## 3 Proposed Current Detection Schemes with Reduced Sensors
### (\(N_{p}+1\))/2 Hall Effect Sensors-Based Current Detection Scheme for OMSRM
\[\mathbf{K}_{\text{5 pABC}}\big{[}i_{\text{A}}\ i_{\text{B}}\ i_{\text{C}}\big{]}^{ \text{T}}=\big{[}i_{\text{HS1}}\ i_{\text{HS2}}\ i_{\text{HS3}}\big{]}^{ \text{T}} \tag{13}\]  
\[\mathbf{K}_{\text{5 pBCD}}\big{[}i_{\text{B}}\ i_{\text{C}}\ i_{\text{D}}\big{]}^{ \text{T}}=\big{[}i_{\text{HS1}}\ i_{\text{HS2}}\ i_{\text{HS3}}\big{]}^{ \text{T}} \tag{14}\]  
\[\mathbf{K}_{\text{5 pCDE}}\big{[}i_{\text{C}}\ i_{\text{D}}\ i_{\text{E}}\big{]}^{ \text{T}}=\big{[}i_{\text{HS1}}\ i_{\text{HS2}}\ i_{\text{HS3}}\big{]}^{ \text{T}} \tag{15}\]  
\[\mathbf{K}_{\text{5 pDEA}}\big{[}i_{\text{A}}\ i_{\text{D}}\ i_{\text{E}}\big{]}^{ \text{T}}=\big{[}i_{\text{HS1}}\ i_{\text{HS2}}\ i_{\text{HS3}}\big{]}^{ \text{T}} \tag{16}\]  
\[\mathbf{K}_{\text{5 pEAB}}\big{[}i_{\text{A}}\ i_{\text{B}}\ i_{\text{E}}\big{]}^{ \text{T}}=\big{[}i_{\text{HS1}}\ i_{\text{HS2}}\ i_{\text{HS3}}\big{]}^{ \text{T}} \tag{17}\]  
where \(\mathbf{K}_{\text{5 pABC}}\), \(\mathbf{K}_{\text{5 pBCD}}\), \(\mathbf{K}_{\text{5 pCDE}}\), \(\mathbf{K}_{\text{5 pDEA}}\) and \(\mathbf{K}_{\text{5 pEAB}}\) are five \(3\ \times\ 3\) submatrices of the matrix \(\mathbf{K}_{\text{5p}}\). \(\mathbf{K}_{\text{5 pABC}}\) consists of the first three column vectors; \(\mathbf{K}_{\text{5 pBCD}}\) consists of the middle three column vectors; \(\mathbf{K}_{\text{5 pCDE}}\) consists of the last three column vectors; \(\mathbf{K}_{\text{5 pDEA}}\) consists of the first, fourth, and fifth column vectors; \(\mathbf{K}_{\text{5 pEAB}}\) consists of the first, second, and fifth column vectors. The phase currents can be uniquely calculated from the measured currents if the five submatrices are full rank as in  
\[rank(\mathbf{K}_{\text{5 pABC}})=rank(\mathbf{K}_{\text{5 pBCD}})=rank(\mathbf{K}_{\text{5 pCDE }})=rank(\mathbf{K}_{\text{5 pDEA}})=rank(\mathbf{K}_{\text{5 pEAB}})=3 \tag{18}\]  
Figure 7: Phase inductance curves. (**a**) Of a three-phase SRM. (**b**) Of a five-phase SRM. where \(rank\left(\cdot\right)\) denotes the function for calculating the rank of a matrix. The original matrix should be row full rank first as in  
\[rank\left(\mathbf{K}_{\text{Sp}}\right)=3 \tag{19}\]  
All coefficient matrices satisfying (19) and (18) can be found directly by evaluating all the possible matrices. However, these matrices differ significantly from each other. Three exemplary matrices are given as follows  
\[\mathbf{K}_{\text{Sp,el}}=\left[ \begin{array}{ccccc}0&0&1&1&1\\ 0&1&1&0&1\\ 1&1&1&1&0\end{array} \right] \tag{20}\]  
\[\mathbf{K}_{\text{Sp,e2}}=\left[ \begin{array}{ccccc}1&0&0&1&0\\ 0&1&1&1&0\\ 0&1&0&0&1\end{array} \right] \tag{21}\]  
\[\mathbf{K}_{\text{Sp,e3}}=\left[ \begin{array}{ccccc}-1&0&0&1&0\\ 0&1&-1&1&0\\ 0&1&0&0&-1\end{array} \right] \tag{22}\]  
We can see that \(\mathbf{K}_{\text{Sp,e1}}\) has the most number of non-zero elements and thus the phase current calculation is the most complicated. More non-zero elements also mean more wires needed to pass through the sensors and higher requirements for the aperture size of the sensor. Therefore, \(\mathbf{K}_{\text{Sp,e1}}\) should be excluded first. The vast majority of hall effect sensors can detect bidirectional currents, and thus the hall effect sensors in most SRM drives are bidirectional though the phase currents are unidirectional. The measuring range of a bidirectional sensor is symmetric with respect to the measured current of 0 A. With \(\mathbf{K}_{\text{Sp,e2}}\), all conductors pass through the sensors in the same direction, and thus only half the range of each sensor is used.