120 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 



type of performance. For development purposes, an approximate solu- 

 tion indicating such optimum conditions is preferred to a rigorous solu- 

 tion in a form too complex for such use. 



Three such approximate solutions for the operate case are descriV)ed 

 below. The first of these, the single stage approximation, is an applica- 

 tion of the solution to the flux rise equation (2) , neglecting the armature 

 motion. It is applicable in cases of slow operation controlled by the rate 

 at which the flux development provides sufficient pull to operate the 

 load. The other two solutions are more generally applicable, particularly 

 for relatively fast operation. One of these, the two-stage approximation, 

 is the simpler in form, while the other, the three stage approximation, 

 is the more accurate. 



3 Single Stage Approximation 



The initial stage of flux development with the armature at rest ends 

 when the pull equals the back tension. During this stage the flux develop- 

 ment is governed wholly by (2), with x = Xi , the initial gap, and 

 (R = (Ri , as given by (4) for x = Xi . Writing tpiiRi for ^s , equation (2) 

 becomes : 



47rG r^ d<p 

 t = 



f 



(Ri Jo <Pi — (P 

 which on integration gives: 



47r(? 1 



where v = (p/tpi , the ratio of the flux attained at time t to the steady state 

 flux (fi or JFs/(Ri . The mmf ratio ^/^s is also equal to v. 



Equation (9) applies rigorously only while the armature is at rest. 

 It is, however, also a close approximation for the initial motion, in which 

 the velocity is small and x differs little from Xi . If the rate of flux de- 

 velopment is slow {G/(Ri large) the armature moves slowly with the 

 pull only slightly in excess of the load curve. In this case the inertia 

 term m-d x/dt in (3) is minor, and the pull F nearly equals the static 

 load dV/dx. Thus the operate time is approximately equal to the time 

 required to develop a pull which exceeds the static load at all points in 

 the travel. This pull is attained at the just operate current or ampere 

 turn value, corresponding to the minimum mmf for static operation, 

 3^0 . Assuming that the armature moves as the flux and pull de\'elop, the 

 operate time is that required for J to equal JFo , and is given by equation 



