1973 Volume 29 Issue 2 Pages 108-114
In a previous report on the same subject, some correction formulas were proposed to determine the electromechanical constants of piezoelectric vibrator with a small coupling factor or a low Q by the resonance-antiresonance method. The formulas for K=C/C_0, γ=1/Q and δ_m=(ω_m/ω_0)^2-1 are given in terms of w=ω_n/ω_m and r=Y_n/Y_m. However, any influence of the external circuit was not considered there. In the present paper, the effect of the external circuit is included by introducing a resistance R_0 in series with the vibrator, as shown in Fig. 1 and Eqs. (1) and (2). The equivalent circuit of the total impedance can be represented by the circuit 1/Y' shown in Fig. 2 (c), where the components are frequency-dependent and indicated in Eqs. (5)〜(7). The effect of the additive R_0 is taken into account by e in (8) and further approximation of neglecting higher order terms with respect to e is made. Then the circuit components are given by (16). The piezoelectric admittance can be differently expressed by Eqs. (17) and (18), which show that the phase angle φ does not vanish at ω_0. As both R'' and φ depend on ω, the through-the-center approximation requires that the admittance Y_m' or Y_n' goes through the center of the respective circle indicating the motional admittance Y_p' at ω_m or ω_n. These situations are shown in Fig. 3, where the notations are given by (20). A geometrical configuration gives us the relations (22), (23) and (26). Following the equations from (24) to (34) successively, we have the approximate formula (35) for δ_m. Rewriting (28), (36) and eliminating A, the formula for γ is given as (38). Lastly, the formula for K is obtained from (36) and shown in (40). The expressions thus determined involve δ_<mn>, an unobservable quantity. We can use α determined by (42) instead of δ_<mn>α is determined by the measurement. Then we have the approximate formulas for K, γ and δ_m, as collected in (IVα. 1)〜(IVα. 3), when δ_m in (40) and the terms involving F and G in (40) and (38) are ignored. Though the quantity e is not observable, the ratio of the current corresponding to Y_m' to the short-circuit current, or κ in (44), gives the value of e by (45). The introduction of κ brings about another set of formulas (Vα. 1)〜(Vα. 3). The fractional errors of the formulas are examined in a similar way as in the previous paper. The results are shown in Fig. 4, where the errors are plotted against e for several combinations of different M and Q. They give an evidence for the applicability of the approximate formulas IVα and Vα. The proposition of these formulas is the purpose of the present paper.
