Comparative Characteristics of Synchronous Reluctance Motors
Abstract Generated abstract
This paper critiques published comparisons of synchronous reluctance motors with induction and synchronous motors, arguing that earlier analyses used parameter assumptions that lead to unrealistic current and power estimates. Using equivalent circuit relations and simplified power expressions, it recalculates comparative performance while accounting for rated current limits, overload factor, direct and quadrature axis reactances, and the dependence of air gap choice on the reactance ratio. The authors conclude that synchronous reluctance motors compare less favorably with induction motors under the parameters previously assumed, but that improved rotor design and higher direct-to-quadrature reactance ratios can substantially increase power capability. They emphasize that valid comparisons must use mutually consistent machine parameters rather than assuming equal stators or equal direct-axis reactance without adjustment.
Full Text
COMPARATIVE PROPERTIES OF SYNCHRONOUS-RELUCTANCE MOTORS
E. V. KONONENKO, B. P. GARGANEEV, A. L. KISLITSYN
(Presented by the scientific seminar of the departments of Electrical Machines and General Electrical Engineering)
In connection with the growing interest in synchronous-reluctance motors, a number of works devoted to the study of these motors have recently been published in the journal Elektrichestvo [1—4]. It is impossible to agree with some of the propositions of these works, especially with the results of the comparative analysis of the properties of a synchronous-reluctance motor and a normal asynchronous motor with a squirrel-cage rotor [1].
The comparison of the characteristics of synchronous-reluctance motors with asynchronous machines is based on the use of the equivalent circuit proposed by O. B. Pevzner [2]. However, the results of the comparative analysis of synchronous-reluctance motors with asynchronous ones [1] cannot be regarded as correct for the following reason. In the analysis it was assumed that the machines being compared have identical stators, while the synchronous-reluctance motor has the following parameters:
\[ x_{ad}=1.0;\quad x_s=0.1;\quad r=0.04;\quad \frac{x_{ad}}{x_{aq}}=5 \]
(here and below all quantities are given in relative units). Such parameters cannot be regarded as real and therefore cannot be taken as the basis of a comparative analysis. Indeed, substituting the above parameter values into the equivalent circuit [1], we obtain that the ideal no-load current is \(I_1=I_0=0.92\). As the load is increased, the current rises, and at \(M=M_{\max}\) it will be equal to \(I_1 \cong 2.5\). Suppose that in synchronous-reluctance motors the static overload factor is
\[ K=\frac{M_{\max}}{M_n}=1.5, \]
then the stator current at rated load is equal to \(I_1=1.3 \div 1.4\), which exceeds the permissible value by 30–40%.
In order to determine what power can be developed by a synchronous-reluctance motor built on the basis of the same stator as an asynchronous one, it is necessary to reduce the current at rated load. Assuming that the parameters \(r\), \(x_s\), and also the ratio \(\frac{x_{ad}}{x_{aq}}\) remain unchanged, the reduction of the current can be obtained by increasing the reactance \(x_{ad}\). With the parameter values accepted above, the stator current will be equal to the rated current at rated load if
$x_{ad} \simeq 1.7$. Consequently, when comparing synchronous-reluctance and induction motors, with the parameter values adopted above it is necessary to take $x_{ad}=1.7$, and not $x_{ad}=1.0$ [1]. In this case the maximum torque of the synchronous-reluctance motor will amount to only 45% of the maximum torque of a normal induction machine built on the basis of the same stator, and not 60–70% [1].
The energy performance indices of synchronous-reluctance motors can be improved by increasing the ratio $\dfrac{x_d}{x_q}$. By improving the rotor design one can obtain $\dfrac{x_d}{x_q}=6$ and more [7]. In this case the power of the synchronous-reluctance motor increases considerably, which makes it possible, within the dimensions of an induction motor, to design a synchronous-reluctance motor of the same power [8].
A comparative analysis of synchronous-reluctance and synchronous machines has the same shortcomings. Indeed, when comparing the maximum powers of synchronous-reluctance and synchronous motors, one usually proceeds from equations for electromagnetic power obtained under the condition that the active resistance of the stator winding is neglected [5, 6]. It is assumed here that the synchronous reactances along the direct axis in the motors are equal, which makes the comparative analysis incorrect.
It is known that the maximum electromagnetic power of a synchronous nonsalient-pole motor, under the assumptions made, is equal to
\[ P_{\mathrm{mm}}^{\mathrm{c}}=\frac{mUE_0}{x_d^{\mathrm{c}}}. \tag{1} \]
Under the condition that $U=E_0$,
\[ P_{\mathrm{mm}}^{\mathrm{c}}=\frac{mU^2}{x_d^{\mathrm{c}}}. \tag{2} \]
The maximum electromagnetic power of a synchronous-reluctance motor, under the same assumptions, is equal to:
\[ P_{\mathrm{mm}}^{\mathrm{cp}}= \frac{mU^2}{2} \left( \frac{1}{x_q^{\mathrm{cp}}} - \frac{1}{x_d^{\mathrm{cp}}} \right) = \frac{mU^2}{2x_d^{\mathrm{cp}}} \left( \frac{x_d^{\mathrm{cp}}}{x_q^{\mathrm{cp}}} - 1 \right). \tag{3} \]
On the basis of the assumptions made above concerning the equality of the synchronous reactances, the ratio of the maximum powers of synchronous-reluctance and synchronous motors of identical dimensions will be
\[ C=\frac{P_{\mathrm{mm}}^{\mathrm{cp}}}{P_{\mathrm{mm}}^{\mathrm{c}}} = \frac{1}{2} \left( \frac{x_d^{\mathrm{cp}}}{x_q^{\mathrm{cp}}} - 1 \right). \tag{4} \]
Substituting $\dfrac{x_d^{\mathrm{cp}}}{x_q^{\mathrm{cp}}}=3$ into equation (4), we obtain $C=1;0$. This indicates that the power of a synchronous-reluctance motor in the same dimensions is equal to the power of a synchronous motor. If, however, the ratio $\dfrac{x_d^{\mathrm{cp}}}{x_q^{\mathrm{cp}}}$ is greater than 3, then it follows from equation (4) that, in the same dimensions, a synchronous-reluctance motor can be built for a pow-
is considerably greater than that of a synchronous motor. Thus, for \(\dfrac{x_d^{\mathrm{cp}}}{x_q^{\mathrm{cp}}}=5\), which is quite realistic for synchronous-reluctance motors, \(C=2.0\).
These data indicate that a comparative analysis based on equality of the reactances \(x_d\) [5, 6] cannot be considered correct.
In making the comparison it is necessary to take into account that, in a properly designed synchronous-reluctance motor, the reactance \(x_d^{\mathrm{cp}}\) depends on the ratio \(\dfrac{x_d^{\mathrm{cp}}}{x_q^{\mathrm{cp}}}\). To construct the dependence \(x_d^{\mathrm{cp}}=f\left(\dfrac{x_d^{\mathrm{cp}}}{x_q^{\mathrm{cp}}}\right)\), as a first approximation the active resistance of the stator winding may be neglected. In this case, from the simplified circle diagram of a synchronous-reluctance motor we determine
\[ I_{\mathrm{n}}=\frac{U}{x_d^{\mathrm{cp}}}\sqrt{\left[\left(\frac{x_d^{\mathrm{cp}}}{x_q^{\mathrm{cp}}}\right)^2-1\right]\sin^2\Theta_{\mathrm{n}}+1} \tag{5} \]
or
\[ x_d^{\mathrm{cp}}=\frac{U}{I_{\mathrm{n}}}\sqrt{\left[\left(\frac{x_d^{\mathrm{cp}}}{x_q^{\mathrm{cp}}}\right)^2-1\right]\sin^2\Theta_{\mathrm{n}}+1}. \tag{6} \]
On the basis of equation (6), under the condition of constant dimensions and linear stator loading, it is possible to determine how the synchronous reactance \(x_d^{\mathrm{cp}}\) varies as a function of the ratio \(\dfrac{x_d^{\mathrm{cp}}}{x_q^{\mathrm{cp}}}\).
Fig. 1. Dependence of \(x_d^{\mathrm{cp}}\) on \(\dfrac{x_d^{\mathrm{cp}}}{x_q^{\mathrm{cp}}}\) for a synchronous-reluctance motor, calculated under the condition that the nominal mode corresponds to the mode in which \(M_{\mathrm{n}}=\dfrac{M_{\max}}{1.5}\).
Fig. 2. Ratio of the maximum powers of a synchronous-reluctance and a synchronous motor as a function of \(\dfrac{x_d^{\mathrm{cp}}}{x_q^{\mathrm{cp}}}\).
This dependence, under the condition that
\[ K=\frac{M_{\max}}{M_{\mathrm{n}}}=1.5 \]
(and \(\Theta_{\mathrm{n}}=21^\circ\)), is presented in Fig. 1. From the data presented it follows that, with an increas-
...as the ratio
\[ \frac{x_d^{\mathrm{av}}}{x_q^{\mathrm{av}}} \]
increases, the air gap must be reduced, and vice versa.
Usually, however, the magnitude of the air gap in synchronous-reluctance motors is recommended to be chosen in the same way as in induction motors, independently of the ratio
\[ \frac{x_d^{\mathrm{av}}}{x_q^{\mathrm{av}}} \]
[4, 5], which cannot be regarded as correct.
When carrying out a comparative analysis of synchronous and synchronous-reluctance motors, it is necessary to take into account the obtained dependence of \(x_d^{\mathrm{av}}\) on
\[ \frac{x_d^{\mathrm{av}}}{x_q^{\mathrm{av}}}. \]
Equation (4), taking into account the above, will have the form
\[ C_1=\frac{P_{\mathrm{mm}}^{\mathrm{av}}}{P_{\mathrm{mm}}^{c}} =\frac{x_d^c}{2x_d^{\mathrm{av}}} \left(\frac{x_d^{\mathrm{av}}}{x_q^{\mathrm{av}}}-1\right). \tag{7} \]
Calculations performed according to equation (7), under the assumption that the compared synchronous motor has \(x_d^c=1.0\), are presented in Fig. 2. From the data given it follows that, with an increase in
\[ \frac{x_d^{\mathrm{av}}}{x_q^{\mathrm{av}}}, \]
the active power developed by the synchronous-reluctance motor increases. At
\[ \frac{x_d^{\mathrm{av}}}{x_q^{\mathrm{av}}}=6 \]
the synchronous-reluctance motor will develop a power approximately equal to the power of a synchronous motor built on the basis of the same stator. Taking into account that for small machines \(x_d^c>1.0\), equality \(C_1=1.0\) will be satisfied at
\[ \frac{x_d^{\mathrm{av}}}{x_q^{\mathrm{av}}}<6. \]
Conclusions
The comparative properties of synchronous-reluctance motors available in the literature incorrectly reflect reality, since when they are compared with induction and synchronous machines, unrealistic parameters are taken as the basis of the analysis.
When carrying out a comparative analysis, it is necessary to take into account that the parameters of a properly designed motor, \(x_d^{\mathrm{av}}\), \(x_q^{\mathrm{av}}\) (and, consequently, the magnitude of the air gap), depend on the ratio
\[ \frac{x_d^{\mathrm{av}}}{x_q^{\mathrm{av}}}. \]
References
-
V. M. Kutsevalov. Relative characteristics and comparative properties of a synchronous reluctance machine. Elektrichestvo, No. 10, 1962.
-
O. B. Pevzner. Equivalent circuit and exact circle diagram of a synchronous-reluctance machine. Elektrichestvo, No. 2, 1960.
-
I. M. Postnikov. Hodograph of current and universal equivalent circuit of synchronous salient-pole and parametric machines. Elektrichestvo, No. 8, 1960.
-
I. M. Postnikov, V. V. Ralle. On determining the dimensions of a parametric (reluctance) motor. Elektrichestvo, No. 3, 1963.
-
N. P. Ermolin. Low-Power Electrical Machines. Vysshaya Shkola, 1962.
-
A. Ya. Berger. Synchronous Machines. ONTI, 1938.
-
Kavamura. Synchronous reluctance motors. Toshiba Rebyu, 14, No. 9, 1959.
-
R. J. Dineen. The synduction motor. Allis-Chalmers Electrical Review, 21, No. 4, 1956.