Analysis of MEMS Electrostatic Actuators
Theory: balancing electrostatic and spring forces
Overview
I'm tilting a MEMS mirror with voltage, but past a certain voltage it suddenly stuck down and won't return. Is it broken?
Probably pull-in. Voltage across two electrodes makes them attract, and the spring-supported moving side approaches. But electrostatic force grows with the inverse square of the gap, while spring force grows only linearly. Once the gap has closed by a third, the spring can no longer balance the electrostatic force and the plate snaps in. If the electrodes touch, surface forces may keep them stuck. Designs must either avoid pull-in or deliberately exploit it.
Force balance
$k$ is spring constant, $x$ displacement, $\varepsilon_0$ vacuum permittivity, $A$ electrode area, $g_0$ initial gap and $V$ voltage. The electrostatic force on the right rises steeply as the gap narrows.
Pull-in voltage
The largest voltage for which the balance has a solution is the pull-in voltage, with displacement one-third of the gap. Pull-in voltage scales with gap to the 1.5 power and the square root of spring constant.
Comb drives
$n$ is the number of fingers, $t$ thickness and $g$ the finger gap. Moving along the direction of engagement, the force is nearly independent of displacement, so parallel-plate-type instability is largely avoided.
From the 1960s resonant-gate transistor
A forerunner of electrostatically actuated micromachines was the resonant-gate transistor published by Nathanson and colleagues in 1967, whose analysis already described pull-in. Silicon micromachining of mechanical structures advanced in the 1980s, and Tang and colleagues demonstrated the comb drive in 1989. Today electrostatic MEMS are in smartphone accelerometers and gyroscopes, projector micromirrors, RF switches and many other products.
Worked examples
Example 1: spring constant and pull-in voltage
Electrode 100 μm square, initial gap 2 μm:
| Spring constant | Pull-in voltage |
|---|---|
| 1 N/m | 5.17 V |
| 5 N/m | 11.57 V |
| 10 N/m | 16.36 V |
Example 2: voltage versus displacement (5 N/m)
| Voltage (fraction of V_PI) | Displacement | Fraction of gap |
|---|---|---|
| 2.89 V (25%) | 0.019 μm | 1% |
| 5.78 V (50%) | 0.080 μm | 4% |
| 8.68 V (75%) | 0.207 μm | 10% |
| 10.41 V (90%) | 0.355 μm | 18% |
| 11.45 V (99%) | 0.561 μm | 28% |
Even just before pull-in, it only moves about 30% of the gap.
Right — theoretically a third of the gap is the limit, and in practice you stay further back for scatter and vibration. At low voltage displacement grows slowly with voltage squared, but near pull-in a small voltage change causes a big jump, making control hard. For large travel, widen the gap (pull-in voltage then rises with the 1.5 power) or use a geometry less prone to instability, such as a comb drive.
Example 3: gap scatter
| Gap error | Pull-in voltage | Change |
|---|---|---|
| −10% | 9.88 V | −14.6% |
| ±0 | 11.57 V | — |
| +10% | 13.35 V | +15.4% |
A 10% gap error shifts pull-in voltage by about 15%. Sacrificial-layer etching and warping from residual stress make gaps vary between devices, so leave ample margin in drive voltage or calibrate after manufacture.
Example 4: comb-drive force
| Voltage | Force (100 fingers, 20 μm thick, 2 μm gap) | Displacement with 1 N/m |
|---|---|---|
| 10 V | 0.89 μN | 0.89 μm |
| 30 V | 7.97 μN | 7.97 μm |
| 60 V | 31.9 μN | 31.9 μm |
Comb force scales with voltage squared and doesn't depend on displacement, so travel of tens of micrometres — far beyond the gap — is possible. If lateral force balance breaks, fingers snap sideways (lateral pull-in), so lateral spring stiffness must be high.
How to approach the analysis
- Electrostatic–structural coupling: deformation changes the field and hence the force; iterate by re-solving the field on the deformed shape, or solve monolithically.
- Tracking pull-in: ordinary voltage-stepping fails to converge just before pull-in; prescribe displacement and solve for voltage (displacement control) to trace the curve through the unstable branch.
- Fringing fields: fields wrapping around electrode edges raise force above the parallel-plate formula, more so as gap grows relative to electrode size.
- Air damping: squeeze-film damping from air forced out of the narrow gap dominates dynamic response.
“Measured pull-in is always lower than calculated”
Prototype MEMS often pull in 10–20% below the parallel-plate formula. One cause is fringing, which makes the real force larger; another is warping from residual stress, leaving parts of the gap narrower than designed; trapped charge in dielectric layers can add on top. Reliable analysis includes 3D fields with fringing and initial deformation from residual stress, calibrated against measurement.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Targeting travel beyond a third of the gap | Pull-in and stiction | Wider gap or comb drive |
| Voltage-controlled analysis only | Non-convergence near pull-in | Trace with displacement control |
| Ignoring fringing | Pull-in overestimated | 3D field |
| Ignoring gap tolerance | Device-to-device voltage scatter | Margin or calibration |
| Neglecting lateral stability (combs) | Lateral pull-in | Stiffen lateral springs |
I'd like to try electrostatic force and energy calculations myself.
Try the capacitor energy tool for stored electrode energy and the capacitor charging tool for charging time. Compare other actuation methods in piezoelectric analysis, electroactive polymers, Lorentz force and magnetostriction.
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