Impedance responses and size-dependent resonances in topolectrical circuits via the method of images

TL;DR

This paper investigates size-dependent impedance resonances in topolectrical circuits, revealing how boundary effects and interference modify resonance modes, with analytic expressions derived for various lattice configurations.

Contribution

It introduces the method of images to analyze and derive impedance characteristics in finite topolectrical circuits, highlighting size-dependent anomalous resonances.

Findings

Size-dependent impedance resonances in finite circuits

Analytic expressions for impedance in various lattice geometries

Boundary effects significantly influence resonance modes

Abstract

Resonances in an electric circuit occur when capacitive and inductive components are present together. Such resonances appear in admittance measurements depending on the circuit's parameters and the driving AC frequency. In this study, we analyze the impedance characteristics of nontrivial topolectrical circuits such as one- and two-dimensional Su-Schrieffer-Heeger circuits and reveal that size-dependent anomalous impedance resonances inevitably arise in finite $LC$ circuits. Through the \textit{method of images}, we study how resonance modes in a multi-dimensional circuit array can be nontrivially modified by the reflection and interference of current from the structure and boundaries of the lattice. We derive analytic expressions for the impedance across two corner nodes of various lattice networks with homogeneous and heterogeneous circuit elements. We also derive the irregular dependency of the impedance resonance on the lattice size, and provide integral and dimensionally-reduced expressions for the impedance in three dimensions and above.