IMPROVEMENT OF RECIPROCITY MEASUREMENTS IN ACOUSTICAL SOURCE STRENGTH

Transcription

1 The 21 st International Congress on Sound and Vibration July, 2014, Beijing/China IMPROVEMENT OF RECIPROCITY MEASUREMENTS IN ACOUSTICAL SOURCE STRENGTH Ling Lu, Hongling Sun, Ming Wu, Xiaobin Cheng, Jun Yang, Jing Tian Key laboratory of Noise and Vibration Research, Institute of Acoustics, Chinese Academy of Sciences, Beijing, China Reciprocity theory exists in stable, passive, linear dynamical systems, and has been applied to calibrate the microphones and hydrophones successfully. However, the accuracy of this method is doubtable, which limits its use as a widely-used engineering tool. Reciprocity theory has also been applied to measure the strength of sound source. However, compared with the results of the free field method and the reverberation method, the most differences are almost 5dB. This paper is focused on possible factors that affect the accuracy. It is found that the frequency responses of sound source at some frequencies have large peaks or troughs, just like impulses in the frequency domain. Besides, it is found that the impulses correspond to the coherence functions, and almost all impulses appear at the frequencies with low coherent value. The results of the reciprocity method agree with the results of the free field method very well after correcting the results by polynomial fitting. 1. Introduction As early as 1860, Helmholtz proposed the reciprocity principle in acoustical field. Thirteen years later Rayleigh proposed the theory of general reciprocity, which extended reciprocity to all linear passive stable dynamical systems [1-3]. Wolde is the pioneer applying the reciprocity principle. Between 1970 and 1988, he and his team had developed a lot of applications, one of which is measuring the volume velocity of a monopole sound source based on the electro-acoustical reciprocity. Reciprocity measurements have some advantages compared with the free field method and reverberation method. Firstly the reciprocity method is applicable in any surrounding which is reciprocal and suitable for in-situ source strength measurement. Secondly the method is suitable for low frequency test [1, 3]. These advantages are attractive. However, Wolde given only 1/3 octave bands experimental results. The differences between the results of the reciprocity method and the results of the free field method are quite large in some frequency bands and the largest difference is about 5dB [4]. This paper follows the work of Wolde and focuses on the factors which lead to the large difference. The factors mentioned by Wolde such as the sound source, the auxiliary loudspeaker, the microphone and nonlinearity are considered fully [4]. It is found there are a lot of impulses in the frequency domain of the volume velocity when the sound source is placed in a complex environment such as reverberant environment. Further, it is found the coherence function is related to the impulses. In fact, most of the impulses are caused by the valley of the transfer function, ICSV21, Beijing, China, July

2 which can t be measured accurately. The results agree with the results of the free field method very well after which are corrected by polynomial fitting. 2. Reciprocity principle and coherence In this section, the electro-acoustical reciprocity, the electro-acoustical-electro reciprocity and the definition of the coherence function will be introduced. Moreover the method for measuring the volume velocity of the monopole sound source based on the elector-acoustical reciprocity and elector-acoustical-elector reciprocity is discussed. 2.1 Electro-Acoustical reciprocity In the electro-acoustical reciprocity measurement, the electro-acoustical system contains four poles, as shown in Fig. 1. Q 1 and P 1 stand for the volume velocity and sound pressure of the acoustical port respectively. I2 and E 2 stand for the electrical current and open output voltage of the electric port respectively. The box stands for the entire system. Q1 I2 P1 E2 Position 1 Acoustical Port Position 2 Electrical Port Figure 1. An electro-acoustical system. If the system is reciprocal, the transfer function from position 1 to position 2 is equal to the transfer function from position 2 to position 1. So there is a relationship between the four poles: E2' P1'' Q1' I2 0 I2'' Q1 0 (1) The single dash ( ' ) in Eq. (1) indicates the reciprocal experiment, and the double dash ( '' ) indicates the direct experiment. From Eq. (1), it is easy to get: Q I '' 2 1' E2' (2) P1 '' According to Eq. (2) the source volume velocity can be measured. It should be pointed out that several conditions must be satisfied: the voltage E 2 ' must be measured when the system is open; the directionality of the microphone and the unknown source must be the same; both the auxiliary loudspeaker and the system are reciprocal. It s very easy to satisfy the first condition by cutting off the input. The second condition can also be satisfied. In order to check the last condition, it is necessary to use the electro-acoustical-electro reciprocity. ICSV21, Beijing, China, July

3 2.2 Electro-acoustical-electro reciprocity An electro-acoustic-electro reciprocity system consists of an electrical system, acoustic system and electrical system, which is similar to the electro-acoustical system. It is easy to get such a relationship if the system is reciprocal: I1' I2'' E2' I2 0 E1'' I1 0 (3) Form Eq. (3), it can be known that if the transfer function of direct and reciprocal experiments are equal, the system and the auxiliary loudspeaker are reciprocal. 2.3 Coherence The coherence between two signals is defined as: 2 Gxy ( f) = (4) G ( f) G ( f) xx yy where x() t and y() t are two signals in time domain; f is the frequency; Gxx ( f) and Gyy ( f) are the auto-spectral density of x() t and yt () respectively; Gxy ( f) is the cross-spectral density between x() t and y( t ). Assuming x( t) and y( t) are the input signal and output signal of a dynamical system, y( t) is completely related to x( t ) if =1. On the contrary, y( t) and x( t) are completely unrelated if =0. Therefore reflects the coherence relationship between the drive signal and response signal of the system. It should be pointed out that, the value of coherence not only connects with noise but also connects with the signal strength and the signal-to-noise ratio (SNR). The coherence value will decrease at the resonant frequency and the anti-resonant frequency when x( t ) is white noise signal. 3. Experimental set-up The reciprocity measurement system for sound source volume velocity consists of two parts: direct experiment and reciprocal experiment, which are shown in Fig. 1(a) and 1(b). In the direct experiment, an electrodynamic loudspeaker with a cavity is considered as the unknown sound source. In the frequency range from 160Hz to 1600Hz, the sound source is omnidirectional and could be regarded as a monopole. The measuring system consists of signal generator, signal collection module and signal post-processing module. It is based on B&K LAN-XI data acquisition hardware (type 3160) and PULSE platform. The auxiliary loudspeaker is electrical type, which is reciprocal theoretically [5] and it has been checked in an anechoic chamber. The power amplifier is used to amplify the input white noise signal for driving the unknown sound source. In the reciprocal experiment, a small resistance (0.61 ) is in series with the auxiliary loudspeaker for calculating the input current by measuring the voltage drop. The microphone is placed at the position of the acoustical center of the unknown sound source. The power amplifier is used for driving the auxiliary loudspeaker by amplifying the input white noise signal. The input ICSV21, Beijing, China, July

4 current of the auxiliary loudspeaker can neither be too large nor too small. Too large input current could cause nonlinearity, and too small one may lead to poor SNR. Figure 2. The measuring system of sound source volume velocity. The experiments have been performed in an anechoic chamber and a reverberation chamber. In the anechoic chamber, there are three different experiments. First, two auxiliary loudspeakers with different size are used to carry out the elector-acoustical-elector reciprocity experiment. The results agree with Eq. (3) very well. So, it is proved that the auxiliary loudspeaker is reciprocal. Second, the reciprocity measurement is carried out in order to test the system in ideal condition. The result is used as a reference. Third, the volume velocity of the unknown sound source is measured using the free field method in which the sound pressure is measured by a microphone in front of the sound source. The formula is: 2 ( ) Q( f)= p f r f (5) In Eq. (5), Q( f ) is the volume velocity, p( f ) is the sound pressure, r is the distance between the sound source and the microphone, and is the density of the air. The result of the free field method is also used as a reference. In the reverberation chamber, two experiments are carried out. The first one is the electoracoustical-elector reciprocity experiment. It is found from the experiment that there are impulses at the valley (the anti-resonant frequency) of the transfer function. It means that in the complex environment, the results of reciprocity experiment may be doubtable at the anti-resonant frequency. Second, the elector-acoustical reciprocity experiment is carried out to measure the volume velocity ICSV21, Beijing, China, July

5 of the unknown source. In order to distinguish the effect of the anti-resonant frequency, the reciprocity experiment is carried out three times in the reverberation chamber, and the position of the auxiliary loudspeaker is changed each time. A parameter Q is defined as: Q( f) Q( f) () I f (6) In Eq. (6), Q( f)and ()are I f the volume velocity and the driving current of the sound source (an electrodynamic loudspeaker with a cavity), respectively. Q( f ) is the intrinsic property of the electrodynamic loudspeaker in its linear range. The unit of Q( f ) is m 3 / sa, which means the volume velocity per unit current input. There are two advantages in introducing (). I f Firstly, there is no coherence function in the direct experiment if there is no (). I f On the other hand, some abnormal results could be distinguished by the coherence function of the direct experiment. Secondly, there is no need to measure the value of the resistance if the same resistance is used in the direct experiment and the reciprocal experiment. Therefore the tolerance of the resistance value could be ignored. 4. Experimental results In this section, the experimental results in the anechoic chamber and reverberation chamber are discussed. The parameter Q( f ) of the unknown source measured by the reciprocity measurement and the free field method is shown in Fig. 3. There are five curves in the figure. The first three curves are the results of reciprocity measurement in the reverberation chamber. Each time, the position of the auxiliary loudspeaker is changed. It can be found that the trends of the three curves are almost the same, but there are a lot of impulses. Moreover, the frequencies of those impulses are different in the three curves. The fourth curve is the result of reciprocity measurement in the anechoic chamber. It is smooth and agrees with the trend of the first three curves very well. The fifth curve is the result of the free field method in the anechoic chamber, and it s smooth too. This is not surprising because the frequency response curve of the electrodynamic loudspeaker is smooth in theory. ICSV21, Beijing, China, July

6 Q/dB(ref:1m 3 /sa) Reciprocity measurement in the reverbration chamber-test1 Reciprocity measurement in the reverbration chamber-test2 Reciprocity measurement in the reverbration chamber-test3 Reciprocity measurement in the anechoic chamber The free filed method in the anechoic chamber frequency(hz) Figure 3. Q of the unknown source measured by the reciprocity measurement and the free field method Further, it is found that the impulses are correlated with the coherence function. Almost all the impulses appear at the frequency with low coherence value (below 0.9) as shown in Fig Coherence of direct experiment Coherence of reciprocal experiment Normalized Q Frequency(Hz) Figure 4. The normalized Q and the coherence function of direct experiment and reciprocal experiment Besides, it s found that most of the impulses appear at the anti-resonant frequency. It s easy to know the reason why the impulses are correlated with the coherence function is that the responses ( E ' in the direct experiment and 2 P '' 1 in the reciprocal experiment) are so small at the anti-resonant frequency. So, the results at the frequency with poor coherence value are abnormal, and they should be corrected. There are many methods to achieve the goal, such as taking an average of the results of different experiments, getting rid of the results with low coherence value, polynomial fitting the results. The simplest method is polynomial fitting. The result modified by polynomial fitting is shown in Fig. 5. ICSV21, Beijing, China, July

7 Reciprocity measurement Polynomial fitting -40 Q/dB(ref:1m 3 /sa) frequency(hz) Figure 5. Comparison of the Q measured by reciprocity method in reverberation chamber and corrected by polynomial fitting In this way, the results of reciprocity measurement can avoid the impulse. The results of different reciprocity experiments modified by polynomial fitting are shown in Fig. 6. They agree with each other very well. It demonstrates the elector-acoustical reciprocity measurement system is stable. Besides, the difference between the free field method and the reciprocity measurement is quite small within the whole frequency band. The largest difference is no more than 3dB. It is not true to make a conclusion that the reciprocity method is not as good as the free field method, since the free field method needs a strict environment--an anechoic chamber. Q/dB(ref:1m 3 /sa) Reciprocity measurement in the reverbration chamber-test1 Reciprocity measurement in the reverbration chamber-test2 Reciprocity measurement in the reverbration chamber-test3 Reciprocity measurement in the anechoic chamber The free filed method in the anechoic chamber frequency(hz) Figure 6. The Q modified by polynomial fitting of different reciprocity experiments. 5. Conclusions In this paper, an electro-acoustical reciprocity measuring system is constructed. According to this system, the volume velocity per unit current input of an unknown electrodynamic loudspeaker has been measured in different environments. It s found the result in the anechoic chamber is ICSV21, Beijing, China, July

8 smooth, whereas there are lots of impulses in the results of the reverberation chamber and the impulses are correlated to low coherence value. The reason why the impulses are correlated with the coherence function is that the responses at the anti-resonant frequencies are so small that the coherence function is affected by the noise at these frequencies badly. The results of reciprocity measurement agree with each other very well after correcting the results by polynomial fitting. Moreover the largest difference is no more than 3dB compared with the result of the free field method. REFERENCES Wolde, T. T., Reciprocity Measurements in Acoustical and Mechano-acoustical Systems. Review of Theory and Applications, Acta. Acust. Acust., 96, 1-13, (2010). Wolde, T. T., Reciprocity experiments on the transmission of sound in ships. PhD Thesis, Delft University, (1973). Fahy, F. J., Some Applications of Reciprocity Principle in Experimental Vibroacoustics, Acoustical Physics, 49, , (2003). Wolde, T. T., Reciprocity Measurement of Acoustical Source Strength in an Arbitrary Surrounding, Noise Control Engineering, 7, 16-23, (1976). Ma, D. Y., The Foundation of Modern Acoustic Theory, Science Press, Beijing, (2004). ICSV21, Beijing, China, July