Synthesis of lossless electric circuits based on prescribed Jordan forms

TL;DR

This paper presents an algorithm for designing lossless electric circuits with specific Jordan form matrices, involving LC-loops and gyrators with explicit component formulas, including negative values for certain elements.

Contribution

It introduces a novel synthesis method for lossless circuits with prescribed Jordan canonical forms, detailing explicit component value calculations and the necessity of negative elements.

Findings

Circuits can be synthesized with desired Jordan forms using LC-loops and gyrators.

Explicit formulas for component values, including negative elements, are provided.

At least one negative component is required for nontrivial Jordan blocks.

Abstract

We advance here an algorithm of the synthesis of lossless electric circuits such that their evolution matrices have the prescribed Jordan canonical forms subject to natural constraints. Every synthesized circuit consists of a chain-like sequence of LC-loops coupled by gyrators. All involved capacitances, inductances and gyrator resistances are either positive or negative with values determined by explicit formulas. A circuit must have at least one negative capacitance or inductance for having a nontrivial Jordan block for the relevant matrix.