Influence of Rotor Design and Magnetic Circuit Saturation on the Energy Performance of Synchronous Reluctance Motors
Abstract Generated abstract
This paper examines how rotor geometry and magnetic circuit saturation affect the power factor and electromagnetic torque of synchronous reluctance motors. Using previously derived magnetic field coefficients, the authors analyze simple and improved rotor designs by relating air gap size, pole overlap, synchronous reactance ratio, and saturation coefficients to energy performance. The results indicate that simple motors can gain power factor and torque from saturation but are limited by excessive induction and heating at small pole overlaps, while improved motors are less sensitive to geometric variation and should use pole overlaps of at least 0.55 to 0.60 with lower saturation. Prototype improved motors built in asynchronous motor dimensions achieved comparable rated power, efficiency, and stator temperature rise, though with a lower power factor.
Full Text
THE EFFECT OF ROTOR DESIGN AND MAGNETIC-CIRCUIT SATURATION ON THE ENERGY PERFORMANCE OF SYNCHRONOUS RELUCTANCE MOTORS
E. V. Kononenko, A. N. Aifert, G. I. Lukiyanov
(Presented by the scientific seminar of the departments of electrical machines and general electrical engineering)
The energy performance of synchronous reluctance motors (SRM) depends to a considerable extent on the selected geometry of the air gap. Comprehensive studies of magnetic fields in the air gap of SRMs, both simple [1] and improved [2], have made it possible to determine coefficients characterizing the magnetic field in the air gap of such motors, to calculate the synchronous reactances and the magnetic circuit of the machine. In addition, such studies have made it possible to assess the influence of the air-gap geometry on the magnetic-field coefficients. Using the magnetic-field coefficients determined in the above-mentioned works, it is also possible to select a rotor design that ensures the best energy and weight performance of an SRM.
When choosing the optimal rotor design for an SRM, one may be guided by the fact that it must provide large values of the maximum power factor \((\cos \varphi_{\min})\) and electromagnetic torque \((M_m)\), with the same overall dimensions and stator-winding temperature rises not exceeding the permissible values. Neglecting the influence of the active resistance of the stator winding, the expressions for \(\cos \varphi_{\min}\) and \(M_m\) may be written in the form [3]
\[ \cos \varphi_{\min} = \frac{\dfrac{x_d}{x_q} - 1}{\dfrac{x_d}{x_q} + 1}; \tag{1} \]
\[ M_m = \frac{U^2}{2x_d} \left( \frac{x_d}{x_q} - 1 \right); \tag{2} \]
where \(U\) is the supply-network voltage,
\(x_d, x_q\) are the synchronous reactances along the direct and quadrature axes, respectively.
From these expressions it is evident that \(\cos \varphi_{\min}\) is determined only by the ratio \(\dfrac{x_d}{x_q}\), while \(M_m\), in addition, is inversely proportional to \(x_d\). It is known that the synchronous reactances along the direct and quadrature axes are directly proportional to the field-form coefficients \((k_d, k_q)\), inversely proportional to the air-gap magnitude \((q)\) and to the magnetic-circuit saturation coefficient \((k_{\mu d}\) and \(k_{\mu q})\). Consequently, \(\cos \varphi_{\min}\) and
\(M_m\) are determined to a considerable extent by the design of the SRD rotor. Knowing the dependence of the magnetic-field coefficients on the geometrical dimensions of the rotor, one can estimate the influence of the geometry on \(\cos\varphi_{\min}\) and \(M_m\).
Simple SRDs
Assuming at first that the magnetic circuit of the SRD is not saturated \((k_{\mu d}=k_{\mu q}\approx 1.0)\), and using the results of the investigation of the magnetic field of simple SRDs [1], one can calculate the dependence of \(\cos\varphi_{\min}\) and \(M_m\) on the size of the air gap and the pole overlap. These dependences are presented in Fig. 1. As is seen from Fig. 1a, the ratio \(\dfrac{x_d}{x_q}\) and \(\cos\varphi_{\min}\) decrease with increasing \(\delta\), while \(M_m\) increases. Larger values of \(\dfrac{x_d}{x_q}\), \(\cos\varphi_{\min}\), and \(M_m\) correspond to a smaller value of \(\alpha\) (Fig. 1b).
Using the magnetic-field coefficients [1], one can also estimate the influence of saturation of the magnetic circuit on the operation of simple SRDs. Under the adopted assumption \((r=0)\), the electromagnetic torque is equal to the maximum at \(\Theta=45^\circ\). In this case the magnetic fluxes along the longitudinal and transverse axes are equal to one another. Since \(\alpha_q<\alpha_d\), and \(k_{\phi q}>k_{\phi d}\), the magnetic circuit along the transverse axis proves to be more saturated and \(k_{\mu q}>k_{\mu d}\) [1, 4]. Therefore, when saturation is taken into account, the ratio \(\dfrac{x_d}{x_q}\) increases, while \(x_d\) decreases, which leads to an increase in \(\cos\varphi_{\min}\).
Fig. 1. Dependence of \(\dfrac{x_d}{x_q}\), \(M_m\), and \(\cos\varphi_{\min}\) on
a) the air gap at \(\alpha=0.5\), and b) the pole overlap at \(\delta=0.003\tau\), for simple SRDs.
and \(M_m\). If it is assumed that, at \(\alpha = 0.5\) and \(\delta = 0.003\tau\), the saturation coefficients are \(k_{\mu d}=1.25\) and \(k_{\mu q}=1.5\), then, as a result of taking saturation into account, the ratio \(\dfrac{x_d}{x_q}\) increases from 3.52 to 3.95, and \(M_m\) from 0.58 to 0.84. If it is assumed that, in determining \(\cos\varphi_{\min}\), the ratios \(\dfrac{x_d}{x_q}\) remain the same as in the calculation of \(M_m\), then the maximum value of the power factor will increase from 0.56 to 0.595.
Despite the fact that simple SRMs have larger values of \(\dfrac{x_d}{x_q}\) at a smaller pole overlap, the choice \(\alpha \leq 0.4\) is inadvisable because of the considerable increase in magnetic induction in the air gap and saturation of the magnetic circuit. Thus, at \(\alpha = 0.4\) and \(\delta = 0.003\tau\), the maximum induction in the air gap of an SRM along the longitudinal axis increases by a factor of 1.38 in comparison with the maximum induction of an asynchronous motor, if motors of the same overall dimensions and having identical stators are compared. Modern asynchronous motors are made with a saturated magnetic circuit. An increase in the maximum induction by a factor of 1.38 leads to the no-load current of the SRM increasing almost to the rated value, the iron losses increasing, and the overheating of the stator winding, even under no-load operation, exceeding the permissible value. To reduce overheating it is necessary to reduce the magnetic flux, which is associated with a decrease in \(M_m\).
Improved SRMs
An analysis of the influence of the rotor design of improved SRMs on \(M_m\) and \(\cos\varphi_{\min}\) can be carried out in the same way as for simple SRMs. Figure 2 presents the dependences of \(M_m\), \(\cos\varphi_{\min}\), and \(\dfrac{x_d}{x_q}\) on the air gap (Fig. 2a) and on the pole overlap (Fig. 2b), calculated for an SRM whose magnetic-field coefficients are determined according to [2] under the condition that \(k_{\mu d}=k_{\mu q}\approx 1.0\). Comparing these dependences with those given in Fig. 1, we see that, in improved SRMs, variation of \(\delta\) and \(\alpha\) has a smaller effect on \(M_m\) and \(\cos\varphi_{\min}\) than in simple SRMs.
The smaller effect on \(M_m\) and \(\cos\varphi_{\min}\) in improved SRMs is also produced by taking into account saturation of the magnetic circuit. Despite the fact that \(\alpha_q\) is comparatively small, the saturation coefficient of the magnetic circuit along the transverse axis of such machines is not large, owing to the presence of internal nonmagnetic gaps [2]. Therefore, in calculating \(M_m\) and \(\cos\varphi_{\min}\), one may assume that \(k_{\mu d}\geq k_{\mu q}\). If it is assumed that, at \(\alpha = 0.6\) and \(\delta = 0.003\tau\), the saturation coefficients of the magnetic circuit are \(k_{\mu d}=k_{\mu q}\approx 1.2\), then, as a result of taking saturation into account, the ratio \(\dfrac{x_d}{x_q}\) decreases from 7.8 to 7.4, \(\cos\varphi_{\min}\) from 0.774 to 0.764, and \(M_m\) increases from 1.46 to 1.57.
Taking the foregoing into account, improved SRMs should expediently be made with less saturation of the magnetic circuit in comparison with simple SRMs. Therefore, the pole overlap in improved SRMs should be chosen not less than \(0.55 \div 0.60\).
On the basis of the recommendations set forth above, prototypes of improved SRMs were designed and manufactured at the Special Design Bureau of Electrical Machine Building, on the basis and in the dimensions of A02 asynchronous motors of the third frame size. The principal quantities characterizing the prototype improved SRMs are given in Table 1. Here
Fig. 2. Dependence of \(\dfrac{x_d}{x_q}\), \(M_m\), and \(\cos\varphi_{\min}\) on
a) the air gap for \(\alpha_1=0,\ \alpha=0.6\); and b) the pole overlap for \(\alpha_1=0,\ \delta=0.003\tau\), for improved SRM
For comparison, the corresponding data are also given for asynchronous motors, in whose dimensions the SRM are made. It is seen from the table that the improved SRM have energy characteristics approaching those of such widely used motors as asynchronous motors, with a somewhat smaller value of the power factor.
Table 1
| Motor type | \(P_n\), kW | \(n_n\), rpm | \(\eta_n\), % | \(\cos\varphi_n\) | \(\dfrac{M_{\text{start}}}{M_n}\) | \(\dfrac{M_{\max}}{M_n}\) | \(\dfrac{M_{\text{in}}}{M_n}\) | \(\Delta t\) |
|---|---|---|---|---|---|---|---|---|
| SRO-31-4 | 2.2 | 1500 | 81.3 | 0.655 | 3.4 | 1.7 | 0.99 | 60° C |
| AO2-31-4 | 2.2 | 1430 | 82.0 | 0.83 | 1.8 | 2.15 | — | 54° C |
| SRO-32-4 | 3.0 | 1500 | 84.0 | 0.666 | 4.0 | 1.62 | 1.26 | 59° C |
| AO2-32-4 | 3.0 | 1430 | 84.0 | 0.837 | 1.9 | 2.4 | — | 56° C |
Thus, with the correct choice of the parameters of the air-gap geometry and of the saturation of the magnetic circuit, within the dimensions of asynchronous motors it is possible to produce improved SRM with the same rated power and stator-winding overheating.
References
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A. N. Aifert, E. V. Kononenko. Study of the magnetic field of simple synchronous reactive motors. Izv. TPI, vol. 172, 1967.
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A. N. Aifert, E. V. Kononenko. The magnetic field in the air gap of improved synchronous reactive motors. In: Electronic and Electromechanical Devices, Moscow, Izd. “Energiya,” 1968.
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E. V. Kononenko, B. P. Garganeev, A. L. Kislitsyn. Some questions of the theory and design of three-phase synchronous-reactive motors. In: Questions of the Theory and Design of Electrical Machines, issue 3, Novosibirsk, 1963.
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E. V. Kononenko, A. N. Aifert, B. P. Garganeev. Allowance for the influence of saturation on the parameters of synchronous reactive motors. Izvestiya TPI, vol. 160, 1966.