Page 397 - Microsensors, MEMS and Smart Devices - Gardner Varadhan and Awadelkarim

APPLICATIONS 377

where
0(t) = (w 0 + ut)t+0 0 (13.49)
Consider a simplified SAW device with only two reflectors. The echo from the first (S\)
input to the mixer is the same as the original FM signal, but with a time delay t\ and a
different amplitude

(13.50)
t
0 1 (0 = [w 0 + u(t — 1 )/2](t - t 1) + 0 0 (13.51)

The IF corresponding to S 1 (t) is expressed as
2
I 1(0 = B 1 cos[ut 1t + w 0t 1 - ut 1/ 2] (13.52)
2
Both the frequency ut 1 and the phase shift <p1 = (w nt 1 — ut 1 )/2 depend on the time
delay t 1 . Once again, because w n is usually much greater than ut 1 , the phase shift is more
sensitive than the variation of the frequency.
The total delay t\ depends not only on the travel time of the surface wave that is a
function of the physical parameter being sensed but also the RF propagation time, which
is a function of separation between the sensor and the reader. To eliminate this error, a
second reflector is used that has a different time delay t 2 and results in the following IF.

I 2 (t) = B 2 cos[ut 2t + w 0t 2 —ut 2 /2] (13.53)

The difference between the two phase shifts can be written as

= [w 0 — u /2(t 2 + t 1)](t 2 — t 1) = KT (13.54)
<p d
where
K = w 0— u/2(t 2 + t 1)~ W 0 (13.55)

because W 0 » U / 2(T 2 + t 1) for the values of the present system.

Now r = T 2 - t 1 = 2d /v (13.56)

where r is the total travel time of the surface wave from the first reflector to the second
and back, d is their physical separation, and v is the SAW velocity.
For the design of the humidity sensor, the difference in delay between two reflectors
(r) is 1.40 us. The differential phase shift can be written as

(13.57)
v

The change in the differential phase shift due to a change in humidity is then

(13.58)
u1 U0 V0