Temperature Sensitivity
Temperature dependence of electromechanical spring constants in biased capacitive micro-structures governs frequency drift in resonant micro-devices. When DC bias voltages are applied across capacitive sensing gaps, electrostatic stiffness thermal shift defines the rate of change in overall system stiffness caused by temperature-induced changes in gap geometry and permittivity. This shift acts in parallel with the intrinsic mechanical temperature coefficient of Young’s modulus, modifying the net resonance frequency of the structure.
It sets the baseline drift rate for voltage-tuned resonators operating across wide ambient temperature spans. The shift model ceases to apply when dielectric charging or physical electrode contact alters the bias electric field.
Voltage Interaction
Electrostatic stiffness introduces a negative spring constant component that scales quadratically with applied DC bias voltage and inversely with the cube of the gap width. Because thermal expansion alters the capacitive gap dimension, electrostatic stiffness thermal shift scales strongly with the operational voltage setting. At high bias voltages, the temperature coefficient of electrostatic stiffness can exceed the mechanical temperature coefficient of silicon, reversing the sign of the overall frequency drift.
Precise voltage regulation is required to prevent bias supply ripple from disguising temperature-induced stiffness shifts during sensor characterization.
Frequency Drift
System response tracking across temperature cycles shows that uncompensated electrostatic stiffness variations produce non-linear frequency offsets. Higher ambient temperatures decrease gap dimensions in constrained package structures, enhancing electrostatic spring softening and pulling resonance frequencies downward.
Qualification Testing
Characterization procedures measure resonator frequency across temperature ranges under multiple DC bias configurations to separate mechanical from electrostatic temperature coefficients. Test systems record frequency output using automated counter equipment while maintaining vacuum levels below one millibar to eliminate gas damping effects. Extracted parameters feed temperature-compensation lookup tables stored in sensor readout circuits, allowing real-time adjustment of electrostatic tuning voltages.
Proper calibration maintains output frequency stability within sub-parts-per-million limits over the entire operating temperature range. Qualified components demonstrate reproducible thermal shift curves without dynamic hysteresis across consecutive thermal test runs.
