424 BELL SYSTEM TECHNICAL JOURNAL 



relation between the attenuation and the phase shift of a physical 

 structure can exist. For example, we can always change the phase 

 shift of a circuit without affecting its loss by adding either an ideal 

 transmission line or an all-pass section. Any attenuation characteris- 

 tic can thus correspond to a vast variety of phase characteristics. 



For the purposes of amplifier design this ambiguity is fortunately 

 unimportant. While no unique relation between attenuation and 

 phase can be stated for a general circuit, a unique relation does exist 

 between any given loss characteristic and the minimum phase shift 

 which must be associated with it. In other words, we can always add 

 a line or all-pass network to the circuit but we can never subtract such 

 a structure, unless, of course, it happens to be part of the circuit 

 originally. If the circuit includes no surplus lines or all-pass sections, 

 it will have at every frequency the least phase shift (algebraically) 

 which can be obtained from any physical structure having the given 

 attenuation characteristic. The least condition, since it is the most 

 favorable one, is, of course, of particular interest in feedback amplifier 

 design. 



For the sake of precision it may be desirable to restate the situations 

 in which this minimum condition fails to occur. The first situation is 

 found when the circuit includes an all-pass network either as an indi- 

 vidual structure or as a portion ofja network^which can be replaced by 

 an all-pass section in combination with some other physical structure.* 

 The second situation is found when the circuit includes a transmission 

 line. The third situation occurs when the frequency is so high that 

 the tubes, network elements and wiring cannot be considered to obey 

 a lumped constant analysis. This situation may be found, for example, 

 at frequencies for which the transit time of the tubes is important or for 

 which the distance around the feedback loop is an appreciable part of 

 a wave-length. The third situation is, in many respects, substantially 

 the same as the second, but it is mentioned separately here as a matter 

 of emphasis. Since the effective band of a feedback amplifier is much 

 greater than its useful band, as the introduction pointed out, the con- 

 siderations it reflects may be worth taking into account even when 

 they would be trivial in the useful band alone. 



It will be assumed here that none of these exceptional situations is 

 found. For the minimum phase condition, then, it is possible to derive 



* Analytically this condition can be stated as follows: Let it be supposed that the 

 transmission takes place between mesh 1 and mesh 2. The circuit will include an 

 all-pass network, explicit or concealed, if any of the roots of the minor A12 of the 

 principal circuit determinant lie below the real axis in the complex frequency plane. 

 This can happen in bridge configurations, but not in series-shunt configurations, so 

 that all ladder networks are automatically of minimum phase type. 



