128 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 



FIRST STAGE 



The pull for x = xi is given by (5A), which may be written: 



where Ai is given by: 



^^^/ C^x, + x Y^ (18) 



In the first stage the flux increases to vnpi , where ^'i is given by (17) 

 for F = Fi . Then the time d of the first stage is given by (9) for v = Vi , 

 or by: 



h = (tc + Ie) In — ^ . (19) 



1 - Wl 



The solution for the first stage, therefore determines Vi and ^i . 



SECOND STAGE 



It is convenient to write the equations for the second stage in terms 

 of the ratio z defined by: 



t 



z = 



tc '!' ts 



where t is now measured from the start of the second stage. Neglecting 

 the change in reluctance wdth armature motion, the ratio v^/vi or v is 

 given by (9) , except that for v = Wi at / = the relation must be written : 



V = 1 - (1 - yi)e~' . (20) 



As ^1 = fP^./cRi , or 47rA//(Ri , and ^^ = — ' ^ , 



Oil ti 



<pi = -— 1 R-tc . 



Oil 



Taking the pull in the second stage as given by the expression apply- 

 ing for x ^ x\ , and substituting the preceding expressions for v and 

 ^p^ , there is obtained: 



' = St. (1 - (1 - '")^"') ■ (2') 



For the second stage dV/dx is constant at — Fi throughout, and 

 (3) reduces to m d'x/df = F — Fi , ov: 



