# Numerical Study on the Phase Sensitivity Variation in Low Frequency Primary Microphone Calibrations
## 1 Introduction
The laser pistonphone technique is widely used in National Metrology Institutes (NMIs)--e.g., the National Physical Laboratory (NPL), the National Metrology Institute of Turkey (UME), the National Metrology Institute of Japan (NMII), the National Institute of Metrology, China (NIM), etc.--as a supplement to the reciprocity method to conduct primary microphone calibrations down to even 0.001 Hz [22; 23; 24; 25]. The laser pistonphone, as shown in Figure 1, produces a sinusoidal sound pressure in the chamber. The time relationship between the calibrating sound pressure and the microphone voltage output is the desired phase sensitivity of the microphone. For the primary calibration, the calibrating sound pressure in the chamber is given by [25]  
\[\underline{p}=p_{\mathrm{L}}p_{\mathrm{L}}p_{\mathrm{N}c}p_{\mathrm{w}c}, \tag{1}\]  
in which,  
\[p_{\mathrm{L}}=\kappa P_{0}\Delta V/V_{0}, \tag{2}\]  
\[p_{\mathrm{L}c}=1/(1+1/j\omega T_{a}), \tag{3}\]  
\[\underline{p}_{\mathrm{L}c}=\Big{(}1+(\kappa-1)\underline{E}_{pV}\Big{)}/\kappa, \tag{4}\]  
where \(p_{\mathrm{L}}\) is the lumped pistonphone model, \(\kappa\) is the specific heat ratio, \(P_{0}\) is the static pressure, \(\triangle V\) is the volume variation in the chamber, and \(V_{0}\) is the chamber volume. \(p_{\mathrm{L}}\), \(p_{\mathrm{N}c}\), and \(p_{\mathrm{w}c}\) are corrections  
Figure 1: (**a**) The calibration characteristics of the laser pistonphone technique. (**b**) Prototype of a laser pistonphone. for the pressure leakage, heat conduction, and wave motion effects [20; 21; 22], in which \(\omega\) is the angular frequency, \(T_{\mathrm{a}}\) is the leakage time constant of the pistonphone, \(\underline{E}_{\mathrm{PV}}\) is its temperature transfer function, and \(p_{\mathrm{wc}}\) (wave motion correction) is quite small and can be omitted. It should particularly be noted that the pressure leakage and heat conduction corrections \(\underline{p}_{\mathrm{lc}}\) and \(\underline{p}_{\mathrm{hc}}\) are both complex quantities, which means that the calibrating sound pressure is not synchronized with the piston displacement excitation, and so the phase difference occurs between them. Let us further analyze the calibration process as shown in Figure 1a. The microphone is essentially a parallel-plate capacitor enclosed by a chamber [26]. When the calibrating sound pressure excites the diaphragm, an associating sound pressure will be generated in the back chamber. This sound pressure will also experience pressure leakage and heat conduction effects [27], and the diaphragm deformation of the microphone is then determined by these two sound pressures and its elasticity. Due to the above amplitude attenuation and phase deviation of the generated sound pressures, the microphone voltage output (equivalent to the diaphragm deformation) will not be synchronized with the calibrating sound pressure, and so its phase sensitivity will vary at low frequencies. However, the above considerations are only based on qualitative analysis, and the mechanism and characteristics of the variation phenomena are hard to imagine. In this paper, by taking advantage of the FEA (Finite Element Analysis) technique the independent phase responses of sound pressures both in the pistonphone and the microphone are simulated considering the influence of multi-physical factors. The phase sensitivities of the microphone when its vent is connected to the calibrating sound field or the outside atmosphere are obtained based on a proposed joint simulation method. Both frequency and time domain simulations are conducted to quantify the variation value of the phase sensitivity and reveal its variation mechanism. The results verified severe phase sensitivity variations at low frequencies, which are greatly influenced by the microphone geometries and the venting state during calibrations. It is therefore recommended to obtain the primary phase sensitivity of the microphone before serving.