Ferromagnetic Distortion of a Two-Frequency Wave 



By ROBERT M. KALB and WILLIAM R. BENNETT 



Frequency components are found for the ferromagnetic induction 

 produced by a small magnetizing force of two incommensurable frequencies. 

 Because of hysteresis the results depend intimately upon the ratios of these 

 frequencies and of their amplitudes. With these ratios as criteria, two solu- 

 tions are provided, adequate for most modulation problems of this character 

 occurring in the field of communications. 



The development is based on Madelung's empirical propositions. From 

 these are deduced the forms of complex hysteresis loops occasioned by two- 

 frequency magnetomotive forces, and from the loops sinusoidal components 

 of the flux wave are derived by means of Fourier's series. The various 

 voltages generated in a coil by such a flux are then calculated and next 

 correlated with analyses for a single applied frequency. The resulting 

 changes in the impedances to the two fundamental frequencies are also 

 evaluated. The most important results are given in graphs and tables. 



Experimental data on a number of specimens show close agreement with 

 curves computed by the theory. 



The analysis discloses several interesting features. It is shown that 

 Madelung's conclusions imply Rayleigh's law of loop similarity; as a 

 consequence the parameters of a Rayleigh loop suffice to describe a complex . 

 loop to the extent that it conforms to Madelung's results. Hysteresis 

 suppression is found not to occur at low fields, although harmonic sup- 

 pression may. The generated side frequencies of the flux appear in unequal 

 pairs, the lower one being the stronger in each instance. Such inequality is a 

 general property ascribable to the multivaluedness of the loop. 



FOR precisely evaluating the performance of communication 

 circuits containing ferromagnetic materials, methods for taking 

 into account the non-linear effects of these materials are needed. To 

 this end there have been devoted certain investigations of the behavior 

 of such materials at the low flux densities usual in communication. 

 These early disclosed that hysteresis is a governing factor for weak 

 fields and led to attempts to solve the problem of its bearing on speech. 

 The complexity of speech and of hysteretic phenomena has made 

 desirable the use of simple testing methods, which in turn require for 

 their interpretation a quantitative theory. Since tests of this sort 

 are usually made with one or more sinusoidal test waves, a theory of 

 single-frequency magnetic performance has already been evolved as a 

 first step toward fulfilling this need. For many purposes single- 

 frequency tests are inadequate, and two-frequency waves are often 

 used to obtain better information bearing on the design or performance 

 of communication systems. It is the purpose of the present paper to 

 take a further step by furnishing the theory of magnetic behavior 

 under a two-frequency force. 



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