## 3 Simulation Results and Empirical Results Verifications
### Discussion of Empirical Results  
Figure 10 compares the displacements (direct solution) that were obtained after the integration of the linear parameters (traditional parameter measurement method [1; 3]) and the displacements measured using the capacitance displacement system and laser displacement measurement system. The simulation result is the direct solution that was obtained by substituting the nonlinear parameters identified by the proposed optimization method (Section 2.3). As seen from Figure 10, the laser displacement measurement system deviated from the direct solution, whereas the capacitance displacement system approached the direct solution. In this case, the displacement measurement errors caused by membrane vibration in a miniature transducer with a metal membrane during laser measurements may be due to optical refraction. Hence, it is established that the capacitance displacement system is more applicable and appropriate for the transmission equation calculations (Section 2.1) related to miniature transducers.  
Figure 10: Comparison of displacement data simulated by the model and displacement measured by using the capacitance and laser displacement measurement systems.  
Figure 9: Capacitance displacement measurement system procedure.  
Figure 11a-e compare the results of measurement by the capacitance system and laser displacement measurement system. The results of the proposed system were found to be different from those of the laser displacement measurement system, which could be due to the repeated fitting results of the latter. The changes in nonlinear parameters could be caused by the temperature and long operation of the miniature transducer. However, the impedance curve in Figure 12 shows that all the theoretical solutions fit resonance frequency and peak values. The impedance curve in Figure 12 shows that due to \(R_{m}(\mathbf{x})\) deviation, BFGS influenced the resonance frequency peak values of the impedance curve. As shown in Figure 13, all of the measurement curves that were fitted using different methods were good, indicating that the method proposed in this study had certain accuracy for the miniature transducer response \(R_{m}(\mathbf{v})\) measured by the laser displacement measurement system, which made it impossible to compare the \(R_{m}(\mathbf{x})\) solutions within the same figure. Figure 13 also indicates that the BFGS with constraint method gives more fit to the measurement result, which is probably due to the constraints of the parameters. Hence, BFGS with constraint method is a more realistic and stable method; however, the results of the three methods were similar. The illustration of nonlinear issues can refer to [5], the nonlinear curves as \(Bl(\mathbf{x})\), \(K_{m}(\mathbf{x})\), and \(R_{m}(\mathbf{x})\) in Figure 11 are almost symmetric. The surround that was made of the plastic and metal material causes a minor asymmetry. The \(L_{e}(\mathbf{x})\) remains the asymmetrical shape that is typical for a motor without any shorting material. The result is a benefit to addressing the problem and adjusting the design of miniature transducers to obtain the minimal distortions.  
Figure 11: _Cont._  
Figure 11: Comparison of nonlinear parameter curves obtained using different methods and laser displacement measurement system. (**a**) \(Bl(\mathbf{x})\), (**b**) \(K_{m}(\mathbf{x})\), (**c**) \(L_{e}(\mathbf{x})\). (Note: Laser dependent on right scale), (**d**) \(R_{m}(\mathbf{x})\), and (**e**) \(R_{m}(\mathbf{v})\). (Note: \(R_{m}(\mathbf{v})\) was used in measurement of \(R_{m}(\mathbf{x})\) using the laser displacement measurement system; thus, another comparison is made).  
Figure 12: Comparison of impedance curves obtained using different methods and the laser displacement measurement system.