ESTIMATION AND CONTROL OF OI'KU.VTE TIMi: OF RELAYS 127 



resistivity material. In either case, this iucromeiit would he of little 

 consequence in the low power case, hut would niat(M-ially affect the 

 operate time in the hi^h power case. 



5 TiiREio Stage Approximation 



The following analysis ])r()\'ides oreatei' accuracy in estimating operate 

 time than the two stage approximation, together with a more accurate 

 representation of the relations hetween performance and the contr(jlling 

 \';U'iables. It differs from the two stage approximation in treating the 

 initial motion as a separate stage of operation. The three stages of 

 operation thus become: (1) Increase of the flux to the value at which 

 the pull e(iuals the back tension, with the armature at rest, (2) flux 

 development and concurrent armature acceleration, assuming equations 

 (2) and (3) to apply, with the approximation that the variations of 

 (R and J with x are ignored, and (3) the later motion, which is treated 

 in the same manner as the second stage of the two stage approximation. 



Formally, the second stage should be restricted to a very small part 

 of the total armature motion, in order to minimize the change in (R 

 which is ignored. Relati\'ely minor error, howe\'er, is introduced in ig- 

 noring the variation in (R through as much as half the total travel. The 

 spring force is taken as constant through the second stage. In the relay 

 case, this isapproximatety true for the initial travel prior to the actuation 

 of the contacts, which results in an abrupt increase in the spring load. 

 Thus the second stage can be taken to extend to the tra\^el at which 

 contact actuation first occurs. If all contacts are actuated near the same 

 point in the travel, the end of the second stage coincides with complete 

 operation, and the third stage need not be considered. If the contacts are 

 spread out, the third stage coincides with the stagger time, or time be- 

 tween operation of the first and last contacts. The operate time there- 

 fore consists either entirely or principally of the time for the first two 

 stages. 



As before let : 



Xi be the initial (open) gap, for which (R = (Ri , 



Fi be the back tension (constant load in the first two stages), 



, ,...,.,. 47r A^' 



tc be the initial coil time constant, — • — , 



(Ri Pv 



.... . 47r 



//.; he the initial eddv current time constant, ^ Ge, 



<Hi 



:T.s be the steady state mmf, 47r.\7, and (^i tlie corresponding Hux for 



X = Xi , or 3^iV(Ri . 



