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Using Equations (4.24) to (4.27) all inductance coefficients are calculated for a 9 kVA, three-phase, four-pole salient pole synchronous machine as

an example. Effects of conventional air-gap asymmetries such as static, dynamic, and mixed eccentricities on machine inductances are studied. These inductances are known as static eccentricity (SE), dynamic eccentric- ity (DE), and mixed eccentricity (ME), respectively. It is assumed that the reference points for both SE and DE are the same. This reference point is at zero angle. The middle point of the first stator tooth and first rotor pole are set at zero angles.

The left panel of Figure 4.8 shows air-gap variations seen from the refer- ence point on the rotor side for eccentric and non-eccentric conditions. The right panel of Figure 4.8 shows these variations seen from the reference point on the stator side.

Figure 4.9 depicts the effective air-gap width for the simulated machine. Stator slot openings, rotor saliency effects, and the shape of rotor poles are considered. All these effects are included in the simulation. Maximum width corresponds with the rotor openings. Exact modeling of the air-gap is the

0 90 180 270 360 Angle (degree) 0 90 180 270 360 0.2 0.3 0.4 0.5 0.6 0.7 Angle (degree) Air Gap (mm) a a b b c c FIGURE 4.8

Air-gap curves for (a) symmetric, (b) 30% static eccentric, and (c) 30% dynamic eccentric machine from the viewpoint of a fixed point on the rotor (left) and a fixed point on the stator (right).

0 50 100 150 200 250 300 350 0 10 20 30 40 50

Rotor Angle (degree)

Air Gap (millimeter)

FIGURE 4.9

most important task for calculating machine inductances. Figure 4.10 shows the air-gap variation for mixed air-gap eccentricity. There are 30% static and 30% dynamic eccentricities. The effect of the rotor pole shape can be seen in this figure [2].

In the following we calculate machine inductances for symmetric and non- symmetric air-gap conditions. When there is any type of individual static or dynamic eccentricities, the amount of each eccentricity is considered to be 30%. Therefore, in mixed air-gap eccentricity there exists 30% static and 30% dynamic eccentricities.

Figure 4.11 shows the stator self-inductances for a symmetric air-gap and for a mixed air-gap eccentricity. Dashed curve is related to healthy machine and solid curve is related to faulty machine. Whereas the maximum magnitude of stator self-inductance in this figure mostly depends on the air-gap width under the rotor pole arc, the minimum magnitude depends on the rotor depth between the rotor poles. It can be seen that under mixed air-gap eccentricity, maximum points have more variation with respect to the minimum points. The reason is that the percentage variation of air-gap width under rotor pole arc is much more

0 50 100 150 200 250 300 350 0

1 2 3

Rotor Angle (degree)

Air Gap Length (mm)

FIGURE 4.10

Zoomed effective air-gap width for the simulated machine with mixed air-gap eccentricity.

0 50 100 150 200 250 300 350 20

40 60 80

Rotor Angle (degree)

Inductance in mH

FIGURE 4.11

than the percentage variation of rotor depth between rotor poles. In the case of individual static or dynamic eccentricities no significant variation is observed.

Figure 4.12 shows the rotor self-inductances for a symmetric air-gap, for an individual static or dynamic air-gap eccentricity, and for a mixed air-gap eccentricity. The effects on individual static eccentricity and dynamic eccen- tricity are the same. Both just increase the rotor self-inductance. However, in mixed air-gap eccentricity the rotor self-inductance varies with the rotor position. It is concluded that this may be a suitable measure for the existence of mixed air-gap eccentricity in synchronous machine. The effect of the sta- tor slot openings is apparent in the rotor self-inductance.

Figure 4.13 shows the stator mutual inductances for a symmetric air-gap and for a mixed air-gap eccentricity. A dashed curve is related to a healthy machine and a solid curve is related to a faulty machine. As expected, mutual inductances have negative values. In this figure, the most negative points of the stator mutual inductance are mostly dependent on the air-gap width under rotor pole arc and the least negative points are dependent on the rotor depth between the rotor poles.

0 50 100 150 200 250 300 350 7 7.5 8 8.5 9 9.5

Rotor Angle (degree)

Inductance in H Healthy SE or DE = 30% SE = 30% & DE = 30% FIGURE 4.12

Self-inductance of one of the rotor windings for healthy (SE: 30%, DE: 30%) and ME (SE: 30% and DE: 30%) conditions.

0 50 100 150 200 250 300 350 –60

–40 –20 0

Rotor Angle (degree)

Inductance in mH

FIGURE 4.13

It can be seen that under mixed air-gap eccentricity the most negative points have more variation with respect to the least negative points. The rea- son is the same as for the stator self-inductance. Again the curves of stator mutual inductances for static and dynamic eccentricities are approximately the same and are very close to the healthy case.

Figure 4.14 represents the mutual inductance curves of one of the stator phases and the rotor winding for a symmetric air-gap and for a mixed air- gap eccentricity. Dashed curve is related to healthy machine and solid curve is related to faulty machine. As expected, the mutual inductance has both positive and negative values. To show the effects of air-gap fault on this inductance a zoomed part of top plot is also shown. It is seen that the effect of the same value of mixed eccentricity on stator to rotor mutual inductance is less than the effect on previous inductances.

Skewing of stator or rotor slots is a technical manufacturing method for better machine performance. With skewed stator or rotor slots, machine inductances vary more smoothly. In other words, slot opening degrades the mutual inductances. Skew can recover it again [4]. In the derived equation for calculating machine inductances, skew effect can be included.

In Figure 4.15, the plots of stator to rotor mutual inductances for a skewed and for an unskewed stator slots are shown. Comparison of the plots in Figure  4.15 shows how a skewed slot affects the shape of these mutual

0 50 100 150 200 250 300 350 –0.5

0 0.5

Rotor Angle (degree)

Inductance (H) 60 70 80 90 100 110 120 0.4 0.5 0.6 0.7 0.8

Rotor Angle (degree)

Inductance (H)

Mixed eccentricity

Healthy rotor

FIGURE 4.14

Mutual inductances between one of the stator phases and rotor winding for the healthy machine and faulty machine with mixed air-gap eccentricity (SE: 30% and DE: 30%).

inductances. Magnitude of mutual inductances in the skewed case is a little less with respect to unskewed inductances. However, the ripples due to slot openings are suppressed in this plot. To further study the effect of skew, the derivative of mutual inductance is calculated and plotted in Figure 4.16. Effect of stator slot openings in an unskewed machine is clear. For a skewed slot the derivative of the mutual inductance variation is smooth.

4.2.4 Experimental Measurement of Inductances of