Hysteresis Loops and Lissajous' Figures. 421 



Now yfrft), the function which represents the current-curve, 

 may by Fourier's theorem be expanded into the series 



A x sin + A. s s'm 60 + A b sin 50 + A n sin n0 



+ B X cos 6 + B 3 cos '60 +B 5 cos 50+ B H cos n0. 



Odd terms only are present, as in all alternating current 

 work ; and there is no constant term, because the mean 

 ordinate is already zero. 



The constituent terms in the area of the hysteresis loop 

 correspond therefore to the integrated products of sin into 

 the successive terms of the above series. 



§ 4. To investigate the form of the constituent elements of 

 the loop, let us consider a simple harmonic motion # = X sin 0, 

 where stands for %7rft,f being the fundamental frequency, 

 and X the amplitude. This motion is to be compounded, at 

 right angles, with another simple harmonic motion 



y=Y n sm(n0+fa); 



where Y n is the amplitude, <j> n a possible angle of phase- 

 difference, and n any (odd) integer giving the order of the 

 harmonic. We have then to find an expression for the curve 

 of which x and y are the coordinates. For simplicity we 

 deduce the expressions where w=l, ?z = 3, and w=5, that is 

 for the first, third, and fifth terms of the constituent elements. 



First Term (Fundamental) ; n = l. 

 We have 



^ = sin 0, (1) 



^r = sm(0 + fa) (2) 



Multiplying both sides of (1) by cos fa y we have 



^ cos fa = sin . cos fa. 



Also 



y 



j> — sin . cos <£x + cos . sin (p 1 . 



Subtracting this equation from the preceding, we have 



x v 



Y cos (pi — y = — cos #.sin fa (3) 



