684 BELL SYSTEM TECHNICAL JOURNAL 



reliance is placed upon the nonlinear properties of the circuit to ful- 

 fill the criteria automatically. The procedure is known by experiment 

 to be effective in the usual type of oscillating circuit. In particular 

 forms of feed-back circuits, however, it may be demonstrated that 

 the transfer factor may be made greater than unity without giving rise to 

 oscillations. This situation was investigated experimentally, and found 

 to be in accord with the stability criterion stated by Nyquist. 



Nyquist's Criterion 



The explicit solution of (1) for the pkS> demands an exact knowl- 

 edge of the configuration of the amplifier and feed-back circuits. When 

 the number of meshes is large, the solution involves much labor. If we 

 wish simply to observe whether or not the system Is stable, however, 

 we need not obtain explicit solutions for the roots; in fact, all we need 

 to know is whether or not any one of the pkS has its real part positive. 

 It turns out that when we know the transfer factor as a function of fre- 

 quency, by calculation or by measurement, a simple inspection of the 

 transfer factor polar diagram suffices for this purpose. This diagram is 

 constructed by plotting the imaginary part of the transfer factor 

 against the real part for all frequencies from minus to plus infinity.^ 



To obtain Nyquist's criterion we consider the vector drawn from 

 the point (1, 0) to a point moving along the polar diagram; if the net 

 angle which the vector swings through in traversing the curve is zero, 

 the system is stable; if not, it is unstable. To express it in the terms 

 used by Routh, if we set I — A (jco) = P -\- jQ, and observe the changes 

 of sign which the ratio P/Q makes when P goes through zero as the 

 frequency steadily increases, the system is stable when there are the 

 same number of changes from plus to minus as from minus to plus. 

 It may be demonstrated that these two statements are equivalent. 



The way in which the above procedures may be shown to reveal the 



existence of a root with positive real part may be outlined somewhat 



along the lines followed by Routh in his analysis.^ Since p is a. com- 



* The transfer factor for negative frequencies A (— Jco) is the complex conjugate of 

 that for positive frequencies A{ju). Thus, if 



A(jw) = X +jY, 

 then 



Ai-jo.) = X -jY. 



^ A number of restrictions on the generality of the analysis may be noted. It is 

 assumed that A(p) has no purely imaginary roots, although the result in this case is 

 otherwise evident. Further it is assumed that A{p) goes to zero as \p\ becomes 

 infinite, and that no negative resistance elements are included in the amplifier. 

 Another point which should be mentioned is that the analysis does not apply to the 

 stability in any conjugate paths that may exist. This point may be exemplified by 

 the balanced tube or push-pull amplifier, in the normal transmission path of which the 

 tubes of a stage act in series. When the series output is connected back to the series 



