A Symbolic Analysis of Relay and Switching Circuits

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33 It is easy to sP£w that the sum modulo two obeys the commutative, asso(}iative, and the distrihltive law with resoect to multiplication, i.e. x 1 ex 2 : ~exl (X 1 eX 2 )eX 3 =X 1 8(X 2 eX 3 ) Also: x el : X' 1 1 Since the sum modulo two obeys the associative law, we may omit parentheses in a sum of several terms Without ambiguity_ The sum modulo two of the n var1- ables X 'A ••••X will for convenience be written: 1 2 n Xlex2ex3···e~ n =~Xk Theorem: The two functions of n variables which require the most elements (relay contacts) in a seriesn n parallel realization Bre ~X"and (~X~)I, each of wlUch requires (3·2 n - 1 _2) elements. 12 1
This will be proved by mathematical induction. First note that it is true for n = 2. There Bre 10 fUnctions of 2 variaQles, namely, rl, X+Y, Xty, XI+Y, XY', X+Y' J. X'Y' J XI +Y', XY' + X'Y, XY+X'Y'. All of these but the last two require two elements; the lest two re~lre· four elements and ara XfY and (X8Y)' respectively. Thus the theorem is tru~ for n = 2. NoW 8 SBuming 1 t true for n-l, we shall prove 1 t true fo!' n and thus complete the induetlon • Any function of n variables may be V'/rl ttan by (lOa): l~ow the terms f(1,X 2 •••X n ) and f(O,Xe:> •••X ) are f\1nc" ...., n . tiona of n-l variables, and if they individually require the most elements for n-l varia bles, 'then f will require the most elements for n variables, providing there is no other method of writing f so that less elements ere required. t~Je have assumed that the most elements for these n-l variables are required by ~Xk and (~Xk) f - If we therefore substi tu te for n f{1,X -- .X 2 ) the function ~x n and for f{O,X - _X ) 2 n the ~unction n ..., k (teXk)f we get: 2 n' n n f = Xl.,#Xk + I Xi(f2Xk ) t = (~2Xk) I
Page 1 and 2: A SYl~:BOLIC OF Al~ALYSIS by Claude
Page 3 and 4: ACKNOvVLEDG1ffiNT The author is ind
Page 5 and 6: 2 COL~ac:s be ~ound. Although a sol
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: 32 where (~) : nl/k Hn-k) ~ is the
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

A Symbolic Analysis of Relay and Switching Circuits

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