772 BELL SYSTEM TECHNICAL JOURNAL 



Details of the derivation are given in Appendix 3. Inasmuch as we 

 assume the appHed field to vary as cos pt, the &'s are in phase with the 

 applied field and the a's are in quadrature. The two o's, it is observed, 

 are connected by a constant of proportionality {ax = — Sa^) and 

 depend upon the remanence; or upon what is the same thing, the 

 hysteresis loss divided by the magnetizing force. 



If we expanded the loop equations to higher powers we should find 

 that fli and az cease to be linearly proportional. The coefficient 61 

 is observed to have its first three terms identical with those of the 

 normal magnetization curve, but the fourth term differs by precisely 

 the amount 63. 



The voltage existing across a coil enclosing a core characterized 

 by the above coefftcients is 



E^nAlO-'^, (26) 



where n is the number of turns enclosing the core, A is the core area 

 in cm-, B is the total flux density and E is the total generated potential. 

 Carrying out this operation we have 



E = nAlO'^iptti cos pt + 3pa3 cos 3pt 



— phi sin pt — Zphz sin 3pt) (26a) 



in which the voltage components in phase with the current depend 

 on the hysteresis coefficients, and the quadrature components depend 

 only on the coefficients of the normal magnetization curve. The 

 dependence of each of these components upon the applied field is clear 

 from Equation (25). To take the two third harmonic components it 

 will be observed that as starts to vary with the square, while 63 starts 

 to vary with the cube, of the applied field. It follows therefore that 

 at sufficiently small amplitudes the third harmonic is produced by the 

 az term and not by the 63 term, which means that under the conditions 

 noted, harmonic production is due to hysteresis and not primarily to 

 permeability change. 



From the two fundamental components of voltage across the coil 

 an expression for the inductance and resistance off'ered to the flow of 

 alternating current may be deduced. The inductance, it is easy to 

 see, is obtained directly from the magnetization curve, — the d.c. 

 permeability of the normal magnetization curve therefore coincides 

 with the a.c. permeability at small fields. The resistance may be 

 obtained from the expression derived above for hysteresis loss per 

 cycle per cc. If we multiply that value by the volume of iron in the 

 coil, by the frequency, and by 10~^ to convert ergs to watts we have 

 the loss per second, and this may be equated to the square of the 



