72 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1957 



20 40 60 80 100 120 140 



FREQUENCY IN KILOCYCLES PER SECOND 



Fig. 2 — Required insertion gain. 



160 



180 



Gain Formula 



The circuit of Fig. 1 may be represented by a simplified circuit con- 

 sisting of an input coupling network, a three stage amplifier, an output 

 network and a feedback impedance Z^ as shown in Fig. 3. From this 

 figure it can be shown that the insertion gain of the repeater is given 

 by:« 



e = 



2e'e'Z, 



Pil^iifJmTZlZiZ^ 



— PiPogmTZlZ2Z0_ 



(1) 



when Zp « QmTZpZgZxZi » (Zo + Z^ and where 



Pi = 



Zi + Zg 



and 



po = 



Zo + Zp 



are "potentiometer terms". The gain of the input network is defined as 

 e*i = Y lEi ; where Y is the open circuit voltage of the input network 

 with Ei as the source, and the gain of the output network is defined as 

 e' = ill\. This expression may be put in familiar form by recognizing 

 that pip^gmrZxZiZ^ is jUjS, the feedback around the loop. Hence 



2i'e'Z 



[r 



M^ 



- M/3J 



(2) 



Equation (2) shows that the insertion gain of the I'epeater is the prod- 

 uct of five factors, namel}': 



