HARMONIC PRODUCTION IN MAGNETIC MATERIALS 763 



channels to which the non-Unear element is common. Such elements 

 are, in general, ferromagnetic core coils or transformers. The distor- 

 tion introduced by the non-linear relation between flux density and 

 magnetizing force is therefore of fundamental importance in the design 

 of iron core coils and transformers which carry simultaneously a 

 number of communication channels. 



The use of iron core coils and transformers in communication work 

 is confined to comparatively low flux densities in contrast to the ordi- 

 nary practice in power work, where operation usually occurs above the 

 knee of the normal magnetization curve at a value of the order of a 

 thousand times greater than that used in communication work. There 

 are two main reasons for this restriction: losses are reduced, and the 

 relation between flux density and magnetizing force approaches line- 

 arity so that distortion is minimized. 



Under actual operating conditions in which the more important 

 crosstalk effects arise, we have a complex wave of magnetizing force 

 acting on the magnetic core. Non-linearity in the magnetic circuit 

 gives rise to new frequencies which normally ^ are related to those 

 impressed upon the circuit, being sums and differences of integral 

 multiples of the originally impressed frequencies. These modulation 

 products are all of odd order - when the core is unpolarized. On 

 purely theoretical grounds we would expect to find relations between 

 the amplitudes of the different frequencies resulting from any one 

 order of modulation. To take the third order modulation products of 

 two impressed frequencies (/i, /o) as an example, we would expect the 

 amplitudes of the harmonics 3/i and 3/2 to be related to the amplitudes 

 of the other third order products: 2/i ± /2, 2/2 ±/i. This has been 

 confirmed by direct experimental test, so that we are enabled to use 

 the generated third harmonic voltage as an index of the generated 

 voltages corresponding to the other third order products. Accordingly, 

 we shall deal in the following with the third harmonic produced by a 

 sinusoidal wave of magnetizing force, and so avoid a more involved 

 analysis. 



The fundamental relation in the operation of ferromagnetic appa- 

 ratus is of course the relation between the flux density B and the 

 magnetizing force H. In contrast to the usual behavior of circuit 

 elements, the relation between the two fundamental quantities — the 

 independent and the dependent variables, // and B in this case — is a 

 function not only of the value of the independent variable, but is also 



^ This is true in the low flux density region. At high densities and with highly 

 reactive circuits as in magnetic niodulators, other frequencies are sometimes found 

 which correspond to natural oscillations of the coil and circuit. 



- Bell System Technical Journal, Jan. 1928, pp. 110, 111. 



