電気ひずみ磁器における非線形電界及び分極過程の有限要素解析

概要

論文の詳細を見る
We have devised bolted Langevin-type torsional vibrators for ultrasonic power applications, and also reported conformal mapping analyses on the distribution of equipotential lines and e. m. f. of a ceramic ring under polarization. These analyses are, however, based on a linearized assumption of D-E relation in the ceramic and the effect of adjacent electrodes is neglected. The present paper reports about a finite element approach for nonlinear problems. We have at first derived the expressions necessary for FEM in the nonlinear electrostatic field problems, secondly devised a method to reduce the instability that might occur in Newton-Raphson procedure, and finally obtained the following results after iterative numerical calculations: 1) Maximum electric field (electric field concentration) does not increase monotonously with increase of the external field (cf. Figs. 8 and 10). 2) Almost all parts of the ceramic can be effectively utilized by the "two-round polarization method" that we have developed. As is clearly seen from Fig. 1 or Fig. 3, this allocation of electrodes will provide nonuniform electric field distribution in the ceramic. If the dielectric constant of the ceramic was determined only spatially, which was independent of the electric field (the D-E curve was assumed linear within the triangular elements), the electric potential distribution would be obtained simply by solving simultaneous equations (4); the D-E relation is, however, usually nonlinear as shown in Fig. 4 so that consideration of the nonlinearity is necessary. Silvester et. al. employed Newton-Raphson iteration in the analysis on saturable magnetic field problems. So far as we know, such a method has not yet been applied to electrostatic problems. At first, we have derived the expressions necessary for the finite element method in nonlinear electrostatic problems as their counterpart. The D-E curve is assumed isotropic but nonlinear as shown in Fig. 4. In the nonlinear case, the most likely solution is the one which minimizes the potential energy of the system, too. In order to obtain such a solution, the Newton-Raphson iteration method is used. Results are shown in eq. (8) and eqs. from (13) to (16). In many cases of the electrostatic problems, the stability of Newton-Raphson iteration is more severe than for the magnetic counterpart because of its nature of the D-E relation as seen from Fig. 2. Then we devised a method to eliminate the instability that might occur in the process using relaxation technique to find an optimum acceleration factor ω in eq. (6) for which the potential energy is minimized. Such ω that satisfies the condition is given in eqs. (23) and (24). Figs. 5 and 6 show the calculated equipotential lines of the nonlinear electrostatic field shown in Fig. 3. Figures from 7 to 10 show the residual polarization and the maximum electric field against the applied electric field. Their special features are that the maximum electric field in the ceramic does not increase monotonously with increase of the external field as mentioned above. The regions face to the electrodes do not contribute to the torsional torque because the residual polarization there is not circumferentially directed. This dead-space sometimes amounts up to 30[%] of the whole ceramic. So we devised "Two-Round Polarization Method" making use of the nonlinear nature of the ceramic, which enables to highly utilize almost all parts of the ceramic (Fig. 12). Fig. 15 shows a small amplitude free admittance locus of a tested vibrator with the ceramic rings processed by this method (Fig. 14), for which the torque factor A=11. 7×10^<-3> [Newton meter/Volt]. If the ceramic rings are ideally circumferentially polarized, the torque factor will be 12. 5×10^<-3>, which means that the measured value amounts to about 94[%] of the ideal one, while a conventional "One-Round Polarization Method" will give 8. 1×10^<-3>, that is, 65[%] of the ideal value.
社団法人日本音響学会の論文
1976-02-01

電気ひずみ磁器における非線形電界及び分極過程の有限要素解析

スポンサーリンク

概要

著者

関連論文

スポンサーリンク