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Deadlines
(Previous Conference: PMAMCM 2018, Majorca, Spain, July 14-17, 2018)
PLENARY SPEAKERS:
Prof. Alexander Milnikov, International Black Sea University, GEORGIA, e-mail: alexander.milnikov@gmail.com
Title: "Method of Successive Projections of Synthesis of Passive LC Circuits with Predefined Resonance Frequencies "
Abstract: A new synthesis method for passive electrical LC circuits with predefined resonant frequencies is presented. The method is based on the representation of the main matrices of the electrical circuit theory - the matrices of fundamental loops and cuts - as linear operators of projections acting in linear spaces, with current and voltage vectors represented as vectors and covectors. The G. Kron's conceptions of primitive, pure-loop, pure-node and orthogonal circuits are essentially used. Nonsingular linear transformation operators acting among pure-loop and pure-node circuits, from one side, and primitive circuits from another side are established. It is shown that transformations between pure-loop and pure-node, and k-loop orthogonal circuits can be represented by means of idempotent projections operations. The detailed picture of subspaces of the linear operators, and their transposes and is represented. Proof is provided that all of the pure-loop circuits resulting from the primitive circuit have pairwise equal eigenvalues, equal in turn to eigenvalues of a primitive circuit. The process of successive projections, which consists of imposing of constraints on pure-loop circuits and leads back to the original orthogonal circuit is determined. The latter is the basis for the Method of Successive Projections (SP-method). The method starts with forming the matrix of fundamental loops of a pure-loop circuit and a diagonal matrix of impedances of the primitive circuit. Then, the matrix of impedances of the pure-loop circuit is calculated. Further the process of successively imposing constraints is organized, which generates the finite sequence of linear operators of represented by means of matrices of loop impedances. The last one is the matrix of impedances of the k-loop orthogonal circuit. Each of the operators creates an intermediate series of eigenvalues, which are related by recurrent inequalities. The SP method's process permitted to introduce notion of conservativeness of the pure-loop circuit's eigen values: an eigen value is conservative, if it is the eigenvalue of both the pure-loop and k-contour orthogonal circuits. It is shown that when imposing the current constraint, the multiplicity of an arbitrary eigenvalue does not increase. The all above leads to the following main results: The eigenvalue of a pure-loop circuit with multiplicity r, which is greater than the number of node pairs in the k-contour circuit, is conservative; The presence of identical impedances connected to the fundamental chords of the k-loop circuit, leads to their conservatism. The latter is the basis for the method of synthesizing a passive multi-loop LC circuit with predefined resonant frequencies. The all results are illustrated with practical control of roots of characteristic polynomials (determinants of matrices of impedances) and synthesis examples.