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

The author has p t e d a non- exclusive licence dowing the National Li* of Canada to reproduce, Ioan, distnbute or sell copies of this thesis in microform, paper or eIectronic formats.

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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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