Analysis of a Radial Flux Air-cored Permanent Magnet
Machine with a Double-sided Rotor and
Non-overlapping Windings
by
Peter Jan Randewijk
Dissertation presented for the degree Doctor of Philosophy in the
Faculty of Engineering at Stellenbosch University
Promotor: Prof. Maarten Jan Kamper
Faculty of Engineering
Department of Electrical & Electronic Engineering
March 2012
Stellenbosch University http://scholar.sun.ac.za
Declaration
By submitting this dissertation electronically, I declare that the entirety of the work contained
therein is my own, original work, that I am the sole author thereof (save to the extent explicitly
otherwise stated), that reproduction and publication thereof by Stellenbosch University will not
infringe any third party rights and that I have not previously in its entirety or in part submitted
it for obtaining any qualification.
March 2012
Copyright © 2012 Stellenbosch University
All rights reserved.
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Abstract
Analysis of a Radial Flux Air-cored Permanent Magnet Machine with a
Double-sided Rotor and Non-overlapping Windings
P. J. Randewijk
Promotor: Prof. Maarten Jan Kamper
Faculty of Engineering
Department of Electrical & Electronic Engineering
Dissertation: PhD (Electrical Engineering)
March 2012
In this dissertation a new type of electrical machine, a Radial Flux Air-Cored Permanent Magnet machine with a Double-sided Rotor and utilising concentrated, non-overlapping windings,
is proposed. The concept of the Double-sided Rotor Radial Flux Air-Cored Permanent Magnet
machine, or RFAPM machine for short, was derived from the Double-sided Rotor Axial Flux
Air-Cored Permanent Magnet (AFAPM) machine. One of the problems that AFAPM machines
experience, is the deflection of the rotor discs due to the strong magnetic pull of the permanent
magnets, especially with double-sided rotor machines. The main advantage of a RFAPM machine over a AFAPM machine is that the rotor back-iron is cylindrically shaped instead of disk
shaped. Due to the structural integrity of a cylinder, the attraction force between the two rotors
does not come into play any more.
The focus of this dissertation is on a thorough analytical analysis of the Double-Sided Rotor
RFAPM machine. With the RFAPM being an air-cored machine, the feasibility to develop a
linear, analytical model, to accurately predict the radial flux-density and hence the induced
EMF in the stator windings, as well as the accurate calculation of the developed torque of the
machine, needed to be investigated. The need for a thorough analytical examination of the
Double-Sided Rotor RFAPM machine stemmed from the need to reduce the blind reliance on
Finite Element Modelling (FEM) software to calculate the back-EMF and torque produced by
these machines.
Another problem experienced with the FEM software was to obtain accurate torque results.
Excessive ripple torque oscillations were sometimes experienced which took a considerable
amount of time to minimise with constant refinement to the meshing of the machine parts.
Reduction in the mesh element size unfortunately also added to the simulation time. The requirement for an accurate analytical model of the RFAPM machine was also necessary in order
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to reduce the amount of time spent on successive FEM simulation to obtain the optimum pole
arc width of the permanent magnet in order to minimise the harmonic content of the radial
flux-density distribution in the the stator windings.
In this dissertation, the use of single-layer and double-layer, non-overlapping, concentrated
winding for the RFAPM machine is also investigated. It was decided to include a comparison of
these two non-overlapping winding configurations with a “hypothetical” concentrated, overlapping winding configuration. This would allow us to gauge the effectiveness of using nonoverlapping winding with respect to the reduction in copper losses as well as in the reduction
in copper volume. It would also allow us to investigate the extent of how much the developed
torque is affected by using non-overlapping windings instead of overlapping windings.
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Uittreksel
Analiese van ’n Radiale-vloed Lugkern Permanent Magneet Masjien met ’n
Dubbelkantige Rotor en Nie- oorvleuelende Windings
P. J. Randewijk
Promotor: Prof. Maarten Jan Kamper
Fakulteit Ingenieurswese
Departement Elektries & Elektroniese Ingenieurswese
Proefskrif: PhD (Elektriese Ingenieurswese)
Maart 2012
In hierdie proefskrif word ’n nuwe tipe elektriese masjien, ’n Radiale-vloed Lugkern Permanent Magneet Masjien met ’n dubbelkantige rotor en nie-oorvleuelende Windings voorgestel.
Die konsep vir die Radiale-vloed Lugkern Permanent Magneet Masjien, of RVLPM vir kort,
is afgelei vanaf die Dubbelkantige Rotor, Aksiale-vloed Lugkern (AVLPM) masjien. Een van
die probleme wat met AVLPM masjiene ondervind word, is die defleksie van die rotorjukke as
gevolg van die sterk aantrekkingskragte van die permanente magnete, veral in dubbelkantige
rotor masjiene. Die hoof voordeel wat die RVLPM masjien inhou bo die AVLPM masjien, is
die feit dat die RVLPM se rotorjukke silindries is in plaas van ronde skywe. As gevolg van die
strukturele integriteit van ’n silinders, speel die aantrekkingskrag van die permanente magnete
nie meer ’n rol nie.
Die fokus van die proefskrif gaan oor die deeglike analitiese analise van die dubbelkantige
RVLPM masjien. Weens die feit dat die RVLPM masjien ’n lugkern masjien is, is daar besluit
om ondersoek in te stel na die moontlikheid om ’n lineêre, analitiese model vir die masjien
op te stel waarmee die radiale-vloeddigtheid, teen-EMK asook die ontwikkelde draaimoment
vir die masjien akkuraat bereken kan word. Die behoefde aan ’n akkurate analitiese model
vir die dubbelkantige rotor RVLPM masjien is om die blinde vertroue te elimineer wat daar in
Eindige-Element Modellering (EEM) sagteware gestel word om die teen-EMK en ontwikkelde
draaimoment van die RVLPM masjien uit te werk.
’n Verdere probleem wat daar met EEM sagteware ondervind is, is die akkurate berekening van die ontwikkelde draaimoment. Oormatige rimpel draaimoment ossillasies is soms
ondervind wat heelwat tyd geverg het om te minimeer, deur voortdurende verfyning van die
EEM maas in die verskillende dele van die masjien. Soos die maas egter kleiner word, verleng
dit die simulasie tyd van die EEM aansienlik. Nog ’n rede vir ’n akkurate analitiese model
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van die RVLPM masjien, is om vinnige metode te verkry om die optimale permanente magneet pool hoekwydte te verkry, wat die minste Totale Harmoniese Vervorming (THV) in die
radiale-vloeddigtheidsdistribusie in die statorgebied sal veroorsaak, sonder om herhaaldelike
EEM simulasies te loop.
In die proefskrif word die gebruik van enkellaag en dubbellaag, nie- oorvleuelende, gekonsentreerde wikkelings vir die RVLPM masjien ook ondersoek. Daar is besluit om hierdie
twee nie-oorvleuelende windingskonfigurasies met ’n “hipotetiese” gekonsentreerde, oorvleuelende windingskonfigurasie te vergelyk. Dit behoort ons in staat te stel om die doeltreffendheid van nie-oorvleuelende windings te bepaal, met betrekking tot die afname in koperverliese
asook die afname in kopervolume. Verder sal dit ons in staat stel om ook mate waartoe die ontwikkelde draaimoment deur nie-oorvleuelende windings beïnvloed word, te ondersoek.
Stellenbosch University http://scholar.sun.ac.za
Dedications
Hierdie tesis word opgedra aan my wederhelfte,
Marilie Randewijk,
vir haar ondersteuning, geduld, moed inpraat en liefde
asook vir die hulp met die proeflees van die tesis.
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Contents
Declaration
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Abstract
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Uittreksel
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Dedications
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Contents
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List of Figures
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List of Tables
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Nomenclature
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1
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Introduction
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Background Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Rare-Earth Permanent Magnets . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Slotless, Toothless and Core-less Windings . . . . . . . . . . . . . . . . . .
1.2.3 The Double-sided Rotor Axial Flux Permanent Magnet Machines . . . . .
1.2.4 Non-overlapping, Concentrated Windings . . . . . . . . . . . . . . . . . .
1.2.5 The Double-sided Rotor Radial Flux Air-Cored Permanent Magnet Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Dissertation Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Dissertation Work Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Theoretical Winding Analysis of the Double-Sided Rotor RFAPM Machine
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Terminology used for the Analysis . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Overlapping (Type O) Winding Configuration . . . . . . . . . . . . . . . . . .
2.3.1 Flux-linkage of the Type O Winding Configuration . . . . . . . . . . .
2.3.2 Conductor Density Distribution of the Type O Winding Configuration
2.3.3 Current Density Distribution of the Type O Winding Configuration . .
2.4 Non-Overlapping Single-layer (Type I) Winding Configuration . . . . . . . .
2.4.1 Flux Linkage of the Type I Winding Configuration . . . . . . . . . . . .
2.4.2 Conductor Density Distribution of the Type I Winding Configuration .
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CONTENTS
2.5
2.6
2.7
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2.4.3 Current Density Distribution of the Type I Winding Configuration . .
Non-Overlapping Double-layer (Type II) Winding Configuration . . . . . . .
2.5.1 Flux Linkage of the Type II Winding Configuration . . . . . . . . . . .
2.5.2 Conductor Density Distribution of the Type II Winding Configuration
2.5.3 Current Density Distribution of the Type II Winding Configuration . .
Number of Turns per Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Magnetostatic Analysis due to the Permanent Magnets
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Background on the Magnetic Field Analysis of Electrical Machines . . . . . . . .
3.3 The Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Magnetic Vector Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 The Vector Poisson Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.1 The Vector Poisson Equation in the Different Regions of the RFAPM Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.2 The Vector Poisson Equation in the Non Permanent Magnet Regions . . .
3.6.3 The Vector Poisson Equation in the Permanent Magnet Regions . . . . . .
3.7 Solving the Laplace and Poisson Equations . . . . . . . . . . . . . . . . . . . . . .
3.7.1 Finding the General Solution of the Laplace and Poisson Equations . . . .
3.7.2 Finding the Particular Solution of the Poisson Equation . . . . . . . . . . .
3.8 Boundary Conditions Between the different Regions . . . . . . . . . . . . . . . . .
3.8.1 The Magnetic Flux-density at the Boundary Between the different Regions
3.8.2 The Magnetic Field Intensity at the Boundary Between the different Regions
3.9 Solving the Magnetic Vector Potential for all the Regions of RFAPM machine . .
3.9.1 On the Inner Boundary of Region I . . . . . . . . . . . . . . . . . . . . . . .
3.9.2 On the Boundary Between Region I and II . . . . . . . . . . . . . . . . . .
3.9.3 On the Boundary Between Region II and III . . . . . . . . . . . . . . . . . .
3.9.4 On the Boundary Between Region III and IV . . . . . . . . . . . . . . . . .
3.9.5 On the Boundary Between Region IV and V . . . . . . . . . . . . . . . . .
3.9.6 On the Outer Boundary of Region V . . . . . . . . . . . . . . . . . . . . . .
3.9.7 The Simultaneous Equations in order to solve the Magnetic Vector Potential and the Magnetic Flux-density . . . . . . . . . . . . . . . . . . . . . . .
3.10 Obtaining the Final Solution of the Magnetic Vector Potential and the Magnetic
Flux-density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.10.1 The Magnetic Vector Potential Solution . . . . . . . . . . . . . . . . . . . .
3.10.2 The Magnetic Flux-Density Solution . . . . . . . . . . . . . . . . . . . . . .
3.11 Validation of the Magnetic Vector Potential and the Magnetic Flux-density Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.11.1 The Magnetic Vector Potential and the Magnetic Flux-density Contour
Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.11.2 Radial and Azimuthal Magnetic Flux-density Distribution . . . . . . . . .
3.11.3 2-D Analytical Analysis vs. 1-D Analytical Analysis . . . . . . . . . . . . .
3.11.4 Harmonic Analysis of the Radial Flux-density . . . . . . . . . . . . . . . .
3.12 The Flux-linkage Calculation for the different Winding Configurations . . . . . .
3.12.1 The Flux-linkage Calulation for the Type O Winding Configuration . . . .
3.12.2 The Flux-linkage Calculation for Type I Winding Configuration . . . . . .
3.12.3 The Flux-linkage Calculation for Type II Winding Configuration . . . . .
3.13 The Back-EMF Calculation for the different Winding Configurations . . . . . . .
3.13.1 The Back-EMF Calculation for the Type O Winding Configurations . . . .
3.13.2 The Back-EMF Calculation for the Type I Winding Configurations . . . . .
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CONTENTS
3.13.3 The Back-EMF Calculation for the Type II Winding Configurations . . . .
3.14 Definition of the General Voltage Constant for the RFAPM Machine . . . . . . . .
3.15 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4
Magnetostatic Analysis of the Armature Reaction Fields
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 The Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Finding the General Solutions for all the Regions . . . . . . . . . . . . . . . . . . .
4.4 Finding the Particular Solutions for Region III . . . . . . . . . . . . . . . . . . . .
4.5 Boundary Equations in order to solve the Armature Reaction Fields . . . . . . . .
4.5.1 The Boundary Equations on the Inner Boundary of Region I . . . . . . . .
4.5.2 The Boundary Equations on the Boundary Between Region I and II . . . .
4.5.3 The Boundary Equations on the Boundary Between Region II and III . . .
4.5.4 The Boundary Equations on the Boundary Between Region III and IV . .
4.5.5 The Boundary Equations on the Boundary Between Region IV and V . . .
4.5.6 The Boundary Equations on the Outer Boundary of Region V . . . . . . .
4.6 Simultaneous Equations to solve for the Armature Reaction Fields . . . . . . . . .
4.7 Obtaining the Solution to the Magnetic Vector Potential and Magnetic Flux-Density
4.7.1 The Magnetic Vector Potential Solution . . . . . . . . . . . . . . . . . . . .
4.7.2 The Magnetic Flux-Density Solution . . . . . . . . . . . . . . . . . . . . . .
4.8 Validation of the Type O Winding Configuration Solution . . . . . . . . . . . . . .
4.8.1 Magnetic Field Solutions of the Type O Winding Configuration . . . . . .
4.8.2 Flux-linkage Calculation of the Type O Winding Configuration . . . . . .
4.8.3 Stator Inductance Calculation of the Type O Winding Configuration . . .
4.9 Validation of the Type I Winding Configuration Solution . . . . . . . . . . . . . .
4.9.1 Magnetic Field Solutions of the Type I Winding Configuration . . . . . . .
4.9.2 Flux-linkage Calculation of the Type I Winding Configuration . . . . . . .
4.9.3 Stator Inductance Calculation of the Type I Winding Configuration . . . .
4.10 Validation of the Type II Winding Configuration Solution . . . . . . . . . . . . . .
4.10.1 Magnetic Field Solutions of the Type II Winding Configuration . . . . . .
4.10.2 Flux-linkage Calculation of the Type II Winding Configuration . . . . . .
4.10.3 Stator Inductance Calculation of the Type II Winding Configuration . . .
4.11 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Torque Calculation
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Average Torque Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Ripple Torque Calculation using the Lorentz Method . . . . . . . . . . . . . . . .
5.3.1 The Simplified Lorentz Method for the Ripple Torque Calculation . . . . .
5.3.2 The Exact Lorentz Method for the Ripple Torque Calculations . . . . . . .
5.4 The Effect of a Reduced Subdomain Model on the Torque Calculations . . . . . .
5.4.1 Using a Subdomain Model with the Permeability in the Rotor Yoke taken
as Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 Using a Subdomain model with the Permeability in the Rotor Yoke taken
as Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Investigating the Effect of Rotor Yoke Saturation on the Analytical Torque Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Investigating the Effect of the Recoil Permeability of the Permanent Magnets on
the Analytical Torque Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Machine Performance Comparison Including the End-turns Effects
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CONTENTS
6.1
6.2
6.3
6.4
6.5
6.6
7
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simple Torque Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . .
End-turn Length Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 End-turn Length Calculation for the Type O Winding Configuration
6.3.2 End-turn Length Calculation for the Type I Winding Configuration .
6.3.3 End-turn Length Calculation for the Type II Winding Configuration
Copper Loss Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Performance Comparison with Fixed Copper-loss Value . . . . . . . . . . . .
6.5.1 Current Density Comparison . . . . . . . . . . . . . . . . . . . . . . .
6.5.2 Torque Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.3 Copper Volume Comparison . . . . . . . . . . . . . . . . . . . . . . .
Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusions and Recommendations
7.1 Introduction . . . . . . . . . . . . . . . . . . . . .
7.2 Dissertation Contribution and Original Content
7.3 Conclusions . . . . . . . . . . . . . . . . . . . . .
7.4 Recommendations for Future Work . . . . . . . .
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Appendices
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A Series–parallel coil combination
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A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
A.2 Total inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
A.3 Total flux-linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
B The Surface Magnetisation Current Density Equivalance
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C Boundary conditions
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C.1 For the magnetic flux density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
C.2 For the magnetic field intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
D Doing Fourier Analysis in Degrees
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D.1 Working with electrical degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
D.2 Working with mechanical degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
E Test Machine Data
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E.1 Test Machine I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
E.2 Test Machine II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
F Permanent Magnetic Circuit Analysis
F.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
F.2 Permanent Magnet Fundamentals . . . . . . . . . . . . . .
F.3 Magnetic Circuit Analysis with Coil Excitation . . . . . . .
F.4 Magnetic Circuit Analysis with PM Excitation . . . . . . .
F.5 NdFeB Permanent Magnets . . . . . . . . . . . . . . . . . .
F.6 Magnetic Circuit Analysis with PM Excitation Continue. . .
F.7 Magnetic Circuit Analysis for the RFAPM Machine. . . . .
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G Armature reaction contour field plots
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G.1 Overlapping, Type O, winding configuration . . . . . . . . . . . . . . . . . . . . . 163
G.1.1 Magnetic vector potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
G.2 Non-overlapping, Type I, winding configuration . . . . . . . . . . . . . . . . . . . 166
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xii
CONTENTS
G.3 Non-overlapping, Type II, winding configuration . . . . . . . . . . . . . . . . . . 170
H Simplified Analytical Analysis
H.1 Solving for all the regions of RFAPM machine simultaneously
H.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
H.1.2 On the boundary between region I and II . . . . . . . .
H.1.3 On the boundary between region II and III . . . . . . .
H.1.4 On the boundary between region III and IV . . . . . . .
H.1.5 On the boundary between region IV and V . . . . . . .
H.1.6 Simultaneous equations to solve . . . . . . . . . . . . .
I
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179
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Brief Python™ Code Desciption
J.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
J.2 Python™ Scripts to call Maxwell® 2D via the COM Interface . . . . . . . . . . . .
J.3 Python™ Scripts to do the Analytical Analysis . . . . . . . . . . . . . . . . . . . .
185
185
185
186
Analytical – FEM Comparison for Test Machine II
I.1 Introduction . . . . . . . . . . . . . . . . . . . . .
I.2 Radial and Azimuthal Flux-density Comparison
I.3 Flux-linkage and Back-EMF Comparison . . . .
I.4 Armature Reaction Flux-density Comparison . .
I.5 Output Torque Comparison . . . . . . . . . . . .
References
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187
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List of Figures
1.1
1.2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
An old air-cored, non-overlapping, AFPM type floppy disk drive. . . . . . . . . . . .
A 3D view of a 16 pole RFAPM machine with non-overlapping windings. . . . . . .
Winding layout for a three-phase RFAPM machine with a Type O winding configuration at time, t = t0 , with k q = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A linearised cross section representation of a coils with multiple layers and turns. . .
The inner and outer turns of coil with a distributed coil side. . . . . . . . . . . . . . .
(a)
Inner turn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(b)
Outer turn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The winding distribution and relative coil position for phase a of an overlap machine with k q = 1 and k∆ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The conductor density distribution, n, for phase a of an overlap machine with k q = 1
and k∆ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The “virtual stator slot” size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The current density distribution for phase a of an overlap machine with k q = 1. . . .
The combined current density distribution for all three phases of a Type O machine
with ωt = 0◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The combined current density distribution for all three phases of a Type O machine
with ωt = 15◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The combined current density distribution for all three phases of a Type O machine
with ωt = 30◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Coil configuration for a three-phase RFAPM machine with non-overlapping, singlelayer (Type I) windings, with k q = 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The winding distribution and relative coil position for phase a of a RFAPM machine
with Type I non-overlapping windings with k q = 21 and k∆ = 23 . . . . . . . . . . . . .
The conductor density distribution for phase a of a Type I concentrated coil machine
with k q = 21 and k∆ = 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The current density distribution for phase a of a Type I non-overlapping winding
configuration machine with k q = 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The combined current density distribution for phase all three phases of a Type I
winding configuration at ωt = 0◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The combined current density distribution for phase all three phases of a Type I
winding configuration at ωt = 15◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The combined current density distribution for phase all three phases of a Type I
winding configuration at ωt = 30◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Coil configuration for a three-phase RFAPM machine with Type II windings with
k q = 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
4
5
12
12
13
13
13
15
15
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17
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19
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20
23
24
25
26
26
27
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xiv
LIST OF FIGURES
2.19 The winding distribution and relative coil position for phase a of a RFAPM machine
with a Type II winding configuration and k q = 21 . . . . . . . . . . . . . . . . . . . . . .
2.20 The conductor density distribution for phase a of a RFAPM machine with a Type II
winding configuration with k q = 21 and k∆ = 23 . . . . . . . . . . . . . . . . . . . . . . .
2.21 The current density distribution for phase a of a Type II winding configuration at
ωt = 0◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.22 The combined current density distribution for phase all three phases of a Type II
winding configuration at ωt = 0◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.23 The combined current density distribution for phase all three phases of a Type II
winding configuration at ωt = 15◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.24 The combined current density distribution for phase all three phases of a Type II
winding configuration at ωt = 30◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 A linear representation of the different regions of the RFAPM machine. . . . . . . . .
3.2 The residual magnetisation distribution of a permanent magnet with respect to φ. .
3.3 The derivative of the residual magnetisation distribution of the permanent magnets
with respect to φ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 The derivative of the remanent flux-density distribution of the permanent magnets
with respect to φ for one pole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Contour plot of the magnetic vector potential obtained analytically. . . . . . . . . . .
3.6 Contour plot of the magnetic vector potential simulated in Maxwell® 2D. . . . . . .
3.7 Contour plot of the magnetic flux-density obtained analytically. . . . . . . . . . . . .
3.8 Contour plot of the magnetic flux-density simulated in Maxwell® 2D. . . . . . . . .
3.9 The radial flux-density distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.10 The azimuthal flux-density distribution. . . . . . . . . . . . . . . . . . . . . . . . . . .
3.11 The magnitude, fundamental component and %THD of the radial flux-density in
the centre of the stator vs. the pole embracing factor. . . . . . . . . . . . . . . . . . . .
3.12 The variation in the flux-density in the centre of the rotor yokes. . . . . . . . . . . . .
3.13 The variation in the shape radial flux-density distribution in the stator windings
shown on the outer –, centre – and inner radius of the stator for the analytical analysis method compared the the FEA analysis done using Maxwell® 2D. . . . . . . . .
3.14 The harmonic spectrum of radial flux-density in the centre of the stator winding
with k m = 0,7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.15 Flux-linkage of a single turn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.16 Comparison of the flux-linkage calculations for the Type O winding configuration. .
3.17 Comparison of the flux-linkage calculations for the Type I winding configuration. . .
3.18 Comparison of the flux-linkage calculations for the Type II winding configuration. .
3.19 Comparison of the back-EMF calculations for the Type O winding configuration. . .
3.20 The harmonic spectrum for the back-EMF obtained from the Maxwell® 2D FEA
results for the Type O winding configuration. . . . . . . . . . . . . . . . . . . . . . . .
3.21 Comparison of the back-EMF calculations for the Type I winding configuration. . . .
3.22 Comparison of the back-EMF calculations for the Type II winding configuration. . .
4.1 Radial Flux-Density Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Azimuthal Flux-Density Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Flux-linkages for phase a, b and c for the Type O winding configuration. . . . . . . .
4.4 Radial flux-density distribution for the Type II winding configuration due to armature reaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Azimuthal flux-density distribution for the Type II winding configuration due to
armature reaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Flux-linkages for phase a, b and c for the Type I winding configuration. . . . . . . . .
4.7 Radial Flux-Density Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8 Azimuthal Flux-Density Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 Flux-linkages for phase a, b and c for the Type II winding configuration. . . . . . . .
29
30
31
32
32
33
42
45
45
46
57
57
58
58
60
60
62
62
63
64
65
67
69
70
71
72
73
74
87
87
90
92
92
95
96
97
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LIST OF FIGURES
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
6.1
6.2
6.3
6.4
6.5
6.6
6.7
A.1
B.1
C.1
C.2
F.1
F.2
F.3
F.4
G.1
G.2
G.3
G.4
G.5
G.6
G.7
G.8
The equivalent circuit of the RFAPM machine. . . . . . . . . . . . . . . . . . . . . . .
The phasor diagram of phase a of the RFAPM machine with Ia and Ea in phase for
generator operation (not to scale). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The phasor diagram of phase a of the RFAPM machine with Ia and Ea in phase for
motor operation (not to scale). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The torque waveform using the simplified Lorentz method for the Type O winding
configuration compared to the Maxwell® 2D simulation. . . . . . . . . . . . . . . . .
The torque waveform using the simplified Lorentz method for the Type I winding
configuration compared to the Maxwell® 2D simulation. . . . . . . . . . . . . . . . .
The torque waveform using the simplified Lorentz method for the Type I winding
configuration compared to the Maxwell® 2D simulation. . . . . . . . . . . . . . . . .
The radial flux-density in at the top, centre and bottom of the stator windings. . . . .
The torque ripple waveforms for the Type O winding configuration. . . . . . . . . .
A FFT of the torque ripple harmonics for the Type O winding configuration. . . . . .
The torque ripple waveforms for the Type I winding configuration. . . . . . . . . . .
A FFT of the torque ripple harmonics for the Type I winding configuration. . . . . .
The torque ripple waveforms for the Type II winding configuration. . . . . . . . . . .
The torque ripple waveforms for the Type II winding configuration. . . . . . . . . . .
A FFT of the torque ripple harmonics for the Type II winding configuration. . . . . .
The torque ripple waveforms for the Type II winding configuration with a rotor
yoke thickness of 20 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The torque ripple waveforms for the Type II winding configuration with a rotor
yoke thickness of 20 mm and a relative recoil permeability for the permanent magnets in Maxwell® 2D, mur,recoil = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A typical coil used in an overlapping winding configuration. . . . . . . . . . . . . . .
A typical coil used in a non-overlapping Type I and Type II winding configuration. .
The equivalent circuit of the RFAPM machine together with the main power flow
components.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The current density versus the coils-side width for the non-overlapping winding
configuration stators with different values of ξ. . . . . . . . . . . . . . . . . . . . . . .
The current density versus the coils-side width for the non-overlapping winding
configuration stators with different values of ξ. . . . . . . . . . . . . . . . . . . . . . .
The current density versus the coils-side width for the non-overlapping winding
configuration stators with different values of ξ. . . . . . . . . . . . . . . . . . . . . . .
The current density versus the coils-side width for the non-overlapping winding
configuration stators with different values of ξ. . . . . . . . . . . . . . . . . . . . . . .
Hypothetical per phase interconnected coil layout of a electrical machine. . . . . . .
The equivalent surface magnetisation current distribution with respect to θ. . . . . .
The boundary condition of ~B across an interface boundary. . . . . . . . . . . . . . . .
~ across an interface boundary. . . . . . . . . . . . . . . .
The boundary condition of H
Magnetic circuit with coil excitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Magnetic circuit with permanent magnet excitation. . . . . . . . . . . . . . . . . . . .
Demagnetisation curve for NdFeB magnets. . . . . . . . . . . . . . . . . . . . . . . . .
Magnetic circuit of the RFAPM machine for one pole pair. . . . . . . . . . . . . . . . .
Contour plot of the magnetic vector potential. . . . . . . . . . . . . . . . . . . . . . .
Contour plot of the magnetic vector potential as simulated in Maxwell® 2D. . . . . .
Contour plot of the magnetic flux density. . . . . . . . . . . . . . . . . . . . . . . . . .
Contour plot of the magnetic flux density as simulated in Maxwell® 2D. . . . . . . .
Contour plot of the magnetic vector potential . . . . . . . . . . . . . . . . . . . . . . .
Contour plot of the magnetic vector potential as simulated in Maxwell® 2D. . . . . .
Contour plot of the magnetic flux density . . . . . . . . . . . . . . . . . . . . . . . . .
Contour plot of the magnetic flux density as simulated in Maxwell® 2D. . . . . . . .
xv
102
103
104
108
109
110
110
112
113
113
114
115
116
116
119
121
126
127
129
132
133
134
135
143
146
148
149
158
159
160
162
163
164
164
165
166
167
168
169
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xvi
LIST OF FIGURES
G.9 Contour plot of the Magnetic Vector Potential. . . . . . . . . . . . . . . . . . . . . . .
G.10 Contour plot of the magnetic vector potential as simulated in Maxwellr 2D. . . . .
G.11 Contour plot of the magnetic flux density. . . . . . . . . . . . . . . . . . . . . . . . . .
G.12 Contour plot of the magnetic flux density as simulated in Maxwellr 2D. . . . . . . .
I.1 The radial flux-density distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I.2 The azimuthal flux-density distribution. . . . . . . . . . . . . . . . . . . . . . . . . . .
I.3 The variation in the flux-density in the centre of the rotor yokes. . . . . . . . . . . . .
I.4 The variation in the shape radial flux-density distribution in the stator windings
shown on the outer –, centre – and inner radius of the stator for the analytical analysis method compared the the FEA analysis done using Maxwell® 2D. . . . . . . . .
I.5 Comparison of the flux-linkage calculations for the Type II winding configuration. .
I.6 Comparison of the back-EMF calculations for the Type II winding configuration. . .
I.7 The radial flux-density distribution due to armature reaction. . . . . . . . . . . . . .
I.8 The azimuthal flux-density distribution due to armature reaction. . . . . . . . . . . .
I.9 The torque ripple waveforms for the Type II winding configuration. . . . . . . . . . .
I.10 A FFT of the torque ripple harmonics for the Type II winding configuration. . . . . .
170
171
172
173
180
180
181
181
182
182
183
183
184
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List of Tables
2.1
3.1
4.1
4.2
4.3
4.4
5.1
5.2
6.1
6.2
6.3
F.1
Comparison between the number of coils, q, the coil-side width factor, k∆, and the
number of turns per coil, N, for the different winding configurations with the same
rotor geometry and stator current and current density. . . . . . . . . . . . . . . . . . .
The governing equations for solving the magnetic vector potential in the different
regions of the RFAPM machine when employing permanent magnet excitation. . . .
The governing equations for the different regions during stator excitation. . . . . . .
The flux-linkage component for each space-harmonic of the Type O winding configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The flux-linkage component for each space-harmonic for the Type I winding configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The flux-linkage components for each space-harmonic for the Type II winding configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison between the result of the various torque calculation methods. . . . . . .
Analytical results with the permeability of the rotor yoke taken as infinity. . . . . . .
Comparison between the Type O, Type I and Type II winding configurations with
the same rotor geometry, stator current and stator current density. . . . . . . . . . . .
A comparison between normalised developed torque for the Type O, Type I and
Type II winding configurations with the same rotor geometry and stator current
density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison between stator copper losses for the different winding configurations
with the same rotor geometry and stator current density. . . . . . . . . . . . . . . . .
Specification of different NdFeB magnets. . . . . . . . . . . . . . . . . . . . . . . . . .
xvii
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42
78
89
94
97
118
118
125
126
130
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Nomenclature
Abbreviations
2-D
Two Dimensional
3-D
Three Dimensional
AC
Alternating Current
AFPM
Axial Flux Permanent Magnet
AFAPM Axial Flux Air-cored Permanent Magnet
CLUI
Command Line User Interface
CVW
Coulomb Virtual Work
DC
Direct Current
EMF
Electromotive Force
EV
Electric Vehicle
FE
Finite Element
FEA
Finite Element Analysis
FEM
Finite Element Modelling
GUI
Graphical User Interface
LHS
Left Hand Side
MST
Maxwell Stress Tensor
[V]
NdBFe Neodymium-Boron-Iron
PCB
Printed Circuit Board
PM
Permanent Magnet(s)
RFAPM Radial Flux Air-cored Permanent Magnet
RHS
Right Hand Side
THD
Total Harmonic Distortion
Roman Symbols
a
number of parallel circuits per phase
A
magnetic vector potential
Am
cross sectional area of each magnet
[m2 ]
Ag
cross sectional area of the airgap
[m2 ]
Awire
cross sectional area of each coil turn
[m2 ]
Bc
flux density in the iron core
xviii
[Wb/m]
[T]
Stellenbosch University http://scholar.sun.ac.za
NOMENCLATURE
xix
Bg
flux density in the airgap
[T]
Bm
flux density in the permanent magnet
[T]
B p1
peak flux density of the first harmonic
[T]
Brem
remanent flux density of the permanent magnets
[T]
Bθ
azimuthal flux density component
[T]
CM
machine constant
dr
magnetic or d-axis of the rotor
ds
magnetic or d-axis of the stator
D
electrical flux density
Ep
peak sinusoidal phase voltage
[V]
h
height/thickness of the stator coils
[m]
hm
magnet height/thickness
[m]
hy
yoke height/thickness
[m]
Hc
magnetic field intensity in the iron core, or,
coercivity of a permanent magnet
[At/m]
Hg
magnetic field intensity in the airgap
[At/m]
Hm
magnetic field intensity in the permanent magnet
[At/m]
Ip
peak sinusoidal phase current
k∆
coil side-width factor
kf
fill factor
km
the magnet (or pole) width to pole pitch ratio
kE
voltage constant
kq
number of coils per pole (per phase)
kQ
number of coils per pole (for all three phase)
kT
torque constant
kw
general winding factor
k w,pitch
winding pitch factor
k w,slot
winding slot width factor
M
magnetisation
[A/m]
M0
residual magnetisation
[A/m]
Mi
induced magnetisation
[A/m]
n
conductor density
N
number of turns per coil
p
number of pole pairs
Pcu
total copper losses
[W]
Pe
electrical power
[W]
Pm
mechanical power
[W]
q
number of coils per phase
Q
total number of coils ( Q = 3q)
Rcoil
total copper resistance per coil
[Ω]
re
the average end-turn conductor radius
[m]
r m|c
the radius as measured to the centre of the magnets
[m]
nominal stator radius
[m]
rn
[C/m2 ]
[A]
[V/rad/s]
[Nm/A]
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xx
Rs
VCu
w
wip
wp
NOMENCLATURE
total copper resistance per phase
total copper volume of the stator
coil side-width of the stator coils
inter-pole gap width
pole width
Other Symbols
ℓ
active copper length of the stator conductors
ℓc
mean magnetic flux path length in the iron core
ℓe
end-turn length of the stator conductors
ℓg
airgap length
ℓn
the average coil-turn length
Greek Symbols
χm
magnetic susceptibility
δ
the angle measured from the centre of the coil side
1
∆
2 coil side-width angle of the stator coils
γ
the angle between the magnetic axis of the rotor and that of the stator
λ
flux-linkage of a single turn
Λ
total flux-linkage per phase
µrecoil
recoil permeability (µ0 µrrecoil )
µrrecoil
relative recoil permeability
φ
azimuthal axis in cylindrical coordinates
ϕ
average flux-linkage per coil
ρcu
copper resistivity
τp
pole pitch angle
τp,res
resultant pole pitch angle
τq
coil pitch angle
τq,res
resultant coil pitch angle
θip
inter-pole width angle
ξ
stator length, ℓ, to the nominal stator radius, rn , ratio
θm
pole/magnet width angle
θp
pole/magnet width angle
θq
coil pitch (or coil span) angle
θr
resultant coil pitch angle
θs
slot width angle
Constants
µ0
permeability of free space (4π × 10−7 )
Accents or attributes
~a
a unit vector
~A
a vector field
Â
amplitude or peak value
[Ω]
[m3 ]
[m]
[m]
[m]
[m]
[m]
[m]
[m]
[m]
[rad]
[rad]
[rad]
[Wb-turns]
[Wb-turns]
[Wb]
[Ωm]
[rad]
[rad]
[rad]
[rad]
[rad]
[rad]
[rad]
[rad]
[rad]
[rad]
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NOMENCLATURE
Subscripts
AR
a quantity, e.g. A, B or H, due to the effect of the Armature Reaction alone
PM
a quantity, e.g. A, B or H, due to the effect of the Permanent Magnets alone
1|2| . . .
specific harmonic component with “1” being the fundamental
a|b|c
phase a, b or c components
|O
|I
|I I
n
t
x|y|z
r|θ|z
for the overlap winding configuration
for the non-overlap Type I winding configuration
for the non-overlap Type II winding configuration
the normal component
the tangential component
the Cartesian coordinates components
the cylindrical coordinates components
Superscripts
I|I I| . . .
the region or domain for which the quantity is valid for
xxi
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C HAPTER 1
Introduction
It is not knowledge, but the act of learning, not possession but the act of
getting there, which grants the greatest enjoyment.
K ARL F RIEDRICH G AUSS
1.1
Introduction
In this dissertation a new type of electrical machine, a Radial Flux Air-Cored Permanent Magnet machine with a Double-sided Rotor and utilising concentrated, non-overlapping windings,
is proposed. The RFAPM machine is a culmination of an electrical machine that
• makes use of the reduction in the cost of rare-earth permanent magnets1 ,
• utilises the increase in strength of rare-earth permanent magnets in order to
• utilise the advantages arising from having air-cored stator windings,
• applies the concept of using concentrated, non-overlapping windings instead of overlapping windings and finally
• addresses the deflection problems experienced by the disc-shaped rotor yokes of Doublesided Rotor Axial Flux Air-Cored Permanent Magnet machines, due to the strong magnetic pull between the permanent magnets situated on the rotor yokes.
1.2
Background Information
The concept of the Double-sided Rotor Radial Flux Air-Cored Permanent Magnet machine, or
RFAPM machine for short, was derived from the Double-sided Rotor Axial Flux Air-Cored Permanent Magnet (AFAPM) machine. The realisation of the RFAPM machine however, would
1 This
point was valid at the start of this research. Since 2011 however, there has been a sharp increase in the price
of rare-earth permanent magnets due to new legislation passed in China, which is the main source of all rare-earth,
permanent magnets in the world.
1
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2
1.2. BACKGROUND INFORMATION
not have been possible without utilising concentrated, non-overlapping windings. The advantages of air-cored windings, specifically with regard to the superior efficiency of air-cored
machines, would also not have been realised without the advancements made in the strength
of rare-earth permanent magnets over the last decade. All of these contributed to the development of the RFAPM machine and will briefly be discussed in this section.
1.2.1 Rare-Earth Permanent Magnets
The discovery of samarium cobalt (SmCo5 ), the first generation of the rare-earth permanent
magnets, paved the way for the manufacturing of extremely compact motors and turbine generators, Gieras and Wing [1, sec. 2.3.3]. In comparison with the original Alnico permanent
magnets, SmCo5 has the advantage of,
•
•
•
•
a high remanent flux density,
an extremely high coercivity,
a linear demagnetisation curve and
a low temperature coefficient
Unfortunately SmCo5 magnets are fairly expensive and thus their application was limited
to “low volume” usage. The second generation of rare-earth permanent magnets, based on
the fairly inexpensive Neodymium (Nd), reduced the cost of NdFeB permanent magnets significantly compared to SmCo5 permanent magnets, whilst retaining all the benefits of SmCo5
as mentioned above, Gieras and Wing [1, sec. 2.3.3]. The reduction in cost of NdFeB permanent magnets, combined with their high remanent flux-density and coercivity values, presented
the opportunity for their usage in the manufacturing of medium to large sized motors and
generators.
1.2.2 Slotless, Toothless and Core-less Windings
Due to the high flux-densities achievable in the air-gap using SmCo5 permanent magnets, the
idea of simply removing the stator teeth to render the machine toothless, or more commonly
known as slotless, was first mentioned in the literature in 1982 by Hesmondhalgh and Tipping
[2]. The main reasons for pursuing the design of slotless machines were, firstly to simplify
the insertion of the stator winding especially for small, high-speed machines and secondly to
eliminate tooth tip saturation and hence lower flux-densities in the stator backing iron.
Due to the high flux-density levels attainable with SmCo5 permanent magnets, Hesmondhalgh and Tipping also investigated the torque generated when removing the stator backing
iron and thus having a completely iron-less stator design. They found that, “. . . the stator iron
gives no inherent benefit as regards the absolute production of torque.” Hesmondhalgh and Tipping,
however, did not recommend the usage of an iron-less stator for practical purposes as the iron
provides a magnetic screen to reduce the eddy current losses in nearby conducting bodies.
In 1988, England [3] proposed a surface-wound, or slotless, permanent magnet motor in
order to, “. . . completely eliminate [the] cogging torque since there are no reluctance change in the
magnetic circuit.’’ England presented a brushless servo motor utilising NdFeB permanent magnets with a ripple torque component of less than 1,5 %.
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CHAPTER 1. INTRODUCTION
3
The idea of utilising slotless, permanent magnet machines for a direct-drive servo application, was proposed by Kaddouri and Le-Huy [4] in 1992, due to the low torque ripple of a these
slotless machines. Their key conclusions, regarding the suitability of these type of machines
for direct-drive applications, can be summarised as follows:
• Although the air-gap flux-density in the slotless motor is less, compared to the slotted motor, it is possible to increase the electrical loading of the machine because of the additional
space available for more copper in order to achieve the same copper current density.
• The comparable efficiency for large diameter, “pancake-like”, slotless motors are similar,
but there is a marked decrease in efficiency for slotless motors as the diameter becomes
smaller.
• Due to the higher electrical loading and the absence of any close thermal contact between
the stator teeth and the stator windings in the slotless machine, particular attention has
to be paid to the thermal design in slotless machines.
The effect of eddy current losses in the stator windings of a toothless (i.e. slotless) machine
was first investigated by Arkadan et al. [5]. The reason for eddy current losses occurring in
toothless machines are that instead of the flux flowing through the stator teeth, the flux now
flows through the windings. Arkadan et al. however found that for a similar 75 kVA conventional designed permanent magnet motor, the core losses in the stator were more than the eddy
current losses in the stator windings for a properly designed toothless machine.
The complete analytical analysis on the armature reaction field and the winding inductance
for a slotless machine was first published by Atallah et al. [6] in 1998. The analytical analysis
method proposed, used of a 2-D distributed current model for the armature reaction field and
winding inductance calculation. This provided a more accurate representation than the 1-D
current sheet method used for conductors situated in stator slots, Zhu and Howe [7].
The advantages and the disadvantages of slotless (or core-less) machines, can thus briefly
be summarised as:
Advantages:
• no cogging torque
• no magnetic pull between the rotor and stator
• lower inductance and hence better voltage regulation for generator applications
• more accurate analytical analysis of the armature reaction field and winding inductance
calculations, are possible
Disadvantages:
• lower flux density and hence a lower power density for the same amount of copper used
• low inductance for drive applications, resulting in a higher switching frequency required
to minimise the ripple current drawn by the machine
• low inductance resulting in high fault currents in generator application
• eddy current losses in the copper windings
1.2.3 The Double-sided Rotor Axial Flux Permanent Magnet Machines
Axial Flux Permanent Magnet (AFPM) Machines are well documented in the literature and
numerous patents exist for the various “flavours” of AFPM machines, Gieras et al. [8]. Of particular interest to us is the Axial Flux Air-cored Permanent Magnet (AFAPM) machine, Wang
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4
1.2. BACKGROUND INFORMATION
[9]. Here the term “air-cored” is yet another term used to describe a slotless, core-less or ironless stator machines, as discussed in section 1.2.2.
One of the problems that AFAPM machines experience, is the deflection of the rotor discs
due to the strong magnetic pull of the permanent magnets, especially with double-sided rotor
machines. This is discussed at length in Wang et al. [10] and Gieras et al. [8, sec. 8.10].
1.2.4 Non-overlapping, Concentrated Windings
The idea of using non-overlapping windings or coils for permanent magnet machines was
first mentioned in the literature by Cros and Viarouge [11] in 2002 for sub-fractional machines.
However these type of sub-fractional machines have been around for much longer, as can be
seen from Figure 1.1, Wong [12], which shows a PCB mounted air-cored, non-overlapping,
AFPM type floppy disk drive.
Figure 1.1: An old air-cored, non-overlapping, AFPM type floppy disk drive, Wong [12].
The main advantage of using non-overlapping windings is the fact that the end-winding
length is reduced. This results in a reduction in the end-winding copper volume and thus of
the total copper volume. Furthermore, non-overlapping coils are easier to manufacture and the
manufacturing process is much simpler to automate. In 2008 Kamper et al. [13] demonstrated a
1 kW Double-Sided Rotor AFAPM machine making use of non-overlapping coils or as Kamper
et al. preferred to call it, “non-overlapping concentrated’ windings, to prevent any ambiguity
in the description of the stator windings.
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5
CHAPTER 1. INTRODUCTION
1.2.5 The Double-sided Rotor Radial Flux Air-Cored Permanent Magnet Machine
The Double-Sided Rotor RFAPM machine was first presented in 2007 by Randewijk et al. [14].
No other reference to this type of machine could be found in the literature prior to this date.
The RFAPM machine is in essence a dual of the AFAPM machine of Kamper et al. [13] as can be
seen in Figure 1.2, that shows a 16 pole RFAPM machine designed for a small wind generator
application, Stegmann [15].
Rotor Outer Magnets
Stator Yoke & Backplate
Rotor Outer Yoke
Stator Coils
Rotor Backplate
Rotor Inner Yoke
Rotor Inner Magnets
Figure 1.2: A 3-D view of a 16 pole RFAPM machine with non-overlapping windings, Stegmann [15].
The main advantage of a RFAPM machine over a AFAPM machine is that the rotor backiron is cylindrically shaped instead of disk shaped. Due to the structural integrity of a cylinder,
the attraction forces between the two rotors does not come into play any more. It was found
instead, that the thickness of the rotor yoke is now determined by the amount of saturation
allowed for in the rotor yoke in order to maintain the desired flux-density in the air-gap, Stegmann and Kamper [16]. Furthermore, Stegmann and Kamper also found that a Double-Sided
Rotor RFAPM machine is approximately 30% lighter than a comparable AFAPM machine. This
makes it ideally suited for low to medium power direct drive wind generators.
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6
1.3
1.3. DISSERTATION PROBLEM STATEMENT
Dissertation Problem Statement
The focus of this dissertation is on a thorough analytical analysis of the Double-Sided Rotor
RFAPM machine proposed by Randewijk et al. [14]. With the RFAPM being an air-cored machine, the feasibility to develop a linear, analytical model, to accurately predict the radial fluxdensity and hence the induced EMF in the stator windings, as well as the accurate calculation
of the developed torque of the machine, needed to be investigated.
The need for a thorough analytical examination of the Double-Sided Rotor RFAPM machine
stemmed from the need to reduce the blind reliance on FEM software to calculate the backEMF and torque produced by these machines. This requirement especially came to the fore in
two parallel MScEng research projects that were started to investigate the feasibility of using
RFAPM machines for small wind turbines, Stegmann [15], as well as the feasibility of using the
RFAPM machine for the primary drive chain in a small Electric Vehicle (EV), Groenewald [17].
The machine parameters for these two RFAPM machines are given in Appendix E, sections E.1
and E.2 respectively.
In these two MScEng research projects, problems were experienced to obtain accurate torque
results using FEM. Excessive ripple torque oscillations were sometimes experienced which took
a considerable amount of time to minimise with constant refinement to the meshing of the machine parts. Reduction in the mesh element size unfortunately also added to the simulation
time, [18]. It was however uncertain as to the exact reason of the remaining ripple torque component specifically with regard to the uncertainty of whether any of the ripple torque could
still be attributed to “computational noise” in the FEM result.
Also, it was disputable which portion of the ripple torque could be attributed to the harmonic content of the radial-flux density distribution and which portion to the slotting effect of
the double-layer, non-overlapping, concentrated coil layout. The requirement for an accurate
analytical model of the RFAPM machine was also necessary in order to reduce the amount of
time spent on successive FEM simulation to obtain the optimum pole arc width of the permanent magnet in order to minimise the harmonic content of the radial flux-density distribution in
the the stator windings.
Although both MScEng research projects used double-layer non-overlapping concentrated
winding designs, it was decided to investigate the use of single-layer, non-overlapping, concentrated winding as well, as the end-windings of the single-layer winding configuration is
even shorter than that of the double-layer winding configuration. Furthermore it was decided
to include a comparison of these two non-overlapping winding configurations with a “hypothetical” concentrated, overlapping winding configuration. This would allow us to gauge the
effectiveness of using non-overlapping winding with respect to the reduction in copper losses
as well as in the reduction in copper volume. It would also allow us to investigate the extent
of how much the developed torque is effected by using non-overlapping windings instead of
overlapping windings.
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CHAPTER 1. INTRODUCTION
1.4
7
Dissertation Work Layout
The layout of the dissertation can be summarised as follows:
Chapter 1: This chapter presents a brief introduction and some background information on
the the Double-Sided Rotor RFAPM machine.
Chapter 2: This chapter will describe the construction of the Double-Sided Rotor RFAPM machine in more detail, especially with regard to the hypothetical overlapping as well
as the single-layer and double-layer non-overlapping winding designs. The winding distribution factors, the winding factors as well as the current density distribution functions for the overlapping, single-layer non-overlapping and double-layer
non-overlapping winding designs, will be deduced.
Chapter 3: In this chapter, the magnetic vector potential field produced by the permanent
magnets of the RFAPM machine will calculated from first principles. For the analytical analysis the machine will be divided into a number of annuli-shaped subdomain regions. The governing Laplace and Poisson equation for each region will
then be solved from the boundary condition values in order to obtain the complete
magnetic vector potential solution for the machine. Maxwell® 2D, a commercial
Finite Element Modelling (FEM) package from Ansys (formerly Ansoft) will be
used to verify the analytical solution. The comparison between the analytical and
the FEM solution will focus on the radial – and azimuthal flux-density distribution
in the stator region. The flux-linkage and induced voltage (or back-EMF) for the
different winding configurations, will also be compared.
Chapter 4: In this chapter, the armature reaction field produced by the different winding configurations will be analysed, using the same methodology as in Chapter 3, but
with the permanent magnets “switched off”. This will also allow us to calculate
the three-phase inductances of the different winding configurations. Due to the
fact that the analytical analysis will be done in 2-D, the effect of the end-windings
on the synchronous inductance will not be taken into account. Again Maxwell® 2D
simulations will be used to verify the analytical results obtained.
Chapter 5: This chapter will focus on the analytical calculation of the torque produced by the
RFAPM machine, with specific emphasis on the ripple torque component of each
winding configuration. In this chapter two FEM packages will be used to verify
the analytical solution for the developed torque, Maxwell® 2D, as well as SEMFEM, which is a proprietary FEM package developed at Stellenbosch University
and written in Fortran with some pre– and post processing done in Python™.
Chapter 6: In this chapter the end-winding effect will be investigated with regard to the difference in the winding resistance and hence copper losses for the different winding
configurations. The effect of varying the current density in the different winding
configurations to obtain equal copper losses will be investigated as well as the
effect this has on the developed torque, copper volume, as well as the torque-percopper-volume ratio as a function of the stator length and the coil-side width.
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8
1.4. DISSERTATION WORK LAYOUT
Chapter 7: The dissertation will conclude with the contributions made by this research, the
conclusions drawn, as well as recommendations for future work to be done on this
topic.
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C HAPTER 2
Theoretical Winding Analysis of the
Double-Sided Rotor Radial Flux
Air-Cored Permanent Magnet Machine
My object has been, first to discover correct principles and then to suggest
their practical development.
J AMES P RESCOTT J OULE
2.1
Introduction
In this chapter, three different winding configurations for the double-sided rotor RFAPM machine will be discussed. These are,
• the overlapping winding configuration, or Type O for short,
• the non-overlapping single-layer winding configuration, or Type I for short, and
• the non-overlapping double-layer winding configuration, or Type II for short.
Although the overlapping winding configuration is technically impossible for the rotor
design shown in Figure 1.2, the discussion thereof will be done as a reference against which
the Type I and Type II winding configurations can be bench-marked.
Our discussion will focus on the differences in the winding factors, the conductor density
distributions and the current density distributions. The fundamental winding factor for each
winding configuration will first be deduced, from first principles, from the flux-linkage calculation. The conductor density distribution will then be used to calculate the winding factors
for all the winding space harmonics. It will be shown that, for the overlapping winding configuration, the working harmonic is the fundamental, or first harmonic of the conductor density
9
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10
2.2. TERMINOLOGY USED FOR THE ANALYSIS
distribution, whereas for the non-overlapping winding configurations, the working harmonic
is the second harmonic of the conductor density distribution.
The conductor density distribution will also be used in Chapters 3 and 4 to calculate the
flux-linkage for the different winding configurations, from the magnetic vector potential caused
by the permanent magnets and the armature windings respectively. This is necessary in order
to calculate the induced voltage, or back-EMF, for the different winding configurations as well
as the different three-phase synchronous inductances respectively.
Although it might not be apparent to the reader why the current density and specifically
the three-phase current density distributions are deduced in this chapter, the reason for that is
that it follows logically from the conductor density distribution calculations. The three-phase
current density distribution will be used in Chapter 5 where the torque calculation, using the
Lorentz method will be discussed, for the different winding configurations.
2.2
Terminology used for the Analysis
Before we begin our analysis of the double-sided rotor RFAPM machine, it is necessary to
define a few quantities with which to describe the working and layout of the machine.
The speed by which the magnetic flux cuts the windings, is determined by the mechanical
speed at which the rotor is turning, ωmech . The frequency, ω in [rad/s], of the induced voltage
and that of the subsequent phase current, is thus determined by the number of pole pairs of
the machine, p, and the speed at which the rotor is turning and can be calculated as follows,
ω = pωmech .
(2.1)
The North (N) and South (S) poles of the RFAPM machine are equally spaced, which implies
◦
around the circumference
that the magnetic field produced by the poles are repeated every 360
p
of the rotor. With the magnetic field being periodic, it is easier to refer to the magnetic field as
being repeated every 360◦ electrical. The relationship between the electrical degrees and the
actual mechanical degrees is therfore determined by the number of pole pairs. Expressed in
radians,
2π
[elec.] =
2π
p
[mech.].
(2.2)
With the N– and S–poles equally spaced, the pole pitch of a machine can thus be described
in terms of either mechanical or electrical degrees as follows
τp =
π
p
[mech.]
or
= π [elec.] ,
(2.3)
(2.4)
with the resultant pole pitch, i.e. the angle between adjacent N– or S–poles,
τp,res = 2τp
= 2π [elec.] .
(2.5)
(2.6)
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CHAPTER 2. THEORETICAL WINDING ANALYSIS OF THE DOUBLE-SIDED ROTOR RFAPM MACHINE
11
The number of coils-per-phase per pole pair, is given by,
q = kq p .
(2.7)
The total number of coils, for a three-phase system is, thus, simply,
Q = 3q .
(2.8)
The coil pitch of the machine is defined as
τq =
π
q
[mech.]
= π [elec.]
(2.9)
(2.10)
and the resultant coil pitch as
τq,res =
2π
q
[mech.]
= 2π [elec.] .
(2.11)
(2.12)
A coil that spans 180◦ electrical, or put differently, if the coil pitch is equal to half the resultant
coil pitch, it is known as a full-pitch winding, Fitzgerald et al. [19, chap. 4].
2.3
Overlapping (Type O) Winding Configuration
For a RFAPM machine with overlapping (Type O) windings, the number of coils per phase
and the number of permanent magnet pole pares will be the same so that from (2.7), we can
say k q = 1. This implies that a complete winding section will be equal to 360◦ electrical1 . A
RFAPM machine with overlapping windings’ winding configuration is thus such that we have
one phase coil per pole pair and thus for a three-phase machine, there will be three coils in total
per pole pair.
In Figure 2.1 a linearised representation of two complete winding sections of a RFAPM
machine with overlapping windings is shown relative to the permanent magnet poles at time,
t = t0 . Furthermore, the magnetic or d-axis of the rotor, dr , is shown relative to the magnetic
axis, or d-axis of phase a of the stator, ds , with γ the electrical angle between the two.
2.3.1 Flux-linkage of the Type O Winding Configuration
For air cored machines, the coil-side widths are usually much wider than for iron cored machines2 . Thus, when the flux-linkage of an air cored coil is calculated, the distributed nature of
the coil sides needs to be take into account, similar to the calculations for the flux-linkage in an
AFAPM machine, see Kamper et al. [13].
1 For
2 For
the non-overlapping windings we will see that this is not the case.
non-overlapping winding configurations, this is even more so.
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12
2.3. OVERLAPPING (TYPE O) WINDING CONFIGURATION
ds
dr
B̂r1 cos( pφ − pωmech t − γ)
ωmech
2∆
q
S
3
2
a
1
3’
2’
3
1’
c′
2
1
N
1’
2’
a′
b
3’
1
2
3
1’
2’
3’
b′
c
N
S
3
2
1
2’
c′
a
S
1’
3
2
1
1’
2’
a′
b
3’
1
2
3
c
1’
2’
3’
φ
b′
N
γ
p
π
q
3’
N
S
π
p
π
q
Figure 2.1: Winding layout for a three-phase RFAPM machine with a Type O winding configuration at
time, t = t0 , with k q = 1.
In Figure 2.1, the coils are depicted as if they consist of multiple layers with a single turn
per layer, while is fact there are actually multiple turns per layer, as shown in Figure 2.2. In
order to simplify our analysis, however, we assume that the radial flux density “seen” by each
turn in a specific layer is the same and equal to the radial flux density in the centre of the coil
in the radial direction (i.e. at r = rn ). We therefore assume that the coils are infinitely thin with
all the layers of the coil located at rn when we calculate the total flux-linkage for each coil.
Furthermore, if we assume that the machine is rotating at a constant speed, the flux-linkage
can be considered to be sinusoidal. It is therefore only necessary to calculate the maximum,
or peak value of the flux-linkage that would occur when the magnetic axis of the coil, i.e. ds ,
is aligned with the magnetic axis of the permanent magnets, i.e. dr . In order to calculate the
total flux-linkage of each phase, we first need to calculate the flux-linkage of a single turn. With
each side of the turn at an arbitrary angle, say δ, measured from the centre of each coil side in
“n” turns per layer
2∆
q
2∆
q
h
“m” layers
rn
“m” layers
Figure 2.2: A linearised cross section representation of a coils with multiple layers and turns.
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CHAPTER 2. THEORETICAL WINDING ANALYSIS OF THE DOUBLE-SIDED ROTOR RFAPM MACHINE
13
the azimuthal direction as shown in Figure 2.3 (a) and (b) for an “inner” and an “outer” turn
respectively, the flux-linkage for a single turn can be calculated as follows,
τq
2
−δ
λ |O ( δ ) =
Z ℓZ
=
rn B̂r1
p
=
rn B̂r1
p
=
2rn B̂r1
p
=
2rn ℓ B̂r1
cos( pδ)
p
0
τ
− 2q
+δ
B̂r1 cos( pφ)rn dφdz
Z ℓ
0
Z ℓ
0
Z ℓ
0
(2.13)
pπ
pπ
sin( 2q − pδ) − sin(− 2q + pδ) dz
(2.14)
sin( π2 − pδ) − sin(− π2 + pδ) dz
(2.15)
cos( pδ)dz
(2.16)
(2.17)
with B̂r1 the peak value of the fundamental component of the radial flux-density distribution1
as shown in Figure 2.1.
ℓ
ℓ
δ
δ
δ
π
q
rn
δ
π
q
rn
(a) Inner turn.
(b) Outer turn.
Figure 2.3: The inner and outer turns of coil with a distributed coil side.
Next the average flux-linkage for each coil is calculated, which in effect is equal to the
average flux-linkage of each turn over the distributed coil side starting from the minimum
“inner” turn to the maximum “outer” turn as shown in Figure 2.3 (a) and (b) respectively. The
average flux-linkage for each coil can therefore be calculated as follows,
1 It
∆
q
ϕ |O =
q
2∆
Z
=
q
2∆
Z
=
2qrn ℓ B̂r1
p∆
sin( q ) .
2
∆p
− ∆q
∆
q
− ∆q
λ|O (δ) dδ
(2.18)
2rn ℓ B̂r1
cos( pδ) dδ
p
(2.19)
(2.20)
will be shown in Chapter 3 that the flux-density distribution in the centre of the windings can indeed be approximated to be sinusoidal with careful selection of the permanent magnet’s pole arc width.
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2.3. OVERLAPPING (TYPE O) WINDING CONFIGURATION
By multiplying the average flux linking for each coil with the total number of turns in each
coil, N, we get the total flux-linkage per coil. The total flux-linkage per phase will however deq
pend on the number of parallel connected coils, a, and the number of series connected coils, a ,
which is obtained by dividing the total number of coils by the total number of parallel circuits.
From Figure 2.1 we can see that due to the symmetrical spacing of the coils, each coil in the
phase winding layout will have the same flux-linkage. Thus, if each coil also have the same
current passing through it, it will have the same inductance. From Appendix A we can see that
the total flux-linkage per phase only depend on the number of series connected coils, so that
for the overlapping winding configuration the total flux-linkage per phase can be expressed as,
Λ a,b,c|O =
q
· N ϕ |O
a
∆
2rn ℓ Nk q B̂r1 sin( kq )
=
·
,
a
( k∆q )
(2.21)
(2.22)
with k q as defined in (2.7).
From (2.22), the coil side width – or “virtual” slot width factor for a RFAPM machine with
overlapping winding, can be defined as
k w,slot|O =
sin( k∆q )
( k∆q )
,
(2.23)
so that (2.22) can be rewritten as
Λ a,b,c|O =
2rn ℓ N B̂r1 k q k w,slot|O
.
a
(2.24)
2.3.2 Conductor Density Distribution of the Type O Winding Configuration
The winding distribution for phase a of an overlap RFAPM machine is shown in Figure 2.4.
From the winding distribution and with the coil-side width equal to 2∆
q , the conductor density
in each “virtual” stator slot will be equal to
|na | =
qN
.
2∆
(2.25)
This allows us to define a conductor density distribution, see Slemon [20, section 5.1], for
phase a of an overlap RFAPM machine as is graphically shown in Figure 2.5. Here a positive
conductor density represents conductors coming out of the page and a negative conductor
density, conductors going into the page.
The conductor density distribution provides an alternative way in which to calculate the
flux-linkage directly from the magnetic vector potential, as will be shown in Chapter 3, section
3.12 due to the permanent magnets and in Chapter 4, section 4.8.2 due to the armature reaction.
The former will be used to calculate the back-EMF of the machine, see section 3.13 and the latter
to calculate the three-phase synchronous inductance of the machine, section 4.8.3.
The Fourier expansion for the conductor density distribution for phase a can be written as
∞
n a |O ( φ ) =
∑
m =1
bmn|O sin(mqφ)
(2.26)
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CHAPTER 2. THEORETICAL WINDING ANALYSIS OF THE DOUBLE-SIDED ROTOR RFAPM MACHINE
ds
15
2∆
q
3 2 1
1’ 2’ 3’
3 2 1
1’ 2’ 3’
3 2 1
1’ 2’ 3’
a
a′
a
a′
a
a′
π
q
φ
2π
q
Figure 2.4: The winding distribution and relative coil position for phase a of an overlap machine with
k q = 1 and k∆ = 1.
n
qN
2∆
2∆
q
a
a
a
φ
a′
a′
π
q
a′
2π
q
Figure 2.5: The conductor density distribution, n, for phase a of an overlap machine with k q = 1 and
k∆ = 1.
with
b m n |O = −
2qN
sin m π2 sin(m∆) .
mπ∆
(2.27)
From (2.27) as well as for the subsequent conductor density distributions for the nonoverlapping winding configurations that will be defined later, we are able to define a number of
winding factors that takes cognisance of all the harmonics present in the winding distribution,
similar to Holm et al. [21, Appendix A]. The first winding factor for the overlapping winding
configuration is the pictch factor and is defined as,
k w,pitch,m|O = sin m π2
.
(2.28)
With m = 1, k w,pitch,m|O = 1 which is consistent with (2.24) where only the fundamental component of the radial flux density was considered.
The second winding factor for the overlapping winding configuration, the coil side width
– or “virtual” slot width factor, is defined as
k w,slot,m|O =
sin(m∆)
.
m∆
(2.29)
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2.3. OVERLAPPING (TYPE O) WINDING CONFIGURATION
The slot width factor, k w,slot,m|O with m = 1 is the exactly the same as slot width factor in (2.23)
where, once again, only the fundamental component of the radial flux density was considered.
In terms of the winding factors, (2.27) can now be written as,
b m n |O = −
2qN
· k w,m|O
π
(2.30)
with
k w,m|O = k w,pitch,m|O · k w,slot,m|O .
(2.31)
Another interesting thing to note, is that if we were able to represent the Fourier expansion
up to infinity, we will note that,
Z − π +∆
2q
q
π
− ∆q
− 2q
n a|O (φ)dφ ≈
Z 0
n a|O (φ)dφ = N
(2.32)
n a|O (φ)dφ ≈
Z τq
n a|O (φ)dφ = − N .
(2.33)
−τq
and
Z
π
∆
2q + q
π
∆
2q − q
0
which is as expected.
2.3.3 Current Density Distribution of the Type O Winding Configuration
In this section the current density distribution and more specifically, the three-phase current
density distribution for the Type O winding configuration will be deduced. This will be used in
Chapter 4 to calculate the armature-reaction field, Zhu and Howe [7] and Zhu et al. [22], using
subdomain analysis, Zhu et al. [23]. The three-phase current density distribution will also be
used to calculate the developed torque for the Type O winding configuration in Chapter 5, by
making use of the Lorentz’ law method, similar to what was done by Holm [24].
The cross section of each “virtual” stator slot or coil-side, is shown in Figure 2.6 and can be
calculated as follows,
Aslot =
=
Z ro Z + ∆
q
ri
− ∆q
rφdrdφ
(ro2 − ri2 )∆
,
q
(2.34)
(2.35)
or in terms of the nominal radius and slot height, as
Aslot =
2rn h∆
.
q
(2.36)
This implies that the magnitude of the current density in each “virtual stator slot”, i.e. the
“slot” current density,
Ni a (t)
aAslot
qNi a (t)
=
2arn h∆
| Jza | =
(2.37)
(2.38)
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CHAPTER 2. THEORETICAL WINDING ANALYSIS OF THE DOUBLE-SIDED ROTOR RFAPM MACHINE
2∆
q
17
h
ro
ri
Figure 2.6: The “virtual stator slot” size.
or in terms of the conductor density,
| Jza | =
|n a | i a (t)
·
.
rn h
a
(2.39)
The current density distribution for phase a of an overlap machine with k q = 1 is shown in
Figure 2.7.
~J
qNi
2arn h∆
2∆
q
a
a
a
φ
a′
π
q
a′
a′
2π
q
Figure 2.7: The current density distribution for phase a of an overlap machine with k q = 1.
The Fourier expansion for the current density distribution for phase a can thus be written
as
Jza|O =
1 ∞
bmn|O i a (t) sin(mqφ) .
arn h m∑
=1
(2.40)
This allows us to define the resultant current density distribution for all three phase, as
Jz|O = Jza|O + Jzb|O + Jzc|O
h
1 ∞
b
=
mn|O sin mqφ i a ( t ) + sin mq ( φ −
∑
arn h m=1
(2.41)
i
4π
2π
3 ) ib ( t ) + sin mq ( φ − 3 ) ic ( t ) .
(2.42)
In order to lower the acoustic noise associated with air cored PM machines, large external
inductors are required, Rossouw and Kamper [25]. Only the fundamental component of the
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18
2.3. OVERLAPPING (TYPE O) WINDING CONFIGURATION
current will therefore be considered. Thus, with (say)
i a (t) = I p cos(ωt)
(2.43)
and for a perfectly balanced three-phase load,
ib (t) = I p cos(ωt −
ic (t) = I p cos(ωt −
2π
3 )
4π
3 )
and
(2.44)
,
(2.45)
the steady state solution of the current density at ωt = 0◦ can be written as,
i a (0) = I p and
(2.46)
i b (0) = i c (0)
(2.47)
Ip
.
(2.48)
2
This will produce a three-phase current density distribution at ωt = 0◦ , as depicted in
Figure 2.8. Unfortunately the three-phase current density distribution is time dependent so
that the spacial current density distribution will vary with time. To illustrate this, the spacial
distribution at ωt = 15◦ and ωt = 30◦ is shown in Figure 2.9 and 2.10 respectively.
=−
~J
qNi
2arn h∆
2∆
q
a
b′
c′
a
b′
c
b
b′
c′
c
b
a′
π
q
a
c′
c
b
a′
φ
a′
2π
q
Figure 2.8: The combined current density distribution for all three phases of a Type O machine with
ωt = 0◦ .
By substituting (2.43), (2.44) and (2.45) in (2.42) we get
h
Ip ∞
sin(mqφ) cos(ωt) + sin mq(φ −
Jz|O =
b
m
n |O
arn h m∑
=1
i
4π
sin mq(φ − 4π
)
cos
(
ωt
−
)
3
3
2π
3 )
cos(ωt −
which, after some careful manipulation, will simplify to
∞
3I p
2arn h ∑ bmn|O sin mqφ + ωt for m = 3k − 2, k ∈ N1
m =1
Jz|O =
∞
3I
p
2arn h ∑ bmn|O sin mqφ − ωt for m = 3k − 1, k ∈ N1
m =2
2π
3 )+
,
(2.49)
(2.50)
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CHAPTER 2. THEORETICAL WINDING ANALYSIS OF THE DOUBLE-SIDED ROTOR RFAPM MACHINE
19
~J
qNi
2arn h∆
a
2∆
q
c′
a
b′
a
c′
b′
c′
b′
b
b
a′
c
a′
π
q
φ
b
c
a′
c
2π
q
Figure 2.9: The combined current density distribution for all three phases of a Type O machine with
ωt = 15◦ .
~J
qNi
2arn h∆
a
2∆
q
c′
b′
a
c′
a
b′
b
c′
b′
b
b
φ
a′
a′
c
π
q
a′
c
c
2π
q
Figure 2.10: The combined current density distribution for all three phases of a Type O machine with
ωt = 30◦ .
or in terms of the winding factors, to
Jz|O =
3qI p N
− arn hπ
3qI p N
− arn hπ
∞
∑
m =1
∞
∑
m =2
k w,m|O sin mqφ + ωt
k w,m|O sin mqφ − ωt
for m = 3k − 2, k ∈ N1
.
(2.51)
for m = 3k − 1, k ∈ N1
From (2.51) it can be seen that the m = 3k − 2, k ∈ N1 time harmonics, which also includes
the fundamental component, will rotate in the abc direction (appear to be moving to the “right”
in Figures 2.8, 2.9 and 2.10) were as the m = 3k − 1, k ∈ N1 time harmonics will rotate in the
acb direction.
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2.4. NON-OVERLAPPING SINGLE-LAYER (TYPE I) WINDING CONFIGURATION
2.4
Non-Overlapping Single-layer (Type I) Winding Configuration
The three-phase coil configuration for a RFAPM machine with non-overlapping single-layer
(Type I) windings is shown in Figure 2.11. The only difference between a RFAPM machine with
overlapping, Type O, windings and that with non-overlapping single-layer, Type I, windings
is with respect to the winding configuration, i.e. the stator layout. The pole pitch and the
resultant pole pitch for a Type I machine is therefore similar to that of a Type O machine, see
equations (2.4) and (2.6) respectively.
ds
dr
B̂r1 cos( pφ − pωmech t − γ)
ωmech
2∆
q
S
4
3
2
1
1’
2’
N
3’
4’
4
a′
a
2
1
1’
S
γ
p
3’
2’
c′
c
N
π
Q
3
S
4’
4
3
N
2
b
N
1
1’
2’
3’
4’
φ
b′
S
π
p
π
Q
Figure 2.11: Coil configuration for a three-phase RFAPM machine with non-overlapping, single-layer
(Type I) windings, with k q = 12 .
However for the Type I windings configuration, we now have, from (2.7), that
q
p
1
= .
2
kq =
(2.52)
(2.53)
This implies that we now not only have half the number of coils per phase as compared
to the Type O winding configuration, but also that, for the Type I winding configuration, each
winding section now spans 720◦ electrical. This can also be seen from the definition of the
resultant coil pitch of the Type I winding configuration which, although defined similarly to
the Type O winding’s resultant coil pitch, yields twice the angle.
τq,res| I =
2π
q
[mech.]
= 4π [elec.] .
(2.54)
(2.55)
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CHAPTER 2. THEORETICAL WINDING ANALYSIS OF THE DOUBLE-SIDED ROTOR RFAPM MACHINE
21
In order to do a full comparative analysis of the winding configuration of the Type I windings, we define a new set of “degrees” namely the winding section degrees, or “wsec.” for short.
This allows us to redefine the resultant coil pitch angle as,
τq,res| I = 2π
[wsec.] .
(2.56)
For the Type I winding configuration the definition for the coil pitch (or maybe less ambiguous, the average coil span) is defined slightly different from the Type O winding configuration,
as
π
[mech.] ,
Q
2π
=
[elec.] or
3
π
[wsec.] .
=
3
τq| I =
(2.57)
(2.58)
(2.59)
The Type I winding configuration can thus be regarded as having fractional-pitch coils, Fitzgerald et al. [19, Appendix. B], as τq| I < 12 τq,res| I .
An important difference between a RFAPM machine with overlapping windings shown in
Figure 2.1 and that of a RFAPM machine with non-overlapping windings, e.g. the single-layer
(Type I) and the double-layer (Type II) shown in Figure 2.11 and Figure 2.181 respectively, is
that the whole surface area of the stator in the overlapping winding configuration is utilised by
the stator windings compared to only ≈ 32 (for illustration purposes only) of the Type I and II
winding configuration.
The maximum coil-side width in winding section degrees (i.e. terms of the resultant coil
pitch), can thus be calculated as
2π 1
2∆ max
=
·
q
q 6
(2.60)
so that
π
[wsec.]
6
π
=
[elec.]
3
π 1
[mech.]
= ·
6 q
∆ max =
(2.61)
(2.62)
(2.63)
The coil-side width factor, k∆, is now defined as,
k∆ =
∆
,
∆ max
(2.64)
and can be used not only to describe the coil-side width angle of the winding configuration,
but also to describe the space utilisation of the stator’s winding configuration.
Another important difference is the phase sequence of the different phase windings. If one
were to compare the Type O winding configuration shown in Figure 2.1 with that of the Type I
winding configuration shown in Figure 2.11, one would see that the sequence of the different
1 See
section 2.5.
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2.4. NON-OVERLAPPING SINGLE-LAYER (TYPE I) WINDING CONFIGURATION
phase windings’ magnetic axes for the Type O windings is ABC compared to that of the Type I
windings, which is ACB. However, on closer inspection we can see that although phase C’s
magnetic axis is located at 120 [wsec. ◦ ] from phase A, this corresponds to 240 [elec. ◦ ]. The
same goes for phase B’s magnetic axis which is located at 240 [wsec. ◦ ] which corresponds to
480 [elec. ◦ ] which is actually the same as 120 [elec. ◦ ].
2.4.1 Flux Linkage of the Type I Winding Configuration
The total flux-linkage per phase for the Type I winding is calculated in exactly the same manner
as for the Type O winding. Once again we need to align the magnetic– or d-axis of the coil with
the magnetic– or d-axis of the magnets in order to calculate the maximum or peak value of
flux-linkage. This allows us to calculate the flux-linkage of a single turn with each turn side at
an angle of δ from the centre of each coil side as follows,
π
2Q − δ
λ| I (δ) =
Z ℓZ
=
rn B̂r1
p
=
rn B̂r1
p
=
2rn B̂r1
p
=
2rn ℓ B̂r1
sin( π3 ) cos( pδ)
p
0
B̂r1 cos( pφ)rn dφdz
(2.65)
pπ
pπ
sin( 6q − pδ) − sin(− 6q + pδ) dz
(2.66)
sin( π3 − pδ) − sin(− π3 + pδ) dz
(2.67)
π
+δ
− 2Q
Z ℓ
0
Z ℓ
0
Z ℓ
sin( π3 ) cos( pδ)dz
0
(2.68)
(2.69)
again assuming a sinusoidal radial flux-density distribution.
Taking into account the number of turns per coil, as well as the total number of series and
parallel connected coils, the total flux-linkage per phase can be calculated as follows,
Λ a,b,c| I =
=
q qN Z
a
2∆
q qN Z
a
2∆
∆
q
− ∆q
∆
q
− ∆q
λ| I (δ) dδ
(2.70)
2rn ℓ B̂r1
sin( π3 ) cos( pδ) dδ
p
(2.71)
q qN 4r ℓ B̂
n
r1
p∆
sin( π3 ) sin( q )
=
2
a
2∆
p
sin( k∆q )
2rn ℓ Nk q B̂r1 sin π3
=
·
.
a
( k∆ )
(2.72)
(2.73)
q
If we compare the total flux-linkage in (2.73) for the Type I winding configuration with that
of the Type O winding configuration given by (2.22), we note the additional sin( π3 ) term. This
is due to the fact that the Type I windings are a fractional pitch windings compared to the full
pitch windings of the Type O winding configuration. We can therefore define the winding pitch
factor for the Type I (non-overlapping) winding configuration, as
k w,pitch| I = sin
π
3
.
(2.74)
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CHAPTER 2. THEORETICAL WINDING ANALYSIS OF THE DOUBLE-SIDED ROTOR RFAPM MACHINE
23
This implies that for the Type O (overlapping) winding configuration we can define the
winding pitch factor, as
k w,pitch|O = 1,0 ,
(2.75)
as was also obtained in (2.28) with m = 1.
Furthermore, it is interesting to note that the “virtual” slot width factor for the non-overlapping
Type I winding is exactly the same as for the overlapping winding configuration, so that in general the winding slot factor could be expressed as
sin k∆q
k w,slot = k w,slot| I = k w,slot|O =
.
(2.76)
∆
kq
Taking into account the winding factors mentioned above for the various winding configurations, we can rewrite (2.24) and (2.73) into a general format, so that
Λ a,b,c|O = λ a,b,c| I = 2rn ℓ N B̂r1 k q k w,pitch k w,slot .
(2.77)
2.4.2 Conductor Density Distribution of the Type I Winding Configuration
The winding distribution for phase a of a RFAPM machine with Type I windings is shown
in Figure 2.12. From the winding distribution and the coil-side width, the conductor density
distribution can be obtained as shown in Figure 2.13. Once again, a positive conductor density
represents conductors coming out of the page and a negative conductor density, conductors
going into the page.
ds
2∆
q
4 3 2 1
1’ 2’ 3’ 4’
4 3 2 1
1’ 2’ 3’ 4’
a
a′
a
a′
π
Q
φ
2π
q
Figure 2.12: The winding distribution and relative coil position for phase a of a RFAPM machine with
Type I non-overlapping windings with k q = 12 and k∆ = 32 .
As mentioned in section 2.3.2, the conductor density distribution allows us to calculate the
flux-linkage due to the permanent magnets, as well as the flux-linkage created by the armature
reaction. Again the former will be used to calculate the back-EMF of the machine and the
latter to calculate the three-phase synchronous inductance of the machine when using a Type I
winding configuration.
The Fourier expansion for the conductor density distribution for phase a can be written as
∞
n a| I ( φ ) =
∑
m =1
bmn| I sin(mqφ)
(2.78)
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2.4. NON-OVERLAPPING SINGLE-LAYER (TYPE I) WINDING CONFIGURATION
n
qN
2∆
2∆
q
a
a
φ
a′
a′
π
Q
2π
q
Figure 2.13: The conductor density distribution for phase a of a Type I concentrated coil machine with
k q = 12 and k∆ = 32 .
with
2qN
sin(m π6 ) sin(m∆)
mπ∆
2qN
=−
· k w,m| I
π
bm n | I = −
(2.79)
(2.80)
which is similar to (2.30) with
k w,m| I = k w,pitch,m| I · k w,slot,m| I ,
(2.81)
where
k w,pitch,m| I = sin(m π6 )
(2.82)
and
k w,slot,m| I = k w,slot,m|O =
sin(m∆)
.
m∆
(2.83)
If we now compare (2.82) with (2.74) we see that they are equal only when m = 2. The same
is true for (2.83) compared to (2.76). This can be attributed to the fact that for a Type I RFAPM
machine, the resultant coil pitch, see (2.55) is twice the resultant pole pitch (2.6). Thus when
we only consider the fundamental radial flux density component, only the second harmonic of
the conductor density distribution, i.e. m = 2, will contribute to the flux-linkage.
2.4.3 Current Density Distribution of the Type I Winding Configuration
In this section the three-phase current density distribution for the Type I winding configuration will be deduced. As was mentioned in section 2.3.3, this will be used in Chapter 4 to
calculate the armature-reaction field as well as the developed torque for the Type I winding
configuration, in Chapter 5.
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CHAPTER 2. THEORETICAL WINDING ANALYSIS OF THE DOUBLE-SIDED ROTOR RFAPM MACHINE
25
With the “virtual stator slot” defined similarly to that shown in Figure 2.61 the current
density distribution for phase a of a RFAPM machine with Type I windings, will have the form
as shown in Figure 2.14 with k q = 12 .
~J
qNi
2arn h∆
2∆
q
a
a
φ
a′
π
Q
a′
2π
q
Figure 2.14: The current density distribution for phase a of a Type I non-overlapping winding configuration machine with k q = 12 .
Take note that the only difference between the magnitude of the conductor density distribution in Figure 2.13 and the current density distribution in Figure 2.14, is the “factor” rni h . This
implies that the Fourier expansion for the current density distribution for phase a of a RFAPM
machine with a Type I winding configuration can be written as:
Jza| I =
1 ∞
bmn| I sin(mqφ)i a (t)
arn h m∑
=1
(2.84)
Using the same approach as with the Type O machine and taking cognisance of the fact
that the phase sequence is now ACB, the resultant steady-state three phase current density
distribution from (2.51) will be equal to,
∞
3qI p N
−
arn hπ ∑ k w,m| I sin mqφ + ωt for m = 3k − 1, k ∈ N1
m =1
(2.85)
Jz| I =
∞
3qI
N
p
k
sin
mqφ
−
ωt
for
m
=
3k
−
2,
k
∈
N
−
1
arn hπ ∑ w,m| I
m =2
and is shown for ωt = 0◦ in Figure 2.15.
The main difference between the three-phase current density distribution for the Type O
winding configuration, equation (2.51) and and the three-phase current density distribution
for the Type I winding configuration shown in (2.85), is that for the Type O winding configuration, the working harmonic is the fundamental component where as for the Type I winding
configuration, the working harmonic is actually the second harmonic. This implies that for the
Type O winding configuration, the 3k − 2 harmonics, which include the fundamental, is moving to the “right” as was shown in Figures 2.8, 2.9 and 2.10 (i.e. rotating counter clockwise). For
1 See
section 2.3.3.
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2.5. NON-OVERLAPPING DOUBLE-LAYER (TYPE II) WINDING CONFIGURATION
the Type I winding configuration, however, the 3k − 1 harmonics, which include the working
(or second) harmonic, will be moving to the “right” as shown in Figures 2.15, 2.16 and 2.17.
~J
qNi
2arn h∆
2∆
q
a
a
b′
c′
c
φ
b
a′
π
Q
a′
π
Q
Figure 2.15: The combined current density distribution for phase all three phases of a Type I winding
configuration at ωt = 0◦ .
~J
qNi
2arn h∆
2∆
q
a
a
c′
b′
φ
b
c
a′
π
Q
a′
π
Q
Figure 2.16: The combined current density distribution for phase all three phases of a Type I winding
configuration at ωt = 15◦ .
2.5
Non-Overlapping Double-layer (Type II) Winding
Configuration
As shown in Figure 2.18, the pole pitch, τp and the resultant pole pitch, τp,res , of a RFAPM
machine with Type II winding configuration is similar to that of a RFAPM machine with a
Type O winding configuration. Also for the Type II winding configuration the number of coils
per phase is half the number of pole pair, so that k q = 21 , which is the same as for the Type I
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CHAPTER 2. THEORETICAL WINDING ANALYSIS OF THE DOUBLE-SIDED ROTOR RFAPM MACHINE
27
~J
qNi
2arn h∆
2∆
q
a
c′
a
b′
b
a′
φ
a′
c
π
Q
π
Q
Figure 2.17: The combined current density distribution for phase all three phases of a Type I winding
configuration at ωt = 30◦ .
winding configuration. This is also true for the resultant coil pitch of the Type II winding
configuration,
τq,res| I I = 4π
[elec.]
(2.86)
= 2π [wsec.]
(2.87)
which is the same as for the Type I winding configuration.
The main difference between the Type I and the Type II winding configurations is that
the Type I windings are non-overlap, single-layer windings which are symmetrically spaced,
while the Type II windings are non-overlapping, double-layer windings, which are placed such
ds
dr
B̂r1 cos( pφ − pωmech t − γ)
ωmech
2∆
q
S
4
3
2
1
N
1’
2’
a′
a
N
γ
p
3’
4’
4
3
2
1
S
1’
3’
2’
c′
c
S
4’
4
3
N
2
b
N
1
1’
2’
3’
4’
φ
b′
S
π
p
2π
Q
Figure 2.18: Coil configuration for a three-phase RFAPM machine with Type II windings with k q = 12 .
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2.5. NON-OVERLAPPING DOUBLE-LAYER (TYPE II) WINDING CONFIGURATION
so as to maximise the coil pitch or coil span of each coil. This implies that each coil of the
Type II winding configuration will be touching their neighbouring coils on either side. By
increasing the coil span, the flux-linkage can be increased at the expense of increasing the endturn length. The coil span for the Type II concentrated coil RFAPM machine can be calculated
from Figure 2.18 as,
τq| I I =
2π 2∆
−
.
Q
q
(2.88)
2.5.1 Flux Linkage of the Type II Winding Configuration
The maximum or peak value of the flux-linkage for the Type II winding configuration is calculated again in exactly the same manner as that for the Type O and the Type I winding configuration. With the d-axis of the coil aligned with the d-axis of the magnets in order to calculate
the peak value of the flux-linkage, we again start by finding the flux-linkage of a single turn,
with each turn side at an angle of δ from the centre of each coil side. The flux-linkage of a single
turn for the Type II winding configuration can thus be calculated as follows,
λ| I I (δ) =
Z ℓZ
0
+δ
π
− Q
− ∆q +δ
rn B̂r1
=
p
rn B̂r1
=
p
Z ℓ
0
Z ℓ
2rn B̂r1
=
p
=
π
∆
Q− q
0
Z ℓ
0
sin
sin
B̂r1 cos( pφ)rn dφdz
pπ
3q
−
− pδ − sin −
pπ
3q
2π
3
2π
3
2π
3
− 2∆ − pδ − sin −
− 2∆ cos( pδ)dz
sin
2rn ℓ B̂r1
sin
p
(2.89)
2π
3
p∆
q
−
p∆
q
+ pδ
dz
− 2∆ + pδ dz
− 2∆ cos( pδ) .
(2.90)
(2.91)
(2.92)
(2.93)
Again taking into account the number of turns per coil, as well as the total number of series
and parallel connected coils, the total flux-linkage per phase can be calculated as follows,
Λ a,b,c| I I =
=
=
q qN Z
a
2∆
q qN Z
a
2∆
∆
q
− ∆q
∆
q
− ∆q
λ| I I (δ) dδ
2rn ℓ B̂r1
sin
p
q qN 4r ℓ B̂
n
r1
sin
a
2∆
p2
2rn ℓ Nk q B̂r1 sin
=
a
2π
3
2π
3
(2.94)
2π
3
− 2∆ cos( pδ) dδ
− 2∆ sin( p∆
q )
∆
− 2∆ sin( kq )
·
( k∆q )
(2.95)
(2.96)
(2.97)
The flux-linkage of the Type II winding configuration can be written into a general format,
so that equation (2.77) can be expanded to include the the Type II winding configuration as
well. The general flux-linkage equation for the RFAPM machine can thus be written as
Λ a,b,c|O = Λ a,b,c| I = Λ a,b,c| I I = 2rn ℓ N B̂r1 k q k w,pitch k w,slot
(2.98)
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CHAPTER 2. THEORETICAL WINDING ANALYSIS OF THE DOUBLE-SIDED ROTOR RFAPM MACHINE
29
again with the general format of the slot width factor given by (2.76), so that we can write
sin k∆q
k w,slot = k w,slot|O = k w,slot| I = k w,slot| I I =
,
(2.99)
∆
kq
with the only difference being the winding pitch factor, k w,pitch , which for the Type II winding
configuration would be equal to
2π
3
k w,pitch| I I = sin
− 2∆ .
(2.100)
2.5.2 Conductor Density Distribution of the Type II Winding Configuration
From the winding distribution for phase a of the Type II winding configuration as shown in
Figure 2.19 and with the coil-side width equal to 2∆
q , the conductor density distribution will
have the form as shown in Figure 2.20. As was mentioned in section 2.3.2, the conductor density distribution allows us to calculate the back-EMF of the machine as well as the three-phase
synchronous inductance of the machine.
ds
2∆
q
4 3 2 1
1’ 2’ 3’ 4’
4 3 2 1
1’ 2’ 3’ 4’
a
a′
a
a′
2(π −3∆)
Q
φ
2π
q
Figure 2.19: The winding distribution and relative coil position for phase a of a RFAPM machine with a
Type II winding configuration and k q = 21 .
From Figure 2.20 and with the use of a few trigonometrical identities, the Fourier expansion
of the conductor density distribution for phase a can be written similar to (2.78) as
∞
n a| I I ( φ ) =
∑
m =1
bmn| I I sin(mqφ)
(2.101)
with
bm n | I I = −
2qN
sin m( π3 − ∆) sin(m∆)
mπ∆
(2.102)
Once again, as was shown in section 2.4.2, it is possible to write the Fourier series expansion
coefficient, bmn| I I , in terms of a number of winding factors, so that we can write
bm n | I I = −
2qN
· k w,m| I I
π
(2.103)
with the general winding factor expanding to
k w,m| I I = k w,pitch,m| I I · k w,slot,m| I I ,
(2.104)
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2.5. NON-OVERLAPPING DOUBLE-LAYER (TYPE II) WINDING CONFIGURATION
n
qN
2∆
2∆
q
a
a
φ
a′
a′
2(π −3∆)
Q
2π
q
Figure 2.20: The conductor density distribution for phase a of a RFAPM machine with a Type II winding
configuration with k q = 12 and k∆ = 32 .
with the pitch factor,
k w,pitch,m| I I = sin m( π3 − ∆)
(2.105)
and the “virtual” slot width factor,
k w,slot,m| I I = k w,slot,m| I = k w,slot,m|O =
sin(m∆)
.
m∆
(2.106)
Furthermore, with the resultant coil pitch equal to twice the resultant pole pitch, i.e. τq,res =
2τp,res , and considering only the fundamental component of the radial flux-density distribution, again only the second harmonic component of the conductor density distribution will
contribute to the flux-linkage. Thus with m = 2, it follows that
k w,pitch,m=2| I I = k w,pitch| I I
(2.107)
k w,slot,m=2| I I = k w,slot| I I .
(2.108)
and
2.5.3 Current Density Distribution of the Type II Winding Configuration
In this section, the three-phase current density distribution for the Type II non-overlapping
winding configuration will be deduced. The three-phase current density distribution will be
used in Chapter 4 to calculate the armature-reaction field for the Type II winding configuration
as well as for the calculation of the developed torque of a RFAPM machine with a Type II
winding configuration in Chapter 5.
The area of the “virtual stator slot” for a RFAPM machine with Type II winding configuration is again exactly the same as for the Type O and Type I winding configurations. The current
density distribution can thus directly be obtained from 2.19 as is shown in Figure 2.21.
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CHAPTER 2. THEORETICAL WINDING ANALYSIS OF THE DOUBLE-SIDED ROTOR RFAPM MACHINE
31
~J
qNi
2arn h∆
2∆
q
a
a
φ
a′
2(π −3∆)
Q
a′
2π
q
Figure 2.21: The current density distribution for phase a of a Type II winding configuration at ωt = 0◦ .
The Fourier expansion of the current density distribution for phase a of the concentrated
Type II RFAPM machine can be written as
Jza| I I =
1 ∞
bmn| I I sin(mqφ)i a (t) .
arn h m∑
=1
(2.109)
Using the same approach as with the Type O, equation (2.51) – and the Type I winding
configuration, equation (2.85), the resultant steady state current density distribution for all
three phases can be calculated as,
∞
3qI p N
−
arn hπ ∑ k w,m| I I sin mqφ + ωt for m = 3k − 1, k ∈ N1
m =1
(2.110)
Jz| I I =
∞
3qI p N
k
sin
mqφ
−
ωt
for
m
=
3k
−
2,
k
∈
N
−
1
arn hπ ∑ w,m| I I
m =2
as shown for ωt = 0◦ in Figure 2.22, ωt = 15◦ in Figure 2.23 and ωt = 30◦ in Figure 2.24. As
was the case for the Type I winding configuration, the 3k − 1 harmonics, which includes the
working (second) harmonic is moving to the “right” (i.e. rotating counter clockwise) in these
figures.
2.6
Number of Turns per Coils
Before we conclude this chapter, it is important to note that for the different winding configurations under consideration, although the number of turns were always shown as N during
the analysis of each winding configuration, the actual number of turns may vary between the
different winding configurations. It was shown, that the overlapping (Type O) winding configuration will always have twice the number of coils than either of the non-overlapping winding configurations. This implies that it would theoretically be possible for the non-overlapping
windings to have twice the number of turns than for either the Type I or Type II non-overlapping
winding configuration. In practise however it turns out to be closer to 1,5 times due to the fact
that we require the turns of each coil to be as wide as possible to maximise the flux-linkage. If
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2.6. NUMBER OF TURNS PER COILS
~J
qNi
2arn h∆
2∆
q
a
a
b′
c′
c
φ
b
a′
a′
2(π −3∆)
Q
2π
q
Figure 2.22: The combined current density distribution for phase all three phases of a Type II winding
configuration at ωt = 0◦ .
~J
qNi
2arn h∆
2∆
q
a
a
c′
b′
φ
b
a′
2(π −3∆)
Q
c
a′
2π
q
Figure 2.23: The combined current density distribution for phase all three phases of a Type II winding
configuration at ωt = 15◦ .
the turns become too narrow due to an increase in the number of turns, these additional turns
will contribute little to the increase in flux-linkage and will only add to the copper losses. The
difference in the stator copper losses between the different winding configurations with the
end-turn effect for the different winding configurations also taken into account, will be further
discussed in Chapter 6.
In order to calculate the number of turns that can fit on a machine, we start by calculating
the “virtual slot area”, Aslot . From (2.36), in terms of the machine parameters,
Aslot =
2rn h∆
q
(2.111)
with
∆ = k∆∆ max
(2.112)
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CHAPTER 2. THEORETICAL WINDING ANALYSIS OF THE DOUBLE-SIDED ROTOR RFAPM MACHINE
33
~J
qNi
2arn h∆
2∆
q
a
c′
a
b′
b
a′
2(π −3∆)
Q
φ
a′
c
2π
q
Figure 2.24: The combined current density distribution for phase all three phases of a Type II winding
configuration at ωt = 30◦ .
as was shown in (2.64), and
∆ max =
π
6q
[mech.]
(2.113)
as defined in (2.63).
This implies that the “virtual” slot area in terms of the coil side-width factor, k∆, can be
written as
Aslot =
πrn hk∆
.
3q
(2.114)
With the “virtual” slot area known, the maximum number of turns can be calculated with
the area of the copper wire, Awire and the fill factor, k f , known.
N=
Aslot
k
Awire f
(2.115)
The fill factor is a function of various aspects, most notably the shape of the wire, i.e. round
or rectangular, whether Litz wire is used and the craftsmanship employed. A good estimate of
the value is usually known from previous experience.
The thickness of the copper wire used, and hence the cross-sectional area thereof, is usually
a trade of between the copper losses allowed for and the copper utilisation with the latter usually dominating. Traditionally a wire current density, Jwire , to be in the range of 4 – 6 A/mm2
is considered to be a good trade off between copper utilisation and copper losses. The choice
of the current density is also based on how easy or difficult it is to extract the heat out of the
stator windings.
For air cored stators, a lower value of current density is recommended, as there is no iron
in contact with copper wires to conduct the heat away from the windings. Thus based on
the r.m.s. current, Irms , the number of parallel circuits, a, and the selected current density, the
cross-sectional area of the copper wire required can be calculated as
Awire =
Irms
,
aJwire
(2.116)
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2.6. NUMBER OF TURNS PER COILS
so that with (2.116) and (2.36) substituted into (2.115), the number of turns per coil can be
calculated in terms of the various machine parameters as
N=
aπrn hk f k∆ Jwire
.
3qIrms
(2.117)
The parameters of two RFAPM non-overlapping, double-layer (Type II) test machines that
were built and tested, are given in Appendix E, section E.1 for Stegmann [15] and Stegmann
and Kamper [16, 26], and in section E.2 for Groenewald [17]. In this dissertation this data for
E.1 will be used for both the analytical and Finite Element (FE) analysis of the RFAPM Type II
machine. Although a Type O and a Type I RFAPM machine were not built, the rotors of these
two “hypothetical” machines were taken as identical to the Type II machine, with the only
difference being in the layout of the stator windings.
With the 32–pole (i.e. 16 pole pairs) rotor being the same for all three machines, it implies
that both the Type I and II machines will only employ 8 coils per phase, whilst the Type O
machine will have the advantage of using 16 coils per phase. The design in section E.1 of
Appendix E uses a coil side-with factor, k∆ = 0,74 which was based on calculation done to
maximise the torque and minimise the copper volume of the stator windings, Randewijk et al.
[14] and Stegmann and Kamper [16, 26]. For comparison purposes it was decided to use the
same value for the Type I machine. The Type O machine has the advantage of utilising the
maximum available coil side-width available in order maximise its flux-linkage and ultimately
the torque produced. The choice of the coil side-width factor, k∆, for the Type I and II machines
will be discussed in more detail in Chapter 6 together with the difference in end-turn lengths
between the three different winding configurations.
Using the same coil side-width factor and the same number of coils per phase for both the
Type I and II winding configuration, it implied that the number of turns per phase would also
be the same for both. From (2.115) the number of turns per coil for these two winding configurations were calculated at 96 turns each with a current density of 5 A/mm2 for both winding
configurations. For the Type O winding configuration, although k∆ =1,0 , the maximum allowable coil-side width is half that of the non-overlap winding configuration’s maximum coil-side
width, due to the fact that twice the number of coils are being used. This resulted in the maximum number of turns per coils being equal to 64 turns, also with a current density value of
5 A/mm2 . In Table 2.1 the key differences between the different winding configurations are
highlighted on which the comparative analytical and FEA of Chapters 3, 4 and 5 will be based
on.
Table 2.1: Comparison between the number of coils, q, the coil-side width factor, k∆, and the number
of turns per coil, N, for the different winding configurations with the same rotor geometry and stator
current and current density.
q
k∆
N
Type O
Type I
Type II
16
1,0
64
8
0,74
96
8
0,74
96
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CHAPTER 2. THEORETICAL WINDING ANALYSIS OF THE DOUBLE-SIDED ROTOR RFAPM MACHINE
2.7
35
Summary and Conclusions
In this chapter the winding factors, conductor density distribution and the three-phase current density distribution for the different winding configurations under consideration were
deduced. As distributed windings were not employed for either winding configuration, the
winding factors consisted of only a pitch factor and a “slot” factor. Being a slot-less machine
the latter could also be called a “coil-side width” factor.
It was seen that due to the fact that for the overlapping winding configuration there is
only one coil per pole pair, the fundamental component of the conductor density distribution
was the working harmonic. This is in contrast to the working harmonic of the Type I and
Type II winding configurations which was seen to be associated with the second harmonic of
the conductor density distribution. This is due to of the fact that for both non-overlapping
winding configurations, there is only one non-overlapping coil for each two rotor pole pairs.
For the overlapping winding configuration the coil sequence were also seen to be ABC as
opposed to the non-overlapping winding configurations’ coil sequence that were ACB – again
due to the fact that there is only one non-overlapping coil for each two rotor pole pairs.
In Figures 2.8 to 2.10 it was also clearly shown how the three-phase current density distribution’s waveform “moves to the right” (i.e. rotating counter clockwise) at synchronous
speed with time. For the Type I and II non-overlapping winding configurations it is however
more difficult to see the working (second) harmonic “moving to the right” (i.e. rotating counter
clockwise) but was non the less shown in Figures 2.15 to 2.17 and Figures 2.22 to 2.24 for the
Type I and Type II winding configurations respectively.
Also in this chapter, the conductor density distribution was derived. This will be used in
Chapter 3 to calculate the back-EMF of the machine as well as the three-phase synchronous
inductance of the machine in Chapter 4, for the three different winding configuration under
consideration.
Finally, the three-phase current density distributions for the three different winding configuration were also derived. This will be used in Chapter 4 to calculate the armature-reaction
field, as well as the developed torque of the RFAPM machine for the different winding configurations in Chapter 5.
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C HAPTER 3
Magnetostatic Analysis of the Magnetic
Fields due to the Permanent Magnets
Finally, two days ago, I succeeded – not on account of my hard efforts, but
by the grace of the Lord. Like a sudden flash of lightning, the riddle was
solved. I am unable to say what was the conducting thread that connected
what I previously knew with what made my success possible.
K ARL F RIEDRICH G AUSS
3.1
Introduction
In this chapter the 2-D analytical analysis of the magnetic fields produced by the permanent
magnets situated on the double-sided rotor of the RFAPM machine will be discussed. After a
short background on the different analysis techniques that exist to model electrical machines,
the magnetic field inside the RFAPM machine, produced by the permanent magnets on the
double-sided rotor, will be solved from first principles. It will be shown how the permanent
magnets’ pole arc width influences the fundamental component and total harmonic distortion1
(THD) of the radial flux-density distribution inside the stator of the machine.
It will also be shown how the flux-linkage and the induced voltage for the different winding configurations can be analytically calculated from the magnetic vector potential for the different winding configurations, using the conductor density distributions derived in Chapter
2. Furthermore, it will be shown that a good approximation of the flux-linkage and induced
voltage can be obtained by simply using the fundamental component of the radial flux-density
distribution in the centre of the stator region.
1 Calculated
with respect to the space harmonics of the radial flux-density distribution.
37
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3.2. BACKGROUND ON THE MAGNETIC FIELD ANALYSIS OF ELECTRICAL MACHINES
3.2
Background on the Magnetic Field Analysis of Electrical
Machines
2-D analytical analysis of permanent magnet machines was initially done by modelling the
permanent magnet’s MMF as current sheets on the surface of the permanent magnets, Boules
[27]. The machine was then linearised and the governing quasi-Poisson – and Laplace equations for the permanent magnets and the air-gap regions respectively, were written in terms of
the magnetic vector potential, ~A. These equations were then solved in the Cartesian coordinate
system from the boundary conditions with the air-gap extended using Carter factors and the
stator represented by an equivalent current sheet on the boundary between the air-gap and the
stator.
In [28], Boules extended the modelling of permanent magnets, by using the magnetisation
~ , to model both parallel – and radially magnetised permanent magnets. The govvector, M
erning equations were however solved using the scalar vector potential, Ω or ϕ, in the polar
coordinate system. For both these modelling techniques, the recoil permeability of the permanent magnets were take to be unity. In [29], Zhu et al. extended the modelling technique further
to include the recoil permeability of permanent magnets and applied this theory to slotless
machines with radially magnetised permanent magnets. This modelling technique was extended even further in [22] by Zhu et al. to include parallel-magnetised permanent magnets and
non-overlapping winding configurations.
Holm [24, chap. 5] and Holm et al. [21] proposed a way in which the analytical field calculations can be simplified even further and will be explained in section 3.7.2 for to the special
case were “mp = 1”. Holm also reverted back to using the magnetic vector potential for the
magnetic field calculations. In this dissertation, this approach will also be used.
For all the above modelling techniques, the rotor – and stator yokes’ permeabilities were
considered to be infinity. Kumar and Bauer [30] included the rotor – and stator yokes’ permeability into the solution for the machine’s magnetic field solution by assigning a specific
permeability to the iron yokes. This allowed the flux-density in the yokes to be calculated in
order to determine the risk of possible yoke saturation to occur. Although Kumar and Bauer’s
method assumes constant permeability and hence does not take saturation into account, it still
is able to give an indication as to whether magnetic saturation is possible. In this dissertation a
specific permeability value will also be assigned to the rotor yokes in order to calculate the fluxdensity inside the yokes and determine the subsequent risk of magnetic saturation occurring
inside the rotor yokes.
In Chapter 1 it was mentioned that the RFAPM machine is the dual of the Axial Flux Aircored Permanent Magnet (AFAPM) machine. The first occurrence in the literature on the analytical analysis of AFAPM were in two papers by Virtic et al. [31, 32]. The first paper focused
on the analytical analysis of the magnetic field caused by the permanent magnets in order to
solve the flux-density distribution in the machine and to calculate the back-EMF in the coils
produced by the permanent magnets. In the second paper, the flux-density distribution caused
by the armature coils were calculated. The torque produced by the machine were then calculate
using the Maxwell Stress Tensor.
An AFAPM machine with a double-sided rotor and single side permanent magnets em-
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CHAPTER 3. MAGNETOSTATIC ANALYSIS DUE TO THE PERMANENT MAGNETS
39
ploying a core-less stator with full-pitched, double-layer windings was analytically analysed
by Chan et al. [33]. Although a RFAPM dual of this machine was considered, it was found
to have a very low radial flux-density distribution with severe flux-leakage between adjacent
magnets and as such was not considered further.
All the analytical modelling techniques mentioned above and more were reviewed in Zhu
et al. [23]. These techniques were subsequently called “subdomain modelling” techniques referring to the different regions or domains into which the machine is subdivided for analysis
purposes. Incidentally, this term was first coined in the classical paper by Abdel-Razek et al.
[34] on the Air-gap Element.
3.3
The Maxwell Equations
We will start our 2-D analytical analysis from first principles with the Maxwell equations. The
Maxwell equations,
∇ × ~E = −
∂~B
,
∂t
~ = ~J +
∇×H
~ = ρv
∇·D
~
∂D
,
∂t
and
∇ · ~B = 0 .
(3.1)
(3.2)
(3.3)
(3.4)
These four equations are the mathematical representation in differential form, of Faraday’s
law, Ampere’s law and Gauss’s laws on the conservation of electrical and magnetic flux respectively and are the four governing equations that completely describe time-varying electromagnetic fields in free space, Cheng [35, sec. 7.3] and Guru and Hiziroglu [36, sec. 7.4, 7.9 and
7.10].
If the wavelength of the time-varying electromagnetic fields are much larger than the phys~
ical dimensions of the device being analysed, the displacement current portion of (3.2), ∂∂tD , will
be negligible compared to the free current density,~J, Binns et al. [37, p. 4], so that equation (3.2)
reduces to the original Ampère for, of
~ = ~J .
∇×H
3.4
(1.2’)
Magnetic Materials
The magnetic polarisation, Stratton [38, p. 128], magnetisation vector, Cheng [35, p. 250] or
the magnetic moment per unit volume, Guru and Hiziroglu [36, p. 206] for a linear, isotropic
medium can be written in terms of the magnetic field intensity as
~ = χm H
~ .
M
(3.5)
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3.5. MAGNETIC VECTOR POTENTIAL
This allow us to define the magnetic flux-density of the medium in terms of the field intensity, as
~ +M
~
~B = µ0 H
(3.6)
~
= µ0 (1 + χ m )H
~
= µ0 µr H
(3.8)
~ .
= µH
(3.9)
(3.7)
For a permanent magnet material, the magnetisation vector is equal to the sum of the residual magnetisation, and the induced magnetisation, i.e.
~ =M
~ 0+M
~i
M
~ 0 + χm H
~ ,
=M
(3.10)
(3.11)
Boules [28], with specific reference to Stratton [38, p. 129].
The flux-density in the permanent magnet material can thus be calculated as,
~B = µ0 H
~ + µ0 M
~
~ + µ0 χ m H
~ + µ0 M
~0
= µ0 H
(3.12)
(3.13)
~ + µ0 M
~0
= µ0 (1 + χ m )H
~0
~ + µ0 M
= µ0 µr H
(3.14)
~ + µ0 M
~ 0.
= µH
(3.16)
(3.15)
With the residual magnetisation
~
~ 0 = Brem ,
M
µ0
(3.17)
the flux-density in the permanent magnet material can also be expressed as
~B = µH
~ + ~Brem ,
(3.18)
as the remanent flux-density is usually the value supplied by the permanent magnet manufacturer in the data sheets and not the residual magnetisation.
3.5
Magnetic Vector Potential
The divergence-free postulate of ~B by (3.4) assures that ~B is solenoidal. Consequently ~B can be
expressed as the curl of another vector field, say ~A, Cheng [35, sec. 6.3], i.e.
~B = ∇ × ~A .
(3.19)
The vector field ~A is called the vector magnetic potential (Cheng [35, sec. 6.3] and Reece and
Preston [39, sec. 1.4]) or the magnetic vector potential (Binns et al. [37, sec. 1.2.1] and Guru and
Hiziroglu [36, sec. 5.6]) or just the vector potential, for short.
In the Cartesian coordinate system, the magnetic vector potential can be written as
~A = A x~ax + Ay~ay + Az| PM~az ,
(3.20)
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CHAPTER 3. MAGNETOSTATIC ANALYSIS DUE TO THE PERMANENT MAGNETS
41
and in the cylindrical coordinate system, as
~A = Ar~ar + Aφ~aφ + Az| PM~az .
(3.21)
If the current density is only in the z direction, i.e. Jx = Jy = 0, it can be shown that the
vector potential will also only be in the z direction, i.e. A x = Ay = 0, Reece and Preston [39,
sec. 2.1.2].
3.6
The Vector Poisson Equations
3.6.1 The Vector Poisson Equation in the Different Regions of the RFAPM
Machine
In order to analyse the RFAPM machine, the machine is divided into several regions or subdomains, Zhu et al. [23]. The regions are chosen in such a way that for each region a separate
equation can be found to describe the magnetic vector potential everywhere inside that region,
as well as a unique set of boundary conditions that exist between the regions, Binns et al. [37,
sec. 4.3]. The regions must also be chosen in such a way, in what would become clearer in
subsequent sections, that the permeability in each region is constant. The magnetic vector potential for each region can be solved more easily if we write it in a vector Poisson equation form
for each region, i.e.
∇2~A = F (r, φ) .
(3.22)
The only logical way to subdivide the machine is into “ring” regions, with concentric circular boundaries analogous to Atallah et al. [6], Zhu et al. [29] & [22] and Holm et al. [21], to
name a few. The different regions for the RFAPM machine is shown in a linear representation
in Figure 3.1 with rn , the nominal radius measured form the centre of the machine to the centre
of the stator region. The azimuthal axis of the machine, φ, is shown in electrical radians.
The different regions of the RFAPM machine are thus:
I–
II –
III –
IV –
V–
the inner yoke region
the inner yoke permanent magnet region
the airgap region (including the air cored stator region)
the outer yoke permanent magnet region
the outer yoke region
The governing equations for solving the magnetic vector potential in the different regions
of the RFAPM machine, is shown in Table 3.1. These equations will be further expanded in
sections 3.6.2 and 3.6.3.
3.6.2 The Vector Poisson Equation in the Non Permanent Magnet Regions
From (3.9), we can write:
~
~ =B
H
µ
(3.23)
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3.6. THE VECTOR POISSON EQUATIONS
r
V
hy
IV
hm
ℓg
III
h
ℓg
II
hm
I
hy
rn
π
p
2π
p
φ
Figure 3.1: A linear representation of the different regions of the RFAPM machine.
Range for r
Region
I
II
rn +
h
2
+ ℓ g + hm + hy ≥ r ≥ rn + + ℓ g + hm
rn +
h
2
+ ℓ g + hm ≥ r ≥ rn + + ℓ g
III
rn +
IV
rn −
V
h
2
h
2
h
2
h
2
h
2
rn −
h
2
h
2
h
2
+ ℓ g ≥ r ≥ rn − − ℓ g
− ℓ g ≥ r ≥ rn − − ℓ g − hm
− ℓ g − hm ≥ r ≥ rn − − ℓ g − hm − hy
µr
Governing equation
µy
∇2~A = 0
~0
∇2~A = −µ0 ∇ × M
1
1
1
µy
∇2~A = 0
~0
∇2~A = −µ0 ∇ × M
∇2~A = 0
Table 3.1: The governing equations for solving the magnetic vector potential in the different regions of
the RFAPM machine when employing permanent magnet excitation.
Substituting this in (1.2’), and assuming that the medium is homogeneous we get
∇ × ~B = µ~J .
(3.24)
Now substituting (3.19) into the above equation, it expands to
∇ × ∇ × ~A = µ~J ,
(3.25)
or if we further expand the left hand side of the equation,
∇(∇ · ~A) − ∇2~A = µ~J .
(3.26)
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CHAPTER 3. MAGNETOSTATIC ANALYSIS DUE TO THE PERMANENT MAGNETS
43
In order to simplify (3.26) we choose
∇ · ~A = 0 ,
(3.27)
which is called the Coulomb gauge, Binns et al. [37, p. 6], so that that (3.26) reduces to
∇2~A = −µ~J ,
(3.28)
which is called the vector Poisson equation, or simply the Poisson equation for short.
In a current-free region, (3.28) reduces to
∇2~A = 0 ,
(3.29)
which is known as the Laplace – or homogeneous version of the Poisson equation and can only
be solved if the boundary conditions are non– or inhomogeneous, Gockenbach [40, sec. 8.1.8].
For 2-D analysis, with Jx = Jy = 0, (3.28) can be written in partial differential form in
Cartesian coordinates, as
∂2 Az| PM
∂2 Az| PM
+
= −µJz ,
∂x2
∂y2
(3.30)
or in cylindrical coordinates as
∂2 Az| PM
1 ∂Az| PM
1 ∂2 Az| PM
+
+
= −µJz .
∂r2
r ∂r
r2 ∂φ2
(3.31)
For the magnetostatic analysis, we ignore the currents in the stator as we are only interested
in the flux-density distribution in the airgap as a result of the permanent magnets alone. The
windings are effectively “switched off”1 . Thus for both the yoke regions, regions I and IV and
the stator/airgap region, region III, we can take Jz = 0. This implies that the Poisson equation
(3.31) reduces to a simple Laplace equation,
∂2 Az| PM
1 ∂Az| PM
1 ∂2 Az| PM
+
+
=0,
∂r2
r ∂r
r2 ∂φ2
(3.32)
in all the non-permanent magnet regions.
3.6.3 The Vector Poisson Equation in the Permanent Magnet Regions
A similar approach can be followed for the permanent magnet regions. The only difference
between a permanent magnet region and a non permanent magnet region is that in the permanent magnet region the magnetic field intensity is also dependent on the residual magnetisation
of the permanent magnets used. Thus from (3.16) and (3.19),
~
~
~ = B − µ0 M0
H
µ
~0
∇ × ~A − µ0 M
=
.
µ
1 In
(3.33)
(3.34)
Chapter 4 we will look at the armature reaction fields produces by the stator windings alone with the permanent
magnets “switched off”.
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3.6. THE VECTOR POISSON EQUATIONS
Substituting the above into (1.2’),
!
~0
∇ × ~A − µ0 M
∇×
= ~J ,
µ
(3.35)
and multiplying through with µ and then expanding the left hand side of the equation, gives
2~
~
~
∇ ∇ · A − ∇ A − µ0 ∇ × M0 = µ~J .
(3.36)
Using the Coulomb gauge again, the above equation simplifies to
~0 .
∇2~A = −µ~J − µ0 ∇ × M
(3.37)
When working in cylindrical coordinates, radially magnetised permanent magnets will
have a residual magnetisation component only in the ~ar direction,
~ 0 = M0| ~ar ,
M
r
(3.38)
where as for parallel magnetised permanent magnets, there will also be a ~aφ component.
~ 0 = M0| ~ar + M0| ~aφ .
M
φ
r
(3.39)
This implies that in general
~0=1
∇×M
r
~ar
r~aφ
~az
∂
∂r
∂
∂φ
∂
∂z
M0|r rM0|φ _
"
#
1 ∂(rM0|φ ) ∂M0|r
−
~az
=
r
∂r
∂φ
"
#
M0|φ
∂M0|φ
1 ∂M0|r
=
+
−
~az ,
r
∂φ
r ∂φ
(3.40)
(3.41)
which reduces to,
~ 0 = − 1 ∂M0~az
∇×M
r ∂φ
(3.42)
with
~ 0| = M0 ,
M
r
(3.43)
for radially magnetised permanent magnets.
The residual magnetisation distribution for a radially magnetised permanent magnet can
be represented by a periodic function as shown in Figure 3.2.
The derivative of the residual magnetisation distribution of the permanent magnets with
respect to φ will therefore also be periodic and can be determined directly from Figure 3.2
as shown in Figure 3.3. An alternative approach using the equivalent surface magnetisation
current distribution to model the permanent magnets as current sheets, Boules [28], is shown
in Appendix B.
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45
CHAPTER 3. MAGNETOSTATIC ANALYSIS DUE TO THE PERMANENT MAGNETS
r
M0
k m πp
π
p
2π
p
3π
p
5π
p
4π
p
φ
Figure 3.2: The residual magnetisation distribution of a permanent magnet with respect to φ.
With
f (φ) =
∂M0
∂φ
(3.44)
being a periodic function, it can be represented by a Fourier series expansion,
∞
f ( φ ) = a0 +
∑
[ am cos(mpφ) + bm sin(mpφ)] .
(3.45)
m =1
Because f (φ) has no DC component,
a0 = 0 ,
(3.46)
and with f (φ) being an even function,
bm = 0 ,
(3.47)
therefore only am has to be calculated.
The calculation of am can further be simplified by setting
1 − km π
β=
,
2
p
(3.48)
r
∞
∞
M0
∞
M0
π
p
M0
∞
M0
M0
2π
p
M0
−∞ −∞
∞
3π
p
M0
M0
5π
p
4π
p
M0
−∞ −∞
φ
M0
−∞
Figure 3.3: The derivative of the residual magnetisation distribution of the permanent magnets with
respect to φ.
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46
3.6. THE VECTOR POISSON EQUATIONS
as shown in Figure 3.4, with k m , the magnet – or pole width defined as a function of the pole
pitch, τp , as shown in Figure 3.2 with τp = πp .
k m πp
r
∞
M0
π
p
−β
π
p
β
φ
M0
−∞
Figure 3.4: The derivative of the remanent flux-density distribution of the permanent magnets with
respect to φ for one pole.
From Figure 3.4, am can now easily be calculated as
am =
=
=
=
π
2p p
f (φ) cos(mpφ)dφ
π 0
Z π
i
2p p h
δ(φ − β) M0 − δ φ − ( πp − β) M0 cos(mpφ)dφ
π 0
2pM0
[cos(mpβ) − cos(mπ − mpβ)]
π
4pM0
cos(mpβ) ,
π
Z
(3.49)
(3.50)
(3.51)
(3.52)
which implies that
f (φ) =
4pM0 ∞
∑ cos(mpβ) cos(mpφ) ,
π m=
1,3,5,...
(3.53)
or from (3.44),
4pM0 ∞
∂M0
=
∑ cos(mpβ) cos(mpφ) .
∂φ
π m=
1,3,5,...
(3.54)
Substituting the above into (3.42), yields
∞
~ o = − 1 4pM0 ∑ cos(mpβ) cos(mpφ)~az
∇×M
r π m=1,3,5,...
(3.55)
for radially magnetised permanent magnets.
Then finally by substituting the above into (3.37) with Jz = 0 in the PM region, the Poisson
equation for region II and IV for the RFAPM machine is given by
∇2~A = µ0
4pM0 ∞
∑ cos(mpβ) cos(mpφ)~az
πr m=
1,3,5,...
(3.56)
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CHAPTER 3. MAGNETOSTATIC ANALYSIS DUE TO THE PERMANENT MAGNETS
47
or in partial differential format in cylindrical coordinates, as
∂2 Az| PM
1 ∂Az| PM
4pµ0 M0 ∞
1 ∂2 Az| PM
+
=
+
∑ cos(mpβ) cos(mpφ) .
∂r2
r ∂r
r2 ∂φ2
πr m=
1,3,5,...
3.7
(3.57)
Solving the Laplace and Poisson Equations
The Laplace equation consists only of a general (or “homogeneous”) solution whereas the Poisson equation consists of a general solution and a particular (or “non-homogeneous”) solution.
3.7.1 Finding the General Solution of the Laplace and Poisson Equations
For simple structures such as rectangles and annuli, the general solution to both the Laplace
and the Poisson equation can be obtained by the method of separation of variables. The solution for an annulus with periodic boundary conditions will be in the form of harmonic functions. Thus the general solution of (3.32) for all the regions will be of the form,
∞
Az| PM (r, φ) = Az,0| PM (r, φ) +
∑
m =1
Az,m| PM (r, φ) ,
(3.58)
Bleecker and Csordas [41, sec. 6.3], with
Az,0| PM (r, φ) = c0 + d0 ln(r ) for m = 0 , and
Az,m| PM (r, φ) = (cm r m + dm r −m )( am cos mφ + bm sin mφ) for m ≥ 1 .
(3.59)
(3.60)
With the flux-density only dependent on the derivatives of the magnetic vector potential
and not on the magnitude, Reece and Preston [39, sec. 2.1.2] we can set
c0 = 0 .
(3.61)
If there is no DC component,
d0 = 0 .
(3.62)
From Zhu et al. [29] and [22] it is evident that if F (r, φ) is an even function, the general
solution will also be even, and if F (r, φ) is odd, the general solution will be odd.
∞
Az,gen| PM (r, φ) =
∑(Cm|PM rmp + Dm|PM r−mp ) cos mpφ
(3.63)
m=1,3,5,...
3.7.2 Finding the Particular Solution of the Poisson Equation
From (3.17), the governing equation (3.57) for regions II and IV, can be written in terms of the
remanent flux-density of the permanent magnets used as,
∂2 Az| PM
1 ∂Az| PM
1 ∂2 Az| PM
4pBrem ∞
+
+
=
∑ cos mpβ cos mpφ .
∂r2
r ∂r
r2 ∂φ2
πr m=
1,3,5,...
(3.64)
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3.7. SOLVING THE LAPLACE AND POISSON EQUATIONS
In order to find the particular solution, one usually makes an intelligent guess as to the
form of the particular solution. The solution is then substituted back into the equation being
solved to see if it indeed satisfies the equation. The following solution to particular solution
was proposed by Zhu et al. [29] and [22],
∞
Az,part| PM (r, φ) =
∑
Gm| PM r cos mpφ
m=1,3,5,...
.
(3.65)
By substituting this solution into the left hand side (LHS) of (3.64),
∞
∂Az| PM
= ∑ Gm| PM cos mpφ ,
∂r
m=1,3,5,...
∂2 Az| PM
=0,
∂r2
∞
∂Az| PM
= − ∑ mpGm| PM r sin mpφ and
∂φ
m=1,3,5,...
∞
∂2 Az| PM
=
−
∑(mp)2 Gm|PM r cos mpφ ,
∂φ2
m=1,3,5,...
(3.66)
(3.67)
(3.68)
(3.69)
we are able to verify the validity of the solution. This results in
Gm| PM =
4pBrem cos mpβ
π (1 − (mp)2 )
for mp 6= 1 .
(3.70)
The problem with this particular solution, is that it is not valid for mp = 11 and thus another
solution for the case mp = 1 has to be obtained for the permanent magnet region.
In order to simplify the analytical solution, Holm et al. [21] proposed that if the Brem were
proportional to 1r , it would yield a unique particular solution for all values of m. Thus the
“new” remanent flux-density value used for the permanent magnet regions is given by the
following equation,
′
Brem =
rcm
Brem .
r
(3.71)
The error introduced is very small as the magnet’s radial thickness, hm , is much smaller
than the radius measured to the centre of the magnets, rcm . Thus with,
hm ≪ rcm
(3.72)
it follows that
r ≈ rcm
(3.73)
Brem ≈ Brem .
(3.74)
and
′
1 This
implies that for a two pole machine, i.e. with p = 1, this particular solution can not be used to solve the
fundamental space harmonic component, i.e. for m = 1.
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CHAPTER 3. MAGNETOSTATIC ANALYSIS DUE TO THE PERMANENT MAGNETS
′
Substituting Brem for Brem in (3.64) gives
∂2 Az| PM
1 ∂Az| PM
1 ∂2 Az| PM
4prcm Brem ∞
+
+
=
∑ cos mpβ cos mpφ
∂r2
r ∂r
r2 ∂φ2
πr2 m=
1,3,5,...
(3.75)
Again we choose a particular solution,
∞
Az,part| PM (r, φ) =
∑ Gm|PM cos mpφ .
(3.76)
m=1,3,5,...
Then by calculating the LHS of (3.75),
∂Az| PM
=0,
∂r
∂2 Az| PM
=0,
∂r2
∞
∂Az| PM
= − ∑ mpGm| PM sin mpφ and
∂φ
m=1,3,5,...
(3.78)
∞
∂2 Az| PM
=
−
(mp)2 Gm| PM cos mpφ ,
∑
∂φ2
m=1,3,5,...
(3.80)
(3.77)
(3.79)
and substituting it back into (3.75), results in
Gm| PM = −
4rcm Brem cos mpβ
.
m2 pπ
(3.81)
which is valid for all m ≥ 0.
3.8
Boundary Conditions Between the different Regions
~
In order to solve Cm| PM and Dm| PM for the different regions, we need to know how the ~B and H
vector field would behave across the boundaries between the different regions, as graphically
depicted in Appendix C.
3.8.1 The Magnetic Flux-density at the Boundary Between the different Regions
From the definition of the magnetic flux-density, (3.19), in the cylindrical coordinate system,
~B = ∇ × ~A
1
=
r
(3.82)
~ar
r~aφ
~az
∂
∂r
∂
∂φ
∂
∂z
Ar rAφ Az| PM
∂Az| PM
∂Aφ
1 ∂(rAφ ) ∂Ar
∂Ar
1 ∂Az| PM
~ar +
~aφ +
~az .
−r
−
−
=
r
∂φ
∂z
∂z
∂r
r
∂r
∂φ
(3.83)
(3.84)
For 2-D analysis (with Jr = Jφ = 0) the above equation reduces to
∂A
∂Az| PM
~B = 1 z| PM~ar −
~aφ
r ∂φ
∂r
= Br~ar + Bφ| PM~aφ . .
(3.85)
(3.86)
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3.9. SOLVING THE MAGNETIC VECTOR POTENTIAL FOR ALL THE REGIONS OF RFAPM MACHINE
Thus for a concentric circular boundary between regions, say, (v) and (v + 1),
(v)
( v +1)
(v)
∂Az| PM
Br| PM = Br| PM
∂Az| PM
∂φ
(3.87)
( v +1)
=
(3.88)
∂φ
3.8.2 The Magnetic Field Intensity at the Boundary Between the different Regions
From (3.9), for a linear isotropic non permanent magnet region,
~
~ =B,
H
µ
(3.89)
and from (3.18), for a linear isotropic permanent magnet region,
~ ~
~ = (B − Brem ) .
H
µ
(3.90)
Thus from (3.71) and (3.86) on a boundary between a non permanent magnet region, say,
(v) and a permanent magnet region, say, (v + 1),
(v)
( v +1)
(v)
Bφ| PM
Hφ| PM = Hφ| PM
Bφ| PM
µ(v)
(3.91)
( v +1)
=
µ ( v +1)
−
Bremφ
µ ( v +1)
.
(3.92)
With radially magnetised permanent magnets, the azimuthal component of ~Brem is zero, so that
the above equation reduces to
(v)
( v +1)
1 ∂Az| PM
1 ∂Az| PM
= − ( v +1)
.
− (v)
∂r
∂r
µ
µ
3.9
(3.93)
Solving the Magnetic Vector Potential for all the Regions of
RFAPM machine
In the non permanent magnet regions, i.e. regions I, III and V, we have
Az| PM (r, φ) = Az|gen (r, φ)
(3.94)
∞
= ∑(Cm| PM r mp + Dm| PM r −mp ) cos mpφ
(3.95)
m=1,3,5,...
so that
1 ∂Az| PM (r, φ)
·
r
∂φ
∞
1
= − · ∑ mp(Cm| PM r mp + Dm| PM r −mp ) sin mpφ
r m=1,3,5,...
Br| PM (r, φ) =
(3.96)
(3.97)
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CHAPTER 3. MAGNETOSTATIC ANALYSIS DUE TO THE PERMANENT MAGNETS
51
and
1 ∂Az| PM (r, φ)
·
µ
∂r
∞
1
= − · ∑ mp(Cm| PM r mp−1 − Dm| PM r −mp−1 ) cos mpφ .
µ m=1,3,5,...
Hφ| PM (r, φ) = −
(3.98)
(3.99)
In the permanent magnet region, i.e. for regions II and IV, we have
Az| PM (r, φ) = Az|gen (r, φ) + Az| part (r, φ)
(3.100)
∞
∞
= ∑(Cm| PM r mp + Dm| PM r −mp ) cos mpφ + ∑ Gm| PM cos mpφ
(3.101)
= ∑(Cm| PM r mp + Dm| PM r −mp + Gm| PM ) cos mpφ
(3.102)
m=1,3,5,...
∞
m=1,3,5,...
m=1,3,5,...
so that
1 ∂Az| PM (r, φ)
·
r
∂φ
∞
1
= − · ∑ mp(Cm| PM r mp + Dm| PM r −mp + Gm| PM ) sin mpφ
r m=1,3,5,...
Br| PM (r, φ) =
(3.103)
(3.104)
and
1 ∂Az| PM (r, φ)
·
µ
∂r
∞
1
= − · ∑ mp(Cm| PM r mp−1 − Dm| PM r −mp−1 ) cos mpφ .
µ m=1,3,5,...
Hφ| PM (r, φ) = −
(3.105)
(3.106)
3.9.1 On the Inner Boundary of Region I
With r = rn −
h
2
− ℓ g − hm − hy ,
AzI | PM (r, φ) = 0
∴
(3.107)
I
−mp
CmI | PM r mp + Dm
=0.
| PM r
(3.108)
3.9.2 On the Boundary Between Region I and II
With r = rn −
h
2
− ℓ g − hm ,
BrI (r, φ) = BrI I (r, φ)
∴
(3.109)
I
−mp
CmI | PM r mp + Dm
= CmI I| PM r mp + DmI I| PM r −mp + GmI I| PM
| PM r
(3.110)
and
HφI (r, φ) = HφI I (r, φ)
∴
−mp−1
I
CmI | PM r mp−1 − Dm
| PM r
µI
=
(3.111)
−mp−1
II
CmI I| PM r mp−1 − Dm
| PM r
µI I
.
(3.112)
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52
3.9. SOLVING THE MAGNETIC VECTOR POTENTIAL FOR ALL THE REGIONS OF RFAPM MACHINE
3.9.3 On the Boundary Between Region II and III
With r = rn −
h
2
− ℓg ,
BrI I (r, φ) = BrI I I (r, φ)
∴
(3.113)
II
−mp
CmI I| PM r mp + Dm
+ GmI I| PM = CmI I|IPM r mp + DmI I|IPM r −mp
| PM r
(3.114)
and
HφI I (r, φ) = HφI I I (r, φ)
∴
−mp−1
II
CmI I| PM r mp−1 − Dm
| PM r
µI I
=
(3.115)
−mp−1
III
CmI I|IPM r mp−1 − Dm
| PM r
µI I I
(3.116)
3.9.4 On the Boundary Between Region III and IV
With r = rn +
h
2
+ ℓg ,
BrI I I (r, φ) = BrIV (r, φ)
∴
(3.117)
III
−mp
CmI I|IPM r mp + Dm
= CmIV| PM r mp + DmIV| PM r −mp + GmIV| PM
| PM r
(3.118)
and
HφI I I (r, φ) = HφIV (r, φ)
∴
−mp−1
III
CmI I|IPM r mp−1 − Dm
| PM r
µI I I
=
(3.119)
IV
−mp−1
CmIV| PM r mp−1 − Dm
| PM r
µ IV
(3.120)
3.9.5 On the Boundary Between Region IV and V
With r = rn +
h
2
+ ℓ g + hm ,
BrIV (r, φ) = BrV (r, φ)
∴
(3.121)
IV
−mp
V
mp
V
−mp
CmIV| PM r mp + Dm
+ GmIV| PM = Cm
+ Dm
| PM r
| PM r
| PM r
(3.122)
and
HφIV (r, φ) = HφV (r, φ)
IV
−mp−1
CmIV| PM r mp−1 − Dm
| PM r
µ IV
=
(3.123)
V
mp−1 − D V
Cm
r −mp−1
| PM r
m| PM
µV
(3.124)
3.9.6 On the Outer Boundary of Region V
With r = rn +
h
2
+ ℓ g + hm + hy ,
AV
z| PM (r, φ ) = 0
∴
V
mp
V
−mp
Cm
+ Dm
=0
| PM r
| PM r
(3.125)
(3.126)
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53
CHAPTER 3. MAGNETOSTATIC ANALYSIS DUE TO THE PERMANENT MAGNETS
3.9.7 The Simultaneous Equations in order to solve the Magnetic Vector Potential
and the Magnetic Flux-density
From (3.108), (3.110), (3.112), (3.114), (3.116), (3.118), (3.120), (3.122), (3.124) and (3.126) the
following ten equations have to be solved for m = 1, 3, 5, → ∞,
mp
CmI | PM ri
mp
−mp
+ DmI | PM ri
−mp
I
CmI | PM rii + Dm
| PM rii
mp−1
µ I I CmI | PM rii
II
CmI I| PM riii + Dm
| PM riii
(3.127)
−mp
mp
− CmI I| PM rii − DmI I| PM rii
−mp−1
− µ I I DmI | PM rii
−mp
mp
=0
mp−1
− µ I CmI I| PM rii
−mp
mp
− CmI I|IPM riii − DmI I|IPM riii
= GmI I| PM
(3.128)
−mp−1
+ µ I DmI I| PM rii
=0
= − GmI I| PM
(3.130)
mp−1
−mp−1
mp−1
−mp−1
µ I I I CmI I| PM riii
− µ I I I DmI I| PM riii
− µ I I CmI I|IPM riii
+ µ I I DmI I|IPM riii
=0
mp
−mp
mp
−mp
III
CmI I|IPM riv + Dm
− CmIV| PM riv − DmIV| PM riv
= GmIV| PM
| PM riv
mp−1
−mp−1
−mp−1
mp−1
− µ IV DmI I|IPM riv
− µ I I I CmIV| PM riv
+ µ I I I DmIV| PM riv
µ IV CmI I|IPM riv
=0
mp
−mp
mp
−mp
IV
V
V
CmIV| PM rv + Dm
− Cm
= − GmIV| PM
| PM rv
| PM rv − Dm| PM rv
mp−1
µV CmIV| PM rv
mp
−mp−1
− µV DmIV| PM rv
−mp
V
V
Cm
| PM rvi + Dm| PM rvi
−
mp−1
V
µ IV Cm
| PM rv
=0
−mp−1
V
+ µ IV Dm
| PM rv
(3.129)
=0
(3.131)
(3.132)
(3.133)
(3.134)
(3.135)
(3.136)
with
ri = r n −
rii = rn −
riii = rn −
riv = rn +
rv = rn +
rvi = rn +
h
2
h
2
h
2
h
2
h
2
h
2
− ℓ g − hm − hy
(3.137)
− ℓ g − hm
(3.138)
− ℓg
(3.139)
+ ℓg
(3.140)
+ ℓ g + hm
(3.141)
+ ℓ g + hm + hy
(3.142)
and
− ℓ g − h2m ) Brem cos mβ
m2 π
4(rn + 2h + ℓ g + h2m ) Brem cos mβ
=−
m2 π
GmI I| PM = −
GmIV| PM
4(r n −
h
2
(3.143)
(3.144)
By writing equations (3.127) to (3.136) in matrix format, it allows us to solve the Cm| PM
and Dm| PM coefficients for the different regions, for the different values of m, much simpler, as
shown in (3.145).
3.9. SOLVING THE MAGNETIC VECTOR POTENTIAL FOR ALL THE REGIONS OF RFAPM MACHINE
54
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mp
ri
mp
rii
−mp
ri
−mp
rii
mp
−mp
−rii
−rii
−mp−1
µ I I r mp−1 −µ I I r −mp−1 −µ I r mp−1
I
µ
r
ii
ii
ii
ii
mp
−mp
mp
−mp
r
r
−
r
−
r
iii
iii
iii
iii
−mp−1
mp−1
−mp−1
mp−1
I
I
I
I
I
I
I
I
I
I
−
µ
r
µ
r
µ
r
−
µ
r
iii
iii
iii
iii
·
−mp
mp
−mp
mp
r
−
r
−
r
r
iv
iv
iv
iv
mp−1
−mp−1
mp−1
−mp−1
IV
IV
I
I
I
I
I
I
µ riv
−µ riv
−µ riv
µ riv
mp
−mp
mp
−mp
rv
rv
−r v
−r v
−
mp
−
1
mp
−
1
−
mp
−
1
mp
−
1
−µ IV rv
µ IV rv
− µV r v
µV r v
mp
−mp
rvi
rvi
I
Cm| PM
0
I
I
I
Dm| PM
Gm| PM
CII
m| PM
0
II
D
− G I I
m| PM
m| PM
III
Cm| PM
0
=
(3.145)
D I I I G IV
m| PM m| PM
IV
C
0
m| PM
IV
IV
Dm| PM − Gm| PM
CV
0
m| PM
0
DV
m| PM
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CHAPTER 3. MAGNETOSTATIC ANALYSIS DUE TO THE PERMANENT MAGNETS
3.10
Obtaining the Final Solution of the Magnetic Vector Potential
and the Magnetic Flux-density
3.10.1
The Magnetic Vector Potential Solution
The solution of the magnetic vector potential for the whole machine can be obtained by substituting the appropriate values of Cm| PM and Dm| PM as obtained in (3.145) into (3.95) and (3.102)
for the different regions.
Az| PM (r, φ) =
3.10.2
∞
∑ (CmI |PM rmp + DmI |PM r−mp ) cos mpφ
m=1,3,5,...
∞
(CmI I| PM r mp
m=1,3,5,...
∞
(CmI I|IPM r mp
m=1,3,5,...
∞
(CmIV| PM r mp
m=1,3,5,...
∞
V
mp
(Cm
| PM r
m=1,3,5,...
for Region I
∑
+ DmI I| PM r −mp + GmI I| PM ) cos mpφ for Region II
∑
+ DmI I|IPM r −mp ) cos mpφ
∑
+ DmIV| PM r −mp + GmIV| PM ) cos mpφ for Region IV
∑
V
−mp
+ Dm
) cos mpφ
| PM r
for Region III
for Region V
(3.146)
The Magnetic Flux-Density Solution
To obtain the solution of the magnetic flux-density for the whole machine is a little bit more
difficult. The first step is to obtain the solution for the radial component of the magnetic fluxdensity. This is done by substituting the appropriate values of Cm| PM and Dm| PM from (3.145)
into (3.97) and (3.104) for the different regions.
∞
I
mp
1
+ DmI | PM r −mp ) sin mpφ
−
·
r ∑ mp (Cm| PM r
m=1,3,5,...
∞
− 1r · ∑ mp(CmI I| PM r mp + DmI I| PM r −mp + GmI I| PM ) sin mpφ
m=1,3,5,...
∞
III
mp
1
·
−
+ DmI I|IPM r −mp ) sin mpφ
Br| PM (r, φ) =
r ∑ mp (Cm| PM r
m=1,3,5,...
∞
IV
−mp
1
·
mp(CmIV| PM r mp + Dm
+ GmIV| PM ) sin mpφ
−
∑
| PM r
r
m=1,3,5,...
∞
−mp
mp
V
1
− · mp(CV
) sin mpφ
+ Dm
r ∑
| PM r
m| PM r
for Region I
for Region II
for Region III
for Region IV
for Region V
m=1,3,5,...
(3.147)
The next step is to obtain the solution for the azimuthal component of the magnetic fluxdensity. This is done by once again using the linear relationship between B and H (i.e. B = µH)
and then substituting the appropriate values of Cm| PM and Dm| PM from (3.145) into (3.99) and
(3.106) for the different regions.
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3.11. VALIDATION OF THE MAGNETIC VECTOR POTENTIAL AND THE MAGNETIC FLUX-DENSITY
SOLUTIONS
56
∞
−
∑ mp(CmI |PM rmp−1 − DmI |PM r−mp−1 ) cos mpφ
m=1,3,5,...
∞
−
∑ mp(CmI I|PM rmp−1 − DmI I|PM r−mp−1 ) cos mpφ
m=1,3,5,...
∞
III
mp−1
− DmI I|IPM r −mp−1 ) cos mpφ
Bφ| PM (r, φ) = − ∑ mp(Cm| PM r
m=1,3,5,...
∞
− ∑ mp(C IV r mp−1 − D IV r −mp−1 ) cos mpφ
m| PM
m| PM
m=∞1,3,5,...
−mp−1
mp−1
V
V
) cos mpφ
− Dm
− ∑ mp(Cm| PM r
| PM r
for Region I
for Region II
for Region III
(3.148)
for Region IV
for Region V
m=1,3,5,...
From (3.147) and (3.148) the magnitude of the B can now be obtained as
q
B| PM (r, φ) = Br2| PM (r, φ) + Bφ2 | PM (r, φ) ,
(3.149)
from which the flux-density contour plot can be obtained as will be shown in the next section.
3.11
Validation of the Magnetic Vector Potential and the Magnetic
Flux-density Solutions
In order to test the validity of the analytical solution presented above, the analytical solution was benchmarked against a Finite Element Analysis (FEA) solution using Ansoft’s Maxwell® 2D. The benchmarking was performed on the test machine data given in section E.1 of
Appendix E.
3.11.1
The Magnetic Vector Potential and the Magnetic Flux-density Contour Plots
A Matplotlib, see Hunter [42], contour plot of the analytical solution to the magnetic vector
potential solution is shown in Figure 3.5 with the contour plot of the FEA solution using Maxwell® 2D, shown in Figure 3.6. The “filled” or “shaded” contour plots of the magnetic fluxdensity for the analytical and FEA solutions are shown in Figure 3.7 and 3.8 respectively. The
contour plot of the magnetic flux-density for the analytical solution was again done using Matplotlib whilst the contour plot of the magnetic flux-density for the FEA solution was obtained
directly from Maxwell® 2D.
3.11.2
Radial and Azimuthal Magnetic Flux-density Distribution
It is clearly evident that it is extremely difficult to compare the analytical and FE solutions
from the Matplotlib and Maxwell® 2D contour plots. In Figure 3.5, the contour plot’s scale
varies between -12 and +12 mWb/m were as the scale on the contour plot generated by Maxwell® 2D in Figure 3.6 varies between −13,26 and 13,22 mWb/m. This is even worse for the
magnetic flux-density plots. In Figure 3.7 the Matplotlib contour plot’s scale varies between 0
and ≈ 2,88 T were as the scale on the contour plot generated by Maxwell® 2D in Figure 3.8
varies between ≈ 0 and 1,936 T.
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CHAPTER 3. MAGNETOSTATIC ANALYSIS DUE TO THE PERMANENT MAGNETS
57
0
4
8
8.8
9.6
7.2
6.45.6
4.8
2.4
0.8
-1.6
0, 02
14.4
13.6
12.8
11.2
13.6
12
10.4
3.2
1.6
+12.8
-3.2
-4
-7.2
-11.2
+3.2
+0.0
-12
-13.6
-12.8
-9.6
-6.4
-6.4
-9.6
-2.4
0
0.8
3.2
12
7.2
11.2
9.6
12.8
6.45.6
8.8
10.4
12
13.614.4
0
12
.8
13.6
-12.8
-0.8
1.6
2.4
4.8
4
8
-3.2
-14.4
-8.8
-5.6
−0, 04
+6.4
-12
8
-12.
-14.4
-13.6
y [m]
0, 00
−0, 02
+9.6
-8
-10.4
-13.6
-4.8
Magnetic vector potential, Az| PM [mWb/m]
12
12.8
0, 04
0, 21 0, 22 0, 23 0, 24 0, 25
x [m]
Figure 3.5: Contour plot of the magnetic vector potential obtained analytically.
From the contour plots of the magnetic flux-density, it appears that there is a huge discrepancy in the analytical and FE solutions, especially with regard to the flux-density in the rotor
yokes. The higher values of flux-density in the rotor yoke is produced by the analytical solution is due to fact that for the analytical solution a linear relationship between the magnetic
Figure 3.6: Contour plot of the magnetic vector potential simulated in Maxwell® 2D.
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58
3.11. VALIDATION OF THE MAGNETIC VECTOR POTENTIAL AND THE MAGNETIC FLUX-DENSITY
SOLUTIONS
2.56
0, 02
2.24
y [m]
1.92
1.60
0, 00
1.28
0.96
0.64
0.32
−0, 02
Flux density, B| PM [T]
0, 04
0.00
−0, 04
0, 21 0, 22 0, 23 0, 24 0, 25
x [m]
Figure 3.7: Contour plot of the magnetic flux-density obtained analytically.
field intensity, H, and the magnetic field density, B, is assumed. For the FE solution, the actual,
non-linear, B–H relationship is used. For our analytical analysis of the RFAPM machine with
regard to the induced voltage and torque generated, we are however more interested in the
flux-density in the airgap, than in the rotor yokes.
Figure 3.8: Contour plot of the magnetic flux-density simulated in Maxwell® 2D.
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CHAPTER 3. MAGNETOSTATIC ANALYSIS DUE TO THE PERMANENT MAGNETS
59
If we were to consider only the flux-density in the centre of the airgap, (3.147) and (3.148)
reduces to
Br| PM (rn , φ) = −
1 ∞
mp
−mp
· ∑ mp(CmI I|IPM rn + DmI I|IPM rn ) sin mpφ
rn m=1,3,5,...
(3.150)
and
∞
mp−1
Bφ| PM (rn , φ) = − ∑ mp(CmI I|IPM rn
m=1,3,5,...
−mp−1
− DmI I|IPM rn
) cos mpφ
(3.151)
for the radial and azimuthal components of the magnetic flux-density respectively.
If we now compare the analytical solution with the FE solution produced by Maxwell® 2D
for the radial and azimuthal components of the magnetic flux-density, we can see that the solutions are matching closely as shown in Figures 3.9 and 3.10 respectively, using Matplotlib. Also
shown in in Figure 3.9 and 3.10 are the radial and azimuthal components of the magnetic fluxdensity in the centre of the both the inner – and outer magnets. The Fourier series expansion of
the analytical solution is only done up to the 27th harmonic. The lack of higher order harmonics
is evident in Figure 3.9 which clearly shows that the analytical solution is finding it difficult to
track the large step changes in the radial flux-density distribution in the permanent magnets.
An interesting observation, is the computational “noisiness” of the FE solution for the azimuthal flux-density distribution as shown in Figure 3.10. This most probably has to do with
the mesh density on the “line” on which the azimuthal flux-density distribution was calculated.
With close-up scrutiny of Figure 3.9, the FE solution’s segmentation of the radial flux-density
distribution in the centre of the stator can also be observed, due to the mesh density on the
centreline of the stator.
3.11.3
2-D Analytical Analysis vs. 1-D Analytical Analysis
From Figure 3.9 it can be seen that the radial flux-density in the centre of the magnets (in a
radial direction, i.e. at −5,625 ◦ and 5,625 ◦ ) are almost exactly the same inside the airgap as it
is inside the permanent magnet on either the inner rotor yoke or the outer rotor yoke. From the
2-D analytical calculations of Test Machine I (see Appendix E.1) as was done in section 3.9, the
value of the flux-density inside the airgap, Bg , was calculated as 0,779 T with a fundamental
component of 0,824 T.
The question arises, “Can we obtain the same result using simple equivalent magnetic circuit analysis?” A brief introduction to magnetic circuit analysis using permanent magnets is
given in Appendix F, with the analysis of a RFAPM machine given in section F.7. In this analysis the assumption was made that fringing could be ignored, i.e. Bm ≈ Bg , that the permeability of the iron yoke is infinity and that average flux path will be in the centre of the permanent
magnets and in the centre of the iron yoke. From the equivalent magnetic circuit analysis, the
flux-density inside the airgap, Bg , was calculated as 0,788 T which correlates very closely with
the 2-D analysis results.
From this result it is however impossible to predict the exact shape of the flux-density distribution and hence to calculate the fundamental component of the flux-density distribution.
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60
3.11. VALIDATION OF THE MAGNETIC VECTOR POTENTIAL AND THE MAGNETIC FLUX-DENSITY
SOLUTIONS
1, 5
Br|rcim ,PM
Radial flux density, Br| PM [T]
1, 0
Br|rn ,PM
Br|rcom ,PM
Br|rcim ,PM (Maxwell® 2D)
0, 5
Br|rn ,PM (Maxwell® 2D)
Br|rcom ,PM (Maxwell® 2D)
0, 0
−0, 5
−1, 0
−1, 5
−11, 2500
−8, 4375
−5, 6250
−2, 8125
0, 0000
2, 8125
5, 6250
8, 4375
11, 2500
Angle, φ [◦ ]
Figure 3.9: The radial flux-density distribution.
This is due to the fact that the 1-D approach of the equivalent magnetic circuit analysis does
0, 3
Azimuthal flux density, Bφ| PM [T]
Bφ|rcim ,PM
Bφ|rn ,PM
0, 2
Bφ|rcom ,PM
Bφ|rcim ,PM (Maxwell® 2D)
Bφ|rn ,PM (Maxwell® 2D)
0, 1
Bφ|rcom ,PM (Maxwell® 2D)
0, 0
−0, 1
−0, 2
−0, 3
−11, 2500
−8, 4375
−5, 6250
−2, 8125
0, 0000
Angle, φ
2, 8125
5, 6250
[◦ ]
Figure 3.10: The azimuthal flux-density distribution.
8, 4375
11, 2500
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CHAPTER 3. MAGNETOSTATIC ANALYSIS DUE TO THE PERMANENT MAGNETS
61
not take any cognisance of the magnetic pole width.1
In Figure 3.11 the peak value of the radial flux-density and the associated peak value of the
fundamental component as a function of k m , the ration between the pole arc width and the pole
pitch, is shown. It can be seen that although the value of the peak radial flux-density increases
only slightly with an increase in pole width, the increase in the fundamental component is
much larger.
Stegmann and Kamper [26] used the equivalent magnetic circuit analysis technique to calculate the yoke thickness of the rotor. The calculation is based on the assumption that the ratio
of the flux-density in the poles and in the yoke is inversely proportional to the ratio of the pole
width and yoke thickness, i.e.
hy
Bm
=
By
wp
2phy
=
πrn k m
(3.152)
(3.153)
assuming that the area under the poles in the air gap is the same as in the magnets, so that the
nominal radius, rn , could be used in the calculation.
Thus from Appendix E.1, the flux-density in the centre of the rotor yoke for Test Machine
I can be calculated as 1,55 T. If we however compare this with the results obtained either 2D analytical analysis, or that obtained using Maxwell® 2D, Figure 3.12 shows that the fluxdensity in the yokes is not constant and vary from approximately 0,4 T directly behind the
magnets to approximately 1,75 T between the magnets.
3.11.4
Harmonic Analysis of the Radial Flux-density
The shape of the radial flux-density, Br| PM , distribution throughout the stator region is not
completely sinusoidal as the distribution of the flux-density in the centre of the stator, Br|,rn ,PM ,
i.e. at r =rn (see Figure 3.9). In Figure 3.13 the variation in the shape of Br| PM is shown at the
outer surface –, r = rn + 2h , the centre –, r =rn and the inner surface of the stator, r = rn − 2h .
The harmonics present in the radial airgap flux-density can be obtained directly from (3.147)
without the need to perform a FFT, by setting r =rn and substituting h for m, so that we can
write,
Br,h|rn ,PM = −
hp I I I hp
−hp
I
(Ch| PM rn + DhI |IPM
rn ) .
rn
(3.154)
The total harmonic distortion (THD) of the radial flux-density in the airgap can be calculated as follows:
q
2
∑∞
h=2 Br,h
%THDBr|r=rn ,PM =
× 100%
(3.155)
Br,1
By varying the width of the permanent magnets, the %THD of the radial flux-density can
be reduced to such a level that the radial flux-density distribution could be regarded as quasisinusoidal. With the magnet width or pole embracing factor, k m = 0,687 5 , the %THD reduces
1 The
flux-density values will be exactly the same, whether the magnetic poles span the entire pole pitch or only half
the pole pitch.
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1, 0
20
0, 8
16
0, 6
12
0, 4
8
0, 2
4
0, 0
0, 5
0, 6
0, 7
0, 8
%THD ( Br| PM )
3.11. VALIDATION OF THE MAGNETIC VECTOR POTENTIAL AND THE MAGNETIC FLUX-DENSITY
SOLUTIONS
Radial flux density, Br| PM [T]
62
0
1, 0
0, 9
Pole embracing factor, k m
Figure 3.11: The magnitude, fundamental component and %THD of the radial flux-density in the centre
of the stator vs. the pole embracing factor.
to its lowest value at just over 5 % as was shown in Figure 3.11 in the previous section, together
1, 8
Bmag| PM,rciy
1, 6
Bmag| PM,rcoy
Bmag| PM,rciy (Maxwell® 2D)
Flux density, Bmag| PM [T]
1, 4
Bmag| PM,rcoy (Maxwell® 2D)
1, 2
1, 0
0, 8
0, 6
0, 4
0, 2
−11, 2500
−8, 4375
−5, 6250
−2, 8125
0, 0000
Angle, φ
2, 8125
5, 6250
8, 4375
11, 2500
[◦ ]
Figure 3.12: The variation in the flux-density in the centre of the rotor yokes.
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CHAPTER 3. MAGNETOSTATIC ANALYSIS DUE TO THE PERMANENT MAGNETS
1, 0
Br|rn −h/2| PM
Br|rn | PM
Radial flux density, Br [T]
Br|rn +h/2| PM
0, 5
Br|rn −h/2| PM (Maxwell® 2D)
Br|rn | PM (Maxwell® 2D)
Br|rn +h/2| PM (Maxwell® 2D)
0, 0
−0, 5
−1, 0
−11, 2500
−8, 4375
−5, 6250
−2, 8125
0, 0000
2, 8125
5, 6250
8, 4375
11, 2500
Angle, φ [◦ ]
Figure 3.13: The variation in the shape radial flux-density distribution in the stator windings shown on
the outer –, centre – and inner radius of the stator for the analytical analysis method compared the the
FEA analysis done using Maxwell® 2D.
with the magnitude and fundamental component of the radial flux-density. It was thus decided that a k m ≈ 0,7 would be a good compromise between a slightly higher fundamental
component of the radial flux-density, but still with a rather low %THD value of 5,3 %. This
however also implies a slight increase in the cost1 of the permanent magnet material used.
In Figure 3.14 the harmonic spectrum of the radial flux-density in the middle of the airgap
with k m = 0,7 is shown. The 5th harmonic although small (< 5 %) is the most prominent,
followed by the 3rd and the 7th .
If we compare this with the harmonic spectrum of radial flux-density as simulated using
Maxwell® 2D also shown in Figure 3.14, we can see that it is almost identical, with the fundamental component slightly lower than that of the fundamental component which was analytically calculated. The fundamental component of the analytically calculated radial flux-density
distribution were found to be only 3% higher than those of the FEM solution. The reason for
this can be attributed to the saturation occurring in the rotor yoke which will result in a higher
reluctance value, and hence a lower flux-density value. What is interesting however, is that
the % THD is exactly the same for both at 5,3 %, implying that the shape of the flux-density
distribution waveforms are similar.
1 Although not the focus of this dissertation, it would be quite interesting to do an analysis of the trade off between the
%THD of the radial flux-density, the cost of the permanent magnets and (say) the torque or torque ripple produced
by the machine as a function of k m . The analytical analysis method presented in the chapter would provide the
ideal tool for such an analysis.
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3.12. THE FLUX-LINKAGE CALCULATION FOR THE DIFFERENT WINDING CONFIGURATIONS
0, 9
Analytical
Maxwell® 2D
0, 8
Radial flux density, Br,h| PM [T]
0, 7
0, 6
0, 5
0, 4
0, 3
0, 2
0, 1
0, 0
0
5
10
15
20
Harmonic Number
Figure 3.14: The harmonic spectrum of radial flux-density in the centre of the stator winding with
k m = 0,7 .
3.12
The Flux-linkage Calculation for the different Winding
Configurations
In Figure 3.15 a single turn of a RFAPM machine is shown. The flux-linkage of this single turn
positioned at an arbitrary angle φ can be calculated as,
λ1|O,PM (r, φ) =
=
Z
Z
S
S
~B · d~s
(3.156)
∇ × ~A| PM · d~s .
(3.157)
Applying Stokes’ Integral Theorem to (3.157), this equation can be rewritten into a line
integral form as,
λ1|O,PM (r, φ) =
=
I
Z
C
~A| PM · d~ℓ
C12
~A| PM · d~ℓ +
(3.158)
Z
C23
~A| PM · d~ℓ +
Z
C34
~A| PM · d~ℓ +
Z
C41
~A| PM · d~ℓ .
(3.159)
With ~A = Az| PM~az , the line integral reduces to
λ1|O,PM (r, φ) =
Z
C12
~A| PM · d~ℓ +
= ℓ Az| PM r, φ −
with
Z
C23
~A| PM · d~ℓ =
Z
C41
Z
C34
τq
2
~A| PM · d~ℓ = 0 .
~A| PM · d~ℓ
− Az| PM r, φ +
(3.160)
τq
2
,
(3.161)
(3.162)
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CHAPTER 3. MAGNETOSTATIC ANALYSIS DUE TO THE PERMANENT MAGNETS
65
3
C23
ℓ
C34
2
C12
4
r
C41
τq
1
φ
~az
Figure 3.15: Flux-linkage of a single turn.
The maximum flux-linkage will occur when the centre of coils are aligned with the centre
of the magnets, or more formally put, when the d-axis of the coil align with the d-axis of the
magnets. This will result in
τ
τ
Az| PM r, φ − 2q = − Az| PM r, φ + 2q
(3.163)
which enables us to simplify the flux-linkage distribution, (3.161), to
λ1|O,PM (r, φ) = 2ℓ Az| PM (r, φ) .
3.12.1
(3.164)
The Flux-linkage Calulation for the Type O Winding Configuration
The maximum or peak value of the total flux-linkage per phase for the Type O winding configuration, can now be calculated by multiplying the flux-linkage distribution, (3.164) with the
conductor density distribution, (2.26), then integrating over one resultant coil pitch period, and
q
finally multiplying by the total number of series connected circuits1 , a .
The conductor density distribution that was deduced in section 2.3.2 is only a 1-D approximation in the azimuthal direction and assumes that all the conductors are situated in the centre
of the winding, i.e. with r =rn . This implies that we only need to use a 1-D representation of
the flux-linkage distribution. This is done by setting r =rn in (3.164).
If a more accurate result needs to be obtained, we would have to redefine the conductor
density distribution of (2.26) so that it is a 2-D distribution. This 2-D distribution will then
have to be multiplied with the 2-D flux-linkage distribution of (3.164) and then integrated, not
only over one resultant coil pitch in the azimuthal direction, but also over the entire height of
the coil in the radial direction. Holm [24, chap. 5] however showed that the flux-linkage and
the resulting back-EMF can be calculated quite accurately by only using the magnetic vector
potential in the centre of the windings.
1 As explained in Appendix A we don’t have to divide by the number of parallel circuit as we assume that the current
divides equally between the number of parallel branches
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3.12. THE FLUX-LINKAGE CALCULATION FOR THE DIFFERENT WINDING CONFIGURATIONS
The peak value of the flux-linkage per phase, can thus be calculated as follows,
Λ a,b,c|O,PM
q
=
a
=
Z
1
2 τq,res|O
− 12 τq,res|O
2qℓ
a
Z
n a|O (φ)λ1|O,PM (r, φ)dφ
1
2 τq,res|O
− 12 τq,res|O
(3.165)
n a|O (φ) Az| PM (rn , φ)dφ .
(3.166)
However, as was mentioned in the previous section, (3.166) and (3.164), this will only be
true if the d-axis of the conductor density distribution, (2.26), is aligned with that of the permanent magnets. We therefore need to redefine (3.146) from a cosine series to a sine series in
order to facilitate the alignment of the magnetic vector potential with that of the conductor
density distribution, as defined in (2.26). We are also only interested in the solution of Az| PM in
the stator region, i.e. region III. Furthermore, by substituting ‘n’ for ‘m’, we can write
∞
I
AzI |IPM
(r n , φ ) =
∑ bnI I I |
Az PM
n =1
sin(npφ)
(3.167)
with
bnI IAI
z| PM
= anI IAI
z| PM
(3.168)
−np
np
I
I
rn
= CnI I| PM
rn + DnI I| PM
(3.169)
which allows us to write (3.166) as
Λ a,b,c|O,PM =
2qℓ
a
Z
1
2 τq,res|O
∞
∑
− 12 τq,res|O m=1
∞ ∞
∞
bmna|O sin(mqφ)
2qℓ
=
∑ bmna|O bnI IAIz|PM
a m∑
=1 n =1
Z
1
2 τq,res|O
− 21 τq,res|O
∑ bnI I I |
n =1
Az PM
sin(npφ)dφ
sin(mqφ) sin(npφ)dφ .
The integral term evaluates to zero except where n = m, where it evaluates to
implies that (3.171) can be simplified to
Λ a,b,c|O,PM =
Nℓ ∞
III
b m n a |O b m
,
Az| PM
πa m∑
=1
(3.170)
(3.171)
π
2q .
This
(3.172)
or from (2.30) in terms of the winding factor as
Λ a,b,c|O,PM = −
2qN ℓ ∞
III
k w,m|O bm
.
Az| PM
a m∑
=1
(3.173)
With the rotor rotating at a constant speed of ωmech , we can write,
λ a|O,PM (t) = Λ a,b,c|O,PM cos ( pωmech t) ,
λb|O,PM (t) = Λ a,b,c|O,PM cos pωmech t −
λc|O,PM (t) = Λ a,b,c|O,PM cos pωmech t −
(3.174)
2π
3
4π
3
and
(3.175)
.
(3.176)
The analytically calculated flux-linkages above is virtually identical to that obtain using
FEA in Maxwell® 2D as shown in Figure 3.16. The problem however with (3.173), if we compare it to the more simpler (2.98), is that although the former equation is very accurate, it is
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CHAPTER 3. MAGNETOSTATIC ANALYSIS DUE TO THE PERMANENT MAGNETS
2, 0
λ a| PM (t)
λb| PM (t)
1, 5
Flux linkage, λ| PM [Wbt]
λc| PM (t)
λ a| PM (t) (Approx.)
1, 0
λb| PM (t) (Approx.)
λc| PM (t) (Approx.)
0, 5
λ a| PM (t) (Maxwell® 2D)
λb| PM (t) (Maxwell® 2D)
0, 0
λc| PM (t) (Maxwell® 2D)
−0, 5
−1, 0
−1, 5
−2, 0
0, 00
1, 25
2, 50
3, 75
5, 00
6, 25
7, 50
8, 75
10, 00
11, 25
12, 50
Time, t [ms]
Figure 3.16: Comparison of the flux-linkage calculations for the Type O winding configuration.
not a simple – but computational intensive design equation. There also seem to be no relation between the two equations. There is also no clear evidence that (2.98) is indeed a good
approximation of (3.173).
However on closer inspection, if we compare (3.169) with (3.146) and (3.147) and remembering (3.154) with n = h, we see that
rBr,h
2qN ℓ ∞
(3.177)
−
k
λ a|O,PM (r ) = −
w,h|O
a h∑
hp
=1
and if we are only interested in the peak fundamental component, i.e. with h = 1 and setting
r =rn , it allows us to express
Λ a,b,c|O,PM ≈
2rn ℓ N
k q k w,m=1|O Br,1
a
(3.178)
with
Br,1 = B̂r1
(3.179)
which is identical to (2.24)
The approximated flux-linkage calculations are also shown in Figure 3.16 and correlate extremely well with the more accurate analytical calculation method.
3.12.2
The Flux-linkage Calculation for Type I Winding Configuration
The peak value of the flux-linkage for a Type I winding configuration can be calculated in
exactly the same manner as was done in section 3.12.1 for the Type O winding configuration,
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3.12. THE FLUX-LINKAGE CALCULATION FOR THE DIFFERENT WINDING CONFIGURATIONS
so that (3.171) becomes,
Λ a,b,c| I,PM =
2qℓ ∞ ∞
∑ bmn|I bnI IAIz|PM
a m∑
=1 n =1
Z
1
2 τq,res| I
− 12 τq,res| I
sin(mqφ) sin(npφ)dφ .
(3.180)
q
π
With p =k q = 12 , it implies that the integral in (3.180) will only evaluate to 2q
when n = 2m,
similar to that mentioned in section 2.4.2, but now including only the even harmonics. Thus,
with (2.80) redefined to
bm̈n| I = −
2qN
· k w,m̈| I
π
(3.181)
and with
m̈ = 2m .
(3.182)
(3.180) reduces to
Λ a,b,c| I,PM = −
2qN ℓ ∞
k w,m̈| I bnI IAI
z| PM
a m∑
=1
(3.183)
when
m̈ = n .
(3.184)
Thus, with the rotor turning at a constant speed of ωmech , the flux-linkage of each phase in
terms of (3.183), can be written as
λ a| I,PM (t) = Λ a,b,c| I,PM cos pωmech t ,
λb| I,PM (t) = Λ a,b,c| I,PM cos pωmech t −
λc| I,PM (t) = Λ a,b,c| I,PM cos pωmech t −
(3.185)
2π
3
4π
3
and
(3.186)
.
(3.187)
The flux-linkages calculated above is again virtually identical to that obtain using FEA in
Maxwell® 2D as shown in Figure 3.17.
Similar to equation (3.178), a good approximation of equation (3.183), when considering
only the fundamental component of the radial flux-density, would be
Λ a,b,c| I,PM ≈
2rn ℓ N
k q k w,2| I Br,1 ,
a
(3.188)
with only the second harmonics of the winding factor being considered, as was discussed in
section 2.4.2 and is shown in Figure 3.17.
3.12.3
The Flux-linkage Calculation for Type II Winding Configuration
Again, for a Type II winding configuration, the peak value of the total flux-linkage per phase
can be calculated similarly to (3.183), so that,
Λ a,b,c| I I,PM = −
2qN ℓ ∞
k w,m̈| I I bnI IAI
z| PM
a m∑
=1
(3.189)
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CHAPTER 3. MAGNETOSTATIC ANALYSIS DUE TO THE PERMANENT MAGNETS
2, 0
λ a| PM (t)
λb| PM (t)
1, 5
Flux linkage, λ| PM [Wbt]
λc| PM (t)
λ a| PM (t) (Approx.)
1, 0
λb| PM (t) (Approx.)
λc| PM (t) (Approx.)
0, 5
λ a| PM (t) (Maxwell® 2D)
λb| PM (t) (Maxwell® 2D)
0, 0
λc| PM (t) (Maxwell® 2D)
−0, 5
−1, 0
−1, 5
−2, 0
0, 00
1, 25
2, 50
3, 75
5, 00
6, 25
7, 50
8, 75
10, 00
11, 25
12, 50
Time, t [ms]
Figure 3.17: Comparison of the flux-linkage calculations for the Type I winding configuration.
and, assuming a constant rotor speed of ωmech , it allows us to write the flux-linkages of each
phase as
λ a| I I,PM (t) = Λ a,b,c| I I,PM cos pωmech t ,
λb| I I,PM (t) = Λ a,b,c| I I,PM cos pωmech t −
λc| I I,PM (t) = Λ a,b,c| I I,PM cos pωmech t −
(3.190)
2π
3
4π
3
and
(3.191)
.
(3.192)
The time plots of the flux-linkages is again virtually identical to the FEA results obtain using
Maxwell® 2D as is shown in Figure 3.18.
The approximation of lambda a,b,c| I I,PM when only considering the fundamental component
of the radial flux-density, similar to that done in (3.178) and (3.188), can be expressed as
Λ a,b,c| I I,PM ≈
2rn ℓ N
k q k w,2| I I Br,1 ,
a
(3.193)
with only the second harmonics of the winding factor being considered, as was discussed in
section 2.5.2 and is shown in Figure 3.18.
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3.13. THE BACK-EMF CALCULATION FOR THE DIFFERENT WINDING CONFIGURATIONS
2, 0
λ a| PM (t)
λb| PM (t)
1, 5
Flux linkage, λ| PM [Wbt]
λc| PM (t)
λ a| PM (t) (Approx.)
1, 0
λb| PM (t) (Approx.)
λc| PM (t) (Approx.)
0, 5
λ a| PM (t) (Maxwell® 2D)
λb| PM (t) (Maxwell® 2D)
0, 0
λc| PM (t) (Maxwell® 2D)
−0, 5
−1, 0
−1, 5
−2, 0
0, 00
1, 25
2, 50
3, 75
5, 00
6, 25
7, 50
8, 75
10, 00
11, 25
12, 50
Time, t [ms]
Figure 3.18: Comparison of the flux-linkage calculations for the Type II winding configuration.
3.13
The Back-EMF Calculation for the different Winding
Configurations
3.13.1
The Back-EMF Calculation for the Type O Winding Configurations
The open circuit voltages or back-EMF can be calculated directly from the flux-linkages by
noting that
λt
.
dt
Thus for the Type O winding configuration, we will have
e(t) =
λ a|O,PM (t)
dt
= − pωmech Λ a|O,PM sin pωmech t
= Ea,b,c|O,PM sin pωmech t
ea|O,PM =
with
Ea,b,c|O,PM = − pωmech Λ a|O,PM
≈−
(3.194)
(3.195)
(3.196)
(3.197)
(3.198)
2qωmech rn ℓ N
k w,1|O Br,1
a
(3.199)
and
eb|O,PM = Ea,b,c|O,PM sin pωmech t −
eb|O,PM = Ea,b,c|O,PM sin pωmech t −
2π
3
4π
3
(3.200)
.
(3.201)
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CHAPTER 3. MAGNETOSTATIC ANALYSIS DUE TO THE PERMANENT MAGNETS
The three-phase open circuit voltages for the Type O winding configuration is shown in
Figure 3.19, where it is compared with the open circuit voltages simulated in Maxwell® 2D
using FEA. It is evident that there is some difference between the two sets of results and that
the FEA results contains a lot of harmonics.
1000
ea| PM (t)
eb| PM (t)
ec| PM (t)
ea| PM (t) (Approx.)
Back EMF, e| PM [V]
500
eb| PM (t) (Approx.)
ec| PM (t) (Approx.)
ea| PM (t) (Maxwell® 2D)
eb| PM (t) (Maxwell® 2D)
0
ec| PM (t) (Maxwell® 2D)
−500
−1000
0, 00
1, 25
2, 50
3, 75
5, 00
6, 25
7, 50
8, 75
10, 00
11, 25
12, 50
Time, t [ms]
Figure 3.19: Comparison of the back-EMF calculations for the Type O winding configuration.
The FFT result of the FEA voltages is shown in Figure 3.20. Although the third and fifth
harmonics are slightly larger than the rest, the harmonics are quite evenly spread. The % THD
of the the FEA simulated voltages amounts to 14,53 % which is quite substantial.
The problem with the analytical calculations for the flux-linkage and the induced voltages
is that it assumes that the radial flux-density in the centre of the stator is constant through out
the entire stator region, which in fact it is not, as was shown in Figure 3.13.
Although the analytical calculations of the radial flux-density is similar to that calculated
in Maxwell® 2D using FEA, the solution obtained in (3.147) is not a closed-form solution. It
would be possible to perform numerical integration on the radial flux-density solutions over
the entire stator height in order to obtain a more accurate analytically calculated flux-linkage
and hence induced voltage result. This would however not be useful as “design equation” nor
a “control equation”, for the Double-sided Rotor RFAPM machine and hence the derivation of
a set of approximated analytical equations is proposed.
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3.13. THE BACK-EMF CALCULATION FOR THE DIFFERENT WINDING CONFIGURATIONS
3.13.2
The Back-EMF Calculation for the Type I Winding Configurations
For the Type I winding configuration, the three-phase open-circuit induced voltages can be
approximated with the following equations,
ea| I,PM = Ea,b,c| I,PM sin pωmech t
(3.202)
eb| I,PM = Ea,b,c| I,PM sin pωmech t − 2π
(3.203)
3
4π
(3.204)
eb| I,PM = Ea,b,c| I,PM sin pωmech t − 3
with
2qωmech rn ℓ N
k w,2| I Br,1 .
(3.205)
a
The analytical solution for the three-phase, open-circuit induced voltages for the Type I winding configuration are shown in Figure 3.21 together with the Maxwell® 2D simulated waveforms.
Ea,b,c| I,PM ≈ −
3.13.3
The Back-EMF Calculation for the Type II Winding Configurations
The analytical approximation of the three-phase open-circuit induced voltages for the Type II
winding configuration, can be given by the following set of equations,
ea| I I,PM = Ea,b,c| I I,PM sin pωmech t
(3.206)
2π
(3.207)
eb| I I,PM = Ea,b,c| I I,PM sin pωmech t − 3
eb| I I,PM = Ea,b,c| I I,PM sin pωmech t − 4π
(3.208)
3
900
800
Back EMF, Ea,h| PM [V]
700
600
500
400
300
200
100
0
0
5
10
15
20
Harmonic Number
Figure 3.20: The harmonic spectrum for the back-EMF obtained from the Maxwell® 2D FEA results for
the Type O winding configuration.
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CHAPTER 3. MAGNETOSTATIC ANALYSIS DUE TO THE PERMANENT MAGNETS
1000
ea| PM (t)
eb| PM (t)
ec| PM (t)
ea| PM (t) (Approx.)
Back EMF, e| PM [V]
500
eb| PM (t) (Approx.)
ec| PM (t) (Approx.)
ea| PM (t) (Maxwell® 2D)
eb| PM (t) (Maxwell® 2D)
0
ec| PM (t) (Maxwell® 2D)
−500
−1000
0, 00
1, 25
2, 50
3, 75
5, 00
6, 25
7, 50
8, 75
10, 00
11, 25
12, 50
Time, t [ms]
Figure 3.21: Comparison of the back-EMF calculations for the Type I winding configuration.
with
Ea,b,c| I I,PM ≈ −
2qωmech rn ℓ N
k w,2| I I Br,1 .
a
(3.209)
These approximated waveforms together with the Maxwell® 2D simulated waveforms are
shown in Figure 3.22.
3.14
Definition of the General Voltage Constant for the RFAPM
Machine
In section 3.13, the existence of a common factor in the simplified, or approximated, back-EMF
equation for the Type O, Type I and Type II could be observed. This allows us to define, for the
RFAPM machine, a voltage constant, k E , similar to that done for traditional sinusoidal permanent magnet AC (or commonly referred to as brushless DC) machines, Mohan [43, Chapter 10].
The peak value of the back-EMFs for the Type O, Type I and Type II winding configurations
can thus be defined as,
Ê f |O = k E|O ωmech ,
Ê f | I = k E| I ωmech
Ê f | I I = k E| I I ωmech
(3.210)
and
(3.211)
(3.212)
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74
3.15. SUMMARY AND CONCLUSIONS
1000
ea| PM (t)
eb| PM (t)
ec| PM (t)
ea| PM (t) (Approx.)
500
Back EMF, e| PM [V]
eb| PM (t) (Approx.)
ec| PM (t) (Approx.)
ea| PM (t) (Maxwell® 2D)
eb| PM (t) (Maxwell® 2D)
0
ec| PM (t) (Maxwell® 2D)
−500
−1000
0, 00
1, 25
2, 50
3, 75
5, 00
6, 25
7, 50
8, 75
10, 00
11, 25
12, 50
Time, t [ms]
Figure 3.22: Comparison of the back-EMF calculations for the Type II winding configuration.
respectively, with the back-EMF values
Ê f |O = Ea,b,c|O,PM ,
Ê f | I = Ea,b,c| I,PM
(3.213)
and
Ê f | I I = Ea,b,c| I I,PM
(3.214)
(3.215)
and the voltage constants,
2qrn ℓ N
k w,1|O Br,1 ,
(3.216)
a
2qrn ℓ N
k E| I =
k w,2| I Br,1 and
(3.217)
a
2qrn ℓ N
k E| I I =
k w,2| I I Br,1
(3.218)
a
with the units of the voltage constants equal to V/rad/s.
The only difference in the definition of the voltage constant between the different winding configuration is that for the overlapping (Type O) winding configuration, the fundamental
space harmonic component of the winding factor is used, as opposed to the second order space
harmonics component of the winding factor for the two non-overlapping winding configurations.
k E |O =
3.15
Summary and Conclusions
In this chapter it was shown how the electromagnetic field produced by the permanent magnets of a RFAPM machine can be calculated in 2-D using subdomain analysis. The radial- and
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CHAPTER 3. MAGNETOSTATIC ANALYSIS DUE TO THE PERMANENT MAGNETS
75
azimuthal flux-density distributions inside the stator region were almost exactly the same as
that produced using Maxwell® 2D. The slightly higher (approximately 3 %) value can be attributed to the fact that for the analytical subdomain analysis method the permeability of the
iron yoke were taken to be constant, thus ignoring saturation in the yokes. This could clearly
be seen in the flux-density contour plots where the highest analytical value was in the order of
2,7 T whereas the highest Maxwell® 2D values were calculated to be just below 2,0 T.
Furthermore was it shown, that by adjusting the pole arc width of the permanent magnets,
the radial flux-density distribution in the centre of the airgap could be made quasi-sinusoidal.
The ability to quickly simulate the radial flux-density for any array of magnetic pole width
values, compared to the time it would take to perform the same task using FEA, illustrates the
power of the analytical analysis technique. We were able to sweep the pole embracing factor,
k m , from 0,5 to 1,0 in 40 intervals steps to obtain the value of k m which would produce the
lowest %THD for the radial flux-density in the centre of the air-gap in a matter of seconds,
as compared to a FEA sweep which would take a few hours. Also the FEA solution requires
successive redrawing of the RFAPM model and to be comparable it requires the model to be
redrawn 40 times for each value of k m which can be quite computational intensive, unless the
FEA solution is properly scripted. Also, to change an azimuthal dimension in Maxwell® 2D is
much more difficult than to change a radial dimension due to the segmentation of arcs being
used by Maxwell® 2D. One has to be careful of the segmentation of the machine to ensure that
neighbouring arcs not overlapping.
In the calculation of the flux-linkage and the induced voltage (or back-EMF) for the different
winding configurations under consideration, it was found that by using only the fundamental
component of the radial flux-density distribution in the centre of the stator, an extremely good
approximation of the flux-linkage and back-EMF could be obtained, especially for the nonoverlapping, Type I and Type II, winding configurations.
Furthermore, it is interesting to note that if we compare (3.199), (3.205) and (3.209) with
one another, the magnitude of the back-EMF is directly proportional to the number of coils
per phase, q and the winding factor. Without considering the winding factor, it would at first
appear that the magnitude of the back-EMF for the non-overlapping windings will always be at
least half of that of the overlapping winding. However if we compare Figure 3.19, Figure 3.21
and Figure 3.22 we see that this is clearly not the case. Furthermore, if we remember that
the Type O winding configuration uses full-pitch windings compared to the fractional pitch
windings employed by the non-overlapping winding configurations, it is clear that the slot
width or coil-side width factor, k w,slot , plays a very important factor in machines with nonoverlapping winding configurations. This will be discussed in more detail in Chapter 6.
Finally, in this chapter, the voltage constants for the the different winding configurations
of the RAPM machine were deduced. The voltage constants for the different winding configurations are essentially the same, with the only difference that for the overlapping winding
configuration, the fundamental space harmonic component of the winding factor needs to be
used, as opposed to the second order space harmonic component of the winding factor for
the non-overlapping winding configurations. The voltage constant is similar to the voltage
constant of a permanent magnet DC machine. This is the reason why permanent magnet AC
machines are sometimes referred to as “Brushless DC” machines.
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C HAPTER 4
Magnetostatic Analysis of the Armature
Reaction Fields
In mathematics you don’t understand things. You just get used to them.
J OHANN VON N EUMANN
4.1
Introduction
The analytical analysis of the armature reaction field, using subdomain analysis, was first done
by Zhu and Howe [7] for a slotted stator machine. The analysis was later extended, in Zhu et al.
[22], to include slotless machines. Both these papers made use of the magnetic scalar potential
to solve the reaction field produced by the stator. Atallah et al. [6], on the other hand, used
of the magnetic vector potential to solve the reaction fields of slotless PM machines. In this
chapter, the magnetic vector potential, similar to Atallah et al., will also be used. By calculating
the magnetic vector potential, the flux-linkage for the windings can easily be calculated as was
shown in section 3.12.
With the permanent magnets effectively “switched off” during the armature reaction calculation, the calculated flux-linkages can be used to obtain the inductances of the different
winding configurations, similar to the method showed in Holm [24, chap. 6]. The inductances
calculation will be done assuming a sinusoidal phase current so that only the space-harmonics
of the windings will be considered. The variation in the inductance due to the time harmonics
of the phase current, will thus be ignored. Unfortunately the analytical method used in the calculation of the phase winding inductances does not take the end-turn effect into consideration,
as it is essentially a 2D analysis method. The additional inductance due to the end-turns can be
calculated with the use of empirical formulas, as was investigated by Rossouw [44]. This will,
however, be left for future work and will not form part of the analysis done in this chapter.
77
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78
4.2. THE GOVERNING EQUATIONS
4.2
The Governing Equations
With the current density distribution known, as was calculated in Chapter 2, it is possible to
solve the magnetic field caused by the stator windings. In order to do this, we once again
need to solve the magnetic vector potential, using a similar process to that done in Chapter 3.
Once again we subdivide the RFAPM machine into different regions similar to that shown in
Figure 3.1. The only difference now is that permanent magnets are ignored, so that the “new”
regions II and IV includes the airgaps on either side of the stator windings respectively. The
different regions to be solved and their governing equations for the calculation of the magnetic
vector potential, are shown in Table 4.1.
Range for r
Region
rn +
I
II
III
IV
V
h
2
+ ℓ g + hm + hy
rn + 2h + ℓ g + hm
rn + 2h
rn − 2h
rn − 2h − ℓ g − hm
≥r
≥r
≥r
≥r
≥r
≥ rn + 2h + ℓ g + hm
≥ rn + 2h
≥ rn − 2h
≥ rn − 2h − ℓ g − hm
≥ rn − 2h − ℓ g − hm − hy
µr
Governing equation
µy
∇2~A| AR
∇2~A| AR
∇2~A| AR
∇2~A| AR
∇2~A| AR
1
1
1
µy
=0
=0
= −µJz
=0
=0
Table 4.1: The governing equations for the different regions during stator excitation.
4.3
Finding the General Solutions for all the Regions
Similar to sections 3.7.1 and 3.7.2 we again need to find the general solution for each region, as
well as the particular solution for the stator region, i.e. region III. Both the general – and the
particular solution will consist of the same space – and time harmonics present in the current
density distribution.1 . From (2.51), the general solution for regions I – V will therefore have the
form,
∞
mq
−mq
) sin(mqφ + ωt) for m = 3k − 2, k ∈ N1
∑ (Cm| AR r + Dm| AR r
Az,gen| AR (r, φ) = m∞=1
,
mq
−mq
) sin(mqφ − ωt) for m = 3k − 1, k ∈ N1
∑ (Cm| AR r + Dm| AR r
m =2
(4.1)
and the particular solution for region III will have the form,
Az,part| AR (r, φ) =
∞
∑ Gm| AR sin(mqφ + ωt)
m =1
∞
∑ Gm| AR sin(mqφ − ωt)
m =2
1 This
for m = 3k − 2, k ∈ N1
.
(4.2)
for m = 3k − 1, k ∈ N1
implies either Jz|O for the Type O winding configuration, Jz| I for the Type I winding configuration or Jz| I I for
the Type II winding configuration
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79
CHAPTER 4. MAGNETOSTATIC ANALYSIS OF THE ARMATURE REACTION FIELDS
From Table 4.1 it is clear that the solution to the governing equations, consist of only Laplace
and Poisson equations and are therefore not time dependant. Thus the following shorthand
notation will be used for (4.1) and (4.2) in order to simplify the writing,
∞
Az,gen| AR (r, φ) =
∑ (Cm| AR rmq + Dm| AR r−mq ) sin(mqφ ± ωt)
(4.3)
m =1
and
∞
Az,part| AR (r, φ) =
4.4
∑
m =1
Gm| AR sin(mqφ ± ωt) .
(4.4)
Finding the Particular Solutions for Region III
The current density distributions of the Typ O –, the Type I – and the Type II winding configurations, equations (2.51), (2.85) and (2.110) respectively, all have the same basic form. The
general format of the expanded governing equation for region III can therefore be written the
following form,
∂2 Az| AR 1 ∂Az| AR
1 ∂2 Az| AR
+
+
= −µ0 Jz
∂r2
r ∂r
r2 ∂φ2
3µ0 qI p N
=−
arn hπ
(4.5)
∞
∑
m =1
k w,m sin mqφ ± ωt
(4.6)
(4.7)
Substituting the particular solution of (4.4) into (4.6), we get
−
∞
3µ0 qI p N
1
2
(
mq
)
G
sin
(
mqφ
±
ωt
)
=
−
∑
m
|
AR
r2
arn hπ
m =1
∞
∑
m =1
k w,m sin mqφ ± ωt
so that
Gm| AR (r ) =
(4.8)
3µ0 I p Nr2
k w,m
aπrn hqm2
(4.9)
which now, unlike (3.81), is a also a function of r, the radial component. We will subsequently
see that in order to solve the boundary condition equations involving Hφ (r, φ), we will need to
calculate
dGm| AR (r )
.
r
Thus, in order to simply the writing, we define,
dGm| AR (r )
dr
6µ0 I p Nr
k w,m .
=
aπrn hqm2
′
Gm
| AR (r ) =
4.5
(4.10)
(4.11)
Boundary Equations in order to solve the Armature Reaction
Fields
In all the regions, except for region III, we have
Az| AR (r, φ) = Az,gen| AR (r, φ)
(4.12)
∞
=
∑ (Cm| AR rmq + Dm| AR r−mq ) sin(mqφ ± ωt)
m =1
(4.13)
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80
4.5. BOUNDARY EQUATIONS IN ORDER TO SOLVE THE ARMATURE REACTION FIELDS
so that
1 ∂Az| AR (r, φ)
·
r
∂φ
∞
1
= · ∑ mq(Cm| AR r mq + Dm| AR r −mq ) cos(mqφ ± ωt)
r m =1
Br| AR (r, φ) =
(4.14)
(4.15)
and
1 ∂Az| AR (r, φ)
·
µ
∂r
∞
1
= − · ∑ mq(Cm| AR r mq−1 − Dm| AR r −mq−1 ) sin(mqφ ± ωt) .
µ m =1
Hφ| AR (r, φ) = −
(4.16)
(4.17)
However in region III we have
Az| AR (r, φ) = Az,gen| AR (r, φ) + Az,part| AR (r, φ)
(4.18)
∞
=
∞
∑ (Cm| AR rmq + Dm| AR r−mq ) sin(mqφ) + ∑
m =1
m =1
Gm| AR (r ) sin(mqφ ± ωt)
(4.19)
∞
=
∑
m =1
Cm| AR r mq + Dm| AR r −mq + Gm| AR (r ) sin(mqφ ± ωt)
(4.20)
so that
1 ∂Az| AR (r, φ)
·
r
∂φ
∞
1
mq
−mq
= · ∑ mq Cm| AR r + Dm| AR r
+ Gm| AR (r ) cos(mqφ ± ωt)
r m =1
Br| AR (r, φ) =
(4.21)
(4.22)
and
1 ∂Az| AR (r, φ)
·
µ
∂r
∞
1
= − · ∑ mq Cm| AR r mq−1 − Dm| AR r −mq−1 +
µ m =1
Hφ| AR (r, φ) = −
(4.23)
′
1
mq Gm| AR (r )
sin(mqφ ± ωt) .
(4.24)
4.5.1 The Boundary Equations on the Inner Boundary of Region I
With r = rn −
∴
h
2
− ℓ g − hm − hy ,
AzI | AR (r, φ) = 0
(4.25)
I
−mq
CmI | AR r mq + Dm
=0
| AR r
(4.26)
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CHAPTER 4. MAGNETOSTATIC ANALYSIS OF THE ARMATURE REACTION FIELDS
81
4.5.2 The Boundary Equations on the Boundary Between Region I and II
With r = rn −
h
2
∴
− ℓ g − hm ,
BrI| AR (r, φ) = BrI|IAR (r, φ)
(4.27)
I
−mq
CmI | AR r mq + Dm
= CmI I| AR r mq + DmI I| AR r −mq
| AR r
(4.28)
HφI | AR (r, φ) = HφI I| AR (r, φ)
(4.29)
and
∴
I
−mq−1
CmI | AR r mq−1 − Dm
| AR r
µI
=
−mq−1
II
CmI I| AR r mq−1 − Dm
| AR r
(4.30)
µI I
4.5.3 The Boundary Equations on the Boundary Between Region II and III
With r = rn −
h
2
− ℓg ,
I
BrI|IAR (r, φ) = BrI|IAR
(r, φ)
(4.31)
II
−mq
= CmI I|IAR r mq + DmI I|IAR r −mq + Gm| AR (r )
CmI I| AR r mq + Dm
| AR r
∴
(4.32)
and
I
HφI I| AR (r, φ) = HφI I| AR
(r, φ)
∴
II
−mq−1
CmI I| AR r mq−1 − Dm
| AR r
µI I
=
(4.33)
I I I r −mq−1 +
CmI I|IAR r mq−1 − Dm
| AR
µI I I
1
′
mq Gm| AR (r )
(4.34)
4.5.4 The Boundary Equations on the Boundary Between Region III and IV
With r = rn +
h
2
+ ℓg ,
I
BrI|IAR
(r, φ) = BrIV
| AR (r, φ )
∴
(4.35)
III
−mq
CmI I|IAR r mq + Dm
+ Gm| AR (r ) = CmIV| AR r mq + DmIV| AR r −mq
| AR r
(4.36)
and
I
HφI I| AR
(r, φ) = HφIV| AR (r, φ)
∴
I I I r −mq−1 +
CmI I|IAR r mq−1 − Dm
| AR
µI I I
1
′
mq Gm| AR (r )
=
(4.37)
IV r −mq−1
CmIV| AR r mq−1 − Dm
| AR
µ IV
(4.38)
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82
4.6. SIMULTANEOUS EQUATIONS TO SOLVE FOR THE ARMATURE REACTION FIELDS
4.5.5 The Boundary Equations on the Boundary Between Region IV and V
With r = rn +
∴
h
2
+ ℓ g + hm ,
V
BrIV
| AR (r, φ ) = Br | AR (r, φ )
(4.39)
−mq
mq
V
−mq
V
IV
+ Dm
= Cm
CmIV| AR r mq + Dm
| AR r
| AR r
| AR r
(4.40)
HφIV| AR (r, φ) = HφV| AR (r, φ)
(4.41)
and
IV r −mq−1
CmIV| AR r mq−1 − Dm
| AR
µ IV
=
V
mq−1 − D V
Cm
r −mq−1
| AR r
m| AR
(4.42)
µV
4.5.6 The Boundary Equations on the Outer Boundary of Region V
With r = rn +
∴
4.6
h
2
+ ℓ g + hm + hy ,
AV
z| AR (r, φ ) = 0
(4.43)
V
mq
V
−mq
Cm
+ Dm
=0
| AR r
| AR r
(4.44)
Simultaneous Equations to solve for the Armature Reaction
Fields
From (4.26), (4.28), (4.30), (4.32), (4.34), (4.36), (4.38), (4.40), (4.42) and (4.44) the following ten
equations have to be solved for m = 1 → ∞,
mq
−mq
=0
(4.45)
mq
−mq
=0
(4.46)
−mq−1
=0
(4.47)
= Gm| AR (riii )
(4.48)
I
CmI | AR ri + Dm
| AR ri
−mq
mq
I
CmI | AR rii + Dm
| AR rii
mq−1
µ I I CmI | AR rii
−mq−1
− µ I I DmI | AR rii
−mq
mq
II
CmI I| AR riii + Dm
| AR riii
mq−1
µ I I I CmI I| AR riii
mq−1
− µ I CmI I| AR rii
+ µ I DmI I| AR rii
−mq
mq
− CmI I|IAR riii − DmI I|IAR riii
−mq−1
− µ I I I DmI I| AR riii
− CmI I| AR rii − DmI I| AR rii
mq−1
− µ I I CmI I|IAR riii
−mq−1
+ µ I I DmI I|IAR riii
= µI I
mq
−mq
III
CmI I|IAR riv + Dm
| AR riv
mq
−mq
− CmIV| AR riv − DmIV| AR riv
′
Gm
| AR (riii )
mq
= − Gm| AR (riv )
(4.49)
(4.50)
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CHAPTER 4. MAGNETOSTATIC ANALYSIS OF THE ARMATURE REACTION FIELDS
mq−1
µ IV CmI I|IAR riv
−mq−1
− µ IV DmI I|IAR riv
mq−1
− µ I I I CmIV| AR riv
−mq−1
+ µ I I I DmIV| AR riv
= −µ IV
mq
−mq
IV
CmIV| AR rv + Dm
| AR rv
mq−1
µV CmIV| AR rv
−mq−1
− µV DmIV| AR rv
′
Gm
| AR (riv )
mq
−mq
=0
(4.52)
−mq−1
=0
(4.53)
−mq
=0
(4.54)
mq
V
V
− Cm
| AR rv − Dm| AR rv
mq−1
V
− µ IV Cm
| AR rv
V
+ µ IV Dm
| AR rv
mq
(4.51)
V
V
Cm
| AR rvi + Dm| AR rvi
with
ri = r n −
rii = rn −
riii = rn −
riv = rn +
rv = rn +
rvi = rn +
h
2
h
2
h
2
h
2
h
2
h
2
− ℓ g − hm − hy
(4.55)
− ℓ g − hm
(4.56)
(4.57)
(4.58)
+ ℓ g + hm
(4.59)
+ ℓ g + hm + hy
(4.60)
′
and Gm| AR and Gm
| AR as defined in (4.9) and (4.11) respectively.
By replacing k w,m with k w,m|O , k w,m| I or k w,m| I I , the simultaneous equations to solve the armature reaction fields for the Type O, Type I and II winding configuration respectively, can be
obtained. In (4.61), the equations of (4.45) to (4.54) are written in matrix format. This allows us
to solve the Cm and Dm coefficients for the different regions and for the different values of m
much easier, for the armature reactions fields of the different winding configurations.
4.7
Obtaining the Solution to the Magnetic Vector Potential and
Magnetic Flux-Density
4.7.1 The Magnetic Vector Potential Solution
The final solution for the magnetic vector potential due to the armature reaction, for the different regions of the machine, is thus obtained by substituting the appropriate values of Cm| AR
and Dm| AR as obtained in (4.61) into (4.13) and (4.20), as shown in (4.62).
4.7. OBTAINING THE SOLUTION TO THE MAGNETIC VECTOR POTENTIAL AND MAGNETIC
FLUX-DENSITY
84
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mq
ri
mq
rii
−mq
ri
−mq
rii
mq
−mq
−rii
−rii
−mq−1
µ I I r mq−1 −µ I I r −mq−1 −µ I r mq−1
I
µ
r
ii
ii
ii
ii
mq
−mq
mq
−mq
r
r
−
r
−
r
iii
iii
iii
iii
−mq−1
mq−1
−mq−1
mq−1
I
I
I
I
I
I
I
I
I
I
−
µ
r
µ
r
µ
r
−
µ
r
iii
iii
iii
iii
·
−mq
mq
−mq
mq
r
−
r
−
r
r
iv
iv
iv
iv
mq−1
−mq−1
mq−1
−mq−1
IV
IV
I
I
I
I
I
I
µ riv
−µ riv
−µ riv
µ riv
mq
−mq
mq
−mq
rv
rv
−r v
−r v
−
mq
−
1
mq
−
1
−
mq
−
1
mq
−
1
−µ IV rv
µ IV rv
− µV r v
µV r v
mq
−mq
rvi
rvi
I
Cm| AR
0
I
Dm| AR
0
CII
0
m| AR
II
D
Gm| AR (riii )
m| AR I I
III µ ′
Cm| AR mq Gm| AR (riii )
D I I I =
m| AR − Gm| AR (riv )
IV µ IV ′
C
− G
(
r
)
m| AR mq m| AR iv
IV
Dm| AR
0
CV
0
m| AR
0
DV
m| AR
(4.61)
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CHAPTER 4. MAGNETOSTATIC ANALYSIS OF THE ARMATURE REACTION FIELDS
∞
I
mq
I
−mq
sin(mqφ ± ωt)
C
r
+
D
r
∑ m| AR
m| AR
m =1
∞
II
mq
II
−mq
sin(mqφ ± ωt)
C
r
+
D
r
∑ m| AR
m| AR
m
=
1
∞
III
−mq
CmI I|IAR r mq + Dm
r
+
G
(
r
)
·
∑
m
|
AR
| AR
Az| AR (r, φ) = m=1
sin(mqφ ± ωt)
∞
IV
mq
IV
−mq
sin(mqφ ± ωt)
C
r
+
D
r
∑
m| AR
m| AR
m
=
1
∞
−mq
mq
V
V
sin(mqφ ± ωt)
r
r
+
D
C
∑
m| AR
m| AR
for Region I
for Region II
for Region III
(4.62)
for Region IV
for Region V
m =1
4.7.2 The Magnetic Flux-Density Solution
To obtain the final solution of the magnetic flux-density, for the different regions of the RFAPM
machine, due to armature reaction, is a little bit more difficult. This is done by substituting the
appropriate values of Cm| AR and Dm| AR from (4.61) into (4.15) and (4.22) to obtain the radial
component of the magnetic flux-density, as shown in (4.63).
− 1r
− 1r
1
−r
Br| AR (r, φ) =
− 1r
1
− r
∞
·
·
·
∑
m =1
∞
∑
m =1
∞
∑
m =1
∞
·
·
∑
m =1
∞
∑
m =1
I
−mq
mq CmI | AR r mq + Dm
cos(mqφ ± ωt)
| AR r
II
−mq
cos(mqφ ± ωt)
mq CmI I| AR r mq + Dm
r
| AR
III
−mq
mq CmI I|IAR r mq + Dm
r
+
G
(
r
)
·
m| AR
| AR
cos(mqφ ± ωt)
IV
−mq
cos(mqφ ± ωt)
mq CmIV| AR r mq + Dm
r
| AR
−mq
mq
V
V
r
r
+
D
mq Cm
cos(mqφ ± ωt)
m| AR
| AR
for Region I
for Region II
for Region III
for Region IV
for Region V
(4.63)
Assuming a linear relationship between the flux-density, B, and the magnetic field intensity,
H, (i.e. B = µH) the azimuthal component of the magnetic flux-density, due to armature
reaction, for the different regions of the RFAPM machine can be obtained by, again, substituting
the appropriate values of Cm| AR and Dm| AR from (4.61) into (4.17) and (4.24) as shown in (4.64).
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4.8. VALIDATION OF THE TYPE O WINDING CONFIGURATION SOLUTION
∞
I
mq−1
I
−mq−1
−
mq
C
r
−
D
r
sin(mqφ ± ωt)
∑
m| AR
m| AR
m =1
∞
II
mq−1
II
−mq−1
−
sin(mqφ ± ωt)
mq
C
r
−
D
r
∑
m| AR
m| AR
m
=
1
∞
′
1
− ∑ mq CmI I|IAR r mq−1 − DmI I|IAR r −mq−1 + mq
(
r
)
·
Gm
| AR
Bφ| AR (r, φ) =
m =1
sin(mqφ ± ωt)
∞
IV
mq−1
IV
−mq−1
−
sin(mqφ ± ωt)
mq
C
r
−
D
r
∑
m| AR
m| AR
m
=
1
∞
−mq−1
mq−1
V
V
−
r
r
−
D
mq
C
sin(mqφ ± ωt)
∑
m| AR
m| AR
for Region I
for Region II
for Region III
for Region IV
for Region V
m =1
(4.64)
From (4.63) and (4.64) the magnitude of the magnetic flux-density can now be obtained as
follows
q
B| AR (r, φ) = Br2| AR (r, φ) + Bφ2 | AR (r, φ) .
(4.65)
The magnetic flux-density contour plots will be discussed in the following sections and are
shown in Appendix G together with the magnetic vector potential contour plots.
4.8
Validation of the Type O Winding Configuration Solution
4.8.1 Magnetic Field Solutions of the Type O Winding Configuration
The Matplotlib contour plot of the analytically calculated magnetic vector potential, is shown
in Figure G.11 , with the FEA solution obtained from Maxwell® 2D shown in Figure G.2. In
Figure G.3 the Matplotlib contour plot of the analytically calculated magnetic flux-density is
shown, whilst the FEA solution obtained from Maxwell® 2D, is shown in Figure G.4.
As was mentioned in section 3.11.1, it is very difficult to compare the analytical solution
with the FEA solution from the contour plots. The best way to do it, is to take slices through
the machine at different radius values and compare the radial – and azimuthal flux-density
distributions between the analytical and FEA results on these “cut lines”. This is shown in
Figure 4.1 and Figure 4.2, where the radial – and azimuthal magnetic flux-density distributions
for the analytically and FEA solution obtained using Maxwell® 2D, are compared.
4.8.2 Flux-linkage Calculation of the Type O Winding Configuration
The peak value of the total flux-linkage due to the armature reaction can be calculated in exactly
the same way as was done in section 3.12 for the calculation of the peak value of the total fluxlinkage per phase due to the permanent magnets. This implies that equation (3.164), which was
derived for the flux-linkage in a single turn in terms of the magnetic vector potential caused by
the permanent magnets, could be rewritten as
λ1|O,AR (r, φ) = 2ℓ Az|O,AR (r, φ) ,
1 The
(4.66)
magnetic vector potential en magnetic flux-density contour plots are shown in Appendix G on pages 163 and
page 164 respectively.
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CHAPTER 4. MAGNETOSTATIC ANALYSIS OF THE ARMATURE REACTION FIELDS
0, 03
Br|rcim ,AR
Radial flux density, Br| AR [T]
0, 02
Br|rn ,AR
Br|rcom ,AR
Br|rcim ,AR (Maxwell® 2D)
0, 01
Br|rn ,AR (Maxwell® 2D)
Br|rcom ,AR (Maxwell® 2D)
0, 00
−0, 01
−0, 02
−0, 03
−11, 2500
−8, 4375
−5, 6250
−2, 8125
0, 0000
2, 8125
5, 6250
8, 4375
11, 2500
Angle, φ [◦ ]
Figure 4.1: Radial Flux-Density Distribution
0, 006
Azimuthal flux density, Bφ| AR [T]
Bφ|rcim ,AR
Bφ|rn ,AR
0, 004
Bφ|rcom ,AR
Bφ|rcim ,AR (Maxwell® 2D)
Bφ|rn ,AR (Maxwell® 2D)
0, 002
Bφ|rcom ,AR (Maxwell® 2D)
0, 000
−0, 002
−0, 004
−0, 006
−11, 2500
−8, 4375
−5, 6250
−2, 8125
0, 0000
Angle, φ
2, 8125
5, 6250
8, 4375
11, 2500
[◦ ]
Figure 4.2: Azimuthal Flux-Density Distribution
with Az|O,AR (r, φ) the vector potential caused only by the three-phase currents flowing in the
windings with the permanent magnets “switched off”.
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4.8. VALIDATION OF THE TYPE O WINDING CONFIGURATION SOLUTION
The total flux-linkage per phase can thus be calculated by multiplying the flux-linkage distribution in (4.66) with the conductor density distribution given in (2.26) (see also Figure 2.5);
then integrating over one resultant coil pitch period and finally multiplying by the number of
q
series connected circuits1 , a . Also, as discussed in section 3.12.1, we assume that the magnetic
vector potential throughout the windings is equal to the magnetic vector potential at the centre
of the windings, at r =rn .
Λ a,b,c|O,AR
q
=
a
=
Z τq|O
2
τq|O n a|O ( φ ) λ1|O,AR (rn , φ ) dφ
− 2
Z τq|O
2qℓ
2
τq|O n a|O ( φ ) Az|O,AR (rn , φ ) dφ
a − 2
(4.67)
(4.68)
From (4.62) and only using the solution for the vector potential in the stator region, i.e.
region III, for a Type O winding configuration and substituting ‘n’ for ‘m’, we can write
∞
I
AzI |IO,AR
(r n , φ ) =
∑
m =1
III
bm
A
z|O,AR
sin(nqφ)
(4.69)
with
III
bm
A
−nq
mq
z|O,AR
= CmI I|IAR rn + DmI I|IAR rn
+ Gm|O,AR (rn )
(4.70)
and from (4.9), using the winding factor, k w,m|O , for the Type O winding configuration as
defined in (2.31), Gm|O,AR (rn ) can be expanded to
Gm|O,AR (rn ) =
3µ0 I p Nrn
k
.
aπhqn2 w,n|O
(4.71)
From (2.26) and (4.69), (4.68) can now be written as
Λ a,b,c|O,AR
2qℓ
=
a
Z τq|O
∞
2
b m m a |O
τq|O
− 2 m =1
∑
∞
sin(mqφ)
2qℓ ∞ ∞
=
∑ bmma|O bmI IAI z|O,AR
a m∑
=1 m =1
∑
m =1
Z τq|O
2
τq|O
− 2
III
bm
A
z|O,AR
sin(nqφ)dφ
sin(mqφ) sin(nqφ)dφ .
(4.72)
(4.73)
With the coil-pitch of the Type O winding configuration, τq|O = πq , the integral term evalπ
uates to zero except where m = n when it evaluates to 2q
. This implies that (4.73) can be
simplified to
Λ a,b,c|O,AR =
πℓ ∞
III
b m m a |O b m
,
Az|O,AR
a m∑
=1
(4.74)
or from (2.30) in terms of the winding factors, as
Λ a,b,c|O,AR =
1 As
2qN ℓ ∞
III
k w,m|O bm
,
Az|O,AR
a m∑
=1
(4.75)
explained in Appendix A we don’t have to divide by the number of parallel circuits as we assume that the
current divides equally between the number of parallel branches
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CHAPTER 4. MAGNETOSTATIC ANALYSIS OF THE ARMATURE REACTION FIELDS
89
The peak value of the total flux-linkage per phase, Λ a,b,c|O,AR , for the Type O winding configuration, using the same rotor configuration as given in Appendix E, section E.1, is calculated
as
Λ a,b,c|O,AR = 48,045 mWbt .
(4.76)
If we compare the peak value of the flux-linkage in (4.76) with the results in Table 4.2 were
the flux-linkage of each space-harmonic component were calculated separately, we can see
that the main contribution of the flux-linkage results from the fundamental space-harmonic
component. This implies that (4.75) may be simplified to
2qN ℓ
III
k w,m=1|O bm
=1 Az|O,AR
a
≈ 47,827 mWbt .
Λ a,b,c|O,AR ≈
(4.77)
(4.78)
which would yield a good approximation of the flux-linkage for the Type O winding configuration.
Due to the fact that the N48 Neodymium-Boron-Iron (NdBFe) surface mounted permanent
magnets that are used in the RFAPM machine’s recoil permeability, µrecoil ≈ µ0 , see also [45],
the flux-linkage will be the same regardless of the rotor position. The magnitude of the fluxlinkage will thus only depend on the value of the phase current, so that for a perfectly balanced
three-phase stator current, we can write,
λ a|O,AR (t) = Λ a,b,c|O,AR cos(ωt)
λb|O,AR (t) = Λ a,b,c|O,AR cos(ωt −
λc|O,AR (t) = Λ a,b,c|O,AR cos(ωt −
(4.79)
2π
3 )
4π
3 )
and
(4.80)
.
(4.81)
In Figure 4.3 the analytically calculated flux-linkages for the Type O winding configuration
is compared with that calculated by Maxwell® 2D, using FEA. The difference between the
Table 4.2: The flux-linkage component for each space-harmonic of the Type O winding configuration.
Space harmonic, m
1
2
3
4
5
6
7
8
9
10
11
12
13
Λ a,b,c|O,AR,m [mWbt]
47,827
–
–
–
0,157
–
0,045
–
–
–
0,008
–
0,004
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4.8. VALIDATION OF THE TYPE O WINDING CONFIGURATION SOLUTION
analytically calculated flux-linkage values and that of the FEA values can be contributed to the
fact that only a 1–D conductor density distribution was used together with the assumption that
the average magnetic vector potential distribution values in the windings remains equal to the
values in the centre of the windings.
0, 06
λ a| AR (t)
λb| AR (t)
0, 04
λc| AR (t)
Flux-linkage, λ [Wbt]
λ a| AR (t) (Maxwell® 2D)
λb| AR (t) (Maxwell® 2D)
0, 02
λc| AR (t) (Maxwell® 2D)
0, 00
−0, 02
−0, 04
−0, 06
0, 00
1, 25
2, 50
3, 75
5, 00
6, 25
7, 50
8, 75
10, 00
11, 25
12, 50
Time, t [ms]
Figure 4.3: Flux-linkages for phase a, b and c for the Type O winding configuration.
4.8.3 Stator Inductance Calculation of the Type O Winding Configuration
With the flux-linkages for the Type O winding configuration known, it is now possible to calculate the inductances of the stator windings for this winding configuration. If we assume the
machine to be perfectly symmetrical, the self inductance for each phase as well as the mutual
inductance between any two phases will be exactly the same. This implies that the inductance
matrix, in terms of the flux-linkage for each phase winding (with the magnets “switched off”)
and the phase current, can be written as,
L |O
λ a|O,AR
λb|O,AR = M|O
M |O
λc|O,AR
M |O
L |O
M |O
ia
M |O
M |O i b
ic
L |O
(4.82)
with LO the self inductance of each phase winding and M|O the mutual inductance between
any two phase windings.
For a balanced, three-phase supply (in the case of motor operation), or for a balanced, threephase load (in the case of generator operation) the inductance matrix in terms of the flux-
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CHAPTER 4. MAGNETOSTATIC ANALYSIS OF THE ARMATURE REACTION FIELDS
linkages and three-phase currents, can be reduced to
ia
L |O − M |O
0
0
λ a|O,AR
0
L |O − M |O
0
λb|O,AR =
ib
ic
0
0
L |O − M |O
λc|O,AR
or simply as
L s |O
λ a|O,AR
λb|O,AR = 0
0
λc|O,AR
0
L s |O
0
ia
0
0 ib
ic
L s |O
91
(4.83)
(4.84)
with Ls|O the three-phase synchronous inductance of the machine.
Assuming a sinusoidal current and flux-linkage, the synchronous inductance, Ls|O , can be
thus be approximated by
L s |O =
Λ a,b,c|O,AR
,
Ip
(4.85)
with I p the peak value of the phase current, see (2.43) to (2.45). This implies that for the test
machine in Appendix E, section E.1, the three-phase stator inductance, Ls|O , for the Type O
winding configuration, with the end-turn effect ignored, would be equal to
Ls|O = 6,55 mH .
4.9
(4.86)
Validation of the Type I Winding Configuration Solution
4.9.1 Magnetic Field Solutions of the Type I Winding Configuration
The Matplotlib contour plot of the magnetic vector potential is shown in Figure G.5 with the
FEA solution obtained from Maxwell® 2D shown in Figure G.6. In Figure G.7 the Matplotlib
contour plot of the analytical solution to the magnetic flux-density is shown. The FEA solution
obtained from Maxwell® 2D shown in Figure G.8.
In Figure 4.4 and Figure 4.5 which shows the Matplotlib plots of the radial – and azimuthal
flux-density distributions respectively, the analytically calculated flux-density values are compared with the results obtained from the Maxwell® 2D FEA simulation.
4.9.2 Flux-linkage Calculation of the Type I Winding Configuration
The flux-linkage for the Type I winding configuration can be calculated in a similar fashion as
for the Type O winding configuration. The flux-linkage in a single coil turn, due to the magnetic
vector potential caused only by the current flowing in the windings, again with the permanent
magnets “switched off”, can be written as,
λ1| I,AR (r, φ) = 2ℓ Az|O,AR (r, φ) .
(4.87)
The total flux-linkage per phase can be calculated by multiplying the flux-linkage distribution in (4.87) with the conductor density distribution given by (2.78)1 ; then integrating over one
q
coil pitch period and finally multiplying by the number of series connected circuits2 , a .
1 Graphically
2 See
illustrated in Figure 2.13
explanation in Appendix A.
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4.9. VALIDATION OF THE TYPE I WINDING CONFIGURATION SOLUTION
0, 02
Br|rcim ,AR
Radial flux density, Br| AR [T]
0, 01
Br|rn ,AR
Br|rcom ,AR
Br|rcim ,AR (Maxwell® 2D)
0, 00
Br|rn ,AR (Maxwell® 2D)
Br|rcom ,AR (Maxwell® 2D)
−0, 01
−0, 02
−0, 03
−0, 04
−22, 500
−16, 875
−11, 250
−5, 625
0, 000
5, 625
11, 250
16, 875
22, 500
Angle, φ [◦ ]
Figure 4.4: Radial flux-density distribution for the Type II winding configuration due to armature reaction.
0, 006
Azimuthal flux density, Bφ| AR [T]
Bφ|rcim ,AR
Bφ|rn ,AR
0, 004
Bφ|rcom ,AR
Bφ|rcim ,AR (Maxwell® 2D)
Bφ|rn ,AR (Maxwell® 2D)
0, 002
Bφ|rcom ,AR (Maxwell® 2D)
0, 000
−0, 002
−0, 004
−0, 006
−22, 500
−16, 875
−11, 250
−5, 625
0, 000
Angle, φ
5, 625
11, 250
16, 875
22, 500
[◦ ]
Figure 4.5: Azimuthal flux-density distribution for the Type II winding configuration due to armature
reaction.
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CHAPTER 4. MAGNETOSTATIC ANALYSIS OF THE ARMATURE REACTION FIELDS
93
Again, with the 1–D conductor density distribution taken at r =rn , we can assume that the
flux-density distribution in the centre of the windings, at r =rn , is the same through out. This
implies that the peak value of the flux-density distribution for the Type I winding configuration
can be calculated as,
Λ a,b,c| I,AR
q
=
a
=
Z τq| I
2
τq| I n a| I ( φ ) λ1| I,AR (rn , φ ) dφ
− 2
Z τq| I
2qℓ
2
τ n ( φ ) Az| I,AR (rn , φ ) dφ
a − q2| I a| I
(4.88)
.
(4.89)
From (2.78) and (4.62) and substituting ‘n’ for ‘m’, we can write
∞
I
AzI |II,AR
(r n , φ ) =
∑
m =1
III
bm
A
z| I,AR
sin(nqφ)
(4.90)
with
III
bm
A
−nq
mq
z| I,AR
= CmI I|IAR rn + DmI I|IAR rn
+ Gm| I,AR (rn )
(4.91)
and, from (4.9), using the winding factor for the Type I winding configuration as defined in
(2.81), the value of Gm| I,AR (rn ) will expand to
Gm| I,AR (rn ) =
3µ0 I p Nrn
k
.
aπhqn2 w,n| I
(4.92)
From (2.78) and (4.90), equation (4.89) can now be written as
Λ a,b,c| I,AR
2qℓ
=
a
Z τq| I
∞
2
bm m a | I
τq| I
− 2 m =1
∑
∞
∑
sin(mqφ)
2qℓ ∞ ∞
=
∑ bmma|I bmI IAI z|I,AR
a m∑
=1 m =1
m =1
Z τq| I
2
τq| I
− 2
III
bm
A
z| I,AR
sin(nqφ)dφ
sin(mqφ) sin(nqφ)dφ .
(4.93)
(4.94)
π
, the integral term in (4.94)
With the coil-pitch of the Type I winding configuration, τq| I = Q
π
, so that (4.94) can be
evaluates to zero except where m = n, when it evaluates once again to 2q
simplified to,
Λ a,b,c| I,AR =
πℓ ∞
III
bm m a | I bm
.
Az| I,AR
a m∑
=1
(4.95)
This has the same form as the flux-linkage equation of the Type O winding configuration
calculated in (4.74), but differs slightly from the flux-linkage of the Type I windings due to
effect of the permanent magnets, as calculated in (3.183). For the latter, only the second order
space-harmonics were present, but for armature reaction, the first, fifth, seventh, eleventh and
thirteenth harmonics are also present as shown in Table 4.3. Only the space-harmonic fluxlinkage values up to the thirteenth harmonic are shown and are calculated from (4.95) for the
Type I winding configuration.1 .
1 Again
the same rotor configuration as in Appendix E, section E.1 was used
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4.9. VALIDATION OF THE TYPE I WINDING CONFIGURATION SOLUTION
Table 4.3: The flux-linkage component for each space-harmonic for the Type I winding configuration.
Space harmonic, m
1
2
3
4
5
6
7
8
9
10
11
12
13
Λ a,b,c| I,AR,m [mWbt]
23,195
17,487
–
2,936
0,392
–
0,024
–
–
0,052
0,022
–
0,013
The total flux-linkage per phase for the Type I winding configuration, using (4.95), is calculated as Λ a,b,c| I,AR =44,165 mWbt. If we compare this with the values in Table 4.3 as well
as with the result for the Type O winding configuration given in Table 4.2, we can see that
the fundamental space-harmonic is not dominant and that the peak value of flux-linkage per
phase has to be takes as the sum of all the space-harmonics components. The flux-linkage calculation method for the armature reaction for the Type I winding configuration also differs from
the method used to calculate the flux-linkage due to the permanent magnets acting alone (see
(3.188)) where the flux-linkage were approximated by using only the second space-harmonic
(or working harmonic) value.
As mentioned in section 4.8.2, due to the fact that surface mounted permanent magnets
are used with a relative recoil permeability close to unity, the peak value of the flux-linkage
will be the same regardless of the rotor position. The magnitude of the flux-linkage will thus
only depend on the value of the phase current in each phase winding, so that for a perfectly
balanced, three-phase stator current, we can write,
λ a| I,AR (t) = Λ a,b,c| I,AR cos(ωt)
λb| I,AR (t) = Λ a,b,c| I,AR cos(ωt −
λc| I,AR (t) = Λ a,b,c| I,AR cos(ωt −
(4.96)
2π
3 )
4π
3 )
and
.
(4.97)
(4.98)
In Figure 4.6 the analytically calculated flux-linkage waveform for the Type I winding configuration is compared with the waveform obtained from FEA simulations using Maxwell® 2D.
Here the analytically calculated flux-linkages are a much closer match than for the values obtained for Type O winding configuration that was shown in Figure 4.3.
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CHAPTER 4. MAGNETOSTATIC ANALYSIS OF THE ARMATURE REACTION FIELDS
0, 06
λ a| AR (t)
λb| AR (t)
0, 04
λc| AR (t)
Flux-linkage, λ [Wbt]
λ a| AR (t) (Maxwell® 2D)
λb| AR (t) (Maxwell® 2D)
0, 02
λc| AR (t) (Maxwell® 2D)
0, 00
−0, 02
−0, 04
−0, 06
0, 00
1, 25
2, 50
3, 75
5, 00
6, 25
7, 50
8, 75
10, 00
11, 25
12, 50
Time, t [ms]
Figure 4.6: Flux-linkages for phase a, b and c for the Type I winding configuration.
4.9.3 Stator Inductance Calculation of the Type I Winding Configuration
Again assuming perfectly balanced, three-phase, sinusoidal currents and flux-linkages, the inductance matrix can be approximated, similarly to section 4.8.3, in terms of the three-phase
currents and flux-linkages, as
ia
Ls| I
0
0
λ a| I,AR
(4.99)
0 ib
λb| I,AR = 0 Ls| I
λc| I,AR
0
0
Ls| I
ic
with the three-phase stator inductance
Ls| I =
Λ a,b,c| I,AR
.
Ip
(4.100)
This implies that for the test machine in Appendix E, section E.1 with non-overlapping,
single-layer (Type I) winding configuration, the synchronous inductance, with the end-turn
effect ignored, would be equal to
Ls| I = 6,02 mH .
4.10
Validation of the Type II Winding Configuration Solution
4.10.1
Magnetic Field Solutions of the Type II Winding Configuration
(4.101)
The Matplotlib contour plot of the magnetic vector potential is shown in Figure G.9 with the
FEA result from Maxwell® 2D shown in Figure G.10. In Figure G.11 the contour plot of the
magnetic flux-density is shown with the FEA result from Maxwell® 2D shown in Figure G.12.
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4.10. VALIDATION OF THE TYPE II WINDING CONFIGURATION SOLUTION
In Figure 4.7 and Figure 4.8 Matplotlib plots of the radial and azimuthal flux-density analytical calculations are compared with Maxwell® 2D finite element calculations.
0, 02
Br|rcim ,AR
Br|rn ,AR
Radial flux density, Br| AR [T]
0, 01
Br|rcom ,AR
Br|rcim ,AR (Maxwell® 2D)
Br|rn ,AR (Maxwell® 2D)
0, 00
Br|rcom ,AR (Maxwell® 2D)
−0, 01
−0, 02
−0, 03
−0, 04
−22, 500
−16, 875
−11, 250
−5, 625
0, 000
5, 625
11, 250
16, 875
22, 500
Angle, φ [◦ ]
Figure 4.7: Radial Flux-Density Distribution
4.10.2
Flux-linkage Calculation of the Type II Winding Configuration
The flux-linkage calculations for the Type II winding configuration will be similar to that of the
Type I winding configuration, so that
Λ a,b,c| I I,AR =
πℓ ∞
III
bm m a | I I bm
Az| I I,AR
a m∑
=1
(4.102)
with
III
bm
A
mq
z| I I,AR
−nq
= CmI I|IAR rn + DmI I|IAR rn
+ Gm| I I,AR (rn )
(4.103)
and
Gm| I I,AR (rn ) =
3µ0 I p Nrn
k
.
aπhqn2 w,n| I
(4.104)
The peak value of flux-linkage per phase for the Type II winding configuration can thus
be calculated as λ a| I I,AR,m =57,885 mWbt. If we compared the peak value to the values of the
individual space-harmonic components given in Table 4.4, or with the approximation for the
flux-linkage due to the permanent magnets alone, (3.193), it is clear neither the first nor the
second harmonic is dominant and that all the space-harmonics needs to be taken into account
for the accurate calculation of the total flux-linkage per phase for armature reaction.
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CHAPTER 4. MAGNETOSTATIC ANALYSIS OF THE ARMATURE REACTION FIELDS
0, 006
Azimuthal flux density, Bφ| AR [T]
Bφ|rcim ,AR
Bφ|rn ,AR
0, 004
Bφ|rcom ,AR
Bφ|rcim ,AR (Maxwell® 2D)
Bφ|rn ,AR (Maxwell® 2D)
0, 002
Bφ|rcom ,AR (Maxwell® 2D)
0, 000
−0, 002
−0, 004
−0, 006
−22, 500
−16, 875
−11, 250
−5, 625
0, 000
5, 625
11, 250
16, 875
22, 500
Angle, φ [◦ ]
Figure 4.8: Azimuthal Flux-Density Distribution
Table 4.4: The flux-linkage components for each space-harmonic for the Type II winding configuration.
Space harmonic, m
1
2
3
4
5
6
7
8
9
10
11
12
13
λ a| I I,AR,m [mWbt]
34,854
21,873
–
0,908
0,038
–
0,096
–
–
0,007
0,061
–
0,030
For a perfectly balanced, three-phase, sinusoidal stator current, we will be able to write,
λ a| I I,AR (t) = Λ a,b,c| I I,AR cos(ωt)
λb| I I,AR (t) = Λ a,b,c| I I,AR cos(ωt −
λc| I I,AR (t) = Λ a,b,c| I I,AR cos(ωt −
(4.105)
2π
3 )
4π
3 )
and
.
(4.106)
(4.107)
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4.10. VALIDATION OF THE TYPE II WINDING CONFIGURATION SOLUTION
In Figure 4.9 the analytically calculated flux-linkages waveforms for the Type II winding
configuration are compared with FEA simulated waveforms, using Maxwell® 2D.
0, 06
λ a| AR (t)
λb| AR (t)
0, 04
λc| AR (t)
Flux-linkage, λ [Wbt]
λ a| AR (t) (Maxwell® 2D)
λb| AR (t) (Maxwell® 2D)
0, 02
λc| AR (t) (Maxwell® 2D)
0, 00
−0, 02
−0, 04
−0, 06
0, 00
1, 25
2, 50
3, 75
5, 00
6, 25
7, 50
8, 75
10, 00
11, 25
12, 50
Time, t [ms]
Figure 4.9: Flux-linkages for phase a, b and c for the Type II winding configuration.
4.10.3
Stator Inductance Calculation of the Type II Winding Configuration
Assuming perfectly balanced, three-phase, sinusoidal currents and flux-linkages, the inductance matrix can be approximated, similar to sections 4.8.3 and 4.9.3, in terms of the three-phase
currents and flux-linkages, as
Ls| I I
λ a| I I,AR
λb| I I,AR = 0
0
λc| I I,AR
0
Ls| I I
0
ia
0
0 ib
ic
Ls| I I
(4.108)
with the three-phase stator inductance
Ls| I I =
Λ a,b,c| I I,AR
.
I Ip
(4.109)
This implies that for the test machine in Appendix E, section E.1 with a non-overlapping,
double-layer (Type II winding configuration, the synchronous inductance, with the end-turn
effect ignored, would be equal to
Ls| I I = 7,896 mH .
(4.110)
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CHAPTER 4. MAGNETOSTATIC ANALYSIS OF THE ARMATURE REACTION FIELDS
4.11
99
Summary and Conclusions
In this chapter the flux-linkages for the different winding configuration from the solution of the
magnetic vector potential in the centre of the stator region. The flux-linkage values were found
to be almost identical to FEA simulation results. With the flux-linkages virtually sinusoidal
and in-phase with the sinusoidal phase currents, the synchronous inductances were easy to
calculate for the different winding configurations.
For the Type O winding configuration, it was possible to simplify the flux-linkage equation and hence the synchronous inductance equation, due to the dominant first order spaceharmonic in the flux-linkage solution. The flux-linkage equation for the Type I and Type II
winding configuration, however, could not be simplified, although it would be possible to
only consider the dominant first, second and fourth order space-harmonics components.
From the calculation of the synchronous inductances, the value of the Type II winding configuration’s synchronous inductance was, as expected, significantly higher (≈ 31 %) than that
of the Type I winding configuration’s synchronous inductance, due to its wider coils and hence
higher flux-linkage value. The interesting finding, however, was that the synchronous inductance of the Type II winding configuration was also higher (≈ 21 %) than that of the Type O
winding configuration, in spite of the fact that the Type II winding configuration has only half
the number of coils per phase. The higher synchronous inductance however can be attributed
to the higher number of turns per coil (see section 2.6).
Finally it must be emphasized that the calculation of the synchronous inductances for the
different winding configurations excluded the additional inductance that would be caused
by the end-turn windings. A method for including these additional end-turn inductances
(whether it can be done analytically or empirically), will be left for future work.
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C HAPTER 5
Torque Calculation
In speaking of the Energy of the field, however, I wish to be understood
literally. All energy is the same as mechanical energy, whether it exists in
the form of motion or in that of elasticity, or in any other form. The energy
in electromagnetic phenomena is mechanical energy.
J AMES C LERK M AXWELL, 1876
5.1
Introduction
The average torque produced by an electrical machine under steady state conditions be expressed as the average power delivered (for motor operation) or absorbed by the machine (for
generator operation) divided by the angular speed of the machine,
Tmech =
Pmech
.
ωmech
(5.1)
The calculation of the ripple torque under steady state conditions requires the detailed calculation of the magnetic field produced by the rotor and stator in order to calculate the electromagnetic forces vectors acting on the different parts of the electrical machine. From the different force vector, the resultant torque, including the ripple torque component, produce by the
machine can be calculated. The four major methods used for calculating these electromagnetic
forces, see Benhama et al. [46], are
• the Lorentz method
• the Maxwell stress tensor (MST) method
• the classical virtual work method and
• the Coulomb virtual work (CVW) method.
From the above methods, only the Lorentz method and the MST method can be used to
calculate the torque analytically. These two methods are not usually preferred for FEA. The
101
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5.2. AVERAGE TORQUE CALCULATION
problem with the Lorentz method is that it can only calculate the electromagnetic forces that
act upon the current carrying conductors. This makes it unsuitable for the calculation of cogging torque, or for the calculation of reluctance torque. The Lorentz method is therefore not
usually implemented in commercial FE packages. A variation of the Lorentz method was proposed by Kabashima et al. [47] which calculates an equivalent current density flowing in a
thin layer of thickness around the magnetised element under consideration. In a comparison
between the equivalent current density method, the equivalent magnetic charge method, the
classical virtual work method and the MST method, Muller [48] found that the accuracy of the
equivalent current density method is heavily reliant on the accuracy of the permeability of the
magnetic material under consideration.
A generalised equivalent magnetising current method, based on the equivalent current
density calculation method, has recently been proposed by Choi et al. [49]. The generalised
equivalent magnetising current method addresses the problem of calculating the force acting
on a magnetised body that is not completely surrounded by air. The MST method suffers form
exactly the same problem. Also when working in 3D, the surface integration required by the
MST method is a lot more complex to perform compared to the classical virtual work method
and the Coulomb virtual work method.
The classical virtual work method, or the more refined version of the classical virtual work
method that makes use of co-energy calculations, is only suited for FEA programs. The CVW
method is an enhancement of the classical virtual work method, Coulomb [50] and [51] and
also uses the co-energy in its calculations. The CVW is the method employed by Ansys’ Maxwell® 2D, Lebedev et al. [52], to calculate force and hence torque values.
As the RFAPM machine is an air cored machine, either the Lorentz method, or the MST
method could be used for an analytical solution of the torque, with the analytical analysis
limited to two dimensions only.
5.2
Average Torque Calculation
The average torque can be calculated directly from the equivalent circuit of the RFAPM machine as shown in Figure 5.1. In order to correlate the equivalent circuit with the work done
in Chapters 3 and 4, it is important to note that Ea f in Figure 5.1, is the RMS value of the
back-EMFs as defined in equations (3.210) to (3.212), for phase a.
Rs
Ia
Ls
+
+
Ea f
Pmech
Tmech
Va
−
−
ωmech
Figure 5.1: The equivalent circuit of the RFAPM machine.
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CHAPTER 5. TORQUE CALCULATION
Similarly, Ls corresponds to the inductance calculated in (4.85), (4.100) and (4.109) for the
Type O, Type I or the Type II winding configurations respectively. Thus with,
Xs = ωLs ,
(5.2)
we can say that Xs Ia is equal to the voltage induced in the stator windings1 , due to the phase
current, Ia , flowing in the windings. In order to simplify the torque calculations, Kamper et al.
[13] and Randewijk et al. [14], we assume that the phase current, Ia is in phase with Ea . From
the phasor diagram shown in Figure 5.2, for generator operation, we can see that the phase
current, Ia , is actually 180◦ degrees out of phase with the back EMF, Ea , hence power is being
delivered by the machine with the torque produced being negative.
Ea f
Ia
Φ AR
γ
jXs Ia
jXs Ia
R s Ia
Va
ΦPM
Figure 5.2: The phasor diagram of phase a of the RFAPM machine with Ia and Ea in phase for generator
operation (not to scale).
The phasor diagram for motor operation is shown in Figure 5.3. Here, with Ia and Ea in
phase, power will be absorbed by the machine and the power delivered will be positive.
This allows us to define the torque produced by the RFAPM machine, while taking cognisance of the direction of power flow, as
Tmech =
3Ea f Ia
,
ωmech
(5.3)
with Ia the RMS values of the phase current for phase a. From (3.210), (3.211) and (3.212),
the average torque can be expressed in terms of the approximated amplitude of the back-EMF
voltages, so that we can write,
Tmech|O = 23 k E|O Îa
(5.4)
Tmech| I =
(5.5)
Tmech| I I =
3
2 k E| I Îa
3
2 k E| I I Îa
(5.6)
for the Type O, Type I and Type II winding configurations respectively, with Îa the amplitude
of the phase current for phase a.
1 Assuming
a perfectly sinusoidal phase current.
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5.2. AVERAGE TORQUE CALCULATION
Va
jXs Ia
jXs Ia
Ia
γ
Φ AR
R s Ia
Ea f
ΦPM
Figure 5.3: The phasor diagram of phase a of the RFAPM machine with Ia and Ea in phase for motor
operation (not to scale).
For both motor and generator operation, the angle, γ, between the magnetic axis of the
rotor, ΦPM , and that of the stator, ΦAR , must be maintained at 90◦ , for equations (5.4), (5.5)
and (5.6) to be true. This is usually achieved by field orientated control for either motor or
generator mode of operation. As mentioned in Chapter 2, the ΦPM and ΦAR axes correspond
to the d-axes, dr and ds respectively as shown in Figures 2.1, 2.11 and 2.18 for the Type O, Type I
and Type II winding configurations respectively.
With γ = 90◦ , we can also define the torque for a Double-sided Rotor RFAPM machine
generically in terms of the peak value of the stator current space vector, Îs , Slemon [20] and
Mohan [43], as
Tmech = k T Îs
(5.7)
with the torque constants defined as
2qrn ℓ N
k w,1|O Br,1 ,
a
2qrn ℓ N
= k E| I =
k w,2| I Br,1 and
a
2qrn ℓ N
k w,2| I I Br,1
= k E| I I =
a
k T |O = k E |O =
(5.8)
kT| I
(5.9)
kT| I I
(5.10)
from (3.216), (3.217) and (3.218) for the Type O, Type I and the Type II windings respectively,
and the stator current space vector
Îs = 23 Îa .
(5.11)
We can see that the torque constants are exactly the same as the voltage factors defined in
Chapter 3, with the only difference that the units for the torque constants are Nm/A versus
the V/rad/s used for the voltage factors. This is consistent with the definition of the voltage
and torque constant for Sinusoidal Permanent Magnet AC (commonly referred to as brushless
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CHAPTER 5. TORQUE CALCULATION
105
DC) Machines, given by Mohan [43, Chapter 10]. This implies that the same control principles
applicable to brussless DC machines, can be used for the control of the Double-side Rotor
RFAPM machine.
5.3
Ripple Torque Calculation using the Lorentz Method
The Lorentz method provides a quick and easy way to calculate the torque ripple under steady
steady state conditions. With ωm constant, the torque is calculated by using only the analytical solution for the magnetic fields produced by the permanent magnets in the stator region,
together with the current density distribution function of the stator windings, Holm [24].
From Lorentz’s law, the volumetric force density, as defined by Muller [48], can be calculated as follows
~fv = ~J × ~B .
(5.12)
With the current density distribution in the stator region,
~Jz = Jz~az
(5.13)
only defined in the axial or z-axis direction and the flux density defined only in the radial and
azimuthal, or r – and φ-axis directions, (5.12) can be written as
~fv = − Jz Bφ~ar + Jz Br~aφ
= f r~ar + f φ~aφ
(5.14)
(5.15)
Only the azimuthal component of the volumetric force density, f φ will contribute to “useful” torque. The volumetric torque density distribution developed by the electrical machine
will thus be equal to
~ρv =~r ×~fv
(5.16)
= rJz Br~az .
(5.17)
with~r the radial vector from the centre of the machine.
The nett electromagnetic or mechanical torque, Tmech , can then be calculated by integrating
the volumetric torque density distribution over the entire stator volume, so that
Tmech =
Z
=ℓ
v
ρv,z dv
(5.18)
Z rn + h Z 2π
2
rn − 2h
0
r2 Jz Br dφ dr .
(5.19)
5.3.1 The Simplified Lorentz Method for the Ripple Torque Calculation
The calculation of to the torque and specifically the torque ripple in the machine can be simplified, if we, once again, assume that the radial flux-density in the centre of the stator windings,
i.e. at radius r =rn , can be considered to be constant throughout the whole stator region. This
implies that (5.19) can be simplified to
Tmech =
rn2 hℓ
Z 2π
0
III
Jz Br,m
|r =rn ,PM dφ .
(5.20)
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5.3. RIPPLE TORQUE CALCULATION USING THE LORENTZ METHOD
This assumption was also made in both sections 3.12 and 3.13 where both the flux-linkage
and back EMF due to the permanent magnets were calculated respectively and yielded good
correlation with the results obtained using Maxwell® 2D.
5.3.1.1 The Simplified Lorentz Method for the Ripple Torque Calculation for the Type O
Winding Configuration
From (2.51), for the three phase current density distribution of the Type O winding configuration and substituting “n” for “m” and “pωmech ” for ”ω” we have
Jz|O =
3qI N
− arn phπ
3qI p N
− arn hπ
∞
∑ kw,n|O sin
nqφ + pωmech t
n =1
∞
∑ kw,n|O sin
n =2
nqφ − pωmech t
for n=3k−2, k ∈ N1
.
(5.21)
for n=3k−1, k ∈ N1
For our analysis we will, similar to that used in Chapter 4, adopt the following short hand
notation for (5.21)
Jz|O = −
3qI p N
arn hπ
∞
∑ kw,n|O sin
n =1
nqφ ± ωmech t
(5.22)
in order to reduce the writing.
From the three phase current density distribution shown in Figure 2.8, it is clear that the d–,
or magnetic axis of (5.22), is located at 0◦ [electr.], whereas the d–, or magnetic axis of the radial
magnetisation distribution, as shown in Figure 3.2, is located a 90◦ [electr.].
Thus from (3.147), the rotating radial flux-density in the centre of the stator windings with
r =rn written in terms of the speed of rotation, ωmech and the relative angle between the d-axis
of the rotor (i.e. the rotating radial flux-density) and stator (i.e. the rotating three phase current
density distribution), γ, can be give by the following equation
∞
III
BrI|IrI=rn ,PM (rn , φ) = − ∑ Br,m
|r =rn ,PM cos( mpφ + mpωmech t + mγ )
(5.23)
m=1,3,5,...
with
III
Br,m
|r =rn ,PM = −
mp I I I
mp
−mp
III
Cm| PM rn + Dm
r
.
| PM n
rn
(5.24)
However, with the requirement that γ=90◦ [elect.] as was mentioned in section 5.2, (5.23)
can be simplified to
∞
I
BrI|IPM
(r n , φ ) =
III
∑ Br,m
|r =r ,PM sin( mpφ + mpωmech t ) ,
m=1,3,5,...
n
(5.25)
which will be equal to the rotating radial flux-density for generator operation. For motor operation, the rotating radial flux-density will just be the negative of (5.25).
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The simplified torque equation, using the Lorentz method, can thus be obtained by substituting (5.21) and (5.25) into (5.20):
Tmech|O = −
3qrn ℓ N I p
aπ
Z 2π
0
∞
∑ kw,n|O sin
n =1
∞
nqφ ± pωmech t ·
(5.26)
III
∑ Br,m
|r =r ,PM sin( mpφ + mpωmech t ) dφ
=−
3qrn ℓ N I p
aπ
∞
m=1,3,5,...
∞
n
III
∑ ∑ kw,n|O Br,m
|r =r ,PM ·
n=1 m=1,3,5,...
Z 2π
0
n
(5.27)
sin nqφ ± pωmech t sin(mpφ + mpωmech t) dφ
With p=q, the integral term evaluates to zero everywhere except where n=m. This allows us
to define a function, say Sm ,
cos (m−1) pω
mech t
Sm =
sin (m+1) pω
t
mech
for m=6k +1, k ∈ N0
for m=6k −1, k ∈ N1
(5.28)
so that (5.27) can be simplifies to:
Tmech|O = −
3qrn ℓ N I p ∞
III
∑ kw,m|O Br,m
|r =rn ,PM Sm .
a
m=1,5,7...
(5.29)
If we only consider the fundamental component of the winding factor and the radial flux density, i.e at m=1, we find that Sm =1 and equation (5.29) simplifies to be equal to (5.4) as expected.
In Figure 5.4, a Matplotlib plot of the torque waveform, for the Type O winding configuration as a function of time, calculated using the simplified Lorentz method, compared with the
torque waveform obtained using Maxwell® 2D, together with the calculated average torque
component. From the plot it is clear that although the average torque value calculated using
the simplified Lorentz method and that obtained using Maxwell® 2D are almost equal,1 the
shape of the torque ripple waveform is however totally different.
5.3.1.2 The Simplified Lorentz Method for the Ripple Torque Calculation for the Type I
Winding Configuration
The simplified Lorentz method yields the following equation for the torque develped by the
Type I winding configuration,
Tmech| I =
−3qrn ℓ N I p ∞
III
∑ kw,2m| I Br,m
|r =rn ,PM Sm .
a
m=1,3,5,...
(5.30)
Considering only the fundamental component of the radial flux density, i.e for m=1, (5.30)
q
simplifies, as would be expected, to (5.5). It is also interesting to note that with k q = p = 21 , only
the even harmonic components of the winding factor, k w,m| I , contribute to the torque ripple.
1 The
average torque value of the Type O winding configuration calculated using the simplified Lorentz method is
≈ 5 % higher than that obtained from the FEA using Maxwell® 2D.
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5.3. RIPPLE TORQUE CALCULATION USING THE LORENTZ METHOD
340
Using the Simplified Lorentz Method
Using Maxwell® 2D
Mechanical Torque, Tmech [Nm]
Using the Torque Constant, k T
330
320
310
300
290
0, 00
1, 25
2, 50
3, 75
5, 00
6, 25
7, 50
8, 75
10, 00
11, 25
12, 50
Time, t [ms]
Figure 5.4: The torque waveform using the simplified Lorentz method for the Type O winding configuration compared to the Maxwell® 2D simulation.
Figure 5.5 shows the torque waveforms as a function of time calculated with the simplified Lorentz method, obtained using FEA in Maxwell® 2D, as well as the simplified average
torque component. Once again it can be seen that although the average values of the torque
waveforms compare well1 , the shape of the torque waveforms differs.
5.3.1.3 The Simplified Lorentz Method for the Ripple Torque Calculation for the Type II
Winding Configuration
The torque waveform calculated using the simplified Lorentz method, for the Type II winding
configuration can be given by the following equation,
Tmech| I I =
−3qrn ℓ N I p ∞
III
∑ kw,2m| I I Br,m
|r =rn ,PM Sm .
a
m=1,3,5,...
(5.31)
(5.31) also simplifies to (5.6) at the fundamental component of the radial flux-density, i.e. with
q
m=1. Similar to the Type I winding configuration, when k q = p = 21 , only the even harmonic
components of the winding factor, k w,m| I I , contribute to the torque ripple.
In Figure 5.6, the output torque waveforms, all as a function of time, is shown for the simplified Lorentz method calculation, together with torque waveform obtained in FEA, using
Maxwell® 2D, as well as the simplified average torque component. Again, the average value
of the torque waveform calculated using the simplified Lorentz method, closely matches the
1 The
average value of the torque for the Type I winding configuration using the simplified Lorentz method is only
≈ 1,0 % higher than that calculated using Maxwell® 2D.
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CHAPTER 5. TORQUE CALCULATION
200
Using the Simplified Lorentz Method
Using Maxwell® 2D
Mechanical Torque, Tmech [Nm]
Using the Torque Constant, k T
195
190
185
180
175
0, 00
1, 25
2, 50
3, 75
5, 00
6, 25
7, 50
8, 75
10, 00
11, 25
12, 50
Time, t [ms]
Figure 5.5: The torque waveform using the simplified Lorentz method for the Type I winding configuration compared to the Maxwell® 2D simulation.
average of the simulated Maxwell® 2D waveform1 , with only the shape of the waveforms that
differs.
5.3.1.4 Conclusions Regarding the Simplified Lorentz Method for the Ripple Torque
Calculation
Although (5.20) yields three elegant looking equations, (5.29), (5.30) and (5.31) that closely
matches (5.4), (5.5) and (5.6) for the Type O, Type I and Type II winding configurations respectively, the shape of the torque waveform obtained using these equations does not compare
favourably with the waveforms of the torque simulated in Maxwell® 2D. This can be attributed
to the radial flux density which is not constant throughout the stator region.
In Figure 5.7, the analytically calculated radial flux-density, as calculated using (3.147)2 , is
compared with the simulated radial flux-density values obtained using Maxwell® 2D. Where
can clearly see that the flux-density distribution varies from sinusoidal in the centre of the
stator windings, to trapezoidal on the inner and outer boundary between the stator windings
and the two air gaps.
5.3.2 The Exact Lorentz Method for the Ripple Torque Calculations
From the conclusion of the section 5.3.1.4, it stands to reason that the torque could be calculated
more accurately using the analytical solution for the radial flux density obtained in (3.147) as a
1 The
average value of the torque for the Type II winding configuration using the simplified Lorentz method is only
≈ 1,3 % higher than that calculated using Maxwell® 2D.
2 Only using the equation for Region III, i.e. the stator region.
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5.3. RIPPLE TORQUE CALCULATION USING THE LORENTZ METHOD
225
Using the Simplified Lorentz Method
Using Maxwell® 2D
Mechanical Torque, Tmech [Nm]
Using the Torque Constant, k T
220
215
210
205
200
0, 00
1, 25
2, 50
3, 75
5, 00
6, 25
7, 50
8, 75
10, 00
11, 25
12, 50
Time, t [ms]
Figure 5.6: The torque waveform using the simplified Lorentz method for the Type I winding configuration compared to the Maxwell® 2D simulation.
function of the radius, r, instead of only considering the solution at r =rn .
∞
I
BrI|IPM
(r, φ) =
I
sin(mpφ + mpωmech t)
∑ BrI|IPM
(5.32)
m=1,3,5,...
1, 0
Br|rn −h/2| PM
Br|rn | PM
Radial flux density, Br [T]
Br|rn +h/2| PM
0, 5
Br|rn −h/2| PM (Maxwell® 2D)
Br|rn | PM (Maxwell® 2D)
Br|rn +h/2| PM (Maxwell® 2D)
0, 0
−0, 5
−1, 0
−11, 2500
−8, 4375
−5, 6250
−2, 8125
0, 0000
2, 8125
5, 6250
8, 4375
11, 2500
Angle, φ [◦ ]
Figure 5.7: The radial flux-density in at the top, centre and bottom of the stator windings.
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CHAPTER 5. TORQUE CALCULATION
with
I
BrI|IPM
=−
mp I I I
III
Cm| PM r mp + Dm
r −mp .
|
PM
r
(5.33)
As were are only interested in the solution to the radial flux-density in the stator region,
III
i.e. Region III, only coefficients CmI I|IPM and Dm
| PM needs to be considered. The solution for the
radial flux density is then integrated over the entire stator region as is required for the exact
solution using the Lorentz method, see (5.19).
5.3.2.1 The Exact Lorentz Method for the Ripple Torque Calculation for the Type O
Winding Configuration
Substituting (5.21) and (5.32) into (5.19), the following equation is obtained
Tmech|O = −
3qℓ N I p
arn hπ
Z rn + h Z 2π
2
rn − 2h
0
∞
∑ kw,n|O sin
n =1
∞
nqφ ± pωmech t ·
(5.34)
III
mp
III
−mp
mpr
C
r
+
D
r
∑
m| PM
m| PM
m=1,3,5,...
sin(mpφ + mpωmech t) dφ dr ,
which can be rewritten as
Tmech|O
3qℓ N I p
=−
arn hπ
∞
∞
∑ ∑
k w,n|O
n=1 m=1,3,5,...
Z rn + h
2
rn − 2h
Z 2π
0
mpr
CmI I|IPM r mp
+
III
−mp
Dm
| PM r
dr ·
sin nqφ ± pωmech t sin(mpφ + mpωmech t) dφ .
(5.35)
Integrating with respect to r yields
Tmech|O
3qℓ N I p
=
arn hπ
∞
∞
∑ ∑
k w,n|O
n=1 m=1,3,5,...
mp
"
Z 2π
0
CmI I|IPM r mp+2
mp + 2
−
−mp+2
III
Dm
| PM r
mp − 2
#rn + 2h
rn − 2h
·
(5.36)
sin nqφ ± pωmech t sin(mpφ + mpωmech t) dφ
and noting that the integral with respect to φ only has a solution when n=m, the solution to
the exact analytical torque calculation using the Lorentz methods reduces to
Tmech|O = −
3qℓ N I p ∞
k w,m|O Rm Sm
arn h m∑
=1,3,5,...
(5.37)
with
Rm =
Z rn + h
2
rn − 2h
= mp
"
r2 ·
−mp )
III
mp(CmI I|IPM r mp + Dm
| PM r
CmI I|IPM r mp+2
mp + 2
r
−
−mp+2
III
Dm
| PM r
mp − 2
#rn + 2h
rn − 2h
dr
(5.38)
(5.39)
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5.3. RIPPLE TORQUE CALCULATION USING THE LORENTZ METHOD
and Sm as defined in (5.28).
In Figure 5.8 the output torque waveform obtained using (5.37) is shown compared to that
obtained using Maxwell® 2D. It can be seen that the torque ripple component has the same
shape and magnitude for both the analytical and FEM solutions. This implies that the harmonic contents of both should be similar. This is confirmed in Figure 5.9, where the harmonic
components of the analytical solution is compared to the solution obtained using Maxwell® 2D.
It is interesting to note the dominance of the sixth order harmonics for both solutions (i.e
6, 12 and 18). Maxwell® 2D is also sporting a second order harmonic which, is absent in the
analytical solution, but this will be discussed in more detail in section 5.3.2.3.
340
Using the Lorentz Method
Using FEM
Mechanical Torque, Tmech [Nm]
Using the Torque Constant, k T
330
320
310
300
290
0, 00
1, 25
2, 50
3, 75
5, 00
6, 25
7, 50
8, 75
10, 00
11, 25
12, 50
Time, t [ms]
Figure 5.8: The torque ripple waveforms for the Type O winding configuration.
5.3.2.2 The Exact Lorentz Method for the Ripple Torque Calculation for the Type I
Winding Configuration
Using the same approach as in section 5.3.2.1 above, the torque calculation for the Type I winding configuration will yield the following equation,
Tmech| I = −
3qℓ N I p ∞
k w,2m| I Rm Sm
arn h m∑
=1,3,5,...
(5.40)
with Sm and Rm as defined in (5.28) and (5.39) respectively.
In Figure 5.10 the output torque waveform of a RFAPM machine with a Type I winding
configuration, calculated using the Lorentz method is compared with the simulated torque
waveform obtained using Maxwell® 2D, as well as with the average torque calculated using
equation (5.5).
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CHAPTER 5. TORQUE CALCULATION
0, 9
Analytical
FEM
0, 8
Tmech|h
Tmech| ave.
× 100 %
0, 7
0, 6
0, 5
0, 4
0, 3
0, 2
0, 1
0, 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Harmonic Number
Figure 5.9: A FFT of the torque ripple harmonics for the Type O winding configuration.
Figure 5.10 shows that, although the average torque value of the analytically calculated
torque waveform using the Lorentz method is slightly higher than that of the torque calculated
Mechanical Torque, Tmech [Nm]
200
195
190
185
180
Using the Lorentz Method
Using FEM
Using the Torque Constant, k T
175
0, 00
1, 25
2, 50
3, 75
5, 00
6, 25
7, 50
8, 75
10, 00
11, 25
12, 50
Time, t [ms]
Figure 5.10: The torque ripple waveforms for the Type I winding configuration.
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5.3. RIPPLE TORQUE CALCULATION USING THE LORENTZ METHOD
using Maxwell® 2D, the shape of the torque ripple is almost exactly the same. This is confirmed
by the harmonic content of the two solutions, as shown in Figure 5.11.
0, 30
Analytical
FEM
0, 20
0, 15
Tmech|h
Tmech| ave.
× 100 %
0, 25
0, 10
0, 05
0, 00
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Harmonic Number
Figure 5.11: A FFT of the torque ripple harmonics for the Type I winding configuration.
5.3.2.3 The Exact Lorentz Method for the Ripple Torque Calculation for the Type II
Winding Configuration
Again using the same approach as in section 5.3.2.1, the torque calculation for the Type I winding configuration is given by the following equation,
Tmech| I I = −
3qℓ N I p ∞
k w,2m| I I Rm Sm
arn h m∑
=1,3,5,...
(5.41)
with Sm and Rm once again as defined in (5.28) and (5.39) respectively.
In Figure 5.12 the analytically calculated torque waveform using the Lorentz method is
compared with that using Maxwell® 2D. Once again the Lorentz method yield a slightly higher
average value compared to the Maxwell® 2D simulation. The shape of the torque ripple is almost exactly the same, except for a second order harmonic component present in the Maxwell® 2D solution as shown in Figure 5.14. This second order harmonic was also present
in torque waveform of the Maxwell® 2D simulation for the Type O winding configuration,
shown in Figure 5.9, but not in torque waveform of the Type I winding configuration, shown
Figure 5.11.
Torque result of the Type II winding configuration differs the most compared to the Maxwell® 2D simulation. In order to validate the the Type II winding configuration’s torque result,
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CHAPTER 5. TORQUE CALCULATION
Mechanical Torque, Tmech [Nm]
225
220
215
210
205
Using the Lorentz Method
Using FEM
Using the Torque Constant, k T
200
0, 00
1, 25
2, 50
3, 75
5, 00
6, 25
7, 50
8, 75
10, 00
11, 25
12, 50
Time, t [ms]
Figure 5.12: The torque ripple waveforms for the Type II winding configuration.
another FEM simulation was done in SEMFEM, see Gerber et al. [18]. SEMFEM, short for Stellenbosch Electrical Machines Finite Element Method, is based on Abdel-Razek et al. [53] and
utilises a combination of FEA and an air gap element (AGE) for the simulation of electrical
machines. The AGE is an analytical description of the air gap of the machine and results in a
faster solution of the magnetic fields in the machine for a transient analysis as it is not necessary to continuously re-mesh the air gap(s) between the rotor and stator of the machine being
analysed.
In Figure 5.13 the torque waveform of the analytically calculated and the torque waveform
obtained using Maxwell® 2D and SEMFEM are shown. We can clearly see that the two FEM
solutions are almost exactly the same, given the resolution of the vertical (y) axis. In Figure 5.14,
the FFT of the torque waveform simulated using SEMFEM is shown. The SEMFEM solution
confirms that there is indeed no second order harmonic present in the torque waveform of
the Type II winding configuration. From communications with Ansoft/Ansys support, the
reason for the second order harmonic in the torque waveform of Maxwell® 2D simulation for
the Type II winding configuration can be attributed to “computational oscillations” due to the
Master and Slave boundaries touching the coils of the machine.
In Figure 5.14, the FFT of the torque waveform obtained using a “tweaked” Maxwell® 2D
FEA model of the Type II winding configuration is also shown. In this “tweaked” Maxwell® 2D
FEA model, the coil sides of the Type II windings that were originally touching the Master and
Slave boundaries, were shifted by 0,05 radians away from the Master and Slave boundaries.
This unfortunately reduces the flux-linkage of the windings marginally and result in a slightly
lower average torque value. The most important aspect however, is that the second order
“computational oscillation” disappeared.
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5.3. RIPPLE TORQUE CALCULATION USING THE LORENTZ METHOD
Mechanical Torque, Tmech [Nm]
225
220
215
210
205
Using Maxwell® 2D
Using the Lorentz Method
Using SEMFEM
200
0, 00
1, 25
2, 50
3, 75
5, 00
6, 25
7, 50
8, 75
10, 00
11, 25
12, 50
Time, t [ms]
Figure 5.13: The torque ripple waveforms for the Type II winding configuration.
0, 25
Analytical
FEM
FEM (Tweaked)
SEMFEM
Tmech|h
Tmech| ave.
× 100 %
0, 20
0, 15
0, 10
0, 05
0, 00
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Harmonic Number
Figure 5.14: A FFT of the torque ripple harmonics for the Type II winding configuration.
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CHAPTER 5. TORQUE CALCULATION
5.4
117
The Effect of a Reduced Subdomain Model on the Torque
Calculations
In this chapter, the torque produced by the different winding configurations were analytically
calculated using different calculation methods. All the torque calculation methods however
relied on the radial flux-density solution obtained in Chapter 3 using the subdomain analysis
method. The subdomain model used in Chapter 3, assumed a constant value of 1 000 for the
relative permeability in the rotor yokes as the subdomain method is a linear approximated
to the magnetic field solution. In this section we will investigate the effect of increasing the
permeability of the rotor yoke to infinity, have on the torque calculation results.
5.4.1 Using a Subdomain Model with the Permeability in the Rotor Yoke taken as
Constant
In Table 5.1 the torque results for the three different winding configurations, calculated with
the different analysis methods, i.e. the simplified average torque calculation of section 5.2, the
simplified Lorentz method discussed in section 5.3.1 and the “exact” Lorentz method of section
5.3.2, are shown. These calculated torque value a shown together with the torque values obtained from the FEM results using Maxwell® 2D and SEMFEM1 . All the analytical calculations
were done assuming a constant value of 1 000 for the permeability in the rotor yokes.
The “simplified average torque calculation method”, calculates the average torque component from the torque constants, as defined in (5.8), (5.9) and (5.10) for the Type O, Type I
and Type II winding configurations respectively. The torque constant uses the peak value of
the radial flux-density distribution in the centre of the stator windings and assumes that the
back-EMF and phase current are not only sinusoidal, but also in phase.
The “simplified Lorentz” method also uses the fundamental component of the radial fluxdensity in the centre of the stator region for the torque calculation, but considers the space
harmonics components of the current density distribution of the different winding configurations as well. The ripple torque calculated, using the simplified Lorentz method is therefore
only that component of the total ripple torque that can be associated with the space harmonics
of the current density distribution.
The “exact” Lorentz method uses the complete solution of the radial flux-density distribution over the entire stator area, in order to calculate ripple torque component due to the
combined effect of the variation in the radial flux-density distribution in the stator region and
the variation in the current density distribution.
5.4.2 Using a Subdomain model with the Permeability in the Rotor Yoke taken as
Infinity
In Table 5.2 the differences between the different torque calculation methods are shown with
the yoke’s relative permeability approximated as being infinity (∞) instead of 1 000 to calculate
the radial flux-density distribution using the subdomain analysis method. By doing this, the
number of governing equations that needs to be solved reduces from five to three, as shown
1 SEMFEM
is only used to model the Type II winding configuration.
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5.4. THE EFFECT OF A REDUCED SUBDOMAIN MODEL ON THE TORQUE CALCULATIONS
Calculation Method
Simplified Average Torque
µr|yoke = 1000
Simplified Lorentz
µr|yoke = 1000
Lorentz
µr|yoke = 1000
Maxwell® 2D
“Tweaked”
SEMFEM
Machine Type
Average
Ripple
%Ripple
%THD
O
I
II
310,29 Nm
192,03 Nm
214,55 Nm
–
–
–
–
–
–
–
–
–
O
I
II
313,05 Nm
192,24 Nm
215,02 Nm
6,33 Nm
1,83 Nm
1,23 Nm
2,02 %
0,95 %
0,57 %
0,51 %
0,24 %
0,14 %
O
I
II
319,27 Nm
196,08 Nm
219,32 Nm
11,13 Nm
2,19 Nm
2,10 Nm
3,49 %
1,12 %
0,96 %
0,87 %
0,28 %
0,24 %
O
I
II
II
302,47 Nm
189,94 Nm
212,12 Nm
211,58 Nm
11,74 Nm
2,13 Nm
2,71 Nm
2,32 Nm
3,88 %
1,24 %
1,28 %
1,10 %
0,89 %
0,27 %
0,25 %
0,24 %
O
I
II
–
–
212,62 Nm
–
–
2,21 Nm
–
–
1,04 %
–
–
0,25 %
Table 5.1: Comparison between the result of the various torque calculation methods.
in appendix H. This results in the coefficients matrix that needs to be solved for each value
of m reducing from a 10 × 10 matrix to a 6 × 6 matrix, without compromising the accuracy of
the results significantly. Although the analytical calculation method is already two orders of
magnitude faster than the FEM solution using Maxwell® 2D1 , the reduced matrix solution did
not prove to be much faster2 using Python™.
Calculation Method
Simplified Average Torque
µr|yoke = ∞
Simplified Lorentz
µr|yoke = ∞
Lorentz
µr|yoke = ∞
Machine Type
Average
Ripple
%Ripple
%THD
O
I
II
310,95 Nm
192,44 Nm
215,00 Nm
–
–
–
–
–
–
–
–
–
O
I
II
313,69 Nm
192,65 Nm
215,47 Nm
6,32 Nm
1,83 Nm
1,23 Nm
2,01 %
0,95 %
0,57 %
0,50 %
0,23 %
0,14 %
O
I
II
319,94 Nm
196,49 Nm
219,78 Nm
11,13 Nm
2,19 Nm
2,10 Nm
3,48 %
1,12 %
0,95 %
0,86 %
0,28 %
0,24 %
Table 5.2: Analytical results with the permeability of the rotor yoke taken as infinity.
1 Approximately,
2 It
≈ 6 s for the analytical solution versus ≈ 890 s for the Maxwell® 2D solution
was only ≈ 1 s faster – both measured using the time function in IPython
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CHAPTER 5. TORQUE CALCULATION
From Table 5.2 we can see that as a result of the permeability of the rotor yoke being regarded as infinity, the MMF drop across the rotor yokes have effectively been ignored, so that
the radial flux-density values as well as the calculated average torque values will being slightly
higher than the values showed in Table 5.1. The difference is only in the order of 0,2 % which
is so small that we can safely ignore the reluctance of the rotor yoke in our torque ripple calculations.
5.5
Investigating the Effect of Rotor Yoke Saturation on the
Analytical Torque Calculation
In section 3.11.4 it was mentioned that the fundamental component of the analytically calculated radial flux-density distribution was 3 % higher than of the FEA solution. This was attributed to saturation in the rotor yokes as could be observed in the flux-density contour plot of
Figure 3.8. In this section, we will investigate the effect of saturation in the rotor yokes region
on the torque calculations. Our investigation will be limited to the Type II winding configuration.
We decided to again use the machine data of Test Machine I in section E.1 of Appendix
E but with the rotor yoke thickness of 20 mm instead of 8 mm. From the FEA solution using
Maxwell® 2D, we saw that peak flux-density values in the centre of rotor yokes reduced from
≈1,7 T to ≈0,67 T. The FEA solution to the torque output were however only slightly higher,
as shown in Figure 5.15, compared to the FEA solution shown in Figure 5.12 with a saturated
8 mm thick rotor yokes.
Mechanical Torque, Tmech [Nm]
225
220
215
210
205
Using the Lorentz Method
Using Maxwell® 2D
Using the Torque Constant, k T
200
0, 00
1, 25
2, 50
3, 75
5, 00
6, 25
7, 50
8, 75
10, 00
11, 25
12, 50
Time, t [ms]
Figure 5.15: The torque ripple waveforms for the Type II winding configuration with a rotor yoke thickness of 20 mm.
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5.6. INVESTIGATING THE EFFECT OF THE RECOIL PERMEABILITY OF THE PERMANENT MAGNETS ON
THE ANALYTICAL TORQUE CALCULATION
120
From Figure 5.15 and Figure 5.12 it is clear that saturation of the rotor yokes reduces the
output torque of this Type II RFAPM machine with only an ≈0,5 %. There is however still
a difference in the analytically calculated torque output and that calculated using FEA. This
clearly implies that the difference in the torque calculation can not be attributed to rotor yoke
saturation.
5.6
Investigating the Effect of the Recoil Permeability of the
Permanent Magnets on the Analytical Torque Calculation
With saturation of the rotor yokes ruled out as a possible cause to the difference between the
FEA solution and analytically calculated solution to the radial flux-density distribution in the
stator region, it was decided to investigate the effect the approximation of the recoil permeability to unity, in our subdomain model, have on the accuracy of the analytically calculated radial
flux-density. We again made used of the machine data of Test Machine I in section E.1 of Appendix E, but now, not only with a rotor yoke thickness equal to 20 mm, but also with the N48
NdBFe permanent magnets’ coercivity modified from 1 050 000 A/m to 1 114 084 A/m to effectively change the value of the relative recoil permeability of the permanent magnets to unity
in our FEA simulation.
The result of the FEA solution to the torque, as well as the analytically calculated value
of the torque using the exact Lorentz method, are shown in Figure 5.16 with both solution
assuming a relative recoil permeability for the permanent magnets of unity. We can see that
both solutions to the torque are an allmost exact match. The lower value of the radial fluxdensity in the FEA solutions where the the recoil permeability were not taken as unity, could
thus attributed to larger inter-pole leakage flux due to the slightly higher recoil permeability of
the permanent magnets compared to the permeability in the surrounding air gaps and stator
region.
5.7
Summary and Conclusions
In this chapter, different calculation methods were employed to calculate the torque produced
by a Double-sided Rotor RFAPM machine. A torque constant, k T , for the RFAPM was derived.
This torque constant is equal to the voltage constant, k E , defined in section 3.14. The torque
constant relates the torque produced by the RFAPM machine to the peak value of stator current
space vector. In order for this to be valid, the phase current and the back-EMF of the RFAPM
machine needs to be in phase. This requires the machine to be operated under field orientated
control to ensure that the d-axis of the rotor and the d-axis of the stator are orthogonal (i.e.
at 90◦ ) to one another. The similarity to a DC machine, where the armature – and the field
flux are also orthogonal to one another and where the torque is directly proportional to the
armature current, is the reason why the Double-side Rotor RFAPM machine can therefore also
be classified as a classical brushless DC machine.
The average, steady state torque produced by the RFAPM machine, can therefore be calculated from the torque constant. The accuracy of the torque constant (and hence the voltage
constant) is directly proportional to the accuracy of radial flux-density value used in the calcu-
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Mechanical Torque, Tmech [Nm]
225
220
215
210
205
Using the Lorentz Method
Using FEM
Using the Torque Constant, k T
200
0, 00
1, 25
2, 50
3, 75
5, 00
6, 25
7, 50
8, 75
10, 00
11, 25
12, 50
Time, t [ms]
Figure 5.16: The torque ripple waveforms for the Type II winding configuration with a rotor yoke
thickness of 20 mm and a relative recoil permeability for the permanent magnets in Maxwell® 2D,
mur,recoil = 1.
lation of torque constant. By using the peak value of the fundamental component of the radial
flux-density distribution in the centre of the stator region as calculated in Chapter 3, the average torque values for the RFAPM machine with either a Type O, Type I or Type II winding
configuration, could be calculated to within ≈1 % of the average value for the torque obtained
using FEA. Based on the the simplicity of the calculation, the accuracy of the average torque
result is truly remarkable and emphasises the importance of the calculation of the pole width
angle to minimise the %THD in the radial flux-density distribution, as discussed in Chapter 3.
The simple Lorentz calculation method used to calculate the torque ripple component was
in essence just a way to see what the effect of the stator slotting is. The simplified Lorentz
calculation method assumed that the radial flux-density distribution in the stator region is constant and sinusoidal. In order to accurately calculate the ripple torque component, the complete
solution to the radial flux-density distribution in the stator region from the subdomain analysis
method has to be used. This is to account for the variation in the radial flux-density distribution
in the stator region.
It was found that the torque ripple component could be calculated very accurately using the
Lorentz method when compared to FEA results. The Lorentz method also gave very accurate
torque ripple result with a simplified subdomain model being employed that assumes the rotor
yokes’ permeability to be infinity. It was however found that the average value of the the
torque calculated using the Lorentz method was ≈3,3 % higher than the average calculated
using FEA. This coincided with the fact that the radial flux-density values, calculated using the
subdomain method, and used in the Lorentz method calculation was found in Chapter 3 to be
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5.7. SUMMARY AND CONCLUSIONS
≈3 % higher. The difference in the radial flux-density values were initially attributed to the fact
that the subdomain analysis method is a linear approximation to the solution of the magnetic
fields and does not take the saturation of the rotor yokes into account.
On closer inspection however, it was found that the main difference in the torque calculated
analytically using the Lorentz method and that calculated using FEA were not as a result of the
rotor yoke saturation, but due to the fact that the subdomain analysis method assumes the
recoil permeability of the permanent magnets to be unity. With the reason for this difference
between the analytically calculated results and that using FEA known, it was decided to leave
the solution to this problem for further research, as the analytical solution at hand is a adequate
engineering solution for the time being.
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C HAPTER 6
Machine Performance Comparison as a
Function of Winding Configuration
Including the End-turns Effects
There are no foolish questions and no man becomes a fool until he has
stopped asking questions
C HARLES P. S TEINMETZ
6.1
Introduction
The end-turns of a coil are defined as those parts of the coil that connect the two coil-sides
of the coil in order to maintain electrical continuity. The end-turns do not contribute to the
induced voltage in the coil, nor do they partake in producing any torque. In this chapter the
end-turns effect of the different winding configurations, i.e. Type O, Type I and Type II, on
the performance of the RFAPM machine will examined. This will be done in order to obtain
a more realistic performance comparison between the different winding configurations, than
possible with the 2-D analytical analysis, done in Chapters 3 to 5. The empirical analysis of the
end-turns done in this chapter, is an enhancement of the analysis done in Randewijk et al. [14]
which in turn was based on the analysis done by Rossouw and Kamper [25] and Kamper et al.
[13] on the end-turn effect in Axial Flux Permanent Magnet (AFPM) machines with air-cored,
non-overlapping, concentrated stator windings.
Although the end-turns will add to the machine’s synchronous inductances, as was calculated in Chapter 4, this effect will be ignored. This is due to the fact that for the Type O, Type I
and Type II air-cored winding configurations, the slight increase in the inductance due to the
end-turns will be minimal and will have almost no effect on the performance of the machine.
123
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6.2. SIMPLE TORQUE COMPARISON
We will rather focus on the effect the end-turns have on the winding resistance, as this has a
direct influence on the losses in different winding configurations and hence the efficiency of the
RFAPM machine. The end-turn length adds to the total length of the coil and thus also adds to
the resistance of the coil.
In this chapter the performance of the RFAPM machine will be measured in terms of the
amount of torque produced by the Type I and Type II non-overlapping windings configurations, compared to the Type O overlapping winding configuration used as benchmark. Although technically not a performance issue, the effect of the end-turn length on the total copper
volume of the different winding configurations and thus on the cost of the machine, will also
be discussed.
6.2
Simple Torque Comparison
Before analysing the effect of the end-turns, we will consider a simple comparison for the average torque produced by a RFAPM machine, as a function of various winding configuration
parameters. From (5.7) and (5.11), the general format of the steady state torque equation for
the RFAPM machine can be written as
Tmech = 23 k T I p
=
√3 k T Irms
2
(6.1)
(6.2)
with the general torque constant, from (5.8), (5.9) and (5.10),
kT =
2qrn ℓ N
k w B̂r1 ,
a
(6.3)
and the winding factor,
k w = k w,slot k w,pitch .
(6.4)
For the same rotor design, the peak value of the fundamental component of the radial fluxdensity, B̂r1 , will be the same for all three winding configurations. If we further ensure that the
nominal radius, rn , and the active stator length, ℓ are also the same for the Type O, Type I and
Type II winding configurations, the difference in torque produced for the different winding
configurations will depend solely on the differences in the stator designs. It is clear that for the
stator designs of the Type O, Type I and Type II winding configurations, the is torque directly
proportional to the number of coils per phase, q; the number of turns per coil, N; and the
winding factor, k w ; and inversely proportional to the number of parallel circuits, a.
To summarise briefly from Chapter 2, the winding factor consists of the coil slot-width
factor,
k w,slot =
sin( k∆q )
∆
kq
(6.5)
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CHAPTER 6. MACHINE PERFORMANCE COMPARISON INCLUDING THE END-TURNS EFFECTS
125
and the coil pitch factor
k w,pitch = 1
=
=
sin( π3 )
sin( 2π
3 − 2∆)
– Type O
(6.6)
– Type I
(6.7)
– Type II ,
(6.8)
q
with ∆, half the coil-side with angle1 and k q , the coil factor equal to p .
The numerical values of the winding factors for the Type O, Type I and Type II winding
configurations are given in Table 6.1, using the machine parameters for Test Machine I from
Appendix E, section E.1. Table 6.1 also shows, the number of coils, q, the coil factor, k q , the
coil-side width factor, k∆ and the number of turns per coil, N, for the different winding configurations.
Table 6.1: Comparison between the Type O, Type I and Type II winding configurations with the same
rotor geometry, stator current and stator current density.
q
kq
k∆
N
k w,slot
k w,pitch
Type O
Type I
Type II
16
1
1,0
64
0,955
1,0
8
0,5
0,74
96
0,903
0,866
8
0,5
0,74
96
0,903
0,969
In order to obtain a comparison between the torque produced by the RFAPM machine for
the Type O, Type I and Type II winding configurations, we start by substituting (2.117), the
equation for the number of turns per coil into (6.3) and expanding (5.7) so that we can write
Tmech =
√
√
2πrn2 hℓk f k∆ Jwire k w B̂r1
(6.9)
2πrn2 hℓk f k∆ Jwire k w,slot k w,pitch B̂r1
(6.10)
k π
√
= 26rn2 hℓk f k q Jwire sin ∆ k w,pitch B̂r1
6k q
(6.11)
=
= C1 K1
(6.12)
√
(6.13)
with
C1 =
26rn2 hℓk f Jwire B̂r1
defined as a constant factor, common to the torque expression for the Type O, Type I and Type II
winding configurations and
K1 = k q sin
1 The
k π
∆
k
6k q w,pitch
coil-side width angle was defined in Chapter 3 as 2∆.
(6.14)
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6.3. END-TURN LENGTH CALCULATION
defined as a variable factor based on the differences in the different stators’ winding design
parameters. Assuming that the current density is the same for the Type O, Type I and Type II
winding configurations, we only need to consider K1 when comparing the developed torque
for the different winding configurations as is shown in Table 6.2.
Table 6.2: A comparison between normalised developed torque for the Type O, Type I and Type II
winding configurations with the same rotor geometry and stator current density.
K1
Normalised Torque
Type O
Type I
Type II
0,5
100 %
0,303
60,59 %
0,339
67,76 %
From Table 6.2, we can see that the torque produced by the two non-overlapping winding
configurations does not compare well with that of the Type O winding configuration. This
comparison is however very simplistic as it does not take into account the losses in the copper
windings associated with the torque delivery. In the next section, section 6.3, we will first
calculate the end-turn lengths for the different winding configurations and then in section 6.4
we will calculate the total copper loss of the windings, taking the effect of the end-turn length,
into account.
6.3
End-turn Length Calculation
The end-turn lengths for the Type O, Type I and Type II winding configurations will be estimated from their respective 3-D winding geometries.
6.3.1 End-turn Length Calculation for the Type O Winding Configuration
In Figure 6.1, the 3-D representation of a typical coil of an overlapping winding configuration
is shown. Although drawn as a solid copper bar, it actually consists of a bundle of conductors.
πrn
q
ℓ
h
ℓe1
ℓe2
Figure 6.1: A typical coil used in an overlapping winding configuration.
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CHAPTER 6. MACHINE PERFORMANCE COMPARISON INCLUDING THE END-TURNS EFFECTS
From Figure 6.1 it can be seen that the end-turn length of each coil1 , ℓe|O , for the Type O
winding configuration, is equal to
ℓe|O = ℓe1 + 2ℓe2 ,
(6.15)
with ℓe1 and 2ℓe2 two sections of the end-turn length as defined in Figure 6.1.
It is however quite difficult to calculate ℓe1 and ℓe2 accurately and therefore a conservative
estimation of the different sections of the end-turn length can be approximated by
ℓe1 ≈
πrn
q
and
(6.16)
ℓe2 ≈ h
(6.17)
with rn the nominal radius measured from the centre of the machine to the centre of the windings, q the number coils per phase and h the stator height. This implies that the end-turn length
can therefore be approximated by
ℓ e |O ≈
πrn
+ 2h .
q
(6.18)
6.3.2 End-turn Length Calculation for the Type I Winding Configuration
The 3-D representation of a typical coil used in a Type I winding configuration is shown in
Figure 6.2.
ℓe = πre
re
ℓ
Figure 6.2: A typical coil used in a non-overlapping Type I and Type II winding configuration.
The average end-turn conductor radius, from Figure 2.11 for the Type I winding configuration can be approximated as
re| I ≈
1 The
πrn
.
2Q
(6.19)
end-turn length defined in this dissertation is the average end-turn length between the centres of the coil’s two
coil-sides.
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6.4. COPPER LOSS CALCULATION
A good approximation of the end-turn length, ℓe| I can therefore by calculated as follows,
ℓe| I ≈ πre| I
(6.20)
n
(6.21)
≈
π2r
2Q
with Q the total number of coils for all three-phases, i.e. Q = 3q.
This assumes that the shape of the end-turn winding approximates a half-circle, which is a
good approximation for a machine with a large number short pitched coils. This would imply
that the average bending radius of the coil conductors is approximately equal to the radius of
the end-turn conductors.
6.3.3 End-turn Length Calculation for the Type II Winding Configuration
The 3-D representation of the Type I coil shown in Figure 6.2 is just as applicable for the Type II
winding configuration. Although this would be regarded as a more conservative estimation
of the end-winding length, we could again assume that the end-turn winding approximates a
half-circle and that the average bending radius of coil conductors can be taken as equal to the
radius of the end-turn conductors.
From Figure 2.18 it can be seen that for the Type II winding configuration, the average endturn conductor radius re| I I will be slightly larger compared to the Type I winding configuration
and will be equal to
re| I I ≈ rn
∆
−
,
Q
q
π
(6.22)
with ∆, half the coil-side width angle, so that the end-turn length can be calculated as
ℓe| I I ≈ πre
(6.23)
π
∆
−
Q
q
πrn (π − 3∆)
.
≈
Q
≈ πrn
6.4
(6.24)
(6.25)
Copper Loss Calculation
The main loss component that would effect the performance of the RFAPM machine is the copper losses in the stator windings. This is graphically depicted in Figure 6.3 with the power flow
shown for generator operation. The copper losses will be the only loss component considered
for our analysis. The reason is due to the fact, that with the RFAPM machine being an air-cored
machine, there are no core-losses in the stator of the machine. Furthermore, due to the relatively low flux densities produced be the stator coils, see Chapter 4, we can safely ignore the
core-losses in the rotor caused by the stator MMF harmonics. Also, Arkadan et al. [5] and Holm
[24, sec. 7.6] calculated the eddy current losses in high speed toothless stator machines with
Litz wire winding and found that these losses were very small, compared to the stator back
(or yoke) iron losses. It is therefore safe to assume that the eddy-current losses in the air-cored
stator windings, using Litz wire, can be ignored.
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CHAPTER 6. MACHINE PERFORMANCE COMPARISON INCLUDING THE END-TURNS EFFECTS
129
Pcu
Rs
Ia
Ls
+
+
Ea f
Pmech
Tmech
−
ωmech
Va
Ps
−
Figure 6.3: The equivalent circuit of the RFAPM machine together with the main power flow components..
In order to calculate the equivalent stator resistance per phase, Rs , we first need to calculate
the average coil-turn length,
ℓn = 2(ℓ + ℓe )
(6.26)
with ℓ, the active stator length1 and ℓe , the end-turn length as calculated in the previous section.
The equivalent resistance of each coil can now be calculated as follows,
Rcoil = N
ρCu ℓn
Awire
(6.27)
with N the number of turns per phase; Awire the area of the copper conductors and ρCu the
resistivity of copper. With the number of conductors per phase, q, and the number of parallel
circuits, a, known, the equivalent resistance per phase can now be calculated as,
Rs =
qRcoil
.
a2
(6.28)
This allows us to calculate the total stator copper loss as
2
PCu = 3Rs Irms
(6.29)
qNρCu ℓn 2
I
a2 Awire rms
qNρCu ℓn
Jwire Irms
=3
a
=3
(6.30)
(6.31)
or from (2.116), only in terms of the current density, Jwire , instead of Jwire and Irms ,
2
PCu = πrn ℓn hk f k∆ ρCu Jwire
= 2πrn (ℓ +
= C2 K2
1 The
2
ℓe )hk f k∆ρCu Jwire
(6.32)
(6.33)
(6.34)
active stator length is also sometimes referred to as the active copper length, or the active stack length in
electrical machines that utilises laminated iron-cored stators.
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6.5. PERFORMANCE COMPARISON WITH FIXED COPPER-LOSS VALUE
with
2
C2 = 2πrn hk f ρCu Jwire
(6.35)
defined as a constant factor, common to the different winding configurations and
K2 = (ℓ + ℓe )k∆
(6.36)
defined as a variable factor based on the differences between the different winding configurations in terms of the nominal turn length, that includes the end-turn length of each coil and the
relative coil-side width factor, k∆.
In Table 6.3, a comparison between the Type O, Type I and Type II winding configuration
is given with the stator current density, Jwire , considered the same for all the winding configurations. If we compare this to Table 6.2 we can see that the relative torque-to-copper-loss ratio
of the non-overlapping Type I and Type II winding configurations compares favourably1 to
the overlapping (Type O) winding configuration. However, the ratio for both the Type I and
Type II are still slightly lower thant the “1,0 ” ration of the overlapping winding configuration.
Table 6.3: Comparison between stator copper losses for the different winding configurations with the
same rotor geometry and stator current density.
K2
Normalised Copper Losses
Type “O”
Type “I”
Type “II”
0,141 6
100 %
0,091 5
64,67 %
0,100 7
71,15 %
The comparison given in Table 6.3 was based on using the same value for the current density, in each of the three winding configuration. In the next section we will discuss the performance comparison between the different winding configurations based on a fixed stator
copper-loss value in the Type O, Type I and Type II winding configurations.
6.5
Performance Comparison with Fixed Copper-loss Value
In this section we will compare the performance of the RFAPM machine with the different
winding configurations, presuming that the stator copper losses in the Type I and Type II winding configurations are equal to that of the Type O winding configuration, PCu|O . From (6.33) the
current density in the Type I winding configuration can thus be calculated as
s
PCu|O
Jwire| I =
(6.37)
2πrn (ℓ + ℓe| I )hk f k∆ ρCu
and in the Type II winding configuration as
Jwire| I I =
1 The
s
PCu|O
.
2πrn (ℓ + ℓe| I I )hk f k∆ ρCu
(6.38)
Torque to copper-loss ratio is calculated at 0,937 and 0,952 for the Type I and Type II winding configurations
respectively.
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CHAPTER 6. MACHINE PERFORMANCE COMPARISON INCLUDING THE END-TURNS EFFECTS
With the current densities in the different winding configurations not equal any more, it
requires us to redefine (6.11) as
Tmech = C1′ K1′
(6.39)
with
C1′ =
√
26rn2 hℓk f B̂r1
(6.40)
the revised constant factor, common to each of the different winding configurations and
k π
∆
′
K1 = k q sin
k
Jwire
(6.41)
6k q w,pitch
the revised variable factor, unique to each of the three different winding configurations, based
on the differences in the winding designs. If follows that (6.33) also needs to be redefined, so
that
PCu = C2′ K2′
(6.42)
with
C2′ = 2πrn hk f ρCu
(6.43)
defined as the revised constant factor common to the different winding configurations and
2
K2′ = (ℓ + ℓe )k∆ Jwire
(6.44)
the revised variable factor based on the differences between the three different winding configurations.
Other than the fact that the non-overlapping (Type I and II) winding configurations hold
the advantage in shorter end-turn lengths, these winding configurations also have the advantage that their coil-side widths can be varied, compared to the overlapping (Type O) winding
configuration, where the coil-side width is usually kept at the maximum coil-side width value.1
The coil-side width factor, k∆, as defined in (2.64), will be used as the abscissae against which
the performance parameters will be plotted for the Type I and Type II winding configurations.
From equations (6.33) and (6.44), we can see that with the active stator length, ℓ, the same
for all three winding configurations, the performance difference between the different winding
configurations will be more pronounced in a short stator machine, i.e. with ℓ ≈ ℓe , than in a
long stator machine, i.e. with ℓ≫ℓe . In order to illustrate this, we will define the ratio ξ as the
ratio between the active stator length, ℓ, and the nominal stator radius, rn , so that
ℓ
.
(6.45)
rn
The performance comparison graphs that will be presented in the follow pages, will be
done for different values of ξ. With the nominal stator radius constant, it implies that the
comparison will effectively be done for different “per unit” stator length values2 , expressed
in terms of the nominal stator radius taken as “base length”. The machine parameters used,
except for the stator length, are based on the double-layer (Type II) machine data shown in
Appendix E, section E.2.
ξ=
1 This
2 This
is best illustrated in the difference between Figure 2.1, Figure 2.11 and Figure 2.18.
is also sometimes referred to as the aspect ratio of an electrical machine.
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6.5. PERFORMANCE COMPARISON WITH FIXED COPPER-LOSS VALUE
6.5.1 Current Density Comparison
In Figure 6.4 the current density for the Type I and II winding configurations are shown as
a function of the coil-side width that would yield the same copper loss as the Type O winding configuration with the current density, Jwire = 5 A/mm2 and the coil-side width factor,
k∆ = 1,0 . With the different winding configurations all working at the same phase current of
7,33 A, we can see that the Type I and Type II winding configurations are able to operate at
a much higher current density for the same amount of copper losses. Even with the coil-side
width factor at 100 %, the current density in the non-overlapping winding configurations will
be slightly higher that the 5 A/mm2 at which the Type O winding configuration is operating
at, due to the lower end-turn length and hence lower coil –, or phase resistance. We can also
see that for different stator lengths, i.e. different values of ξ, there is not much difference in the
current density values for a specific value of k∆.
10
Stator current density, Jwire [A/mm2 ]
9
8
7
6
5
4
3
2
1
0
0, 0
Type I: ξ =0.1
Type II: ξ =0.1
Type I: ξ =0.2
Type II: ξ =0.2
Type I: ξ =0.3
Type II: ξ =0.3
Type I: ξ =0.4
Type II: ξ =0.4
Type I: ξ =0.5
Type II: ξ =0.5
0, 1
0, 2
0, 3
0, 4
0, 5
0, 6
0, 7
0, 8
0, 9
1, 0
Stator coil-side width factor, k∆
Figure 6.4: The current density versus the coils-side width for the non-overlapping winding configuration stators with different values of ξ.
6.5.2 Torque Comparison
The current density curves showed in Figure 6.4 however, needs to be put in perspective with
the relative output torque curves for the non-overlapping winding configurations, as shown
in Figure 6.5. We can see that for ξ = 0,1 , i.e. for a short stator machine, the double-layer
non-overlapping winding configuration (Type II) can deliver almost 90 % of the torque of a
overlapping winding configuration, but at a relative coil-side width of only ≈ 80 %. We can
furthermore see that although the end-turn length of the double-layer winding configuration
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CHAPTER 6. MACHINE PERFORMANCE COMPARISON INCLUDING THE END-TURNS EFFECTS
133
is slightly more than that of the single-layer winding configuration (Type I), and thus will have
a lower current density for the same amount of copper losses, see Figure 6.4, it still is able to
produce more torque due to its wider coil pitch configuration.
1, 0
0, 9
Relative torque output
0, 8
0, 7
0, 6
Type I: ξ =0.1
Type II: ξ =0.1
Type I: ξ =0.2
Type II: ξ =0.2
Type I: ξ =0.3
Type II: ξ =0.3
Type I: ξ =0.4
Type II: ξ =0.4
Type I: ξ =0.5
Type II: ξ =0.5
0, 5
0, 4
0, 3
0, 2
0, 1
0, 0
0, 0
0, 1
0, 2
0, 3
0, 4
0, 5
0, 6
0, 7
0, 8
0, 9
1, 0
Stator coil-side width factor, k∆
Figure 6.5: The current density versus the coils-side width for the non-overlapping winding configuration stators with different values of ξ.
6.5.3 Copper Volume Comparison
The advantage of the non-overlapping winding configurations in terms of a shorter end-turn
lengths, coupled with the fact of being able to vary their coil-side widths, result in a much
lower copper volume, VCu , when compared to the overlapping winding configuration. The
total copper volume for the different winding configurations can be calculated as
VCu = 2(ℓ + ℓe ) N Awire
(6.46)
or in terms of the phase current and current density, as
VCu =
2(ℓ + ℓe ) N Irms
.
aJwire
(6.47)
In Figure 6.6 the relative copper volume of the two non-overlapping winding configurations with respect to that of the overlapping winding configuration is shown. We can clearly see
that the copper volume of the double-layer (Type II) winding configuration is always higher,
e.g. at k∆ = 0,7 for a ξ = 0,1 (i.e. short stator) machine, the copper volume is 15 % higher. For
longer stator machines (i.e. a higher ξ value) the difference becomes less.
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6.5. PERFORMANCE COMPARISON WITH FIXED COPPER-LOSS VALUE
1, 0
0, 9
Relative copper volume
0, 8
0, 7
0, 6
0, 5
Type I: ξ =0.1
Type II: ξ =0.1
Type I: ξ =0.2
Type II: ξ =0.2
Type I: ξ =0.3
Type II: ξ =0.3
Type I: ξ =0.4
Type II: ξ =0.4
Type I: ξ =0.5
Type II: ξ =0.5
0, 4
0, 3
0, 2
0, 1
0, 0
0, 0
0, 1
0, 2
0, 3
0, 4
0, 5
0, 6
0, 7
0, 8
0, 9
1, 0
Stator coil-side width factor, k∆
Figure 6.6: The current density versus the coils-side width for the non-overlapping winding configuration stators with different values of ξ.
From Figure 6.6 we can also see that even with a k∆ = 1,0 , i.e. the coil-side width at an absolute maximum, the copper volume for the double– and single-layer winding configurations
are essentially the same, but still less than that of the overlapping winding configuration. This
is due to the reduced end-turn length of the non-overlapping winding configurations. It is also
clear that for higher ξ values (i.e. the longer the stator becomes) the less the difference between
the overlapping and non-overlapping winding’s copper volumes become, due the the fact that
ℓ≫ℓe .
A further decrease in the copper volume can be obtained by decreasing the coil-side width.
For the single-layer (Type I) winding configuration, the relationship between the copper volume
and coil-side width is linear, where as for the double-layer (Type II) winding configuration, the
relationship is more parabolic. In Figure 6.5 we saw that the maximum torque produced by the
double-layer windings were obtained at k∆ ≈ 0,8 . At this value of k∆, the copper volume of
the double-layer windings is ≈ 75 % (depending on the value of ξ) of the overlapping winding configuration’s copper volume. At this value of k∆, the single-layer winding configuration
produced less torque however, but also requires less copper.
In Figure 6.7 the torque produced per copper volume as a function of k∆ is shown. The
interesting thing to note is that except for ξ = 0,1 and 0,2 , the torque per copper volume
rations for the other values of ξ at k∆ between 0,7 and 0,8 are virtually the same for both the
non-overlapping winding configurations and yields an approximate 5 % to 15 % better result
than for the overlapping winding configuration.
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CHAPTER 6. MACHINE PERFORMANCE COMPARISON INCLUDING THE END-TURNS EFFECTS
135
Relative torque per copper volume
2, 0
Type I: ξ =0.1
Type II: ξ =0.1
Type I: ξ =0.2
Type II: ξ =0.2
Type I: ξ =0.3
Type II: ξ =0.3
Type I: ξ =0.4
Type II: ξ =0.4
Type I: ξ =0.5
Type II: ξ =0.5
1, 8
1, 6
1, 4
1, 2
1, 0
0, 0
0, 1
0, 2
0, 3
0, 4
0, 5
0, 6
0, 7
0, 8
0, 9
1, 0
Stator coil-side width factor, k∆
Figure 6.7: The current density versus the coils-side width for the non-overlapping winding configuration stators with different values of ξ.
6.6
Summary and Conclusions
In this chapter we have discussed the effects the end-turn lengths of the Type O, Type I and
Type II winding configurations have on the performance of the RFAPM machine. Due to the
longer end-turn length of the overlapping winding configuration compared to that of the nonoverlapping winding configurations, the overlapping winding configuration has to operate at
a lower stator current density in order to produce the same amount of stator copper losses.
The single-layer (Type I) configuration has the lowest end-turn length, resulting in it being able
to operate at a slightly higher current density than that of the double-layer (Type I) winding
configuration, for the same amount of stator copper losses.
With the copper losses the same for all three winding configurations, the increase in current density for the non-overlapping winding configurations result in their maximum torque
being much closer to that of the overlapping winding configuration than initially predicted in
Table 6.2. This was shown in Figure 6.5 with the double-layer winding configuration holding
the edge over the single-layer winding configuration.
It is however in the torque per copper volume ratio where the non-overlapping winding
configurations really outperformed the overlapping winding configuration. This was especially prevalent for “pancake like” short stator machines, i.e. low values of ξ, where the endturn effect is more dominant, as shown in Figure 6.7.
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C HAPTER 7
Conclusions and Recommendations
The greatest reward lies in making the discovery; recognition can add little
or nothing to that.
F RANZ E RNST N EUMANN
7.1
Introduction
The work presented in this dissertation focused on the complete analytical analysis of the
Double-sided Rotor Radial Flux Air-cored Permanent Magnet (RFAPM) machine. The Doublesided Rotor RFAPM machine is a new type of electrical machine that was first introduced by
Randewijk et al. [14] in 2007. Two subsequent articles on the RFAPM machine were published
by Stegmann and Kamper [16, 26] and focused specifically on the design aspect of the Doublesided Rotor RFAPM machine pertaining to their use in medium power wind generator applications.
The aim of this research was to obtain a detailed analytical solution of the electromagnetic
fields inside the RFAPM machine. Of specific interest was an analytical solution that could
quickly and accurately calculate the radial flux-density in the stator region of the RFAPM machine. This was required to determine the influence, the permanent magnet pole arc width
have on the shape of the radial flux-density distribution.
Furthermore, an analytical solution to the steady state induced phase voltages (or backEMF) as well as the average steady state torque of the RFAPM machine in terms of key machine parameters were required that would be suitable for implementation in an optimisation
algorithm. The reason for this was to reduce the reliance of the RFAPM machine optimisation
on extremely slow FEM solutions, as well as to supplement the design procedure presented in
Stegmann and Kamper [26] for wind generator applications.
An analytical solution for the ripple torque produced by the RFAPM machine was also
required. The reason for this was to address the severe torque oscillation problems experienced
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138
7.2. DISSERTATION CONTRIBUTION AND ORIGINAL CONTENT
in the FEM calculation of the ripple torque component of the RFAPM machine as mentioned by
Gerber et al. [18], especially when using Maxwell® 2D and to a lesser extend with SEMFEM. In
order to solve the oscillation problems in the FEM torque calculation, the mesh in the air gap
and stator region had to be refined considerable, resulting in very long simulation times. It was
also debatable if the final solution was indeed correct, or still contained some “computational
noise”.
The research also considered the pro’s and con’s of using single-layer, non-overlapping,
concentrated (Type I) winding configuration as well as the usually more preferred, doublelayer, non-overlapping, concentrated (Type II) winding configuration, that was used by Stegmann and Kamper [16, 26]. An obvious advantage of the Type I winding configuration over
the Type II winding configuration, is its shorter end-turn length. However its short coil-pitch
length implies a lower back-EMF value and hence a lower torque value. I was also decided to
benchmark these non-overlapping winding configurations with a classical overlapping (Type
O) winding configuration, albeit difficult to implement in the Double-side Rotor RFAPM machine, due to the overlapping end-turn windings. The reason for this comparison was to prove
the effectiveness of non-overlapping winding configurations for the RFAPM machine, compared to the more traditional overlapping winding configuration found in classical electrical
machine text books.
7.2
Dissertation Contribution and Original Content
In this dissertation, the following new and original contributions were made:
• A complete set of winding factors for the concentrated, overlapping (Type O) –; the concentrated, single-layer, non-overlapping (Type I) – and the concentrated, double-layer,
non-overlapping (Type II) winding configuration were derived. The winding factors include the space harmonic components. It was shown that if the flux-density distribution
in the stator region is sinusoidal, only the fundamental space harmonic component of the
winding factor for the Type O winding configuration will contribute to the flux-linkage.
For the non-overlapping, Type I and Type II winding configurations, only the second order space harmonic component of the winding factor will contribute to the flux-linkage.
• A subdomain analytical model of the RFAPM machine was developed whereby the magnetic vector potential as well as the radial – and azimuthal flux-densities in the RFAPM
machine could be calculated. It was shown how a sweep of the permanent magnets’ pole
arc width, using the analytical subdomain model, could be done to quickly determine
the optimum value of the pole arc width angle that would result in the lowest percentage
Total Harmonic Distortion (%THD) value for the radial flux-density distribution in the
centre of the stator region.
• With the radial flux-density distribution in the centre of the stator region considered to
be quasi-sinusoidal, a voltage constant and torque constant for the RFAPM machine were
derived. The voltage constant, k E , gives the relationship between the back-EMF and the
speed of the RFAPM machine. The torque constant, k T , which is equal to the voltage
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CHAPTER 7. CONCLUSIONS AND RECOMMENDATIONS
139
constant and gives the relationship between the torque produced by the RFAPM machine
and the peak value of stator current space vector.
• An equation was derived from the analytical subdomain model of the RFAPM machine
using of the Lorentz method, whereby the torque ripple component produced by the
Type O, Type I and Type II winding configurations could be calculated quickly and fairly
accurately. We were also able to distinguish between the ripple torque component caused
by the winding layout, and that caused by the variation in the radial flux-density distribution in the stator region.
• A comprehensive comparison of the Type O, Type I and Type II winding configurations
was done relating to the effect, the end-turn windings of the different winding configurations have on the performance of the RFAPM machine, in terms of torque, copper losses
and copper volume.
7.3
Conclusions
The linear subdomain model of the Double-sided Rotor RFAPM machine as presented in this
dissertation, provide a sufficiently accurate analytical model of the RFAPM machine from
which the radial flux-density, flux-linkage, back-EMF and the developed torque of the machine can be calculated much faster, especially when compared to a FEA solution. With the
subdomain model of the RFAPM, the optimum pole arc width of the permanent magnet, that
would yield the lowest %THD for the radial flux-density distribution in the stator windings,
can be calculated within a matter of seconds. This is a further prove to value this subdomain
model has as a design tool for RFAPM machines.
The importance of having a quasi-sinusoidal radial flux-density distribution in the stator
region of the RFAPM machine can not be over emphasised, as it provides us with a simple
voltage – and a torque constant, similar to that of a normal sinusoidal permanent magnet AC
or “brushless DC” machine. These constants simplify, not only the design aspect of RFAPM
machine, but also from an application aspect, in that it provides the engineer with a set of
simplified equations for the RFAPM machine’s back-EMF and developed torque in terms of
the rotational speed and phase current, respectively. However, as was mentioned in Chapter 5,
the torque constant will only be valid with the back-EMF and phase current in phase. This is
achieved using field orientated control.
In this dissertation the accuracy of the subdomain model for the RFAPM machine, as well as
the accuracy of the voltage – and torque constant were benchmarked against a FEA simulation
of a RFAPM machine that was designed for a small, low speed, direct drive wind generator
by Stegmann [15]. The parameters for this machine (Test Machine I) is given in section E.1 of
Appendix E.
The same benchmarking was also done against a FEA simulation of a RFAPM machine that
was designed by Groenewald [17], as direct replacement for an Internal Combustion Engine
(ICE) in an Electric Vehicle (EV) project at Stellenbosch University. The parameters for this
machine (Test Machine II) is given in Appendix E, section E.2. The results of the benchmarking
tests showing the analytically calculated waveforms together with the waveforms obtained
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140
7.4. RECOMMENDATIONS FOR FUTURE WORK
using FEA in Maxwell® 2D, are shown in Appendix I. From the comparison for the radial and
azimuthal flux-density in section I.2, the flux-linkage and back-EMF comparison in section I.3
and the comparison of the armature reaction fields in section I.4, it is clear that the analytical
model derived in this dissertation for the RFAPM machine, is indeed valid.
However in the comparison between the analytically calculated ripple torque and the solution provided by Maxwell® 2D as shown in Figure I.9, it obvious that the Maxwell® 2D solution is clearly wrong. This can also be seen from the harmonic spectrum of the torque waveforms as shown in Figure I.10. The torque waveform of the Maxwell® 2D solution should only
contain 6th order harmonics, similar to that of the analytical solution, as was shown in equation
(5.28).
This emphasises the problems experience with the FEA modelling of RFAPM machines. For
the FEA model of Test Machine II, the mesh element size in the different regions of the machine
was chosen, using exactly the same relationship between the machine dimensions and that of
element size as was done for Test Machine I.
7.4
Recommendations for Future Work
The analytical analysis method presented in this dissertation, although sufficiently accurate,
can be improved further, by taking the recoil permeability of the permanent magnets into consideration. The method can also be extended to include parallel-magnetised permanent magnets, instead of only radially-magnetised permanent magnets. Furthermore, the effect of the
end-windings can be added to the calculation of the synchronous inductances.
In this dissertation, all the scripting to perform the analytical analysis, as well the Maxwell® 2D simulations, was done using Python™. The Python™ scripts that were used, are
briefly discussed in Appendix J. At the moment the Python™ scripts make use of a Command
Line User Interface (CLUI) which, although very powerful, are not very user friendly.
It is recommended that a Qt based Graphical User Interface (GUI), e.g. PyQt41 or PySide2 ,
be developed for the Python™ scripts, for people unfamiliar with Python™. The integration of
OpenOpt3 , a numerical optimisation package written in Python™, into the GUI should also be
investigated, in order create an analytical based design tool for the design and optimisation of
RFAPM machines.
1 http://www.riverbankcomputing.co.uk/software/pyqt
2 http://www.pyside.org/
3 http://openopt.org
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A PPENDICES
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A PPENDIX A
Series–parallel coil combination
A.1
Introduction
When calculating the total flux-linkage of series–parallel combination of coils, it is important
not to note that the flux-linkages does not add together as inductances would. This is best
explained using an example as shown in Figure A.1 for a phase grouping of a number of coils
in, say, phase a of an hypothetical machine.
Ia
L a5
L a6
L a7
L a8
L a1
L a2
L a3
L a4
Ia
a
Ia
a
Figure A.1: Hypothetical per phase interconnected coil layout of a electrical machine.
A.2
Total inductance
The inductance of each coil in the phase grouping can be calculated as
L aj =
=
Λ aj
Îaj
aΛ aj
Îa
with j = 1 . . . q
,
(A.1)
(A.2)
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144
A.3. TOTAL FLUX-LINKAGE
with Îa the peak value of the phase current Ia and assuming that the phase current divides
equally between the two branches due to all the coils having the same inductance and hence
the same flux-linkage, Λ aj .
The total inductance of phase a can now be calculated as follows
q
a
· L aj
a
qL aj
= 2 .
a
La =
A.3
(A.3)
(A.4)
Total flux-linkage
From (A.4) it is easy to make the wrong conclusion regarding the total flux-linkage. It must be
emphasised that the total flux-linkage for phase a,
Λ a 6=
qΛ aj
.
a2
(A.5)
Substituting (A.2) into (A.4), we get
La =
=
∴
q
aΛ aj
Îa
a2
qΛ aj
a Îa
qΛ aj
L a Îa =
a
(A.6)
(A.7)
(A.8)
so that we can write
Λa =
qΛ aj
a
which is the total flux-linkage “seen” by phase a.
(A.9)
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A PPENDIX B
The Surface Magnetisation Current
Density Equivalance
The effect of the magnetisation vector is equivalent to a volumetric – combined with a surface
magnetisation current density, Cheng [35, sec. 6.6]. If the magnetisation vector is uniform inside
the permanent magnet material, as is the case for a radial magnetised permanent magnet in
cylindrical coordinates, the currents of the neighbouring atomic dipoles that flow in opposite
directions inside the permanent magnet will effectively cancel out, leaving no net current in
the interior but only on the surface of the permanent magnets.
The equivalent surface magnetisation current distribution of the permanent magnets is
shown in Figure B.1 and can be calculated as follows,
~Jms = M
~ ×~an
(B.1)
= M0~ar ×~aθ
(B.2)
= M0~az .
(B.3)
In this dissertation the derivative of the residual magnetisation distribution is used, as
shown in Figure 3.3 and not the equivalent surface magnetisation current distribution as shown
in Figure B.1. This comparison merely shows that they are indeed equivalent. We can also see
that this equivalent surface magnetisation current distribution is basically equivalent to the
current sheets used in the analysis of permanent magnet machines employed by Boules [28].
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146
z
∞
∞
∞
∞
∞
~Jms
M0
M0
π
M0
M0
M0
2π
M0
−∞ −∞
3π
M0
M0
4π
M0
−∞ −∞
5π
M0
−∞
Figure B.1: The equivalent surface magnetisation current distribution with respect to θ.
θ
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A PPENDIX C
Boundary conditions
C.1 For the magnetic flux density
If we write (3.4) in integral form,
I
s
~B · d~s = 0
(C.1)
and we choose the Gaussian surface s in the form of a flat cylinder (e.g. a pillbox) with an
infinity small thickness, to span across the boundary between to regions, we can expand (C.1)
to
Z
s1
~B · d~s +
Z
s2
~B · d~s = 0 ,
(C.2)
with s1 the top portion of the Guassian surface and s2 the bottom portion of the Guassian
surface.
With s1 = s2 , the above equation reduces to
B1n = B2n ,
(C.3)
Figure C.1 or in vector form,
~an · (~B1 − ~B2 ) = 0 ,
(C.4)
which implies that the normal components of the flux density is continuous across a boundary,
Cheng [35, sec. 6.10].
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148
C.1. FOR THE MAGNETIC FLUX DENSITY
B2t
~B2
B2n
~B1
B1n
B1t
Figure C.1: The boundary condition of ~B across an interface boundary.
Thus from the general solution for the magnetic vector potential, the fluxdensity can be
calculated in the Cartesian coordiante systems as follows
~B = ∇ × ~A
(C.5)
~ax
~ay
~az
∂
∂x
∂
∂y
∂
∂z
=
(C.6)
A x Ay Az
∂Ay
∂Ay
∂Az
∂A x
∂Az
∂A x
=
~ax +
~ay +
~az .
−
−
−
∂y
∂z
∂z
∂x
∂x
∂y
(C.7)
For 2D analysis (with Jx = Jy = 0), this reduces to
~B = ∂Az~ax − ∂Az~ay .
∂y
∂x
(C.8)
Thus if the boundary between region (say) (v) and (v + 1) is parallel with the x-axis,
(v)
( v +1)
By = By
(v)
∂Az
∂x
(C.9)
( v +1)
=
∂Az
∂x
(C.10)
In the cylindrical coordinate system,
~B = ∇ × ~A
1
=
r
(C.11)
~ar
r~aθ
~az
∂
∂r
∂
∂θ
∂
∂z
Ar rAθ Az
1 ∂Az
∂Aθ
∂Ar
∂Az
1 ∂(rAθ ) ∂Ar
=
−r
−
−
~ar +
~aθ +
~az .
r
∂θ
∂z
∂z
∂r
r
∂r
∂θ
(C.12)
(C.13)
For 2D analysis (with Jr = Jθ = 0) the above equation reduces to
~B = 1 ∂Az~ar − ∂Az~aθ .
r ∂θ
∂r
Thus for a concentric circular boundarie between regions, say, (v) and (v + 1),
(v)
Br
( v +1)
= Br
(C.15)
( v +1)
(v)
∂Az
∂θ
(C.14)
=
∂Az
∂θ
(C.16)
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APPENDIX C. BOUNDARY CONDITIONS
C.2 For the magnetic field intensity
If we write (1.2’) in differential form,
I
c
~ · d~ℓ =
H
Z
s
~J · d~s
(C.17)
with c the contour spanning the surface s. If we choose c to span accross the boundary between
two regions and we make the thickness infinite small, and if we further assume that the surface
current density on the boundary is zero, we can write
Z
c1
~ · d~ℓ +
H
Z
c2
~ · d~ℓ = 0
H
(C.18)
with c1 the top portion of the contour and c2 the bottom portion of the contour.
With c1 = c2 the above equation reduces to,
H1t = H2t ,
∴
(C.19)
see Figure C.2, or in vector format for the general case,
~ 1−H
~ 2 ) = ~Js ,
~an × (H
(C.20)
which implies that the tangential component of the magnetic field intensity is continuous across
a boundary if and only if the free surface current density on the boundary is zero.
H2t
H2n
~1
H
~2
H
H1n
H1t
~ across an interface boundary.
Figure C.2: The boundary condition of H
From (3.9), for a linear isotropic non permanent magnet region,
~
~ =B,
H
µ
(C.21)
and from (3.18), for a linear isotropic permanent magnet region,
~ ~
~ = (B − Brem ) .
H
µ
(C.22)
Thus from (3.71) and (C.14) on a boundary between a non permanent magnet region, say,
(v) and a permanent magnet region, say, (v + 1),
(v)
Hθ
(v)
Bθ
µ(v)
( v +1)
= Hθ
=
( v +1)
Bθ
µ ( v +1)
(C.23)
−
Bremθ
.
µ ( v +1)
(C.24)
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150
C.2. FOR THE MAGNETIC FIELD INTENSITY
With radial magnetised permanent magnets, the azimutial component of ~Brem is zero, so that
the above equation reduses to
(v)
1 ∂Az
µ(v) ∂r
( v +1)
∂Az
= ( v +1)
∂r
µ
1
.
(C.25)
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A PPENDIX D
Doing Fourier Analysis in Degrees
D.1
Working with electrical degrees
The Fourier series expansion for am is calculated as follows,
4
am =
T
Z
T
2
0
f (t) cos(mω0 t)dt .
(D.1)
The Fourier series of an angular distribution, can also directly be calculated, by setting,
θ = ω0 t ,
∴
dθ
= ω0 ,
dt
dθ
dt =
ω0
dθ
=T
2π
(D.2)
(D.3)
(D.4)
(D.5)
with
ω0 =
2π
.
T
(D.6)
This also affects the integration boundaries, so that,
ω0 t
t =0
=0
(D.7)
=π,
(D.8)
Z π
(D.9)
and
ω0 t
t= T2
resulting in
4
am =
2π
0
f (θ ) cos(mθ )dθ
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152
D.2. WORKING WITH MECHANICAL DEGREES
D.2
Working with mechanical degrees
The same can also be done when working in mechanical degrees, by setting:
ω0 t
,
p
ω0
dθ
=
,
dt
p
dθ
dt = p
ω0
dθ
= pT
,
2π
θ=
∴
(D.10)
(D.11)
(D.12)
(D.13)
with
ω0 =
2π
.
pT
(D.14)
Again the integration boundaries change, so that
ω0 t
t =0
=0
(D.15)
and
ω0 t
t= T2
=
π
.
p
(D.16)
resulting in
am =
4p
2π
Z
π
p
0
f (θ ) cos(mpθ )dθ
(D.17)
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A PPENDIX E
Test Machine Data
E.1
Test Machine I
For more information regarding this machine refer to Stegmann [15] and Stegmann and Kamper
[16, 26].
p
kq
q
km
rn
l
w
=
=
=
=
=
=
=
=
=
=
∆
=
Q
=
=
∆ max
=
=
=
k∆
=
h
hm
=
=
=
– the total number of pole (magnets) pairs on the inner/outer rotor yoke
– coils-per-phase to pole-pair ratio
– number of coils per phase
16
1
2
pk q
8
3q
24
0,7
232
76
22,473
w
2rn
48,433 · 10−3
2,775
π
2Q
65,45 · 10−3
3,75
∆
∆ max
0,74
10,0
8,2
– total number of coils
[mm]
[mm]
[mm]
– the magnet angle to pole-pitch angle ratio
– nominal stator radius
– active stack/copper length
– coil side width
– coil side width angle
[rad]
[◦ ]
– maximum coil side-width angle
[rad]
[◦ ]
– coil side-width factor
[mm]
[mm]
– height/thickness of the stator coils
– magnet height/thickness
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154
E.2. TEST MACHINE II
hy
ℓg
N
a
n
=
=
=
=
=
f
=
Hc
Brem
Ip
=
=
=
=
E.2
8,0
1,0
96
1
300
np
120
80,0
1 050 000
1,4
7,33
[mm]
[mm]
[rpm]
– yoke height/thickness
– air gap length
– number of turns per coil
– number of parallel branches
– rated speed
– rated frequency
[Hz]
[A/m]
[T]
[A]
– the maximum coercivity force
– the remanent flux density of the permanent magnets
– rated peak current value per phase
Test Machine II
For more information regarding this machine refer to Groenewald [17].
p
kq
q
km
rn
l
w
=
=
=
=
=
=
=
=
=
=
∆
=
Q
=
=
∆ max
=
=
=
k∆
=
h
hm
hy
ℓg
N
a
n
=
=
=
=
=
=
=
=
– the total number of pole (magnets) pairs on the inner/outer rotor yoke
– coils-per-phase to pole-pair ratio
– number of coils per phase
12
1
2
pk q
6
3q
18
– total number of coils
2
3
125
80
13,96
w
2rn
55,84 · 10−3
3,2
π
2Q
87,266 · 10−3
5,0
∆
∆ max
0,64
8,0
4,0
5,0
2,0
33
6
4800
[mm]
[mm]
[mm]
– the magnet angle to pole-pitch angle ratio
– nominal stator radius
– active stack/copper length
– coil side width
– coil side width angle
[rad]
[◦ ]
– maximum coil side-width angle
[rad]
[◦ ]
– coil side-width factor
[mm]
[mm]
[mm]
[mm]
[rpm]
– height/thickness of the stator coils
– magnet height/thickness
– yoke height/thickness
– air gap length
– number of turns per coil
– number of parallel branches
– rated speed
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155
APPENDIX E. TEST MACHINE DATA
f
=
Hc
Brem
Ip
=
=
=
=
np
120
960,0
923 000
1,4
152,08
– rated frequency
[Hz]
[A/m]
[T]
[A]
– the maximum coercivity force
– the remanent flux density of the permanent magnets
– rated peak current value per phase
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A PPENDIX F
Permanent Magnetic Circuit Analysis
F.1 Introduction
In this appendix magnetic circuit analysis of circuits with permanent magnets will be discussed. In the literature magnetic circuit analysis is also sometimes referred to as 1-D analytical
analysis, Zhu et al. [23].
F.2 Permanent Magnet Fundamentals
All materials are composed of atoms with a positively charged nucleus and a number of orbiting negatively charged electrons, Cheng [35, chap. 6]. The orbiting electrons cause a “circulation current” and form microscopic magnetic dipoles. In the absence of an external magnetic
field, the magnetic dipoles of most materials, except that of permanent magnets, have random
orientation resulting in no net magnetic moment. The application of an external magnetic field
causes both an alignment of the magnetic moments of the spinning electrons and an induced
magnetic moment due to the change in the orbital motion of the electrons.
At a macroscopic level, a magnetic material is usually represented by a magnetisation vec~ , which is defined as the vector sum of the magnetic dipole moments per unit volume
tor, M
on a microscopic level. It can be proved that the magnetisation vector is equivalent to both a
volumetric current density
~Jm = ∇ × M
~
(F.1)
and a surface current density
~Jms = M
~ ×~an .
(F.2)
If the magnetisation vector of a material is uniform inside the material, the currents of
the neighbouring atomic dipoles that flow in opposite directions will cancel, leaving no net
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F.3. MAGNETIC CIRCUIT ANALYSIS WITH COIL EXCITATION
~ will be
current in the interior. This implies that inside the material, the curl of the constant M
zero and from (F.1), ~Jm will be zero. A uniformly magnetised permanent magnet can therefor
be represented by an equivalent surface current density or a fictitious winding, Nm carring a
current im so that the MMF produced by the permanent magnet,
F = Nm im
(F.3)
= Hm hm
(F.4)
with Hm the coercivity of the permanent magnet and hm the height or thickness of the permanent magnet, Fitzgerald et al. [19, chap. 1 & 3].
F.3 Magnetic Circuit Analysis with Coil Excitation
To start the analysis, we will only consider a magnetic circuit with a single coil excitation,
Figure F.1.
φ
ℓc + ℓ g
i
Ag
N
ℓg
Ac
Figure F.1: Magnetic circuit with coil excitation.
Basic magnetic circuit analysis is usually done using Ampère’s law. Applying Ampère’s
law to the above circuit yield:
Hc ℓc + Hg ℓ g = Ni
(F.5)
If the relative permeability of the iron core tends towards infinity, i.e. µrc → ∞ or µrc ≫ 1,
the magnetic field intensity in the airgap can now easily be calculated from (F.5).
Hg =
Ni
ℓg
(F.6)
With,
Bg = µ0 Hg ,
(F.7)
the airgap fluxdensity will be equal to:
Bg = µ0
Ni
ℓg
(F.8)
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APPENDIX F. PERMANENT MAGNETIC CIRCUIT ANALYSIS
F.4 Magnetic Circuit Analysis with PM Excitation
We now replace a portion of the iron core with a permanent magnet as shown in Figure F.2.
φ
ℓc + ℓ g
Ag
hm
ℓg
Ac
Figure F.2: Magnetic circuit with permanent magnet excitation.
According to Ampère’s law:
Hm hm + Hc ℓc + Hg ℓ g = 0
(F.9)
If we once again take the permeability of the iron core as striving toward infinity, (F.9)
reduces to,
Hm hm + Hg ℓ g = 0 ,
(F.10)
which is impossible to solve as both Hm and Hg are unknowns.
However in terms of flux, we know that,
Φm = Φg
(F.11)
because all the flux that goes through the magnet must pass through the airgap.
∴
Bm Am = Bg A g
(F.12)
Unfortunately the relationship between the fluxdensity and the magnetic field intensity in
the magnet are as such undefined as opposed to the simple relationship that excists in the
airgap, (F.7).
F.5 NdFeB Permanent Magnets
The advantages of using “Rare-earth” permanent magnets, e.g. Samarium-Cobalt (SmCo5 ) or
Neodymium-Iron-Boron (NdFeB), are not only that they have a much higher coercivity than
Alnico magnets but also that they have a linear demagnetisation curve. The demagnetisation
curve for NdFeB magnets can thus be represented by a straight line as shown in Figure F.3.
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F.6. MAGNETIC CIRCUIT ANALYSIS WITH PM EXCITATION CONTINUE. . .
Brem
B [T]
∆B
∆H
Hc
= µrrecoil
H [kA/m]
Figure F.3: Demagnetisation curve for NdFeB magnets.
For N48 NdFeB magnets, this straight line approxamation is however only valid at 20◦ C,
[45]. For higher temperatures, a higher temperature grade magnets, e.g. N48H, should be used
which has a linear demagnetisation curve up to 50◦ C, [54].
The recoil permeability, µrecoil , of a permanent magnet is defined as
µrecoil = µ0 µrrecoil
=
∆B
.
∆H
(F.13)
(F.14)
For a NdFeB permanent magnet with a linear demagnetisation curve, the recoil permeability will thus be equal1 to,
µrecoil =
Brem
,
Hc
(F.15)
with the equation for the demagnetisation line given by
Bm = µrecoil Hm + Brem .
(F.16)
From the specifications for the different grade NdFeB magnets as given in Table F.1, the
linear equation for the different grades of NdFeB magnets kan easily be found.
F.6 Magnetic Circuit Analysis with PM Excitation Continue. . .
From (F.7), the magnetic field intensity in the airgap can be written as
Hg =
Bg
,
µ0
(F.17)
and from (F.16), the magnetic field intensity in the permanent magnet can be written as
Hm =
1 The
Bm − Brem
.
µ0 µrrecoil
symbol used for the magnetic field intensity in F.4 is the same as that used for the magnetic coersivity.
(F.18)
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APPENDIX F. PERMANENT MAGNETIC CIRCUIT ANALYSIS
Table F.1: Specification of different NdFeB magnets.
Grade
Brem [mT]
Hcb [kA/m]
Hcj [kA/m]
N27
N30
N33
N35
N38
N40
N42
N45
N48
N50
N52
1030–1080
1080–1130
1130–1170
1170–1220
1220–1250
1250–1280
1280–1320
1320–1380
1380–1420
1400–1450
1430–1480
≥ 796
≥ 796
≥ 836
≥ 868
≥ 899
≥ 907
≥ 915
≥ 923
≥ 923
≥ 796
≥ 796
≥ 955
≥ 955
≥ 955
≥ 955
≥ 955
≥ 955
≥ 955
≥ 955
≥ 876
≥ 876
≥ 876
( BH )max [kJ/m3 ]
199–23
223–247
247–271
263–287
287–310
302–326
318–342
342–366
366–390
382–406
398–422
Tw
80 ◦ C
80 ◦ C
80 ◦ C
80 ◦ C
80 ◦ C
80 ◦ C
80 ◦ C
80 ◦ C
80 ◦ C
60 ◦ C
60 ◦ C
Substituting (F.17) and (F.18) into (F.10) to solve the Ampère’s law equation, yield:
Bg
Bm − Brem
· hm
· ℓg = −
µ0
µ0 µrrecoil
Brem − Bm
hm
∴
Bg =
·
µrrecoil
ℓg
(F.19)
From (F.12), if the area of the airgap and that of the magnets are the same, i.e. if we ignore
fringing, Bg = Bm which implies that (F.19) can further be simplified to,
Bg =
Brem
1 + µrrecoil ·
ℓg
hm
,
(F.20)
which is similar to (2.14) in [1] except that here we have a double airgap.
From (F.20) the fluxdensity in the airgap can now easily be calculated from the specifications
of the NdFeB magnet used and the airgap length.
Alternatively, (F.20) allows one to calcualte the required length, or thickness, of the NdFeB
magnet required for a required airgap fluxdensity as follows:
hm =
ℓ g µrrecoil
Brem
−1
Bg
(F.21)
F.7 Magnetic Circuit Analysis for the RFAPM Machine.
In Figure F.4 the magnetic equivalent circuit for one pole pair of a RFAPM machine is shown.
The equivalent circuit differs slightly from the simplified one shown in Figure F.2.
The RFAPM machine have two airgap, two sets of magnets as well as a set of air cored
windings. This implies that for the RFAPM machine (F.20) will change to,
Bg =
Brem
1 + µrrecoil ·
2ℓ g + h
2hm
,
(F.22)
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F.7. MAGNETIC CIRCUIT ANALYSIS FOR THE RFAPM MACHINE.
Φm
hy
hm
ℓg
h
ℓg
hm
hy
Φm
rn
0
π
2
π
θ
Figure F.4: Magnetic circuit of the RFAPM machine for one pole pair.
and (F.21) to
(ℓ g + 2h )µrrecoil
.
hm =
Brem
−1
Bg
(F.23)
From this figure, using (F.19) the fluxdensity in the airgap for test machine I, see Appendix
E.1, can be calculated as,
Bg = 0,788 T ,
(F.24)
and that for test machine II, see Appendix E.2, as,
Bg = 0,498 T .
(F.25)
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A PPENDIX G
Armature reaction contour field plots
G.1
Overlapping, Type O, winding configuration
G.1.1
Magnetic vector potential
0.18
0.280.3
+0.16
0.2
0.16
0.12
0, 00
+0.24
0.26
+0.08
0.08
0.04
+0.00
0
y [m]
0.02
0, 02
0.22
0.24
-0.04
-0.08-0.06
-0.1
-0.08
−0, 02
-0.02
16.18
-0.-0
-0.3
-0.2
6-0.24
-0.24
-0.22
-0.14
-0.12
0
−0, 04
-0.16
-0.28-0.2
Magnetic Vector potential, Az| AR [mWb/m]
0
0.06
0.1
0.14
0, 04
0, 21 0, 22 0, 23 0, 24 0, 25
x [m]
Figure G.1: Contour plot of the magnetic vector potential.
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G.1. OVERLAPPING, TYPE O, WINDING CONFIGURATION
Figure G.2: Contour plot of the magnetic vector potential as simulated in Maxwell® 2D.
32.0
0, 02
28.0
y [m]
24.0
20.0
0, 00
16.0
12.0
8.0
4.0
−0, 02
Flux density, B| AR [mT]
0, 04
0.0
−0, 04
0, 21 0, 22 0, 23 0, 24 0, 25
x [m]
Figure G.3: Contour plot of the magnetic flux density.
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APPENDIX G. ARMATURE REACTION CONTOUR FIELD PLOTS
Figure G.4: Contour plot of the magnetic flux density as simulated in Maxwell® 2D.
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166
Non-overlapping, Type I, winding configuration
0
0
0.20.225
0.25
0.175
0.275
0.35
0.0
25
0, 05
5
0.37
.12
5 0
5
0.4
5
5 0.1
0.1
+0.40
25
0.4
+0.30
0.325
+0.20
0.3
+0.10
0.05
0
0, 00
-0.075
+0.00
-0.1
-0.10
-0.20
-0.35
-0.4
-0.30
-0.
4
25
-0.40
-0.3
7
5
-0.0
−0, 05
25 -0
.05
-0.1
2
5
-0.17
-0.45
5
-0.475
Magnetic Vector potential, Az| AR [mWb/m]
0.07
0.475
0.4
y [m]
-0.275
-0.225
-0.15
-0.325
-0.3
-0.25
-0.2
0
0
G.2
G.2. NON-OVERLAPPING, TYPE I, WINDING CONFIGURATION
0, 20
0, 21
0, 22
0, 23
0, 24
0, 25
x [m]
Figure G.5: Contour plot of the magnetic vector potential
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APPENDIX G. ARMATURE REACTION CONTOUR FIELD PLOTS
Figure G.6: Contour plot of the magnetic vector potential as simulated in Maxwell® 2D.
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168
G.2. NON-OVERLAPPING, TYPE I, WINDING CONFIGURATION
54.0
48.0
y [m]
42.0
36.0
30.0
0, 00
24.0
18.0
12.0
6.0
Flux density, B| AR [mT]
0, 05
0.0
−0, 05
0, 20
0, 21
0, 22
0, 23
0, 24
0, 25
x [m]
Figure G.7: Contour plot of the magnetic flux density
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APPENDIX G. ARMATURE REACTION CONTOUR FIELD PLOTS
Figure G.8: Contour plot of the magnetic flux density as simulated in Maxwell® 2D.
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170
Non-overlapping, Type II, winding configuration
0
0
0.24
0.32
0.28
0.36
0.4
0, 05
0.12
0.5
6
0.16
0.52
+0.48
0.44
+0.32
+0.16
0.2
0.08
0
0, 00
0.04
+0.00
-0.16
-0.16
-0.04
-0.32
-0.48
2
-0.5
-0.48
-0
.
-0.0
8
-0.1
2
56
Magnetic Vector potential, Az| AR [mWb/m]
0.48
y [m]
−0, 05
-0.4
-0.44
-0.32
-0.2
-0.36
-0.24
-0.28
0
0
G.3
G.3. NON-OVERLAPPING, TYPE II, WINDING CONFIGURATION
0, 20
0, 21
0, 22
0, 23
0, 24
0, 25
x [m]
Figure G.9: Contour plot of the Magnetic Vector Potential.
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APPENDIX G. ARMATURE REACTION CONTOUR FIELD PLOTS
Figure G.10: Contour plot of the magnetic vector potential as simulated in Maxwellr 2D.
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G.3. NON-OVERLAPPING, TYPE II, WINDING CONFIGURATION
0, 05
64.0
56.0
y [m]
48.0
40.0
0, 00
32.0
24.0
16.0
8.0
Flux density, B| AR [mT]
72.0
0.0
−0, 05
0, 20
0, 21
0, 22
0, 23
0, 24
0, 25
x [m]
Figure G.11: Contour plot of the magnetic flux density.
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APPENDIX G. ARMATURE REACTION CONTOUR FIELD PLOTS
Figure G.12: Contour plot of the magnetic flux density as simulated in Maxwellr 2D.
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A PPENDIX H
Simplified Analytical Analysis
H.1
Solving for all the regions of RFAPM machine simultaneously
H.1.1
Introduction
By taking the permeability of the yoke iron ≈ ∞, we only need to solve for regions II, III and
IV.
H.1.2
On the boundary between region I and II
With r = rn −
∴
H.1.3
h
2
− ℓ g − hm ,
HφI I (r, φ) = 0
(H.1)
II
−mp−1
CmI I| PM r mp−1 − Dm
=0
| PM r
(H.2)
On the boundary between region II and III
With r = rn −
h
2
− ℓg ,
BrI I (r, φ) = BrI I I (r, φ)
∴
CmI I| PM r mp
+
II
−mp
Dm
| PM r
+
GmI I| PM
=
CmI I|IPM r mp
(H.3)
+
III
−mp
Dm
| PM r
(H.4)
and
HφI I (r, φ) = HφI I I (r, φ)
∴
−mp−1
II
CmI I| PM r mp−1 − Dm
| PM r
µI I
=
(H.5)
−mp−1
III
CmI I|IPM r mp−1 − Dm
| PM r
µI I I
(H.6)
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H.1. SOLVING FOR ALL THE REGIONS OF RFAPM MACHINE SIMULTANEOUSLY
H.1.4
On the boundary between region III and IV
With r = rn +
h
2
+ ℓg ,
BrI I I (r, φ) = BrIV (r, φ)
(H.7)
III
−mp
CmI I|IPM r mp + Dm
= CmIV| PM r mp + DmIV| PM r −mp + GmIV| PM
| PM r
∴
(H.8)
and
HφI I I (r, φ) = HφIV (r, φ)
∴
H.1.5
CmI I|IPM r mp−1
− DmI I|IPM r −mp−1
µI I I
(H.9)
CmIV| PM r mp−1
=
− DmIV| PM r −mp−1
µ IV
(H.10)
On the boundary between region IV and V
With r = rn +
h
2
+ ℓ g + hm ,
HφIV (r, φ) = 0
(H.11)
IV
−mp−1
CmIV| PM r mp−1 − Dm
=0
| PM r
(H.12)
H.1.6
Simultaneous equations to solve
From (H.2), (H.4), (H.6), (H.8), (H.10) and (H.12) the following ten equations have to be solved
for m = 1, 3, 5, → ∞,
mp−1
−mp−1
CmI I| PM rii
− DmI I| PM rii
−mp
mp
II
CmI I| PM riii + Dm
| PM riii
mp−1
µ I I I CmI I| PM riii
mp
=0
mp
−mp
−mp−1
− µ I I CmI I|IPM riii
− CmI I|IPM riii − DmI I|IPM riii
− µ I I I DmI I| PM riii
−mp
III
CmI I|IPM riv + Dm
| PM riv
(H.13)
mp
mp−1
−mp−1
µ IV CmI I|IPM riv
− µ IV DmI I|IPM riv
mp−1
−mp−1
CmIV| PM rv
− DmIV| PM rv
=0
−
mp−1
−mp
− CmIV| PM riv − DmIV| PM riv
= − GmI I| PM
(H.14)
−mp−1
+ µ I I DmI I|IPM riii
=0
= GmIV| PM
mp−1
µ I I I CmIV| PM riv
−mp−1
+ µ I I I DmIV| PM riv
(H.15)
(H.16)
=0
(H.17)
(H.18)
with
h
2
h
riii = rn −
2
h
riv = rn +
2
h
rv = rn +
2
rii = rn −
− ℓ g − hm
(H.19)
− ℓg
(H.20)
+ ℓg
(H.21)
+ ℓ g + hm
(H.22)
and
− ℓ g − h2m ) Brem cos mβ
m2 π
4(rn + 2h + ℓ g + h2m ) Brem cos mβ
=−
m2 π
GmI I| PM = −
GmIV| PM
4(r n −
h
2
(H.23)
(H.24)
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APPENDIX H. SIMPLIFIED ANALYTICAL ANALYSIS
177
mp−1
−mp−1
rii
−rii
mp
−mp
mp
−mp
riii
riii
−riii
−riii
µ I I I r mp−1 −µ I I I r −mp−1 −µ I I r mp−1
I I r −mp−1
µ
iii
iii
iii
iii
mp
−mp
mp
−mp ·
riv
riv
−riv
−riv
mp−1
−mp−1
mp−1
−mp−1
I
I
I
IV
IV
I
I
I
µ
r
−
µ
r
−
µ
r
µ
r
iv
iv
iv
iv
mp−1
−mp−1
rv
−r v
CmI I| PM
0
II
D
− GmI I| PM
m| PM
III
Cm| PM
0
=
(H.25)
D I I I G IV
m| PM m| PM
IV
C
0
m| PM
IV
0
Dm| PM
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A PPENDIX I
Analytical – FEM Comparison for Test
Machine II
I.1 Introduction
In this section the validity of the analytical model derived in this dissertation is tested against
against the Maxwell® 2D FEA model of Test Machine II, given in section E.2 of Appendix E.
Good correlation between the analytical and FEA results were obtained as is shown in the
following sections. One exception is however that of the torque output as shown in Figure I.9.
The ripple torque of the FEA simulation is clearly not correct as can be seen from harmonic
spectrum shown in Figure I.10. There is clearly some sort of “computational resonance” present
at the twelfth harmonic in the Maxwell® 2D result.
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I.2. RADIAL AND AZIMUTHAL FLUX-DENSITY COMPARISON
I.2 Radial and Azimuthal Flux-density Comparison
1, 0
Br|rcim ,PM
Br|rn ,PM
Radial flux density, Br| PM [T]
Br|rcom ,PM
0, 5
Br|rcim ,PM (Maxwell® 2D)
Br|rn ,PM (Maxwell® 2D)
Br|rcom ,PM (Maxwell® 2D)
0, 0
−0, 5
−1, 0
−15, 00
−11, 25
−7, 50
−3, 75
0, 00
3, 75
7, 50
11, 25
15, 00
Angle, φ [◦ ]
Figure I.1: The radial flux-density distribution.
0, 3
Azimuthal flux density, Bφ| PM [T]
Bφ|rcim ,PM
Bφ|rn ,PM
0, 2
Bφ|rcom ,PM
Bφ|rcim ,PM (Maxwell® 2D)
Bφ|rn ,PM (Maxwell® 2D)
0, 1
Bφ|rcom ,PM (Maxwell® 2D)
0, 0
−0, 1
−0, 2
−0, 3
−15, 00
−11, 25
−7, 50
−3, 75
0, 00
Angle, φ
3, 75
7, 50
[◦ ]
Figure I.2: The azimuthal flux-density distribution.
11, 25
15, 00
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APPENDIX I. ANALYTICAL – FEM COMPARISON FOR TEST MACHINE II
1, 6
Bmag| PM,rciy
Flux density, Bmag| PM [T]
1, 4
1, 2
Bmag| PM,rcoy
Bmag| PM,rciy (Maxwell® 2D)
Bmag| PM,rcoy (Maxwell® 2D)
1, 0
0, 8
0, 6
0, 4
0, 2
−15, 00
−11, 25
−7, 50
−3, 75
0, 00
3, 75
7, 50
11, 25
15, 00
Angle, φ [◦ ]
Figure I.3: The variation in the flux-density in the centre of the rotor yokes.
0, 6
Br|rn −h/2| PM
Radial flux density, Br [T]
0, 4
Br|rn | PM
Br|rn +h/2| PM
Br|rn −h/2| PM (Maxwell® 2D)
0, 2
Br|rn | PM (Maxwell® 2D)
Br|rn +h/2| PM (Maxwell® 2D)
0, 0
−0, 2
−0, 4
−0, 6
−15, 00
−11, 25
−7, 50
−3, 75
0, 00
3, 75
7, 50
11, 25
15, 00
Angle, φ [◦ ]
Figure I.4: The variation in the shape radial flux-density distribution in the stator windings shown on
the outer –, centre – and inner radius of the stator for the analytical analysis method compared the the
FEA analysis done using Maxwell® 2D.
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I.3. FLUX-LINKAGE AND BACK-EMF COMPARISON
I.3 Flux-linkage and Back-EMF Comparison
0, 20
λ a| PM (t)
λb| PM (t)
0, 15
Flux linkage, λ| PM [Wbt]
λc| PM (t)
λ a| PM (t) (Approx.)
0, 10
λb| PM (t) (Approx.)
λc| PM (t) (Approx.)
0, 05
λ a| PM (t) (Maxwell® 2D)
λb| PM (t) (Maxwell® 2D)
0, 00
λc| PM (t) (Maxwell® 2D)
−0, 05
−0, 10
−0, 15
−0, 20
0, 000
0, 104
0, 208
0, 312
0, 417
0, 521
0, 625
0, 729
0, 833
0, 938
1, 042
Time, t [ms]
Figure I.5: Comparison of the flux-linkage calculations for the Type II winding configuration.
1000
ea| PM (t)
eb| PM (t)
ec| PM (t)
ea| PM (t) (Approx.)
Back EMF, e| PM [V]
500
eb| PM (t) (Approx.)
ec| PM (t) (Approx.)
ea| PM (t) (Maxwell® 2D)
eb| PM (t) (Maxwell® 2D)
0
ec| PM (t) (Maxwell® 2D)
−500
−1000
0, 000
0, 104
0, 208
0, 312
0, 417
0, 521
0, 625
0, 729
0, 833
0, 938
1, 042
Time, t [ms]
Figure I.6: Comparison of the back-EMF calculations for the Type II winding configuration.
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APPENDIX I. ANALYTICAL – FEM COMPARISON FOR TEST MACHINE II
I.4 Armature Reaction Flux-density Comparison
0, 03
Br|rcim ,AR
0, 02
Br|rn ,AR
Radial flux density, Br| AR [T]
Br|rcom ,AR
0, 01
Br|rcim ,AR (Maxwell® 2D)
Br|rn ,AR (Maxwell® 2D)
0, 00
Br|rcom ,AR (Maxwell® 2D)
−0, 01
−0, 02
−0, 03
−0, 04
−0, 05
−30, 0
−22, 5
−15, 0
−7, 5
0, 0
7, 5
15, 0
22, 5
30, 0
Angle, φ [◦ ]
Figure I.7: The radial flux-density distribution due to armature reaction.
0, 006
Azimuthal flux density, Bφ| AR [T]
Bφ|rcim ,AR
Bφ|rn ,AR
0, 004
Bφ|rcom ,AR
Bφ|rcim ,AR (Maxwell® 2D)
Bφ|rn ,AR (Maxwell® 2D)
0, 002
Bφ|rcom ,AR (Maxwell® 2D)
0, 000
−0, 002
−0, 004
−0, 006
−30, 0
−22, 5
−15, 0
−7, 5
0, 0
Angle, φ
7, 5
15, 0
22, 5
[◦ ]
Figure I.8: The azimuthal flux-density distribution due to armature reaction.
30, 0
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I.5. OUTPUT TORQUE COMPARISON
I.5 Output Torque Comparison
58, 0
Mechanical Torque, Tmech [Nm]
57, 5
57, 0
56, 5
56, 0
55, 5
55, 0
Using the Lorentz Method
Using Maxwell® 2D
54, 5
Using the Torque Constant, k T
54, 0
0, 000
0, 104
0, 208
0, 312
0, 417
0, 521
0, 625
0, 729
0, 833
0, 938
1, 042
Time, t [ms]
Figure I.9: The torque ripple waveforms for the Type II winding configuration.
0, 25
Analytical
Maxwell® 2D
Maxwell® 2D (Tweaked)
Tmech|h
Tmech| ave.
× 100 %
0, 20
0, 15
0, 10
0, 05
0, 00
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Harmonic Number
Figure I.10: A FFT of the torque ripple harmonics for the Type II winding configuration.
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A PPENDIX J
Brief Python™ Code Desciption
J.1 Introduction
In this Appendix, the Python™ code1 that was developed for the analysis of the Double-sided
Rotor Radial Flux Air-cored Permanent Magnet (RFAPM) machine will be briefly discussed.
The Python™ code makes use of the Numpy module developed by Oliphant [55] specifically
for “Scientific Computing”. The Numpy module adds multi-dimensional array capabilities to
Python™. This give Python™ similar features to Matlab®, just using multi-dimensional arrays
instead of two-dimensional matrices.
For visualisation of the results, the excellent add-on package, Matplotlib by Hunter [42] is
used. The pylab library inside the Matplotlib package could be described as a “Matlab®–like
plotting library for Python™ “ with near identical keyword used, e.g. plot, xlabel, ylabel,
axis, grid, linspace, etc.
The interactive Python™ environment of IPython created by Perez and Granger [56], provides
the ideal interactive scripting, testing and debugging environment for the implementation of
the analytical solution into Python™ code.
J.2 Python™ Scripts to call Maxwell® 2D via the COM Interface
test_machine_i.py – The data for Test Machine I.
test_machine_ii.py – The data for Test Machine I.
poly.py – A collection of polygon functions.
farey.py – An implementation of the Farey function, implemented in Python™.
1 This
code is available for download at http://staff.ee.sun.ac.za/pjrandewijk/downloads/phd/.
185
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J.3. PYTHON™ SCRIPTS TO DO THE ANALYTICAL ANALYSIS
pyMaxwell2D.py – Some Python™ wrapper classes for the ActiveX or COM functions of
Maxwell® 2D.
rfapm.py – The main Python™ script that “draws” and simulates the RFAPM machine in
Maxwell® 2D depending on whether test_machine_i.py or test_machine_ii.py
were imported..
J.3 Python™ Scripts to do the Analytical Analysis
test_machine_i.py – The data for Test Machine I, same file as in section J.2.
test_machine_ii.py – The data for Test Machine I, , same file as in section J.2.
mxwltxt.py – Contains the loadmxwltxt function to to convert the Maxwell® 2D output
*.txt file into a Numpy record array for easier plotting with Matplotlib.
analytical_solution_i.py – The main script that does the analytical calculations for Test
Machine I
analytical_solution_ii.py – The main script that does the analytical calculations for
Test Machine II, essentially the same script as the above, with the only difference in the
scaling of the Matplotlib graphs
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Anto Mastepan