On the calculation of the xd, xq reactances of synchronous reluctance motors
Abstract Generated abstract
This paper examines the calculation of direct- and quadrature-axis reactances in synchronous reluctance motors and argues that some published methods incorrectly substitute armature-reaction coefficients for field-form coefficients. Treating the reluctance motor as a special case of a salient-pole synchronous machine, the authors compare the meanings of these coefficients and derive relations showing the resulting error when the wrong coefficients are used. Using analytical formulas and field-pattern based coefficient ratios, they show that the error in calculated reactances can be substantial, reaching about 34 percent for a pole-arc coefficient of 0.4, and is nearly the same for both axes.
Full Text
ON THE CALCULATION OF THE REACTANCES \(x_d, x_q\) OF SYNCHRONOUS RELUCTANCE MOTORS
E. V. Kononenko, B. P. Garganeev, A. N. Aifert
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
In calculating the characteristics of synchronous reluctance motors, the correct determination of the synchronous reactances \(x_d, x_q\) is of great importance. However, in [1] the determination of the reactances \(x_d, x_q\) is carried out, in our opinion, incorrectly. This is explained as follows. In [1] \(x_{ad}, x_{aq}\) are calculated on the basis of the formulas:
\[ x_{ad} = x_m \frac{k_d}{k_{\text{н}d}}; \tag{1} \]
\[ x_{aq} = x_m \frac{k_q}{k_{\text{н}q}}, \tag{2} \]
where \(k_d, k_q\) are the coefficients of the direct- and quadrature-axis armature reactions (given in the form of curves determined on the basis of constructing field diagrams);
\(k_{\text{н}d}, k_{\text{н}q}\) are the saturation coefficients in the direct and quadrature axes;
\[ x_m = \frac{4}{\pi} \cdot f \cdot \psi_0 \frac{mW^2}{p} \cdot k_w^2 \frac{\tau l}{\delta} \tag{3} \]
is the inductive reactance under the assumption that the rotor is cylindrical, expressed through the design data of the machine.
The use of the armature-reaction coefficients in equations (1), (2) is inadmissible. It is known that the armature-reaction coefficients are understood as the reduction factors of the m.m.f. (magnetomotive force) of the direct and quadrature armature reactions to the m.m.f. of the excitation winding [2].
When determining the reactances \(x_{ad}, x_{aq}\) in salient-pole synchronous machines, the theory of two reactions is used. Owing to the nonuniformity of the air gap caused by saliency, the magnetic fluxes along the direct and quadrature axes, produced by the stator winding, are reduced. The coefficients determining the degree of reduction of the amplitudes of the fundamental field harmonics are called the field-form coefficients \(k_{ad}, k_{aq}\) along the direct and quadrature axes.
Therefore the reactances \(x_{ad}, x_{aq}\), which are proportional to the magnetic flux, in a salient-pole machine will be reduced propor-
proportionally increase \(k_{ad}, k_{aq}\) in comparison with a machine having a cylindrical rotor.
It follows from the foregoing that, when applying equations (1), (2), it is necessary to substitute into them not the armature-reaction coefficients \(k_d, k_q\), but the field-form coefficients \(k_{ad}, k_{aq}\).
An analogous error is made in the calculation of the reactances \(x_{ad}, x_{aq}\) in the method for calculating synchronous reluctance motors set forth in [3].
Such an inadmissible inaccuracy in calculating the parameters of synchronous reluctance motors has become possible because insufficient attention is paid to questions of their theory. A synchronous reluctance machine is a special case of a synchronous salient-pole machine. Therefore, in studying the operation and calculation of synchronous reluctance motors, the conclusions of the theory of the salient-pole synchronous machine are often used without sufficient grounds, under the assumption that the field winding on the rotor is open-circuited.
In synchronous machines, however, in calculating \(x_{ad}, x_{aq}\), transformed formulas (1), (2) in relative units are often used, in which the reactances \(x_{ad}, x_{aq}\) are expressed through the p.u. armature reactions and air gap, or through the linear loading and the induction in the air gap [4, 5]. In this case, the coefficients of armature reaction \(k_d, k_q\) enter the calculation equations.
On this basis, in [1, 3], when calculating the reactances \(x_{ad}, x_{aq}\) of synchronous reluctance motors by equations (1), (2), the coefficients \(k_d, k_q\) are erroneously introduced, despite the fact that \(x_m\) is expressed through the design data of the machine.
The field-form coefficients \(k_{ad}, k_{aq}\) and armature-reaction coefficients \(k_d, k_q\) for a uniform and very small gap under the pole shoe \((\delta = 0)\) are determined by the equations [2]:
\[ k_{ad}=\frac{\alpha\pi+\sin\alpha\pi}{\pi}, \tag{4} \]
\[ k_{aq}=\frac{\alpha\pi-\sin\alpha\pi}{\pi}, \tag{5} \]
\[ k_d=\frac{\alpha\pi+\sin\alpha\pi}{4\sin\frac{\alpha\pi}{2}}, \tag{6} \]
\[ k_q=\frac{\alpha\pi-\sin\alpha\pi}{4\sin\frac{\alpha\pi}{2}}. \tag{7} \]
From these equations one can find that
\[ \frac{k_d}{k_{ad}}=\frac{k_q}{k_{aq}}=\frac{\pi}{4\sin\frac{\alpha\pi}{2}}. \tag{8} \]
Equation (8) characterizes the magnitude of the error obtained when the coefficients \(k_d, k_q\) are substituted into equations (1), (2) instead of \(k_{ad}, k_{aq}\).
In those cases where the air gap under the pole is of finite magnitude, the values of the coefficients \(k_d, k_q, k_{ad}, k_{aq}\) can be determined on the basis of constructing field patterns [6].
In Figs. 1 and 2 the ratios \(\dfrac{k_d}{k_{ad}}\), \(\dfrac{k_q}{k_{aq}}\) are presented as functions of the pole-arc coefficient \(a\), for various ratios of the air gap under the pole \(\delta\) to the pole pitch \(\tau\) (for the case \(\dfrac{\delta_{\max}}{\delta}=1.0\)).
As follows from the curves given, the error in calculating the reactances \(x_{ad}, x_{aq}\) when the coefficients \(k_d, k_q\) are incorrectly used instead of \(k_{ad}, k_{aq}\) is considerable and, at \(a=0.4\), reaches 34%. From equation (8), and also from a comparison of the curves shown in Figs. 1 and 2, it follows that the error in determining \(x_{ad}, x_{aq}\) is practically the same.
Fig. 1. Curves for determining the ratio
\[
\frac{k_d}{k_{ad}}
\]
for salient-pole machines at
\[
\frac{\delta_{\max}}{\delta}=1.0.
\]
Fig. 2. Curves for determining the ratio
\[
\frac{k_q}{k_{aq}}
\]
for salient-pole machines at
\[
\frac{\delta_{\max}}{\delta}=1.0.
\]
In the practice of designing electrical machines, all calculations are usually performed taking into account only the fundamental harmonics. The influence of the higher harmonics is then taken into account independently. Therefore
In the foregoing it was assumed that the calculation of the coefficients \(k_d\), \(k_q\), \(k_{ad}\), \(k_{aq}\) and of the reactances \(x_{ad}\), \(x_{aq}\) is performed for the fundamental harmonics.
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