Inverted circle diagram of a synchronous reluctance motor
Abstract Generated abstract
This paper develops an inverted circle diagram for the synchronous reluctance motor using the load angle between the current vector and the rotor direct axis as the argument. From the motor vector diagram, the author derives an equivalent impedance with active and reactive components expressed through the direct-axis and quadrature-axis reactances, leakage reactance, stator resistance, and mismatch angle, then shows that inversion of the impedance locus gives the current circle diagram. The method is presented as a simpler engineering calculation procedure than earlier diagrams based on voltage or emf angles, requiring directly determined motor parameters and giving a clearer interpretation of torque and operating characteristics. The paper also discusses practical corrections for copper and iron losses, leakage flux, and magnetic saturation, including approximate procedures for selecting parameters from no-load and short-circuit conditions.
Full Text
Inverse Circle Diagram of a Synchronous Reluctance Motor
B. P. Garganeev
(Recommended by the scientific seminar of the departments of electrical machines and apparatus and of general electrical engineering.)
Synchronous reluctance motors (SRMs), having certain techno-economic advantages and high operational reliability, are finding wide application in industry [1 et al.]. Until recently, their spread was mainly limited to low-power drives. In recent years, extensive work has been carried out to improve the operating characteristics of SRMs and to bring them closer to asynchronous motors. Improved SRMs abroad successfully compete with other types of synchronous motors at powers up to 30–35 kW [2 et al.].
However, in the theory and design of SRMs many questions remain unresolved. A number of works [1–5 et al.] have been devoted to the construction of circle diagrams for the synchronous operating mode of an SRM (hereinafter in this article simply circle diagrams), as the most visual and comprehensive form of calculation. The possibility of constructing a circle diagram of the current of an SRM has long been known. Since the torque of an SRM is rigidly connected with the angle of displacement of the longitudinal axis of the salient-pole rotor relative to the magnetic flux, all investigations, especially in the initial period of the development of SRM theory, were carried out as functions of the angle of displacement of the longitudinal axis of the rotor and the current vector. In this case, as shown by the author [3], the circle diagram of the SRM current vector is obtained in inverted form. In [3] only the basic solution of the problem is presented; investigations taking into account the special features of SRMs were not carried out. Of great practical importance in calculations of SRMs is allowance for losses in copper and steel, for saturation of the steel, and also for leakage fluxes.
A circle diagram of the SRM of the type widely used in asynchronous machines is obtained if the angle between the e.m.f. (complete or without allowance for leakage flux) and the transverse axis, or the voltage and the transverse axis, is taken as the argument. The most fundamental circle diagram of this type, allowing for losses in copper, steel, and leakage fluxes, was developed in [4]. The authors obtained comparatively simple expressions for the ordinates of the center of the circle and for the radius vector. But since they took as the argument the angle \(\Theta'\) (with inductive resistances, including also the reactive resistances due to leakage fluxes), a certain inaccuracy is admitted in the calculations, because in this case the line of zero torques does not pass through the points \(\Theta' = 0\) and \(\Theta' = \pi/2\). In fact, in the case of ideal no-load operation, when the rotor is located strictly along the magnetic flux, and still more in the “short-circuit” mode, when the rotor is located perpendicular to the magnetic flux and the current reaches its maximum value, there is always a flux
leakage, and, consequently, the EMF as well, and the angle \(\Theta'\). In this work, iron losses are taken into account by a vector parallel to the EMF, and not to the voltage vector, which also introduces a certain additional error.
The subsequent works [1, 2, 5, etc.] introduced nothing fundamentally new. Certain inaccurate assumptions in [5], due to the angle \(\Theta\) being taken as the argument (between the transverse axis and the voltage vector), were pointed out in a discussion by the author of [1]. In addition, in our opinion, the introduction of a proportional dependence of the iron losses on the current magnitude, as was done in [5] by a corresponding change in the active resistance of the stator winding, cannot give correct results for an SRM, where the magnetic flux changes only slightly. Work [1] has no such shortcomings, but the complexity of the calculations does not make it possible to apply it as an engineering method.
The theory of the SRM was developed most fundamentally in [6], where the following is stated: “...a substantial shortcoming... of the analysis is the circumstance that all quantities... are found as functions of... the time angle between the vector \(\dot U\) and \(j I_d x_{ad}\). However, for judging the behavior of the machine... what is of substantial importance is not the angle \(v_e\), but the spatial angle \(\vartheta\) between the axes of the stator field and the rotor poles, i.e., the load angle.”
Fig. 1. Vector diagram of an SRM.
In the present work a method is proposed for constructing the circle diagram of an SRM in which the load angle between the current vector and the direct axis of the rotor is taken as the argument. This made it possible to obtain a simpler equivalent circuit of the SRM and, in addition, to simplify the calculations and to represent more clearly the operating characteristics and certain physical processes occurring in the SRM when the load changes.
The vector diagram of an SRM is presented in Fig. 1. The stator resistance and current, taking into account the active phase resistance and the resistance
the leakage can be written as
\[ \bar z=(R' + r)+j(x' + x_s). \tag{1} \]
\[ \dot I=\frac{\dot U}{(R' + r)+j(x' + x_s)}. \tag{2} \]
Here \(r; x_s\) are the active resistance and leakage reactance of the phase. \(R'; x'\) are the active and reactive resistances of the stator phase, dependent on the load.
When the real axis is aligned with the current vector, the e.m.f. from the main flux is determined as
\[ \dot E=E_a+jE_x=I(R'+jX'). \tag{3} \]
The components \(E_a\) and \(E_x\) are readily determined from the vector diagram, as the e.m.f. from the direct and quadrature fluxes through the angle \(\beta\).
\[ E_a=E_d\cos\left(\frac{\pi}{2}-\beta\right)+(-E_q)\cos\beta. \tag{4} \]
\[ E_x=E_d\cos\beta+E_q\sin\beta. \tag{5} \]
\(E_d\) and \(E_q\) are determined as in synchronous salient-pole machines:
\[ E_d=x_aK_dI\cos\beta=x_{ad}I\cos\beta=x_{ad}I_d, \tag{6} \]
\[ E_q=x_aK_qI\sin\beta=x_{aq}I\sin\beta=x_{aq}I_q, \tag{7} \]
where \(x_a\) is the inductive reactance of the stator-winding phase under the assumption that the air gap is uniform and equal to the minimum; \(K_d; K_q\) are the field-form coefficients along the direct and quadrature axes. Taking equations (6, 7) into account, from equations (4, 5) we obtain
\[ E_a=\frac{1}{2}I(x_{ad}-x_{aq})\sin 2\beta, \tag{8} \]
\[ E_x=I(x_{ad}\cos^2\beta+x_{aq}\sin^2\beta). \tag{9} \]
Accordingly,
\[ R'=\frac{E_a}{I}=\frac{x_{ad}-x_{aq}}{2}\sin 2\beta, \tag{10} \]
\[ X'=\frac{E_x}{I}=x_{ad}\cos^2\beta+x_{aq}\sin^2\beta. \tag{11} \]
According to equation (2), the current of the SRM, taking equations (10, 11) into account, is determined as
\[ \dot I= \frac{\dot U} {r+\dfrac{x_{ad}-x_{aq}}{2}\sin 2\beta +j\left(x_{ad}\cos^2\beta+x_{aq}\sin^2\beta+x_s\right)}. \tag{12} \]
Thus, the complete complex resistance of equation (12) contains only two series-connected resistances—active and inductive:
\[ R=r+\frac{x_{ad}-x_{aq}}{2}\sin 2\beta, \tag{13} \]
\[ X=x_{ad}\cos^2\beta+x_{aq}\sin^2\beta+x_s. \tag{14} \]
When the real axis is aligned with the voltage vector, equation (12) is simplified. By simple mathematical transforma-
transformations the denominator of equation (12) is reduced to the form:
\[ \bar Z = Z_1 e^{j\tau_1} + \frac{x_{ad}+x_{aq}}{2} e^{j\tau_2} + \frac{x_{ad}-x_{aq}}{2} e^{j(\tau_2-2\xi)} = \bar A+\bar B e^{-j2\beta}. \tag{15} \]
Here
\[ \bar Z_1=r+jx_s=Z_1 e^{j\tau_1}, \tag{16} \]
\[ \bar A = Z_1 e^{j\tau_1} + \frac{x_{ad}+x_{aq}}{2} e^{j\tau_2}, \tag{17} \]
\[ \bar B = \frac{x_{ad}-x_{aq}}{2} e^{j\tau_2}. \tag{18} \]
From equation (15) it is evident that the complete resistance complex represents the sum of two vectors, one of which, \(\bar A=\mathrm{const}\), while the second, \(\bar B\), is constant in modulus but, depending on \(\beta\), the end of the vector describes a circle in the complex plane with its center at the end of vector \(\bar A\); moreover, when \(\beta\) changes from \(0\) to \(360^\circ\), the radius vector \(\bar B\) traverses the circle twice, i.e., from \(0\) to \(720^\circ\). From the theory of alternating currents it is known that if the inverse curve of the resistance is constructed in the same complex plane, then the new curve will be nothing other than the admittance. Its form will be similar, i.e., the end of the admittance vector \(\bar Y\) will also describe a circle, but in a mirror image with respect to the axis of reals, with the center of inversion at the origin. Figure 2 shows the inversion carried out graphically, in which the starting point was the known condition:
Fig. 2. Inversion in the complex plane.
\[ \bar Z \tilde Z' = s \quad \text{or, equivalently,} \quad \bar Z \bar Y = s, \tag{19} \]
where \(s\) is the degree of inversion.
Let \(\bar Z=Ze^{j\alpha};\ \bar Y=Ye^{j\alpha'}\), then
\[ \bar Y=\frac{s}{\bar Z}=\frac{s}{Z}e^{-j\alpha}, \tag{20} \]
i.e., the distance of the sought point from the center of inversion is determined by the ratio of the degree of inversion to the modulus of the vector, but this point will be
lie on the ray that is the mirror image, in the axis of real quantities, of the principal ray. If the axis of real quantities is made to coincide with the voltage vector, then the inverted curve will represent, on a certain scale, at the same time the circle diagram of the current, since
\[ \dot I = U \tilde Y . \tag{21} \]
Since we are interested only in the circle diagram of the current, the intermediate diagrams \(\tilde Z\) and \(\tilde Y\) need not be constructed; but in this case, when constructing the circle diagram of the current, in order to determine the center of the circle and its radius we may assume that all three diagrams \((\dot I;\tilde Z;\tilde Y)\) are superposed in one plane, while being drawn to different scales. Having determined the latter, one can find the degree of inversion \(s\), and then the inverted vector \(\tilde A'\) and radius vector \(\tilde B'\). This method of construction is complicated. The simplest construction method consists in determining the ray passing through the center of the circle and two points lying on the circle. These points are easy to find, for example, from the no-load current and the “short-circuit” current:
a) the direction of the ray-vector passing through the center of the circle is determined from equation (17)
\[ \operatorname{tg}\alpha_{\mathrm{c}} = -\frac{2x_3+x_{ad}+x_{aq}}{2r} = -\frac{x_d+x_q}{2r}; \tag{22} \]
b) the magnitude of the vector of “ideal” no-load is determined from equation (12) for the case \(\beta=0\), and the direction from the ratio of reactive and active resistances for this mode
\[ I_0=\frac{U}{\sqrt{x_d^2+r^2}}; \qquad \operatorname{tg}\alpha_0=-\frac{x_d}{r}; \tag{23} \]
c) the magnitude of the “short-circuit” current vector and its direction are determined similarly, but for the case \(\beta=90^\circ\)
\[ I_{\mathrm{sc}}=\frac{U}{\sqrt{x_q^2+r^2}}; \qquad \operatorname{tg}\alpha_{\mathrm{sc}}=-\frac{x_q}{r}. \tag{24} \]
Fig. 3. Circle diagram of the current of a synchronous reluctance motor.
Figure 3 shows the circle diagram of the current constructed by the method indicated above. Attention should be paid to certain features when working with the inverted circle diagram. Since
the principal dependence on the argument \(\beta\) is obtained for the complex \(\overline{Z}\), then on the inverse circle diagram of current, which in essence is the diagram of conductance \(\overline{Y}\), as is seen from equation (21), the angle \(2\beta\) must be measured at the end of the vector \(\widetilde{Z}'\) (see Fig. 2), while the magnitude of the current is measured along this same ray, but on the conductance circle. For example, the magnitude of the motor current at the torque corresponding to the mismatch angle \(\beta_1\) is found as a vector with its origin at the center of rotation (the origin of coordinates) and its end at the point of intersection of the straight line with the conductance circle at the point \(\delta_y\). Moreover, the point \(a_z\) is located at the angle \(2\beta_1\), measured counterclockwise from the line connecting the center of the circle \(O'\) with the point \(\beta = 0\), taken on the circle \(\widetilde{Z}'\). Thus, the readings of the angle and of the current magnitude are taken up to the points of intersection of the ray with the circle, always on different sides.
Fig. 4. Operating characteristics of an SRM obtained from the inverse circle diagram.
Figure 4 presents the operating characteristics of an SRM with a reluctance power of \(3.5\ \text{kW}\), obtained from the circle diagram. The allowance for saturation and losses in the steel, as well as for higher harmonics, has a major influence on the accuracy of the calculations. In an SRM, depending on the mismatch angle \(\beta\), the magnitude of the components of the magnetic flux along the longitudinal and transverse axes and the form of the induction distribution in the air gap change. Therefore it cannot be assumed that the machine parameters remain
unchanged. At the same time, the circle diagram is constructed for unchanged parameters. Consequently, it is necessary to construct a family of circle diagrams or to select certain averaged parameters for different operating conditions that give the closest agreement, i.e., correspond to actual operation.
Allowance for saturation of the steel. As comparison of the results of an experimental study of the synchronous reluctance motor with calculated characteristics has shown, neglecting the effect of saturation leads to an inadmissible calculation error. Since the magnetic flux changes its position relative to the axes of the poles depending on the load, and considering that the magnetic reluctance along the direct and quadrature axes is different, the saturation coefficients of the steel along the direct and quadrature axes cannot be identical and, moreover, vary depending on the load. Therefore, for more accurate calculations it is necessary to construct the circle diagram for no-load operation and then for “short circuit.” An approximate calculation method may also be recommended, which gave satisfactory results. It consists in determining the no-load point and the short-circuit point for different operating conditions (i.e., for different \(\beta\)), namely:
a) When determining the no-load current \(I_0\) from equation (23), assuming that at small angles \(\beta\) the entire magnetic flux passes along the direct axis, in calculating \(x_{ad}\) the coefficient of magnetic saturation is determined as
\[ \kappa_{\mu d} = \frac{(\Sigma A W_0 - A W_{\delta d}) + A W'_{\delta d}} {A W'_{\delta d}} . \tag{25} \]
Here \(A W_{\delta d}\); \(\Sigma A W_0\) are determined from the no-load characteristic for the no-load emf (i.e., \(E \simeq (0.8 \div 0.95)U\)),
\[ A W'_{\delta d} = \frac{A W_{\delta d}}{K_d} \]
is the air-gap magnetomotive force, reduced to the first harmonic of the flux (allowing for the increase in magnetic reluctance due to the nonuniform gap).
The inductive reactance along the direct axis is
\[ x_{ad} = \frac{x_a K_d}{\kappa_{\mu d}}, \qquad x_d = x_{ad} + x_s; \tag{26} \]
b) When determining the “short-circuit” current from equation (24), assuming that at large angles \(\beta\) the main part of the magnetic flux passes along the quadrature axis, in calculating the coefficient of magnetic saturation we determine it as
\[ \kappa_{\mu q} = \frac{(\Sigma A W_0 - A W_{\delta q}) + A W'_{\delta q}} {A W'_{\delta q}} . \tag{27} \]
Here \(A W_{\delta q}\); \(\Sigma A W_0\) are determined from the no-load characteristic for the emf corresponding to maximum load (i.e., \(E \simeq U - (I_{\mathrm{kz}}\sqrt{x_s^2 + r^2})\));
\[ A W'_{\delta q} = \frac{A W_{\delta q}}{K_q} \]
is the air-gap magnetomotive force, reduced to the first harmonic of the flux.
The inductive reactance along the quadrature axis is
\[ x_{aq} = \frac{x_a K_q}{\kappa_{\mu q}}; \qquad x_q = x_{aq} + x_s. \tag{28} \]
Allowance for iron losses. The iron losses in an SRM when operating from a network with $U=\mathrm{const}$ may be regarded as constant, since $\Phi \approx \mathrm{const}$. Therefore, after determining them for the rated mode, the refinement of the circle diagram is carried out by shifting the input-power line $P_1=0$, as is done in constructing circle diagrams of asynchronous motors.
When constructing operating characteristics, the circle diagram is used by a method similar to that described for asynchronous machines, i.e., $\cos\varphi$, $P_1$, $P_2$, $\eta$, etc., are found in the same way. For example, the line of zero torques is found by connecting the ends of the current vector at no-load and at short circuit, and the line of zero power $P_2=0$ by drawing a parallel line at a distance equal, on the power scale, to the mechanical losses. To determine the angle, a scale similar to the slip scale of asynchronous machines can be constructed; but since this scale is essentially $\tg\beta$, and the rated mode and maximum torques are usually in the zone $\beta>60^\circ$, where $1.5<\tg\beta<\infty$, construction of the scale is impractical. It is more accurate to determine $\beta$ from the central angle of the circle.
Conclusions
The circle diagram of a synchronous-reluctance motor, in which the angle of misalignment between the longitudinal axis of the rotor and the current vector is taken as the argument, has a number of advantages:
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The equivalent circuit is simplified, since it has no magnetizing circuit. The equivalent resistances acquire a physical meaning ($R'$ is proportional to the electromagnetic torque).
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The number of mathematical operations is reduced. In the calculation, the directly determined motor parameters $r$, $x_s$, $x_{ad}$, $x_{aq}$ are required.
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The characteristics obtained illuminate the physical picture of SRM operation more realistically and clearly.
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Allowance for losses in iron and copper, for iron saturation, and for leakage fluxes does not cause any special complication in the calculation of the circle diagram.
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