Analysis of a symmetrically stabilized three-phase oscillator and some of its applications

Daboul, Jamil; Kaplan, B.Z.; Kottick, D.

doi:10.1109/tcs.1987.1086167

Cited by 8 publications

(2 citation statements)

References 9 publications

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“…It can be shown that the following solution is a possible steady state solution of (4): X k = A c o s [ w t + ( k -l ) n / 2 + $ ] , k = 1,2,3,4(modulo4) ( 5 ) where A = 111~. On one hand ( 5 ) satisfies the linear conservative parts of (4), namely it satisfies (1). On the other hand it nulls the terms of the square brackets in (4).…”

Section: Introductionmentioning

confidence: 99%

Development of a symmetrically stabilized four‐phase oscillator and some implications

Kaplan

Yardeni

1990

Circuit Theory & Apps

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SUMMARYThis paper treats the development of a non-linearly stabilized oscillator model which generates four sine waves in quadrature. This system is dealt with in spite of the fact that quadrature signals can be generated by a simple two-phase oscillator. The latter generates only two of the signals needed in a four-phase system. The rest of the phases signals could nevertheless be obtained by inventing the originally existing signals. It appears, however, that for certain applications the generation of all four signals in a completely cyclic and symmetrical manner (the one described here) is preferable. It is envisaged that one such application is related to recent methods of actively feeding phased array antennas, where each element in the array is connected to an appropriate oscillator phase stage.Most of the paper deals with the development of an appropriate generator model. The non-linear oscillator dynamics is treated comprehensively. The peculiar behaviour associated with the limit cycle dynamics and with other manifestations of the system dynamics is investigated. It appears from the detailed simulation work that there exist regions of initial conditions where the system solutions are expected to be relatively sensitive to initial conditions. As a result, it is believed that the system with certain additions may reveal chaotic dynamic behaviour. The realization of the system in electronic hardware is also discussed.

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Section: Introductionmentioning

confidence: 99%

Development of a symmetrically stabilized four‐phase oscillator and some implications

Kaplan

Yardeni

1990

Circuit Theory & Apps

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“…Except for the change of notation and a change arising from a different sign convention for the unperturbed terms, this is the solution found by Daboul et ai. 6 Note that (a) the coefficients of all the perturbation terms in equations (15a-c) are real, so there is no frequency (b) the unperturbed mode zl = A1 exp(j w , f ) , zo = 0, with a particular value of I A1 1, is a solution.…”

Section: Selection Rule For Terms Of Zrmentioning

confidence: 99%

Exact solutions of cyclically symmetric oscillator equations with non‐linear coupling

Fox

1994

Circuit Theory & Apps

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SUMMARYKaplan and Yardeni have found very simple exact limit cycle solutions in cyclically symmetric systems of N oscillator equations with linear coupling in zero order of a perturbation parameter and non-linear coupling in first order. In contrast with such solutions in other non-linear systems, each of these limit cycles is a normal mode of the unperturbed equations, with no change in frequency. The sources of this simple behaviour are studied here with the equations expressed in terms of the normal mode co-ordinates of the unperturbed system rather than in the original co-ordinates. It is found that the simplicity of the Kaplan-Yardeni solutions arises partly from an additional symmetry of the perturbation terms beyond the cyclic symmetry and partly from the specific choices of the perturbations. Extension of their systems to arbitrary N leads to the result that all such sets of equations have similar simple limit cycles. More general cyclically symmetric sets of equations are also discussed, with limit cycle solutions whose frequencies are shifted from the zero-order values by easily calculated amounts or with solutions which are linear combinations of zero-order normal modes.The results are exact, obtained without the use of small-perturbation theory.

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Exact Solutions of Cyclically Symmetric Oscillator Equations With Non-Linear Coupling. Part Iii: Globally Stable Single-Frequency Limit Cycles

Fox

1997

Int. J. Circ. Theor. Appl.

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Analysis of a symmetrically stabilized three-phase oscillator and some of its applications

Cited by 8 publications

References 9 publications

Development of a symmetrically stabilized four‐phase oscillator and some implications

Development of a symmetrically stabilized four‐phase oscillator and some implications

Exact solutions of cyclically symmetric oscillator equations with non‐linear coupling

Exact Solutions of Cyclically Symmetric Oscillator Equations With Non-Linear Coupling. Part Iii: Globally Stable Single-Frequency Limit Cycles

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