124 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 



which holds the armature at rest until it attains the flux level Vip\ . The 

 approximation will most nearly approach the true relation if the effect 

 of this restraint is minimized, and v taken at the value which minimizes 

 ^0 . This minimum value can be determined by trial, using the curves of 

 Fig. 5 to compute /o for several values of v, and taking the minimum from 

 the resulting plot of U versus v. 



In development studies, the operate time of interest is usually not 

 that for an assigned winding, but that for the winding giving minimum 

 time for a given power input. Substituting in (14) the expression for 

 Gc given by (12), there is obtained the equation: 



where Ie is written for 47rC7ij/(Ri , the eddy current time constant. As 

 Gc remains to be determined, this expression contains two independent 

 variables, v and T, which are to be so chosen as to make /o a minimum. 

 Equating the partial derivative Avith respect to T to zero, there is ol> 

 tained the following expression for the value of T for- which the time 

 is a minimum: 



7 = 2m{.v-i — X2) I 



4C,r In ^ 



1 - v/ 

 On substituting this expression for T, the preceding equation becomes: 



\ V- PR/ I — V \ V- \ — V PR / 



If the value of v which minimizes t^^ is determined, the resulting value 

 of to is the minimum operate time attainable by optimum coil design. 

 In particular, if Ie is negligible, this minimum corresponds, from Fig. 5, 

 to y = 0.715, for which 



- In = 2.5. 



v^ 1 — y 



The corresponding value of Gc , the coil constant value making to a 

 minimum, is given by (12). In this use of equation (12), r and T are 

 taken at the optimum values obtained as indicated above. 



OPTIMUM COIL CONSTANT 



As in the case of the single stage approximation, the optimum coil 

 constant corresponds to a value for v of 0.715 when Ie is negligible. From 



I 



