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Circuit Synthesis based on Prescribed Lagrangian

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Document pages: 14 pages

Abstract: We advance here a synthesis algorithm for electric circuit based onprescribed quadratic Lagrangian, that is the circuit evolution equations areequivalent to the relevant Euler-Lagrange equations. The proposed synthesis canbe viewed as a systematic approach in a realization of any finite dimensionalphysical system described by a Lagrangian in a lossless electric circuit sothat their evolution equation are equivalent. The synthesized circuit iscomposed of (i) capacitors and inductors of positive or negative values for therespective capacitances and inductances, and (ii) gyrators. The circuitdynamics is governed by the relevant Euler-Lagrange (EL) equations. The circuittopological design is based on the set of LC fundamental loops (f-loops) thatare coupled by GLC-links each of which is a serially connected gyrator,capacitor and inductor. The set of independent variables of the underlyingLagrangian is identified with f-loop charges defined as the time integrals ofthe corresponding currents. The EL equations for all f-loops account for theKirchhoff voltage law whereas the Kirchhoff current law holds naturally asconsequence of the setup of the coupled f-loops and the corresponding chargesand currents. The proposed synthesis provides for efficient implementation ofthe desired spectral properties in an electric circuit. The synthesis providesalso a way to realize arbitrary mutual capacitances and inductances throughelementary capacitors and inductors of positive or negative respectivecapacitances and inductances.

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