Transient Processes within Parametrically Excited Linear Electric Circuits

Purely reactive and high-quality circuits with one changing reactive parameter is considered. Two base types of parametric interference causing the transient processes of a base type (analogous for transition and pulse characteristics within externally excited circuits) is proposed. The time-dependent Hamiltonian of a special type has been recommended for use as mathematical model of purely reactive circuit. The canonical transformation of the Hamiltonian to new generalized coordinate and momentum is carried out using the generating function depending on explicitly input time. These transformations have allowed obtaining the averaged Hamiltonian equations, which has determined asymptotically valid impulse (action) and coordinate (phase) and also as a consequence the base type transients have been described within the initial variables of a state. Transients for the losses circuits cannot be obtained in the same path that for the purely reactive circuits. Another approach based on the existence of two large parameters fixing the smoothness of capacity changing and the high-quality circuit is proposed.

References

[1] Migulin V.V., Medvedev V.I., Mustel’ E.R., Parygin V.N., eds. Osnovy teorii kolebaniy (pod red. V.V. Migulina). [Fundamentals of the oscillations theory]. Moscow, Nauka Publ., 1988. 392 p.

[2] Bogolyubov N.N., Mitropol’skiy Yu.I. Asimptoticheskie metody v teorii nelineynykh kolebaniy. [Asymptotic methods in the theory of nonlinear oscillations]. Moscow, Nauka Publ., 1974. 408 p.

[3] Markov F.P., Sokolov V.A. Tsepi s peremennymi parametrami [Circuit with variable parameters]. Moscow, Energiya Publ., 1976. 448 p.

[4] Volosov V.M., Morgunov B.I. Metod osredneniya v teorii nelineynykh kolebatel’nykh sistem [The averaging method in the theory of nonlinear oscillatory systems]. Moscow, MGU Publ., 1971. 507 p.

[5] Rabinovich M.I., Trubetskov D.I. Vvedenie v teoriyu nelineynykh kolebaniy i voln [Introduction to the theory of nonlinear oscillations and waves]. Moscow, Nauka Publ., 1984. 432 p.

[6] Heading J. An introduction to phase-integral methods. Methuen, 1962. 160 p. (Russ. Ed.: Khedding Dzh. Vvedenie v metod fazovykh integralov (metod VKB). Moscow, Mir Publ., 1965. 238 p.).

[7] Nayfeh Ali H. Introduction to Perturbation Techniques. 1st ed. Wiley, 1981. 536 p. (Russ. Ed.: Nayfe A.Kh. Teoriya vozmushcheniy. Moscow, Mir Publ., 1976. 446 p.).

[8] Kamke E. Differentialgleichungen: Loosungsmethoden und Loosungen. Akademische Verlagsgesellschaft, Leipzig, 1967. (Russ. Ed.: Kamke E. Spravochnik po obyknovennym differentsial’nym uravneniyam. Per. s nem. 4-e izd., ispr. [Handbook on ordinary differential equations]. Moscow, Nauka Publ., 1971. 388 p.)

[9] Arnol’d V.I. Matematicheskie metody klassicheskoy mekhaniki [Mathematical methods of classical mechanics]. Moscow, Nauka Publ., 1974. 432 p.