522 BELL SYSTEM TECHNICAL JOURNAL 



Since the gain varies with frequency, the amplifier will give approx- 

 imately constant response only within a certain range of frequencies. 

 The band of the amplifier is defined as that frequency interval within 

 which the magnitude of the gain is constant within some specified toler- 

 ance; the bandwidth is the size of this interval. We wish to express the 

 gain of the amplifier in terms of its bandwidth, in the following way: 



The voltage gain of this amplifier has a maximum, called To , at band 

 center frequency /o . Take the band of the amplifier Bk{A) as that interval 

 within which the voltage gain is within a factor of 1/N times the maxi- 

 mum. 



r(co) 



We can analogously define the band of a simple circuit J5„(C) by the 

 relation 



> i- defines Bn{A) (Al-9) 



"~ N 



^^2«(t0o) 



> i defines 5„(C). (Al-9) 



n 



It follows directly that 



BniC) = -^ Vn^ - 1 . (Al-10) 



Since the amplifier gain is inversely proportional to the product of the 

 circuit admittances, it follows that «i n2 = N. 



The intrinsic bandwidth resulting from the tube admittance may not 

 be suitable for the intended application. In that case the band may be 

 widened by increasing Gu or Gjp with a corresponding decrease in gain. 

 We have then the problem of adjusting Gu and G^e for greatest band ef- 

 ficiency, i.e., maximum gain for a given bandwidth, with synchronous 

 tuning. It turns out that if the bandwidth is less than that needed, then 

 the circuit of higher Q should be lowered until either (a) the band be- 

 comes wide enough, or (b) the Q's become equal. In case (b), both Q's 

 should then be lowered, maintaining equality, until the band is wide 

 enough. 



Two important limiting cases are to be considered: (a) Qu = Qie , i.e. 

 the band is shaped equally by the input and output circuits; and (b) 

 Qu << Qif , i-e. the band is shaped by only the output circuit. In the 

 equal-Q case we have 



G\e _ Gie 



n = N (Al-11) 



Bs(A) = ^ Jpp^N-^ 



1 



