794 BELL SYSTEM TECHNICAL JOURNAL 



Upon integration of (23) the coefificients for the fundamental and third 

 harmonic flux components are found to be as follows : 



a, ^ -{A - C/3), bx = B, 



4/^,3C\ , „ ^^^^ 



which, by reference to (20), may be put in terms of the branch coeffi- 

 cients. 



Appendix 4 — Impedance Reaction to a Small Third Harmonic 

 IN THE Presence of a Large Fundamental 



We have for the two hysteresis branch equations from Eqs. (4a), 

 (5a) 



B,{h, H) = B{0, H) + m + 7/i' + a,,li^ + • • •, 

 B^{h., H) = - B{0, H) + ^h - yJf^ + a^oh' + •■-, 



(30) 



in which 



i3 = aio + anil + auIP, 

 7 = — (ao2 + aoill). 



Putting (29) in (30) we get 



(31) 



B{h, II) = A + B cos pt + C cos 2pt + D cos 3pt + F cos npt 

 + G[cos (n + \)pt + cos {n - Vjpf] 



+ J[cos (w + 2)pt + cos (h - 2)pQ (32) 



in which the coefficients have the following significance 



-/l3/2, 



{33) 



A = B{0, II) + Ih^yil, F = I31h + 3azdhnhl2, 



B ^ I3H, + 3a3o//i'/4, G = yllJh, 



C = yHi'/l, J = 3azoHi'H3/4. 



D = asolli'/i, 



The coefficients of the Fourier Series for the output wave may 

 now be obtained as before by combining the two equations (30) since 

 each one is operative during one-half the cycle. There results an 

 expression similar to the one obtained in the single frequency case, 

 and since we have 



/ = boj2 + 2a/i sin kpt -\- l^bk cos kpt 



the coefficients are evaluated from the expressions 



