On the Theory of Synchronous Reluctance Machines
Abstract Generated abstract
This paper examines the theoretical description of synchronous reluctance machines and argues that treating them simply as salient-pole synchronous machines with an open rotor excitation winding gives an inaccurate physical interpretation. Using vector diagrams and power equations, it distinguishes the time angle between terminal voltage and induced electromotive force from the spatial load angle between the rotor pole axis and the resultant magnetomotive force, showing that these angles generally cannot be equated. The analysis derives electromagnetic power expressions for constant stator current and constant supply voltage, including the effects of different direct- and quadrature-axis reactances and stator resistance. It concludes that maximum power occurs at different load angles under different operating conditions, and reports experimental data that support the derived angle relationship.
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Toward the Theory of Synchronous-Reluctance Machines
E. V. Kononenko
(Presented by the scientific seminar of the Electromechanical Faculty)
Despite the fact that in recent years so-called synchronous-reluctance motors have found wide application in low-power drives, the question of their theory has not received sufficient attention. The theory currently in existence does not explain with complete precision the physical aspect of the operation of these machines.
A synchronous-reluctance machine is a special case of a synchronous machine with a salient-pole rotor on which there is no direct-current excitation winding. Therefore, when studying the operation of a synchronous-reluctance machine, one usually proceeds from the theory of the salient-pole synchronous machine, regarding the excitation winding on the rotor as open-circuited. Such an approach to the study of the operation of a synchronous-reluctance machine leads to an incorrect conception of the physical processes taking place in it.
The electromagnetic power of synchronous machines is expressed as a function of the angle of time displacement of the e.m.f. vectors along the longitudinal axis of the rotor and the voltages at the machine terminals. In order to explain, from the physical standpoint, the process of operation of a synchronous machine in parallel with the network, this angle is equated to the spatial angle of displacement between the axis of the rotor pole and the axis of the resultant magnetizing force (m.f.), which produces the resultant magnetic flux in the air gap. Such an assumption is valid only for synchronous machines with a cylindrical rotor, if the voltage drops in the active resistance \(r\) and the leakage reactance \(x_s\) of the stator winding, as well as saturation of the machine steel, are neglected [1].
Such an assumption is still less valid for a synchronous salient-pole machine.
Owing to the difference in magnetic permeances along the longitudinal and transverse axes of the rotor, the sinusoidal wave of the armature-reaction m.f. creates a nonsinusoidal magnetic flux, which, according to the theory of two reactions, is resolved into components along the longitudinal and transverse axes. On the basis of such a resolution, the voltage diagram for a salient-pole synchronous machine is constructed.
Neglecting the active resistance of the stator winding and saturation of the machine steel, from the diagram (Fig. 1) one can determine the electromagnetic power:
\[ P_{\mathrm{em}} = m \frac{U E_0}{X_d} \sin \Theta + m \frac{U^2}{2} \left( \frac{1}{x_q} - \frac{1}{x_d} \right) \sin 2\Theta, \tag{1} \]
where
\[ \Theta \]
is the angle of time displacement between the voltage vector \(U\) and the e.m.f. vector produced by the rotor flux \(E_0\),
\[ x_d = x_{ad} + x_s \]
is the total reactance along the longitudinal axis,
\[ x_q = x_{aq} + x_s \]
is the total reactance along the transverse axis.
The first term of equation (1) is due to excitation from the rotor, the second to the magnetic asymmetry of the rotor.
The spatial angle of displacement between the axis of the rotor pole and the resultant m.m.f. is equal to \(\beta\) for a two-pole machine. In the general case this angle is equal to \(p\beta\), where \(p\) is the number of pole pairs.
From the diagram (Fig. 1) it is seen that, for \(x_d \ne x_q\), the resultant e.m.f. due to the armature-reaction flux,
\[ E_a=\sqrt{E_{ad}^{\,2}+E_{aq}^{\,2}}, \]
will not be perpendicular to the current vector \(\dot I\). Consequently, in order to allow the equality of the angles \(\Theta=\beta\), in a salient-pole synchronous machine it is necessary to neglect not only the resistances \(r\), \(x_s\) and the saturation of the machine steel, but also to assume that \(x_d=x_q\).
Fig. 1.
Fig. 2.
In synchronous-reluctance machines the electromagnetic power is determined only by the second term of equation (1):
\[ P_{\mathrm{em}}=m\,\frac{U^2}{2}\left(\frac{1}{x_q}-\frac{1}{x_d}\right)\sin 2\Theta . \tag{2} \]
In this case, equating the angles \(\Theta=\beta\) [2; 3; 4; 6] is completely unjustified and leads to erroneous conceptions of the operation of synchronous-reluctance machines.
Let us consider the operation of a synchronous-reluctance motor, using the vector diagram presented in Fig. 2.
The electromagnetic power of a synchronous-reluctance motor is equal to
\[ P_{\mathrm{em}}=m I E \cos(\widehat{EI}). \tag{3} \]
From the vector diagram of Fig. 2 it follows that
\[ E\cos(\widehat{EI})=E_{ad}\sin\beta-E_{aq}\cos\beta, \tag{4} \]
where
\[ E_{ad}=I_d x_{ad}=I x_{ad}\cos\beta \]
\[ E_{aq}=I_q x_{aq}=I x_{aq}\sin\beta . \]
Consequently,
\[ P_{\mathrm{em}}=m\frac{I^2}{2}\left(x_d-x_q\right)\sin 2\beta . \tag{5} \]
In this case, and in what follows, we assume that the resistances \(x_d\) and \(x_q\) are constant and do not depend on the load angle \(\beta\).
It follows from equation (5) that, at constant current in the stator windings, the electromagnetic power varies with the load angle according to a sinusoidal law.
The maximum power will correspond to the angle \(\beta=45^\circ\).
However, the case \(I=\mathrm{const.}\) is not typical in practice, since normally synchronous reluctance motors under normal conditions operate from a network with constant voltage.
The expression for the electromagnetic power as a function of the voltage \(U\) and the load angle \(\beta\) can also be obtained from the diagram (Fig. 2). Indeed,
\[ AO=I(x_d\cos\beta+r\sin\beta)=U\cos\Theta, \]
\[ AB=I(x_q\sin\beta-r\cos\beta)=U\sin\Theta \]
and
\[ I^2=\frac{2U^2}{x_d^2+x_q^2+(x_d^2-x_q^2)\cos 2\beta+2r(x_d-x_q)\sin 2\beta+r^2}. \tag{6} \]
Substituting equation (6) into (5), we obtain:
\[ P_{\mathrm{em}}= \frac{mU^2(x_d-x_q)\sin 2\beta} {x_d^2+x_q^2+(x_d^2-x_q^2)\cos 2\beta+2r(x_d-x_q)\sin 2\beta+r^2}. \tag{7} \]
It follows from equation (7) that the law of variation of the electromagnetic power with the load angle \(\beta\) will not be sinusoidal, and the maximum power will not correspond to the angle \(\beta=45^\circ\).
Neglecting the active resistance of the stator winding, we obtain:
\[ P_{\mathrm{em}}= \frac{mU^2(x_d-x_q)\sin 2\beta} {x_d^2+x_q^2+(x_d^2-x_q^2)\cos 2\beta}. \tag{8} \]
Equations (8) and (2) are equivalent, since they define one and the same power. Solving these equations simultaneously, one can determine the relationship between the angles \(\Theta\) and \(\beta\).
However, a simpler relation between these angles can be obtained directly from the vector diagram (Fig. 2). Indeed,
\[ \tg\Theta=\frac{AB}{AO}= \frac{x_q\sin\beta-r\cos\beta}{x_d\cos\beta+r\sin\beta}. \tag{9} \]
Equation (9) gives a simple relationship between the angles as a function of the machine parameters.
Neglecting, as before, the active resistance of the stator winding, we obtain
\[ \tg\Theta=\frac{x_q}{x_d}\tg\beta. \tag{10} \]
For hydrogenerators and other synchronous salient-pole machines it can usually be assumed [7] that \(x_d=2x_q\).
Then
\[ \tg\Theta=0.5\,\tg\beta . \]
It follows from this that the angle \(\beta=45^\circ\) corresponds to the angle \(\Theta=26.5^\circ\). Consequently, by equating the angles, we allow an error of 40%.
Substituting the value \(x_d=2x_q\) into equation (8), we obtain
\[ P_{\mathrm{em}}=\frac{mU^{2}\sin 2\beta}{x_q(5+3\cos 2\beta)}. \]
The value of the angle \(\beta\) corresponding to \(P_{\max}\) is
\[ \beta=63.5^\circ. \]
This angle will correspond to the angle \(\Theta=45^\circ\), as is seen from equation (2).
Equation (10), for \(x_d=2x_q\), gives the same relationships between the angles.
Since synchronous-reluctance motors are usually made for small powers, of the order of not more than \(1\)—\(2\) kW, then, neglecting the active resistance of the stator winding, we obtain an inaccurate relationship between the angles. Indeed, in the case of ideal no-load operation, substituting the value \(\beta=0\) into equation (9), we obtain:
\[ \tg \Theta=-\frac{r}{x_d}. \tag{11} \]
Consequently, in studying the steady-state operating conditions of synchronous-reluctance machines, two cases are of interest.
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The current in the stator windings remains constant as the load changes. The maximum power developed by the machine in this case will correspond to the instant when the rotor lags behind (or leads) the stator n.s. axis by 45 electrical degrees. With a further increase of load the machine falls out of synchronism, as follows from equation (5);
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When operating from a network with constant voltage, the maximum power will correspond to the angle \(\Theta=45\) electrical degrees [equation (2)]. In this case the rotor will lag behind (or lead) the stator n.s. axis by an angle greater than 45 electrical degrees, as follows from equations (7) and (8). The relationship between the angles \(\Theta\) and \(\beta\) depends on the parameters of the machine and can be determined from equations (9) or (10).
Fig. 3.
The theoretical statements set forth above are confirmed by the experimental data shown in Fig. 3, where curve 1 is calculated from equation (9), and curve 2 is plotted from experimental data.
LITERATURE
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