THE BIASED IDEAL RECTIFIER 145 



at the ends of the y-axis. In Case (3), —1 < ^o + ^i < 1, ^o — ^i < —1, 

 a single closed curve is obtained. The limits of integration must be chosen 

 to fit the proper case. The Fourier series expansion of g{x,y) may be 

 written : 



00 00 



g(^) y) = zL ^ O'mn COS mx cos ny (1.9) 



where amn is found from integrals of the form: 



A = -^^ / dy I j{P cos X -\- Q cos y — b) cos mx cos ny dx (1.10) 



Here, as usual, «„ is Neumann's discontinuous factor equal to two when m 

 is not zero and unity when m is zero. The values of the limits for the dif- 

 ferent cases are : 

 Case I, flmn = Ai-\- A2 



({xi = 0, X2 = arc cos (—^0 — ki cos y) 

 1-/^0 I (^-^^^ 



yi = arc cos — , y2 — tt 



(ari = 0, :i:2 = X 

 1 _ ^j, I (1.12) 

 yi = 0, ^2 == arc cos — — 



Xo = arc cos ( — ^0 ~ ^1 cos y 



y2 = TT 



X2 = arc cos (—^0 — ^1 cos y) 



(1.13) 



y2 = arc cos 



{-'^) 



(1.14) 



For a considerable variety of rectifier functions/, the inner integration may 

 be performed at once leaving the final calculation in terms of a single definite 

 integral. 



A somewhat different point of view is furnished by evaluating the integral 

 (1.4) for the biased single-frequency harmonic amplitude, and then replacing 

 the bias by a constant plus a sine wave having the second frequency. When 

 each harmonic of the first frequency is in turn expanded in a Fourier series 



