Some topologico‐dynamical properties of linear passive reciprocal networks

Abstract

The paper considers the complete solvability and the order of complexity of passive RLCT ( T = multiwinding ideal transformer) networks. A topological approach based on the determinant polynomial of the matrix of hybrid equations, formed as a set of 1st-order differential and algebraic equations, reveals the structure of the formulation tree and the subnetworks accountable for degeneracies. Topological and algebraic degeneracies are defined. Necessary and sufficient conditions for complete solvability are derived, and two algorithms are given to determine the order of complexity topologically, i.e. without having an explicit state-space representation.

Bibo Stability of a Class of Linear Time‐variable Semi‐state Systems and Application in the Analysis of RLC Electrical Networks

“…In the modelling of LTV RLC forced networks the input vector u may be composed of currents and voltages of independent sources and also of their time derivatives [8,10,11]. However, we need not deal with such details and will turn our attention to the form of matrices A and P of eq.…”

“…The semi-state variables are selected to be the voltages across the capacitors and the currents through the inductors, and are represented by vectors v c and i L , respectively. According to [8][9][10][11] and our notation in eq. (la) one has Here, C, L, G, R and H are the appropriate capacitance, inductance, conductance, resistance and connectivity matrices, respectively.…”

“…matrix, P 2 is n 2 x n 2 p.s.d. matrix, then n 1 ≤ rank P < n, "t Є T. (c) The SSS (1) appearst in the process of modelling of LTV RLC networks [8][9][10][11].…”