# A Novel Method for Modeling the Electromagnetic Characteristics of Switched Reluctance Motors
## 3 Modeling the Entire Flux-Linkage Characteristics
### Five-Order Fourier Series Flux-Linkage Model  
The stator phase inductance of the SRM can be modeled by the following Fourier series [22]:  
\[L(i,\theta)=\sum_{n=0}^{\infty}L_{n}(i)cos(nN_{r}\theta) \tag{7}\]  
By multiplying Equation (7) with the phase current \(i\), the flux-linkage expression can be derived as follows:  
\[\psi(i,\theta)=\sum_{n=0}^{\infty}iL_{n}(i)cos(nN_{r}\theta)=\sum_{n=0}^{ \infty}\lambda_{n}(i)cos(nN_{r}\theta) \tag{8}\]  
where \(\lambda_{n}(i)\) is the coefficient, which can be calculated based on the flux-linkage curves of different positions. Based on the five flux-linkage curves at specific positions, Equation (8) can be simplified as a five-order Fourier series:  
\[\psi(i,\theta)=\lambda_{0}(i)+\lambda_{1}(i)cos(N_{r}\theta)+\lambda_{2}(i) cos(2N_{r}\theta)+\lambda_{3}(i)cos(3N_{r}\theta)+\lambda_{4}(i)cos(4N_{r}\theta) \tag{9}\]  
Obviously, only to determine the Fourier series coefficients \(\lambda_{0}(i)\), \(\lambda_{1}(i)\), \(\lambda_{2}(i)\), \(\lambda_{3}(i)\) and \(\lambda_{4}(i)\), the entire flux-linkage characteristics of SRM can be obtained. Using (9), the flux-linkage curves of five special positions are described as follows:  
\[\Psi_{5\times 1}=M_{5\times 5}\Lambda_{5\times 1} \tag{10}\]  
Figure 6: Comparison of the estimated and measured flux-linkage curves at \(11.25^{\circ}\). where \(\Psi_{5\times 1}=[\psi_{0}(i)\;\psi_{7.5}(i)\;\psi_{11.25}(i)\;\psi_{15}(i)\;\psi_{22.5 }(i)]^{T}\), \(\Lambda_{5\times 1}=[\lambda_{0}(i)\;\lambda_{1}(i)\;\lambda_{2}(i)\;\lambda_{3}(i) \;\lambda_{4}(i)]^{T}\), \(M_{5\times 5}=:\)  
\[\left[ \begin{array}{cccc}1&1&1&1&1\\ 1&cos(\pi/3)&cos(2\pi/3)&cos(\pi)&cos(4\pi/3)\\ 1&cos(\pi/2)&cos(\pi)&cos(3\pi/2)&cos(2\pi)\\ 1&cos(2\pi/3)&cos(4\pi/3)&cos(2\pi)&cos(8\pi/3)\\ 1&cos(\pi)&cos(2\pi)&cos(3\pi)&cos(4\pi)\\ \end{array} \right].\]  
Meanwhile, a high order polynomial with a constant term of zero is used to fit the flux-linkage curves of the above five special positions:  
\[\Psi_{5\times 1}=A_{5\times 7}I_{\gamma\times 1} \tag{11}\]  
where \(I_{7\times 1}=[i\;i^{2}\;i^{3}\;i^{4}\;i^{5}\;i^{6}\;i^{7}]^{T}\) and the coefficient matrix \(A_{5\times 7}\) is determined by the least square method, as shown in Table 1.  
According to Equations (10) and (11), the Fourier series coefficient matrix can be calculated as follows:  
\[\Lambda_{5\times 1}=M_{5\times 5}^{-1}A_{5\times 7}I_{\gamma\times 1} \tag{12}\]  
To demonstrate the accuracy of the modeling, comparative studies are carried out between the proposed analytical method and the rotor clamping method. The comparison results of the characteristics between the flux-linkage with current and flux-linkage with rotor position are shown in Figure 7a,b, respectively. There was a good agreement between the calculated and measured flux-linkages. Figure 7: Entire flux-linkage characteristics. (**a**) Comparison of calculated and measured flux-current (\(\psi\) - \(i\)) characteristics; (**b**) comparison of calculated and measured flux-position (\(\psi\) - \(\theta\)) characteristics; (**c**) error between the calculated and measured flux-linkage; (**d**) root-mean-square error (RMSE).