Single tree-finding program for digital computer evaluation of topological formulas for network analysi

Chan, S. P.

Rode, F.

English

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COKPUTZ3 LVkLb'IIT i G:: OF TO?OI,CC Z C A L  FOi'JiLJLAS 
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France Rode 
E l e c t r i c a l  Zngineer 
H e w l c t  t -?ac %aril Cocipany 
Palo Alto,  C a l i f o r n i a  . 
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and ' N.68-29357 
. S. Park  Cnan 
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Q C  Unive r s i ty  of Santa Clara 8 .: 
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Ass oc i a t  e P r  o€e s s or of  E i  c c t r  i c a l  Engineering r a l  
'c 5)  San ta  Clara, C a l i f o r n i a  
P P :  
m .- = 
m 
L n  (D A3STRACT = 
Kost of t h e  e x i s t i n e  t o p o l o g i c a l  for ra las  developed for network 
a n a l y s i s  a r e  not  well s u i t e d  for  d i g i t a l  computation, 
Chan n o d i f i e d  Kayeda' s formulas 
Recent ly  Chan and 
I , .I. 
L I -- I 
s o  t h a t  t L e  t o p o l o g i c a l  terinr :an be conputed by means of a s i n g l e  tree- 
finciing program. 
I n  i ~ l s  work it is S~,OT,JC -:?,at t h c  cc. x c r a t i o n a l  p rocess  lor e v a l u a t i n g  
(1) and ( 2 )  can be shorzened cons ide rab iy  'c_. xhe use of a new t r e e - f i n d i n g  
program. S p c i f i c a l l y  it i s  shown t h a t  1; i s  equa l  t o  t h e  trees of a 
subnetwork ;{ of tne given network 11, and t h a t  W 
can 'os a u t o x i t i c a l l y  computed from t h e  set  0:- 2 - t r e e s  requirec f o r  t h e  
eya luh t  i o n  of Wl,l, .) 
of Char, and Chan s i n c e  t h e  t r e e  f i n d i n g  progran i s  a p p l i e d  on ly  f o u r  t imes,  
i n s t e a d  of n i n e .  
12 ,1 2 1 -5 2 I , 1 2 11' ,22'  
Th i s  r e p r e s e n t s  a s i g n i f i c a n t  improvenent of t h e  work 
A prograx €or cv:.l-,iatj.ng t h e  v a r i o u s  t o p o l o g i c a l  tcrrns i n  (1) and ( 2 )  
has been w r i t t e n  tase; .  on tLesc p r i n c i p l e s  u s i n g  t h e  new tree findin,o: submomam 
ueveloped by t h e  m i J ~ > . x s e  .
v L .  
\ 
~ -------1- 
*:"his Gl-k was supnorted i n  p a r t  by t h e  I ; ~ ? t i o ~ i ~ l  Aeronautics and Space 
Adminis t ra t i  on, Grant No. NCf\-05--017-012 e 
https://ntrs.nasa.gov/search.jsp?R=19680019885 2020-03-12T09:51:40+00:00Z
1. 
2 1  z 
Franc e Rode 
E l e c t r i c a l  Xiiginecr 
Hewlett-Pac%ai-d Coxpany 
Palo Al to ,  C a l i f o r n i a  
and 
S. Park Chan 
AssociaLe Professor  of E i e c t r i c a L  Xngineering 
, Univers i ty  of S a n t a  Clara 
, S a n t r  Ciara, C a l i f o r n i a  
209 ma Al1113Vd 
Topolocical  f o r n u l a s  f o r  network analysis have been developed by 
numerous au tho r s .  [1]-[ 1 0  j 
of ih5ayeCa 2nd Seshu [ 3 j f o r  d i g i t a l  c o q x ' t a t i o n  u s i n g  n ine  s e t s  of 
t r e e s  f o r  o b t a i n i n g  a l l  t h e  necessary f a c t o r s  i n  t ne  expres s ions .  
t h i s ,  s t udy ,  t h e s e  t o p o l o g i c a l  forrnulas a r e  f u r t h e r  developed so that 
only  four  s e t s  of t r e e s  are r equ i r ed ,  u s i n g  on ly  one t r e e  f i n d i n g  p r o g r m .  
I n  a d d i t i o n ,  a t r e e  f i n d i n g . a l g o r i t h m  i s  developed f o r  computing a11 
t r e e s  of a network wi thout  dup l i ca t ion .  
Recerltly. Cilan an2 Chan n o d i f i e d  t h e  f o r n u l a s  
I n  
, 
'Jfi r c r,< 
2 Oi'EIi-CIRCUIT IlfZ'EiI$2fC2 
lJ&rCi)! 
, zoc 
The t o p o l o g i c a l  e q r e s s i o n  f o r  t h e  open-c i r cu i t  impedance, Zoc of a 
p a s s i v e  two-port 
Nayeda and Sesku [l] , is: 
?: w:-chout m t u a l  inductances  ( F i g .  11, as given by 
- 
zoc - 
1 
v - 
i - 
whcre V i s  a sun of  all t ree-admi t tance  'produczs of t h e  network N, 
as?. 
s a x  ne.-l;trork. 
z&C;; . 'J, 
corxiected. p a r t ,  and t h e  v e r t i c e s  on t h e  other s i d e  i n  t h e  o t h e r  connected 
p a t  of the 2- t r ee .  V can be generated from t h e  graph of N by epply ing  
E 'iree fincling p r o g a m .  As known, lJi o f  r\T can be obta ined  by 
c o d e s c i n e  v e r t i c e s  i and j of N '  to form sub-network il' [U]. 
\.I i n d i c a t e s  t h e  sun?riiztion of 2-tree adriii'ctance products  of t,he 
The t y p e s  of 2- t rees  are de f ined  by t h e  subscr ip- i s  of 
nnrx ly ,  t h e  v e r t i c e s  on one  s i 6 e  of t h e  coimnc are ir, o m  
, J  
i j  
I_- .-- 
*This work w a s  su.pporied i n  p a r t  by t he  National Aeronaut ics  and Space 
' 
Adminis t ra t  ion  ~ G r a n t  No. llGi?-O5-017-0l2. ' 
11' ' i s  t h e r e f o r e  t h e  sum of all t r e e  s d n i t t a n c e  products  o f  N 5,l' 
and w 2,2' i s  t h e  sum o f , a l l  t r e e  admit tance products  of  N22' *. 
and \J can be found w12,1 I 2' 12' ,.l'2 
Iiere we s h a l l  shor.r t h a t  
N Observe t h a t  11' by s e l e c t i n g  t h e  t r e e s  .of 
-t w - + '(J 1,1'2 - w122' ,1' + w12',1'2' 12' , 1 1 2 .  + Wl,J'22' 1,l' - w12,1' - w 
w12 ,1' 2' 
52,1 '2 I - 5 2  ' ,1' 2 
Thus, when a tree of 
if it has v e r t i c e s  1 and 2 i n  one par t  and v e r t i c e s  1' and 2 '  j n  t h e  nt.her 
p a r t  of t h e  2-tree of  
of  t rees  i s  much s imple r  and faster t h a n  e s , t a b l i s h i n g  
by us ing  any t r e e  f i n d i n g  process  f o u r  times as done by Chan and Chan [IO]. 
3. . SXOZT-CIRCUIT ADXiTT1II;K!E , 
Nil, 
,N . 
i s  found, it i s  a l s o  p l aced  i n  sum 
i s  generated similarly.  Such s o r t i n g  . w12',1'2 
M/tn!l)c 
> %c 
Assuming t h a t  Zoc i s  a l r eady  known, t h e  on ly  te rm t o  be determined i n  
t h e  f o r n u l a  f o r  Y *  [ i l l  sc 
w12 ,1 I 2 I -52 11' 2 - 
ysc - 
52,1'2' 412' ,112 . w l , l a  
i s  XU, where 
x u  = 3 
12',2,1' + 5,2,1'2' + %2,2' ,1' u1,2',1'2' ( 3 )  
The syrfibol U , i n d i c a t e s  t h e '  sumnation o f  3-tree , a & n i t t a n c e  products  where' 
groups of i t s  s u b s c r i p t s  i n d i c a t e  t h e  r e l a t i o n s h i p  of four v e r t i c e s  (1,l' ,2,2' ) .  
Iiere w e  s h a l l  s t a t e  and prove t h e  fo l lowing  theorem. 
n Lheorerr, 1. The expres s ion ,  Z3 ,  i s  equal t o  t h e  sum of a l l  t ree-admit tance 
( F i g .  2). 11' ,22' produc t s  of t h e  subnetwork N 
- Proof. 2y combining v e r t i c e s  w and 2' i n t o  a s i n g l e  v e r t e x  2" a subnetwork 
N 
2-tree a 6 x i t t h n c e  p roduc t s  Ii o f  t h e  'network X 
of  a l l  t r e e  admit tance products  of I\: 111,221 . 
used to ei.n?h~tsir/ l~ +;hat 
r e su i t s  ( F i g .  3 ) .  S i m i l a r l y ,  as shown earlier for w t h e  sum of i , J '  22 ' 
i s  equal  t o  t h e  sui 22 ' 
d i f f e ren t  ~ ~ * ~ ~ ~ ~ ~ w - ~ y ~  i s  used.  ) 
1,l' 
(Here I(, i n s t e a d  of W ,  i s  
. . .  
. .  5,l' = K 1(2,2'), 1' + 5,1q2,21) 
- ,I' + IC2,12' ,1' + K1,1'2,2~ + K1,2,1f2" ( 5 )  
i( being merely a symbol of summation of t h e  k - t r e e  admit tance p roduc t s .  
Conseqwntly,  i f  K i s  r ep laced  by U, 3 i t h  t h e  terms rea r ranged  ( 5 )  
be c o m  s 
+ u  + u  + u  
' .  
( 6 )  
5,l' cu = 
t h e  proof i s  complete. and 
(Of 
From t h e  above theorem, it is evident  t h a t  on ly  one s c t  of t r e e s  
) i s  r e q u i r e d  i n  ob ta in ing  CU . -  Nu' ,22' 
The foilobring example w i l l  i l l u s t r a t e  t h a t  t h e  s e t  of  all t rees  of 
corres?onds t o  t h e  s e t  of a l l  t e rns  i n  
W 
CU . Also, it w i l l  shov 
,22' 
t h a t  t h e  t e r n s  involved i n  
f o r  W 
ZxorLigle 1 . Find all t h e  terms f o r  Zoc an6 Ysc for two-port N of  
Fig.  4. 
corqxwison).  E s t a b l i s h  a l s o  t h e  sun of a l l  admit tance p roduc t s ;  each 
a c h i t t a n c e  was given t h e  value of u n i t y .  Tie r e s u l t  i s  p r i n t e d  as t h e  
o u t p u t  of  t h e  computer program. 
can be  extracteci  from t h c  expres s ion  12,l' 2 ' 
1,l' 
(Th i s  exarfiple i s  from Chan and Chan [ 2 ]  i n  o r d e r  t o  make a 
, /' 
Tnis  a lgo r i thm i s  dcvcl'oped t o  compute t h e  tcrrns i n  t h e  t o p o l o g i c a l  
f o r a u l a s  $resented i n  i r e c e d i n g  s e c t i o n s .  I n  t h i s  summary only a b r l e f  
d e s c r i p t i o n  of t h e  b a s i c  p r i n c i p l e  involved and some of t h e  advantages 
of t he  a lgo r i thm a r e  included.  The a lgo r i thm i s  based on t h e  foi lowing 
d e f i n i t i o n  of a t r ee :  
Basica1l.y , .  t h i s  algorithm c o n s i s t s  of  t v o  s t e p s  : 
' *.. 
S t q  1. Form all subgra$is of  G 
( i s o l a t e d  v e r t i c e s  are  not i' p e r m i t t e d ) .  
of G with v-1 e d g e s  and v v e r t i c e s  
Ste? 2. 
Somc of t h e  advantages of t h i s  a1t;orithn are: 
S e l e c t  t h c  connected subgraphs. 
a> ! h e  determinat ion of each t r e e  i s  done' independent ly  so  , t h a t  t h e  
t r e e s  need not  be s t o r e d  but car, be p r i n t e d  i m m d i a t e l y  upon 
being found--this being an advantage over  t h e  a l g o r i t h m  developed 
by KayeGa and Seshu [12] .  . 
b )  Mo e x t r a  s t o r a g e  i s  needed for subgraphs as r e q u i r e d  i n  Minty 's  
a lgo r i thm [ 13). 
c )  The s e l e c t i o n  of subgraphs i s  not based on c i r c u i t  comparison, and 
hence it i s  no t  necessary t o  gene ra t e  (and hence t o  s t o r e )  any 
c i r c u i t s  i n  t h e  e n t i r e  process .  
6. CONCLUSIO~~S 
Tie example i l l u s t r a t e s  ' t h a t  Zoc and . Ysc can b e  eva lua ted  by 
&encra t ing  only fou r  sets  of t r e e s .  
used wi th  t h e  fol lowipg networks : 
The sane t r e e  f i n d i n g  a lgor i thm i s  
1. Xettrork K for  c o q u t i n g  V 
2. Xetwork (ob ta ined  from N wi th  v e r t i c e s  1 and 1' s h o r t  c i r c u i t e c j  
3. laetwork N ' (ob ta ined  from N wi th  v e r t i c e s  2 and 2 '  s h o r t  c i r c u i t e d )  22' 
for computing bJ2,21 Y ' 
(obtained from N w i t h  v e r t i c e s  1 and 1' s h o r t  11' '22' 4 .  !Jetwork N 
c i r c u i t e d  t o g e t h e r )  for  computing CU . 
Although t h e  example i s  simple,  t h i s  program can  be used fo r  a two- 
p o r t  nexwork with up t o  23 e lexen t s  and f i f t e e n  v e r t i c e s .  
e a s i l y  e q a n d e d  i f  t h e  speed of t h e  corfiputer p e r n i t s  so .  
It can  be 
I t  should a lso be  po in ted  out t h a t  t h e  same p r i n c i p l e s  can be extended 
to t h e  a n a l y s i s  of a c t i v e  networks w i t h  mutual inductances.  
[l] V.S. P e r c i v a l ,  "The Graph of Act ive  Networks," ?roc.  IEEE, vol .  102,  
p t  C ,  pp 270-2'78; Septa 1955. 
- ,  
. [ 2 ]  S. 5. Nason, "Topological  Analysis  of Linear  Ronrec iproca l  Networks , ' I  
.Proc. IRE, v o l  45,  pp 829-835; June 1957. 
[ 3 ]  W. Xsyeda and  S. Seshu, "Topological  Formulas for  Network Funct ions , ' I  
Univ. of Ill. Bul . ,  v o l  55, No. 23, Urbana, Ill., Mov. 1957. 
It. Kayeda, "Topological  Formulas f o r  Act jve Networks," I n t .  Tech Rt., 
'Ito. 8,  U . S .  A r m y  Contract  Eo. DA-11-022-O2D-1983, Univ. of I l l . ,  
Urbana, Ill., Jan .  1958. 
C.L. Coates, "General Topological  Forinulas fo r  Linear  Network Funct ions , I '  
I R X  T r a n s .  on C i r c u i t  T'neory, CY-5, pp 30-42; March 1958. 
[4] 
153 
[ 61 \!.I{. Chen, "Topological  Analysis  for' Active Ifetworks , I 1  IFXE Trans .  on 
C i r c u i t  Theory, CT-12, pp 85-91; >larch 1965. 
[ 7 ]  A .  Talbot, "Topological  Analysis  of General Linear  Networks , ' I  IEZE 
1-1 l r a n s .  O n  C i r c u i t  Theory, CT-12, 92 170-180; June 1965. 
{ a ]  2 . A .  Calahan, "LirLear Settrork Analys is  and R e a l i z a t i o n  D i g i t a l  Computer 
Programs: An I n s t r u c t i o n  Iv!anual," Univ. of'I11. Bul. Vol. 62 ,  
Eo. 58, Urbana ,  111.; Feb. 1965. 
191 F.F. Kuo, "$!etworlc Analys is  by D i g i t a l  Conputer," Proc .  IEEE,  v o l .  54, 
Xo. 6, pp 820-829; June 1966. 
.. 
1103 S.P. Chan and  S.G. ~ h m ,  "%lopological Forrnulas for D i g i t a l  Computation," 
Proc. l s t  Annual. P r ince ton  Conf. on I n f o r m t i o n  Sc iences  and S y s t e m ,  
pp 319-323, Xarch 1967. 
S.P. Chan, In t roudc to ry  Toaol.oqical :..-.,-pis of E l e c t r i c a l  Xctworks, 
t o  be publ i shed  by Iiolt ,  Eiinehari and Winston, i n  1968. 
[111 
[12 ]  \.I. Xayeda and S. Seshu, "Generation of Trees  without  Dup l i ca t ion , "  
IEJZE Transac t ions  on C i r c u i t  Theory, v o l  CT-12, ILo. 2 ,  June 1965. 
G.J. Mirky, "A Sirn;>le Algorithm for  L i s t i n g  A l l  t h e  Trees  of a Graph," 
IEEZ 'i'ransactic,:.~ on C i r c u i t  Theory, Idarch 1965, p. 120. 
[13] 
. I  . .  
' a .  
. .
I c 
l !( i/26/67 * * .  
F I N O  Z O C  A N D  Y S C  O F  RETWORK N WITH 6 V E R T e p 6  EDG.,AkD 
IKCIDENCE K E T i i l X  A 
1 
2 
3 
4 
2'  
I '  
1 2 3 4 5 6  
1 0 0 0 0 0  
c 0 0 0 0 1  
1 1 1 0 0 0  
G G ; O l l  
0 0 0 1 1 0  
0 1 0 1 0 0  
C 
,.--. 
i 
r' -. 
\ .  
\ .  
. .  
F O U R  S E T S  O F  T R E E S  
~ 
TREES O F  N T H E E S  GF T R E E S  O F  Nz2J 
1 1 2 3 4 6  .1  1 3 4 6  1 1 2 3 4  
2 1 2 3 5 6  2 1 3 5 6  2 1 2 4 5  2 1 3 6  3 1 3 4 5 6  
4 1 2 4 5 6  
3 1 4 5 6  3 1 2 4 6  3 2 3 5  
4 2 - 3 4 6  4 i 2 3 5  4 2 3 6 .  
5 .2  3 . 5  6 5 1 3 4 5  5 1 3 4  
6 2 4 5 6  6 1 2 3 6  6 1 4 5  
7 3 4 5 6  7 1 3 4 6  7 1 4 6  
8 2 3 4  
9 2 4 5  
IO 2 4 6  
11 3 4 5  
1 2  3 4 6  
SUM U = Y 1 Y 3 Y 5 + Y 1 Y 3 Y 6 . ~ Y 2 Y 3 Y 5 ~ ~ 2 Y 3 Y 6 + - Y l Y 3 Y 4 ~ f - y l Y 4 Y 5 +  
+ Y 1  Y 4  Y 6 + Y 2 Y 3  Y 4 + Y  2Y4Y 5+ Y 2  Y ~ Y ~ + Y ~ Y ~ Y ~ G Y ~ Y ~ Y ~  
.r i t  r Yl- lObM VZ=lF  Y3=1CHI'I Y4=lOHM Y5=1F .Y6=1CHM 



Computer evaluation of topological formulas for network analysis

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