## 2 Theoretical Model of an SL  
The layout of a typical SL is shown in Figure 1 and the technical date of the loudspeaker used in the experiment are listed in Table 1. The SL with an NIC is assembled at the end of the impedance tube, and the effective absorption can be achieved after reasonable adjustment of the electrical parameters. From an energy perspective, it can be understood that the sound energy is dissipated in the form of mechanical and electrical energy, reducing the reflected sound energy and achieving the purpose of sound absorption. When the SL is in an open-circuit state, it can be considered a single-degree-of-freedom, second-order system, which consists of stiffness, mass and damping [21]. The mechanical impedance under the case of an air cavity of depth \(D\) can be expressed as:  
\[Z_{m}=\delta_{m}+j\omega M_{m}+\frac{1}{j\omega K_{m}}+\frac{\rho_{0}c_{0}A}{i \tan(kD)} \tag{1}\]  
where \(\omega\) is the angular frequency, \(c_{0}\) is the speed of sound in air, \(Mm,Km,\delta m\) are the moving mass, suspension diaphragm stiffness and damping of the moving-coil loudspeaker, respectively. \(Z_{air}=\rho_{0}c_{0}A\) is the acoustic impedance of air, and A is the cross-section area of the impedance tube. \(k\) is the wavenumber and \(D\) is the cavity depth, where a more specific impedance expression can be obtained after making a second-order approximation to \(\tan\left(kD\right)^{-1}\)[22]:  
\[Z_{m}(\omega)=\delta_{m}+j\omega(M_{m}+\frac{\rho_{0}AD}{3})+(K_{m}+\frac{ \rho_{0}c_{0}^{2}A}{D})/j\omega \tag{2}\]  
From Equation (2), it can be seen in the acoustic model of the SL with an air cavity that the total stiffness is the sum of the suspension diaphragm stiffness and the air spring of the air cavity, where the second item dominates. For example, a cylindrical cavity with a depth of 10 cm and radius of 5 cm can produce a stiffness of 11 KN/m, while a four-inch loudspeaker's diaphragm stiffness is generally 1 KN/m. The total vibrating mass is the sum of the mechanical vibrating mass and one-third of the cavity air mass; the latter is usually negligible. An NIC can generate the equivalent value of a negative electrical parameter between the in-phase input and ground [23]. It can flexibly adjust the impedance caused by a larger resistance and inductance of the loudspeaker itself, enabling impedance of the SL to match with air over a wider frequency band. When connecting the SL with an NIC, the impedance of the circuit is: \begin{table}
\begin{tabular}{c c c c c c} \hline \(BI\) & \(R_{\epsilon}\) & \(L_{\epsilon}\) & \(M_{\text{m}}\) & \(K_{\text{m}}\) & \(\delta_{\text{m}}\) \\ \hline
3.1 Tm & 6.5 \(\Omega\) & 0.5 mH & 4.8 g & 926 N/m & 1.74 sN/m \\ \hline \end{tabular}
\end{table} 
Table 1: Technical data for the loudspeaker Hivi–M4N. Figure 1: Schematic of the shunted loudspeaker with an NIC. \[Z_{e}(\omega)=(R_{c}-R)+j\omega(L_{c}-L)+\frac{1}{j\omega C} \tag{3}\]  
The following is a derivation of the electrical force and impedance analogy. When the sound waves are transmitted to the loudspeaker's diaphragm, it will produce a vibration with speed of \(v\). The loudspeaker's coil will cut the magnetic field of the permanent magnet, producing an induced electrical potential \(Blv\). As the induced current is \(Blv/Z_{e}\), the electromagnetic force applied to the coil is \(F_{\rm e}=B^{2}l^{2}v/Z_{e}\). The equivalent mechanical impedance induced by the circuit can be obtained by \(Z_{\Delta m}(\omega)=F_{\rm e}/v=(Bl)^{2}/Z_{e}(\omega)\). Here, the total impedance of the SL can be expressed as:  
\[Z_{sys}(\omega)=Z_{m}(\omega)+(Bl)^{2}/Z_{e}(\omega) \tag{4}\]  
The normal incident absorption coefficient of the SL is:  
\[\alpha(\omega)=1-|\frac{Z_{sys}-Z_{air}}{Z_{sys}+Z_{air}}|^{2}=\frac{4Z_{air} \text{Re}(Z_{sys})}{\left[Z_{air}+\text{Re}(Z_{sys})\right]^{2}+\text{Im}(Z_{ sys})^{2}} \tag{5}\]  
Equation (5) shows that the sound absorption of the SL depends on the acoustic impedance of the system.