# Numerical Study on the Phase Sensitivity Variation in Low Frequency Primary Microphone Calibrations
## 2 Independent Pistonphone Simulation  
### Model Description  
A membrane sealing structure-based pistonphone is designed, as shown in Figure 2[28]. The cavity is sealed by a silicone membrane, and the piston drives the membrane to vibrate. The cavity wall is made of a seamless steel tube, thus satisfying the rigid wall condition, and produces stable heat conduction. A tiny equalizing hole is opened at the end cover to achieve a static pressure equilibrium. The lower limiting frequency of the pistonphone is about 0.01 Hz and can be further extended by replacing the equalizing hole. All the specific parameters are given in Table 1.  
Figure 2: Schematic diagram of the pistonphone chamber. In the calibration process, except at constant amplitude and zero initial phase with frequency-varying piston displacement excitation, there are pressure leakage, heat conduction, and wave motion effects that affect the phase response between the calibrating sound pressure and the piston displacement. Therefore, four models corresponding to the heat conduction and sealing condition, adiabatic and sealing condition, adiabatic and leaking condition, and heat conduction and leaking condition are respectively established in COMSOL 5.3, as shown in Figure 3. The thermoviscous acoustics modules of both the frequency domain and the transient analysis are used to respectively get the frequency response of the phase difference and the waveform difference between the calibrating sound pressure and the piston displacement [29]. It can be calculated that the temperature change caused by the generated sound pressure is less than 0.2 \({}^{\circ}\)C. Thus, only the effective area of the air in the pistonphone is modeled as shown in Figure 4. Five boundary layers are refined at the surrounding walls to simulate the heat conduction and viscosity loss, and the other remaining area is freely divided into tetrahedral meshes. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Symbol** & **Value** \\ \hline Chamber radius & \(R_{1}\) & 65 mm \\ Chamber length & \(L_{1}\) & 393 mm \\ Chamber volume & \(V_{0}\) & \(5.3\times 10^{6}\) mm\({}^{3}\) \\ Diaphrag radius & \(R_{2}\) & 35 mm \\ Sealing seat length & \(L_{2}\) & 24.2 mm \\ Piston radius & \(R_{3}\) & 20 mm \\ Piston internal length & \(L_{3}\) & 6 mm \\ Piston displacement & \(\xi_{\text{pa}}\) & 5 mm \\ Equalizing hole radius & \(R_{4}\) & 0.205 mm \\ Equalizing hole length & \(L_{4}\) & 20 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of the laser pistonphone. Figure 4: **(a)** The FEA model of the pistonphone. (**b**) Sound pressure distribution for the heat conduction and leaking condition at 0.1 Hz. Figure 3: Flow chart of the pistonphone simulation. For two models considering the pressure leakage effect, the outer end of the equalizing hole is connected to the atmosphere and the boundary condition is set as \(p_{\text{end}}=0\) Pa. Meanwhile, for two sealing models the outer end is closed. For the other aspect, for two models considering the heat conduction effect, the surrounding walls are set as isothermal--that is, the temperature boundary \(T_{t}\) is always 0. Meanwhile, for two adiabatic models the chamber walls are set as adiabatic--that is, the temperature gradient \(-\mathbf{n}\cdot(-k\
abla T_{t})=0\), where \(\mathbf{n}\) is the normal vector. At the initial moment, the piston stays at its equilibrium position and outputs a sinusoidal vibration. The amplitude of the piston displacement is selected to be 5 mm, corresponding to a 150 Pa calibrating sound pressure at ideal conditions. In the frequency domain module of COMSOL 5.3, only the amplitude of piston velocity and not the displacement can be applied, and the sine function is set in its background. However, the piston velocity follows the cosine law, and thus will cause severe high-order noise at the time \(t=0^{+}\).