HARMONIC PRODUCTION IN MAGNETIC MATERIALS 795 



2 r^" 



ak = - \ (^ + C cos 2pl + Gicos Apt + cos 2pt)) sin kpt d{pt), 

 ttJo 



2 r'" (34) 



^^ = ± (5 cos ;/?/ + (i^ + /^) cos 3pt 

 ttJo 



+ /(cos 5_p/ + cos pt)) cos /c^/ d{pt). 



Upon integration we find 



h, = B, bs = F. (35) 



Comparing these two coefficients we see that at low amplitudes we 

 may write 



bi = ixHi, bs = 1J.H3, 



in which the permeability is the same to the two components, and is 

 determined by the fundamental amplitude. For the dissipative 

 terms we find 



ai = -{A- C/3 - 2G/5), 



TT 



4 (^50 



as = - (Al3 - 3C/5 + 6G/35), 



TT 



but some care is required in interpreting these expressions. Inasmuch 

 as we are primarily interested here in determining the dissipative 

 component to a third harmonic magnetizing force of amplitude H3, 

 we are required to select from as only those terms containing H3, 

 which means the single term 24G/357r. The other terms take care 

 of the harmonic producing properties of the core and do not affect 

 the impedance to the third harmonic. The impedance term for the 

 third harmonic comes down to 



24 



OOTT 



which may be written as 



24 B{0, H) 

 35t Hi 



This may be compared with the corresponding term for the funda- 

 mental given by (25). 



Appendix 5. Effect of Air-Gap by Vacuum Tube Analogy 



In the elementary treatment of non-linear two element vacuum 

 tube circuits, approximate solutions are obtained in the form 



J = Ji -\- J2 -^ ' ' ' Jn, 



