# An Overview of Position Sensorless Techniques for Switched Reluctance Machine Systems
## 3 An Overview of Recent Development in Position Estimation Methods of SRM
### Magnetic-Model-Based Position Sensorless Methods
#### 3.1.2 Based on the Inductance Model Methods  
Inductance and flux linkage are the same as the most important essential characteristics of electric machines. Therefore, they have been extensively studied to promote the development of indirect position estimation. The calculation of the inductance has a simpler calculation method than the flux linkage, as can be seen from Equation (3). Inductance has a direct balance relationship with voltage and current, so various inductance-based methods have been proposed. These methods are intensively studied in inductance modeling, inductance acquisition methods, and considering the inductance-position relationship to indirectly obtain the rotor position.  
As shown in Figure 6, the phase inductance is the largest at the aligned position \(\theta_{a}\) of the stator and the contact and the smallest at the misaligned position \(\theta_{u}\). Generally, three special positions are selected for parameter fitting to establish a mathematical model of the inductance. The inductance model based on the Fourier series is shown in (6).  
\[L_{k}(\theta_{ph},i_{k})=L_{0}(i_{k})+L_{1}(i_{k})\cos(N_{r}\theta_{ph})+L_{2 }(i_{k})\cos(2N_{r}\theta_{ph}) \tag{6}\]  
where \(L_{0}\), \(L_{1}\), and \(L_{2}\) are the parameters to be fitted and \(N_{r}\) represents the number of rotor poles.  
Based on the inductance model of first switching harmonics via Fourier series to reduce the need for controller memory and interpolation [85; 86]. The online calibration [87; 88], considering mutual inductance [89], and considering magnetic saturation [90; 91] were used to enhance the accuracy of the inductance model.  
In terms of the method of inductance acquisition, the rotor estimation method based on the inductance modeling-current model is incremental inductance [92] utilizing the conduction phase measurement, and the motor performs well in the low speed range. Phase inductance information is obtained based on pulse injection, and then the combined vector quadrature decomposition method is combined with the inductance partition method to eliminate position sensors [93]. Further, many scholars have found various relationships between the inductance characteristics and the rotor position to effectively detect the rotor position. After rotor position failures are detected, an inductance slope-based method is used to supplement the missing signal [94; 95]. After that, a method based on phase-inductance vector coordinate transformation was proposed and improved [96; 97]. Unbalanced inductance will cause the traditional inductance feature-based and inductance modeling-based methods to reduce the accuracy of position estimation, and even fail to  
Figure 6: Current waveforms for different control algorithms and inductor cycles.  
drive the motor, as shown in Figure 7. Further, to improve the general applicability of rotor position estimation, a detection method considering inductance imbalance is applied [98]. As shown in (7), by establishing the relationship between the inductance and the current, the zero-crossing law of the slope at the inflection point of the inductance can be found [99; 100; 101]. Such a feature is efficient for localization of the rotor position \(\theta_{ox}\) without the need for additional sensors and additional circuitry.  
\[\left\{\begin{array}{l}U_{0-}=\omega L_{0-}\frac{d\theta_{0-}}{d\theta_{ph}}+ \omega i_{0-}\frac{d\theta_{0-}}{d\theta_{ph}}\\ U_{0+}=\omega L_{0+}\frac{d\theta_{0+}}{d\theta_{ph}}+\omega i_{0+}\frac{d \theta_{0+}}{d\theta_{ph}}\end{array}\right. \tag{7}\]