THE BIASED IDEAL RECTIFIER 147 



oscillating cosine function. That is, if the input signal can be repre- 

 sented as 



E= A (0 COS0 (/), (1.15) 



where .1 (/) is never negative and has a spectrum confined to the frequency 

 range in which lirfC is negligibly small compared with 1/7?, while cos 0(/) 

 has a spectrum confined to the frequency range in which \/R is negligibly 

 small compared with 2irfC, we divide the components in the detector output 

 into two groups, viz.: 



1. A low-frequency group /;/ containing all the frequencies comparable 

 with those in the spectrum of .1 (/). The components of this group flow 

 through R. 



2. A high-frequency group Ihf containing all the frequencies comparable 

 to and greater than those in the spectrum of cos (f) (/). The components 

 of this group flow through C and produce no voltage across R. 



The instantaneous voltage drop across R is therefore equal to Ii/R, and 

 hence the bias on the rectifier is 6 + Ii/R. If .1 and </> were constants, we 

 could make use of (1.3) and (1.4) to write: 



.arc cos [(b+Ii/R)/A] 



I If + hf = :^" + 2 <^n cos nd (1.16) 



rt pare COS 1(0-1-1 If a )i A i 



On = - I f{A cos X — b — IifR) cos nx dx (1.17) 



TV Jo 



If .4 and (f) are variable, the equation still holds provided Ii/R < .1 at all 

 times. Assuming the latter to be true (keeping in mind the necessity of 

 checking the assumption when /;/ is found), we note that terms of the form 

 fln cos n d consist of high frequencies modulated by low frequencies and hence 



; the main portion of their spectra must be in the high-frequency range. 



I Hence we must have as a good approximation when the envelope frequencies 



ii are well separated from the intermediate frequencies, 



\ ■> /«arc cos [(6+/;/K)/4] 



1 hf = ^ = - \ f{A cosx - b - IifR)dx (1.18) 



1 I TT Jq 



jl This equation defines /;/ as a function of A, and if it is found that the 

 ! condition b -\- IifR<\ is satisfied by the resulting value of Ii/, the problem 

 j is solved. If the condition is not satisfied, a more complicated situation 

 , exists requiring separate consideration of the regions in which b + Ii/R < A 

 ' and 6 -f IifR > A . 



I To be specific, consider the case of a linear rectifier wnth forward con- 

 ductance a = l/R, and write V — Ij/R. Then 



'^'V ^ Va - {b a- Vy- - (b A- V) arc cos ^-^tZ (1.19) 

 XV A 



