AMATCHING ACOUSTIC PORT PORT 2 ACOUSTIC PORT 1 Z。k PORT 2 ACOUSTIC BEAM ELECTROD ACTIVE REGION FIGURE 48.2 Prototype transducer geometry. FIGURE 48.3 Model of active region. in shape. Connections to these electrodes form the electrical port for the transducer and the voltage between them creates a spatially uniform electric field in the active region, and this time-varying electric field couples to the acoustic waves propagating between the electrodes. If the planar electrodes are many wavelengths in ansverse dimensions and the active region is much thinner, and if the axial direction is a pure mode direction for the piezoelectric, the waves in the active region can be considered as plane waves. We then have the one- dimensional geometry considered earlier. The transducer may be in contact with another elastic medium on either side, as indicated in Fig. 48. 2, so that the plane waves propagate in and out of the active regions in the cross-sectional region shown. Thus, the transducer has in general two acoustic ports for coupling to the outside world as well as the electrical port. In the absence of piezoelectric coupling, the active region could be represented by a one-dimensional transmission line as discussed in the previous section and as indicated by the heavy lines in Fig. 48.3. with zoelectricity there will be the stiffening of the appropriate stiffness constants as discussed in Eq (48.9)with the concomitant perturbation of the characteristic impedance Zo and the phase velocity Vp but more important there will also be coupling to the electrical port. One model including the latter coupling is shown in Fig. 48.3 in which the parameters are defined by sin(to/Oo) 2e/∈sin(o/200) (48.13) OAZO Here Co is the capacity that would be measured between the electrodes if there were no mechanical strain on he piezoelectric, A is the cross-sectional area of the active region, and X is an effective reactance. The quantity r is the transformer ratio(with dimensions)of an ideal transformer coupling the electrical port to the center of the acoustic transmission line. K is the electromechanical coupling constant for the material as defined in Eq(48.9). The so-called resonant frequency oo is that angular frequency at which the length d of the active region is one-half of the stiffened wavelength, wo=VId. In the physical configuration of Fig. 48. 4(a), the transducer has zero stress on the surfaces of the active region and hence both acoustic ports of Fig 48.3 are terminated in short circuits and the line is mechanically resonant at the angular frequency oo At this frequency the secondary of the transformer of Fig 48.3 is open circuited if there are no losses, and thus the electrical input impedance is infinite at this frequency and behaves like a parallel resonant circuit for neighboring frequencies. This configuration can be used as a high-Q resonant circuit if the mechanical losses can be kept low, as they are in single crystals of such piezoelectric materials as quartz. It should be noted, however, that the behavior is not as simple as that of a simple L-C parallel resonant circuit, primarily because of the frequency dependence of the effective reactance X and of the transformer ratio in the equivalent circuit. The electrical input impedance is given by oC /1-k2 tan kd/2 (48.14) kd/2 c 2000 by CRC Press LLC© 2000 by CRC Press LLC in shape. Connections to these electrodes form the electrical port for the transducer and the voltage between them creates a spatially uniform electric field in the active region, and this time-varying electric field couples to the acoustic waves propagating between the electrodes. If the planar electrodes are many wavelengths in transverse dimensions and the active region is much thinner, and if the axial direction is a pure mode direction for the piezoelectric, the waves in the active region can be considered as plane waves. We then have the one￾dimensional geometry considered earlier. The transducer may be in contact with another elastic medium on either side, as indicated in Fig. 48.2, so that the plane waves propagate in and out of the active regions in the cross-sectional region shown. Thus, the transducer has in general two acoustic ports for coupling to the outside world as well as the electrical port. In the absence of piezoelectric coupling, the active region could be represented by a one-dimensional transmission line as discussed in the previous section and as indicated by the heavy lines in Fig. 48.3. With piezoelectricity there will be the stiffening of the appropriate stiffness constants as discussed in Eq. (48.9) with the concomitant perturbation of the characteristic impedance Z0p and the phase velocity Vp, but more important there will also be coupling to the electrical port. One model including the latter coupling is shown in Fig. 48.3 in which the parameters are defined by (48.13) Here C0 is the capacity that would be measured between the electrodes if there were no mechanical strain on the piezoelectric, A is the cross-sectional area of the active region, and X is an effective reactance. The quantity r is the transformer ratio (with dimensions) of an ideal transformer coupling the electrical port to the center of the acoustic transmission line. K is the electromechanical coupling constant for the material as defined in Eq. (48.9). The so-called resonant frequency v0 is that angular frequency at which the length d of the active region is one-half of the stiffened wavelength, v0 = pV/d. In the physical configuration of Fig. 48.4(a), the transducer has zero stress on the surfaces of the active region and hence both acoustic ports of Fig. 48.3 are terminated in short circuits and the line is mechanically resonant at the angular frequency v0. At this frequency the secondary of the transformer of Fig. 48.3 is open circuited if there are no losses, and thus the electrical input impedance is infinite at this frequency and behaves like a parallel resonant circuit for neighboring frequencies. This configuration can be used as a high-Q resonant circuit if the mechanical losses can be kept low, as they are in single crystals of such piezoelectric materials as quartz. It should be noted, however, that the behavior is not as simple as that of a simple L-C parallel resonant circuit, primarily because of the frequency dependence of the effective reactance X and of the transformer ratio in the equivalent circuit. The electrical input impedance is given by (48.14) FIGURE 48.2 Prototype transducer geometry. FIGURE 48.3 Model of active region. C A d jX j C K r e AZ 0 0 2 0 0 0 0 2 = = = 2 e e ; sin( / ) / / sin( / ) w pw w pw w w pw w Z j C K kd kd in = Ê Ë Á ˆ ¯ ˜ 1 1 2 2 0 2 w – tan / /