HARMONIC PRODUCTION IN MAGNETIC MATERIALS 775 



In the case of a sinusoidal magnetizing force the fundamental and 

 third harmonic flux densities were derived with the aid of the trigono- 

 metric series development. These were shown to depend in a simple 

 way at low fields upon the normal magnetization and hysteresis loss 

 characteristics. The in-phase fundamental voltage component de- 

 pends directly upon the hysteresis loss per unit current, while the 

 fundamental component in quadrature varies with the magnetization 

 curve at low forces. The in-phase third harmonic varies with the 

 in-phase fundamental at low forces, and the third harmonic in quadra- 

 ture comes in only with larger forces in a manner which depends upon 

 the magnetization characteristic. The range of forces over which our 

 equations are valid depends simply upon the number of terms taken 

 in the development of the branch equations, and is not necessarily 

 restricted to small forces. The hysteresis loop coefficient enters into 

 the impedance offered to the fundamental frequency by a coil enclosing 

 the core material in question in such a way that the third harmonic 

 produced may be deduced from the change of resistance with current. 

 Further, the ratio of the change of reactance with current to the 

 change of resistance with current may be used to provide a test of 

 Rayleigh's relation, which is found to hold for some materials, while 

 it is invalid for others. The precision obtainable in the evaluation of 

 hysteresis loop coefficients is much greater by the a.c. bridge measure- 

 ment under proper experimental conditions than by the analysis of 

 ballistic loops. Incidentally attempts have been made to obtain B-H 

 loops by ^C methods with the aid of the Braun tube, for example, ^^ 

 but the precision attainable is not sufficiently high for our purpose. 



Complex Magnetizing Force. Our preceding analysis has furnished 

 us with the fundamental voltage drop across a coil enclosing the iron 

 core under consideration, together with the third harmonic voltage 

 generated in the coil winding due in general partly to the non-linear 

 B-H relation and partly to the effect of hysteresis. The generated 

 ^•oltages corresponding to the fifth, seventh, and higher orders are 

 also calculable by the same methods but will not be specifically con- 

 sidered since they are smaller than the third harmonic at low fields. 

 This generated third harmonic voltage exists in its entirety across the 

 coil winding only when the impedance of the external circuit is much 

 higher than that of the coil at the harmonic frequency, a condition 

 not usually satisfied in telephone circuits. A current of the third 

 harmonic frequency then flows in the circuit, its amplitude and phase 

 depending evidently upon the generated third harmonic voltage — 

 that is, the coil structure and core material — as well as upon the total 



" Peterson, Phys. Rev., Vol. 27, No. 3, p. 320. 



