Design of a Ferrite Permanent Magnet Rotor for a Wind Power

UPTEC ES13018

Examensarbete 30 hpJuni 2013

Design of a Ferrite Permanent Magnet Rotor for a Wind Power Generator

Petter Eklund

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Design of a Ferrite Permanent Magnet Rotor for aWind Power Generator

Petter Eklund

Due to the insecurity of the supply of raw materials needed forneodymium-iron-boron magnets, typically used in permanent magnet generators, theuse of ferrite magnets as an alternative was investigated. The investigation wasconducted by attempting to redesign a generator that previously usedneodymium-iron-boron magnets for use with ferrite magnets. The major part of theredesign was to find an alternate rotor design with an electromagnetic design adaptedto the characteristics of the ferrite magnets.It was found that ferrite magnets can be used to replace neodymium-iron-boronmagnets with changes to the electromagnetic design of the rotor. The changes of theelectromagnetic design increase the amount of magnetically active material in therotor and, therefore, require the mechanical design of the rotor to be changed. Thenew rotor design also requires some changes to the generator support structure. Adesign for a replacement rotor, using ferrite magnets, along with the required changesto the support structure, is presented.

ISSN: 1650-8300, UPTEC ES13 018Examinator: Kjell PernestålÄmnesgranskare: Sandra ErikssonHandledare: Stefan Larsson

Populärvetenskaplig sammanfattningPå senare tid har priset på och tillgången till råmaterialen som behövs föratt tillverka neodym-järn-borpermanentmagneter blivit osäkra. Denna typav magneter används i bland annat permanentmagnetiserade generatorer.På grund av osäkerheterna har det blivit önskvärt att byta ut neodym-järn-bormagneterna mot andra typer av permanentmagneter när man bygger gen-eratorer. I det här arbetet har därför möjligheterna att bygga om en gener-ator, som ursprungligen byggdes för att använda neodym-järn-bormagneter,så att den använder en annan typ av magneter undersökts. Denna andra typav permanentmagneter som användes var ferritmagneter.

För att anpassa generatorn efter ferritmagneterna måste den roterandedelen som magneterna sitter på, rotorn, byggas om eller bytas ut. Att denstillastående delen av generatorn, statorn, ska lämnas oförändrad är en be-gränsing i konstruktionsarbetet från uppdragsgivaren. Den nya rotorn måsteanpassas till att ferritmagneterna är mycket svagare än neodym-järn-bor-magneterna. Svagare magneter innebär att större volym magneter behövs,vilket gör rotorn tyngre. Dessutom måste magnetfältet förstärkas för attgeneratorn ska ge tillräcklig elektrisk utspänning.

Eftersom magnetfältet måste förstärkas, fungerar inte längre den rotortypsom tidigare använts. I den rotortypen sitter magneterna på utsidan av enstålcylinder med ena polen mot och andra polen bort från cylindern. I dennatyp av rotor sammanfaller rotorns magnetiska poler med magneternas. Enrotortyp som förstärker magnetfältet kan fås genom att placera magneternai en ring med stålstycken emellan och magneternas poler mot stålstyckena.Magneternas poler som sitter mot samma stålstycke ska vara av samma typ,alltså antingen två nordpoler eller två sydpoler mot samma stålstycke. Stål-styckenas funktion är att samla ihop magnetfältet från en större yta av enmagnets poler än vad som kan vändas mot rotorns utsida i det utrymme somfinns för en av rotorns poler. Denna ihopsamling av magnetfältet gör attrotorns poler kan bli starkare än magneterna som driver dem.

Eftersom den nya rotortypen har mer material i den magnetiska kretsen,det vill säga stålstyckena och magneterna, blir den tyngre och kräver enkraftigare stödstruktur. Till stödstrukturen kan man inte använda vanligtkonstruktionsstål i närheten av den magnetiska kretsen. Det beror på attstål är magnetiskt och kan skapa en oönskad väg för magnetfältet mellanmagneternas poler. Därför har aluminium, som inte är magnetiskt, använtstill den stödstruktur som behövs för att hålla ihop ringen av stålstycken ochmagneter och förbinda den med generatorns axel. Rostfritt stål kunde ocksåha använts men det är svårare än aluminium att bearbeta och tyngre. Efteratt konstruktionen gjorts färdig kontrollerades att alla delar var rimliga att

tillverka och det gjordes en studie för att kontrollera att konstruktionen gickatt montera.

Slutsatsen av arbetet blev att det i permanentmagnetiserade generatorergår att ersätta neodym-järn-bormagneter med ferritmagneter. För att göradetta måste en annan rotorkonstruktion användas och den elektromagnetiskaprestandan kan bli något sämre.

Contents

Executive summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 Introduction 41.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Why Replace the Neodymium-Iron-Boron Magnets . . 61.1.2 Earlier Usage of Ferrite Magnets in Permanent Magnet

Synchronous Generators . . . . . . . . . . . . . . . . . 61.1.3 Earlier Usage of Ferrite Magnets in Other Kinds of

Rotating Electrical Machines . . . . . . . . . . . . . . . 71.2 Scope and goals of the present work . . . . . . . . . . . . . . . 8

2 Theory 92.1 Simulation of Electrical Machines Using Two-Dimensional Fi-

nite Element Methods . . . . . . . . . . . . . . . . . . . . . . 92.2 Methods of Calculating Magnetic Forces on Magnetic Materials 10

2.2.1 Analytic Model for Verification of Magnetic Force Cal-culation . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Bolted Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Design Process 173.1 Method and Materials . . . . . . . . . . . . . . . . . . . . . . 173.2 Design Requirements . . . . . . . . . . . . . . . . . . . . . . . 173.3 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Results 214.1 Description of components . . . . . . . . . . . . . . . . . . . . 22

4.1.1 Rotor End Plates . . . . . . . . . . . . . . . . . . . . . 224.1.2 Pole Shoe Holder . . . . . . . . . . . . . . . . . . . . . 234.1.3 Pole Shoe . . . . . . . . . . . . . . . . . . . . . . . . . 234.1.4 Permanent Magnet . . . . . . . . . . . . . . . . . . . . 254.1.5 Magnet Holder Bar . . . . . . . . . . . . . . . . . . . . 254.1.6 Inner Support . . . . . . . . . . . . . . . . . . . . . . . 25

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4.1.7 Large Support Ring . . . . . . . . . . . . . . . . . . . . 274.1.8 Flange . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1.9 Small Support Ring . . . . . . . . . . . . . . . . . . . . 284.1.10 Shaft . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.1.11 Generator End Board . . . . . . . . . . . . . . . . . . . 30

4.2 The New Magnetic Circuit . . . . . . . . . . . . . . . . . . . . 304.3 Magnetic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 334.4 Calculation of Bolted Joints . . . . . . . . . . . . . . . . . . . 344.5 Static Stiffness of the Rotor . . . . . . . . . . . . . . . . . . . 354.6 Study of Natural Frequencies and Modes of Vibration . . . . . 364.7 Comparison of New and Old Design . . . . . . . . . . . . . . . 384.8 Unfinished Parts of the Design . . . . . . . . . . . . . . . . . . 38

5 Discussion of results 40

6 Conclusion 43

Bibliography 44

A Drawings 47A.1 Rotor bottom end plate . . . . . . . . . . . . . . . . . . . . . 48A.2 Rotor top end plate . . . . . . . . . . . . . . . . . . . . . . . . 49A.3 Pole shoe holder . . . . . . . . . . . . . . . . . . . . . . . . . . 50A.4 Pole shoe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51A.5 Magnet holder bar . . . . . . . . . . . . . . . . . . . . . . . . 52A.6 Inner support . . . . . . . . . . . . . . . . . . . . . . . . . . . 53A.7 Large support ring . . . . . . . . . . . . . . . . . . . . . . . . 54A.8 Flange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55A.9 Small support ring . . . . . . . . . . . . . . . . . . . . . . . . 56A.10 Shaft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57A.11 Generator end board . . . . . . . . . . . . . . . . . . . . . . . 58

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Executive SummaryUsing electromagnetic simulations, the magnetic circuit of a neodymium-iron-boron, NdFeB, magnetised permanent magnet generator was redesignedto use ferrite permanent magnets. A support structure was designed for thenew magnetic circuit using computer-assisted design tools and finite elementsfor structural mechanics analysis.

The result of the investigation was that in order to replace NdFeB magnetswith ferrite magnets, a new rotor has to be designed. A design for such arotor, including other required design changes, is presented here. The designhas some minor details that need to be finalised.

The proposed future action is to finalise the design and build the rotor.

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Chapter 1

Introduction

Wherein the background to the project along with the scope and goals arepresented.

1.1 BackgroundThis project is part of the wind power research program at the Division ofElectricity at Uppsala University, and the goal is to design a new rotor forthe generator used. The wind power concept studied in the program usesa vertical axis wind turbine that drives a permanent magnet synchronousgenerator mounted on a common shaft. This allows the generator to beplaced on the ground which allows a heavier generator to be used, and thedirect drive keeps the number of moving parts down. To allow variable speedoperation, the generator is connected to the grid via a full converter.

The existing generator, built using the old design, was completed in 2006,has a rated power of 12 kW at 127 rpm, and has 32 poles giving an electricalfrequency of 33.9 Hz at rated speed. The generator has an outer statordiameter of 886 mm, and the stator stack has a nominal length of 222 mm[1]. A summary of the characteristics of the old design is given in Table 1.1,and two photographs of the existing generator are shown in Figure 1.1.

In the existing generator, the rotor is magnetised using surface mountedNeodymium-Iron-Boron, NdFeB, permanent magnets, PMs. Due to the re-cent increase in price of NdFeB PMs, it has become interesting to investigatethe possibility of using other, less expensive, PM materials. One possiblecandidate for PM material is ceramic ferrites, which will be investigated inthis project.

Replacing the NdFeB PMs with ferrite PMs will require a new design ofthe rotor. This is because of the very different characteristics of the NdFeB

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Table 1.1: Characteristics of the generator built using the old design. Thistable is based on Table 4.1 of [2].

Rated power 12.0 kWNo load phase voltage (rms) 161 V

Armature current density (rms) 1.6 A/mm2

Electrical frequency 33.9 HzRotational speed 127 rpmNumber of poles 32

Number of slots per pole and phase 5/4Stator inner diameter 760 mmStator outer diameter 886 mm

Air gap width 10 mmGenerator length 222 mm

Figure 1.1: Two photographs showing the existing generator, with surfacemounted NdFeB PMs. To the left, the entire machine is shown, and to theright, is a close-up of the air gap.

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PMs and the ferrite PMs: the NdFeB PMs have significantly higher remanentflux density and coercivity than the ferrite PMs. While the remanence of theNdFeB magnets is sufficient to give an acceptable air gap flux, even whensurface mounted, the ferrite magnets require some kind of flux concentratingscheme to achieve acceptable air gap flux densities. One possibility is tomount the magnets in a so-called spoke configuration. A spoke configurationmeans that magnets with alternating tangential magnetisation, poles of thesame kind facing each other, are placed with pole shoes in between. Thisallows the flux from a large area of the PM to be directed into a rotor poleand is known as flux concentration.

1.1.1 Why Replace the Neodymium-Iron-Boron Mag-nets

To make NdFeB PMs, you need the rare earth metals neodymium and dyspro-sium, which make up about 30% and 3% of the magnet’s weight, respectively.There are issues with dependence on these metals. One is the price insta-bility: between August 2009 and 2011, the price of neodymium increased byover 1000%. As neodymium and dysprosium accounts for 60% and 35% ofthe material cost, respectivly, changes in the prices of these metals affect theprice of magnets significally. Another issue is the insecurity of supply. Cur-rently, China controls 97% of the worldwide production of rare earth metals.Due to an increase in domestic demand, exports can be expected to decrease.The cheaper and more commonly available ferrite PMs can become a viablesubstitute for the NdFeB PMs, despite their lower performance, in light ofthese problems [3].

1.1.2 Earlier Usage of Ferrite Magnets in PermanentMagnet Synchronous Generators

The use of ferrite permanent magnets, PMs, in PM magnetised synchronousmachines is not a new idea. In 1979, Binns et al. [4] proposed and built asynchronous machine using ferrite PMs. The rotor used a configuration withaxially magnetised PMs placed between flux guides that guide the flux intoa radial direction in the air gap. The article also mentions a machine witha spoke design. A spoke design means the magnets in the rotor are placedlike spokes and are tangentially magnetised with two poles of the same kindfacing each other.

In the mid 1990s, Spooner et al. [5, 6, 7] published articles describing amodular design of a PM synchronous generator. Various generator topologies

6

were considered, but the radial flux topology was chosen for further develop-ment. This generator design concept used ferrite magnets and was intendedto be used as part of a direct driven, variable speed wind power concept.Since the generator is synchronous, a full converter is used for grid connec-tion. Another modular design, using an axial flux topology, is presented byMuljadi et al. [8]. The design is supposed to be usable both with ferrite andNdFeB PMs, but the test prototype was built using NdFeB PMs.

One important precursor to the current work was the Windformer™ con-cept presented by ABB in 2000. In this concept, a direct driven generatorwound with high voltage cable and magnetised with ferrite magnets formedthe core of a wind power system. The magnets were mounted in a spokeconfiguration, and the flux was directed into the air gap using pole shoes [9].

In 2005, Kim et al. presented results on a generator magnetised using fer-rite PMs with a rotor that is significantly longer than the stator. This enablesflux concentration in the axial direction as well as in the plane perpendicu-lar to the axis of rotation. The flux concentration in the axial direction isbeneficial for ferrite magnetised generators. The axial flux concentration isbeneficial because it allows the air gap flux to be raised higher above the re-manent flux density than with flux concentration only in the plane of rotation[10].

In 2012, Seok-Myeong Jang et al. published results showing that a ferritePM generator could be built with the same diameter and length and similarperformance as a NdFeB PM generator. To achieve this, a redesign of themagnetic circuit was necessary [11].

1.1.3 Earlier Usage of Ferrite Magnets in Other Kindsof Rotating Electrical Machines

Also in electric motor design, there have been efforts to replace rare earthmagnets with ferrite magnets. In 2010, Miura et al. suggested using an axialflux machine with ferrite PMs to achieve the same power density as in a rareearth PM machine. The ferrite magnets would be magnetised in an axialdirection and mounted in a disk shaped rotor sandwiched between two diskshaped stators. To achieve the same power density, a higher rotational speedwould be required as the ferrite motor gives lower torque [12].

In 2012, multiple authors published results relating to ferrite PMs in syn-chronous motors. Dorrell et al. [13] investigated the possibility of using aferrite magnet rotor in the motor of the Toyota Prius hybrid electric vehicle.It was concluded that such substitution could be possible, and that the mag-nets should be put in a spoke arrangement to avoid demagnetisation and for

7

flux concentration. In the case of interior PM rotors, Barcaro et al. foundthat a slight lengthening of the machine could make a ferrite PM machine avalid replacement for a rare earth PM machine [14].

Ferrite magnets can also be used to construct PM reluctance generators.This gives comparable performance as a rare earth magnetised machine ofthe same size but at a reduced material cost [15].

In 2010, Kurihara et al. suggested a design for a PM reluctance generatorusing ferrite PMs mounted between the stator poles. The function of thePMs was to induce a sufficient residual flux to allow a small voltage to beinduced. This voltage was then rectified and used to drive an excitationcurrent through the field windings of the machine, allowing the machine tostart without an external source of excitation current [16].

1.2 Scope and goals of the present workThe goal of the project was to redesign a permanent-magnet synchronousgenerator to use ferrite permanent magnets instead of the neodymium-iron-boron permanent magnets. The stator is to be reused and can not, therefore,be changed.

The new design was to give a similar magnetic flux density in the air gapas the old rotor. Furthermore, the new rotor should be easy to assemble andinstall and provide secure mounting for the permanent magnets. The scopeof the project, therefore, included electromagnetic and mechanical design.Thermal design was excluded as it is not likely to be of concern to a perma-nent magnetised rotor. The thermal design of the stator was not regarded asthe stator design would not be changed. The electromagnetic and mechanicaldesign processes have been coupled since they provide constraints for eachother.

Building the rotor and determining what price of neodymium-iron-boronpermanent magnets will be needed to make the ferrite more economical wereboth outside the scope of this project.

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Chapter 2

Theory

Wherein some of the theory used in the project is presented briefly.

2.1 Simulation of Electrical Machines Using Two-Dimensional Finite Element Methods

The generator has been simulated using finite element software to solve theelectromagnetic field equations. The theory for how this is done is presentedbelow.

Using magnetic vector potential, ~A, which is defined by ~B = ∇ × ~A,where ~B is the magnetic flux density, the electromagnetic field equations canbe formulated as

∇ · ν∇~A = σ∂ ~A

∂t− ~J (2.1)

where ν is the magnetic reluctivity, the inverse of permeability, σ is theelectrical conductivity and ~J is the current density [17].

Let the axis of rotation of the machine coincide with the z-axis of acylindrical coordinate system and the ends at z = ±l/2 with l being thelength of the machine. Then the magnetic fields in the z = 0 plane can beused to approximate the fields in the whole length of the machine, given thatthe length of the machine is much larger than the size of the air gap and otherfeatures in the geometry in the plane. In the z = 0 plane, the z-componentof ~B is Bz = 0 and also ∂ ~A/∂z = 0 due to symmetry. This allows us toget the fields in the plane as the z-component of eqn. (2.1), which simplifiesthe solution to solving for only Az. Further simplification can be made byusing sector symmetry and periodic boundary conditions to reduce size ofthe domain over which calculations are made [2, 17].

9

A full account of the finite element method is beyond the scope of thisthesis, but a brief overview will be given. The basic idea is that you have apartial differential equation, PDE, on a domain Ω with boundary conditionson the boundary ∂Ω. Then there exists a vector space of functions on Ω thatfulfils the boundary conditions on ∂Ω called V , and the true solution is inthis space.

The next step is to rewrite the PDE using multiplication with an arbitraryelement of V , called the test function, using the L2 inner product on Ω andGreen’s identities into a integral equation. The integral equation is called theweak form of the PDE. Once the weak form of the PDE has been derived, Ω istriangulated and a subspace Vh of V that has a finite number of basis functionis defined. One possible choice of basis function is the hat-function. The hat-functions takes the value of one in one of the node of the triangulation andzero in all other nodes; between nodes it is linear. Since the basis functionsspan Vh, any function in Vh can be written as a linear combination of them.

Finally, the basis functions are used as the test function and the solu-tion to the PDE is approximated as an unknown linear combination of thetest functions. This transforms the integral equation into a system of lin-ear equations that can be solved by various methods. It can be shown thatthe approximation of the solution is the orthogonal L2-projection of the truesolution on Vh. This means that it is the best approximation of the truesolution that can be made on Vh measured in the L2-norm [18].

2.2 Methods of Calculating Magnetic Forces onMagnetic Materials

There are two methods that can be used to calculate magnetic forces onmagnetic materials that have been found in literature and used in this project.One is based on the conservation of energy and the principle of virtual workand is described by Hayt and Buck [19, p. 290], and the other is the Maxwellstress tensor that can be derived from the Lorentz force using Maxwell’sequations [20, p. 193].

The virtual work approach is limited to linear media and cases wheremovement does not cause any change in magnetisation or induce eddy cur-rents. In this case, linear media means that ~B is related to the magneticfield, ~H, through the relation

~B = µ ~H (2.2)

where the magnetic permeability µ is a constant scalar. Under these circum-stances, one can assume that the energy stored in the magnetic field in a

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volume, V , is given by

Wm =1

2

∫∫∫V

~B · ~H dV (2.3)

and, furthermore, that the change in energy dWm can be equated with mech-anical work done by moving part of the magnetic circuit a directed distance,l dl , while a magnetic force, ~Fm, acts upon it giving

dWm = ~Fm · l dl (2.4)

giving the l component of ~Fm as

Fm,l =dWm

dl. (2.5)

The Maxwell stress tensor approach is more general. Its derivation startswith the Lorentz force on a point charge

~F = q( ~E + ~v × ~B) (2.6)

where q is the charge, ~E is the electrical field, and ~v is the velocity of thecharge. The point charge is then replaced with a charge distribution to getthe force per volume. Using Maxwell’s equations together with some vectoralgebraic manipulations, the force density is given by

~f + µ0ε0∂~S

∂t= ∇ ·T (2.7)

where ~S = 1µ0~E× ~B is the Poynting vector, and T is the second order tensor-

dyadic known as the Maxwell stress tensor with elements given by

Tij = ε0[EiEj −1

2E2δij] +

1

µ0

[BiBj −1

2B2δij] (2.8)

where E and B are the components of corresponding vector quantities alongdirection indicated by the index, no index signifying absolute value, and δijis the Kronecker’s delta function.

The left hand side of eqn. (2.7) represents change of momentum density,both mechanical and in the field. By taking the integral of eqn. (2.7) over avolume, the force on the volume is given. Using the divergence, theorem itcan be shown that the normal component of the Maxwell stress tensor is theflux of momentum through a surface.

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In the case of a generator the electric fields in the air gap are negligible,E ≈ 0, and the frequencies low, ∂~S

∂t≈ 0, and the force on the rotor becomes

~F =

∮S

n ·T dS (2.9)

where n is the outward normal unit vector of a closed surface S enclosing therotor but not the stator. Likewise, the torque on the rotor is given by

~M =

∮S

~r × (t ·T) dS (2.10)

where t is the tangent unit vector to S that is also in the plane of rotationand ~r the location relative to the axis of rotation.

2.2.1 Analytic Model for Verification of Magnetic ForceCalculation

The purpose of this model is not to determine the exact force but rather toverify that the force obtained from the FEM model has the right order ofmagnitude. The model uses the principle of virtual work.

The magnetic circuit of the generator is modelled in 32 pieces as twohalf poles with a permanent magnet, PM, between them. The geometry ofthe model is given in Figure 2.1. The out-of-plane thickness is l. The ironparts are approximated as a perfect magnetic conductor, which simplifies themagnetic circuit to a PM and two air gaps. This should be an acceptableapproximation as the iron parts have a relative permeability in the order ofat least 103, making the reluctance in the iron negligible compared to thatof the air gap. In the air gap, the permeability is that of free space, denotedµ0.

Recall the virtual work approach to calculating magnetic force, outlinedin section 2.2, and especially eqn. (2.3). Then assume that ~B in the air gapis perpendicular to the cross section area, Ag, of the air gap associated withhalf a pole with amplitude B⊥g. Also assume that ~B in the magnets is alsoperpendicular to the cross section of the PMs, area bl, with amplitude B⊥pm.Then eqn. (2.3) evaluates to

Wm = 2AggB2⊥g

2µ0

+ (bl)hB2⊥pm

2µ0

(2.11)

where everything but B⊥g and B⊥pm is known. To find B⊥g and B⊥pm, amagnetic reluctance circuit model can be used.

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g

πDsi

32

M

b

h

Iron

Air gap

PM

Direction of M

Figure 2.1: Geometry of the two-half-pole-model used for validation of forcecalculations. Dsi is the inner diameter of the stator. The out-of-plane thick-ness is denoted l.

In a magnetic reluctance circuit model, the PM is modelled as a magneto-motive force, mmf, in series with a reluctance and the air gaps as two equalreluctances in series. The mmf of a PM is given by the product of theremanence, M , and the height, h, of the magnet along the direction of mag-netisation

F = Mh . (2.12)

Magnetic reluctance is given by

R =d

µA(2.13)

where d is the length of the magnetic flux tube, and A is the cross sectionarea of the magnetic flux tube. For the PM, this gives

RPM =h

µ0lb(2.14)

and for the air gapRg =

g

µ0Ag(2.15)

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where the area Ag is the average area of half the pole pitch and half the poleshoe width given by

Ag = l

(πDsi

64− h

4

). (2.16)

From Hopkinson’s law, we know that

F = Φ(RPM + 2Rg) (2.17)

where Φ is the magnetic flux. Recalling the assumption that ~B is uniform,and perpendicular to the cross sections in the air gap and in the PM, themagnetic flux in the circuit is then given by

Φ = AgB⊥g = bl B⊥pm (2.18)

which can be rewritten as

B⊥g =Φ

Ag, B⊥pm =

Φ

bl(2.19)

and inserted into eqn. (2.11). This gives

Wm = 2Agg

(ΦAg

)2

2µ0

+ (bl)h

(Φbl

)2

2µ0

(2.20)

that can be simplified to

Wm = Φ2

(g

µ0Ag+

h

2µ0bl

)= Φ2

(g

µ0Ag+RPM

2

)(2.21)

Inserting eqn. (2.21) into eqn. (2.5), the magnetic force on one two-half-poleunit can be expresses as

Fm(g) =dWm

dg=−1

µ0Ag

(F

RPM + 2gµ0Ag

)2

(2.22)

with the positive direction of the force in the direction of increasing g. Thesame expression can be derived using the magnetism-only case of the MaxwellStress Tensor on the magnetic circuit.

To calculate the force on the rotor, define a starting air gap g0 and a rotordisplacement ∆g. Then the total force on the rotor, for a displacement ∆gfrom perfectly concentric with the stator, is calculated by summing the forceon each unit. The force from each unit is calculated using an air gap of

g = g0 + ∆g cos(θ) (2.23)

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where θ is the angle between the line through the middle of the unit andthe direction of the displacement. The forces are then summed up, the crossdisplacement component cancels out, and the along displacement componentis

Frotor(∆g) =32∑i=1

[Fg(g0 + ∆g cos

(2π

32i+ φ

)) · cos

(2π

32i+ φ

)](2.24)

where φ is the angle between the line of the displacement and the line throughthe middle of the nearest pole.

2.3 Bolted JointsFor bolted joints, there are two different mechanisms by which the joint canwithstand shearing forces. One way is to have the bolt fitted tightly into thehole, transferring the force between the parts via contact forces and shearstresses in the bolt. The other way is to have the bolt press the two partstogether with such a force that the friction between the surfaces of the partsis sufficient to withstand the shear force. To use friction instead of shearingto transfer the force generally requires stronger bolts [21, p. 2:41]

In the current design, it was not possible to use tightly fitted bolts inall places; therefore, friction was used. The theory developed to calculaterequired number of bolt is described below.

To calculate how many bolt were needed for each joint, it was assumedthat the bolts would have to press the parts together with such force thatthe friction forces would be able to withstand the loads on the joint. Thecommon model for friction between hard surfaces, such as metal, given inliterature gives the maximum friction force as

Ff = fFn (2.25)

where µ is the coefficient of friction and Fn the normal force. The coefficientof friction depends on the surfaces [22, p. E4.1].

The force Fn in this case is the axial force on the bolts when loads in theaxial direction have been subtracted. Let Fla be the load forces along theaxis of the bolt, Fls the load forces shearing the bolt, Fas the axial force onthe bolt at suitable level of prestressing and N the number of bolts. Then acondition can be formulated as

NFas ≥ s

(Flsµ

+ Fla

)(2.26)

15

where s is a factor of safety. When transferring torques, the force is heldto act upon the radius of the cylinder in the case of cylindrical surfaces, oralong the circle centred upon the axis of the torque and going through themid-point of the bolts in the plane of the torque.

16

Chapter 3

Design Process

Wherein the design process and its results are described.

3.1 Method and MaterialsThe main tools in the design process have been computer modelling software.The computer assisted design software package SolidWorks®1 has been usedboth to define the geometry of all parts and to check that they fit together.The simulation package in SolidWorks has been used for structural finite el-ement analysis. For electromagnetic simulation, COMSOL Multiphysics®2

has been used. To check the results from COMSOL, the in-house electro-magnetic simulation software KALK has also been used [2].

In addition, some simpler models of bolted joints and the magnetic circuithave been derived and implemented in MATLAB®. These are described insections 2.2.1 and 2.3, respectively.

3.2 Design RequirementsThere were a number of requirements on the design. Since the stator was tobe reused, it could not be changed. This fixed the inner stator diameter at760 mm and stator length at 224 mm. The number of poles was also fixedat 32.

Whenever possible, the use a standard size of magnets was another re-quirement. Additionally, the air gap should be at least 7 mm.

1www.solidworks.com2www.comsol.com

17

To be able to fasten the pole shoes, there needs to be room for bolts. Thematerial surrounding a threaded bolt hole needs to be at least twice as thickas the diameter of the bolt as a rule of thumb. It was judged that at leastan M4 bolt would be needed. This meant that the inner end of the pole shoehad to be at least 8 mm wide.

Finally, it was desirable to keep the electric characteristics of the newdesign as close to the characteristics of the old design as possible.

3.3 ProcedureThe first step of the design process was to find the requirements and con-straints imposed on the new rotor by the parts of the generator that wouldbe reused. For instance, the inner diameter of the stator and the minimumrequired air gap limit rotor diameter and the stator stack height determinesthe height of the rotor. Also, the new rotor will have to supply approximatelythe same magnetic flux density if the electrical properties of the generatorare to remain comparable.

Once the basic requirements and limitations were made clear, a verysimple model of the magnetic circuit, just an air gap and a permanent magnetrepresenting a single pole, was made to get a rough estimate of what size ofpermanent magnet was required. Approximate magnetic properties of thePM were taken from a textbook [23]. This information was used to chooserotor topology. The two topologies of interest were surface mounted, used inthe old rotor, and the spoke configuration with pole shoes; see Figure 3.1.

(a) (b)

Figure 3.1: Two possible topologies for a permanent magnetised rotor; (a)shows a surface mounted configuration, and (b) shows a spoke configurationwith pole shoes. The arrows indicate the direction of magnetisation of themagnets. Gray areas indicate soft magnetic materials.

Once the topology was chosen, the spoke configuration, a parametric, twodimensional model of the cross sectional geometry of the generator, was con-structed in COMSOL Multiphysics®. In this model, the permanent magnets

18

were modelled as a region of constant magnetisation. Armature phase volt-ages were calculated by integrating the electric field induced perpendicular tothe plane of rotation for each phase, with corrections for winding direction,dividing by the cross sectional area of the conductor, and multiplying by thelenght of the machine. The armature currents were then applied as an ex-ternal current density in the conductors. The current density in a conductorwas calculated as

Jz =Vph

(Rwind +Rload)A(3.1)

where Vph is the induced voltage in a phase, Rwind the winding resistance,Rload the load resistance, and A the area of the conductor cross section. Forthe soft magnetic materials, the steel in stator and pole shoes, one-to-oneBH-curves implemented as interpolation tables were used to relate ~B and ~H.Out of plane conductivity was set to zero to avoid unphysical eddy currents.

When all the material parameters were set, the magnetic circuit wassimulated using finite element methods, FEM, to solve the electromagneticfield equations. The underlying theory for the simulations is presented insection 2.1.

From the FEM simulations, the desired size of PMs was determined. Someconstraints on the geometry were given; for instance, a minimum width ofthe inner end of the pole shoe was required so that bolts to hold it in placecould fit. To find out if there were standard sizes of PMs close to the desiredsize, manufacturers of ferrite PMs were contacted. Using standard sizesis advantageous as it avoids the costs and longer delivery times associatedwith sizes made to measure. The sizes and material grades offered by themanufacturers were then inserted into the simulation to check if they gavesufficient performance.

In parallel with the process of finding PMs to use, simulations were per-formed to determine the magnetic forces and torques that would be generatedin the new machine. Also, the forces on the PM during mounting were exam-ined through simulation. For these simulations, magnetisation was assumedto be at the upper bound of the offered magnetisation of the material gradewith the highest magnetisation. This was done to ensure that the calculatedforces will be at least as large as the actual forces.

Once an upper bound on the forces from the magnetic circuit was deter-mined, design of the support structure could start. Due to problems causedby welds in the old rotor, welded joint were avoided, and bolted joints wereused instead. The bolted joints were designed with a clamp load to avoidsubjecting the bolt to shearing forces. Details which needed lots of machiningwere avoided when possible, and parts made from sheet metal or prefabri-cated profiles were favoured.

19

The criterion for static stiffness was formulated as follows: Given a rotoreccentricity of ∆g there will be a net force on the rotor due to unbalancedmagnetic pull from the magnetic circuit. If the design is stiff enough, thisforce should not be large enough to pull the rotor ∆g toward the stator fromits original position, where it is concentric with the stator.

When a suitable ferrite magnet had been found and the rotor was foundto be stiff enough against static deformation, simulations were run to exam-ine the natural modes of vibration. Some changes to the design were thenmade to attempt to shift the natural frequencies associated with the modesfound upward. Shifting the frequencies upward moves them away from themost prominent frequencies emanating from the electromagnetic parts of themachine. Most of these changes were additions of stronger supports to stiffenthe rotor against vibrations in the natural modes of vibration.

As a final step in the design process, the design was reviewed to makesure all bolts could be accessed for tightening, and a rough plan for how toassemble the rotor was made to ensure it could be done. This also led tosome minor changes. Due to time constraints, the selection of bearings andthe design of the bearing unit housing were omitted, but some considerationhave been given to the matter.

20

Chapter 4

Results

Wherein the results from the design process are presented. First, an overviewof the whole assembly is given, and then the parts are presented one at thetime.

A section view of the new rotor design is shown in Figure 4.1. Aluminiumhas been the material of choice for much of the support structure due to itslow weight and nonmagnetic nature. All parts are joined by bolts to avoidwelds, which can cause deformity in the structure.

The rotor has two plates in the plane of rotation (see Section 4.1.1), oneat the bottom and one at the top. Between the two plates, there are holders(see Section 4.1.2) for the pole shoes (see Section 4.1.3) and supports (seeSection 4.1.6) to stiffen the structure. The pole shoes are fastened to theirholders with bolts. The PMs (see Section 4.1.4) are held in place betweenthe pole shoes by the bottom plate, bars that are bolted into the top of thepole shoes after mounting the magnet, the pole shoe holders, and the ridgeson the corners of the pole shoes. To attach the rotor to the shaft, two flangesare used (see Section 4.1.8). Each flange is made up of two halves that arebolted together. The flanges transfer the torque, using friction both at theinterface with the shaft and with the rest of the rotor. For extra stiffness, athick ring of aluminium (see Section 4.1.7) is fastened to both of the rotorend plates using the same bolts that are used for fastening the pole shoeholders. The ring is also fastened to the supports with bolts going throughthe end plates.

A more detailed description of all the rotor parts is presented in Section 4.1.

21

4.1.4

4.1.3

4.1.5

4.1.2

4.1.10

4.1.8

4.1.94.1.1

4.1.7

Figure 4.1: Section view of the new rotor, the central angle of the sectorshown is 99°. The cut planes meet at the axis of rotation. The rotor shafthas been cropped at the lower edge of the drawing. One inner support joiningthe upper and lower plate of the rotor is obscured by the shaft but is presentedin Section 4.1.6. The new generator end boards have also been omitted forclarity but are presented in Section 4.1.11. The number on the parts is thenumber of the section describing that part more in-depth.

4.1 Description of componentsA more in depth presentation of each component and the factors consideredwhen designing them is presented below. Drawings with measurements canbe found in Appendix A.

4.1.1 Rotor End Plates

The end plates of the rotor are shown in Figure 4.2. The purpose of therotor end plates is to keep the rotor together in the plane of rotation. Thebottom rotor end plate also provides a good starting point during assemblyof the rotor. The thickness of 10 mm of aluminium was chosen as an initialapproximation, based on the fact that the thickness of the plate with the samefunction in the old rotor was 5 mm of steel. When the natural frequencies ofthe new rotor were investigated, it was noted that making the plates thickerwould improve the frequencies of some modes, but due to lack of space, thiscould not be done. The small holes have clearances for M4 bolts and thelarge for M8 bolts.

22

Figure 4.2: Top view of the rotor end plates. The bottom plate is to theleft, and the top plate is to the right. The plates are made from 10 mmthick aluminium and have outer diameters of 685 mm and 484 mm, with thebottom plate being the larger of the two.

4.1.2 Pole Shoe Holder

The pole shoe holder is shown in Figure 4.3. The purpose of the pole shoeholder is to provide a mounting place for the pole shoes and to connect therotor end plates. At an earlier stage, mounting the pole shoes on a pipe wasconsidered, but due to lack of available pipes of the required diameter, thecurrent design with individual holders was chosen instead.

The pole shoe holder is made from aluminium, and is intended to bemanufactured by cutting an extruded U-profile into pieces 36.2 mm wide.The profile chosen is 220 mm by 60 mm, and it has a material thicknessof 5 mm. The width of the pieces is determined by the pole pitch angle,360/32 = 11.25, and the radial position of the pole shoe holder. The holeshave clearance for M4 bolts.

4.1.3 Pole Shoe

The pole shoe is shown in Figure 4.4. The purpose of the pole shoe is tofocus the magnetic flux from the permanent magnets into the air gap.

Due to difficulties with fitting a dovetail key into the inner end of thepole shoes, a solid pole shoe was chosen instead of a stacked pole. This willintroduce some extra machining cost but was judged to be the only workableapproach. A possible alternative, brought to notice too late in the designprocess to be properly evaluated, was to stack the poles using a plate thickenough to accommodate a threaded bolt hole in the inner end.

To conduct magnetic flux efficiently, the pole shoe will have to be madefrom a ferromagnetic material. Ordinary carbon steel was chosen for itsmechanical properties and low cost. The pole shoe has threaded holes for

23

Figure 4.3: Isometric view of the pole shoe holder. The outer sides are220 mm by 60 mm by 36.2 mm and material thickness is 5 mm.

Figure 4.4: Isometric view of a pole shoe. Height of the pole shoe is 224 mm.

24

M4 bolts at both the lower and upper end as well as on the backside. Theseare for fastening the pole shoe in the rest of the rotor and for fastening themagnet holder bars in the pole shoes.

The shape of the cross section has, at the end away from the air gap,been designed to fill out the space between the permanent magnets and tohave a flat face against the pole shoe holder. At the end toward the air gap,the shape has been designed to create a high peak amplitude of the radialair gap flux density. This was done by having a constant air gap of 7 mmin the middle of the pole shoe for the midmost 8.44 mm and then taperingthe pole shoes away from the stator. On the corners sticking out into the airgap, ridges have been placed to prevent the PMs from moving out into theair gap. No special effort was made to optimise the shape of the pole shoefor low harmonic content in the output voltage waveform or to reduce torqueripple.

4.1.4 Permanent Magnet

The purpose of the permanent magnet is to magnetise the rotor. It is arectangular cuboid with sides 226.5±2.5 mm, 122±0.2 mm, and 38±0.1 mmwith magnetisation parallel to the shortest side. The cuboid is assembledfrom smaller pieces that have been bonded together prior to magnetisation.The material is ceramic ferrite in a grade called Y40 with a remanent mag-netisation of 0.45 T. The longest side is 226.5 mm rather than 224 mm toensure that the poles always get the magnetisation from the full height of aPM, even if the side is on the low end of the tolerance. The two other sideswere chosen as stated because there was already a tool for manufacturingblock magnets of that size available, which made them less costly. While theused size of the PMs is not strictly a standard size of PMs, it has most of theadvantages of a standard size of PMs and no usable, standard size of PMswas available.

4.1.5 Magnet Holder Bar

The magnet holder bar can be seen in Figure 4.5. The purpose of the mag-net holder bar is to prevent the PMs from moving upward out of the slotbetween the pole shoes. The bar is made of non-magnetic aluminium to avoidinterfering with the magnetic circuit. The holes have clearance for M4 bolts.

4.1.6 Inner Support

The inner support is shown in Figure 4.6. The purpose of the inner support

25

Figure 4.5: Top view of the magnet holder bar. The bar is made from 4 mmthick aluminium and has an overall length of 104 mm.

Figure 4.6: Isometric view of the inner support. The height is 220 mm,thickness 30 mm, and the material is aluminium.

26

is to make the rotor more rigid. This prevents deformation due to the weightof magnetically active material in the rotor part of the magnetic circuit. Italso makes the rotor stiffer against vibrations in some of the natural modesof vibration, as presented in Section 4.6. The inner support can be cut froma 30 mm aluminium plate. The holes at the ends are threaded for an M8bolt, while the holes on the ledges have clearance for an M8 bolt.

4.1.7 Large Support Ring

The large support ring is shown in Figure 4.7. Its main purpose is to stiffenthe rim of the rotor against some natural modes of vibration; see Section 4.6.The outer diameter is 468 mm, and the part is made from a 30 mm aluminiumplate.

Figure 4.7: Isometric view of the large support ring. The thickness is 30 mm,the outer diameter 468 mm, and the material is aluminium.

The inner diameter is large enough to allow two inner supports (seeSection 4.1.6) to be cut from the piece of metal removed from the centreof the ring and, the material thickness is the same. By manufacturing thering and two supports from the same piece of metal, the amount of scrap canbe reduced. Holes on the rim have clearance for M4 bolts, and the holes onthe inward teeth have clearance for M8 bolts.

4.1.8 Flange

The flange is show in Figure 4.8. The purpose of the flange is to attach therotor to the shaft. This is done by clamping the flange onto the shaft withbolts and bolting the rotor end plates to the flange. The flange is divided

27

Figure 4.8: Isometric view of the flange with the two halves separated. Theouter radius is 92.5 mm, and the height is 53 mm. Material thickness variesbetween 5 mm and 10 mm. The flange is made of steel.

into two parts to make assembling the rotor easier. The flange is made ofsteel, since it is far from the magnetic circuit, and will it be subject to higherstresses than most of the structure. All holes have clearance for M8 bolts.

4.1.9 Small Support Ring

The small support ring is shown in Figure 4.9. The purpose of the smallsupport ring is to stiffen the inner rim of the end plates and smooth out thepressure applied by the bolts that bond the end plates with the flange. Italso makes the connection between the flange and the inner support stronger.The small support ring is made from steel. Holes have clearance for M8 bolts.

4.1.10 Shaft

The shaft is shown in Figure 4.10. The purpose of the shaft is to hold therotor in place relative to the static parts of the machine and to transfer thetorque from the primary mover. It is far from the magnetic circuit and canbe expected to be subjected to among the highest stresses in the structure.Therefore, steel was chosen as the material for the shaft. Length and diameterare both inherited from the old design, since both affects the connection tothe prime mover. The section with larger diameter will be inside the rotor. Itwas added to make the shaft stiffer against vibrations in some of the vibrationmodes presented in Section 4.6. The bearings will be mounted using pressfits, which means parts of the shaft will be machined with tight tolerances.

28

Figure 4.9: Isometric view of the small support ring. Outer diameter is185 mm, thickness 10 mm, and it is made from steel.

Figure 4.10: Isometric view of the shaft. The length is 990 mm, and thediameter of the narrower parts is 95 mm. Parts of the shaft will have to bemachined with high tolerance required for press fitting the bearings onto theshaft.

29

4.1.11 Generator End Board

The generator end board is shown in Figure 4.11. The purpose of the gen-

Figure 4.11: Isometric view of the generator end board. The board is1000 mm across, 25 mm thick and made from a glass fibre–epoxy composite.

erator end board is to hold the bearings that keep the shaft in place andstabilise the generator. The main loads are forces from unbalanced magneticpull and the weight of the rotor.

New end boards will be needed, since the new rotor has a larger mass thanthe old one. The generator will mostly be used for research and, therefore,larger openings for inserting measuring equipment are desired. The materialis a glass fibre reinforced epoxy board with an elasticity modulus of 24 GPaand tensile strength of 300 MPa, according to material data supplied by themanufacturer.

4.2 The New Magnetic CircuitFor the new magnetic circuit, a spoke type topology with pole shoes waschosen. This topology was chosen in order to get magnetic flux concentration,since the air gap flux density needs to be larger than the remanent flux densityof the ferrite PMs. The pole shoes are made from solid steel. Stacked poleswere ruled out, due to difficulties with fitting dovetail keys at the inner endand also because of difficulties with compressing the stack once in place. The

30

pole shoes are mounted on a non-magnetic support structure, represented inthe electromagnetic model and in Figure 4.12 as an aluminium tube.

Magnetisation is provided by ferrite block magnets. The PM is magne-tised along the shortest side, which has a length of 38 mm. The nominalremanent flux density is 450 mT, according to specifications provided by themanufacturer [24].

CableAluminiumFerrite PMSolid steelLaminated silicon steelAir30 mm

Figure 4.12: Cross sectional geometry of the new magnetic circuit with thevarious parts indicated. Eight instances of the sector shown together form acomplete cross section. Parts indicated as air can also contain magneticallyinactive support structure.The ferrite PMs are magnetised along their sidetangential to the rotation of the rotor with alternating polarity. The out ofplane thickness is 224 mm.

A cross section of the geometry of the magnetic circuit is shown inFigure 4.12. The stator is to be reused from the old design and can not,therefore, be changed. This fixes the stator inner diameter at 760 mm. Theair gap is 7 mm at its narrowest by design requirements. The face of the poleshoe parallel to the stator has a width of 8.44 mm, and the rest of the outerface of the pole shoe tapers away from the stator. On the side of the poleshoe are ridges that prevent the PM from moving outward into the air gap.The inner parts of the pole shoe are shaped so they fill the space between thePMs and make contact with the pole shoe holder. For detailed measurementson the pole shoes, see drawings in Appendix A.4. Since the exact steel used

31

in the pole shoes has not yet been determined, a generic BH-curve from thematerial library of COMSOL called “soft iron” was used for the steel used inthe pole shoes. For the stator steel a BH-curve was available and was used.

In Figure 4.13, the magnetic flux density, ~B, for a static simulation hasbeen plotted. There are regions where ~B is very concentrated, mainly around

[T]

[mm]

[mm

]

Figure 4.13: The magnetic flux density from a static simulation. The colourscale is | ~B| in tesla. The thin lines are flux lines of ~B.

the outward corners of the ridges on the side of the pole shoes but also outsidethe corners of the stator teeth. The radial component of the air gap fluxalong a line 3 mm from the inside of the stator is plotted in Figure 4.14. Themaximum amplitude of the radial air gap flux is 0.71 T and the fundamentalof the waveform has an amplitude of 0.66 T. At a nominal speed of 127 rpm,this would induce a no load, line to neutral, RMS voltage of 144 V.

The design was put into the in house FEM software, KALK, and thesimulations were remade. The simulations made with KALK showed goodagreement with the simulations made with COMSOL. The amplitude of theair gap flux density only differed by 3% despite the fact that the geometry didnot match entirely. Also, the general distribution of magnetic flux density wassimilar, both with respect to direction and amplitude, in both simulations.The geometry could not be made exactly the same, due to the geometryinput feature of KALK being rather constrained. Agreement of results from

32

0 10 20 30 40−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Mechanical angle [°]

~B · r( ~B · r)1

Mag

neticflu

xdensity

[T]

Figure 4.14: The radial component of ~B in the air gap, for the two polepairs in the sector used in the simulation along a circular arc 3 mm insidethe inner perimeter of the stator. Maximum amplitude is 0.71 T, and thefundamental of the wave form is added as a dotted line. The amplitude ofthe fundamental is 0.66 T.

KALK and the model made in COMSOL can be seen as a verification of thelatter because KALK has been verified by experiment [25].

4.3 Magnetic ForcesThe forces on a misaligned rotor were computed with the methods discussedin Section 2.2, using the FEM model for field calculations. The forces werealso calculated with the simplified model derived in Section 2.2.1 for verifi-cation. The calculations using integration of the Maxwell Stress tensor andthe calculations with the virtual work gave a resultant force, due to an un-balanced magnetic pull of 10 kN at 3 mm displacement of the rotor. The ver-ification model, derived in Section 2.2.1, gave 28 kN at 3 mm displacement,but since it is very simplified, anything within the same order of magnitudecan be considered acceptable agreement. The maximum torque occurring ata two-phase short circuit was determined to approximately 8300 Nm, whichwas judged to be the highest torque the machine might encounter. The max-imum force a single pole shoe might be subjected to, occurring at 3 mmeccentricity of the rotor, was 2 kN.

33

4.4 Calculation of Bolted JointsThere are four different bolted joints in the design. These are the jointsbetween the pole shoe and its holder, the holder and the top or bottomplate, the flanges and the top and bottom plates, and the flanges and theshaft.

From section 4.3, it was known that each pole shoe will be subjected toan outward electromagnetic pull of about 2 kN at most, and that the torqueon the shaft would be 8.3 kNm at most. There are inertial forces due to therotation of the rotor in addition to the electromagnetic forces. The inertialforces were calculated to a total of 570 N per pole for both the pole shoeand one PM at the nominal speed of 127 rpm. This gives the design loads as2.6 kN, pulling the individual pole toward the stator and a torque of 8.3 kNm.

The coefficients of friction, f , used in the calculation were chosenas foundin literature or slightly lower to be on the safe side. For two aluminiumsurfaces rubbing against each other, 1.3 was stated in literature and this wasrounded to fAl−Al = 1, which was used. For an aluminium surface, againsta steel surface fAl−Steel = 0.5 was given in literature and used. For two steelsurfaces against each other, 0.8 was given in literature, and for steel againstcast iron, 0.4 was given in literature. The coefficient of friction fSteel−Steel =0.4 was used for one steel surface against another, due to uncertainties inmaterial composition [22, p. E5.1].

Two kinds of bolts were necessary, M4 in A2-70 ISO standard strengthclass to be used near the magnetic circuit, and M8 in 8.8 ISO standardstrength class to be used away from the magnetic circuit. The A2-70 class ismade from stainless steel, and it was chosen because it is not ferromagnetic.The 8.8 class was chosen as it is the default choice of strength class, andthere were no special requirements motivating the use of another strengthclass. The M4 bolt should be prestressed with a force of 2.6 kN, and the M8should be prestressed with 15.2 kN [26]. These forces have been used whendetermining the number of bolts required, using the criterion in eqn. (2.26)of Section 2.3.

Given the above assumptions, the force required to fix the pole shoe tothe pole shoe holder was calculated. This force should both resist radialforces, both inertial and electromagnetic, and provide enough normal forceto resist any shearing forces through friction. Six M4 bolts were used. Thisgives a factor of safety, FoS, of 3.3 against the design loads given earlier. Tofix the pole shoe holder to the top and bottom plates of the rotor, four M4are used for each joint. This gives a FoS of 7.4 against the design loads.

For the flange, only the forces required for transferring the torque needto be considered. Both the weight of the rotor and the forces due to unbal-

34

anced magnetic pull are an order of magnitude less than the force requiredto transfer the torque at the radii in question, 47.5 mm for the shaft–flangejoint and 80 mm for the flange–plate joint, and are therefore ignored whendesigning the joints. The bolts chosen to fasten the flange to the shaft areten M8 per flange, five on each side of the shaft, for an FoS of 1.4 againstdesign loads. To fasten the flange to the top and bottom plate of the rotor,nine M8 per flange part were chosen, giving an FoS of 2.6 against designloads.

4.5 Static Stiffness of the RotorFrom the electromagnetic simulations, see Section 4.3, it was determinedthat displacing the rotor 3 mm from the centre of the stator would causean unbalanced magnetic pull of 10 kN. To investigate if the design was rigidenough, a static linear simulation was made using the simulations package ofSolidWorks.The basis for the geometry used in the simulation was the CADmodel of the new generator with some simplifications made. The simplifica-tions were to remove all bolt holes and to model the joints as if the parts hadbeen bonded together.

The stator was chosen as a reference point and considered rigid, since itwas judged to be much more rigid than the rest of the generator. A force of10 kN was applied to the outward face of a single pole shoe, pulling it towardthe stator, which is a simplification that leads to slight exaggeration of thedeformations.

The result was that the maximum deformation in the design was about0.20 mm at the top of the pole shoe to which the force was applied. Of this,0.18 mm was in the plane of rotation.

Some of the stresses computed in the simulation exceeded the yield strengthof the material in the pole shoe holders, resulting in a local FoS of 0.76. Amore accurate analysis was made in which the force on each individual polewas applied to the air gap face of the pole shoe, 2.6 kN per pole toward thestator. Then, the resultant force of the unbalanced magnetic pull, 10 kNat an arbitrary angle in the plane of rotation, was superimposed on the perpole forces. This analysis showed that the minimum FoS was 1.53 for thepoint with the lowest FoS. The point with the lowest FoS is a point wherethe material is compressed.

35

4.6 Study of Natural Frequencies and Modesof Vibration

All mechanical structures have natural frequencies with associated modes ofvibration. If the frequencies are close to the frequency of a load, energy can bestored in vibration in the corresponding mode. Unless sufficient dampeningis present in the structure, the energy can build up and cause large repeateddeformations, either resulting in direct failure or premature material fatigue.

To investigate the natural frequencies and their associated modes of vi-bration, simulations were made. Commercially available software packageSolidWorks Simulation was used. After considering the result of the first sim-ulation, changes were made to shift the natural frequencies to above 200 Hz,which was the frequency of the cogging at nominal speed.

All joints were modelled as bonded, and solid geometry was used for allcomponents. Components were simplified by removing bolt holes and fillets.The stator was simplified by making the yoke slightly thicker and removingthe teeth, preserving the mass of the stator as closely as possible. The statorwas also simplified by modelling it as a solid piece of steel rather than a stackof plates.

The ten lowest natural frequencies according to the simulation are pre-sented in Table 4.1. The modes of vibration associated with the ten lowestnatural frequencies are presented schematically in Figure 4.15. The frequencyvalues are likely to be highly inaccurate, and it is mainly the modes of vi-bration that are of interest.

Table 4.1: The ten lowest natural frequencies of the design. The type ofmode shape is referring to illustrations in Figure 4.15.

Mode Frequency Mode shape- [Hz] n/a1 43 a2 77 b3 144 c4 145 c5 186 d6 192 d7 291 e8 295 e9 295 f10 361 f

36

(a) (b) (c)

(d) (e) (f)

Figure 4.15: Schematic illustrations of the natural modes of vibration associ-ated with the ten first natural frequencies, according to simulation. Some ofthe modes have the same shape but are rotated compared to each other. Ar-eas with diagonal lines are the stator. Mode shapes a, c, d, and e are shownin a cross section through the axis of rotation, b is shown from above, andf at an angle without the stator. Dashed lines show the deformed geometryand the dashed arrows indicate direction of movement.

37

The second mode is most likely an artifact, due to how the simulation wasmade. It arises because the bearings are modelled as rings of steel, bondedboth to the stationary part of the generator and to the shaft. A similar modewill be present, but it will depend on the rest of the drive train, which is notmodelled in the simulation.

4.7 Comparison of New and Old DesignIt was desirable to preserve the properties of the generator as far as possible.In Table 4.2, a brief comparison of the new and the old design is given.

Table 4.2: A comparison of the new and old designs. The roman numeralswithin parentheses indicate the source of the data for the old design. Thesources are: (I) is from internal, unpublished, documents at the Division ofElectricity, (II) is taken from [2], and (III) is taken or estimated from datain [27]. Data for the new design is taken from the simulations made duringthe design process.

Quantity Old design New designAmplitude of air gap flux densityfundamental [T] (I)

0.79 0.66

Phase voltage, no load [Vrms] (II) 161 144Armature winding current density,rated load [Arms/mm2] (II)

1.6 1.8

Rated power [kW] 12 12Minimum air gap [mm] (II) 10 7Mass of rotor [kg] (I) 130 402Mass of PM [kg] (III) 41 158Moment of inertia [kgm2] (III) 16.9 33.9

4.8 Unfinished Parts of the DesignDue to the time constraints associated with a master’s thesis, some parts ofthe design were left unfinished. The most important of these was the exactchoice of bearings and the design of the bearing housing. Some initial studieswere made, and space to accommodate tapered roller bearings, capable ofsustaining both one direction axial loads and large radial loads, as well ashousing for them has been left in the design. The shaft will also have to beslightly modified to accommodate the bearing mountings. The lower bearingwill be fastened by an interface fit and an abutment, and the upper bearing

38

by use of an adaptor sleeve, both requiring narrow tolerances. Also, theexact design of the top and bottom composite boards will be dependent onthe bearing housing for the central hole and holes for fastening the bearinghousing to the board.

In addition, mechanical tolerances of all parts have to be determined toensure that everything fits together.

39

Chapter 5

Discussion of results

A few aspects of the design are still unfinished, due to time constraints.Additional research on the tolerances required for interference fits with thebearings as well as the design of the bearing housing will be needed before thedesign can be completely finalised. A final choice of bearings also needs tobe made. Additionally, tolerances on all other parts need to be determined.Doing this, however, should not be difficult when time is available.

The FEM simulations have simplifications. Approximating the generatorwith a two-dimensional cross section of the geometry and then multiplyingby machine length is a well established method, and while it is not perfectat modelling the effects at the end of the generator, it gives good results.

The PMs were modelled as regions with constant magnetisation, and thisshould be a good approximation. There are other ways to model PMs, suchas a region with near unity relative permeability and two out of plane currentsheets along the sides along the magnetisation.

The soft magnetic materials were not modelled with hysteresis loops butrather a one-to-one relationship, BH-curve, between ~B and ~H. Also, for thepole shoes, the magnetic properties of the steel were approximated with ageneric soft iron material from COMSOL’s material library, since the exactsteel grade had not yet been decided. This should not be a problem sincemost irons will have permeabilities that are orders of magnitude larger thanthat of free space. This is true unless the steel is saturated and magneticflux densities high enough to saturate the steel occur, mostly in the statorfor which a proper BH-curve was available.

Due to the above simplifications, there are some uncertainties in the re-sults of the electromagnetic simulations. It is, however, clear that the airgap flux density amplitude will be lower with the new design and, therefore,the output voltage will decrease. The rated power can be maintained by in-creasing the current density in the stator winding. A higher current density

40

will cause increased resistive losses, which in turn lowers the efficiency of themachine.

The structural mechanics calculations had simplifications. The largestsimplification was that parts joined by bolts were modelled as bonded. Thisshould, however, be a valid approximation as long as the forces do not loadthe bolts outside their elastic domain. In the static simulations, the poleshoe holders were in places subjected to stresses larger than half of theiryield strength (FoS less than two), which indicates that it would be a goodidea to use thicker material. There are, however, few U-profiles of suitabledimensions with greater material thickness than those used. A possible wayaround this problem would be to change the design to use two L-profiles formaking the pole shoe holder as there seems to be more L-profiles with thickermaterial available on the market.

Another concern is the first natural mode of vibration, mode a of Figure 4.15at a frequency of about 43 Hz. Even though it is not expected to be excitedduring normal operation, it could become so. Should the mode become aproblem, there are ways to shift the frequency upward so much it will not bea problem by adding steel beams between the bearing housing and the barsholding the stator together.

There are five other natural frequencies below the cogging frequency of200 Hz. The lowest of these, the second natural frequency, is most likely anartifact of how the simulation was made and should not be a problem. Thefour remaining frequencies have such shape that it seems likely that they willnot be excited during normal operation. The bolted joints in the rotor mightalso provide some dampening. However, the vibrations should be carefullymonitored during initial testing of the new design once it has been built.

Laminated poles were opted against, due to problems with fitting keys.However, if a much thicker plate is used for lamination, threaded bolt holescould be fitted in the face of the plates facing the pole shoe holder. This wasrealised too late in the design process to be properly evaluated. This couldpossibly could simplify production of the pole shoes significantly by allowingthe pole shoes to be cut from a plate. The pole shoes in the current designwill require a lot of milling to shape from the raw material. Drawbacks withlaminated poles are that holes going through the entire length of the poleshoe would be required, disrupting the magnetic flux, and that more boltswould be needed to fasten the pole shoe to the holder.

The design has been strongly influenced by the requirement to leave thestator intact. The lack of space for the ferrite PMs has been a problem andhas prevented the new design from achieving the same performance as theold design. A larger stator inner diameter or larger pole pitch angle wouldprobably have been used if both the rotor and stator had been designed

41

together. An outer pole machine, with the PMs mounted on a ring shapedrotor outside a cylindrical stator, could have been an interesting possibility.

42

Chapter 6

Conclusion

The new design, as presented in chapter 4, fulfils the requirements given atthe start of the project, stated in Section 1.2.

The electric performance will not be the same as with the old rotor.The air gap flux density and output voltage will both have lower amplitude.The rated power is possible to maintain by increasing the current density inthe armature windings. The overall performance of the new design will besimilar to the performance of the old design, making the new design a viablesubstitute.

Assembly of the new design has been investigated. The investigationindicated no major difficulty with the assembly process. The new design willalso be able to withstand the mechanical loads it will be subjected to.

The design has been constrained by the requirement to keep the stator de-sign unchanged. If the whole machine had been designed at once, a differentstator design would have been used and better performance achieved.

The conclusion of this report is that the new rotor design suggested, oncefinalised, will be a viable substitute for the old rotor.

43

Bibliography

[1] A. Solum, P. Deglaire, S. Eriksson, M. Stålberg, M. Leijon, and H. Bern-hoff. Design of a 12kw vertical axis wind turbine equipped with a directdriven PM synchronous generator. In EWEC 2006-European Wind En-ergy Conference & Exhibition, Athens, Greece, 2006.

[2] Sandra Eriksson. Direct Driven Generators for Vertical Axis Wind Tur-bines. Acta Universitatis Upsaliensis, Uppsala, Sweden, 2008.

[3] S. Eriksson and H. Bernhoff. Rotor design for PM generators reflectingthe unstable neodymium price. In Proceedings of the 2012 XXth In-ternational Conference on Electrical Machines (ICEM), pages 1419–23.IEEE, 2012. 2012 XXth International Conference on Electrical Machines(ICEM), 2-5 Sept. 2012, Marseille, France.

[4] K.J. Binns and A. Kurdali. Permanent-magnet a.c. generators. ElectricalEngineers, Proceedings of the Institution of, 126(7):690–696, july 1979.

[5] Z. Chen and E. Spooner. A modular, permanent-magnet generator forvariable speed wind turbines. In Electrical Machines and Drives, 1995.Seventh International Conference on (Conf. Publ. No. 412), pages 453–457, sep 1995.

[6] E. Spooner and A.C. Williamson. Direct coupled, permanent magnetgenerators for wind turbine applications. Electric Power Applications,IEE Proceedings -, 143(1):1–8, jan 1996.

[7] E. Spooner, A.C. Williamson, and G. Catto. Modular design ofpermanent-magnet generators for wind turbines. Electric Power Ap-plications, IEE Proceedings -, 143(5):388 –395, sep 1996.

[8] E. Muljadi, C.P. Butterfield, and Yih-Huie Wan. Axial-flux modularpermanent-magnet generator with a toroidal winding for wind-turbineapplications. Industry Applications, IEEE Transactions on, 35(4):831–836, jul/aug 1999.

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[9] Mikael Dahlgren, Harry Frank, Mats Leijon, Fredrik Owman, and LarsWalfridsson. Windformer – wind power goes large scale. ABB Review,3:31–37, 2000.

[10] Ki-Chan Kim and Ju Lee. The dynamic analysis of a spoke-type perma-nent magnet generator with large overhang. Magnetics, IEEE Transac-tions on, 41(10):3805 – 3807, oct. 2005.

[11] Seok-Myeong Jang, Ho-Jun Seo, Yu-Seop Park, Hyung-Il Park, andJang-Young Choi. Design and electromagnetic field characteristic anal-ysis of 1.5 kW small scale wind power generator for substitution of Nd-Fe-B to ferrite permanent magnet. IEEE Transactions on Magnetics,48:2933–6, Nov. 2012.

[12] T. Miura, S. Chino, M. Takemoto, S. Ogasawara, A. Chiba, andN. Hoshi. A ferrite permanent magnet axial gap motor with segmentedrotor structure for the next generation hybrid vehicle. In Electrical Ma-chines (ICEM), 2010 XIX International Conference on, pages 1 –6, sept.2010.

[13] D.G. Dorrell, M. Hsieh, and A.M. Knight. Alternative rotor designsfor high performance brushless permanent magnet machines for hybridelectric vehicles. Magnetics, IEEE Transactions on, 48(2):835–838, feb.2012.

[14] M. Barcaro and N. Bianchi. Interior PM machines using ferrite to sub-stitute rare-earth surface PM machines. In Electrical Machines (ICEM),2012 XXth International Conference on, pages 1339 –1345, sept. 2012.

[15] K. Nakamura, J. Yoshida, and O. Ichinokura. A novel high power per-manent magnet reluctance generator using ferrite magnet. In PowerElectronics and Applications, 2009. EPE ’09. 13th European Conferenceon, pages 1–8, sept. 2009.

[16] K. Kurihara, T. Kubota, T. Kosaka, and T. Nakamura. A single-phasereluctance generator with permanent magnets between stator teeth. InElectrical Machines (ICEM), 2010 XIX International Conference on,pages 1–6, sept. 2010.

[17] A. Hannalla. Analysis of transient field problems in electrical machinesallowing for end leakage and external reactances. Magnetics, IEEETransactions on, 17(2):1240–1243, 1981.

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[18] Mats G. Larson and Fredrik Bengzon. The Finite Element Method:Theory, Implementation, and Practice. Springer, 2010.

[19] Jr. William H Hayt and John A Buck. Engineering Electromagnetics.McGraw-Hill, 7 edition, 2006.

[20] John David Jackson. Classical Electrodynamics. John Wiley & Sons,Inc., 1962.

[21] Gunnar Dahlvig. Konstruktionselement och maskinbyggnad. Liber,Stockholm, Sweden, 5th edition, 1988.

[22] Michael J. Neale, editor. Tribology Handbook. Elsevier, 2nd edition,1995.

[23] David Jiles. Introduction to Magnetism and Magnetic Materials. CRCTaylor & Francis, Florida, 2nd edition, 1998.

[24] Bob johnson associates - ceramic/ferrite magnets - sintered grades.Retrieved from http://www.bjamagnetics.com/html/ceramic_ferrite_magnets_-_sint.html on 2013-05-07.

[25] Sandra Eriksson, Andreas Solum, Mats Leijon, and Hans Bernhoff. Sim-ulations and experiments on a 12 kw direct driven PM synchronousgenerator for wind power. Renewable Energy, 33(4):674 – 681, 2008.

[26] Magnus Carlunger, Carl-Gösta Dock, Torsten Friedler, and IngvarIsaksson. Bultens teknikhandbok, 1999. Retrieved from http://www.bufab.com/Portals/0/090330Teknikhandboken_72_dpi_090327.pdfon 2013-05-13.

[27] Fredrik Bülow, Sandra Eriksson, and Hans Bernhoff. No-load core lossprediction of PM generator at low electrical frequency. Renewable en-ergy, 43:389–392, 2012.

46

Appendix A

Drawings

In this appendix the drawings of the parts are given. The order of the partsis the same as in section 4.1. The part names given on the drawing are thenames used for the CAD program files and a table relating them to the namesused in the report is given table A.1.

Table A.1: Table for translating the part names on the drawings to the partnames used in the report. In order of appearance in Section 4.1.

In report On drawing

Bottom rotor end plate rotorbottenskivaTop rotor end plate rotortoppskivaPole shoe holder polskohållarePole shoe polsko_omparamMagnet holder bar ändstoppInner support innerstöd_v3Large support ring stödringFlange krageSmall support ring liten_stödringShaft axel2Generator end board generatorlock

47

112

R80

R11

1,75

R14

3,25

R174,7

5

R206,16

R222,14

R330

686

16

16

16

4,5

0

9

120

97,

50 ±

0

10

rotorbottenskivaWEIGHT:

A4

SHEET 1 OF 1SCALE:1:5

DWG NO.

TITLE:

REVISIONDO NOT SCALE DRAWING

MATERIAL:

DATESIGNATURENAME

DEBUR AND BREAK SHARP EDGES

FINISH:UNLESS OTHERWISE SPECIFIED:DIMENSIONS ARE IN MILLIMETERSSURFACE FINISH:TOLERANCES: LINEAR: ANGULAR:

Q.A

MFG

APPV'D

CHK'D

DRAWN

484

112

R80

R11

1,75

R143,25 R

174,

75

R206,16

R222,14

4,50

9

484

120

97,

50

143

,25

10

484

rotortoppskivaWEIGHT:

A4


DWG NO.

TITLE:


MATERIAL:

DATESIGNATURENAME



Q.A

MFG

APPV'D

CHK'D

DRAWN

5 5

50

220

5

13

38,

80

38,

80

38,

80

38,

80

38,

80

13

36,18

4,50

21 13

10,

09

10,

09

36,

18

polskohållareWEIGHT:

A4


DWG NO.

TITLE:


MATERIAL:

DATESIGNATURENAME



Q.A

MFG

APPV'D

CHK'D

DRAWN

27 60

9,6

8

129,98

40,

72

11,

25°

11,25°

125,20

34,

23

3,

30

224

3,30

13

38,

80

38,

80

38,

80

38,

80

38,

80

17

60 27

polsko_omparamWEIGHT:

A4


DWG NO.

TITLE:


MATERIAL:

DATESIGNATURENAME



Q.A

MFG

APPV'D

CHK'D

DRAWN

83,81

R10

4,5

0

4

ändstoppWEIGHT:

A4


DWG NO.

TITLE:


MATERIAL:

DATESIGNATURENAME



Q.A

MFG

APPV'D

CHK'D

DRAWN

220

125

95

5

30

140

3

0 1

0

95

9 16

31,50 31,50 16

30

R2

15,75

30 1

5 6,80

R2

innnerstöd_v3WEIGHT:

A4


DWG NO.

TITLE:


MATERIAL:

DATESIGNATURENAME



Q.A

MFG

APPV'D

CHK'D

DRAWN

468

38

8

R206,16

R222,14

4,5

0

16 16

9

174,75

30 468

stödringWEIGHT:

A4


DWG NO.

TITLE:


MATERIAL:

DATESIGNATURENAME



Q.A

MFG

APPV'D

CHK'D

DRAWN

R47,50

R57,50

R92,50

R80 9

11,

50

10

1

R0,25

R0,50

R0,50

TRUE R0,25

R0,50 R0,50

9

7,80

8

13 16

16

16

16

8

krageWEIGHT:

A4


DWG NO.

TITLE:


MATERIAL:

DATESIGNATURENAME



Q.A

MFG

APPV'D

CHK'D

DRAWN

18

5

122

R80

9 18° 18°

7

185

liten_stödringWEIGHT:

A4


DWG NO.

TITLE:


MATERIAL:

DATESIGNATURENAME



Q.A

MFG

APPV'D

CHK'D

DRAWN

990

240

53

95

53

35

39 1

45

425

110

97

97

95

95

55

axel2WEIGHT:

A4


DWG NO.

TITLE:


MATERIAL:

DATESIGNATURENAME



Q.A

MFG

APPV'D

CHK'D

DRAWN

100

0

R455 R485

13,50

1000

11

22°

1

1,50

°

12

0

R16

R34,01

395

280

230

25

generatorlockWEIGHT:

A4


DWG NO.

TITLE:


MATERIAL:

DATESIGNATURENAME



Q.A

MFG

APPV'D

CHK'D

DRAWN

Design of a Ferrite Permanent Magnet Rotor for a Wind Power

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