Selection of the Optimal Rotor Geometry for Synchronous Reluctance Motors

E. V. Kononenko, G. I. Lukiyanov

Presented by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering

In connection with the expansion of the field of application of synchronous reluctance motors (SRMs), studies of optimal ratios in their design and the development of practical recommendations are becoming especially relevant.

In most cases SRMs are designed on the basis of induction motors. In this case the stator geometry is given. The required characteristics can be obtained by changing the winding data of the stator and the rotor design.

The geometry of an SRM rotor is characterized by the size of the air gap between stator and rotor, the pole-arc coefficient, the number, dimensions, and arrangement of additional nonmagnetic slots, and the number and dimensions of the slots of the starting winding. In designing an SRM it is necessary to have recommendations for choosing the indicated elements of the rotor design.

The synchronous reactances \(x_d\) and \(x_q\) depend on the rotor design. The latter are proportional to the field-form coefficients \(k_d\), \(k_q\) and inversely proportional to the size of the air gap and to the saturation coefficients of the magnetic circuit.

The influence of rotor geometry on the coefficients \(k_d\) and \(k_q\) was investigated in [1]. On this basis it is recommended to choose the pole height no less than \(0.1\tau\), and the width of the additional magnetic slot on the pole \((\Delta)\) no less than \((0.05\text{–}0.07)\tau\). When choosing the distance between nonmagnetic slots, one should restrict oneself to the range of values \((0.05\text{–}0.15)\tau\). The height of the additional nonmagnetic slot is recommended to be chosen as small as possible, not more than \(0.5\tau\). The most favorable values of the pole-arc coefficient for improved SRMs lie within the range \(0.55\text{–}0.65\). The air gap is chosen equal to \((0.003\text{–}0.005)\tau\).

The rotor of improved SRMs may be made with a different number of additional nonmagnetic slots. The number of these slots and their width have a substantial influence on the energy performance. Table 1 gives the maximum values of useful power, efficiency, and power factor as functions of the number of nonmagnetic slots on the pole. The experimental specimens were made with a pole-arc coefficient \(\alpha_p = 0.8\). The rotor laminations have the designs shown in Fig. 1. As can be seen from the figure, the rotor designs have different total widths of nonmagnetic slots per pole. From the data in the table it follows that the best energy indicato—

Table 1

The rotor design with three nonmagnetic slots per pole has the highest power factor and maximum power (Fig. 1, c). If the central nonmagnetic slot is removed and only two nonmagnetic slots are left on the rotor (Fig. 1, b), this leads to a decrease in \(P_{2m}\) by 29%, in the power factor by 14.7%, and in efficiency by 4.6%. If only the central nonmagnetic slot is left on the rotor (Fig. 1, a), the motor characteristics deteriorate still more, since in this case the total width of the nonmagnetic slots is reduced. However, taking into account that there is not always sufficient space in the rotor yoke for placing a large number of additional slots, preference should be given to a rotor design with two additional slots. Only for small rotor diameters may a design with one slot per pole be selected.

a   b   c

Fig. 1. Rotor laminations of a synchronous reluctance motor with different numbers and arrangements of nonmagnetic slots

After the rotor design, the pole-arc coefficient, and the air-gap size have been selected, the next stage in designing the rotor of a synchronous reluctance motor is determining the optimal geometry of the tooth zone, i.e., selecting the dimensions and dimensional ratios of the teeth and slots of the starting winding.

In a synchronous reluctance motor, owing to the presence of additional nonmagnetic slots on the rotor and to the higher air-gap induction in comparison with induction motors, the magnetic circuit proves to be more saturated than in induction motors. Analysis of the designed machines shows that the magnetomotive force (m.m.f.) of the rotor amounts to from 10 to 25% of the m.m.f. of the entire magnetic circuit. If the geometry of the tooth zone is chosen incorrectly, the rotor m.m.f. may increase by a factor of 2–5 or more in comparison with the indicated value. Therefore, when choosing the geometry of the rotor tooth zone, one should strive to ensure a minimum value of the m.m.f.

If the area of the slot of the starting winding is specified and remains constant, then when the height or width of the rotor tooth is varied, the inductions in the yoke and in the rotor teeth, as can be seen from Fig. 2, change in opposite directions. Therefore, for certain ratios of tooth and slot dimensions, the total rotor m.m.f. will have a minimum (Fig. 3). Consequently, if such ratios of the tooth-zone geometry are chosen for which the rotor m.m.f. is minimal, then the resultant m.m.f. will also be minimal. This should lead to an increase in the power factor. Therefore, when selecting the width and height of the rotor tooth, one should proceed from the minimum rotor m.m.f. In doing so, let us assume that the height of the pole is equal to the height of the slot of the starting winding and to the height of the interpolar space. When determining the slot pitch of the starting winding, we assume that the slots are uniformly distributed over the entire circumference of the rotor.

Fig. 2. Dependences of the inductions in the rotor teeth and yoke and of their ratio on the tooth height

The magnetomotive force of the rotor is equal to the sum of the m.m.f. of the teeth and the yoke

\[ F_2 = 2h_z \cdot H_z + l_c \cdot H_c, \tag{1} \]

where \(H_z\) and \(H_c\) are the magnetic-field intensities in the teeth and in the rotor yoke, respectively,

\(h_z\) is the height of the pole or of the rotor tooth.

The mean length of the magnetic lines of force in the rotor yoke, \(l_c\), is determined by well-known formulas, and the mean height of the rotor yoke is

\[ h_{\mathrm{cp}}=\frac{1}{2}(D_2-d_{\mathrm{в}})-h_z-\delta_{\mathrm{в}}, \tag{2} \]

where \(d_{\mathrm{в}}\) is the shaft diameter,

\(\delta_{\mathrm{в}}\) is the width of the additional nonmagnetic slots in the rotor yoke.

Then, taking (2) into account, the rotor m.m.f. referred to the pole pitch is determined from the following expression:

\[ F'_2=2h'_z \cdot H_z + H_c \cdot \frac{\pi}{4p} \left( \frac{2p}{\pi}+d'_{\mathrm{в}}+2h'_z+2\delta'_{\mathrm{в}} \right), \tag{3} \]

where

\[ h'_z=\frac{h_z}{\tau}, \qquad d'_{\mathrm{в}}=\frac{d_{\mathrm{в}}}{\tau}, \qquad \delta'_{\mathrm{в}}=\frac{\delta_{\mathrm{в}}}{\tau}, \]

and it is assumed that \(D_2 \approx D\), where \(D\) and \(D_2\) are the stator and rotor bore diameters, respectively.

The minimum value of the m.m.f. and the corresponding value of \(h'_z\) can be determined from the condition that the first derivative of expression (3) is equal to zero. To differentiate this expression, the magnetization curve of the steel must be represented in analytical form. In the present case, however, it is simpler and more convenient to determine the minimum m.m.f. by direct calculation using formula (3). This method also makes it possible to establish the permissible limits of deviation from the optimum values of \(h'_z\).

The inductions in the yoke and in the teeth, by which the corresponding magnetic-field intensities are determined, are found from the expressions

\[ B_c = \frac{B_{md}\cdot \alpha_d} {k_c \cdot \left(\dfrac{2p}{\pi}-d'_{\mathrm{в}}-2\delta'_{\mathrm{в}}-2h'_z\right)} ;\qquad B_z=\frac{B_{md}\cdot t_2}{k_c \cdot b_z}, \tag{4} \]

\(\alpha_d\) is the calculated coefficient of the pole arc along the longitudinal axis. We express the width of the rotor tooth through the variable \(h'_z\). In low-power synchronous reluctance motors, oval slots are most often used on the rotor. For the area of an oval slot, the expression is valid

\[ Q_n=-\frac{b_z^2\cdot Z^2}{(Z-\pi^2)}\cdot K-\frac{b_z\cdot Z}{(Z-\pi)^2}\cdot(2MK+h_1\cdot N)+ \frac{M^2K+h_1MN-h_1^2\pi N}{(Z-\pi)^2}, \tag{5} \]

where

\[ K=-\frac{\pi(Z^2+\pi^2)}{4(Z+\pi)^2}-\frac{Z^2}{(Z+\pi)^2},\quad M=\pi(D_2-2h_{\mathrm{sh}}),\quad h_1=h_z-h_{\mathrm{sh}}; \]

\[ N=\frac{2Z-\pi^2}{2}, \]

\(Z\) is the number of rotor slots, on the assumption that the slots are distributed uniformly around the entire circumference,

\(h_{\mathrm{sh}}\) is the height of the rotor-slot slit.

From expression (5) we find the tooth width

\[ b_z= \frac{(2MK+h_1N)\pm\sqrt{(h_1N)^2+4K\left[h_1^2\pi N+Q_n(Z-\pi)^2\right]}} {2KZ}. \tag{6} \]

Substituting (6) into (4), we then obtain

\[ B_z= \frac{B_{md}} {k_c\left\{ 1.975K+\frac{h'_zN}{2p} \pm \sqrt{ \left(\frac{h'_zN}{2p}\right)^2+ 4K\left[\left(\frac{h'_z}{2p}\right)^2\pi N+ \frac{Q'_n(Z-\pi)^2}{\pi Z}\right] } \right\}}, \tag{7} \]

where

\[ Q'_n=\frac{Q_n\cdot Z}{\pi D^2}, \]

and \(Q_n\) is the area of one rotor slot.

In deriving formula (7) it was assumed that

\[ \frac{4h_{\mathrm{sh}}}{D_2}=0.025, \]

which is the case in actual motors; moreover, in view of the small value of \(h_{\mathrm{sh}}\) in comparison with the slot height, it was assumed that \(h'_1\approx h'_z\).

By varying \(h'_z\) within practically acceptable limits and under the condition that \(\delta'_B\), \(d'_B\), \(Q'_n\), and \(B_{md}\) are constant, formulas (3), (4), and (7) can be used to calculate the dependence \(F'_2=f(h'_z)\). Figure 3 gives such dependences for an SRM designed on the basis of the asynchronous series A02 and having

Fig. 3. Dependences of the rotor m.m.f. on tooth height for various values of \(B_{md}\).

the following initial data: \(2p=6\), \(Z=48\), \(d'_B=0.55\), \(\delta'_B=0.14\), \(B_{md}=0.75\ \mathrm{T}\), \(Q'_n=0.04\). From the curves of this figure it is evident that the rotor m.m.f., for example at \(B_{md}=0.75\ \mathrm{T}\), has a minimum at \(h'_{z\,\mathrm{opt}}=0.24\). If \(h'_z\) is chosen 29% smaller than the optimum value, the rotor m.m.f. increases by a factor of 2.57 compared with the minimum possible value. From the standpoint of reducing the coefficient \(k_q\), it is sufficient to choose the pole (tooth) height to be not less than \(0.1\tau\). Calculations show that with such

values of \(h'_z\) for the given motor, the rotor m.m.f. increases by tens of times. Consequently, when choosing the pole height it is also necessary to be guided by considerations of reducing the m.m.f.

The rotor m.m.f. is affected by the width of the additional nonmagnetic slot \(\delta'_v\). Calculations have shown that, as \(\delta'_v\) increases, the rotor m.m.f. increases, which is associated with an increase in the induction in the yoke, and reaches a minimum at smaller values of \(h'_z\).

The magnitude of the maximum induction in the air gap has a significant influence on the rotor m.m.f. As \(B_{md}\) increases, the rotor m.m.f. increases (Fig. 3), and its minimum shifts toward larger values of \(h'_z\).

The influence of the relative slot area on the m.m.f. and \(h'_{z\,\mathrm{opt}}\) is shown in Fig. 4. From the curves it is evident that, at small values of \(Q'_n\), the rotor m.m.f. changes little over a sufficiently wide range of variation of \(h'_z\). As \(Q'_n\) increases, the m.m.f. increases, since for one and the same slot height the tooth width decreases with increasing \(Q'_n\), which leads to an increase in the induction in the teeth. As \(Q'_n\) increases, larger values of \(h'_z\) should be selected so that the m.m.f. is minimal.

Fig. 4. Dependences of the rotor m.m.f. on tooth height, for various values of \(Q'_n\) (\(B_{md}=0.75\))

On the basis of the calculations, the optimal values of the slot height as a function of the slot area have been determined for motors of 0.6–10 kW, designed on the basis of the A02 series; these are given in Fig. 5. These data are convenient to use in the preliminary selection of the slot height.

Fig. 5. Optimal values of the rotor slot height as a function of the relative slot area (\(B_{md}=0.75\))

Similar calculations were performed for a series of SynRMs designed on the basis of the new 4A series of asynchronous motors with shaft heights of 80–132 mm. Preliminary calculations of these motors showed that the maximum induction in the air gap lies within the range 0.95–1.00 T. For \(B_{md}=0.95\) T and \(Q'_n=0.05\), the optimal values of slot height, tooth width, induction in the teeth, and rotor yoke are determined; they have the following values:

\[ h'_{z\ \mathrm{opt}}=0.18\text{--}0.20;\qquad b_{z\ \mathrm{opt}}=(0.47\text{--}0.53)t_2 \]

\[ B_{z\ \mathrm{opt}}=2.1\text{--}2.0\ \mathrm{T};\qquad B_{c\ \mathrm{opt}}=1.6\text{--}1.8\ \mathrm{T}. \]

The diameters of the oval rotor slot are determined from the expressions

\[ d_2=\frac{\pi(D_2-2h_{\mathrm{sh}})-Z\cdot b_z}{Z+\pi},\qquad d_1=\frac{d_2(Z+\pi)-2\pi(h_z-h_{\mathrm{sh}})}{Z-\pi}. \tag{8} \]

The distance between the centers of the slot circles is

\[ h_{12}=\frac{(h_z-h_{\mathrm{sh}})\cdot Z-d_2\cdot Z}{Z-\pi}. \tag{9} \]

The advisability of choosing the rotor slot geometry on the basis of the minimum of the no-load current is confirmed by the results of calculating the operating characteristics. Table 2 gives the rated data and maximum values of efficiency, power factor, and useful power for the SRO-31-4 motor with rated power of 2.2 kW, calculated for a slot height \(h'_z=0.206\), which is close to optimal, and with a 22% reduced height, while keeping the rotor slot area unchanged. If the geometric ratios of the rotor tooth zone are selected on the basis of the minimum no-load current, then, as can be seen from the data presented, not only does the power factor increase, but the efficiency also increases.

Table 2

In two-pole machines, because of the comparatively small rotor diameter, the possibilities for varying slot dimensions and for searching for their optimal values are limited. Therefore they have to be chosen on the basis of the maximum permissible inductions in the teeth and yoke, which are approximately 2.1 T.

Designing the geometry of the rotor tooth zone also includes the selection of the slot area. From the standpoint of reducing the no-load current and increasing the energy indices, it is necessary to choose it as small as possible. However, the starting characteristics also depend on the value of \(Q'_n\). Thus, for example, to increase the pull-in torque it is expedient to reduce the active resistance of the starting winding [2].

Analyzing the starting characteristics of the machines constructed and tested, one can recommend values of \(Q_n\) equal to 0.045–0.055, or take it equal to \((0.8\text{--}1.0)Q'_n\) of the corresponding asynchronous motor. With such values of \(Q'_n\), the SynRM has satisfactory starting characteristics.

The number of rotor slots is selected depending on the number of stator slots according to the recommendations of [3]; moreover, it must be a multiple of \(2p\). However, satisfaction of the latter condition is not always possible. In this

in this case, first one should determine an approximate value of the pole width from the condition

\[ b_p = (0.55—0.65)\tau, \]

and then it is also possible to determine the number of bars per pole, which must be an integer,

\[ n_c = \frac{b_p}{t_2} - 1. \]

The pitch of the bars of the starting winding is selected from the same conditions as for ordinary synchronous motors [3]

\[ t_2 \ne t_1, \qquad t_2 \geq 0.8t_1. \]

Then the dimensions of the rotor slots are determined. Thus, the described method for determining the geometry of the tooth zone of the rotor can be used in designing synchronous reluctance motors.

REFERENCES

1. E. V. Kononenko. Synchronous reluctance machines. Moscow, “Energiya,” 1970.

2. E. V. Kononenko, G. I. Lukiyanov. On the design of optimal synchronous reluctance motors. Izvestiya TPI, vol. 242, Tomsk, 1972.

3. P. S. Sergeev, N. V. Vinogradov, F. A. Goryanov. Design of electrical machines. Moscow, “Energiya,” 1969.