A New Model of Fringing Capacitance and its Application to the Control of Parallel-Plate Electrostatic Micro Actuators

Fringing field has to be taken into account in the formulation of electrostatic parallel-plate actuators when the gap separating the electrodes is comparable to the geometrical dimensions of the moving plate. Even in this case, the existing formulations often result in complicated mathematical models from which it is difficult to determine the deflection of the moving plate for given voltages and therefore to predict the necessary applied voltages for actuation control. This work presents a new method for the modeling of fringing field, in which the effect of fringing field is modeled as a serial capacitor. Numerical simulation demonstrates the suitability of this formulation. Based on this model, a robust control scheme is constructed using the theory of input-to-state stabilization (ISS) and back-stepping state feedback design. The stability and the performance of the system using this control scheme are demonstrated through both stability analysis and numerical simulation.

💡 Research Summary

The paper addresses a critical modeling challenge in electrostatic parallel‑plate micro‑actuators: when the electrode gap becomes comparable to the plate dimensions, fringe fields significantly increase the capacitance and consequently the electrostatic force. Traditional models that ignore fringe effects (C = εWL/G) become inaccurate, while existing analytical fringe‑field formulas are highly nonlinear and unsuitable for control‑oriented design because they do not provide an explicit voltage‑deflection relationship.

To overcome this, the authors propose a novel representation of the fringe field as a single time‑varying series capacitor. The total capacitance is expressed as the sum of an over‑estimated “substitute” parallel‑plate capacitor (with a larger effective area so that it matches the real capacitance at the initial gap) and a series compensation capacitor that accounts for the excess capacitance due to fringing. The series capacitor is infinite at the initial gap and decreases as the gap closes, approaching the real capacitance. Importantly, the exact functional dependence of this series capacitor on the gap is not required for control; only known bounds on its variation are needed. Finite‑element simulations (using CoventorWare) provide these bounds (minimum series capacitance ≈ 6.47 pF, parallel parasitic ≈ 0.226 pF for the studied geometry).

The mechanical‑electrical dynamics are derived using Kirchhoff’s laws and the standard mass‑spring‑damper model for the movable plate. The resulting equations reveal that parallel parasitics affect only the electrical dynamics (slowing the response), whereas series parasitics alter both static equilibrium (shifting the pull‑in position) and dynamics. To facilitate analysis, the authors normalize the system variables (voltage, charge, displacement) and obtain a compact nonlinear state‑space model (equation 6) that captures the influence of the bounded parasitic terms.

For control, the paper adopts the Input‑to‑State Stability (ISS) framework, treating all uncertainties (parasitics, damping variations, resistance changes) as external inputs. Under three explicit assumptions—bounded parasitic capacitances, bounded damping ratio, and known resistance limits—the system is shown to be ISS. A back‑stepping design is employed: a virtual control law is first defined for the transformed subsystem, then the actual voltage input is derived by inverting the electrical dynamics. The final control law (equations 16‑18) includes several gain parameters (k₁, k₂, k₃, κ₁…κ₄). By selecting these gains sufficiently large, the tracking error can be made arbitrarily small, and the closed‑loop system remains ISS despite the uncertainties. The only singularity occurs at the exact initial position where the control input would blow up; this is mitigated by applying zero voltage at that instant, after which the system stabilizes.

Reference trajectories are generated using a fifth‑order polynomial that satisfies initial and final position and velocity constraints, ensuring smooth motion. The authors provide explicit coefficients for a set‑point move from 0 % to 100 % of the full gap.

Simulation studies use a 600 µm × 300 µm plate with an initial gap of 30 µm. Nominal parameters are chosen (ζ₀ = 0.1, r₀ = 1 kΩ, pρ₀ = 0, sρ₀ = 0) and then perturbed (Δζ = 0.02, Δr = 0.1 kΩ, Δpρ = 0.2 pF). The series parasitic lower bound (6.47 pF) and parallel parasitic (0.226 pF) derived from FEM are incorporated. For set‑point commands at 20 %, 40 %, 60 %, 80 % and 100 % of the full deflection, the controller drives the actuator to the desired positions with negligible steady‑state error and voltage requirements well below the pull‑in voltage. The results confirm that the proposed fringe‑field model together with the ISS‑based back‑stepping controller provides accurate, robust performance even in the presence of significant modeling errors and parameter variations.

In conclusion, the paper delivers a practical modeling simplification for fringe fields, turning a complex spatial effect into a manageable series capacitor with known bounds. Coupled with an ISS‑oriented back‑stepping control strategy, this enables robust voltage control of electrostatic MEMS actuators without requiring precise knowledge of the fringe‑field function. The approach is computationally light, suitable for real‑time implementation, and paves the way for extensions to multi‑axis or higher‑order MEMS devices. Future work is suggested to include experimental validation and application to more complex MEMS architectures.