For the calculations and simulations, the authors used OPERA 3D commercial software, which guarantee the solution of the non-linear magnetic problem, taking into account the eddy currents induced in the rotor cage as well as rotor movement. The use of 3D model resulted from the existence of rotor slots’ skew. The authors are familiar with 2D solutions, which include, in an approximate way, this parameter. But we decided to use a full 3D model, realizing the extension of computing time. Built models took into account the dynamic phenomena occurring in the electromagnetic field, in electrical circuits and mechanical system of the motor. The built 3D FEM model is described by very well known Maxwell’s equations

$$\begin{array}{}\mathrm{\nabla}\times \overrightarrow{H}=\overrightarrow{J}\end{array}$$(1)

$$\begin{array}{}{\displaystyle \mathrm{\nabla}\times \overrightarrow{E}=-\frac{\mathrm{\partial}\overrightarrow{B}}{\mathrm{\partial}t}}\end{array}$$(2)

$$\begin{array}{}\mathrm{\nabla}\cdot \overrightarrow{B}=0\end{array}$$(3)

$$\begin{array}{}\overrightarrow{J}=\sigma \left(\overrightarrow{E}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}\overrightarrow{u}\times \overrightarrow{B}\right)\end{array}$$(4)

$$\begin{array}{}\overrightarrow{B}=\mathrm{\nabla}\times \overrightarrow{A}\end{array}$$(5)

where *H⃗*,*J⃗*,*B⃗*,*E⃗* are magnetic field strength, current density, flux density and electric field strength vectors, respectively. Moreover, *u⃗* and *A⃗* are velocity and magnetic potential vectors, respectively.

In regions where the field is only derived from a vector potential, the following equations, as a combination of Eq. 1-5, is used

$$\begin{array}{}\mathrm{\nabla}\times \frac{1}{\mu}\mathrm{\nabla}\times \overrightarrow{A}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}-\sigma \frac{\mathrm{\partial}\phantom{\rule{thinmathspace}{0ex}}\overrightarrow{A}}{\mathrm{\partial}t}\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\sigma \mathrm{\nabla}V\end{array}$$(6)

where *V* is an electric scalar potential, *μ* is the magnetic permeability, and *σ* is the electric conductivity.

In the analyzed FEM model, in the rotor there are conductive elements leading to induced eddy currents. The dimensions of these elements are so large that it is necessary to solve the problem of taking into account the uneven distribution of induced currents. This is achieved by solving an extra equation for each region which leads to induced currents.

$$\begin{array}{}\begin{array}{l}\mathrm{\nabla}\times \frac{1}{\mu}\mathrm{\nabla}\times \overrightarrow{A}\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\mathrm{\nabla}\frac{1}{\mu}\mathrm{\nabla}\cdot \overrightarrow{A}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\sigma \phantom{\rule{thinmathspace}{0ex}}\left(\overrightarrow{u}\times \mathrm{\nabla}\times \overrightarrow{A}\right)\\ {\int}_{\mathrm{\Omega}}\left(-\sigma \frac{\mathrm{\partial}\overrightarrow{A}}{\mathrm{\partial}t}\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{\partial}\mathrm{\Omega}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}{\int}_{\mathrm{\Omega}}{J}_{S}\overrightarrow{\mathrm{\partial}}\mathrm{\Omega}\end{array}\end{array}$$(7)

where *Ω* is the volume where eddy current exists, *S* is the cross section of the element.

Together with the field equations the circuit and mechanical equations were solved. The voltage equation for the stator winding can be written as:

$$\begin{array}{}{\displaystyle {U}_{ph}=R\phantom{\rule{thinmathspace}{0ex}}i\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}N\phantom{\rule{thinmathspace}{0ex}}\frac{d\mathit{\Psi}}{dt}}\end{array}$$(8)

where *U*_{ph} is the instantaneous phase voltage on the stator winding, *R* is the winding resistance, *i* is the instantaneous current in the winding, *N* is the number of turns, *Ψ* is the magnetic flux linkage (calculated by field equations). Taking into account the wire diameter of the stator winding, the skin effect was ignored when calculating the resistance.

Equation (8) can be written into a new form, using the calculated vector potential *A⃗*.

$$\begin{array}{}{\displaystyle {U}_{ph}=R\phantom{\rule{thinmathspace}{0ex}}i\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}N\phantom{\rule{thinmathspace}{0ex}}\frac{1}{{S}_{c}}\underset{{S}_{c}}{\int}\frac{\mathrm{\partial}{A}_{z}}{\mathrm{\partial}t}ds}\end{array}$$(9)

where *S*_{c} is the cross section of the stator winding, *A*_{z} is the z-component of vector potential.

The rotor speed can be calculated based on the mechanical equation

$$\begin{array}{}\stackrel{}{T}=J\frac{{d}^{2}\mathit{\Theta}}{d{t}^{2}}\end{array}$$(10)

where *T* is the sum of the electromagnetic torque computed by the program, friction and load torques, *J* is the polar moment of inertia, *θ* is the position angle.

The authors considered many rotor’s structures (the stator geometry and the stator winding parameters remained constant), took into consideration the minimization of tools cost. Examples of analyzed rotor geometry are shown in Fig. 2.

Figure 2 The analyzed rotor geometries. a) reference one, b) additional small teeth, c) wider outer teeth, d) additional horizontal teeth. Regions marked gray are made of aluminum

The study was conducted by energizing the motor by 50 Hz sinusoidal voltage. The authors studied not only the impact of the rotor geometry on the motor dynamics, but also the possibility of a different magnetic material usage, with respect to the commonly used one (M600-50A). During simulations the SMC material (Somaloy500) available on the market was used. This material has “worse” magnetization curve with respect to the “original” material (see Fig. 3), but thanks to it, we have the possibility to influence the ratio of *d*-axis and *q*-axis reluctances. As it is known, this ratio determines the reluctance torque of the motor, but as shown in the paper, also affects the reduction of parasitic torques that occur during motor start-up (this is the Gorges phenomenon, where there is a significant decrease in the asynchronous torque generated by the motor, existing at half synchronous speed). The proposal to use a SMC material does not increase too much the whole motor cost [15]. Unfortunately, the use of the SMC material significantly increased the magnetizing component of the current, which is of great importance for fractional power motors, where the magnetizing component is a dominant in the consumed current – see . To reduce this negative effect, we increased the core length by 10% and reduced the air gap thickness by 20%, with respect to the reference motor. In this way we limited the increase in current consumption to 6%, while improving the motor dynamics. A comparison of the rotor speed during start-up, for the geometries shown of Fig. 2a and 2b, is presented in Fig. 4.

Figure 3 The *BH*-curves for used magnetic materials. 1 – M600-50A, 2 – SOMALOY500

Figure 4 The comparison of the start-up rotor speed vs. time. 1 – the reference motor, 2 – the proposed solution

Table 1 Comparison of consumed currents at nominal load

A problem, which exists in the test motor, is currents flowing in the rotor cage, even for a synchronous speed – . They cause Joule losses both in the rotor cage and in the stator winding. Since the rotor cage has a specific structure, so the authors determined current waveforms for the bar located in the “classical” rotor slot, and for the bar located in a “wide” slot. Examples of waveforms are shown in Figs. 5-6.

Figure 5 The current waveforms of the bar located in “classical” slot. 1 – the reference motor, 2 – the proposed solution

Figure 6 The current waveforms of the bar located in “wide” slot. 1 – the reference motor, 2 – the proposed solution

The dynamics of the reference and the proposed motors can be compared using dynamic curves – see Fig. 7, where faster access to the synchronous speed by motor having the proposed structural solution (specified geometry-Fig. 2b, the SMC material in the rotor, the package longer of 10%, the air-gap width reduced by 20%) is clearly visible. According to the authors, this is the best solution among the analyzed variants.

Figure 7 The dynamic curves electromagnetic torque vs. rotor speed

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