INDIRECT METHODS FOR MEASURING AIR-GAP NONUNIFORMITY IN INDUCTION MOTORS

E. K. STRELBITSKII

(Presented by the scientific seminar of the Electromechanical Faculty)

It is known that, owing to technological and operational causes, the air gap of induction motors is not uniform. Two types of air-gap nonuniformity are distinguished. First, nonuniformity caused by displacement of the shaft axis relative to the axis of the stator bore—static nonuniformity; second, nonuniformity caused by eccentric mounting of the rotor core on the shaft—rotating nonuniformity. Of course, in reality both of these types of nonuniformity occur, but usually attention is given to the type of nonuniformity that predominates. In addition to the two indicated types of air-gap nonuniformity, there is also nonuniformity caused by the elliptical shape of the stator bore or by the elliptical shape of the rotor core. However, the air-gap nonuniformity produced by these causes has the same effect on the magnetic field of the machine in the air gap as the nonuniformity caused by eccentricity [1]. Nonuniformity caused by the elliptical shape of the stator bore produces static nonuniformity and is analogous to the nonuniformity caused by displacement of the rotor axis relative to the axis of the stator bore. Nonuniformity caused by the elliptical shape of the rotor core produces rotating nonuniformity of the air gap and is analogous to the nonuniformity caused by eccentric mounting of the rotor on the shaft.

It should be noted that static nonuniformity occurs more often and has a greater magnitude. This is explained by the technological features of the production of induction machines. Indeed, the rotor core is turned relative to the shaft, which is fixed in the centers of the machine tool. Thus, the eccentricity of the rotor core relative to the shaft is determined by only one operation, which moreover introduces comparatively little runout. The displacement of the shaft axis relative to the axis of the stator bore is determined by errors arising in a whole series of operations, namely: boring of the bearing seats in the two end shields, boring of the bearing locks

shields, boring of the stator-frame locks, and boring of the frame for seating the stator core. Moreover, the operations listed introduce a larger absolute error than boring of the rotor core. For the reasons set forth, technological inspection is mainly concerned with static nonuniformity, and in what follows we shall have in mind only this type of nonuniformity.

Nonuniformity of the air gap causes the appearance of an unbalanced magnetic force, which is always directed toward the smallest air gap and has constant and alternating components. The frequency of the fundamental harmonic of the alternating component of the force is equal to twice the frequency of the supply network. The magnitude of this force for induction motors of frames 3…6 of a single series lies within the range from tens to hundreds of kilograms, depending on the magnitude of the eccentricity, induction, principal dimensions, and number of pole pairs of the machine [1]. Nonuniformity of the air gap adversely affects the performance indices of electrical machines. The energy indices deteriorate somewhat; a redistribution of current occurs among the parallel branches of the stator winding and, as a result, a redistribution of temperatures [6]. At a high rotor temperature, large local overheating of the slot-wedge zone of the stator winding may occur at the location of the minimum air gap. Wear increases and the service life of the bearings is shortened. The alternating component of the unbalanced magnetic force causes vibration of the frame at a frequency equal to twice the frequency of the supply network [2, 3, 4]. A large number of works have been devoted to the calculation of magnetic fields and unbalanced magnetic forces with a nonuniform air gap. It should be noted, however, that the magnitudes of the forces obtained from the formulas of different authors sometimes differ quite substantially from one another.

In view of the foregoing, the need has arisen to inspect the air gap during manufacture and repair of induction motors.

Measurement of the air gap is usually carried out with feeler gauges, if the design of the machine permits this. The design of a number of machines greatly complicates such inspection (enclosed machines, in which for purposes of measurement special holes must be provided in the bearing end shields, which after measurement must be sealed), or completely precludes the use of this method (explosion-proof machines). Therefore there arises a need for indirect measurement of air-gap nonuniformity. The development of such a method will make it possible to ensure inspection of air-gap nonuniformity in enclosed machines and to improve the quality and reliability of machines.

We investigated a number of variants of indirect measurement of air-gap nonuniformity. A brief description and assessment of them are given below.

Measurement by shaft deflection

One manifestation of air-gap nonuniformity is the deflection of the shaft under the action of the unbalanced magnetic force (Fig. 1).

By measuring, with the aid of an indicator, the displacement of the free end of the shaft, one can judge the magnitude of the eccentricity. The displacement $\Delta h$ can be calculated if the force and the design data of the shaft with the rotor core are known. In induction motors of frames 3…6, the displacement amounts to several tens of microns. Measurement

is carried out as follows. The indicator is brought up to the free end of the shaft. The scale mark is set to zero. Then a direct current approximately equal to the no-load current is supplied to the stator winding. The end of the shaft is displaced and the indicator reading is recorded.

The disadvantages of this method are:

1. The necessity of measuring in at least two mutually perpendicular directions, since we do not know the orientation of the plane of eccentricity, while the displacement lies in this plane.

2. The large error of the method, caused by the radial play of the ball bearings, which is of the same order as the magnitude of the displacement and amounts to 20–50 microns for motors of frames 3–6 [5, p. 153].

3. The small magnitude of the displacement and, consequently, the need to use an indicator with a scale division of 0.002 mm. The indicated shortcomings greatly hinder the application of this method.

Fig. 1.

Measurement by the friction moment produced
in the bearings by the unbalanced magnetic force

It is not possible to measure the unbalanced magnetic force directly, since the rotor and stator form a closed system with respect to this force. However, it is possible to measure the friction moment in the ball bearings arising from this force. The dependence of this moment on the force is expressed by the formula [5, p. 318]:

\[ M = 1.3\, K R \frac{D_0}{d_{\mathrm{sh}}}\ \text{g·cm}, \tag{1} \]

where \(K\) is the coefficient of rolling friction; \(R\) is the radial load on the bearing, in g; \(D_0\) is the diameter of the circle on which the balls are located, in cm; \(d_{\mathrm{sh}}\) is the diameter of the balls, in cm.

The disadvantages of this method are:

1. When the rotor rotates in a direct-current magnetic field, the electromagnetic torque of the squirrel-cage rotor is a strong interference in the measurements.

2. There is a rather large scatter of friction moments in assembled machines.

1. The presence of additional torques caused by magnetic attraction between the rotor and stator teeth, as illustrated in Fig. 2. Such nonuniformity of the torque does not allow sufficient measurement accuracy to be obtained; therefore this method is unacceptable.

Fig. 2. Dependence of the friction torque on the shaft rotation angle when the stator winding is supplied with direct current.

Measurement Based on Distortion of the Magnetic Field in the Air Gap

With a uniform air gap between stator windings arranged at an angle of 90 electrical degrees to one another, there is no magnetic coupling. In the case of a nonuniform air gap, the curve of magnetic induction in the air gap is distorted and, for a two-pole machine, has the form shown in Fig. 3.

Fig. 3. Distribution of induction around the circumference of the air gap of a two-pole machine for the instant when the axis of the magnetic flux makes an angle of \(50^\circ\) with the plane of eccentricity; \(e=0.8\).

Thus, the axis of the phase magnetic flux deviates from the geometrical axis of the winding, and a flux component appears in the perpendicular direction, which links with the winding located at an angle of 90 electrical degrees.

The measurement was carried out as follows. An alternating voltage was applied to two phases of the three-phase stator winding connected in an incomplete star (the excitation winding). To the third phase,

to which a voltmeter was connected; we shall call it the measuring winding (Fig. 4). It may be assumed that the distribution of induction along the circumference of the air gap from the first harmonic of the magnetomotive force is expressed by formula (2) [2].

\[ B_\theta = \frac{B_m}{\psi_\alpha} \left[ \frac{\cos p(\theta+\alpha)-\beta_\alpha}{1-e\cos\theta} \right]\sin\omega_1 t . \tag{2} \]

where \(B_\theta\) is the instantaneous induction in the air gap at angular coordinate \(\theta\);

\(B_m\) — maximum value of the magnetic induction;

\(\varepsilon\) — eccentricity;

\(e\) — relative eccentricity;

\(p\) — number of pole pairs;

\(\alpha\) — eccentricity angle, equal to the angle between the plane of eccentricity and the central line of the magnetic field.

Fig. 4.

Schematically, the nonuniform air gap of an electric motor is shown in Fig. 5. Obviously, an e.m.f. will be induced in the measuring winding,

\[ E_{\text{meas}} = 4.44\, w_{\text{meas}}\, k_{w,\text{meas}}\, f_1\, \Phi_{\text{meas}}\cdot 10^{-8}\ \text{V}, \]

\[ \Phi_{\text{meas}}=\int_{\text{n.o.}} B_\theta\, d\theta, \tag{3} \]

The calculations performed showed that, when \(e\) and \(\alpha\) are varied, the e.m.f. induced in the measuring winding changes. This e.m.f. amounts to approximately \(1\ldots3\%\) of the voltage on the excitation winding. Experiments carried out on a machine with a smooth rotor fully confirmed the correctness of the theoretical conclusions.

However, the magnitude of the e.m.f. induced in the case of a toothed rotor depends strongly on the angle of rotation. Figure 6 shows the variation of the e.m.f. as a function of the angle of rotation of the rotor of an asynchronous—

Fig. 5. Diagram of a transverse section of an induction motor with a nonuniform air gap.

Fig. 6.

of the A31/4 motor. The presence of strong slot interference precludes the use of this method for machines with a toothed rotor.

Measurements Based on Rotor Vibration under the Action of the Variable Component of the Magnetic Unbalanced Force

As was stated above, magnetic unbalanced forces have constant and variable components. It is known that, for a single-phase machine, the magnitude of the constant component and the amplitude of the variable component depend substantially on slip [3, 4]:

\[ T'_{\text{a per}} = T_{\text{sr}} \frac{s(2-s)}{2} + T_{\text{a per}} \frac{2-2s+s^2}{2}, \]

\[ T'_{\text{sr}} = T_{\text{sr}} \frac{2-2s+s^2}{2} + T_{\text{a per}} \frac{s(2-s)}{2}, \tag{4} \]

\(T'_{\text{sr}}\) — constant component of the force of a single-phase machine;
\(T'_{\text{a per}}\) — amplitude of the variable component of a single-phase machine;
\(T_{\text{sr}}\) — constant component of the force of a three-phase machine;
\(T_{\text{a per}}\) — amplitude of the variable component of a three-phase machine.

Since \(T_{\text{a per}}\) is small and amounts to approximately 15% of \(T_{\text{sr}}\) for a two-pole machine and approximately 5% for a four-pole machine, it may be assumed that, in the case of a stationary machine,

\[ T'_{\text{a per}} \approx T'_{\text{sr}} \approx \frac{1}{2} T_{\text{sr}}. \tag{5} \]

Thus, it may be stated that, under single-phase short circuit, the amplitude of the variable component is equal to the constant component of the force, i.e., it has a considerable magnitude. In Fig. 7, as an illustration, the dependence of the force on the current of a three-phase short circuit, borrowed from [1], is presented for the A52/4 motor.

The variable component of the force causes transverse bending oscillations of the shaft in the plane of eccentricity. The amplitude of oscillations of the free end of the shaft can be calculated if the magnitude of the force and the design data of the shaft with the rotor core are known. By measuring the amplitude of vibrations of the free end of the shaft, one can judge the magnitude of the eccentricity. However, calculation of the magnitude of the magnetic unbalanced force, performed using the formulas of various authors, gives a rather large scatter. In addition, under short circuit the current depends strongly on the ratio of the total resistances of the stator and rotor windings. Therefore one should proceed by experimentally taking the dependence of the vibration amplitude on the magnitude of the eccentricity.

In practice this is carried out as follows. For the given machine, the bearing end shields must be modified in such a way as to make it possible to vary smoothly and control the magnitude of the eccentricity: measurement windows are selected, and the diameters of the bearing seats are increased by an amount equal to the air-gap value. The indicated dependence is taken under single-phase short circuit of the motor. The current in the stator winding may be selected within the range from 100 to 200% of nominal. The current value during calibration and control measurements must, without fail, be one and the same.

In the absence of saturation of the magnetic circuit, which is the case under short-circuit conditions, the relation between the magnitude of the magnetic unbalanced force and the eccentricity may be considered linear. Therefore, during calibration it makes sense to take only one point, namely at the maximum eccentricity (of the order of \(e = 0.8\)). Measurement at this point gives a smaller error, since the vibration amplitude is then at its maximum.

This method is free of a number of shortcomings of the preceding methods. Indeed, the magnitude of the force is practically independent of the orientation of the rotor and stator teeth, since the force is determined by the total magnetic flux of the motor. Some change in the force with the angle

Fig. 7.

of rotation of the rotor may occur because of different damping, by the rotor cage, of the stator magnetic flux caused by poor casting of the rotor. However, rotors with poor casting should be rejected at the preceding inspection.

The error caused by radial play of the rolling bearings is also eliminated in this method, since there is a constant pressing of the inner ring against the outer ring by the constant component of the magnetic unbalanced force.

The disadvantages of this method are:

1. The necessity of measurement in at least two mutually perpendicular directions.
2. The necessity of using a vibrometer.

Nevertheless, this method can be recommended for introduction into the practice of acceptance testing.

REFERENCES

1. A. P. Voskresenskii, Investigation of the Influence of Nonuniformity of the Air Gap on the Characteristics of an Induction Motor. Dissertation. Moscow, 1957.
2. A. Covo, “Unbalanced Magnetic Force in Induction Motors with an Eccentric Rotor,” Tr. A. I. E. E., vol. 73, pt. III B.

1. E. K. Strelbitskii. Unbalanced magnetic forces in single-phase induction motors with a nonuniform air gap. Izv. VUZ, collection “Electromechanics,” No. 10, 1961.

2. E. K. Strelbitskii. Investigation of vibrations of low-power single-phase induction electric motors. Dissertation, Tomsk, 1961.

3. N. A. Spitsyn, A. I. Sprishevskii. Rolling Bearings. Mashgiz, 1961.

4. Cho-Khon-El. Investigation of the operation of induction machines with a nonuniform air gap. Dissertation, MEI, Moscow, 1957.

Received by the editors
in June 1962.