# An NUTSF at Sub-Region for Suppression of Torque Ripple in Switched Reluctance Motors
## 2 Mathematical Model of Switched Reluctance Motor  
Due to the special doubly salient structure of an SRM, the nonlinear factors, such as high saturation of the magnetic circuit and eddy current, will affect the performance of the SRM when its motor operates. If all the nonlinear factors are considered, the relevant calculation will be very complicated, and it is difficult to establish a unified and simple mathematical model for the SRM. The SRM can be regarded as the combination of a mechanical and an electrical port. The electromechanical energy conversion formulas between the two ports include mainly a voltage formula, mechanical formula and torque formula. The phase voltage formula can be expressed as follows  
\[U_{k}=R_{k}i_{k}-e_{k}=R_{k}i_{k}+\frac{d\varphi_{k}}{dt} \tag{1}\]  
where \(U_{k},R_{k},i_{k},e_{k}\) and \(\varphi_{k}\) are the voltage, resistance, current, induction electromotive force and flux linkage of the \(k\)-th phase windings, respectively. The mechanical formula can be expressed as follows  
\[T_{e}=J\frac{d^{2}\theta}{dt}+D\frac{d\theta}{dt}+T_{L} \tag{2}\]  
where \(T_{e}\), \(J\), \(D\) and \(T_{L}\) are the electromagnetic torque, rotary inertia, rubbing factor and load torque, respectively. The torque formula can be expressed as follows  
\[T_{x}=\frac{\partial W^{\prime}}{\partial\theta}\Big{|}_{i=const}=-\frac{ \partial W}{\partial\theta}\Big{|}_{\varphi=const} \tag{3}\]  
where \(T_{x}\), \(W^{\prime}\) and \(W\) are the instantaneous electromagnetic torque, magnetic co-energy and the stored energy, respectively. The SRM has strong nonlinear characteristics, so it is difficult to analyze and calculate the magnetic co-energy and stored energy, which makes it difficult to build an accurate and practical mathematical model of an SRM. This paper adopts a lookup table method to establish an SRM mathematical model in order to simulate and test the model more accurately and reliably. In order to build an SRM mathematical model, the data of torque \(T(i,\theta)\) and flux linkage \(\varphi(i,\theta)\) are obtained by a simulation calculation for a 6/20 SRM in finite element software (ANSYS). Figure 1 shows the structure of a 6/20 motor and the data of \(T(i,\theta)\) and \(\varphi(i,\theta)\). According to Formula (1), Formula (2) and the data of \(T(i,\theta)\) and \(\varphi(i,\theta)\) obtained from finite element analysis, the simulation model is built, as shown in Figure 2.  
The first part in Figure 2 is constructed according to the voltage Formula (1). Firstly, the phase voltage \(U\) is subtracted by the winding voltage \(iR\) to obtain the change rate of flux linkage \(d\varphi/d\theta\), which is integrated to obtain the flux linkage \(\varphi\) afterwards. Secondly, the phase current \(i(\varphi,\theta)\) is obtained by calculating the reverse interpolation for the lookup table \(\varphi(i,\theta)\) where the flux linkage is simulated by finite element analysis. The flux linkage  
Figure 1: Structure of 6/20 SRM and its simulation results: (**a**) structure; (**b**) torque \(T(i,\theta)\); (**c**) flux linkage \(\varphi(i,\theta)\). \(\varphi\) and rotor position angle \(\theta\) are used as inputs of the lookup table \(i(\varphi,\theta)\), and the current \(i\) is output. The second part in Figure 2 is constructed according to the mechanical Formula (2). The electromagnetic torque \(T_{e}\) of each phase winding is obtained using the lookup table \(T(i,\theta)\) where the phase current \(i\) and rotor position angle \(\theta\) are used as inputs and the \(T(i,\theta)\) is simulated by finite element analysis. The angular velocity \(w\) or \((\frac{d\theta}{dt})\) can be obtained based on the electromagnetic torque \(T_{e}\), load torque \(T_{L}\), friction factor \(D\) and rotary inertia \(J\) in Formula (2). However, the linear model of electromagnetic torque is usually adopted in the qualitative analysis for the torque ripple control strategy of an SRM in order to highlight the basic torque characteristics of the SRM and simplify the calculation process.