Jassal_PhD_thesis_final.
Eddy Current Loss Modeling for
Design of PM Generators
for
Wind Turbines
Proefschrift
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op donderdag 16 oktober 2014 om 10:00 uur
door
Anoop JASSAL
Master of Science in Electrical Engineering
geboren te Hoshiarpur, India
Dit proefschrift is goedgekeurd door de promotor:
Prof.dr.eng. J.A. Ferreira
Copromotor:
Dr.ir. H. Polinder
Samenstelling promotiecommissie:
Rector Magnificus, voorzitter
Prof.dr.eng. J.A.Ferreira, Technische Universiteit Delft, promotor
Dr.ir. H.Polinder, Technische Universiteit Delft, copromotor
Prof.(emer).dr. J. Tapani, Aalto University, Finland
Prof.dr.ir. L. Dupre, Ghent University, Belgium
Prof.dr. A. Neto, Technische Universiteit Delft
Prof.dr. G.J.W. van Bussel, Technische Universiteit Delft
Dr.ir. D.J.P. Lahaye, Technische Universiteit Delft
ISBN: 978-94-6203-677-2
Printed at: CPI - Koninklijke Wöhrmann B.V., The Netherlands
Copyright © 2014 by Anoop Jassal
All rights reserved. No part of the material protected by this copyright notice may be
reproduced or utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and retrieval system
without permission from the publisher or author.
To my family and friends for their love and support
Acknowledgements
“It took time which was running too fast, I thank my lord it has been done at last.”
As I am writing this acknowledgement, the two lines above describe my state
of mind. It is a milestone in my life and culmination of an effort not only
from my side but also from many individuals who have made it possible. I
would like to thank and acknowledge their support here.
I would first and foremost like to thank my supervisor Dr. Henk Polinder and
promoter Prof. Dr. J.A. Ferreira who have shown immense patience and
support for me during this research. I appreciate their zeal, passion and
understanding of technical as well as non-technical aspects of a research. A
special thanks to Dr. Domenico Lahaye for his support and motivation for
both analytical and computational parts of this research.
My research was supported by VWEC B.V. (now XEMC-Darwind) and the
team there has been a constant motivation for me. I would take this
opportunity to thank Mr. Kees Versteegh and Mr. Hugo Groenman who
despite the financial crisis and management changes kept me aloof from the
worries about funding. A heartfelt thanks to Mr. A.S. Karanth, Michiel,
Robert, Chen, Kejia, Manoj and all those involved in A-90/XV-90 project for
technical support, discussions and maintaining a great working environment.
Many thanks to our wonderful, diverse and cooperative EPP group who
made me feel at home at TU Delft. Special thanks to Ghanshyam, Deok-Je,
Alex, Martin (for helping me with Matlab, FE simulations and Dutch
translations), Johan, Zhihui, Marcelo, Dalibor, Ivan, Milos, Ilija, Todor, Rick,
Yi Wang, Yeh Ting and all other colleagues whom I have worked with. Many
thanks to Ir. Dong Liu for carrying out useful experiments in the lab.
Heartfelt thanks to Mr. Rob Schoevaars, Harry, Kasper and all laboratory
staff for their prompt help and support during experimental work.
I would also like to acknowledge the support, friendship, encouragement and
immeasurable help offered by Nada, Balazs, Kostas, Ivo, Fadiah and Marcela.
Thanks guys for standing by my side during thick and thin of life !
I can’t thank enough to my family back in India who have been my
supporting pillars. It is because of their vision and hard-work that I could
even think about doing higher studies. Thanks Mom-Dad and sister Renu for
your patience and everything you have given me. Thanks to all my Indian
friends who have supported me throughout my life especially Navdeep,
Saurabh, Neeraj, Abhishek, Amarsha, Nitin, Shuchi, Tanu, all GEROH
members and many more.
Last but not the least I would thank my loving, bubbly and immensely
supportive wife, Harsh, without whom this whole research would have been a
dry-grinding task.
-
Anoop Jassal
Table of Contents
1. Introduction.......................................................................... 1
1.1. Wind – A Renewable Energy Source .................................................................... 1
1.2. The Offshore Trend: Conditions and Challenges .............................................. 2
1.3. Wind Energy Conversion ....................................................................................... 2
1.3.1 Types of Generators in Wind Turbines ..................................................... 4
1.3.1.1 Squirrel Cage Induction Generators (SCIG) ................................ 4
1.3.1.2 Wound Rotor Induction Generators (WRIG) ............................... 5
1.3.1.3 Synchronous Generator (SG) ....................................................... 7
1.4. Concentrated windings..........................................................................................10
1.5. Research Focus .......................................................................................................11
1.6. The Thesis Objective.............................................................................................13
1.7. Thesis Outline.........................................................................................................13
2. State of Art in PM Generator Manufacture ....................... 17
2.1. Permanent Magnet Direct Drive Generators ....................................................17
2.1.1 Specifications of the 2 MW Generator ....................................................18
2.2. Stator Construction................................................................................................19
2.2.1 Stator Housing .............................................................................................19
2.2.2 Stator Yoke - Laminates .............................................................................20
2.2.3 Coils and Winding .......................................................................................22
2.2.3.1 Common Winding Types in Large Electrical Machines ............ 22
2.2.3.2 Winding of Reference Machine: 2 Layer Lap Type .................. 24
2.3. Rotor Construction ................................................................................................28
2.3.1 PM Assembly on Rotor ..............................................................................28
2.3.2 Rotor Back-iron, Shaft and Support Structure .......................................29
2.3.3 Bearings .........................................................................................................29
2.4. Limitations Posed by PMDD Generators .........................................................30
2.5. Summary ..................................................................................................................31
3. Eddy Current Losses.......................................................... 33
3.1. Eddy Current Loss – Physics ...............................................................................33
3.2. Eddy Current Losses in Concentrated Windings..............................................34
3.3. Eddy Current Losses in Electrical Machines – A Survey ................................36
3.3.1 Stage 1: 1892~ 1950 – Experiments and Formulas ...............................38
3.3.2 Stage II: 1951-1990- Analytical Methods .................................................40
3.3.3 Stage III: 1991 Onwards-FE, Numerical and Analytical Methods ......44
3.4. Results and Trends.................................................................................................48
3.5. Summary ..................................................................................................................50
4. Analytical Modeling ........................................................... 59
4.1. Analytical Models ...................................................................................................59
4.2. Modeling Approach for PMDD machines ........................................................59
4.3. Assumptions ...........................................................................................................60
4.4. Derivation of Partial Differential Equation (PDE) ..........................................62
4.5. Boundary Conditions.............................................................................................65
4.5.1 Boundary Condition 1 ................................................................................65
4.5.2 Boundary Condition 2 ................................................................................66
4.6. General Solution of the Partial Differential Equation (PDE) ........................66
4.6.1 The General Solution for Laplace’s equation..........................................66
4.6.2 General Solution of Poisson’s Equation and Treatment of Time .......69
4.7. Field Due to Stator Currents Only ......................................................................71
4.7.1 Excitation for the field ................................................................................71
4.7.2 Effect of motion ..........................................................................................74
4.7.3 Some Important Observations – Current Sheet Excitation .................75
4.8. Field Due to Magnets Only ..................................................................................77
4.8.1 Excitation for the field of PMs..................................................................78
4.8.2 Some Important Observations – Field of PM ........................................79
4.9. The Combined Magnetic Field ............................................................................80
4.10. Derived Quantities from Az ...............................................................................81
4.10.1 Magnetic Flux Density from Az ..............................................................81
4.10.2 Induced Current Density from Az ..........................................................82
4.10.3 Eddy Current Losses from Induced Current Density .........................84
4.11. Application of Analytical Model........................................................................85
4.12. Summary ................................................................................................................86
5. Finite Element (FE) Modeling.......................................... 87
5.1. Introduction ............................................................................................................87
5.2. Finite Element Method .........................................................................................88
5.3. COMSOL 3.5a General Environment ...............................................................88
5.3.1 Application Modes and PDE ....................................................................89
5.3.2 FE Model Setup ...........................................................................................89
5.3.3 General Procedure for Problem Setup.....................................................91
5.4. Machine Model .......................................................................................................91
5.4.1 Geometry Drawing and Symmetry ..........................................................92
5.4.2 Meshing .........................................................................................................93
5.4.3 Boundary Conditions ..................................................................................94
5.4.4 Physics and Material Settings .....................................................................95
5.4.5 Motion ...........................................................................................................96
5.4.6 Visualization and Post-processing ............................................................97
5.5. Results of FE Modeling ........................................................................................99
5.5.1 Validation of Analytical Model ..................................................................99
5.5.2 Eddy Current Loss Calculation Using FE Models .............................. 101
5.5.2.1 Variations for models ............................................................... 103
5.5.2.2 Results for Eddy Current Losses from FE Models .................. 103
5.6. Summary ............................................................................................................... 107
6. Experimental Analysis ...................................................... 109
6.1. Introduction ......................................................................................................... 109
6.2. Experimental Setup............................................................................................. 109
6.3. Static Tests ........................................................................................................... 110
6.3.1 Static Tests – Procedure .......................................................................... 110
6.3.2 Static Tests – Apparatus .......................................................................... 111
6.3.3 Results Static Tests ................................................................................... 113
6.4. Rotary Tests – Main Principle ........................................................................... 121
6.4.1 Rotary Tests – Apparatus ........................................................................ 122
6.4.2 Variations in Rotary Tests ....................................................................... 123
6.4.2.1 Measurement of the prime mover and mechanical losses ........124
6.4.3 Rotary Tests Case I: PM excitation but no stator current.................. 126
6.4.3.1 Procedure for the no-load test (Case I) .....................................127
6.4.3.2 Separation of Stator and Rotor Iron Losses ..............................128
6.4.3.3 Results – Case I ........................................................................132
6.4.4 Rotary Tests Case II: PM excitation with stator current .................... 134
6.4.4.1 Procedure for the on-load test (Case II) ...................................134
6.4.4.2 Results – Case II: Stator currents and PM excitation ...............135
6.4.5 Rotary Tests Case III: No PM excitation, only stator current .......... 139
6.4.5.1 Procedure for Case III: Only stator currents no PM field.........140
6.4.6 Results – Case III: Stator current excitation only................................ 141
6.5. Summary ............................................................................................................... 143
7. Trends and Design Guidelines ......................................... 145
7.1. Introduction ......................................................................................................... 145
7.1.1 Slot-pole Combinations Used ................................................................. 147
7.2. Results: Eddy Current Loss Trends ................................................................. 148
7.3. Other Design Considerations ........................................................................... 151
7.3.1 Winding Factor ......................................................................................... 151
7.3.2 Balanced Set of Windings........................................................................ 153
7.3.3 Cogging Torque ........................................................................................ 153
7.3.4 Rotor Force Balance................................................................................. 154
7.4. Summary ............................................................................................................... 154
8. Conclusions & Recommendations ................................... 157
8.1. Conclusions .......................................................................................................... 157
8.2. Thesis Deliverables Revisited ............................................................................ 160
8.3. Recommendations for Further Research ........................................................ 161
9. Appendices ........................................................................ 163
9.1. Example of Solution of Partial Differential Equation .................................. 163
9.2. Some Settings of the Used FE Software ......................................................... 169
9.2.1 Solver Settings ........................................................................................... 169
Summary ............................................................................... 175
Samenvatting ........................................................................ 179
List of Publications ............................................................... 183
Biography .............................................................................. 185
List of Figures
Fig.1.1: a) Energy sources trend; b) Breakup of energy sources ...................................................... 1
Fig.1.2: Wind turbine components ....................................................................................................... 3
Fig.1.3: Fixed speed geared concept for SCIG ................................................................................... 4
Fig.1.4: Variable speed geared concept for SCIG .............................................................................. 5
Fig.1.5: Variable speed, direct drive concept for SCIG ..................................................................... 5
Fig.1.6: Limited variable speed geared concept for WRIG (OptiSlip) ........................................... 6
Fig.1.7: Variable speed geared concept for WRIG (DFIG) ............................................................. 6
Fig.1.8: Variable speed, direct driven, electrically excited concept for SG .................................... 7
Fig.1.9: Variable speed, direct driven, PM excited concept for SG ................................................ 8
Fig.1.10: Variable speed, geared, PM excited concept for SG ......................................................... 8
Fig.2.1: Major components of a PMDD generator for wind turbine ........................................... 18
Fig.2.2: Stator housing for the 2 MW generator for wind turbine ................................................ 20
Fig.2.3: Insulated sheet-steel strips...................................................................................................... 21
Fig.2.4: Generator Housing laminate assembly ................................................................................ 21
Fig.2.5: Stator yoke manufacture - a) Lamination pressing with hydraulic cylinders for
compression b) Pressed stack to form stator teeth and yoke ............................................. 21
Fig.2.6: Single layer concentric winding: a) Cross section b) Isometric View [14] .................... 22
Fig.2.7: Double layer lap winding: a) Cross section b) Isometric View [14] .............................. 23
Fig.2.8: Concentrated coil winding: a) Cross section b) Isometric View .................................... 24
Fig.2.9: Coil manufacturing precedure - a) Individual wires joined together to form conductor
b) Looping of conductor to form rough shape on a winding-jig c) Finished spool held
together with tape d) The spool stretched to create the diamond coil shape .................. 25
Fig.2.10: Machine winding a) Loose coil ends after coil layout in slots b) Brazing of the end
connectors for coil interconnection ......................................................................................... 26
Fig.2.11: Machine winding a) Coil end connections brazed b) Insulation of end connection
manually c) terminal preparation .............................................................................................. 26
Fig.2.12: Machine impregnation – Resin being pumped into the stator assembly ..................... 27
Fig.2.13: Machine impregnation –Baking for resin hardening in a large oven ............................ 27
Fig.2.14: Generator assembly – pre assembled rotor with magnets inserted into stator bore . 28
Fig.2.15: Rotor assembly – magnet insertion on to rotor ............................................................... 29
Fig.2.16: Bearings used in wind turbines: single, double and triple bearing [5] .......................... 30
Fig.2.17: Bearings used in wind turbines: a) Single layer tapered bearing b) Double layer
tapered bearing [5] ...................................................................................................................... 30
Fig.3.1: Eddy currents induced in a conductor ................................................................................. 34
Fig.3.2: A concentrated winding former and after winding the coil around the former ........... 34
Fig.3.3: Harmonic content of concentrated winding mmf for an arbitrary phase distribution 35
Fig.3.4: Eddy currents loss analysis methods in Electrical machines............................................ 49
Fig.3.5: Eddy currents loss analysis – Focus on type of machine ................................................. 50
Fig.4.1: Actual geometry of a PM machine ....................................................................................... 61
Fig.4.2: Simplified geometry for analytical model ............................................................................ 61
Fig.4.3: Flux density waveform and current/m length for a concentrated winding machine .. 72
Fig.4.4: Surface current density (A/m) decomposition for 9-8 combination concentrated
winding machine ......................................................................................................................... 74
Fig.4.5: Simplified geometry for analytical model ............................................................................ 76
Fig.4.6: Effect of motion (arbitrary model) a) Flux lines without motion b) Flux lines with
motion........................................................................................................................................... 77
Fig.4.7: Geometry for field due to magnets only ............................................................................. 78
Fig.4.8: Excitation for field due to magnets only ............................................................................. 78
Fig.4.9: Flux lines due to permanent magnets only ......................................................................... 80
Fig.4.10: Flux lines due to both current sheet and magnets ........................................................... 80
Fig.4.11: Flux density due to magnets for the case of PM excitation, Br = 1.2 T ..................... 81
Fig.4.12: Flux density due to current sheet for fundamental harmonic ....................................... 82
Fig.4.13: Induced current density due current sheet excitation of one harmonic ...................... 83
Fig.4.14: Induced eddy current losses due to current sheet excitation of one harmonic ......... 84
Fig.4.15: Variation of eddy current losses due to current sheet excitation for various slot-pole
combinations ............................................................................................................................... 85
Fig.5.1: General procedure to setup a problem ................................................................................ 91
Fig.5.2: Main dimensions of a concentrated winding machine...................................................... 92
Fig.5.3: Symmetry in a machine over one quarter ............................................................................ 93
Fig.5.4: Meshing over a machine model 3 slots per 4 poles combination ................................... 93
Fig.5.5: Boundary conditions shown here for a 3- 4 combination ............................................... 94
Fig.5.6: Material setting shown here for a 3- 4 combination ......................................................... 95
Fig.5.7: Sub-domains which are assigned motion for a 3- 4 combination................................... 96
Fig.5.8: A contour plot of magnetic potential and surface plot of magnetic flux density plotted
together ......................................................................................................................................... 97
Fig.5.9: a) Flux density (y-component), derived from surface plot at one time instant using
cross-section plot b) Eddy current losses magnets by using sub-domain integration of
resistive losses over magnets real time c) Cogging torque calculated by user defined
global variable and its evolution in time. ................................................................................ 98
Fig.5.10: Flux lines for simplified FE model used for validation .................................................. 99
Fig.5.11: Induced current density for a harmonic a) Analytical b) FE ....................................... 100
Fig.5.12: Magnetic flux density shown for a 9-8 slot-pole combination with active magnets 101
Fig.5.13: Induced current density for 9-8 combination ................................................................ 102
Fig.5.14: Eddy current losses in various parts of machine ........................................................... 102
Fig.5.15: Eddy current losses in different slot-pole combinations (scale on each figure is
different) ..................................................................................................................................... 104
Fig.5.16: Eddy current losses with slot opening (including PM excitation as well) ................. 105
Fig.5.17: Eddy current losses with slots per pole (Only stator current excitation to compare
with analytical models) ............................................................................................................. 105
Fig.6.1: Static tests - schematic of the arrangement ....................................................................... 111
Fig.6.2: Power Supply (with inbuilt voltage, current and power measurement) ....................... 111
Fig.6.3: Stators a) open slots b) semi-closed slots .......................................................................... 112
Fig.6.4: a) copper ring b) aluminum ring c) ST37 steel ring ......................................................... 113
Fig.6.5: Experimental setup ............................................................................................................... 113
Fig.6.6: Comparison of Losses for a) Copper ring b) Aluminum ring c) ST37 steel ring for
open slot machine ..................................................................................................................... 115
Fig.6.7: Comparison of Losses for a) Copper ring b) Aluminum ring c) ST37 steel ring for
semi-closed slot machine .........................................................................................................117
Fig.6.8: Effect of frequency on a) resistance b) inductance of the open slot machine ............118
Fig.6.9: Effect of frequency on a) resistance b) inductance of the semi-closed slot machine 119
Fig.6.10: Thermal pictures of the rings immediately after tests. Note higher temperature on
the edges of the ring .................................................................................................................120
Fig.6.11: Rotary tests - schematic of the arrangement ...................................................................123
Fig.6.12: Stator and mechanism of torque measurement for the 9 kW PM machine ..............123
Fig.6.13: Frictional losses for open-slot and semi-closed slot machine .....................................126
Fig.6.14: Effect of slotting on flux density: Pulsation under teeth and air part of stator
alternately ....................................................................................................................................127
Fig.6.15: Eddy current loss as a function of frequency and flux density ...................................129
Fig.6.16: Flux density in various parts of stator for estimating stator losses .............................130
Fig.6.17: No load losses (due to slotting) for open slot machine ................................................132
Fig.6.18: No load losses (due to slotting) for semi-closed slot machine ....................................132
Fig.6.19: Flux density change in the air-gap of a) semi-closed slot machine has very low
change of flux density b) open slot machine has much larger change in flux density ..133
Fig.6.20: Losses due to both stator currents and PMs in open slot machine for a current of
a) 4A b) 7A c) 10A d) 13A .....................................................................................................136
Fig.6.21: Losses due to both stator currents and PMs for semi-closed slot machine for a
current of a )4 A b) 7 A c) 10 A d) 13 A ...............................................................................137
Fig.6.22: Loss Calculation Procedure: a) Total Iron Losses from Experiments b) Rotor Iron
Losses deduced after estimating stator iron losses..............................................................138
Fig.6.23: Eddy current loop in 3d and 2d treatment of induced current. ..................................139
Fig.6.24: Effect of phase angle of current on eddy current loss calculation..............................139
Fig.6.25: Losses calculation (shown for the case of open slot machine)....................................141
Here, Rotor Losses = Input power – Copper losses – Stator iron losses; ................................141
Fig.6.26: Losses due to stator current excitation only and no active PM field for open slot
machine .......................................................................................................................................142
Fig.6.27: Losses due to stator excitation only and no active PM field for semi-closed slot
machine .......................................................................................................................................142
Fig.7.1: Losses in the solid rotor back-iron due to stator currents only (I), PMs only (Br) and
combined field (I and B) ..........................................................................................................148
Fig.7.2: Losses in the magnets due to stator currents only (I), PMs only (Br) and combined
field (I and B) .............................................................................................................................149
Fig.7.3: Total losses in the machine due to combined field .........................................................150
Fig.7.4: Mechanical power as a function of q for chosen topologies .........................................150
Fig.7.5: Total rotor losses as a fraction of mechanical power ......................................................151
Fig.7.6: Variation of winding factor with slots per pole per phase ‘q’ [3] ..................................152
Fig.7.7: Cogging elements contributing to the cogging torque ....................................................153
List of Tables
TABLE 1-1: GENERATOR SYSTEMS IN WIND TURBINES............................................................................ 9
TABLE 2-1: GENERATOR PARAMETERS ....................................................................................................19
TABLE 3-1: SOME FORMULAE USED FOR EDDY CURRENT LOSSES CALCULATIONS...........................39
TABLE 3-2: SOME ANALYTICAL FORMULATIONS USED FOR EDDY CURRENT LOSS
CALCULATION.....................................................................................................................................43
TABLE 3-3: SOME NUMERICAL FORMULATIONS USED FOR EDDY CURRENT LOSS
CALCULATION.....................................................................................................................................46
TABLE 4-1: MODELING CONSTANTS AND VARIABLES ..........................................................................85
TABLE 5-1: PHYSICS SETTINGS ...................................................................................................................95
TABLE 5-2: SOME USEFUL POST-PROCESSING OPTIONS.........................................................................97
TABLE 5-3: LOSS COMPARISON FOR 9-8 COMBINATION MACHINE.................................................. 100
TABLE 6-1: MACHINE PARAMETERS: DIMENSIONS ............................................................................. 112
TABLE 6-2: MACHINE PARAMETERS: MATERIAL PROPERTIES .......................................................... 112
TABLE 6-3: EXCITATION CASES .............................................................................................................. 124
TABLE 6-4: PEAK FLUX DENSITY IN VARIOUS PARTS OF STATOR .................................................... 130
TABLE 6-5: MASS OF VARIOUS PARTS OF STATOR ................................................................................ 130
TABLE 6-6: TEMPERATURE OF MAGNETS IN DIFFERENT EXPERIMENTS ....................................... 135
TABLE 6-7: FREQUENCY AND CORRESPONDING ROTATIONAL SPEED .......................................... 141
TABLE 7-1: SPECIFICATION OF ANALYZED MACHINE ....................................................................... 146
TABLE 7-2: DIFFERENT SLOT-POLE COMBINATIONS ANALYZED ................................................... 147
List of Symbols and Abbreviations
Latin Letters
A
A
A
A
B
B
b
b
C
E
f
g
g
g
H
h
h
I
J
j
K
K
k
k
l
m
m
n
n
n
P
P
p
Magnetic vector potential
Surface current density
Area
Phase A of 3 phase balanced system
Phase B of 3 phase balanced system
Magnetic flux density
Height of 2d simplified analytical geometry
Width in geometrical dimensions
Phase C of 3 phase balanced system
Electrical field strength
Electrical frequency
Constant for solution of partial diff. equation
Mechanical air gap
Acceleration due to gravity
Magnetic field intensity
Constant for solution of partial diff. equation
Height in machine geometry
Current
Electrical current density
Operator for imaginary part
Surface current density
Loss coefficient
Constant for spatial distribution
Constant used in variable separation
Length/thickness in machine geometry
Constant used in variable separation
Mass
Rotational speed
Number of harmonic
Constant used in variable separation
X coordinate dependent part of partial diff. eqn.
Power loss/Power
Number of poles
[Wb.m-1]
[A. m-1]
[m2]
[T]
[m]
[m]
[V.m-1]
[Hz]
[m]
[m/s2]
[A.m]
[m]
[A]
[A.m-2]
[A.m-1]
[m]
[kg]
[rad.sec-1]
[W]
Q
q
R
S
S
T
T
t
U
V
V
X
Y
Y coordinate dependent part of partial diff. eqn.
Slots per pole per phase
Electrical resistance
Space dependent part of partial diff. equation
Number of slots
Time dependent part of partial diff. equation
Torque
Time
Voltage
Electrical scalar potential/Voltage
Linear speed of harmonic
X coordinate in Cartesian system
Y coordinate in Cartesian system
[ohm]
[N]
[s]
[V]
[V]
[m/s]
[m]
[m]
Greek Letters
α
β
γ
δ
θ
λ
μ
ν
π
ρ
σ
τ
ω
Constant used in PDE solution
Constant used in PDE solution
Constant used in PDE solution
Skin depth
Spatial angle
Wavelength
Permeability
Velocity
Constant
Electrical resistivity
Electrical conductivity
Distance/Pitch
Rotational speed/Electrical frequency
[m]
[radians]
[m]
[H.m-1]
[m/s]
[Ohm.m]
[S.m-1]
[m]
[Radian.s-1]
Subscripts
0
1,2,3,..
a,b,c
h
ext
m
n
p
r
r
r, rem
ry
s
sy
t
x,y,z
Pertaining to free space or initial condition
Region of PDE solution in simplified geometry
Electrical Phase A, Phase B and Phase C
Harmonic number in rotor reference frame
external
Pertaining to magnet
Harmonic number
Pertaining to pole dimensions
Pertaining to rotor
Relative
Magnetic Remanance
Pertaining to rotor yoke
Stator/Slot quantity
Pertaining to stator yoke
Pertaining to tooth dimension
Component in x, y or z direction
Subscripts for Power/Power Loss
_
ac
Cus
dc
e, eddy
Fe
Fes
Fer
gen
in
load
m, mech
out
Pm
r
Combines two subscripts
Alternating current
Stator copper loss
DC quantity
Eddy current loss
Total iron loss
Stator iron loss
Rotor iron loss
Generator power
Input power
Load
Mechanical
Output
Prime mover
Static ring region / Rotor region
Superscripts and Accents
s
2,3,…
e
→
Stator quantity
Raised to power (square, cube, …)
External quantity
Peak value
Vector Quantity
Abbreviations
DFIG
EESG
FE
GCD
LCM
MMF,
mmf
ODE
PDE
PM
PMDD
PMSG
SCIG
SG
WRIG
Doubly Fed Induction Generator
Electrically Excited Synchronous Generator
Finite Element
Greatest Common Divisor
Least Common Multiple
Magneto Motive Force
Ordinary Differential Equation
Partial Differential Equation
Permanent Magnets
Permanent Magnet Direct Drive
Permanent Magnet Synchronous Generator
Squirrel Cage Induction Generator
Synchronous Generator
Wound Rotor Induction Generator
1. Introduction
This chapter presents an introduction to the field of research. The need for such a research justified by
recent trends and developments is explained. After a bird’s eye view of the whole field, the problem
statement is formulated. The chapter culminates in a brief outline of the thesis.
1.1. Wind – A Renewable Energy Source
Our ever increasing energy need is prompting us to look for alternative energy
sources. At present most of the energy is generated using fossil fuels which contribute
to CO2 concentration in atmosphere and hence global warming. Moreover, the amount
of available fossil fuels is limited. Therefore, to circumvent the problem of global
warming and to fulfill the energy demand, renewable energy sources are being targeted
as an alternative. It is clear from fig. 1.1(a) that gradually, the composition of energy
sources is changing and this change is likely to continue in the near future.
a)
b)
Fig.1.1: a) Energy sources trend; b) Breakup of energy sources
Source a): International Energy Agency, 2002; Source b): Earth Trends 2008, using data from IEA 2007
From fig. 1.1(b) it can be easily observed that wind energy is a highly productive
source of renewable energy. Wind is abundant, clean and everlasting whereby utilizing it
fully to its capacity is imperative. In order to further utilize the potential of this
renewable source, there is a trend to go offshore i.e. further from coast towards deeper
sea.
1
1.2. The Offshore Trend: Conditions and Challenges
This recent trend of installing wind turbines further from shore towards deeper sea
came into picture because of:



High availability of wind
High wind speed
Low visual impact
The possibility of much more energy yield is a reasonable driver to go offshore.
Moreover, experience in offshore installations for example oil rigs, shipping etc. already
exists in industry. Nevertheless there are many challenges to be met due to different
ambient conditions compared to wind turbines inland/onshore. There are also
challenges which can arise due to wind energy injection into the existing system of
transport of electricity. These challenges are highlighted below.




The sea environment is one of the most corrosive environments.
Maintenance in offshore region is a challenge because of limited accessibility.
Erection and commissioning of wind turbines in deep sea is difficult.
The whole concept has to be economical to keep a reasonable price per unit of
energy.
 Connection of converted wind energy to existing on-shore power system and to
bring it to the end user.
 Technical dynamics i.e. integration of wind energy with existing electrical network.
 Economical dynamics: i.e. electricity unit pricing, commitment and trade between
different countries. An understandable challenge is to keep the price of energy
from renewable sources competitive with other sources of energy (like hydro,
thermal etc.)
From the perspective of wind turbines, the main constraints for an offshore scheme
are of high reliability, high efficiency and reasonable cost. At present there is a lot of
discussion in industry as well as academia regarding which type of generators are most
suitable for offshore conditions? The answer is not straight forward as each type has its
pros and cons. Therefore this research becomes even more interesting and reasonable as
it is a step towards the answer to this question.
1.3. Wind Energy Conversion
Wind has a lot of energy content as such but this energy is required to be converted
into some useful form like mechanical motion or electricity. In order to convert the flow
of wind into electrical energy, the following main components are needed:
a) Wind turbine rotor i.e. blades connected to a hub to convert wind flow into
rotary motion.
2
b)
c)
d)
e)
f)
g)
h)
i)
j)
Support structure i.e. tower, nacelle, hub etc.
Generator to convert rotary motion into electricity.
Pitch system for blades to harness the wind energy efficiently.
Yaw system to orient the wind turbine rotor in the direction of wind.
Foundation on which the whole structure is erected.
Power electronic converter to ensure smooth power supply
Cabling, lightning protection and grounding.
Transformer for grid connection (if required)
Cooling mechanism and auxiliaries.
In this thesis, we focus only on the generator part of the wind turbine. The generator
part is where the mechanical power is converted into useful electricity. The generator is
a very critical part and in some cases (as in a direct-drive topology) most expensive
single component of the wind turbine after the rotor blade assembly. However,
expensive or not, the generator is the heart of the wind turbine.
Generator
Lightning
Protection
Nacelle
Turbine Hub
Ventilator
Power Cables
Turbine Blades
Tower
Fig.1.2: Wind turbine components
3
1.3.1 Types of Generators in Wind Turbines
There are many types of wind turbines in the market these days and many ways to
classify them. Interested readers can have a look at [1]-[3] and [5]-[10]. Here we will
focus on classification based on generator systems
1.3.1.1 Squirrel Cage Induction Generators (SCIG)
The basic advantage of squirrel cage induction generators is ruggedness due to
simplicity in mechanical construction. A gearbox is often used to run the generator at
high speeds however a direct driven version has also been investigated [4]. The SCIG
are further divided into three types:
A. Fixed speed geared and directly connected to the grid
This concept is one of the oldest, cheapest and simplest concepts for generator
system of a wind turbine. The scheme shown in fig. 1.3 was popular in early nineties but
new installations are rather rare these days because of poor power quality. There is no
converter connection but to compensate for reactive power drawn from the grid,
capacitor banks are used. In some configurations, a soft starter is employed to improve
power quality.
Fig.1.3: Fixed speed geared concept for SCIG
B. Variable speed geared and connected to grid with full converter
The aim of this concept, shown in fig. 1.4 is to utilize ruggedness of SCIG with
improvement in power quality. The full converter used in this concept is able to
maintain good power quality but adds appreciable cost to the generator system.
4
Fig.1.4: Variable speed geared concept for SCIG
C. Variable speed direct driven and connected to grid with full converter
The concept of using SCIG and direct drive topology has been developed keeping
offshore scenario into perspective [4]. Due to high reliability of the SCIG, this concept
has some advantages but the mass of a direct driven SCIG is very large which limits its
utility. This scheme shown in fig. 1.5 has not been applied and hence not in market yet.
Fig.1.5: Variable speed, direct drive concept for SCIG
1.3.1.2 Wound Rotor Induction Generators (WRIG)
A. Limited variable speed, geared and directly connected to grid
The limited variable speed concept is also known as “OptiSlip” concept depicted in
fig. 1.6. This concept has been primarily used by Vestas (V66-1.65 MW) and Suzlon (2
MW) [6].
5
Fig.1.6: Limited variable speed geared concept for WRIG (OptiSlip)
The generator system for this concept is a WRIG and has a variable external rotor
resistance. The external resistance is changed via a power electronic converter which is
controlled optically and is mounted on the generator rotor (hence the name OptiSlip).
Consequently, the output power fluctuations of the generator system can be controlled.
This scheme gets rid of brushes and slip-rings, the maintenance issues can be negotiated
well. On the downside, the rotational speed can be controlled only upto 10% above the
synchronous speed.
B. Variable speed, geared and connected to grid with partial converter (popularly known as DFIG)
The wound rotor induction generators are most popular generator systems in the
market till date. A partial converter is utilized to supply magnetizing current at a
particular frequency to the wound rotor of the induction generator so as to maintain the
output frequency at the stator terminals. Stator can be directly connected to the grid or
via a transformer. The configuration is shown in fig. 1.7. Due to double connection with
grid these are popularly known as Doubly Fed Induction Generators (DFIG).
Fig.1.7: Variable speed geared concept for WRIG (DFIG)
6
The use of partial converter, which is about 20-30% of the rated power of the
generator, makes it cost effective and power quality efficient. However complex control
system, mechanical gearbox and use of brushes makes the system prone to failures.
1.3.1.3 Synchronous Generator (SG)
Synchronous generators are the conventional power generators. However in order to
use these generators for wind energy, they require either mechanical speed control or
full converter at their grid interface. According to the method of field excitation, these
can be classified into electrically excited and PM excited machines. A system which aims
at using advantages of both PMSG and geared systems is also present in the market.
A. Variable speed direct drive, electrically excited and connected to grid with full converter
These generators are the conventional synchronous generators with separately
excited field winding on the rotor as shown in fig. 1.8. These machines are well known,
rugged and provide high degree of voltage control. However, because of low rotational
speed, these machines are very large. Another disadvantage is additional losses in the
rotor winding which carries DC current. Currently these machines are well established in
the market.
Fig.1.8: Variable speed, direct driven, electrically excited concept for SG
B. Variable speed direct drive, Permanent Magnet (PM) excited and connected to grid with full
converter
The variable speed PM excited generator systems offer highest efficiency because of
absence of rotor copper losses (as compared to EESG). The PMs ensure that the
machine is very rugged and has high power density. The minimal moving parts promise
7
high reliability which makes these machines a candidate for offshore installations. On
the other hand, PMs are expensive and need additional mechanical protection against
the environment. As shown in fig. 1.9, Absence of gearbox means that the machine
ends up large in size, massive and expensive. Nevertheless these machines are very
popular and gaining considerable market share. This thesis deals with PMSG type of
machines in detail.
Fig.1.9: Variable speed, direct driven, PM excited concept for SG
C. Variable speed, geared, Permanent Magnet (PM) excited and connected to grid with full converter
In order to overcome the disadvantages of PMSG’s large size, weight and cost, an
interesting topology has gained attention recently. This topology requires a gearbox in
addition to the PMSG system as shown in fig. 1.10. High speed of operation makes the
generator smaller and supposedly cost effective. Thus a cheaper generator can be
specified by using a 1, 2 or 3 stage gearbox and increasing the speed of rotation of
generator. The advantages are the same as in PMSG type system. Addition of gearbox
adds some reliability constraints to overall system.
Fig.1.10: Variable speed, geared, PM excited concept for SG
8
We can summarize the most important types of generator systems used in wind
industry as shown in table 1-1.
TABLE 1-1: GENERATOR SYSTEMS IN WIND TURBINES
Generator system
Squirrel Cage Induction
Generator
(SCIG)
Wound Rotor Induction
Generator
(WRIG)
Electrically
excited
field
Synchronous
(EESG)
Generator
(SG)
PM field
(PMSG)
Speed
control
Fixed
Variable
Limited
Variable
Variable
Variable
Variable
Variable
Variable
Drive
train
Geared
Direct
Driven
Geared
Geared
Direct
Driven
Geared
Direct
Driven
Geared
Converter
C
W
M
E
No
++
++
+/-
--
Full
+
--
++
-
++
++
-
--
++
++
--
+/-
Full
+
--
-
+
Full
+
-
-
+
Full
--
--
++
++
Full
++
++
-
+
Partial
(optislip)
Partial
C = Cost; W = Weight; M = Maintenance; E = Efficiency
So far we have seen that each generator type used in the industry has its own
advantages and disadvantages. In order to reach a decision regarding offshore wind
energy, we must focus on a certain type of generator and see how we can make it more
suitable for power generation. PMSG seems to be very promising in terms of reliability
and efficiency. The main manufacturers for this technology are Goldwind, Vensys,
Scanwind, EWT, STX with General electric (GE) and Siemens also entering the field.
Looking at the generator system, at present, most popular machine technology in
this field is distributed winding, radial-flux synchronous machine. If we look at
manufacturing cost breakup (see fig. 1.11) of direct-drive wind turbines, the generator
cost is substantial [21]. A deeper analysis reveals that apart from PM material cost, an
important part of generator cost is stator coil manufacture, its assembly into winding
with insulation and impregnation. These costs are high because of excessive manual
labor involved in winding the generator. Thus concentrated windings which can
substantially reduce the manual labor in winding of generator (due to possibility of
automation) have been considered in this thesis.
9
distribution of costing
2%
0%
rotor :
15%
drivetrain:
29%
hydraulic:
nacelle:
cover :
yaw mechanism:
tow er:
generator:
25%
E-system/converter:
10%
transformer :
auxiliary equipment:
12%
1%
3%
3%0%
Fig.1.11: PM Generator – cost breakup [21]
1.4. Concentrated windings
In order to overcome the shortcomings of PMDD generators, while still using PM
for field excitation, many solutions and concepts have been proposed in literature. PM
machines with fractional pitch double layer concentrated windings have been proven
successful for low power machines. These windings have a coil mounted on each tooth
of the stator rather than being distributed in a number of slots as shown in fig. 1.12.
These machines offer smaller sizes, high power densities and high efficiencies [13],[14]
and [15]. The concept has already been applied to wind energy generators by many
researchers [8],[11],[15] and [19] though for smaller power ratings. We can say that the
concept has a lot of potential however much more effort is required to establish this
technology in the market. There are some design issues/pitfalls while using concentrated
windings which have been brought out by [12]-[20]. The important issues which need
attention at design stage are:
-
10
Winding factor can be low compared to the distributed windings.
Cogging torque can be high due to slots being open type/rectangular.
Eddy current losses in solid conductive parts due to space harmonics and sub
harmonics in the magnetic field of concentrated windings.
Fig.1.12: Cross section of a concentrated winding topology
A lot of research has been done in order to make concentrated windings a viable
option for the goals of cost and weight reduction for wind energy generators. The above
mentioned disadvantages have been tackled adequately to facilitate design process.
Although still in an early phase, research is proving that concentrated windings can be
employed for electrical generators for wind energy applications. The winding factor and
cogging torque can be successfully controlled by selection of appropriate slot pole
combination. Also the method of calculation of these parameters can be found in [13][19].
The Eddy current losses however are still very little known in the field of large direct
drive wind turbines. A lot of literature is available for small high speed machines. Large
synchronous machines with wound field poles have also been dealt with since early
1940’s. The research in the field of large direct drive machines with PMs is relatively
new. In [22] analytical and FE methods to calculate eddy current losses in the solid
conductive parts of wind turbines are compared. Thus it is evident that concentrated
winding topology can be applied to wind turbine generators. The problem remaining is
“how to define the slot pole combination such that it leads to low eddy current losses?”
Jassal and Polinder [19] have proposed to distribute the coils around the periphery of
the machine to reduce the eddy current losses however firm guidelines can’t be deduced.
1.5. Research Focus
This thesis exclusively focuses on the electrical generator part of the whole wind
turbine system. It is justified because generator is one of the most expensive
components of a wind turbine. Out of the many types of wind turbine generators
possible, this thesis deals with Permanent Magnet Direct Drive (PMDD) or PMSG type
of generators. This type of generator has been chosen because it fits the criteria for
offshore applications viz. high energy yield and low number of components (therefore
high reliability).
However, these generators are expensive due to distributed windings (high number
of coils) as shown in fig. 1.13(a), PM material cost and a full converter. Therefore, it is
evident that we need to do some research regarding the cost of these generators. In that
respect, an important feature of this whole research is to use fractional pitch
concentrated windings (fig. 1.13(b) ) instead of distributed windings in stator of PMDD
11
generator so as to lower manufacturing costs. This is possible because of winding
automation in case of concentrated windings. The resulting winding is wound over each
tooth resulting in much smaller overhangs compared to a distributed winding.
There is another drawback of this winding scheme in the form of induced losses in
solid parts of the machine. The losses are present because of eddy currents induced by
the high harmonic content of the winding magneto motive force or mmf as shown in fig.
1.14 (more details are presented in chapter 3, sections 3.1 and 3.2).
Wind turbines are high torque low speed machines which results in solid rotor yoke
because it is the torque carrier. A straight forward solution is to laminate the rotor (like
stator) but this will lead to additional manufacturing and assembly costs while putting a
compromise on strength of torque carrier. Due to this reason, this option has not been
considered for analysis.
a)
b)
Fig.1.13: a) Distributed winding normally used b) Concentrated winding topology
Concentrated winding mmf
Distributed winding mmf
Ideal sinusoidal mmf
Fig.1.14: Concentrated winding mmf compared with distributed winding mmf
12
1.6. The Thesis Objective
As mentioned in section 1.4, there are some additional eddy current losses associated
with the choice of concentrated windings. Since high performance magnets used in
PMDD generators are also conductive, the eddy current losses in the magnets can be so
high that the magnets get permanently de-magnetized due to temperature rise.
Therefore in order to design a good machine, it is imperative to take these eddy current
losses into account. Therefore the main objective of this research can be formulated as:
“Eddy current loss modeling for design of permanent magnet concentrated winding generators for wind
turbines”
The subsequent deliverables pertaining to the main objective can be defined as:
 A simple and generic analytical model for predicting eddy current losses.
 Validation of the analytical model using FEM; Validation of analytical and FE
models formulation using experiments to bring out effect of simplifications
used for models.
 Deduction of trends in eddy current losses for various slot-pole combinations.
 Design guidelines for PMDD generators with respect to eddy current losses.
1.7. Thesis Outline
The thesis has been organized into 8 chapters.
Chapter 1 introduces the field of application and defines the objective of thesis.
Chapter 2 gives an overview of the state of art in manufacturing of PM direct drive
generators. This chapter also forms the basis of reason for research.
Chapter 3 gives the background of research done in the field of eddy current loss
analysis. An extensive literature survey is conducted on methods and developments in
eddy current loss analysis in electrical machines. More details of the problem taken up as
the main scientific contribution are presented.
Chapter 4 is dedicated to analytical modeling of electromagnetic field and analysis of
eddy current losses in solid parts. For analytical modeling:
2d model is formulated because axial length of machine is much larger than the
pole-pitch whereby end effects are neglected.
Rectangular coordinate system is used because diameter of machine is very
large and a section of machine is almost rectangular.
All materials are chosen as linear and isotropic.
Motion is included in analytical model.
13
Chapter 5 presents Finite element (FE) modeling for eddy current losses in PMDD
generators for wind turbines.
First a simplified time-harmonic model is presented for validation of analytical
model.
Full transient simulations are performed for analysis of losses with all
geometrical effects taken into account.
The model is applied to a number of possible configurations
Chapter 6 contains experimental analysis and results. These experiments are conducted
on 9 kW machines with concentrated windings. The aim of the chapter is to validate the
analytical and FE models developed for large direct drive machines.
Static tests are performed to validate analytical models.
Rotary tests are performed to validate FE models.
Chapter 7 compares modeling results to generate trends and design guidelines for
designing large wind energy generators with concentrated windings.
Chapter 8 summarizes the conclusions and recommendations.
14
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Technology, Göteborg, Sweden, 2000.
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turbines”, Ph.D. dissertation, Delft University of Technology, Delft, The Netherlands, 2004.
[4] M.Henriksen, “Feasibility study of induction generators in direct drive wind turbines”, MSc
thesis, Technical University of Denmark (DTU), May 2011.
[5] P. Lampola, “Directly driven, low-speed permanent-magnet generators for wind power
applications”, Ph.D. dissertation, Helsinki University of Technology, Finland, 2000.
[6] D.J.Bang, “Design of transverse flux permanent magnet machines for large direct drive wind
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725-733, September 2006.
[8] S. Widyan, “Design, optimization, construction and test of rare-earth permanent magnet
electrical machines with new topology for wind energy applications”, Ph.D. dissertation
Technische Universität Berlin, Berlin, Germany, 2006.
[9] H. Polinder and J. Morren, “Developments in wind turbine generator systems”, Electrimacs
2005, Hammamet, Tunisia.
[10] H. Polinder, S.W.H. de Haan, M.R. Dubois, J.G. Slootweg, ‘Basic operation principles and
electrical conversion systems of wind turbines'. In EPE Journal, December 2005 (vol. 15, no.
4), pp. 43-50.
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[15] H. Polinder, M.J. Hoeijmakers, M. Scuotto, “Eddy-current losses in the solid back-iron of
permanent-magnet machines with concentrated fractional pitch windings,” in Proc. of the
2006 IEE International Conference on Power Electronics, Machines and Drives, Dublin, 46 April 2006, pp. 479-483.
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15
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[19] Jassal, A., Polinder, H., Shrestha, G., Versteegh, C., 2008. “Investigation of Slot Pole
Combinations and Winding Arrangements for Minimizing Eddy Current Losses in Solid
Back-Iron of Rotor for Radial Flux Permanent Magnet Machines”, Proceedings of the
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16
2. State of Art in PM
Generator Manufacture
The aim of this chapter is to present the modern methods used in manufacture of PM distributed
winding generators for wind turbines. This chapter primarily targets the students/readers who don’t have
sufficient background of the electrical machine manufacturing technologies. For an expert, these processes
are very well known. The chapter highlights the labor intensive aspect of the distributed windings which
adds to the cost and manufacturing time of a machine. These aspects establish the reasons for looking
into alternate winding topologies such as concentrated windings to lower the cost and manufacturing time.
A 2 MW wind turbine generator, designed built and tested forms the basis of description. This
generator was developed in the Netherlands and manufactured in Germany whereby the author was
involved in the electromagnetic design of the generator.
2.1. Permanent Magnet Direct Drive Generators
Permanent Magnet Direct Drive (PMDD) wind energy generators are gaining
popularity amongst manufacturers as they have high energy density and low
maintenance. Unfortunately these types of generators are heavy and expensive. A lot of
research aimed at weight and cost optimization is going on in the field of PM generators
[1]-[4]. PMDD generators have come of age with the invention of high performance
magnets, like Samarium Cobalt and Neodymium-boron-iron which has made it possible
to design high performance generators. In this chapter, the aim is to highlight the
limitations of the distributed winding type PMDD type machines used for wind turbine
application.
Developments in electrical machine technology have led to evolution of
manufacturing methods of electrical machines as well. Manufacturing methods in
principle are old but owing to technology, scale of manufacture and standards, the
manufacturing processes have become complex. It is important to know how are these
machines manufactured and where can the process be improved further. Starting with
the general construction of a PMDD generator, the major parts as shown in fig. 2.1 are:
1.
2.
3.
4.
5.
6.
Stator Housing
Stator iron - laminates
Coils and winding
PM assembly
Rotor back-iron, shaft and support structure
Bearings
17
Fig.2.1: Major components of a PMDD generator for wind turbine
2.1.1 Specifications of the 2 MW Generator
The PM direct drive generator was designed to operate in a warm tropical climate
which poses the following design constraints:
a) Design wind class: IEC 3A according to IEC 61400
b) Maximum Outer Lamination Diameter: 3.9 m (Transportation constraint)
c) Power output: 2 MW at grid
d) Rotational speed: 18 rpm
e) Voltage output: As per Converter Requirement (Line voltage ~ 540 V)
f) Ambient Temperature: -200C to 500C for partial loading
400C for full load
g) The heat dissipation should be ~ 7 kW/m2 (for an area of ~ 18m2)
18
TABLE 2-1: GENERATOR PARAMETERS
Nominal power at grid
Nominal voltage
Nominal current
Outer diameter
Axial length
Nominal Speed
Input Torque
Cooling air gap
Cooling outside
Surface heat dissipation
Bearing configuration
Winding
2 MW
537 V
2450 A
3.9 m
1.5 m
18 rpm
1.18 MNm
Forced air
Natural air
~7 kW/m2
Double bearing
Distributed
2.2. Stator Construction
This section deals with the stator or stationary part of the generator only. Various subparts of the stator and their construction is explained further.
2.2.1 Stator Housing
The stator housing is the outermost covering of the electrical machine which holds
all the electromagnetically active parts [9]-[10]. In most general terms, stator housing is
an inactive part and plays the role of keeping all assemblies in place. The size of large
machines and the requirement of a high mechanical strength normally mean that the
housing is casted as one part. However in very large machines generally housing is
casted in large sections which are welded together (fig. 2.2). The material normally used
for housing is cast steel.
The inner size of the housing is machined thereafter to house the stator laminates.
The housing also carries the mechanism of fixing the laminated steel (for construction
of stator yoke) to itself thereby maintaining the air gap radius.
In natural air cooled systems, such as the one employed in the presented generator,
housing also aids in cooling the machine. External cooling fins were mounted on the
outside of the housing. In case generator is water cooled or force air cooled, this cooling
function of the stator housing is limited. Nevertheless it can be seen that generator
housing is one of the critical components of the generator assembly because of multiple
roles it play and the required mechanical precision.
19
Fig.2.2: Stator housing for the 2 MW generator for wind turbine
2.2.2 Stator Yoke - Laminates
The stator yoke is in form of thin sheets of magnetic steel called “steel laminates”
shown in fig. 2.3. The laminates are available in form of rolled thin sheets (~1mm). The
sheets are electrically insulated to prevent flow of eddy currents. The shape of stator
teeth/slots including stator back iron are either laser-cut (for small amount) or punched
out (for large amounts) from this sheet. The maximum width of this electrical sheet
steel is about 1.25m. This presents a limit on the maximum length of a section of stator
yoke obtainable from the sheet. Consequently, a number of smaller segments are
arranged within the generator housing to form the complete round stator. The stator
yoke is built (axially) as a stack of these sheets cut into the shape of the sections of the
machine as shown in fig. 2.4. These sheets are placed along the supports mounted on
the stator housing. The stacked sheets are then pressed together (see fig. 2.5) to form
the axial length of the machine. These individual segments are interleaved to maintain
20
equal pressure during laminate pressing whereby the sheets are equally strained and
breakage can be avoided.
Fig.2.3: Insulated sheet-steel strips
Laminates cut from the
sheet steel roll
Fig.2.4: Generator Housing laminate assembly
a)
b)
Fig.2.5: Stator yoke manufacture - a) Lamination pressing with hydraulic cylinders for compression b) Pressed stack to
form stator teeth and yoke
21
2.2.3 Coils and Winding
The following sections give a brief overview of the winding types used in machine
manufacture followed by a detailed description of the lap winding manufacture.
2.2.3.1 Common Winding Types in Large Electrical Machines
The electrical machine winding can be done in many ways and there are several
different methods to classify them. Detailed winding overviews can be found in
literature and interested readers are referred to [6]-[13]. The purpose of this section is to
introduce only the main winding types commonly used in large electrical machines and
bring out their advantages and disadvantages. The common winding types can be
classified as:
A) Distributed type: In this winding, the coils of same phase are distributed in a
number of slots and then interconnected to produce the required MMF. The distributed
type can be further classified as single layer and double layer. There are many other types
of windings which can fall under this category such as wave winding, mush winding, hair
pin windings etc. [7] and [8] but we will restrict ourselves to the single layer concentric
and double layer lap windings. This is because these winding types are most commonly
used for large machines such as wind turbine generators.
Single Layer Concentric Winding
The single layer concentric winding has only one coil side in each slot and therefore
only one layer of conductors in a slot as shown in fig. 2.6.
a)
b)
Fig.2.6: Single layer concentric winding: a) Cross section b) Isometric View [14]
The advantages of this type of winding are:
a) High fill factor i.e. more copper conductor area in a slot.
b) No half-filled slots after finishing the winding
22
The main limitations of this type of winding are
a) Complex end windings as different shapes are needed for coil crossover.
b) Winding short-pitching for harmonic elimination not possible.
Double Layer Lap Winding
The double layer lap winding is the most commonly used distributed winding type
for large machines. This type of winding has two coil sides in each slot and therefore
two layers of conductors in a slot as shown in fig. 2.7.
a)
b)
Fig.2.7: Double layer lap winding: a) Cross section b) Isometric View [14]
The advantages of this type of winding are:
a) Symmetric end windings and only one shape of coil is needed.
b) Coils can be short pitched to eliminate harmonics.
The main limitations of this type of winding are
a) Lower fill factor (compared to single layer winding) because of inter-layer
insulation within top and bottom coil-side.
b) Half-filled slots obtained after completion of winding.
B) Concentrated type: This winding is also known as tooth-wound winding and
fractional slot winding (slots/pole/phase is a fraction). In this winding each coil of a
phase is wound around a tooth. These windings are not very popular design option in
large machines because of the harmonic rich mmf content they produce as discussed in
sections 1.4 and 1.5. These windings can also be single or double layer [11]-[13] but in
the context of thesis, we will elaborate only on the double layer type as shown in fig. 2.8
In smaller machine size (~ few kW) this type of winding has been very successful
[11] and [12]. The main reason is that for smaller machine sizes, the labor intensive
winding process can be automated while utilizing semi-closed slots. For larger machines,
23
the logical choice for winding automation is by using pre-formed coils. This invariably
requires open slots which in turn contribute to mmf harmonics and therefore eddy
current losses in solid conductive parts.
Stator
Coil
a)
b)
Fig.2.8: Concentrated coil winding: a) Cross section b) Isometric View
The advantages of this type of winding are:
a) Short and symmetric end windings and only one shape of coil is needed.
b) Coil winding can be automated whereby labor, cost and time for production can
be reduced.
The main limitations of this type of winding are
a) Lower fill factor* due to necessity of workable mechanical gap between coil
sides in same slot.
b) Presence of slot harmonics in flux which can lead to eddy current losses.
*Note: Single layer concentrated windings can have high fill factor but since they are known to have
higher eddy current losses [13] these are not considered in this thesis.
2.2.3.2 Winding of Reference Machine: 2 Layer Lap Type
The method of distributed lap winding with preformed coils used in this machine is
well established [6]-[10]. In general the text-books dealing with the design of electrical
machines gives details of mmf production from the distributed winding as well as some
schematics of how a distributed lap winding is laid into slots. However, the description
given in text books is usually not sufficient to form clear picture of the manufacturing
process. This makes it difficult to foresee the problems of manufacture of such windings.
Therefore this section has been included in this thesis.
The coils and winding arrangement is the most difficult and labor intensive part of
an electric machine. The need for special skill-set and workmanship restricts the
assembly process of the machine.
As shown in fig. 2.9, the whole process of machine winding starts with insulated
copper conductors on spools. First of all, the strands from various spools are tightly
24
wrapped to form a simple loop. Thereafter, the loop is insulated with an insulating
cotton tape to prepare the loop for coil formation. This step can be done automatically
or manually depending on the coil size. Once the loop has been insulated, it is stretched
hydraulically to form the coil shape for a distributed winding. The shape is such that one
side of coil can be placed in top side of the slot and another coil can be placed in the
bottom side of the slot.
a)
b)
c)
d)
Fig.2.9: Coil manufacturing precedure - a) Individual wires joined together to form conductor b) Looping of conductor to
form rough shape on a winding-jig c) Finished spool held together with tape d) The spool stretched to create the
diamond coil shape
The next step is placement of coils in slots and interconnection to form the machine
winding. This step (fig. 2.10) is also very labor intensive because of distributed windings
and hence large number of coils to be interconnected.
25
a)
b)
Fig.2.10: Machine winding a) Loose coil ends after coil layout in slots b) Brazing of the end connectors for coil
interconnection
The interconnection is done by brazing the coil ends. The brazed coil ends are then
insulated manually and cable connections for terminals are prepared. This process is
shown in fig. 2.11. Finally the machine winding is checked for insulation strength
according to various standard tests and sent for insulation impregnation.
a)
b)
c)
Fig.2.11: Machine winding a) Coil end connections brazed b) Insulation of end connection manually c) terminal
preparation
The most popular impregnation method is vacuum pressure impregnation where the
whole machine after winding is placed in a steel chamber which creates vacuum inside.
26
Then the insulating material is allowed to flow and due to negative pressure, the
insulating material goes on to the surface of generator. After a sufficiently thin and
uniform layer of insulating material is deposited on the stator, the machine is taken out
heated in an oven to harden the insulating material.
Another method is resin impregnation where viscous resin is allowed to flow through
the stator at a slow pace, whereby the coil insulation absorbs the resin. The machine is
then heated in an oven to dry and harden the resin. This method was used for the
machine under discussion due to unavailability of a vacuum impregnation tank which
was large enough to accommodate the machine. The steps of resin pumping and resin
baking are shown in fig. 2.12 and fig. 2.13 respectively.
Stator Bore
sealed to
contain resin
Resin Pumped
Fig.2.12: Machine impregnation – Resin being pumped into the stator assembly
Oven
Fig.2.13: Machine impregnation –Baking for resin hardening in a large oven
27
2.3. Rotor Construction
Different manufacturers use different generator rotor topologies (having different
consequences). Since the main objective of the thesis is to calculate eddy current losses
due to concentrated windings, this topic is a side-track. This section has been
nevertheless included to keep the completeness in this overview of state of the art PM
generators manufacture. Consequently, the section is very short and general.
2.3.1 PM Assembly on Rotor
A very challenging task in assembly of the whole generator is assembling active
magnets on to the rotor. In one case, the whole rotor can be assembled with unmagnetized magnets and then the magnets can be magnetized later. This method poses
important challenges in maintaining the quality of magnetization and requirement of
special equipment to magnetize the magnets. In context of the machine in discussion,
we focus on assembly of already magnetized magnets (as these were used). There are
basically two ways to achieve this. First method is to pre-assemble the whole rotor and
magnets then lower the assembly into stator bore as shown in fig. 2.14.
Crane
Rotor
Stator
Fig.2.14: Generator assembly – pre assembled rotor with magnets inserted into stator bore
Second method is to insert and align rotor back iron in stator bore and insert
magnets on the rotor surface thereafter. Both these methods require negotiation of
attractive magnetic forces between stator yoke and the magnets. The second method i.e.
magnets insertion on to the rotor was chosen to avoid risks during assembly. This
method is depicted in fig. 2.15.
28
Fig.2.15: Rotor assembly – magnet insertion on to rotor
2.3.2 Rotor Back-iron, Shaft and Support Structure
The rotor back-iron is cast steel machined to the dimensions of the rotor. The
tolerance on manufacture of this part is very high because this part ensures that the
mechanical air gap between PMs and stator yoke is maintained under all conditions.
Such high accuracy demands strong support structure which adds to weight and cost of
the machine. The shaft is also casted and machined accurately to fit bearings and rotor
structure.
2.3.3 Bearings
The choice and placement of bearings has major consequences on overall size,
weight and mechanical design of the machine. The bearings are often custom made. In
general, for large PMDD machines, there is a choice of single or double bearing system
however some triple bearing systems are also reported [5]. In the present design double
bearings were used because of long axial length. Fig. 2.16 and 2.17 show important
bearing placement concepts and types used in wind turbines.
29
Fig.2.16: Bearings used in wind turbines: single, double and triple bearing [5]
a)
b)
Fig.2.17: Bearings used in wind turbines: a) Single layer tapered bearing b) Double layer tapered bearing [5]
2.4. Limitations Posed by PMDD Generators
A conventional PMDD generator is heavy, large and costly. The reasons for this are
emanating from the following properties of the machine.




30
In direct drive topology, rotational speed of the generator rotor is low whereas
torque is high. Therefore a large diameter and higher number of pole pairs is
required resulting in a large machine.
It uses Permanent magnets for creating excitation field. These magnets are still
costly as a material, adding to the overall cost.
Depending on the design, fixation of PM on rotor might require innovative
techniques to negotiate magnetic force between the iron and PM.
A small air gap has to be maintained to maximize air gap flux due to which
rotor support structure becomes very heavy.



A large and heavy machine increases logistics costs (costs of transport, hoists
and cranes etc.)
PM have a danger of demagnetization due to high temperature. Cooling of
these generators is a major issue especially at sites which have high ambient
temperatures. This necessitates margins to be taken in design, for example,
lowering the current density, over-sizing the magnets, increasing surface area of
machine etc.
Permanent Magnets are also prone to corrosion and special protection is needed
to ensure their long life.
It is with this background that new methods of machine manufacture are being
considered. We will restrict our scope of interest to PM direct drive machines for wind
energy applications.
2.5. Summary
In this chapter, the manufacturing methods for large PM machines used in the
industry today have been explained. Manufacturing experience and some design details
for an actual 2 MW PM generator are shared. The procedure and therefore the
difficulties in manufacture of the labor intensive distributed windings are explained. The
limitations posed by such PM direct drive generators have been documented based on
the experience.
31
Bibliography
[1] E. Spooner, P. Gordon, C.D. French, “Lightweight, Ironless-Stator, PM Generators for
Direct-Drive Wind Turbines” Electric Power Applications, IEE Proceedings, Volume 152,
Issue 1, pp. 17 – 26, 7 Jan. 2005.
[2] McDonald A.S., Mueller M.A., Polinder H., “Comparison of generator topologies for directdrive wind turbines including structural mass” Proc. Int. Conf. Electrical Machines
(ICEM),Crete, Greece, September 2006.
[3] McDonald A.S., “Structural analysis of low-speed, high torque electrical generators for direct
drive renewable energy converters”, PhD Thesis, School of Engineering & Electronics,
University of Edinburgh, UK, 2008.
[4] S.Engström, B.Hernnäs, C.Parkegre, S. Waernulf , “Development of NewGen - a new type
of direct-drive generator”, proceedings of EWEC, London, UK, 2004
[5] J.N.Stander, G.Venter, M.J. Kamper, “Review of direct-drive radial flux wind turbine
generator mechanical design”, Research article, Wind Energy magazine pp. 459–472, July
2011, DOI:10.1002/we.484.
[6] M.Kostenko and L.Piotrovsky, “Electrical Machines – II: Alternating current machines”, 3rd
edition 1974, pp. 51-102.
[7] A.K.Sawhney, “A course in Electrical Machine Design”, 5th edition, Dhanpat Rai and Co.
Publishers, pp.230, 289-295.
[8] M.G. Say, “The performance and design of alternating current machines” 3 rd edition, CBS
Publishers, pp. 196-224.
[9] P.C.Sen, “Principles of Electrical machines and power electronics”, 2 nd edition, Wiley
Publishers, pp. 132-136 and pp. 569-578.
[10] A.E.Fitzgerald, C.Kingsley and S.D.Umans, “Electric Machinery”, 6th edition, Tata McGrawHill publishers, pp.173-212 and pp.644-656.
[11] J. Cros, P. Viarouge, “Synthesis of high performance pm motors with concentrated
windings”, IEEE Transactions on Energy Conversion, vol. 17, pp. 248–253 (2002).
[12] F. Magnussen, C. Sadarangani, “Winding factors and Joule losses of permanent magnet
machines with concentrated windings,” in Proc. of the 2003 IEEE International Electric
Machines and Drives Conference, 2003, pp. 333 – 339, vol.1.
[13] H. Polinder, M.J. Hoeijmakers, M. Scuotto, “Eddy-current losses in the solid back-iron of
permanent-magnet machines with concentrated fractional pitch windings,” in Proc. of the
2006 IEE International Conference on Power Electronics, Machines and Drives, Dublin, 46 April 2006, pp. 479-483.
[14] H.Polinder “Lecture Notes for the course AC Machines”, TU Delft, 2008.
32
3. Eddy Current Losses
This chapter gives an overview of the research done in the field of eddy current losses in
permanent magnet machines and details of the problem taken up as the main scientific contribution i.e.
eddy current loss modeling of large direct drive wind turbines with concentrated windings. The contents of
this chapter have been published in electrical power applications journal, IET [115].
3.1. Eddy Current Loss – Physics
Eddy currents or Foucault currents were discovered by French physicist Léon
Foucault in 1851[114]. The name eddy current comes from the fact that the circulation
of currents resembles vortices caused in fluids by turbulence of flow. The physics of
eddy currents follows from Lenz’s law which states that:
"An induced current is always in such a direction as to oppose its cause"
Eddy current is induced in a conductor to oppose the change in flux that generates
it. The eddy currents are caused when a conductor is exposed to a changing magnetic
field due to relative motion of the magnetic field source and conductor or due to
variations of the field with time.
This can cause a circulating flow of electrons, or a current, within the body of the
conductor. These “eddies” of current create induced magnetic fields that oppose the
change of the original magnetic field due to Lenz’s Law. The phenomenon of eddy
currents has many physical effects some of which are undesirable while some can be
used. The two major effects of eddy currents are:
a) Magnetic field opposition: The eddy currents generate their own field so that
they counteract the primary field that produced them. This property is used in
eddy current braking, crack testing etc.
b) Heating: The circulation of eddy currents leads to ohmic heating of the
conductor they are induced in. This can be a desirable property in heating
applications while it is a drawback in electrical machines.
33
Fig.3.1: Eddy currents induced in a conductor
Source http://www.spacialenergy.com/images/etfield.gif
3.2. Eddy Current Losses in Concentrated Windings
It is known thus far that eddy current losses are induced in solid conductive parts of
machines when concentrated windings are used. The main reason for induction of eddy
currents is the space harmonic content in mmf produced by the concentrated windings.
Time harmonics due to converter operation also induce eddy current losses but in this
thesis attention has been focused on space harmonics as they are inherent to the
generator while time harmonics are external to the generator. In a machine with
concentrated windings, each winding is wound around a tooth of machine.
Fig.3.2: A concentrated winding former and after winding the coil around the former
34
The mmf produced by concentrated windings is not sinusoidal but trapezoidal. In
many cases it can be treated as rectangular wave. These waveforms can be decomposed
into a number of harmonics using Fourier series. Some of these harmonic fields rotate
at same speed as rotor and these are called the torque producing harmonics. However
there can be significant harmonic fields which rotate at a speed different than the rotor
giving rise to a relative motion between a conductive surface and a magnetic field. These
conditions are responsible for induction of eddy currents in the conductive body.
Fig.3.3: Harmonic content of concentrated winding mmf for an arbitrary phase distribution
The dissociation of these fields into their harmonic components depends strongly on
the winding distribution and slot-pole combination used in a concentrated winding
machine.
The problem of eddy current losses in electrical machines with concentrated windings is
not new. However, in the field of large low speed electrical machines such as wind
turbine generators, it is relatively new. Amongst wind turbines also we are focussing on
large direct drive machines with PM excitation. Though PM machines have all the
advantages regarding force density and efficiency, the PM themselves are very sensitive
to temperature. If due to any reason, temperature increases to a certain value, the
magnets can be permanently de-magnetized.
Another problem associated with high performance PMs is that they are electrically
conductive. Eddy currents can be induced within the PMs and the heating effect of
these induced eddy currents can raise the magnet temperature. It is imperative to study
the effect of induced eddy current losses in case of concentrated winding machines. PM
35
machines can’t be exposed to high temperatures due to risk of demagnetization. To limit
the temperature rise, what has been done in the past is to reduce current density,
increase surface area, add fins to the machine housing etc. The additional cooling fans
form an obvious choice. This subject of machine cooling still has some scope wherein it
might be useful to try to inculcate machine cooling in the concentrated winding
topology.
With respect to cooling also, this topology has potential because we can have more
surfaces exposed for cooling (for example if we blow air in between the two coils
mounted on stator teeth).
3.3. Eddy Current Losses in Electrical Machines – A Survey
Electrical machines are one of the prime inventions of mankind. Their utility
and importance need no introduction. Like many engineering utilities, electrical
machines have been evolving over time. In the past 150 odd years, we have been able to
develop a whole branch of electrical engineering dealing specifically with electrical
machines. One of the most important subjects falling under the category of electrical
machine design is iron losses in the electrical machines. The complex geometry, material
properties, presence of time varying electromagnetic fields amidst electrically conducting
parts have made the study of iron losses rather challenging and interesting. There are
many general iron loss formulations which are used to estimate and group the iron
losses in electrical machines. According to most accepted view, the Iron losses in
electrical machines can be broadly classified into three categories:
a) Eddy current losses: The resistive losses caused due to induced electric currents
which are produced due to change in flux density.
b) Hysteresis Losses: The losses within the structure of magnetic material (at
domain level) due to changes in flux density.
c) Excess Losses: The excess losses arise because of internal correlation fields
between magnetic domains. These fields, together with other effects such as
eddy currents, act as a damping field opposing any changes in the external
magnetizing field hence the excess loss is produced.
Triggered by experiments and empirical relations, the early research focused on first
two types of losses i.e. eddy current and hysteresis losses. Steinmetz classified that
energy lost in iron losses is composed of two components [1].
E   Bˆ1.6   NBˆ 2
(3-1)
Where,
E = Energy Lost per cycle;  = the coefficient of hysteresis;  = the coefficient of eddy
currents; N= frequency; B̂ is the peak flux density.
36
The first part of this expression is the hysteresis loss contribution and the second part
presents eddy current loss contribution. Jordan proposed a similar formulation but with
different exponents [2].
Pfe  Chyst fBˆ 2  exc Cec f 2 Bˆ 2
(3-2)
Here, Pfe = Iron losses; f is electrical frequency; B̂ is the peak flux density; Chyst and Cec
are coefficients of hysteresis and eddy current losses; exc is the excess or anomalous
loss factor.
With further experience, another loss term called “excess losses” was added to account
for discrepancies in measurements and calculations [3].
Pfe  Chyst fBˆ 2  Cec f 2 Bˆ 2  Cex f 1.5 Bˆ1.5
(3-3)
Another formulation for total iron losses is by dissociating losses caused by linear
magnetization, rotational magnetization and losses cause by higher harmonics [4]. This
development has its roots in increased use of power electronics and hence increased
harmonic content in power supply or generated voltage.
Pfe  C1Plin  C2 Prot  C3 Ph
(3-4)
Here, C1, C2 and C3 are empirical coefficients and Plin, Prot and Ph denote power loss
due to linear magnetization, rotation magnetization and higher harmonics respectively.
However, in this particular research we are going to focus on eddy current loss analysis
in electrical machines. Eddy current or Foucault current is a well-known phenomenon in
electromagnetics. There are numerous eminent people who contributed to this field of
research and it is not possible to name them all here. In order to limit the scope, only
rotating machinery and journal publications were considered in the survey. This is
because the trends in the field of development of eddy current loss analysis can be
projected easily even with this restricted scope. The research on this subject can be
divided into three stages based on the most used/popular method of analysis.
a) Stage I: Early phase 1892~1950
b) Stage II: Middle phase 1951~1990
c) Stage III: Modern phase 1990 onwards
This demarcation in stages is porous and has been done to arrange the evolution of
this research systematically. It is important to note that in many cases more than one
method is employed to reach a conclusion. There is a primary analysis which is a
37
contribution and a secondary analysis which is used to validate the model or
experiments. The phases and their segregation according to the methods are based on
the primary method of analysis used by the authors.
The chapter starts with a general introduction to iron losses in electrical machines and
then a fundamental description of eddy current loss phenomenon are presented. After
that, the three stages of development are described. Some formulations used during the
development stage have been summarized. The formulations are chosen to represent a
variety of effects which scientists/engineers were trying to capture for example, popular
formulations with 2d/3d analysis, linear/non-linear materials, scalar/vector potential
etc. are chosen. It might be pointed out that the problem formulations are not limited to
those mentioned in the survey but it is not possible to include all formulations. Based on
the literature material studied in this research, certain trends have been brought out.
3.3.1 Stage 1: 1892~ 1950 – Experiments and Formulas
Like many physical discoveries, the eddy currents were “observed” first and then the
science able to describe the phenomenon was built around those observations.
Therefore, initial studies were done on giving a theoretical basis for the phenomenon.
The scientists of those times were trying to understand the effects produced by these
intriguing currents. From the perspective of electrical machines, the losses due to eddy
currents were first studied seriously in 1892 by J.J.Thomson [5] where he analyzed
heating effects of eddy currents in an iron plate.
Amongst the rotating machines, DC machines were the most popular machines until
the invention of AC machines by Tesla and Ferraris almost simultaneously around 1888
[6] and [7]. In DC machines, eddy current losses were studied in conductors and
armature iron especially due to slotting of armature [8]-[12]. However, eddy current
losses were not given that much importance before late 1800s. The invention of AC
current, induction machines and transformers triggered a lot of interest in analysis of
eddy current losses. It was quickly understood that these losses depend on material
conductivity and surface area where current could flow. Consequently, sheet steel
laminations were being used to restrict eddy current losses in cores of machines since
(atleast) as early as 1900 [13]. Thereafter, the research on materials and methods to
reduce such losses was carried out. Meanwhile, electromagnetic field modeling was also
getting more elaborate and accurate. Following these developments, M.B.Field wrote
one of the first analytical papers on eddy current losses in solid and laminated
conducting materials [14].
From about 1904 onwards, research in induction machines picked up but it saturated
soon [15]-[17]. This could be because induction machines reached an optimum
performance level suitable enough for applications of that time. The same trend can be
seen in eddy current loss analysis in the field of power transformers. From the study of
methodology employed by most authors of this time, it can be said that initial research
(late 1800’s to early 1900’s) was done to observe the effects of the eddy currents.
Therefore, most of the research was based on performing experiments and deriving
38
empirical formulas to calculate eddy current losses. This practice continued for almost
three decades. During this period, some publications on induction machines and
transformers can be seen as in [13], [14] and [18]. Considerable interest was also shown
in material characterization [19]-[24]. Around 1937, engineers and scientists started
analyzing losses in cores of synchronous machines analytically [24] and [25]. This work
was one of the first scientific works on eddy current losses in AC machines.
TABLE 3-1: SOME FORMULAE USED FOR EDDY CURRENT LOSSES CALCULATIONS
First Author
Formula
Methodology
1.5 ˆ 2
Pe  kv B 
Bˆ  Flux  density ( peak );   tooth  pitch
F.W.Carter [28]
v  velocity
1.5
v
Pe  ke m2 Bˆ 2  

M.B. Field [14]
here, m  thickness(lam.);
 sinh mh  sin mh 
Pe  0.315  mHˆ 2s 

 cosh mh  cos mh 
  resisivity; Hˆ  Field  strength : surface
s
m  2 104
Pe 
C.A.Adams [9]
Theoretical
Theoretical
Analytical
f
; h  thickness;   permeability
10 
2.65  B 


104  103 
2.5
1.55
 v 
 3
 10 
q1.88 
slot  opening
(cm); v  velocity (cm / s)
airgap
  tooth  pitch(cm); B  Flux  density
Experimental
q
Pe  ke B x v1.75 ( g  0.05)
K.Aston [26]
T.Spooner [17]
s3
 4.9
 0.0207  s3
Pe  ke B x v1.7 1 

g   3.1

v  speed ( ft / min); s,  , g , ke are constants
Pe  kBˆ 2.6
v1.6  s 
 
 0.3  g 
Experimental
Experimental
2.2
Experimental
39
Till that time linear formulation (in Cartesian coordinates) of machines was popular
[26] and [27]. This was followed by some more experimental work in 1941 by Rao and
Aston who summarized previously used formulas in literature of the time. Readers
interested in knowing about these formulas can refer to [26]. The formulas thus
developed were used for refining the design of machines.
A loss calculation comparison using different formulae has been presented by Aston
and Rao in fig. 20 of [26]. This type of loss comparison however is not very scientific.
This is because of complexity of problem we can’t determine a common reference for
all the formulae. Different scientists at different times have different correlations within
their findings. Hence details on this quantitative comparison of losses have been omitted.
It may also be highlighted here that the above table doesn’t list all the available
formulations for calculation of eddy current losses which is virtually impossible to
include in a single paper. However, the purpose here is to show the empirical nature of
the results. It was the lack of consistency in expressions and measurements that led to
further improvement in the eddy current loss formulation in stage II. At this stage
analytical, semi-empirical and empirical analysis’ availability was a trigger for other
applications for eddy currents as well. Heating, braking and crack detection are just a
few to name. In short, we can say that this period presents observation of phenomenon
of eddy currents followed by keen pursuit of understanding it according to the means
available at that time. This laid ground for the next period which dealt extensively with
the scientific development of methods for analysis of eddy current and losses thereof
especially in rotating electrical machines.
3.3.2 Stage II: 1951-1990- Analytical Methods
The 2d analytical methods for determination of electro-magnetic fields had already
developed during stage I of the research on eddy current losses. B.Hague wrote a book
on the electromagnetic field modeling in engineering applications as early as 1929 [29].
Till the end of stage I, rectangular coordinates, linear materials and harmonic
formulations were popular. In stage II, as we will see later in this section, material nonlinearity, effects of curvature and 3d effects were taken into account in analytical
modeling.
The most popular machines of stage II were the conventional synchronous machines.
This can be directly linked to development of many hydro-power and thermal-power
plants erected during this period especially 1965-90. This resulted in an increased use of
synchronous machines for generation of electricity. In this period significant research
was done on sheet steel used in electrical machines and eddy current loss analysis
thereof [30]-[32]. It is already known and acknowledged that an appreciable amount of
research was done in Germany and France related to electrical machines, materials and
electromagnetic field modeling. The utility of these works is somewhat restricted due to
the language and hence not considered for formulating trends in this survey.
40
A. General Formulation for Eddy Current Losses
The general formulation which almost all the authors followed consisted of magnetic
field solution starting from Maxwell’s equations. This type of analysis existed in stage I
and scientists were building upon that knowledge to include more complex geometries,
material properties and 3d effects. Here in this section, we briefly explain the general
formulation of eddy current based problems. The initial formulations of eddy current
losses in this stage were in 2 dimensions. The following assumptions were used to setup
the problem.
-
Cartesian coordinates were used in 2d analysis
Materials were assumed linear wherein B-H curve was a straight line.
The end effects were neglected.
The stator was assumed slotless and the winding was replaced by a current sheet
placed at an effective air gap. Fourier series was commonly used to take into
account space harmonics.
Using these assumptions, the analytical models could then be solved by employing
boundary conditions. The problem was solved for magnetic potential. Both scalar and
vector magnetic potential were used but vector magnetic potential seemed more popular.
The solution of magnetic potential leads to calculation of an induced current density and
then material resistivity is used to calculated eddy current losses. This analysis is very
common and therefore details of analysis are not presented here. Interested readers can
refer to [110] and [111] for details. Some important formulations are presented in table
3-2.
B. Developments in eddy current loss analysis- Stage II
H.Bondi and K.C.Mukherji presented a thorough analytical model for tooth ripple
losses in pole shoes of synchronous generators in 1957 [33]. Following this,
P.D.Agarwal wrote a benchmark paper on eddy current losses in solid and laminated
iron in 1959 [34]. Apart from that, A.J.Wood and C.Concordia wrote a series of four
papers on machines with solid rotors in 1958-1960 [35]-[38]. These papers were
completely analytical with rigorous mathematical theory. Effects of curvature, 3d effects
and material non-linearities were explored analytically. This was also one of the first
detailed analytical works in polar coordinates. Although these papers were focusing
more on field modeling but eddy current losses could be analyzed from these results.
From 1961-1970 we can see a steep rise in analytical work done on magnetic field
modeling and eddy current losses in electrical machines. In 1962, P.Hammond and
N.Kesavamurthy wrote on eddy current losses in solid conductive slabs followed by
research on eddy current losses in thin sheets [39] and [40]. This analysis was taken a
step further by SubbaRao by including effect of saturation analytically in 1964 [30] and
41
[31]. Effect of eddy current reaction field in analytical eddy current loss calculation was
aptly done in the same period by Mukherji [32].
Although technically possible, the analytical modeling started becoming very complex
at this stage. The results thus obtained were also not very accurate (compared to
measured experimental results) owing to a number of assumptions required to solve
such complex problems analytically. The engineers and scientists were looking for
somewhat better methods to formulate and solve eddy current loss problems. One of
the approaches used at that time was the circuit approach linked to analytical modeling
[41]. R.L.Stoll and P.Hammond generalized the theory of loss modeling due to eddy
currents in 1965-66 [42] followed by the first attempt of using numerical methods for
analyzing eddy current losses in 1967 [43]. K.Oberretl while working on pole losses in
synchronous generators presented an innovative analog circuit methodology for
inculcating the effect of variable permeability (saturation) in loss calculation [44]. This
method received some attention but never became main-stream. From 1967-1970,
pioneers like P.J.lawrenson, C.J. Carpenter and R.A.Jamieson worked extensively on
theory of eddy current losses in conductive media including 3d effects [45]-[48]. In 1970,
K.K.Lim and P.Hammond developed universal loss charts for eddy current losses in
thick conductive plates serving as a guide to machine designers [49]. They used analytical
as well as experimental results to derive those charts. The 1970’s also saw the first major
deviation from analytical to numerical methods for electrical machines. R.L.Stoll
presented general solution method for eddy current losses numerically in 1970[50]
followed by a power series method together with J.Muhlhaus [51] in the same year. Soon
after that, Finite Element (FE) method appeared on the scene. In the field of electrical
machines, M.V.K.Chari, P.Reece and C.J.Carpenter were the first proponents of this
method and it gained immediate popularity [52]-[54]. The reason for this deviation was
the capability of FE method to model difficult geometries and inclusion of material
properties [55]. The popularity of FE and numerical methods is evident from a steep
decline in the usage of analytical modeling (see fig. 3.4). There were also attempts to
include circuital analysis along-with numerical methods to simulate eddy current losses
in a more physical sense [56].
A sharp decline in research on eddy current losses in rotating machines in 1980’s was
noticed. One reason could be that most commonly used machines already reached
somewhat optimal performance levels by iterating on design and experience. There was
hardly any new research focusing on eddy current loss analysis on induction machines or
synchronous machines in 1980’s. However, modeling of magnetic fields was still going
on [57]-[61]. The FE method for eddy current loss calculation did get some attention
[60]-[62].
We can term this period as a period of analysis and formulation of new tools for
problems related to eddy currents in rotating electrical machines. The analytical methods
matured and the FE methods came into existence during this period. By the late 1980’s,
FE methods were still limited by the computing capabilities and numerical algorithms
available at that time. This shortcoming was soon overcome assisted by a revolution in
computing in the early 1990’s.
42
TABLE 3-2: SOME ANALYTICAL FORMULATIONS USED FOR EDDY CURRENT LOSS CALCULATION
First Author
H.Bondi [33]
N.Kesavamurthy
[40]
V.Subbarao [31]
R.L.Stoll [42]
P.J.Lawrenson
[47]
Problem
Formulation
- 2d Cartesian
coordinates.
- Scalar
potential
- Linear
- End effects
neglected
- 2d Cartesian
coordinates.
- Vector
potential
- Linear
- end effects
neglected
- 2d Cartesian
coordinates.
- Vector
potential
- non-linear
material
- end effects
neglected
- 2d Cartesian
coordinates.
- Vector
magnetic
potential
- linear
material
- end effects
neglected
- 3d polar
coordinates.
- Vector mag.
potential
- linear
material
Loss Calculation
Pe 
 2 / 
( J  H ).ds .dt
2 0 s
Pe  eddy current losses;
J  current density;
H  magnetic field intensity
1
 ( Eˆ x 0r Hˆ y 0r cos 10r Eˆ y 0r Hˆ x 0r cos 20r )
4
Pe  eddy current losses;
Eˆ x 0 r , Eˆ y 0 r  Electric field peak x,y comp.;
Pe 
Hˆ x 0 r , Hˆ y 0 r  Magnetic field intensity x,y comp.
 10 r , 20 r  Respective phase angles
Pe 
4
2
 abm H m2  2
  electrical frequency
a, b  dimensions in m
m  normalized permeability
H m  normalized field intensity
 2  normalized flux (complex)
Pe 
1  j  qp 2 ( Kˆ z e  qb ) 2 (  j ) 

Re 
2
2
2 
(



)





r

 
{1  (1  p 4 )}
{1  (1  p 4 )}
; 
2
2
p
0  r 
; q   / g;
q2 
g  pole pitch; Other symbols have usual meanings
L
a 2 /  2 a
Pe 
 Sz z 0   Sz z  L .dr 0  Sr r a .dz d dt
2 0 0 0


a  radius of cylinder
Sr , S z  radial and axial component of poynting vector
L  Length of cylinder
Other symbols have usual meanings
43
3.3.3 Stage III: 1991 Onwards-FE, Numerical and Analytical Methods
The 1990’s saw regenerated interest in eddy current losses owing to developments
in Permanent Magnet (PM) machines, power electronics and concentrated windings. A
significant increase in utility of PM machines and combination of power electronic
converters with Induction machines brought a second wind to eddy current loss analysis.
G.R.Slemon and X.Liu proposed an approximate modeling of eddy current losses based
on analytical flux variation calculation in 1990[63]. Z.Liu et al. presented a rather useful
analytical model to predict eddy current losses analytically [64]. This was the outcome of
past developments on analytical methods and ready verification available in form of FE
analysis. Some research on materials and geometrical research is reflected in [65] and
[66]. The analytical methods developed thus far are still being used today with minor
improvements. The most important developments during this stage came from
numerical and FE methods for eddy current loss calculations.
In 1992-1993, B.C.Mecrow and A.G. Jack showed improved possibilities with nonlinearity and 3d FE modeling [67]. Further, electro-thermal coupling for eddy current
loss calculation using analytical and numerical methods was proposed [68].
The later part of 90’s saw 3d FE modeling techniques being developed as
importance of end-effects in PM machines was being realized [69]. Japanese had been
working steadily on material properties of PM material, laminations and steel during the
90’s but much of useful results from Japanese research started appearing in early 2000’s
probably because of extensive use of 3d FE analysis which was better equipped to deal
with eddy current problems now. Many PM machines were used with power electronic
converters which added to eddy current losses as well. F.deng, T,Nehl and H.Polinder
have analyzed these effects in detail [70]-[74].
A. Numerical Methods and FE Formulation for Eddy Current Losses
There are many numerical methods which have been used to solve electrical
engineering problems. Power series or successive approximation method and finite
difference method were the first major numerical methods to be used for eddy current
loss problems [111]. The power series method however failed to take into account the
reaction field of eddy current losses properly which led to its decline [112]. On the other
hand the finite difference method, based on Taylor’s series was very popular before the
advent of FE method. The finite difference method was limited because of large
number of simultaneous equations required to be solved. Unavailability of computing
power available at that time (stage I) led to gradual decline in popularity of this method.
Interested readers can refer to [111] for details of this method. However, from the
literature survey, the primary and most popular numerical method used in the field of
electrical machines is Finite Element method which we will discuss here briefly. FE
method uses numerical techniques to solve partial differential equations but the problem
is divided into a “finite number of elements” and requires minimization of energy
44
functional solved locally for each element. For eddy current problems the energy
functional can take the form [60]:
 B 2 

  J . A  i  f  A 2 dv
F   
 2  

V



B= magnetic field; A= mag. vector potential; μ= permeability; v = volume;
(3-5)
σ= electrical conductivity; J= applied current density; f= frequency
The mathematical problem to be solved is then minimization of energy functional F i.e.
F
0
A
(3-6)
The theory of finite elements is well known and since the paper is focused on literature
survey of eddy current losses, the solution methods are not explained here. After
solution of A , the induced eddy current density can be found from
Jind ( x, y, z)  i.2 f  ( x, y, z ) A( x, y, z )
(3-7)
Eddy current losses can be calculated easily by integrating this current density over an
area
J 2 ( x, y, z )
Pe   ind
dV
(3-8)
V

Pe is the eddy current loss ; J ind is the induced current density; V is volume; σ is
conductivity
The overall solution is compiled and can be easily displayed for a chosen quantity as
well as some other derivable quantities from the original variable solved for. It may be
mentioned that because of discretization of the domain in FE method, complex
geometries and material non-linearities can be easily handled. FE method has developed
a lot since its inception into electrical engineering. Various new algorithms, mathematical
techniques have been implemented to improve the modeling and post-processing of FE
method. Description of various innovations in numerical methods is out of scope for
this survey.
45
TABLE 3-3: SOME NUMERICAL FORMULATIONS USED FOR EDDY CURRENT LOSS CALCULATION
First Author
R.Stoll [51]
Problem
Formulation
- 2d
Cartesian
- Vector
potential
Loss Calculation
Method
Akl (1)  ( K k 0  K k 1 y  K k 2 y 2  ...  K kk y k )  e y / g e  j x / g
Ak  k th order term in power series for A (mag. vector potential);
K k = coefficient
g  pole pitch;
Power
Series
*Eddy current losses are then calculated from solution of A
B.C.Mecrow
[67]
- 3d
Cartesian
- Scalar
mag.
potential
- End
effects
included
- Linear
Material
d 

(TA  TB )   
dt  2

TA , TB  Potential functions in adjacent bodies

1

  TA   
W.N.Fu
[113]
  TB  
  Scalar magnetic potential
TA , TB and  are solved to get eddy current density and losses
A 
 umn    (vBr )
t lM
2
M

 A u 
   lM     mn  d  

mn
Pe    m 1

 t lM 
n 1 

2
2
2
 i1n Rkn  (i1n  iMn ) Rkn

Pe  Eddy current losses in magnets
N
A  magnetic vector potential axial component
v  reluctivity of the material
M , N  slices and bars cut out of the domain
umn  voltage between nodes of the n th bar and m th slice
lM  Axial length per slice of magnet
Rkn  Terminal resistance between adjacent bars
46
FE
  Electrical conductivity
  Permeability
  (v  A)  
- 2d
Cartesian
- Vector
mag.
potential
- linear
material
- end effects
neglected
1

Circuit
element
coupled
with FE
1

 A
 1
     A  Ja   
      M


t

 0


K.Yamazaki
[97]
- 3d
cartesian
- Vector
mag.
potential
- linear
material
  A

  
     0

t

 
d
va  voa  Ra ia 
 Ra ia
dt
b a
2
2d
2
Pe 
  J x ( x, y)  J y ( x, y) dxdy

0 0


A,   magnetic vector and electric scalar potential resp.
 ,   permeability and conductivity resp.
FE
+
Analytical
J a  Armature current density; M  Magnetization of PM
va , Ra , ia  Armature voltage, resistance and current resp.
  Flux linkage of armature coil
a, b  Dimensions;
J x ( x, y ), J y ( x, y )  x,y comp. of induced current density
calculated analytically
B. Developments in eddy current loss analysis- Stage III
Just as we observed a rapid increase in analytical methods in 1960’s, another
phenomenon that virtually exploded in research during 2000-2010 was the PM machines.
Almost 70% of the research of eddy current losses was done on PM machines of one
type or another. The Neodymium magnets were available for cheaper prices and it led to
a great impetus on development of PM machines. However, this time around the tools
i.e. methods to deal with eddy current loss problems were at hand. Therefore, in this
period we see extensive use of analytical methods and FE methods to reach good
machine designs [75]-[81]. Z.Q.Zhu, D.Ishak and D.Howe wrote many papers related to
modeling of magnetic fields and eddy current losses in PM machines [82]-[85]. Although
the techniques used were not new but the application was rather new. It led to
tremendous interest in PM machines for large power applications especially in the field
of renewable energy like Archimedes Wave Swing (AWS), PM Direct drive generators
for wind turbines etc. [86].
Many new applications were identified with the advent of PM machines. High
speed machines without slots, fault tolerant machines and switched reluctance machines
etc indicated a new breed of application where eddy current loss analysis was deemed
necessary [87]-[90]. At the same time a lot of research on investigating the eddy current
losses in conductive Neodymium magnets themselves was gathering momentum. 3d FE
is the favorite tool when it comes to material characterization because of difficulty in
modeling material anisotropy and end effects analytically. PMs are conductive and
therefore modeling eddy current losses in magnets was imperative [91]-[93]. K.Yamazaki
and Y.Fukushima et al. have done a lot of work on eddy current loss analysis in PMs
47
[94]-[97]. In Europe also, significant amount of work on material characterization for
steel laminations solid iron samples etc. has been done [98]-[101].
With all these advancements, modeling of machines for new applications benefitted
and we can see a lot of publications on analysis of these special machines. M.Markovic
and Y.Perriard have presented generalized models for slotless PM machines in 2007-09
respectively [102] and [103]. The low power machines (fractional kW – few kW) can be
made cheaper using concentrated windings with fractional pitch because of winding
automation possibility. However, these types of windings also have eddy current losses
associated with them [104]-[108]. There is a good literature survey on analytical machine
modeling mentioned in [109]. A very thorough literature survey on practical design
aspects of concentrated winding machines can be found in [110].
Therefore, any development in the sector of fractional pitch concentrated winding
machines led to analysis of eddy current losses associated with them. The latest
additions in such machines are the ultra-high speed machines (~105 rpm or more) and
PM direct drive wind turbines. These machines utilize concentrated windings for cost
benefits, and high power densities. For some applications where 3d effects play a major
role, analytical methods are rather limited whereby it becomes necessary to resort to the
3d FE calculations. It is important to note that in this period, both the analytical and FE
methods have been used extensively. The probable reason could be that analytical
methods give a quick insight into dependence of design on various parameters which is
good for optimization. FE methods are more accurate however they take longer time to
solve. Thus a combination of these methods is the preferred way at present.
3.4. Results and Trends
From literature survey, it is clear that the main methodologies used in analysis
of eddy current losses are:
a) Experimental: The analysis is done by experimental observations and empirical
formulas derived for eddy current losses thereof.
b) Analytical: Starting from Maxwell’s equations magnetic and electric field quantities
are derived and used for evaluating eddy current losses.
c) Numerical: This method is basically same as analytical method with the exception
that instead of a mathematical expression, the derived partial differential equations
are solved numerically.
d) Finite Element (FE): This is an advancement of numerical methods where the
whole space is divided into very fine mesh and the quantities are derived by solving
for each region numerically. A big advantage is that any geometry, motion and nonlinearity can be incorporated at the same time whereas a probable disadvantage is
long time taken to solve and requirement of specialized software which is rather
expensive.
e) Equivalent Circuits: The machine is modeled in terms of lumped elements arranged
in an equivalent circuit depending on machine geometry. The eddy current losses
are then derived from this equivalent circuit.
48
Each of these methods has played an important part in development of theory of
eddy current loss analysis. However the popularity and utility of methods have been
changing over time. These trends have been deduced based on this extensive literature
survey. The utility percentage shown in fig.3.4 is based on the number of papers used
for this survey. It is not an exact number however the trend projected by these numbers
is indicative of the actual utility.
Utilization of Methods for Eddy Current Loss Analysis
100
Experimental
Analytical
Numerical
Finite Element
Equivalent Circuit
90
80
70
Utility in %
60
50
40
30
20
10
0
1950
1960
1970
1980
Year
1990
2000
2010
Fig.3.4: Eddy currents loss analysis methods in Electrical machines
Further, it can be seen from the literature survey that the research focus on eddy
current losses in the field of electrical machines shifted in different time periods. The
main trend of research and the types of machines studied can be grouped as:
a) General: This means development of methodology and analysis tools related to
eddy current loss analysis.
b) Materials: This indicates the research which didn’t focus on any particular type of
machine but on the materials which can be used in the machines. For example,
research on eddy current losses in sheet steel laminations and solid iron.
c) Conventional DC machines: These machines need no introduction but in this
survey, these machines refer to electrically excited DC machines with rotating
armatures and commutator.
d) Conventional Induction machines: These include squirrel cage and wound rotor
type of induction machines and with or without power electronic drive connected
to them.
e) Conventional Synchronous machines: This term refers to the synchronous
machines with electrically excited field winding on the rotor and distributed stator
windings in the stator.
49
f)
g)
PM machines: This is a general term and there can be many different types of PM
machines. In this survey this refers to general class of PM machines which use PM
as the field excitation.
Special: These machines appeared in late 1990’s and switched reluctance motors,
single phase motors, printed circuit board motors, flux switching machines etc. falls
under this category.
It may also be mentioned here that the percentage of focus on research is relative
and based on the number of papers used in this survey only. The purpose here is to
present trends rather than numbers. Figure 3.5 shown below projects the focus of
research during a given time period relative to other machine types in question.
Research Focus on Machine Type
100
General
Materials
Conv. DC
Conv. Induction
Conv. Synchronous
Permanent magnet
Special
95
90
85
80
75
70
Focus on research in %
65
60
55
50
45
40
35
30
25
20
15
10
5
0
1950
1960
1970
1980
Year
1990
2000
2010
Fig.3.5: Eddy currents loss analysis – Focus on type of machine
3.5. Summary
The basic physics behind Eddy current losses is briefly explained, followed by details
of eddy current loss generation in concentrated windings. The historical development of
the Eddy current loss analysis in rotating electrical machines has been documented.
Trends in development of Eddy current loss analysis are derived based on an extensive
literature survey.
50
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58
4. Analytical Modeling
This chapter presents the analytical model used for calculation of eddy current losses in the
solid conductive parts of the PMDD machines. Analytical models are very useful for quick comparative
studies on various topologies. The accuracy of such models is rather limited but the process of developing
such models gives great insight into physics of the problem. Besides, this chapter also sets up ground for
further research in terms of quantifying effects of various assumptions.
4.1. Analytical Models
Analytical models are not new in the field of eddy current loss analysis. A lot of
research on developments of analytical models specifically for eddy current loss
calculation was done in the 1960-70 [2] and [4]. Readers interested in knowing more
about historical development of the analytical methods can refer to [5] and [7]. The
strengths of analytical methods are speed of solution, insight into physics of the
problem and their generic nature. The limitations of these methods are the assumptions
and resulting inaccuracies. Another drawback is long time taken to formulate the model.
However, once formulated it takes very short time for solution.
From this thesis’ point of view it may be mentioned that this research is not entirely
based on analytical modeling but utilizes FE method for calculation of eddy current
losses as well. This is because electrical machines have a complex geometry and the
analytical methods are limited in capturing the geometrical effects such as slotting of
stator and magnets. As mentioned in chapter 3 (see fig. 3.4), the analytical methods still
remain very popular because of the insight into problems they offer. This means that the
problem features can be easily linked to the design outputs whereby quick comparative
studies can be easily performed. Moreover the analytical models don’t require very
specific and expensive software to perform FE calculations. This is the motivation for
doing analytical modeling for this research as well.
4.2. Modeling Approach for PMDD machines
The field of large PMDD machines is rather new and very interesting from research
point of view. When starting with such a detailed analysis on PMDD machines, there are
59
many options and methodologies which can be applied. This section gives a brief
motivation behind the choices made for analysis.
Analytical or Finite Element Method?
This is a prime choice which has to be made but for this research both approaches
were used. The reason is that both the methods have their own advantages and
limitations. The idea is to use analytical models to select the promising topologies and
then use FE models to calculate eddy current losses more accurately. In the process,
effects of various assumptions made in analytical models can be brought out.
2d or 3d analysis ?
The choice between 2d or 3d is rather easy to make in case of large PMDD
machines. 2d analysis was chosen in this case because the axial length of such machines
is much larger than the air gap whereby 3d effects can be neglected.
Polar or Cartesian/Rectangular coordinate system ?
For this analysis, Cartesian coordinates were chosen because the radius of large
PMDD machines is very large whereby a small section of the machine can be treated as
linear. This also makes the mathematics simpler. Later it has been shown that curvature
doesn’t have substantial effect on the eddy current loss calculation.
Scalar or Vector magnetic Potential ?
The choice between these formulations was made based on some advantages of the
use of vector magnetic potential. The flux lines can be directly visualized as lines of
constant magnetic vector potential. Many useful quantities such as flux density, electric
field, induced current density and other useful quantities can be directly computed from
vector magnetic potential. Therefore, magnetic vector potential was chosen. It may be
mentioned that either choice will anyways lead to good results.
Linear or non-linear iron ?
Linear formulation for rotor and stator parts was chosen to avoid mathematical
complications. Later on it is shown that the effect of material non-linearity is not that
prominent on the calculation of eddy current losses. Another reason for such a choice
was that there are two fields active in the machine. One is the field due to armature
current and the other is the field of PMs. The resultant field is the sum of the two fields
if the materials are linear.
4.3. Assumptions
It is not easy to include all the geometrical details in an analytical model. Some
simplifications are needed in order to follow the analytical procedure as mentioned in
[1],[3] and [4]-[6]. The difficulty is to form and solve the partial differential equations
60
(valid Maxwell’s Equations) for each region of the machine including its geometry
effects. The assumptions used are:
1.
2.
3.
4.
5.
6.
7.
8.
The stator is slot less, placed at effective air gap from the rotor surface and incurs
no losses due to eddy currents.
Windings are replaced by an equivalent current sheet travelling along the stator bore
surface and varying in time.
The relative permeability of stator iron is infinite.
The relative permeability of rotor back iron is realistic.
All materials are linear and isotropic.
The magnets are adjacent to each other so that they occupy the whole surface of
rotor and have electrical conductivity so that eddy currents can be induced.
2d modeling in Cartesian coordinates is selected with the assumption that
everything remains constant in the z-direction.
The end effects are neglected.
Fig.4.1 shows a real PM machine main dimensions and Fig.4.2 shows the simplified model
Stator yoke
Slot
Windings
Tooth
PM
Rotor back-iron
Air gap
Fig.4.1: Actual geometry of a PM machine
The simplified analytical model of machine consists of four layers and some boundaries.
B1
Region
Region1;1;Stator
Iron with
yoke Permeability
with Permeability
μ1 μ1
B4
B4
Stator
Current Sheet with Sinusoidal surface Current = K sin (kx) at (Boundary)
b
Region 2; Air with Permeability μ2
Y
B2
B5
B6
Region 3; Magnet region Permeability μ3 and conductivity 3
B7
Z
X
Region 4; Rotor back Iron with Permeability μ4 and conductivity 4
B3
l
Fig.4.2: Simplified geometry for analytical model
Here B1 – B7 are the boundaries where boundary conditions will be applied.
On B1 and B3, magnetic insulation condition is applied.
61
On all other boundaries, flux continuity and equivalence of tangential component of
magnetic field intensity (H-field) boundary condition is applied. These boundary
conditions are explained further in section 4.5.
4.4. Derivation of Partial Differential Equation (PDE)
The primary aim is to compute losses in the back-iron and the magnet region (regions 3
and 4). Starting from Maxwell’s equations for quasistatic fields:
  H  J  J ext
(4-1)
H is the magnetic field strength (A/m);
J and J ext are induced and external current density respectively (A/m2)
 E 
B
0
t
(4-2)
E is the electric field (V/m); B is the magnetic flux density (T)
 J ext  0
(4-3)
 B  0
(4-4)
The constitutive relations for the material (assumed isotropic) are:
J E
B   H  Brem
(4-5)
(4-6)
Here  is the electrical conductivity;  is the magnetic permeability
Brem is the remanent flux density of the magnet.
H is the magnetic field strength.
From (4-6), we can write for H as:
H
1

( B  Brem )
Substituting this value of H in (4-1) we get
62
(4-7)
1

   ( B  Brem )   J  J ext


  B   ( J  J ext )    Brem
(4-8)
Since the permeability is different for different materials, this equation holds true for
each region having a constant permeability and not for all regions in general.
From vector calculus we know that   (  F )  0 for any vector F . Also from (4-4)
we see that  B  0 and it may be written as:
B   A
(4-9)
Here A is called the vector potential. Substituting for B from (4-9) in (4-8), we get
  (  A)   ( J  J ext )    Brem
(4-10)
and
  (  A)  (  A)   2 A
Further, using  A  0 for a Gaussian surface, the left hand side of (4-10) can be
written as:
2 A   J   J ext   Brem
(4-11)
Using constitutive relations of (4-5) and (4-6) we can write
J E
E
A
A
or J  
t
t
(4-12)
Substituting for J in (4-11) and re-arranging terms we arrive at the following equation:
2 A  
A
  J ext   Brem
t
(4-13)
This equation is Poisson’s equation and will be the main equation to be used in
evaluating eddy current losses. This equation represents a general case and holds true for
vectors in 3 dimensions. However, we made a few assumptions to simplify the field
calculation and using the assumptions, we can reason:
63
a) The stator currents flow only in z direction (perpendicular to plane of reference)
b) Brem has only y component and its curl is in z direction only.
c) Only z component of vector potential i.e. Az is present. We assume 2d
geometry and everything is constant in z direction thus Az depends only on x
and y space coordinates (because nothing changes in z direction)
We can rewrite (4-13) in its component form (using our assumptions) as below:
By ,rem
 2 A 2 A 
A
  2z  2z    z   J zext 
zˆ
y 
t
x
 x
(4-14)
J zext is the z component of the external current density, By ,rem is the y component of
remanent flux density such that its curl points in z direction. Here xˆ, yˆ , zˆ are unit
vectors for x, y, z coordinates. We have assumed that nothing varies in z direction and
that Brem has only a y component. Therefore when we take curl of Brem there is a
variation along x direction signifying the change of polarity of magnet (from North to
South and vice-versa) and the direction of curl is along z axis.
Further, there are two magnetic fields interacting with each other. One is the field
produced by the permanent magnets and the other is the field produced by the current
sheet (representing effect of a winding). The approach is to calculate corresponding
vector magnetic potential (for current sheet and permanent magnets) and add together
to get the net result. The magnets are considered inactive (but conductive) for
calculating the eddy current losses.
The other quantities can be derived further from the vector magnetic potential. For
each region of the machine the Poisson’s equation (4-13) can be simplified depending
on which excitations are present in a particular region. We define the next steps for the
case only current sheet is present and magnets are “inactive”. However due to current
sheet, there is an eddy current loss in the magnets owing to their conductivity.
Equations for each region of Fig.4.2 can be written as:
Stator and air gap:
2 Az  0
Magnets and rotor back iron:
64
 2 Az  
(4-15)
Az
t
(4-16)
4.5. Boundary Conditions
There are two types of boundary conditions applied for modeling. The arrangement
of the machine geometry for application of analytical model results in the geometry as
shown in fig. 4.2.
4.5.1 Boundary Condition 1
The first boundary condition is implied by Ampere’s Law and states that the
tangential component of the magnetic field intensity on one side of the boundary is
equal to that of the other side with a surface current density added to it. Mathematically,
this translates to:
nˆ  ( H1  H 2 )  K
(4-17)
H1 is the magnetic field intensity in region 1 on one side of boundary
H 2 is the magnetic field intensity in region 2 on other side of boundary
K is the surface current density at the boundary.
n̂ is the unit normal vector to the boundary pointing outwards for each region.
In our coordinate system normal translates to y component and tangential translates to x
component of the quantity.
Equation (4-17) can be written as H x1  H x 2  K and using B   H we can write:
1
1
Bx1 
1
2
Bx 2  K
(4-18)
Using B   A , we can write
1 Az1 1 Az 2

K
1 y 2 y
(4-19)
Here, Az1 is the z component of magnetic vector potential in region 1,
Az 2 is the z component of magnetic vector potential in region 2.
65
4.5.2 Boundary Condition 2
The other boundary condition is implied by conservation of magnetic flux. The
boundary condition states that the normal component of the flux density on one side of
boundary interface is equal to that on the other side. Mathematically,
nˆ  ( B1  B2 )  0
(4-20)
This means that By1  By 2 .Using B   A , we can write for By1 and By 2
Az1 Az 2

x
x
(4-21)
4.6. General Solution of the Partial Differential Equation (PDE)
The stator and air gap have been assumed non-conductive whereby the equation to be
solved is Laplace’s equation. Due to eddy currents induced in the conductive rotor backiron and magnets, the equation to be solved gets modified to Poisson’s equation.
4.6.1 The General Solution for Laplace’s equation
In Cartesian coordinates Laplace’s equation can be written as
 2 A 2 A
  2z  2z
y
 x

0

(4-22)
Assuming the solution to be of the form Az ( x, y)  P( x)Q( y) , where P is a function
of x only and Q is a function of y only. For simplicity, we can say
Az  PQ
(4-23)
Calculating derivatives and putting in (4-22), we get
 2 P
 2Q 
Q

P
0

2
2 

x

y


Dividing both sides by PQ , we can write
66
(4-24)
 1  2 P 1  2Q 


0
2
Q y 2 
 P x
or
(4-25)
1 P
1Q


2
P x
Q y 2
2
2
The constant  is the separation constant as we have obtained P as a function of x
only and Q as a function of y only. Partial derivatives can be treated as actual derivatives
and we get two Ordinary Differential Equations (ODEs) which can be solved as shown
below.
We will solve for P only as the solution for Q would be similar. Before actual solution
we will consider the three cases that arise for different values of.
Case 1: If  = 0
The Laplace’s equation becomes
d 2P
0
dx 2
(4-26)
Thus P = Ax + B (on integrating twice)
Case 2: If > 0
rx
Let P = e then putting the values of P and its double derivative in
1 2 rx
r e 
e rx
(4-27)
 r   
If  and   are both solutions, then their sum is also a solution. This can be
written as:
P  m'e
x
 n ' e
x
(4-28)
Similarly, we can write for Q,
Q  g 'e
 y
 h ' e
 y
 g 'e j
y
 h ' e j
y
(4-29)
67
Here m’, n’, g’, h’ are arbitrary constants.
Putting =k2 where k is a real number, we can re-write (4-28) and (4-29) as:
P  m ' ekx  n ' e kx
(4-30)
Q  g ' e jky  h ' e jky
(4-31)
Using the identities
e  cosh   sinh 
e   cosh   sinh 
e j  cos   j sin 
(4-32)
e  j  cos   j sin 
We can also write the solution in hyperbolic sine and cosine terms as
P  m cosh kx  n sinh kx
(4-33)
Q  g cos ky  h sin ky
(4-34)
Here m, n, g and h are arbitrary constants.
From (4-23) , (4-30) and (4-31) we get the complete solution as:
Az ( x, y)  (me jkx  ne jkx )( geky  he ky )
(4-35)
Case 3: If < 0
In this case, we can write  = -k2. We will get a similar result like (4-35) but P will
contain the real exponent terms whereas Q will contain imaginary exponent terms. The
complete solution can then be written as:
Az ( x, y)  (mekx  ne kx )( ge jky  he jky )
(4-36)
We are using a periodic boundary condition on left and right side of our domain i.e.
periodicity along x axis, (see fig 4.2) we proceed with the solution where P is having
cosine and sine terms. This selection is made owing to known periodicity of sine and
cosine terms.
68
Therefore we choose < 0 and equation (4-36) as the general solution
4.6.2 General Solution of Poisson’s Equation and Treatment of Time
Since the Poisson’s equation we formulated has time dependent terms, we will first
show how periodic time variation is treated in analytical models. Therefore to start with,
Az is a function of space in 2 dimensions (x, y) and time t. Since the field is spatially
dependent on x and y coordinates and in addition on time t, we can write the solution of
Az to consist of a part ‘T’ dependent on time t and a part ‘S’ dependent on space (x,y)
only [4].
Az ( x, y, t )  S ( x, y).T (t )
(4-37)
Substituting this result in (4-16) leads to:

( ST )
t
T
 T ( 2 S )   S
t
2
 S  T


 k2
S
T t
 2 ( ST )  
(4-38)
2
Here, k is constant of separation. It must be independent of variations in both time
and space. Thus from (4-37), we can separate the time dependent part as:
1 T k 2

T t 
T k 2


T 0
t 
k2
T  Ke

(4-39)
t
Here, K is a constant depending upon a particular field.
Similarly, for the space dependent part, from (4-38), we can write
2 S  k 2 S  0
(4-40)
69
This equation is known as Helmholtz’s equation. Helmholtz’s equation needs a more
rigorous solution compared to Laplace’s equation. However, there is a possible
simplification for the case when:
- All field quantities in steady state have amplitudes which are constant and
have the time variations which are at same frequency as the excitation
- The media under observation is linear.
If we call the frequency of excitation as , then (4-16) can be written as:
Az ( x, y, t )  S ( x, y)e jt
(4-41)
Now if we substitute (4-41) into (4-16) we can write:
e jt  2 S  j Se jt
  2 S  j S
(4-42)
Thus the explicit appearance of time can be eliminated and the solution reduces to
solution of (4-42) where if:
 2  
(4-43)
Helmholtz equation (4-40) can be solved by separation of variables. Expanding in
Cartesian coordinates, we can write:
2S 2S
 2  j 2 S  0
2
x
y
(4-44)
Then we can write the solution as two parts, P(x) dependent on x only and Q(y)
dependent on y only.
S  P( x)Q( y )
1  2 P 1  2Q


 j 2
2
2
P x Q y
(4-45)
Since the right hand side of (4-45) is a constant, we can say that each of the terms on left
hand side of (4-45) is a constant as well. Now, assuming
1 2 P
  2
2
P x
70
(4-46)
We know that solution to such an equation is:
P  m sin  x  n cos  x
Further, we can write from
1  2Q
 j 2   2
2
Q y
The solution to this equation is:
Q  g sinh

(4-47)

 2  j 2 y  h cosh


 2  j 2 y
(4-48)
Therefore the general solution can be written as:
Az  (m sin  x  n cos  x).( ge


 2  j 2 y
 he



 2  j 2 y
)
(4-49)
Here    and  is a constant defining variation along x-axis
m,n, g and hare constants to be found by using boundary conditions
σ is the conductivity; μ is the permeability and ω is the electrical frequency
This solution is basically a snapshot of the entire picture because there is no mention of
time. If we want to know the solution at any time instant t then we should multiply the
jt
solution with e .
4.7. Field Due to Stator Currents Only
As mentioned before, there are two fields present in a PMDD machine. In this section
we deal with the field existing due to stator currents only.
4.7.1 Excitation for the field
The machine in question is a concentrated winding machine as shown in fig. 4.1. The
mmf produced by concentrated windings comprises of a number of harmonics and is
far from sinusoidal. However, the problem setup and general solution we have
developed consists of sine and cosine functions or their harmonic form. Therefore, the
stator current density has to be expressed a sum of sine and/or cosine functions i.e. as a
Fourier series. Thus, in order to apply any arbitrary excitation, we have to decompose
the excitation into sine and cosine components. The excitation of the model is deduced
using the following steps:
71
a) First of all, the surface current density waveform is determined as shown in fig.
4.3. The tall rectangular blocks represent the magnitude of current.
b) The amplitudes for various harmonics are deduced from Fourier decomposition
of this waveform for one phase. Then the net effect of 3-phases is taken into
account by summing the individual phase waveforms.
c) Each of the resultant amplitudes is applied separately as excitation of the current
sheet and losses are evaluated.
d) The total losses are then sum of losses computed by individual harmonic
excitation.
Fig.4.3: Flux density waveform and current/m length for a concentrated winding machine
Each harmonic found after Fourier decomposition of phase A alone, is also present
in other phases but displaced and forms a complete set of 3-phase sinusoidal system.
Thus for each phase, the surface current density can be written as a sum of harmonics.
We can write an expression of the form:
72

K A   Kˆ n cos(nkx) cos(t )
n 1
2
2
K B   Kˆ n cos(nkx  ) cos(t  )
3
3
n 1

4
4
K c   Kˆ n cos(nkx  ) cos(t  )
3
3
n 1

(4-50)
Here,
K A , K B , K C are surface current densities of phase A,B and C respectively (A/m)
Kˆ n is amplitude of nth harmonic’s surface current density (A/m)
ω is the electrical frequency(rad/s); t is time (s)
k is a constant for space distribution ; x is a space coordinate
The net effect is summation of the three separate phase contributions. It can be easily
verified that after summation, the current sheet for nth harmonic can be written as:
3
Kn  Kˆ n cos(nkx  t )
2
for n = 1,4,7,…
(4-51)
3
Kn  Kˆ n cos(nkx  t )
2
for n = 2,5,8,…
(4-52)
Kn  0
for n = 3,6,9,…
(4-53)
This represents the actual excitation in the form of a travelling wave which is both
function of space and time. Fig.4.4 shows an example of decomposition of surface
current density. Each harmonic forms an excitation for the analytical model.
73
5
6
Waveform of current sheet
x 10
4
Magnitude
2
0
-2
-4
-6
0
0.1
0.2
0.3
0.4
Distance along x
0.5
0.6
0.7
Fig.4.4: Surface current density (A/m) decomposition for 9-8 combination concentrated winding machine
4.7.2 Effect of motion
The excitation for the magnetic field in the model is a travelling wave of current
density. The rotor of a machine moves at constant speed but each harmonic field
(produced by stator mmf decomposition) except the torque producing harmonic, rotates
at a different speed w.r.t rotor. Furthermore, the direction of rotation is also different
for different space harmonics.
This relative difference in speed of harmonics and rotor is responsible for inducing
eddy current losses in the solid conductive parts of the machine. The effect of motion
results in a modification of the frequency induced by a harmonic. The speed of n th
harmonic can be calculated as:
vn 
1 f
n
th
vn is travelling velocity of n harmonic.
1 is fundamental wavelength.
f is the fundamental electrical frequency.
74
(4-54)
The relative speed of the harmonic w.r.t. the moving part can be calculated as
vh  vn  vr
(4-55)
Here,
vh is the relative speed of harmonic
vr is the speed of rotor
 is a + if the harmonic moves in opposite direction to the rotor and – if the harmonic
moves in same direction as rotor.
The equivalent change of frequency can be calculated from (4-54) by replacing vn with
vh and calculating the frequency as:
nv
f h  h and h  2 f h
1
(4-56)
Consequently, the excitation in (4-51) and (4-52) can now be expressed as
3 ˆ
K n (cos nkx cos ht  sin nkx sin ht )
2
3Kˆ
j t
 jht 
K n  n (cos nkx  j sin nkx).e h  (cos nkx  j sin nkx).e
4 

Kn 
(4-57)
It must be noted that the frequency ω in (4-57) has been modified to h .
Equation (4-57) also indicates that there is a space dependent part and a time
dependent part defining K. Since we know that variation in time is sinusoidal, we can
solve for space part alone. Further, (4-57) shows that there are two excitations possible,
one with +ω i.e. (cos kx  j sin kx) and another with negative -ω i.e. (cos kx  j sin kx) .
Therefore if we apply either of these excitations actual losses will then be twice the
losses calculated due to such an excitation.
4.7.3 Some Important Observations – Current Sheet Excitation
The reference figure for general solution is fig. 4.2 which has been reproduced here for
ease of reference
75
B1
Region
Region
1;1;
Stator
Iron with
yoke Permeability
with Permeability
μ1 μ1
B4
B4
Stator
B2
Current Sheet with Sinusoidal surface Current = K sin (kx) at (Boundary)
b
B5
Region 2; Air with Permeability μ2
Y
B6
Region 3; Magnet region Permeability μ3 and conductivity 3
B7
Z
X
Region 4; Rotor back Iron with Permeability μ4 and conductivity 4
B3
l
Fig.4.5: Simplified geometry for analytical model
a) The solution for each harmonic in the regions without conductivity and eddy
current losses is given by (4-36) which is reproduced here.
Az ( x, y)  (mekx  ne kx )( ge jky  he jky )
(4-36)
b) The general solution for regions with conductivity and eddy current losses has
been derived in (4-49) which is reproduced here.
Az  (m sin  x  n cos  x).( ge


 2  j 2 y
 he



 2  j 2 y
)
(4-49)
c) Since we can choose the origin of excitation in fig. 4.4 for excitation, we can get
either an even or an odd function whereby we know that variation along x-axis
is either a sine or a cosine. Therefore, knowing the excitation, we get rid of
either m or n of the general solution.
d) The excitation for the model is taken into account in the boundary condition 1,
applied on boundary B5.
Afterwards we can apply boundary conditions to all other boundaries and solve for
find particular solution in this configuration. The details of solution are presented in the
appendix in form of a sample problem.
76
Flux Lines
Flux
0.4
0.4
Stator μr = 4000
0.35
0.3
Air μr = 1
0.25
Distance along y-axis
Distance along y-axis
0.3
0.2
Magnet μr = 1
0.15
0.1
0.25
Air μr = 1
0.2
Magnet μr = 1
0.15
0.1
Rotor μr = 200, σ = 5e6
0.05
0
Stator μr = 4000
0.35
0
0.1
0.2
0.3
Rotor μr = 200, σ = 5
0.05
0.4
0.5
0.6
Distance along x-axis
0.7
0.8
0.9
0
1
0
0.1
a)
Flux Lines
Flux Lines
0.4
Stator μr = 4000
0.35
Distance along y-axis
0.3
0.25
Air μr = 1
0.2
Magnet μr = 1
0.15
0.1
σ = 5e6
0.4
0.5
0.6
Distance along x-axis
Rotor μr = 200, σ = 5e6
0.05
0.7
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Distance along x-axis
0.7
0.8
0.9
1
b)
Fig.4.6: Effect of motion (arbitrary model) a) Flux lines without motion b) Flux lines with motion
4.8. Field Due to Magnets Only
In this section, the field due to presence of PM alone has been presented. The
analytical calculation is based on the assumptions resulting in the geometry of fig. 4.7
shown below.
77
0.2
0.3
0.4
0
Distance a
Y
B1
Region 1; Iron with Permeability μ1
b
Z
B4
X
Region 2; Air with Permeability μ2
B5
B6
B2
Region 3; Magnet region Permeability μ3 and Remanant Flux Density of 1.2 T
B7
Region 4; Iron with Permeability μ4
B3
l
Fig.4.7: Geometry for field due to magnets only
Region 1 is iron having permeability μ1Region 2 is air, having permeability μ2
Region 3 is iron, having permeability μ3 and conductivity 3
Region 4 is iron, having permeability μ4 and conductivity 4
Subscripts 1, 2, 3 and 4 define the region where the quantity is present.
4.8.1 Excitation for the field of PMs
The Flux density due to the magnets (in region 3) is defined as a square wave. The
expression for this wave is represented by a summation of Fourier series expansion. The
summation of harmonics leads to a flux density waveform as shown in fig. 4.8.
Fig.4.8: Excitation for field due to magnets only

4 Br
sin(nkx)
n 1,3,5... n

Here, k 
78
2
l
(4-58)
4.8.2 Some Important Observations – Field of PM
a) Since the waveform is an odd function, a sum of sine components only exists.
Therefore, we know that variation along x-axis is sinusoidal.
b) There is no excitation source in regions 1,2 and 4. Therefore, we have to solve
Laplace equation in these regions.
c) In region 3 we have a flux density of 1.2 T as shown in fig. 4.8. This is given as
a sum of harmonics. In this region we solve the equation
2 A   Brem
d) For regions 1,2 and 4 the general solution will be of the form:


n 1,3,5,...
sin(nkx)( g m e jnky  hm e jnky )
(4-59)
(4-60)
k  2
m = 1,2,3 or 4 denote the region of interest
e) For region 3, the general solution is obtained by adding a particular integral of
to the general solution of Laplace equation as explained in [4]. This particular
integral can be easily found out to be

(2n )4 Br
sin(nkx)
n3k 2
(4-61)
Thus the general solution of (4-29) in region 3 is given by:
(2n )4 Br 

sin(nkx)  g m e jnky  hm e jnky 
n3k 2 

n 1,3,5,...
k  2


(4-62)
m = 1,2,3 or 4 denote the region of interest
After that we can apply boundary conditions to find particular solution in this
configuration. The flux lines are shown in fig. 4.9.
79
Flux lines
0.4
0.35
0.1
Stator μr = 4000
Distance along y-axis
0.3
0.05
Air μr = 1
0.25
0.2
0
Magnet μr = 1, Br = 1.2 T
0.15
-0.05
0.1
Rotor μr = 200
0.05
0
0
0.1
0.2
-0.1
0.3
0.4
0.5
0.6
Distance along x-axis
0.7
0.8
0.9
1
Fig.4.9: Flux lines due to permanent magnets only
4.9. The Combined Magnetic Field
The combined magnetic field is found by superposition i.e. the summation of the field
produced by the stator currents and the field produced by the permanent magnets. This
summation is valid because the materials have been assumed to be linear. As shown in
fig. 4.10, the effect of high frequency stator currents can be seen as some skin effect in
the back-iron region belonging to the rotor of the machine.
Flux lines
0.4
0.15
0.35
Stator μr = 4000
0.1
Distance along y-axis
0.3
0.05
Air μr = 1
0.25
0.2
0
Magnet μr = 1, Br = 1.2 T
0.15
-0.05
0.1
-0.1
Rotor μr = 200
0.05
-0.15
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Distance along x-axis
0.7
0.8
Fig.4.10: Flux lines due to both current sheet and magnets
80
0.9
1
4.10. Derived Quantities from Az
There are many quantities which can be easily derived from magnetic vector
potential. From this research’s point of view, the most important derived quantities are:
1. Magnetic flux density B: Serves as a visual aid to analyze the results.
2. Induced current density J: The induced current density is used to compute the
eddy current losses in solid conductive parts of the machine.
4.10.1 Magnetic Flux Density from Az
The magnetic flux density can be derived by taking curl of Az. Basically we use (4-9)
and since A has only a z-component, B can have only x and y components.
Therefore, using rectangular coordinates, we can write (4-9) as:
B
Az
A
xˆ  z yˆ
y
x
(4-63)
The flux densities of the PM and stator field are shown in fig. 4.11 and 4.12 respectively.
Stator
PM
Back Iron
Fig.4.11: Flux density due to magnets for the case of PM excitation, Br = 1.2 T
81
Stator
PM
Back Iron
Fig.4.12: Flux density due to current sheet for fundamental harmonic
4.10.2 Induced Current Density from Az
As shown in (4-12) (reproduced here again), the induced current density can be
calculated from the calculation of time derivative of A
J E
E
A
A
or J  
t
t
(4-12)
We have already expressed general solution of Az ( x, y) in (4-22) and (4-23). Further,
because the mmf of concentrated windings is not sinusoidal, we have Fourier series
decomposition of current sheet wave as excitation. This implies that each excitation will
be a pure sine or cosine. The treatment of time variation is taken into account by
82
utilizing the periodicity of sine and cosine functions (see section 4.6.2 of this chapter) by
making Az ( x, y) as a sinusoidal time varying function expressed as:
Az ( x, y, t )  S ( x, y)e jt
(4-64)
Here S ( x, y) represents the space dependent part of the solution which is solved using
(4-45). Therefore, (4-12) can be written as:
A
A ( x, y , t )
  z
t
t

J z ( x, y )    S ( x, y )e jt 
t
J z ( x, y )   j  S ( x, y )e jt 
J z ( x, y )  
(4-65)
The induced current density is shown in fig. 4.13.
Stator
PM
Back Iron
Fig.4.13: Induced current density due current sheet excitation of one harmonic
83
4.10.3 Eddy Current Losses from Induced Current Density
The eddy current loss is assumed to be limited by material resistance whereby loss
can be easily calculated from induced current density using:
Peddy
J z2 ( x, y)
 
dA

s
(4-66)
Here,
Peddy is the eddy current losses in W; Jzis the induced current density in A/m2
A is area in m2; σ is conductivity in Siemens
Equation (4-66) represents eddy current loss due to one harmonic. The total loss can be
calculated by adding the loss due to individual harmonic component. The loss
distribution can be visualized as shown in fig. 4.14.
Stator
PM
Back Iron
Fig.4.14: Induced eddy current losses due to current sheet excitation of one harmonic
84
4.11. Application of Analytical Model
The analytical model is used for determining eddy current loss trends in different
slot-pole combinations of concentrated winding machines. It is clear that the
assumptions used in analytical model bring some inaccuracies in the eddy current loss
estimation. However, the speed of analytical modeling and independence from any
special software requirement makes it very useful for short-listing and qualitatively
comparing various slot-pole combinations. Modeling variables and constants are listed
in table 4-1.
TABLE 4-1: MODELING CONSTANTS AND VARIABLES
Quantities kept constant
Quantities allowed to vary
a)
b)
c)
Mechanical speed of rotor
and air gap.
Amplitude of the stator
current sheet
Materials and properties
a)
Pole-pitch as per
combination
b) Winding factor as per slotpole combination.
c) Harmonic excitations as per
Fourier decomposition
Using analytical modeling, various slot-pole combinations are modeled and
dependencies on slots per pole per phase are established as shown in fig. 4.15. More
results of comparisons and trends are presented in chapter 7.
Fig.4.15: Variation of eddy current losses due to current sheet excitation for various slot-pole combinations
85
4.12. Summary
A generic analytical model has been developed starting from the basic Maxwell’s
equations. Certain assumptions have been made based on properties of large generators.
A useful contribution of such a generic model is the ability to qualitatively compare
various topologies having different slot-pole combinations. The analytical model is used
to qualitatively compare some useful slot-pole combinations and derive trends in
relation to eddy current losses.
Bibliography
[1]
[2]
[3]
[4]
[5]
[6]
[7]
86
S.R.Holm, “Modelling and optimization of a permanent magnet machine in a flywheel,”
Ph.D. dissertation, pp. 62-66, Dept. Electrical Power Engineering, TU Delft, The
Netherlands, 2003.
P.D.Agarwal, "Eddy-Current Losses in Solid and Laminated Iron", AIEE Transactions, vol.
78, p.169 , 1959.
H.Polinder, “On the losses in a high-speed permanent-magnet generator with rectifier,”
Ph.D. dissertation, pp. 12-16, Dept. Electrical Power Engineering, TU Delft, The
Netherlands, 1998.
K.J.Binns,P.J.Lawrenson and C.W. Trowbridge, “The analytical and numerical solution of
electric and magnetic fields”, 1995 edition, John Wiley & Sons publisher, pp. 3-6 and pp. 9597.
Pfister, P.-D.; Perriard, Y.; , "Slotless Permanent-Magnet Machines: General Analytical
Magnetic Field Calculation," Magnetics, IEEE Transactions on , vol.47, no.6, pp.1739-1752,
June 2011.
Zhu, Z.Q.; Ng, K.; Schofield, N.; Howe, D.; , "Improved analytical modelling of rotor eddy
current loss in brushless machines equipped with surface-mounted permanent magnets,"
Electric Power Applications, IEE Proceedings - , vol.151, no.6, pp. 641- 650, 7 Nov. 2004.
Jassal, A.; Polinder, H.; Ferreira, J.A.; , "Literature survey of eddy-current loss analysis in
rotating electrical machines," Electric Power Applications, IET , vol.6, no.9, pp.743-752,
November 2012.
5. Finite Element (FE)
Modeling
This chapter introduces the FE software used and the modeling procedure for large direct drive
electrical machines. The chapter gives a brief introduction to the theory behind FE software and general
procedure for problem setup and solution. Intermediate checks have been specified where necessary. The
chapter ends with some results obtained from FE modeling
5.1. Introduction
The Finite Element (FE) method originated from the need for solving complex
problems related to elasticity and structural analysis in civil and aeronautical engineering.
Its development can be traced back to the work by Richard Courant [1]. Another
pioneer in the field was Hrennikoff [2].Finite Element (FE) methods are being used
extensively these days to simulate complex geometries and scenarios which are difficult
to solve analytically. Rigorous mathematics was involved in solving electromagnetic
problems when FE was not available. Earlier research in electromagnetics shows
frequent use of conformal transformations, method of images etc. These days, the
rigorous mathematical techniques are seldom taught in engineering disciplines.
Therefore, the students from engineering back-ground find it difficult to grasp the
analytical method. The analytical methods are limited in capturing the following effects:
a) Complexity of geometry
b) Material non-linearity and anisotropy
c) Interlinked transient effects like thermo-electric, motional effects etc.
The mathematical complexity of analytical models and the inaccuracy due to
simplifying assumptions has made FE methods much popular. Improved graphics in FE
software have made better visualization and plotting possible. Consequently, numerical
analysis division of mathematics has gained tremendous momentum and a number of
FE solvers are available in the market. Continuous improvements in computing speed
and improved processing power have aided in overall development of FE methods.
87
Research is still going on for improving the quality of algorithms and making the
software more user-friendly. Some of the most popular FE softwares used now-a-days
are:
- ANSYS
- Vector Field’s OPERA
- FLUX from CEDRAT
- FEMM – Foster Miller
- Infolytica MAGNET,
- COMSOL
All these FE software are efficient, useful and well known and it is not the purpose
of this chapter to comment on features of the software. So we shift focus on the
software used for this research. COMSOL Multiphysics 3.5a was used as the FE
software environment. COMSOL provides an easy coupling with MATLAB. In fact
COMSOL was sold as FEMLAB from MATLAB and later on developed independently
as COMSOL software. Many versions of this software exist and the latest at the time of
this research was COMSOL 3.5a (later versions are now available).
5.2. Finite Element Method
In essence, the Finite Element Method (FEM) is a numerical technique to solve
Partial Differential Equations (PDEs) and Integral Equations (IEs). Since PDEs are easy
to handle PDE’s solutions are more common. In FEM, the solution of PDEs is based
on either complete elimination of equations or breaking them into Ordinary Differential
Equations (ODEs). These ordinary differential equations are then solved numerically by
discretizing the space into smaller sub-domains and solving locally. FE formulation of a
problem consists of the following basic steps:
1. Weak Formulation of the original boundary value problem.
2. Discretization of the selected domain space into cells or elements (meshing).
3. Choice of the basic function. This function is a piecewise linear function to get sparse
matrix for the vector space.
4. Formulation of the system to resolve the field problem.
5. Solution of the problem. The solution is obtained by solving the resulting system of
equations.
We will not discuss the mathematics behind these methods here as it has been widely
published. Interested readers can check the following references for more details [4]-[8].
5.3. COMSOL 3.5a General Environment
The software COMSOL Multiphysics has a number of application modes and
depending upon the application, a suitable mode can be chosen. Each application mode
88
is further sub-divided into application modules. For this research, the following
application modes and modules were used:
5.3.1 Application Modes and PDE
a) COMSOL multiphysics mode and AC Power Electromagnetics module: This
module is essentially a time harmonic solver. In present research this module was
used to verify the solution of analytical model. The time-harmonic mode uses the
following equation as the basic PDE.
 V 
e
 (  1 1 A )   v  ( A )   
  Jz
0 r
z
z
 L 
(5-1)
Here,
Az is the z- component of vector magnetic potential
is the electrical frequency
σ is the electrical conductivity
Vis the scalar electric potential
νis the velocity of the subdomain
Jzeis the external current density
 0 is permeability of free space
 r is relative permeability of material
b) AC/DC mode with rotating machinery module was used: The AC/DC rotating
machinery module is a transient solver which can take into account motion as well
as material properties in real time. This inbuilt module has predefined conditions
for modeling physical rotation and applicable boundary conditions. The transient
mode uses the following equation as the basic PDE.
 A
z
 t



   (  1 1 A )  J e
0 r
z
z


(5-2)
The symbols have the same meaning as in (5-1).
5.3.2 FE Model Setup
The purpose of using this software is to model a concentrated winding machine with
permanent magnets which has design attributes similar to a generator used in a wind
89
turbine. In the selected topology i.e. concentrated windings, there is a problem of eddy
current losses in solid conductive parts of the machine such as magnets and back iron of
rotor. We have already seen in analytical modeling (chapter 4) that analytical model can’t
capture all the effects. On the other hand, FE software can model slotting, saturation
and motion all at once. However, in order to capture eddy current losses, we need to
setup the model in a useful way. The basic requirements are:
a) Symmetry - Since the machine in question is a large machine so modeling whole
machine takes too long to solve. Thus we should use symmetry to model only a part
of the machine and extrapolate the results thus obtained.
b) Mesh size - The phenomenon of eddy current losses has a certain skin depth and this
is in the order of fraction of millimeters (mm). This means that we need a rather fine
mesh in the region where we expect eddy currents to occur.
c) Geometry accuracy - The accuracy with which we build our geometry plays a very
important part in convergence of a solution. This is critical since we are using only a
part of machine to model and periodic boundary conditions. If the geometry is not
accurate enough, the solver might fail to start during transient simulations.
d) Time step and relative tolerances - The smaller the time step, the better the results but
also more the time taken for solution. Thus in order to reach an acceptable solution,
the step size should not be too large to get inaccurate results and it should not be
too small that it takes forever to solve the problem. There are different tolerances
on different parameters and one can get a better estimate of tolerances by hit and
trial.
e) Initial values - In order to define a good solution in transient case, a relatively accurate
static model has to be solved first. The transient model gets its initial values from
this solution.
f) Boundary conditions - The boundary conditions are of course the basis on which the
solver solves the PDE. Modeling of just a section of the machine requires the use of
periodic boundary conditions on east and west and symmetry/anti-symmetry can be
chosen depending on geometry and physics. Similarly on the boundary between
stator and rotor (which is moving), the boundary condition used is either symmetry
or anti-symmetry. This condition takes the number of sections of machine into
account.
g) Physics - This simply means the materials that we are going to use in the model. A lot
of common materials are there in the library but if there are some specific materials,
these can be added into the library.
h) Motion - Motion to simulate transient conditions is done using frames of reference.
In COMSOL, there is two frames of reference viz. reference or stationary frame and
moving or ALE (Arbitrary Langrangian Eulerian) frame []. Sub-domains which
move during simulation are assigned movement in ALE frame and they move with
respect to the fixed parts. The two reference frames are coupled using coupling
90
variables (called lm1, lm2 etc.). These coupling variables are linked to periodic
conditions at the interface boundaries. More details will be explained later.
5.3.3 General Procedure for Problem Setup
Since COMSOL is closely coupled with MATLAB, almost all of the steps can be
done in MATLAB interface as well. Drawing of geometry is for example much easily
done with MATLAB as compared to the Graphic User Interface (GUI) in COMSOL.
Post-processing is also very convenient with MATLAB especially for quantities which
have to be derived from primary post-processing data. These are the basic advantages
with COMSOL. It leads to a much better insight into the working of FE program and
hence the results can be intuitively verified.
The general procedure to set-up a problem is shown in the flow chart of fig. 5.1.
Choose
Dimensions
Select Application
Mode
Draw Geometry
Material Properties
Create Mesh
Solve
Visualization and post-processing
.
Fig.5.1: General procedure to setup a problem
5.4. Machine Model
There can be a number of useful slot-pole combinations for designing a machine
with concentrated windings. The eddy current loss in solid conductive parts is one of
91
the major criteria to determine utility of a combination. Winding factor is the other
important criterion [9]-[11].
Therefore we aim to compare a number of possible combinations and devise
guidelines for selection. In order to keep the comparison fair, the stator dimensions,
current loading, air gap velocity, thickness of magnets and rotor back-iron as well as
materials has been kept same for all combinations. The dimensions that vary are length
of a section containing a given number of slots and poles along with magnet span.
Figure 5.2 shows the dimensions used for modeling.
Fig.5.2: Main dimensions of a concentrated winding machine
Here the dimensions have the following meaning and values:
bt = Tooth width
bs= Slot width
τs = Slot pitch
hs = Slot height
τp = Pole pitch
lm = Magnet thickness
g = Mechanical air gap
The air gap velocity of the rotor surface has been fixed for comparison between
different slot-pole combinations. Some useful combinations have been selected based
on a good winding factor and these will be compared for eddy current losses in solid
magnets and rotor back-iron or rotor yoke. Further, in chapter 7, a whole range of
useful slot-pole combinations are analyzed and trends are formulated using FE software.
5.4.1 Geometry Drawing and Symmetry
Although it seems obvious but when drawing geometry it is better to work with
relations and formulas rather than numerical values. The use of symmetry greatly
92
reduces the computation time. Symmetry can be defined in boundary conditions as
shown in fig. 5.3. It can be periodic, anti-periodic or some potential function.
Fig.5.3: Symmetry in a machine over one quarter
Accuracy in drawing should be such that when the machine is fully completed there
should be no gaps between sections anywhere.
5.4.2 Meshing
The mesh size is very important because of very small skin depth of the eddy current
losses. The mesh should be fine and regular enough to simulate the skin depth where
required as shown in fig. 5.4
Fig.5.4: Meshing over a machine model 3 slots per 4 poles combination
There can be many possible shapes for meshing a sub-domain but triangular mesh is
most common because it has the smallest possible area for any closed surface.
COMSOL gives control over the size of mesh triangle in any sub-domain or along a
93
boundary and the rate at which the size grows within a sub-domain. There are predefined mesh options from extremely coarse mesh to extremely fine mesh as well.
5.4.3 Boundary Conditions
The boundary conditions are used for solution of PDE and connect various subdomains together in solution. These boundary conditions are described using an
example of 3-4 slot-pole combination (see fig. 5.5). Main conditions available to be used
are:
a) Magnetic Insulation: A  0 where A is the vector magnetic potential.
b) Continuity: n  ( H1  H 2 )  0 where H1 and H 2 represent the magnetic field
on either side of boundary.
c) Periodic condition: Asource   Adestination where + means symmetry and –
means anti-symmetry and A is the vector magnetic potential.
d) Periodic point condition: This is a special condition used on the points which
connect the extent of the moving boundary. Coupling variables are also defined
at these points.
Magnetic insulation
Continuity
Periodic condition
Sector symmetry
Periodic point condition
Fig.5.5: Boundary conditions shown here for a 3- 4 combination
94
5.4.4 Physics and Material Settings
The physics of the model can be set either by using inbuilt library or by using userdefined materials. The material and physics settings are shown in table 5-1. This can be
done using Matlab or GUI but from experience, it is easier done by GUI. The material
properties for each of sub-domain are shown in fig. 5.6. The material properties can be
modified and a new material can be added to the existing library for future use as well.
Stator Iron
Winding
Tooth
Air
N
N
S
S
Rotor Iron
Fig.5.6: Material setting shown here for a 3- 4 combination
TABLE 5-1: PHYSICS SETTINGS
Sub-domain
Relative
permeability
Electrical
Conductivity
(S/m)
B-H Relation
External
Current
Density
(A/m2)
Stator Iron
r  f ( B)
0
B  0 r H
-
Tooth
r  f ( B)
0
B  0 r H
-
Winding
1
0
B  0 r H
3.5x106
B  0 r H  Br
Magnets*
1
1x106
Br   Br
X2
-
X 2 Y2
Rotor Iron
r  f ( B)
5x106
B  0 r H
-
Air
1
0
B  0 r H
-
* For magnets, X and Y are spatial coordinates to set radial flux density distribution.
95
5.4.5 Motion
COMSOL contains a module called AC/DC module which contains a sub-module
called rotating machinery which is used for transient modeling. The effect of motion is
taken into account by creating super-imposed reference frames within the rotating
machinery module. These reference frames are:
a) Stationary or Reference frame – This frame shows the original position of the
parts when there is no motion present. This is basically a time harmonic solver.
b) Moving or ALE reference frame – In this frame of reference, the relative
motion between parts is assigned. The ALE algorithm is used to couple the
solution on moving part to the solution on static part whereby computation
time is lowered [12]. Since the mathematics of ALE is very specific, the details
are not presented here and this is viewed just as a tool for analysis. One
disadvantage is that the rotating machinery module can’t model linear motion.
The rotation is given by the following reference frame conversion:
 X '   cos(2n t )  sin(2n t )  X 
 
 
 Y '   sin(2n t ) cos(2n t )  Y 
Here
n is rotational speed in rpm; t is time
X ' and Y ' are transformed coordinates at any time t
X and Y are the fixed coordinates
Fig.5.7: Sub-domains which are assigned motion for a 3- 4 combination
96
5.4.6 Visualization and Post-processing
The general post processing window gives a choice of plots which one needs to
visualize on one plot for example contour plot of vector magnetic potential and surface
plot of magnetic flux density can be plotted together in one figure. Further, the
quantities to be plotted and the other settings of plots like color, number of lines, scaling
etc. can be set from the specific visualization settings of the plot.
There are also a lot of options for defining variables and expressions using basic
quantities. The post processor is coupled with Matlab and therefore the plotting options
are further enhanced. Some capabilities used in this thesis are shown in table 5-2. A
sample plots of useful values (w.r.t. this thesis) are shown in fig. 5.8 and fig. 5.9.
Option
Cross-section
Plot
Sub-domain
integration
Global
variables plot
TABLE 5-2: SOME USEFUL POST-PROCESSING OPTIONS
Function
Plot of quantity along a user-defined line.
Plot of pre-defined or user-defined quantity
integrated over a sub-domain and its
evolution in time.
Plot of user-defined variable, coupled to
sub-domain or boundary.
Example
Flux density plot at any ycoordinate for all xcoordinates.
Eddy current losses, induced
current density etc.
Torque acting on the subdomain.
Fig.5.8: A contour plot of magnetic potential and surface plot of magnetic flux density plotted together
97
a)
b)
c)
Fig.5.9: a) Flux density (y-component), derived from surface plot at one time instant using cross-section plot b) Eddy
current losses magnets by using sub-domain integration of resistive losses over magnets real time c) Cogging torque
calculated by user defined global variable and its evolution in time.
98
5.5. Results of FE Modeling
5.5.1 Validation of Analytical Model
The analytical model and solution of PDE was validated with a simplified model
made in COMSOL 3.5(a) FE software. This model was prepared with the same
assumptions as that of analytical model to verify analytical solution itself. The geometry
consists of four layers as in the analytical model. On the boundary between stator and
air gap, there is a surface current density, resulting in a travelling wave of magnetic flux
density in the air gap. The current sheet is responsible for flux density and losses. It is
placed at the stator surface as in the analytical model. The four layers have the same
material properties as in analytical model. Actual motion is present in the FE model
whereas for analytical model, a modified frequency with complex harmonic excitation
was used to simulate motion. The material settings for four layers are:
1. A layer with perfect iron. New material was defined in material library and material
properties were chosen such that the losses in stator part can be neglected. The
properties were defined as electrical conductivity  = 0; Relative permeability μr =
4000.
2. A layer of air with  = 0 S; Relative permeability μr = 1.
3. A layer of magnets with  = 1x106 S; Relative permeability μr = 1.
4. A layer of solid back iron. A new material was defined having the following
properties  = 5x106 S; μr = 200.
The results are shown in table 5-3 and fig. 5.10 and 5.11. The losses for simplified
models in table 5-3 do not match exactly because of numerical integration used for loss
calculation in FE program.
Stator
Effective
Air-Gap
Magnets
Back Iron
Fig.5.10: Flux lines for simplified FE model used for validation
Another reason for deviation could be the geometry curvature required to inculcate
motion in the FE program. As a check, the induced current density (from which losses
are calculated) for both analytical and simplified FE model was compared and found to
match exactly. After initial validation of analytical model, the model was compared for
eddy current loss calculation with transient FE simulations. Further, both the analytical
99
and FE models were compared with experiments performed on a 9 kW PM machine.
These comparisons are presented in chapter 6.
TABLE 5-3: LOSS COMPARISON FOR 9-8 COMBINATION MACHINE
Harmonic 1
2
5
7
8
Losses from Analytical model (W)
Magnets
26
74
311 0.34 1.90
Back Iron 15
28
68 0.09 0.2
Losses from simplified FE model (W)
Magnets
24
68
301 0.14 1.40
Back Iron 18
20
74 0.06 0.6
a)
b)
Fig.5.11: Induced current density for a harmonic a) Analytical b) FE
100
5.5.2 Eddy Current Loss Calculation Using FE Models
The modeling method was applied on various combinations of slots and poles with
many variations. There can be many useful and interesting output quantities but in
context of this research, we limited to following quantities:
a) Surface plot of magnetic flux density to see if flux density is within limits.
b) Surface plot of induced current density to visualize where losses are present
c) Eddy current losses in magnets, back-iron and total using sub-domain
integration
Sample output of these plots is shown in fig. 5.12-5.14. All these plots will not be shown
for all the machines because plots are similar.
Flux Density
Fig.5.12: Magnetic flux density shown for a 9-8 slot-pole combination with active magnets
101
Induced Current Density
Fig.5.13: Induced current density for 9-8 combination
Total eddy current losses
Eddy current losses: magnets
Eddy current Losses back -iron
Fig.5.14: Eddy current losses in various parts of machine
102
5.5.2.1 Variations for models
Extensive FE modeling was done to study effects of variations in model on the
eddy current losses. These variations play an important part in comparison with
analytical model. The most important variations are:
a) Effect of slot opening (for ¾ slot open and ½ slot open).
b) Effect of magnetic saturation.
c) Effect of presence and absence of magnetic field of magnets.
All models are compared for these variations and then some trends have been
brought out for comparison between various models.
5.5.2.2 Results for Eddy Current Losses from FE Models
Some useful combinations are analyzed for eddy current losses using the
methodology explained in this chapter. The results for the loss calculations are shown in
fig. 5.15 and 5.17.
The FE modeling is used for comparison and validation of the analytical model. The
analytical model gives a fair match in trend of losses when compared to FE simulations.
Total losses compare well in case where a higher order harmonic is responsible for
losses as in 9-8, 9-10 and 12-10 combinations. The eddy current losses in the analytical
model are up to 1.4 times higher than values in FE models in case of 3-4 combination.
This is because losses in 3-4 combinations are produced by first harmonic which
penetrates the whole conductive region. In general, the analytical model over-estimates
total losses in the case where a lower order harmonic produces losses. This is because
the total conductive area is higher in the analytical model and the effective air-gap
calculated using carter’s correction factor is estimation only.
Since magnets are represented as one region, losses in magnets are much higher than
losses in back-iron as some shielding is provided by the conductive magnets.
The results for the non-linear iron case are very similar to those in linear iron case.
This is due to low flux density ~ 1.2 T setup by stator currents in absence of PM
magnetization whereby the skin depth increases and hence little effect of saturation
comes into play.
The machines are further compared for effect of slotting by modeling semi-closed
slots. The excitation conditions were same as in previous case. The rectangular slots are
compared with ½ slot closed and ¾ slot closed. Use of semi-closed slots immensely
reduces the losses as shown in fig. 5.16.
The computation time for the FE program in case of linear iron is in the range of
120-180 seconds and for non-linear case is 200-360 seconds. The computation time for
the analytical model is 0.06 seconds. For the case when PMs are active, the computation
time of FE model is about 600 seconds.
103
104
Fig.5.15: Eddy current losses in different slot-pole combinations (scale on each figure is different)
Fig.5.16: Eddy current losses with slot opening (including PM excitation as well)
Fig.5.17: Eddy current losses with slots per pole (Only stator current excitation to compare with analytical models)
105
It is noteworthy to compare fig. 5.17 with fig. 4.15 at this point. It is shown by
plotting both graphs in same figure. The comparison brings out a limitation in analytical
modeling where a lower order harmonic is responsible for the eddy current losses. It can
be seen that in 3-4 slot-pole combination, the analytical model over-estimates the losses
while FE model predicts lower losses. In this particular combination, 2nd harmonic
produces torque whereas 1st harmonic is mainly responsible for losses. The low
frequency of 1st harmonic allows it to penetrate deeper in the conducting surfaces. Since
the analytical model assumes magnets as a continuous layer (and not separate magnets),
the area used for integrating induced current density is larger. This leads to higher loss
estimation. In other cases also, the analytical model predicts higher losses however the
deviation is smaller. It can be seen that the results (atleast trends) are comparable.
Fig.5.18: Eddy current loss calculation with slots per pole per phase: A comparison between analytical and FE methods
for the case of only stator current excitation
106
5.6. Summary
Finite Element modeling methodology has been explained and some results thus
obtained have been presented. The FE method is more accurate than the analytical
method. However, similar trends from analytical and FE models could be derived with
some exceptions. The analytical model overestimates losses in case where a lower order
harmonic is responsible for losses (explained in description of fig. 5.18). The effect of
slotting in presence of the PM field and motion produces appreciable losses which can’t
be ignored. On the other hand these losses are expected to be similar for different slotpole combinations as long as the stator dimensions are similar.
Bibliography
[1] Pelosi, G.; , "The finite-element method, Part I: R. L. Courant [Historical Corner]," Antennas
and Propagation Magazine, IEEE , vol.49, no.2, pp.180-182, April 2007.
[2] Hrennikoff, H., “Solutions of Problems in Elasticity by the Framework Method”, Journal of
Applied Mechanics, A 169-175, 1941.
[3] Courant, R., “Variational Methods for the Solution of Problems of Equilibrium and
Vibration", Bulletin of American Mathematics Society, Vol. 49, 1-23, 1943.
[4] Oliveira, E., "Theoretical Foundation of the Finite Element Method", International Journal
of Solids and Structures", Vol. 4, pp. 926-952. 1969.
[5] A.B.J Reece, T.W. Preston, “Finite Element Methods in Electrical Power Engineering”,
Oxford University Press, 2000.
[6] Nicola Bianchi, “Electrical Machine Analysis Using Finite Elements”, CRC Press, 2005.
[7] A.Bondeson, T.Rylander, P. Ingelstrom, Computational Electromagnetic, Springer, 2005.
[8] F. Vermolen, D.Lahaye, “Introduction to Finite Elements”, Lecture Notes, TU Delft, 2008.
[9] J. Cros, P. Viarouge, “Synthesis of high performance pm motors with concentrated
windings”, IEEE Transactions on Energy Conversion, vol. 17, pp. 248–253 (2002).
[10] F. Magnussen, C. Sadarangani, “Winding factors and Joule losses of permanent magnet
machines with concentrated windings,” in Proc. of the 2003 IEEE International Electric
Machines and Drives Conference, 2003, pp. 333 – 339, vol.1.
[11] Bianchi, N.; Bolognani, S.; Fornasiero, E.; , "An Overview of Rotor Losses Determination in
Three-Phase Fractional-Slot PM Machines," Industry Applications, IEEE Transactions on ,
vol.46, no.6, pp.2338-2345, Nov.-Dec. 2010.
[12] J. Donea et.al. , “Encyclopedia of computational mechanics”, chapter 14, vol. 1, John-Wiley
publishers 2011.
107
108
6. Experimental
Analysis
This chapter aims at validating the FE and analytical modeling. The concept and methodology
are explained first. Thereafter, the framework for carrying out various types of experiments is presented.
The equipment and their specifications are briefly explained. This is followed by detailed explanation of
procedure of the experiments. The deviations in results between various methods are documented and the
reasons for deviation have been brought out.
6.1. Introduction
Experimental analysis is the most trusted tool when it comes to practicality of eddy
current loss analysis. Analytical modeling involves a lot of assumptions whereby the
results are inaccurate. FE modeling can take into account many non-linearities but
generally 2d analysis is done and there are material anisotropies, edge effects, physical
non-uniformities in dimensions etc. which might lead to deviation between experimental
and FE results. Further, even the experiments are not perfect owing to measuring
inaccuracies. However, the experiments bring us closest to the real physical scenario.
The aim of the experiments performed is to validate various eddy current loss models.
6.2. Experimental Setup
The experimental setup aims at validating both analytical and FE models developed
in the course of this research. Although FE modeling can take into account more 2d
effects than the analytical models (slotting, material non-linearity etc.) but it is difficult to
include material anisotropies, edge effects, physical non-uniformities etc. These 3d
effects are difficult to incorporate even in 3d FE analysis because the material properties
differ from specimen to specimen. Therefore experiments are needed to validate the
models and bring out deviations. For the present research, the experiments are
conducted in two phases:
I. Static tests – Tests without actual motion
II. Rotary tests – Tests where real machine motion is utilized.
109
The theory and rationale behind tests is explained further in the following sections.
6.3. Static Tests
Static tests imply that there is no moving part involved in the experimental setup. The
test is conducted for three different metallic rings which are made from aluminum,
copper and mild-steel. The purpose of these tests is to validate the formulation of the
analytical model as the test setup represents simplest rotor structure (only a ring) and no
physical motion. The schematic for these tests is shown in fig. 6.1.
6.3.1 Static Tests – Procedure
The following steps explain the procedure for conducting static tests.
1. A fixed voltage and frequency is applied to the stator (alone) of machine to
create a rotating field.
2. The power taken to set-up the flux in air is measured. This power can be
expressed as: Pin1 = PFes +PCus ; where PFes is stator iron loss and PCus is copper
loss in the stator.
3. Now the stator is covered by a metallic ring and power input is measured again
keeping current same as before. This power can be expressed as:
Pin2 = PFes +PCus+Pr ; where Pr is the power lost in the ring.
4. The power input will be different in the two cases (without and with ring). This
difference in power indicates eddy current losses in the ring. Therefore power
loss in the ring Pr = Pin2-Pin1
It is assumed that the stator iron losses are same for both operations which is not
exactly true. This is because the magnetic field setup by the stator current changes when
there is ring compared to the case without ring. Copper and Aluminum ring have very
low relative permeability. Therefore inductance remains fairly constant when compared
with the case without the ring. Thus we can say that this assumption of constant stator
losses is valid. In case of mild steel ring, the permeability is very high. Therefore the
change in inductance compared with the case without any ring is large. In this case the
assumption is expected to bring more deviation in results. However since it is difficult to
estimate stator iron losses especially for low flux densities, the assumption is made
anyways.
The assumption is verified by measuring the inductance and resistance of the setup
with and without the rings and explained in section 6.3.3 on the results of the static tests
(see fig. 6.8 and 6.9)
110
Fig.6.1: Static tests - schematic of the arrangement
6.3.2 Static Tests – Apparatus
The apparatus for the static tests consists of the following equipment:
a) Power supply – 6 kW, Variable frequency (17-1000) Hz, with inbuilt voltage,
current and power measurement (see fig. 6.2).
b) Stator with 3-phase winding and open slots (fig. 6.3)
c) Stator with 3-phase winding and semi-closed slots (fig. 6.3)
d) Rings of copper, aluminum and ST37 steel (fig. 6.4)
e) Voltage, current and power measurement
The machine parameters are shown in table 6-1 and 6-2. The total setup is depicted in fig. 6.5.
Fig.6.2: Power Supply (with inbuilt voltage, current and power measurement)
111
a)
b)
Fig.6.3: Stators a) open slots b) semi-closed slots
TABLE 6-1: MACHINE PARAMETERS: DIMENSIONS
Parameter
Nt
Np
Nc
rs
rsi
hs
hso
bt
btt
hsy
s
lg
rm
lm
bp
ls
Description
Open slot
number of tooth
number of pole
number of turns for each coil
air gap radius
inner radius stator
slot height
slot opening height
tooth width
tooth width with edges
height of stator yoke
sloth pitch
air gap length
permanent magnet surface radius
magnet length
magnet width
stack length
Rated Current
Rated Speed
Rated Power
27
18
15
91 mm
46 mm
30 mm
8 mm
10 mm
21.17 mm
2 mm
93 mm
5 mm
24 mm
56 mm
10 A
3600 rpm
9 kW
Semi-closed
slot
27
18
15
91 mm
46 mm
26 mm
4 mm
8 mm
18 mm
10 mm
21.17 mm
2 mm
93 mm
5 mm
24 mm
56 mm
10 A
3600 rpm
9 kW
TABLE 6-2: MACHINE PARAMETERS: MATERIAL PROPERTIES
Parameter
Description
Value
back iron resistivity
2x10-7 Ω.m
fe
magnet resistivity
1.3 x10-6 Ω.m
m
Brm
magnet remanent flux density
1.2 T
µrfe
relative permeability back iron
200
µrm
relative permeability magnet
1.05
112
a)
b)
Fig.6.4: a) copper ring b) aluminum ring c) ST37 steel ring
c)
Power Supply
Stator (Open slots)
Aluminum
Ring
Steel Ring
Copper
Ring
Stator (Semi-closed)
Fig.6.5: Experimental setup
6.3.3 Results Static Tests
The results for the static tests are shown in this section in fig. 6.6 for open slot
machine and in fig. 6.7 for semi-closed slot machine. The eddy current loss calculation
for the three rings of different materials and stators with open slots as well as semiclosed slots are presented.
113
a)
b)
114
c)
Fig.6.6: Comparison of Losses for a) Copper ring b) Aluminum ring c) ST37 steel ring for open slot machine
In case of aluminum and copper, the experiments show higher losses. This is because
of end effects where the eddy currents get crowded at the edges of the ring. Since the
axial length of the machine is very small compared to pole pitch, the end effects are
rather pronounced. The FE and analytical models are for large machines where axial
length is much larger than the pole pitch therefore 2d only. The experimental losses
include end effects. It can be seen that both analytical and FE models overestimate
losses at lower frequencies while they underestimate losses at higher frequencies.
115
a)
b)
116
c)
Fig.6.7: Comparison of Losses for a) Copper ring b) Aluminum ring c) ST37 steel ring for semi-closed slot machine
117
In order to understand the deviation, the impedance of stator with ring mounted is
measured for difference frequencies and results are shown in fig. 6.8 and 6.9.
a)
b)
Fig.6.8: Effect of frequency on a) resistance b) inductance of the open slot machine
118
a)
b)
Fig.6.9: Effect of frequency on a) resistance b) inductance of the semi-closed slot machine
At low frequencies, the reaction field is negligible and therefore the circuit’s
inductance is high and resistance is low as observed from the machine terminals.
119
However the resistance in actual machine is higher than in FE or analytical models
because of end effects. This phenomenon is verified by using a thermal camera.
Immediately after the tests, the setup is put inside a cardboard box and pictures are
taken with the thermal camera through a slit sized to the camera sensor. This is done to
eliminate any error in thermal image due to reflection of light. The pictures shown in fig.
6.10 clearly show end effects i.e. the edges of the rings have higher loss concentration
than the central part of the ring.
Aluminum ring @ 1 kHz
Initial condition
Edges hotter
than the rest
Copper ring @ 1kHz
Initial condition
Edges hotter
than the rest
Fig.6.10: Thermal pictures of the rings immediately after tests. Note higher temperature on the edges of the ring
120
The over-estimation of eddy current losses for low frequencies and under-estimation
at high frequencies can be further explained by taking increased resistance due to end
effects into account.
Since losses are limited by resistance, power measured at stator terminals is given by:
P
U2
R
P is power measured; U is voltage and R is resistance.
Thus increased resistance due to end effects leads to lower losses in experiments for low
frequencies.
At higher frequencies, the reaction field is strong and thus inductance reduces and
resistance increases as seen at machine terminals. The current flow seen from machine
terminals is governed by inductance and power is given by:
P  I 2R
Thus, an increase of resistance leads to higher losses. It can be seen from fig. 6.6 and
6.7 that the models still predict correct order of magnitude of losses.
The losses in case of the non-linear ST-37 steel are higher in analytical model compared
to experimental and FE (which also takes non-linearity into account). This is because of
local saturation at high frequencies which has not been taken into account in analytical
model.
6.4. Rotary Tests – Main Principle
Rotary tests imply that there is a rotating part involved in the experimental setup.
The principle of power balance is used to calculate losses in the rotor back-iron and
magnets. This method involves measurement of input power supplied to the prime
mover and the output power of the concentrated winding machine.
The machine considered here, is a generator and feeds a known load. If the machine
is operating at synchronous speed, the losses due to torque-producing component are
not present therefore we can assume that all the losses are occurring due to harmonics.
These losses comprise of copper and iron losses. We can separately calculate copper
losses for the machine and the remaining losses are the iron losses.
The power balance equation can be written as
Pin  Ppm _ m  Pmech  Pgen  Pload
(6-1)
Here,
121
Pin = Electrical power input
Ppm _ m = Power lost in prime-mover
Pmech = Mechanical power at shaft
Pload = Power taken by load
Pgen  PCus  PFes  PFer = Power loss in generator
PCus is copper loss in stator; PFes is iron loss in stator; PFer is iron loss in rotor
The total losses can be separated to obtain the copper and Iron losses. The
manufacturer’s data for stator can be used to estimate the iron losses in the stator
laminations. The rest of losses can be separated assuming that eddy current losses are
the major contribution towards iron losses in solid rotor. In order to separate iron losses
in rotor due to stator currents and PM magnetization, the same test is performed when
magnets are replaced by un-magnetized magnets. This ensures that the losses are only
due to stator excitation.
6.4.1 Rotary Tests – Apparatus
The apparatus for rotary tests shown in fig. 6.11 and 6.12 consists of the following
main components:
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
122
DC Power supply (600 V, 60 A, 36 kW) for armature
DC Power supply (340 V, 1.92 A) for field excitation
DC machine as prime mover (30 kW)
Stator with 3-phase winding and open slots
Stator with 3-phase winding and semi-closed slots
Rotor with magnets
Rotor with un-magnetized magnets
Voltage, current and power measurement
Spring balance for torque measurement and fixing mechanism
Resistive load for loading the machine
Speed measurement (Encoder fixed to DC machine)
3 Phase
Connection
DC
Connection
Fig.6.11: Rotary tests - schematic of the arrangement
Force
from
spring
balance
Stator can
rotate around
this assembly.
Stator can rotate
around this
assembly over
shaft
The bolt is attached to
spring
balance which counters
rotation of stator and reads
force
Fig.6.12: Stator and mechanism of torque measurement for the 9 kW PM machine
6.4.2 Variations in Rotary Tests
The rotary tests are conducted on the machine and compared with FE simulations to
bring out the differences. The method of power balance and direct torque measurement
are used to measure losses in the machine. The separation of iron losses into stator and
rotor losses is carried out by calculating the stator iron losses from manufacturer’s
datasheet. A correction factor of 1.55 (to account for punching and processing of
123
laminations) is used to estimate losses in the stator. Same rotary tests are conducted on
both open slot and semi-closed slot machine. Further, the rotary tests are conducted for
the cases tabulated in table 6-3.
TABLE 6-3: EXCITATION CASES
Case
I
II
III
Excitation
PM but no stator currents
Stator currents + PM
Stator currents only
Validation
Losses: PM only (slotting)
Losses: Combined
Losses: Stator Currents
The detailed account of procedure for measurement is given in the subsequent sections
6.4.2.1 Measurement of the prime mover and mechanical losses
It is necessary to measure the prime mover and mechanical losses before we proceed
with loss measurements with different cases mentioned in table 6-3. The prime mover
and mechanical losses are two separate entities and therefore calculated from the
experimental results. These losses are calculated for three different cases viz. unloaded
shaft, shaft with stator only and shaft with rotor only. The case with both stator and
rotor can’t be considered because that would lead to some electromagnetic power
exchange and extra losses.
The input power is the power absorbed from the DC power supply (connected with the
armature) by the DC motor which acts the prime mover.
Pin _ dc  U dc I dc
Here Pin _ dc is the input DC power;
(6-2)
U dc is the DC armature voltage;
I dc is the armature current.
Out of this input power, some power is lost in the DC motor armature. This power is
simply calculated by
2
Pcu _ dc  I dc
Rdc
Rdc 
124
U dc
I dc
(6-3)
Here Pcu _ dc is the copper loss of the prime mover; Rdc is the armature resistance. The
armature resistance is also verified by measuring after running the machine for some
time
The power balance can now be established as:
Pin _ dc  Pcu _ dc  Pm _ pm
(6-4)
Pm _ pm is the mechanical power lost due to friction and other rotational losses in prime
mover and assembly.
Further, the mechanical power loss Pm _ pm is calculated from three different
measurements:
1. First the shaft is not loaded and rotated at different speeds (100-1500 rpm in steps of
100 rpm). The voltage, current and power of the DC motor is recorded at all
different speeds. The mechanical power lost in this operation is calculated using (63). This power is called Pm _ pm0 i.e. the mechanical loss without the influence of the
mass of rotor and stator.
2. Secondly, only the rotor is mounted onto the shaft and the voltage, current and power
of the DC motor is recorded at the same speeds in as in 1.The mechanical power lost
is calculated using (6-4). This power is called Pm _ pm1 i.e. is the mechanical loss with
only the influence of the mass of rotor.
3. Step 2 is repeated using the stator instead of the rotor. The mechanical power loss is
called Pm _ pm 2 i.e. the mechanical loss with only the influence of the mass of stator.
Finally the mechanical loss at a certain speed is calculated by using
Pm _ pm  Pm _ pm1  Pm _ pm2  Pm _ pm0
(6-5)
These measurements provide us with the amount of prime mover and mechanical losses
at various speeds. This result is independent of the load so it can be used in both noload and on-load tests.
Results: The measurement results for both open-slot and closed-slot machines at
different speeds are shown in fig. 6.13.
125
Fig.6.13: Frictional losses for open-slot and semi-closed slot machine
The mechanical loss measured with the semi-closed-slot machine is a little bit higher
than with the open-slot machine. This could be due to the slightly higher mass of the
semi-closed-slot stator. Each curve shows an almost linear relationship. This implies that
most of the mechanical loss comes from bearing losses, because bearing losses vary
linearly with speed while air friction losses have a quadratic or even cubic variation with
speed.
6.4.3 Rotary Tests Case I: PM excitation but no stator current
There are mainly two sources of magnetic field excitation. The first source is stator
currents and the second source is the presence of permanent magnets (PMs). Even in
the absence of stator currents, there can be eddy current losses in the machine stator
and rotor due to effect of slotting (see fig. 6.14). The losses exist because of change of
flux density in stator iron and rotor iron as the magnets pass under stator teeth and slot
(air).
126
Fig.6.14: Effect of slotting on flux density: Pulsation under teeth and air part of stator alternately
In order to capture these losses, the machine is run on different speeds with no
current in the stator (as in a no load test) and losses are estimated.
6.4.3.1 Procedure for the no-load test (Case I)
In this test both the stator and rotor are mounted on the setup. The terminals of
PMSG remain open circuited and therefore no current flows in the stator winding. The
rotor is driven by the prime mover and the stator is kept stationary by the spring balance
arrangement which also measures force. The power balance for this condition can be
expressed by modifying (6-1):
PFer  Pin  Pm _ pm  PFes
Here, the meaning of symbols is same as in eq. (6-1)
(6-6)
Equation (6-6) can also be written as:
PFer  Pmech  PFes
(6-7)
Here Pmech is the mechanical power at the shaft measured by using the spring balance.
The spring balance measures mass in kg. The mass in kg is converted into force by
multiplying it by g (acceleration due to gravity). This force is then multiplied by the
perpendicular distance between the center of the stator and the point of contact of the
spring balance hook. The mechanical power can be calculated by multiplying torque
with mechanical rotational speed.
127
Tmech  mgr
Pmech  Tmechm
(6-8)
Here Tmech is the mechanical torque; m is the mass shown by spring balance;
g is acceleration due to gravity; m is the mechanical rotational speed
r is the perpendicular distance from center of stator to spring balance hook
In this case the electromagnetic power is converted into iron losses (both stator iron
and rotor iron) and magnet losses in the machine. It is evident that in the power flow
model we have to know the stator iron loss, otherwise only the total iron loss in the
machine can be found out. The power is measured at different speeds. The speed range
is chosen from 100rpm to 1500rpm with a step size of 100rpm.
6.4.3.2 Separation of Stator and Rotor Iron Losses
The iron losses measured in the tests are the sum of stator and rotor iron losses.
Here in this chapter, a method utilizing finite element simulations is used to give an
estimation of stator iron losses. The fact that the iron losses in ferromagnetic materials
depend on the flux density and its frequency is utilized. Eddy current loss per unit
volume can be written as:
Pe  Ke f 2 Bm2
(6-9)
K e is the eddy current loss coefficient; f is the frequency; Bm is the peak magnetic flux
density which is a sinusoidal function.
The eddy current loss coefficient is usually hard to determine as it is not a constant.
This makes the application of (6-9) rather difficult. Manufacturers of ferromagnetic
materials therefore provide the users with curves or look-up tables. The curves or tables
show the iron losses as a function of the peak value and the frequency of flux density as
shown in fig. 6.15. The material used for the stators in this thesis is M-19-29-Ga fully
processed non-oriented silicon steel. Its dependence on peak flux density at various flux
densities is provided by the manufacturer.
128
Fig.6.15: Eddy current loss as a function of frequency and flux density
The unit for the flux density is Gauss and for the iron loss is Watt/ Pound. The
conversion of these units to SI units follows:
1 Gauss= 0.0001 Tesla ; 1 Watt/Pound=2.20462 Watt/kg
Now the peak values of flux density in the stator iron is estimated. Finite element
simulations are used again to solve for the flux density. Using the same geometry and
the same parameters of the actual machine, the finite element simulation can provide a
relatively precise prediction of flux density in various parts of the stator i.e.in the stator
teeth, the tooth caps and the stator yoke (see fig. 6.16). The frequency is the frequency
of operation obtained according to the number of poles and the rotational speed of
machines. The frequency is calculated as:
f 
m p
120
(6-10)
f is the electrical frequency; m is the mechanical speed in rpm and p is the number of
poles
With the peak values of flux density written in Table 6-4, the iron loss per unit mass can
be found in each region
129
TABLE 6-4: PEAK FLUX DENSITY IN VARIOUS PARTS OF STATOR
Stator Part
Open slot machine
Semi-closed slot machine
Tooth
1.5 T
1.65 T
Yoke
0.9 T
0.95 T
Tooth cap
NA
1.40 T
Fig.6.16: Flux density in various parts of stator for estimating stator losses
The mass used is the amount of stator iron material found after subtracting the mass of
copper used in winding from total stator mass. Further, since flux density in different
parts is different, the mass is calculated using following steps:
1. The total mass of the stator is measured.
2. The total stator iron mass is obtained after the mass of copper windings is
deducted.
3. The mass of teeth, tooth caps or yoke is calculated from the geometrical volume
after multiplying the volume with the mass density of the stator iron which is
uniform.
4. The part of stator yoke in which the flux density hardly varies is deducted. The
effective yoke mass contributing to iron losses is approximately 0.6 time of the
total mass of yoke. The mass of each region is summarized in Table 6-5.
TABLE 6-5: MASS OF VARIOUS PARTS OF STATOR
Stator Part
Open slot machine
Semi-closed slot machine
130
Tooth
2.55 kg
2.02 kg
Effective Yoke
0.99 kg
0.89 kg
Tooth cap
NA
0.56 kg
5. Thereafter the iron loss is calculated in each region with its own mass and iron
loss per unit mass. The total stator iron loss is subsequently obtained by adding
the losses in all regions of the stator iron.
It should be noticed that this stator iron loss is not absolutely accurate but is an
estimate. The accuracy is dependent on several factors. One decisive factor is the
deterioration of the iron material during its production process. Some processes such as
punching and heat treatment can lead to higher iron losses in the iron. A correction
factor is thus introduced to take the increase of iron losses due to production process
into account. According to experience, the correction factor for iron losses of the openslot stator is 1.45 and for the semi-closed-slot stator is 1.55. This difference exists
because the latter stator has tooth caps which complicate its production and bring about
relatively more iron losses than the former.
At this stage, we have the total iron losses measured from the power balance method. In
order to get the rotor iron losses, we can simply subtract the estimated stator losses
from the total iron losses.
131
6.4.3.3 Results – Case I
The results for the no load tests i.e. case I are shown in the following figures.
Fig.6.17: No load losses (due to slotting) for open slot machine
Fig.6.18: No load losses (due to slotting) for semi-closed slot machine
The two machines behave quite opposite to each other in terms of the eddy current
132
losses in the rotor. In case of the open slot machine, most of iron losses occur in the
rotor whereas in case of semi-closed slot machine, losses are higher in the stator. This is
because of very little amplitude of flux density pulsations in case of semi-closed
machine.
a)
Flux density
change due
to slotting
b)
Fig.6.19: Flux density change in the air-gap of a) semi-closed slot machine has very low change of flux density b) open
slot machine has much larger change in flux density
133
6.4.4 Rotary Tests Case II: PM excitation with stator current
In order to find the influence of armature currents together with magnets on iron losses,
we connect the machine used as a generator (PMSG) to a purely resistive load.
Therefore there is a current flow in the stator windings due to generated voltage and
connection of load. The resistors are adjusted for various values and the PMSG is
rotated at different speeds. The idea is to keep the current constant while the speed
changes.
6.4.4.1 Procedure for the on-load test (Case II)
For each load condition (current at a particular value) and speed, the total iron loss is
measured by using the power balance method. The power flow can be written as:
PFe  Pmech  PCus  PLoad
(6-11)
Here, PCus is the copper loss in the stator; PLoad is the power consumed by load.
The copper loss in the PMSG is calculated as
PCus  ( I a2  Ib2  I c2 ) Rs
(6-12)
Here Rs is the resistance of each phase of the stator winding. Rs at different frequencies
is found by looking up the results in fig 6.8 and 6.9. These resistance values are adjusted
for the temperature rise using the standard temperature coefficient of resistance of
copper at 0.00393 ohm/0C.
I a , I b and I c are the currents in each phase
The output power is calculated as the sum of power lost in all three-phase loads:
PLoad  U a I a  Ub Ib  U c I c
(6-13)
U a , U b and U c are the voltages across the load resistors
We do not use the load resistance to calculate the power dissipated here because:
1. The load resistance can be measured only when the PMSG is offline while voltages
can be measured on line.
134
2. The load resistance is quite sensitive to temperature. Precise measurement is difficult
especially when the resistor is offline, because the temperature of the resistor
decreases fast.
The RMS values of current for the both PMSGs are selected as 4A, 7A, 10A and
13A. The chosen speeds for both PMSGs are 600rpm, 800rpm, 1100rpm, 1200rpm,
1300rpm, 1400rpm and 1500rpm. Too low speeds (e.g. 100 rpm, 300rpm) could lead to
unacceptable errors in measurement so they are not used.
The rotor iron losses are calculated by estimating the stator losses and subtracting them
from the total iron losses measured experimentally. The details of this procedure are
already explained in section 6.4.3.2 (separation of stator and rotor iron losses).
A note on temperature of magnets: The magnets used in the rotor of the PMSGs are rareearth type. The remanent magnetic flux density is dependent on temperature. Therefore,
it is important to adjust the value of remanent flux density (Br) for modeling purposes. It
becomes all the more important in the on-load test because different values of current
lead to different values of losses and hence different temperature of magnets as shown
in table 6-6.
TABLE 6-6: TEMPERATURE OF MAGNETS IN DIFFERENT EXPERIMENTS
Current
No Load
4A
7A
10 A
13 A
Temperature of Magnets in
Open Slot Machine (0C)
70
70
75
80
90
Temperature of Magnets in
Semi-closed Slot Machine (0C)
70
70
75
75
75
6.4.4.2 Results – Case II: Stator currents and PM excitation
This case represents the actual scenario which brings out deviations from FE models.
Since analytical model is limited by complex geometry and FE model is 2d, this case
presents the most “close to reality” scenario. The losses due to end effects and other
material non-linearities are also present in these measurements. The results of these
experiments are shown in fig. 6.20 for open slot machine and in fig. 6.21 for semi-closed
slot machine.
135
a)
b)
c)
d)
Fig.6.20: Losses due to both stator currents and PMs in open slot machine for a current of a) 4A b) 7A c) 10A d) 13A
The FE method predicts higher losses than the experimental measurements. As we
load the stator more, the difference between the experimental and FE values of the
losses increases. It may be noted that the FE model predicts almost same amount of
losses for all different current loading whereas the experimental values decrease as the
current loading increases. In general, the eddy current losses increase with speed. This is
expected because the eddy current losses increase as square of frequency.
136
a)
c)
b)
d)
Fig.6.21: Losses due to both stator currents and PMs for semi-closed slot machine for a current of
a)4 A b) 7 A c) 10 A d) 13 A
In case of the semi-closed slot machine, the FE method predicts lower losses than
the experimental measurements. As we load the stator more, the difference between the
experimental and FE values of the losses decreases. It may be noted that the FE model
predicts almost the same amount of losses for all different current loadings except when
the current increases to 13 A. As seen in the case of open-slot machine, the
experimental loss values decrease as the current loading increases. This difference in loss
prediction can be explained by:
a) The stator iron loss: The FE model is a 2d model whereby, the laminations and
current induced in laminations can’t be modeled. Therefore, the stator losses are not
included in the model. This results in almost constant loss prediction by FE model
although current is varied. On the other hand, the iron losses in the stator are estimated
and subtracted from the total iron loss in experimental results. The results from the
experiments show that as we increase the stator current, losses in the stator increase
whereas the iron losses in the rotor decrease. This is a direct consequence of using
137
power balance. Since the quantity of interest is resulting from difference of two large
quantities, the measurements are very sensitive. The actual power values and resulting
stator iron loss calculation is shown in fig. 6.22 for the case of 4 A loaded condition of
semi-closed slot machine. This case is chosen because the rotor iron losses are very low
as compared to the other power measurements.
a)
b)
Fig.6.22: Loss Calculation Procedure: a) Total Iron Losses from Experiments b) Rotor Iron Losses deduced after
estimating stator iron losses
In fig. 6.22 (a), the total iron losses (curve 1) are deduced by subtracting power
dissipated in load + copper losses (curve 2) from input power at shaft curve 3. This is
shown as curve 1 = curve 3 – curve 2 in the figure.
Similarly, fig. 6.22 (b) shows the rotor losses (curve 1) deduced by subtracting stator
losses calculated (curve 2) from total iron losses (curve 3). Notice the scale change from
fig. 6.22 (a) to fig. 6.22 (b).
b) The inaccuracy in torque measurement: The inaccurate measurement of torque to
calculate input mechanical power to the generator can be a reason for deviation. This is
all the more important in the case of semi-closed slot machine because the space
harmonic content of mmf is low and therefore the induced eddy current losses are
low(~10 W) compared to losses in case of open slot machine (~100 W).
c) End effects: The FE model being 2d, is not capable of handling all the end effects
which can be prominent for the machines used for experiments (axial length comparable
to pole pitch). The end effect means that the induced current circulates in loops in the
solid conductive parts as shown in fig. 6.23. The induced current circulates along x- axis
as well as along z-axis. The 2d model doesn’t consider the current (and hence the loss)
along the path of x-direction. This factor might be ignored if we are comparing larger
PM machines having axial length >> pole pitch so that the current loop has a much
138
larger length than width.
Fig.6.23: Eddy current loop in 3d and 2d treatment of induced current.
d) Effect of Phase angle: It has been assumed in the FE model that the phase angle
between current and EMF is zero. In reality, there can be a phase difference due to
different loading conditions. A fictitious phase angle was given to see the effect of such
a variation. Since normally this angle is small for purely resistive load, even if it is a
factor responsible for deviation, the magnitude is not considerable.
Fig.6.24: Effect of phase angle of current on eddy current loss calculation
6.4.5 Rotary Tests Case III: No PM excitation, only stator current
This case is a special case which is supposed to validate the analytical model because
these are the conditions actually used in the analytical modeling. The PMs are replaced
by un-magnetized magnets of the same material. This leads to the condition that eddy
139
current losses are present but the effect of PM field is not present. The analytical model
is based on this scenario. The underlying idea is that the effect of slotting is primarily the
same for different slot-pole combinations as long as stator geometry is same. This
means that the eddy current losses due to PM field and slotting are more or less
constant across the range of slot-pole combinations. It is the stator current excitation
that defines the trend.
6.4.5.1 Procedure for Case III: Only stator currents no PM field
In this test both the stator and rotor (with un-magnetized PMs) are mounted on the
setup. The terminals of the machine are connected to a variable frequency power supply
such that a constant current of 9 A is made to flow through the stator at a particular
frequency.
The rotor is driven by the prime mover and the purpose is to rotate the rotor at
corresponding synchronous speed so that load torque is zero. The machine acts as a
motor and all the mmf in the airgap is due to stator currents. The rotor tries to rotate
under the effect of the torque producing harmonic but the prime mover rotates the
rotor at synchronous speed and counters the load torque.
Therefore the electromagnetic energy exchange can be seen as limited to the machine
part where iron losses are induced by the stator field (space harmonics). These iron
losses are responsible for the counter-torque at synchronous speed. The power flow can
be simply written as:
PFe  Pac  PCus
(6-14)
Here, PCus is the copper loss in the stator; Pac is the power supplied to stator,
measured directly from the power supply. The copper loss in the PM machine is
calculated as:
PCus  ( I a2  Ib2  I c2 ) Rs
(6-15)
Here Rs is the resistance of each phase of the stator winding. I a , I b and I c are the
currents in each phase. The main steps for experiment are:
The stator is supplied by variable frequency supply and the rotor is rotated at
synchronous speed shown in table 6-7. The total iron losses are estimated from the
power balance. Then, the rotor iron losses are calculated by estimating the stator losses
and subtracting them from the total iron losses measured experimentally. The details of
this procedure are already explained in section 6.4.3.2 (separation of stator and rotor
iron losses).
140
TABLE 6-7: FREQUENCY AND CORRESPONDING ROTATIONAL SPEED
Frequency (Hz)
45
90
135
180
225
Rotation speed (rpm)
300
600
900
1200
1500
6.4.6 Results – Case III: Stator current excitation only
As discussed in sections 3.1 and 3.2, the concentrated windings induce losses in the
solid conductive parts of the machine. In order to capture these losses, the machines’
PMs are replaced with un-magnetized magnets so that eddy current losses due to stator
currents can still exist while losses due to PM field are absent.
In principle, this could lead to validation of loss calculation in the analytical model.
However because of only stator currents present and a relatively large air gap, lot of flux
doesn’t cross the air gap. Thus the losses in rotor are expected to be rather low.
Consequently in order to proceed with meaningful results very sensitive torque
measurements are required. The results are presented in figs. 6.25-6.27. No definite
validation can be confirmed.
Fig.6.25: Losses calculation (shown for the case of open slot machine)
Here, Rotor Losses = Input power – Copper losses – Stator iron losses;
Fig. 6.25 shows the calculation process used to find out the rotor losses. The input
power is measured at different speeds and copper losses are subtracted. The remaining
losses are mainly iron losses in stator and rotor. From the total iron losses, the stator
iron losses are subtracted to find out rotor iron losses. It can be seen that the copper
141
losses are dominant and stator losses are under-estimated. Therefore, the rotor losses
calculated are over-estimated. However, if we look at fig. 6.26 and fig. 6.27, the
experimental losses are still higher. This discrepancy is the result of very low torque
available to measure and hence measurement inaccuracy. Hence both measurements and
calculations are uncertain and nothing concrete can be concluded from these set of
experiments.
Fig.6.26: Losses due to stator current excitation only and no active PM field for open slot machine
Fig.6.27: Losses due to stator excitation only and no active PM field for semi-closed slot machine
142
The reasons for deviation between FE, analytical and experimental results can be:
a) Inaccuracy in the torque measurement.
b) Under-estimation of stator iron losses because of 2d analysis used whereas
in reality, end effects are considerable.
6.5. Summary
A framework for carrying out experiments in order to validate eddy current modeling
techniques has been presented. These experiments are conducted in various stages
leading to comparison of various aspects of models in each stage. The method of
measurements and the results thereof are documented. Two sets of experiments are
conducted i.e. stationary for validating analytical models and rotary for validating both
FE and analytical models. The static set of experiments fairly validates the analytical
model formulation. In the rotary tests, the eddy current losses in the rotor due to
different excitations of the machine are separately analyzed and compared with models
for validation. The excitations used for loss calculation are stator slotting effect, winding
space harmonics and both excitations combined.
.
143
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[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
144
S.R.Holm, “Modelling and optimization of a permanent magnet machine in a flywheel,”
Ph.D. dissertation, pp. 62-66, Dept. Electrical Power Engineering, TU Delft, The
Netherlands, 2003.
H.Polinder, “On the losses in a high-speed permanent-magnet generator with rectifier,”
Ph.D. dissertation, pp. 12-16, Dept. Electrical Power Engineering, TU Delft, The
Netherlands, 1998.
K.J.Binns,P.J.Lawrenson and C.W. Trowbridge, “The analytical and numerical solution of
electric and magnetic fields”, 1995 edition, John Wiley & Sons publisher, pp. 3-6 and pp. 9597.
J. Cros, P. Viarouge, “Synthesis of high performance pm motors with concentrated
windings”, IEEE Transactions on Energy Conversion, vol. 17, pp. 248–253 (2002).
F. Magnussen, C. Sadarangani, “Winding factors and Joule losses of permanent magnet
machines with concentrated windings,” in Proc. of the 2003 IEEE International Electric
Machines and Drives Conference, 2003, pp. 333 – 339, vol.1.
H. Polinder, M.J. Hoeijmakers, M. Scuotto, “Eddy-current losses in the solid back-iron of
permanent-magnet machines with concentrated fractional pitch windings,” in Proc. of the
2006 IEE International Conference on Power Electronics, Machines and Drives, Dublin, 46 April 2006, pp. 479-483.
A.K.Sawhney , A course in Electrical Machine Design, 5th edition, Dhanpat Rai and Co.
Publishers, pp.122-127.
Haylock, J.A.; Mecrow, B.C.; Jack, A.G.; Atkinson, D.J.; , "Operation of fault tolerant
machines with winding failures," Energy Conversion, IEEE Transactions on , vol.14, no.4,
pp.1490-1495, Dec 1999.
7. Trends and Design
Guidelines
This chapter aims at bringing out trends in eddy current loss depending upon the slot-pole
combination used. Since the eddy current loss analysis is not the only major factor influencing the design,
other indicators of machine performance such as cogging torque, balanced winding layout etc are briefly
explained. The purpose is to present a more complete picture of design selection. This information might
be trivial for electrical machine design experts but for a general reader and students this additional
information is useful and hence included. The FE models are used to determine effect of selecting a slotpole combination on eddy current losses in rotor of machine. The overall goal is to draw some trends and
guidelines on selecting slot-pole combinations for concentrated winding topology for wind turbines.
7.1. Introduction
The performance of electrical machines is governed by many factors. Different types
of machines have different sensitivities to machine parameters. As discussed throughout
this thesis, eddy current losses are of prime importance when we discuss PM machines
with concentrated windings. There are many other design characteristics which define
machine performance in general. Some of the important aspects which need due
attention at an initial stage of design are:
a)
b)
c)
d)
Winding factor
Balanced windings
Cogging Torque
Rotor force balance
These are not all the parameters which have to be considered for an optimal machine
design but are some important parameters for an initial design. There is a lot of literature
available on utility of these design parameters. Therefore, these are just introduced here
to give an overview. Once these parameters are chosen, a unique slot-pole combination
145
geometry can be obtained. The weight, cost, reliability and other performance
parameters can then be derived for a particular design.
The primary comparison is still the eddy current loss analysis in this particular
chapter. The trend analysis is based on a stator chosen to represent approximately a 1.25
MW wind turbine generator. The specifications for this configuration are mentioned in
table 7.1. The number of rotor poles was varied keeping the ratio of pole-pitch to
magnet span constant. Further, the air-gap velocity for the rotor surface was kept
constant for all the slot-pole combinations listed in table 7-2.
TABLE 7-1: SPECIFICATION OF ANALYZED MACHINE
Stator Parameters
1527.9 mm
(NB: specified: circumference 9.6 m)
Yoke thickness
55 mm
Slot height
80 mm
Slot width
50 % of slot pitch
Coil current
2750 A (peak)
Current angle
0°
Stator μr
4000
Airgap
5 mm
Stack length
1m
Power range
1.1-1.3 MW
Surface speed
4 m/s (~27.5 RPM)
Rotor Parameters
Magnet thickness
20 mm
Magnet span
70 % of pole pitch
Yoke thickness
35 mm
Magnet resistivity
1 μΩm
Yoke resistivity
200 nΩm
Yoke μr
200
Magnet remanent flux density Br 1.2 T
Magnet μr
1
Outer radius
146
7.1.1 Slot-pole Combinations Used
TABLE 7-2: DIFFERENT SLOT-POLE COMBINATIONS ANALYZED
S No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
Combination
slot
pole
3
2
3
4
9
8
9
10
12
10
12
14
18
14
18
22
24
22
24
26
36
26
36
34
36
38
36
46
48
34
48
38
48
46
48
50
48
58
48
62
72
50
72
58
72
62
72
70
72
74
72
82
72
86
72
94
144
98
144
106
144
110
144
118
144
122
144
130
144
134
144
142
144
146
144
154
144
158
144
166
144
170
144
178
144
182
144
190
Winding Number of poles for Slots/pole/phase
Factor
144 slots
q
0.866
96
0.50
0.866
192
0.25
0.945
128
0.38
0.945
160
0.30
0.933
120
0.40
0.933
168
0.29
0.902
112
0.43
0.902
176
0.27
0.949
132
0.36
0.949
156
0.31
0.867
104
0.46
0.953
136
0.35
0.953
152
0.32
0.867
184
0.26
0.859
102
0.47
0.907
114
0.42
0.954
138
0.35
0.954
150
0.32
0.907
174
0.28
0.859
186
0.26
0.848
100
0.48
0.912
116
0.41
0.933
124
0.39
0.955
140
0.34
0.955
148
0.32
0.933
164
0.29
0.912
172
0.28
0.848
188
0.26
0.837
98
0.49
0.874
106
0.45
0.89
110
0.44
0.917
118
0.41
0.928
122
0.39
0.944
130
0.37
0.95
134
0.36
0.955
142
0.34
0.955
146
0.33
0.95
154
0.31
0.944
158
0.30
0.928
166
0.29
0.917
170
0.28
0.89
178
0.27
0.874
182
0.26
0.837
190
0.25
147
7.2. Results: Eddy Current Loss Trends
The results for the eddy current loss analysis are presented in this section. There are
two different excitations responsible for the eddy current losses in solid conductive parts
of the analyzed machine. The first excitation is due to the current in the stator winding.
The second excitation source is the pulsation of magnetic field due to effect of slotting.
The effect of these different excitations is elaborated on the conductive parts of the
machine. Three simulations are executed for each slot-pole combination:
1. One with the rated remanent flux density of the PMs and no stator currents,
thus providing the losses due to slotting.
2. One without PM magnetization, but with nominal stator currents, thus
providing the losses due to space harmonics due to the stator current.
3. One with both magnets and currents applied, to confirm that the superposition
of 1 and 2 applies.
In figures 7.1 and 7.2 the back-iron and magnet eddy current losses are presented.
Fig.7.1: Losses in the solid rotor back-iron due to stator currents only (I), PMs only (Br) and combined field (I and B)
148
Fig.7.2: Losses in the magnets due to stator currents only (I), PMs only (Br) and combined field (I and B)
A brief inspection reveals that superposition principle can be applied for the two
different excitations. In the case without currents but with PM active, some shielding of
the magnet eddy current losses due to back iron eddy-currents can be observed for low
values of slots per pole per phase i.e. q.
In the losses due to space harmonics no shielding effects are visible and the losses
increase with decreasing q (i.e. increasing electrical frequency). As a result, the relative
variation in magnet losses is 22 % over the full range of q, while the minimum and
maximum iron losses differ by more than 10 times.
It can be seen from fig. 7.3 that the trend for loss analysis is mainly governed by the
rotor back-iron losses and total magnet losses are more or less comparable for different
slot-pole combinations. This means that if we want to find trends within slot-pole
combinations, then analytical model which gives a qualitative comparison of losses due
to stator currents alone is sufficient.
149
Fig.7.3: Total losses in the machine due to combined field
For q > 0.35 the losses remain relatively constant and the decision for a certain slotpole combination should be based on other criteria. One possibility is to consider the
mechanical power, shown in fig. 7.4 which shows a peak around q = 0.35.
Fig.7.4: Mechanical power as a function of q for chosen topologies
150
The rotor losses relative to the mechanical power are presented in fig. 7.5 again
showing that any slot-pole combination with q > 0.35 performs similar from a rotor loss
perspective.
Fig.7.5: Total rotor losses as a fraction of mechanical power
7.3. Other Design Considerations
As mentioned in the introduction section, in a complete design of machine, eddy
current losses are not the only design criteria. There are many other design
considerations which result in an optimal design. The important ones are mentioned in
this section as a general guideline. These have been introduced to lay emphasis on
totality of eddy current loss analysis in context of a complete machine design.
7.3.1 Winding Factor
There are many definitions of winding factor, available in literature [7]. In very
simple words, to form a concept, winding factor can be defined as:
“The ratio of the phasor sum of emf generated by individual coils of a distributed winding to
maximum emf that can be generated with a full pitch, concentrated winding”.
151
Winding factor is in a way an indicative of how “efficiently” emf can be generated by a
given sequence of connected coils. If all coils were full pitch and concentrated, it would
mean a winding factor of 1. The winding factor is made up of three components
a) Distribution factor: To take into account the fact that the coils are distributed
along the stator periphery and not at one place.
b) Pitch factor: This reduction factor takes into account the effect of short-pitching
of coils i.e. if the coil may doesn’t span full pole pitch or 180 degrees electrical.
c) Skewing factor: To take stator and/or rotor skewing into account. In this thesis
this factor is ignored for all relevant calculations.
The winding factor is then given as multiplication of the distribution factor, pitch
factor and skewing factor. The significance of winding factor lies in the fact that average
electromagnetic torque of a machine is directly proportional to the winding factor. A
lower winding factor results in a lower torque or in other words, to get the same torque,
we need a bigger machine. As shown in fig. 7.6, the winding factor for fractional slot
windings tends to decrease substantially for q<0.25 and for q>0.5 [2] and [3]. Due to
this reason, in the trend analysis, the q values were chosen to vary from 0.25-0.5.
It may be noted that for q = 1, the winding factor is only 0.5. This is because of the
assumption made in [3] that the phase spread is assumed to be 60 electrical degrees
(which is common). For a 3 phase system and q =1, there are 3 slots under one pole.
Since one pole spans 180 electrical degrees, one coil will spread 60 electrical degrees.
Therefore as shown in equations (5)-(9) in [3], the distribution factor =1 however the
pitch factor = 0.5 resulting in a winding factor of 0.5.
Fig.7.6: Variation of winding factor with slots per pole per phase ‘q’ [3]
152
7.3.2 Balanced Set of Windings
In most cases of electrical machine design, we use 3-phase windings. The 3-phase
windings of a machine must be balanced in order to avoid noise, torque-ripple and
unbalanced forces on machine structural parts.
“A balanced winding has coils arranged in such a way to produce a symmetrical system of equally
time-phase displaced emfs of identical magnitude, frequency and waveform.”
It is quite easy to know if a slot-pole combination will yield a balanced winding or not
because it depends only on the number of slots and poles.
In order to obtain a balanced winding:
“The slots per phase for each periodic part of machine should be an integer number and not a fraction.”
Periodic part of the machine simply means the set of slots and poles which repeats
itself as we go around machine periphery. The number of periodicities of a slot-pole
combination is given by Greatest Common Divider (GCD) of number of slots and
number of pole-pairs. Mathematically, the condition for balanced winding may be
expressed as:
1
S
 Integer
3 GCD (S , p)
Here, S is the number of slots, p is the number of pole pairs, 3 is number of phases.
7.3.3 Cogging Torque
The cogging torque is produced by the interaction of PMs with the stator teeth. The
PMs always try to align themselves in such a position that magnetic reluctance is
minimized.
Fig.7.7: Cogging elements contributing to the cogging torque
153
The magnets on the rotor interact with stator teeth, trying to position themselves in
the lowest magnetic reluctance path or in other words they have a preferred position.
This phenomenon is called cogging. Figure 7.7 shows that when a magnet edge is near a
tooth and approaching, it experiences a pull towards the tooth. This leads to a sudden
torque in direction of movement. Similarly when a magnet edge leaves a tooth, it
experiences a pull creating a torque against the movement.
Each of this interaction leading to a sudden torque is called a cogging torque element.
Net Cogging torque can be expressed as the sum of all such interactions around the
rotor periphery. If all of these individual contributions add up, it can lead to a very high
cogging torque.
Cogging torque presents problems during start-up of the machine. It may also
introduce a torque ripple in total electromagnetic torque. In case a long shaft is used in a
machine, and cogging torque is high, it can lead to shaft fatigue. The cogging torque can
be easily calculated and negotiated as explained in [4]-[6]. It is shown in these references
that magnet span relative to pole-pitch, slot-pole combination and positioning of
magnets are the most important factors in negotiating cogging torque.
7.3.4 Rotor Force Balance
The rotor force here refers to the radial outward pull caused on the rotor surface due
to magnetic force of attraction between PMs and the stator iron. If due to any reason,
the PMs are placed on the rotor in such a way that the distribution of magnets on rotor
surface is unsymmetrical, it can lead to local deformation of the rotor and unbalanced
magnetic pull. This leads to undue stress on the shaft and bearings.
Such unbalanced pull can also result from manufacturing tolerances, especially parts
which define air-gap and low shaft stiffness. More information regarding calculation of
this unbalance force is available in [7]
7.4. Summary
The chapter presents trends in eddy current loss evaluation for surface mounted PM
machines. The transient 2d FE model developed in chapter 5 is used for that purpose.
Some other basic guidelines regarding design of machines are then presented. These
other basic guidelines are known in literature but are presented here to put eddy current
loss analysis in the overall perspective of machine design. All the guidelines together
can be used while designing surface mounted PM machines with fractional slot windings.
154
Bibliography
[1] J. Cros, P. Viarouge, “Synthesis of high performance pm motors with concentrated
windings”, IEEE Transactions on Energy Conversion, vol. 17, pp. 248–253 (2002).
[2] F. Magnussen, C. Sadarangani, “Winding factors and Joule losses of permanent magnet
machines with concentrated windings,” in Proc. of the 2003 IEEE International Electric
Machines and Drives Conference, 2003, pp. 333 – 339, vol.1.
[3] Skaar, S.E., Krovel, O., Nilssen, R.: "Distribution, coil span and winding factors for PM
machines with concentrated windings", ICEM-2006, Chania (Greece), Sept.2006, paper 346.
[4] Z. Q. Zhu and D. Howe, “Analytical prediction of the cogging torque in radial-field
permanent magnet brushless motors.” IEEE Trans. Magn., vol. 28, no. 2, pp. 1371–1374,
Mar. 1992.
[5] T. Li and G. Slemon, “Reduction of Cogging Torque in Permanent Magnet Motors”, IEEE
Transactions on Magnetics, Vol. 24, No.6, Nov. 1988.
[6] N. Bianchi and S. Bolognani, “Design Techniques for Reducing the Cogging Torque in
Surface-Mounted PM Motors”, IEEE Transactions on Industry Applications, Vol. 38, No.5,
Sep. 2002.
[7] A.K. Sawhney “A Course in Electrical Machine Design”, Dhanpat Rai and Sons,
edition 1997.
155
156
8. Conclusions &
Recommendations
This chapter summarizes the findings and contributions of the thesis.
8.1. Conclusions
Electrical machine design is a complex and iterative process. There are many
different points of view which define the framework of a machine design. A particular
design might be best for an application while redundant for another application.
Given this background, we can say that we don’t generally go for the technically best
possible design but for the most optimum design to suit a requirement. In order to
simplify this process, a design engineer might have to shuffle through a variety of
designs before selecting the one which is eligible for further optimization. This is where
the modeling process is used and having analytical and FE models help during this stage.
This thesis is focused on one aspect of evaluating a design i.e. eddy current loss
evaluation in double layer fractional slot windings. However, this thesis highlights other
aspects of machine design such as history and trends, manufacturing technology,
validation of models and some aspects of machine performance as well. In the following
text, we will go through the main interest points presented in each chapter and also
highlight contributions of this research work.
Chapter 2: In this chapter, the manufacturing methods for large PM machines
used in the industry today are explained. Manufacturing experience for an actual 2 MW
PM generator is shared. This knowledge of manufacturing processes is valuable as it
provides the reader with details which are not very obvious. This chapter brings out
limitations and challenges of current technology which are helpful in forming the thesis
objective.
 The large number of coils and manual labor involved in winding manufacture
make assembly of generator rather tedious. More time spent on machine
manufacture also means higher cost and lower through-put.
 In addition to the material costs, the present winding technology adds a large cost
to an already expensive machine. Therefore, it makes sense to explore other
methods to wind the generator.
157
Chapter 3: Eddy current loss analysis in rotating electrical machines is an old and
intriguing field. This chapter presents an interesting literature survey on historical
development of eddy current loss analysis in rotating electrical machines. This is an
important contribution of this thesis as such a survey has not been done before. The
survey is completed with many trends in development of analysis methods for the eddy
current loss analysis resulting in a journal paper (Chapter 3, [115]).
Chapter 4: In this chapter, an analytical model is developed starting from the basic
Maxwell’s equations. The strength of this analytical model is its generic nature and
simplicity. Although the analytical modeling theory for the eddy current loss analysis
already exists but in the field of large direct drive wind turbines the application is rather
new. Therefore this work in that sense is a contribution. Due to large size, many safe
assumptions can be made which simplifies the model and makes it very generic. Some
important conclusions from this chapter are:
 The solution time for the analytical model is in the order of 1-10 seconds
depending on the accuracy desired. The model can be adapted very quickly for
various slot-pole combinations and an easy comparative study or trend analysis
can be performed.
 The solution of this model doesn’t require any special expensive software which
is a distinct practical advantage. The model can be programmed even in
Microsoft Excel. Here however, Matlab was used.
 The eddy current loss trends for various useful slot-pole combinations are
derived in this chapter which is a contribution.
 The model also has some limitations (as in over-prediction of losses) especially
when a low order harmonic produces eddy current losses.
Chapter 5: In this chapter Finite Element modeling is explained. Some useful slotpole combinations are compared for eddy current losses and the results thus obtained
are presented. The conclusions drawn from the chapter are:




158
The FE method is more accurate than the analytical method as it includes the
effects of slotting and magnetic saturation.
In general, analytical model predicts similar trends for eddy current losses when
compared with FE model. However in slot-pole combinations where a low order
harmonic is responsible for losses, analytical model predicts higher losses.
The effect of slotting in presence of the PM field and motion produces
appreciable losses. These losses can’t be ignored but are rather easy to include in
a FE calculation.
A mixed approach of use of analytical and FE models can produce useful results
in short time.
Chapter 6: Experiments to validate analytical and FE models developed in chapter 4
and 5 are presented in this chapter. Since experiments are the most close to actual
situation in the real machine, they can be used for validation. The extent of deviation
from experimental values is documented and reasons are deduced. This is a contribution
of this chapter. The eddy current losses are studied for three different excitation cases i.e.
PM field only, bringing out effect of slotting; Stator field and PM field bringing out the
combined effect of slotting and winding harmonics; Only stator field and no PM field
bringing out the effect of only winding harmonics. The following conclusions may be
drawn:
 In case of the open slot machine, the eddy current losses due to PM field only
(slotting) as well as combined field (PM and stator current) are appreciable and
therefore are easier to measure. Both analytical and FE models can be used to
predict the approximate amount of losses.
 In case of semi-closed slot machine, the eddy current losses are very low. The
present setup can’t measure the losses accurately enough therefore the analytical
and FE models can’t be fully validated for this case.
 In case of excitation with stator current only and no PM field, the experimental
results remain inconclusive due to very low amount of losses and measurement
inaccuracies.
 3d effects contribute to substantial mismatch between models and experimental
measurements.
 Very sensitive torque measurements are needed to validate the loss models for
cases where losses are very low (~10 W)
Chapter7: The purpose of this chapter is to apply the FE modeling method to a
number of useful slot-pole combinations and bring out trends in eddy current losses
w.r.t. slot-pole combination. The main contribution of this chapter comes from the
validation of assumptions used for analytical modeling. It can be concluded that:




Superposition principle can be used to find total eddy current losses due to
slotting effect and winding mmf space harmonics.
The slotting effect is more or less constant for a given stator geometry, type of
magnet and air-gap velocity.
The power loss due to the winding mmf space harmonics sets the trend for
variation of eddy current losses amongst various slot-pole combinations.
The trend analysis shows that the value of slots per pole per phase (q), around
0.35-0.40 presents a very good operating region in terms of low eddy current
losses and high winding factor.
159

The slot-pole combinations 12-10 and 9-8 have acceptable eddy-current losses
and very high winding factor. The combination 3-2 has very low eddy current
losses but also a lower winding factor.
8.2. Thesis Deliverables Revisited
In the section 1.6, the thesis objective, some deliverables from the thesis are listed. Here
we take a look at the final status of these deliverables.
a.
A simple and generic analytical model for predicting eddy current losses.
This deliverable was completely achieved and a generic analytical model has
been developed in chapter 4.
b.
Validation of the analytical model using FEM; Validation of analytical and FE models
formulation using experiments to bring out effect of simplifications used for models.
The analytical model formulation was validated in chapter 5 with FEM and with
experiments in form of static tests in chapter 6. The analytical loss calculation
for the experimental machines matched fairly with FEM. However there was
significant deviation from the experimental results. Especially in the case where
eddy current losses were low, analytical and FE models could not be fully
verified experimentally. The main reasons for deviation are 3d effects and
inaccurate torque measurements. Therefore this deliverable was partially
achieved.
c.
Deduction of trends in eddy current losses for various slot-pole combinations
Eddy current loss trends were drawn for a large span of useful slot-pole
combinations both analytically and with FEM. Apart from some overestimation on lower order harmonics, similar trends were obtained by both
analytical and FE methods.
d.
Design guidelines for PMDD generators with respect to eddy current losses.
Based on the trend analysis for eddy current losses done in chapter 4, 5 and 7,
useful slot pole combinations with acceptable losses and winding factors have
been identified and recommended. Some other design guidelines are explained
briefly to present a perspective of loss analysis in overall design.
160
8.3. Recommendations for Further Research
This research deals with a very complex phenomenon of eddy current loss modeling.
The diffused nature of the induced currents, material anisotropies, measurement
inaccuracies and the fact that losses occur inside the material (hence difficult to measure)
are some of the main aspects which make this topic very complex. This research is an
attempt to provide the machine designer with some modeling methodology to select
some useful topologies (at an early design stage) for more rigorous design optimization.
However, there are many topics for further research which have been recognized during
this work which are mentioned below:



Since many discrepancies/mismatches amongst models in prediction of losses
have been attributed to 3d effects, a natural first recommendation is to
investigate eddy current losses in 3 dimensions.
The experiments can be performed on a more elaborate setup with very
accurate torque measurement to gain more confidence in experimental values.
This can lead to a better model validation.
It is also recommended to conduct experiments on a larger number of slotpole combinations to validate the trend analysis as well.
161
162
9. Appendices
9.1. Example of Solution of Partial Differential Equation
B1
Region
Region1;1;Stator
Iron with
yoke Permeability
with Permeability
μ1 μ1
B4
B4
Stator
Current Sheet with Sinusoidal surface Current = K sin (kx) at (Boundary)
b
Region 2; Air with Permeability μ2
Y
B2
B5
B6
Region 3; Magnet region Permeability μ3 and conductivity 3
B7
Z
X
Region 4; Rotor back Iron with Permeability μ4 and conductivity 4
B3
l
Fig.9.1: Sinusoidal current sheet exitation
Region 1 is iron having permeability μ1
Region 2 is air, having permeability μ2
Region 3 is magnets, having permeability μ3 and conductivity 3
Region 4 is iron, having permeability μ4 and conductivity 4
The boundary B1 is located at y = b; B3 is at y = 0; B5 is at y = 3b/5; B6 is at y = 2b/5
and B7 is at y = b/5
Subscripts 1, 2, 3 and 4 define the region where the quantity is present.
Now using boundary condition on boundary
B1  Az1( x, y b )  0
Az1  sin kx( g1eky  h1e  ky )  0
 g1ekb  h1e  kb  0
Using boundary condition on boundary
(9-1)
B3  Az 4( x , y 0)  0
163
When y=0
g 4 e ky  h4e  ky  0
 g 4  h4  0
(9-2)
On boundary B5, when y = 3b/5, we know that
Az1( x ,3b / 5)  Az 2( x ,3b / 5)
g1e
 3 kb 


 5 
 h1e
 3 kb 


 5 
 g 2e
 3 kb 


 5 
 h2e
 3 kb 


 5 
(9-3)
On boundary B5 where y = 3b/5, we get
1  Az1  1  Az 2 

 K sin kx
1  y  2  y 
1
 3kb 
 3kb   1 
 3kb 
 3kb   K
  g1 sinh 
g 2 sinh 
  h1 cosh 
 
  h2 cosh 
 

1 
 5 
 5  2 
 5 
 5  k
(9-4)
On boundary B6, when y=2b/5, we know that
Az 2( x ,2b / 5)  Az 3( x ,2b / 5)


 2 3b 
 2 3b  
 2kb 
 2kb  
sin(kx)  g 2 cosh 
  h2 sinh 
   sin( x).  g3 cosh 
  h3 sinh 

 5 
 5 
 5 
 5 


Here,   k 2  j  2 ; k   (
Excitation is same)
Thus,

 2 3b 
 2 3b  
 2kb 
 2kb   
 g 2 cosh  5   h2 sinh  5     g3 cosh  5   h3 sinh  5  



 





(9-5)
Also on boundary B6,
164
1  Az 2  1  Az 3 

0
2  y  3  y 

k 
 2 3b 
 2 3b  
 2kb 
 2kb    
g 2 sinh 
  h2 cosh 
    g3 cosh 
  h3 cosh 
  0

2 
 5 
 5   3 
 5 
 5 
On boundary B7, when y=b/5, we know that
(9-6)
Az 3( x ,b / 5)  Az 4( x,b / 5)


  b 
 kb 
 kb  
sin(kx)  g3 cosh    h3 sinh     sin( x).  h4 sinh  4  
 5
 5 
 5 


Here,   k 2  j  2 ; k   (
Excitation is same)
Thus,

  3b 
  3b  
  4b 

 g3 cosh 
  h3 sinh 
   h4 sinh 
 5 
 5 
 5 

(9-7)
Also on boundary B7,
1  Az 3  1  Az 4 

0
3  y  4  y 

3 
  3b 
  3b    4 
  4b  
  0
 g3 sinh 
  h3 cosh 
    h4 cosh 
3 
 5 
 5 
 5   4 
(9-8)
Equations (9-1) to (9-8) can be written in matrix form as shown below:
165
sinh kb
0
0
 cosh kb

 cosh  3kb  sinh  3kb   cosh  3kb   sinh  3kb 









 5 
 5 
 5 
 5 

 sinh  3kb  cosh  3kb   sinh  3kb   cosh  3kb 









 5 
 5 
 5 
 5 

1
1
2
2


 2kb 
 2kb 
0
0
cosh 
sinh 



5


 5 


 2kb 
 2kb 
k sinh 

 k cosh 

5 

 5 

0
0

2
2


0
0
0
0




0
0
0
0


0
0
0
 2 b 
 cosh  3 
 5 
 2 b 
 3 sinh  3 
 5 
3
 b
cosh  3 
 5 
 b
 3 sinh  3 
 5 
3



0
0



0 

0
0
  g1   0 
  h1   
   K 
 2 3b 
 sinh 
0
  g2   k 

 5 
 h   0 
 2   
 2 3b 
 3 cosh 
  g3   0 

 5 
   
0
  h3   0 
3
 h  0
  3b 
  4b    4   
sinh 
 sinh 
 


 5  
 5 

 b
 b
 3 cosh  3   4 cosh  4  
 5 
 5 

3
4

0
0
(9-9)
We have the following physical constants for the geometry
K = 2 x 106 A/m
μ0 = 4 .10
μ1 = 4000 μ0
μ2= μ0
μ3=1.09 μ0
μ4=200 μ0
3 =5 x 106
4 =3 x 106
7
After solution, we can find out the constants to define Az everywhere in the domain.
166
 g1   - 0.7365 + j0.6381 
  

 h1   0.8907 - j0.7717 
 g 2   - 0.1042 + j0.2675 
  

 h2    - 0.1480 - j0.1629 
 g   - 0.1676 + j0.2150 
 3 

 h3   0.0883 - j0.0181 
   - 0.0035 - j0.0013 

 h4  
(9-10)
Thus, the complete solution can be written as:
Az1  sin kx ((- 0.7365 + j0.6381) cosh ky  (0.8907 - j0.7717)sinh ky )
Az 2  sin kx ((- 0.1042 + j0.2675) cosh ky  (- 0.1480 - j0.1629)sinh ky )
Az 3  sin kx((- 0.1676 + j0.2150) cosh  3 y  (0.0883 - j0.0181)sinh  3 y )
Az 4  sin kx((- 0.0035 - j0.0013)sinh  4 y)
(9-11)
To verify the results, a case where we know k was taken up. The value of k can be
found out easily by comparing the periodicity of sine with length of the domain l
If there is one period of sine within length l then:
2
l
If there are n periods of sine within length then
k
k
2n
l
(9-12)
n=3 was taken for analysis and comparison with FEM. The FEM model results and
the analytical results are compared below.
The electrical frequency for this simulation was 10 Hz as the depth of penetration was
clearly visible at this frequency.
167
Fig.9.2: Plot for Az for x = 0.84 m and f = 10 Hz
Fig.9.3: Plot for Az for x = 0.84 m and f = 10 Hz from FE software
168
DC
5 Hz
10 Hz
20 Hz
30 Hz
40 Hz
50 Hz
Fig.9.4: Plot for Az for x = 0.84 m and different frequencies
9.2. Some Settings of the Used FE Software
9.2.1 Solver Settings
Once the problem is set-up as explained in chapter 5, the solver is configured for
solution. There are two solvers used during set up of transient problems pertaining to
machines. These solvers are static (time harmonic) solver and transient solver. The
important settings for each of these solvers are mentioned below:
a) Static Solver: The most important settings are the relative tolerance and the number
of iterations allowed for solution. Generally, smaller the relative tolerance, higher
the number of iterations required to reach the desired accuracy.
b) Transient Solver: The transient solver requires more settings because the results
from the static solver are used as initial conditions for the transient solver. The first
step is to define how to obtain initial conditions.
The important settings for configuring the transient solver are:
1) The time step size, the relative tolerance and the absolute tolerances: This is very
critical in determining the solution accuracy. There is a global relative tolerance and
169
there is an absolute tolerance for the solution. Different variables to be solved must
be assigned individual absolute tolerance. The tolerance for the variable (depending
upon the values that variable can assume) is assigned as product of relative
tolerance and absolute tolerance. For example the values that Az can take are very
small as compared to the values taken by coupling variable lm1 therefore different
absolute tolerances are assigned to these variables.
2) The time steps stored in the output and restrictions on time steps taken by the
solver.
3) The number of iterations required for each time step.
4) The update of Jacobian matrix: This means refresh rate of the Jacobian used to
solve the PDE during iterative solution process. The higher refreshing rate means
more accurate solution. This parameter can be set to update at each of the iterations,
at each time step or minimal update. The time taken is inversely proportional to the
accuracy.
The settings for the solvers are shown here. These settings are not the only possible
settings but just serve as an example. For different machines and different mesh sizes,
these settings can be different.
Fig.9.5: Settings for stationary solver
170
Fig.9.6: Settings for the initial conditions for the transient solver
171
Fig.9.7: Settings for the initial conditions for the transient solver
Fig.9.8: Settings for time stepping, non-linear solver and Jacobian matrix update
172
While the solver is running, it displays the reciprocal of step size and relative error
during convergence of iteration. These quantities give useful information for
determining if the solution is converging or not. If the solver is changing step size very
frequently, it means that the solver is not able to solve the problem. This instability of
solution is seen as peaks in the reciprocal of step size and very low slope of the relative
error for the iteration. If such a situation occurs, it is better to look for geometrical
inaccuracies, insufficient meshing of sub-domains and periodic boundary conditions as
check. Visualization in FEM is an added advantage during analysis. This helps in
understanding the physics better. The effects that can’t be captured by analytical
methods are very well captured and can be seen without any further imagination.
Visualization also serves as a check on the solution. If there is something wrong, it is
visible. A sample of good and bad solutions is shown in the figure below:
a)
Fig.9.9: a) Good solution b) Unstable solution
b)
173
174
Summary
Eddy current loss modeling for design of PM generators for wind turbines
PhD Thesis
By Anoop Jassal
Background
The main motivation for this thesis comes from manufacturing experience of a 2
MW direct drive Permanent Magnet (PM) generator for wind turbine. The
reference generator uses a distributed winding for stator and an inner rotor with
surface mounted permanent magnets. It was found that a substantial amount of
cost for such a generator comes from manufacturing process of distributed
windings. The use of concentrated windings can reduce the winding cost and
hence overall cost of generator. Another motivation for this research comes
from the trend of wind turbines being installed in large offshore wind turbine
parks. These wind turbine parks require lighter, cheaper and more efficient
generators which are more reliable and modular in construction to lower overall
installation costs.
However, the use of concentrated windings introduces additional losses in the
solid conductive parts of the rotor due to eddy currents induced by harmonics of
winding mmf. The eddy current losses need to be analyzed so that the
concentrated windings can be safely installed in the generator.
Main Goal
This thesis focuses on the modeling, analysis and validation of the eddy current
loss models suitable for PM generators for wind turbines. The modeling
approach is a mixed use of analytical and Finite Element (FE) methods. A
simplified generic analytical model for predicting eddy current losses at an early
design stage is formulated. However, due to simplifying assumptions, analytical
method leads to a qualitative analysis rather than quantitative. The detailed
analysis for the most promising topologies selected by analytical method is
carried out with Finite Element (FE) method. Both analytical and FE methods
175
need some validation/verification whereby experiments are performed for
comparison. Design guidelines are developed thereafter.
Literature Survey
A detailed literature survey outlining the development of various
methodologies concerning eddy current loss calculation in rotating electrical
machines was conducted. The idea of this survey is to answer two basic
questions viz.
 What type of work has already been done in the field of eddy current loss
calculation?

Which fields of rotating electrical machines have been thoroughly
covered and in which fields can we contribute?
This survey summarizes useful contributions from eminent scientists and brings
out various trends in the methods used for calculation of eddy current losses.
Modeling of Eddy Current Losses
Two modeling methodologies are used to evaluate eddy current losses.
A) Analytical modeling
B) Finite Element (FE) modeling
Analytical model is based on simplifying assumptions which are frequently
used in literature. Maxwell’s equations are solved for magnetic vector potential
on a simplified geometry using proper boundary conditions. The induced current
density is calculated from the magnetic vector potential and then the eddy
current losses are calculated from the induced current density. The input for the
model is derived from the Fourier decomposition of the linear current density in
the stator (which is assumed slot-less). The utility of the analytical model is a
quick qualitative comparison for eddy current losses amongst various possible
slot-pole combinations. No expensive software is needed to solve the analytical
model which is an advantage.
FE modeling is done in 2 dimensions with COMSOL software version 3.5a. The
FE model includes all geometrical effects and material non-linearities except 3d
effects. This might be valid for machines with long axial length and small pole
pitch (which is generally the case with large direct drive generators). Many useful
slot-pole combinations over the whole range of slots/pole/phase were analyzed
and trends regarding eddy current losses were deduced. Assumptions used in
176
analytical models were verified with FE analysis. It was deduced that analytical
model over-estimates the eddy current losses for the case when a lower order
harmonic is responsible for eddy current losses. The assumption that losses due
to slotting are more or less similar for same geometry of stator slots was
validated. This means that the eddy current loss trend amongst various slot-pole
combinations is governed by stator current induced eddy current losses.
Experimental Analysis
The experimental analysis aims at validating both analytical and FE models.
The experiments were conducted on two 9 kW PM machines, one with open
slots and another with semi-closed slots. The principle of power balance is used
to deduce combined rotor losses (rotor back-iron and magnets). Two different
sets of experiments were conducted.
A) Static experiments: To validate the analytical model formulation and
results
B) Rotary experiments: To validate FE model formulation and results
Further, in the set of rotary experiments, different excitations were used to
validate parts of the model. These include PM excitation without stator
excitation; PM excitation with stator excitation; only stator excitation without
PM excitation. It was deduced that both FE and analytical models can be used
for predicting eddy current losses in case of open slot machines (which is
generally the case for large machines with preformed coils). In case of semiclosed slot machine, the losses are very low and require very sensitive torque
measurements in order to use power balance method. The 3d effects,
measurement inaccuracies and ambiguity in calculation of stator iron losses are
the important contributors to the difference in modeling results and
experimental results.
Trend Analysis
Since FE method is more accurate, it was used to calculate eddy current losses
for 44 useful slot-pole combinations spanning over whole useful range of
winding factors and slots/pole/phase. From this analysis, useful trends were
established and also some assumptions regarding modeling were verified.
177
From the perspective of eddy current losses, the most useful combinations lie
within the range of slots/pole/phase ranging from 0.35 to 0.5. The 9-8 and 1210 slot-pole combinations have high winding factor and low eddy current losses.
The 3-2 slot pole combination had lowest losses but has also lowest winding
factor.
In Conclusion
Concentrated windings have potential to lower cost of PM generators for
wind turbines by utilizing winding automation processes known for this topology.
The use of concentrated winding poses a challenge of keeping the additional
eddy current losses in solid conducting parts of the machine to a minimum. The
slot-pole combination plays a very vital role in generation and induction of eddy
current losses. Therefore a suitable slot-pole combination at an early design stage
promises a good electromagnetic design. In order to analyze the eddy current
losses in various slot-pole combinations, an analytical model was formulated and
verified with FE analysis as well as experiments. Analytical method is
recommended for shortlisting useful slot-pole combinations and FE method
being more accurate is recommended for actual loss calculation. Both the FE
and Analytical models were compared against experimental results and the
discrepancies were documented. Further research on more accurate experiments
performed on a larger number of machines together with 3d FE modeling of
eddy current losses is recommended.
178
Samenvatting
Eddy current loss modeling for design of PM generators for wind turbines
Proefschrift
van Anoop Jassal
Achtergrond
De voornaamste motivatie voor dit proefschrift komt voort uit de fabricageervaringen van een 2 MW direct aangedreven permanent-magneetgenerator voor
windturbines. Het referentieontwerp heeft een verdeelde wikkeling in de stator
en een interne rotor met permanente magneten aan het oppervlak. Het bleek dat
een aanzienlijk deel van de kosten voor zo een generator in het fabricageproces
van de verdeelde wikkelingen zitten. Het gebruik van geconcentreerde
wikkelingen kan de wikkelkosten en daarmee de totaalkosten verlagen. Een
tweede motivatie komt voort uit de trend om windturbines in grote offshore
windparken te plaatsen. Deze windparken vereisen lichtere, goedkopere en
efficiëntere generatoren die betrouwbaarder en modulair zijn om de totale kosten
te verlagen.
Echter, het gebruik van geconcentreerde wikkelingen leidt dat extra verliezen
in de massieve geleidende delen van de rotor ten gevolge van wervelstromen
geïnduceerd door harmonischen in de stator-mmk. De wervelstroomverliezen
moeten geanalyseerd worden zo dat geconcentreerde wikkelingen veilig kunnen
worden toegepast in de generator.
Hoofddoel
Dit proefschrift richt zich op het modelleren, analyseren en valideren van de
wervelstroomverliesmodellen geschikt voor PM-generatoren voor windturbines.
De modelleringsmethode is een mengvorm tussen analytische methodes en
eindige-elementenmethodes. Een vereenvoudigd generiek model wordt
opgesteld om wervelstroomverliezen in een vroeg ontwerpstadium te
voorspellen. Echter, door vereenvoudigende aannames is de analytische methode
eerder een kwalitatieve dan een kwantitatieve analyse. De veelbelovendste
topologieën volgens het analytische model worden in detail geanalyseerd met de
179
eindige-elementenmethode. Zowel de analytische methode als de eindigeelementenmethode moeten gevalideerd/geverifieerd worden waarvoor
experimenten gedaan worden ter vergelijking. Ontwerprichtlijnen worden daarna
opgesteld.
Literatuuronderzoek
Een gedetailleerd literatuuronderzoek naar de ontwikkeling van methodes
voor wervelstroomverliesberekeningen in roterende elektrische machines is
uitgevoerd. Het idee van dit onderzoek is het beantwoorden van basale vragen,
namelijk
 Welk soort werk is reeds gedaan op het gebied van
wervelstroomverliesberekeningen?

Welke onderzoeksgebieden in roterende elektrische machines zijn
uitgebreid beschreven en op welke gebieden kunnen we bijdragen?
Dit onderzoek vat nuttige bijdragen van eminente wetenschappers samen en
brengt verscheidene trends naar voren op het gebied van methodes voor
wervelstroomverliesberekeningen.
Modelleren van wervelstroomverliezen
Twee modelleringsmethodes worden gebruikt om wervelstroomverliezen te
bepalen.
A) Analytische modellen
B) Eindige-elementenmodellen (FE)
Het analytische model is gebaseerd op vereenvoudigende aannames die vaak
gebruikt worden in literatuur. De maxwellvergelijkingen worden opgelost naar de
magnetische vectorpotentiaal op een vereenvoudigde geometrie met geschikte
randvoorwaarden. De geïnduceerde stroomdichtheid wordt berekend uit de
vectorpotentiaal waarna de wervelstroomverliezen worden berekend met de
geïnduceerde stroomdichtheid. De invoer voor het model is afgeleid uit de
Fourieranalyse van de lineaire stroomdichtheid in de stator (die groefloos
verondersteld wordt). Het nut van het analytische model is een snelle kwalitatieve
vergelijking van de wervelstroomverliezen voor verschillende mogelijke
combinaties van groeven en polen. Dure software is niet nodig voor het
oplossen van het analytische model, wat een voordeel is.
180
FE-modellering wordt in 2 dimensies gedaan met COMSOL software versie 3.5a.
Het FE-model bevat alle geometrische effecten en niet-lineaire
materiaaleigenschappen, behalve 3d-effecten. Dit kan geldig zijn voor machines
met een lange axiale lengte en een korte poolsteek (wat in het algemeen het geval
is voor grote direct aangedreven generatoren). Een groot aantal nuttige groefpoolcombinaties over het gehele bereik aan groeven per pool per fase zijn
geanalyseerd en trends voor de wervelstroomverliezen zijn afgeleid. Aannames
gebruikt in de analytische modellen zijn geverifieerd met FE-analyses. Hieruit
bleek dat analytische modellen de wervelstroomverliezen overschatten wanneer
een harmonische van lagere orde de verliezen veroorzaakt. De veronderstelling
dat verliezen door vertanding min of meer gelijk blijven bij een zelfde geometrie
van de statorgroeven is bevestigd. Dit betekent dat de wervelstroomverliestrend
voor verschillende groef-poolcombinaties bepaald wordt door de verliezen
geïnduceerd door statorstromen.
Experimentele Analyse
De experimentele analyse dient ter validatie van zowel de analytische als de FEmodellen. De experimenten zijn uitgevoerd met twee 9 kW PM-machines, een
met open groeven en een met half gesloten groeven. Het
vermogensbalansprincipe wordt gebruikt voor de bepaling van de
gecombineerde rotorverliezen (rotorjuk en magneten). Twee verschillende sets
proeven zijn uitgevoerd.
A) Statische proeven, voor de validatie van het analytische model en
resultaten
B) Draaiende proeven, voor de validatie van het FE-model en resultaten
Verder zijn bij de draaiende proeven verschillende stimuli gebruikt om gedeelten
van het model te valideren. Deze omvatten gemagnetiseerde magneten zonder
statorstromen; gemagnetiseerde magneten met statorstromen; statorstromen
zonder gemagnetiseerde magneten. Het is bepaald dat zowel de FE als de
analytische modellen gebruikt kunnen worden voor het voorspellen van de
wervelstroomverliezen voor machines met open groeven (wat in het algemeen
het geval is voor grote machines met vormspoelen). In de machine met half
gesloten groeven zijn de verliezen zeer laag en is een zeer gevoelige
koppelmeting nodig om het vermogensbalansprincipe te kunnen gebruiken. De
3d-effecten, meetonzekerheden en ambiguïteit in de berekening van de
ijzerverliezen zijn de belangrijkste oorzaken voor verschillen tussen de
gemodelleerde resultaten en experimentele resultaten.
181
Trendanalyse
Omdat de FE-methode accurater is, is deze gebruikt voor de berekening van
de wervelstroomverliezen voor 44 realistische groef-poolcombinaties uit het
bruikbare bereik van wikkelfactoren en groeven per pool per fase. Met deze
analyse zijn nuttige trends bepaald en enkele modelaannames geverifieerd.
Qua wervelstroomverliezen liggen de nuttigste combinaties tussen de 0.35 en 0.5
groeven per pool per fase. De 9-8 en 12-10 combinaties hebben een hoge
wikkelfactor en lage wervelstroomverliezen. De 3-2 combinatie heeft de laagste
verliezen maar ook de laagste wikkelfactor.
In Conclusie
Geconcentreerde wikkelingen hebben potentie om de kosten van PMgeneratoren voor windturbines te verlagen door gebruik van automatische
wikkelprocessen. Het gebruik van geconcentreerde wikkelingen vormt een
uitdaging door de bijkomende wervelstroomverliezen in de geleidende delen van
de machine. De groef-poolcombinatie speelt een sleutelrol in het opwekken en
induceren van wervelstroomverliezen. In een vroeg stadium een geschikte groefpoolcombinatie selecteren belooft een goed elektromagnetisch ontwerp. Om de
wervelstroomverliezen te analyseren voor verschillende groef-poolcombinaties is
een analytisch model opgesteld en geverifieerd met zowel FE-analyse als
experimenten. De analytische methode wordt aanbevolen voor een eerste selectie
van bruikbare groef-poolcombinaties en de accuratere FE-methode wordt
aanbevolen voor de daadwerkelijke verliesberekening. Zowel de FE als de
analytische modellen zijn vergeleken met experimentele resultaten en de
verschillen zijn beschreven. Verder onderzoek naar accuratere experimenten met
een groter aantal machines en 3d FE-modellering van wervelstroomverliezen
wordt aanbevolen.
182
List of Publications
As Author:
Journal Publications
1.
Jassal, A; Polinder, H; Ferreira, J.A., “Literature survey of eddy current loss
analysis in rotating electrical machines”, Electric Power Applications, IET, vol.6,
no.9.pp. 743-752, Nov. 2012.
2.
Jassal, A.; Polinder, H.; Damen M.E.C.; Versteegh, K.;, “Design Considerations
for Permanent Magnet Direct Drive Generators for Wind turbines”, International
Journal of Engineering and Technology ( IJET) Vol.4(3): pp. 253-257 ISSN: 17938244, 2012.
Conference Publications
1.
2.
3.
4.
Jassal, A.; Polinder, H.; Shrestha, G.; Versteegh, C.;, “Investigation of Slot Pole
Combinations and Winding Arrangements for Minimizing Eddy Current Losses
in Solid Back-Iron of Rotor for Radial Flux Permanent Magnet Machines”,
Electrical Machines (ICEM), 2008 XVIII International Conference on , 6-9 Sept. 2008.
Jassal, A.; Polinder, H.; Shrestha, G.; Versteegh, K.;, “Closed Slot Topology for
Reduction of Cogging and Noise in Permanent Magnet Direct Drive Generator
for Wind Turbines” Proceedings of European Wind Energy Conference EWEC,
Marseille, France, 16-19 March 2009.
Jassal, A.; Polinder, H.; Lahaye, D.; Ferreira, J.A.; , "Comparison of analytical
and Finite Element calculation of eddy-current losses in PM machines,"
Electrical Machines (ICEM), 2010 XIX International Conference on , vol., no., pp.1-7,
6-8 Sept. 2010.
Jassal, A.; Polinder, H.; Lahaye, D.; Ferreira, J.A.; , "Analytical and FE
calculation of eddy-current losses in PM concentrated winding machines for
wind turbines," Electric Machines & Drives Conference (IEMDC), 2011 IEEE
International , vol., no., pp.717-722, 15-18 May 2011.
Talks and Seminar
1.
2.
Jassal, A.; Polinder, H.; “Cost Optimization of Permanent Magnet Direct Drive
Generators”, Dutch Wind Workshops, October 2010.
Jassal, A.; Polinder, H.; Kirschneck, M. “Generator Systems for Wind
Turbines”, Workshop at International Quality and Productivity Center (IQPC), 22-23
May, Bremen, Germany 2012.
183
Books/Chapters
1.
Jassal, A.; Versteegh, K.; Polinder, H.; “Case study of the permanent magnet
direct drive generator in the Zephyros wind turbine”, Chapter in Electrical drives
for direct drive renewable energy systems, woodhouse publishing ISBN 1845697839 and
ISBN-13: 9781845697839, March 2013.
As Co-Author:
1. G.Shrestha; D.Bang; A.Jassal; H.Polinder; J.A.Ferreira;, “Energy Converters for
Future Wind Turbines”, Proceedings of Global Wind EnergyConference (GWEC),
Beijing, China,28-30 Sept. 2008.
2. G.Shrestha; D.Bang; A.Jassal; H.Polinder; J.A.Ferreira;, “Investigation on the
Possible Use of Magnetic Bearings in Large Direct Drive Wind Turbines”,
Proceedings of European Wind Energy Conference EWEC, Marseille, France ,16-19
March 2009.
3. Firmansyah, M.; Jassal, A.; Polinder, H.; Lahaye, D., "Eddy current loss
calculation in rotor back iron for concentrated winding PM generator," Electrical
Machines (ICEM), 2012 XXth International Conference on , pp.2666,2670, 2-5 Sept.
2012.
4. Imoru O.; Jassal A.; Polinder H.; Nieuwkoop E.; Tsado J.; and Jimoh A.A.; “An
inductive power transfer through metal object” Proceedings of IEEE
International Future Energy Electronics Conference (IFEEC), pp. 246-251, Nov. 3-6,
2013.
5. Liu D.; Jassal A.; Polinder H.; Ferreira J.A., “Validation of Eddy Current Loss
Models for Permanent Magnet Machines with Fractional-Slot Concentrated
Windings” Proceedings of IEEE Electric Machines & Drives Conference (IEMDC),
pp.678-685, 12-15 May 2013.
184
Biography
Anoop Jassal was born in Punjab, India in 1983. He graduated from Punjab
Engineering College, Chandigarh, India in 2003. After graduation, he worked at
Vardhman Spinning and General Mills (VSGM) at Baddi, India as Electrical Engineer
for three years.
He joined Delft University of Technology in 2006 for his MSc. Thereafter, in August
2008 he started working towards his PhD at Electrical Power Processing (EPP) group of
Delft University of Technology. His area of research is permanent magnet direct drive
generator design for wind turbines.
He worked part time at VWEC (now XEMC-Darwind), at Hilversum, Netherlands
since August 2008 till July 2012. Since August 2012, he is working at GE Global
Research at Munich, Germany as Research Engineer. His present work is focused on
design of various electro-mechanical devices and large machines.
Anoop Jassal is geboren in Punjab, India in 1983. Hij is afgestudeerd aan het Punjab
Engineering College, Chandigarh, India in 2003. Hierna werkte hij drie jaar bij
Vardhman Spinning and General Mills (VSGM) in Baddi, India als Electrical Engineer.
In 2006 begon hij aan de technische universiteit Delft zijn MSc.-opleiding. Daarna, in
augustus 2008, begon hij een promotietraject in de Electrical Power Processing-groep
aan de technische universiteit Delft op het gebied van direct aangedreven permanentmagneetgeneratoren voor windturbines. Ook werkte hij parttime bij VWEC (nu XEMCDarwind) in Hilversum tussen augustus 2008 en juli 2012. Sinds augustus 2012 werkt hij
bij GE Global Research in München, Duitsland als Research Engineer. Hierbij richt hij
zich op het ontwerp van uiteenlopende elektromechanische apparaten en grote
machines.
185
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