Page 222 - An Introduction to Microelectromechanical Systems Engineering

Microelectromechanical Resonators 201

bandwidth, which is the distance in frequency between the points of response that
are1 ( 2 − 3dB )below the response maximum. Bandwidth is also given a percentage
of the center frequency.
Quartz crystals are presently at the core of every electrical resonant circuit
because, historically, integrated electronic oscillators have not been able to achieve
the large quality factors necessary for the stable operation of frequency-selective
communications systems. A typical quartz crystal has a Q that reaches 10,000 or
even higher. By comparison, the quality factor of an electrical filter consisting of a
network of inductors, capacitors, and resistors (RLC network) is typically far less
than 1,000, limited by parasitic resistive losses in the circuit. The quality factor has
also an effect on insertion loss. For example, a simple bandpass filter consisting of a
series inductor and capacitor, with parasitic resistance, in series with an output
resistor, which has a center frequency at 16 MHz, a Q of 100, and a bandwidth of
2.9 MHz (18% of the center frequency) has an insertion loss of 0.8 dB—in other
words, the signal suffers an undesirable attenuation of about 9%. The insertion loss
increases further as the Q decreases. Quality factors above 1,000 are generally con-
sidered high for many electronic and RF applications. If micromechanical resona-
tors can demonstrate high Q over a wide range of tunable frequencies, then
integrating them with electronics will consequently lead to system miniaturization.
The frequencies of interest cover the range between 800 MHz and 2.5 GHz for
front-end wireless reception, as well as the intermediate frequencies at 455 kHz
1
and above.
Based on the equation for resonant frequency, it follows immediately that a
reduction in size, which brings about a decrease in mass and stiffening of the spring,
increases the resonant frequency. This is the basic argument for the micromachining
of resonators. The various designs differ in their implementation of excitation and
sense mechanisms.

Comb-Drive Resonators
One of the earliest surface-micromachined resonator designs [15], which is now
commonly used in various MEM devices, is the interdigitated-finger comb-drive
structure developed at the University of California, Berkeley, California (see Figure
7.9). This structure is comprised of folded springs supporting a shuttle plate that
oscillates back and forth in the plane of the wafer surface. The folded springs relieve
residual stress and give a more compact layout. An applied voltage, either positive
or negative, generates an electrostatic force between the left anchor comb and
shuttle comb that pulls the shuttle plate to the left in Figure 7.9. This electrical force
2
F is given by ½(dC/dx) V , where V is the applied voltage, and dC/dx is the rate of
e
increase in capacitance as the finger overlap increases and is constant for a given
design. Because the voltage is squared, the force is always attractive. When a
sinusoidal ac voltage ν cos(ωt) is applied, where ν is the amplitude and ω is the
a a

1. A receiver converts the frequency of a selected incoming RF signal to a fixed intermediate frequency by het-
erodyning the signal with the local oscillator. This allows the remaining circuits in the receiver to remain
precisely tuned to the intermediate frequency regardless of the frequency of the incoming signal. The follow-
ing frequencies are generally considered intermediate frequency: 50 kHz, 100 kHz, 262 kHz, 455 kHz, 500
kHz, 9 MHz, 10.7 MHz, 45 MHz, and 75 MHz.