It is important to note that the obtained ˆ is an approximated solution if J 0

# It is important to note that the obtained ˆ is an

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It is important to note that the obtained ˆ is an approximated solution if J ≈ 0, i.e. 1,3 , 0 ) ( k f k . This final value of the criterion corresponds to “small residual”. B. Stability analysis For the stability analysis of the synchronized states, it is important to consider an initial differential system describing the amplitudes 2 1 , A A and phases 2 1 , dynamics and the coupling current in Cartesian format cy cx A A , . Thus, dynamic equations developed in [11] have been adapted to the studied case and give the following differential relations: 02 01 . , , , a x x f (29) where T cy cx A A A A x 2 2 1 1 and
1 1 2 2 0 0 1 1 2 2 0 0 2 2 2 02 2 2 2 2 2 1 1 1 01 1 1 2 1 1 02 01 sin sin - cos cos cos sin sin cos 4 3 cos sin sin cos 4 3 , , , A A A A A A A A A A A A A G G bA a A A A A A A G G bA a A x f ac cx c cy ac ac cy c cx ac cy cx a cy cx a L L a cy cx a cy cx a L L a a with c L c cx G I A cos , c L c cy G I A sin and c is the phase of the coupling current. These differential equations are nonlinear in states x , and unfortunately, the stability theory developed for the linear problem does not apply directly to this system. In practice, we typically linearize this system around a synchronized solution noted 0 x and consider the eigenvalues of the Jacobian matrix A so that: x A x x f x x x . . 0 (30)
where 0 x x x are the small variations of the synchronized states x and 0 6 6 x x x f A is ac ac ac ac ac ac cx a cy cx a a L L a cy cx a cy cx a cy cx a L L a A A A A A A A A bA G a G A A A A A A A A bA G a G A 2 0 1 1 0 1 0 2 0 1 1 0 1 0 2 2 2 2 2 2 2 1 1 1 1 1 2 1 1 1 2 1 sin cos sin cos sin cos cos sin 0 0 4 9 0 0 0 sin cos cos sin 0 cos sin 4 9 Synchronized states are asymptotically stable if and only if all the eigenvalues of the Jacobian matrix A have negative real parts. Thus, after each estimation of variables values , the resulting Jacobian matrix is evaluated and these six eigenvalues, noted i , are computed.

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• Spring '16
• LC circuit, R. A. York

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