THE BIASED IDEAL RECTIFIER 149 



Equation (1.23) expresses the fact that we may calculate the signal com- 

 ponent in the output of a half-wave linear rectifier by taking I/tt times the 

 envelope. Equation (1.24) shows that we may calculate the response of 

 an envelope detector by taking t times the low-frequency part of the 

 Fourier series expansion of the linearly rectified input. Thus two procedures 

 are in general available for either the envelope detector or linear rectifier 

 solution, and in specific cases a saving of labor is possible by a proper choice 

 between the two methods. The final result is of course the same, although 

 there may be some difficulty in recognizing the equivalence. For example, 

 the solution for linear rectification of a two-frequency wave P cos pt -^ Q 

 cos qt was given by the author in 1933', while the solution for the envelope 

 was given by Butterworth in 1929^ Comparing the two expressions for 

 the direct-current component, we have: 



- 2P o 



Elf = -y[2E — (1 — k") K], where K and £are complete elliptic integrals 



of the first and second kinds with modulus k = Q/P 



— 2P 



A {t) = — (1 + k) El, where Ei is a complete elliptic intregal of the 



TT 



second kind with modulus ki = 2 \/k/{\ + k). Equation (1.24) implies 

 the existence of the identity 



(1 +k)Ei^ 2E- {\ - k') K (1.25) 



The identity can be demonstrated by making use of Landen's transforma- 

 tion in the theory of elliptic integrals. 



2. Single-Frequency Signal 



The expression for the harmonic amplitudes in the output of the rectifier 

 can be expressed in a particularly compact form when the conducting part 

 of the characteristic can be described by a power law with arbitrary ex- 

 ponent. Thus in (1.4) if /(c) = az\ we set X = b/P and get 



•arc cos X 



2 73" /<arc cos a 

 aP i , ^ y , 



I fln = / (cos X — A) COS nx ax 



TT Jo 



! 



2^T{p + DaPW - X)"^^ 



I 



* S. Butterworth, Apparent Demodulation of a Weak Station by a Stronger One 

 Experimental Wireless, Vol. 6, pp. 619-621, Nov. 1929. 



