ESTIMATION' AXD COXTUOI. OK Ol'KIlATK TIMH OF RKLAYS 123 



the flux remains coiistaiit at the \alu(> attained at tlic end of tlic first 

 stage. 'I'his approximation necessarily over-estimates tiie opei'ate lime, 

 as it takes the two stages as occurring in sequence, whereas lliey ac- 

 tually oxerlap. 



Let /i he the time of the first stage, and /•> the time of the second stage, 

 w here ^> is the time for the motion from Xi to some other armature posi- 

 tion .r_. , measiiringthe completion of ojXM-ation. The first stage ends when 

 the flux has attained some value r<pi , wliere ^i = ;T^■/(Rl , the steady state 

 flux for the open gap, and v remains to be determined. Then ^i is given 

 hy (9) for this value of r. 



By hypothesis, the flux remains constant at Vipi throughout the second 

 stage, and the corresponding pull is then given by (5A) for ip = Vipx . 

 'llie work T' + T is the integral of F-dr from .V] to .(••^ , gi\-en l)y th(> 

 integral of (')A). Hence: 



^ttA {ClXo + Xi)(ClXo + X2) ' 



In this equation .r,, .1 may be replaced by the expression for (}^o 

 given by (4A) for .r = Xi , and Gc-I R substituted for (NI)' in the re- 

 sulting equation, which then becomes: 



where d, is given b}^: 



(Cl - l)a;o(.Ti - ^2) * 



The time ^2 of the second stage depends upon the difference between 

 the pull and load curves. As a representative condition, this may be 

 taken as approximately constant. For such uniform acceleration of the 

 effective mass jn through the distance Xi — x^ , the kinetic energy T 

 equals 2wCri — .r->)'/75, giving an expression for ^2 in terms of T. As the 

 time /i of the first stage is given by (9), the total operate time to , or 

 ^1 + /l> is given by: 



, MG, + Gc), 1 , f2m{x, - X2)y ,. .. 



'" ^ — w, — ^^r^". + V — r — ;• ^^^^ 



In this approximate expression for the op(M'ate time, v and T are as 

 yet undetermined quantities, related by ecjuation (12), in which Gc 

 is the value of N' R for the coil used. The two stage operation assumed 

 in deriving this expression corresponds to the existence of a restraint 



