# Analysis and Identification of Nonlinear Acoustic Damping in Miniature Loudspeakers
## 2 Miniature Loudspeaker Model Considering Nonlinear Acoustic Damping
### Theoretical Analysis of Nonlinear Acoustic Damping in Miniature Loudspeakers  
When a miniature loudspeaker works under a large signal condition, the air velocity through the orifice on the front cover is sufficiently large. It is well-known that the acoustic damping of an orifice becomes nonlinear with high-amplitude air velocity. Flow separation occurs near the orifice, resulting in nonlinear acoustic damping in a miniature loudspeaker.  
Figure 1: Cross-sectional view of a moving-coil miniature loudspeaker.  
As depicted in Figure 2, the deviation in the nonlinear acoustic damping can be obtained by applying the conservation of momentum to a control volume around the orifice of the miniature loudspeaker [7; 9]:  
\[\rho_{0}S_{0}l_{eq}\frac{dv_{0}}{dt}+\rho_{0}(v_{l}^{2}S_{l}-v_{0}^{2}S_{0})+ \tau_{\omega}S_{\omega}=(p_{0}-p_{l})S_{0} \tag{1}\]  
where \(\rho_{0}\) denotes the air density, \(l_{eq}\) represents the equivalent length of the orifice, and \(\tau_{\omega}\) is the frictional wall shear stress distributed over the orifice-wetted area \(S_{\omega}\); \(v_{0}\) and \(P_{0}\) denote the air velocity and the sound pressure at the starting surface of the control volume, respectively; and \(v_{l}\) and \(P_{l}\) denote the air velocity and the sound pressure at the ending surface of the control volume, respectively. The terms in Equation (1) describe the inertia force, momentum convection, momentum loss caused by the sheer force, and pressure forced on the control volume, respectively. The air velocity through the starting and ending surfaces can be related using the discharge coefficient \(C_{D}\) with \(v_{l}=v_{0}/C_{D}\). The discharge coefficient represents the average volume flow rate entering/exiting the control volume [9]. Another relationship between \(v_{0}\) and \(v_{l}\) can be obtained from the conservation of mass within the control volume, \(v_{0}S_{0}=v_{l}S_{l}\). By dividing both sides of Equation (1) by \(S_{0}\), the equation can be rewritten as:  
\[\Delta p=\rho_{0}l_{eq}\ \frac{dv_{0}}{dt}+R_{l}v_{0}S_{0}+\rho_{0}\frac{1-C_{D} }{C_{D}}v_{0}\bigg{|}v_{0}\bigg{|} \tag{2}\]  
where \(R_{l}=\frac{8\pi\mu l_{eq}}{S_{0}^{2}}\) denotes the equivalent linear acoustic damping coefficient caused by \(\tau_{\omega}S_{\omega}\) with \(\mu\) being the viscosity coefficient of air [18]. The absolute value of \(v_{0}\) is used to ensure that Equation (2) is applicable to both the in-flow and out-flow half cycles.  
From Equation (2), the nonlinear acoustic damping in miniature loudspeakers can be obtained as:  
\[R_{a}=R_{l}+\rho_{0}\frac{1-C_{D}}{C_{D}}\frac{|v_{0}|}{S_{0}} \tag{3}\]  
which clearly shows that the acoustic damping of an orifice contains a linear part engendered by viscous losses and a nonlinear part resulting from flow separation. Notably, the discharge coefficient \(C_{D}\) is known to be sensitive to the shape of the orifice. In this study, \(C_{D}\approx 0.68\) was used in the theoretical calculations with consideration of the squared-edge orifice. Some details about the discharge coefficients are discussed in Section 4.