With the bearing aging, installation error, manufacturing error, and other reasons, mixed eccentricity phenomenon often inevitably appears in large-scale motor equipment. Mixed eccentricity is a combination of static and dynamic eccentricities. This paper mainly studies the characteristics of unbalanced magnetic pull caused by eccentricity and the factors that affect its size. Since one moment of dynamic eccentricity is equal to static eccentricity, therefore, for simplicity we only discuss the situation of static eccentricity and do not discuss dynamic and mixed situations.

For static eccentricity, the geometric center of the rotor and the rotation center of the rotor overlap but there is a deviation between the geometric center of the stator and the rotation center of the rotor, as shown in Fig. 1. In the health condition of the motor, the stator geometric center O₁, the rotor geometric center O₂, and the rotor rotation center O_r are all concentric. Since there is no deviation between the geometric center of the rotor and the rotation center of the rotor under static eccentricity, the air gap length between the inner circle of the stator and the outer circle of the rotor does not change as the rotor rotates but it changes in the circumferential direction.

Fig. 1.

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	Fig. 1. (color online) Geometric locations of static eccentricity.

In order to conveniently represent the air gap length expression, a relative polar coordinate system is set up in which the horizontal rightward direction is the positive direction of the polar axis and the counterclockwise direction is the positive direction of the polar angle. In practice, since the rotor outer radius R₂ is much larger than the normal air gap length δ, ignoring a small amount (δ/R₂)², the air gap length under static eccentricity is usually expressed as 1 where e represents the offset of O₁ and O_r under static eccentricity and θ₀ is the polar angle where the air gap length is the smallest. 2 For ease of description, eccentricity ratio ξ is defined as Eq. (2).

Eccentricity destroys the uniform distribution of air gap length between stator and rotor. For the rotating pole of an induction motor, figure 2 shows the magnetic circuit during actual operation. Its simplified magnetic circuit diagram is shown in Fig. 3. In the design of the motor, the magnetic reluctance of silicon steel sheet in the yellow zone in Fig. 3 is far less than the air gap reluctance at the maximum magnetic density moment. Under the condition of ensuring sufficient model accuracy, the loop flux expression is expressed as 3 where the excitation MMF F_m refers to the amplitude of the MMF of the largest loop in Fig. 2. R_s is the magnetic reluctance of silicon steel sheet in the yellow zone in Fig. 3 and R_a1, R_a2 are the magnetic reluctance of air gaps.

Fig. 2.

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	Fig. 2. (color online) Magnetic path of a 6-pole induction motor.

Fig. 3.

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	Fig. 3. (color online) Simplified magnetic circuit of one loop.

According to Eq. (3), we obtain the maximum air gap flux density as 4 Taking static eccentricity as an example, we substitute two air gap length values of the largest loop of one magnetic pole into Eq. (4). In Eq. (5), it can be seen that under the same magnitude of excitation MMF, fluctuation of effective air gap length in the denominator becomes larger along with rising pole numbers, and uneven space distribution of air gap flux density is intensified. When the pole pair is one, deterioration coefficient K (6) becomes zero, i.e., there is no fluctuation of effective air gap length. 5 6 In addition, it should be noted that the MMF generated by different stator windings in the eccentric state does not completely change with the air gap length, resulting in asymmetry flux density distribution on the circumference. According to previous studies, the number of parallel branches of stator windings is an important factor affecting the distribution of magnetic flux density under eccentricity.

In order to accurately calculate the magnitude of the MMF at the circumferential position under eccentricity, the finite element software is used. The areas covering the stator pole and the rotor pole are drawn for calculation respectively. The deep color is the stator pole and the light color is the rotor pole as shown in Fig. 4. These areas are fixed and do not rotate with the rotor. For each of these areas, the following integral of the current density 7 can be used to obtain the MMF per pole at each location.

Fig. 4.

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	Fig. 4. (color online) Computing regions of stator poles and rotor poles.

According to the Maxwell tensor method, in order to make the force on the interface and other problems with ponderomotive force easy to be calculated, ponderomotive force (volume force) is often attributed to an equivalent group of tensions (area forces). The so-called equivalent means that the resultant force and moment acting on the magnetic substance of a given volume V is exactly equal to the resultant force and moment of the tension on the surface S. S is the surface of volume V. In the process of calculating the unbalanced magnetic pull, a circle close to the outer circle of the rotor is taken as the calculation curve in the two-dimensional (2D) model. The expression of magnetic tension is the same as in Eq. (8), where the flux density B refers to the flux density perpendicular to the interface, called radial flux density acquired from the simulation model using the finite element package. Figure 5 shows a schematic diagram of unbalanced magnetic pull calculation. In this paper, the force in the direction of rotor deflection is only calculated. Equation (9) and the discrete magnetic tensions on the calculation circles are summed up to obtain the total unbalanced magnetic pull (10). Where μ₀ is air permeability, R₂ is the outer radius of the rotor, l is the axial length of the motor, α is the circumferential angle between the two points in Fig. 5, and i is the number of the point on the calculation circle. 8 9 10

Fig. 5.

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	Fig. 5. Calculation of unbalanced magnetic pull.

The direct cause of inhomogeneous space distribution of flux density is the MMF. Figures 6 and 7 show the stator MMF, rotor MMF, and resultant MMF for different loads and different numbers of parallel branches under static eccentricity. The static eccentricity here refers to 60% static eccentricity, which is biased to 0 degree.

Fig. 6.

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	Fig. 6. (color online) MMFs of parallel branch a = 1 (left) and a = 6 (right) under static eccentricity (at no load).

Fig. 7.

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	Fig. 7. (color online) MMFs of parallel branch a = 1 (left) and a = 6 (right) under static eccentricity (at full load).

It can be seen from Fig. 6 that the rotor MMFs in each zone with a = 1 under no-load conditions have a big difference while the rotor MMF in each zone with a = 6 has a little difference. The increase in the number of parallel branches makes the stator MMF compensate for the rotor MMF to cope with the static eccentricity caused by the uneven spatial air gap, but the final resultant MMF has no significant difference. In Fig.7, the stator and rotor MMFs with a = 1 and a = 6 under full-load condition do not change much, but the harmonics of the resultant MMF are high and the stator slot harmonics are mostly obtained through decomposition. Since the change in the resultant MMF of each state is not obvious, in order to show the difference, the concept of a per-unit (P.U.) value is introduced. The P.U. value of the excitation MMF is based on the MMF per pole under health, and that of the air gap length is based on the air gap length under health. Figure 8 gives the P.U. values of flux density for each state, in which the excitation MMF is substituted for root mean square. The farther the curve value is from the health P.U. value 1, the greater the change in flux density. It can be clearly seen from this figure that the a = 1 curve has a greater change in flux density than the a = 6 curve, i.e., the unbalanced magnetic pull is greater; and the unbalanced magnetic pull becomes greater with load.

Fig. 8.

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	Fig. 8. (color online) Comparison of P.U. values at different loads and different parallel branches under the same static eccentricity.

According to the above-mentioned calculation method, the unbalanced magnetic pull in the deviation direction is calculated for different loads and different numbers of parallel branches under 0%, 20%, 40%, and 60% static eccentricity to verify the above-mentioned criteria. Figures 9–10 show the calculation result of UMPs under different situations. Figure 11 gives the main-frequency amplitudes of UMPs in Figs. 9–10. It can be seen from the figure that the 0-Hz UMP with a = 1 is much larger than that of a = 6 and full load increases the 0-Hz UMP. These results are consistent with the above conclusions. It has also been found that as the eccentricity ratio increases, the stator slot harmonic components (11) of the UMP increase. The amplitude of twice power frequency component generated after the square of flux density does not change much as the ratio of eccentricity increases, as shown in Fig. 11.

Fig. 9.

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	Fig. 9. (color online) UMPs of parallel branch a = 1 under static eccentricities at no load (a) and at full load (b).

Fig. 10.

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	Fig. 10. (color online) UMPs of parallel branch a = 6 under static eccentricities at no load (a) and at full load (b).

Fig. 11.

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	Fig. 11. (color online) Main-frequencies amplitude of UMP.

The slip s at no load is 0.02%, and the slip s at full load is 0.8%. The gravity of the rotor of the motor reaches 3200 N. Compared to the UMP growth rate of the eccentricity ratios at zero frequency, the amplitude of the stator slot harmonics increases much more slowly but only the high-frequency component (slot harmonics) makes sense on the vibration and noise of the motor. In addition, it should be noted that the direction of the force is from the position with large magnetic permeability to the position with small permeability. The direction of the unbalanced magnetic force calculated in this paper is the positive direction of the eccentric direction, i.e., the deviation is far from the center of the circle. The unbalanced magnetic pull acts as the result of deteriorating eccentricity. 11

According to Eq. (5), the degree of deterioration due to eccentricity is related to the number of pole pairs of the motor. In particular, when the number of pole pairs is 1, the maximum value of the air gap flux density does not change theoretically, and the deterioration of the inner magnetic circuit increases. This section changes the parameters of the simulation motor and designs a motor with a pole pair of 1 to calculate the unbalanced magnetic pull under static eccentricity. The model motor used in this section is basically the same as the parameters in Table 1, except that the stator winding and the stator outer diameter are changed. If the size of the motor is unchanged, the number of poles becomes smaller, and the number of magnetic circuits included in each pole region increases a lot. While the stator yoke width is not increased, the stator yoke is severely saturated. Therefore, the width of the stator yoke to a suitable value is increased. In order to ensure fair comparison, the air gap flux density of the design is basically the same as that of the 6-pole motor. The specific modification parameters are shown in Table 3.

Table 3.

Table 3.

Table 3. Comparative parameters of p = 1 and p = 3. . p = 1 p = 3 Number of conductors per slot 2 6 Number of strands 54 18 Wire diameter 1.4 mm 1.4 mm Pitch 30 10 Stator outer diameter 600 mm 520 mm

Table 3. Comparative parameters of p = 1 and p = 3. .

Figure 12 shows the calculation area for the stator and rotor poles of a two-pole motor. Figure 13 shows the results of the calculation of the MMF under different loads. It can be seen from the figure that the change of the air gap has little effect on the MMF of the two-pole motor. Figure 14 shows the calculating result of the unbalanced magnetic pull and the amplitudes of the major frequencies of unbalanced magnetic pull under 60% static eccentricity. The average amount of unbalanced magnetic pull caused by the equivalent static eccentricity of a motor with a pole pair of 1 is much smaller than the amplitude of parallel branch a = 1 or 6 in a 6-pole motor as shown in Figs. 9–10, which is in accordance with Eq. (5).

Fig. 12.

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	Fig. 12. (color online) Computing regions of stator poles and rotor poles (p = 1).

Fig. 13.

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	Fig. 13. (color online) Computing MMFs under 60% static eccentricity (p = 1) at no load (a) and at full load (b).

Fig. 14.

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	Fig. 14. (color online) UMPs of different loads and its spectrum under 60% static eccentricity (p = 1).