324 BELL SYSTEM TECHNICAL JOURNAL 



method has yet been developed for handling this problem in a manner 

 analogous to that applicable for a single frequency, practically im- 

 portant cases are solved here by means extensible to more complicated 

 ones. 



Considerations of Wave Shape and Loop Forms 

 Scope of the Two- Frequency Analysis 



The envelope of a wave affords a means for classification. Waves 

 whose envelopes change gradually during any oscillation form a class 

 apart from those waves which have envelopes subject to abrupt 

 changes. Members of each class can be segregated into those with 

 envelopes nearly symmetrical with respect to the average magnetizing 

 force, those with envelopes of almost uniform width, and so on. One 

 of the two last-mentioned properties in a wave with a gradually 

 changing envelope is essential to successful analysis by the methods 

 about to be detailed. 



The cases of magnetization analyzed, which include all the two- 

 frequency wave shapes that can qualify under the foregoing criterion 

 of tractability to analysis, are the following: 



Case 1. The ratio between the geometric and arithmetic means of the 

 amplitudes much smaller than the ratio between the sum and 

 difference of the frequencies. 



Case 2. One fundamental frequency high relative to the other; the pro- 

 duct of the higher frequency with its amplitude large relative to 

 the product of the lower frequency with its amplitude. 



Between the two cases there exist intermediate ratios of frequencies 

 and amplitudes over which the theory does not extend; however, the 

 most frequent problems are usually entirely within the domain of a 

 single case. The inequalities involved in the foregoing case definitions 

 are not susceptible to simple explicit statement as limiting numerical 

 ratios, in advance of the development of the theory. Much depends 

 upon the accuracy required in predicting performance. From data 

 supplied in the paper, following the theory, it is possible to determine 

 the practical limitations of the mathematical treatment. 



Formation of Complex Loops 



Considering the complex hysteresis loops arising from multi-fre- 

 quency magnetizing forces to be many-valued characteristics for 

 determining the flux density, it is pertinent to study their formation 

 and to correlate their parameters in so far as may be possible with 

 those of single-frequency loops. The latter at low fields are known 



