Page 407 - Microsensors, MEMS and Smart Devices - Gardner Varadhan and Awadelkarim

APPLICATIONS 387

The transmission matrix T can be related to the incoming and outgoing waves as

(13 77)
-

Equation (13.77) can be further simplified using SAW amplitudes and the electrical output
from the IDT.
a i [ T i ] (13.78)

where a i is the input electrical signal at the ith plane and the acoustic submatrix can be
written as
, . , (13.79)
-t\2

and
( t 13 )
(13.80)
—(t 23)

Also, space between the IDTs and reflectors is represented by

[ eB d e 0 —iBd 1 e —iBd (13 81)
'
0 u
The total acoustic matrix [M] for a two-port resonator can be now obtained as

[
[M] = G l ][D 2 ][t3][D 4 ](t 5 ][D 6 ][G 7 ] (13.82)
Here, [G 1] and [G 7 ] are related to two SAW reflectors at the end and [D 2] and [D 6] are the
spacings between the gratings and adjacent IDTs. [D 4] is the separation between the IDTs.
[t 3] and [t 5] are the acoustic submatrices as shown in Equation (13.79). Equation (13.82)
can be solved for the frequency response of the resonator by applying the boundary
conditions.
w 0+ = w7-=0 (13.83)

which assumes that there is no external SAW at the input reference side.
It is also noted that the source and load impedances determine the electrical properties
of the transmission line. It can be seen from Figure 13.22 that

t
[W 2] = 3[W 3] + a 3T 3 (13.84)
Applying the foregoing boundary conditions for the transducer [t 3], the first-order response
of two-port resonator can be obtained by solving the following equations:

[W 5] = [D 6][G 7][W 7] (13.85)