Multimodal Resonance in Strongly Coupled Inductor Arrays

TL;DR

This work develops a circuit-theory framework to predict and analyze multi-modal resonance in magnetically over-coupled inductive arrays of parallel LC coils. By deriving two-coil and ND multi-coil models and formulating an eigenvalue problem, the authors show that over-coupled arrays exhibit multiple resonant frequencies corresponding to distinct magnetic excitation modes, which are validated with 2D FEM simulations and experimental measurements on 3- and 5-coil configurations. The results reveal mode-count, dispersion, and excitation-pattern dependencies on array geometry and measurement location, including realistic asymmetries. The approach enables designing inductive sensing arrays and potential magnetic-field shaping without multiplexing, with implications for near-field sensing and WPT-based sensing applications. Key equations, such as $\omega'_{\pm}$ and the ND eigenproblem $\mathbb{\Omega}\vec{v}_i = \omega_i^2 \vec{v}_i$, underpin the predictive framework and its practical utility.

Abstract

Magnetic resonance coupling (MRC) is widely used for wireless power transfer (WPT) applications, but little work has explored how MRC phenomena could be exploited for sensing applications. This paper introduces, validates and evaluates the unique multi-resonant phenomena predicted by circuit theory for over-coupled inductive arrays, and presents eigen-formulae for calculating resonant frequencies and voltage modes within passively excited arrays. Finite-element simulations and experimental results demonstrate the validity of the multi-modal resonant principles for strongly-coupled inductor arrays. The results confirm the distinctive multi-modal resonant frequencies these arrays exhibit, corresponding to the specific magnetic excitation "modes" (comparable to vibrational modes in multi-degree-of-freedom systems). The theoretical and finite element models presented offer a framework for designing and optimizing novel inductive sensing arrays, capitalizing on the unique resonant effects of over-coupling and exploiting their potential magnetic field shaping.

Figures (8)

• Figure 1: Equivalent circuit simulated impedance spectra for coupled resonant inductor circuits showing a) the dual resonant peaks of the frequency spectra and b) coupling coefficient verses frequency dispersion curves, representing the resonant vibrational modes of the system.
• Figure 2: Equivalent circuit diagram of an N-dimensional array
• Figure 3: Circuit model predicted impedance magnitude profiles of a 3-coil linear array systems of identical coil inductance ($10~\mu H$) and capacitance ($150~pF$) with varying series resistance, $R$, showing the resonant damping effect and disparity between the eigen-matrix calculated resonant frequencies (black dotted lines) and the circuit predicted peak frequencies (red solid lines).
• Figure 4: Predicted impedance spectra for three 3-coil over-coupled arrays configurations of identical coil inductance ($10~\mu H$), capacitance ($150~pF$), series resistance ($10~\Omega$) and coupling coefficient ($k = 0.14$) showing i) impedance magnitude spectra $|Z(f)|$ for three coil measurement configurations; Linear (A and B), and iii) Close-packed (C) arrays. Plots in blue and red represent measurements of a linear array from the end and centre coils respectively. The yellow curve shows the measurement for a close-packed coil configuration (C). Dotted lines correspond to the theoretically predicted resonant eigen-frequencies with colours corresponding to the predicted eigen-modes shown in the ii) and iii) for linear and close-packed arrays respectively.
• Figure 5: 2D FEM impedance spectra for two linear 3-coil over-coupled arrays configurations of identical resonance showing impedance magnitude $|Z|$ spectra and the spatial distribution of the magnetic flux density at each resonant frequency (i-iii). Showing a) linear array measured from active end coil, and b) measured from an active central coil.