Trajectories of nonlinear RLC networks: A geometric approach

The response of a nonlinear time-varying coupled network starting from a given operating point is considered. We view the response as motion occurring in a differentiable manifold in , where 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 . 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 . 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 if and only if the capacitor voltages, inductor currents, and time constitute a parametrization for . Finally, we show that our conditions include as special cases the determinateness conditions previously obtained by several authors.