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

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.

arXiv comment24 pages, 10 figures

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• Two-point spectroscopy of Fibonacci topoelectrical circuits
• Protected chaos in a topological lattice
• Observation of edge solitons and transitions between them in a trimer circuit lattice
• One-dimensional topology and topolectrics of nonsymmorphic Kramers degenerate systems
• General theory of perturbation of infinite resistor networks
• Line defects in infinite networks of resistors