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Paths of magnetic flux harmonics in a machine stator core
Journal
PATHS OF MAGNETIC
FLUX HARMONICS
of Magnetism
IN A MACHINE
and Magnetic Materials 26 (1982)50-52 North-Holland Publishing Company
STATOR CORE
G.S. RADLEY Robert Gordon’s Institute of Technologv, Aberdeen,
Scotlund,
UK
and A.J. MOSES Wolfson Centre for Mugnetic Muteriuls
Measurements fifth harmonics
Technologv,
lJnioersi(v College. CmdifJ
Wules, UK
of local magnetic flux density have been made on a stator core model. Plots of the fundamental. third and are presented. The third and fifth harmonic patterns suggest that flux circulates in local closed paths.
1. Introduction Distributions of magnetic flux have previously been studied in model transformer cores [ 11, where local closed paths apparently exist for individual harmonics. This work has been extended to the stator core of a large machine where similar patterns have been found in an experimental model.
2. Apparatus The model is described in detail elsewhere [2]. It consists of a segmented stack of grain-oriented steel laminations of 1.1 m outer diameter and 0.5 cm depth, excited by a winding fed from a threephase supply. Local in-plane flux density within the core is measured from sets of mutally orthogonal single-turn search coils wound through holes in the test segment, at 35 test points. By inverting the segment a total of 61 points may be studied per segment position in the stack. 9 of these points are duplicated by this process as a check on local effects of the steel. These 9 points can be seen in the figures as those with two vectors radiating from the common test point. The magnitude and phase of the harmonic components of e.m.f. from the search coils were measured using a Bruel and Kjaer frequency analyzer 0304~8853/82/0000-0000/$02.75
0 1982 North-Holland
and phase meter. Flux density magnitude and spatial angle were calculated and plotted directly by computer.
3. Experimental results Fig. 1 shows the fundamental component of flux density for successive increments of time over the cycle, wt being the phase angle. The six diagrams are laid out in the form of a semicircle for convenience and to demonstrate the smooth change in spatial angle that occurs with time. The pattern does not represent the overall distribution in the machine at a single instant. The figure clearly shows the dipole pattern of a 2-pole machine. The agreement both in magnitude and phase between the 2 vectors for each of the 9 duplicated points can be seen. The pattern is almost continuous over butt joints. At wt = 60” (near the interpolar axis) over the centre line of the segment where it bridges butt joints in adjacent layers the vectors are some 15% greater in magnitude than those away from the centre line. The vectors near the butt joints are correspondingly reduced. No measurements of the component of flux normal to the plane of the lamination were made in this study, but it is assumed that there is a small transfer of funda-
G.S. Radley, A.J. Moses / Paths of magnetic flux harmonics
Fig. I. Fundamental intervals in time.
component
of in-plane flux density at 30”
mental between adjacent layers caused by the joints. Fig. 2 shows patterns of the third harmonic at intervals of wt of 15 fundamental degrees. The pattern for ot = 0” shows two regions where the flux is apparently circulating in a closed path in the plane of the lamination. Using the convection of tooth and slot numbering given, these occur (a) behind tooth 2 and slot 3, where the circulation is clockwise, and (b) behind tooth 4 and slot 4, being there anticlockwise. The pattern at wt = 0” can be seen in reverse half a cycle later at wt = 60’. At wt = 75’ the vectors for the lower four positions on the centre line are directed to the right. Conversely the vectors flanking the two butt joints are at this time directed to the left. Both sets appear to peak at this instant, but there is no
Fig. 2. Third harmonic 15” intervals.
component
of in-plane
flux density at
return path in the plane of the sheet for either set. This corresponds approximately to the time when the interpolar axis of the fundamental component is &eeping the segment. Thus the centre line third harmonic fhtx is in the opposite direction to the fundamental at this time. It is assumed that the steel is close to magnetic saturation at this location and time and that the third harmonic is generated locally due to the non-linearity of the steel there. A closed path for this third harmonic is assumed to exist with a radial axis so that the return path is first in the normal direction into adjacent layers and then across the butt joints of these layers. This
G.S. Rudley, A.J. Moses / Paths of mugnetic flux harmonics
52
Table Spread
I
Order
of harmonic
in flux density
measurement Magnitude
fundamental third fifth
due to segment (%)
30 50 50
inversion
Phase (/360”) 140
I IO” 75”
gradient in permeability of the steel is the dominant factor in the generation of these local circulations. Table1 compares magnitude and phase of the two vectors for each of the nine duplicated tests points as the segment is inverted. The inferior agreement for the higher harmonics cannot simply be explained as poorer measurement repeatability due to smaller flux densities compared with the fundamental, because the measured e.m.f. values for all three harmonics were of similar order of magnitude. It seems more likely that local effects, such as permeability, influence the harmonic paths. Fig. 3. Fifth harmonic intervals.
component
of in-plane
flux density
at 6’
is consistent with the left ward flow of the third harmonic near the butt joints of the test layer at wt = 15”. The pattern of the fifth harmonic is shown in fig. 3. Here 72” represents a complete cycle so that half a cycle is shown with increments of 6’. Agam local in-plane circulation can be seen, (a) at LOI= 12”, behind tooth 2, clockwise, and (b) at 18’ and 24’, behind tooth 3, anticlockwise. Jackson [3] has examined the flux distribution in a stator core theoretically and obtained patterns of local closed paths of in-plane third harmonic flux somewhat similar to those for the third and fifth harmonic found here. He suggests [4] that a
Acknowledgements The authors wish to thank the British Steel Corporation for permission to publish this work, which was carried out as part of a BSC contract awarded by the European Coal and Steel Community.
References [I] A.J. Moses and B. Thomas, IEEE Trans. Magn. MAG-IO (1974) 148. [2] G.S. Radley and A.J. Moses, IEEE Trans. Magn. MAG-17 (1981) 1311. [3] R.L. Jackson. Compumag Conf., Grenoble (1978) paper
1.4. [4] R.L. Jackson,
private
communication.
PATHS OF MAGNETIC
FLUX HARMONICS
of Magnetism
IN A MACHINE
and Magnetic Materials 26 (1982)50-52 North-Holland Publishing Company
STATOR CORE
G.S. RADLEY Robert Gordon’s Institute of Technologv, Aberdeen,
Scotlund,
UK
and A.J. MOSES Wolfson Centre for Mugnetic Muteriuls
Measurements fifth harmonics
Technologv,
lJnioersi(v College. CmdifJ
Wules, UK
of local magnetic flux density have been made on a stator core model. Plots of the fundamental. third and are presented. The third and fifth harmonic patterns suggest that flux circulates in local closed paths.
1. Introduction Distributions of magnetic flux have previously been studied in model transformer cores [ 11, where local closed paths apparently exist for individual harmonics. This work has been extended to the stator core of a large machine where similar patterns have been found in an experimental model.
2. Apparatus The model is described in detail elsewhere [2]. It consists of a segmented stack of grain-oriented steel laminations of 1.1 m outer diameter and 0.5 cm depth, excited by a winding fed from a threephase supply. Local in-plane flux density within the core is measured from sets of mutally orthogonal single-turn search coils wound through holes in the test segment, at 35 test points. By inverting the segment a total of 61 points may be studied per segment position in the stack. 9 of these points are duplicated by this process as a check on local effects of the steel. These 9 points can be seen in the figures as those with two vectors radiating from the common test point. The magnitude and phase of the harmonic components of e.m.f. from the search coils were measured using a Bruel and Kjaer frequency analyzer 0304~8853/82/0000-0000/$02.75
0 1982 North-Holland
and phase meter. Flux density magnitude and spatial angle were calculated and plotted directly by computer.
3. Experimental results Fig. 1 shows the fundamental component of flux density for successive increments of time over the cycle, wt being the phase angle. The six diagrams are laid out in the form of a semicircle for convenience and to demonstrate the smooth change in spatial angle that occurs with time. The pattern does not represent the overall distribution in the machine at a single instant. The figure clearly shows the dipole pattern of a 2-pole machine. The agreement both in magnitude and phase between the 2 vectors for each of the 9 duplicated points can be seen. The pattern is almost continuous over butt joints. At wt = 60” (near the interpolar axis) over the centre line of the segment where it bridges butt joints in adjacent layers the vectors are some 15% greater in magnitude than those away from the centre line. The vectors near the butt joints are correspondingly reduced. No measurements of the component of flux normal to the plane of the lamination were made in this study, but it is assumed that there is a small transfer of funda-
G.S. Radley, A.J. Moses / Paths of magnetic flux harmonics
Fig. I. Fundamental intervals in time.
component
of in-plane flux density at 30”
mental between adjacent layers caused by the joints. Fig. 2 shows patterns of the third harmonic at intervals of wt of 15 fundamental degrees. The pattern for ot = 0” shows two regions where the flux is apparently circulating in a closed path in the plane of the lamination. Using the convection of tooth and slot numbering given, these occur (a) behind tooth 2 and slot 3, where the circulation is clockwise, and (b) behind tooth 4 and slot 4, being there anticlockwise. The pattern at wt = 0” can be seen in reverse half a cycle later at wt = 60’. At wt = 75’ the vectors for the lower four positions on the centre line are directed to the right. Conversely the vectors flanking the two butt joints are at this time directed to the left. Both sets appear to peak at this instant, but there is no
Fig. 2. Third harmonic 15” intervals.
component
of in-plane
flux density at
return path in the plane of the sheet for either set. This corresponds approximately to the time when the interpolar axis of the fundamental component is &eeping the segment. Thus the centre line third harmonic fhtx is in the opposite direction to the fundamental at this time. It is assumed that the steel is close to magnetic saturation at this location and time and that the third harmonic is generated locally due to the non-linearity of the steel there. A closed path for this third harmonic is assumed to exist with a radial axis so that the return path is first in the normal direction into adjacent layers and then across the butt joints of these layers. This
G.S. Rudley, A.J. Moses / Paths of mugnetic flux harmonics
52
Table Spread
I
Order
of harmonic
in flux density
measurement Magnitude
fundamental third fifth
due to segment (%)
30 50 50
inversion
Phase (/360”) 140
I IO” 75”
gradient in permeability of the steel is the dominant factor in the generation of these local circulations. Table1 compares magnitude and phase of the two vectors for each of the nine duplicated tests points as the segment is inverted. The inferior agreement for the higher harmonics cannot simply be explained as poorer measurement repeatability due to smaller flux densities compared with the fundamental, because the measured e.m.f. values for all three harmonics were of similar order of magnitude. It seems more likely that local effects, such as permeability, influence the harmonic paths. Fig. 3. Fifth harmonic intervals.
component
of in-plane
flux density
at 6’
is consistent with the left ward flow of the third harmonic near the butt joints of the test layer at wt = 15”. The pattern of the fifth harmonic is shown in fig. 3. Here 72” represents a complete cycle so that half a cycle is shown with increments of 6’. Agam local in-plane circulation can be seen, (a) at LOI= 12”, behind tooth 2, clockwise, and (b) at 18’ and 24’, behind tooth 3, anticlockwise. Jackson [3] has examined the flux distribution in a stator core theoretically and obtained patterns of local closed paths of in-plane third harmonic flux somewhat similar to those for the third and fifth harmonic found here. He suggests [4] that a
Acknowledgements The authors wish to thank the British Steel Corporation for permission to publish this work, which was carried out as part of a BSC contract awarded by the European Coal and Steel Community.
References [I] A.J. Moses and B. Thomas, IEEE Trans. Magn. MAG-IO (1974) 148. [2] G.S. Radley and A.J. Moses, IEEE Trans. Magn. MAG-17 (1981) 1311. [3] R.L. Jackson. Compumag Conf., Grenoble (1978) paper
1.4. [4] R.L. Jackson,
private
communication.