周辺電極を有する薄円板圧電性磁器振動子の解析

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The piezoelectric ceramic vibrator of thin disc type with surrounding electrodes on both surfaces as shown in Fig. 1 presents a unique phenomenon that the first higher order vibration in the radial direction can be excited more strongly than the fundamental vibration. The fundamental equations of stress components and electric flux density for the symmetrically extended vibration of the above vibrator are expressed by Eq. (1) in cylindrical coordinates. In order to carry out the analytical calculation, the differences of sound velocities in the parts of ceramics with and without electrodes were considered, as Tachibana had already tried in the analysis of ceramic vibrator with dot-shaped electrodes. Young's modulus and Poison's ratio in the part with electrodes are denoted by Y_0 and σ_0, and those in the part without electrodes by Y and σ, respectively. By considering the boundary conditions that (1) displacement component [U_r]_<r=0> = 0, (2) stress component [T_<rr>]_<r=a> = 0 and (3) U_r and T_<rr> are continuous at r=b, the constants A, B and D are determined by Eq. (15), (16) and (17). Two-terminal admittance of the vibrator is calculated by Eq. (23). Furthermore, resonant frequency fR, antiresonant frequency f_A and Δf/f_R are expressed by Eqs. (25), (27) and (29), respectively. Fig. 2 shows the equivalent circuit of the vibrator in the neighborhood of resonant frequency. Equivalent constants C_0, L_1 and C_1 are calculated by Eqs. (30), (31) and (33), respectively. In order to ascertain whether or not the above calculated results coincide with the measured results, experiments were made for PZT disc ceramics by varying the diameter of surrounding electrodes. σ_0 was 0. 295 which was measured for the unsymmetrical vibration. σ was 0. 36 which was determined by extrapolating the curve of the ratio of the fundamental resonant frequency to the first higher order resonant frequency vs. p=b/a to p=1 (which corresponds to the part without electrodes). As the ratio of sound velocities λ a value of 0. 982 was used which was calculated by Eq. (34). The electromechanical coupling coefficient k_p of the used specimens was about 28. 15%. Measured results of resonant and antiresonant frequencies for several specimens are shown in Table 3. Fig. 3 shows the roots of resonant frequency equation H(R, λ, p, σ_0, σ)=O (Eq. (24)) as a function of p by varying the parameter λ under the condition of σ_0=σ=0. 295. Fig. 4 gives the calculated curves of X_R vs. p and X_A vs. p where X_R and X_A are the roots of Eq. (24), and Eq. (26), respectively. Calculated and measured results of resonant frequency are shown in Fig. 5 as a function of p. In the same way, antiresonant frequency, Δf/f_R, equivalent inductance and capacitance are shown in Figs. 6, 7, 8 and 10, respectively. From these results it was concluded that the calculated values coincide with the measured ones considerably well. From the relation of Eq. (32), it is understood that the equivalent inductance L_1 is, proportional to the thickness of the ceramic disc (t). Calculated and measured results of L_1 are shown in Fig. 9 as a function of t. Though the gradient of the measured L_l vs. t curves coincided with the calculated one, fairly large differences was found in the absolute values. Measured results of equivalent resistance at resonant frequency for various surrounding electrodes are shown in Table 4. As mentioned in the first place of this paper, this result shows that the first higher order vibration can be excited more strongly compared with the fundamental vibration. Consequently, ceramic disc vibrators of several MHz class can be obtained easily by applying the above-mentioned fact. Such vibrators are interesting from the view point of practical applications for filter element and so on.
社団法人日本音響学会の論文
1969-11-10

周辺電極を有する薄円板圧電性磁器振動子の解析

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