A Symbolic Analysis of Relay and Switching Circuits

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I Introduction: Types of Prob'lams In the control and protective circuits of complex electrical systems it is frequeutly necessary to make intricate interconnections of relay con~acts and switches. Examples of these circuits occur in automa.tic telephone exchanges, industr5.al motor control equipment and in almost any circuits designed to gerform cornplex operations a.utomatically. Twu problems that occur in connection with such ne~works of switches will be treated here. ·rb.e first, which will be c9.11ed analysis, is to cietermine the opera~ing characterj.. s tics of a given circuit. It is, of course, alwa.ys possible to analyze any given circuit by setting up all possible sets of initial conditions (positions of switches and relays) and following through the chain of event£ so instigated. This method is, however, very tedious and open GO frequent error. The second problem is that of synthesis. Given certain characteristics, it is required to find 9. circuit irLcorporating these characteristics. ~rhe solution of this type of problem is not unique and it is -cI-J.erefore additiollally desira.ble that the circuit requiring the least number of sVii te h blades and relay
2 COL~ac:s be ~ound. Although a solution can usually be obtained. by a lieu t and try" m_e thod , fi rs t satis fyi ng one requirement and then making additions until all are satisfied, the circuit so obtained will seldom be the simples~. '.mis method 9.1so ha.s the disadvantages of being long, and the resulting design often contains hidden "sneak circuits." The method of solution of these problems which will be developed here may be described briefly as follows: An;! cireui t is represented by a set of equations, the terms of the equations representing the variot~s relays and switches of the circuit. A calculus is developed for manipulating these equations by simple mathematical processes, most of which are .. similar to ordinary algebraic algorisms. This Ca.lculus is shown to be exactly analogous to the QalculUs of P,ropositions \lsed in the s'ymbolic study of logic. For the synthesis problem the desired characteristics are f~rst written as a s~rs"tem of eauations, and the equations are then ffianipulated into tha form representing the simplest circuit. The circuit may then be inwediately drawn from the equations. By this nethod it is always possible to find the simplest I circ~it containing only series and 9arallel connections,
Page 1 and 2: A SYl~:BOLIC OF Al~ALYSIS by Claude
Page 3: ACKNOvVLEDG1ffiNT The author is ind
Page 7 and 8: • 4 II ~eries-parallel Two Termin
Page 9 and 10: • 6 3. a~ 0 + 0 = 0 A closed ~irc
Page 11 and 12: 8 equals 1 .. If X is the }1-ind-=J
Page 13 and 14: 10 Ole to thi s analogy any theorem
Page 15 and 16: 12 first note the following equatio
Page 17 and 18: l~ be obtained by replacin~ each va
Page 19 and 20: 16 y as the "XU of 17b. }IO\iV mult
Page 21 and 22: 18 III Multi-terminal end Non-aerie
Page 23 and 24: 20 The wye to delta tran sformation
Page 25 and 26: 22 Fig. 12 'The hinderance function
Page 27 and 28: This 8~a1n gives for the hinderance
Page 29 and 30: 26 I I I 1 X 12 X 13 • • •
Page 31 and 32: 28 Xi matrix the oroduct of the ele
Page 33 and 34: 30 50m~~~mes a relay X 1s to operat
Page 35 and 36: 32 where (~) : nl/k Hn-k) ~ is the
Page 37 and 38: This will be proved by mathematical
Page 39 and 40: 36 Dual l{et\vorks. The ne~et1ve of
Page 41 and 42: 38 In a eonstant-voltage relay syst
Page 43 and 44: identically the same. Thus XY + XZ
Page 45 and 46: 42 zero if either factor is zero, w
Page 47 and 48: varia "bles of Fig. 26, the termina
Page 49 and 50: 46 the circuits are- as shown in Fi
Page 51 and 52: !8 a time delay relB;T of sona so r
Page 53 and 54: 50- on the in genu~..ty 0 f the de
Page 55 and 56:
52 A Selective Circuit A relay 1:\
Page 57 and 58:
54 Zl z .~ "f'l, +--........~---" A
Page 59 and 60:
56 11Vhen z has opersted the lock u
Page 61 and 62:
58 A Vote Counting Circuit A circui
Page 63 and 64:
60 starting from the right, So is o
Page 65 and 66:
62 A Factor Table Machine A machine
Page 67 and 68:
ep11ssent1ng the numbe'!' being con
Page 69 and 70:
66 Usin~ the method of factorj.ng o
Page 71 and 72:
This design requires tnat t~e prime
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A Symbolic Analysis of Relay and Switching Circuits

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