Rapid 2-D Complex Capacitance Estimation Enables Real-Time Design of Impedance-Based Sensors

Capacitance underpins a wide spectrum of impedance-based sensors, yet the usual method of using iterative finite-element modelling of frequency-dependent, fringing-field geometries is prohibitively slow. We present a rapid, fully 2-D framework that computes the complex capacitance of arbitrary, polygonal cross-sections by combining conformal-mapping theory with a generalised boundary criterion based on material admittance. The method (i) partitions multi-layer structures into homogeneous sub-domains, (ii) assigns mixed Dirichlet–Neumann boundaries that capture both permittivity and conductivity, and (iii) assembles the global capacitance through simple series-and-parallel rules applied to the conformal moduli. Benchmarking against both FEM (COMSOL) and Boundary Element Methods (BEM) shows excellent agreement (< 3 % deviation). Crucially, our method is orders of magnitude faster (5 s) than FEM (233 s) and BEM (418 s) for a representative coplanar sensor. Canonical test cases with angled or nested interfaces further confirm that the approach remains accurate when conventional partial-capacitance formulas fail. By decoupling speed from mesh density, our technique enables real-time optimisation, inverse design and uncertainty analysis for next-generation touch, flow-cytometry and quantum-device sensors.