OPEN-CONTACT PERFORMANCE OF TWIN CONTACTS 1377 



(1 — /„) of the opens will persist for a relatively large number of opera- 

 tions. A study of a variety of these clearing characteristics generally 

 indicates a rapid rise to the asymptotic value /„ in less than 100 opera- 

 tions, and to / = 0.5 in less than 10 operations. As will be shown the 

 fractional persistency (1 — /«,) is of major importance in determining 

 twin-contact performance. In general, ho^^•ever, opens on twin-contacts 

 are due to both the persistent half-opens and the temporary half-opens 

 that might develop into twin-opens before clearing takes place. 



DEVELOPMENT OF THE THEORY 



Consider a large number of contacts operating at steady conditions. 

 After N operations, let s^ be the fraction of the contacts that is per- 

 manently half -open and Sn be the fraction that is temporarily half -open. 

 As discussed, the number of operations necessary to clear a half-open is 

 not constant and, for the majority of the contacts, is of the order of a 

 few operations. To simplify the treatment, it is assumed that each 

 temporary half-open \n\\ clear in an average of n operations from the 

 time it first occurred. The fraction sn must, therefore, have been produced 

 during the n operations directly preceding the time t. Since n is usually 

 relatively small, the universe can be assumed to have had a negligible 

 change during the operations n. Hence, 



Sn = (rate of formation of temporary half -opens) X n 



= 77[2(1 - s)Ps - (1 - s)P:']f^ 



where s = total fraction of half-opens = s^i + s«, • Substituting /3 = 

 fif^Ps{2 - Ps) one gets 



s-n = 5-^ (1 - sj (2) 



Also, after A'' operations, the incremental change ds^ due to dN opera- 

 tion is: 



ds^ = [2(1 - s)P. - (1 - s)P/](l - U dN - s^Ps dN 



where the second term is the reduction in s^ due to occurrence of twdn- 

 opens. Substituting a = (2 — Ps)(l — /«) and combining with equation 

 2 to eliminate s-^ give: 



ds^ = 



'" 1 + a + /3 



