## 3 Equivalent Circuit Model of the N-TWSRIM  
The equivalent circuit is the main method used to study the electromagnetic theory, working characteristics, and parameter design of SRIM. The impedance of the equivalent circuit parameters of SRIM is critical to the design calculations for the motor, whose ECM is shown in Figure 5. The equivalent circuit can be used to represent the voltage, current, and energy conversion relationship between the stator and the rotor. In this model, \(\mathrm{U_{1}}\) is the input phase voltage, \(\mathrm{I_{1}}\) is the stator current, \(\mathrm{R_{1}}\) is the stator winding resistance, \(\mathrm{X_{1}}\) is the stator reactance, \(\mathrm{E}\) is the electromotive force, \(\mathrm{X_{mg}}\) is the excitation reactance, \(\mathrm{I_{r}}\) is the rotor current, \(\mathrm{R_{cu}}\) is the equivalent resistance of the copper layer, \(\mathrm{R_{steel}}\) is the equivalent resistance of the steel layer, \(\mathrm{X_{cu}}\) is the equivalent reactance of the copper layer, \(\mathrm{X_{steel}}\) is the equivalent reactance of the copper layer, \(\mathrm{R_{r}}\) is the rotor equivalent resistance, and \(Z_{r}\) is the rotor equivalent reactance. Figure 4: Air-gap flux density and magnetic field distribution at different times: (**a**) 0\({}^{\circ}\) times, (**b**) 30\({}^{\circ}\) times, (**c**) 60\({}^{\circ}\) times, (**d**) 75\({}^{\circ}\) times, (**e**) 90\({}^{\circ}\) times. Figure 5: Equivalent circuit model of N-TWSRIM. In order to analyze the complex eddy current field distribution in a solid rotor and calculate the rotor equivalent impedance, the following basic assumptions are used:  
1. The Carter coefficient \(\mathrm{K_{c}}\) is used to consider the stator core slotting effect. 2. The motor is deployed in the circumferential direction and the stator and rotor extend infinitely along the axial direction. 3. The harmonic components in the field are not considered and the displacement current is ignored. 4. The stator winding current is equivalent to the current sheet at the interface of the air gap and stator. The current layer is infinitely long along the circumference but the current layer thickness is ignored. 5. The end effect is expressed as end effect coefficient \(\mathrm{K_{o}}\). On the basis of the traditional multilayer model [20], Figure 6 shows the composite multilayer model of the N-TWSRIM, in which the rotor is divided into \(N\) layered regions. The \(1\sim N-2\) layers represent the steel layer, the s \(N\) layer represents the air gap, and the \(N-1\) layer represents the copper layer. In general, the more the number of layers is divided, the higher the corresponding calculation accuracy. \(\sigma_{i}\) is the conductivity of each layer and \(\mu_{i}\) is the permeability of the \(i\) layer. X is the circumferential direction, Y is the radial direction, and Z is the axial direction. The stator winding current is equivalent to the current sheet at the interface of the stator and air gap, as shown in Figure 6. The internal surface current \(J\) of the stator is represented by Equation (1). \[J=J_{0}e^{i(\omega_{1}t-ax)} \tag{1}\]  
where \(J_{0}\) is the magnitude of the stator interface current density, \(a=\pi/\tau\), \(\tau\) is the pole pitch, and \(\omega_{1}\) is the angular frequency. \(J_{0}\) can be expressed as  
\[J_{0}=\frac{2\sqrt{2}m_{1}K_{dp}W}{\pi D_{11}}I_{1} \tag{2}\]  
where \(m_{1}\) is the stator phase number, \(K_{dp}W\) is the number of effective conductors in series per phase, \(I_{1}\) is the current of the stator phase, and \(D_{\mathrm{i1}}\) is the diameter of the stator inner. According to this assumption, the electromagnetic field relationship can be expressed as  
\[\left\{ \begin{array}{c}\dot{B}=B_{x}\dot{e}_{x}+B_{y}\dot{e}_{y}\\ \dot{E}=E_{z}\dot{e}_{z}\end{array} \right. \tag{3}\]  
where \(\dot{B}\) is the magnetic density, \(\dot{H}\) is the magnetic field intensity, and \(\dot{e}_{x},\dot{e}_{y},\dot{e}_{z}\) represent the unit vectors in the X, Y, and Z directions, respectively.