## 2 Mathematical Model  
### Transmission Equation  
The suspension system of a miniature transducer comprises a surround and a diaphragm. The behaviors of these two parts are described by the mechanical mass, stiffness, and damping. The nonlinear vibration characteristics of the suspension and the magnetic system occur with the instantaneous excursion of voice-coil displacement from the mean position. To represent this nonlinear vibration characteristics in modeling, the nonlinear parameter \(K_{m}(\mathbf{x})\) as a form of the non-linear mechanical stiffness is used to describe the nonlinear vibration characteristics [8]. The parameters \(K_{m}(\mathbf{x})\) and \(R_{m}(\mathbf{x})\) indicate that the change in the value of the mechanical stiffness depends on the displacement \(\mathbf{x}\), while \(L_{e}(\mathbf{x})\) represents the voice-coil inductance of the magnetic system, and \(Bl(\mathbf{x})\) represents the nonlinearity in the energy transfer between the electrical and mechanical domains. Figure 1a shows a schematic representation of the miniature transducer under large-signal conditions:  
Figure 1: Schematic representations of (**a**) A miniature transducer. (**b**) Diagram showing and key parameters, and (**b**) Diagram of the equivalent circuit.  
The equivalent circuit of an electrical domain (Figure 1b) can be derived based on [PERSON]'s voltage law (Equation (1)):  
\[L_{e}(\mathbf{x})\frac{d\mathbf{i}(t)}{dt}+\left[\frac{dL_{e}(x)}{d\mathbf{x}} \frac{d\mathbf{x}(t)}{dt}+R_{e}\right]\mathbf{i}(t)+Bl(\mathbf{x})\frac{d \mathbf{x}(t)}{dt}=\mathbf{e}(t) \tag{1}\]  
where \(\mathbf{e}(t)\) is the input voltage; \(\mathbf{i}(t)\) is the voice coil current; and, \(\mathbf{x}(t)\) is the voice coil displacement. Nonlinear parameters include voice coil _inductance_\(L_{e}(\mathbf{x})\) and the force factor \(Bl(\mathbf{x})\). With regard to the mechanical domain, [PERSON]'s second law of motion can be used to derive the current, as shown in the following equation:  
\[M_{m}\frac{d^{2}\mathbf{x}(t)}{dt^{2}}+R_{m}(\mathbf{x})\frac{d\mathbf{x}(t)}{ dt}+K_{m}(\mathbf{x})\mathbf{x}(t)=\left(Bl(\mathbf{x})-\frac{dL_{e}(\mathbf{x})}{d \mathbf{x}}\frac{\mathbf{i}(t)}{2}\right)\mathbf{i}(t) \tag{2}\]  
where \(M_{m}\) is the mass of transducer diaphragm; \(R_{m}(\mathbf{x})\) is the damping coefficient of the suspension system; and, \(K_{m}(\mathbf{x})\) is the nonlinear coefficient of the stiffness of the suspension system. All nonlinear parameters, including \(K_{m}(\mathbf{x})\), \(R_{m}(\mathbf{x})\), \(Bl(\mathbf{x})\), and \(L_{e}(\mathbf{x})\), are related to displacement \(\mathbf{x}\). In the air load system model, the mass \(M_{m}\) in Equation (2) includes the mechanical mass of the diaphragm and voice coil. The interaction of air load mass \(M_{rad}\) and resistance \(R_{rad}\) in acoustic radiations must be also considered [1]. Therefore, the air load in the mechanical and acoustic fields is given, as follows:  
\[\left(Bl(\mathbf{x})+\frac{dL_{e}(\mathbf{x})}{d\mathbf{x}}\frac{\mathbf{i}(t )}{2}\right)i(t)=(M_{m}+M_{rad}S_{d})\ddot{\mathbf{x}}(t)+R_{m}(\mathbf{x}) \dot{\mathbf{x}}(t)+K_{m}(\mathbf{x})\mathbf{x}(t) \tag{3}\]  
\[M_{rad}\mathbf{u}(\mathbf{t})=R_{rad}[S_{d}\dot{\mathbf{x}}(t)-\mathbf{u}( \mathbf{t})] \tag{4}\]  
Under the initial conditions:  
\[\mathbf{x}(0)=0,\ \mathbf{i}(0)=0,\ u(0)=0,\ \text{and}\ \frac{d\mathbf{x}(t)}{dt} \bigg{|}_{t=0}=0 \tag{5}\]  
where \(S_{d}\) is the diaphragm area. The \(\mathbf{u}(t)\) is the shunt volume velocity related to the acoustic radiation mass \(M_{rad}\) and the acoustic radiation resistance \(R_{rad}\) in Equation (4).