## 2 Mathematical Model of the Motor  
The transient processes for the LSPMSM start-up are calculated using the LSPMSM mathematical model. When modelling the LSPMSM, it is assumed that:  
* The magnetic fields generated by the stator and rotor windings have a sinusoidal spatial distribution;
* The magnetic permeability of the steel is constant;
* The stator and rotor windings are symmetrical;
* Each winding is powered by a separate source;
* The mains voltage phasor is constant throughout the entire starting process, and an increase in the motor current does not cause a decrease in its amplitude;
* The magnetic core losses are not taken into account. The system of ordinary differential LSPMSM equations to be solved is represented as:  
\[ \begin{array}{l}\mathrm{d}\lambda_{\mathrm{sd}}/dt-p\cdot\lambda_{\mathrm{ sq}}\cdot\mathrm{d}\varphi/dt+R_{\mathrm{s}}\cdot I_{\mathrm{sd}}=U_{\mathrm{sd}};\\ \mathrm{d}\lambda_{\mathrm{sq}}/dt+p\cdot\lambda_{\mathrm{sd}}\cdot\mathrm{d} \varphi/dt+R_{\mathrm{s}}\cdot I_{\mathrm{sq}}=U_{\mathrm{sq}};\\ \mathrm{d}\lambda^{\prime}{}_{\mathrm{rd}}/dt+r^{\prime}{}_{d}\cdot I^{\prime }{}_{\mathrm{rd}}=0;\\ \mathrm{d}\lambda^{\prime}{}_{\mathrm{rd}}/dt+r^{\prime}{}_{d}\cdot I^{\prime }{}_{\mathrm{rd}}=0;\\ I^{\prime}{}_{\mathrm{rd}}=(\lambda^{\prime}{}_{\mathrm{rd}}-\lambda_{\mathrm{ sd}})/L_{\mathrm{vd}};\\ I^{\prime}{}_{\mathrm{rq}}=(\lambda^{\prime}{}_{\mathrm{rq}}-\lambda^{\prime}{}_{ \mathrm{sq}}-\lambda^{\prime}{}_{\mathrm{0}})/L_{\mathrm{sq}};\\ I_{\mathrm{sd}}=\lambda_{\mathrm{sd}}/L_{\mathrm{sd}}-l^{\prime}{}_{\mathrm{rd }};\\ I_{\mathrm{sq}}=(\lambda_{\mathrm{sq}}-\lambda^{\prime}{}_{\mathrm{0}})/L_{ \mathrm{sq}}-l^{\prime}{}_{\mathrm{rq}};\\ T=3/2\cdot Z_{\mathrm{p}}(\lambda_{\mathrm{sd}}\cdot I_{\mathrm{sq}}-\lambda_{ \mathrm{sq}}-I_{\mathrm{sd}});\\ J\cdot\mathrm{d}^{2}\varphi/dt^{2}=T-T_{\mathrm{load}},\end{array}  \tag{1}\]where \(U_{sd}\) and \(U_{sq}\) are the stator voltages along the \(d\) and \(q\) axes; \(I_{sd}\), \(I_{sq}\), \(I^{\prime}_{rd}\), and \(I^{\prime}_{rq}\) are the stator and rotor currents; \(L_{sd}\), \(L_{sq}\), \(L_{rd}\), and \(L_{rq}\) are the stator and rotor total inducances; \(L_{vd}\) and \(L_{rq}\) are the rotor leakage inductances; \(\lambda_{\rm sd}\), \(\lambda_{\rm sq}\), \(\lambda^{\prime}_{\rm rd}\), and \(\lambda^{\prime}_{\rm rq}\) are the stator and rotor flux linkages; \(\lambda^{\prime}_{0}\) is the permanent magnet flux linkage; \(R_{s}\) is the stator resistance; \(Z_{p}\) is the number of motor pole pairs; \(r^{\prime}_{nd}\) and \(r^{\prime}_{rq}\) are the rotor resistances; \(\varphi\) is the mechanical rotational angle equal to the integral of the motor speed \(\omega\); \(T\) is the motor torque; \(T_{load}\) is the loading torque; \(j\) is the total moment of inertia. All initial conditions are equal to zero, except for the stator flux along the \(q\)-axis \(\lambda_{\rm sq0}=\lambda^{\prime}_{0}\). Figure 2 shows the implementation of equation system (1) in Simulink. According to Figure 2a, the LSPMSM fed directly from the mains (\"Discrete 3-phase Source\"), was mechanically coupled to the load. The load was simulated as a summation of a constant torque \(T_{c}\) and component \(\omega\)-\(F\) that represented a bearings frictional torque, which depended on speed linearly, as is shown in Figure 3b. Figure 2: Simulink model of the LSPMSM motor in the d-q axes. (**a**) General view of the model; (**b**) calculation of stator currents; (**c**) calculation of rotor currents; (**d**) torque calculation. The LSPMSM parameters (rated power 550 W, rated speed 1500 rpm) given in Table 1 were taken as the motor parameters. It is known that the operating temperature significantly affects the resistance and rotational speed of induction motors [41]. At the same time, the heating of the LSPMSM is much less due to lower losses, and it does not have any slip. This makes it possible to not consider the influence of temperature when identifying its parameters [34].