## The order of complexity of electrical networks

The natural frequencies of an electrical network are defined, and the number of these natural frequencies is called the *order of complexity* of the network. *RLC* networks are considered, and the order of complexity, σ, is shown to be given byσ = *B _{L}* +

*N*+

*S*−

*S*−

_{C}*S*.Here,

_{CR}*B*is the number of inductors in the network,

_{L}*N*is the number of nodes,

*S*,

*S*and

_{C}*S*are the connectivities, i.e. the number of separate parts of, respectively, the given network and those subnetworks formed of the capacitors only and of the capacitors and resistors only. Other expressions for σ are also obtained.It is shown that this order of complexity is also the number of arbitrary integration constants in the complete solution of the network equations, and the number of dynamically-independent network variables. Complete sets of such dynamically-independent variables are obtained by a process of elimination from the network equations. A particular type of complete set is classified topologically, such sets being made up of voltages across capacitors forming a forest of the capacitor-only network obtained by open-circuiting all the resistors and inductors, together with the currents through inductors forming a set of chords of the inductor-only network obtained by short-circuiting all the capacitors and resistors.

_{CR}