I
IF (A L P H I I )) 2 , 2 , 3
3 I Fl B E T A ! I )- A L P H I I ) ). 2 , 2 , 4
4 . J= I
1= 1+ 1
IF(ALPH(I)-BETAtJ)) 2,2,5
5 IF(I-NO) 3, 8, 3
8 NCON=O
RETURN
2 NCON=I
RETURN
END
ZZJOB 5
ZZFOR 5
# L D I SK
SU BROUTINE DMPOT
D I M E N S I O N L D ( 3 0 ) , N T Y P t 3 0 ) , V T R ( 5 0 0 ) , S E L M ( I I ♦ 3 0 ) , S R ( I I )• W E L M ( I I ,3 0) *
1 E L E M 1 3 0 ) , D T ( I l ) , W K ( I l ) » W Y ( 1 1 ) , A L B A R I 1 0 ) , B T B A R ( 1 0 ) , G R A D ( 2 0 ) , I F R I 30)
113
ZZJOB 5
ZZFOR 5
* L D IS K
SUBROUTINE CONSTR
D I M E N S I O N L D (30) , N T Y P I 3 0 ) , V T R ( S O O ) , S E L M l 1 1 , 3 0 ) , S P ( I l ) , W E L M I 1 1 , 3 0 ) »
I E L E M ( 3 0 ) , D T ( I l ) , W K ( I l ) , W Y ( 1 1 ) , A L B A R t 1 0 ) , B T B A R I 1 0 ) , G R A D ( 2 0 ) , I F R (30)
2, ITOt 30) , I T R ( I O O O ) , F N C O ( H ) ,FDC OI 11) , G T ( I l ) , A L P H I 10) , B ET A! 10') , F T C O
3( 1 1 )
2 , ( T O U T ) » IT R I I J O ) , r N C CI 11 I , FD C O f 11) , B T ( H ) ,A L P H ( 10 ), EETAI 10) , F T C O
3(11)
C O M M O N N O D , N T T f N Z L P , N T S , NvO,.’-ISP,LI/,N T Y P , V T R t S E L M , S P , W E L M , W C C tW R D , W C
IS, W C R ,WRR, E L E M tB T ,WK ,'VY,NVB , A L B A R , B T B A R , V O P A R , G R A D , G R M A X ,NGR I , S TP,
2 N 0 M I ,.NOuT , N CO N, IJ , IG , X B , I F P , I T O ,N E L M , N ITR , I T R , N F A R , F N C O , F D C O , B T , AL
. 3 P H , 3 E T A , F T C 0 , C M A X , C-*. I N tK M A X , R M IN, C D , RB , C S U M tV O B J
D I M E N S I O N M R E l 5 ) , J R C l D ) , V AL ID )
P R I N T 13
H F O R M A T Il H , / , I H , 1 3 X , 1 4 H Z E R 0 L O C A T I O N S ,2 6 X , 1 4 H P 0 L E L O C A T I O N S / )
I
F IS E N S E S W I T C H I) 2 0 1 , 2 0 2
2T1 P U N C H 13
2 '2
CONTINUE
D O 14 1 = 1 , N O N I
I F IS E N S E S W I T C H I) 2 0 3 , 1 4
2 1 3 P U N C H I 5 ,ALPHI I ) ,B E T A I I )
14 P R I N T 15 ,ALPl-H I ) ,BETA! I )
15 F O R M A T I F2 fi,«20 , F 3 9 • 20,)
PRINT 16,ALPH(NO)
16 F O K M A T ( F 2 8 , 2 0 » / )
- P R I N T 17
17 F O R M A T (2 0 X ,3 9 H C 0 E F F IC I E N T S OF IN PU T AD MI T T A N C E Y ( I , I ) , /9X» 2 2 H N U M E R
1 A T 0 R C O E F F I C I E N T S , 1 8 X , 2 4 H D E N 0 M IN A T O R C O E F F I C I E N T S / )
I F (S E N S E S W I T C H I) 2 0 4 , 2 0 5
2 0 4 P U N C H 1 6 , A L P r U NO).
P U N C H 17
205 C O NT IN UE
D O 21 I = 1 , N 0
IF ( S E N S E S W I T C H I) 2 0 6 , 2 1
2 0 6 P U N C H 1 5 , F T C O H ) ,FDCOI I)
21 P R I N T 1 5 , FTCOI I ) ,FDCOI I)
PRINT 16 ,FTCO(NSP)
P R I N T 24
24 F O R M A T I2tiX,23H O P T I M I Z h T ION P h R a M l T E R S / I N , 6 2 H S E N S IT IV I TY O F C O E F F I
ICIENT M WITH RESPECT TO ELEMENT NUMBER J. /1 H )
M= I
J= I
IF (S E N S E S W I T C H I) 2 0 7 , 5 5
207 PUNCH 16,FTCO(NSP)
P U N C H 24
55 D O 27 1 = 1 , 5
M R E ( I I=M
J R E ( I I=J
V A L ( I I = S E L M ( M tJ)
IF(J-NELP) 2 b , 29,29
28 J = J + I
G O T O 27
29 IF(M-NSP) 30,31,31
30 M = M + I
J= I
27 CO NT IN UE
31 IF (I -S ) 3 2 , 3 3 , 3 3
32 D O 3 5 L= I ♦5
M R l (L ) = O
JRE(L)=O
35 V A L ( L ) = O .
33 P R I N T 3 6 , 1 M R E ( I ) » J R C ( I ) , 1 =1 ,5 )
36 F O R M A T ( 3 X » 5 ( 2 H M = , I 2 » 3 H » J = » I 2 » 7 X ) I
P R I N T 3 7 , ( V A L I D , 1 = 1,5)
37 FO RM AT(5E16.5,/)
I F t S E N S E S W I T C H I) 2 0 8 , 2 0 9
11 2 F O R M A T (2 3 H N E T W O R K E L E M E N T V A L U E S )
I F t S E N S E S W I T C H I) 2 1 6 , 2 1 7
2 1 6 P U N C H 112
217 CONTINUE
D O 111 1 = 1 , N E L P
I F t S E N S E S W I T C H I) 2 1 8 , 1 1 1
2 1 8 P U N C H 1 1 3 , I , E L E M t I)
111 P R I N T 1 1 3 , 1 ,ELE Mt 11
11 3 F O R M A T (9 H E L E M E N T ( , 1 2 , 2 H ) = , E l 8.8)
7 7 P R I N T 8 0 ,V O B J
Bv F O R M A T t 2 8 X , 2 4 H 0 B J E C T I V £ F U N C T I O N V A L U E ,Z , 2 8 X , E 2 4 . 8 , Z I H » Z , 1 H ,1 QH *
I * * * * * * * * * ,60 X ,9H**#-******)
I F (S E N S E S W I T C H I) 2 1 9 , 2 2 0
219 PUNCH 80,VOBJ
220 CONTINUE
RETURN
END
ZZJOB 5
ZZFOR 5
•LDISK
SU BROUTINE EVGRAD
D I M E N S I O N L D ( 30) , N T Y P t 3 0 ) , V T R ( S O O ) , S E L M t 1 1 , 3 0 ) , S P ( I l ) , W E L M (11 ,30 ) ,
I E L E M t 3 0 ) , D T t 1 1 ) , W K l 1 1 ) , W Y ( 1 1 ) , A L B A R ( I O ) , B T 3 A R ( 1 0 ) , G R A D ( 2 0 ) ,I FR (S O)
2 , 1 T O ( 3 0 ) , I T R ( I O O O ) , F N C O ( I l ) , F D C O t 1 1 ) » B T ( 11) , A L P H ( I O ) , B E T A ( I O ) , F T C O
3( 11 )
COMMON NOD,NTT,NE LP ,NT S,N O , NSP,LD,NTYP,VTR,SELM,SP,WELM,WCD,WRDtWC
IS,WC Ri WR R,ELEM,DT,WK,WY ,N VB,ALBAR,BTBAR,VOBAR,GRAD,GRMAXiNGRI.STP,
2 N C M I , N C M T f N C C N f I J t I 3 , K B , IF R , I T O , N E L M f N I T R , I T R , N F A R , F N C O , F D C O , B T e A L
3PH,BETA,FTC0,CMAX,CMIN,RMAX.RMIN,CD,RD,CSUM,V03J
DO I I=ItNO
1 ALPHI D = A L B A R l 1 1
00 2 1 = 1,NOMT
2 Ij ET A( I ) = B T B A R ( I )
1 =I
5 A L P H l I )= 1 . 0 0 1 * A L B A R ( I I
7 CALL SYNTHZ
CALL EVLVTR
CALL SENSIT
CALL EVOB
IF(I-NO) 3,3,4
3 G R A D l D = ( ( V O B J - V O B A R ) Z t A L P H t I I - A L d A R l I ) ) ) * A L B A R ( I )**2
ALPH(I)=ALBARI I)
1=1 + 1
IF(I-NSP) 5,6,5
6 M=I-NO
B E TA (M)= 1 . 0 0 1*BT8AR(M)
GO TO 7
4 GR A D t 11 = ( ( V O B J - V O b A R ) Z t S E T A ( M ) - B T B A R ( M ) ) ) * B T B A R ( M ) * * 2
B E T A (M )= B T B A R (M )
IF(I-NVB) 10,33,10
10 1= 1+ 1
GO TO 6
33 R E T U R N
END
114
20 8 P U N C H 3 6 » ( M R E ( I ) , J R E U )*1=1*5)
P U N C H 3 7 , ( V A L ( I ) *1 =1 ,5 )
209 CO NT IN UE
IF(J-NELP) 55,53*53
53 IF(M-NSP) 56 ,5 7* 57
56 M = M + I
J=I
G O T O 55
57 CR =C MA X/ CM IN
RR=RMAXZRMIN
P R I N T 3 8 ,C D *RD, C S 1J M i C R »RR
3 8 F O R M A T ( 3 6 H C A P A C I T A N C E D I S T R I B U T I O N P A R A M E T E R = * E 1 7 . 8 , Z * 3 5 H R E S ISTA
I N C E D I S T R I B U T I O N P A R A M E T E R = ,E l 7 . 8 * Z * 2 9 H T O T A L C A P A C I T A N C E P A R A M E T E
2 R = * E 1 7 . 8 , Z * 1 9 H C A P A C I T A N C E R A T I O = ,E l 7 * 8 » Z » 1 8 H R E S I S T A N C E R A T I O = , El
3 7 . 8 * Z * 7 0 H S E N S I T I V I T Y OF C O E F F I C I E N T M W I T H R E SP EC T TO V O LT AG E FOL
4L0WER GAIN K.*Z*1H )
M=I
I F f S E N S E S W I T C H I) 2 1 0 , 4 5
2 1 0 P U N C H 3 8 , C D ,R D , C S U M , C R ,RR
4 5 D O 3 9 1= 1 , 5
M R E ( I )= M
V A U II=FNCO(M)*.Ol
IF(M-NSP) 39,41,41
39 M = M + I
4 1 IF ( I - S ) 4 2 , 4 3 , 4 3
42 D O 40 L = I ,5
' MRE(L)=O
4 0 V A U I )=0.
4 3 P R I N T 4 4 , ( M R E ( I) ,1 = 1.5)
4 4 F O R M A T t 6 X , 5 ( 2 H M = , 1 2 , 12'X) )
P R I N T 37 , (V A L ( I) ,1 = 1,5)
I F t S E N S E S W I T C H I) 2 1 1 , 2 1 2
21 1 P U N C H 44-,(MRE( I) ,1 = 1,5)
P U N C H 3 7 , tV A Lt I ),1 = 1,5)
21 2 C O N T I N U E
IF(M-NSP) 45 ,47,47
47 M=I
P R I N T 18 9
1 8 9 F O R M A T ( 7 9 H S E N S I T I V I T Y OF C O E F F I C I E N T M W I T H R E S P E C T T O V O L T A G E FO
I L L O W E R IN P U T A D M I T T A N C E , Z , I H )
I F t S E N S E S W I T C H I)' 2 1 3 , 7 5
2 1 3 . P U N C H 18 9
75 D O 69 1 = 1 , 5
MREt D = M
V A U I I = F D C O ( M ) * . 01
IF(M-NO) 69,71,71
69 M= M+ 1
71 1= 1 + 1
I F t 1-5) 7 2 , 7 3 , 7 3
72 D O 7 0 L = I , 5
MRE(L)=O
70 V A U U = O .
73 P R I N T 4 4 , (MREt I ) ,1 =1,5)
P R I N T 3 7 , (V A L ( I ) ,1 = 1,5)
I F (S E N S E S W I T C H I) 2 1 4 . 2 1 5
2 1 4 P U N C H 4 4 , (MREt I ) ,1 = 1,5)
P U N C H 3 7 , (V A L ( I) ,1 = 1,5)
215 CONTINUE
IF(M-NO) 7 5 ,9 7, 97
9 7 P R I N T 112
ZZJ08 5
ZZFOR 5
*LDISK
SUBROUTINE EVUVTR
D I M E N S I O N L D { 3 0 ) »NTY.P(30) , V T R ( S O O ) »SEL M( 1 1 , 3 0 ) ,S P ( I l ) ♦ W E L M ( 11 ,30 I »
I E L E M ( 3 0 ) , D T ( I l ) , W K ( I l ) , W Y ( I l ) , A L B A R t 1 0 ) , B T B A R ( 1 0 ) , G R A D (20 I , I F R ( 3 0 )
2 , 1T O I 3 0 ) , I T R ( I O O O ) , F N C O ( I l ) , F D C O ( I l ) , B T ( I l ) , A L P H ( I O ) , B E T A t 1 0 ) , F T C O
3( 1 1 )
COMMON NOD,NTT,NELP,NTS,NO,NSP,LD,NTYP,VTR*SELM,SP,WELM,WCD,WRDfWC
I S , W C R »W R R , E L E M » D T » W K » W Y , N V B , A L B A R , B T B A R , V O B A R , G R A D , G R M A X » N G R I , S T P ,
2 N 0 M I , N O M T ,N C O N , IJ , I B ,K B , I F R , I T O , N E L M , N IT R , IT R * N F A R , F N C O * F D C O » B T ,A L
3 P H , B E T A , F T C O ’, C M A X ,CM IN , R M A X , R M I N , C D , R D , C S U M ,V O B J
M=-I
D O 18 1 = 1 , N T S
DO" 2 N = 1 , N F A R , 2
M=M+2
K= 5 * ( N - I )+ 1
LD(K)=ITRtMIZlOOO
L D ( K + 1 ) = ( I T R t M )- 1 0 0 0 * L D (K ) JZlO
L D (K + 2 ) = ( IT R (M 1 * 1 0 0 0 0 ) Z 1 0 00
IF(N-NFAR) 4,5,5
4 L D (K + 2 I=L D t K + 2 I+ I T R ( M + 1 )Z 1 0 0 0 0
L D tK+ 3 ) = ( ITR(M+l')*.iO) Z l O O O
2 L D IK + 4 )= ( IT R (M + 1 ) * 1 0 0 0 I Z l O O O
5 VTR(I)=I.
D O 18 L= Ii NT T
J=LD(L)
IF(J-NO) 21 ,2 1, 19
21 V T Rl I)= V T R ( I ) Z E L E M ( J )
G O T O 18
19 V T R ( I l = V T R t I ) * E L E M ( J )
18 C O N T I N U E
RETURN
END
ZZJOB 5
ZZFOR 5
* L D IS K
SU BR OU TI NE EVOB
D I M E N S I O N L D ( 3 0 ) , N T Y P ( 3 0 ) , V T R ( 5 0 0 ) , S E L M t 1 1 , 3 0 ) , S P I 1 1 ) , W E L M ( I I ,3 0) ,
1 E L E M ( 30 I , D T ( I l ) » W K ( 1 1 ) , W Y ( I l ) , A L t i A R ( I O ) , B T B A R ( I O ) , G R A D t 2 0 ) , IFRt 30)
2 , 1 T O (30) , I T R ( I O O O ) , F N C O ( I l ) , F D C O t 1 1 ) , B T t 1 1 ) , A L P H ( I O ) , B E T A t 1 0 ) , F T C O
3( 1 1 )
C O M M O N N O D •N T T , N E L P , N T S i N O «N S P , L U , N T Y P » V T R , S E L M , S P , W E L M » W C D » W R D » W C
1S,WCR,WRRfELEM,DT,WK,WY,NVB,ALBAR,BTBAR,VOBARfGRAD,GRMAXiNGRIfSTPf
2 N 0 M I , N O M T , N C O N f I J , I B i K B , IF R , I T O , N E L M , N I T R f I T R , N F A R , F N C O f F D C O , B T , A L
3 P H, BE TA ,FT CO ,C M A X . C M I N , R M AX ,RM IN,CD,RD,C S UM ,VOBJ
R M A X = EL E M t I )
RMIN=RMAx
RSUm =RMAX
DO I 1=2,NO
IFIELEMt 11-RMAX) 2,2,3
3 RMAX=ELEM(I)
2 IFtELEMt D - R M I N I 4, 5, 5
4 RMIN=ELEMt I)
5 RSUM=RSUM+ELEMt11
I CONTINUE
I =NO+1
8
7
9
10
6
11
CMAX=ELEMU)
CMIN-CMAX
CSUM=CMAx
M = 1+ 1
DO 6 I=M,NELP
I F ( EL E M I I J - C M A X l 7 , 7 , 8
CMAX=ELEM(I)
IF(ELEM(I)-CMIN) 9,10,10
C M I n =ELEM(I)
CSUM=CSUM+ELEM(I )
CONTINUE
RBAR=NO
RBAReRSUMZRBAR
I=NELP-NO
CBAR=I
CB AR =C SU MZCBAR
RD=O.
D O 11 ! - ! , N O
R D = R D + ( E L E M ( I )-RtiAR)**2
CD=O.
I=NO
12 1= 1+1
C D = C D + (E L E M t I ) - C B A R ) * * 2
IF(I-NELP) 12,13,12
13 V O B J = W R D * R D + W C D * C D + W C S * C S U M + W C R * (C M A X Z C M I N ) + W R R * (R M A X Z R M I N I
DO 14 J= 1 , N E L P
D O 14 M = 2 , N SP
1 4 V O B J = V O B J + W E L M ( M , J ) * ( A B S F ( S E L M ( M , J ) ) ).
D O 1 5 M = I 1NO
V O B J = V O B J + W K (M ) * F N C O ( M ) * , O l
15 V O B J = V O B J + W Y ( M I* F D C O (M )* ,O l
RETURN
END
ZZJO B- 5
ZZFOR 5
*LDISK
S U B R O U T I N E P O A D (P O C , N O C , P O D , N O D , R P O 1N P O )
D I M E N S I O N L D I 3 0 ) , N T Y P t 3 0 ) , V T R ( 5 0 0 ) , S E L M I 1 1 , 3 0 ) , S P ( I l ) , W E L M t 11 , 3 0 ) ,
I E L E M (30) ,DT( 11) »WK( 11) »WY( 1 1 ) , A L B A R ( I O ) ,BTBA'RI 10 ) , G R A D I 20 ) , IFR( 30)
2 » I T O ( 30 ) ,ITRt 10 00 ) , F N C O (11 )•, FD CO l 1 1 ) ,BT (1 1 ) , A LP Ht 10 ) , B ET At 10 I , F T C O
3( 11 )
C O M M O N N O D , N T T , N E L P e N T S , N O , N S P , LD i N T Y P , V T R , S E L M , S P , W E L M , W C D , W R D , WC
I S i W C R , W R R , E L E M , D T , W K , W Y , N V B , A L B A R , B T 8 A R , V O B A R , G R A D , G R M A X f N G R I,ST P,
2 N 0 M I , N O M T f N C O N , I J , I S , K B , I F R » I T O , N E L M f N I T R , IT R , N F A R , F N C O , F D C O , B T ,AL
3 P H , B E T A , F T C O , C M A X i C M I N , R M A X , RM I N , C D , R D , C S U M , V O B J
D I M E N S I O N P O C ( 1 1 ) , P O D t 1 1 ) , R P O I 11)
IF(NOC-NOD) 11,12,12
11 N = N O D
G O T O 14
12 N = N O C
14 IF(N) 4 , 4 , 2
2 J=NOC+I
K=NOD+!
D O 8 I=Jfll8 POC(I)=O.
DO 9 I=Kfll
9 P O D ( I )=0.
DO 31 =1 ,N
RPO< I )=PO C( I H-PO Dl I).
NPO=N
RETURN
END
ZZJOB 5
ZZFOR 5
•Ld i s k
S U B R O U T I N E P O M U L l A N t I A N iP O D >N O D ♦P O C iN O C iKERR)
D I M E N S I O N LD l 3.>) t N T Y P O V ) t V T R I 5 0 0 > t S E L M U I #3 0 > t S P (11 1 > W E L M 111 >30) >
I E L E M l 3 0 > tD T I 1 1 ) tWKl 1 1 ) tWYI 1 1 ) t A L B A R l 1 0 ) t B T B A R l 1 0 ) » G R A D ( 2 0 ) 11 FR I 30)
2» ITOl 30.) i ITRl 10 00 ) t F N C O l 11) t F D C O ( I l H B T l l l ) , A LP Hl 10) ,BET A! 10) t F T C O
3( 11 )
COMMON NODtNTT,NELP,NTS,NO,NSPtLDtNTYP,VTRtSELM,SR,WELM,WCD,WRD,WC
IStWCRtWRRtCLEMtDT tWKtW Y tNVBtALGARtBTBARtVOBAR,GRADtGRMAXtNGRItSTPt
2N0MI,NOMTtNCONt IJ 11G ,KU,IF R ,ITOtNELM,NIT R 11T R tNFAR,FNCOtFDCOtBT,AL
SPHtBETA,FTCC,CMAXtCMINtRMAXtRM IN,CD,RDtCSUMtVOGJ
DIMENSION ANl 1 1 ) ,PODl 1 1 1 ,POCI 1 1 ),DPC(Il),DPD(Il),DRP(Il)
ZZJOB 5
Z7 F O R 5
*LDI5K
S U BR OU TI NE RDIN
D I M E N S I O N LU I 30) , N T Y P l 3 v ) , V T R ( S O O ) tS E L M I 1 1 , 3 0 ) , S P ( I l ) i W E L M ( l l , 3 0 J t
I E L E M t 3 0 ) t D T ( 1 1 ) t W K ( I D t W Y ( 1 1 J ,A L U A R ( 1 0 ) , B T d A R t l O J , G R A D ( Z O ) t J F R ( 3 0 )
2 t I T O l 3 0 ) , I T R l U O O ) , F NC Ol I D , F D C O ( I l ) , J T C 1 1 > , A L P H I 1 0 ) , B E T A l 1 0 ) , F T C O
3( 11 )
C O M M O N NOD >N T T , N E L P , N T S , N O , N S P t L D i N T Y P , V T R , S E L ‘I,SP,V.'ELM,W C D t W R D , W C
I S , W C R tW R R t E L E M t D T , K K ,W Y , N V B , A L G A R , U T t i A R ,V O U A R , G R A D , G R M A X t N G R I t ST P,
Z N O M I , N O M T t N C O N t I J t I t i t K B , I F R , I T C t N E L M t N I T R t I T R t N F A R , F N C 0 , F D C 0 , 3 T ,AL
3 P H tB E T A , F T C 0 1C M A X ,C M IN ,R M A X ,R M IN ,C D ,R D i C S U M ,V O B J
READ 11,HOD,NCLP,N O 1 IiitU,KB ,NELM
11 F O R M A T I 712)
R E A D 1 2 , IN T Y P ( I ) t I= I , 3 0
R E A D 12, I I F R l D ,1 = 1,30)
READ 13,WCOIWRDtWCStWCR,WRR
13 F O R M A T 1 9 F 1 0 « 2 )
NSP=NOtl
NCARD=I I INELP*NSPl-l)/8)tl
DI ME NS IO N WTPEtai
MV=O
D O 31 I = H N C A R D
R E A D 1 7 , I W T P E l M ) ,M = l » 8 )
1 7 F O R M A T I 8 F 1 0 e2 )
D O 31 J = I ,8
MV=MVtl
MI=(MV-I)ZNELPtl
MJ=MV-IMi-IHNELP
31 W E L M ( M H M J ) = W T P E ( J )
R E A D 1 7 , ( W K ( I ) , 1=1,8)
IF(NSP-B) 41 ,41,42
4 2 R E A D i 8 , W K ( 9 ) ,W K( IO )
4 1 R E A D 1 7 , "IWYl I ) ,1 = 1,8)
IF IN S P - 8 ) 4 3 , 4 3 , 4 4
4 4 R E A D 1 8 , W Y l 9 ) ,W Y(IO)
' 43 NCARD=INOZBHl
R E AD 1 7 , I S P l 11,1=1,8)
IF(NCARD-I) 26,26,27
2 7 R E A D 1 8 , S P ( 9 ) ,S P( IO )
18 FO RMAT(2F10,2)
2 6 R E A D 1 7 , IA LB AR I I ) , 1 = 1,8)
I F IN O - 8 ) 3 3 , 3 3 , 3 4
3 4 R E A D 1 8 , A L B A R I 9 ) , A L B A R l 10)
3 3 R E A D 1 7 , IB TB AR l 1 1 ,1 = 1,8)
NTT=NOD-I
NVB=2*NO-2
NOMI=NO-I
N0MT=N0-2
N F A R = I I I4 * N T T ) - l )Z l O l t l
RETURN
END
ZZJOB 5
ZZFOR 5
#LDISK
SU BR OU TI NE SENSIT
D I M E N S I O N L D I 3 0 ) , N T Y P I 3 0 ) , V T R I 50 0> t S E L M l l l , 3 0 ) ,SP 11 1 ) tW E L M l 1 1 , 3 0 ) ,
I E L E M I 3 0 ) ,DT I 1 1 ) tW K l 1 1 ) tW Y ( I l ) , A L G A R l 1 0 ) tB T B A R ( 1 0 ) , G R A D I2 0 ) 11 F R l 30)
2 t I T O l 3 0 ) , I T R I 1 0 0 0 ) , F N C O ( I l ) , F D C O ( I l ) , B T ( I l ) , A L P H 11 0 ) tBETAl 1 0 ) , F T C O
3( 1 1 )
C O M M O N N O D ,N TT , N E L P , N T S ,N O , N S P , L D t N T YPt V T R, SELIlt SPt W E L M , W C D , W R D t W C
IS,WC R, WR R,E L E M tD Tt WK ,W Yt NV Bt AL dA Rt oT BA R, VO BA R,G RA D,GRMAXtNGRItSTPt
ZNOMltNOMTtNCONtIJtIBtKBtIFR1ITOtNELMtNITRtITRtNFARtFNCOtFDCOtBTtAL
S P H t B E T A t F T C O t C M A X t C M I N ,R MA Xt RM IN,C D , RDtC SU Mt VO BJ
D I M E N S I O N S T O T l 11)
ST0T(1)=ELEM(1)
D O 21 I = Z t N O
21 S T O T ( I ) = S T O T ( I ) K E L E M l I)
STOT(I)=I.ZSTOT(I)
O O 22 I = 2 , N S P
22 S T O T ( I ) = O .
116
IFl IA N * N O D ) 1 , 1 , 2
1 NOC=O
RETURN
2 N=IANtNOD-I
I F IN - i I ) 3 , 3 , 4
4 KERR=3
RETURN
3 DO 8 I=ItN
8 D R P { I )=0.
D O 9 I = I , IAN
9 D P Cl I )= A N I I I O O 10 I = I i N O D
10 DP D l D = P O D l I )
DO 5 I=ItIAN
DO 5 J=ItNOD
K= I+J-l
5 D R P ( K ) = D R P I K ) + D P C i n * D P D I J)
NOC =N
DO 6 I=ItNOC
6 P O C l I ) = DR P( 11
RETURN
END
RE AD 1 2 , U T O U ) , 1*1,30)
12 F O R M A T I 3 0 1 2 1
ZZJOB 5
ZZFOR 5
*LDISK
SUBROUTINE SEOGEN(NEL)
D I M E N S I O N L U (30) tN T Y P I 3 0 ) t V T R I 5 0 0 ) tS E L M ( 1 1 1 3 0 ) t S P ( 1 1 ) » W E L M I 1 1 1 3 0 ) t
I E L E M O O ) t D T( ll ) t'.VKUl) t W Y ( l l ) tALI3AR (10) tIiTBAR ( 10 ) t G R A D (20 ) 11 FR (30)
2 t I T O ( 3 0 ) t l T R ( l O O O ) . F N C O ( I l ) tFD C O f 1 1 ) t B T ( 1 1 ) tA L P H I 1 0 ) t B E T A ( 1 0 ) tF f CO
3( 11 )
C O M M O N N O D t N T T t N E L P t MT S . N O . N S P t L D t NT Y P t V T R t S E L M t S P t W E L M t W C D t W R D t W C
I S . W C R . W R R . E L C M t D T tW K iW Y t N V B . A U i A R v i T ^ A R t V O B A R . G R A D t G R M A X t N G R I .STPt
2 N 0 M I t N O M T t H C O N t I J t I U t K J 1 1F R . I T O t H E L M t N I T R t I T R t N F A R . F N C O t F D C O . t i T t A L
3 P H t B E T At F T C O i C M A X t C M I N t R M A X t R M I N . C D t R D t C S U M t V O B J
I F ( N E L - I ) I t I t2
1 NEL=2
IF(LD(I)-NELM) 3t2t2
3 LD(I)=LD(I)+!
RETURN
2 J=NELM+1-NEL
10 IF ( LDI N E D -J ) 6 t 5 1 5
5 IF(NEL-NOD) 7t6t7
7 NEL=NEL+!
J=J-I
G O T O 10
6 J=LD(NEL)+NEL
L=O
15 I F ( L - N E L ) 1 3 » 1 2 t l 2
13 L = L + I
LD(L)=J
J=J-I
G O T O 15
8 NEL=O
12 R E T U R N
END
ZZJOB 5
ZZFOR 5
♦LDISK
SUBROUTINE STCOMP
D I M E N S I O N L D ( 3 0 ) t N T Y P U G ) . V T R ( S O O ) . S E L M C 1 1 1 3 0 ) t S P (1 1 ) t W E L M ( 1 1 . 3 0 ) •
I E L E M O O ) t D T U D .W K ( I l ) . W Y d D t A L B A R I 10 T t B T B A R I 10) t G R A D (20 I 11 FR (30)
2 t I T 0 ( 3 0 ) t ITR ( 1 0 0 0 I I F N C O I 11 Jt F D C O f 11 ) , B T ( l l ) t A L P H d O J t B E T A d O J t F T C O
3( 1 1 )
COMMON NODtNTTiNELP.NTStNOtNSPtLDtNTYPtVTRtSELMtSPtWELMtWCDtWRDtWC
I S t W C R t W R R t E L E M t D T t W K tW Y t N V B t A L B A R t B T B A R t V O B A R t G R A D t G R M A X t N G R I t S T P t
2N O M I tNOMTtNCONtIJt IB tK Bt IF Rt IT Ot NE LM tN IT Rt IT Ri NF AR tF NC Ot FD COi BT iA L
SP Ht BE TA tF TC Ot CM AX tCM INtRMAXtRMINtCDtRDtCSUMtVOBJ
GRMAX=O.
D O 4 -I= I . N VB
IF ( I - N O ) 2 12 16'
2 COMP=ABSF(GRAD(I)ZALBARd))
GO TO 5
6 J=I-NO
C O M P = A B S F (G R A D ( I I Z ti TB AR (J ))
5 IF (GRMAX-COMP) 3i4t4
3 GRMAX=COMP
4 CONTINUE
STP=*01/GRMAX
RETURN
END
ZZJOB 5
ZZFOR 5
♦LDISK
SUBROUTINE SYNTHZ
D I M E N S I O N L D I 30 I t N T Y P O O " ) . V T R ( S O O ) t S E L M I l I .30) t S P d l ) t W E L M I 1 1 . 3 0 ) .
I E L E M O O ) . D T d D .WK( I D t W Y d l ) . A L B A R I 10 ) t B T B A R (10 ) t G R A D ( 20 ) 11 FR! 30)
2 t ITOC 30) 1 1 T R (100 0) t F N C O d D . F D C O d I ) tBT (I U . A L P H I 10) .BET A! 10) t F T C O
3dl)
COMMON N O D iNTT.NELPtNTSiNO.NSPtLDtNTYPtVTRtSELMtSPtWELMtWCDtWRDtWC
I S tW C R t W R R t E L E M t D T tW K t W Y t N V t i tA L B A R i B T B A R . V O B A R . G R A D i G R M A X i N G R I . S T P t
2N 0M IiNOMTtNCON.IJtIBtKBtIFRtITOtNELMtNITRtITRiNFARtFNCOtFDCOtBTtAL
3 P H. BE TA .FTCOtCMAXtCM IN.R M A X t R M INtCDiRDtCSUM.VOBJ
D I M E N S I O N C A P d O ) t C A P D d O ) . R E S d O I . A N d D tP O D d D tPOC( 11) . Q T d D
NRTN=NOD
BETA(NOMI)=SP(NO)
DO 3 I=ItNOMT
3 B E T A (N O M I )= B E T A l N O M I )- B E T A l I )
AN(2)=1.
AN(D=ALPHd)
P0D(2)=1.
11?
DO I MsltMOD
D O I J = I l MELP.
1 SELM(MtJ)=O.
M=-I
DO 8 I=ItNTS
D O 2 N = I t N F A R 12
M=M+2 •
K= 5 * ( N - I )+ 1
LD(K)=ITR(M)ZlOOO
L D (K + l ) = ( I TR (M )- 1 0 0 0 * L D (K ) )/10.
L D (K + 2 ) = ( I T R ( M m O O D O ) / 1 3 0 0
I F ( N - N F A R ) 4 15 t 5
4 L D ( K + 2 ) = L D ( K + 2 ) + I T R ( M + l 1/10000
L D (K + 3 ) = ( I T R t M + I ) # 1 0 ) / 1 00 3
2 L D (K + 4 ) = ( I T R ( M + 1 ) * 1 3 0 D ) / 1 0 0 0
5 K= I
DO 6 N=ItNTT
L=LD(N)
6 K=K+NTYP(L)
STOT(K)=STOTJK)+VT R(I)
DO 7 L=ItNTT
J=LD(L)
7 S E L M ( K t J ) = S C L M I K tJ ).+VTR ( I )
8 CONTINUE
DO 9 J=ItNELP
D O 9 M = 2 t N SP
IF(J-NO) 27t2 7t 2U
2 7 S E L M (MtJ) = I S E L M (.Mt J ) # ( - 3 . 9 9 0 0 9 9 0 0 9 9 ) ! / S T C T ( M )
GO TO 9
28 SC LM (M tJ)=SELM(MtJ)ZSTOT(M)
9 CONTINUE
RETURN
END
9
6
4
5
7
8
10
928:1
I=I
P O D( U = A L P H l
I+ D
C A L L P O M U L I A N i I A N t P O D iN O D tP O C tN O C tKERR)
L= I
AN(U=POClL)
I F ( L - N O C ) 4 t 5, 4
L=L+I
GO TO 6
IAN=NOC
IFl I- N O M I) 7 , 8 , 7
1= 1+ 1
GO TO 9
D O 10 I = 1 , I A N
DTl I )= A N 11 I
NDT=IAN
AN(Z)=I.
ANl U = B E T A ( I )
IAN=Z
I=I
C A P D ( i S ) = sJ T( NQT) ZJiT CL. T )
A N ( Z ) = - I . S C A P C - 1)
IAN = Z
C A L L P O 1VJLlAXt IAN t PT , N B T , r o C , I1IOC, K C R fU
CALL- P O A D l O O C t N O C t J T ,N D T ,A N , IAN)
I
J
DO I V I = I t I A N
DT(I ) -AN( I I
NDT=IAN-;
AN(A)=-CAPDl N
ZZJOB 5
ZZFOR 5
SLDISK
SU BR OU TI NE TKSTP
D I M E N S I O N L D (30 I , N T Y P ( S O ) , V T R ( S O C ) , S E L M I 1 1 , 3 0 ) ,S P (11) , W E U 1U 11 ,30 ) ,
I E L E M 1I 30) ,D T ( I l ) ,WKl 11) , W Y ( I l ) ,A L B A R t 10) ,U T B A R l 10 ) , G R A D (20) ,I FR ('30)
2 , 1 T O (30) , I T R ( I O O O ) , F N C O I 11) , FD CO l I U ,BT 111)., A L P h (10 ), B E T A I 10) ,F TC O
3(11)
C O M M O N N O D , N T T , N E L P , N T S , N O t N S P f L D , N T Y P , V T R , S E L M t SP,WELM',WCDtWRDtWC
1 $ , W C R » W R R , E L E M , D T , W K , W Y , N V B f A L B A R , U T B A R t V O r i M R , G R A D , G R M A X , N G R J ,STP,
2 N 0 M I * N O M T » N C O N , I J , I 6 » K B , I F R , I T O , N E L M , N I T R , I T R i N F A R , F N C O , F D C O » U T ,AL
3 P H , B E T A f F T C O , C M A X t C M I N , R M A X i R M I N , C D , R D fC S U M , V O B J
DO I I=ItNO
1 ALPH(I)=ALBAR(I)-STPSGRADm
DO 2 - I=I,NOMT
M = I + NO
2 BETA(I)=BTBAR(I)-SfPSGRAD(M)
3 CALL CONSTR
CALL SYNTHZ
CALL EVLVTR
CALL SENSIT
C A LL EVOB
RETURN
END
ZZJOB 5
ZZFOR 5
SLDISK
SUBROUTINE TRCUICLS)
118
13 P O D ( I ) = B E T A l I+ U
C A L L P O M U L (A N ♦ I A N , P O D , N O D ,P O C t N O C ,K E R R )
D O il L = I t N O C
11 A N ( L ) = P O C ( L )
IAN=NOC
I F ( I - N O M T ) 1 2 , 1 8 , IZ
IZ 1= 1+ 1
G O T O 13
18 D O 14 1 = 1 , IAN
U 3 T I I I=ANl I )
NRT=IAN
D O 15 I = U N O
L = N D T + I-I
J = L- I
D T ( L ) = I D T ( L ) Z D T (I I )
UT(J)=BT(J)ZoT(I)
FDvU(J)=OT(J)
15 F T C O ( L ) = D T ( L )
UT(I)=I.
FTCO(I)=-I.
DT(I)=I.
F D C O t I )= 1.
D O 16 I = Z t N D T
16 IT (I I=DTt U - G P t I )
V< U=D.
NOT=NDT
X O = CiiT
D O 17 X = I f 1IO
CAPm=DT(NUT)ZBT(NJT)
AN(I)=O.
IAN»2
CALL POMUL(AN,IANiBTtNBT,POCtNOCtKERR)
CALL POADlPOC,NOC,QT,NQT,AN,IAN)
DO 20 I-IiIAN
IF(M-I) 20,317,20
317 FNCO(I)=AN(I)
20 OT(I)=AN(I)
NQT=IAN-I
RES(M)=BTlNBT>ZDT(NDT)
DO 21 I=IiNDT
21 BT(I)=BT(I)-RES(M)SDTin
CAP(M)=CAP(M)-CAPDlM)
17 NBT=NBT-I
DO 22 I=I,NO
22 ELEM(I)=RES(I)
I=NO
ELEMl1+1)=CAP(I)
ELEM(1+2)=CAPDl2)
1=1+3
DO 23 J=3,N0
ELEM(I)=CAP(J)
ELEMl !+U=CAPD(J)
23 1=1+2
NOD=NRTN
RETURN
END
ZZJOB 5
ZZFOR 5
* L D ISK
SU BR OU TI NE TRCOM
D I M E N S I O N L D ( 3 0 ) , N TY P( 3 0 ) , V T R ( 5 0 0 ) ,S EL Ml1 1 , 3 0 ) , S P ( I l ) , W E L M l 11,301»
I E L E M (.30) ,D T ( I l ) ,W K ( I l ) ,WY( 11) »A L B A R (IsO ) , B T B A R d O ) , G R A D ( Z O ) , I F R (.30)
2 , if 0 ( 3 0 ) , I T R ( I O O O ) , F N C O d i ) , F D C O l 1 1 ) , B T ( 1 1 ) , A L P H I 1 0 ) , B E T A l 1 0 ) ,F T C O
3(11)
C O M M O N N O D » N T T , N E L P , N T S , N O , N S P , L D ,N T Y P ,V T R »S E L M ,S P » W E L M » W C D , W R D » W C
I S t W C R t W R R t E L E M t D T ♦W K »W Y tN VI l, A L B A R iJ T B A R , V O B A R , G R A D , G R M A x , N GR I ,STPt
2 N O M I , N O M T t N C O N t I J i I B t K B , IF R , I f 0 , N E L M , N I T R , I T R t N F A R t F N C O t F D C O t B T t A L ’
3 P H , B E T A ,F T C O , C M A X t C M I N ,R M A X ,R M I N , C D ,R D , C S U M , V 0 3 J
NITR=O
KCR =O
NEL=C
DO 3 I=ItNOD
L = N O D + I-I
3 LD(I)=L
4 C A L L T R D E T ( N E L j KE R)
IF(KER) 5,6,5
5 C A L L E X IT
6 IF(NEL) 40 2, 7, 40 2
7 CALL TRCLIICLSl
IF ( I C L S - 6 ) 4 0 2 , 8 , 4 0 2
8 DO 9 I=ItNOD
IF(LD(I)-NELP) 9,9,408
408 IF(LD(I)-IB) 402,9,402
9 CONTINUE
NITR=NfTR+!
I=I
32 L= I
31 I F ( L D ( I ) - I B ) 3 3 , 3 4 , 3 3
34 L=L-I
G O T O 19
33 IF(L-Z) 10,11,12
11 I T R ( N I T R ) = I T R t N I T R I+ 1 0 * L D ( I )
G O T O 19
10 I T R ( N I T R ) = L D t I ) * 1 0 0 0
G O T O 19
12 I F (L - 4 ) 1 3 , 1 4 , 1 5
14 I T R ( N I T R ) = I T R l N I T R ) + L D ( 1 1 * 1 0 0
G O T O 19
1 5 I T R ( N I T R ) = I T R ( N I T R ) + L D ( I)
G O T O 19
13 I T R ( N I T R ) = I T R l N I T R ) + L D ( D Z l O
NITR=NITR+1
IT R( N I T R ) = I L D ( I ) - I O * ( L D ( I ) Z l O ) 1*10000
19 I F ( I - N O D ) 2 6 , 4 0 2 , 4 0 2
2 6 i= I + l
L=L+1
IF (N IT R - 9 9 8 ) 5 1 , 5 2 , 5 2
52 P R I N T 53
53 FO RM A T ( 2 6 H N O . OF TREES EX CE ED S LIMIT)
CA LL EXIT
51 IF ( L - S ) 3 1 , 3 1 , 3 2
402 CA LL SEQGEN(NEL)
IF ( N E L ) 4 , 5 0 1 , 4
50 1 N T S = N I T R Z N F A R
RETURN
END
ZZJOB 5
ZZFOR 5
* L D ISK
SUBROUTINE TRDET(NELtKER)
D I M E N S I O N L D (30) , N T Y P ( 3 0 ),, V T R ( 5 0 0 ) ,SELMI 1 1 »30 ) *S P (1 1 ) »W E L M I 11 , 3 0 ) ,
I E L E M (30 I ,DT( 11) ,WK( 1 1 1 ,WY( 1 1 ) , A L B A R CIO ) , B T B A R (10 ) , G R A D ( 20 ) 11 FR (.30)
2 » I T0( 3 0 ) , I T R I 10 00 ) , F N C O ( I l ) ,FDCOI I D t B T ( H ) , A L P H (10 ) ,BET A! 10) ,F T C O
3( 11 )
C O M M O N N O D t N T T , N E L P ,N T S , N O , N S P , L D , N T Y P , V T R , S E L M , S P t W b L M t W C D , W R D t W C
I S , W C R t W R R t E L E M . D T , W K , W Y , N V B t A L B A R , B T B A R , V O B A R , G R A D , G R M A X t N G R I , ST P,
2 N 0 M I , N O M T , N C O N , I J , I B , K B , I F R , I T O , N E L M , N O T R d T R , N F A R , F N C O t F O C O t B T , AL
3PH,BETA»FTC0,CMAX»CMIN,RMAX,RMIN»CD»RD,CSUM,V0BJ
D I M E N S I O N N D E ( 30)
MT=NOD+!
DO I 1=1,MT
I NDE(I)=O
NSET=I
M=NOD
601 J=LD(M)
K=IFR(J)
L=ITO(J)
IF(NDEtK)-NDE(L)) 4,2,3
4 IF lN DE (K )) 676,646,650
119
2 t I T 0 ( 3 0 ) W T R ( I C O O ) , F N C O ( I l ) , F O C O ( I l ) , B T ( I l ) , A L P H ( I O ) »B E T A t 1 0 ) ,F T C O
3( 11 )
C O M M O N N O D ,TjTT , N E L P ,((T S ,NO , N S P , L O ,NT Y P , V T R , S E L M , S P , W E L M , W C D , W R D ♦ WC
I S , W C R , ‘, / R R » E L E M , D T , W K »V,'Y»NVS,A L BA R, .i T B A R * V O B A R , G R A D t G R M A X , N G R I »STP»
2.N0MI ,N O M T , N C O N , IJ , IQ ,KB , I F R , I T O , N E L M , N I T R ♦ I T R t N F A R , F N C O t F D C O ,BT ,AL
3 P H , B E T A , F T C O , C M A X , C M I N , RM AX tR M IN,C D , N D , CS UM tV OB J
ICLS=I
DO I I=ItNOD
IF(LD(I)-IB) 2,3,2
3 IK=I
ICLS=ICLS+!
2 I F ( L D l I ) -KB) '1 ,1 7, 1
17 I C L S = I C L S + 2
I CONTINUE
I F ( IC L S - 2 ) 5 , 6 , 5
C KSV=LD(IK)
L D ( IKI=KB
CALL TRDET(NELtKER)
L D ( IK l = K S V
IF(NEL) 5,7,5
7 K S V = L D l IK)
LD(IK)=IJ
CALL TRDET(NbLtKER)
L D ( IKI=KSV
IF(NEL) 51,52,51
52 I C L S = I C L S - 1
51 I C L S = I C L S + A
5 RETURN
END
120
2 IF(NDElL)) 676,644,653
3 I F t N D E l U ) 676,645,647
676 KER=6
RETURN
644 NDE(K)=NSET
•ND E(L)=NSET
NSET=NSET+!
60 TO 637
645 NDE(L)=NOE(K)
GO T O 637
646 NDE(K)=NDE(L)
GO TO 637
647 NEL=NDE(K)
IF=O
25 IF(IF-MT) 2 6 ,6 37 ,6 37
•26 I F = I F + !
I F t N D E t IFl-NEL) 25 ,27,25
27 NDElIFl=NDE(L)
G O T O 25
650 NEL=NDE(L)
IF =O
35 I F ( I F - M T ) 3 6 , 6 3 7 , 6 3 7
36 IF=IF+!
IF(NDE(IR)-NEL) 35,37,35
37 N D E ( IFJ=NDE(K)
G O T O 35
653 NEL=N
RETURN6 3 7 IF ( M - I ) 4 1 , 4 1 , 4 2
42 M= M- I
GO T O 601
41 NE L= O
RETURN
'
END
A P P E N D I X . II
O P T I M I Z A T I O N COMPUTER P R O G R A M INPUT A N D
OUTPUT DATA P O R E X A M P L E P R O B L E M
122
ZZJOB 5
Z Z XEQSCHONE
5100411051212
101010101010000
1 0 2030401020303 0'4040 105
20304000500050005000000
100.
Io
Ic
IO
I.
Io
I.
I.
Io
Io
Io
I.
Io
Io
Io
Io
IO
Io
IO
I.
I.
Io
IO
10.
10.
IOo
50.
50.
50 O
3.4143
2.6132
Io
.35
o05
.25
.3
ol
I.
I.
IO
I.
IO
I.
IO
10.
I.
I.
Io
I.
I.
I.
10.
50.
2.6132
4.
10.
50 .
I.
Ie
I.
IO
I.
I.
I.
I.
IO
I.
Ie
I.
I.
Ie
i.
I.
I.
Ie
I.
INITIAL NETWORK
SENSITIVITY OF COEFFICIENT M WITH RESPECT TO VOLTAGE FOLLOWER INPUT ADMITTANCE
POLE LOCATIONS
•05000000
,25000000
•35000000
4,00000000
,30000000
2,21320000
C O E F F I C I E N T S O F IN P U T A D M I T T A N C E Y l l , l )
NUMERATOR COEFFICIENTS
DENOMINATOR COEFFICIENTS
1,00000000
1,00000000
27,10714200
155,28571000
265,71428000
57,14285700
13.78516700
39,35779200
15,06114800
OPTIMIZATION PARAMETERS
S E N S I T I V I T Y O F C O E F F I C I E N T M WITH. R E S P E C T T O E L E M E N T N U M B E R J,
M= I,J= 2
000.OOOOOE-99
M = I ,J= 3
000,OOOOOE-99
M = I ,J = 4
00 0 . OOOOOE-99
M= I,J= 5
00 0.OOOOOE-99
M= I,J= 6
,OOOOOE-99
M = I ,J= 7
000.OOOOOE-99
M= I,J= 8
00 0.OOOOOE-99
M= I,J= 9
0 0 0 . OOOOOE-99
M= I , J = IO
O O O oO O O O O E - 9 9
M= 2 , J= I
-986.843656-03
M= 2, J= 2
- 9 7 8 . 1 5 7 3 4 E - 03
M= 2 , J= 3
-807.85073E-O3
M= 2,J= 4
-197.445356-03
M = 2 , J= 5
254.07932E-04
M= 2 , J= 6
.OOOOOE-99
M= 2, J= 7
244.34303E-03
M= 2 , J= 8
000.OOOOOE-99
M= 2,J= 9
730.249246^03
M= 2,J=IO
00 0.OOOOOE-99
M= 3,J= I
-932.013Y3E-03
M = 3» J = 2
- 8 0 7 . 3 1 1 1 4 E - 03
M= 3 , J= 3
- 114.477646-03
M= 3 , J= 4
-126.39550E-03
M= 3 , J= 5
2 6 8 . 0 7 2 6 2 E — 03
M= 3,J= 6
979.53544E-05
M= 3,J= 7
7 3 9 . 4 2 8 8 1 E-03
M = 3, J= 8
I57.09939E-03
M= 3 , J= 9
291.261546-03
M= 3,J=IO
534.343T3E-03
M= 4 , J= I
-654.85183E-03
M= 4, J= 2
- 2 0 1 . 0 6 6 7 2 E - 03
M= 4 » J= 3
— 6 4 1 . 3 6 3 6 8 6 —04
M= 4 , J= 4
-700.44388E-04
M= 4,J= 5
100.00000E-02
M= 4,J= 6
I58•30040E-03
M= 4»J= 7
42 6, 13 20 2 E-04
M= 4 , J= 8
896.66212E-03
M= 4 , J= 9
318.29200E-03
M= 4,J=IO
583.93274E-03
M= 5,J= I
.OOOOOE-99
M = 5, J = 2
000,OOOOOE-99
M = 5 iJ = 3
000.OOOOOE-99
M= 5,J= 4
000.OOOOOE-99
M= 5,J= 5
100.00000E-02
M= 5,J= 6
100.00000E-02
M = 5, J = 7
4 5 5 . 7 1389E-04
M= 5,J= 8
954.426616-03
M= 5,J= 9
352.78571Er03
M= 5,J=IO
6 4 7 . 2 1 4 2 8 E — 03
CAPACITANCE DISTRIBUTION PARAMETER=
.11601307E+04
RESISTANCE DISTRIBUTION PARAMETER=
.8 6115 8 75 E- 01 '
TOTAL CAPACITANCE PARAMETER=
,89351085E+02
CAPACITANCE RATIO=
,50874323E+03
RESISTANCE RATIO=
.43371771E+01
S E N S I T I V I T Y O F C O E F F I C I E N T M W I T H R E S P E C T T O V O L T A G E F O L L O W E R G A I N K,
M= 2
2 0 7 , 6 6 2 8 1 E-03
M= 3
100.484^85-02
M= 4
I 16.38857E-02
M= 2
137.85167E-03
M= 3
393,57?92E-03
M= 4
150,6U48E-03
M= 5
20 0.OOpOOE-IO
NETWORK ELEMENT VALUES
,12940401E+00
ELEMENT ( D =
E L E M E N T I 21 =
,47469814E+00
•28644922E+00
E L E M E N T ! 31 =
EL EM EN T! 4)=
,10944864E+00
•66396000E-01
E L E M E N T ! 51 =
.44711095E+01
E L E M E N T ! 61 =
.16128330E+01
E L E M E N T ! 7) =
ELEMENT! 8)=
•33778516E+02
E L E M E N T ! 9) =
•17435457E+02
.31986774E+02
E L E M E N T (1 0 ) =
OBJECTIVE FUNCTION VALUE
,18845128E+04
**********
M= I,J= I
.OOOOOE-99
M= I
.OOOOOE-99
M= I
100.00000E-04
,10000000
M= 5
200.00000E-10
GZT
ZERO LOCATIONS
FINAL NETWORK
SENSITIVITY OF COEFFICIENT M WITH R ES PE C T TO VOLTAGE FOLLOWER INPUT ADMITTANCE
M= I
100.00000E-04
.28234783
.89603857
1.43481370
.06435969
.44318273
1.04377740
1.54673050
C O E F F I C I E N T S O F IN PU T A D M I T T A N C E Y ( I f l )
DENOMINATOR COEFFICIENTS
NUMERATOR COEFFICIENTS
1.00000000
1.00000000
5.35470860
7.19889770
2.75482070
19.39866400
64.23079800
67.27739100
21.71604200
OPTIMIZATION PARAMETERS
S E N S I T I V I T Y O F C O E F F I C I E N T M W I T H R E S P E C T T O C L E M E N T N U M B E R J.
M= IfJ= 2
COO.OOOOOE-99
M= IfJ= 3
00 0 . OO uO OE-99
M= IfJ= 4
OOO.OOOOOE-99
M= ItJ= 5
OOO.OOOOOE-99
M = I 1J = 6
.OOOOOE-99
M= IfJ= 7
000.OOOOOE-99
M= IfJ= 8
000.OOOOOE-99
M= IfJ= 9
OOO.OOOOOE-99
M= IfJ=IO
OOO.OOOOOE-99
M = 2 , J= I
- 9 5 4 . 1 1 4 3 IE-03
M= 2♦ J= 2
-9 49 . 4 6 9 9 2 E-03
M = 2 fJ = 3
-689.555396-03
M = 2 fJ= 4
-377.13733E-O3
M= 2f J= 5
138.91015E-03
M= 2,J= 6
.OOOOOE-99
M = 2 fJ = 7
463.20888E-03
M= 2 f J= 8
OOO.OOOOOE-99
M= 2fJ= 9
397.88097E-03
M= 2fJ=IO
OOO.OOOOOE-99
M = 3 fJ = I
-672.93738E-03
M= SfJ= 2
- 7 4 0 . 7 0 0 9 0 E- 0 3
M = 3 fJ = 3
-319.39494E-O3
M = 3 fJ = 4
- 2 4 7 . 1 6 4 7 5 E - 03
M = 3 fJ= 5
569•29967E-03
M= 3»J= 6
150.43165E-03
M = 3 . J= 7
502.36999E-r03„
M = 3 tJ = 8
165.61801E-O3
M = 3 fJ = 9
3 0 3 • 12235E-03
M= 3tJ=IO
309.15830E-03
M= 4 . J= I
-197.71502E-03
M = 4 f J= 2
- 3 1 O 7 4 2 O E -0 3
M = 4 1 J= 3
-294.9052 5E- O3
M = 4 fJ = 4
-180.40453E-03
,M= 4f J = 5
100.00000E-02
M = 4 fJ = 6
623.355806-03
M= 4«J= 7
2 0 8 . 6 9 9 9 1 E-03
M = 4 tJ = 8
462.73674E-03
M = 4 t J= 9
3 3 9 . 12733E-03
M= 4fJ=IO
345.88022E-03
M= SfJ= I
.OOOOOE-99
M= SfJ= 2
OOO.OOOOOE-99
M= SfJ= 3
000.00000E-99
M= SfJ= 4
OOO.OOOOOE-99
M= 5.J= 5
100.00000E-02
M= SfJ= 6
100.00000E-02
M= SfJ= 7
302.03707E-03
M= SfJ= B
697.96294E-03
M= SfJ= 9
495.07091E-03
M= SfJ=IO
504.92908E-03
CAPACITANCE DISTRIBUTION PARAMETER=
.33868742E+02
RESISTANCE DISTRIBUTION PARAMETER=
.62202557E-02
TOTAL CAPACITANCE PARAMETER=
.27975746E+02
CAPACITANCE RATIO=
.20176902E+02
RESISTANCE RATIO=
.15807672E+dl
S E N S I T I V I T Y O F C O E F F I C I E N T M W I T H R E S P E C T T O V O L T A G E F O L L O W E R G A I N K.
M= I
•OOOOOE-99
M= 2
926.55410E-04
M= 3
205.49498E-03
M= 4
105.29029E-03
M= 5
100.OOOOOE-IO
M= 3
719.88977E-04
M= 4
275.48207E-04
M= 5
100.OOOOOE-IO
NETWORK ELEMENT VALUES
ELEMENT ( D =
•26164052E+00
E L E M E N T ( 21=
.29526399E+00
E L E M E N T ( 31 =
.25631027E+00
• 18678524E+00
E L E M E N T ( 4) =
E L E M E N T ( 51 =
.36300010E+00
. 7 3 2 4 2 1 7 8 E + 31
E L E M E N T ( 61 =
•27318214E+01
E L E M E N T I 71 =
E L E M E N T ( 8) =
.6 3 128343E+Q1
•55665150E+01
E L E M E N T t 9) =
ELEMENTdOI =
.56773586E+01
OBJECTIVE FUNCTION VALUE
•12658011E+03
**********
M = I 1J = I
.OOOOOE-99
M= 2
535.47086E-04
*********
irST
POLE LOCATIONS
ZERO LOCATIONS
A P P E N D I X III
NIMERIQAL COMPUTATIONS
-126Thls a p p endix presents the numerical computations for
the networks of Figures 1.2 and 13»
A.
F U M E R I G A L COMPUTATIONS F O R N E T W O R K OF F I G U R E 1.2.
Guil l e m i n presents a meth o d for Synthesizing the t r a n s ­
fer fun c t i o n
(s) =Eou^.(s)/I^n (s) j where
is m i n i m u m
phase and stable, "by means of a lossless n e t w o r k terminated
b y resistors
(see pages 535-537 of Reference 25)»
In the
p r e sent case, we desire to synthesize
Z1 2 (S)=E ^ + 2 - 8 2 8 8 + 4
(III.I).
s +1.414-s+l
Sep a r a t i n g the n u m e r a t o r into its even and odd parts, we
have
K(s *4)
Z1 2 =Z1 2 a +Z1 2 b =
. + K (2.828s)
s +1.414s+l
(m.2)
s +1.414s+l
Guillem i n points out that since the numerator of Z-^ is a
H u r witz polynomial,
its even and odd parts hav e simple zeros
restricted to the j-axis.
Chapter Il of Refe r e n c e 25 shows
that tr a n s f e r functions of the for m of Z ^ a and Z^g^ m a y be
realized b y a lossless n e t w o r k terminated b y a. single r e ­
sistor.
T h u s , the two t r a n s f e r functions are realized s e p a ­
r a t e l y and connected in series to f orm the total transfer
function.
Util i z i n g K = X and a p p lying G u i l l e m i n 8s procedure
to Z^ 2 a and Z-^-J3 separately.
-127-
s 2 +4
J12 a
„ 1.414s
(III.3)
12a"
1+z22a
1+
s2 +l
1.414s '
A n e t w o r k p o s s e s s i n g the above z parameters consists of a
two section ladder n e t w or k whose input shunt arm is the
impedance Z^pa and whose output series arm is the impedance
z2 2 a _ z 1 2 a ‘
Thus 9
2
'12a= ^ I .414s = 0<>177s+ T T T O s
(III.4)
wh i c h is a 0.177 h e n r y coil in series with a 1.414 farad
capacitor.
The series arm is
22a
12a
s2 +l
1.414s
1
A s 2 +!
As'
1.414s “ 1.414s
0.530s
(III.5)
w h i c h is a 0.530 h e n r y inductor.
Si m i l a r l y
is realized as
z12b
V 2.828s
1
A — o---s^+1
(III.6)
12b= l+sgzb = 1+
s^+1
and
o7 0 7 s .
'12b"
s 2 +l
(III.7)
1.414s+
,707s
w h i c h is a 1.414 farad capa c i t o r in parallel w i t h a 0.707
h e n r y coil.
The series arm is
-128-
= .707s =
.707s
1.414s
g2 +1-
z22b z 12Td =
s2 +l
s 2 +l
I_____
1.4 1 4 s +
I
.707s
(III.8)
w h i c h is a repeat of
^12a an ^ ^12h are n o w r e a l i z e d b y connecting I ohm r e ­
sistors across the output t e rminals of the above ladders.
These two networks are t h e n connected in series as shown in
Figure 1.2 to realize Z - ^ C s ) «
B.
M J M E R I O A L COMPUTATIONS F O R N E T W O R K OF F I G U R E 1.3«
Kartell's method
(41) is to be applied to the transfer
function
8^+2.828s+4
(III.9)
s2 + 1 . 4 1 4 s +1
Pantell shows that a m i n i m u m ph a s e and stable T
V
m a y al-
ways be realized w i t h resistive terminations at each end
provide d T
j
axis.
( )=0 or T
( )=0 and no poles are located on the
o
U t i l i z i n g the surplus factor s +8.4 8 s +4 and let t ing
K = K ^ k c,, Tv m a y be put into the form
T_(s)= k1
s +1.414s+l
k2 S2 ^ a8f8^/4 = fIof2
s +8.48s+4
(III.10)
-129where b o t h
and. fg are p o s i t i v e real.
Set t i n g k^=l and
k 2=0.2 and f o l l o w i n g P a n t e l l 1s procedure
Z-] = __ 1 _ _k ' f = s 2 -i-1.414s-fl _ 0 . 2 s 2 + 5656 s +Q.S
^2 2
I" ^ f 1
s +8.48s+4
s +8.48s+4
"
0.8s +0.848s+0.2
s2 +8.48s+4
(III.11)
and
a + 8 .48s +4
"I" kgf g
(III.12)
s +2.8 2 8 s +4
P a n t e l l 's method relies on the fact that the surplus factor,
k^, and kg m a y always be chos e n such that f-^, fg, Z^, and
are pos i t i v e real.
ladder n e t w o r k and
en across Y^.
in Figure 1.3.
b ecomes the input series arm of a
its f o l l o w i n g shunt arm w i t h
tak­
Z^ and Y^ are realized w i t h the results shown
L I T E R A T U R E CONSULTED
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7o
B. R 0 Myers, "Contribution to transistor RC network
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19»
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41
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MONTANA STATE UNIVERSITY LIBRARIES
I i i i i r
762 10005632
III
D378
Hogin, J. L.
Active RC networks utilizing
the voltage follower__________
IM A M K A N D
/Z-V- / . PvtnC
A P D ftg a a
_____T
/
2)378
/iG78