On the Selection of Main Dimensions of Improved Synchronous Reluctance Motors
Abstract Generated abstract
The paper addresses the preliminary design of improved synchronous reluctance motors, focusing on selection of the stator bore diameter and stack length. It proposes a simplified method based on electromagnetic loadings and the standard machine constant formula, deriving expressions that relate calculated power, winding and saliency coefficients, air-gap induction, linear loading, and the main dimensions. The authors give practical guidance for choosing coefficients, loadings, power factor, efficiency, diameter-to-length ratio, and standard stator outside diameters, with adjustments relative to comparable induction motors. A worked 3 kW, four-pole example illustrates the calculation and refinement of the principal dimensions.
Full Text
On the Selection of the Main Dimensions of Improved Synchronous Reluctance Motors
E. V. Kononenko, G. I. Lukiyanov
(Presented by the scientific seminar of the Department of Electrical Machines)
In designing new electric motors, such as improved synchronous reluctance motors (SRM), it is first of all necessary to determine the principal geometric dimensions—the bore diameter \((D)\) and the stack length \((l)\) of the stator. As in other electrical machines, the weight, cost, and techno-economic characteristics of an SRM depend on the relationship between \(D\) and \(l\).
At present, in the technical literature there are a number of methods for determining the main dimensions of an SRM [1, 2]. In [1] a method is proposed for selecting the main dimensions according to a specified value of the maximum torque in synchronous operation or according to a specified value of overload capacity. In [2] it is proposed to determine the main dimensions according to the base power, which is taken as
\[ P_{\text{base}}=\frac{3}{2}\cdot \frac{U^{2}}{X_m}, \]
where, in order to determine the latter, the induction \(B_\delta\) is specified and, instead of a linear load, the relative air gap
\[ \delta'=\frac{\delta}{\tau}. \]
The application of these methods presupposes the use of experimental data available for motors with power up to 600 W. In addition, calculations by these methods require a considerable expenditure of time in determining the main dimensions, especially at powers above 0.6 kW.
In the present work a method is proposed for determining the main dimensions of an SRM on the basis of the electromagnetic loadings \(A\) and \(B_{md}\), using the well-known machine-constant formula for an ordinary synchronous or asynchronous motor.
It is known that the calculated power of an SRM can be determined as follows:
\[ P_p=m\cdot E\cdot I_1 \tag{1} \]
Let us express the calculated power through the electromagnetic loadings and the main dimensions.
According to [3], the first harmonic component of the phase EMF, producing the magnetic flux along the direct axis, is
\[ E_{aq1}=\sqrt{2}\pi\cdot f\cdot W_1\cdot K_w\cdot \Phi_{d1}, \tag{2} \]
where
\[ f=\frac{p\cdot n}{60} \]
is the frequency of the current in the network;
\(W_1\) — number of turns of a phase of the stator winding;
\(K_w\) — winding factor;
\(\Phi_{d1}\) is the first harmonic component of the magnetic flux along the direct axis.
The current of a phase of the stator winding is
\[ I_1=\frac{A\cdot \pi \cdot D}{2\cdot m\cdot W_1}, \tag{3} \]
where \(A\) is the linear loading.
Substituting formulas (2) and (3) into (1), we obtain
\[ P_p=\frac{\sqrt{2}\pi^2}{2}\,K_w\cdot p\cdot n\cdot \Phi_{d1}\cdot A\cdot D. \tag{4} \]
Since
\[ \Phi_{d1}=\alpha_d\cdot \tau \cdot l\cdot B_{ad1}, \tag{5} \]
where \(\alpha_d\) is the calculated pole-arc coefficient;
\(\tau\) is the pole pitch;
\(B_{ad1}\) is the first harmonic induction along the direct axis,
then
\[ P_p=\frac{\sqrt{2}\pi^3}{4}\cdot K_w\cdot \alpha_d\cdot D^2\cdot l\cdot n\cdot A\cdot B_{ad1}. \tag{6} \]
Usually, when determining the main dimensions, one operates with the maximum induction in the air gap. According to [4], the latter is equal to
\[ B_{md}=\frac{B_{ad1}}{K_d}, \tag{7} \]
where \(K_d\) is a coefficient characterizing the degree of reduction of the amplitude of the first harmonic of the field along the direct axis due to the saliency of the rotor.
Taking this into account, expression (6) can be written as follows:
\[ P_p=\frac{\sqrt{2}\pi^3}{4}\,K_w\cdot \alpha_d\cdot K_d\cdot D^2\cdot l\cdot n\cdot A\cdot B_{md}. \tag{8} \]
From equation (8) we obtain the expression for the machine constant
\[ C_A=\frac{D^2\cdot l\cdot n}{P_p} =\frac{4}{\sqrt{2}\pi^3\cdot K_w\cdot \alpha_d\cdot K_d\cdot A\cdot B_{md}}, \tag{9} \]
whence
\[ D=\sqrt[3]{C_A\cdot \frac{P_p\cdot \gamma}{n}}, \tag{10} \]
where
\[ \gamma=\frac{D}{l}. \]
Expression (10) makes it possible to determine the main dimensions of an SRM. For this purpose it is first necessary to specify the electromagnetic loadings and a number of coefficients. In determining \(C_A\), one may take \(K_w=0.92\text{--}0.93\) for a two-layer winding with a shortened pitch and \(K_w=0.96\) for a single-layer winding with diametral pitch; the values of the coefficients \(\alpha_d\) and \(K_d\) are determined from the curves in [4].
The power of an SRM in nominal operation is determined by the permissible overheating of the stator winding [5]. In an SRM there are no electrical losses in the rotor. Consequently, in order that the overheating of the stator winding be the same as in the corresponding induction motor, it is necessary to increase the linear loading of the SRM in comparison with the linear loading of an induction motor of the same design and dimensions by 10–15%.
Usually, for better utilization of electrical steel, the magnetic system of an SRM is made more saturated than in induction motors. When selecting the induction in the air gap, this circumstance must be taken into account and \(B_{md}\) increased by 10–15% in comparison with \(B_\delta\) of an induction motor.
The calculated power can be determined by the formula
\[ P_p=\frac{K_{Ed}\cdot P_n}{\eta_n\cdot \cos\varphi_n}. \tag{11} \]
Approximate values of the coefficient \(K_{Ed}\):
\[ \begin{aligned} K_{Ed}&=0{,}93\text{—}0{,}96 &&\text{for } 2p=2,\\ K_{Ed}&=0{,}86\text{—}0{,}96 &&\text{for } 2p=4,\\ K_{Ed}&=0{,}86\text{—}0{,}94 &&\text{for } 2p=6,\\ K_{Ed}&=0{,}86\text{—}0{,}87 &&\text{for } 2p=8. \end{aligned} \]
The nominal values of efficiency are selected the same as for an induction motor of the corresponding power. \(\cos\varphi_n\) is selected 15–20% lower than \(\cos\varphi_n\) of an induction motor of the corresponding power. The linear loading, maximum induction in the air gap, \(\cos\varphi_n\), and efficiency of induction motors are determined according to [6].
The ratio \(\gamma=\dfrac{D}{l}\), when designing an SRM on the basis of an induction motor, is taken the same as for an induction motor. In the general case it is desirable to increase this ratio, especially in 2- and 4-pole designs in comparison with an induction motor. This is due to the fact that improved SRMs have additional nonmagnetic slots on the rotor, the width of which, in design, should be chosen as large as possible from the standpoint of saturation of the rotor yoke, but not less than \((0{,}05\text{—}0{,}075)\tau\).
The outside diameter of the stator \(D_a\) is determined from the relation
\[ D_a \approx D+0{,}8\tau, \]
or from the relations:
\[ \begin{aligned} D_a&=(1{,}95\text{—}1{,}85)D &&\text{for } 2p=2,\\ D_a&=(1{,}61\text{—}1{,}56)D &&\text{for } 2p=4,\\ D_a&=(1{,}44\text{—}1{,}41)D &&\text{for } 2p=6,\\ D_a&=(1{,}39\text{—}1{,}34)D &&\text{for } 2p=8. \end{aligned} \]
The diameter \(D_a\) is rounded to the nearest standard value according to Table 1.
Table 1
| 1st size | 2nd size | 3rd size | 4th size | 5th size | 6th size | 7th size | 8th size |
|---|---|---|---|---|---|---|---|
| mm 133 |
mm 153 |
mm 180 |
mm 208 |
mm 243 |
mm 291 |
mm 343 |
mm 393 |
After this, \(D\) and \(l\) are refined.
Example of determining the main dimensions
- Initial data:
\[ P_n=3\ \text{kW};\qquad 2p=4;\qquad f=50\ \text{Hz}, \]
- Determine \(P_p\) from (11)
\[ P_p=\frac{0.86\cdot 3000}{0.85\cdot 0.71}=4270\ \text{(VA)}, \]
where
\[ K_{Ed}=0.86;\quad \eta_n=0.85;\quad \cos\varphi_n=0.71. \]
- Determine \(C_A\) from (9).
\[ C_A=\frac{4\cdot 10^4}{\sqrt{2\pi^3\cdot 0.96\cdot 0.55\cdot 0.92\cdot 265\cdot 0.81}} =8.8\ \text{(cm}^3/\text{J)}, \]
where
\[ K_w=0.96;\quad \alpha_d=0.55;\quad K_d=0.92;\quad A=265\ \text{(A/cm)},\quad B_{md}=8100\ \text{(G)}. \]
- Taking \(\gamma=0.97\), determine \(D\) from (10)
\[ D=\sqrt[3]{8.8\,\frac{4270\cdot 0.97}{50/2}}=11.3\ \text{(cm)}. \]
- Determine \(D_a\)
\[ D_a=(1.61\text{--}1.56)D=18.2\div 17.2\ \text{(cm)}. \]
We choose the standard value \(D_a=18\ \text{(cm)}\).
- Refine \(D\) and determine \(l\)
\[ D=\frac{D_a}{1.61}=\frac{18}{1.61}=11.2\ \text{(cm)}; \]
\[ l=\frac{D}{\gamma}=\frac{11.2}{0.97}=11.5\ \text{(cm)}. \]
REFERENCES
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I. M. Postnikov, V. V. Ralle. On the calculation of three-phase synchronous reluctance motors. In: Problems of Technical Electrodynamics, Publishing House “Naukova dumka,” Kiev, 1965.
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E. V. Kononenko, A. N. Aifert. Investigation of the magnetic field of simple synchronous reluctance motors. Proceedings of TPI, vol. 172, 1967.
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E. V. Kononenko. Comparison of synchronous-reluctance, synchronous, and asynchronous motors. Izvestiya Vuzov, “Electromechanics,” No. 4, 1965.
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Winding Data of Asynchronous Motors. Edited by P. I. Tsibulevsky, “Energiya,” 1966.