# A Multi-Tone Sound Absorber Based on an Array of Shunted Loudspeakers
## 4 Experimental  
The experimental setup in an acoustic impedance duct is shown in Figure 5, where the array of four shunted loudspeakers is placed at the right end. The cross section of the duct is about 0.40 m \(\times\) 0.40 m and the cut-off frequency is 430 Hz. A source loudspeaker is located at the left end of the duct (It is about 7 m away from the shouted loudspeaker array and is not seen in Figure 5) and driven through the amplifier. The DC voltage source is used to supply the power for the negative impedance converters in the shunt circuits. The sound absorption coefficient was measured using the two-microphone transfer function method according to ISO 10534-2 with a B&K PULSE 3560B analyzer [20]. The parameters to set up are as follows: the sampling rate is 25.6 kHz, and the frequency resolution is set as 1 Hz; the microphone spacing is 0.35 m; the distance from the right microphone to the sample is 1.32 m; the distance from the left microphone to the source is 3.64 m; a random signal is set as the input to the source with the voltage level of 0.07 Vrms to ensure that the signal to noise ratio is above 10 dB.  
An optimal system was designed based on the analysis in Section 3. The Thiele-Small parameters and dimensions of each loudspeaker are listed in Tables 3-6, where the cavity depth is optimized at 7.5 cm. However, the mechanical resistance \(R_{\mathrm{ms}}\) is not equal to the optimal value \(\rho_{0}c_{0}{S_{0}}^{2}/S\) for the maximal sound absorption peak, because the number of the loudspeaker samples we have is limited. The shunt circuit configuration for the four loudspeakers is listed in Table 7. Theoretically, the resonance of each shunted loudspeaker unit could be adjusted to any frequency. However, considering the nonlinearity of the loudspeaker [21], a more accurate formulation of the impedance \(Z_{\mathrm{e}}\) is \(B^{2}l^{2}/S_{0}/[R_{\mathrm{E}}+\mathrm{j}\omega L_{\mathrm{E}}+1/(1/R_{2}+ 1/L_{2})+Z_{\mathrm{s}}]\), where \(R_{2}\) is the electrical resistance due to the eddy current losses and \(L_{2}\) is the para-inductance of the voice coil. Therefore, the adjustment of the resonance frequency of the shunted loudspeaker has constraints. In practical, the inductance element in the shunt circuit generates parasitic resistance. By using the ohmmeter to measure the value, negative resistance is employed to cancel the parasitic resistance.  
Figure 5: Measurement setup of the shunted loudspeaker array.  
Take the shunt circuit parameters in Table 7 and the TS parameters and dimensions in Tables 3-6 into Equation (1), the specific acoustic impedances of each shunted loudspeaker are then calculated. It should be noted that the measured sound absorption is contributed to by both the loudspeaker diaphragm and the front surface of the closed-box, because the diaphragm area is smaller than the cross section of the duct. Therefore, the impedance \(Z_{\text{i}}\) is corrected by the area ratio factor \(a^{2}/4S_{0}\). Substituting the corrected impedance \(Z_{\text{i}}(a^{2}/4S_{0})\) into Equation (5), the sound absorption coefficient can be calculated according to Equation (15) and shown as the dashed line in Figure 6.  
0.80, and 0.84 respectively, with the resonance peaks being at 107 Hz, 208 Hz, 302 Hz, and 395 Hz respectively. The frequency shift of the absorption peaks between the simulation and the experiment results is not more than 11 Hz, which may be caused by the measurement precision of the loudspeaker TS parameters and the electronic components in the shunt circuit in the experiments. The differences of the peak values of the sound absorption coefficient between experimental and simulation results may be due to extra parasitic resistances of the electronic components.