# The Cluster Computation-Based Hybrid FEM-Analytical Model of Induction Motor for Fault Diagnostics
## 2 The Motor's Model  
The voltage equations of a squirrel-cage induction motor with a stationary stator and short-circuited rotor cage can be described using magnetic coupled circuits theory as follows:  
\[\mathbf{V}_{s}=\ \mathbf{I}_{s}\mathbf{R}_{s}+\ \frac{d}{dt}(\mathbf{L}_{ss}\mathbf{I}_{s}+\ \mathbf{L}_{sr}\mathbf{I}_{r}) \tag{1}\]  
\[\mathbf{0}=\ \mathbf{I}_{r}\mathbf{R}_{r}+\ \frac{d}{dt}(\mathbf{L}_{rs}\mathbf{I}_{s}+\ \mathbf{L}_{rr}\mathbf{I}_{r}) \tag{2}\]  
where \(\mathbf{V}_{s}\), \(\mathbf{I}_{s}\), \(\mathbf{I}_{r}\), \(\mathbf{R}_{s}\), \(\mathbf{R}_{r}\), \(\mathbf{L}_{ss}\), \(\mathbf{L}_{sr}\), and \(\mathbf{L}_{rr}\) are the vectors containing the machine's voltage, currents, resistances, and inductances respectively. The stator-stator, stator-rotor, rotor-stator, and rotor--rotor self and mutual inductance matrices (\(\mathbf{L}_{ss}\), \(\mathbf{L}_{sr}\), \(\mathbf{L}_{rs}\), and \(\mathbf{L}_{rr}\)) can be defined as follows;  
\[\mathbf{L}_{ss}=\left[ \begin{array}{ccc}L_{\text{\small{a}ss}}&L_{\text{\small{a}bs }}&L_{\text{\small{a}cs}}\\ L_{\text{\small{b}ss}}&L_{\text{\small{b}bs}}&L_{\text{\small{b}cs}}\\ L_{\text{\small{c}ss}}&L_{\text{\small{c}bs}}&L_{\text{\small{c}cs}}\\ \end{array} \right], \tag{3}\]\[\mathbf{L}_{sr}=\left[ \begin{array}{cccccc}L_{ar1}&L_{ar2}&\ldots&L_{ar1}&\ldots&L_{ arm}&L_{are}\\ L_{br1}&L_{br2}&\ldots&L_{bri}&\ldots&L_{lm}&L_{bre}\\ L_{cr1}&L_{cr2}&\ldots&L_{cri}&\ldots&L_{cm}&L_{ce}\end{array} \right] \tag{4}\]  
\[\mathbf{L}_{rr}=\left[ \begin{array}{cccccc}L_{r1r1}&L_{r1r2}&\ldots&L_{r1 ri}& \ldots&L_{r1m}&L_{r1r}\\ L_{r2r1}&L_{r2r2}&\ldots&L_{r2 ri}&\ldots&L_{r2m}&L_{r2r}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ L_{rr1}&L_{rr2}&\ldots&L_{rriri}&\ldots&L_{rim}&L_{rrir}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ L_{mr1}&L_{mr2}&\ldots&L_{mr1}&\ldots&L_{mm}&L_{rrne}\\ L_{rr1}&L_{rr2}&\ldots&L_{rriri}&\ldots&L_{rem}&L_{rere}\end{array} \right] \tag{5}\]  
\[\mathbf{R}_{rr}=\left[ \begin{array}{cccccc}2(R_{b}+r_{e})&-R_{b}&0&0&\cdots&0&- R_{b}&-r_{e}\\ -R_{b}&2(R_{b}+r_{e})&-R_{b}&0&\cdots&0&0&-r_{e}\\ 0&-R_{b}&2(R_{b}+r_{e})&-R_{b}&\cdots&0&0&-r_{e}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&0&\cdots&2(R_{b}+r_{e})&-R_{b}&-r_{e}\\ -R_{b}&0&0&0&\cdots&-R_{b}&2(R_{b}+r_{e})&-r_{e}\\ -r_{e}&-r_{e}&-r_{e}&-r_{e}&\cdots&-r_{e}&-r_{e}&n_{b}r_{e}\end{array} \right] \tag{6}\]  
The last rows and columns in \(\mathbf{L}_{sr}\), \(\mathbf{L}_{rr}\), and \(\mathbf{R}_{rr}\) correspond to the end ring values, which can be neglected in case of a perfect symmetrical machine, as the net end ring current is always zero. In unsymmetrical machines, these entries are important to simulate the end ring faults and to avoid the singularity problems while taking the inverse of inductance matrices. For the ease of implementation, all these matrices can be grouped. \[\mathbf{V}_{s}=\left[ \begin{array}{cccc}v_{as}&v_{bs}&v_{cs}\end{array} \right]^ {T} \tag{7}\]  
\[\mathbf{I}_{s}=\left[ \begin{array}{cccc}i_{as}&i_{bs}&i_{cs}\end{array} \right]^ {T} \tag{8}\]  
\[\mathbf{I}_{r}=\left[ \begin{array}{cccc}i_{r1}&i_{r2}&\cdots&i_{m}i_{re}\end{array} \right] ^{T} \tag{9}\]  
\[\mathbf{L}=\left[ \begin{array}{cccc}L_{ss}&L_{sr}\\ L_{rs}&L_{rr}\end{array} \right] \tag{10}\]  
The currents, torque, and speed can be calculated as:  
\[\left[ \begin{array}{c}\mathbf{I}_{s}\\ \mathbf{I}_{r}\end{array} \right]=\left[ \begin{array}{cc}\mathbf{L}_{ss}&\mathbf{L}_{sr}\\ \mathbf{L}_{rs}&\mathbf{L}_{rr}\end{array} \right]^{-1}\int\!\!\left[\left[ \begin{array} []{c}\mathbf{V}_{s}\\ 0\end{array} \right]-\left[ \begin{array}{c}\mathbf{R}_{s}&0\\ 0&\mathbf{R}_{r}\end{array} \right]\left[ \begin{array}{c}\mathbf{I}_{s}\\ \mathbf{I}_{r}\end{array}