roebel windings for hydro generators -...
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ROEBEL WINDINGS FOR HYDRO GENERATORS
Thesis for the Master of Science (MSc) degree Poopak Roshanfekr Fard
Division of Electric Power Engineering Department of Energy & Environment CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 2007
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ROEBEL WINDINGS FOR HYDRO GENERATORS
Poopak Roshanfekr Fard
Division of Electric Power Engineering Department of Energy & Environment
CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden, 2007
Department of Energy & Environment Chalmers University of Technology Göteborg, Sweden, 2007 Proposed & Sponsored by: ABB Performed at: Chalmers University of Technology Göteborg, Sweden Supervisor & Examiner: Dr. Sonja Tidblad Lundmark Division of Electric Power Engineering Department of Energy & Environment Chalmers University of Technology
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ACKNOWLEDGEMENTS This thesis work has been carried out at the Division of Electric Power Engineering, Chalmers University of Technology. I would like to express my special thanks to my supervisor, Dr. Sonja Tidblad Lundmark who has not only enlightened me with her courses at Chalmers University of Technology but also actively participated to complete this thesis work. Her technical insight, advice and patient guidance are gratefully appreciated. The financial support granted by ABB AB Corporate Research is gratefully acknowledged. I am also indebted to Mr. Bengt Rothman and Dr. Waqas Arshad at ABB for their help. Many thanks go to all the members of the Division of Electric Power Engineering at Chalmers University of Technology who shared their knowledge with me. Last but not the least, I wish to thank my family, in particular my dear brother Siamak for his special support during the period of study at Chalmers University of Technology. For all your help I need to say:
Thank You.
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ABSTRACT Large-scale generators tend to have high power. Normally, the armature winding of these generators consist of multiple strands insulated separately and transposed (using Roebel transposition) in order to suppress losses caused by eddy currents and circulating currents. As the generator reaches high power density, circulating currents in the armature winding are large, and hence the loss must be estimated accurately in generator design. Calculation of such losses requires the distribution of circulating currents, which differs from strand to strand. The Roebel bar optimum structure allows increasing machine efficiency, and consequently energy savings. In this thesis, the circulating currents in a Roebel transposed diamond coil have been studied. The Roebel transposition in the active part of the generator have been 180° and 360°. Moreover the circulating currents for a traditional Roebel bar (no transposition at the end region) and for a Roebel bar with transposition at the end region has been carried out. Finally the values of circulating currents and the losses due to these currents for these Roebel bars have been compared. The study has been carried out using models of the winding developed in a FEM-program package. A 2D model of the active part has been drawn in MagNet. From the flux linkages that are taken from the program the circulating currents has been calculated analytically. A 3D model of the end region of the winding has been drawn in MagNet and the fields of this region has been studied. The circulating currents in half a coil has been investigated from the results obtained from the 2D and 3D simulations. The conclusions of the study are that using 360° transposition in the active part reduces the circulating currents almost to zero. 180° transposition reduces the circulating current. The study further concludes that using transposition in the end region reduces the circulating currents in the bar. Finally the study concludes that the minimum circulating current loss is achieved in case of 360° transposition in the active part (slot region) and transposition in the end region. It is also shown a cost-effective well chosen manufacturing process. Key Words: Roebel bar, Circulating currents, 360° transposition, 180° transposition, active part (slot region), end region, 2D finite element method (FEM), 3D finite element method (FEM)
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Table of Contents ACKNOWLEDGEMENTS............................................................................................... iii ABSTRACT.......................................................................................................................... v Table of Contents............................................................................................................... vi 1. INTRODUCTION .......................................................................................................... 1
1.1. Objective of the Thesis ............................................................................................ 1 1.2. Outline of the Thesis Report .................................................................................... 1
2. THEORETICAL BACKGROUND................................................................................ 3 2.1. Armture Winding Configuration.............................................................................. 3 2.2. Roebel Bars.............................................................................................................. 3
2.2.1. Transposition in the slot region......................................................................... 4 2.2.2. Transposition in the end region......................................................................... 5
2.3. Analysis of strand currents in slot region ................................................................ 5 2.3.1. Analysis of strand currents in slot region at 360° transposition ....................... 6 2.3.2. Analysis of strand currents in slot region at 180° transposition ....................... 9
2.4. Strand circuit .......................................................................................................... 10 2.5. Circuit equations .................................................................................................... 10
2.5.1. Unknown being strand current........................................................................ 11 2.5.2. Unknown being the circulating current........................................................... 12 2.5.3. Selection of unknowns in circuit equations .................................................... 13
2.6. Magnetic Flux Linkage in the strands.................................................................... 15 2.7. Analysis of strand currents in half coils................................................................. 15 2.8. Resistance .............................................................................................................. 16 2.9. Inductance .............................................................................................................. 16
3. SIMULATION OF SLOT REGION ............................................................................ 17 3.1. Simulated model .................................................................................................... 17
3.1.1. Boundary conditions ....................................................................................... 18 3.2. Inductance calculation ........................................................................................... 18
3.2.1. Phase current ................................................................................................... 18 3.3. Resistance .............................................................................................................. 19 3.4. External flux........................................................................................................... 19
3.4.1. Radial flux....................................................................................................... 19 3.4.2. Transversal flux .............................................................................................. 20 3.4.3. Circulating current due to radial and transversal flux before transposition.... 20 3.4.4. Circulating current after transposition ............................................................ 22
3.5. Internal flux............................................................................................................ 24 3.6. External & Internal flux ......................................................................................... 25
3.6.1. Circulating current due to internal and external flux after applying transposition.............................................................................................................. 25
4. SIMULATION OF END REGION .............................................................................. 27 4.1. End winding without transposition ........................................................................ 27
4.1.1. Boundary conditions ....................................................................................... 28 4.1.2. Inductance calculation .................................................................................... 28 4.1.3. Resistance ....................................................................................................... 28 4.1.4. External flux.................................................................................................... 29
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4.1.5. Circumferential external flux.......................................................................... 29 4.1.6. Internal flux..................................................................................................... 30 4.1.7. Internal & External flux .................................................................................. 31
4.2. End winding with transposition ............................................................................. 31 4.2.1. Boundary conditions ....................................................................................... 32 4.2.2. Inductance calculation .................................................................................... 32 4.2.3. Resistance ....................................................................................................... 32 4.2.4. External flux.................................................................................................... 33 4.2.5. Internal flux..................................................................................................... 34 4.2.6. Internal & External flux .................................................................................. 34
5. CALCULATION OF CIRCULATING CURRENT IN COIL..................................... 35 5.1. Circulating current in case of 180° transposition in the slot region....................... 35 5.2. Circulating current in case of 360° transposition in the slot region....................... 37 5.3. Loss situations........................................................................................................ 38
6. CONCLUSIONS........................................................................................................... 39 7. FUTURE WORK.......................................................................................................... 41 APPENDIX 1. THE SCRIPT FILE OF ONE END COIL IN VISUAL BASIC FORMAT........................................................................................................................................... 42 APPENDIX 2. MATLAB PROGRAM DEVELOPED FOR CALCULATION OF CIRCULATING CURRENTS IN THE ACTIVE PART IN CASE OF 360° TRANSPOSITION ........................................................................................................... 48 APPENDIX 3. MATLAB PROGRAM DEVELOPED FOR CALCULATION OF CIRCULATING CURRENTS IN THE ACTIVE PART IN CASE OF 180° TRANSPOSITION ........................................................................................................... 54 REFERENCES ................................................................................................................. 59
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1. INTRODUCTION Large-scale generators tend to have high power density. Normally, the armature winding of these generators consist of multiple strands insulated separately and transposed (using Roebel transposition) in order to suppress losses caused by eddy currents and circulating currents. As the generator reaches high power density, circulating currents in the armature winding can be large, and hence the loss must be estimated accurately in generator design, Because the circulating current losses may lead to an overheating. Calculation of such losses requires the knowledge distribution of circulating currents, which differs from strand to strand. The Roebel bar optimum layout allows increasing machine efficiency, and consequently energy savings.
1.1. Objective of the Thesis The main objective of the thesis work presented in this report is to compare the circulating current losses between the different winding layouts. The impact of transposition in the active part as well as transposition in the end region are of interest. This study focuses on comparing the circulating current losses between two different Roebel bars. The first bar is a traditional Roebel bar (no transposition in the end region) and the second bar is a bar for which transposition in the end region is applied. The study further assesses the impact of transposition in the slot region. To study circulating current losses, a winding model was developed in a FEM-program package. A 2D model of the active part has been drawn in MagNet (Infolytica 2007). From the flux linkages that are taken from the program the circulating currents has been calculated analytically. Two geometries of the end region has been drawn in 3D MagNet and the fields of this region has been studied. The results obtained from the 2D and 3D simulations were then used to evaluate the circulating currents losses for the two different Roebel bars layouts.
1.2. Outline of the Thesis Report This report is arranged in seven chapters. Chapter 1, which constitutes the introduction, provides the background to the thesis work and sets out objectives, which the study aims to accomplish. Chapter 2 discusses the theoretical background relevant to this thesis work. Chapter 3 primarily focuses on the slot region, the active part of the generator. The model is simulated in MagNet and the flux linkages due to external and internal flux in the
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strands are outputs from the software. Circulating current in case of 180° transposition and 360° transposition are calculated analytically. Chapter 4 present the simulation of the end region. In Chapter 5 the results of chapter 3 and 4 are used to calculate the circulating current in one half bar. Finally chapter 6, aimed at detailing overall conclusions of the thesis work, and chapter 7 presents some future work, forms the last chapter of the report.
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2. THEORETICAL BACKGROUND
This chapter gives the theoretical background necessary for the study of the circulating currents that occur in the strands of a Roebel bar.
2.1. Armture Winding Configuration The armatures of nearly all synchronous machines, and of medium-large induction machines, are wound with double-layer windings. A double layer winding is one in which there are two separate coil sides in any one slot; a given coil has one coil side in the upper half of a slot (i.e. in the top layer nearest to the air gap), and its return coil side lies in the bottom half of another slot (i.e. in the bottom layer). The end winding arrangement of the double-layer winding is illustrated in Fig. 2.1.
Figure 2.1: End winding of a double-layer winding.
2.2. Roebel Bars The stator coils of large generators are built from a large number of strands of rectangular cross section. These strands or subconductors are insulated from each other. This is because with fine conductor division, eddy current losses are reduced (Fujita, Kabata et al. 2005) compared to a massive bar. The problem with induced eddy currents is however still existing and because of magnetic flux linkage, circulating currents will flow in the winding strands, where the value differs from strand to strand. As power density is high in large-scale generators, circulating currents in the armature winding are large. These circulating currents produce circulating current losses. Circulating current losses may lead to overheating. In order to reduce these losses Roebel transposition is applied. In a Roebel transposition the strands are transposed along the machine’s length in the slot’s area. Figure 2.2 shows construction of a Roebel bar with 360° transposition in the slot area.
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Figure 2.2: Construction of classic Roebel bars (360° transposition) (Sequenz 1973).
2.2.1. Transposition in the slot region In Roebel transposition the angle may be 90°, 180°, 360°, 450°, 540° and so on, and the circulating current loss depends on the angle. In this study, angles of 180°, and 360° are examined. Figure 2.3 and 2.4 show the transposition in the slot region (with 10 strands) due to 180° and 360°, where the transposition of strand number eight is highlighted.
Figure 2.3: 180° transposition in slot region.
Figure 2.4: 360° transposition in slot region.
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2.2.2. Transposition in the end region At the end of a traditional Roebel bar all the strands are short-circuited within a box known as a clip. Figure 2.5 shows the end region of a traditional Roebel bar. If the clip is replaced with a curve then it’s said that there is transposition at the end of the bar. Figure 2.6 shows the end region of a Roebel bar with transposition.
2.3. Analysis of strand currents in slot region In a Roebel bar each coil contains of several layers in two rows; as can be seen in figure 2.7. The real model of this generator contains 14 layers, although in this study 5 layers has been considered in the simulation. The current within each strand is the sum of two components:
1) Input current that flows uniformly into every strand. 2) Circulating current that differs from strand to strand which sums to zero over the
cross-section of all the strands in one bar (Macdonald 1971).
enlarged view of bar section
enlarged view of bar section Figure 2.6: End winding
with transposition.
Figure 2.5: End winding without transposition.
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Figure 2.7: Copper wire arrangement in the simulated model.
2.3.1. Analysis of strand currents in slot region at 360° transposition Figure 2.8 shows the strand transposition in the slot region at 360°. As is evident from Fig. 2.8(a), a strand is slanted inside the slot because of the Roebel transposition. The analytical model shown in Fig. 2.8(b) approximates this slant stepwise. In this huge hydro generator, the stator core length is long compared to the slot depth (almost 10 times bigger); therefore, the strand slant is small, and the stepwise approximation employed for the analytical model seems appropriate. Strand current analysis reduces to the solution of Eq. (2.1), with Z denoting the impedance matrix (including resistance and inductance), I the strand current vector, and V the voltage drop difference vector (magnetic flux linkage between strands differentiated with respect to time):
[Z]{I}= {V} (2.1) Incidentally, V includes magnetic coupling between the top and bottom coils in the same slot. The variation of this magnetic coupling between the top and bottom coils due to strand current nonuniformity is assumed to be small, and therefore, Eq. (2.1) is here solved independently for top and bottom coils.
Top coil
Bottom coil
Insulation
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Figure 2.8: Transposition of strand (360°): (a) actual strand and (b) analytical model
(Kazuhiko Takahashi 2003).
Z and V in Eq. (2.1) are obtained with regard for transposition at the slot region, as shown in Fig. 2.8. Z and V are found per unit length, with 0Zl and 0Vl pertaining to the slot region. The superscript denotes the number of transpositions. The number of transpositions is taken to be the number of layers by which a strand in the slot is shifted. Supposing that the number of strand layers is n/2 in case of 360° transposition, the Roebel transposition is completed at n because of the two-row configuration. In this simulation the number of strand layers is 5, so in the case of 360° transposition the Roebel transposition is completed at 10. Zlk and Vlk at k transpositions can be found by permutation of the rows and columns in 0Zl , and rows in 0Vl , according to the changed position (matrix is shown in figure 2.9). With dlc denoting the strand length within one transposition interval, the following equations for Z and V are obtained in case of 360° transposition:
[ ] [ ] [ ] [ ]
{ } { } { } { }ncc
c
ncc
c
Vldl
VldlVldl
V
Zldl
ZldlZldl
Z
22
22
10
10
+++=
+++=
�
�
(2.2)
In Eq. (2.2), the strand length pertaining to 0Zl , nZl and 0Vl , nVl is dlc/2. This transposition interval corresponds to the number of transpositions 0 and n in the slot region. dlc is the core length divided by the number of transposition. As in this study the
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number of transposition is 10 in case of 360° transposition, the value of dlc is tenth of the core length.
��������������
�
�
��������������
�
�
=
0000000001100000000001000000000010000000000100000000001000000000010000000000100000000001000000000010
P
Figure 2.9: Permutation matrix and strand enumeration of Roebel Bar (Haldemann 2004).
The 1 in the first row and in column two of the permutation matrix moves a strand from position 1 to position 2; the 1 in the second row in column three moves a strand from position 2 to 3, etc. Figure 2.10 shows the strand position in transposition 1.
Figure 2.10: Strand enumeration of Roebel Bar after first transposition.
As it was mentioned before, permutation of the rows and columns in the basic impedance matrix (P* 0Zl *P´) leads directly to 1Zl and so forth. By permutation the rows of 0Vl (P* 0Vl ) the second voltage matrix 1Vl can be obtained and so on.
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2.3.2. Analysis of strand currents in slot region at 180° transposition Figure 2.11 shows the strand transposition in slot region at 180°. Supposing that the number of strand layers is n/2 in case of 180° transposition, the Roebel transposition is completed at n/2. In this simulation the number of strand layers is 5, so in the case of 180° transposition the Roebel transposition is completed at 5. In case of 180°, the equation 2.2 can be modified as shown below:
[ ] [ ] [ ] [ ]
{ } { } { } { }2/10
2/10
22
22
ncc
c
ncc
c
Vldl
VldlVldl
V
Zldl
ZldlZldl
Z
+++=
+++=
�
�
(2.3)
Zlk and Vlk at k transpositions can be found by permutation of the rows and columns in
0Zl , and rows in 0Vl , according to the changed position (matrix is shown in figure 2.9). dlc denoting the strand length within one transposition interval, and is the core length divided by the number of transposition. As in this study the number of transposition is 5 in case of 180° transposition, the value of dlc is fifth of the core length. In Eq. (2.3), the strand length pertaining to 0Zl , 2/nZl and 0Vl , 2/nVl is dlc/2. This transposition interval corresponds to the number of transpositions 0 and n/2 in the slot region.
Figure 2.11: Transposition of strand (180°): (a) actual strand and (b) analytical model.
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2.4. Strand circuit First the strand circuit has been considered in order to derive the circuit equations. Suppose that the number of winding strand layers is n/2 (in the simulation 5); because of the two-row configuration, the total number of strands is n (which is 10 in the simulated model; see figure 2.7). As shown in Fig. 2.12, the strand circuit includes resistances R and inductances lkL , . Resistance R is normally the same for every strand. lkL , stands for
mutual inductance between strands k and l , becoming the self-inductance of strand k when lk = . Only self inductances are shown in Fig. 2.12, but all mutual inductances are actually considered in the analysis. Here I is the input current in the stator winding strand; the strand current flowing through strand k is ki + I/n. The strand current ki + I/n is the sum of the stator winding input current I distributed uniformly among the strands (that is, I/n), and the nonuniformity component ki . The nonuniformity ki occurs due to the circulating current.
exkΦ is the external magnetic flux produced in strand k . It could be due to rotor currents, or can be produced by other windings. The magnetic flux produced by the current I/n in all strands is referred to as the internal magnetic flux.
Figure 2.12: Strand circuit (Kazuhiko Takahashi 2003).
2.5. Circuit equations Two types of circuit equations are considered. First circuit equations with the strand current being unknown and second circuit equations with the circulating current being unknown which is used in the calculation.
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2.5.1. Unknown being strand current Let us consider circuit equations in which the unknown is the strand current. The voltage drops of strands 1 to n are shown in Eqs. (2.4) to (2.7) (Kazuhiko Takahashi 2003):
�=
+++=n
kkk n
IiLj
nI
iRV1
,111 )()( ω (2.4)
�=
+++=n
kkk n
IiLj
nI
iRV1
,222 )()( ω (2.5)
. . . �
=−−− +++=
n
kkknnn n
IiLj
nI
iRV1
,111 )()( ω (2.6)
�=
+++=n
kkknnn n
IiLj
nI
iRV1
, )()( ω (2.7)
Now if we assume that the strands are joined together at the end of slot region:
11 exjVV Φ−=− ω (2.8)
22 exjVV Φ−=− ω (2.9) . . .
11 −− Φ−=− exnn jVV ω (2.10)
exnn jVV Φ−=− ω (2.11) V is the unknown voltage along the bar in the slot region. From equations (2.4) to (2.11), the following relation can be derived:
������
������
�
Φ−=−+++
Φ−=−+++
Φ−=−+++
Φ−=−+++
�
�
�
�
=
−=
−−
=
=
exn
n
kkknn
exn
n
kkknn
ex
n
kkk
ex
n
kkk
jVnI
iLjnI
iR
jVnI
iLjnI
iR
jVnI
iLjnI
iR
jVnI
iLjnI
iR
ωω
ωω
ωω
ωω
1,
11
,11
21
,22
11
,11
)()(
)()(
)()(
)()(
� (2.12)
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The strand current kI and the circulating current ki in strand k are related in the following way:
nI
iI kk += (2.13)
On the other hand, from the current conservation law, the sum of strand currents kI is equal to the input current I:
IIn
kk =�
= 1 (2.14)
Thus the following simultaneous equations with strand currents as unknown are obtained in combination of Eq. (2.12) with Eq. (2.14):
�������
�������
�
=
Φ−=−+
Φ−=−+
Φ−=−+
Φ−=−+
�
�
�
�
�
=
=
−=
−−
=
=
II
jVILjRI
jVILjRI
jVILjRI
jVILjRI
n
kk
exn
n
kkknn
nex
n
kkknn
ex
n
kkk
ex
n
kkk
1
1,
)1(1
,11
21
,22
11
,11
ωω
ωω
ωω
ωω
�
(2.15)
2.5.2. Unknown being the circulating current Now let us consider circuit equations with the circulating current considered to be unknown, as is used for calculations in this study (Kazuhiko Takahashi 2003). Since the input current I/n in equation (2.12) is known for every strand, it is transformed to the right-hand side:
nI
LjnI
RjViLjRin
kkex
n
kkk ��
==
−−Φ−=−+1
,111
,11 ωωω (2.16)
The last term on the right-hand side of equation (2.16), other than ωj is the flux produced in strand 1 when the current I/n flows through the strand, that is, the internal flux. The internal flux in strand 1 can be expressed as follows:
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�=
=Φn
kkin n
IL
1,11 (2.17)
Equation (2.16) can be rewritten to obtain the following equation for circulating current
1i :
nI
RjViLjRi inex
n
kkk −Φ+Φ−=−+ �
=
)( 111
,11 ωω (2.18)
On the other hand, the current conservation law states that the sum of the circulating current ki is zero:
01
=�=
n
kki (2.19)
Thus the following simultaneous equations with circulating currents as unknown are obtained in combination of Eq. (2.18) with Eq. (2.19):
�������
�������
�
=
−Φ+Φ−=−+
−Φ+Φ−=−+
−Φ+Φ−=−+
−Φ+Φ−=−+
�
�
�
�
�
=
=
−−=
−−
=
=
0
)(
)(
)(
)(
1
1,
)1()1(1
,11
221
,22
111
,11
n
kk
innexn
n
kkknn
ninnex
n
kkknn
inex
n
kkk
inex
n
kkk
i
nI
RjViLjRi
nI
RjViLjRi
nI
RjViLjRi
nI
RjViLjRi
ωω
ωω
ωω
ωω
�
(2.20)
2.5.3. Selection of unknowns in circuit equations The circuit equations derived in Sections 2.5.1 and 2.5.2 are compared here to decide which unknown should be chosen, the strand current or the circulating current. Comparing circuit equation (2.15) with the strand current as the unknown, and circuit equation (2.20) with the circulating current as the unknown, the coefficients on the left are the same. As regards the right-hand side, Eq. (2.15) involves only voltages related to the external magnetic flux, while Eq. (2.20) involves voltages related to the resistance as well as to both the internal and external magnetic flux. In addition, Eqs. (2.15) and (2.20) differ in the current conservation terms. Now it must be considered whether the strand current in Eq. (2.15) or the circulating current in Eq. (2.20) should be employed when calculating the magnetic flux by FEM magnetic field analysis.
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In conventional magnetic field analysis using FEM, the magnetic flux linkage in the strands is the sum of the internal flux produced by the current I in the winding under consideration, and the external flux, which corresponds to the right-hand side of Eq. (2.20). Therefore the circulating current is treated as an unknown in this study. Thus, Eq. (2.20) is employed for strand current calculation on the model coils. Incidentally, in case of no-load operation with input current I and the internal flux being zero, the substitution of I= 0 and 0=Φ ink into Eqs. (2.13) and (2.20) gives kk iI = ; that is, Eqs. (2.15) and (2.20) become identical. The matrix form of Eq. (2.20) that was used in MATLAB (MathWorks 2006) to calculate the circulating currents is shown in Eq. (2.21).
������
�
�
������
�
�
−−
−−−−
=
������
�
�
������
�
�
������
�
�
������
�
�
−
−−
001111
11
10
2
1
10
2
1
1010102101
2102221
1101211
Uj
Uj
Uj
V
i
i
i
ZZZ
ZZZ
ZZZ
ωλ
ωλωλ
��
�
�
�����
�
�
(2.21)
exkinkk Φ+Φ=λ (2.22)
U
nI
R = (2.23)
Circulating currents can be obtained by:
������
�
�
������
�
�
−−
−−−−
������
�
�
������
�
�
−
−−
=
������
�
�
������
�
�−
001111
11
10
2
11
1010102101
2102221
1101211
10
2
1
Uj
Uj
Uj
ZZZ
ZZZ
ZZZ
V
i
i
i
ωλ
ωλωλ
�
�
�
�����
�
�
� (2.24)
����
�
�
����
�
�
+
++
=
����
�
�
����
�
�
=
1010102101
2102221
1101211
1010102101
2102221
1101211
LjRLjLj
LjLjRLj
LjLjLjR
ZZZ
ZZZ
ZZZ
Z
ωωω
ωωωωωω
�
����
�
�
�
����
�
�
(2.25)
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[ ]����
�
�
����
�
�
−−
−−−−
=
Uj
Uj
Uj
V
10
2
1
ωλ
ωλωλ
� (2.26)
Permutation is applied to Z and [V] matrix.
2.6. Magnetic Flux Linkage in the strands The magnetic flux linkage on the right-hand side of Eq. (2.20) is calculated by FEM magnetic field analysis. In slot region FEM-based 2D magnetic field is used and FEM-based 3D magnetic field for the end region.
2.7. Analysis of strand currents in half coils Equations (2.2) and (2.3) derived in the slot region can be modified to obtain the equations regarding to half coils. 0Zl and 0Vl pertains to the slot region, and eZ and eV to the end region. Eqs. (2.27) and (2.28) are shown, respectivly, for the case of 360° transposition and 180° transposition in the slot region.
[ ] [ ] [ ] [ ] [ ]
{ } { } { } { } { }enc
cc
enc
cc
VVldl
VldlVldl
V
ZZldl
ZldlZldl
Z
++++=
++++=
22
22
10
10
�
�
(2.27)
[ ] [ ] [ ] [ ] [ ]
{ } { } { } { } { }enc
cc
enc
cc
VVldl
VldlVldl
V
ZZldl
ZldlZldl
Z
++++=
++++=
2/10
2/10
22
22
�
�
(2.28)
16
2.8. Resistance The resistance can be calculated from :
NAl
Rρ= (2.29)
Where � is the resistivity (16.7 n�m for copper), N is the number of turns, l is the mean length of one turn, and A is the conductor area. For this model, the number of turns is 1.
2.9. Inductance The self-inductances of the strands can be calculated by:
I
L kkk
λ=, (2.30)
kλ is the flux linkage in the strand k due to the current I in strand k, which is obtained from the results of the FEM calculation. Here current I is the input current (rated current of the generator) divided by the total number of strands (in this study 10 strands), which was called I/n in section 2.4. The current value of the other strands is zero. The mutual-inductances of the strands, can be calculated by
IL l
kl
λ=, (2.31)
lλ is the flux linkage in the strand l due to the current I in strand k, which is obtained
from the results of the FEM calculation. As mentioned previously, current I is the input current (rated current) divided by the total number of strands that was called I/n in section 2.4. The current value of strand l as well as the other strands except strand k is zero.
17
3. SIMULATION OF SLOT REGION This chapter describes the winding model developed in MagNet and used for studying the flux linkages in the slot region. The results of the calculated circulating currents in the slot region are given in this chapter as well.
3.1. Simulated model Figure 3.1 shows the simulated model of the slot region in MagNet 2D. One slot, two half slots and two teeth are considered. As it is shown in the figure, the coil close to the air gap is called top coil, and the coil for which the position is toward the slot bottom is called bottom coil. Top coil and bottom coil are insulated from each other. The slots are surrounded by saturable iron. As in this study the true shape of the rotor is not simulated, a permanent magnet is placed close to the top coil to produce the flux, which comes from the rotor.
Figure 3.1: Simulated model of slot region.
Bottom coil
Top coil
Air gap
Magnet to produce radial flux
18
3.1.1. Boundary conditions The boundary conditions of the model are shown in figure 3.2.
Figure 3.2: Boundary conditions of the 2D model.
3.2. Inductance calculation Inductances are calculated as it was explained in section 2.9. The current that is applied to each strand is the rated current divided by the number of strands. This current is calculated in section 3.2.1.
3.2.1. Phase current
IVP L3=
AKV
MVAI ph 6.1574
11*3
30 ==
AI 46.15710
6.1574 ==
I is the current in each strand.
19
3.3. Resistance Resistance is calculated as it was explained in section 2.8. For this model the conductor area (the area of each strand) is 57.9415 2mm , and as in this study Z is found per unit length, therefore the value of resistance per unit length is 288n�/mm.
3.4. External flux There are two types of external flux in the slot region:
1. Radial flux 2. Transversal flux
3.4.1. Radial flux Radial flux is the flux entering the slot from the air gap. It will be shown later that this flux can produce very high circulating currents in the top coil, especially in the strands that are close to the air gap. As in this study the rotor is not simulated, the radial flux is produced with a magnet. The air gap flux density is considered to be 0.9 T. Figure 3.3. shows radial flux distribution in the slots.
Figure 3.3: Radial flux distribution.
20
3.4.2. Transversal flux Flux produced by other bars in the same slot is called transversal flux. Figure 3.4. shows the transversal flux in the top coil, which is produced by the bottom coil. In contrast to the top coil, in the bottom coil it is the transversal flux that is the main source of producing circulating current.
Figure 3.4: Transversal flux distribution in top coil due to bottom coil.
3.4.3. Circulating current due to radial and transversal flux before transposition Figure 3.5 and 3.6. show the circulating current produced by external (radial and transversal) flux if there is no transposition. It gives emphasis to the need of transposition.
21
Absolute Value of Circulating Currents in Top Coil due to External Flux
0
200
400
600
800
1000
1200
1400
1600
1 2 3 4 5 6 7 8 9 10
strand number
circ
ulat
ing
curr
ent (
A)
radial flux
transversal flux
Figure 3.5: Circulating current in top coil before transposition.
Absolute Value of Circulating Current in Bottom Coil due to External Flux
0
100
200
300
400
500
600
1 2 3 4 5 6 7 8 9 10
strand number
circ
ulat
ing
curr
ent (
A)
radial flux
transversal flux
Figure 3.6: Circulating current in bottom coil before transposition.
22
3.4.4. Circulating current after transposition In this study two different transpositions are considered in the slot region: 180° transposition and 360° transposition. The equations that are used to calculate circulaing current in the slot region due to transposition were explained in sections 2.3.1, 2.3.2 and 2.5.3. By applying these equations, the circulating current after transposition are obtained. Figure 3.7 shows the circulating currents, which are produced by radial flux in top coil after transposition.
Absolute Value of Circulating Current due to Radial Flux in Top Coil after Transposition
0
20
40
60
80
100
120
140
1 2 3 4 5 6 7 8 9 10
strand number
circ
ulat
ing
curr
ent (
A)
180° transposition
360° transposition
Figure 3.7: Circulating current due to radial flux in top coil after transposition. Figure 3.8 shows the circulating currents, which are produced by transversal flux in top coil after transposition.
23
Absolute Value of Circulating Current due to Transversal Flux in Top Coil after Transposition
0
50
100
150
200
250
1 2 3 4 5 6 7 8 9 10
strand number
circ
ulat
ing
curr
ent (
A)
180° transposition
360° transposition
Figure 3.8: Circulating current due to transversal flux in top coil after transposition.
Figure 3.9 and 3.10 show the circulating current in the bottom coil due to radial and transversal flux respectively after applying transposition.
Absolute Value of Circulating Current due to Radial Flux in Bottom Coil after Transposition
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
1 2 3 4 5 6 7 8 9 10
strand number
circ
ulat
ing
curr
ent (
A)
180° transposition
360° transposition
Figure 3.9: Circulating current due to radial flux in bottom coil after transposition.
24
Absolute Value of Circulating Current due to Transversal Flux in Bottom Coil after Transposition
0
50
100
150
200
250
300
350
400
450
500
1 2 3 4 5 6 7 8 9 10
strand number
circ
ulat
ing
curr
ent (
A)
180° transposition
360° transposition
Figure 3.10: Circulating current due to transversal flux in bottom coil after transposition.
3.5. Internal flux Internal flux is produced by the current I in the windings. In this study rated current is applied to the windings as an internal flux. The value of this current for each strand is the rated current divided by the number of the strands. This current was calculated in section 3.2.1. Figure 3.11 shows the internal flux distribution in the top coil.
Figure 3.11: Internal flux distribution in top coil.
25
3.6. External & Internal flux Now by applying external and internal flux the circulating current due to these fluxes can be obtained. Figure 3.12 shows the internal and external flux distribution in the coils.
Figure 3.12: Internal flux distribution in top coil.
3.6.1. Circulating current due to internal and external flux after applying transposition The absolute value of circulating current due to internal and external flux for the top coil and the bottom coil are shown in figures 3.13 and 3.14 respectively in case of 180° transposition and 360° transposition.
26
Absolute Value of Circulating Current due to Internal & External Flux in Top Coil after Transposition
0
50
100
150
200
250
300
350
1 2 3 4 5 6 7 8 9 10
strand number
circ
ula
tin
g c
urr
ent
(A)
180° transposition
360° transposition
Figure 3.13: Circulating current due to internal and external flux in top coil after applying transposition.
Absolute Value of Circulating Current due to Internal & External Flux in Bottom Coil after Transposition
0
50
100
150
200
250
300
350
1 2 3 4 5 6 7 8 9 10
strand number
circ
ulat
ing
curr
ent (
A)
180° transposition
360° transposition
Figure 3.14: Circulating current due to internal and external flux in bottom coil after applying transposition.
27
4. SIMULATION OF END REGION In Chapter 3, a 2D model of the slot region is developed for the purpose of studying the circulating current in case of 180° and 360° transposition in the active part. In this chapter the focus is on the presentation of the simulation of the end winding. A 3D model is developed in Magnet to study the end region fields. This chapter begins by presenting two different models of the end region:
1. End winding without transposition (traditional Roebel bar) 2. End winding with transposition
4.1. End winding without transposition At the end of a traditional Roebel bar all the strands are short-circuited within a box known as a clip. In this study to simulate this type of Roebel bar, each strand in the top coil is connected electrically to the strand of the bottom coil in the same place. Only the coil in which the end region fields are studied (and later circulating currents calculated) is represented with strands. The other coils are modelled as whole conductors. Figure 4.1 shows the end winding without transposition.
enlarged view of bar section
Figure 4.1: End winding without transposition.
28
4.1.1. Boundary conditions Figure 4.2 shows the boundary conditions in the end region without transposition. The coil is surrounded by air. The boundary between the active part and the end region is represented with the “field normal” (Dirichlet) boundary condition.
Figure 4.2: Boundary conditions of the end region without transposition.
4.1.2. Inductance calculation Inductances are calculated as it was explained in section 2.9. The current that is applied to each strand is the rated current divided by the number of strands. This current was calculated in section 3.2.1.
4.1.3. Resistance Resistance is calculated as it was explained in section 2.8. For this model the conductor area (the area of each strand) is 57.9415 2mm , the length of the strand is 2*367 mm. With the given winding data R=211.55 ��.
Active part Air
Active part
29
4.1.4. External flux There are two types of external flux in the slot region:
1. Circumferential flux 2. Radial flux
Circumferential external flux is produced by the other coils in the end region. Radial flux is the flux entering the end region from the rotor. In the coil-end, magnetic flux is assumed to be much larger in the circumferential direction than the radial direction (Kazuhiko Takahashi 2003), therefore radial external flux in the end region is not considered in this study.
4.1.5. Circumferential external flux In this study for the end region a part of the generator with 19 coils is considered. Figure 4.3 shows this part of the machine. As it is shown in the figure, the coil in the middle is simulated with strands. This is the coil in which the end fields are studied. The other coils that produce the circumferential external flux are simulated with one conductor. A three-phase current is applied to the windings. The boundary conditions are similar to those used in section 4.1.1.
Figure 4.3: Studied part of the end region.
30
Circumferential flux distribution is shown in Figure 4.4.
Figure 4.4: Circumferential external flux distribution in the studied coil.
4.1.6. Internal flux Internal flux is produced by the current I in the windings. In this study rated current is applied to the windings to produce the internal flux. The value of this current for each strand is the rated current divided by the number of the strands. This current was calculated in section 3.2.1. Figure 4.5 shows the internal flux distribution in the end region.
Figure 4.5: Internal flux distribution in the end region.
31
4.1.7. Internal & External flux Now by applying external and internal flux, the flux linkage for each strand in the end region can be obtained in MagNet. Figure 4.6 shows the internal and external flux distribution in the studied coil.
Figure 4.6: Internal & External flux distribution in the end region.
4.2. End winding with transposition As it was mentioned before, one of the objectives of this thesis is to study the effect of the transposition in the end region on circulating current. Therefore in this section the end widing with transposition is modeled in 3D MagNet. As it’s shown in figure 4.7 a curve is used to create transposition in the end region. Transposition in the end region means that if the strand is placed in the upper layer of the top coil, it will be in the lower layer of the bottom coil, in the same row, as shown in figure 4.7.
enlarged view of bar section
Figure 4.7: End winding with transposition.
32
4.2.1. Boundary conditions The boundary conditions in the end region with transposition are shown in figure 4.8. The coil is surrounded by air. As it was mentioned in section 4.1.1 the boundary between the active part and the end region is represented with the “field normal” (Dirichlet) boundary condition.
Figure 4.8: Boundary conditions of the end region with transposition.
4.2.2. Inductance calculation Inductances are calculated as it was explained in section 2.9. The current that is applied to each strand is the rated current divided by the number of strands. This current was calculated in section 3.2.1.
4.2.3. Resistance Resistance is calculated as it was explained in section 2.8. For this model the conductor area (the area of each strand) is 57.9415 2mm . The length of the strand in the straight part is 2*367 mm. The length of the strands in the curved part is different. Radius of each curve is calculated, and the length will be obtained by multiplying the radius with �. With the given winding data the resistances are shown in table 4.1 from the highest value to lowest value:
Air Active part
Active part
33
R1 245.42 �� R2 238.5 �� R3 231.57 �� R4 224.65�� R5 217.73 ��
Table 4.1: Resistances regarding to end region with transposition.
4.2.4. External flux The external flux that is applied in this section is similar to the one, which was explained in section 4.1.4. This means that only circumferential flux is considered and radial flux is not investigated in this study. The part of the machine with 19 coils is simulated and the circumferential external flux in one coil due to 18 coils is investigated. Figure 4.9 and 4.10 show the studied model and the circumferential external flux in one coil due to the other coils respectively.
Figure 4.9: Studied part of the end region with transposition.
Figure 4.9: Circumferential external flux distribution in the studied coil.
34
Only the coil in which the end fields are studied is simulated with strands. The other coils that produce the circumferential external flux are simulated with one conductor. The boundary conditions are similar to section 4.2.1.
4.2.5. Internal flux Internal flux is produced by the current I in the windings. Similar to the previous part, rated current is applied to the windings to produce the internal flux. The value of this current for each strand is the rated current divided by the number of the strands. This current was calculated in section 3.2.1. Figure 4.10 shows the internal flux distribution in the end region with transposition.
Figure 4.10: Internal flux distribution in the end region with transposition.
4.2.6. Internal & External flux Now by applying external and internal flux, the flux linkage for each strand in the end region with transposition can be obtained in MagNet. Figure 4.11 shows the internal and external flux distribution in the studied coil.
Figure 4.11: Internal & External flux distribution in the end region with transposition.
35
5. CALCULATION OF CIRCULATING CURRENT IN COIL In this chapter the results of chapter 3 and 4 are used to calculate the circulating current. Circulating currents are calculated due to internal and external flux in the slot and end region. In the slot region, the top coil is investigated. Equations 2.27 and 2.28 are used respectively to calculate the circulating current in case of 360° transposition and 180° transposition in the slot region.
5.1. Circulating current in case of 180° transposition in the slot region In this section circulating current in the half coil are calculated by equation 2.28. 180° transposition in slot region and two different end winding (with and without transposition) is considered and the circulating current are studied. Table 5.1 shows the circulating current in case of 180° transposition in the active part and no transposition in the end region. ki indicates the circulating current in strand k.
1i =771.4158 A, ∠ 243.6679 ° 6i =96.8669 A, ∠ 162.9009 °
2i =241.2256 A, ∠ 76.4936 ° 7i =287.1597 A, ∠ 264.7503 °
3i =286.6432 A, ∠ 76.7233 ° 8i =738.3202 A, ∠ 52.4681 °
4i =633.1083 A, ∠ 242.7126 ° 9i =616.5572 A, ∠ 45.1463 °
5i =246.5435 A, ∠ 189.7432 ° 10i =21.2069 A, ∠ 126.2331 ° Table 5.1: Circulating current in half coil in case of 180° transposition in slot region,
without transposition in the end region. Now the coil with transposition at the end is considered. Circulating currents in case of 180° transposition in the slot region and transposition in the end region are shown in table 5.2. ki indicates the circulating current in strand k.
1i = 472.0746 A, ∠ 235.8022° 6i = 608.1988 A, ∠ 57.2230°
2i = 177.1128 A, ∠ 167.0777° 7i = 202.0501 A, ∠ 335.5651°
3i = 216.3982 A, ∠ 233.2272° 8i = 171.9715 A, ∠ 22.3232°
4i = 202.7888 A, ∠ 188.0267° 9i = 233.9665 A, ∠ 49.4290°
5i = 222.3012 A, ∠ 269.7712° 10i = 117.7545 A, ∠ 117.9953° Table 5.2: Circulating current in half coil in case of 180° transposition in slot region, with
transposition in the end region.
36
Figure 5.1 compares the results in table 5.1 and 5.2. Current base is 157.46A, which is the strand current.
0
1
2
3
4
5
6
1 2 3 4 5 6 7 8 9 10
strand number
circ
ulat
ing
curr
ent (
p.u)
180° transposition in slotregion withouttransposition at the end
180° transposition in slotregion with transpositionat the end
Figure 5.1: Circulating current in half coil in case of 180° transposition in slot region.
37
5.2. Circulating current in case of 360° transposition in the slot region In this section circulating current in the half coil are calculated by equation 2.27. 360° transposition in slot region and two different end winding (with and without transposition) is considered and the circulating current are studied. Table 5.3 shows the circulating current in case of 360° transposition in slot region and no transposition in the end region. ki indicates the circulating current in strand k.
1i = 664.4100 A, ∠ 247.9831° 6i = 197.1298 A, ∠ 195.4412°
2i = 379.1074 A, ∠ 56.7523° 7i = 419.1161 A, ∠ 245.9141°
3i = 473.5002 A, ∠ 60.7037° 8i = 532.4697 A, ∠ 57.1162°
4i = 448.5372 A, ∠ 247.2537° 9i = 443.5876 A, ∠ 45.7125 °
5i = 231.5530 A, ∠ 190.3761° 10i = 29.4936 A, ∠ 156.5411°
Table 5.3: Circulating current in half coil in case of 360° transposition in slot region, without transposition in the end region.
Now the coil with transposition at the end is considered. Circulating currents in case of 360° transposition in the slot region and transposition in the end region are shown in table 5.4. ki indicates the circulating current in strand k.
1i = 381.7879 A, ∠ 242.8974° 6i = 517.7096 A, ∠ 63.3898°
2i = 141.0659 A, ∠ 102.1985° 7i = 169.6186 A, ∠ 285.9037°
3i = 12.7318 A, ∠ 21.8306° 8i = 56.0112 A, ∠ 288.8226°
4i = 83.3784 A, ∠ 170.2658° 9i = 89.0117 A, ∠ 60.1534°
5i = 233.2236 A, ∠ 270.2632° 10i = 114.5843 A, ∠ 126.2220° Table 5.4: Circulating current in half coil in case of 360° transposition in slot region, with
transposition in the end region. Figure 5.2 compares the results in table 5.3 and 5.4. Current base is 157.46A, which is the strand current.
38
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
1 2 3 4 5 6 7 8 9 10
strand number
circ
ulat
ing
curr
ent (
p.u)
360° transposition in slotregion withouttransposition at the end
360° transposition in slotregion with transpositionat the end
Figure 5.2: Circulating current in half coil in case of 360° transposition in slot region.
5.3. Loss situations In this section the losses of all the strands are added up together with resistance being unknown. The results are shown in table 5.5.
180° transposition in slot region without transposition in end region
2.21e6*R
180° transposition in slot region and transposition in end region
0.85e6*R
360° transposition in slot region without transposition in end region
1.56e6*R
360° transposition in slot region and transposition in end region
0.55e6*R
Table 5.5: Losses due to different transposition.
39
6. CONCLUSIONS The general aim of the work was to investigate the effect of the transposition in the end region on circulating current and circulating current loss in one turn diamond coils. This study also focused on circulating current in the slot region in case of a 180° transposition and a 360° transposition. The study was carried out using models developed in a FEM-program package (MagNet). In the active part, the model was developed in 2D MagNet. The end region simulation was carried out in 3D MagNet. The results of these two parts were used to calculate the circulating current in the half coil. The results show that high circulating current will be produced in the slot region if there is no transposition, especially external radial flux can produce very high circulating currents in strands close to the air gap of the top coil. Therefore, in the active part the transposition is necessary. Otherwise, high circulating current will burn up the machine. 180° transposition in the active part reduces the circulating current. But circulating current can reach zero in case of a 360° transposition in the active part. The calculated values of circulating currents in the bar are higher than expected. However, as the objective of this thesis is to compare the circulating current losses between the different winding layouts, the results are considered useful. One of the reasons of the high values of circulating current, can be due to considering less number of layers. The real model of the considered Roebel bar contains 14 layers, but in this study 5 layers has been considered in the simulation. Probably, with more strands (and thus more layers), the circulating currents would be smaller. It should be noted that the high values of circulating currents may influence the assumption made; that the used matrix-equation, [Z]{I}= {V}, could be solved independently for top and bottom coils. This independancy between top and bottom coils is true if the variation of magnetic coupling between the top and bottom coils due to strand current nonuniformity is assumed to be small (although V includes magnetic coupling between the top and bottom coils in the same slot). After obtaining high values of circulating currents, the magnetic coupling between the top and bottom coils due to the circulating currents can not be assumed small, and thus the circulating currents of one bar is likely to impact the circulating currents of the other bar in the same slot. Further, it was found that transposition in the end region reduces the circulating currents in the bar. Losses of the winding with transposition in the end region are less than the windings without transposition in the end region. This means that in case of 180° transposition in the active part and transposition in the end region the circulating current loss is less than the case of 360° transposition in the active part without transposition in the end region. Thus, it will be cost-effective to use a winding with 180° transposition in the active part and transposition in the end region instead of a winding with 360° transposition in the active part without transposition in the end region. It might even be
40
cost effective to use the winding with 180° transposition in the active part and transposition in the end region compared to the best (low loss) case: The minimum circulating current and losses is achieved in case of 360° transposition in the slot region with transposition in the end part. Even though this last case leads to somewhat lower losses, the higher cost of assembly of the Roebel bars with 360° transposition compared to the Roebel bars with 180° transposition could show an overall advantage of the case of 180° transposition in the active part together with transposition in the end region.
41
7. FUTURE WORK It was earlier explained that the radial external flux in the end region is not considered as it is small compared to the circumferential external flux. To verify this assumption, radial external flux in the end region should be considered and the effect of this flux on circulating current studied. Further, the rotor can be modelled with solid pole-shoes and a rotor winding, giving sinusoidal flux, instead of the simplified rotor representation in form of a magnet with uniform magnetisation. In this study the part of the machine of 19 coils are considered. However, a larger part of the machine could be considered, for instance circulating currents for a one pole-model could be calculated, including more coils and using periodic boundary conditions. The model should also contain the correct number of layers of the coil, consequent to the discussion on the previous page. Also, as the transposition of a Roebel bar may be 90°, 180°, 360°, 450°, 540°, and the circulating current loss depends on the angle, also the 90°, 450° and 540° transposition in the slot region can be investigated. However, this is more interesting for turbo generators. Further, core losses within the end parts of the stator core (represented as a conductive material) could be investigated. Such a model could also include supportive parts like press fingers and the clamping plate. Finally, verification of the calculated values of circulating currents with measured values should be done, ideally on the actual machine, otherwise on an experimental model.
42
APPENDIX 1. THE SCRIPT FILE OF ONE END COIL IN VISUAL BASIC FORMAT Dim MN6, Dev, Con, device As Object Dim StringRotationAngle As String Dim rowvalue, columnvalue, RColumn, compno, Counter As Integer Dim RotationAngle As Variant Dim RotationAxis(2), Center(2) Dim ArrayOf1Value(0), ArrayOf2Values(1), ArrayOf3Values(2), ArrayOf4Values(3),
ArrayOf5Values(4), ArrayOf6Values(5) Dim ArrayOf7Values(6), ArrayOf8Values(7), ArrayOf9Values(8), ArrayOf10Values(9),
ArrayOf11Values(10) Dim ArrayOf12Values(11), ArrayOf13Values(12), ArrayOf14Values(13),
ArrayOf15Values(14), ArrayOf16Values(15) Dim ArrayOfValues1(14), ArrayOf2Values2(2) Dim ArrayOfValues9(3) Dim ArrayOfValues10(3), ArrayOfValues11(3) Dim ArrayOfValues111(2) Dim ArrayOfValues(0), ArrayOfSeveralValues(1) As Variant 'THIS VERSION DRAWS THE Private Sub CommandButton1_Click() Application.DisplayAlerts = False Range("C5").Select columnvalue = ActiveCell.Value StartMagnet DrawWindingModel (columnvalue) WindingFileName = "Y:\Students\Poopak\Thesis\winding" & columnvalue WindingFileName = WindingFileName & ".mn" Call MN6.saveDocument(WindingFileName) CloseMagnet Analyse (columnvalue) CloseMagnet Application.DisplayAlerts = True End Sub Sub StartMagnet()
43
'Opening of Magnet from Excel: Set MN6 = CreateObject("Magnet.Application") Set Dev = MN6.newDocument Set Con = MN6.GetConstants MN6.Visible = True 'Change unit to millimeters and set the frequency RatedFrequency = Worksheets("Sheet1").Cells(12, columnvalue).Value Call Dev.beginUndoGroup("Set Properties", True) Call Dev.setDefaultLengthUnit("Millimeters") Call Dev.setSourceFrequency(RatedFrequency) Call Dev.endUndoGroup End Sub Sub CloseMagnet() 'If the save routine is inactivated then magnet will ask for each model 'if the model is to be saved Call MN6.saveDocument("Y:\Students\Poopak\Thesis\temp.mn") Call MN6.CloseDocument(False) Call MN6.Exit Set MN6 = Nothing End Sub Sub DrawWindingModel(columnvalue) NumberOfTeeth = Worksheets("Sheet1").Cells(32, columnvalue).Value NumberOfPoles = Worksheets("Sheet1").Cells(22, columnvalue).Value statorInnerRadius = Worksheets("Sheet1").Cells(29, columnvalue).Value statorOuterRadius = Worksheets("Sheet1").Cells(30, columnvalue).Value AxialLengthOfStatorCore = Worksheets("Sheet1").Cells(31, columnvalue).Value NumberOfTeeth = Worksheets("Sheet1").Cells(32, columnvalue).Value EndCoil = Worksheets("Sheet1").Cells(33, columnvalue).Value ConnectionEndCoilLowerBar = Worksheets("Sheet1").Cells(34, columnvalue).Value ConnectionEndCoilUpperBar = Worksheets("Sheet1").Cells(35, columnvalue).Value HalfCoilPitch = Worksheets("Sheet1").Cells(36, columnvalue).Value PitchOfLowerBar = Worksheets("Sheet1").Cells(37, columnvalue).Value PitchOfUpperBar = Worksheets("Sheet1").Cells(38, columnvalue).Value RightMiddleConnectionOfLowerBar = Worksheets("Sheet1").Cells(39,
columnvalue).Value LeftMiddleConnectionOfLowerBar = Worksheets("Sheet1").Cells(40,
columnvalue).Value RightMiddleConnectionOfUpperBar = Worksheets("Sheet1").Cells(41,
columnvalue).Value LeftMiddleConnectionOfUpperBar = Worksheets("Sheet1").Cells(42,
columnvalue).Value copperheight = Worksheets("Sheet1").Cells(43, columnvalue).Value
44
copperwidth = Worksheets("Sheet1").Cells(44, columnvalue).Value BarSlotLenghtBottom = Worksheets("Sheet1").Cells(45, columnvalue).Value BarSlotLenghtTop = Worksheets("Sheet1").Cells(46, columnvalue).Value CoilPitch = Worksheets("Sheet1").Cells(47, columnvalue).Value LenghtOfBar = Worksheets("Sheet1").Cells(48, columnvalue).Value WindingMaterial = Worksheets("Sheet1").Cells(24, columnvalue).Value WindingMaterial = "Name=" & WindingMaterial ' "Name=Copper: 5.77e7 Siemens/meter" AirInnerRadius = Worksheets("Sheet1").Cells(51, columnvalue).Value AirOuterRadius = Worksheets("Sheet1").Cells(52, columnvalue).Value AirLenght = Worksheets("Sheet1").Cells(53, columnvalue).Value 'lines of first comp. Call Dev.newConstructionSliceLine(0, 135.3, 0, 135.3 + copperheight) Call Dev.newConstructionSliceLine(0, 135.3 + copperheight, copperwidth, 135.3 +
copperheight) Call Dev.newConstructionSliceLine(copperwidth, 135.3 + copperheight, copperwidth,
135.3) Call Dev.newConstructionSliceLine(copperwidth, 135.3, 0, 135.3) 'first component: 'center point x = copperwidth / 2 y = 135.3 + copperheight / 2 Call Dev.selectAtWithObjectCode(x, y, Con.infoSetSelection, Con.infoSliceSurface) ArrayOfValues(0) = "halfwinding" 'multi sweep. 'First row: ArrayOf6Values(0) = "Frame" ArrayOf6Values(1) = "Cartesian" ArrayOf3Values(0) = 0 ArrayOf3Values(1) = 0 ArrayOf3Values(2) = 0 ArrayOf6Values(2) = ArrayOf3Values ArrayOf3Values(0) = 1 ArrayOf3Values(1) = 0 ArrayOf3Values(2) = 0 ArrayOf6Values(3) = ArrayOf3Values ArrayOf3Values(0) = 0 ArrayOf3Values(1) = 1 ArrayOf3Values(2) = 0 ArrayOf6Values(4) = ArrayOf3Values ArrayOf3Values(0) = 0 ArrayOf3Values(1) = 0
45
ArrayOf3Values(2) = 1 ArrayOf6Values(5) = ArrayOf3Values ArrayOfValues1(0) = ArrayOf6Values 'Second row: ArrayOf1Value(0) = "Start" ArrayOfValues1(1) = ArrayOf1Value 'Third row: ArrayOf2Values(0) = "Line" ArrayOf2Values(1) = 63 ArrayOfValues1(2) = ArrayOf2Values 'Fourth row: ArrayOf2Values(0) = "Blend" ArrayOf2Values(1) = "Automatic" ArrayOfValues1(3) = ArrayOf2Values 'Fifth row: (sweeping) ArrayOf2Values(0) = "Line" ArrayOf3Values(0) = x ArrayOf3Values(1) = -33 + copperheight / 2 ArrayOf3Values(2) = 277 ArrayOf2Values(1) = ArrayOf3Values ArrayOfValues1(4) = ArrayOf2Values 'Sixth row: ArrayOf2Values(0) = "Blend" ArrayOf2Values(1) = "Automatic" ArrayOfValues1(5) = ArrayOf2Values 'Seventh row: (sweeping) ArrayOf2Values(0) = "Line" ArrayOf3Values(0) = x ArrayOf3Values(1) = -33 + copperheight / 2 ArrayOf3Values(2) = 367 ArrayOf2Values(1) = ArrayOf3Values ArrayOfValues1(6) = ArrayOf2Values 'Eigth row: Blends automatically ArrayOf2Values(0) = "Blend" ArrayOf2Values(1) = "Automatic" ArrayOfValues1(7) = ArrayOf2Values 'Ninth row: ArrayOf4Values(0) = "Arc" ArrayOf4Values(1) = 180
46
ArrayOf3Values(0) = 41.22 ArrayOf3Values(1) = -33 + copperheight / 2 ArrayOf3Values(2) = 367 ArrayOf4Values(2) = ArrayOf3Values ArrayOf3Values(0) = 0 ArrayOf3Values(1) = 1 ArrayOf3Values(2) = 0 ArrayOf4Values(3) = ArrayOf3Values ArrayOfValues1(8) = ArrayOf4Values 'Tenth row: Blends automatically ArrayOf2Values(0) = "Blend" ArrayOf2Values(1) = "Automatic" ArrayOfValues1(9) = ArrayOf2Values 'Eleventh row: (sweeping) ArrayOf2Values(0) = "Line" ArrayOf3Values(0) = 41.22 + (41.22 - x) ArrayOf3Values(1) = -33 + copperheight / 2 ArrayOf3Values(2) = 277 ArrayOf2Values(1) = ArrayOf3Values ArrayOfValues1(10) = ArrayOf2Values 'Twelveth row: ArrayOf2Values(0) = "Blend" ArrayOf2Values(1) = "Automatic" ArrayOfValues1(11) = ArrayOf2Values 'Thirteenth row: (sweeping) ArrayOf2Values(0) = "Line" ArrayOf3Values(0) = 41.22 + (41.22 - x) ArrayOf3Values(1) = -201.3 + copperheight / 2 ArrayOf3Values(2) = 83 ArrayOf2Values(1) = ArrayOf3Values ArrayOfValues1(12) = ArrayOf2Values 'Fourteenth row: ArrayOf2Values(0) = "Blend" ArrayOf2Values(1) = "Automatic" ArrayOfValues1(13) = ArrayOf2Values 'fiveteenth row: (sweeping) ArrayOf2Values(0) = "Line" ArrayOf3Values(0) = 41.22 + (41.22 - x) ArrayOf3Values(1) = -201.3 + copperheight / 2 ArrayOf3Values(2) = 0
47
ArrayOf2Values(1) = ArrayOf3Values ArrayOfValues1(14) = ArrayOf2Values Call Dev.getCurrentView.makeComponentInAMultiSweep(ArrayOfValues1,
ArrayOfValues, " Name=Copper: 5.77e7 Siemens/meter", infoMakeComponentUnionSurfaces Or infoMakeComponentRemoveVertices)
48
APPENDIX 2. MATLAB PROGRAM DEVELOPED FOR CALCULATION OF CIRCULATING CURRENTS IN THE ACTIVE PART IN CASE OF 360° TRANSPOSITION clear close all clc %rated current I=157.46; % inductances of active part L11=0.0002696663847727/I L12=0.0002696292315019/I; L13=0.0002695690959058/I; L14=0.0002695124007238/I; L15=0.0002694562736687/I; L16=0.0002694561157492/I; L17=0.0002695118054042/I; L18=0.0002695663093384/I; L19=0.0002696188996202/I; L110=0.0002696425010106/I; L21=0.000269632677542/I; L22=0.0002696581993071/I L23=0.0002696209929717/I; L24=0.0002695615307046/I; L25=0.0002695044655222/I; L26=0.0002695039608182/I; L27=0.000269559644359/I; L28=0.000269612139528/I; L29=0.0002696372384525/I; L210=0.000269620732435/I; L31=0.0002695762229986/I; L32=0.0002696246744957/I; L33=0.0002696556754481/I; L34=0.0002696139482471/I; L35=0.0002695527274356/I; L36=0.0002695498830943/I; L37=0.0002696034150876/I; L38=0.0002696314942568/I; L39=0.0002696139922385/I; L310=0.0002695762285665/I; L41=0.0002695234679382/I; L42=0.0002695691529666/I; L43=0.0002696178890892/I; L44=0.0002696411920334/I; L45=0.00026960309196/I;
49
L46=0.0002695939815653/I; L47=0.0002696195985998/I; L48=0.0002696049614919/I; L49=0.00026956691847/I; L410=0.0002695259375733/I; L51=0.0002694715826204/I; L52=0.0002695163303039/I; L53=0.0002695609110175/I; L54=0.0002696073348031/I; L55=0.0002696299287232/I; L56=0.0002696098624963/I; L57=0.0002695938169567/I; L58=0.000269557476717/I; L59=0.000269516146253/I; L510=0.0002694746503081/I; L61=0.0002694714257058/I; L62=0.0002695158266146/I; L63=0.0002695580676962/I; L64=0.000269598225434/I; L65=0.0002696098635239/I; L66=0.0002696300330446/I; L67=0.0002696072297174/I; L68=0.0002695604360928/I; L69=0.0002695169031622/I; L610=0.0002694748746503/I; L71=0.0002695228740723/I; L72=0.0002695672680793/I; L73=0.000269607357391/I; L74=0.0002696196000635/I; L75=0.0002695895755753/I; L76=0.0002696029873104/I; L77=0.0002696406319705/I; L78=0.0002696161289169/I; L79=0.0002695697253362/I; L710=0.000269526760072/I; L81=0.0002695734392827/I; L82=0.000269615823905/I; L83=0.0002696314971123/I; L84=0.0002696010234981/I; L85=0.0002695492959654/I; L86=0.0002695522543214/I; L87=0.0002696121894622/I; L88=0.0002696557156757/I; L89=0.000269627039545/I; L810=0.0002695800106166/I; L91=0.0002696223965893/I; L92=0.0002696372893783/I; L93=0.0002696103616084/I; L94=0.0002695593470203/I; L95=0.000269504332159/I; L96=0.0002695050880551/I;
50
L97=0.0002695621524317/I; L98=0.0002696234060691/I; L99=0.0002696598272067/I; L910=0.0002696362020305/I; L101=0.000269642770654/I; L102=0.0002696175559009/I; L103=0.0002695693706477/I; L104=0.0002695151389906/I; L105=0.0002694596092833/I; L106=0.000269459832627/I; L107=0.0002695159600519/I; L108=0.0002695731498905/I; L109=0.0002696329746397/I; L1010=0.0002696711209443/I; %frequency f=50; w=2*pi*f; %resistance of active part R=(1.67e-8*1e-3)/(7.58e-3*7.644e-3) %voltage drop of resistance constant=R*I %flux linkages of active part taken from MagNet due to internal & external flux Flux1=0.0003419381887571; Flux2=0.0003418579960511; Flux3=0.0003422281557179; Flux4=0.0003425983952953; Flux5=0.0003428119796131; Flux6=0.0003428049242101; Flux7=0.0003425623401431; Flux8=0.0003420852192332; Flux9=0.0003412861295266; Flux10=0.0003398538076865; %impedance matrix of active part in transposition 0 L0=[R+j*w*L11,j*w*L12,j*w*L13,j*w*L14,j*w*L15,j*w*L16,j*w*L17,j*w*L18,j*w*L19,j*w*L110 j*w*L21,R+j*w*L22,j*w*L23,j*w*L24,j*w*L25,j*w*L26,j*w*L27,j*w*L28,j*w*L29,j*w*L210 j*w*L31,j*w*L32,R+j*w*L33,j*w*L34,j*w*L35,j*w*L36,j*w*L37,j*w*L38,j*w*L39,j*w*L310 j*w*L41,j*w*L42,j*w*L43,R+j*w*L44,j*w*L45,j*w*L46,j*w*L47,j*w*L48,j*w*L49,j*w*L410
51
j*w*L51,j*w*L52,j*w*L53,j*w*L54,R+j*w*L55,j*w*L56,j*w*L57,j*w*L58,j*w*L59,j*w*L510 j*w*L61,j*w*L62,j*w*L63,j*w*L64,j*w*L65,R+j*w*L66,j*w*L67,j*w*L68,j*w*L69,j*w*L610 j*w*L71,j*w*L72,j*w*L73,j*w*L74,j*w*L75,j*w*L76,R+j*w*L77,j*w*L78,j*w*L79,j*w*L710 j*w*L81,j*w*L82,j*w*L83,j*w*L84,j*w*L85,j*w*L86,j*w*L87,R+j*w*L88,j*w*L89,j*w*L810 j*w*L91,j*w*L92,j*w*L93,j*w*L94,j*w*L95,j*w*L96,j*w*L97,j*w*L98,R+j*w*L99,j*w*L910 j*w*L101,j*w*L102,j*w*L103,j*w*L104,j*w*L105,j*w*L106,j*w*L107,j*w*L108,j*w*L109,R+j*w*L1010] %permutation matrix P=[0,1,0,0,0,0,0,0,0,0 0,0,1,0,0,0,0,0,0,0 0,0,0,1,0,0,0,0,0,0 0,0,0,0,1,0,0,0,0,0 0,0,0,0,0,1,0,0,0,0 0,0,0,0,0,0,1,0,0,0 0,0,0,0,0,0,0,1,0,0 0,0,0,0,0,0,0,0,1,0 0,0,0,0,0,0,0,0,0,1 1,0,0,0,0,0,0,0,0,0]; %impedance matrix of active part in transposition 1 L1=P*L0*P' %impedance matrix of active part in transposition 2 L2=P*L1*P' %impedance matrix of active part in transposition 3 L3=P*L2*P' %impedance matrix of active part in transposition 4 L4=P*L3*P' %impedance matrix of active part in transposition 5 L5=P*L4*P' %impedance matrix of active part in transposition 6 L6=P*L5*P' %impedance matrix of active part in transposition 7 L7=P*L6*P' %impedance matrix of active part in transposition 8 L8=P*L7*P' %impedance matrix of active part in transposition 9 L9=P*L8*P' %impedance matrix of active part in transposition 10 L10=P*L9*P' %strand length within one transposition in case of 360 dl=820/10
52
%impedance matrix after transposition L=((dl/2)*L0)+(dl*L1)+(dl*L2)+(dl*L3)+(dl*L4)+(dl*L5)+(dl*L6)+(dl*L7)+(dl*L8)+(dl*L9)+((dl/2)*L10) %final impedance matrix finalL=[L(1,1) L(1,2) L(1,3) L(1,4) L(1,5) L(1,6) L(1,7) L(1,8) L(1,9) L(1,10) -1 L(2,1) L(2,2) L(2,3) L(2,4) L(2,5) L(2,6) L(2,7) L(2,8) L(2,9) L(2,10) -1 L(3,1) L(3,2) L(3,3) L(3,4) L(3,5) L(3,6) L(3,7) L(3,8) L(3,9) L(3,10) -1 L(4,1) L(4,2) L(4,3) L(4,4) L(4,5) L(4,6) L(4,7) L(4,8) L(4,9) L(4,10) -1 L(5,1) L(5,2) L(5,3) L(5,4) L(5,5) L(5,6) L(5,7) L(5,8) L(5,9) L(5,10) -1 L(6,1) L(6,2) L(6,3) L(6,4) L(6,5) L(6,6) L(6,7) L(6,8) L(6,9) L(6,10) -1 L(7,1) L(7,2) L(7,3) L(7,4) L(7,5) L(7,6) L(7,7) L(7,8) L(7,9) L(7,10) -1 L(8,1) L(8,2) L(8,3) L(8,4) L(8,5) L(8,6) L(8,7) L(8,8) L(8,9) L(8,10) -1 L(9,1) L(9,2) L(9,3) L(9,4) L(9,5) L(9,6) L(9,7) L(9,8) L(9,9) L(9,10) -1 L(10,1) L(10,2) L(10,3) L(10,4) L(10,5) L(10,6) L(10,7) L(10,8) L(10,9) L(10,10) -1 1 1 1 1 1 1 1 1 1 1 0] %voltage in active part in transposition 0 V0=[-j*w*Flux1-constant;-j*w*Flux2-constant;-j*w*Flux3-constant;-j*w*Flux4-constant;-j*w*Flux5-constant;-j*w*Flux6-constant;-j*w*Flux7-constant;-j*w*Flux8-constant;-j*w*Flux9-constant;-j*w*Flux10-constant] %voltage in active part in transposition 1 V1=P*V0 %voltage in active part in transposition 2 V2=P*V1 %voltage in active part in transposition 3 V3=P*V2 %voltage in active part in transposition 4 V4=P*V3 %voltage in active part in transposition 5 V5=P*V4 %voltage in active part in transposition 6 V6=P*V5 %voltage in active part in transposition 7 V7=P*V6 %voltage in active part in transposition 8 V8=P*V7 %voltage in active part in transposition 9 V9=P*V8 %voltage in active part in transposition 10 V10=P*V9 %voltage vector after transposition
53
V=((dl/2)*V0)+(dl*V1)+(dl*V2)+(dl*V3)+(dl*V4)+(dl*V5)+(dl*V6)+(dl*V7)+(dl*V8)+(dl*V9)+((dl/2)*V10) %final voltage vector finalV=[V(1,1);V(2,1);V(3,1);V(4,1);V(5,1);V(6,1);V(7,1);V(8,1);V(9,1);V(10,1);0] %calculation of circulating current icirculate=inv(finalL)*finalV
54
APPENDIX 3. MATLAB PROGRAM DEVELOPED FOR CALCULATION OF CIRCULATING CURRENTS IN THE ACTIVE PART IN CASE OF 180° TRANSPOSITION clear close all clc %rated current I=157.46; % inductances of active part L11=0.0002696663847727/I L12=0.0002696292315019/I; L13=0.0002695690959058/I; L14=0.0002695124007238/I; L15=0.0002694562736687/I; L16=0.0002694561157492/I; L17=0.0002695118054042/I; L18=0.0002695663093384/I; L19=0.0002696188996202/I; L110=0.0002696425010106/I; L21=0.000269632677542/I; L22=0.0002696581993071/I L23=0.0002696209929717/I; L24=0.0002695615307046/I; L25=0.0002695044655222/I; L26=0.0002695039608182/I; L27=0.000269559644359/I; L28=0.000269612139528/I; L29=0.0002696372384525/I; L210=0.000269620732435/I; L31=0.0002695762229986/I; L32=0.0002696246744957/I; L33=0.0002696556754481/I; L34=0.0002696139482471/I; L35=0.0002695527274356/I; L36=0.0002695498830943/I; L37=0.0002696034150876/I; L38=0.0002696314942568/I; L39=0.0002696139922385/I; L310=0.0002695762285665/I; L41=0.0002695234679382/I; L42=0.0002695691529666/I;
55
L43=0.0002696178890892/I; L44=0.0002696411920334/I; L45=0.00026960309196/I; L46=0.0002695939815653/I; L47=0.0002696195985998/I; L48=0.0002696049614919/I; L49=0.00026956691847/I; L410=0.0002695259375733/I; L51=0.0002694715826204/I; L52=0.0002695163303039/I; L53=0.0002695609110175/I; L54=0.0002696073348031/I; L55=0.0002696299287232/I; L56=0.0002696098624963/I; L57=0.0002695938169567/I; L58=0.000269557476717/I; L59=0.000269516146253/I; L510=0.0002694746503081/I; L61=0.0002694714257058/I; L62=0.0002695158266146/I; L63=0.0002695580676962/I; L64=0.000269598225434/I; L65=0.0002696098635239/I; L66=0.0002696300330446/I; L67=0.0002696072297174/I; L68=0.0002695604360928/I; L69=0.0002695169031622/I; L610=0.0002694748746503/I; L71=0.0002695228740723/I; L72=0.0002695672680793/I; L73=0.000269607357391/I; L74=0.0002696196000635/I; L75=0.0002695895755753/I; L76=0.0002696029873104/I; L77=0.0002696406319705/I; L78=0.0002696161289169/I; L79=0.0002695697253362/I; L710=0.000269526760072/I; L81=0.0002695734392827/I; L82=0.000269615823905/I; L83=0.0002696314971123/I; L84=0.0002696010234981/I; L85=0.0002695492959654/I; L86=0.0002695522543214/I; L87=0.0002696121894622/I; L88=0.0002696557156757/I; L89=0.000269627039545/I; L810=0.0002695800106166/I; L91=0.0002696223965893/I; L92=0.0002696372893783/I; L93=0.0002696103616084/I;
56
L94=0.0002695593470203/I; L95=0.000269504332159/I; L96=0.0002695050880551/I; L97=0.0002695621524317/I; L98=0.0002696234060691/I; L99=0.0002696598272067/I; L910=0.0002696362020305/I; L101=0.000269642770654/I; L102=0.0002696175559009/I; L103=0.0002695693706477/I; L104=0.0002695151389906/I; L105=0.0002694596092833/I; L106=0.000269459832627/I; L107=0.0002695159600519/I; L108=0.0002695731498905/I; L109=0.0002696329746397/I; L1010=0.0002696711209443/I; %frequency f=50; w=2*pi*f; %resistance of active part R=(1.67e-8*1e-3)/(7.58e-3*7.644e-3) %voltage drop of resistance constant=R*I %flux linkages of active part taken from MagNet due to internal & external Flux1=0.0003419381887571; Flux2=0.0003418579960511; Flux3=0.0003422281557179; Flux4=0.0003425983952953; Flux5=0.0003428119796131; Flux6=0.0003428049242101; Flux7=0.0003425623401431; Flux8=0.0003420852192332; Flux9=0.0003412861295266; Flux10=0.0003398538076865; %impedance matrix of active part in transposition 0 L0=[R+j*w*L11,j*w*L12,j*w*L13,j*w*L14,j*w*L15,j*w*L16,j*w*L17,j*w*L18,j*w*L19,j*w*L110 j*w*L21,R+j*w*L22,j*w*L23,j*w*L24,j*w*L25,j*w*L26,j*w*L27,j*w*L28,j*w*L29,j*w*L210 j*w*L31,j*w*L32,R+j*w*L33,j*w*L34,j*w*L35,j*w*L36,j*w*L37,j*w*L38,j*w*L39,j*w*L310
57
j*w*L41,j*w*L42,j*w*L43,R+j*w*L44,j*w*L45,j*w*L46,j*w*L47,j*w*L48,j*w*L49,j*w*L410 j*w*L51,j*w*L52,j*w*L53,j*w*L54,R+j*w*L55,j*w*L56,j*w*L57,j*w*L58,j*w*L59,j*w*L510 j*w*L61,j*w*L62,j*w*L63,j*w*L64,j*w*L65,R+j*w*L66,j*w*L67,j*w*L68,j*w*L69,j*w*L610 j*w*L71,j*w*L72,j*w*L73,j*w*L74,j*w*L75,j*w*L76,R+j*w*L77,j*w*L78,j*w*L79,j*w*L710 j*w*L81,j*w*L82,j*w*L83,j*w*L84,j*w*L85,j*w*L86,j*w*L87,R+j*w*L88,j*w*L89,j*w*L810 j*w*L91,j*w*L92,j*w*L93,j*w*L94,j*w*L95,j*w*L96,j*w*L97,j*w*L98,R+j*w*L99,j*w*L910 j*w*L101,j*w*L102,j*w*L103,j*w*L104,j*w*L105,j*w*L106,j*w*L107,j*w*L108,j*w*L109,R+j*w*L1010] %permutation matrix P=[0,1,0,0,0,0,0,0,0,0 0,0,1,0,0,0,0,0,0,0 0,0,0,1,0,0,0,0,0,0 0,0,0,0,1,0,0,0,0,0 0,0,0,0,0,1,0,0,0,0 0,0,0,0,0,0,1,0,0,0 0,0,0,0,0,0,0,1,0,0 0,0,0,0,0,0,0,0,1,0 0,0,0,0,0,0,0,0,0,1 1,0,0,0,0,0,0,0,0,0]; %impedance matrix of active part in transposition 1 L1=P*L0*P' %impedance matrix of active part in transposition 2 L2=P*L1*P' %impedance matrix of active part in transposition 3 L3=P*L2*P' %impedance matrix of active part in transposition 4 L4=P*L3*P' %impedance matrix of active part in transposition 5 L5=P*L4*P' %strand length within one transposition in case of 180 dl=820/5 %impedance matrix after transposition L=((dl/2)*L0)+(dl*L1)+(dl*L2)+(dl*L3)+(dl*L4)+((dl/2)*L5) %final impedance matrix finalL=[L(1,1) L(1,2) L(1,3) L(1,4) L(1,5) L(1,6) L(1,7) L(1,8) L(1,9) L(1,10) -1
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L(2,1) L(2,2) L(2,3) L(2,4) L(2,5) L(2,6) L(2,7) L(2,8) L(2,9) L(2,10) -1 L(3,1) L(3,2) L(3,3) L(3,4) L(3,5) L(3,6) L(3,7) L(3,8) L(3,9) L(3,10) -1 L(4,1) L(4,2) L(4,3) L(4,4) L(4,5) L(4,6) L(4,7) L(4,8) L(4,9) L(4,10) -1 L(5,1) L(5,2) L(5,3) L(5,4) L(5,5) L(5,6) L(5,7) L(5,8) L(5,9) L(5,10) -1 L(6,1) L(6,2) L(6,3) L(6,4) L(6,5) L(6,6) L(6,7) L(6,8) L(6,9) L(6,10) -1 L(7,1) L(7,2) L(7,3) L(7,4) L(7,5) L(7,6) L(7,7) L(7,8) L(7,9) L(7,10) -1 L(8,1) L(8,2) L(8,3) L(8,4) L(8,5) L(8,6) L(8,7) L(8,8) L(8,9) L(8,10) -1 L(9,1) L(9,2) L(9,3) L(9,4) L(9,5) L(9,6) L(9,7) L(9,8) L(9,9) L(9,10) -1 L(10,1) L(10,2) L(10,3) L(10,4) L(10,5) L(10,6) L(10,7) L(10,8) L(10,9) L(10,10) -1 1 1 1 1 1 1 1 1 1 1 0] %voltage in active part in transposition 0 V0=[-j*w*Flux1-constant;-j*w*Flux2-constant;-j*w*Flux3-constant;-j*w*Flux4-constant;-j*w*Flux5-constant;-j*w*Flux6-constant;-j*w*Flux7-constant;-j*w*Flux8-constant;-j*w*Flux9-constant;-j*w*Flux10-constant] %voltage in active part in transposition 1 V1=P*V0 %voltage in active part in transposition 2 V2=P*V1 %voltage in active part in transposition 3 V3=P*V2 %voltage in active part in transposition 4 V4=P*V3 %voltage in active part in transposition 5 V5=P*V4 %voltage vector after transposition V=((dl/2)*V0)+(dl*V1)+(dl*V2)+(dl*V3)+(dl*V4)+((dl/2)*V5) %final voltage vector finalV=[V(1,1);V(2,1);V(3,1);V(4,1);V(5,1);V(6,1);V(7,1);V(8,1);V(9,1);V(10,1);0] %calculation of circulating current icirculate=inv(finalL)*finalV
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REFERENCES Fujita, M., Y. Kabata, et al. (2005). "Air-cooled large turbine generator with multiple-pitched ventilation ducts." Electric Machines and Drives, 2005 IEEE International Conference on: 910. Haldemann, J. (2004). "Transpositions in stator bars of large turbogenerators." Energy Conversion, IEEE Transaction on 19(3): 553-560. Infolytica (2007). Documentation Center. www.infolytica.com. Kazuhiko Takahashi, M. T., Masaki Sato (2003). "Calculation Method for Strand Current Distributions in Armature Winding of a Turbine Generator." 143(2): 50-58. Macdonald, D. C. (1971). "Circulating-current loss within Roebel bar stator windings in hydroelectric alternators." 118: 689-697. MathWorks (2006). The Language of Technical Computing. www.mathworks.com. Sequenz, H. (1973). Herstellung der Wicklungen elektrischer Maschinen. Wien/New York, Springer.