Cause of anomalous extra attractive force in charged resonant capacitors

Determine the physical mechanism underlying the additional attractive force F_un between the plates of charged parallel-plate resonant capacitive transducers (with gaps of 10–26 μm) that produces an anomalous negative stiffness and a divergent tuning of the resonant frequency with applied voltage, beyond the quadratic dependence predicted by standard electromechanical theory under constant-charge operation.

Background

The authors measure the resonant frequency of large-area, small-gap, parallel-plate capacitors where one plate vibrates at around 1 kHz under d.c. bias. Classical electromechanical modeling predicts a quadratic tuning with voltage due to an electric stiffness term arising from stray capacitance. However, all datasets exhibit substantially larger tuning than predicted and, at high fields, a faster-than-exponential (divergent) behavior indicative of an additional attractive force.

They systematically assess and dismiss conventional explanations (gap shrinkage beyond pull-in limits, mechanical spring softening, fringe fields, Casimir and patch effects, PTFE nonlinearity, residual gas, electromagnetic currents and fields). They introduce an empirical model featuring a divergent susceptibility localized near the plate edge and speculate that vacuum fluctuations might act as dielectric inclusions, but the physical origin remains unresolved.