# Numerical Study on the Phase Sensitivity Variation in Low Frequency Primary Microphone Calibrations
## 1 Introduction  
In the areas of sound source localization, photo-acoustic imaging, and sound intensity measurement, the phase characteristics of microphones are as important as their modulus characteristics [1; 2; 3; 4]. Nowadays, the study of low-frequency sound and infrasound is developing rapidly, as promoted by the monitoring requirements for a wide variety of large-scale natural and artificial activities--e.g., earthquakes, volcanos, tsunamis, nuclear explosions, supersonic jets, rocket launchings, duct transportation, and industrial production [5; 6; 7; 8]. All the above studies require simultaneous measurement using the microphone monitoring system; by resolving the time differences in acoustic signals from different microphones, the spatial localization and separation are determined. Apparently, the phase characteristics of microphones are key to low-frequency sound and infrasound measurements [9]. The techniques used for calibrating the phase response of the microphone mainly include the in situ method, the loudspeaker or exciter-based calibrator, the wave guide method, the newly proposed pressure gauge comparison method, the reciprocity method, and the laser pistonphone method. The in situ method is widely used to get the phase consistency of microphones in the array [10; 11]. However, the calibration results are only applicable to the current system layout. Once the installation position is changed or the microphone is replaced, a recalibration is required. Moreover, due to the free field requirement and the wind noise in the infrasound measurement, the use of the in situ method in the infrasound range is quite limited. When multiple microphones are connected to the loudspeaker or exciter-based calibrator, their phase consistency can be calibrated in the frequencyrange of 0.01 to 100 Hz [12; 13; 14; 15]. Y.C. Yang of the Institute of Acoustics, Chinese Academy of Sciences, has proposed a calibration system to calibrate the designed infrasound microphone down to 0.001 Hz. The microphones are arranged along a circumference and connected to the sound source by pipes of equal length so that their phase consistency is obtained [16]. More importantly, F. Larsonner et al. studied the phase sensitivities of MB 2005 and MB3 microbarometers in the frequency range of 0.02 to 20 Hz and reported the phase leading phenomenon [15]. The wave guide method and the pressure gauge comparison method require reference sensors to be the comparison standard [17; 18]. Depending on that, the calibrating sound pressure in the above methods is obtained experimentally from a reference sensor but not theoretically, and thus these methods all belong to secondary calibration techniques. The above infrasound calibration studies mainly focus on getting the phase consistency of infrasound sensors based on the comparative (secondary) calibrations, but the primary calibration technique is little studied. The reciprocity method as standardized in IEC (International Electrotechnical Commission) 61094-2 [19] is the most widely accepted primary calibration technique. R. Nel et al. studied the modulus and phase sensitivities of microphones based on the reciprocity method in the frequency range of 1 to 100 Hz and found that the low frequency phase sensitivity is influenced more by the coupler and microphone geometries [20]. R. Jackett et al. determined the phase sensitivity uncertainty and found that the low-frequency uncertainty caused by the pressure leakage effect cannot be ignored [21]. However, due to the great signal attenuation and the model inapplicability, this method is not applicable to the wide infrasound frequency range below 1 Hz.