Trajectories of nonlinear RLC networks: A geometric approach

The response of a nonlinear time-varying coupled <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RLC</tex> network starting from a given operating point is considered. We view the response as motion occurring in a differentiable manifold <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Σ</tex> in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R²ᵇ × R₊</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</tex> is the number of branches. We impose two basic manifold conditions (MC) on the network. First, the resistor characteristics are required to be a manifold <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Λ</tex> . Second, the resistor characteristics and their connections are such that the set of branch voltages and branch currents satisfying both the Kirchhoff laws and the resistor characteristics is a manifold <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Σ</tex> . We then show that under the conditions imposed on the RLC elements and the topology of the network, the network has a unique response specified by a flow on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Σ</tex> if and only if the capacitor voltages, inductor currents, and time constitute a parametrization for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Σ</tex> . Finally, we show that our conditions include as special cases the determinateness conditions previously obtained by several authors.