1234 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER, 1953 



and (1) 



(0 



. i(n) = cot"^ (n - 1)^^' (b) 



The Felation between the number of discharges n and time t is given by 

 t =lZ cot"^ in - lf'\ 



03 n=l 



Only an empirical expression for this summation was obtained with a 

 maximum error less than 10 per cent: 



The current build-up in the circuit is, therefore, expressed as a function 

 of time by the following relation: 



m =^{Vo- v)t (2) 



In other words, the current is independent of the contact capacitances 

 and increases hnearly with time at a rate inversely proportional to the 

 circuit inductance. t As soon as the circuit current reaches the arc 

 initiation current /jj, a steady arc is established. The initiation time of 

 the steady arc is therefore given by: 



U = I- ^. (3) 



2 Vo — V 



For /, = 0.5 ampere, L = 10"^ henry and Fo - i; = 30 volts, U = 2.6 

 X 10"* second. 



C, L and Contact, Fig. 2 



If the battery is replaced by a condenser C where the ratio a == c/C 

 is much less than 1.0 an analysis similar to the above can be made. In 



^ • The assumption that all discharges will take place from the same voltage Vq 

 w only true if: the motion of the contact is negligibly small, the discharges do not 

 cbiinKe the contact geometry to the extent of materially changing the contact 

 separation, and if the effect of the residual ions is negligible. 



t This is onlv true if the circuit resistance is zero. For a finite circuit resistance 



H, and ^ Wj much less than 1 ,0, it can be shown that the current approaches 

 the asymptotic value " — . 



1 It is shown later by measurement, that the initiation current /» is essentially 

 the same as the arc terminating current I^. 



