Page 53 - MEMS Mechanical Sensors

42 MEMS Simulation and Design Tools

electrodes is energized. These feedback forces are summed with any external inertial
force acting on the proof mass.
The model allows the optimization of many design parameters such as the elec-
trode area, spring constant, proof mass, the required electronic pick-off gain, and
the sampling frequency. Predictions on the control loop stability can be made and
the signal-to-quantization noise ratio can be derived. Additional effects such as
inherent noise sources (Brownian or thermal noise) can be simulated by adding ran-
dom number generators, or unwanted electrostatic forces due to the electrical exci-
tation voltage required for the electronic interface circuitry can be added to the
model and their influence on the performance of the sensor can be studied [2, 3].
Modeling of these second order effects obviously increases the simulation time con-
siderably. On a modern computer a simulation run with the basic model presented
in Figure 3.2 may only take seconds to a few minutes; if the other effects are added
the simulation time may increase to a few hours. A typical methodology is to start
with a basic model, capturing only first order effects, then adding various second
order effects and evaluating their influence on the performance of the device. Those
that have a negligible effect on the sensor can subsequently be discarded again to
speed up the simulation.
The accuracy and merits of such an approach obviously rely on the analytical
understanding of the underlying physics of the sensor to be simulated. The modeling
process as such is done analytically by the designer, often by hand calculations. Cer-
tain FEM software tools automate this process by performing, for example, a full
mechanical modal analysis, and then extracting a lump parameter model that is suit-
able for implementation in a system simulation tool.

3.2.1.2 Spice
Spice is typically an electronic circuit simulator. It can also, however, be used to
simulate parts from another physical domain. Two approaches are possible: one can
map electrical quantities to equivalent ones in the physical domain to be considered
and build an equivalent electrical circuit. If a mechanical part is to be considered,
then the mapping is as follows [4]:

Mass == Inductance; Damping == Resistance; Stiffness == 1/Capacitance; Force ==
Voltage; Position == Charge.

A similar mapping process can be derived for other physical domains, for exam-
ple, thermal processes. This allows the simulation of the dynamics of mechanical
structures such as resonators, accelerometers, and pressure sensors. Even more com-
plex phenomena such as squeeze film damping can be modeled in such a way [5].
The second approach is to make use of the analog behavioral library most com-
mercial Spice packages include [6]. This library contains models for system level
blocks such as integrator, transfer functions, look-up tables, summers, and gain
blocks. It allows dynamic models of many physical sensors to be developed. In
Figure 3.3, a system level model implemented in OrCad PSpice of a closed loop
accelerometer is shown.
The main advantage of both approaches is obviously that in Spice the interface
and control electronics of the sensor can be easily simulated.