Resonant frequency temperature coefficient is dependent on material properties, resonator dimensions and vibration mode. It could be effectively tuned by the resonator dimensions or by the domain structure in ferroelectric crystals. Optimum dimensions for the zero temperature coefficient resonator are calculated for ring radial vibration mode as a function of resonator dimensions (r 2 /r 1 =3.8 for hard PZT ceramics, APC841 type). There are similar results of the temperature coefficient calculations for PZT ceramics and crystal resonators. The temperature coefficient is generally smaller for higher overtones of resonant mode.

Electromechanical properties have been measured by a resonant method for ''very soft'' (APC856), ''soft'' (APC850), and ''hard'' (APC840, APC841, and APC880) lead zirconate-lead titanate (PZT) ceramics as a function of poling temperature and poling electric field. Quantitative results are presented for piezoelectric coefficients d 33 , d 31 , k 31 , k p , and k t ; for elastic stiffness c E 33 , elastic compliance s E 11 , and dielectric permittivity " T 33 . An elevated temperature has a significant effect on the poling efficiency for hard PZT ceramics -in particular, piezoelectric coefficients could be substantially (by 50% of their saturated value) increased by the appropriate choice of poling temperature. A similar effect is observed for the poling electric field for hard PZTs at lower temperatures, but the electric field intensity is less significant at higher poling temperatures. The longitudinal and transversal components of material tensors change in different extent as a result of poling.

Bar-shaped piezoelectric ceramic transformers (PTs) working in the longitudinal vibration mode (k31 mode) were studied. Two types of the transformer were designed--one with the electrode divided into two segments of different length, and one with the electrodes divided into three symmetrical segments. Parameters of studied transformers such as efficiency, transformation ratio, and input and output impedances were measured. An analytical model was developed for PT parameter calculation for both two- and three-segment PTs. Neither type of bar PT exhibited very high efficiency (maximum 72% for three-segment PT design) at a relatively high transformation ratio (it is 4 for two-segment PT and 2 for three-segment PT at the fundamental resonance mode). The optimum resistive loads were 20 and 10 kΩ for two- and three-segment PT designs for the fundamental resonance, respectively, and about one order of magnitude smaller for the higher overtone (i.e., 2 kΩ and 500 Ω, respectively). The no-load transformation ratio was less than 27 (maximum for two-segment electrode PT design). The optimum input electrode aspect ratios (0.48 for three-segment PT and 0.63 for two-segment PT) were calculated numerically under no-load conditions.

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