非線形最適化手法による位相反転形スピーカシステムの設計

抄録

In this paper a new design method of a phase inverter loudspeaker system, which uses for the first time the non-linear optimization method, is discussed. From circuit analysis of Fig. 1 (b), the sound pressure transfer function of a phase inverter loudspeaker system can be obtained as shown in Eq. (1) and its frequency response function G(X, ω) is given in Eq. (3). The frequency response of G(X, ω) is controlled by the system parameters vector X(Eq. (2)), whose elements X_i, i=1, 2, ・・・, 8, are composed of r_0, s_0, m_0, r_B, s_B, r_P, s_P and m_P, respectively, in Fig. 1(b). In order to realize flat response and at the same time extension of the low frequency response, we define an evaluation f(Eq. (4)). The value of f decrease as G(X, αω_i) approaches to the target-level G_i, where α is a positive parameter and has a function of extending the low frequency response, and where ω_i is the freqency point of target. By minimization (optimization) of f we can obtain X which realizes flat response and extension of the low frequency response. In Fig. 2 one point of evaluation is considered to explain the process of extending the low frequency response. Point A on the initial response reaches point B whose level is close to the target-level G_1 after k_1-th trial of minimization. Then the value of α is changed from the initial value α_0 into α_1(α_1&ltα_0), and the evaluation point moves to point C from point B. Again point C reaches point D whose level is close to G_1 after k_2-th trial of minimization. The value of α is again changed from α_1 into α_2(α_2&ltα_1), and the evaluation point moves to point E from point D. The value of element X_i has the upper and lower limits within which X_i varies, or a specified value. This constraint on the value of X_i is decided by a designer's intension, and makes the process of minimization complicated. Therefore X_i is transformed into Y_i which has no constraint as shown in Eq. (6) and Fig. 3, and f is minimized, by the Davidon-Fletcher-Powell method, with respect to Y whose element is Y_i. An example of the minimization process is shown in Fig. 4 for two variables. When Y realizing minimization of f is obtained, Y is transformed back into X in Eq. (6). The flow diagram of this design method is shown in Fig. 5. A model of evaluation function vs. the number of trial times of minimization is shown Fig. 6. In this paper three design examples are given. The constraint and the frequency response of target for the design examples are given in Eqs. (19) and (20), respectively. Design example 1 is for a drone cone type system in the case when all elements of X are variables (Figs. 7, 8 and Table 1). Design example 2 is for the drone cone type system in the case when the volume of box is specified (Fig. 9, Table 2). Design example 3 is for a port type system in the case when the driver speaker is specified (Fig. 10, Table 3). This design method makes it possible to realize flat response and at the same time extend the low frequency response under the constraints.