Page 211 - An Introduction to Microelectromechanical Systems Engineering

190 MEM Structures and Systems in RF Applications

Basic electromagnetic theory teaches that when the signal wavelength is on the
order of the size of the system through which it flows, it is necessary to use a special-
ized type of electrical connections called transmission lines to carry the electrical sig-
nal from one point to another [1]. Transmission lines have a conductor for the signal
and one or multiple nearby ground conducting lines running parallel to the signal
line. The familiar coaxial lines used for cable television are one example where the
signal conductor is in the center of a hollow cylindrical ground conductor. On cir-
cuit boards, strip lines, which have a signal line sandwiched between two ground
planes and separated by a dielectric, are common. Another form that is more easily
implemented on circuit boards, and especially on chips, is the coplanar waveguide.
This has a central strip of metal for the signal, with ground strips on both sides. The
whole structure resides on a dielectric, with air or vacuum above. Devices that trans-
mit the RF signal, such as switches, must match the characteristic impedance of the
transmission line to avoid signal reflections. Similarly, devices at the output termi-
nals of a transmission line must be impedance-matched to collect the full signal
strength and avoid undesirable reflection.
Losses fall into two categories: conductor loss and dielectric loss. Conductor
loss is due to the nonzero resistance of the materials used, resulting in heating, and is
modeled as an equivalent resistance in series with the signal path. Low-resistivity
metals such as gold, copper, and aluminum are therefore commonly used for the
conductors in RF MEMS. Contacts in a switch and even between layers of different
materials also add resistance, which must be considered. Dielectric loss is due to
atomic-scale dipoles excited in the dielectric material, also resulting in heating, and
can be modeled as an equivalent parallel conductance. Eddy currents induced in an
underlying conducting substrate are also modeled as a parallel conductance. Insulat-
ing or semiinsulating substrates are often necessary for RF-MEMS devices to
minimize the loss due to eddy currents. High-resistivity (>5,000 Ω•cm) silicon sub-
strates are acceptable but semi-insulating gallium arsenide (GaAs) is preferred when
possible. Gallium arsenide is already in common usage for microwave integrated cir-
cuits for its high electron mobility. Insulators such as glass and alumina are preferred
from a low-loss standpoint, although other considerations such as process compati-
bility, cost, and thermal coefficient of expansion mismatch factor into substrate
selection.

Passive Electrical Components: Capacitors and Inductors

Quality Factor and Parasitics in Passive Components
All capacitors and inductors have parasitics associated with them that limit their
performance. Two parameters that describe their performance and enable compari-
sons between devices are the quality factor Q and the self-resonance frequency f .
SR
The quality factor Q is a measure of loss in a linear-circuit element and is defined
as the maximum energy stored during a cycle divided by the energy lost per cycle.
For reactive components such as capacitors and inductors, it is equal to the absolute
value of the ratio of the imaginary part of the impedance to the real part of the
impedance: for a capacitor C with series resistance R , Q = 1/(2πfCR ); for an induc-
s S
tor L with series resistance R , Q =2πfL/R [see Figure 7.1 (a, b)]. In both cases, a
s S