THE BIASED IDEAL RECTIFIER 151 



in which F and p are variable, provided that P is always positive. We thus 

 can apply the results to detection of an ordinary amplitude-modulated wave 

 or to the detection of a frequency-modulated wave after it has passed through 

 a slope circuit. 



A case of considerable practical interest is that of an amplitude-modulated 

 wave detected by a diode in series with a parallel combination of resistance 

 R and capacitance C. The value of C is assumed to be sufficiently large so 

 that the voltage across R is equal to the ao/2 component of the current 

 through the diode multiplied by the resistance. This is the condition for 

 envelope detection mentioned in Part 1. The diode is assumed to follow 

 Child's law, which gives v = 3/2. We write 



V _ r(5/2)(l -X^aP^'' (, ...l-A .2-. 



where X = V/P. Note that K is a constant equal to the direct-voltage 

 output if P is constant. If P varies slowly with time compared with the 

 high-frequency term cos pt, V represents the slowly varying component of 

 the output and hence is the recovered signal. 

 But 



Hh, h 3; k') = i|^ [2(2^=^ - 1)£ + (2 - 3k'){l - k')K] (2.8) 



where A' and E are complete elliptic integrals of the first and second kind 

 with modulus k. Hence 



37r (1 -t- 3X)(1 -f X) 



^:vp = ^ = — I — ^-^^ <"' 



where the modulus of A' and E is \/(l — X)/2. This equation defines p 

 as a function of X, and hence by inversion gives X as a function of p. The 

 resulting curve of X vs. p is plotted in Fig. 6 and may be designated as the 

 function X = g (p). If we substitute X = V/P we then have 



V = P g{3Tr/Ra V2P) (2.10) 



This enables us to plot V as a function of P, for various values of Ra, Fig. 7. 

 Since P may represent the envelope of an amplitude-modulated (or diflf- 

 erentiated FM) wave, and V the corresponding recovered signal output 

 voltage, the curves of Fig. 7 give the complete performance of the circuit 

 as an envelope detector. In general the envelope would be of form P — 

 Po[l -f c s{l)\, where s{t) is the signal. We may substitute this value of P 

 directly in (2.10) provided the absolute value of c s{t) never exceeds unity. 



