1972 年 28 巻 10 号 p. 553-564
In the measurement of a piezoelectric vibrator having a small figure of merit by the resonance-antiresonance method, the high minimum-to-maximum-admittance-ratio is obstructive to an accurate determination of the electromechanical constants. In the present paper, various approximate formulae of correction for the ratio are derived and examined. The present treatment is always based on the through-the-center approximation, as described by Martin. The constants to be determined are K, γ andω_0 in eq. (2). When the values of M and Q(or K andγ)are given, the squares of the reduced frequencies z_m and z_n are determined by solving eq. (9)and the value of r is exactly evaluated. In paragraph 2. 1, z_<m, n> in eq. (11)are used, whereγ^2/K is ignored in comparison with γ^2/K^2 in(10)given by Martin, and thus the approximations (I. 1)〜(I. 4)are obtained. From the graphical treatment on the basis of Fig. 1 are derived the formulae (II. 1)〜(II. 4). The derivation is understood by following the sequence from(13)to(22)and seems most faithful to the through-the-center approximation. Ignoring the term δ_m in(II. 1)and(II. 4)provides the approximation III. Using the frequencies(25)instead of (11), the other sets of formulae can be derived, but are not described here. The accuracies of the present approximate formulae are examined in §3, compared with those given by Martin. In 3. 2, Δ_<mn> and r are calculated using the approximate formulae for M and Q given, and they are compared with the exact solution on the Δ-r chart after Martin, as shown in Fig. 2. Such indications, however, are not suited for examining the accuracy of the approximation. The fractional errors are estimated by(29)for each approximation and the distributions are shown on the Δ-r diagram, as in Fig. 3. The approximation III seems to be the best one. Actually, the squared frequency difference δ_<mn> is not an observable. Substituting (I. 3) for δ_m in(6), we have α_1 in (30), which is measurable and can be used instead of δ_<mn> in the case I. Similarly α_2 in(33)is obtained in the cases II and III. Uually, we often use β or 2Δ_<mn> in place of δ_<mn>. The errors for IIIα, IIIβ and IIIΔ are shown in Fig. 4. When no corrections are taken into account in respect of r, as described in (0β. 1)or(0Δ. 1), the errors for K are not ignorable, as shown in Fig. 6, whereas that of K_<IIIα> is negligible. The errors of the present approximations always increase in the upper left range of Δ-r chart. These defects are not dissolved within the through-the-center approximation. In §4, another approximation is considered rather in an empirical way. Examining the plot of the fractional errors of IIIα versus Δ_<mn> in Fig. 5, new parameters ξ and ζ are introduced for the readjustment errors, as in eqs. (34)〜(37). The errors of IIIα are plotted against ξ or ζ, and these plots suggest a way of decreasing the errors. Thus the improved formulae(*. 1)〜(*. 3)are found. The errors are within 3, 3 and 1% for K^*, γ^* and δ_m^* respectively. Shibayama early reported a graphical method of evaluating small coupling factor. The present approximation covers the entire range of his chart with sufficient accuracy. The proposition of the two sets of the approximate formulae, IIIα and *, is the purpose of the present paper.