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368 IDT MICROSENSORS
with different amplitudes B and angular frequencies. The frequencies and phases of these
two signals depend on the delay times. The two signals can be separated in the frequency
domain. Because (wo is usually much greater than , the phase shift is more sensitive to
the variation of the delay time than the frequency.
The difference of the two phases can be written as
<P = (<Pi- <P2) = N) - M('i + '2)/2][/ 2 - ti] (13.16)
where the extra delay time of the second echo compared with the first is equal to the
round trip time of the acoustic wave traveling from the first reflector to the second and
is given by
T = (t 2-ti) = — (13.17)
v
where d is the distance between the two reflectors.
The phase difference is sensitive to the change in the delay time. The variation of the
phase difference because of the change in the delay time is expressed as
and Ar = 2AJ/U (13.18)
Because COQ is usually much larger than the other term, the equation reduces to
(13.19)
The wave traveling time T is proportional to the distance between the two reflectors and
is inversely proportional to the velocity. If the possible velocity variation of the SAW
under stress is neglected and we take into account only the direct effect on the distance,
we have
2sd
(13.20)
where e is the mechanical strain (Ad/W) and r 0 is the traveling time when the strain
is zero.
Thus, the sensitivity of this wireless sensor system depends on the operating frequency
and the round traveling time of the SAW between the two reflectors.
13.4.1.1 Sensor dimensions and reading system
A strain sensor was developed by Varadan et al. (1997), wherein YZ-cut lithium niobate
(LiNbO 3) crystal of dimensions 3.7 mm x 6.7 mm x 0.5 mm was used. An IDT was
designed with a central frequency of 912.5 MHz. The IDT is connected directly to a
small antenna of dimensions 25 mm x 75 mm x 0.3 mm. The distance between the two
reflectors was 0.15 mm and the round trip traveling time of the surface waves was 0.1 us
at zero stress. The wireless reading system had a patch antenna to send out FM microwave
signals and to receive the echoes coming from the reflectors through the antenna-IDT.
The transmitted signal was linearly modulated from 905 to 920 MHz in a pulse lasting
1760 th of a second. The resolution of the phase detector of the system is 1°. According