REGENERATION THEORY AND EXPERIMENT 681 



It is interesting to compare the criterion with those derived for the 

 mechanical systems of classical dynamics. In his Adams Prize Paper 

 on "The Stability of Motion," ■* and again in his "Advanced Rigid 

 Dynamics," ^ Routh investigated the general problem of dynamic 

 stability and established a number of criteria based upon various 

 properties of dynamical systems. When applied to the problem of 

 feed-back amplifiers, keeping Nyquist's result in mind, one of them is 

 found to be equivalent to Nyquist's criterion, although expressed in 

 different terms and derived in a different way. 



To provide a background for the experiments, we propose to state 

 some of the criteria for stability which are to be found in the literature 

 of vacuum tube oscillators, and to compare them with Nyquist's or 

 Routh's criterion, the development of which is most conveniently de- 

 scribed somewhat along the lines followed by Routh. Following this 

 we shall deal with the experimental methods and apparatus which were 

 used in testing the criterion, and conclude with some extensions of the 

 criterion. 



Circuit Analysis and Stability 



Conditions required for the starting of oscillations in linear feed- 

 back circuits, corresponding to instability, are to be found in the litera- 

 ture of vacuum tube oscillator circuits, expressed in a number of os- 

 tensibly different forms. These are usually based upon the familiar 

 mesh differential equations for the system which involve differentia- 

 tions and integrations of the mesh amplitudes with respect to time. 

 Using the symbol p to denote differentiation with respect to time, each 

 mesh equation becomes formally an algebraic one in p, involving the 

 circuit constants and the mesh amplitudes. The solution of this sys- 

 tem of equations is known to be expressible as the sum of steady state 

 and transient terms. The transient terms are each of the form Bk 

 eP*', the BkQ being fixed by initial conditions, and the pkS being de- 

 termined from the circuit equations. If we set up the determinant of 

 the system of equations — the discriminant — and equate it to zero, the 

 roots of the resulting equation are the ^^'s above. In general each 

 mesh equation involves p to the second degree at most, and with n 

 meshes the discriminant is of degree 2n at most. Accordingly we may 

 express the determinantal equation as 



F{p) = 0= K(P- P,){P - P2) ■■■ (P- p2n). (1) 



As for the steady state term, in the simplest case in which a sinu- 

 soidal wave of frequency wjlir is impressed, it is equal to the impressed 



4 Macmillan, 1877. 



^ Macmillan, 6th edition, 1905. 



