Geometric properties of nonlinear networks containing capacitor-only cutsets and/or inductor-only loops. Part I: Conservation laws

This paper is the first in a two part sequence which studies nonlinear networks, containing capacitor-only cutsets and/or inductor-only loops from the geometric coordinate-free point of view of differentiable manifolds. Given such a nonlinear networkN, with °0 equal to the sum of the number of independent capacitor-only cutsets and the number of independent inductor-only loops, we establish the following: (i) circuit theoretic sufficient conditions to guarantee that the set Σ0, of equilibrium points is a δ0-dimensional submanifold of the state space Σ ofN; (ii) circuit theoretic sufficient conditions for the condition thatN has δ0 independent conservation laws and hence that through each pointσ of the state space Σ ofN, there passes a codimension δ0 invariant submanifold ∑σ* of the network dynamics; (iii) circuit theoretic sufficient conditions to guarantee that the manifolds ∑σ* and Σ0 intersect transversely.

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