160 BELL SYSTEM TECHNICAL JOURNAL 



1 [ (3.15) 



An = jr {kiAio — yloi — ^0^11) 



^30 = —^0^20 — ^1^21 1 



1 (3.16) 



^03 = — r (^0^02 + ^112) 

 ki J 



The third-order product A21 is of considerable importance in the design 

 of carrier ampHfiers and radio transmitters, since the (2/> — 9)-product is 

 the cross-product of lowest order falling back in the fundamental band when 

 overload occurs. Figure 14 shows curves of .I21 calculated by Mr. J. O. 

 Edson from Eq. (3.12) by mechanical integration. 



We point out also that the Unear-rectifier coefficients give the Fourier 

 series expansion of the admittance of a biased square-law rectifier when two 

 frequencies are applied. 



We shall next discuss the problem of reduction of the integrals appearing 

 above to a closed form in terms of tabulated elliptic integrals^. This can 

 be done for all the coefficients above except the d-c for the zero-power law 

 and for the d-c and two fundamentals for the linear rectifier. These contain 

 the integral 



H(i^o , ^1) = / arc cos (^0 + ^1 cos y) dy (3.17) 



which has been calculated separately and plotted in Fig. 22. When the 

 arc cos term is accompanied by cos wy as a multiplier with m ?^ 0, an integra- 

 tion by parts is sufficient to reduce the integrand to a rational function of 

 cos y and the radical \^\ — {ko + ki cos yY, which may be reduced at once 

 to a recognizable elliptic integral by the substitution z = cos y. It is 

 found that all the integrals except that of (3.17) appearing in the results 

 can be expressed as the sum of a finite number of integrals of the form: 



• cos 9 gm ^2 



By differentiating the expression z""" V'(l — z)^[l — {ko -f- kiz"-] with 

 respect to 2, we may derive the recurrence formula: 



^rn = —7 7Tr2 K2W — 3)^0^12™-! 



[m — l)ki 



+ (w - 2){kl - k\- 1)Z^_2 (3.19) 



- (2W - S)hkiZra-3 + (W - 3)(1 - kl)Z^-i\ 



' Power series expansions of coefficients such as treated here have been given by A. G. 

 Tynan, Modulation Products in a Power Law Modulator, Proc. L K. E., Vol. 21, pp. 

 1203-1209, Aug. 1933. 



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