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 pteda non- L'auteur a accordé une Licence non exclusive licence dowing the exclusive permettant à la National Li* of Canada to Bibliothèque nationde du Canada de reproduce, Ioan, distnbute or sell reproduire, prêter, distriilmer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or eIectronic formats. la forme de microfiche/film, de reproduction sur papier ou sur format electronique.

The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qai protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reprodnced without the author's ou autrement reproduits sans son pdssi01l. autorisation. 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 les computation time and are readily available du~gthe 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 cmbe 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 dts

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 thaWto 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 Mtute 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 deredduring 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 hm 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 ...... 11 2.3.3 Two Dimensional FEM Formulation of Magnetostatic Roblems ...... 12 2.3.4 Magnetization ModeIs of Magrets...... 14 2.4 Time-Stepped FE'of PM Synchronous Motors...... 15 2.401 C0mcti0~1of 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 CumtLoss ...... 23 2.5.2 Hysteresis Los...... 24

Chapter 3 Shpiified Lroa Loss Mode1 ...... 25 3.1 Review of Analfical Iron Loss Mode1 ...... 26 31Test 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...... C53 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 CmtLos Mode1...... 69 3.5.1 Eddy Current Los 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 ...... 74 3 .5.7 Effkct of Tooth Width 6thFixed Slot Pitch ...... 74 3.5.8 Eddy CmtLoss 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 Cmt...... 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 hnLosses ...... 102 4.6.4 Comparison of Iron Losses ...... 103 4.7 Lctsses Measurement of the Wole Motor ...... 104 4.7. I Simulatecl Iron Losses ...... 104 4.7.2 Cornparison to Measured Iron Lasses ...... 105 4.8 Discussion ...... ~107 Chapter 5 Mhhbtion of Iron Losses in PM Synchronous mot ors...... ^...... 109 5.1 Optimization of Electrical Machines...... 109 5.2 Design of Magnets ...... 110 52.1 Magnet Edge Beveling ...... 110 5.2.2 Shaping of Magnets ...... 112 5.2.3 Magnet Width and Magnet Coverage ...... 113 5.3 Design of Slots ...... 114

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 ...... ~...... ~...... 118 5.4.3 Optimal Number of Poles ...... 119 Chapter 6 Conclusions ...... ~...... 121 6.1 Contributions and Conclusions ...... 121 6.2 Future Work ...... ~...... 1...... 124 LIST OF FIGURES

Fig.2.l Rotor structures of PM synchronous motors ...... 7 Fig.2.2 Magnetization models of permanent magnets ...... 14 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 los...... 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 10svs . 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 hction 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 hedpole 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 ...... 55 Fig.3.22 Radial component of tooth ffwc 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 circ~erentialcomponent 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 Werfunction 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.6Shulated cumnt response of the 2.5hp motor with blocked shaft ...... 92 Fig.4.7 Meadcurrent 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-4-10SampIed 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 ...... 19 Table II-3 Parameters and dimensions of the 8-pole motor ...... 21

Table III-1 Distance needed for tooth flux to rise hmzero 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 (~6ds)...... 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 cmt10s vs .tooth width at 12OHz...... SI Table III-9 Eddy current los 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 120Hz ...... 72 TabIe III44 Effect of number of dots with fked pole pitch and variabIe dot pitch at 220HZ ...... *...... 73 Table III-15 Effit of magnet thickness on yoke eddy current loss at 120Hz ...... 73 Table III-16 Effect of airgap length on yoke eddy current los 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 los 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 Theconstants of the 8-pole motor ...... -91 Table N-2Measurements 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-7Ratù5gs 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-4Torque 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 los constant Empiricaiiy determined hysteresis 10sconstant 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 Ecircumferential 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 thestepped FEM Mitlcimum operation speed Total -ber of dots used in hestepped FEM for halfperîod Nmnber of poIes Coppa loss Eddy curent loss density Tooth Eddy current 10s 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 los induced by radïaVnormai component of tooth flux density calcuIated by Approximation mode1 PetrFEMEddy cunent loss induced by radiaUnorma1 component of tooth flux density calculated by FEM Pq Yoke eddy cunent loss P, Eddy cmtloss 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 herradius of stator Stator phase resistance Out radius of stator Magnetic field region Number of stator dots The Time interval used in timc stepped FEM Rise time of tooth or yoke flux density Period EIectricai time constant Samphg paiod Theconstant of cunent PZ Mechanical time constant Theconstant 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 Slot openings Tooth width Subscripts for x and y component respectively Distance needed for flux density to rise hmzero 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~moantedpermanent mapet (PM) synchronous motors.

PM synchronous motors have higher efficiency than standard induction machines with the same power rating and the same he.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 hite 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 bnlosses 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 dyticai 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 los is proportional to the square of thechange 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.

Badon 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~oachto 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 meadwhen 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 necvto 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 circumfdai 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 ddty. 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~chireto 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 rare4PM materiai, neodymium-iron-boron (NdFe-B), has 1.4T of cesidual flux, I IkAh 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 spdIn 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 show 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 daceif 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 sevdtimes the magnets Qing off the rotor surfaces during the operation of PM motors. One effivebut 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 doy 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 dacemomted magnets.

It is possîle to build the motor with cin=urnférentially magnetized buried magnets as shown in Fig.2.1(d). The ovdcost 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 los (fiction and windage 10s); iron los (open circuit cmlos); 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 los is due to the existence of a hysteresis loop in the core B/Hcharacteristics. 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 los 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 fhn 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 dyby the fimdamentai firesuency variation of the magnetic field [21]. The measured Bon losses in electncd machines are usually much larger than that calculateci hmthe Smusoidai assumption. The thesis wüi be focuseci on the evduation ofiron losses of PM synchronous motors. Massive experïmental dtsshow 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 hystds loss and the latter is usually Med 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 @roportionaIto 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; osis the anguiar fkquency of the magnetic field; Rh, k, and P are empirically determined constants which can be obtained by curve fitting hmmanuf~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, kd.01-0.07.

An expression for the classical eddy cment loss can also be developed based on the resistivity of the core material [21]. The resuit is gendyfond to be less than that hmthe 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 domnor 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~ltIoss. 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 cmtloss 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 nmahthe most powerfil tool to calculate electromagnetic field distributons and field-related parameters [26].

Thestepped 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 ail loss predictions with experiments.

The 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 Md2's Equations. The differential form of Md2'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 cmbe 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 cm 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 hm(2.15). In region S enclosed by unary boundary II 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 me[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 hmuifolytica 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 dacemounted machines is les 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 pdelmagnetized 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 cmbe 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 sWgof meshes simulates the actdmotion 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 berstator 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 eqdto 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- a = O. This is the naîural botmdary condition which wiii be satisfied 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 refdto 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 hear PM synchronous motor

Fig24 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 Mycovered 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 Worm 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 dace- 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 hm the radial direction to the circulzlferential direction. As the magnetic flux penetrates Merinto the yoke, the circumferential component begins to dominate and cm be regarded as being essentially uniform in the circUIllferentia1 direction. The cin=umferentiai flw seems to be dodydistriiuted 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 meum when the tooth is above the middie of the magnet-

The magnetic flux is also mdyin 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 hm(2.2):

Eddy cumnt loss cm be obtained by discretking (2.24) in the the 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 theinterval between two consecutive FEM steps is

where fis 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 hmexperirnents by [22]:

By performing the-stepped FEM anaiysis, the peak value of flux density Be,- du~g a period in each element can be found. The hysteresis loss in the stator is:

2.5.3 Total lron Losses

The total hnlosses 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 dtscan then be used to develop and validate SimpIified iron losmodeis. 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 prebarydesign 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 dyavaiiable 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 hnlosses 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 dts and adfical 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 dacemounted 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 hmzero to a maximum and then remain essddly constant while the magnet passes. At the Iagging edge, the tooth flux drops hmmaximum 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 los 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.

methe ovdapplication 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 hmthe middle of tooth to the middle of the space between the two poles. It can be seen hmFig.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 hntedge of the magnet appmaches the tooth (d6) 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 hntedge 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

Ii A6 @). When the mapet reaches the middle of the tooth

I &Y (c). When the tooth is covered by magnet

I d2i (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 hmFig.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 the 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 as the actual flux detywavefom. The rise thne of flux density for this approximation can be obtamed by comparing the eddy current loscalcuiated 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 pa 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 ofthe tooth flwt density.

A conclusion can also be drawn hm(3.1) that the eddy current los is inverseIy proportional to the rise the ofthe hear approximated flux density.

Dis tancelpole pitch

Fig.3.4 Approximation of tooth aior density wavefonn

The rise thefor the hear flux can be determitmi by (3.1) using the eddy current 10s 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 cmbe obtained. Since it takes halfa period for the magnet to traverse a pole-pitch, a halfperiod in the thedornain 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 hmzero 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 papole. So one slot pitch is 0.167 pole-pitch. It can be seen hmTable IIi-1 that the eq@vaieat distance for the normal component of tooth flux density to rise hmzero 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 hmzero to plateau

Method Obtained by (3.4) Mcasurcd on Fig.39 Suggested in [8]

Wr O. 1600 0.1420 0.0833

hxlh 0.960 0.852 0.500

Table III-2 shows the eddy cunent los induced by the normal component of tooth flux density calculateci by diffefent methods. Table III-2 Eddy current loss induced by the nodcomponent of tooth flux density

Use dot Use actuai Use Suggested Method used pitch slope sinusoicial in [8] Eddy current loss at 72.7 70.4 82.4 29.0 142.1 l20Hz or 12.lm/s (W)

Emr (%) - -3.1 13.4 -60.1 95.5

Et can be seen fiom Table III-2 that

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 (mrof 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.

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 los is almost Wce of that calculated by FEM.

Therefore, it can be concluded hmthe above observations that:

The waveform of the normal component of tooth flux density can be appmxhaîed by a lin- trapezoidal waveform.

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 hmzen, to plateau value is approximately one dot pitch, 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 hmzero to plateau is appmnimateiy the thefor 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 los cm be Merwritten 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 hmFig.3.7 that when sht closure or dot opening changes, the achieved plateau value of the normal component oftooth 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. Badon 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 mitmagnet 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 cmtIoss m the tmth is consistent with the srnaIl change of slope of the nomai component of tooth flux density as seen hm 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. t I 0 l 1 Closed dot. Open dot i 2 t I I 6 60 c...... 1 w 1 Q, i 5 ! O l io 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.167 to c~û.990.The pdelLmes are drasvn to show the dope of the flux. It can be seen hmFig.3.10 that, when the magnet coverage is within the range of adl.167 to d.833, the major part ofthe 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 hmzero to plateau value. Although the slope is the same, the flux density does not mach the plateau value when the rnagnet coverage fdsto a~û.167or 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 papole). 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~entmagnet widths, the flux waveform foilows a lin- pattern with the same dope except for the widest magnet coverage d.860. 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.405.

It can be concluded hmtht 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 hm1 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 mencyand nxed speed respectively.

The wavefoms of the normal component of the tooth flux density for eland q==2are shown in Fig.3.14 when plotted as a finiction of pole pitch. The distance needed for the flux to rise cmbe 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

Slots per pole per phase q 1 2 3 4 Distance needed (in pole-pitch) 0,284 O, 142 0.095 0.071

Distance needed (m dot-pitch) 0.852 0,852 0.852 0.852 It can be seen hmTable Iü-3 that, since the two cases have diffkrent pole-pitches, the distances are différent for ciiffit q when expresseci as a btion of pole pitch. However, when expressed as a Monof dot pitch, the distances needed for the normal component to nse hm zero to plateau is seai to be the same. Therefore, it is dcsirable to plot the flux waveforms as hctions 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 ddtyhave the same dope. It can also be seen hmboth 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, equhn (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 paphase q

The tooth eddy cumnt loss is also calculated at fied speed as shown in Table ïïI-5. It is showin Table III-5 that the ratio of tooth eddy current losto the product of number of poles squared and the nimiber of slots papole 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 mapdthickness 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 hmFig3.16 haî, with constant slot pitch,

Gien the ratio of magnet thickness to akgap length, proportionally bcreasing @ap length and magnet thickness WUdecnase 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 udyconstrained 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 derniabove 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 nodcomponmt of the flux density in the center of a tooth are shown in Fig.3.18. It cm be seen hmFig3.18 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 hmTable 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

SIot per pole per phase q I 2 3 4 Distance (in pole pitch) 0230 0,142 0.101 0,087

Although the magnet to airgap length ratio was kept constant, the ratio of tooth width

49 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 dtsare listeci in Table III-7. B can be seen hmTable 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 curwt loss calculateci by the approximation model and that cdculated by FEM. It can be seen hmTable m-8 that when tooth width is changed hm36% 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 hedslot pitch has Iittle effect to the approximation model dgother 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 cmtloss by FEM (W) 63.6 72.7 61.6 60.9 Eddy cmtloss by approximation 60.7 70.4 60.3 mode1 (W) / 1 1

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 los Ïnduced by the normal component. It was found that the eddy current los 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 los 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 hmthe 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 02 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 cmtIoss 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 papole paphase 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 los 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 losmode1 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 hmzero to maximum is 0.3 pole-pitch or 0.9 slot-pitch. In 0thwords, 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 los 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-piîch

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 1mm, 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 hm Fig.3.16 and Fig.3.20 respectively (1,16 = 3, XI6 = 16.8, y = 0, kq=1.03 and k=1.15). The ~roximatedtooth 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 hm3mm 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 hm3mm 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 hmthose 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 hmthose 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. 11, 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 los 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 III Case VI

- 1 Motor type Linear Linear I Linear Linear 1 Slot pitch I (mm) 1 Tooth width w~mm)

lot openings w, (mm)

1 Slob per pole per phase q 1 Tooth £indnuity Bth (T)

Eddy loss in the teeth by FEM (W) Eddy loss in the teeth by Ipropcwed 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: 0.113 yoke thickness fiom the daceof 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 hmFig.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 hmthe 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 hmzero tu plateau or fd hmplateau 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 hmzero to plateau or 0.58 pole-pitch hm 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. Redthat 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 longitu~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 hmzero 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% les 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

Use magnet Use actuai Method FEM Sinusoidd coverage siope Eddy current los (W) 73.3 70.1 80.6 57.7 at 120Hz or 12.1ds

Ermr (%) 0.0 -4.4 10.0 -2 1.3

It cm be seen firom Table III4 1 that

When the distance needed for longitudinal yoke flux density to nse hmnegative 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.

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:

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 hm 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 hmnegative plateau to positive plateau is appmximately the magnet width. 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 hmthe surfiice of the stator O,rnidcüe of the yoke 0,lB yoke thickness above the teeth (m)and lm. 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 daceof 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 dBmtplateau 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, hmFig.327 and Fig.328

where y is the distance hmthe 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 dacein the yoke, r, and & are projected pole pitch and projected slot pitch which are meadin the yoke. For linear motors, r, is the same as r and A, is the same as X. In rotary machines A, and .c, cmbe 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 mersimplified:

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 dkty 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

[ 3qn;

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 hntedge 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 hm-$ 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 los induced by the longitudindcirculliferential component ffux is

It can be inferred fiom (3.25) that the eddy current los 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 los by FEM and the approximation model is shown in Table III-12.

It can be seen hmTable 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 los 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 mon(3.25) and compared to that obtallied by FEM as shown in Table Lü-13. It cm be seen hmTable III43 that for the given range of magnet coverage, the difference between the estimaîed yoke eddy cmtIoss 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 los 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 cmbe drawn hmthis 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 FEMO

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 hear PM rnotor was varied hm2Smm 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 Yoke eddy nment loss by IApproximation mode1 (W) 3.5.6 Effect of Airgap Length

In order to test whehthe approximated yoke eddy current los 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 hmTable 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 hm4.4mm 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 hm Table III-15 that the yoke eddy nment los 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 cmbe 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 as 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 nodand 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 cmtloss is:

Therefore, yoke eddy cumnt los 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 hmzero to the plateau. The nse thecdculated by (3.14) is 0306 pole-pitch. These dtscoincide with each other.

The nomial component of yoke flux density takes 0.13 pole-pitch to rise hmzero to plateau. RecalI that the normal component oftooth flux daisity of this motor takes 0.128 pole pitch to rise hmzero 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.72hm 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 19.03W (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 OZpole-pitch to rise hm zero to plateau. The meadnse 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 hmzero to plateau.

By using the dimensions of the motor and kq=l.1 hmTable III-9,correction factor k, can be fouad to be 1.1 1. The predicted loss is 5.8 1W by the proposed approximation mode1 comparing to 5.75W caiculated by FEM (-1 -1% discrepancy).

Case III: A Linear Motor with Open Slot

The hem PM motor of Fig2.4 is imrestigated with FEM and the appro~ationmodel. 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

Badon Case III, the rnagnet coverage a was change hm0.659 to 0.532 and yoke thickness was changed km20mm to 16mm but keeping other dimensions the same. kris found to be 1.26. The yoke eddy current los is 90.3W 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 hm 8.4m.m to 3.2mm, magnet thickness was changed from 3mm to lOmm and airgap length were changed hm lmm to 3.2mm. Correction factor k, was fond to be 1.23. The yoke eddy cunent los cm 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 mrof 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 hm Table Ili-19 that the yoke eddy current loss cdculated by the proposed approximation model are genedy acceptable when comparing with the FEM dts.The error is aiways within f6%. Table III49 Eddy cucrent los m the yoke caiculated by FEM and proposed mode1

Case Ili Case IV

1 Machine type

Slot pitch (mm) kz 1 Tooth width (mm) 1 Mapathickness (mm)

.. .- . 1 Slots per pole per phase q Inner radius of the stator r (mm) 1 Tooth height d, (mm)

-- 1 Yob thichess d, (mm)

1 Correction factor k, 1 Correction factor k, Total yoke eddy current loss by FEM (W) TOMyoke eddy loss by Iprop~ed mode1 (W) 3.6 Simplified Hysteresis Loss Model

When the 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 los 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): by average flux (xl0-3m3) 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 coktrepeated 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 losis negligr'ble. For inset and burieci PM synchronous motors, the rotor iron Ioss mut also be considered.

In IEEE STd 115-1 995 [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 mdinput powa, with copper los 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 hmcomplete and the data available is subjected to a high degree of lmcertainty [36].

In order to avoid the mea~urementof 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 codg,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 cmt,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:

T, = -P 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 cmbe 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 Cmare the machine constants.

If the fiction is neglected, ail 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 Tmand Teare the mechanical and electrical constants respectively:

Since Te is much mderthan 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, 1Ra 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 hmthe 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 12.6~1 Mechanical theconstant Tm 1 1.5s I Time constant of current PI TI 1 5ms Theconstant of speed PI Tp 1.04s 1 Gain of cumnt regulator KI 0.2 Gain of speed regulator kp 0.3 L P 1 Cumnt sampling p&od Ti ( 0.5s 1 Speed sarnpling period Ts 1 3rns 1

It cm be seen hmFig.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 192L

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 theconstants which were neglected in the transfer fimction of the motor control system.

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 hmFig.4.9 that the meadspeed 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.

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 Meadspeed 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 cmnot 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& AD 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 EJ 20

u O 3 0) g -20 at -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, cmtand 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 mirent 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-2and (4.15). The calculateci phase voltage is 3N 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 0thmeasunmmts. Therefore, it can be concludeci that the PWM wavefom caused the muitimeters and voltmeter hction 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 meadlos 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 bm2hp (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 meadbefore 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 adfiiction los of the motor is shown in Fig.4.13.

Fig.4.13 Measined fiction and windage los 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 dycontnbued 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 hmFig.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. Fimeasurement of torqne is dyless 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 ofkt 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-4Parameters needed to calculate the iron losses (W)

Table N-5 Iron losses pdcted by FEM for the €!-pole rnotor(W)

Table IV-6Iron 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& muenc~(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.erentfkom that of the &pole motor due to the change of lamination thickness.

Table N-8Parameters and coefficients needed to predict iron losses (W)

C e Yoke volume Vy 0.8382xiO-~m~ Yoke flnx densiîy Byk 12827T h 1 Eddy 10s constant R, 1 0.07 1 Hysteresis constant 1 44 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 motor0

Teeth, eddy cunent los 0.5 1.9 4.3 7.7 12.0 17.3 Teeth, hystds 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, hystdsloss 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, hystds los 3.7 7.4 11.2 14.8 18.6 22.3 Total iron Iosses by 6-6 26.7 40.1 55.7 73.6 &nplifi.ed mode1 l 1 1 1 1 1

It can be sen that the predicted iron losses of the motor are m good agreement. The emrs are within 7% or les than 0.2% of the rated power.

4.7.2 Cornparison to Measured lron Losses

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-11 Meadmechanicd 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 meamd 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 les 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 dtsand the experimental data The error is less than 0.2% of the rated POW~,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 dlbe 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, fba the engineering point of view, is to explore the mechanism, tendency and fht indication of the associated problems, bring domthe 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 corneonly 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 Ofien 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 flof 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 dts,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 1 Bevel degrees beveiing) 1 Tooth eddy current 10s (W) 1 Tooth hysteresis loss(W) 1 Yoke eddy current Ioss (W) 1 Yoke hysteresis los (W) 1 Total iron los (W) Total airgap flux per pole 1.3773 1.3777 1.3789 1.3820 1 (x1OJWb) III Reduction of tooth eddy Icmt loss (%) Total reductioa of ùon Losses(%)

It cm be seen hmTable V-1 that when the bevel of the magnet edge is increased, the tooth curent losdemases. 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 ohcomponents 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 cmbe 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 cmbe 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 hmtwo 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 wt 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 los is

Smce yoke eddy cmrent Ioss Ïs approxbmtely mversely proportionai to magnet coverage, magnet coverage should be large enough in ordato 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 papole 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 ophaî 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 hmthe number of poles constraint Unposeci on ac motors operated at nxed fresuency. The mverter cm operate at fkquencies hma 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 hmedue to the reduction of yoke thickness and/or an increase in the inner radius of the stator.

Siuce the operathg fhquency incfea~esproportïonally 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 los and tooth hysteresis los 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 cmbe seen hmTable V-2 that the iron losses of the motor increase by 24W if the nmber of poles increases by a factor of two.

Table V-2 Iron los 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 herradius 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 hm(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 ~tifiedtorque 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 hmTable V-3 that the torque increases by 40% and 10% respectively when the number of poles increases from 2 to 4 and hm4 to 6, while the torque oniy inmases 8.6% when the number of poles inmases hm6 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 ph, 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-4Torque 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 £iwdensity 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 hmzero 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 daceof the yoke and maximum near the tooth.

The eddy current 10s in the yoke can be approxhated by first considerhg the circUaLfere~ltialcomponent 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 hmthe 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 duce 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. [Il. Bose B. K., Power Electronicr und Variable Frequency Drives: Technology and Applications, IEEE Press, 1997

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