## 2 Modeling of Five-Phase Induction Motor
### Mathematical Model of Five-Phase Induction Motor in Normal Operation  
The mathematical description of multiphase machines is an essential element of their research. The mathematical description of a five-phase induction motor in a star connection will be the subject of this subsection. For a motor model in a pentagon or pentacle connection, phase changes between the motor windings must be considered, depending on the connection. As a consequence, the resultant voltage and current amplitude change directly on the motor stator windings without affecting the supply voltage, as seen in Figure 3.  
Using (1) is obtained voltages for a symmetrically distributed and balanced five-phase machine:  
\[U_{a}=\sqrt{2}U_{rms}sin(\omega t), \tag{2}\]  
\[U_{b}=\sqrt{2}U_{rms}sin\bigg{(}\omega t-\frac{2\pi}{5}\bigg{)}, \tag{3}\]  
\[U_{c}=\sqrt{2}U_{rms}sin\bigg{(}\omega t-\frac{4\pi}{5}\bigg{)}, \tag{4}\]  
\[U_{d}=\sqrt{2}U_{rms}sin\bigg{(}\omega t+\frac{4\pi}{5}\bigg{)}, \tag{5}\]  
\[U_{e}=\sqrt{2}U_{rms}sin\bigg{(}\omega t+\frac{2\pi}{5}\bigg{)}. \tag{6}\]  
Decoupling transformation matrix of the five-phase induction motor model:  
\[\left[ \begin{array}{c}U_{d}\\ U_{q}\\ U_{x}\\ U_{y}\\ U_{0}\end{array} \right]=\frac{2}{5}\left[ \begin{array}{ccc}1&cos\big{(} \frac{2\pi}{5}\big{)}&cos\big{(}\frac{4\pi}{5}\big{)}&cos\big{(}\frac{6\pi}{5 }\big{)}&cos\big{(}\frac{8\pi}{5}\big{)}\\ 0&-sin\big{(}\frac{2\pi}{5}\big{)}&-sin\big{(}\frac{4\pi}{5}\big{)}&-sin\big{(} \frac{6\pi}{5}\big{)}&-sin\big{(}\frac{8\pi}{5}\big{)}\\ 1&cos\big{(}\frac{6\pi}{5}\big{)}&cos\big{(}\frac{12\pi}{5}\big{)}&cos\big{(} \frac{18\pi}{5}\big{)}&cos\big{(}\frac{24\pi}{5}\big{)}\\ 0&-sin\big{(}\frac{6\pi}{5}\big{)}&-sin\big{(}\frac{12\pi}{5}\big{)}&-sin\big{(} \frac{18\pi}{5}\big{)}&-sin\big{(}\frac{24\pi}{5}\big{)}\\ \frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\end{array}  \right]\left[ \begin{array}{c}U_{a}\\ U_{b}\\ U_{c}\\ U_{d}\\ U_{e}\end{array} \right]. \tag{7}\]  
Voltage stator and rotor equations in reference \(d_{\text{f}}\) frame:  
\[U_{ds}=R_{s}i_{ds}-\omega_{a}\psi_{qs}+\rho\psi_{ds}; U_{dr}=R_{r}i_{dr}-(\omega_{a}-\omega)\psi_{qr}+\rho\psi_{dr}, \tag{8}\]  
\[U_{qs}=R_{s}i_{qs}-\omega_{a}\psi_{ds}+\rho\psi_{qs}; U_{qr}=R_{r}i_{qr}+(\omega_{a}-\omega)\psi_{dr}+\rho\psi_{qr}, \tag{9}\]  
\[U_{xs}=R_{s}i_{xs}+\rho\psi_{xs}; U_{xr}=R_{r}i_{xr}+\rho\psi_{xr}, \tag{10}\]  
\[U_{ys}=R_{s}i_{ys}+\rho\psi_{ys}; U_{yr}=R_{r}i_{yr}+\rho\psi_{yr}, \tag{11}\]  
\[U_{0s}=R_{s}i_{0s}+\rho\psi_{0s}; U_{0r}=R_{r}i_{qr}+\rho\psi_{0r}. \tag{12}\]  
Figure 3: Phase diagram of a five-phase motor: (**a**) pentagon connection; (**b**) pentacle connection. Flux linkages of stator/rotor:  
\[\psi_{ds}=(L_{ls}+L_{m})i_{qs}+L_{m}i_{qr}; \psi_{dr}=(L_{lr}+L_{m})i_{qr}+L_{m}i_{qs}, \tag{13}\] \[\psi_{qs}=(L_{ls}+L_{m})i_{ds}+L_{m}i_{dr}; \psi_{qr}=(L_{lr}+L_{m})i_{dr}+L_{m}i_{ds},\] (14) \[\psi_{xs}=L_{ls}i_{xs}; \psi_{xr}=L_{lr}i_{xr},\] (15) \[\psi_{ys}=L_{ls}i_{ys}; \psi_{qr}=L_{lr}i_{yr},\] (16) \[U_{qs}=R_{s}i_{qs}-\omega_{a}\psi_{ds}+\rho\psi_{qs}; U_{qr}=R_{r}i_{qr}+(\omega_{a}-\omega)\psi_{dr}+\rho\psi_{qr}, \tag{9}\]0  
Torque calculation:  
\[U_{qs}=R_{s}i_{qs}-\omega_{a}\psi_{ds}+\rho\psi_{qs}; U_{qr}=R_{r}i_{qr}+(\omega_{a}-\omega)\psi_{dr}+\rho\psi_{qr}, \tag{9}\]1  
Relation for rotor speed calculation:  
\[U_{qs}=R_{s}i_{qs}-\omega_{a}\psi_{ds}+\rho\psi_{qs}; U_{qr}=R_{r}i_{qr}+(\omega_{a}-\omega)\psi_{dr}+\rho\psi_{qr}, \tag{9}\]2  
where \(R_{s}\) represents the stator resistance; \(R_{r}\)--rotor resistance; \(L_{l}\)--leakage inductance; \(L_{m}\)-- represents the maximum mutual inductance of the stator to the rotor; \(U\)--voltage; \(i\)--current; \(\psi\)--flux linkages; \(I\)--a moment of inertia; \(p\)--number of poles; \(T_{L}\)--load moment; \(T_{e}\)--electromechanical moment; \(\omega_{r}\)--angular velocity of the rotor; \(\omega\)--angular velocity; \(\omega_{a}\)--the angular velocity at which the machine equations of any reference frame are transformed; and \(\rho F\) represents \(d\)/\(dt\)[32; 33].