328 BELL SYSTEM TECHNICAL JOURNAL 



be taken as zero, equivalent to measuring the time from a point where 

 the two components of the magnetizing force become maximum simul- 

 taneously. Then 



h = P cos pt -\- Q cos qt (2) 



is the instantaneous magnetizing force. Phase angles can be made 

 arbitrary by replacing pt and qt by pt + dp and qt + Og, respectively, 

 in this equation and the subsequent results. 



The configurations of complex loops can be followed by altering 

 single-frequency loops to accord with the results of investigations of 

 loop formation. If to a single-frequency magnetizing force a relatively 

 small one of slightly lower frequency is added, the resultant is an 

 oscillatory force whose peaks undulate around the value reached by 

 those of the original wave. Their maximum will be the sum of the 

 amplitudes of the two components and their minimum the difference. 

 On a hysteresis loop (Fig. la) this means that portions between suc- 

 cessive reversal points will differ slightly from one another, being 

 formed approximately as if belonging to successively smaller loops 

 until a minimum peak is passed, thenceforth as if belonging to suc- 

 cessively larger loops, and so on cyclically. Such behavior is sketched 

 in the figure. 



As the amplitude of the lower frequency component of the mag- 

 netizing force is increased the undulations become more pronounced 

 and the preceding picture more inexact. When both amplitudes are 

 equal the envelope of the resultant magnetizing force vanishes period- 

 ically and the portion of the hysteresis loop formed while this envelope 

 goes from its maximum to zero is in the nature of a spiral (Fig. lb), a. 

 similar curve being developed outwardly as the envelope increases 

 again to its maximum. Provided only that successive peaks of the 

 magnetizing force do not differ greatly in magnitude, each portion of 

 such a loop between adjacent reversal points may be assumed to 

 have the form of a branch of a single-frequency hysteresis loop having 

 its point of origin coincident with that of the portion of the complex 

 loop. Then the induction may be derived from the magnetizing force 

 by the use of single-frequency data in accordance with the known 

 manner of formation of complex loops. 



If now the amplitude of the higher frequency component be decreased 

 to a relatively small value, the undulations in the envelope subside, 

 and a condition similar to the original one is seen to obtain. This 

 time, however, the amplitude of the lower frequency will be found to 

 be the one about which these undulations occur, and the characteristic 

 will again look like that in Fig, la. 



