**Manu Agarwal – Doctoral Research Summary**

**Title: Advanced Considerations in Electrostatic
Transduction & Nonlinearities in Resonant Microstructures**

Electrostatic
sensing and actuation (transduction) have been well explored and modeled for
different kinds of displacement sensing, viz., detecting changes in capactive-gap width. These include devices like
electrostatic accelerometers where the acceleration information is contained in
the change of the capacitive gap size, induced from the acceleration induced
displacement of a proof mass mounted on a mechanical spring.

With the advent of microresonator technology for
replacing quartz in timing/frequency references and filters and also resonant
sensors such as resonant- mass, strain or acceleration sensors, there has been
an increased interest in using electrostatic transduction for interfacing with
these resonant microstructures. These microstructures are different from those
used for displacement sensing in several ways. These have higher quality
factors and are generally operated on or very near their resonance frequency,
typically using a self-oscillating circuit. The information of interest is
contained in the resonant frequency, rather than change in gap, and hence the
limits to resolution are caused by the phase (frequency) noise in the
oscillator.

In this new paradigm, we need to revisit the physics of electrostatic
transduction. E.g., electrostatic spring softening, or the frequency dependence
on the DC polarization voltage (or DC bias voltage, needed for electrostatic
transduction) would cause a small and generally negligible change in the
sensitivity of a capacitive displacement sensor. However, in resonant
microstructures this phenomenon causes undesirable resonant frequency shifts
which could throw off a calibration, or it could also couple in frequency
(phase) noise in the presence of voltage noise in the DC voltage supply.

Higher order nonlinearities in the stiffness due to mechanical
(geometrical or material) properties or the electrostatics lead to modification
of the 2^{nd} order linear differential equation of motion into a Duffing equation. For
high Quality factor (Q) systems, such as the resonators in question, a closed
form solution can be obtained for it by treating it as a perturbed simple
harmonic oscillator. This leads to what is popularly known as the
amplitude-frequency dependence (A-f) effect, where the eigen-frequency
of the system becomes dependent on the amplitude of oscillation. This effect
limits the linearity range of the resonator and hence dictates the sustainable
power or current handling. This nonlinear A-f effect in MEMS resonators is one
of the focal points of this research. Models for this A-f effect have been developed
and scaling and optimization of current handling by optimization of both DC bias voltage as well as the structural design and
scaling are being studied.