FEEDBACK AMPLIFIER DESIGN 



425 



a large number of relations between the attenuation and phase char- 

 acteristics of a physical network. One of the simplest is 



f 



U — 



Bdu 



{A^ - ^o), 



(1) 



where u represents log///o, /o being an arbitrary reference frequency, 

 B is the phase shift in radians, and A% and A^ are the attenuations in 

 nepers at zero and infinite frequency, respectively. The theorem 

 states, in effect, that the total area under the phase characteristic 

 plotted on a logarithmic frequency scale depends only upon the differ- 

 ence between the attenuations at zero and infinite frequency, and not 

 upon the course of the attenuation between these limits. Nor does it 

 depend upon the physical configuration of the network unless a non- 

 minimurn phase structure is chosen, in which case the area is necessarily 

 increased. The equality of phase areas for attenuation characteristics 

 of different types is illustrated by the sketches of Fig. 1. 



Fig. 1 — Diagram to illustrate relation between phase area and change in attenuation. 



The significance of the phase area relation for feedback amplifier 

 design can be understood by supposing that the practical transmission 

 range of the amplifier extends from zero to some given finite frequency. 

 The quantity Ao — A„ can then be identified with the change in gain 

 around the feedback loop required to secure a cut-off. Associated with 

 it must be a certain definite phase area. If we suppose that the maxi- 

 mum phase shift at any frequency is limited to some rather low value 

 the total area must be spread out over a proportionately broad interval 

 on the frequency scale. This must correspond roughly to the cut-off 

 region, although the possibility that some of the area may be found 

 above or below the cut-off range prevents us from determining the 

 necessary interval with precision. 



A more detailed statement of the relationship between phase shift 

 and change in attenuation can be obtained by turning to a second 



