1378 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 



Avhere K is an integration constant. Let at .V = 0, s^ = So , which is a 

 description for the initial conditions of the contacts. The sokition be- 

 comes : 



s^ = 



1 + " + ^ „ \ -(l+a+^)/{l+^)P3.V 



1 - 1 - — '-^ So ]e 



(5) 



1 + « +/3 L 

 The rate, or probabiUty, of twin-contact failure is determined from: 



P, = sP« + (1 - s)P/ (6) 



Substituting from 2 and 5 into 6 and reducing gives: 



P, = P/ 4- P.(l - Ps) Y^ 



l+a+iS\\ OL 7 /_ 



where, one may repeat for convenience: 



a = (2 - P.)(l - fj and /3 = 7y^P,(2 - P.) 



For all practical cases, Pg « 1, a = 2(1 — /«,) and /3 = 2nf^Ps < 1. Sub- 

 stituting in 7 gives 



P, = P/ -f 2P, 



1 fx, I -, I 1 3 ~ 2/^ ^ ^ _(3_2/ )p^A^ 



«/„P.+ _^^^l-^l___^^,,. 



(70 



This is a general expression, relating the expected performance of twin- 

 contacts to that of single contacts. It is evident that the idealistic per- 

 formance of Pt = Ps", i.e., the probability of a twin-contact failure is the 

 square of that for smgle contacts, can only be achieved if: (a) f„ = 1.0, 

 i.e., persistent half-opens never occur, (b) il = 0, i.e., each temporary 

 half-open occurring during one operation will clear during the subse- 

 quent operation, and (c) So = 0, i.e., there is no initial contamination. 

 These conditions are never obtained in practice and generally Pt is much 

 greater than P/. 



Equation 7' also indicates that at the beginning of operation, when 

 the exponent is much less than 1.0, Pt is given by: 



(Pt)o = P/(l + 2nU + PsSo (8) 



Numerically if il = 50, /„ = 1.0 and So = 0, Pj = 101 P/ which is 101 

 times worse than the idealistic performance of Pt = P^. The initial rate 



