Page 405 - Microsensors, MEMS and Smart Devices - Gardner Varadhan and Awadelkarim

APPLICATIONS 385

between the electric and acoustic parameters, and D matrix represents the acoustic space
between the IDTs and the reflectors. The boundary conditions are applied assuming that
there are no incoming waves from outside the reflectors and output impedances are
matched at the electrical terminal. The frequency response can be computed from trans-
mission line matrices for appropriate sections. The amplitudes of the waves are represented
by scalar electric potential 3>, which can be written in terms of SAW velocity v(x) as

(13 63)
. 2 2 ^ I ^ v V A — • "-^ /
dx |_u (x)
where w is the radiation frequency. The wave velocity v(x) is perturbed sinusoidally about
while passing through the metallic strips and can be written as
V 0
Au
v(x) = V 0 cos(Ax) (13.64)

where A is equal to 2n/L, and L is the period.
The solution of Equation (13.63) gives a pair of coupled-wave equations that can be
represented as forward and backward SAW waves with amplitudes R and S.

—R—j8R = jkS (13.65)
—S—j8S = jkR (13.66)

The grating matrix, G, for an array can be written as shown in the following equation
(Campbell 1998). The equation for this matrix and that for T are provided in Appendix J
together with a list of symbols. We repeat the equations here for the sake of clarity.

a /<$ — ja \ 1
T + tanh(crL)
k ) J
= C
<* — • 7~ ,
c>
tanh(crL)
(13.67)
where k is the coupling coefficient, a is the attenuation constant, L is the period, B 0 is
the unperturbed phase constant, and

. k
2
a = Jk — (8 —ja) 2 and C = - cosh(<rL)
v
n
Similarly, the transmission matrix T, which relates the electrical and acoustic parameters
of IDTs, as shown in Figure 13.22, can be obtained from the knowledge of the scattering
matrix of an IDT (Cross and Schmidt 1977).

\t rl
(13.68)
=
U '33 J