Harmonic Balance, Melnikov method and nonlinear oscillators under resonant perturbation

Bonnin, Michele

doi:10.1002/cta.423

Cited by 20 publications

(15 citation statements)

References 22 publications

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“…Nonlinear circuits may exhibit interesting phenomena such as sub-harmonic oscillations and coupled selfoscillations. These phenomena are analyzed by the HB method based on the Melnikov method [36] and modified describing function method [37] respectively. Another frequency-domain method is Volterra series [38,39] and perturbation methods [40,41].…”

Section: Q(x (T))+ T Q(x (T))+ F(x (T)) = B(t)mentioning

confidence: 99%

Envelope analysis of nonlinear electronic circuits based on harmonic balance method

Kawata

Kinouchi

Yamagami

et al. 2012

Circuit Theory & Apps

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SUMMARYWe propose here a Spice-oriented envelope analysis based on the HB (harmonic balance) method, where Fourier coefficients are assumed to be slowly varying. The Fourier expansions of nonlinear devices are executed by MATLAB in the symbolic forms. In this time, the nonlinearities need to be approximated by the polynomial functions. The determining equation of the HB method is formulated as Sine-Cosine circuit in the form of schematic diagram using ABMs (analog behavior models) of Spice. Each sub-circuit corresponding to the higher harmonic component is almost the same circuit topology as the original one and has dynamic elements such as capacitors and inductors. The Sine-Cosine circuit can be solved by the transient analysis of Spice. Thus, our method is rather a symbolic approach in the meaning that the HB determining equation is given by the schematic diagram of Spice. Our method can be easily applied to the analysis of middle order of nonlinear communication circuits such as mixers and amplitude modulators and to the analysis of interesting phenomena in the nonlinear oscillations. After many simulation experiments, the results show that our envelope analysis is about 50 times faster than the direct transient analysis.

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Section: Q(x (T))+ T Q(x (T))+ F(x (T)) = B(t)mentioning

confidence: 99%

Envelope analysis of nonlinear electronic circuits based on harmonic balance method

Kawata

Kinouchi

Yamagami

et al. 2012

Circuit Theory & Apps

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“…(i) Let V (e) = 1/2e e be a Lyapunov function candidate for Equation (17). Taking into account that B(·) is skew-symmetric, one obtains thaṫ…”

Section: (T)+ B(ū(t))]e + B[u(t E)−ū(t)]ementioning

confidence: 99%

“…The HB method [4,17] allows an N th harmonic, T -periodic assignmentx i N (t) forx i (t), i = 1, 3 to be found. For this, let us denotē…”

Section: Harmonic Balance-based Selection Of Current Referencesmentioning

confidence: 99%

Harmonic balance‐based control of a boost DC/AC converter

Olm

Biel

Spinetti-Rivera

et al. 2011

Circuit Theory & Apps

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SUMMARYThe achievement of step-up inversion with a boost DC/AC converter requires appropriate periodic references for inductor currents, which have to satisfy ordinary differential equations (ODE) of the Abel type. These are equations with highly unstable solutions for which the existence of periodic solutions remains unproved. Hence, the studies reported so far in this subject obtain periodic output voltages that approximately track the expected profile using different periodic current references that do not exactly satisfy the Abel ODE. However, neither an explanation of why are periodic output voltages still obtained, nor an assessment of the output voltage error is provided. This paper analyzes the effect of using periodic current references in a Lyapunov-based controlled boost DC/AC converter performing step-up inversion tasks. It is shown that, for sufficiently accurate current references, the system exhibits asymptotically stable periodic solutions with bounded error. Moreover, the paper propounds the use of Harmonic Balance (HB)-based techniques to obtain such current references. Simulation and experimental results confirm that this choice yields periodic output voltages with an error that may be lowered using higher HB approximations.

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“…The analysis of the Melnikov function is a separate mathematical problem and there exist various mathematical techniques for its solving [7][8][9][10][11][12]. In the case of a polynomial perturbed Hamiltonian system the Melnikov function is generally an Abelian integral.…”

Section: Introductionmentioning

confidence: 99%

Analysis and synthesis of oscillator systems described by perturbed single well Duffing equations

Georgiev

2010

Nonlinear Dyn

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Abstract:The paper presents an analysis as well as a synthesis of oscillator systems described by single well Duffing equations under polynomial perturbations of fourth degree. It is proved that such a system can have a unique hyperbolic limit cycle. There has been obtained an analytical condition for the arising of a limit cycle and an equation giving the parameters of this limit cycle. There has been proposed a method for the synthesis of oscillator systems of the considered type, having preliminarily assigned properties. The synthesis consists of an appropriate choice of the perturbation coefficients in such a way, that the oscillator equation should have a preliminary assigned limit cycle. Both the analysis and the synthesis are performed with the help of the Melnikov function.

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Harmonic Balance, Melnikov method and nonlinear oscillators under resonant perturbation

Cited by 20 publications

References 22 publications

Envelope analysis of nonlinear electronic circuits based on harmonic balance method

Envelope analysis of nonlinear electronic circuits based on harmonic balance method

Harmonic balance‐based control of a boost DC/AC converter

Analysis and synthesis of oscillator systems described by perturbed single well Duffing equations

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