Shannon - A Symbolic Analysis of Relay and Switching Circuits

7/30/2019 Shannon - A Symbolic Analysis of Relay and Switching Circuits

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A S Y l ~ : B O L I C A l ~ A L Y S I SOF

by

Claude Elwood ShannonB.S. , University o f E ichigan

Submitted in Par t i a l Fulfi l lment of theRequirements for the Degree of

lLASTER OF SCIE1JCEfrom tile

: ' :assachusetts Inst i . tute of Technolcgy1940

Sigl18.ture 01-- Author _Department of Electr ical Engineering, August 10, 1937

S i g n a t ~ r e of Professorin Char38 of R esearch- - - - - - - - - - - - - - - -

S i ~ Y J . c l t u r e of Cnclirman of Deuartment- - (COfficittee on Graduate Students ~ L ~ - ~ - - ~ - ~ - _ - . ~ , ~ ~ ~ , - - - -Jy


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,. f'; \~ \ . ....

TABLE OF C O ~ T T E N T Spage

Int roduct ion; Types of Problems - - - - - - - - 1II ser ies-Para l le l Two-Terminal Circui ts

Fundamental Def in i t ions and Postulates - - - -Theorems - - - - - - - - - - - - - - - - - - - - - 6Analogue With the c alcu lu s o f Proposit ions - - - 8

I I I Mul t l - Irarminal and Non-Saries-Paral le l Networks - - -18Equivalence of n-Term1nel Networks - - - - - - - 18star-Ivlash and Delts- 'Nye Transformations - - - - - - 19Hinderance Function of a Non-Sarles-Paral le l Network 21S1roul t anaous Equa t ions - .. - _. - - - - . - - - - - 2 ~Matrix Methods - - - - - - - - - - - - - - - - - 25Spec ia l Types o f Relays and S ~ i t o h e s ... - 28

\

IV ~ ~ t h a s 1 s of Networks - - - - - - - - - - - - - - - 31General Theorems on Networks and Functions - - - - 31Due 1 ,N etworks - - - - - - - - - - - - - - - - - - - 36synthesis of the Gapersl Symmetric Function - - - - 39Equations from Given Operating Charac t e r i s t i c s - - - 47

v I l lus t r a t ive Examples - - - -A select ive Ci r cu i t -An Elec t r ic Comb1nat1.on Lock -A Vote C O U ~ t1ng C1rcu1 tAn Adder to the Ba sa Two

- - - - - - - - - -51- - - 52

555859

A FactorReferences 69


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ACKNOvVLEDG1ffiNT

The au thor i s indebted to PrOfeSSOI' F. L.Hitchcock, who supervised the thesis , for helpfulcri t icism and advice.


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I In t roduc t ion : Types o f Prob'lams

In the control and protect ive c i rcu i t s of complex e lec t r i ca l systems it i s frequeutly necessary tomake in t r i ca te interconnections of relay c o n ~ a c t s andswi tches . Examples o f these c i r cu i t s occur in au to-ma.tic te lephone exchanges , indust r5 .a l motor con t ro lequipment and in almost any c i r c u i t s designed to gerform cornplex opera t ions a.utomatical ly. Twu problemstha t occur in connect ion with such n e ~ w o r k s o f switcheswi l l be t r e a t ed he r e . rb.e f i r s t , which wi l l be c9.11edana lys i s , i s to cietermine th e o p e r a ~ i n g charac te r j.. s-t i c s of a given c i r cu i t . It i s , of course , alwa.ys poss ib le to analyze any given c i r cu i t by set t ing up a l lpossible sets of i n i t i a l condi t ions (posi t ions ofswitches and re lays) and following through the chainof event so ins t iga ted . This method i s , however ,very tedious and open GO frequent error .

The s ec ond p roblem i s tha t o f syn thes i s .Given ce r t a i n cha r ac t e r i s t i c s , it i s required to f ind9. c i r cu i t irLcorporating these cha r a c t e r i s t i c s . ~ r h esolut ion of th i s type of problem i s not unique and iti s -cI-J.erefore addi t io l la l ly desi ra .ble t h a t the c i r c u i tr equ i r ing the l e a s t number o f sVii te h blades and r e l ay


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C O L ~ a c : s be ~ o u n d . Although a solut ion can usual ly beobta i ned. by a lieu t and t ry" m_e tho d , f i rs t sa ti s fy i ngone requirement and then making addi t ions un t i l a l lare sa t i s f i ed , the c i rcu i t so obtained wi l l seldombe the s i m p l e s ~ . ' .mis method 9.1so ha.s the disadvan-tages of being long, and the resul t ing design oftenconta ins hidden "sneak c i r cu i t s . "

The method o f so lu tio n o f these problems whichwi l l be developed here may be described br ie f ly asfo l lows: An;! c i r eu i t i s represented by a se t o f equa-t i ons , the terms of th e e qu atio ns represent ing the

v a r i o t ~ s re lays an d sw itch es o f the c i r c u i t . A ca l -culus is developed for manipulating these equationsby simple mathematical processes , most of which are

.. s imi l a r to ordinary a lgebra ic a lgor i sms . This Ca.l-culus i s shown to be exact ly analogous to the Qalcu-lUs o f P,ropos i t ions \ l sed i n the s 'ymbolic s tudy o flog ic . For the synthesis problem the desired charac-t e r i s t i c s a re f ~ r s t wri t ten as a s ~ r s " t e m o f e au atio ns,and the equat ions are then ffianipulated into tha formrepresent ing the simplest c i rcu i t . The c i r c u i t maythen be inwediately drawn from the equat ions. Byth i s nethod it i s always possible to find the simplestc i r c ~ i t containing only series and 9ara l l e l connec t ions ,


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and for ce r t a in types of funct ions it i s 903sib le tofind the simplest c i r c u i t conta in ing any type of con-nec tion . In the analys is problem the equations r e ~ r e -sen t ing th e given c i r c u i t a re wri t t en and may then bein terpre ted in terms of the ogerating cha r a c t e r i s t i c so f th e c i r c u i t . It i s alGo possible with the ca lcu lusto ob ta in any number o f c i r cu i t s equ iva len t to a givenc i r c u i t .

phraseology wil l be borrowed frJm ordinarynetwork theory fo r con06pts in swi tch ing c i r cu i t stha t are r o u g h ~ y a L ~ l o g o u s to those of iffipedencJnetworks.


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I I ~ e r i e s - p a r a l l e l Two Terminal J i rcu i t s

F u n d a ~ e n t a l Definit ions and Postula tes . Tve sha 11l imi t our tre atrn8n t to c i rcu i t s conta ining only re lay contacts and switches, and therefore a t any givent ime the c i r cu i t between any tVifo termi nals must bee i the r open ( inf in i te impedance) or Closed (zeroimpedance). Let us associa te a symbol Xao or moresimply X, with the terminals a ana b. This var iable ,a. funct ion o f t ime, lNill be ca l l ed the hinderanceof the two terminal ~ i r c u i t a-b. The symbol 0 (zero)wil l be used to represent th e h inde ranc e of a closedc i r cu i t , and th e s-ymbol 1 (un i ty ) to represent thehinderance o f an open c i r c u i t . trhus when the c i r -\ cu l t a-b i s open Xab = 1 and when closed Xah =O.Two hinderances Xab and Xed will oe said to be equalif \'Vhenever the c i r c u i t a -b i s o'pen, tl1.8 c i r cu i t c -di s open, qnd. v'Jhenevc:r a-b i s c losed , c -d i s c losed .NoW l e t the symbol + (plUS) be defined to mean theser ies connect ion o f the tvYO terr!l inal c i r c u i t s whosehinderanc as a re ad1ied toge"thel' Thus Xa b i ~ c d i:3the Ilderanc e 0 f the e i r e u i t a.cd when b a n d c a r e

c o ~ n e c t e d together . Similarly the product of twohinderances (Xab Xed) wil l be defined to mean the


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hinderance o f the c i r ~ u i t formed by connecting thec i r c u i t s a - b and c - d i n p a r a l l e l . A r e l a y c o n t a c to r swi tch w i l l be r e p r e s e n t e d i n a c i r c u i t by thesymbol i n F i g . 1 , the l e t t e r being the correspondingh i n d e r a n c e f u n c t i o n . F i g . 2 shows t h e i n t e r p r e t a -t i o n o f t h e plus s i g n and F i g . ! the r:lul t i p l i c q t i o n s i g n .

Xaba ..._IIIlIftIO Do---..b x y _ (X+Y)....... 0-0'- ' - - - e o - - - x-Ci}. _ Xy- - - -0 . . . . . . .Fi g. 1 F i g . 2 Fig. :3

This c h o i c e o f symbols makes t h e m a n i p u l a t i o n o fhinderancea very s i m i l a r to ordinary numerical a l g e ~b r a .

It i s evident tha"G \vi t h the a b o ~ l e d e f i n i t i o n st h e f o l l u w i n g p o s t u l a t e s w i l l h o l d :

Po s tula. te s1 . a . 0 - 0 = 0

b. 1 at 1 =12. a . 1 0 = 0 1 =1

o. 0-1 ~ ~ . O =0

A closed C i r c u i t i n p a r a l l e lvii th a c los ed c i rcu i tis 8.c l o s ed e i r e ui t An aoen c i r c u i t i n s e r i e svv1th an open c i r c u i t i s anopen c ir'c u i t.An o?en ~ i r c u i t i n s e r i e sw i t h a c l o s e d c i r c u i t i ne i ~ h e r o r d e r i s an ogene i r e u i "t .A c l o s e d c i r c u i t i n p a r a l l e lw i t h a n open ~ i r c u i t i ne i t h e r o r d e r is a c l o s e d

~ i r c u i t .


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3 . a ~ 0 + 0 = 0 A closed ~ i r c u i t in 8erieswith a closed c i rcu i t i s ac l as ad rC\J . i t.

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b. 1-1 = 1 An ODen c i rcu i t in para l le lwith an open c i r cu i t i s anopen c i r eu i t .4: At a n ~ r g i van tj.me e l t h e r X =0or X : 1 .

These are suf f ic ien t to develop a l l the theo-rems ";J"hich wi l l be \ l sed in connect ion \vith c ir 4cui t3con ta in ing only se r i e s and ' para l le l connect ions . The

p o s ~ L 1 1 a t e s a re ar ranged in pa irs to emphasize a dua l i tyre la t ionsrJ . ip betweerl the opera t ions o f addi t ioI l andml11t ipl icat ion ar.Ld th e q .u an ti t ie s zero C:ind o n ~ . Thus

yif in any of the BpOs tu la t e s the zero ' s are replacedt;y one ' s and the mul t ip l i ca t ions by add i t i ons and vicev9rs a , tb_e corresponding bi pos tu la te wil l . reS'"tllt.This fgc t i s of great importance. I t gives e ach theoreme d t l a l ~ it being necessa ry to prove only o'ne to es ta -bl i sh both. T ~ e only one of these postula tes whichdi f fe r s from ordinary algebra i s lb . However, th isens b l e s g J ~ e f - 1 t s impl i f i os ti onsin th_e me!lipula t i on of

Theorems. Irl. t h i s sec t ion a numbar of theorems gov-erning the combination of hinderances wi l l be ~ i v e n .Ina smucl:1 a s an;T of the theorems ma,,- he Druved by avery s imple process , tl1.e proofs wi l l not be g:iver


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except fo r an i l l u s t r ~ t i v e example. Tbe method ofProof i s t h a t of' "pe r fec t induc t ion ," i.e., the ve r i -f ica t ion o f the theorem fo r a l l possi ble cases . Since

P o s ~ l l p t e 4 each var iab le i s l imi ted to the valueso Bn d 1 , th i s i s B simple rna t t e r . Some of the the oremsmay be Droved more e legant ly by recourse to p ~ e v i o u stheorems, Cut the method o f pe rfec t induct ion i s so un i -versal tha t i t i s pro ba b ly to be pre fe r red .

1 . 8 . X + Y y + xb . xy =yx

2 . a . x + (y + &) = (x + y) + gb. X(YIi) .. (xy)&.

3 . a x (y + ii ) .. xy + X5-b. x + yfll - (x + y) (x + a-)

4. a l-x .. x-b. 0 + x - x-5. a 1 + x 1b. O x = 0

For example, to Drove theo rem 4A, note the tX i s e i t he r o r 1. If it i s 0, the theorem f'ollowsfrom 1)ostll1ate 2b; if 1 , it folLOWS from rOs ' tulate 3b.

'Je sha l l now def ine a new ope! 'ation to beoal l ed negs t ion . rhe negat ive o f a ~ i n d e r a n c e X Wil lbe Writ ten Xt and i s defined as a var iable Which i sequa l to 1 When X equals 0 and equal to 0 When X


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equals 1 .. If X is the }1-ind-=Jr'3TIce of the make contac t so f a rela:T:i then XI i s trte hinderance o f the break con-t ac t s of the same re lay. The def in i t ion of the nega-t ive of 8 hinder8nce gives th e fo llowing theorems:

6. 8 X + XI - 1b. V "" - 0n . ~ \ . ...

7 . a . 0 ' - 1-b. 1 1 - 0-

8 . (X t ) I - X-Analogue ~ f l i t h the ca l cu lus o f P r o p o s 1 t i o t L ~ s . '.Te Rrenow in A posi t ion to demonstrate the equ iv alence o fth i s calculus vvith cer ta in elementary p3r t s of th ecalCtl1u_s of propos i t i ons . Yne a lgebra o f l o ~ c (1) I(2 ) , (3 ) or ig ina t ed G e o r ~ e Ecole , i s a symoolicmethod of i nve st ig a ti ng l og ic a l r e la ti on sh i p s. Thesymbols o f Boolean algebra admit of two log ica l i n t e r -pl"e ta t i6ns . If i n t e rp r e t ed in terms of c lasses , th evarta b1 '3 3 are no t l im i t ed to tn.G p os sib le v alu eso and 1 . This in te rp re ta t ion is k n o ~ v n as the algebraof c la s se s . I f , hoVJ8Ver, the terms a re taken to r e p r ~ e -sen t propos i t ions , \ve have th e calClll1 . .1S o f p,roposi t i onsin W ~ i c h var iables are l imi ted to the values 0 and 1*,*I'his r e f 3 r s on ly to the c18 s s i c8 l theory o f the oa1cul'J.s o f Propos i t ions Recent ly some ~ v o r k ha s beendone vvi trt l og i ca l systems in vVhich pro posi t ion s ma:Tha ve more than tvvo t1 t ru th vallIe s ."


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9as are the hinderance functions above. Usually t ~ e twosUbjects are developed simultaneously from tae Same se t01 postula tes , except for the a dd itio n in t ~ e case ofthe Cctlculus of Proposit ions of a postulate e ~ u i v a l e n tto postulate 4 aoove. S.V. Huntington (4) gives tnefollowin5 se t of postula tes for symbolic logic :

1. Tne class K contains a t lea s t two dis t inc telements.

2. I f a and b are in tne class K tnen a+ b i sin tl1.e class K.

3- a + b z b + a4. ( a . b) + C = a + (b ... c)5. a+a:a6. ab + ab ' :: a where ab is defined as (a'+ b ' ) '

I f we l e t t ~ e class K be t ~ e class consist ing of thetwo e l e m ~ n t s 0 and 1 , taen tnese postulates follow fromthose given on pages 5 and 6. Also postulates 1 , 2,and 3 given tnere can be deduced from Huntington'spostulates . Aduing 4 and res t r ic t ing our discussionto tile CEi.lculus of propos i t ions , i t is evident t ha t aperf 'ect tine.logy exis ts between tne calculus fo r swi tcn-ing c i r eu i ts B.Jlli t I l i s br2J1Ch of symbolic loSlc . * Thetwo in terpre tc t ions of t ~ e symbols are sh:wn in Table 1.

*This 8.nalogy lllay also be seen from a s l i sh t l y d. i f fe rentviewpoint. Instead of associat ing Xab di rec t ly wltfi thec i rcu i t a-b l e t Xab represent t ~ e grooosi t ion tnat the


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Ole to t h i s analogy any theorem 0 f tre calculuso f Proposit ions i s also a t rue theorem if i n t e r p r e te d i nterms of ~ a l a y c i r c u i t s . The remaining theorems in t h i ssection are taken d i r e c t l y from t h i s f i e l d .

IDe M o r ~ 8 n s theorem:9 . 8. (X + Y + ) t =X t . y ' . Z '

b. (X.Y.Z )' =X' + y, + Z1 + This theorem gives the negative o f a sum or product interms of the negatives o f t h e summands or f a c t o r s . I tmay be e a s il y v e r if ie d for two terms by ~ ~ b s t l t u t 1 n ga l l possible values and then extended to any n u m b e ~ nof variables by mathematicsl induct ion.

A function of c e r t a i n varia b l e S X l , ~ 2 " - - - -Xx,. 1sany express ion formed fl'om t h e v a r i a b l e s w i t h the opara-t10n S 0 f a ddi t1on, mul t i p l i ca t1on, and ne ga t1on. Then o t a t i o n t'(Xl , X2 , Xu) w i l l be used to r e p r e s e n t a

f l J . : 1 ~ t 1 o n . Thus we m1@:ht have f{X, Y, Z ) ) =XY + X' ( y ' + Z ' ) .In i n f i n i t e s i m a l calculus it i s shown t h a t any r u n ~ t 1 o n(providing i t i s continuous and a l l d e r i v a t i v e s era eon-

, t inuous) may be expanded i n 8 T8:rlor Ser ie s A somewha tsimilar expansion i s possible i n the calculus of propos1-t i o n s . To develop the s e r i e s expansion of funct ions

(Foo tnote con tin ued from preceding page)c i r c u i t a - b i s open. Then a l l the symbols are d i r e c t l yi n t e r p r e t e d a s P:--'oposit1ons and the o p e r a t i o n s o f a d d i t i o nand ~ u l t i p l l c a t 1 o n w i l l seen to represent s e r i e s andp a r a l l e l c o n ~ e c t i o n s ~


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TABLE I

A n a l o ~ u e Between the Calculus of Proposit ionsand the Symbolic Relay Anslysis

symbol

xo1

X +Y

XY

x'

=

In terpre ta t ion in relayclrcu1csThe c1rcul t X.The c1rcl.A.i t 1 s c losed .

The c l r ru1 t i s open.fhe ser ies connection ofc1reu1 t s X and YThe para l le l connectionof c1reu1ts X and Y'rhe cireu1 t whic h 1 S 0 penwhen X i s closed, andclosed when X i s open.The a t rcu1 t s open andclose simultaneously.

In terpre ta t ion in thecalculus o f P ro po sitio nsThe proPosit ion X.The proposition 1sfa l se .The proposit ion 1st rue .The proposit ion which1 s t rue if a1 the r X o:rY 1s true. ,The proposit ion Which1s true if both X an.dy a r e t rue .The contradictory o fproposi t ion X.Each propositionimplies the other .


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10.

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f i r s t note th e fo llowing equations:a . f (X l X2 Y ) :: ~ i{,l X2 y ) + X' r(0 , X2 Xn )" "'-0, f '" , 11 I ,b. f(Xl Xn > [f(O,X2 Xn) + x1l.(f(l,X2 Xn)+Xi)

These reduce to iden t i t i e s i f . we le t Xl aqual ei thero or 1 . In these equations the fUnction f is sa id tobe expanded arout Xl. The coeff ic ients of X end Xi

i l l ~ 1 1in ~ a r e func tio ns of the (n- l ) variati les X2 XUand may thus be expanded smu t any o.f these var iablesin the same manner. The addi t ive terms in \a:ke1so maybe exnanded in th i s manner. Thus we get :11. a . f(XI Xn ) =XIX2 f ( l ~ l , X 3 Xn) + X I X ~ f(I,O,X3 Xn)

+ X1X2 f(O,1,X3 Xn ) + X ! X ~ f(O,O,X5 Xn )b. f(Xl Xn ) e [Xl + X2 + f(O.O,X3 Xn ) ] [Xl +

X ~ + f(O, l l .Xn ) ] - [Xi + X2 + f(l.O Xn ) ] lX' + XI + f ( l , l ,X x )]1 2 3 n

Continuing th is process n t imes we wil l arive a t thecomplete ser ies expansion haVing the fo I'm:

1 1) XtX2 X + + f(O,O,O O)1 nXIX' x'1 2 nb. f(XI Xn ) : [Xl + X2 + xn + f(O,O,O O)]- [X i + X2 + Xn + f(l,O!O O' [Xl+ Xl + xt + f ( l , l , l ) ]2 n


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By 129, f 1s equal to the sum o f t'he produ c ts formedby permuting primes on the terms of X1X2 Xn in a l lPossi ble ways and giVing each product a coe f f i c ien tequal to the value of the fU.nction when tha t producti s 1 . Similarly fo r 12b.

As an appl icat ion of the series expansion itshould be noted tha t if we wish to f ind a c i rcu i trepresent ing any gi van function we can always expandthe function by a t t he r lOa o r lOb in such a way t ha tany given variable appears a t most twice, once as amake contact and once 8 S 8 break contac t . This i sshown in F i f l;. 4:.

x'1

x,={Fig . ~

Simi la r ly by 11 any o the r var iab le need apJEsr no morethan ~ times (two make and two break contacts) e tc .

,A general izat ion of De Morgans theorem i srepresented symbolically in the following equation:

13. [ r (Xl,x21 ~ I + , .)]1. = f ( X i ' X ~ Xri ' ,+)By th is we mean t ha t the negative of any function may


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be obtained by r e p l a c i n ~ each var iab le by i t s negat iveand ln tarchanging the + and s'YUlools. EXPlic i t andimpl ic i t parentheses wi l l , of course, remain in thesame plaees . For example, th e n eg ativ e of X + y .(Z + ' fIX') wi l l be XI (y ' + Z' ( ' ~ l l + X).

Soma other theorems usefUl in s impl iyingexpress ions e re g1 van below:

14. 8 X - X + X =X + X + X - e tc .- -b. X - X X - X X X - etc .- - -

15a 'S . X + XY =Xb. X(X + y) =X

16. 8. XY+ X ' ~ - X:Y + x t ~ + 'ye-b. (X + Y)(Xf + 8 ) = (X + Y) (XI + ~ } ( y + ~ )

17 . a Xf(X) =Xf(l)b. X + f(X) =X + f{O)

18. 8. X'f(X). =X1f(O)b X' + f (X) =XI -+. f ( l )...

A n ~ express ion formed with the opera t ions o fadd i t i on , mult ip l ica t ion, and n eg atio n representseXPlic i t ly a c 1 r ~ ~ 1 t containing only ser ies andpa r a l l s l connec tior!. s Su ch a c i rou1 t wi l l be ca l l a da ser ies-para l le l c i r cu i t . Each l e t t e r in an axpres-sian of th i s sor t represents a make or break r e l ayconts c t , or e swi toh blade and conts c t To f ind thec i r c u i t r equ i r i ng the l e a s t number o f contacts , it i s


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therefore n ec es sa ry to manipulate the expression intothe form in which the l e a s t n u m b ~ r o f l e t t e r s appear .The theorems ~ i v e n above are always su f f i c i en t to dot h i s . A ].1 t t l e pract ice in t he m a ni pu la tio n of theses;rm1:o1s i s 811 t ha t i s requ i red . For tuna te ly most ofthe theorems are ex ac tly the same as those o f numeri-cal a l ~ e o r a - - t h e associa t ive , commutative, and d i s t r i b -ut1ve laws of algebra hold here . The wri te r has foundtheorems 3 , 6, 9 , 1 4 , 15, 16a, 17, and 18 to be es-peclal ly usefu l in the s impl i f ica t ion of complex ax-pres s ian s .

AS a n examp l e of the 81mp11 f ica t ion of ex-press ions consider the c i rcu i t shoWn in F i ~ . 5 .

5 '~ . - .v y_- . . . . .- -0 0---0 ~ WI

xF i Q;. 5

o

z'. . . . . . '0Z

The hind'3rance funct ion Xab fo r t h i s c i rc l l i t wi l l be:Xab =W+\\II(X+Y) + ( X + ~ H S + W ' + e ) ( ~ ' + Y + S ' V ),"i\[= ~ + X + Y + ( X + ~ ) ( S + l + g ) ( g l + Y + s t V ) v= W + X + y + g ( ~ ' + S ' V )

lthesa reduct ions walee made \'V1"tth 17b us ing f i r s t then X and


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y as the "XU o f 17b . }IO\iV mul t ip ly ing ou t :Xab =W + X + Y + gel + ~ ~ S I V

: W+ x + y + ~ S I VThe c i rcu i t corresponding to th is expression

i s shown in F ig . 6 . Note the la rge reduct ion i n thenumber o f elemen ts.

zw X Ya _.- l IDO ViO----00 DlO- -_ .n .o a----.....- - a

Fig . 6It i s convenient in drawing c i r c u i t ~ to l abel

a re lay With the same l e t t e r as the hinderance ofmake contacts of the re lay . Thus i f a relay i s con-neoted to e source of voltBQ:6 through a network whosehlnder8nce funct ion i s X, the relay and any make con-t ec ta on- i t would be labeled X. Break aontects wouldbe labeled XI. This assumes t r ~ t the relay operatesinstarl t lY and tha t the make oontacts close end thebreak contacts open s1multaneousl .. y . Cases in whichthere 1s 8 time delay wi l l be t rea ted l a t e r .

It i s also poss ib le to use th e analogy betweenBooleian algebra and re lay c i r c u 1 ~ s in th9 oppositedi rec t ion , i . e . , to represent logical re la t ions by


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means of e le c tr ic c ir cu its . Some interest ing resul tzhave been obtained a l o n ~ th i s l i ne , h l t are of no im-portance here .


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I I I Multi-terminal end Non-aeries-paral lel Circui tsEquivalence o f n-Tetlminal Networks.control c i r r n ~ 1 t wil l take the form of Fig. 7 , wherex ,X X are re lays or other devices controlled1 2 nby the e i r eu i t and N i s a network o f relay- contacts andsw1 t ches .

+

Fig . 7

--

I t i s desirable to f ind t ransformations tha t ~ e y beapplied to N which wil l keep the operation of a l lthe rela:vs Xl Xn the same. So fa r he. ve onlycons ide red t rans fo rma ti on s Which may be apPl ied toa two..;terminel ne twork keeping the opera t ion 0 f onere lay in s - 3 1 " ~ i e s With th is network the sama. Tot h i s en.d we s h al l de fine equivalence o f two n--term ..ina 1 networks as fol lows:Def in i t i on : TvvQ n-termina 1 networks hI an d N wi11be sa id to be equivalen t wi th respec t to th e se


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t e rmina l s if and only if X jk =Y jk .j, k = 1 , 2 , :3,-. n' r . 7 r ~ e r e X i s the hinderance on network 1-T b e t ~ v e e n t e rmi -jkne ls ,1 and k , and Y i s tha t fo r 11 between the co r -jk

r e s p o n d i n ~ terminals .Thus under t h i s de f in i t i on th e equ_ivelenc3s

o f the preceding sec t ions were \\'1 th respec t to two

Star-Mesh end Delta-vVye Transformat ions .. - As in ordi -nary network theory there ex ie t :3 l;l:1r to me fJh 2nd de l t ato vvy-e t ransforms t i on s . The de l t a to wye t l ~ 8 n sforms-t ion i s shown in Fig . 8 . These two n e t w o ~ k s areequivalent with respect to the three t e r m i n a l ~ a ,b, and c , since by the d i s t r l l n t i v e law Xab = R(S + T)=RS + RT and s imi lar ly for the o the r pai rs of termi-nels a-c end b-c.

b- - b1R S RS--

ReT STa/ 'cTFig . 8


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The wye to de l ta t ran sformation i s shown inFig . 9 . This follovJS from the f ac t tha t Xab = R + S :(R + S)(R + T + T + S).

( R ~ S )

a

(Tot-a)

c

An n po in t s ta r a1 so he s a me s h e qu 1va 1 en tw ith the cent ra l node el iminated. This i s formedaxe c t ly 8 s in the simple th ree pain t s ta r , by Con-nect1ng each pair of terminals of the mesh through8 h1nderan ce which i s the sum 0 f the co z : ~ e s p o n d i n g

For n :: 5 t h i s i s s h..Q\vn i n ~ F i g . 10 .b

arms of the s t a r .b

&1R

e

c/

F ig . 10

ac____ - - - - . ~ ~ ..... t I ! I ~ - - - . . ,

e


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Hinderance Function of a Non-Ser ies-paral le l Network,The methods of par t I I were not suf f ic ien t to handlec i r cu i t s which contained connections other than thoseof a se r ie s -pa ra l l e l type . The bridge o f Fig. 11, fo rexamPle, i s a non-aer ies-para l le l network. These ne t works v ' i l l lJe handled by reducing to en equiva len ts e r i e s -pa r a l l e l c i r eu i t . ' rhree methods have baendeveloped fo r f inding the equivalent of a networksuch e s the br idge .

b

v

s

u

R

Fig .The f i r s t is the obVious method of aPPlying

the t ransformst ions un t i l the network i s o f these r ie s -para l l e l type an d then wr1 t1ng the h1nderan cefunction bv inspect ion. This process i s exact lythe same a s i s used in s impl i fy ing complex impedal1 c enetworks. To apply t h i s to the c i r cu i t of Fig . 11 ,f i r s t el iminate the node c , by applying the s ta rto mesh t ransformation to the s t a r a-c , b-c, d-c.This ~ i ves the network o f F ig . 12 .


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Fig . 12'The hinderance function may be wri t ten dovvn frominspect ion for t h i s network.

Xab = (R + S)[U(R + T) + V (T + S)}S l m p 1 1 f y i n ~ by the theorems gives :

x = RU + S'J + RTV + SIDabThe second method of anal:rs1 s i s to draw

sll Pas 81 ble paths between the points under oonsid-ere t ion throu.gh the network. These paths ere drawn

B l o n ~ the l i n e s represent ing the component hinder-.,.,Bnce eleJllents o f the c i r cu i t . I f anyone of thesepa ths h8 s zero hinderen as , the requ i red :f\ln et lenmust be zero. Hence if the ~ e s u l t is writ ten 8Sa product , the hirlderanes o f each path vl i l l be af ac to r of t h i s product. The required resul t maytherefore be wr1 t t an as the product of the hlnder-ances o f a l l pass i b le pe th s b9tween the two po in t s .P ath s whioh touCh the sarna po in t more than once need


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no t be con sid-3red. In F i ~ . 13 t h i s method i s apPl iedto the b r idge .

8..

The paths are marked in red.

~ _ b

Fig . 13The :f\.lnction 1s therefore g1 van by:

Xab = (R + sHu + V)(R + T + V)(U + T + S)v:; RU + SY + RTV + UTSThe same re su l t is th us obta in ed as with the f i r s tmethod.

The tl1.ird method, the dual o f the second, isto draw a l l poss ib le l ines Which VJould br-e8k t he c i r -cui t between the point s under cons! dara t i on , makingthe l i ne s go through th e hinderances of the c i rcu i t .The ras111t i s writ ten as 8 sum, each term corres -pending to a Qdrtain l i n e . The sa t arms a re the J:Jrod-ucts of a l l the hinderances on the l i ne . This methodi s ap nlied to the bridge in F 1 ~ . 14, the l ines b e 1 n ~drawn in red .

-. .b

F ig . 14


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This 8 ~ a 1 n gives for the hinderance of the network:-t-Xab = RUt SV + RTV + sm

The th i rd method is usual ly the most convenientand rapid , fo r it ~ v e s the re su l t d i rec t ly 8S a sum.It seems much eas ie r to p..8ndle S'lms than products due,no dOUbt, to the fact tha t in ordinary algebra wehave the distr11:utlve law X{Y + Z) =n + XZ, bUt no ti t s dual X + yz = (X + Y)(X + Z). I t i s , however,sometime s d1 ff ' lcul t to 8 pply th.e th i rd me thod to non-planar networks (networks which cannot be drawn on 8plane without cross ing l i n e s ) and in th is case one ofthe other t'iVO methods may 1Je used.Simultaneous Equations. I f there are n dependentvar iab les , there wi l l be n simultaneous equations da-f in ing the system. Any addit1 va tsrms which 8 re Commonto several o f the funct ions may be factored out in themanner 1.11ustrated by the fol lowing example. Theseterms need only be real ized onoe to take ears of a l l

cB=A + B + OW ~X =A + B + \VXY A + CY A~ = E ~ + f

'ft[A +IB + I , ~Xy Cy

~ - E ~ + f-

the funct ions in which they epte 81 ' .

Fig . 15


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25

Sometimes the r e l a t i o n ab t =0 o b t a i n s "betweentwo r e l a y s 8 end b. This 1 s t r u e , f o r example, i n asequ ent1Bl sy stem whe re ee ch r e l a y o f the sequen ce1.oCks i t s e l f i n and a precedes b i n t he sequence .'Nhenever b i s o'perated 8 i s operated. I n such a casethe following s i m p l i f i c a t i o n s may be made:If a b l = 0Then a ' b t = s , b l + a b ' - b l-

ab - e b + a b l = 8-8 I + b = 1(a ' + b l ) - (a l + b'){e' + b) - a l- -( a + b) = (e + b) ( 8 ' + b) = b

Matrix Methods. I t i s also p o s s i b l e to t r s a t mult1-terminal networks by means of m a t r i c e s . Although u s e -f u l f o r t h e o r e t i c a l work t h e method i s cumbersome foxap r a c t i c a l problems and w i l l th'3r l3fore o n l y be b r i e f l ysketched . ?/e s h a l l a s mma t h e same ru19s o f m B n i p u l a t ~ o no f m a t r i c e s as usuell-:T d e f i n e d in v/orks on h i g h e r a l g e -bra, the only difference b a 1 n ~ t h a t the elements ofour matr ices w i l l be hinderance funct ions r a t h e r thano r d i n a r y a l g e b r a i c numbers o r v a r i a b l e s . The XI m a t r i xo f 8 n e t w o r k w i t h n nodes w i l l be d e f i n ed a s t h e f o l -


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I I I1 X12 X13 X1nI , IX21 1 X23 X2n . . . . .. .. . . . . . . .. . . . .. . . . . ... . . ... . . . . . . . . , .

X I . 1n2

where X'j is the negative of the hinderance common to~ knodes j and k .Theorem: The X' matrix of a network formed by connecting two n node networks in p a ~ 8 1 1 e l (oorrespond-ing nodes togethsI ' ) i s the sum of tr18 1nd1Vidual XImatr ices . This theorem 1s more general thaD mightappear a t f i r s t since any interconnection of two net-works may be oonsidered 8S a paral le l connection oftwo networks wi th the same numb3r o f nodes by addingnodes 8S ~ e d e d whose mutual hinderances to the othernodes i s one.

Now define s matrix to be cel led the U' mB-t r ix of a network 8 S fol lows:

~ U12 . . . . . UlnU21 1 ....... U2n


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WhA-rA TIl" 1-g the n e ~ a t i v e of trle hinderance function.- . - - jk ~from node j to k, the network considered 8 S a two t e ~ -1nal c i re l l i t . Thus fo r the three node network ofFig. 16 the X' and TIl matrices are as shown a t ther igh t .

2xl\y 1 Xl z' 1 X '+y ' Z I z'+x1y'x' 1 y' x '+y ' z ' 1 y'+X'7. 'l L ~ 3 Zl y ' 1 z '+x'y ' :f l+X'Z' 1z

Fig . 16 X' Matrix U' MatrixT h e o ~ e m : Any p O ~ e r of the XI matrix of 8 networkgives a netvlork which i s equivalent With respectto 811 nodes. The matrix is ra ised to a powsr bythe usual ru le for mult ipl icat ion o f ma tr ic es .Theorem:

I t , I1 U12 .... U1n 1 X12 ... X1n sI t , XlU21 1 ... . U2n X21 1 ...- 2rl-... . . . . . . . ......

U ~ l ,...... " .,X1n ..... 1 II

s ~ n-l

Theorem: Any node., say the kth , may be alirnina tedles' tling the network equivalent with respec t to allremaining no des by adding to each eleraent X ~ s of the


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Xi matr ix the oroduct of the elemoo.ts X'k and Xk , andr _8s 'c r ik ing aut th e k th rowan d column.

Thu s el imina t in ~ the 3rd node of Fig . 16 we p;et:

L+z' z t x'+z'y 'l+y'y'

I: 1x'+J1z'

X'+y1zl

1

The proofs of these theorems are of a simplena tu re , but qui te l o n ~ end wi l l no't be given.Spec ia l TyP3 s o f Relays Band SVrl t c he s . In cer ta in type sof c i rcui t s it i s necessary to preserve 8 def in i tesequent ia l re la t ion in t'he operat ion of the con ' tactso f a re lay . This is done with make-barare-break (o rcon t i nu i ty ) and brae k-make (ot ' t r a nsfer ) con ta c t s .In hand11np; t h i s type o f 01 rout t th e simple s t me thadseems to be to assume in se t t ing up the equationstha t the make and break contaots opera te s1multane-ous ly , and aft:3I' a l l s impl i f i c a t i ons o f th e equa t ionshave been made end the re su l t ing ciI 'cul t drawn therequi red type of contact sequence i s found from in -spec t ion .

Relays haVing a t ime delay in onsra t1ng o rdeoner8t ing may be t reated s imi l a r l y o r by s h i f t i n ~


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tha t iu!" t ix is . 'rnus if 8 ! 'elay co i l i s Con_naeted to 8battery through a hinderance X, and the re lay has ade lay of p seconds in opere t ing and r e l ea s ing , thenth a h in deran ce fUnction of the eontacts o f the re laywi l l also be X, but a t a time p seconds l a t e r . Thismay be ind ica ted by wri t ing X( t ) for the hinderance inse r i e s With the r e l ay , and X(t-p) fo r t h a t o f the r e l a t oontacts .

There Bre many spec ia l types of re lays endsWitches fo r pa r t i cu l a r Plr-poses, such as the s t e p p 1 n ~switches and selector switches of various s o ~ t s ,multi-winding re lays , cross-bar switches, e t c . Theopera t ion of a l l these types may be descr ibed withthe words "or , " "and,n " i f , " l1oparated," and "notopera ted." This i s a su f f i c i en t eondi t lon t ha t maybe desc r i bed in terms o f hinderance fUnctions withthe operat ions of addi t ion , mul t iP l ica t ion , negat i on , end equa l i ty . Thus 8 two w i n d i n ~ re lay mightbe So const ructed tha t it i s operated if the f i r s tor the second winding i s opera ted (ac t iva ted) andthe f i r s t ~ the seeond windings are not operated.Usual ly , however, these specia l relays occur only a tthe end of a complex a i rcu i t and may be omitted ent i re ly from the o alc ule tio ns to be added a f t e r there s t of the c i r cu i t i s designed.


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5 0 m ~ ~ ~ m e s a relay X 1s to opera te when B c i r cu i t r closes and to remain closed independent o f run t i l a c i r cu i t S opens . Suoh e c i r cu i t i s known ase lock- in c i r cu i t . I t s equat ion i s :

X = rX + SReplacing X by X. ~ ves:

XI = rX' + Sor

X : ( r l + X)S'In t h i s ca sa X i s opened when r closes and remainsopen unt i l S opens.


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IV Synthes is o f NetworksSome General Theorems on }letworks and FUnctions. Ithe 3 bee'n shown _that any furl e t t o n may be expanded i n aser1e s con 81 st1n g 0 f a sum of produ e t a , each prodU a tbeinp; o f t h e form XlX2 Xn wi t h some permutat ion ofprimes on the l e t t e r s , and each p roduct hav ing the coe f f i c i e n t 0 o r 1 . Now since each of the n variablesmayor may not have a prime, there 1s 8 t o t a l of 2nd i f f e r e n t products of t h i s form. Similar ly each produ c t may have the c o e f f i ~ 1 a n t 0 o r t h e c o e f f i c i e n t 1

22n t ho t h e r e a r e p o s s i b l e sums o f 1 s S O I ' t . Each oft h e s e sums w i l l represent a unique funct ion, but someof t h e funct ions may a c t u a l l y involve l e s s than n v a r i -a b l e s ( i . e . , t h e y ere of such a form thQ t fo r one o rmore o f t h e n v a r i a b l e s , say X ~ , we have i d e n t i o a l l yf(Xl, X k ~ l ' 0, Xk+l' Xn ) =f(X1.Xk-1J 1 , Xk+1 ,Xn ) so the t under no oo.ndi t10n s do as the va lue ofthe fu nc tio n depend on the value o f X k

Hence we have the theorem:Theorem: The number o f functions of n variables or

2nl e s s i s 2 To f i n d the number o f funct ions Which 80tua l l y

involve n v a r i a b l e s we proceed as f o l l o w s . Let r/(n) bethe required number. Then Q1 the theorem j u s t given:


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where ( ~ ) : nl /k Hn-k) i s the number of comb1.nation sof n t h ings taken k a t a t ime . S o l v 1 n ~ fo r ~ ( n ) gives:

2n n ~; (n) = 2 - ~ (R);{k)k=OBy 5Ubst1 tu t ing fo r ,(n-l) on th e r i g h t the s imi la rexpression found by replacing n by n -l in th is equation,x then s imi lar ly sUbsti tut ing fo r ~ ( n - 2 ) in the expres-s ian thus obta ined , e tc , an equat ion may be obtainedinvolv ing only ~ ( n ) . This equat ion may than be slmp11 f i ad to t he form:

~ n 2k n-k;(n) : [ (k)2 (-1) ]k :As n increases th i s e x ~ e s s i o n approaches i t s leading

nterm 22 asymptot ical ly . The e r ro r in uSing o nly t h i sterm fo r n : 5 i s l e ss than .01%.

We sha l l now determine those fUnctions of nvert.s ble s which requi re tb.e mo s t re lay con tac ts to re -e l iza , and find th e number of contacts r equ i r ed . Ino rde r to do t h i s , it 1s necessary to define a func t ionof two var iab les krtown as th e sum modulo two o r d i s -junct of the ~ l a r i e b l e s . This funct ion 1s wr i t t enx.l ex2 end i s def ined by the equa t ion:

X l ~ X 2 =X1X2 + X1X2


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It i s easy to sPw tha t the sum modulo two obeys thecommutat ive , asso(}ia t ive , and the d i s t r i h l t i v e lawwith resoect to mult ip l ica t ion, i . e .x1ex2 : ~ e x l

(X1eX2 )eX3 =X18(X2eX 3 )

Also:

x el : X'1 1Since the sum modulo two obeys the assoc ia t ive law,we may omit parentheses in a sum of several termsWithout ambiguity_ The sum modulo two of th e n var1-ables X1 'A2 Xn wi l l fo r convenience be wri t t en :

nX l e x 2 e x 3 e ~ = ~ X k

Theorem: The two funct ions of n variables which re -quire the most elements ( relay contacts) in a ser ies -

n npa r a l l e l r ea l i za t ion Bre ~ X " a n d ( ~ X ~ ) I , each o f wlUch12 1requi res (32n -1_2) elements .


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This w i l l be proved by mathematical induct ion.F i r s t note t h a t it i s t r u e f o r n = 2 . There Bre 10fUnctions o f 2 variaQles, namely, r l , X+Y, Xty, XI+Y,XY', X+Y' J. X'Y' J XI +Y', XY' + X'Y, XY+X'Y'. All ofthese but the l a s t two require two elements; the l e s ttwo r e ~ l r e four elements and ara XfY and (X8Y)'r e s p e c t i v e l y . Thus the theorem i s t r u ~ f o r n = 2 .NoW 8 SBuming 1 t true f o r n - l , we s h a l l prove 1 t t r u efo!' n and thus complete the i n d u e t l o n Any functiono f n v a r i a b l e s may be V'/rl t t a n by ( l O a ) :

l ~ o w t h e terms f (1 ,X2 Xn ) and f(O,Xe:> X ) are f\1nc"...., n .t i o n a of n - l v a r i a b l e s , and if t h e y i n d i v i d u a l l y r e -quire the most elements for n - l varia b l e s , ' then f w i l lrequire the most elements fo r n v a r i a b l e s , providingthere i s no other method of writ ing f so t h a t l e s selements ere r e q u i r e d . t ~ J e have assumed t h a t the mostelements f o r these n - l v a r i a b l e s a r e required by

~ X k and ( ~ X k ) f - I f we therefore su b s t i tu t e f o rnf{1,X2 -- .Xn ) t h e f u n c t i o n ~ x and f o r f{O,X2 - _Xn )n ..., k

the ~ u n c t i o n ( teXk) f we get:2n ' n nf =Xl.,#Xk + I X i (f2Xk ) t = ( ~ 2 X k ) I


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From the symetry of th i s funct lon there is no other Vv8yo f e x p a n d i ' n ~ "vhich v ~ i l l reduce the number of elements .If the r11nctions ere s t lbs t i tu ted in the o th er o rd er ,we ~ e t :

This oomvletes the proof tha t these functions requirethe most elements- To show that each requires (3_2n - 1 _2)elements, l e t the number of elements required be denoted by s (n ) . Then from (19) we ~ a t the differenoeequat ion:

s(n) : 2s(n-l) + 2With s (2) = 4 . This i s l i n ea r , ~ v 1 t h cons tan t coe f f i c i en t s , end may be solved by the usua l me thods (5 ) .The solution i s :

n - ls(n) = 3 .2 -2a s may be ~ a sUy verlf1 ad by su bst1 tu t in g in the d l f-te rence equation and toundary condi t ion .

Note t"hat the above only apPl ies to 8 s s r i e s -para l le l r e a l i z a t i on . In a l a t e r sect ion it Wil l benshown t h a t the f l l n c t i o n ~ X k and i t s negative may be1r es l i zed with 4:(n-l) elements u s i n ~ 8 more p:eneraltype of c i r c u i t . The fUnction requir ing the mostelements u s i n ~ any type of c i r c u i t has no t as yetbeen determined.


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Dual l{et\vorks.

36

The n e ~ e t 1 v e of any network m87 befound b;y De ~ l o r ' g e n l s theorem, bu t tm ne t v ~ o r k mustf i r s t be transformed into an eQUivalent ser ies-para l le lc i r c u i t (un less it i s a l ready o f t h i s t ype ) . A theoremWill be developed With which th e nega t ive o f any planartwo-terminal c i rou i t may be round di rec t ly . As B corol l a ry a method o f f ind ing a constsn t current 01 rcu i tequivalent to e ~ i v e n constant voltage c i rcu i t andvice versa Wil l be g i van.

Let N represent a planer network of hindersnoas , With the function Xab between th e terminalsa and b Which are on the outer edge of the network.For def in i teness e on sld er th e netwo!'k of Fig . 17(here the hinderances are shown merely as l ine s ) .NoW l e t M rep rese nt th e dual of N, as found by thefollow! np; pro cess; fo r as ch c on tour or me sh 0 f Nassign a n9de o r junction point of M. For eaohelement of ' N say Xk l s e p i r 8 t i n ~ the contours r a nds there corresponds an e l ~ m e n t Xk connecting thenodes r a n d s o f M. The area e x t 3 r 1 o ~ to N i s tobe considered as tVlQ meshes, c and d, correspondingto nodes c end d of M. Thus th e dual o f F 1 ~ . 17 i sthe network of F i ~ . 18.


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a

m.esh cs

mesh dFig . 17

b

Fig .

c37

Theorem: I f M and N bear th i s dua l i ty re la t ionship ,then Xa b = X ~ d

To pro va th i S J le t t he networks M end N besuperimposed, the nodes o f M within the correspondingmeshes of M and corresponding elements cross ing. Forthe network of Fig . 17 , th i s 1s shoWn in Fig. 19,With N in black and M in red . Inc identa l ly , thesa s i e s t me thad 0 f f inding the dual of a ne two rk .(Whether of t h i s type or an 1mpedlnce nstwork) 1s todraw the required ne two rk superlmpo sed on t h.e g1.vannetwortk. Now, i f ' Xab : 0 , then there must be some-path fI'om 8 to b a l o n ~ the l ines of N such t ha t everyelement on th i s path equals z ero . Bu t th i s path repre-sents a pa th across M d1 v i ding the c i r cu i t from c to dalong w n i ~ h every element of M 1s ona. Hence Xed =1 .Similar ly , i f Xed =0, then Xab =1 , and it follows tha tX -V I8b - "''"''ad-

a

Fig .

b


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In a eonstan t-vo lt age re lay system a l l there lays are in paral le l across the l ine . To open arelay a ser ies conneotion 1s o p e n ~ d . The general con-s tant-vol tage system 1s shown in Fig. 20. In a constant-currant system the re lay s a re a l l in se r ies in the l i ne .To d e ' ~ o p e r a t e a re lay , it i s shor t c l rou i tad . The gen-e ra l constant-current c i rcu i t corresponding to Fig. 20i s shown in F 1 ~ . 21. I f the relay Yk of F i ~ . 21 isto be operated whenever the relay X

kof F i ~ . 20 i s

opera ted and not otherwi se , than eVidentl y the hin-der8tlCe in pa ra l l e l wi th Yk whi_ch Shorts 1.t out mus tbe the na ga tiva 0 f the hinderan ce . .,- in s a r i as vii thXk Which connects it across the vol tage sou rc e. I ft h i s i s t rue fo r a l l th e re lay s, we sha l l say t ha t theoonstant-currant and constant-voltage systems are equiv-a l en t . The 8 Cove theorem may "be used 'to f ind equivalentC ircuits of th is sor t . For, if we make the networksN end M of Figs. 20 and 21 duels in the sense described,than the condit ion wi l l be sa t i s f i ed .

Econstant voltagesource.

Fig . 20

ConstantI current t1source.

~ Ynl ~ - " " ' ' ' ' I- - - - - - - - - "Fig . 21


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39

A simple example of th is i s shown in Figs. 22 and 23.

E

R~ - - - ..

Fig . 22

I R'

F 1 ~ . 23

y:3

, Synthes i s o f the General S x m e t ~ 1 c FUnction. As ha sbeen shown, any funct ion represents exp l i c i t l y aser ies-paral le l c i rcui t . The ser ies-paral le l ~ e 8 1 1 z 8 -t ion may requi re mo re elements J howev61", the n someother c i rcu i t representing.the same funct ion. Inth i s section a method wi l l be given fo r f inding a c i r oui t representing B certain type of fUnct ion whichin ~ e n e r a l i s much more economical o f elements thanthe bes t se r ie s -pe ra l l e l c i r cu i t . This type of fUnc-t10n f requen t ly appears in r e l ay ci rcui 'Cs and i s ofmuch importance.

A func t ion 0 f the n var ia bl es X l ' X2 , .Xni s sa id to be symmetric in these var iab les if anytn t e r c h a n ~ e of the sa v aria b les lea ve s th e funct ion


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ident ica l ly the same. Thus XY + XZ + yz i s symmetr1ai"n th e var iab les X, Y, and Z. Since any permutat iono f var iab les may be obtained by sucaess ive 1n te rchengaso f two var iab les , a necessary and su f f i c i en t condi t iontha t 8 function be symmetric i s tha t any 1 n t e r c h a n ~ eof tw o var iables leaves the fUnction unal tered.

We now give a theorem Which forms the besisof the method of synthesis to be d ~ a c r 1 b e d .Theorem: The necessary and suf f ic ien t condit iontha t 8 fUnction be symmetric 1 s t ha t it may be spec1-t ied bf s ta t ing a se t of numbers 81 ' 8 2 , 8 k suchthB t i f axa c tl y a j ( j =1 , 2 , :3, k) 0 f t he va r1 a b1 e sare z ero ,th en the fUnction i s zero and not otherwise.This fol lows eas i ly f ~ o m the de f in i t ion . For th e ex-ample g1 van the sa num bars a re 2 and 3 .Theorem: There are 2n +1 symma'tric functions of nvat-1ables. For eV8I ' y se lec tio n o f 8 s e t of numbal'sfrom the numbers 0 , 1 , 2 , n corresponds to one andonly ona s ~ ~ e t r i c function. Since there are n+l numberseach o f Which may be a1 t he r taken o r no t i n our se l ec

n+lt1on, the to ta l number of functions i s 2 Two ofthese fUnctions are t r iv ia l , however, namely the se -la ctia ns in Which none and a l l of the numbers aretaken. These give the "functions"l and 0 respec t ive ly .


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B'! proper se lec t ion of the var ia lbes manyapparent ly u n s ~ r m m e t r i c funct ions may te made symmetric.For e xamp le , XY'Z + X'YZ 1-. X'Y 'Z ' , al though no t symmetricin X ~ Y, end Z, i s symnetl"io in X, Y, and Z '.

~ e se t of ~ m b e r s a l , a2 , sk wil l fo r convenience be cal led the 8-n'umbers of the funct ion.The theorems concerning comtlnations of symmetricfunctions ere most easi ly s ta ted in terms of the018 s sa S 0 f 8 -numbar s For th i s rea son we dena ta thecless of a-numbers by a s 1 n ~ l e l e t t e r A. I f two d i f fe r -ent se ts of a-numbers are under consideration they willbe denoted by A1 and A2 The symmetric function of nvar ia ble s he ving the a -num bel 'S 8 1 , 82 sk wi l l 'bewritten Sn{al , 8 2 ak ) or an(A).Theorem: 3n (A l ) Sn(A2 ) =Sn(Al + A2 )where A1 + A2 means the l06!ca l sum or the classes Aland A2 i.e., the c la ss of tho sa numbers which B re membersof e i ther Al o r A2 or both. Thus 36(1, 2, 3 ) . 86 (2 , 3 , 5)i s equa 1 to S6 (1 , 2 , 3 , 5).

Theorem: 3n (Al ) + 5n(Aa) ~ Sn(A1 ,A2)where AlwA2 is the logical product of the mlassas i . e . ,the a la s s 0 f numbers Which are common to A1 and A2. Thus56 (1 , 2 , 3) + S6 (2 , 3 , 5) '"C 36(2 , 3 ) .These theorems follow from the f a c t tha t 8 product is


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zero if e i ther factor i s zero, while 8 sum i s zero onlyif both terTI1S are zex'o. The negs t i va of 8 se t o f a -numberswi l l be \"I1'1tten AI and meBns the c la ss o f a l l the numbersfrom 0 to n 1nclus i va which a re not members o f A. Thusif A i s the se t of numbers 2 , 3 , and 5, and n 6 thenAI i s the set of numbers 0 , 1 , ~ , and 6.Theorem:These thaorams are useful if several symmetric functionsare to be obtained simultaneously ..

Before we study the synthesis of 8 network forthe general symmetric fUnction consider the c i rcu i t 8-bof Fig . 2 ~ . Th1.s c i rcu i t represents 33(2) .

L o X ~XI2

7L :n-3__ 0XI3

The l ine c o m L ~ g in s t a f i r s t encounters a pa i r o fh1nderancas Xl and xl. I f Xl = 0, the l ine i s switched


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43

up to the l e v e l marked 1 , meaning t h a t 1 of the var1eQles1s zero. I f Xl =1 , the l i n e stays on the l e v e l marked0; next hinderanaas X2 and X ~ Bra encountered. I f X2i s ~ e r o , the l i n e i s switched up a l e v e l ; if n o t , its t a y s a t t h e same l e v e l . F i r l s l l y reaching the r i g h thand s e t o f t e r m i n a l s t h e l i n e has been switched upto 8 l e v e l r e p r e s e n t i n g the number of v a r i a b l e s which

;.:..

a r e dqua]. to z e r o . Terminal b 1s connected t o l e v a l2 and t h e r e f o r e the c i r c u i t a - b w i l l be comPleted ifand only if 2 o f t h e variables a r e zero. Thus thef u n c t i o n 33 (2 ) i s r e p ~ e s e n t e d . If 33(0 ,3) had beend e s i r e d , t e r m i n a l b would 'be connected t o both l e v e l so end 3 . In f i g u r e 24 c e r t a i n of the elements Breevlden t l y Sll perf ' luous. The c1 rout t may be s i m p l i f i e dto t h e form o f F i g . 25.

F i g . 25For the general function exact ly t h e same

method i s follov'Jed. Using t h e g e n e r a l CirC1..1it f o r n

b


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var ia "bles o f Fig. 26, the te rminal b i s cOllnected "to thel eve l s c o r r e s - ; ~ o n d i n g to the a-numbers of the desiredsvmmetric funct ion. In Fig . 26 th e h in de ra nc es Brarepresented by simple l i ne s , and the l e t t e ~ s are omittedfrom the c i r cu i t , hut the hinderance of each l ine mayeas i l y be seen by gene r a l i z i ng Fig . 24:.

NOTE: All s loping l i n e shave hinderance of thevar i a ble \vri t ten below;hor izon ta l l i nes havenegative of th is hinder-ance.

~ b2

to a1 numberso

(n-:-l)n

........ ...aFig . 26

Aftar terminal b i s connected, a l l superfluous ele-ments may be deleted.

In cer ta in cases it i s poss i b le to grea t lys impl i fy th e c i r cu i t by sh i f t i ng the l eve l s down.Supnose the rbnction 36 (0 ,3 ,5) 1s des i red . Ins teadof continuing the c i rcu i t up to the 6th l eve l , we


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45

oonnect the 2nd l eve l tsck down to the zero l eve l asshown in F ig . 27. The zero l eve l then a lso becomesthe 3rd leve l and the 6th l eve l .

2 ,51 ,40 ,3 ,6a .... - - . j . . . . - - - - . - - - ~ I 1 1 ' - - - - ~ ~ - - - - - -..------....-.-... b

th t e rmina l b connected to th is le ve l, we have r e a -l1zed the function with a great saving of elements.El iminat ing unnecessary elements tIlo c i r cu i t o f Fig . 281s obtained. This deV ice 1 s e 3p ec ia l ly usefu l if the8-numbers form an a r i t ~ m e t 1 c progression, although itcan sometimes be applied in other cases . The fUnctionsn nl :2Xk end CC2Xk )1 Which were shown to require the most1 io 1elements for a s e r 1 e s ~ p a r a l l e l real izat ion have verysimple c i rcu i t s when developed in t h i s mann,er. It

n/Jan be eas i ly shown t h a t if n i s even, then!:2Xk i s

1the symmetric function with a l l the even numbersfo r a-numbers , if n i s odd it has a l l the odd numbersnfo r a-numbers . The funct ion ~ X k ) I i s , o f course ,

1jus t the opcos1te. Using the sh i f t i ng down process


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46

the c i rcui t s are- as shown in Fig. 29.

a -----......... ~ - - . - . . ~ - - _ w : ~ - - . . . . . _ - - - .......---....... b

n~ 2 Xk for n odd;1

a - - - - - - ~ bXl ~ X3 xn-l Xnn nl:2 Xk for n even; (1;2 Xk)1 fo r n odd.1 1Fig . 29

These circui ts each require 4{n-l) elements. Theywi l l be re co gn iz ed a s the f ami l i a r c i r cu i t fo r cont ro llinJ2; a 11 ght from n pain t s ; I f a t an:, one of thepoints the posi t ion of the switch 1s changed, theto ta l number of vAriables which e q u a l ~ ~ e r o i s changedby one, so t h a t if the l igh t 1s on, it willbe turnedo ff end if a l r / ~ a d ~ r o f f , it wi l l be turned on .


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47

The genera l network o f Fig . 26 conta ins n(n + 1)elements. It can be shown t ha t fo r any ~ i v e n s ~ l e c t i o n'o f a-numbers a t l ea s t n of the elements wil l be super-f luous . I t fol lows t ha t any symmetric function of nvar iab les can be raa l ized With a t most n2 elements .

Equations from Given o p e r a t i n ~ Charac te r i s t i c s . In gen-e ra l , there i s a cer ta in s e t of independent variablesA, B, a, Which may be s \n tches , e x t e ~ l e l l Y operatedor protect ive re lays . Thera i s also a se t of d e p e n d ~ n tvar iables x , y, z Which represent re lays , motors oro the r d e v i ~ e s to be cont ro l l ed by the c1rcu i t . It 1srequired to f ind a network which ~ 1 v a s fo r each poss iblecomtanetion of values of the independent var iab les , thecor rec t value s fo r a l l the dependent va r i ab le s . The

f o l l o w i n ~ pr inc ip l e s g i va the genera l method of s o l u t ion .

1 .- Additional dependent var iables must bein t roduced fo r each added phase o f opera t ion of 8sequen t i a l sys tem. Thus if it i s des i red to construct

.8 system which op era tes in th ree s teps , two addi t iona lv8riables must be introduced to represent t h ~ b e g i n n i r ~of the l a s t t\'Vo st!3ps. These add i t i ona l var iab lesmay r e p ~ e s e n t contacts on a s tepping switch or relaysWhich lock in sequentia l ly . Simi la r ly each requi redt ime d e 1 8 ~ T wl11 reqt1.ire a nevv va r i ab l e , represent ing


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!8

a t ime delay relB;T o f sona so r t Other forms of r e l ayswhich may be uscessary wi l l usua l ly be obvious fromthe nature o f the problem.

2 . The hinderance equat ions fo r each of thedependent var iab les should noW be writ ten down. These

t ~ n c t i o n s may involve any of the var iab les , dependentor independent , including the var iab le whose fUnctioni s being determined (as , for example, in a l o ~ k inc i rCUi t ) . The condit ions may be ei ther condit ionsfc\r operation or fo r non-opera t ion . Equat ions arewri t ten from operat ing charac te r i s t i c s a c c o r d i n ~ toTable I I . TC' i l l u s t r a t e th e use o f t h i s t ab le s t lP -pose a re lay A i s to operate if x 1s o p ~ r 8 t e d andyor z i s operated end x or w or z 1s not o pe ra ted.The express ion fo r A wi l l 1)8:

A = x + yz + X'W'ZILock in r e l ay equations have a lr -gadv been disoussed .It does no t , o f course , matter if the seme condit ionsare p l l t in tb.e expression more than once - - s l l superfluous mater ia l wi l l disappear in the f in a l s im p lif i ca t ion .

3 . The expre s s ian s fo r th e 'va r ious dependentvar iab les should next be s impl i f i ed a s much 8 s poss ib leby means o f the theorems on manipUlat ion o f these quanti t i e s . Jus t ho\v mtlch t h i s 0811 be done depends somewhat

r


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TABLE I I

R E L A T I O ~ : OF OPERATlllG C F ~ R A C T E R I S T I C S AtTD EQUArIOrIS

SymoolX

-XI

+( -- ) ,

In Terms of operationThe switch or re layX is 0 pe ra ted .I f .The switoh or re layX i s no t operated.Or And.The c i rcu i t ( - - ) i s notclosed, or apply DeMorgan's Theorem.

In Terms of Non-operationThe switch or relay X1 s not operated.I f .The swi tch or relay X1 s 0 p3 ra t ad .And.Or.The c irc ui t ( - - ) i sclosed , or apply DeMorgans Theorem.


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50-

on the in g e n u ~ .. ty 0 f the de s1 er 4. The resul t ing c i rcu i t should now be

drawn. Any necessary addit ions dic ta ted by pract ica loonsidera t ions such as current oarrying ab i l i t y , se -quence of contact operat ion, e tc . , shculd be made.


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51

v I l lus t r a t ive ExamplesIn th i s sect ion several problems wil l be

so lv ed with the methods whiah have been developed.The examples are intended more to show the versa t i l -i t y of relay and sWitching c i rcui t s and to i l lus t ra tethe us e o f the ca lcu lus in aotual pro blems than to de s -cr ibe prac t ica l devices.

It i s p oss ib le to ~ r f o r m complex mathematicalor;>eretions by means o f re lay c i r cu i t s . lTumbers may berepresented b1 the pos i t i ons of relays or s t epp ingsWitches , and i n t e rconnec t ions between se t s o f re layscan be made to represent various mathematioal operat ions . In f e c t , any opera t ion t ha t can be completelydescr ibed to the requ i red accuracy ( i f numerical) ina f in i t e number of s teps us ing the words n i f , " nor,"Uand, U e tc . (see Ta b le I I ) CBn be done a utoma t i ce l lywith relayse_ The l e s t two examples are i l lus t ra t ionsc f ma thems - t ical opera t ioris a C; conpl i shed wi th re lay s.


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52

A Select ive Circui tA re lay 1:\ i s to ope ra te when anyone , any

three or \vhen a l l four of the re lays w, x, Y, and zare ope ra t ed . The hinderance funct ion fo r A. wi l leViden t l y be:

A = wxy& + wtxtya + w ' x y l ~ + w ' x y ~ ' + wX'y'& +wX'yii ' + wxylit

Reducing to the s imples t s e r i e s - pa r a l l e l form:A =w[x(ys + yt&l ) + X'(y1& + y ~ t ) + w'[x(y'& + ya')

+ x'ya]Th.le c i r c u i t is sho'lm. in F 1 ~ . 30. It requires 20 e l e -ments.

W WI A+ -.-..---.a e-----e---..... ....---.....--------nIx Xl x Xl

F ig . 30

However, using the 8 : . ~ m e t r i c function method, we maywrite fo r A:


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53

w x yF i g . 31

z

ThjEj c i r c u i t c o nt a i ns O T l l y 1 5 e l e m e n t s . A still f u r -t h e r r e d u c t i o n m 8 ~ T b e made w i t h t h e f ollow 1 ng d e v i c e .F i r s t w r i t e :

A I ~ ( 0 , 2 )l ' h i s he s t h e c i r e u i t o f F i g . 3 2 . ' J ~ l h a tis r e q u i r e d isth e n e g a t i v a o f ttl1 s f u n o t i o n . T hi s i s B p l a n a r n e t w ork a nd we msy a p p l y t h e theorem on t h e d u e l 0 f a n e t -w o r k , t h u s o b t a i n i n g t h e c i r c u i t shown i n F i g . 3 3 .

+

~ ? . ~ ~.c:. ~ . ~ ...--....... O--.-L. .."ArJ'11~ __w x y

F i g . 32z

T h i s c o n t a i n s 1 4 e l e m e n t s a n d i s p r o b a b l y t h e m o s t e c o n o m -1c"81 c i r c u i t o f a n y s o r t .


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+ - - ........ - - - "z

Zl

F ig . 33

Wi

"f'l,

A

54


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55

Desi@l o f an Elect r ic Comtanation Lock

An elec t r ic lock 1s to be constructed withthe following oharac te r i s t i c s . There Bre to be 5 pushbutton swi tche s available on the f ron t of the lock .These wil l be labeled a , b, c , d, 9. To operate thelock the bl t tons must be pressed in the followingorde r - c , 0, e and c s imu lta neou sly , d . l ~ l h e n Op3ratedin t h i s sequence th e lock i s to be unlocked, but if anybutton i s pressed incorrec t ly an alarm U i s to operate .To re lock tIle system 8 sWi tch g mus t be operated . Tore lease the alarm once it has s tar tad)8 sWitch h must-be opera ted . This being a saquen t i e l system a1 ther 8s tepping swi tch o r a dd iti on sl sequent ia l re lays arerequired . Using s eq ue ntia l re la ys l e t them be denotedby w, x, y, and Z oorresponding respect ively to the

c o ! ' ~ e c t seq.uence o f opera t ing the push rot tons . Anaddi t ional - t 1 ~ e delay re lay i s also required due tothe th i rd step in the opera t ion , ObVious ly , evenin co rrec t operat ion a and c cannot be pressed a t ex-act ly the same t ime, but if only one is pressed andheld down the alarm should operate . Therefore assumean auxi l ia ry t ime delay re lay y Which wi l l operateif e i the r a or c alOne is pressed a t the end o f step 2end held down longer than t ime s the d e l ~ Y of the re lay .


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56

11Vhen z has opers ted th e lock unloclcs and a t t h i s poin tl e t a l l the other relays drop out of the c i r cu i t . Theequations of the system may be writ ten down immediatelY:

W cw + + U t ~~x = bx + ~ + & I + UI I - ..... '; ,t' ! .y = (a + c)y + x + 6 ' + U'& = l l . ( ~ + y) + g ' + U 'v = x + y ' + 8e + a 'o ' + a ' + U'U : e(wl + abd)(w + Xl + ad)(X + y l + dv( t -s(y + b)LI

'it' + h I + at./

These express ions ean be s impl i f ied cons ide rab ly , f i r s tby comb1n1nB; th e second and th i rd fac tors in the f i r s tterm o f U J and then by ! B c t o r 1 n ~ ou t the common termsof the severa l funct ions. The f i na l s impli t tad form1s as below:

u : h t + e [ a d ( b+w t ) + X t ] (x 1- yo' + dv ) (y + b )Uw = wx :y :v = =

&t+ 1f1 bx + w\(s+C)yx +

y t + a c + 8 ' C ~

g I + ('1 + 4& + U IThis corresponds to the c ir tcui t on th e f o l . l o l J V i n ~ page.


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tJ

w. -----C:

7 a-d.]-7'. -oXa

~ ' " = o i i i i i i i i 4

u

Zl

_ - - - U' . t_0_---- - - - . CI --e__[7


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58

A Vote Counting Circui tA c i rcu i t i s to be eonstructed with the follow-

ing prope r t i e s . Thera are to be thi r teen l i gh t s , marked0,1,2 12 and tw elve two-posi t ion switches , ~ l ' x2 x12 ' one fo r each voter , each me rked with two possiblevotes , yes or no. There is a lso a control tu t ton c.The l igh ts are to count the number of ' yes ' votes .t t 5 voters move t h9 i r sWitches to the ' yes ' posi t ionand the r e m a i n 1 n ~ 7 vote Ino, ' the l i gh t marked 5 1sto l i g h t up providing the contro l button C i s pressed,and simile rl:r fo r any number of votes .

This i s clearly an app li ca tio n o f symmetricfunc t ionS diseussed prev ious ly . If we r ep resen t th el l ~ h t s hv ' the synltols La ' Ll , L12 , the th'3 equ.st10ns of the system wil l evident ly be:

k = 0, 1 , 12The oi rcu i t representing th i s system according to thesymmetric funct ion development wil l be:

cX2 x3 . Fig * :35

+


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59

Elec t r i c Adder to t"b.e Base T"wvo

A c i rcu i t i s to be designed t h a t wil l 8utomat-i c a l l y add two numbers , u s 1 n ~ only re lays and sw i t che s .

A l t h o u ~ h any numbering base could be used the c i r cu i t i sgreat lY s imp l i f i ed us ing the sca le of two. Each d i g i ti 8 thus e l ther 0 0 r 1 ; the number Who sa di ~ t s in orderare sk ' 8 k _1 , B k _2 , 8 2 ,8 1 , 8 0 has th e valuekL B 2 j Let the tw o numbers whi ch B re to be added bej'";O jrepresented by a se r i es of sWitches ,sk ' a k _1 , ~ l ' 8 0representing the vario us digi t s of one of the numbersand ~ , ' \ : -1 ' ~ , bO the di ~ t s 0 f the other number.The sum vv111 be repres-3nted by the pO sitions o f a se to f re lays sk+1 ' 5 k , ~ - 1 5 1 , SO. A number whichis ca r r i ed to th e j t h column from th e ( j - l ) t h columnwil l be repre sen ted by a re1a Y C j 1ft he value ofany d1 g i t 1-5 zero , the c o r r e s ' P O n d 1 n ~ r e l ay or awi tchwi l l be taken to be in th e posi t ion of zero hinderance;if one , in th e p os i t io n Where th e hinderance i s one .The e c tua l edd i t ion i s sho\:Vn 1:elow:

c k+1 Ok c+1c j 2 c1 carr ied numberst1oak ----- 8 .;+18 8 2 a1 e.o 1 s t number

~ b j ~ l b j b2 ~ be 2nd number

Sum


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60

s ta r t ing from the r igh t , So i s one if 8 0 1 S ona and boi s zero or if 8 0 1s zero and bo one but no t otherwise.Hence:

C1 i s one i f both 8 0 and bo are one but not otherwise.

5j is one if just ana o f 83' b j , Cj 1s one, or if 811three are one.

va r i a b le s l el' b ~ , C,.]C j +1 i s one if two or i f threa of these variatiles ~ r e one.

\Using the method o f symme'tric funct ions, and sh i f t -ing down for s j ~ i ves the ci rout ts of Fig . 36.

j a 1 , -2 , k j 0

+

a-lJ c J

.j+1-

Fig . 36

+


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E 1 1 m i n e t i n ~ ~ l p e r f l u o u s elements we arrive a t F 1 ~ . 37.

j - 1, 2, 3 s k

+b 1j


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62

A Factor Table MachineA machine i s to be designed which wi l l auto-

matical ly pr in t 8 tab le of factors and primes of a l lthe in tegers from 1 to 100 ,000 ,000 . I f 8 number 1 ~prime, it 1s to be so marked; if composite, i t s l ea s tfactor i s to be printed beside it. The pr incipleWhich wi l l be used i s tha t of the sieve of Eratosthenes(6) Let the na t u ra l numbers be wr1 t t en in order :

1 , 2 , 3 , 4, 5, 6,7,8, . . . Now consider the prime numbers in ordar, 2, 3, 5, 7,11 , 1 3 , 1 7 Each 2nd number a f t e r 2 in the row ofnatura l numbers has the l e a s t prime fac tor 2; each'th i rd number af te r 3 Which i s not 8 mult iple o f 2 has thel ea s t prime fac tor 3; each 5th number af te r 5 not div i9ib le by 2 or 3 has the l e a s t prime fector 5, e tc . Anynumber F not haVing a prime factor l ess than i t s e l fi s , of course, e prime. It i s customary in ta blasof t h i s so r t to omit numbers div is ib le by 2 , 3 , o r 5thus reducing the numbar of in tegers which need beQonsidered to 4/15 o f the l a rges t number N (108 inthis case) . I t should also be noted tha t any compositenumber l e s s than o r equal to N has a l e a s t factor lessthan o r equal to IN. Thus in 011 r ca sa only prime sl e ss than 10,000 need be considered in the f 1 1 t e r i n ~


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process d e s c r i b e d . The asymptotic formula N/ln N( f o r the number o f primes l e s s than N) shows t h a tthere are a b o ~ t 1000 p rimes l e s s than 10,0000 Leteach of t h e s e nrimes a f t e r 5 be represented Of acounter Ok w i t h the following proper t ies . There eret h r e e magnets, M2' M ~ , and lvI6 ~ v h e n M2 opere tes a l lthe counters are advanced 2 u n i t s ; M ~ and M6 advancethe oounters and 6 u n i t s ~ e s p s c t i v e l y . The purposeof these magnets i s to automatically omit numbersd i v i s i b l e by 2,3, and 5. l'Jote t h a t s t 8 r t i n ~ w i t h 1the next number not d i v i s i b l e by 2, 3, or 5 i s 7 ,an advance o f 6; the next advanoe i s 4 (to 11) , then2 ( t o 1 3 ) . The t o t a l cycle o f advanees i s a s f o l l o w s :

6, 4: , 2 , ~ , 2 , ~ , 6, 2 ( l)a f t e r which the seme s e ~ 1 a s 15 repeated ( the period1s 30, the l e a s t common mult iple o f 2, 3, and 5 ) .As t h e successive numbers are considered fo r f a c t o r so r p ~ i m a 1 1 t y , t h e aounters w i l l advance a c c o r d i n ~ t ot h i s sequence. When any counter Ok represent ing theprime Pk rep.ches the value o f t h i s prime it is to beso constructed t h a t it automatically makes a connec-t i o n Xke Each counter i s to have a return magnet R ~ JWhich when a c t i v a t e d returns the counter to zero. Thegene ra l ope I's tio n o f the devi cs w i l l then be a s fol loVJs.S t a r t i n g a t t h e number 1 ( t h e counter and p r i n t e r


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rep11ssent1ng the numbe'!' being con sidered se t a t 1) andwith the counters r e p ~ e s e n t l n g the primes l ess than10,000 a l l se t a t z e r ~ th e coun ters are advanced according to the sequenoe (1) . I f fo r any number N, Xk makescont-act, then Pk i s a fac tor of N; the l e a s t Pk beingthe l eas t fac tor . I f no Xk makes con tac t , N 1s 8prime. When any Xk makes oontact , it is to be automatical ly returned to zero by means of Rk To recordth e re su l t s a prain t a r Uk should be a s socia ted wi theach counter which wi l l p r i n t the value o f th e primePk oPPOsite N when ma gnet Uk i s ac t iva ted . If N i s aprime, a Pri n t e r S should prin t a symto 1 -to ca l l 8 t t en t lon to the fac t .

A l t h o u ~ h t h i s en t i r e d esig n c ou ld be es r r1edou t with re lays a lone , it i s probably more economicalto cons t ruc t the counters on mechanical princ i P les ,and 1;herefore only the cont ro l c i r cu i t s vn i l be described. To automatioally advance the numbers Btshor t in tervals some kind of an impulse generator is

n ~ c e s s 8 r y . The simplest method of obtaining th is 1sto use a relay with a small tim e delay 8. I f ther e l ay i s l abe led z( t), then the con t ac t s have 8 hinde:raance funct ion Z( t - 3), and the connect ion Z( t ) - Zt (t- B)Wil l ~ i v e e se r i e s o f impu.lses of J)er1od 21. The s e quence of advances may oe eas i ly obt8ined with an 8


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p O ~ i l l t : po ta ry swi tch. . Let t h i s switch have a magn,et Lv\rliieh ad.vtanoas ttle sV:ii toh one po in t wh.ans o t tva te.d.

T l f t t a l . " ' i 1 f ~ a e o n n ~ l o t , L so the t L = Z( t .. ,.) and Clonnectljite contacts of the rotary s\"J1 tch to. the magnets M2 ,M:;lla!!ld AtlS according to theord,er o f (1 ') , the cOttutersw,tl l a l l b'a a dv,sn cad pe r iod i as l . ly in t h i s S6qtt$nde.

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tna Xk$ wi..*l equal ~ e r o i f the n . U I l ' 1 ~ r 1. s compost te .1.n tltd.sc9se these }tkS sh.ou1d (laU sa t ha s malle $ tfa c'"ta::r to J)J!tnt.and then retuX'n to zero. This conditionW111be sst :1sfied by the follow!.ng equat ions :

I t : ~ l .tJ .: X... + .I./..t~ . X t . . + 0. t (t- 3) k :: 1, 2 , 3.....k k'=l j

l ~ :: ltk +1( t - e ) (2 )y{ t )= .. ("6-6).

tJ:hatis, -0,119 pr1r.lter tT opepates if Xk.... :: 0 and tht:) x.J-9 e,. . . . . .... kj(:k, do nco t . equal z ere> Also a ftera d a1.8 y e to ai.low

. 01: p:rinting, the counter i s rZ"7turned to zero. Ifnone of the Xksmake contact .on a !1l1mbep bT, it is a'P:p:tmeand S should prin. t Phi s a a n be a c co mpl i shed'With th e f o l 1 o ~ v i n g eqt la t ion:

:rnhemain pr in te r B'nd counter should pr in t on each


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U s i n ~ the method o f factorj .ng o f simultaneous equat ionsthe system (2) can be greatlY s impl i f ied as fOllows:

U1 =X1U2 - X- 2U3 - X3 + X'- 1U - X .... \+n ~ - l \.. + X'n

The c i rcu i t of the ent i re device i s sho\m schematical lyin Fig . 38.


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Fig . 38


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This design requires tna t t ~ e primes lessthan 10)000 be known. I f desired, tne machine couldbe made to autoITl9.tic8.11y connect in neVi counters asthe primes were found, but there 8.re many 8.ccurs,tetables of primes up to 10,000 so tha t this would notbe necessary.

As to the pract icabi l i ty of ellen a device, itm i ~ ~ t be s ~ i d tna t J .P . Kulik spent 20 years inconstruct ing a t able of primes up to 100 ,000 ,000 andwhen finished i t was found to contain so many errorst ha t it was not worth publlsning. The macnine describedhere cou ld p robab ly be made to nandle 5 numbers persecond so tna t the table would require only about 2months to construct .


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REFERE1TCES

(1) A complete b i b 1 1 o ~ r a p h J , , -PV .L the l i t e r a tu re ofsym"oo11c log ic i s gi' ,en in the Journal o fsymoolio Log!

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