PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV

Vol. 98　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1960

EXPERIMENTAL DETERMINATION OF PARAMETERS OF SYNCHRONOUS-RELUCTANCE MOTORS

E. V. Kononenko

(Presented by the scientific-methodological seminar of the Electromechanical Faculty)

The experimental determination of the parameters of any electrical machine is of great practical importance. However, the determination of the parameters of synchronous-reluctance motors has not been sufficiently covered in the literature. This article describes methods for determining certain parameters.

Determination of the synchronous reactances \(x_d\) and \(x_q\)

It is recommended that the determination of the synchronous reactances for synchronous-reluctance motors having a starting short-circuited winding on the rotor be carried out by calculation [1].

The usual slip method [4, 5, 6], recommended for the experimental determination of the reactances \(x_d\), \(x_q\), has a number of disadvantages; moreover, this method is suitable only for machines with open rotor windings. For machines with short-circuited starting windings, the slip method gives unsatisfactory results.

When testing synchronous-reluctance motors, the following method [7] may be recommended for determining the synchronous reactances along the direct and quadrature axes \(x_d\), \(x_q\).

Fig. 1.

The synchronous-reluctance motor being tested is mechanically coupled to a synchronous motor. Both machines must have the same number of pole pairs. The machines are set up so that the stator of one of these machines (it is immaterial which) can be rotated in space by not less than 180 electrical degrees. The stator windings are connected to the network so that the magnetic fields of both machines rotate in the same direction. Figure 1 shows a sche-

of the installation assembled for determining the reactances \(x_d, x_q\). Here, 1 is the synchronous-reluctance motor under test, whose shaft is rigidly connected to the shaft of synchronous motor 2. Both machines have the same number of pole pairs. The synchronous-reluctance motor, with the aid of additional bearings 3 at the free ends of the shaft, is mounted so that the stator can be turned in space through not less than 180 electrical degrees.

When the motors are connected to the network, the magnetic fields rotate synchronously in one direction, and the rotors, which are mechanically connected, also rotate at synchronous speed. The synchronous motor is selected so that its maximum torque is greater than the maximum torque of the synchronous-reluctance motor. By turning the stator of the synchronous-reluctance motor, one can obtain a change in the angle between the axis of the magnetic flux and the pole axis of the rotor within 180°. The rotation of the stator in space is measured by means of a scale graduated in electrical degrees. By connecting an ammeter, voltmeter, and wattmeter into the stator-winding circuit of the synchronous-reluctance motor, at each turn of the stator one can determine the value of the synchronous reactance (with the stator winding connected in star)

\[ x=\sqrt{\left(\frac{U}{\sqrt{3}I}\right)^2-\left(\frac{P}{I^2}\right)^2},\quad (\Omega), \]

where \(U\) is the line voltage;

\(I\) is the phase current;
\(P\) is the power per phase.

The maximum value of \(x\) is equal to the synchronous reactance along the direct axis

\[ x_{\max}=x_d. \]

The minimum value of \(x\) is equal to the synchronous reactance along the quadrature axis

\[ x_{\min}=x_q. \]

The method considered above makes it possible to determine the synchronous reactances \(x_d, x_q\) with good accuracy in the case where the rotor has a starting short-circuited winding.

In the absence of a corresponding synchronous motor, the reactance \(x_d\) can be determined from an ideal no-load test of the synchronous-reluctance motor. Ideal no-load operation can be obtained by coupling the synchronous-reluctance motor to a direct-current motor.

In this case

\[ x_d=\sqrt{\left(\frac{U}{\sqrt{3}I_{\min}}\right)^2-\left(\frac{P}{I_{\min}^2}\right)^2}. \]

The reactance \(x_q\) cannot be determined with the aid of an auxiliary direct-current motor, since the direct axis of the rotor during

in determining \(x_q\), it must lag (or lead) the amplitude of e.m.f. by an angle \(\pi/2\), which corresponds to an unstable operating mode. The approximate synchronous reactance along the transverse axis \(x_q\) can in this case be determined by the method of pulling the motor out of synchronism [2].

The determination of the other resistances of a synchronous-reluctance motor presents no difficulty and can be carried out by the usual methods [4, 5].

Experimental determination of the angles \(\beta\) and \(\Theta\)

As was shown in [3], the angles \(\beta\) and \(\Theta\) depend on the load. In studying the synchronous operating mode of synchronous-reluctance machines, determining the permissible load, etc., knowledge of the angles \(\beta\) and \(\Theta\) is of great practical importance.

In the literature known to us, no recommendations have been found for determining these angles in synchronous-reluctance machines. Based on methods for determining the angle \(\Theta\) in synchronous machines [5, 6], simple methods have been developed for determining the angles \(\beta\) and \(\Theta\) in synchronous-reluctance machines.

Oscillographic method for determining the angle \(\beta\)

The angle \(\beta\) is the angle of mismatch between the longitudinal axis of the rotor pole and the axis of the resultant m.m.f. of the stator winding. To determine the angle \(\beta\) in a steady synchronous operating mode, the setup shown in Fig. 2 is used. A permanent magnet 3 is fastened to the shaft of the synchronous-reluctance motor 1; opposite it is placed a coil with a steel core 4, fixed in space. The synchronous-reluctance motor is mechanically connected to an independently excited dc load generator 2.

Fig. 2.

At each passage of the magnet past the coil (during rotation of the motor), an e.m.f. is induced in it. Fig. 3 shows an oscillogram of the current in the stator phase and the e.m.f. in the coil. The maximum value of the current in the phase will occur at the moment when the axis of the resultant m.m.f. coincides with the axis of this phase. The e.m.f. in the coil records the position of the rotor in space and, consequently, also relative to the m.m.f. of the stator winding.

Fig. 3.

To determine the displacement between the curves \(i\) and \(e\), we shall take the points corresponding to the instants at which the curves \(i\) and \(e\) pass through zero; these can be fixed with sufficient accuracy.

The angle of displacement between the curves (Fig. 3) is equal to

\[ \alpha = \frac{x}{T}\,2\pi,\qquad \text{(radian)} \]

where \(x\) is the displacement between the curves in mm,

\(T\) is the period of the current curve in mm.

Since the magnet on the shaft of the synchronous reluctance motor is fixed in an arbitrary position, in order to determine the angle \(\beta\) for a given load it is necessary to know the angle \(\alpha\) for the instant when \(\beta = 0\), i.e., for the case of ideal no-load operation. To obtain ideal no-load operation, the direct-current generator must be driven by the motor. By varying the torque of the direct-current motor, we set the current in the stator winding of the synchronous reluctance motor to a minimum and record oscillograms of the current and the emf in the coil. From the oscillogram we determine the angle of displacement between the curves corresponding to \(\beta = 0\),

\[ \alpha_0 = \frac{x_0}{T_0}\,2\pi,\ \text{(radian)}. \]

When the motor is loaded, the axis of the rotor pole is displaced relative to the axis of the resultant mmf of the stator winding; consequently, the curve \(e\) is also displaced relative to the current in the phase. Suppose that, for some load, the displacement angle has been determined from the oscillogram as

\[ \alpha_1 = \frac{x_1}{T_1}\,2\pi,\ \text{(radian)}. \]

Then the required angle \(\beta\), corresponding to the given load, is equal to

\[ \beta_1 = \alpha_1 - \alpha_0 = \left(\frac{x_1}{T_1} - \frac{x_0}{T_0}\right)2\pi,\ \text{(radian)}. \]

The period \(T\) is determined for each oscillogram taken, since it may turn out that the drum of the oscillograph rotated at different speeds while recording different oscillograms. By varying the load on the shaft of the synchronous reluctance motor, the angle \(\beta\) can be determined at different loads up to the point at which the motor falls out of synchronism.

When setting the minimum current in the stator winding of the synchronous reluctance motor in order to obtain the angle \(\beta = 0\), it must be taken into account that in this case the motor consumes from the network power expended only on losses in the active resistance of the stator winding.

Knowing the active resistance of the stator winding and determining the minimum value of the current, the consumed power can be determined. Thus, the initial position of the rotor corresponding to the angle \(\beta = 0\) can be determined with high accuracy.

Determination of the Angle \(\Theta\)

\(\Theta\) is the angle formed by the voltage vector with the transverse axis of the rotor. It can be determined from the equation [3]

\[ \Theta = \varphi + \beta - \frac{\pi}{2}. \]

Consequently, having determined the angles \(\varphi\) and \(\beta\) for a given load, one can also determine the angle \(\Theta\).

REFERENCES

1. Goldovskii E. M. Reluctance Motors for Sound Motion Pictures. Moscow, 1935.

2. Gorokhov N. V., Zimin V. K., and Petrova L. M. Experimental determination of the parameters of synchronous machines. El-vo, No. 11, 1940.

3. Kononenko E. V. On the theory of synchronous reluctance machines. Proceedings of the Tomsk Polytechnic Institute, Vol. 82, 1956.

4. Kostenko M. P. Electrical Machines, special part, GEI, 1944.

5. Kulebakin V. S. Testing of Electrical Machines and Transformers, ONTI, 1935.

6. Richter R. Electrical Machines, Vol. 2, Moscow–Leningrad, 1936.

7. Hansteen H. B. Modified Slip Test for Experimental Study of Synchronous Reactance. El. Eng., No. 9, 1948.