354 BELL SYSTEM TECHNICAL JOURNAL 



Hysteretic Impedances 



The fundamental voltages are not included in the table. For them 

 each component is separately significant. The singly primed £'s are 

 in phase with the corresponding magnetizing current and the doubly 

 primed £'s in quadrature. Hence, the former are resistance drops 

 and the latter reactance drops, defining incremental components of 

 impedance analogous to those mentioned for a single-frequency input. 



For each of the fundamental frequencies the results developed previ- 

 ously may be used to determine these components. They represent the 

 hysteretic resistance and the hysteretic reactance to the fundamental 

 at hand, specified by a subscript p or q. They are tabulated in Table 

 III for the special cases considered. The total resistance of the coil 



Case \a 

 .«1 

 A«l 



ij I Mo 



. -. iTT N V ^ ^ 



5 i Mo 



A/?9 -ry — gLo/ 



o t Mo 



AXg -^-T — qLoI 



o / Mo 



to either fundamental current can be calculated by adding to the value 

 of AR from the table the resistance of the windings, the eddy current 

 resistance, and the initial (viscosity) resistance, all evaluated for the 

 frequency of the fundamental. The eddy currents must be so small 

 that the flux density is substantially uniform across the cross section 

 of the core. The reactance Xo of the coil can be diminished by the 

 eddy current reduction factor for the fundamental frequency and 

 added to the hysteretic reactance to give the net reactance of the coil 

 under these conditions. 



The table is helpful in evaluating the effect of one fundamental 

 current upon the other. Within the limits of the analysis, which in 

 substance limits the permeability to linear variation with the field 

 intensity, the hysteresis loss at any frequency is either increased or 

 unchanged by the superposition of a second frequency. Nearly equal 

 input currents whose frequencies do not differ greatly share equally 

 the hysteresis loss. This amounts to about twice what it would if 

 either fundamental (lowed alone. If the frequencies differ greatK', the 



