## 3 Equivalent Circuit Parameters Calculation (Rotary Part)
### Rotor Parameters Calculation  
To simplify the calculation, the following assumptions are proposed:  
1. The motor is supplied with balanced three-phase sinusoidal currents. The air gap flux density is constant in the axial direction. 2. The material primary core is isotropic, the permeability is infinite, and the conductivity is equal to zero. 3. The hysteresis, end, and saturation effects are ignored. The magnetic leakage in the rotor slots is neglected. Figure 4: Simplified circuit diagram of SSR. Figure 5: Single phase equivalent circuit of the rotary part. Each field includes a fundamental component, and the time curves of each field are sinusoidal. The rotor is symmetrical in structure, and the air gap magnetic field is sinusoidal in the circumferential direction. The amplitudes of currents in the two adjacent axial slots are equal, and the electric angles differ by \(\alpha_{2}\) in phase,  
\[\alpha_{2}=\frac{p\times 360^{\circ}}{Q_{2}} \tag{2}\]  
\(Q_{2}\) is the number of axis slots under the magnetic field. By combining [PERSON]'s current law and Figure 4, the relationship between axial and circumferential currents is \(\dot{I}_{A1}=\dot{I}_{C11}-\dot{I}_{C12}\), \(\dot{I}_{A2}=\dot{I}_{C12}-\dot{I}_{C13}\), \(\cdots\), and Figure 6 shows the vector diagram of currents [13; 14]. The relationship between axial current \(I_{A}\) and circumferential current \(I_{C}\) is as follows  
\[I_{A}=2I_{C}\sin(\frac{\alpha_{2}}{2}) \tag{3}\]  
Thus, the total losses in one axial slot and the corresponding front and back end slots are calculated by  
\[P_{Ctu(A+C)}=I_{A}^{2}R_{2A}+2I_{C}^{2}R_{C}=I_{A}^{2}\Bigg{(}R_{2A}+\frac{R_{ C}}{2\sin^{2}\frac{\alpha_{2}}{2}}\Bigg{)}=I_{A}^{2}R_{2} \tag{4}\]  
where \(R_{2A}=nR_{A}\) is the sum of axial resistance. \(R_{2(A+C)}\) is the equivalent axial resistance, when the front and back end circumferential slots are incorporated in axial slots. \[R_{2(A+C)}=\Bigg{[}nR_{A}+\frac{R_{C}}{2\sin^{2}(\alpha_{2}/2)}\Bigg{]} \tag{5}\]  
The equivalent axial reactance is calculated as follows:  
\[X_{2(A+C)}=\Bigg{[}nX_{A}+\frac{X_{C}}{2\sin^{2}(\alpha_{2}/2)}\Bigg{]} \tag{6}\]  
The expressions of the resistance and reactance of each slot are as follows:  
\[\left. \begin{array}{c}R_{2}=\frac{\rho L}{A}=\frac{\rho L}{d\varpi}\\ X_{2}=2\pi f\mu_{0}\frac{N^{2}}{pq}L\sum\lambda\end{array} \right\} \tag{7}\]  
where \(\rho\) is the resistivity and \(A\) is the sectional area of each slot. The \(\mu_{0}=4\pi\times 10^{-7}H/m\) indicates the air permeability. \(N\) is the winding turns. The \(q\) represents the number of slots per pole per phase. \(\sum\lambda\) is the sum of each slot leakage permeance. Figure 6: Vector diagram of currents. For cage rotors, the rotor phase number \(m_{2}=Q_{A}\), \(Q_{A}\) refers to the rotor axial slot number, the rotor turns \(N_{2}=1/2\), \(K_{dp2}=1\) represents the rotor fundamental winding and conversion coefficients  
\[K=\frac{m_{1}}{m_{2}}(\frac{N_{1}K_{dp1}}{N_{2}K_{dp2}})^{2}=\frac{4m_{1}(N_{1}K _{dp1})^{2}}{Q_{A}} \tag{8}\]  
where \(m_{1}\) is the stator phase, \(N_{1}\) denotes the number of stator conductors in the series per phase, and \(k_{dp1}\) stands for the stator fundamental winding coefficient. For conventional motors, the expressions of the equivalent rotor resistance and reactance are as follows,  
\[ \begin{array}{l}R^{\prime}_{2s}=\frac{K}{s}R_{2}=\frac{4m_{1}(N_{1}K_{dp1})^{ 2}}{8Q_{A}}R_{2}\\ X^{\prime}_{2s}=KX_{2}=\frac{4m_{1}(N_{1}K_{dp1})^{2}}{Q_{A}}X_{2}\end{array}  \tag{9}\]  
The ECM of the motor with a-parallel windings is as shown in Figure 7.  
The expressions of the equivalent rotor resistance and reactance in each branch are  
\[ \begin{array}{l}R^{\prime}_{2 sn}=aR^{\prime}_{2s}=\frac{aK}{s}R_{2}\\ X^{\prime}_{2 sn}=aX^{\prime}_{2s}=aKX_{2}\end{array}  \tag{10}\]  
where \(1\leq n\leq a\).