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On an equation connected with the theory of triode oscillations

Published online by Cambridge University Press:  24 October 2008

R. A. Smith
R. A. Smith
Affiliation:
St John's CollegeCambridge
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Extract

This discussion is concerned with the periodic solutions of the differential equation

where the parameter k is small. The equation arises from an investigation of the forced oscillations of a simple electrical circuit (6) containing a triode. The function f(x) is derived from the current-voltage characteristic of the triode and it has a roughly ⊂-shaped graph, with a flat bottom and some slight asymmetry. For a regenerative circuit, we have f(0) < 0. Van der Pol (5) considered the equation with f (x) = x2 − 1, and he and later writers (1, 2, 4) have built up an extensive theory of triode oscillations for this ideal case. In practice, the graph of f(x) has a much flatter bottom than that of the function x2 − 1, and it was asked by Cartwright ((3), p. 214) whether this flattening would alter the qualitative behaviour of the circuit near resonance. The main part of this paper is an attempt to answer this question.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 48 , Issue 4 , October 1952 , pp. 698 - 717
Copyright
Copyright © Cambridge Philosophical Society 1952

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References

REFERENCES

(1)Andronow, A. and Witt, A.Arch. Elektrotech. 24 (1930), 99.CrossRefGoogle Scholar
(2)Cartwright, M. L.J. Instn elect. Engrs, 95/iii (1948), 88.Google Scholar
(3)Cartwright, M. L.Contributions to the theory of non-linear oscillations (Ann. Math. Studies, no. 20, Princeton, 1950), pp. 149–241.Google Scholar
(4)Gapanov, V. I.Zh. tekh. Fiz. S.S.S.R. 5 (1935), 821.Google Scholar
(5)van der Pol, B.Phil. Mag. (7), 3 (1927), 65.Google Scholar
(6)van der Pol, B.Proc. Inst. Radio Engrs, N.Y., 22 (1934), 1051.Google Scholar