An approximate analytical (structural) superposition in terms of two, or more, "alfa"-circuits of the same topology: Pt.1 - description of the superposition

TL;DR

This paper introduces an approximate superposition principle for nonlinear circuits composed of similar conductors with polynomial or quasi-polynomial characteristics, revealing a near-linear map between input and characteristic functions.

Contribution

It generalizes the superposition concept to nonlinear alfa-circuits, showing that the input characteristic of combined circuits closely approximates a linear combination of individual characteristics.

Findings

Superposition holds with high precision in nonlinear alfa-circuits

The input characteristic map is approximately linear for combined circuits

Examples include circuits with quadratic and cubic characteristics

Abstract

One-ports named "f-circuits", composed of similar conductors described by a monotonic polynomial, or quasi-polynomial (i.e. with positive but not necessarily integer, powers) characteristic i = f(v) are studied, focusing on the algebraic map f --> F. Here F(.) is the input conductivity characteristic; i.e., iin = F(vin) is the input current. The "power-law" "alfa-circuit" introduced in [1], for which f(v) ~ v^"alfa", is an important particular case. By means of a generalization of a parallel connection, the f-circuits are constructed from the alfa-circuits of the same topology, with different "alfa", so that the given topology is kept, and 'f' is an additive function of the connection. We observe and consider an associated, generally approximated, but, in all of the cases studied, always high-precision, specific superposition. This superposition is in terms of f --> F, and it means that F(.) of the connection is close to the sum of the input currents of the independent "alfa"-circuits, all connected in parallel to the same source. In other words, F(.) is well approximated by a linear combination of the same degrees of the independent variable as in f(.), i.e. the map of the characteristics f --> F is close to a linear one. This unexpected result is useful for understanding nonlinear algebraic circuits, and is missed in the classical theory. The cases of f(v) = D1v + D2v^2 and f(v) = D1v + D3v^3, are analyzed in examples. Special topologies when the superposition must be ideal, are also considered. In the second part [2] of the work the "circuit mechanism" that is responsible for the high precision of the superposition, in the most general case, will be explained.