# An Overview of Position Sensorless Techniques for Switched Reluctance Machine Systems
## 2 The Composition of SRM Drive System
### Mathematical Model of SRM  
The motion state of the SRM can be controlled as long as the windings of each phase are driven according to certain principles. Figure 5 shows the change of the electromagnetic state of a 12/8 three-phase SRM during the rotation of one rotor pole pitch. The first state is that the salient poles of the rotor are aligned with the center of the grooves of the stator as shown in Figure 5a. The rotor turns 22.5\({}^{\circ}\) counterclockwise to reach the second state. At this time, the rotor salient poles are aligned with the center of the stator salient poles. In [0\({}^{\circ}\), 22.5\({}^{\circ}\)], the inductance of this phase gradually increases from minimum to maximum due to the decrease in reluctance. After that, the inductance will decrease until it turns another 22.5\({}^{\circ}\) to reach the third state. As shown in Figure 5b,c, the static curves of flux linkage and torque under different currents can be obtained by finite element analysis. Figure 4: Topology of asymmetric half-bridge inverter. (**a**) Static structure. (**b**) Magnetization. (**c**) Freewheeling. (**d**) Demagnetization. Figure 3: The structure of position sensorless SRM drive system. The motor conforms to the law of electromagnetic induction in the process of operation. The motor has \(m\) phases, and each phase winding satisfies Equation (1). As shown in Figure 5, the flux linkage of SRM is very nonlinear due to its structure. The flux linkage model can be fitted mathematically, but the accuracy of this method is greatly limited. Generally, the magnetic link information is obtained by the three-dimensional look-up table (LUT) method. \[e_{k}=-\frac{d\psi_{k}}{dt} \tag{1}\]  
where \(\psi_{k}\), \(e_{k}\), and \(t\) represent the flux linkage, induced electromotive force, and time of the \(k\)th phase winding, respectively, and \(k=1\), \(2\),..., \(m\). There is a mapping relationship between the flux linkage \(\psi_{k}\), the rotor position angle \(\theta_{ph}\), and the phase current \(i_{k}\). The relationship between the flux linkage and the inductance \(L_{k}\) is shown in the following formula. \[\psi_{k}(\theta_{ph},i_{k})=L_{k}(\theta_{ph},i_{k})i_{k} \tag{2}\]  
where \(L_{k}\) and \(i_{k}\) represent the phase inductance and phase current, respectively. \(\theta_{ph}\) is rotor position. According to [PERSON]'s voltage law, each phase loop conforms the voltage balance equation. \[U_{k}=R_{k}i_{k}+\frac{d(i_{k}\cdot L_{k})}{dt} \tag{3}\]  
where \(U_{k}\) and \(R_{k}\) represent the phase voltage and phase resistance, respectively. The mechanical balance equation can be obtained via the relevant mechanics theory. \[T_{e}=J\frac{d\omega}{dt}+T_{L}+\mathrm{D}\omega \tag{4}\]  
where \(T_{e}\) and \(T_{L}\) represent the electromagnetic torque generated by the motor and load torque, respectively. \(J\) and \(D\) are the const parameters of the moment of inertia and viscous friction coefficient. \(\omega\) is the actual speed of the motor. Figure 5: SRM structure topology of a three-phase \(12/8\) poles and electromagnetic properties. (**a**) Rotor electrical angle. (**b**) Magnetization curve. (**c**) Phase torque curve. The relationship between these physical quantities is the basis of the control research for SRM, not only for the rotor position estimation control, but also the design of the motion control algorithm [52; 53]. The speed control algorithm determines the output performance of the motor, including torque quality, speed regulation range, and robustness. Common speed control methods include current chopping control (CCC), angle position control (APC), voltage chopping control (VCC) [54; 55; 56; 57], direct torque control (DTC) [58; 59], and direct instantaneous torque control based on torque sharing function (TSF) [60; 61; 62]. Another type of control algorithm is the signal fault tolerant and position sensorless control algorithm, which is designed to enhance the stability of the hardware layer. There is an inevitable connection between speed control and position-free fault-tolerant control.