148 BELL SYSTEM TECHNICAL JOURNAL 



When 6 = (the case of no added bias), this equation may be satisfied by 

 setting 



V = cA,() <c < 1, (1.20) 



which leads to 



R yd' 



1 — arc cos c, ^ (1-21) 



defining c as a function of Ro/R- The value of c approaches unity when 

 the ratio of rectifier resistance to load resistance approaches zero and falls 

 off to zero as Ro/R becomes large. The curve may be found plotted else- 

 where . This result justifies the designation of this circuit as an envelope 

 detector since with the proper choice of circuit parameters the output 

 voltage is proportional to the envelope of the input signal. 



The equations have been given here in terms of the actual voltage applied 

 to the circuit. The results may also be used when the signal generator 

 contains an internal impedance. For example, a nonreactive source inde- 

 pendent of frequency may be combined with the rectifying element to give a 

 new resultant characteristic. If the source impedance is a constant pure 

 resistance tq throughout the frequency range of the signal input but is 

 negligibly small at the frequencies of other components of appreciable size 

 flowing in the detector, we assume the voltage drop in ro is roCi cos (/). 

 We then set n — 1 in (1.17) and replace ai by {Aq — A)/rQ, where Aq is 

 the voltage of the source. The value of lu in terms of A from (1.18) is 

 then substituted, giving an implicit relation between A and Ao . 



A further noteworthy fact that may be deduced is the relationship be- 

 tween the envelope and the linearly rectified output. By straightforward 

 Fourier series expansion, the positive lobes of the wave (1.15), may be 

 written as: 



(E, £>0\ p 



£r - =-4(/) - + ' cos 4>{t) 



\ 0, E <0 / 



TT 2 



2 Y^ ( — )"* cos 2m 0(/) 



(1.22) 



7rm=i 4w2 — 1 



Hence if we represent the low-frequency components of Er by Ei/, we have: 



£,/ = ^ (1.23) 



IT 



or 



A (/) = wE,f (1.24) 



* See, for example, the top curve of Fig. 9-25, p. 311, H. J. Reich, Theory and AppHca- 

 tions of Electron Tubes, McGraw-Hill, 1944. 



