## 3 Proposed Current Detection Schemes with Reduced Sensors
### (\(N_{p}+1\))/2 Hall Effect Sensors-Based Current Detection Scheme for OMSRM  
Phase number of an OMSRM can be expressed as  
\[N_{\text{p}} =2Z_{3}+1,\;Z_{3}\geq 1\text{ and }Z_{3}\text{ is an integer} \tag{9}\]  
In combining (2), (3) and (9), we obtain  
\[\theta_{\text{ps}}=\tau_{\text{r}}Z_{1}/(2Z_{3}+1) \tag{10}\]  
Clearly, there exists  
\[\theta_{\text{ps}}\
eq\tau_{\text{r}}/2 \tag{11}\]  
Therefore, the positive torque region of a phase overlaps with that of any other phase, and thus a phase and any other phase could be excited at the same time to generate a positive or negative motor torque. However, it is found that there are at most \((N_{\text{p}}+1)/2\) phase torques in the same direction at any rotor position, and thus at most \((N_{\text{p}}+1)/2\) multiplexed current sensors could be enough to detect all the phase currents. For demonstration, phase inductance curves of a three-phase SRM and a five-phase SRM are, respectively, shown in Figure 7a,b, where \(\theta\) is the rotor position; \(\theta_{\text{u,A}}\), \(\theta_{\text{u,B}}\), \(\theta_{\text{u,C}}\), \(\theta_{\text{u,D}}\), and \(\theta_{\text{u,E}}\) denotes the unaligned positions of phases A, B, C, D, and E, respectively; \(\theta_{\text{a,A}}\), \(\theta_{\text{a,B}}\), \(\theta_{\text{a,C}}\), \(\theta_{\text{a,D}}\), and \(\theta_{\text{a,E}}\) represent the aligned positions; \(\text{Se}_{1}\), \(\text{Se}_{2}\),..., \(\text{Se}_{10}\) indicate the position sectors. Figure 6: Current detection circuit using a shunt resistor with non-isolated single-supply non-inverting amplifier. It can be observed that an electrical cycle of the three-phase SRM is divided into six sectors by the unaligned and aligned positions of all phases, two inductance curves are rising in the three sectors highlighted by gray rectangles and two inductance curves are falling in the other three sectors. Similarly, an electrical cycle of the five-phase SRM is divided into ten sectors, three inductance curves are ascending in the five sectors marked by gray rectangles and three inductance curves are descending in the other five sectors. In addition, the phases with the same torque direction are consecutively adjacent. Therefore, there are only \(N_{\text{P}}\) combinations of the \((N_{\text{P}}+1)/2\) phases with the same torque direction. For the three-phase SRM, the three combinations are phases A and B, phases B and C, as well as phases C and A. For the five-phase SRM, the five combinations are phases A, B, and C; phases B, C, and D; phases C, D, and E; phases D, E, and A; as well as phases E, A, and B. We take the five-phase SRM as an example to demonstrate the process of determining the current sensor multiplexing scheme for an OMSRM. Relationships between the phase currents and the currents measured by the three hall effect sensors (HS\({}_{\text{I}}\), HS\({}_{\text{2}}\), and HS\({}_{\text{3}}\)) in the five-phase SRM drive can be expressed by  
\[ \begin{split}\mathbf{K}_{\text{5p}}\big{[}\ i_{\text{A}}\ i_{\text{B}} \ i_{\text{C}}\ i_{\text{D}}\ i_{\text{E}}\ \big{]}^{\text{T}}=\left[\begin{array}{ccccc}K_{11}&K_{12}&K_{13}&K_{14}&K_ {15}\\ K_{21}&K_{22}&K_{23}&K_{24}&K_{25}\\ K_{31}&K_{32}&K_{33}&K_{34}&K_{35}\end{array}\right]\big{[}\ i_{\text{A}}\ \ \ \ i_{\text{B}}\ \ \ i_{\text{C}}\ \ \ \ i_{\text{D}}\ \ \ \ i_{\text{E}}\ \big{]}^{\text{T}}=\left[\begin{array}{c}i_{\text{HS1}}\\ i_{\text{HS2}}\\ i_{\text{HS3}}\end{array}\right]\end{split}  \tag{12}\]  
where \(\mathbf{K}_{\text{5p}}\) denotes the coefficient matrix, each element of which can be \(-1\), \(0\), or \(1\). The sign of an element indicates the direction of the phase current-carrying conductor through the sensor. For the five combinations, (12) can be simplified to (13)-(17), respectively.