906 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1953 



on the second tube approaches the value Si Pa Z32 . This is a limit, com- 

 mon to multi-loop structures, over and above the usual limit imposed by 

 the Nyquist stabiUty criterion. A similar S P Z limit can be derived for 

 the feedback on each stage. The same multi-loop mechanisms which 

 operate to limit the feedback also result in making the feedback on any 

 given stage relatively insensitive to variations in other transconductances 

 or impedances. In a single loop circuit a one db change in beta circuit 

 impedance or first stage transconductance causes a one db change in the 

 feedback on every tube. Here the feedback on, say, the second stage 

 would generally change only one-half db at most frequencies. While this 

 is an advantage in the sense that transconductance decay does not 

 decrease feedback as rapidly as it would in a single-loop amplifier, it 

 is a disadvantage in that it militates against improving stability margins 

 by shaping the out-band impedance of the beta circuit. 



The capacity distributions within the input and output amplifiers 

 are such that while the S P Z limitations on feedback are closely ap- 

 proached, the most stringent limitation on the feedback obtained is the 

 Nyquist criterion. 



The use of the Nyquist criterion, particularly with respect to defining 

 the margins against singing, is likewise complicated by the multi-loop 

 nature of the circuit. Ignoring some very recent work, the implications of 

 which have not been fully explored at this writing, it can be said of a 

 multi-loop structure that the apparent margins against singing shown 

 by any plot of feedback give no certain information as to how safe from 

 singing the circuit is. The phase, as well as the magnitude, of the feedback 

 on each stage is a function of the magnitude of the other transcon- 

 ductances, and either decay or increase of these transconductances might 

 destroy the phase margin. In these circumstances, it is theoretically 

 necessary to examine for stability every conceivable combination of 

 transconductances. A more practical expedient, of course, is to rely on 

 judgment backed by computation and laboratory experiment on the 

 circuit for a wide but far from infinite number of circuit conditions. 



Another difficulty arises from the fact that the feedback on each 

 tube is different, so that gain and phase margins obtained for one return 

 ratio do not imply equal gain and phase margins for the return ratio on 

 some other tube of the same amplifier. It does not follow, however, that 

 it is necessary to investigate separately the margins on each stage 

 versus circuit element variations. The point to be stressed here is that 

 we are using the behaviour of the return ratio merely as an index of 

 our real concern: the i)c)sition on the p-plane of zeros of the determinant 

 of the circuit, and the dciterminant is the same for both stages. Rather 



