Page 229 - An Introduction to Microelectromechanical Systems Engineering

208 MEM Structures and Systems in RF Applications

linear chain widens the extent of this passband but also increases the number of rip-
ples. In general, the total number of oscillation modes is equal to the number of cou-
pled oscillators in the chain.
Coupled-resonator filters are two-port devices, with a two-lead input and two-
lead output. An ac voltage input drives the filter, while the output is taken in the
same method as that for a single resonator: a dc bias is applied. The current due to
the capacitance change, V dC/dt, is the output, which is typically fed to a transim-
D
pedance amplifier to generate an output voltage. From the perspective of an electri-
cal engineer, a dual electrical network models the behavior of a filter made of
coupled micromechanical resonators. The dual of a spring-mass system is a network
of inductors and capacitors (LC network): The inductor is the dual of the mass (on
the basis of kinetic energy), and the capacitor is the dual of the spring (on the basis of
potential energy). A linear chain of coupled undamped micromechanical resonators
becomes equivalent to an LC ladder network. This duality allows the implementa-
tion of filters of various types using polynomial synthesis techniques, including
Butterworth and Chebyshev common in electrical filter design. Widely available
“cookbooks” of electrical filters provide appropriate polynomial coefficients and
corresponding values of circuit elements [18].

Film Bulk Acoustic Resonators

Another method of creating microelectromechanical bandpass filter is to use a pie-
zoelectric material. By sandwiching a sheet of piezoelectric material with a reasona-
bly high d (see Chapter 3) and low mechanical energy loss between two electrodes,
33
a resonator is created [see Figure 7.14(a)]. When an ac signal is applied across the
piezoelectric, an acoustic wave, traveling at the speed of sound in the material, is
generated. If the top and bottom surfaces of the device are in air or vacuum, there is
an acoustic impedance mismatch, and the wave is reflected back and forth through
the thickness. When the acoustic wavelength is equal to twice the thickness, a stand-
ing wave is formed (mechanical resonance) and the electrical impedance is low [see
Figure 7.14(b)]. The frequency response of such devices is commonly modeled by
the simplified L-C-R electrical network shown in Figure 7.14(c). The series induc-
tance and capacitance in the model represent the kinetic energy of the moving mass
and the stored energy due to compression and expansion of the material, respec-
tively, while the series resistor represents energy loss. This resistance is relatively
small with a good design and process, enabling quality factors of over 1,000 in pro-
duction devices. There is also a significant electrical capacitance between the plates,
represented by the parallel capacitor. The series capacitor and inductor in this sys-
tem have a series resonance—the low impedance in Figure 7.14(b). Due to the paral-
lel capacitor, the system also predicts a separate, parallel resonance—the high
impedance in Figure 7.14(b).
The goal of a bandpass filter, such as those linking the input or output circuitry
to the antenna of a cellular phone, is to transmit a narrow range of frequencies with
low loss and filter out both higher and lower frequencies. To make a bandpass filter,
FBARs are placed in a ladder network such as that shown in Figure 7.14(d) [25]. The
series FBARs are designed to have the same series-resonant frequency and corre-
sponding low impedance, which transmits the desired frequency with low loss [see
Figure 7.14(e)]. These devices do not transmit higher frequencies due to the high