Complex bases and the complex degree of freedom of LCR networks

Abstract

AbstractA complex basis can be defined for a set × of variables such as element currents and voltages (excluding the exciting current and voltage), of a linear lumped‐con‐stant network in the sinusoidal steady state. If a subset Xβ of × is a minimal set that satisfies the condition that any variable in × ‐ Xβ can be expressed as a linear combination of variables in Xβ with constant (not including frequency) coefficients, then Xβ is called a complex basis of X, and the number of variables in, Xβ is called the complex degree of freedom. This paper derives one of the complex bases of the set × of all the element currents and voltages excluding the exciting currents and voltages of the LCR network. Let G be the graph of the network, and consider trees Ti and Tv, together with their cotrees Ki and Kv in G, satisfying the following conditions C1 and C2: Ti ∩ Kv is a union edge set of current sources and voltage sources; Under the condition C1, Ti and Tv contain as many capacitors as possible and Ki and Kv contain as many inductors as possible. Let Tsc be a tree of the graph obtained from G by shorting all the edges other than capacitor edges and let K0z be a cotree of the graph obtained by opening all the edges other than inductor edges. Let Ti, Tv ⊇ Tsc and Ki, Kv ⊇ K0z. Then, capacitor voltages of Tv ‐ Tsc, inductor currents of Ki ‐ K0z, capacitor currents of Tv ∩ Ki and inductor voltages and resistor voltages form a complex basis of X.