MODELLING OF IRON LOSSES OF
PERMANENT MAGNET
SYNCHRONOUS MOTORS
Chunting Mi
A thesis submitted in confollILity with the requirements
for the Degree of Doctor of Philosophy in the
Department of EIectncal and Computer Engineering
University of Toronto
O Copyright by Chunting Mi, 200 1
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MODELLING OF IRûN LOSSES OF PERMANENT
MAGNET SYNCHRONOUS MOTORS
Chunting Mi
A thesis submitted in conformity with the requUements
for the De- of Doctor of Philosophy in the
Department of Electrical and Computer Engineering
University of Toronto
@ Copyright by Chunting Mi, 200 1
ABSTRACT
This thesis proposes a refïned approach to evaluate iron losses of surface-mounted
permanent magnet (PM) synchronous motors.
PM synchronous moton have higher efficiency than induction machines with the same
&me and same power ratnigs. However, in PM synchronous motors, iron Iosses form a larger
portion of the total losses than in induction machines. Therefore, accurate calculation and
. * - muimuzation of iron losses is ofparticular importance in the design of PM synchronous motors.
Aithough time-stepped FEM can provide good estimations of iron Iosses, it requires
hi& effort and is very time-consmning for most designs and particularly for optimidons.
Besides, FEM anaiysis can oniy be performed afta the machine geometrical dimensions have
. * Tl
been detexmined. In contras appmximate anaiyticai loss mode1 yields qyick insight while
requiring much l e s computation time and are readily available d u ~ g the prelhinary stage
when the motor design is being iterated. Therefore, it is desirable to develop iron loss models
that produce an acceptable prediction of the ovedi im losses in a relatively simple way and
provide machine designers a more reliable method to estimate iron losses in the initial design
of PM machines.
In this thesis, simplined models for the estimation of tooth and yoke eddy current losses
are devdoped based on the observations of FEM results and anaiyticai analyses of the magnetic
fields taking into account detailed geometrical effects. The effectiveness and limitations of the
models are investigated. Correction factors are derived for geomehical Muences.
The nature of the flux density waveform has great impact on iron losses. Therefore,
reduction of iron Iosses c m be achieved by properly shaping the magnets, designing the dots
and choosing appropriate number of poles. Some guidelines are provided to minimize iron
losses of PM synchronous motors.
In order to test the proposed simplifieci loss models, measurements were performed on
two dace-mounted PM synchronous motors. Puticular attention was focused on the
measmement of PWM wavefoms and decomposition of no-load losses. Eqcrimental d t s
of the two motors codbmed the validÏty of the proposed iron loss models.
1 am etemally grateful to my supervisors, Professor G. R Slemon and Professor R
Bonert, for their invaluable guidance, encouragement and financiai support throughout the
period of my course study and thesis research, TheY attitude, entbusiasm and dedication towards
scientSc research deeply influenced me and will benefit for al1 rny Me.
1 sincerely th& Professor Lavers and Profesor Dawson for their precious suggestions
and technical discussions.
1 am also t h a W to Dr. KV. Namjoshi for spending much of his spare time reading
the draft of my thesis and giving me a lot of advice. Specid thanks are due to the leaders and
feiIows at Xi'an Petmleum M t u t e for granting me the opportunity to study abmaci. The author
also wishes to extend his gratitude to his Master's thesis supervisor Professor Jiang Zongrong,
one of the pioneers in the field of rare-earth PM machines, for his excellent quality of teaching.
Special appreciation goes to my wife Yuhong for her undetsfalldmg and support during
my Ph.D studies- This thesis is aiso a gift to my lovely daughter Shishi for the many sacrifices
she d e r e d during the years of my research work. This thesis is also dedicated to my parents
and my parents-blaw for their understanding, encouragement and mental support.
The hancial supports h m the Ontario Graduate Scholmhip for Science and
Technology (OGSST), the University of Toronto S.G.S Intemational Student Award, the
University of Toronto Doctorkd Feiiowships and Ontario Shrdent Assistant Program (OSAP),
are gratefully acknowledged.
CONTENTS
........................................................................................................................... CONTENTS .eV
Chapter 1 Introduction .........m....,b...e~~..~.a..~~~~.~~~~~~~~..~.............a.a........a..~m.~mm..........e......ea.. 1
.......... .....*.*.*..................*.*.*.......*................................. 1.1 Thesis Objectives ........ 1
1.2 Thesis Outline ........................................................................................................... 3
........ ... Chapter 2 Iron Losses and TimeStepped FEM of PM Synchronous Motors ., -5
2.1 PM Synchronous Motoa .......................................................................................... 5
.................................................................................... 2.2 Iron Losses in PM Machuies 8
.................................. 2.3 Finite Elment Method of Electromagnetic Field hblems 10
2.3.1 Maxwell's Equations ...................................................................................... 10
2.3.2 The Magnetic Vector Potential ................... ...................................... 1 1
2.3.3 Two Dimensional FEM Formulation of Magnetostatic Roblems ................. 12 2.3.4 Magnetization ModeIs of Magrets ............................................................... 14
2.4 Time-Stepped F E ' of PM Synchronous Motors ................................................... 15
2.401 C0mcti0~1 of Meshe~......+~~~ ........................................................................ 15 2.4.2 Boundary Conditions ..................................................................................... 15
2.4.3 Examples Used in the Thesis .. ...................................................................+... 17
2.5 EvaIuation of Iron Losses with FEM~.~~o.****-.*.--.----*---ti--------*---~~-.~-~~~~~~-~~~--~~---~-~-.+-*r23
.......................................................................................... 2.5.1 Eddy C u m t Loss 23
............................................................................................... 2.5.2 Hysteresis L o s 24
Chapter 3 Shpiified Lroa Loss Mode1 ........... ..................................................................... 25
3.1 Review of Analfical Iron Loss Mode1 .................................................................. 26
................................................. 3 1 Test of Linear Waveforms of Tooth Flux Density 27
3.3 SimpIified Tooth Eddy Cment Loss Model .......................................................... 34
................................ 3.3.1 Eddy Current Loss Induced by the Nomial Component 34
..................................................................................... 3.3.2 Effect of SIot Closure 35
............................................... 3.33 Effect of Magnet Width ................. .......... 38
............................... 3.3.4 Effect of Slots Per Pole Per Phase with Fked Slot Pitch 42
................... 3.3.5 Effect of Airgap Length and Magnet Thiclmess .............. 46
............................ 3.3.6 Effect of SIots Pet Pole Per Phase with Fixed Pole Pitch -47
33.7 Effect of Tooth Width with Fixed Slot Pitch .............................................. 50
33.8 Eddy Current Loss Induced by Circderential Component ......................... 51
3.3.9 Simplified Expression of Tooth Eddy Current Loss ...................si.................C 53
3 .3.10 Case Studies on Tooth Eddy Currait Loss ................... ... ....................... 54
3.4 Wavefoms of Yoke Flux Density ............ .... .................................................. 59
............ 3.4.1 Waveforms of the Longitudinal Component of Yoke FIux Density ..59
3 .4.2 Waveforms of the Normal Component of Yoke Flux Dnisity .................... ..64 ................................................... 3.4.3 Anaiytical Solutions to Yoke Flux Density 65
3.5 Simplified Yoke Eddy C m t L o s Mode1 .......................................................... 69
3.5.1 Eddy Current L o s Induced by the Longitudinal Component ....................... 69
3.5.2 Effect of Slot Closure ..................................................................................... 70
3.5.3 Efféct of Magnet Coverage ............................................................................ 71
3.5.4 Effect of SIots Per Pole Per Phase .................................................................. 72
........................................................................... 3.55 Effect of Magnet Thickness 73
3.5.6 Effect of Airgap Length ............................................................................ 7 4
3 .5.7 Effkct of Tooth Width 6th Fixed Slot Pitch ................................................. 74
................................ 3.5.8 Eddy C m t Loss Induced by the Nomial Componat 75
...................................... 3.5.9 Simplifiecl Expression of Yoke Eddy Current Loss +76
................................................... 3.5.10 Case Studies on Yoke Eddy Current Loss 77
3.6 Simplifieci Hysteresis Loss Mode1 ........................................................................ 82
...... Chapter 4 Measlirement of Iron Losses in PM Synchronotts Moton mm..,mmm.. ~.mmoeammmme83
4.1 Loss Measurement of PM Synchronous Machines ................................................ 83
4.1.1 No-load Loss Measurement of PM Synchronous Machines .......................... 83
4.1.2 Load Loss and Motor Efficiency ......... .... ............................................... 84
............... ..... 4.2 Implementation of the Control of a PM Syncbuous Motor .... 85
42.1 System Description ........................................................................................ 85
....................................... 4.2.2 Mathematical Mode1 of PM synchmnous Moton 0.85
4.2.3 Transfer Function ........................................................................................... 87
4.2.4 Speed Limit for Cumnt Control .................... .. ....................................... 89
4.2.5 Torque and Speed Control ............................................................................. 90
4.2.6 Current and Speed Responses ........................................................................ 91
4.3 Measurement of Voltage, Current and Power ........................................................ 94 ...
4.3.1 Computerized Data Acquisition ..................................................................... 94
4.3.2 Measurement of C m t ................................................................................. 94
4.3.3 Meamernent of Supply Voltage .................................................................... 95
4.4 Vaiidation of Measurements Using Phasor Analyses ............................................ 96 4.5 Decomposition of Iron Losses and Mechanical Losses ......................................... -97
.................................... 4.5.1 With the HeQ of the Rotor of m Induction Machine 98
4.52 With the Help of a Nonmagnetic Stator ......................................................... 98
4.53 Measming before Magnets Assembled ................... .............................. 98
4.6 Loss Measurement of the 8-pole Motor ................................................................. 99 0 .
4.6.1 Description of the Motor ................................................................................ 99
4.6.2 Measwed Losses .......................................................................................... 100
4.6.3 Sirnuiated h n Losses .................................................................................. 102
4.6.4 Comparison of Iron Losses .......................................................................... 103
4.7 Lctsses Measurement of the Wole Motor .................................................... 1 0 4
4.7. I Simulatecl Iron Losses .................................................................................. 104
4.7.2 Cornparison to Measured Iron Lasses .......................................................... 105
4.8 Discussion ...........................................................................................................~ 107
Chapter 5 Mhhbt ion of Iron Losses in PM Synchronous mot ors...................^......... 109
5.1 Optimization of Electrical Machines ............... ................................................. 1 0 9
............................................................................................ 5.2 Design of Magnets 1 1 0
52.1 Magnet Edge Beveling ............................................................................ 1 1 0
................................................................................... 5.2.2 Shaping of Magnets 112
.......................................................... 5.2.3 Magnet Width and Magnet Coverage 113
5.3 Design of Slots .................................................................................................. 1 1 4
5.3.1 Number of Slots ................... ,,., ............................................................... 114
5.3.2 Slot Closure .................................................... .. ................................. 114
................................................................................................... 5.4 Number of Poles 115
5.4.1 Iron Losses and Number of Poles ................... .............. ............... 116
5.4.2 Torque and Nimiber of Poles ....................... .....~....................~............... 1 18
............................................................................. 5.4.3 Optimal Number of Poles 119
Chapter 6 Conclusions ....................................................................................~................. ..121
........................................................................... 6.1 Contributions and Conclusions 121
6.2 Future Work ...........................................................~...........................1.............. 1 2 4
LIST OF FIGURES
................................................................. Fig.2.l Rotor structures of PM synchronous motors 7
Fig.2.2 Magnetization models of permanent magnets ........................................................ 1 4
Fig.2.3 A linear PM machine for the-stepped FEM analysis ................................................ 16
Fig.2.4 Flrix distribution of the lin- PM syacbronous motor ......................... ............ 18
....................... .............. Fig.2.5 Cross-section with meshes of the 4pole PM motor ....... 19
Fig.2.6 Detatled meshes around the airgq of the 4-pole motor ........................................... 20
Fig.2.7 Flux distribution of the &pole motor .......................................................................... 20
Fig.28 Geometry, meshes and flux distribution of the 8-pole PM motor .................... ........... 22
....................................................... Fig.3.1 Change of flux in a tooth of the hear PM motor 28
....... ................... Fig.32 Wavefom of the normal component of tooth flux density .... 29
..................... Fig.3.3 Cornparison of tooth flux density waveform with that suggested in [8] 30
.......................................................... Fig.3.4 Approximation of tooth flux density waveform 31
................................... Fig.3 . 5 Approximation of the normal component of tooh flux density 34
Fig.3.6 Illustration of dot closure ............................................................................................ 36
............................. Fig.3.7 Effect of dot closure on tooth flux density at the center of a tooth 36
............................................... Fig.3.8 Tooth fiw density wavefom for différent slot closure 37
......................................................... Fig.3.9 Effect of dot cfosure on tooth eddy current l o s 38
Fig3.10 Waveform of the normal component of tooth flux density (q=2) ............................. 39
Fig.3.11 Wavefom of the normal component of tooth flux density (FI) ................... .., ...... 40
Fig.3.12 Tooth eddy current 10s vs . magnet width (q=2) ....................................................... 41
Fig.3.13 Configrnation of lhear motors performed by FEM, varying q and p ...... .. ............. -42
Fig.3.14 Tooth flux waveforms for different q with the same slot pitch when plotted as a
fimction of pole-pitch ..... , .......................................~............................................... .43
Fig.3.15 Tooth flux waveforms for diffefent q with the same dot pitch when plotted as a
.................... h c t i o n of dot pitch ..................................... .... Fig.3.16 Relationship ofkq vs . magnet thichess and airgap length ....................................... 47
Fig.3.17 Changing of the number of slots per pole per phase with h e d pole pitch ............... 48
Fig.3.18 Normal component of tooth flux density for different q with variable slot-pitch ..... 49
Fig.3.19 Longitudinal component of tooth flux density in the shoe area ................................ 52
Fig.320 Correction factor kc with regard to dot closure, airgap and tooth width .................. 53
............................ Fig.3.21 Radial component of tooth flux density of the 4-pole PM motor 5 5
................... Fig.3.22 Radial component of tooth ff wc density of the 8-pole PM motor .... -56
Fig.313 Waveforms of the longitudinal component of yoke flux densiSr (above a tooth) ..... 59
Fig.3.24 Waveforms of the longitudinal component of yoke flux density (above a dot) ....... 60
Fig.3.25 Approximation of the longitudinal component of yoke flux density ........................ 61 .......... ................... Fig3.26 Normal component of yoke flux daisity of the linear motor .. 64
Fig.327 Flux distribution in the yoke ..................................................................................... 65
................ Fig.3.28 Approximated normal flux density waveform in the yoke ... ............. 66 ....................... Fig.329 Normal component of yoke flux density as a function of yoke depth 68
................... Fig.3.30 Approximation of the c i rc~erent ia l component of yoke flux density 69
Fig.3.3 1 Yoke flux waveform of the 4-pole PM motor ........................................................... 78
Fig.3.32 Yoke flux wavefom of the &pole PM motor ........................................................... 79
Fig4 . 1 Closed loop control of a PM synchronous motor drive system .................................. 86 FigA.2 Equivalent Circuit of PM synchronous motors ........................................................... 87 Fig.4.3 Simplifiecl W e r function of a PM synchmnous motor ........................................... 89 Fig.4.4 Phasor diagrams of a PM synchronous motor ............................................................ -90 Fig.4.5 BIock diagram of the speed loop ................................................................................. 90
FigzQ.6 Shulated cumnt response of the 2.5hp motor with blocked shaft ............................ 92
Fig.4.7 M e a d current response of the 2.5hp motor with blocked shaft ........................... *.92
Fig.4.8 Shuiated speed response of the 2.5hp motor ............................................................. 93
Fig.4.9 Measined speed response of the 2Jhp motor ............................................................. 93 ................................................................................. Fig-4-10 SampIed phase cunent ..........,.. 95
Fig.4.11 FiIter design ............................................................................................................... 95
................... Fig.4.12 Measured phase voltage .........L.L..................................................... -96
................................. Fig.4.13 Measureù Ection and windage loss of the 8-pole PM motor 100
Fig.4.14 Measured iron losses of the 8-pole PM synchronous motor .................................. .IO1
.......................... Fig.4.15 Cornparison of calculateci and measund loss of the 8-pole motor 103
Fig.4.16 Cornparison of calculateci and measured losses of the 4pole motor ................... ... 106
Fig.S.1 The beveling of magnet edge .................................................................................... 111
............... Fig.5.2 FIwc distrilution of the motor with un-beveled and beveled magnet edge 112
Fig.5.3 PM rotor with beveled or curved magnets ................................................................ 113
.............................................................. Fig.S.4 Flwc distribution with different dot closure 115
LIST OF TABLES
Tabie 11-1 Parameters and dimensions of the linear motor ............................... ., ......... 17
.................................................. Table II-2 Parameters and dimensions of the 4-pole motor 1 9
Table II-3 Parameters and dimensions of the 8-pole motor ..................................................... 21
Table III-1 Distance needed for tooth flux to rise h m zero to plateau ...................... ..... 32
.......... Table III-2 Eddy currcnt loss induced by the normal comportent of tooth flux density 33
.............. Table III-3 Distance needed for the nomal component of tooth flw density to rise 43
...... Table III4 Eddy current loss in the teeth when changing the number of poles (j%OHz) 45
...... Table III4 Eddy current loss in the teeth when changing the number of poles ( ~ 6 d s ) 45
Table III4 Distance needed for tooth flux density to rise at constant dope .................... .... 49
............ . Table III-7 Tooth eddy cunent loss vs number of dots per pole per phase at 120Hz 50
Table III4 Tooth eddy c m t 10s vs . tooth width at 12OHz ................................................ S I
............... Table III-9 Eddy current l o s in the teeth calculated by FEM and proposed mode1 58
Table III-10 Distance x needed for yoke flux to rise ............................................................... 62
Table III-11 Eddy current loss induced by the longitudinal component of yoke flux
density ....................................................................................................................... 63
Table III42 Effect of slot closure on yoke eddy curent loss at 120Hk ................................... 71
........................... Table IiI-13 Effect of rnagnet coverage on yoke eddy curent loss at 120 Hz 72
TabIe III44 Effect of number of dots with fked pole pitch and variabIe dot pitch at
220HZ ...................................................................................*.................................... 73
Table III-15 E f f i t of magnet thickness on yoke eddy current loss at 120 Hz ......................... 73
Table III-16 Effect of airgap length on yoke eddy current l o s at 120Hz ............................... 74
TabIe III-1 7 Effect of tooth width on yoke eddy cunent Ioss at 120Hz .................................. 75
Table III-18 Test of comtion factor kr.. ............................................................................... .76
Table III-1 9 Eddy carrent l o s in the yoke cdcnlated by FEM and proposed mode1 ............. 81
Table III-20 Hysteresis losses calculated by FEM and by average flux density at 120& ...... 82
Table IV-1 T h e constants of the 8-pole motor ..................................................................... -91
................................................................ Table N-2 Measurements of the $-pole PM motor -97
............. Table IV-3 Ratings of the original induction motor and the PM synchronous motor 99
........................................... Table IV-4 Parameters needed to cdculate the iron losses (W) 102
Table N-5 Iron losses predicted by FEM for the %pole motor(W) ...................................... 102
Table IV-6 Iron losses predicted by simplined mode1 for the $-pole motor(W) ................... 102
................... Table IV-7 Ratù5gs of the 5hp induction motor and PM synchronous motor .... 104 ........................... Tabie IV-8 Parameten and coefficients needed to predict iron losses (W) 104
................................... Table IV-9 Iron losses predicted by FEM for the 4-pole motor(W) 105
Table N-10 Iron lossa predicted by simplifiecl mode1 for the Cpole motor(W) ................. 105 .............................. Table N-ll Measured mechanical losses ofthe 4-pole PM motor (W) 106
. .................................... Table V-1 Tooth eddy cment loss vs magnet bevehg at at 120Hz 111
Table V-2 Iron loss as a hction of number of poles .......................................................... 117
.............................................................. Table V-3 Torque as a firnction of number of poIes 119
Table V-4 Torque and efficiency as fmctions of the nrnnber of poles at 1800rprn .............. 120
LIST OF SYMBOLS
Fractional rnagnet coverage (pole enclosure)
Empincaily determined constants for hysteresis loss
Angle of magnet beveling
Abap lmgh
Power angle (angle between back emf E, and terminal voltage LI)
Angle between phase-a current and d-axis
Relative dot closure
Conductivity of the medium of electromagnetic field
Permittivity of the medium of elechomagnetic field
Permeability of the medium of electromagnetic field
Reluctivity of the medium of electromagnetic field
Pole pitch
Power factor angle (angle between phase-a current I, and terminal voltage U)
Slot pitch
Rojected dot pitch at the rnidde of yoke
Total flux linkage per phase
d-axis flux Iinkage
q-axis flux Mage
Angular speed
Anguiar n#luency of the magnetic Eeld
Voimne elecûical charge density
Vector magnetic potentid
Magnetic potentid of node k
Area of elexnent e in FEM
Vector flux density
Maximum flux deflsity in the iron
nns values of the airgap flux density
Plateau value of the circderential component of yoke flw density
Plateau value of the radiaUnormal component of yoke flux deLlSity
Plateau value of the radiai component of yoke flux density above dots
Plateau value of the radial component of yoke flux deasity above teeth
Plateau value of tooth flux density
Plateau value of yoke flux density
Constants represent mary boundary constraint values
Constants represent biaary boundary constraht values
Machine constant
Machine constant
Electrical flux dmsity
Slot height
Yoke depth
Yoke depth of 2-pole motor configuration
Subscript for element e
Electrical field density
Total number of elements of the stator core in FEM
Induced back emf
F=Pency
Magnetic field mtensity
i, jt m Subscripts for nodes i, j, m of eiernent e
1, Phase-a current
Id d-axiscurrent
Iq q-zaiscunent
Electrical density
Inertia
rms values of the linear current density on the stator inner periphery.
Correction factor for effect of circumferential component of tootii flux density
EmpincaUy determuled eddy current l o s constant
Empiricaiiy determined hysteresis 10s constant
Gain of current PI
Magnetic usage
Gain of speed PI
Correction fxtor for effect of magnet thickness and airgap length
Correction factor for effect O E circumferential component of yoke flux deTlSi@
Torque constant
Ratio of yoke eddy current loss of radial component to circumfixential component
Unary boundary enclosed by S
Binary boundary enclosed by S
Stator stack length
d-axis inductance
q-axis inductance
Magnet length
Nurnber of phases
Modulation ratio
Speed
Subscripts used for time step n in t h e stepped FEM
Mitlcimum operation speed
Total -ber of dots used in h e stepped FEM for halfperîod
Nmnber of poIes
Coppa loss
Eddy curent loss density
Tooth Eddy current 1 0 s
Pa= Eddy current loss induced by circumferentiai component of tooth flux deaSity
P, Eddy cumnt loss induced by radiaYnomia1 cornponmt of tooth flux density
Peam Eddy current l o s induced by radïaVnormai component of tooth flux density
calcuIated by Approximation mode1
PetrFEM Eddy cunent loss induced by radiaUnorma1 component of tooth flux density
calculated by FEM
Pq Yoke eddy cunent loss
P , Eddy c m t loss induced by circionferential component of yoke flux density
PWFm Eddy ctrrrent loss induced by circumferential component of yoke flux density
caicuiated by FEM
Eddy current loss Înduced by normal component of yoke flw density
Friction and windage loss
Hysteresis loss density
Hysteresis loss of the teeth
Hysteresis loss of the yoke
Total iron Ioss density
Output power of the motor
Number of dots per pole per phase
h e r radius of stator
Stator phase resistance
Out radius of stator
Magnetic field region
Number of stator dots
T h e
Time interval used in timc stepped FEM
Rise time of tooth or yoke flux density
Period
EIectricai time constant
Samphg paiod
T h e constant of cunent PZ
Mechanical time constant
T h e constant of speed PI
Torque
Phase voltage
DC-iink voltage
Line-üne voltage
Total volume of stator teeth
Total volume of stator yoke
Total number of tums of stator winding
Magnet width
Slo t openings
Tooth width
Subscripts for x and y component respectively
Distance needed for flux density to rise h m zero to plateau
d-axis reactance
q-axis reactance
Base reluctance
Chapter 1
Introduction
1.1 Thesis Objectives
The main objective of this thesis is to develop a simpüfied mode1 to evahate iron
Iosses in surCac~moanted permanent mapet (PM) synchronous motors.
PM synchronous motors have higher efficiency than standard induction machines with
the same power rating and the same h e . However, iron losses in PM syachronous machines
form a larger proportion of the total Iosses than is usud in induction machines. This is partly
due to the etimination of signincant rotor losses and partly due to their operation at near unity
power factor. Thmfore, accurate evaluation and mhhization of iron losses is of particular
importance in PM synchronous motor design [Il. This thesis addresses this issue with partidar
emphasis on the eddy current loss in the stator teeth and stator yoke.
Iron losses m electricai machines are usually predicted by assuming that the magnetic
flux in the magnetic core has only one aiteniating component that is sinusoicial. The Uon losses
evaiuated in this appmach may be 20% lower than the me- values and the discrepmcy is
even Iarger in PM machines [2].
An alternative approach considers the harmonies of the magnetic flux density and
obtains the SUI[L of losses of each component [3-q. The h i t e element method (FEM) is ofien
used to obtain the field distnibution in the motor. Since the FEM is considered as a precise tool
in the caldation of electromagnetic fields, it is cornmody accepteci that a good estimation of
b n losses can be achieved with this approach [7]. However, this approach needs high effort for
generai use in most designs and is not practicd durhg the preliminary design iteration stages.
An d y t i c a i model developed by Slemon and Liu, uses a simple approach to estimate
the iron losses in PM synchronous motors [a]. It assumed that: (1) the tooth and yoke flux
waveforms are trapezoidal with regard to the; (2) the eddy current l o s is proportional to the
square of t h e change rate of the magnitude of flux density. While the overall application of
these approximations produced an acceptable prediction of the total measured iron losses in an
experhental machine, the validity of each approximation remained in doubt.
Deng [9] and Tseng [IO] evaluated the Uon losses of PM synchronous machines in a
similar procedure to [8]. The geornetrical effects were also ignoreci and only the magnitude of
magnetic flux density was used to perform the calculation.
Therefore, there is need of a verified simple approach to evaluate iron losses in PM
synchronous motors for use in the motor design studies including detailed geometrical effects
of the design. This simplified iron loss model should be able to reliably predict the imn losses
of PM synchronous motors but avoid the high effort of FEM tools. The sirnplined
approximation iron loss model will be developed and validated based on these guidelines.
B a d on the pmposed model, design guidehes will be provideci to design low loss PM
synchronous motors including geometrical effects.
To cornplexnent the main objective, the thesis will also mvestigate the measurement of
iron losses of PM synchronous motors.
Loss measurement m eiectrical machines has been a niffidt task for the electricai
mdustry [Il]. There is no effective app~oach to measme the iron losses in PM machmes reiiably
to vaify differaces in designs so fa, 112-141.
Measurement of iron losses involves the decomposition of no-Ioad losses of a PM
synchronous motor, since iron losses and mechanicd losses in a PM synchronous motor are
dways present simuitaneously.
Although no-Ioad loss of a synchronous machine can be m e a d when the machine
is driven by a dynamometer, it is more desirable to measure the losses when the machine is
operafed as a motor. As PM synchronous motors are usually designeci without damping, a close-
loop controlled inverter is n e c v to nm the motor. Therefore, the measurement of losses also
involves the mtasurement of pulse-width-modulated (P WM) wavefoms.
The theoretical and experimental shidies are focused on two PM synchronous motors
with dace-mounted magnets.
1.2 Thesis Outline
This thesis contains six chapters. The contents of the thesis are summarized below.
In Chapter 2, the evaluation of iron losses in PM machines is reviewed and the tirne-
stepped FEM is introduced. FEM andysis is perfiomied on three PM synchronous motors
includmg two existing PM motors. Equatiom are developed to calculate iron losses using the -
stepped FEM.
In Chapter 3, simplifieci Boo loss models are developed In order to achieve the
simplined models, the proposed andytical mode1 in [8] is reviewed and tested with the FEM
for different cases. A refiaed simplified tooth eddy cunent loss mode1 is developed taking into
account the detaüed geometricai shape of diffierent desip . The effects of magnet width, slot
closme, number of dots per pole pet phase, airgap length, tooth width and magnet thickness are
studied. A correction factor is mtroduced to reflect the geometricd innuences. Another
comction fictor is intmduced to reflect the iron Ioss induced by the c i r c u m f d a i component
of the tooth flux density.
A new simplifïed yoke eddy cunent loss model is developed based on the two
orthogonal components of yoke flux density. The wavefom of the circurnférential component
of the yoke flux daisity can be appmràmated by a trapezoidal wavefom. The waveform of the
normal component of the yoke flux density is found to have the same shape as that of the
normal component of tooth flux d d t y . The two components of yoke flux density are related
to each other through the dimensions of the yoke.
Some examples are tabulated for each of the simplined models to demonstrate the use
and validate the effectiveness of the mdel. These models are simple formula for the machine
designer to calculate iron Iosses in PM synchronous moton.
Chapter 4 investigates the meamrement of iron losses in PM synchronous motors.
Measurements of iron losses are performed when the motor is driven by a dynamometer and
when the rnotor is fed by an inverter. Separation of iron losses and mechanical loues is also
investigated. The experimental work of two PM synchronous motors is presented here. The
cornparison between the measured iron losses and the predicted iron losses confhed the
validity of the pmposed iron loss model. Despite the difficulties to measure iron losses, the
experiment provides an indication of the useflilness of the proposed model.
Chapter 5 proposes effective ways to minimize iron losses in PM synchronous motors.
A series of approaches are provided to minimize iron losses of PM synchronous motors
Finaiiy, the conclusions and contributions of the thesis are presented in Chapter 6.
Suggestions for futlire studies are also included
Chapter 2
lron Losses and Time-Stepped FEM of PM
Synchronous Motors
2.1 PM Synchronous Motors
A typical PM synchronous motor dnve system consists of a PM synchronous motor, an
inverter, a shaft encoder and a rnicrocontrolIer.
A PM synchronous motor has a sirnilar str~chire to a DC excited synchronous motor
except that the magnetic field is supplied by PM materiai in a PM machine.
With the latest developrnent in rare-earth PM materid, it is aow possible to develop
high efficiency, high power factor, and high power density PM motors for industrial and
residential piirposes. The modem r a r e 4 PM materiai, neodymium-iron-boron (NdFe-B),
has 1.4T of cesidual flux, I I k A h of coetcive force and 13.6kAlm intrinsic coercive force. The
dropping cost and lower temperature sensitivity have made Nd-Fe-B an ideai material for
developing high performance PM synchnous motors.
The stator structure of a PM synchronous motor is simi1ar to that of an induction
machine. Dependhg on Ïts application, the windmg distniitxtiotl could be near shusoidal or
trapaoidal. It is preferable to build the PM motor windiag with a nmr sinusoida1 distn'bution
if a smoother torque is desired at low s p d In order to avoid cogging torque, skewed stator
slots may be used.
The rotor structure of a PM synchronous motor can be built either with dace-mounted
magnets, inset magnets or circUIILferentia1 rnagnets. Typicai rotor structures are shown in
Fig2.l.
Among the major types of rotor structures, rotors with surface-mounted magnets as
s how in Figl.l(a) are commonly used in PM synchronous motors for their simplicity. The
drawback is that the magnets may easily peel off fiom the d a c e if they are orighally glued
on the rotor surface. Rare-earth PM materials are hgiie and rust and scratch easily. Therefore,
the Life span of the magnets is reduced if they are exposeci directly to air. The author has
experienced s e v d times the magnets Qing off the rotor surfaces during the operation of PM
motors. One e f f i v e but costly way to avoid the mechanical weakness of smfhce-mounted PM
synchronous motors is to bond the magnets with a cylindricai sleeve made of high strength d o y
as shown in Fig.2.l (b) [15-171.
Rotors with inset magnets as shown in Fig.Z.l(c) can provide a more secure magnet
setting. It cm dso produce more maximum torque due to its unequai d-axis and q-axis
reluctance [18]. Both the dace-mounted and the inset rotor stmctures Fig.2.l(a) and (b)
require arc shaped rnagnets which greatly inmeases the manufacturing cost. Segmented
rectanguiar strips may be used to reduce the manuficturing cost.
PM motors with magnets burieci just below the surface have a mechanical advantage
over Sltrface mounted PM motors especidy at high speed. Although the possible use of
rectangular magnets eases the manufacture of magnets, the manufacture of rotor laminations is
more complicated than those for d a c e momted magnets.
It is possîle to build the motor with cin=urnférentially magnetized buried magnets as
shown in Fig.2.1(d). The o v d cost can be significantly reduced by using rectanguiar shaped
magnets. The chmnfîerential structure is normally srntable for machines with six or more poles.
(a). Surface-momted magnets (b). Surfaced rnagnets with rotor sleeve
(c), Inset magnets (d). CircUILlferentiaI magnets
1-magnet 2-core 3-shaft 4-non-magnetic matenal %rotor sleeve
Fig.2.l Rotor structures of PM syncbronous motors
Iron Losses in PM Machines
According to IEEE Std 115-1995 [35], the loues of a synchronous motor are
decomposed into the followhg indMdual components: mechanical l o s (fiction and windage
10s); iron los (open circuit c m los); stray-load los (short circuit core los); stator copper los
and fîeld copper loss. In the case of a PM motor, there is no field copper loss.
The mechanical loss is due to the bearing friction, windage on the rotor, ventilation fan
and coohg pump. These losses are constant at constant speed and vary in direct proportion to
the changes in speed. The iron loss is due to the change of magnetic field in the stator and rotor
core. It usuaiIy decomposed into hysteresis loss and eddy cumnt 10s. Hysteresis l o s is due to
the existence of a hysteresis loop in the core B/H characteristics. Eddy cment loss is due to the
currents induced in the core material, which is also a conductor of electricity. The stray-load is
due to the armature reaction, flux leakage and mechanical imperfections of the motor. The short-
circuit cort loss of a PM synchrmous rnotor includes the extra copper loss (eddy current loss
in the stator winding), the harrnonic iron loss in the teeth and a loss in the end structure of the
machine. The copper l o s is due to the current flow in the winding. It increases slightly with
load due to the skin effect and eddy current in the winding.
The calculation of coppa losses is relatively simple and accurate. The prrdiction of
mechanical loss is usualiy predicted by cornparing the losses with that of a simiIar machine. The
calculation of iron losses and stray-load loss of electrical machines is fat f h n complete. The
procedure sttggested in [19] has been used for decades to calcuiate the stray-Id loss of large
motors. For smaU to medium size electrical machines, the usual practice is to assume a
percentage of the total input power as the stray-load loss [20,49]. In classical motor designs,
the iron losses are viewed as behg caused d y by the fimdamentai firesuency variation of the
magnetic field [21]. The measured Bon losses in electncd machines are usually much larger
than that calculateci h m the Smusoidai assumption. The thesis wüi be focuseci on the evduation
ofiron losses of PM synchronous motors.
Massive experïmental d t s show that the iron losses CO& of two componaits: one
component is proportional the kquency and the other is proportional to the square of the
fkquency. The former is usually Iinked h y s t d s loss and the latter is usually M e d eddy
current Ioss,
For a sinusoiciai flux density, over the working range of fhquency, the total iron losses
can be represented by the sum of the hysteresis loss (proportional to the fkquency) and eddy
current loss @roportionaI to the square of kquency):
where k is the maximum flux density in the iron, ph, is the total iron loss density, ph is the
hysteresis loss density andp, is the eddy current loss density; os is the anguiar fkquency of the
magnetic field; Rh, k, and P are empirically determined constants which can be obtained by
curve fitting h m m a n u f ~ s data measured with sinusoidal flux density. Typicai values for
silicon Uon material fiequently used in induction machines, with the stator fkquency given in
ruus, are kh=44-56, 84.5-2.5, k d . 0 1-0.07.
An expression for the classical eddy cment loss can also be developed based on the
resistivity of the core material [21]. The resuit is g e n d y fond to be less than that h m the
second term of (2.1).
In dàced-mounted PM synchronous machines, the iron Iosses are mainiy in the stator
core. The flux density waveform m the stator core is neither d o m nor sinusoicial. This non-
sinusoicial waveform of flux density doesn't affect the hysteresis loss provided there are no
minor Ioops, but is critical to the eddy ~ ~ l ~ e ~ l t Ioss. It is therefore desirable to restate the eddy
current loss density formula in the foilowing format for generai fiux waveforms [8,21]:
In (23, the tEne derivative of vector mot density 6 is used to evaluate the instantaneow
eddy c m t loss densityp,. It also shows that the kon losses are induced by the field variation,
whether it is pulsating or rotating [29]. The eddy current loss is related not oniy to the
magnitude of the fiw density, but also to the way in which each of the two orthogonal
components of the flux density changes [21-221.
For periodic variable fields, the average eddy current loss density can be obtained by
integrating (2.2) over one period:
2.3 Finite Element Method of Electromagnetic Field Problems
Although it nquires high effort for general use in PM motor designs, tirne-stepped FEM
n m a h the most powerfil tool to calculate electromagnetic field distributons and field-related
parameters [26] .
T h e stepped FEM is aiso used as an effective tool to verify loss calculation based on
Sunpier loss models [9-1423-241, as it is economicaiiy and technicalIy impractical to verify ai l
loss predictions with experiments.
T h e stepped FE. will be used as the tool to develop the simplified loss models and
to validate the proposed Von loss models in this thesis.
2.3.1 Maxweli's Equations
AU electromagnetic phenornena are govemed by M d 2 's Equations. The differential
form of M d 2 's Equations cm be expressed as:
where E is the electncai field intensity, D is the electncal flux density, H is the magnetic field
intensity, J is the eloctricd current dens=ity, p, is the volume electricai charge density.
For isotropie medial material, the constitutive equations to MaxweIi's equations are:
where p, a, and E are the permeability, conductivity and pemiittivity of the medium of
electromagnetic field respectively.
MaxweIl's equations c m be fûrther simplifieci if the dimension of the current carrying
system is mal1 when compareci with the wavelength in fke space. The displacement current
is zero and (2.5) can be re-written as:
2.3.2 The Magnetic Vector Potential
A fimrlamental redt of vector andysis is that the divergence of the curl of any twice
differentiable vector A vanishes identicaily:
Comparing (2-12) to (2.6), it can be concluded th&:
where A is referred to as magnetic vector potential. Of course, the vector potentid is not fWy
dehed by (2.13) because the Helmholtz theorem of vector analysis States that a vector is
uniquely dehed if and only if both its curi and divergence are known, as wekl as its value at
some space points. A cornmon choice for the divergence of A is know as Cmlomb convention,
which is generally used for static field problems and low fkquency electmmagnetic problems:
2.3.3 Two Dimensional FEM Formulation of Magnetostatic Problems
The magnetic field of a no-load PM synchronous motor c m be cousidered as a
mapetostatic problem. In solving this two-dimensionai non-Linear magnetic field problem, the
laminated material can be assumed to be isotropie in the radiai direction. Therefore, substituthg
(2.13) to (2.1 1) leads to:
where v is the magnetic reluctivity of the medium:
Therefore, the two-dimensional non-hear partiai différentia1 equation for a
magnetostatic problem can be derived h m (2.15). In region S enclosed by unary boundary I I
and binary 12, the differential equation is:
where CI and C2 are constants represent tmary and binary boundary constraint values
The field problem of (2.17) cm be expressecl in variational terms as an energy
functiond; the miniminne of which yields the requUed solution:
where
For the first order ûiangdar finite elements of unrestncted geometry, the potaitial
solution is defineci in terms of shape functions of the tnangular geometry and its node values
of potential.
The minhhtion of the fimctionai (2.1 8) is pdormed with respect to each of the nodal
potentials:
When the minimiriltion descrîhed by (2.21) is cmied out for di the tnangies of the field
region, the following matrix equation cm be obtaineù. By solving the following equation, the
unknown vector potential cm be determined.
where [m is a nonlinear symmetric m e [Pl is the excitation matrix. Nkwton-Raphson
iteration cari then be used to solve the above nonlinear equation,
2.3.4 Magnetization Models of Magnets
There are two models commonly used to represent permanent magnets: a magnetization
vector and an equivalent current sheet These two methods have different starting points but lead
to the same set of equations [25]. In the FEM package (MagNet5.1 h m uifolytica Co.) used
in this thesis, the equivalent cunent sheet is used to represent the magnets [26].
With an @valent cunent model, the magnet is represented by a thin Iayer of elements
with current dong the sinface of the magnets. For a rotary PM synchronous motor with arc-
shaped suffie-mounted magnets, there are two possibilities of magnetization for the magnets:
radial magnetization and paraiîel magnetization. The representations are shown in Fig.2.2.
(a). Radial magnetization (b). Parallel magnetization
Fig.2.2 Magnetization models of permanent magnets
It has been shown that the usage of PM material in parallel-magnetized d a c e mounted
machines is l e s than that wah radial magnetization [271. The magnetic usage can be written as
where a is magnet coverage (-1) andp is the total number of poles.
Somehes, rectangular shaped magnet strips are used to replace an: shaped magnets to
reduce the mimufacbiiring cost In this case, mdividual magnet strïp is parallel magnetized while
each of the strips has a different orientation of magnetic field.
In this thesis, one of the experimentd PM motors (&pole) is p d e l magnetized while
the other one (4-pole) is radidy magnetized
2.4 TimeStepped FEM of PM Synchronous Motors
The iron losses and the flux detlsity waveforms in the con of a PM synchronous motor
c m be obtained by the-stepped FEM [SI.
In the analysis of the-stepped FEM, it must accommodate the motion of the rotor or
the stator. This cm be accomplished by developing compatible mesh structures on stator and
rotor respectively and shitting them to simulate the actual motion. At each time step, FEM itseif
is pedionned on static electromagnetic fields. The s W g of meshes simulates the a c t d motion
of the machine.
2.4.1 Constmcüon of Meshes
The PM motor contains a standstiIl stator and a moving rotor. In establishg the meshes
for the anaiysis, the rotor shouid be moved and positioned at each time step such that it does not
disturb the integrity of the me& struchue as it moves. The initial meshes of the stator and the
rotor are geneiated nich that haif of the air gap belongs to the stator and the other haif to the
rotor. The stator mesh and the rotor mesh share the same boundary at the middle of the air gap.
The b e r stator circurnfîerence at the air gap and the outer rotor circderence are divideci into
qua1 steps so that their nodes coincide. To provide movement of the rotor, the tirne step is
chosen so that the angle or length of each step is e q d to the intmal between two neighboring
nodes dong the middle of the air gap. Because ofpexiodicity of the magnetic field, only one pair
of poles need be modeled Similariy, because of the halfwave symmetry of flux density, only
a halfperiod need be calculated.
2.4.2 Boundary Condiüons
There are three types of botmdary constraints in electromagnetic problems. The first
type of botmdary is described as no constraiirts. In which case, the boundary condition is
aA assumed as satîsfying- = O. This is the naîural botmdary condition which wiii be satisfied a automatically by FEM if the potentiai of a boimdary node is not specined.
The second type of boundary is called unary botmdary on which the potential of each
node is specifïed as a constant.
The third kind of boundary is called binary boundary- In this case, it implies that the
potentids on one side of the region are equal or opposite to the potentids on the other side of
the region. This is the case when oniy one pole or a pair of poles of the motor is modeled An
important practicai point is that the model must have an equal number of aodes for binary
constrziints*
If one pair of poles is chosen as a FEM model, the boundaries can be summarized as
shown in Fig.2.3 for a hear PM synchronous motor &SM) with longitudinal laminations.
In Fig2.3, the outer boundaries of the stator Zr and the rotor Zz are unary bomdaries.
Boundaries (YI , Yl ') and (Y3, Y3 ' ) on the cross sections of the LSM are binary boundaries,
Along the middle of the air gap when the rotor is moved out of stator, (Y2 , Y2') are also binary
boundaries, where the dots simulate the moving of meshes.
stator
Y3 rotor
- -
&
Fig.23 A hear PM machine for the-stepped FEM analysis
f6
2.4.3 Examples Used in the Thesis
Three PM synchronous motors were used to evaIuate iron losses and validate the
simplified models. The linear motor is a pure design example and the 4-pole and 8-pole PM
motors are existing motors.
Case I: A Linear PM Synchmnous Motor
FEM analyses were perfonned on a simple idealized linearly arranged 3-phase PM
synchronous motor with longitudinal laminations (the ünear motor) with dimensions given in
Table II- t .
Since a LSM with longitudinal laminations is ais0 caiied a longitudinal flux LSM (the
flux lines lie in the plane which is parailel to the direction of the travelling magnetic field),
"longitudinal flux" is refemd to the flux which is parailel to the travelling magnetic field.
SimiIarIy, 'Normal flux" is r e f d to the flux which is pqendicular to the travelling magnetic
field or the movement of the rotor.
In rotary machines, the circudierential component (equivaient ?O the longinidinai
component in linear machines) and radial (or normal) component will be used to represent the
two orthogonal components of flux density.
The flux distrïïution of the linear PM motor of Fig.2.3 is shown in Fig.24. The
dimensions of this motor will be varied in Chapter 3 to test the geometric effect on iron losses.
Table II-1 Parameters and dimensions of the linear motor
Fig.2.4 Flux distniution of the h e a r PM synchronous motor
F ig24 shows that the magnetic field in the stator teeth over the magnets is essentiaily
rmiform and in the normal direction. The flux density in a tooth is essentially zero when it is
above the middle of the space between the two magnets. The flux density is constant when it
is M y covered by the magnet
In the bottom ngion of the teeth where the magnetic flux passes dong the teeth and
emerges into the yoke, the flux tums from the normal direction into the longitudinal direction.
As the magnetic flux penetrates fiirther into the yoke, the longitudinal component begins to
dominate and cm be regarded as being essentiaily W o r m m the Iongitudinai direction. in other
words, in the yoke where it is above the space between the magnets, the flux density is mainiy
in the longitudinal direction. In the yoke where it is above the magnets, the flux density is
mainly in the normal direction. The longitudinal flux is seen to be uniformly distniiuted over
the thickness of the yoke whiie the nomal flux is not.
Case II: A 4-Pole Shp PM Synchronous Motor
As a second example, an existing 4poIe, 5hp, 1800rpm, surface mounted PM rnotor
(the +pole motor) with dimensions shown in Table II-2 was simuiated through time-stepped
FEM. A cross-section of the motor dong with its mesfies is shown m Fig.2.5. A detailed mesh
around the airgap area is shown in Fig2.6. The field distribution of the motor during one step
of the the-stepped FEM is shown m Fig2.7.
Table II-2 Panmieters and dimensions of the 4-pole motor
1 Slots per pole per phaseq 1 3
1 Magnet thickness 1, 1 6.3~10*~3m
Fig.2.5 Cross-section with meshes of the 4pole PM motor
Slidmg Mface
Fig.2.6 Detailed meshes around the aîrgap of the 4pole motor
Fig.2.7 Flux distri'butÏon of the 4poIe motor
20
The flux patterns in Fig.2.7 shows that the magnetic field in the stator teeth of a d a c e -
mounted PM motor is roughiy Miform and is mainly in the nomal direction. The fiux density
in a tooth remains near zero whm it is not above the magnet. The flux density in a tooth reaches
maximum when it is above the maguet
The magnetic flux is putially in the chumferential direction in the shoe area of the
teeth. In the region ncar the bottom of the teeth whert the magnetic flux passes dong the teeth
and emerges into the yoke, the flux tums h m the radial direction to the circulzlferential
direction. As the magnetic flux penetrates M e r into the yoke, the circumferential component
begins to dominate and cm be regarded a s being essentially uniform in the circUIllferentia1
direction. The cin=umferentiai flw seems to be d o d y distriiuted over the thickness of the
yoke.
Case III: An &pole 2.5hp PM Synchronous Motor
A third example is another avdable PM motor (the bpole motor) with dimensions
given in Table II-3. It is an &pole, 2.5hp. 18ûûrpm, suface mounted PM motor. A cross section
of the motor dong with meshes and flux lines is show in Fig.2.8.
Table II-3 Parameters and dimensions of the 8-pole motor
Fig.2.8 Geometry, meshes and flux distribution of the 8-pole PM motor
By observing the flux patterns in Fig.2.8, it can be concluded that the magnetic field in
the teeth is mainly m the normal direction. Smce this motor has only one dot per pole per phase,
no tooth has zero flux during the rotation of the rotor. The flux in the tooth reaches m e u m
when the tooth is above the middie of the magnet-
The magnetic flux is also m d y in the circumferential direction m the shoe area of the
teah. In the yoke where it is above the -et, the flux density is rnaidy in the radial direction
In the yoke where it is above the space between the magnets, the flux density is mainly in the
circrrmferential direction-
2.5 Evaluation of lron Losses with FEM
2.5.1 Eddy Current Loss
The eddy current loss is induced by the variation of magnetic Qw, whether it is
pulsating or rotating. The flux density in the core is usually decomposed into two orthogonal
components and the eddy current loss can be obtained by considering the eddy current loss
contributecl by each component [2, 28, 29,321. If the magnetic field is periodic in the time
domain, the eddy current loss density can be derived h m (2.2):
Eddy cumnt loss cm be obtained by discretking (2.24) in the t h e domain when
performing FEM. If the two orthogonal components of the flux density of element e in the stator
iron at tirne butant t, are represented by Ban, Ban, the square of the derivative of vector flux
density B can be written as:
where N is the total nrmiber of steps used for the-stepped FEM aualysis in a haifperiod.
When performing thne stepped FEM, the whole stator is ùivided into a nrmiba of
elements. The flux density of each elment at each time instant can be determined. If or@ a half
period is analyzed, the t h e interval between two consecutive FEM steps is
where f is the firesuency of the magnetic field
Substitutmg (2.25) and (226) into (2.241, the total eddy current loss cm be expressed as:
E N N
= 4pNkelfi f C 'e [z (&c - Bcr,n-,)' + C (Bq,,, - &a-r)'] (W) (2.27)
where A, is the area of elment e, E is the total number of elements of the stator core and is
the stack length of the stator.
2.5.2 Hysteresis Loss
Assuming no minor loops, hysteresis loss is proportional to the quasi-static hysteresis
loop area times the hquency and is therefore not flected by the shape of flux density. For a
given material, once the peak flux deasity of each element has been found, the hysteresis loss
cm be obtained. Hysteresis loss can be appmximated h m experirnents by [22]:
By performing the-stepped FEM anaiysis, the peak value of flux density Be,- d u ~ g
a period in each element can be found. The hysteresis loss in the stator is:
2.5.3 Total lron Losses
The total h n losses in the stator are the sum of hysteresis loss and eddy current los:
Actual iron losses can be predicted for different regions of the motor using FEM. The
d t s can then be used to develop and validate SimpIified iron l o s modeis.
Chapter 3
Simplified lron Loss Model
Although time-stepped FEM can provide good estimations of iron losses, time-stepped
FEM itselfranains a culnbersome tool and very tirne-coflsuming for most designs and partidar
for optimi7ations. FEM anaiysis cm only be perfomed after the machine geomeûical
dimensions have been determined. Repeated FEM is not practical for the p r e b a r y design of
an electricai motor with its many iterations in geometry.
In contrast, approximate anaiyticd loss models yield quick insight while rquiriag much
less computation time and are d y avaiiable during the prelimlliary stage of a motor design,
Therefore, it is desirable to develop iron loss models that will produce an acceptable prediction
of overall iron losses in a relatively simple way and provide machine designers with a more
diable method to estimate iron losses in the initial design of PM machines.
In this chapta, the h n losses ofa PM synchronous motor will be decomposed into four
components to develop simplined models respectiveiy: tooth eddy current loss, yoke eddy
cumnt los, tooth hysteresis 10s and yoke hysteresis 10s. SÏmpMed modeis for the estimation
of tooth and yoke eddy c-t losses are deveIoped based on the observations of FEM d t s
and a d f i c a l analyses of the magnetic field taking into accoimt detailed geometncal effects.
The assumptions made in [8] will be examined. The effiveness and limitations of the madels
win be imrestigated. Correction factors wiIl be derived for geometricai infiuences and the losses
indaceci by the miwr component of the flux density.
3.1 Review of Analytical lron Loss Model
A simple andyticd iron loss model, developed by Slemon and Liu [8], has been used
by a few authors to detennuie the bon losses of PM synchronous motors [9-10,30-321.
In the prelunlliary study on iron losses of d a c e mounted PM motors in [8], a number
of tentative conclusions were derived:
The eddy current Ioss was assumeci to be dependent on the square of tune rate of
change of the flux deosity vector in the stator core.
The tooth eddy current loss was assumed to be concentrated in those teeth which are
near the edges of the dace-mounted magnets and thus was independent of the
anguiar width of rnagnets.
The flux deflsity in a tooth was assumed to be approximately uniform.
As the magnet rotates, the flux density in a stator tooth at the leading edge of the
magnet was assumed to rise linearly h m zero to a maximum and then remain
e s sdd ly constant while the magnet passes. At the Iagging edge, the tooth flux
drops h m maximum to zero in the same pattem.
The rise or fd time of the tooth flux density was assumed to be the time mtmal for
the magnet edge to traverse one tooth width.
For a given torque and speed rating, the tooth eddy current l o s was stated as
approximately proportional to the number of poles divided by the width of a tooth.
Alternativeiy, the tooth eddy cnrrent loss was approximately proportional to the
product of poles squared and slots per pole per phase.
For a given ftesuency, the tooth eddy cmrent Ioss was stated to be proportionai to
the number of slots per pole per phase.
The eddy current loss m the stator yoke was approximated using ody the
circUILlfereatid component of yoke flux density.
m e the o v d application of these approximations produced an acceptable prediction
of the total measured losses in an experimental machine, the validity of each individuai
approximation remaineci in doubt. The uncertainties in these models are (1). The validity and
accuracy of the derived flwc waveform, both in the teeth and in the yoke. (2). The mor caused
by oniy using the magnitude of the flux density. (3). The errer caused by neglecting geometry
effects on flux waveforms and eddy current loss of the motors.
In this chapta, these assumptions are examinecl in tum and their resuits are compared
with those obtained by tirne-stepped FEM.
3.2 Test of Linear Waveforrns of Tooth Flux Density
In order to examine and refine the assumptions made in [8], the tooth flux density
waveforms wen obtained by time stepped FEM on the linear PM motor shown in Fig.3.1. The
tooth marked I in Fig.3.l(a) is taken as an example to obtain the tooth flux density waveform.
The flux density in the tooth is obtained as a function of distance x, where x is defineci as the
distance h m the middle of tooth to the middle of the space between the two poles. It can be
seen h m Fig.3.1 that:
The flux density in the tmth rem& zero when the twth is above the midde of the
space between the two poles (A) as shown in Fig.3.l (a).
The flux increases when the h n t edge of the magnet appmaches the tooth ( d 6 )
as show in Fig.3.l(b). The flux keeps mcreasing when the magnet edge passes the
tooth (4) as shown in FigAl(b) and FigD3.l(c)
The mot approaches a ma7cimum when the h n t edge of the magnet Ieaves the tooth
(d) as shown Fig.3.l (c).
The flux in the tooth remains constant when it is above the magnet (d) as shown
in Fig.S.l(d)
(a). When the tooth is above the middle of the two poles (A) l I
I i A 6
@). When the mapet reaches the middle of the tooth
I &Y (c). When the tooth is covered by magnet
I d 2 i (6). When the tooth reaches the middle of the pole
Fig.3.1 Change of flux in a tooth of the lin= PM motor
The dcuiated wavefom by FEM for the normal component of the flux density in the
center of a tooth is shown m Fig.3.2. It cm be seen h m Fig.32 that the rise of the flux density
faiiows almost a linear pattern except at the begirmmg and end of the change.
Fig.3.2 Waveform of the normal component of tooth flux density
If the dope of the linear part of the flux density waveform is taken and extendeci fÎom
zero to maximum flux density as show in Fig.3.2, the distance needed for the flux density to
rise is 0.142 pole-pitch or 0.852 dot-pitch.
ùi [8], it was suggested that the Ne of the tooth flux density takes one tooth width or
0.5 dot-pitch for configurations with equal slot and tooth width. The approximated flux density
wavefom saggested in [8] is compareci with the flux density waveform obtained by FEM as
shown in Fig.3.3. It can be seen that the rise of flux densîty takes much longer t h e than
suggested in 181.
Although the hear approximation in Fig.3.2 better refiects the rise pattem of the tooth
flux density, the cddated tooth eddy current loss by this hear approximation is expected to
be Iarger than tbat caIcuiated by FEM because the acttial flpx rises slower at the beginning and
end of the change.
Fig.3.3 Cornparison of tooth flux density waveform with that suggested in [8]
Fig.3.4 introduces the assumption that the observed flux density waveform fkom FEM
rises in appmximately the time needed for the magnet edge to ûaverse one slot pitch in contrast
to a tooth width as assumeci in [8]. The assumption can be tested as foiiows.
Suppose that the waveform of the normal component of the tooth flux density can be
represented by a hear approximation as shown in Fig.3.4. This hear approximation pmduces
the same eddy current Ioss a s the actual flux d e t y wavefom. The rise thne of flux density for
this approximation can be obtamed by comparing the eddy current l o s calcuiated by FEM and
the eddy current 10s calculated by the linear appmximation.
The rise time of the linear approximation is At and the change happens four tirnes p a
period. The eddy current Ioss induced by the normal component of this approximation is:
where Y, is the volume of the teeth, Bth is the plateau value of the normal component of tooth
flux density, which is essentiaily the same as the magnitude o f the tooth flwt density.
A conclusion can also be drawn h m (3.1) that the eddy current l o s is inverseIy
proportional to the rise t h e o f the hear approximated flux density.
Dis tancelpole pitch
Fig.3.4 Approximation of tooth aior density wavefonn
The rise t h e for the hear flux can be determitmi by (3.1) using the eddy current 1 0 s
caicuiated by FEM:
where P*J= is the eddy current 10s mduced by the normal component of twth flux density
calcdated by FEM.
The distance Ar that the rnagnet travels during At c m be obtained. Since it takes halfa
period for the magnet to traverse a pole-pitch, a halfperiod in the t h e dornain is equivalent to
one pole-pitch in distance domah, e-g.,
Therefore, when scpressed in the distance domain, the distance needed for the normal
component of tooth flux density to rise hm zero to maximum is
In Table KI-1, the distance needed for the normal component of tooth flux density to rise
h m zero to the plateau value is calculateci by (3.4) and cornpared to that measured on Fig.3.2
and that suggested in [8]. There are 6 dots p a pole. So one slot pitch is 0.167 pole-pitch. It can
be seen h m Table IIi-1 that the eq@vaieat distance for the normal component of tooth flux
density to rise h m zero to the plateau value is approximately one dot pitch. Thenfore, the
suggestions in [8] are not appropriate.
Table III-1 Distance needed for tooth fIw to rise h m zero to plateau
Method Obtained by (3.4) Mcasurcd on Fig.39 Suggested in [8]
W r O. 1600 0.1420 0.0833
hxlh 0.960 0.852 0.500
Table III-2 shows the eddy cunent l o s induced by the normal component of tooth flux
density calculateci by diffefent methods.
Table III-2 Eddy current loss induced by the n o d component of tooth flux density
Et can be seen fiom Table III-2 that
Method used
Eddy current loss at l20Hz or 12.lm/s (W)
E m r (%)
When the distance needed for the tooth flux density to rise is assumeci to be one dot
pitch, the calcuiated loss is close to that calculated by FEM ( m r of 3.1%).
When the distance needed for tooth flux density to rise is assumed to be the
extaision of the linear part of tooth flw waveform, the calculated loss is about 13%
more than that caicuiated by FEM.
72.7
-
When the flw density is assumai to be sinusoidal with the plateau flux density as
its magnitude, the eddy loss is Iess than a halfof that caiculated by FEM.
When the distance needed for tooth flw density to nse is assumed to be a tooth
width, the calculated l o s is almost Wce of that calculated by FEM.
Use dot pitch
70.4
-3.1
Therefore, it can be concluded h m the above observations that:
The waveform of the normal component of tooth flux density can be appmxhaîed
by a lin- trapezoidal waveform.
Use actuai slope
82.4
13.4
The rise time of the nomai component of tooth flux density is appmximately the
tmie needed for the magne$ edge to traverse one dot pitch. Altematively, the
distance needed for the tooth fIux density to rise h m zen, to plateau value is
approximately one dot pitch,
Use sinusoicial
29.0
-60.1
Suggested in [8]
142.1
95.5
3.3 Simplified Tooth Eddy Current Loss Model
3.3.1 Eddy Current Loss lnduced by the Normal Component
As it has been shown in the last section, the waveform of the normal component of the
tooth flux density in a die-mounted PM synchronous motor can be approximated by a
simple trapezoidal wavefoxm as shown in Fig.3.5 for a . open dot PM motor.
j Center of a tooh
Fig.3.5 Approximation of the nomal cornponent of tooh flux density
The time interval for the approrcimated normal component of twth flux density to
rise h m zero to plateau is appmnimateiy the t h e for the magnet edge to traverse one dot
pitch. For an m-phase PM motor with q dots per pole per phase, there are mq dots pet pole.
Therefore, the tmie needed for the magnet edge to traverse one dot pitch is
Substitnting (3 -5) to (3. l), the eddy current loss indoced by the normal component of
tooth flux density can be appmtcmiated for three-phase motors:
The total tooth eddy cumnt l o s cm be M e r written as a function of speed n:
It can be inferred fiom (3.6) and (3.7) that:
The eddy cunent loss induced by the nomal component of the tooth flux density is
proportional to the nimiber of dots per pole per phase q for given fkequencies.
increasing q will dramaticaily increase the tooth eddy current Ioss.
For givm torque and speed ratings, the tooth eddy current Ioss induced by the
nomai component is appmximately proportional to the product of poles squared and
dots per pole per phase.
3.3.2 Effect of Slot Closure
The previous mode1 was based on an open-slot configuration. In order to test the hear
assumption of the rise and fiill of the normal component of tooth flux density, the effect of dot
closure is now studied. Fig.3.6 illustrates the slot closure of an electrical machine. The openhg
at the surfiace of the dots is w, The relative closure y can be defhed as the ratio of slot closure
to dot pitch:
FEM was perfomed on the Iinear PM motor by varying slot closure but keeping other
dimensions constant, The wavefonns of the nomal component of the tooth flux density at the
center of the tooth are shown Fig3.7 for différent dot closure.
Fig.3.6 Illustration of slot dosure
Fig.3.7 Effect of slot closure on tooth flux density at the center of a tooth
It can be seen h m Fig.3.7 that when sht closure or dot opening changes, the achieved
plateau value of the normal component of tooth flux deflsity waveform also changes due to the
change of effective airgap reluctance.
In order to compare the rate of rise of the normal component of tooth flux density
against dot closure, the tooth flux density was nomalued to the same value for different dot
closure as show in Fig.3.8. The normalized wavefomis of the nomai component of twth flint
density show that the dope is approximately aidependent of slot closure with only a slight
decrease in slope as the dot is closed.
Fig.3.8 Tooth flux d&ty waveform for different dot closure
The efféct of dot closure was also studied on configurations by changing magnet
thickness and airgap length. B a d on nomaiized plateau flux density, the tooth eddy cunent
losses were cdcdated by FEM for different dot closures and shown in Fig.3.9 for two cases
with mit magnet thickness of 3mm and 6mm. It can be seen that the tooth eddy current Ioss
sIightIy mcreases as the dot closure decreases. The inmase of eddy current los is less than 3%
when dot closure changes ftom ldmmm to 8.4mm based on normaiïzed tooth flux density.
This smaIi increase of eddy c m t Ioss m the tmth is consistent with the srnaIl change of slope
of the nomai component of tooth flux density as seen h m Fig.3.8. Therefore, it can be
concluded that the approximation mode1 is independent of slot closure.
s i Im=3mm&lmm . % 1 = 70 c ................. m ................. nc .................. x ................. K ................... .? ............ c.
l t I
0 1 Closed dot. 2 Open dot i t I
I .............................. 6 60 c .................... ............................................................. 1 w
Q, 1 i 5
O !
io l
I 50 f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +
1
Slot openings Wo (mm)
Fig.3.9 Effect of dot closme on tooth eddy current loss
3.3.3 Effect of Magnet Width
In the approximation model, tooth eddy current loss is assumed to occur only when
there is a change in the normal component of tooth flux density when those teeth pass the
magnet edges. It foflows that the loss would be independent of the angular width of magnet.
In order to test whether the tooth eddy current is independent of the angular width of
rnag.net, FEM analyses were performed on the hear PM motor such that the width of the
magnets was changed by keeping the same number and shape of dots.
The nrst example was the linear PM motor with 2 slots per pole per phase. Fig.3.10
shows a quarter of a period (a haIfpole pitch) of the wavefom of the nomal cornponent of the
flux density in the center of a tooth when the magnet coverage was changed fiom M . 1 6 7 to
c~û.990. The p d e l Lmes are drasvn to show the dope of the flux.
It can be seen h m Fig.3.10 that, when the magnet coverage is within the range of
adl.167 to d . 8 3 3 , the major part of the flux waveform foilows a hear pattern with the same
slope and same plateau. The hear approximation of the flux density needs 0.142 pole-pitch to
rise h m zero to plateau value. Although the slope is the same, the flux density does not mach
the plateau value when the rnagnet coverage f d s to a~û.167 or the magnet width is one dot-
pitch.
The slope of the flux density wavefom starts to increase when the magnet coverage is
increased beyond oV0.833 or the space between the two magnets is less than one dot-pitch. The
increase in sIope of flux density suggests that the simpiified twth eddy current loss mode1 is not
valid beyond aXl.833.
Fig.3.10 Waveform of the nomai component of tooth flux density (q=2)
As a second example, FEM was perfomed on a hear PM machine with q=L (3 dots
p a pole). The wavefomis of the normal mmponent of the tooth flux density at the center of a
tooth are show in Fig.3.11.
Fig.3.11 Waveform of the norrnai component of tooth flux density (q=l)
In Fig.3.11, five parailel lines are also drawn to show the dope of the hear part of the
flux waveforms. It can be seen that, for diffe~ent magnet widths, the flux waveform foilows a
lin- pattern with the same dope except for the widest magnet coverage d . 8 6 0 . The distance
needed for normal component of tooth fiux density to rise at constant is again 0.284 pole-pitch
(or 0.852 dot-pitch). For different magne widths, the twth flux density is seen to have the same
plateau value except for the aarrow magnet widths of d.298 and d . 4 0 5 .
It can be concluded h m tht above observations that over a wide range of magnet width,
the dope of the Iinear part of tooth fiux and the plateau vahie of the tooth fia density are
constant, When the magnet width is Iess than about a dot pitch, the no& component of the
flux density in the tooth wiIL not reach the plateau as otha cases do. When the space between
the two magnets is less than about a dot pitch, the slope of the normal component of tooth flux
density is considerably increased.
The tooth eddy cunent losses for the linear PM motor with Mirent magnet widths
shown in Fig.3.10 were calculateci using FEM and shom in Fig.3.12.
Magnet coverage a
Fig.3.12 Tooth eddy cumnt loss vs. magnet width (q=2)
The losses shown on Fig.3.12 firrther CO& that the tooth eddy cunent loss remauis
substantiaily constant as the magnet width is varied over a wide range.
For the impracticd cases with a magnet coverage less than 0.25, the loss decreases
because the flux density doses not reach a plateau.
When the space between the two magnets is less than about one a slot pitch, the eddy
current 105s is seen to be considerably mcreased, The effect of the adjacent magnets is to
inmase the slope of the nomd component and eliminate the slow rise of flux densïty in the
regîon near zero tooth flux density.
A sigainant conclusion about magnet width is that the tooth eddy current loss is
dramatidy increased as shown in Fig.3.12 when magnet coverage approaches 1.0 because of
the significant increase of the dope of tooth flux density as show in Fig.3.10. The increase in
Ioss is more than 50% with 180' magnets ody for the norxnal component of tooth flux density.
Therefore, motors with 180' magnets should be avoided in the design of a PM motor in contrast
to that suggested in [33].
3.3.4 Effect of Slots Per Pole Per Phase with Fiied Slot Pitch
In order to confirm that the tooth eddy current loss is approximately proportionai to the
nmber of slots per pole per phase for given kquency, FEM was performed on a hear1y
arrangeci PM motor such that the nrnnber of slots per pole per phase is varied with fixai dot
pitch X. In other words, the pole pitch s is changed according to the number of dots per pole as
shown in Fig.3.13 (q is changed h m 1 to 4). To simpiify the maiysis, the magnet coverage is
kept constant at 0.667 and the open slot configuration is used.
(CI- el
Fig.3.13 Configuration of linear motors peifomed by FEM, varying q and p
On a Iinear machine with fixed fime, bmcreasing q means the reduction of nimiber of
poles if the dot pitch is £ked In order to be comparable, the tooth eddy cment loss is
cdculated for fïxed m e n c y and nxed speed respectively.
The wavefoms of the normal component of the tooth flux density for e l and q==2 are
shown in Fig.3.14 when plotted as a finiction of pole pitch. The distance needed for the flux to
rise c m be measured as listed in Table III-3.
Fig.3.14 Tooth flux wavefoms for different g with the same dot pitch when plotted as a
hction of pole-pitch
Table ID-3 Distance needed for the normal component of tooth flux density to rise
2
O, 142
0,852
3
0.095
0.852
Slots per pole per phase q
Distance needed (in pole-pitch)
Distance needed (m dot-pitch)
4
0.071
0.852
1
0,284
0.852
It can be seen h m Table Iü-3 that, since the two cases have diffkrent pole-pitches, the
distances are différent for c i i f f i t q when expresseci as a b t i o n of pole pitch. However, when
expressed as a M o n of dot pitch, the distances needed for the normal component to nse h m
zero to plateau is s e a i to be the same. Therefore, it is dcsirable to plot the flux waveforms as
hct ions of dot pitch as shown in Fig.3.15. It can then be seen that for the two cases, the
waveforms of the tooth flux d d t y have the same dope. It can also be seen h m both Table
III-3 and Fig.3.15 that the linear approximation to the rise of the normal component of the tooth
flux density takes 0.852 dot-pitch w matter how many teeth per pole per phase.
Fig.3.15 Tooth flux waveforms for différent q with the same slot pitch when plotted as a
fùnction of slot pitch
Table III-4 shows the caiculated tooth eddy current loss by FEM with nxed kqpency.
Row 4 of Table El-4 shows that the ratio of tooth eddy cmrent Ioss the number of slots per pole
per phase (Pm lq) is nearly constant. Therefore, e q u h n (3.6) is vaiidated: the tooth eddy
current loss is proportionaï to the nnmber of slots per pole per phase for a given
fiequenq.
Table III-4 Eddy cment loss in the teeth when changing the number of poles wOHz)
Slot number per pole p a phase q
The tooth eddy cumnt loss is also calculated at f ied speed as shown in Table ïïI-5. It
is show in Table III-5 that the ratio of tooth eddy current l o s to the product of number of poles
squared and the nimiber of slots p a pole per phase (P,, l& is nearly constant. Therefore,
equation (3.7) is vaiidated: the tooth eddy current loss is proportionai to the product of
poles squared and the number of slots per pole per phase for a given speed.
Table ïïI-5 Eddy current loss in the teeth when changing the nurnber of poles (v====m/s)
The concIusio11~ made fiun Table III-4 and Table m-5 are true for a given nmnber and
size of slots when the number of poles is to be selected as shown in Fig3.13. The eEects of
nimiber of dots per pole per phase for motors with a given number of poles will be studied in
Section 3.3.6.
3.3.5 Effect of Airgap Length and Magnet Thickness
It is shown in Section 32 that, for the Linear PM motor, using dot pitch as the distance
needed for the rise of the normal component of tooth flux density gave quite good estimation
of the eddy c-t loss induced by the normal component of tooth flux density. In this section,
the effect of variations of airgap length and magnet thickness is investigated. For each
combination of the two dimensions, a correction factor kg is evaluated based on FEM loss
andysis.
Suppose that after introducing the correction factor kg, the approximated eddy current
loss is the same as that obtained by FEM, then
Correction fztor kq is nrSt developed on the linear PM motor with equal dot and tmth
width. A s& of FEM analyses were pdomed such that for a given ratio of magnet thickness
l,, over airgap length 6, both the m a p d thickness and the akgq length were varieci. Then, this
procedure is repeated for different ratios of magnet thickness to airgap length. Slot pitch was
kept constant at aîî times.
The correction factor k, was caicuIated for a wide muge and plotted on Fig.3.16. It cm
be see h m Fig3.16 haî, with constant slot pitch,
Gien the ratio of magnet thickness to akgap length, proportionally bcreasing
@ap length and magnet thickness WU decnase k4.
Given airgap length and dot pitch, increasing magnet thickness wilI decrease
Fig.3.16 Relationship of kp vs. magnet thickness and airgap length
Fig3.16 is plotted for a very wide range of dimension variations. In an actual PM motor
design, the variation of the dimensions is u d y constrained in a smaii range. Typicdiy, airgap
length m small machines is 0Smm to 2.0mrn. To make good use of tooth density, id8 is usually
larger than 2.5 and X/6 is usually within the range of 8 to 24. Therefore, the variation of kq for
PM motors d e r n i above is in the range of 0.85 to 1.15.
3-3.6 Effect of Slots Per Pole Per Phase with Fied Pole Pitch
In section 3.3.4 it was shown that the tooth eddy current loss is proportional to the
number of slots per pole per phase q for givm fkquency. In that case, both the ratio of maguet
thickness to airgap length and the ratio of tooth width to abgap length were kept constant,
Therefore, each case has a same factor kT
In this section, the efféct of the number of dots per pole per phase with fixeci pole pitch
wilI be studied. On the linear arranged PM motor with open slots, the dot pitch is varied such
that the pole pitch and number of poles are kept constant while changing the number of dots per
pole per phase as show in Fig.3.18.
Varyhg q by fixing pole pitch wiiI requin a change to the dot pitch. Note that in the
previous section, slot pitch was kept constant when deriving kq. The fact is, when slot pitch
changes, both the ratio of magnet thickness to airgap length and the ratio of tooth width to
airgap length wiii change. In this section, the efféctiveness of k, wiU also be tested against slot
pitch by kexping a constant magnet thickness and airgap length.
Fig3.17 Changhg of the nmnber of dots per pole per phase with fked pole pitch
48
The flux wavefomis of the n o d componmt of the flux density in the center of a tooth
are shown in Fig.3.18. It cm be seen h m Fig3.18 that when the pole pitch is kept constant and
the dot pitch is inversely proportional to the number of dots per pole per phase q, the nse time
of the linear part of tooth flux density decreases whm q bcreases. The larger the q, the less time
is needed for the magnet edge to traverse one slot pitch as shown in Table M. It can be clearly
seen h m Table m-6 that the decrease m the titne intervai is not proportional to q. A correction
factor is necessary.
Fig.3.18 Normal component of tooth flux density for different q with variable slot-pitch
Table III-6 Distance needed for tooth flux density to rise at constant dope
Although the magnet to airgap length ratio was kept constant, the ratio of tooth width
49
SIot per pole per phase q
Distance (in pole pitch)
I
0230
2
0,142
3
0.101
4
0,087
over airgap length was changed when the tooth width was changed. Therefore, the correction
factor k, is different for each case. Constant can be determined for each case on Fig.3.16. By
using (3.9). the eddy cmrent loss induceci by the normal component of tooth flux density can
be determined. These d t s are listeci in Table III-7. B can be seen h m Table III-7 that the
approximated tooth eddy current losses have good agreement with those cdcuiated by FEM
Therefore, it can be concluded that the variation of R, is also valid whm varying the slot pitch.
Table III-7 Tooth eddy current loss vs. number of dots per pole per phase at 120Hz
3.3.7 Effect of Tooth Width with Fked Slot Pkh
The previous investigation was based on equal slot and tooth width. In this section, the
effect of unequal tooth and slot width wifi be studied. The andysis was pdormed such that the
tooth width was varied but the slot pitch was kept constant. Both the volume of the teeth and
tooth flux density Vary with tooth width by keeping a constant slot pitch. Table III-8 shows the
compzaison between the tooth eddy c u r w t loss calculateci by the approximation model and that
cdculated by FEM. It can be seen h m Table m-8 that when tooth width is changed h m 36%
to 74% dot-pitch, the error between the twth edciy current loss calculateci by the approximation
model and that calcdated by FEM is generaUy less than 5%. Therefore, vvaryig tooth width
within hed slot pitch has Iittle effect to the approximation model d g other dimensions
are kept constant,
Table Ill-8 Tooth eddy current loss vs. tooth width at 120Hz
Tooth width (mm)
Tooth flux density (T) 1.5453 1.1047
Tooth volume (xl w3m3) 0.2627 0.3447 0.4268 0.5089
Tooth eddy c m t loss by FEM (W) 63.6 72.7 61.6 60.9
Eddy c m t loss by approximation mode1 (W) / 60.7 1 70.4 1 60.3
3.3.8 Eddy Current Loss lnduced by Circumferential Component
So €a, only the normallradial component of tooth flux has been considered for the
estimation of eddy current loss in the stator teeth. Since the circumferential/longitudind
component follows a more complex pattern, it is difficult to Iineatly approximate the
longitudinal component of the tooth flux deflsity and the eddy current loss induced by this
component.
When the eddy 10s induced by the circtrmferentid flux is cousïdered, the total eddy
current loss m the teeth can be expressed as:
where Pm is the eddy cment loss induced by circumferentid component of the tooth flux
density and k, is a correction factor.
Fh, for the kear PM motor with open dots, the eddy loss mduced by circumfefentid
or longitudinal flux in the teeth is compareci with the eddy current l o s Ïnduced by the normal
component. It was found that the eddy current l o s induced by the longitudinal component of
tooth fimc density is approximately 15% of that induced by the normal component of tooth flux
density and is constant over a wide range of magnet coverage.
Second, the correction factor k, was tested when chanping the magnet width, magnet
thichess and airgap length for the open dot linear PM motor. It was found that the correction
fêçtor & is not affecteci by the magnet width nor magnet thickness but affited by airgap length.
Finally, the idluence of slot closure on R, was studied. When the slot closure is
inmaseci, the longitudinal flux in the shoe area of the teeth inmeases significantly as show in
Fig.3.19. The eddy currait l o s induced by the longitudinal flux ais0 signincantly increases with
the increase of dot dosure.
Fig.3.19 Longitudinal component of tooth flux density m the shoe area
It was found h m the FEM results that k, is a hction of relative slot closure y and the
ratio of slot-pitch to airgap length. The correction factor k, is graphed in Fig.320 for different
slot closure and slot-pitch to airgap-length ratio.
1 i
O 0.1 0 2 0 3 0 -4 0.5 0.6 Relative siotciosure r
Fig.3.20 Correction faor k, with regard to dot closure, airgap and tooth width
3.3.9 Simplified Expression of 100th Eddy Current Loss
The total eddy current loss in the stator teeth of a PM synchronous motor can be
approxhated es:
where kq and k, are correction factors which cm be obtained through Fig.3.16 and Fig-3-20
respectively.
Therefore* the tooth eddy c m t Ioss is proportional to the nurnber of dots per pole per
phase for a given fkquency.
When expresseci as a hction of speed, the tooth eddy current 105s cm be approximated
as:
Therefore, the tooth eddy current loss is proportional to the product of number of dots
p a pole p a phase and the square of number of poles for a give speed.
3.3.10 Case Studies on Tooth Eddy Cument Loss
The developed tooth eddy currait loss model was applied to some cases to evaiuate the
tooth eddy current loss. The predicted tooth eddy losses are compared to those calculated using
FEM method. Listed in Table III-9 are some of the calcuiation examples.
Case I: The 4-Pole PM Motor
The 4-pole motor was analyzed using the developed tooth eddy current l o s model and
FEM. Fig.3.21 shows the waveform of the radial component of the tooth flux density of the 4-
pole PM synchronous motor of Fig.2.7 obtained by FEM at the center of a tooth. The Iinear part
of the waveform was extended to show the dope of the flux density.
It can be measured from Fig.3.21 that the distance needed for the hear part of the
normal component of the tooth flint density to rise Orom zero to the plateau value is 0.128 pole-
pitch or 1.15 slot-pitch.
From the dimensions of the motor, it can be calculateci that t, I S = 3.15, k16 = 5.1
y = 0.186. From Fig.3.16,4 can be found to be 0.72. From Fig3.20, &cm be found to be 1.1 8.
The tooth eddy current Ioss of this motor was caldated by the approximation model (3.12) as
l8.02W whüe that caldateci by FEM is I73W (3.9% discrepancy). The proposed tooth eddy
current l o s mode1 gave a good approximation of the tooth eddy current Ioss.
Fig.3.21 Radiai component of tooth flux density of the 4-pole PM motor
Case II: The 8-Pole PM Motor
The waveform of the radiai component of the tooth flux density of the &pole PM
synchronous motor of Fig.2.8 is obtained by FEM and shown in Fig.3.22.
The distance needed for the iinear part of the normal comportent of tooth flux density
to rise h m zero to maximum is 0.3 pole-pitch or 0.9 slot-pitch. In 0 t h words, the measured
time on Fig.322 for the rise time of tooth flux is 1.38ms. The time interval for the rnagnet eâge
to traverse one slot pitch 102mm is 1.3Ans at l8OOrpm (or L20Hz).
From the dimensions of this motor, we have 1,16 = 4.3, LI 6 = 25.2 y = 0.356. From
Fig.3.16, it cm be found that kp=l -1. From Fig.3.20, it can be found that &=I .33.
The tooth eddy current l o s calculateci by the appmrcimation mode1 (3.12) is 8.MW
whiie that obtained by FEM is 822W (-2.2% discrepancy).
0.000 0.083 0.167 0.250 0.333 0.41 7 0.500
Dis tancetpole-piîc h
Fig.3.22 Radiai component of tooth flux density of the &pole PM motor
Case III: A Linear Motor with Open Slot
The Iinear PM motor of Fig.2.4 with open dot was investigated with FEM and the
approximation rnodel. The motor has an airgap length of 1 mm, maguet thickness of 3mm, equal
dot and tooth width of 8.4mm and 2 dots per pole per phase. From the dimensions of the motor,
the correction factors can be foimd h m Fig.3.16 and Fig.3.20 respectively
(1,16 = 3, X I 6 = 16.8, y = 0, kq=1.03 and k=1.15). The ~roximated tooth eddy current loss
is 83.4W comparing with 83.7W obtained by FEM (-0.4% discrepancy).
Case IV: Varying Slot Closure and Magnet Thickness
Based on the configurations m Case III, the open dot was change to partly closed slot
wo=l.6mm, and rnagnet thickness was changed h m 3mm to 6mm but keeping otha
dimensions the same. From the dimensions of the motor, 1, / 6 = 6,1/ 6 = 16.8, y = 0.405 , kq
and k, are found to be 0.93 and 1.33 respectively. The tooth eddy current loss is I27.3W
caldated by the approximation mode1 and 1222W calculateci by FEM (4.2% discrepancy).
Case V: Varying Slot Closure, Magnet Thfckness and Airgap Length
Based on the configurations in Case IIII, the dot openings were changed fkom open to
3.2mm, maguet thickness was changexi h m 3mm to lOmm and airgap length was changed
fiom lmm to 3.2mm. From the dimensions of the motor, i, / 6 = 3.13, L16 = 5.25, y = 0.3 1.
Both kq and k, are diffefent h m those in Case m. With kq=û.72 and k,=1.24, the tooth eddy
current loss can be approximated by (3.12) as 78.5W, whiIe that obtained by FEM is 75.9W
(4.3% discrepancy).
Case VI: Varying Slot Width
Based on Case DI, the tooth width and the slot width were changed fiom 8.4m to
42mm but keeping other dimensions constant. Both k, and k, are different h m those in Case
m. From the dimension of the rnotor, 2,16 = 3, A / 6 = 8.4, y = O. With kq=û.8 1 and kc=l. 1 1,
the approximated tooth eddy currait loss has an ermr of 4% when compared with that obtained
by FEM (4.0% discrepancy).
Summary
AU the above six examples are listed in Table DI-9 for cornparison. It can be seen firom
Table m-9 that the tooth eddy cumnt l o s caiculated using the proposed approximation mode1
is m good agreement with those caiculated using the FEM method. The error is always within
a fS% error band.
Table III-9 Eddy current loss in the teeth calculated by FEM and proposeci mode1
Case 1 Case iI Case VI Case III
Linear -
Linear 1 Motor type Linear Linear I 1 Slot pitch I (mm)
1 Tooth width w ~ m m )
lot openings w, (mm)
1 Slob per pole per phase q
1 Tooth £in dnuity Bth (T)
Eddy loss in the teeth by FEM (W)
I Eddy loss in the teeth by propcwed mode1 eV)
3.4 Waveforrns of Yoke Flux Density
It was assumed in [8] that the yoke flux density can be approximated by a trapezoidal
wavefom and the eddy current los in the yoke can be approximated using only the magnitude
of the average yoke flux density. In order to test these assumptions, FEM was perfomed on the
kear anangeci PM motor and the wavefom of the yoke flux density were obtained.
3.4.1 Wavefomis of the Longitudinal Component of Yoke Flux Density
Fig.3.23 shows the wavefom of the longitudinal component of the yoke flux density
of the linear PM motor calculated by FEM obtained above a tooth. The flux densities were
obtained at the following locations in the yoke: 0.1B yoke thichess fiom the surface of the
stator. (II). Middle of the yoke. 0. 1B yoke thickness above the teeth.
Fig3.23 Waveforms of the longihininal component of yoke flux density (above a tooth)
Fig.3.24 shows the waveforms of the longitudinal component of the yoke flux density
of the linear PM motor caiculated by FEM obtained above a dot. The flux densities were
obtained at the foiiowing Iocations in the yoke above a tooth: 0 . 1 1 3 yoke thickness fiom the
d a c e of the stator. (LI). Middle of the yoke. (m. 1B yoke thickness above the teeth.
Fig.3.24 Wavefom of the longitudinal component of yoke flux deasity (above a dot)
It can be seen h m Fig.3.23 that the longitudind component of the yoke flux density
approximately foUows a iinear trapaoidal waveform. The plateau value of this component is
approximately evenly distributeci over the thickness of the yoke.
The original assumption made in [8] assumed that the longitudinal component of yoke
flux density increases approxixnately hear1y h m the midde point of the magnet to the edge
of the magnet and it remains appxhately constant m the yoke sector which is not above the
magnet The thne needed for the rise of the longitudinal component of the yoke fia density
h m zero tu plateau or fd h m plateau to zero is approximately the time intema1 for haif of
the magnet width to traverse.
In order to deteamine the riseffd thne of the flux density, an appmximated trapmidal
wavefonn of the longitudinal component of yoke flux density was plotted in Fig.3.25 with the
average flux densitics obtained above the dot and above the tooth.
Fig.3.25 Approximation of the longitudinal component of yoke flux density
It can be seen nom Fig.325 that the plateau of the waveform is longer than the space
between the two magnets. The length of the plateau is 0.42 pole-pitch and the distance needed
is 0.29 pole-pîtch for the flux density to rise h m zero to plateau or 0.58 pole-pitch h m
negative plateau to positive plateau. The space between the two rnagnets is 0.333 pole-pitch.
The difference between these two is 0.087 poie-pîtch. R e d that one dot-pitch of this machine
is 0.167 pole-pitch. The Iength of the plateau is approximately the sum of the space between the
two magnets and a halfslot pitch. Based on this observation, distance x can be expressed as:
If the nse time of the Iinear part of yoke flux densïty were used to evaluate the eddy
carrent 10% induced by the longitudinal component of yoke flux density, the resuits wodd be
larger than that calculated by FEM. This is due to the fact that the actual flux density rises
slower at the begimimg and the end of the change and that the longituninal tlux density is lower
near the teeth than in other part of the yoke.
Suppose that the waveform of the l o n g i t u ~ component of the yoke flux density can
be represented by a Iinear trapaoidai wavefom and this approximation produces the same eddy
current loss as the achial waveform does. Using a similar procedure to that of the teeth in
Section 3.2, the equivalent distance x for this approximation can be obtained by using (3.2):
where kC is the plateau value of the longitudinal component of yoke flux density, Vy is the
volume of the yoke and PeyCJEM is the eddy current loss of the longitudinal component
cdcuiated by FEM.
In Table ID-10, the distance needed for the longitudinal component of yoke flux density
to rise h m zero to plateau value was calculated by (3.15) and compared to that measured on
Fig.3.25 and that suggested in [8]. The distance obtained by (3.15) is about 4.4% l es than the
magnet width.
Table III-10 Distancex needed for yoke flux to rise
1 Obtainedby 1 Measined on Suggested m [8] (3 -15) Fig.3.25 (magnet width)
Table III-1 1 shows the eddy cmrent loss mduced by the longitudinal component of yoke
flux density caiculated by different methods.
Table III-1 1 Eddy cunent loss induced by the longitudinal component of yoke flux density
It cm be seen firom Table III4 1 that
Method
Eddy current l o s (W) at 120Hz or 12.1ds
Ermr (%)
When the distance needed for longitudinal yoke flux density to nse h m negative
plateau to positive plateau is assumed to be magnet coverage as suggested in [8], the
calcuiated loss is close to that calcuiated by FEM (ermr of -4.4%).
When the distance needed for longitudinal yoke flux density to rise is assumed to
be the extension of the hem part of yoke flux waveform, the calculated loss is
about 10% more than that caicuiated by FEM.
FEM
73.3
0.0
When the flux density is assumed to be sinusoida1 with the plateau flux density as
its magnitude, the eddy loss is 2 1.3% less than that calculated by FEM.
Therefore, it can be concluded fmm the above observations that:
Use magnet coverage
70.1
-4.4
The waveform cf the longitudinal component of yoke flux density can be
approximated by a linear trapezoidal waveform.
The time needed for the longitudinal component of yoke flw density to rise h m
negative plateau to positive plateau is approxhately the time needed for one point
in the yoke to traverse a magnet width. AIternatively, the distance needed for the
yoke flux density to rise h m negative plateau to positive plateau is appmximately
the magnet width.
Use actuai siope
80.6
10.0
Sinusoidd
57.7
-2 1.3
3.4.2 Wavefoms of the Normal Component of Yoke Flux Density
Fig.3.26 shows the waveforms of the normal component of yoke flux density for the
Iinear motor obtained by FEM. The flux densities were obtaiaed at four locations in the yoke:
1B yoke thickness h m the surfiice of the stator O, rnidcüe of the yoke 0 , l B yoke thickness
above the teeth (m) and l m . the teeth 0. The wavefom of the normal component of
tooth flux density gr) is also shown in the same graph for cornparison.
Fig.3.26 Normal component of yoke flux density of the hear motor
It cm be seen fhm Fig.3.26 that
The normal component of yoke flux density is seen to have similar waveforrn as that
of the normal componmt of tooth flux density, particuIarLy m the region just above
the tooth (curve IV on Fig.326) as wouid be expected. The value of the plateau is
dependent on the tooth flux density and rnotor dimension,
The plateau value of the normal component of yoke flux demky is different at each
layer of the yoke. It is maximal near the tooth and drops dramatically penetrating
f.urther into the yoke. It approaches zero near the d a c e of the motor.
3.4.3 Analytical Solutions to Yoke Flux Density
In order to develop the relationship between the normal component and the longituninat
component of yoke flux density, the flux distniution is show again in Fig.3.27.
Cross section A
Fig.3.27 Flux disîrîiution in the yoke
In the section of the yoke not above the magnet, the total flux is assumeci to be constant
and maialy contri'buted by the Iongitudinai component (kcdy). h the section of the yoke above
the magnet, the total flux is contnbaed by both longituninai and normal components of the flux
density.
Smce it has been shown that the normal component of yoke flux density foIlows a
trapezoidal waveform similar to that in the teeth as &own in Fig.3.28 with d B m t plateau at
each Iayer, the rise time of the linear appmximated flux is one dot-pitch divideci by correction
fiictor kq.
Fig.3.28 Approximated normal flux density wavefom in the yoke
The fiux supplied by the magnet is assumed to be in the normal direction when it passes
through the teeth. Part of the flux tums mto longitudinal direction on its way penetrating fhther
into the yoke. In the yoke, the flw that crosses the horizontal Iayer (cross section B on Fig.327)
must be quai to that passes through the vertical laya (cross section A on Fig.3.27), assrmiing
no flux linkage. Therefore, h m Fig.327 and Fig.328
where y is the distance h m the edge of the yoke dong the verticai Iayer, Br ( y ) and hJy) are
the instantaneous value and plateau valw of the normal component of yoke flux density at
distance y ftom the d a c e in the yoke, r, and & are projected pole pitch and projected slot
pitch which are m e a d in the yoke. For linear motors, r, is the same as r and A, is the same
as X. In rotary machines A, and .c, c m be expresseci as:
where dt is the height of teeth, r is the inner radius of the stator.
Since Br is hear in the integral part, (3.16) can be m e r simplified:
The above analytical solution is based on an average basis. In order to validate the
andytical solution, the nomal component of yoke flux density was obtained at the area above
a tooth and the other above a slot respectively. Fig.3.29 shows the normal component of yoke
flux dmsity as a fhction of yoke depth calculateci by FEM and by the andytical method. It can
be seen that
The nomai component of yoke flux d k t y is not evenly distributeci near the
teethlslots area In the yoke where it is above the teeth, the flux density is higher
than the flux densîty in the area where it is above the dots.
Equation (3.19) gives a good estimation of the normal component of yoke flux
density up to 213 of the of the yoke depth.
Since eddy curreflt loss is proportional to the square of flux demi@, it may not
accurate when it is employed to evaluate the eddy current Loss using (3.19).
0.00 0.10 020 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 Yoke depth (per unit)
Fig.329 Normal component of yoke flux dcnsity as a finiction of yoke depth
Based on the above observation, the expression for the nomal component of yoke flux
density is then redeveloped for the yoke above a tooth and the yoke above a dot respectively.
In the yoke above a tooth
[ 3 q n ;
In the yoke above a slot:
where 4, ( y ) and fi, (y) are the plateau values of the normal component of yoke flux density
above a tooth and above a dot respectively.
3.5 Sirnplified Yoke Eddy Current Loss Model
3.5.1 Eddy Current Loss lnduced by the Longitudinal Cornponent
It has been shown that the waveform of the longitudinal component of the yoke flux
density of a hear machme or the circumferential cornponent of the yoke flux density of a rotary
machine can be approxmiated as a ûapezoidd waveform as shown in Fig.330. The longitudinal
component of the flux density at any point in the yoke starts to rise when the middle of the
magnet passes this point. It reaches plateau value when the rear magnet edge reaches this point.
It keeps constant when this point passes the space between the two poles. Then it starts to
decrease when it hits the h n t edge of following magnet. The flux density reaches zero again
when it reaches the middle point of this magnet then it increases in the other direction.
- - - - -- -- --
Fig.3.30 Approxiruation of the cit.cumferentid component of yoke ffim density
69
For rotary machines, the thne interval requirwl for the magnet of width w, to pass one
point in the stator yoke, (e-g the circumferential component of the yoke flux density changing
h m -$ to kc or fiorn kc to -je) is
where a, the magnet coverage, is de- as the ratio of magnet width over pole-pitch:
The tirne rate of change of circderential component of the yoke flux density is
The change of flux happens twice in one period. The total yoke eddy current l o s
induced by the longitudindcirculliferential component ff ux is
It can be inferred fiom (3.25) that the eddy current l o s induced by the circumferentiai
component of yoke flux density is inversely proportional to the magnet coverage. It is not
related to the number of dots pet pole per phase or any dimensions of the motor.
3.5.2 Effect of Slot Closure
In order to test the deriveci approximation equation of yoke eddy current los, the effect
of geometcical variations of the linear PM motor was studied. The yoke eddy cmrent losses
caldatecl by the approximation mode1 are compared with those cddated by FEM.
First, FEM was performed on the hear motor by varying the dot dosure but keepmg
other dimensiorts unchanged. The catculated yoke eddy current l o s by FEM and the
approximation model is shown in Table III-12.
It can be seen h m Table III42 that yoke eddy current 105s increases when slot closure
increases due to the increase of yoke flux density. The yoke eddy current 10% caiculated by the
approximation model is in accordance with that calculated by FEM for différent dot clostue.
The error is within 5%. Thenfore, the efftct of slot closurt on the approlrimation rnodel can be
neglected.
Table III-12 Effect of dot closure on yoke eddy curent loss at 120Hz
Appmximation model 0
Différence (%) -4.4 -3.3 -2.7 -3.0 -1.8
3.5.3 Effect of Magnet Coverage
In order to test that the yoke eddy current l o s is inversely proportional to magnet
coverage, the magnet coverage of the linear PM motor was varieâ fhm 0.532 to 0.833. In some
cases, the yoke thichess of the motor was &O changed according to the magnet coverage to
keep a relatively constant yoke flux density-
For different magnet coverage, the eddy current loss was caIculated by the
approximation m o n (3.25) and compared to that obtallied by FEM as shown in Table Lü-13.
It c m be seen h m Table III43 that for the given range of magnet coverage, the difference
between the estimaîed yoke eddy c m t Ioss by the approxkdon model and that calcdated
by FEM is generaUy less than fi%. Therefore, magnet coverage has Little effect on the
appmxïmatïon model.
Table III-13 Effect of magnet coverage on yoke eddy curent loss at 120Hz
Magnet coverage a 1 0.833 1 0.718 1 0.667 1 0.532
Yoke flux density (T)
Yoke volume ( x lO"m3) 0.9 193 0.9882 0.9 193 0.73544
Yoke eddy cumnt Ioss by FEM (W) 74.2 73.5 73.3 69.9
Yoke eddy current loss by approximation model (W)
3.5.4 Effect of Slots Per Pole Per Phase
Equation (315) shows that the eddy current loss induced by the circinnferentiai yoke
flux density is not effected by the number of slots per pole per phase q. In order to test this
statement, the effect of the nurnber of dots per pole per phase was studied. On the hear PM
motor, within fixed dot pitch, the nimiber of dots per pole per phase was changed and the slot
pitch was changed accordmgiy. The yoke eddy current loss was calculated for each case by
FEM and compated with that dculated by the approximation mode1 as shown in Table Ili-14.
It can be seen from Table III-14 that, although the yoke eddy current l o s inmases with q due
to the increase of yoke flux density, the aror between the yoke eddy current Ioss calcuiated by
the approximation mode1 and that calculated by FEM is Iess than 7%.
Two concIusions c m be drawn h m this test Firsf the effect of number of slot per poIe
per phase on the appmlcimation mode1 is negligiile. Secondly, with equal dot and tooth width,
slot pitch has little effect on the approxhnation model.
Table III-14 Effect of number of slots with fixed pole pitch and variable dot pitch at 120Hz
Slots per pole per phase q
Yoke flux density (T)
Yoke eddy cumnt loss by F E M O
Yoke eddy current loss by approximation model 0
3.5.5 Effect of Magnet Thickness
In order to test if the approximation model is affecteci by magnet thichess, the magnet
thickness of the h e a r PM rnotor was varied h m 2Smm to 6mm. The yoke eddy current loss
induced by the circderentiai component calculated by FEM and by the approximation mode1
is shown in Table ïiï-15. It can be seen fiorn Table III45 that the aror between the yoke eddy
current loss calculated by the approximation model and that calculated by FEM is less than 5%
when magnet thickness changes in a wide range. Therefore, the approximated yoke eddy currait
loss model is effective for different magnet thickness assuming other dimensions are kept
c0IlSfZUlt.
Table m-15 Effect of magnet thickness on yoke eddy current ioss at 120Hz
1 Magnet thickness (mm)
1 Yoke flux densîty (T)
1 Yoke eddy cumnt 10s by FEM
I Yoke eddy nment loss by Approximation mode1 (W)
3.5.6 Effect of Airgap Length
In order to test w h e h the approximated yoke eddy current l o s mode1 is affécted by
airgap length, the airgap Iength of the hear PM motor was varied fbm 0.5- to 2.5mm. The
yoke eddy current loss induced by the circumferentid component caiculated by FEM and
cdcuiated by the approximation model is show in Table III-16.
Table III-16 Effkct of airgap Iength on yoke eddy current loss at 120Hz
Yoke flux densi
-6.6 -4.3 -2.6 -1.3 -1.1
it can be seen h m Table m-16 haî, dthough the yoke eddy curent Ioss decreases as
the airgap Iength is increased due to the decrease of flux density, the error between the loss
calnilateci by the approximation mode1 and that cdculated by FEM is within 7%. Therefore, the
effect of airgap length on the approximated yoke eddy current loss model can be neglected.
3.5.7 Effect of Tooth Width wi€h Fixed Slot Pitch
Equation (3.25) also shows that the approximated yoke eddy current loss model is not
affécted by tooth width. In order to test this statement, the tooth width of the hear PM motor
was vaned h m 4.4mm to 12.4mm for a nxed slot pitch of 16.8mm. The yoke eddy cuirent loss
induced by the cimdierenntial component caidated by FEM and by the approximation model
is shown in Table III-15. It cm be seen h m Table III-15 that the yoke eddy nment l o s
mcreases with tooth width due to the iucrease of flux density and the mæase of tooth vo1ume.
However, the ermr between the 10s calculateci by the two methods is within 6%. Therefore, the
influence of tooth width on the appmximated yoke eddy current loss mode1 c m be negiected.
Table III4 7 Effect of tooth width on yoke eddy current loss at 120Hz
3.5.8 Eddy Current Loss lnduced by the Normal Component
It has been show that the wavefom of the nomal component of the yoke flux density
has the same waveform as that of the normal component of tooth flux density. Therefore, the
approximation formula derived for the tooth eddy current loss density (normal component oniy)
can be used to approximate the eddy cumnt loss density induced by the normal componemt of
yoke flux density at each layer of the yoke:
Since at each layer of the yoke the nomal component of flux d&ty has a diffefent
plateau, it is desirable to integrate the los over the whole yoke to get the totai eddy nment 10s.
The plateau of the normal component of yoke flux density changes hearly with the depth of
yoke as shown m Fig3.29and at each Iaya of the yoke the flux densïty foiiows the hear
pattern shown in Fig.3.28.
By ushg (326), (320) and (321), the average eddy 10s density mduced by the nomial
component of yoke flux density.
The projected dot pitch h, in (3.20) and (3.21) is also a function of y for rotary
machines. In order to simprifL the integration, assume that projected dot pitch )c, can be
obtained a s a constant at the middle of yoke, then:
where kj, is the ratio of eddy current loss induced by the two component of yoke flux density.
The conected expression (3.29) was found to be approximately accurate for a wide
range of configuration variations on the hear PM motor. Table Dl-1 8 shows the calculated 5, for four cases used in Section 3 .5.4.
Table III-18 Test of correction factor k,
3.5.9 Simplified Expression of Yoke Eddy Current Loss
SUmmmg the eddy current losses bduced by n o d and Ionpituciinal components, the
total eddy current loss in the yoke is derive&
where k, is a correction factor for the eddy current loss induced by the normal component of
yoke flux density:
Therefore., yoke eddy currait los is inversety proportional to magnet coverage for given
operational firesuency. When expressed as a fhction of speed, the yoke eddy c m t loss is:
Therefore, yoke eddy cumnt l o s is proportional to the poles quand divideci by magnet
coverage for a given speed.
3.5.10 Case Studies on Yoke Eddy Current Loss
The developed approximation mode1 was applied to some cases to estimate the yoke
eddy cturent los. The &ts are compared to those caldated by FEM method Listed in Table
III49 are the same examples used to calculate tooth eddy current loss.
Case 1: The 4-Pole PM Motor
The waveforms of the two components of the yoke flrm density at middle yoke of the
4-pole PM motor calcuiated by FEM are shown in Fig.3.3 1. It cm be seen that the Luiear part
of the circumferentiai component of yoke flux density bkes 0.3 pde-pitch to rise h m zero to
the plateau. The nse t h e cdculated by (3.14) is 0306 pole-pitch. These d t s coincide with
each other.
The nomial component of yoke flux density takes 0.13 pole-pitch to rise h m zero to
plateau. RecalI that the normal component oftooth flux daisity of this motor takes 0.128 pole
pitch to rise h m zero to plateau. It therefore connmis thai the nonnal component of yoke flux
density has the same shape as that of the normal component of tooth flux density.
Fig.3.3 1 Yoke flux wavefom of the 4-pole PM motor
By using the dimensions of the motor aod v.72 h m Table III-9, correction factor
k, can be found to be 1.14. The yoke eddy current loss of this motor is 18.07W calculated by
FEM whüe the predicted yoke eddy current loss by the proposed mode1 is 19.03 W (discrepancy
(5.3 % discrepancy).
Case II: The 8-Pole PM Motor
The waveforms of the two components of the yoke flw density at middle yoke of the
8-pole PM motor are shown in Fig.3.32. Shce there is only one dot per pole per phase, there
is more distortion in the flux waveforms. However, these wavefom still follow the
approxirnated pattem.
It can be seen that the Iinear approximated ckderentiai component of yoke flux
densîty takes O Z pole-pitch to rise h m zero to plateau. The m e a d nse tirne Fig3.32
comcides with that calculated by (3.14).
Fig.3.32 Yoke flux waveform of the 8-pole PM motor
The nomai component of yoke flw density takes 0.3 pole-pitch to rise fiotn zen, to
plateau. Recall that the normal component of tooth flux density of this motor also takes 0.3
pole-pitch to nse h m zero to plateau.
By using the dimensions of the motor and kq=l. 1 h m Table III-9, correction factor k,
can be fouad to be 1.1 1. The predicted loss is 5.8 1 W by the proposed approximation mode1
comparing to 5.75W caiculated by FEM (-1 -1% discrepancy).
Case III: A Linear Motor with Open Slot
The h e m PM motor of Fig2.4 is imrestigated with FEM and the appro~ation model.
From the dimensions of the motor, the coxrection factors can be found as kr=1.33. The
approximated yoke eddy cunent 10s is 94.3W comparing with 97.4W obtained by FEM
(discrepancy -32%).
Case N: Varying Magnet Coverage and Yoke Thickness
B a d on Case III, the rnagnet coverage a was change h m 0.659 to 0.532 and yoke
thickness was changed k m 20mm to 16mm but keeping other dimensions the same. kr is found
to be 1.26. The yoke eddy current l o s is 90.3 W cdcuiated by the approximation model and
90.8W cdculated by FEM (-0.5% discrepancy).
Case V: Varying Slot Closure, Magnet Thickness and Airgap Length
Based on Case III, the dot opening was changed h m 8.4m.m to 3.2mm, magnet
thickness was changed from 3mm to lOmm and airgap length were changed h m lmm to
3.2mm. Correction factor k, was fond to be 1.23. The yoke eddy cunent l o s c m be
approximated as 97.OW, whüe that obtained by FEM is 93.2W (4% discrepancy).
Case VI: Varying Slot Width
Based on Case III, the tooth width and the dot width were changed fkom 8.4m.m to
42mm but keeping other dimensions constant. Correction factor k, was fond to be 1.51 . The approximated tooth eddy current loss has an m r of 1.6% when compared with that caicuiated
using FEM.
Summary
ALI of the above examples are listeci in Table III-19 for cornparison. It can be seen h m
Table Ili-19 that the yoke eddy current loss cdculated by the proposed approximation model
are genedy acceptable when comparing with the FEM d t s . The error is aiways within f6%.
Table III49 Eddy cucrent los m the yoke caiculated by FEM and proposed mode1
Case I l i Case IV
1 Machine type
Slot pitch (mm) kz
1 Tooth width (mm)
1 M a p a thickness (mm)
.. .- . 1 Slots per pole per phase q
Inner radius of the stator r (mm)
1 Tooth height d, (mm) -- 1 Y o b thichess d, (mm)
1 Correction factor k,
1 Correction factor k,
Total yoke eddy current loss by FEM (W)
I TOM yoke eddy loss by prop~ed mode1 (W)
3.6 Simplified Hysteresis Loss Model
When t h e steppeci FEM is performed, the hysteresis loss in the stator yoke and the
stator teeth cm be easily calculated using (2.29).
The teeth and yoke hysteresis l o s can dso be caiculated using the folIowing equation
respectively :
where Bt and B, are the average of plateau flux densities of al! elements in the teeth and the
average of plateau flux densities of al1 elements in the yoke respeçtively.
Table III-20 shows the cornparison of caiculated hysteresis loss by using FEM and by
using average flux density for the Iinear arranged PM motor at 120Hz with hysteresis loss
constant kh=44 and 8=1.9. It cm be seen that the crror between the two methods is negiigible.
Table DI-20 Hysteresis losses cdculated by FEM and by average flux density at 120Hz
Volume Hysteresis los Hysteresis loss Hm deflSit): (xl 0-3m3)
by average flux by FEM OK)
density (Mi)
Yoke 1 1.2558 ( 09193 1 47.07 1 47-01
Chapter 4
Measurement of lron Losses in PM Synchronous
Motors
In orda to validate the proposed iron loss models, the losses of two existing PM motors
were measured.
4.1 Loss Measurement of PM Synchronous Machines
Loss measurement in electncai machines has been a difficult task for electrical industry.
The electncal industry is spending an enormous amomt of money to improve the accuracy of
the testhg equipment in order to measure motor efficiency within M.5% 1341.
When mcasuring machine Iosses, it is necessary to employ precise instruments and
caliration in order to achieve higher accuracy. It is ais0 desirable to c o k t repeated sarnples
of the data at each point to compute an average value.
4.1.1 Ndoad Loss Measurement of PM Synchronous Machines
No ioad Iosses of a PM synchronous motor mclude stator iron losses, mechanical lasses
(wmdage and fiction losses), and stator copper 10s. For snrf8ce-mounted PM synchronous
motors, rotor iron l o s is negligr'ble. For inset and burieci PM synchronous motors, the rotor iron
Ioss m u t also be considered.
In IEEE STd 1 15-1 995 [35l it States that the core 10% a . each value of armature voltage
is deteRnined by subtracting the fiction and windage loss f?om the total power @ut to the
machine being tested.
When a PM synchmnous machine is operated as a motor, without any mechanical
device connected to its shaft, the m d input powa, with copper l o s subtracted, represents
the combination of no load irm losses and fiction and windage loss of the machine.
Altematively, when a PM synchrmous motor is dnven by a DC motor, with the stator
circuit of the permanent magnet motor opened, the measured shaft torque multiplieci by the
speed represents the combination of no load iron losses and fiction and windage los.
4.1.2 Load Loss and Motor Effïciency
The losses of a loaded PM machine inchde copper losses, Von losses, windage and
fiction losses and stray losses.
It is known that stmy losses are due to leakage flux, mechanicd imperfection of the
airgap, non-sinusoiclal current distniution, slotting and so on. It is commonly agreed that the
investigation of stray load losses in rotahg electrical machines is far h m complete and the
data available is subjected to a high degree of lmcertainty [36].
In order to avoid the mea~urement of stray load los, it is preferred to use direct
input/output measurement [371. Although the accuracy of direct input/out measurement test
has been chaiienged by many authors and is expensive and thne c o d g , it is sa the rnost
accepteci and will be the major method in measMng motor efficiency in the next few years to
corne [37.
Whai a PM synchnous machine is operated as a motor with a mechanicd Ioad
cornecteci to its shaq by directly measming the shaft toque and input power, the efficiency and
total loss can be determined.
PM synchronous motors for variable-speed drives are Usuany designeci without dampmg
cage, Therefore, a PM motor has to be operated with closed loop control. A PWM wave is
inevitable in the control of variable-speed PM synchronous motors. The measurement ofPWM
waves is beyond the capability of traditional electrid instruments, The effectiveness of PWM
wave measurement wiU be addresseci in this chapter.
4.2 lrnplementation of the Control of a PM Synchronous Motor
In order to measure the iron losses of a PM synchronous motor when running it as a
motor, a control system was implemented.
4.2.1 System Description
The system consists of a dace-mounted PM synchronous motor built within a 2hp
induction motor fiame, a VSI and an absolute position encoder. A MC68332 mimcontroller
is used to genaate the gating. Fig.4.1 shows the block diagram of the control system of the PM
synchronous motor drive system. A sinusoidd cunent was implemented to control the motor.
Both torque control and spend control were implemented
4.2.2 Mathematical Model of PM synchronous Motors
The developed torque of a PM synchronous motor cm be expressed as:
where h. 6, yrd and yq are d-axk c m t , q-axis current, d-axis flw linkage and q-axis flux
linkage respectively.
In a balanced system with the phase-a cunent leading rotor d-axis by 8 degrees* the
torqye can be written as:
P T, = - y,l, sine 2
where Ia and y, are magnitude of stator current and flux linkage of each phase respectively.
Since the flux is constant for PM motors, the control of torque cm be achieved by
coatrolling either the amplitude of the stator cuxrent or the angle 0. It can be seen that when
£king 8 torque is proportional to the stator curent- Especiaüy when 0=9OoY torque reaches
maximum with the same supply current.
Surface-mounted cylindricai PM synchronous motors have esentidy smiilar direct-axis
and quadrature-axis synchronous reactance. Therefore, the equivdent circuit c m be shown in
Fig-4.2.
Fig.4.2 Equivdent Circuit of PM synchronous motors
4.2.3 Transfer Function
From Fig.4.2, the @valent circuit equation can be written as:
where E, is the induce phase back d, r, is the stator phase &stance, Lq is the quadratureaxis
inductance and U is the magnitude of phase voltage.
For PM machines, the excitation is fixed, therefore, back emfEo and torque equation
(4.2) cm be expressed as:
where Ce and Cm are the machine constants.
If the fiction is neglected, ai l of the torque is used to develop acceleration:
where J is the inertia of the drive system, a, is the mechanical speed of the motor.
The Laplace transformation of (4.3) through (4.6) yields the transfer fiuiction of the
motor:
where Tm and Te are the mechanical and electrical constants respectively:
Since Te is much m d e r than Tm, (4.7) cm be finther written as:
Based on nomialized parameters, the transfèr hction of the motor is shown in Fig.4.3 where
Ka = 2, 1 Ra and Z', is the base reiuctance.
Fig.4.3 Simplined tramfer function of a PM synchronous motor
4.2.4 Speed Limit for Current Control
The maximal output voltage of the inverter is ümited by the DC link voltage. This
voltage iimit wiIi set a limit to the maximal speed of the motor. The rms value of the line-line
voitage U' of the inverter can be expressed as
where ma is the modulation ratio and Ud is the DC link voltage.
The phasor diagrams of a PM synchronous motor are shown in Fig.4.4 for maximum
toque operaîion and flux weakening operation respectively. The follow equation can be derived
h m the phasor diagram shown in Fig.4.4:
(E, +I,R)' +(I,x,)' =u2
Then the speed limit of maximum torque operation cm be derÏved:
Maximum torve operation @). FIwc weakening operation
Fig.4.4 Phasor diagrams of a PM synchronous motor
4.2.5 Torque and Speed Control
The transfer function of the control system is shown in Fig.4.5.
Fig.4.5 Block diagram of the speed loop
The speed of the motor is measiired and compared to the given speed. The ermr signal
of the speed is fed to the speed PI and a cunent commmd is generated. The current command
is compared to the measiired stator current and the aror signal is fed to the current PL A voltage
command is then generated and forwarded to gengate the gating signai.
4.2.6 Current and Speed Responses
The electricai and mechanical tirne constants of the 8-pole PM motor are listed in Table
IV-1. The simdated and measured current responses are shown in Fig.4.6 and Fig.4.7
respectively. The simulated and measured speed responses are shows in Fig.4.8 and Fig.49
respectively.
Table TV4 The constants of the 8-pole motor
1 Electrical the constant Te 1 1 2 . 6 ~ 1 Mechanical t h e constant Tm 1 1.5s I
1 Cumnt sampling p&od Ti ( 0.5s 1 Speed sarnpling period Ts 1 3rns 1
It c m be seen h m Fig.4.6 and Fig.4.7 that both the simulated and measured current
response has a first overshoot happening at 3.7- with the msucimum current reaching 24A and
23.6A respectively. Both simuiated and rneasured current reach a steady state cumnt at 1 9 2 L
Time constant of current PI TI 1 5ms
While the simulated current response has ody one overshoot, the measured curent
response has some damped oscillation. This is due to the sampling delay of the control system
as well as bigh order t h e constants which were neglected in the transfer fimction of the motor
control system.
L P
T h e constant of speed PI Tp
Gain of speed regulator kp Gain of cumnt regulator KI
It can be seen fkom Fig.4.8 that the simdated speed response has an overshoot at 2.2s
with maximum speed 1225rpm and period of 2.5s. second undershoot at 4.5s with speed
94ûrpm. The steady state speed is 99ûpm. It can be seen h m Fig.4.9 that the m e a d speed
reqonses has an overshoot at 2.3s with maximum speed 1220rpm and period 2.5s, second
undershoot at 49s with speed 928rpm. The steady state speed is also 992rpm.
1.04s 1 0.3 0.2
Thmefore, it can be concIuded that the simulateci speed responses are in gwd agreement
with the measured speed response-
Step Response
Fig.4.6 Simulated cment response of the 2.5hp motor with biocked shaft
-0.01 -0.005 O 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Ume (seconds)
Fig.4.7 Measured current response of the 2.5hp motor with blocked shaft
Step Response
6 8
Time (sec.)
Fig.4.8 Simulated speed response of the 2.5hp motor
O 2 4 6 8 10 12 14 16 tirne (seconds)
Fig.43 M e a d speed response of the 25hp rnotor
4.3 Measurement of Voltage, Current and Power
While doing the experiments of PM synchronous motors, it was fond that measured
voltage, current and power by different instruments have signincant discrepancies. The
instniments employed include traditional meters (Voltmeter, Ammeter and Wattmeter), digital
rnultimeters, digital power meters and direct sampling using a computer.
The discrepancy is generated by PWM waves since PM motors are often fed by PWM
wavefonns. A PWM wave is a series of pules with dinerent widths. Traditional meters are
designed for continuous waveforms whereas PWM waves are discontinuous. Therefore,
traditional designed electricd rneters c m not give precise readings of the quantities of PWM
waves. In order to overcome this difficdty, computer data acquisition as weil as applying of
fikers is suggested to perform the measurements.
4.3.1 Computerized Data Acquisiüon
Since the accuracy of standard voltmeter and ammeta is not adequate for the
rneasurements of PWM waves, it is naessary to use instantaneous values of voltage and current
to caldate their RMS values and power. With f& A D devices and mimprocessors, the
supply voltage and amnt of a PM syncbnous motor can be converted to digital signais. The
cumnt sensor and the voltage sensor should have very low distortion and band width. The mis
value, real power and power factor can be dexived nom these instantaneous values using
computer simulation software such as Matlab.
4.3.2 Measurement of Current
Since the control strategy assures a Smusoidaf cnrreat wave form, the phase current only
consists maînIy the fimdamental. Fig.4.10 shows the measured phase current waveform of the
2.5hp PM motor at 1080rpm.
Sampled phase cunent
Tirne, (Seconds)
Fig.4. 1 O Sampled phase current
Measurement of Supply Voltage
A LC filter between the VSI and the PM motor can filter out most of the high fkquency
hamonics supplied to the PM motor. Fig.4.t 1 shows the filter used in the 2.5hp PM motor.
Fig.4.12 shows the measured phase voltage waveform and filtemi phase voltage (the thick he)
waveform at 1080rpm and 17A The measirred amphde of 8kHz (twice the switching
frequency) at point A is ody about 1% of that measured at the output of the VSI.
Fig.4.1 1 Fiiter design
Sampled phase Voitage 80
60
40
h
J 20 E
O u
3 0)
g -20 t a
-40
-60
-80 0.008 0.01 0.012
Tirne, (Seconds)
Fig.4.12 Measirred phase voltage
4.4 Validation of Measurements Using Phasor Analyses
A Phasor diagram is a powemil tool for analyshg synchronous motors. Since a phasor
analysis is based on the hdarnentai puantities of voltage and cments, it is possible to validate
the measured values of voltage, c m t and power using phasor diagram.
From the phasor diagram shown in Fig.4.4(a),
There fore,
Smce current is sinusoida1 and Xd = Xq in surface rnomted PM synchronous motors,
rneasured cumnt, reactance and rtsistance are accurate. AU mmeasined electrical qyantities of
the motor must satisfy (4.15). Table IV-2 shows a group of data measured on a PM motor with
mi ren t meters.
Table IV-2 Measurements of the 8-pole PM motor
ûther measurements are: Xq = 2.4mH, ra=û.176Ohm, E0=28.6V. The phase voltage can
be caicdated using the data in Table IV-2 and (4.15). The calculateci phase voltage is 3 N and
powa angle is 25.4 degrees. Therefore, the measured values by cornputer are consistent with
each other. The measured voltage by rnultimeters is not consistent with 0 t h measunmmts.
Therefore, it can be concludeci that the PWM wavefom caused the muitimeters and voltmeter
hc t ion improperly.
4.5 Decomposition of lron Losses and Mechanical Losses
In PM synchronous motors, the iron loaes and mechanical losses are always present
smiuitaneous1y. The mechanical losses m permanent magnet motors, which include windage
and fiction losses, mut be known before the iron losses can be obtained. There are three ways
to perform this ta&: with the heIp of an induction motor rotor; with the heip of a replica of a
nonmagnetic stator; and perfomi the measurement before the magnets have been assembleci to
the rotor.
4.5.1 Wrth the Help of the Rotor of an Induction Machine
A PM synchnous motor is usually built within the same fiame as a standard induction
motor. Ifthe stator of a PM motor has the same cikaension as that of an induction machine, the
rotor of the mduction motor cm be inserted and the windage and friction losses measund
In this procedure, the bearing fiction will essentially be the same but not neccssarily the
windage loss. This is because the cap will be different and the dace-mounted PM rotor
does not have a smooth surface.
4.5.2 Wth the Help of a Nonmagnetic Stator
Tseng et al suggested an iron Ioss me;isurement method by replacing the stator with a
nonmagnetic replica made of hard plastic with exactly the same geometricd dimensions as the
originai stator [IO]. When the original stator is assembled, the measrired losses include the stator
iron losses and mechanical losses When the magnetic rotor is assembleci with the non-magnetic
replica, the m e a d l o s only inchdes the mechanical loss. Therefore, the differerzce between
the two will be the stator iron Iosses and rotor iron Ioss.
In this procedure, the bearing fiction loss cm be dinérent if the dummy stator has a
dinerent weight. Besides, manufactitring a dummy stator for Iarge PM motors is not practicd.
4.5.3 Measuring before Magnets Assembled
A practicd and accurate method is to measme the mechanicd loss before the rnagnets
are assanbled. It is preferable to assemble the PM motor with no magnetized maguets mounted
on the srnfkce of the rotor to simulate the magnetic rotor. In this procedure, the dummy motor
has exactly the same configuration as the PM motor without iron Ioss.
4.6 Loss Measurement of the 8-pole Motor
4.6.1 Description of the Motor
A 2.5hp dace-motmted PM synchronous motor was constmcted inside a standard
145T induction motor firame. Both the stator laminations and windings were redesigned. The
ratings of the original induction motor and the redesiped PM motor are given in Table IV-3
[38]. The key distinguishing characteristics of the motor are:
The machine was designed with 8 poles and 1 dot per pole per phase.
Ml9 grade steel with 0.65mm laminations was used to construct the rotor.
MI5 grade low loss steel sheet with 035mm thiclmess laminations was used to build
the stator,
The magnets were dace-mounted.
The rated power was increased b m 2hp (induction) to 2.5hp (PM synchronous).
Table N-3 Ratings of the original induction motor and the PM synchronous motor
( Ratings and parameters 1 Induction motor 1 PM synchronous motor 1
4.6.2 Measured Losses
The mechanicd Ioss of the motor was m e a d before the magnets were assembled.
The shaft torque was recordai when only the DC dynamometer was connected to the torque
sensor at different speeds. Then the PM motor was cormected without PM magnets on the mtor.
The two motors were connected via the torque sensor and the torve for the same speed range
was recorded. The net torque of the PM motor is derived by subtracting the torque of the DC
dynamometer from the combined torque of the two motors. The windage and fiction Ioss is
then derhed. The measured windage a d fiiction los of the motor is shown in Fig.4.13.
Fig.4.13 Measined fiction and windage l o s of the &pole PM motor
The PM motor was then assembleci with magnets on the rotor. The no-load loss is
detennined by measining the voltage and current suppüed to the PM synchronous motor by an
inverter with a near Smusoidd cuuent usïng an mvata, wÏthout connectmg any load to its sha&
The losses caused by the stator currait fed to the PM motor are negligiile at no load. The
measured mput powa is d y contnbued by the stator iron Iosses and fiiction and windage
losses*
The motor was ais0 tested open-circuited, driva by a dynamometer. The toque-speed
product represents the total loss of the motor. The measured iron Iosses by both methods are
shown in Fig.4.14.
Fig.4.14 Measured iron Iosses of the &pole PM synchronous motor
It can be seen h m Fig.4.14 that the measued iron losses using torque-speed product
has a 4W offset comparing with those measured using electrical input power rneasurements.
The possible reason causing the discfepancy is the mechanical setup of the test systern.
F i measurement of torqne is d y less precise than measurements of electncal quantities.
Secody, the DC dynamometer which drives the PM motor bas very large mertia compared to
the PM motor. Thirdly, there was usuaüy a mail o f k t in the torque teadings caused by the
torque rippIe of the PM motor. NewertheIess, the measurements of iron losses are within 0.2%
of the rat4 power.
4.6.3 Simulated lron Losses
In order to test the proposed loss model, FEM analyses were performed on the PM
motor. The coefficients needed to calculate the iron losses are shown in Table IV-4 [3 8,391.
Each component of the iron losses at diffierent speeds was calculated by FEM as shown in Table
IV-5 and was calculated by proposed model as shown in Table IV-6.
Table IV-4 Parameters needed to calculate the iron losses (W)
Table N-5 Iron losses pdcted by FEM for the €!-pole rnotor(W)
Table IV-6 Iron losses predicted by simplined model for the 8-pole motor0
It can be seen that the simulated iron losses by the two methods are in good agreement
and the errors are within 3%.
4.6.4 Comparison of lron Losses
In order to compare the measined and predicted lron losses, the measined and prdcted
iron losses of the motor at different speeds are shown in Fig.4.15.
It can be seen from Fig.4.15 that the iron lasses predicted by both FEM and the
simplified mode1 are consistent with the measured iron losses over the whole speed range, the
difference being aiways within 5%.
Fig.4.15 Comparison of cdcuiated and measiired Ioss of the 8-pole motor
4.7 Losses Measurement of the CPole Motor
A 4-pole 5hp dace-mounted PM synchronous motor was also employed to do the
experiment. The machine was built within the hune of a %p standard induction motor and the
same stator was used for the PM motor. The original induction motor was totally enclosed fan
cooled type motor. The ratmgs ofboth the induction motor and the PM motor are given m Table
IV-7 [40]. The stator laminations are 0.65mm steel sheets with higher Ioss coefficients.
Table N-7 Ratings of the Shp induction motor and PM synchronous motor
1 Ratings and parmeters 1 Induction motor 1 PM motor
Voltage (line-line) (V) 208 183.6
Rated current (A) 16.7 16.7
Rat& m u e n c ~ (Hz) 60 60
4.7.1 Simufated lron Losses
The iron losses of this motor were calculated by FEM and the proposed model. The
coefficients needed to predict the losses are Iisted m Table IV-8 [8,39,40]. Both k, and kh are
d.erent fkom that of the &pole motor due to the change of lamination thickness.
Table N-8 Parameters and coefficients needed to predict iron losses (W)
1 Eddy 1 0 s constant R, 1 0.07 1 Hysteresis constant 1 44
C Yoke volume Vy 12827T
e
h
0.8382xiO-~m~ Yoke flnx densiîy Byk
The pradicted iron losses at different speeds by the two methods are listed in Table IV-9
and Table IV40 respectively.
Table IV-9 Iron losses predicted by FEM for the 4pole m o t o r 0
Teeth, eddy cunent l o s 0.5 1.9 4.3 7.7 12.0 17.3
Teeth, h y s t d s 10s 1.8 3.6 5.5 7.3 9. 1 10.9
Yoke, eddy current Ioss 0.5 2.0 4.5 8.0 12.6 18.1
Yoke, hysteresis loss 3.7 7.5 11.2 14.9 18.6 22.4
Total iron losses by FEM 6.5 15.0 25.5 37.9 52.3 68.7
Table IV40 Iron Iosses predicted by simplified mode1 for the bpole rnotor(W)
s~eed (r~m) 300 600 900 1200 1500 1800
Teeth, eddy current loss 0.6 2.3 5.2 9.3 14.5 20.9
Teeîh, h y s t d s loss 1.8 3.6 5.4 7.2 9.0 10.8
Yoke, eddy currait loss 0.5 2.2 4.9 8.7 13.6 19.6
Yoke, h y s t d s l o s 3.7 7.4 11.2 14.8 18.6 22.3
Total iron Iosses by l 6-6 1 1 26.7 1 40.1 1 55.7 1 73.6
&nplifi.ed mode1
It can be sen that the predicted iron losses of the motor are m good agreement.
are within 7% or l e s than 0.2% of the rated power.
4.7.2 Cornparison to Measured lron Losses
The emrs
The mechanical loss of the motor was measured before the magnets were assembled.
The measured mechanical losses at different speech are shown in Table IV4 1.
Table IV-1 1 M e a d mechanicd losses of the 4-pole PM motor (W)
Total mechanical loss
Then the iron losses were measured by feeding the motor with a inverter and ciriving the
motor with a dynamometer respectively. The measured iron losses of the motor are shown in
Fig.4.16 with predicted iron losses by proposed model and FEM method.
Fig.4.16 Cornparison of calculateci and m e a m d losses of the 4-pole motor
It can be seen that the measured iron Losses by both rnethods, the predicted iron Losses
by FEM and the predicted loss by simplined model are in good agreement. The difference is
aiways within 13.5% or l e s than 0.2% of the rated power.
4.8 Discussion
From the experimental data of the two PM synchronous motors, one can observe a gwd
agreement between the Uon losses predicted by the proposed model, the FEM d t s and the
experimental data The error is less than 0.2% of the rated P O W ~ , which represents a great
improvemeat over the conventional approach.
It cm also be seen that the iron losses predicted by the proposed model are generaliy
Iess than the measured Von losses. This discrepancy may be due to the fact that the rotor iron
losses wexe not taken mto account in the proposed hou loss model.
Both the proposed model and experimental rwults are quoted for the no-load condition
ody. When the motor is loaded, the iron Iosses d l be induced not only by the PM field but
also the field induced by the stator currents. The magnetic flux density in the stator teeth and
stator yoke wiil be changed. The combined field may also cause minor loops of hysteresis.
Hysteresis losses caused by minor loops can be predicted using the simple method proposed in
[411.
Further more, not only the prediction of Uon losses of PM motors, but dso the
measurement of iron losses, codtutes a quite difncult task. Diffetent experimentd methods
yield ciiffirent resuits [37].
As pointed out at the beginniag of this chapter, the improvement of efficiency
measurement of 0.5% in electrical machmes with efficiency of 95% represents an ermr of 10%
in the measured total tosses. Therefore, the confidence in the loss rneasurement in electncal
machinery industry is relatively low regardles the high effort in the process.
The FEM is ais0 a numencd approximation of the magnetic field due to the limitation
computmg resources.
Even if the losses codd be accurate1y predicted and measlired, there wodd be practical
difficdties. For example, iron tosses are dnectly relateci to the characteristics of the
fmmagnetic steel sheets used in the motor, which are different for the same kmd of material
mannfactured at a different time due to mechanical stresses and temperature.
The most important aspect, f b a the engineering point of view, is to explore the
mechanism, tendency and fht indication of the associated problems, bring d o m the complex
problem mto a manageable problem with SUfficient accuracy, not so much to determine the
exact values. The thesis study briugs a closa yet simple prediction towards the insight of Uon
loues in PM synchronous machines.
Chapter 5
Minimiration of lron Losses in PM Synchronous
Motors
5.1 Qptimization of Electrical Machines
Elecîrical motor-drîven equipment utilizes approximately 58% of the electrical energy
consumed [42]. Among this electricai motor usage, 93% is consurneci by mduction motors with
power ratings of 5.1 hp and greater, 5% is consumeci by induction motors with power rathgs of
Shp or less. DC motors and synchronous motors c o r n e only 2% of the total electrical motor
usage [43]. From 1982 to 1992, the cost ofelectrical power has increased 120450% [44]. This
trend has for& many manufacturers to address the problern of improving the efficiency of
electrical motors,
Smce most of the electncai motor usage is induction motors, the focus has been placed
on the opthkation of induction machines. However, induction motors have their own
limitations, e.g., the rotor slip loss always exists. This has lead to a search for alternative
approaches. One important approach among them is to use variable speed PM synchmnous
motors,
W1th proper design, a PM machme will genefauy have higher energy efficiency than any
P
other type of rotahg machines of equivdent power output. This is due to the efimuiatioa of
field ohmic Ioss as compared to DC excited machines, and due to the eIimination of the rotor
loss as compared to induction motors. PM machines not only have the advantage of higher
eniciency, but also have many other advantages such as less volume and weight, higher power
factor and easiex control.
There was vay little interest in PM machines for drive systems two decades ago. It was
only when new rare-earth PM materiai with their high energy products and low cost became
available that ind- recognized PM machines were not just low power devices of minùnal
commercial importance [45]. With the advances and cost reductions in modem rare-earth PM
matai& and power electronics, it is now possible to consida using motors in applications such
as pirmps, faas, and cornpressor drives, where the higher initial cost can be rapidly paid back
by energy savings O fien within a year [46].
The overail optimization of a PM synchronous motor requires balance between the
rnanufactlrring cost, efficiency, power factor and torque capability. In most casa, these
parimieters conflict with each other and a compromise has to be made. The losses of a PM
synchronous motor can be decomposed into four components, namely stator winding los, iron
loss, mechanicd loss and stray load 10s. These losses cm be reduced either by using better
quaiity matenals, or by optimization of the motor design, or by both.
In this Chapter, some design approaches are praposed to reduce the iron losses of PM
synchronous motors by maintaining a maximum torque and maximum efficiency. The proposed
methods include propa design of magnets, dots and number of poles.
5.2 Design of Magnets
5.2.1 Magnet Edge Beveling
Beveling of magnets wiU change the dope of the teeth flux waveform. In order to test
110
this, FEM analysis wexe perfiormed on a 4-pole motor with varying fl of magnet ed& as show
in Fig.S.1. The dashed luie on Fig.S.1 is the origmal magnet with rectanpuiar edges.
Fig.S.1 The beveiing of magnet edge
In order to obtain consistent d t s , the magnet volume is kept constant when changing
the magnet edge. Table V-1 shows the caiculated iron losses of the motor by FEM for different
magnet edges.
Table V-1 Tooth eddy cumnt loss vs. magnet beveling at at 120Hz
90" (no beveiing) 1 Bevel degrees
1 Tooth eddy current 10s (W)
1 Tooth hysteresis loss(W)
1 Yoke eddy current Ioss (W) 1 Yoke hysteresis l o s (W)
1 Total iron l o s (W)
Total airgap flux per pole 1 (x1OJWb) 1.3773 1.3777 1.3789 1.3820 I I I
I Reduction of tooth eddy c m t loss (%)
Total reductioa of ùon Losses(%)
It cm be seen h m Table V-1 that when the bevel of the magnet edge is increased, the
tooth curent l o s demases. When the magnet is beveled at 27", the tooth eddy cmrent loss is
reduced by 9.5%; the total iron losses are reduced by almost 4% while the total airgap flux and
o h components of iron losses are kept nearly constant. Fig.5.2 shows the flux distriiution of
the iavestigated motor with non-beveled and beveled magnet edge respectively.
Fig.5.2 Flux distnhtion of the motor with un-beveled and beveled magnet edge
5.2.2 Shaping of Magnets
Baseû on the above observations, similar but aitemative approaches c m be employed
to achieve the same effect as magnet edge bevehg.
One method is to use cwed magnets with the greatest thichess in the center and the
thinnest at the two ends as shown in Fig.5.3n). This will impmve the mature of the flux
wavefomis both in the teeth and m the yoke. Therefore, the eddy current los c m be reduced.
The drawback of the method is that it will increase the difficdties and cost in manuf'turing
the magnets.
Another approach is to parallei magnetize the mapets or to gradient magnetize the
magnets with the flux mcreasing h m two ends to the middle of the magnets. The drawback of
this appmach is that the magnets are not fbiiy ntillzed.
(a) Beveled magnets (3) Curved magnets
FigS.3 PM rotor with beveled or curved magnets
5.2.3 Magnet Width and Magnet Coverage
In Chapter 3, it has been shown that changing magnet width over a wide range does w t
change the maximum flux density achieved m the teeth, nor does it change the dope of the flw
waveform in the teeth. However, when space between the two magnets is Iess than one dot
pitch, the dope of tooth flux begins to mcrease. EspeciaLiy when magnet coverage approaches
1.0, the dope of the normal flux in the teeth is dramaticaiiy increased thus tooth eddy current
losses are sigdicantly increased. Therefore, for a m-phase PM motor with q dots per pole per
phase, the maximum magnet coverage to have maximum fimc linkage but low tooth eddy
current l o s is
Smce yoke eddy cmrent Ioss Ïs approxbmtely mversely proportionai to magnet
coverage, magnet coverage should be large enough in orda to mimmize yoke eddy current loss.
On the other han& magnet width ais0 has significant effects on cogging torque [47].
According to [473, the optimal rnagnet coverage to achieve the least torque cogging is:
where n is any integer which satisfies acl .
Combining (5.1) and (5.2), the optimal magnet coverage can be obtained. For example,
for rnoton with two dots p a pole per phase (m=3, q=2), magnet coverage that satisfies (5.1)
and (52) is a4.527 or a4.693. In order to minimize yoke eddy current los, the o p h a î
magnet coverage is a4.693.
5.3 Design of Slots
5.3.1 Number of Slots
It was fond in Chapter 3 that for a given supply fiequency, tooth eddy current loss is
proportional to the number of slots per pole per phase. Hence, it is beneficid to use fewer but
wîder teeth. The minimal possiailty is to utilize 1 dot per pole per phase.
On the other han& for a reasonably good approximation to a sinusoidal distribution of
induced emf, the nimiber of slots per pole per phase should be at Ieast 2 [48].
5-3.2 Slot Closure
By increasing slot closure, the maximum vaIue of airgap flux density can be increased
if the same amount of magnet materiai is used The flux distniution of the &pole PM motor
with ciiffirent sIot closure is shown in Fig.5.4. The Qon losses in Fig.3.9 dso show that whai
the airgap flux is kept constant, the iron losses are about the same for different dot closures.
Smali dot openings are widely adopted in most rnotor designs. The reason is that
although reducing dot openings does not reduce the stator iron losses of the motor, it increases
output torque and reduces the rotor eddy current Ioss.
However, totally closed dots, as shown in Fig.5.4, wilI increase leakage flux in the
shoes of the dots. This flux doesn't contribute to the generated toque but wiU increase the iron
loss. Therefore, totally enclosed dots are not recommended.
(a). Totally encloseci dots (b). Partiy closed dots
Fig.5.4 Flux distriion with diffkrent s1ot closure
5.4 Number of Poles
Variable speed drives are fk h m the number of poles constraint Unposeci on ac motors
operated at nxed fresuency. The mverter cm operate at fkquencies h m a few Hertz up to a
few hundred Hertz. This aliows the number of poles to be selected fieely to achieve the best
motor design, By increslsing the number of poles, the length of the end tum windmgs is reduced
and thus the copper usage can be reduced and the stator copper loss is reduced. The thicknes
of the stator yoke is dso reduced and therefore the usage of core material for the yoke is
reduced. More powa output or torque can be ehieved with more poles for the same h m e due
to the reduction of yoke thickness and/or an increase in the inner radius of the stator.
Siuce the operathg fhquency incfea~es proportïonally to the nimiber of poles in order
to keep the desired speed, the eddy cunent Ioss mcreases despite the fact that the mass of the
core material is reduced.
In order to have a monable cornparison base, the foIlowing anaiysis assumes a given
fiame and speed. This assumption results in a fixed machine core length and a constant copper
loss and fiction and windage los.
5.4.1 lron Losses and Number of Poles
Total number of slots is usuaiiy hnited by the frame size. Whm the total nurnbs of
dots is nxed, the number of dots per pole per phase is inversely proportional to the number of
poles. The tooth eddy current loss and tooth hysteresis loss can be wdten as:
where S is the total stots of the stator
S = mpq (5.5)
Therefore, both tooth eddy current l o s and tooth hysteresis l o s is proportiod to
nimiber of poles.
Yoke volume can be expressed as a fiinction of mmiber of poles:
whem 9 is the outer radius of the stator, dy2 is the yoke thickness of the motor with a 2-pole
conngrnation.
Yoke eddy curent loss and yoke hysteresis loss cm be expressed as:
Therefore, yoke eddy current loss is proportional to number of poles while yoke
hysteresis loss remains roughly constant when the number of poles is changed.
Take the onginai fhne of the &pole PM motor as an example. The iron losses are
calcdated and Iisted in Table V-2 as hctions of the number of poles. The total number of slots
is S=36. The possible number of poles is 2,4,6, and 12. It c m be seen h m Table V-2 that the
iron losses of the motor increase by 24W if the nmbe r of poles increases by a factor of two.
Table V-2 Iron l o s as a hction of number of poles
Number of poles 2 4 6 12
30 60 90 180 - -- - - . -
Tooth eddy currekloss 01 8.7 - 17.3 1 26.0 1 52.0
Yoke eddy cunent loss (W) 1 8.6 1 18.1 1 27.5 1 55.9
Tooth hysteresis los (W) 1 5.5 1 10.9 1 16.3 1 32.7
Iron losses 0 1 44.0 1 68.7 1 92.6 1 163.7
5.4.2 Toque and Number of Poles
The torque of a PM motor is [18]
where BIg and KI, are the mis values of the airgap flux and the hear current density on the
stator inner periphery.
The iinear current daisity cm be expressed as a function of stator current [49]:
where W is the total number of tums for each phase, r is the h e r radius of stator.
Therefore, motor torque is a function of rotor radius? airgap flux density and stator
current. If constant airgap flux density and stator current are assumed, then
Assume that the height of dots is kept constant. Than the inner radius of the stator can
be expressai as a fimction of the amber of poIes:
It can be seen h m (5.12) that whenp is mail, the increase of r is significant as p is
increased. Therefore, torque can be signincantly increased by increasing the number of poles
assumingp is smali.
Take the frame of the 4-pole PM motor as an example. The torque is
2 T = k,(rj --d,, - d,) = 332 x (0.095 - 0.0696/ p - 0.0172) P
The ~ t i f i e d torque of this motor as a fiuiction of the nimiber of poies is shown in
Table V-3 .
Table V-3 Torque as a hction of number of poles
It can be seen h m Table V-3 that the torque increases by 40% and 10% respectively
when the number of poles increases from 2 to 4 and h m 4 to 6, while the torque oniy inmases
8.6% when the number of poles inmases h m 6 to 12.
5.4.3 Optimal Number of Poles
The optimal number of poles may Vary for dZEerent applications. The criteria are
whether an optimal efficiency and a maximal torque are achieved. The motor efficiency is:
Output power Pm and iron loss p h , are fiinctions of the number of poles p. Copper
loss and fiction loss are assumeci constant. The optimal efficiency can be found by solving
Take the same example fmm the last section. The combineci copper, fiction and
windage losses are 289W. The efficiency can be expresseci as
The torque and efficiency of the motor are iisted in Table V-4. It cm be seen that the
choices of 4 and 6 poles are optimal amongst di possible choices.
Table V-4 Torque and efficimcy as fimctions of the ninnber of poles at 18Oûrpm
The choices for the numba of poles are different for each application. It requira the
balance ofmataial usage, achieved maximal toque, efficiency and manufacturing cost Usuaily
the best choices are four, six and eight poles among all the possibilities for a PM rnotor design.
In smaii machines, the number of poles is limited by the maximum number of slots which is
limiteci by the minimum practicai width of the stator teeth and slots.
Chapter 6
Conclusions
6.1 Contributions and Conclusions
The contributions of the thesis are:
(1). Developed renned approximation models of iron losses for dace-mounted PM
synchronous motors taking into accomt details of the geometry of the design. Based on the
models, the evaluation of iron losses of surface-mounted PM synchronous motors can be
perforrned without the high effort of a the-stepped FEM. These models also give motor
designers an easier but more precise tool to predict iron losses of dace-mounted PM
synchmnous motors in their initial design depending on the geometry of the design.
From the deveIopment of sirnplined Kon Ioss models of surfiace-mounted PM
synchronous motors, the foiiowùig conclusions an made for tooth eddy current Ioss:
For rectanpuiar edged rnagnets, the £iw density in the teeth of a PM synchronous
motor is approximately udionn. The nomai component of tooth flux density can
be approximated by a trapezoidd waveform.
For the approximated tmpezoidal flux waveform of the normai compoxient of tooth
flux demsity, the time needed to rise fkom zero to the plateau is approxùnately the
time that a magnet edge traverses one slot pitch
The eddy current loss in the teeth of a PM synchronous motor can be estimated by
first considering the normai component of tooth flux density and then a correction
factor for the circumferential component.
For a given operathg frequency, the tooth eddy current loss of a PM synchronous
motor is proportional to the number of dots per pole per phase.
For a given torque and speed rating, the tooth eddy cment loss is approxirnately
proportionai to the prcxiucts of number of poles squared and dots per pole per phase.
The tooth eddy current loss is independent of the angular width of the magnet.
Chanping tooth width over a wide range within the same dot pitch wiII not change
the tooth eddy current loss of a PM motor.
The tooth eddy current loss is not affiected by the slot closure.
A correction factor cm be introduced to reflect the effect of airgap length, slot pitch
and magnet thickness of the motor.
The following conclusions cm be made for yoke eddy current loss:
The circumfefentid component of yoke flux density is approxhately evedy
distn'buted over the thickness of the yoke. It can be approxirnated by a trapezoidal
wavefom. The time needed for this component to rise h m zero to the plateau is the
time that a point m the yoke passes a haifof the magnet width.
The normal component of yoke flux density has the same waveform as that of the
normai component oftooth flux density. The plateau value of the yoke flw density
is IinearIy distn%uted over the thichess of the yoke with zero at the d a c e of the
yoke and maximum near the tooth.
The eddy current 1 0 s in the yoke can be approxhated by first considerhg the
circUaLfere~ltial component and then a correction factor for the nonnai component.
For a given fiequency, yoke eddy current loss is proportional to the inverse of
magnet coverage.
For a given torque and speed rating, yoke eddy current loss is proportional to the
number of poles squared.
Major geometricai variations such as magnet thickness, airgap Iaigth, number of dot
per pole per phase and tooth width of a PM synchronous motor have Iittle effect on
yoke eddy current loss.
(2). Perfiormed measurements of uon losses of surface-mounted PM synchronous
motors. Some methods were suggested to decompose the iron losses h m the total no load
losses of a PM synchronous motor. Measunment of PWM waves was discussed and phasor
analyses were proposed to validate the measurements of a PM synchronous motor.
(3). Roposed general approaches to reduce iron losses of PM synchronous motors. The
iron losses of a PM motor cm be reduced by implernenting one or more of the following
recommendatiom:
Beveled magnet edge, cmed magnets and optimized magnet coverage will d u c e
the iron losses of dace-mounted PM synchronous motors.
Using fewer but wider dots can reduce tooth eddy current los.
Ushg more poles can genedy increase the torque of the motor. Optimal numbers
of poles are 4,6, or 8 dependmg on the appiication.
6.2 Future Work
Suggestions for the futine studies are given below:
(1). Modeling of iron losses in PM synchronous motors with buried magnets, inset
magnets and circdérentid magnets.
(2). Deteduhg the iroa losses of Ioaded PM synchronous motors, including both
calculations and experiments.
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