REGENERATION THEORY 



143 



singularities anywhere including at co, the integral in {ii) may be 

 evaluated by the theory of residues by completing the path of inte- 

 gration so that all the poles of the integrand are included. We then 

 have 





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If the network is made up of a finite number of lumped constants there 

 is no essential singularity and the preceding expression converges 

 because it has only a finite number of terms. In other cases there is an 

 infinite number of terms, but the expression may still be expected to 

 converge, at least, in the usual case. Then the system is stable if all 

 the roots oi \ - w = have x < 0. If some of the roots have x > 

 the system is unstable. 



The calculation then divides into three parts: 



1. The recognition that the impedance function is 1 — w." 



2. The determination of whether the impedance function has zeros 

 for which x ^ 0.^^ 



W -PLANE 



Fig. 10 — Network of loci x = const., and y = const. 



3. A deduction of a rule for determining whether there are roots for 

 which X > 0. The actual solution of the equation is usually too 

 laborious. 



To proceed with the third step, plot the locus x = in the w plane, 

 i.e., plot the imaginary part of zv against the real part for all the 

 values of 3-, - CO < 3, < CO. See Fig. 10. Other loci representing 



X = const. 



and 



y = const. 



11 Cf. H. W. Nichols, Phys. Rev., vol. 10, pp. 171-193, 1917. 

 1^ Cf. Thompson and Tait, "Natural Philosophy," vol. I, § 344. 



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