Hysteresis Loops and Lissajous Figures. 431 



area of the fundamental elliptical constituent. To test this 

 conclusion the areas of the three loops, figs. 18, 19, and 20 

 were planimetered for comparison with the values of A x : 



Planimeter reading-. A x . Ratio. 



Fio-. 18 33-8 0-57 594 



Fio-. 19 140- 2-31 605 



Fig. 20, 167-5 2-74 610 



§ 9. Presence of Eddy- Currents. 



If the hysteresis loop has been produced by some slow 

 process, absence o£ eddy-currents may be assumed. But 

 this is by no means the case when alternating currents of 

 ordinary frequencies are used, even if the iron be finely 

 laminated. It therefore remains to be seen how the presence 

 o£ eddy -currents will affect the size and form of the hysteresis 

 loop. The eddy-current, being a secondary current, will be 

 of pure sinusoidal form ouly if the inducing electromotive 

 force be of a pure sinusoidal form, and if the resistance 

 and permeability be constant also. But it is not necessarily 

 in phase with the impressed electromotive force, but may lag 

 by magnetic reaction ; and indeed, as is already known, lags 

 by different amounts at different depths below the surface of 

 the iron. Assuming equal permeability and resistance in the 

 different layers, the effect of the eddy-current will be repre- 

 sented with sufficient accuracy by a sine-curve lagging by 

 an amount that will depend on conditions into which there 

 is no need here to enter. For here, again, the only effective 

 component — effective "that is in the sense of involving ex- 

 penditure of energy — is the sine-component in phase with 

 the voltage ; and the element which the sine- component con- 

 tributes to the loop is an orthogonal ellipse. So far as it lags 

 it possesses a cosine-component, and this contributes to the 

 loop only an oblique line, shearing the loop over ; but this 

 constituent is wattless. Harmonic analysis cannot of itself 

 distinguish as to how much of the fundamental elliptical 

 constituent of the loop, or of the fundamental sine-component 



I 



°* -.*> " -i Af\. o.qi cr\z 







All the constituent curves belonging to the higher orders 

 have zero areas; the lobes formed by the crossing of their 

 outlines being alternately positive areas and negative areas. 

 This is only another way of saying that the integrals (i.) and 

 (ii.) above are always zero. As for those of form (ii.), they 

 are obviously so, as all cosine constituents shrink up to mere 

 lines. 





