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On the Stability of the Periodic States of the Triode Oscillator

Published online by Cambridge University Press:  24 October 2008

W. M. H. Greaves
W. M. H. Greaves
Affiliation:
St John's College
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Extract

1. Appleton and van der Pol have shown that in a simple Triode or Dynatron generating circuit the anode potential v is related to the time t by a differential equatior of the type

where f (v) is a power series in v, and may be written

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 22 , Issue 1 , February 1924 , pp. 16 - 23
Copyright
Copyright © Cambridge Philosophical Society 1924

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References

* Phil. Mag., Ser. 6, Vol. 43, No. 253, p. 179, 01 1922.Google Scholar

† Phil. Mag., Ser. 6, Vol. 45, No. 267; p. 402, 03 1923.Google Scholar

‡ Phil. Mag., Vol. 45, pp. 401–414, 03 1923.CrossRefGoogle Scholar

§ Proc. Roy. Soc., A., Vol. 103, pp. 516–524, 1923.CrossRefGoogle Scholar

∥ This possibility is not definitely excluded by the first investigation of Appleton and van der Pol, which was intended purely as a first approximation and in which a whole series of periodic terms is neglected.

* For details of this transformation see Proc. Roy. Soc., A., Vol. 103, p. 517, 1923.Google Scholar

* It was explained in one of the previous papers (Proc. Roy. Soc., A, Vol. 103, p. 517, 1923) that a solution of an equation such as (8) is to be regarded as periodic if at the end of each period x resumes its original value and y changes by a multiple of 2π. Such a solution will be strictly periodic when expressed in terms of the original dependent variable v.Google Scholar

† The actual details of the reversion are not needed here. It may be carried out if desired with the aid of Lagrange's formula. (See, for instance, Goursat, , Cours d'Analyse, t. 1, ch. IX, p. 481.)Google Scholar

* See note at end of paper.

* Strictly speaking we have only proved so far that (when c > 0) the periodic solution of (13) corresponding to x=a is stable and we have not proved the stability of the corresponding group of solutions of (8). This, however, will be proved in the following section.

* Loc. cit., equation (10), p. 182.Google Scholar

* This method of approximation is that which ie used in celestial mechanics for the purpose of calculating the secular motions of the major planets. cf.Tisserand, , Mécanique Céleste, t. I, ch. XXVI.Google Scholar

† Cf. Charlier, , Mechanik des Himmels, vol. 1, p. 89,Google Scholar or the original source, Weierstrass, , Ges. Werke, Bd. 11, p. 1.Google Scholar

* Cf. Poincaré, , Lea méthodes nouvelles de la mécanique céleste, t. I, ch. VII, pp. 335 et seq.Google Scholar