A Symbolic Analysis of Relay and Switching Circuits

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27 WhA-rA .- . - TIl" - jk 1-g the ne~ative of trle hinderance function ~ from node j to k, the network considered 8S a two te~- 1nal cirellit. Thus for the three node network of Fig. 16 the X' and TIl matrices are as shown at the right. 2 xl\y lL~3 Zl z 1 Xl z' 1 X '+y' Z I z'+x1y' x' 1 y' x'+y' z' 1 y'+X'7.' y' 1 z'+x'y' :fl+X'Z' 1 Fig. 16 X' Matrix U' Matrix Theo~em: Any pO~er of the XI matrix of 8 network gives a netvlork which is equivalent With respect to 811 nodes. The matrix is raised to a powsr by the usual rule for multiplication of matrices. Theorem: I t , I 1 U12 .... U 1n 1 X 12 ... X 1n s I t , Xl 2rl - U 21 1 .... U 2n X 21 1 ... - ..... ........... U~l· • • • • • · • • • ...... " . , , • X 1n •.•.... 1 II s ~ n-l Theorem: Any node., say the kth, may be alirnina ted les'tling the network equivalent with respect to all remaining no des by adding to each eleraent X~s of the
28 Xi matrix the oroduct of the elemoo.ts X'k and X k ' , and r _8 s'criking aut the kth rowan d column. Thu s elimina tin ~ the 3rd node of Fig. 16 we p;et: L+z' z t x'+z'y' l+y'y' I: 1 , x'+J1z' X'+y1zl 1 The proofs of these theorems are of a simple nature, but quite lon~ end will no't be given. Special TyP3 s of Relays Band SVrl tches. In certain type s of circuits it is necessary to preserve 8 definite sequential relation in t'he operation of the con'tacts of a relay. This is done with make-barare-break (or continuity) and brae k-make (ot' transfer) contacts. In hand11np; this type of 01 routt the simple st me thad seems to be to assume in setting up the equations that the make and break contaots operate s1multaneously, and aft:3I' all simplifications of the equations have been made end the resulting ciI'cult drawn the required type of contact sequence is found from inspection. Relays haVing a time delay in onsrat1ng or deoner8ting may be treated similarly or by shiftin~
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: 26 I I I 1 X 12 X 13 • • •
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

A Symbolic Analysis of Relay and Switching Circuits

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