ESTIMATION" AND t'OXTllOL OF OI'KRATE TIME OF RELAYS 131 



3 2m.4i(Ri(.ri - X2) tc + ts zl ,^.. 



h = ^,-r> -, 77 7- V^'*; 



I-Ii tc hyVi , Z-y) 



As can he seen in Fig. G, /sCci , z) is, to a rough first appi-oxiinatiou, 

 proportional to z'\ To the extent that this approximation holds, [2 is 

 proportional to the cube root of tnAi(Ri{xi — X2)/{I'R), as for the last 

 term in (15) and (16), except that (.Ci — .^2)' is replaced by Ai(Hi(.Ci — .X2). 

 From (-1A) and (18), Ai(Ri is given by: 



(C - l).To 



Here.ro = A(Ro , and (2 is a minimum for that pole face area A for which 

 .li(Ri is a minimum. This condition, as determined by equating the 

 derivative of the preceding expression with respect to Xo to zero, is 

 given by: 



2 n 2 



xt = Ca 



> 



or Xi/A = \/Cl ' ^0 , where A is the optimum pole face area for fast 

 operation. This is a smaller pole face area than that for maximum sensi- 

 tivity, for which Xi/A should equal (Ro , as shown in the companion 

 article cited above. If the pole face area satisfies the preceding condi- 

 tion, the expression for Ai reduces to: 



Vcl- 1 



and the term Ai(Ri in (24) reduces to .Ti(.ri — X2) multiplied by the factor 

 (VCl + l)/('\/Cz, ~ !)• This leakage factor, which increases with the 

 ratio of leakage to useful flux, is then the only term in (24) which varies 

 AATth the design of the electromagnet. 



If f2ivi , z) were strictly proportional to z^, equation (24) would be 

 nearly independent of the coil constant N^/R, to which tc is proportional. 

 As /2(yi , z) departs from this cube law relation, and also varies with Vi, 

 t2 is not independent of the coil constant. The optimum coil constant is 

 that which minimizes /i + to. In any specific case, a succession of values 

 may be assumed for Gc , and ti and ^2 evaluated by the relations given 

 above. The resulting relation between ^1 + ^2 and Gc is similar in character 

 to that described above in connection with the two stage approximation. 



6 Design for Fast Operation 



The approximations discussed above provide a means for estimating 

 the operate time attainable in specific cases, and for selecting certain 



