334 BELL SYSTEM TECHNICAL JOURNAL 



equation (12) is 



/ (I - k){l + k) _ TT 



^^^"^ \(1 -hk)(l - K) 2' 



which can be made small by making k and k each small compared to 

 unity, or each large compared to unity, or both approximately equal. 

 For the previously excluded instance k = 1, this angle can be limited 

 and the last condition fulfilled by keeping k nearly equal to one, as is 

 obvious from the equations. So for each of these three conditions on 

 the parameters, to a definite degree of approximation the envelope at 

 each point of tangency becomes the magnetizing force for the nearest 

 reversal point, a feature useful for the transformation of equation (8) 

 into a function of time. 



Calculation of the Induction — Case 1 



The three conditions are confluent and will be seen to set the 

 limiting bounds for case 1. When the fundamental frequencies lie 

 close together the ratio of their amplitudes is practically unrestricted ; 

 as more widely spaced frequencies are chosen it becomes necessary to 

 require an increased (or diminished) amplitude ratio in order that the 

 phase angle in equation (12) does not exceed the chosen limit. This 

 limit must be such that the cosine of that phase angle is substantially 

 unity. 



The maximum magnetizing force is thereupon 



Hj = (- 1)^P V(l + ky - 4k sin2 St . (13) 



As a periodic even function of st, II j may be expanded in a Fourier's 

 series 



4-"+ i: A, cos ■nst^ , (14) 



//. = (- iy{p + Q) 



ri = l 



where 



4 rv 



TT Jo 



A,= - \ VI - y^i'sin^ XcosrjXrt'X (15) 



in terms of the parameter 



, 2\!PQ 2^k _ 2\\Jk 



(P + (1 + k) (1 + l/k) ' 



which never exceeds unity and which diminishes as k is either increased 

 or decreased from unity. The integral (15) reduces immediately to 

 elliptic form by the substitution z = sin X. All odd order coefficients 

 are zero. For the first three significant values of rj : 



