Page 410 - Microsensors, MEMS and Smart Devices - Gardner Varadhan and Awadelkarim

390 IDT MICROSENSORS

0.6

0.4

^ 0.2
i
1
i» 0
'i
5
> –0.2

–0.4

-0.6
0.05 0.1 0.15 0.2
Time (us)

1.5

1.0-

c 5.0-

£ 0.0
Q.
23 0.05 0.15 02
1-5.0
-1.0-

-1.5
Time (us)

Figure 13.25 Computed velocity of a particle at the middle of the resonator at the resonant
frequency. The negative sign indicates the up and down movement of the particle at the antinode
with reference to the substrate plane

V = - (13.88)

I = v(f) (13.89)

where 0 is related to the turns ratio of an equivalent acoustic-to-electric transformer that
can be written in terms of its electromechanical coupling coefficient k, frequency f 0, and
the total capacitance C of the IDT as

2
0 = j2f 0Ck Apv (13.90)
Hence, the mechanical characteristic impedance Z m = F/v of the substrate can be ex-
pressed as an equivalent transmission line characteristic impedance Z 0. For a substrate of
density p and effective cross-sectional area A, the mechanical impedance for a propagating