II Coupled van der Pol oscillators theory A Overview of R Yorks theory R Yorks

Ii coupled van der pol oscillators theory a overview

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II. Coupled van der Pol oscillators theory A. Overview of R. York’s theory R. York’s work considers the analysis of simple single-ended van der Pol oscillators coupled through either a resistive network or a broadband network that produce a constant amplitude and a phase shift between the oscillators [7, 9]. This analysis is useful in showing the effects of the coupling network parameters but, unfortunately, it is limited to the broadband case. In these conditions, R. York used a generalization of Kurokawa’s method [10] to extend the oscillators dynamics to the case of narrow-band coupling. Using a system made of two parallel resonant circuits containing nonlinear negative conductance devices and coupled through a series resonant circuit, as shown in Fig. 1, and starting from the admittance transfer functions ( Y 1 , Y 2 , Y c ) binding the coupling current ( I c ) to the oscillators’ voltages ( V 1 and V 2 ), J. Lynch and R. York described the oscillators’ dynamic equations, as well as those for the amplitude and phase of the coupling current. Then, by setting the derivatives to zero, the algebraic equations describing the oscillators’ frequency locked states were obtained as follows:
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) ( sin 1 ) ( cos 1 ) ( sin 1 ) ( cos 1 2 1 0 2 0 02 1 0 2 2 2 2 0 1 2 0 2 0 01 2 0 1 2 1 2 0   A A A A A A A A A A a c ac a a c ac a (1) with: c R G 0 0 1 : the coupling constant, where G 0 is the nonlinear conductance at zero voltage; C G a 0 2 : the oscillator bandwidth; c c ac L R 2 : the unloaded coupling circuit bandwidth; 2 1 , A A : the amplitudes of oscillators 1 and 2, respectively; 1 2 : the inter-stage phase shift; 02 01 , : the free-running frequencies or tunings of oscillators 1 and 2, respectively; c 0 : the resonant frequency of the coupling circuit; cos 1 1 2 ac c : the coupling strength scale factor; ac c 1 tan : the coupling phase.
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Furthermore, let us note that in the equations presented above, we refer the oscillators’ free- running frequencies or tunings, 01 and 02 , and the synchronization frequency of the system, , to the coupling circuit resonant frequency, , 0 c using the following substitutions: c 0 01 01 c 0 02 02 c c 0 Thus, a solution to (1) indicates the existence of a frequency-locked state and allows to obtain the amplitudes A 1 and A 2 of the two oscillators as well as the inter-stage phase shift and the synchronization pulsation for a combination ( 01 , 02 ). B. New expression of the equations allowing an accurate prediction of the oscillators’ amplitudes The purpose of this subsection is to present a new formulation of York’s equations describing the locked states of two coupled van der Pol oscillators using an accurate model allowing a good prediction of the oscillators’ amplitudes. Indeed, in [11], the negative conductance presented by the active part has the following expression:
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  • Spring '16
  • LC circuit, R. A. York

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