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

UPTEC ES13018 Examensarbete 30 hp Juni 2013

Petter Eklund Abstract Design of a Ferrite Permanent Magnet Rotor for a Wind Power Generator Petter Eklund

Teknisk- naturvetenskaplig fakultet UTH-enheten Due to the insecurity of the supply of raw materials needed for neodymium-iron-boron magnets, typically used in permanent magnet generators, the Besöksadress: use of ferrite magnets as an alternative was investigated. The investigation was Ångströmlaboratoriet Lägerhyddsvägen 1 conducted by attempting to redesign a generator that previously used Hus 4, Plan 0 neodymium-iron-boron magnets for use with ferrite magnets. The major part of the redesign was to find an alternate rotor design with an electromagnetic design adapted Postadress: to the characteristics of the ferrite magnets. Box 536 751 21 Uppsala It was found that ferrite magnets can be used to replace neodymium-iron-boron magnets with changes to the electromagnetic design of the rotor. The changes of the Telefon: electromagnetic design increase the amount of magnetically active material in the 018 – 471 30 03 rotor and, therefore, require the mechanical design of the rotor to be changed. The

Telefax: new rotor design also requires some changes to the generator support structure. A 018 – 471 30 00 design for a replacement rotor, using ferrite magnets, along with the required changes to the support structure, is presented. Hemsida: http://www.teknat.uu.se/student

Handledare: Stefan Larsson Ämnesgranskare: Sandra Eriksson Examinator: Kjell Pernestål ISSN: 1650-8300, UPTEC ES13 018 Populärvetenskaplig sammanfattning

På senare tid har priset på och tillgången till råmaterialen som behövs för att tillverka neodym-järn-borpermanentmagneter blivit osäkra. Denna typ av 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 generatorer. I det här arbetet har därför möjligheterna att bygga om en generator, som ursprungligen byggdes för att använda neodym-järn-bormagneter, så att den använder en annan typ av magneter undersökts. Denna andra typ av permanentmagneter som användes var ferritmagneter. För att anpassa generatorn efter ferritmagneterna måste den roterande delen som magneterna sitter på, rotorn, byggas om eller bytas ut. Att den stillastående delen av generatorn, statorn, ska lämnas oförändrad är en be- gränsing i konstruktionsarbetet från uppdragsgivaren. Den nya rotorn måste anpassas till att ferritmagneterna är mycket svagare än neodym-järn-bormagneterna. Svagare magneter innebär att större volym magneter behövs, vilket gör rotorn tyngre. Dessutom måste magnetfältet förstärkas för att generatorn ska ge tillräcklig elektrisk utspänning. Eftersom magnetfältet måste förstärkas, fungerar inte längre den rotortyp som tidigare använts. I den rotortypen sitter magneterna på utsidan av en stålcylinder med ena polen mot och andra polen bort från cylindern. I denna typ av rotor sammanfaller rotorns magnetiska poler med magneternas. En rotortyp som förstärker magnetfältet kan fås genom att placera magneterna i 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 en magnets poler än vad som kan vändas mot rotorns utsida i det utrymme som ﬁnns för en av rotorns poler. Denna ihopsamling av magnetfältet gör att rotorns 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 en kraftigare stödstruktur. Till stödstrukturen kan man inte använda vanligt konstruktionsstål i närheten av den magnetiska kretsen. Det beror på att stål är magnetiskt och kan skapa en oönskad väg för magnetfältet mellan magneternas poler. Därför har aluminium, som inte är magnetiskt, använts till den stödstruktur som behövs för att hålla ihop ringen av stålstycken och magneter och förbinda den med generatorns axel. Rostfritt stål kunde också ha använts men det är svårare än aluminium att bearbeta och tyngre. Efter att konstruktionen gjorts färdig kontrollerades att alla delar var rimliga att tillverka och det gjordes en studie för att kontrollera att konstruktionen gick att montera. Slutsatsen av arbetet blev att det i permanentmagnetiserade generatorer går att ersätta neodym-järn-bormagneter med ferritmagneter. För att göra detta måste en annan rotorkonstruktion användas och den elektromagnetiska prestandan kan bli något sämre. Contents

Executive summary ...... 3

1 Introduction 4 1.1 Background ...... 4 1.1.1 Why Replace the Neodymium-Iron-Boron Magnets . .6 1.1.2 Earlier Usage of Ferrite Magnets in Permanent Magnet Synchronous Generators ...... 6 1.1.3 Earlier Usage of Ferrite Magnets in Other Kinds of Rotating Electrical Machines ...... 7 1.2 Scope and goals of the present work ...... 8

2 Theory 9 2.1 Simulation of Electrical Machines Using Two-Dimensional Fi- nite Element Methods ...... 9 2.2 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 17 3.1 Method and Materials ...... 17 3.2 Design Requirements ...... 17 3.3 Procedure ...... 18

4 Results 21 4.1 Description of components ...... 22 4.1.1 Rotor End Plates ...... 22 4.1.2 Pole Shoe Holder ...... 23 4.1.3 Pole Shoe ...... 23 4.1.4 Permanent Magnet ...... 25 4.1.5 Magnet Holder Bar ...... 25 4.1.6 Inner Support ...... 25

1 4.1.7 Large Support Ring ...... 27 4.1.8 Flange ...... 27 4.1.9 Small Support Ring ...... 28 4.1.10 Shaft ...... 28 4.1.11 Generator End Board ...... 30 4.2 The New Magnetic Circuit ...... 30 4.3 Magnetic Forces ...... 33 4.4 Calculation of Bolted Joints ...... 34 4.5 Static Stiﬀness of the Rotor ...... 35 4.6 Study of Natural Frequencies and Modes of Vibration . . . . . 36 4.7 Comparison of New and Old Design ...... 38 4.8 Unﬁnished Parts of the Design ...... 38

5 Discussion of results 40

6 Conclusion 43

Bibliography 44

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

2 Executive Summary

Using electromagnetic simulations, the magnetic circuit of a neodymium- iron-boron, NdFeB, magnetised permanent magnet generator was redesigned to use ferrite permanent magnets. A support structure was designed for the new magnetic circuit using computer-assisted design tools and finite elements for structural mechanics analysis. The result of the investigation was that in order to replace NdFeB magnets with ferrite magnets, a new rotor has to be designed. A design for such a rotor, including other required design changes, is presented here. The design has some minor details that need to be finalised. The proposed future action is to finalise the design and build the rotor.

3 Chapter 1

Introduction

Wherein the background to the project along with the scope and goals are presented.

1.1 Background

This project is part of the wind power research program at the Division of Electricity at Uppsala University, and the goal is to design a new rotor for the generator used. The wind power concept studied in the program uses a vertical axis wind turbine that drives a permanent magnet synchronous generator mounted on a common shaft. This allows the generator to be placed on the ground which allows a heavier generator to be used, and the direct drive keeps the number of moving parts down. To allow variable speed operation, 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 electrical frequency of 33.9 Hz at rated speed. The generator has an outer stator diameter 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 mounted Neodymium-Iron-Boron, NdFeB, permanent magnets, PMs. Due to the re- cent increase in price of NdFeB PMs, it has become interesting to investigate the possibility of using other, less expensive, PM materials. One possible candidate for PM material is ceramic ferrites, which will be investigated in this project. Replacing the NdFeB PMs with ferrite PMs will require a new design of the rotor. This is because of the very diﬀerent characteristics of the NdFeB

4 Table 1.1: Characteristics of the generator built using the old design. This table is based on Table 4.1 of [2]. Rated power 12.0 kW No load phase voltage (rms) 161 V 2 Armature current density (rms) 1.6 A/mm Electrical frequency 33.9 Hz Rotational speed 127 rpm Number of poles 32 Number of slots per pole and phase 5/4 Stator inner diameter 760 mm Stator outer diameter 886 mm Air gap width 10 mm Generator length 222 mm

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

5 PMs and the ferrite PMs: the NdFeB PMs have significantly higher remanent flux density and coercivity than the ferrite PMs. While the remanence of the NdFeB magnets is sufficient to give an acceptable air gap flux, even when surface mounted, the ferrite magnets require some kind of flux concentrating scheme to achieve acceptable air gap flux densities. One possibility is to mount the magnets in a so-called spoke configuration. A spoke configuration means that magnets with alternating tangential magnetisation, poles of the same kind facing each other, are placed with pole shoes in between. This allows the flux from a large area of the PM to be directed into a rotor pole and 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 dysprosium, 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 by over 1000%. As neodymium and dysprosium accounts for 60% and 35% of the material cost, respectivly, changes in the prices of these metals aﬀect the price 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 viable substitute for the NdFeB PMs, despite their lower performance, in light of these problems [3].

1.1.2 Earlier Usage of Ferrite Magnets in Permanent Magnet Synchronous Generators The use of ferrite permanent magnets, PMs, in PM magnetised synchronous machines is not a new idea. In 1979, Binns et al. [4] proposed and built a synchronous machine using ferrite PMs. The rotor used a configuration with axially magnetised PMs placed between flux guides that guide the flux into a radial direction in the air gap. The article also mentions a machine with a spoke design. A spoke design means the magnets in the rotor are placed like spokes and are tangentially magnetised with two poles of the same kind facing each other. In the mid 1990s, Spooner et al. [5, 6, 7] published articles describing a modular 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 intended to 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 connection. Another modular design, using an axial flux topology, is presented by Muljadi et al. [8]. The design is supposed to be usable both with ferrite and NdFeB PMs, but the test prototype was built using NdFeB PMs. One important precursor to the current work was the Windformer™ concept presented by ABB in 2000. In this concept, a direct driven generator wound with high voltage cable and magnetised with ferrite magnets formed the core of a wind power system. The magnets were mounted in a spoke configuration, 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 ferrite PMs with a rotor that is significantly longer than the stator. This enables flux concentration in the axial direction as well as in the plane perpendicular to the axis of rotation. The flux concentration in the axial direction is beneficial for ferrite magnetised generators. The axial flux concentration is beneficial because it allows the air gap flux to be raised higher above the remanent 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 ferrite PM generator could be built with the same diameter and length and similar performance as a NdFeB PM generator. To achieve this, a redesign of the magnetic circuit was necessary [11].

1.1.3 Earlier Usage of Ferrite Magnets in Other Kinds of Rotating Electrical Machines Also in electric motor design, there have been eﬀorts to replace rare earth magnets with ferrite magnets. In 2010, Miura et al. suggested using an axial ﬂux machine with ferrite PMs to achieve the same power density as in a rare earth PM machine. The ferrite magnets would be magnetised in an axial direction and mounted in a disk shaped rotor sandwiched between two disk shaped stators. To achieve the same power density, a higher rotational speed would be required as the ferrite motor gives lower torque [12]. In 2012, multiple authors published results relating to ferrite PMs in synchronous motors. Dorrell et al. [13] investigated the possibility of using a ferrite magnet rotor in the motor of the Toyota Prius hybrid electric vehicle. It was concluded that such substitution could be possible, and that the magnets 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. found that a slight lengthening of the machine could make a ferrite PM machine a valid 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 of the same size but at a reduced material cost [15]. In 2010, Kurihara et al. suggested a design for a PM reluctance generator using ferrite PMs mounted between the stator poles. The function of the PMs was to induce a sufficient residual flux to allow a small voltage to be induced. This voltage was then rectified and used to drive an excitation current through the field windings of the machine, allowing the machine to start without an external source of excitation current [16].

1.2 Scope and goals of the present work

The goal of the project was to redesign a permanent-magnet synchronous generator 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 gap as the old rotor. Furthermore, the new rotor should be easy to assemble and install and provide secure mounting for the permanent magnets. The scope of the project, therefore, included electromagnetic and mechanical design. Thermal design was excluded as it is not likely to be of concern to a permanent magnetised rotor. The thermal design of the stator was not regarded as the stator design would not be changed. The electromagnetic and mechanical design processes have been coupled since they provide constraints for each other. Building the rotor and determining what price of neodymium-iron-boron permanent magnets will be needed to make the ferrite more economical were both outside the scope of this project.

8 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 the electromagnetic field equations. The theory for how this is done is presented below. Using magnetic vector potential, A~, which is defined by B~ = ∇ × A~, where B~ is the magnetic flux density, the electromagnetic field equations can be formulated as ∂A~ ∇ · ν∇A~ = σ − J~ (2.1) ∂t where ν is the magnetic reluctivity, the inverse of permeability, σ is the electrical conductivity and J~ is the current density [17]. Let the axis of rotation of the machine coincide with the zˆ-axis of a cylindrical coordinate system and the ends at z = ±l/2 with l being the length of the machine. Then the magnetic fields in the z = 0 plane can be used to approximate the fields in the whole length of the machine, given that the length of the machine is much larger than the size of the air gap and other features in the geometry in the plane. In the z = 0 plane, the z-component ~ ~ of B is Bz = 0 and also ∂A/∂z = 0 due to symmetry. This allows us to get the fields in the plane as the z-component of eqn. (2.1), which simplifies the solution to solving for only Az. Further simplification can be made by using sector symmetry and periodic boundary conditions to reduce size of the domain over which calculations are made [2, 17].

9 A full account of the finite element method is beyond the scope of this thesis, but a brief overview will be given. The basic idea is that you have a partial differential equation, PDE, on a domain Ω with boundary conditions on the boundary ∂Ω. Then there exists a vector space of functions on Ω that fulfils the boundary conditions on ∂Ω called V , and the true solution is in this space. The next step is to rewrite the PDE using multiplication with an arbitrary element of V , called the test function, using the L2 inner product on Ω and Green’s identities into a integral equation. The integral equation is called the weak form of the PDE. Once the weak form of the PDE has been derived, Ω is triangulated and a subspace Vh of V that has a finite number of basis function is 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 and zero in all other nodes; between nodes it is linear. Since the basis functions span 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 solution to the PDE is approximated as an unknown linear combination of the test functions. This transforms the integral equation into a system of linear equations that can be solved by various methods. It can be shown that the approximation of the solution is the orthogonal L2-projection of the true solution on Vh. This means that it is the best approximation of the true 2 solution that can be made on Vh measured in the L -norm [18].

2.2 Methods of Calculating Magnetic Forces on Magnetic Materials

There are two methods that can be used to calculate magnetic forces on magnetic 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 work and is described by Hayt and Buck [19, p. 290], and the other is the Maxwell stress tensor that can be derived from the Lorentz force using Maxwell’s equations [20, p. 193]. The virtual work approach is limited to linear media and cases where movement does not cause any change in magnetisation or induce eddy currents. In this case, linear media means that B~ is related to the magnetic ﬁeld, H~ , through the relation B~ = µH~ (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

10 volume, V , is given by ZZZ 1 ~ ~ Wm = B · H dV (2.3) 2 V and, furthermore, that the change in energy dWm can be equated with mechanical work done by moving part of the magnetic circuit a directed distance, ˆ ~ l dl , while a magnetic force, Fm, acts upon it giving ~ ˆ dWm = Fm · l dl (2.4) ˆ ~ giving the l component of Fm as

dWm F ˆ = . (2.5) m,l dl The Maxwell stress tensor approach is more general. Its derivation starts with 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 the charge. The point charge is then replaced with a charge distribution to get the force per volume. Using Maxwell’s equations together with some vector algebraic manipulations, the force density is given by ~ ~ ∂S f + µ00 = ∇ · T (2.7) ∂t

S~ = 1 E~ ×B~ T where µ0 is the Poynting vector, and is the second order tensor- dyadic known as the Maxwell stress tensor with elements given by

1 2 1 1 2 Tij = 0[EiEj − E δij] + [BiBj − B δij] (2.8) 2 µ0 2 where E and B are the components of corresponding vector quantities along direction indicated by the index, no index signifying absolute value, and δij is 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 a volume, the force on the volume is given. Using the divergence, theorem it can be shown that the normal component of the Maxwell stress tensor is the ﬂux of momentum through a surface.

11 In the case of a generator the electric ﬁelds in the air gap are negligible, ∂S~ E ≈ 0, and the frequencies low, ∂t ≈ 0, and the force on the rotor becomes I F~ = nˆ · T dS (2.9) S where nˆ is the outward normal unit vector of a closed surface S enclosing the rotor but not the stator. Likewise, the torque on the rotor is given by I M~ = ~r × (tˆ· T) dS (2.10) S where tˆ is the tangent unit vector to S that is also in the plane of rotation and ~r the location relative to the axis of rotation.

2.2.1 Analytic Model for Veriﬁcation of Magnetic Force Calculation The purpose of this model is not to determine the exact force but rather to verify that the force obtained from the FEM model has the right order of magnitude. The model uses the principle of virtual work. The magnetic circuit of the generator is modelled in 32 pieces as two half poles with a permanent magnet, PM, between them. The geometry of the model is given in Figure 2.1. The out-of-plane thickness is l. The iron parts are approximated as a perfect magnetic conductor, which simpliﬁes the magnetic circuit to a PM and two air gaps. This should be an acceptable approximation as the iron parts have a relative permeability in the order of at least 103, making the reluctance in the iron negligible compared to that of the air gap. In the air gap, the permeability is that of free space, denoted µ0. Recall the virtual work approach to calculating magnetic force, outlined in section 2.2, and especially eqn. (2.3). Then assume that B~ in the air gap is perpendicular to the cross section area, Ag, of the air gap associated with ~ half a pole with amplitude B⊥g. Also assume that B in the magnets is also perpendicular to the cross section of the PMs, area bl, with amplitude B⊥pm. Then eqn. (2.3) evaluates to

2 2 B⊥g B⊥pm Wm = 2Agg + (bl)h (2.11) 2µ0 2µ0 where everything but B⊥g and B⊥pm is known. To ﬁnd B⊥g and B⊥pm, a magnetic reluctance circuit model can be used.

12 πDsi 32

M Iron Air gap PM

Direction of M b

Figure 2.1: Geometry of the two-half-pole-model used for validation of force calculations. Dsi is the inner diameter of the stator. The out-of-plane thickness 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 equal reluctances in series. The mmf of a PM is given by the product of the remanence, M, and the height, h, of the magnet along the direction of magnetisation F = Mh . (2.12) Magnetic reluctance is given by d R = (2.13) µA where d is the length of the magnetic ﬂux tube, and A is the cross section area of the magnetic ﬂux tube. For the PM, this gives h RPM = (2.14) µ0lb and for the air gap g Rg = (2.15) µ0Ag

13 where the area Ag is the average area of half the pole pitch and half the pole shoe width given by πDsi h Ag = l − . (2.16) 64 4 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, the magnetic ﬂux in the circuit is then given by

Φ = AgB⊥g = bl B⊥pm (2.18) which can be rewritten as Φ Φ B⊥g = ,B⊥pm = (2.19) Ag bl and inserted into eqn. (2.11). This gives 2 Φ Φ 2 Ag bl Wm = 2Agg + (bl)h (2.20) 2µ0 2µ0 that can be simpliﬁed to 2 g h 2 g RPM Wm = Φ + = Φ + (2.21) µ0Ag 2µ0bl µ0Ag 2 Inserting eqn. (2.21) into eqn. (2.5), the magnetic force on one two-half-pole unit can be expresses as !2 dWm −1 F Fm(g) = = 2g (2.22) dg µ0Ag RPM + µ0Ag with the positive direction of the force in the direction of increasing g. The same expression can be derived using the magnetism-only case of the Maxwell Stress Tensor on the magnetic circuit. To calculate the force on the rotor, deﬁne a starting air gap g0 and a rotor displacement ∆g. Then the total force on the rotor, for a displacement ∆g from perfectly concentric with the stator, is calculated by summing the force on each unit. The force from each unit is calculated using an air gap of

g = g0 + ∆g cos(θ) (2.23)

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

32 X 2π 2π Frotor(∆g) = Fg(g0 + ∆g cos i + φ ) · cos i + φ (2.24) 32 32 i=1 where φ is the angle between the line of the displacement and the line through the middle of the nearest pole.

2.3 Bolted Joints

For bolted joints, there are two different mechanisms by which the joint can withstand shearing forces. One way is to have the bolt fitted tightly into the hole, transferring the force between the parts via contact forces and shear stresses in the bolt. The other way is to have the bolt press the two parts together with such a force that the friction between the surfaces of the parts is sufficient to withstand the shear force. To use friction instead of shearing to transfer the force generally requires stronger bolts [21, p. 2:41] In the current design, it was not possible to use tightly fitted bolts in all places; therefore, friction was used. The theory developed to calculate required number of bolt is described below. To calculate how many bolt were needed for each joint, it was assumed that the bolts would have to press the parts together with such force that the friction forces would be able to withstand the loads on the joint. The common model for friction between hard surfaces, such as metal, given in literature gives the maximum friction force as

Ff = fFn (2.25) where µ is the coeﬃcient of friction and Fn the normal force. The coeﬃcient of 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 the axial direction have been subtracted. Let Fla be the load forces along the axis of the bolt, Fls the load forces shearing the bolt, Fas the axial force on the bolt at suitable level of prestressing and N the number of bolts. Then a condition can be formulated as Fls NFas ≥ s + Fla (2.26) µ

15 where s is a factor of safety. When transferring torques, the force is held to act upon the radius of the cylinder in the case of cylindrical surfaces, or along the circle centred upon the axis of the torque and going through the mid-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 Materials

The main tools in the design process have been computer modelling software. The computer assisted design software package SolidWorks®1 has been used both 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 element analysis. For electromagnetic simulation, COMSOL Multiphysics®2 has been used. To check the results from COMSOL, the in-house electromagnetic simulation software KALK has also been used [2]. In addition, some simpler models of bolted joints and the magnetic circuit have been derived and implemented in MATLAB®. These are described in sections 2.2.1 and 2.3, respectively.

3.2 Design Requirements

There were a number of requirements on the design. Since the stator was to be reused, it could not be changed. This ﬁxed the inner stator diameter at 760 mm and stator length at 224 mm. The number of poles was also ﬁxed at 32. Whenever possible, the use a standard size of magnets was another requirement. Additionally, the air gap should be at least 7 mm. 1www.solidworks.com 2www.comsol.com

17 To be able to fasten the pole shoes, there needs to be room for bolts. The material surrounding a threaded bolt hole needs to be at least twice as thick as the diameter of the bolt as a rule of thumb. It was judged that at least an M4 bolt would be needed. This meant that the inner end of the pole shoe had to be at least 8 mm wide. Finally, it was desirable to keep the electric characteristics of the new design as close to the characteristics of the old design as possible.

3.3 Procedure

The first step of the design process was to find the requirements and constraints imposed on the new rotor by the parts of the generator that would be reused. For instance, the inner diameter of the stator and the minimum required air gap limit rotor diameter and the stator stack height determines the height of the rotor. Also, the new rotor will have to supply approximately the same magnetic flux density if the electrical properties of the generator are to remain comparable. Once the basic requirements and limitations were made clear, a very simple model of the magnetic circuit, just an air gap and a permanent magnet representing a single pole, was made to get a rough estimate of what size of permanent magnet was required. Approximate magnetic properties of the PM were taken from a textbook [23]. This information was used to choose rotor topology. The two topologies of interest were surface mounted, used in the 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 conﬁguration with pole shoes. The arrows indicate the direction of magnetisation of the magnets. Gray areas indicate soft magnetic materials.

Once the topology was chosen, the spoke conﬁguration, a parametric, two dimensional 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 to the plane of rotation for each phase, with corrections for winding direction, dividing by the cross sectional area of the conductor, and multiplying by the lenght of the machine. The armature currents were then applied as an external current density in the conductors. The current density in a conductor was calculated as Vph Jz = (3.1) (Rwind + Rload)A 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. For the soft magnetic materials, the steel in stator and pole shoes, one-to-one BH-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 was simulated using finite element methods, FEM, to solve the electromagnetic field equations. The underlying theory for the simulations is presented in section 2.1. From the FEM simulations, the desired size of PMs was determined. Some constraints on the geometry were given; for instance, a minimum width of the inner end of the pole shoe was required so that bolts to hold it in place could fit. To find out if there were standard sizes of PMs close to the desired size, manufacturers of ferrite PMs were contacted. Using standard sizes is advantageous as it avoids the costs and longer delivery times associated with sizes made to measure. The sizes and material grades offered by the manufacturers were then inserted into the simulation to check if they gave sufficient 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 generated in the new machine. Also, the forces on the PM during mounting were exam- ined through simulation. For these simulations, magnetisation was assumed to be at the upper bound of the offered magnetisation of the material grade with the highest magnetisation. This was done to ensure that the calculated forces will be at least as large as the actual forces. Once an upper bound on the forces from the magnetic circuit was determined, design of the support structure could start. Due to problems caused by welds in the old rotor, welded joint were avoided, and bolted joints were used instead. The bolted joints were designed with a clamp load to avoid subjecting the bolt to shearing forces. Details which needed lots of machining were 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 rotor eccentricity of ∆g there will be a net force on the rotor due to unbalanced magnetic pull from the magnetic circuit. If the design is stiff enough, this force should not be large enough to pull the rotor ∆g toward the stator from its original position, where it is concentric with the stator. When a suitable ferrite magnet had been found and the rotor was found to be stiff enough against static deformation, simulations were run to exam- ine the natural modes of vibration. Some changes to the design were then made to attempt to shift the natural frequencies associated with the modes found upward. Shifting the frequencies upward moves them away from the most prominent frequencies emanating from the electromagnetic parts of the machine. Most of these changes were additions of stronger supports to stiffen the rotor against vibrations in the natural modes of vibration. As a final step in the design process, the design was reviewed to make sure all bolts could be accessed for tightening, and a rough plan for how to assemble the rotor was made to ensure it could be done. This also led to some minor changes. Due to time constraints, the selection of bearings and the design of the bearing unit housing were omitted, but some consideration have been given to the matter.

20 Chapter 4

Results

Wherein the results from the design process are presented. First, an overview of the whole assembly is given, and then the parts are presented one at the time. A section view of the new rotor design is shown in Figure 4.1. Aluminium has been the material of choice for much of the support structure due to its low weight and nonmagnetic nature. All parts are joined by bolts to avoid welds, which can cause deformity in the structure. The rotor has two plates in the plane of rotation (see Section 4.1.1), one at 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 (see Section 4.1.6) to stiffen the structure. The pole shoes are fastened to their holders with bolts. The PMs (see Section 4.1.4) are held in place between the pole shoes by the bottom plate, bars that are bolted into the top of the pole shoes after mounting the magnet, the pole shoe holders, and the ridges on the corners of the pole shoes. To attach the rotor to the shaft, two flanges are used (see Section 4.1.8). Each flange is made up of two halves that are bolted together. The flanges transfer the torque, using friction both at the interface with the shaft and with the rest of the rotor. For extra stiffness, a thick ring of aluminium (see Section 4.1.7) is fastened to both of the rotor end plates using the same bolts that are used for fastening the pole shoe holders. The ring is also fastened to the supports with bolts going through the end plates. A more detailed description of all the rotor parts is presented in Section 4.1.

21 4.1.4 4.1.5

¥ ¥ ¥ ¥ ¥ ¥ ¥ 4.1.3 ¥ ¥ 4.1.1 4.1.2 4.1.9 4.1.7 4.1.8

4.1.10 Figure 4.1: Section view of the new rotor, the central angle of the sector shown is 99°. The cut planes meet at the axis of rotation. The rotor shaft has been cropped at the lower edge of the drawing. One inner support joining the upper and lower plate of the rotor is obscured by the shaft but is presented in Section 4.1.6. The new generator end boards have also been omitted for clarity but are presented in Section 4.1.11. The number on the parts is the number of the section describing that part more in-depth. 4.1 Description of components

A more in depth presentation of each component and the factors considered when designing them is presented below. Drawings with measurements can be 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 the rotor end plates is to keep the rotor together in the plane of rotation. The bottom rotor end plate also provides a good starting point during assembly of the rotor. The thickness of 10 mm of aluminium was chosen as an initial approximation, based on the fact that the thickness of the plate with the same function in the old rotor was 5 mm of steel. When the natural frequencies of the new rotor were investigated, it was noted that making the plates thicker would improve the frequencies of some modes, but due to lack of space, this could not be done. The small holes have clearances for M4 bolts and the large for M8 bolts.

22 Figure 4.2: Top view of the rotor end plates. The bottom plate is to the left, and the top plate is to the right. The plates are made from 10 mm thick aluminium and have outer diameters of 685 mm and 484 mm, with the bottom 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 shoe holder is to provide a mounting place for the pole shoes and to connect the rotor end plates. At an earlier stage, mounting the pole shoes on a pipe was considered, but due to lack of available pipes of the required diameter, the current design with individual holders was chosen instead. The pole shoe holder is made from aluminium, and is intended to be manufactured 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 thickness of 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 holes have 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 to focus the magnetic flux from the permanent magnets into the air gap. Due to difficulties with fitting a dovetail key into the inner end of the pole shoes, a solid pole shoe was chosen instead of a stacked pole. This will introduce some extra machining cost but was judged to be the only workable approach. A possible alternative, brought to notice too late in the design process to be properly evaluated, was to stack the poles using a plate thick enough to accommodate a threaded bolt hole in the inner end. To conduct magnetic flux efficiently, the pole shoe will have to be made from a ferromagnetic material. Ordinary carbon steel was chosen for its mechanical 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 are 220 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. These are for fastening the pole shoe in the rest of the rotor and for fastening the magnet 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 to have 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 radial air gap flux density. This was done by having a constant air gap of 7 mm in the middle of the pole shoe for the midmost 8.44 mm and then tapering the pole shoes away from the stator. On the corners sticking out into the air gap, ridges have been placed to prevent the PMs from moving out into the air gap. No special effort was made to optimise the shape of the pole shoe for low harmonic content in the output voltage waveform or to reduce torque ripple.

4.1.4 Permanent Magnet The purpose of the permanent magnet is to magnetise the rotor. It is a rectangular cuboid with sides 226.5±2.5 mm, 122±0.2 mm, and 38±0.1 mm with magnetisation parallel to the shortest side. The cuboid is assembled from smaller pieces that have been bonded together prior to magnetisation. The material is ceramic ferrite in a grade called Y40 with a remanent magnetisation of 0.45 T. The longest side is 226.5 mm rather than 224 mm to ensure that the poles always get the magnetisation from the full height of a PM, even if the side is on the low end of the tolerance. The two other sides were chosen as stated because there was already a tool for manufacturing block magnets of that size available, which made them less costly. While the used size of the PMs is not strictly a standard size of PMs, it has most of the advantages of a standard size of PMs and no usable, standard size of PMs was available.

4.1.5 Magnet Holder Bar The magnet holder bar can be seen in Figure 4.5. The purpose of the magnet holder bar is to prevent the PMs from moving upward out of the slot between the pole shoes. The bar is made of non-magnetic aluminium to avoid interfering 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 mm thick 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 weight of magnetically active material in the rotor part of the magnetic circuit. It also makes the rotor stiﬀer against vibrations in some of the natural modes of vibration, as presented in Section 4.6. The inner support can be cut from a 30 mm aluminium plate. The holes at the ends are threaded for an M8 bolt, 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 stiﬀen the 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 aluminium plate.

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 (see Section 4.1.6) to be cut from the piece of metal removed from the centre of the ring and, the material thickness is the same. By manufacturing the ring and two supports from the same piece of metal, the amount of scrap can be reduced. Holes on the rim have clearance for M4 bolts, and the holes on the 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 the rotor to the shaft. This is done by clamping the flange onto the shaft with bolts 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. The outer radius is 92.5 mm, and the height is 53 mm. Material thickness varies between 5 mm and 10 mm. The flange is made of steel. into two parts to make assembling the rotor easier. The flange is made of steel, since it is far from the magnetic circuit, and will it be subject to higher stresses 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 small support ring is to stiffen the inner rim of the end plates and smooth out the pressure applied by the bolts that bond the end plates with the flange. It also 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 the rotor in place relative to the static parts of the machine and to transfer the torque from the primary mover. It is far from the magnetic circuit and can be expected to be subjected to among the highest stresses in the structure. Therefore, steel was chosen as the material for the shaft. Length and diameter are both inherited from the old design, since both affects the connection to the prime mover. The section with larger diameter will be inside the rotor. It was added to make the shaft stiffer against vibrations in some of the vibration modes presented in Section 4.6. The bearings will be mounted using press fits, 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 is 185 mm, thickness 10 mm, and it is made from steel.

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

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 is 1000 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 and stabilise the generator. The main loads are forces from unbalanced magnetic pull and the weight of the rotor. New end boards will be needed, since the new rotor has a larger mass than the old one. The generator will mostly be used for research and, therefore, larger openings for inserting measuring equipment are desired. The material is a glass ﬁbre reinforced epoxy board with an elasticity modulus of 24 GPa and tensile strength of 300 MPa, according to material data supplied by the manufacturer.

4.2 The New Magnetic Circuit

For the new magnetic circuit, a spoke type topology with pole shoes was chosen. 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 density of the ferrite PMs. The pole shoes are made from solid steel. Stacked poles were ruled out, due to difficulties with fitting dovetail keys at the inner end and also because of difficulties with compressing the stack once in place. The

30 pole shoes are mounted on a non-magnetic support structure, represented in the electromagnetic model and in Figure 4.12 as an aluminium tube. Magnetisation is provided by ferrite block magnets. The PM is magnetised along the shortest side, which has a length of 38 mm. The nominal remanent ﬂux density is 450 mT, according to speciﬁcations provided by the manufacturer [24]. 30 mm Air Laminated silicon steel Solid steel Ferrite PM Aluminium Cable

Figure 4.12: Cross sectional geometry of the new magnetic circuit with the various parts indicated. Eight instances of the sector shown together form a complete cross section. Parts indicated as air can also contain magnetically inactive support structure.The ferrite PMs are magnetised along their side tangential to the rotation of the rotor with alternating polarity. The out of plane thickness is 224 mm.

A cross section of the geometry of the magnetic circuit is shown in Figure 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. The air gap is 7 mm at its narrowest by design requirements. The face of the pole shoe parallel to the stator has a width of 8.44 mm, and the rest of the outer face of the pole shoe tapers away from the stator. On the side of the pole shoe 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 the PMs and make contact with the pole shoe holder. For detailed measurements on 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 the material library of COMSOL called “soft iron” was used for the steel used in the 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 has been 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 colour scale 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 outside the corners of the stator teeth. The radial component of the air gap flux along a line 3 mm from the inside of the stator is plotted in Figure 4.14. The maximum amplitude of the radial air gap flux is 0.71 T and the fundamental of 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 the simulations were remade. The simulations made with KALK showed good agreement with the simulations made with COMSOL. The amplitude of the air gap flux density only differed by 3% despite the fact that the geometry did not match entirely. Also, the general distribution of magnetic flux density was similar, both with respect to direction and amplitude, in both simulations. The geometry could not be made exactly the same, due to the geometry input feature of KALK being rather constrained. Agreement of results from

32 B~ · rˆ ~ (B · rˆ)1 − − Magnetic ﬂux density [T] − − Mechanical angle [°] Figure 4.14: The radial component of B~ in the air gap, for the two pole pairs in the sector used in the simulation along a circular arc 3 mm inside the inner perimeter of the stator. Maximum amplitude is 0.71 T, and the fundamental of the wave form is added as a dotted line. The amplitude of the fundamental is 0.66 T.

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

4.3 Magnetic Forces

The forces on a misaligned rotor were computed with the methods discussed in Section 2.2, using the FEM model for field calculations. The forces were also calculated with the simplified model derived in Section 2.2.1 for verifi- cation. The calculations using integration of the Maxwell Stress tensor and the calculations with the virtual work gave a resultant force, due to an unbalanced 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 magnitude can be considered acceptable agreement. The maximum torque occurring at a two-phase short circuit was determined to approximately 8300 Nm, which was judged to be the highest torque the machine might encounter. The maximum force a single pole shoe might be subjected to, occurring at 3 mm eccentricity of the rotor, was 2 kN.

33 4.4 Calculation of Bolted Joints

There are four different bolted joints in the design. These are the joints between the pole shoe and its holder, the holder and the top or bottom plate, the flanges and the top and bottom plates, and the flanges and the shaft. From section 4.3, it was known that each pole shoe will be subjected to an outward electromagnetic pull of about 2 kN at most, and that the torque on the shaft would be 8.3 kNm at most. There are inertial forces due to the rotation of the rotor in addition to the electromagnetic forces. The inertial forces were calculated to a total of 570 N per pole for both the pole shoe and one PM at the nominal speed of 127 rpm. This gives the design loads as 2.6 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 found in literature or slightly lower to be on the safe side. For two aluminium surfaces rubbing against each other, 1.3 was stated in literature and this was rounded to fAl−Al = 1, which was used. For an aluminium surface, against a steel surface fAl−Steel = 0.5 was given in literature and used. For two steel surfaces against each other, 0.8 was given in literature, and for steel against cast 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 in material composition [22, p. E5.1]. Two kinds of bolts were necessary, M4 in A2-70 ISO standard strength class to be used near the magnetic circuit, and M8 in 8.8 ISO standard strength class to be used away from the magnetic circuit. The A2-70 class is made 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, and there were no special requirements motivating the use of another strength class. The M4 bolt should be prestressed with a force of 2.6 kN, and the M8 should be prestressed with 15.2 kN [26]. These forces have been used when determining 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 to the pole shoe holder was calculated. This force should both resist radial forces, both inertial and electromagnetic, and provide enough normal force to resist any shearing forces through friction. Six M4 bolts were used. This gives a factor of safety, FoS, of 3.3 against the design loads given earlier. To fix the pole shoe holder to the top and bottom plates of the rotor, four M4 are 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 need to 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 required to transfer the torque at the radii in question, 47.5 mm for the shaft–flange joint and 80 mm for the flange–plate joint, and are therefore ignored when designing the joints. The bolts chosen to fasten the flange to the shaft are ten M8 per flange, five on each side of the shaft, for an FoS of 1.4 against design 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 design loads.

4.5 Static Stiﬀness of the Rotor

From the electromagnetic simulations, see Section 4.3, it was determined that displacing the rotor 3 mm from the centre of the stator would cause an unbalanced magnetic pull of 10 kN. To investigate if the design was rigid enough, a static linear simulation was made using the simulations package of SolidWorks.The basis for the geometry used in the simulation was the CAD model of the new generator with some simplifications made. The simplifications were to remove all bolt holes and to model the joints as if the parts had been bonded together. The stator was chosen as a reference point and considered rigid, since it was judged to be much more rigid than the rest of the generator. A force of 10 kN was applied to the outward face of a single pole shoe, pulling it toward the stator, which is a simplification that leads to slight exaggeration of the deformations. The result was that the maximum deformation in the design was about 0.20 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 strength of the material in the pole shoe holders, resulting in a local FoS of 0.76. A more accurate analysis was made in which the force on each individual pole was applied to the air gap face of the pole shoe, 2.6 kN per pole toward the stator. Then, the resultant force of the unbalanced magnetic pull, 10 kN at an arbitrary angle in the plane of rotation, was superimposed on the per pole forces. This analysis showed that the minimum FoS was 1.53 for the point with the lowest FoS. The point with the lowest FoS is a point where the material is compressed.

35 4.6 Study of Natural Frequencies and Modes of Vibration

All mechanical structures have natural frequencies with associated modes of vibration. If the frequencies are close to the frequency of a load, energy can be stored in vibration in the corresponding mode. Unless sufficient dampening is present in the structure, the energy can build up and cause large repeated deformations, either resulting in direct failure or premature material fatigue. To investigate the natural frequencies and their associated modes of vibration, simulations were made. Commercially available software package SolidWorks Simulation was used. After considering the result of the first simulation, 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 all components. Components were simplified by removing bolt holes and fillets. The stator was simplified by making the yoke slightly thicker and removing the teeth, preserving the mass of the stator as closely as possible. The stator was also simplified by modelling it as a solid piece of steel rather than a stack of plates. The ten lowest natural frequencies according to the simulation are presented in Table 4.1. The modes of vibration associated with the ten lowest natural frequencies are presented schematically in Figure 4.15. The frequency values are likely to be highly inaccurate, and it is mainly the modes of vibration that are of interest.

Table 4.1: The ten lowest natural frequencies of the design. The type of mode shape is referring to illustrations in Figure 4.15. Mode Frequency Mode shape - [Hz] n/a 1 43 a 2 77 b 3 144 c 4 145 c 5 186 d 6 192 d 7 291 e 8 295 e 9 295 f 10 361 f

36 (a) (b) (c)

(d) (e) (f)

Figure 4.15: Schematic illustrations of the natural modes of vibration associated with the ten ﬁrst natural frequencies, according to simulation. Some of the 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 shown in a cross section through the axis of rotation, b is shown from above, and f at an angle without the stator. Dashed lines show the deformed geometry and the dashed arrows indicate direction of movement.

37 The second mode is most likely an artifact, due to how the simulation was made. It arises because the bearings are modelled as rings of steel, bonded both to the stationary part of the generator and to the shaft. A similar mode will be present, but it will depend on the rest of the drive train, which is not modelled in the simulation.

4.7 Comparison of New and Old Design

It 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 numerals within parentheses indicate the source of the data for the old design. The sources are: (I) is from internal, unpublished, documents at the Division of Electricity, (II) is taken from [2], and (III) is taken or estimated from data in [27]. Data for the new design is taken from the simulations made during the design process. Quantity Old design New design Amplitude of air gap ﬂux density 0.79 0.66 fundamental [T] (I) Phase voltage, no load [Vrms] (II) 161 144 Armature winding current density, 1.6 1.8 2 rated load [Arms/mm ] (II) Rated power [kW] 12 12 Minimum air gap [mm] (II) 10 7 Mass of rotor [kg] (I) 130 402 Mass of PM [kg] (III) 41 158 Moment of inertia [kg m2] (III) 16.9 33.9

4.8 Unﬁnished Parts of the Design

Due to the time constraints associated with a master’s thesis, some parts of the design were left unfinished. The most important of these was the exact choice of bearings and the design of the bearing housing. Some initial studies were made, and space to accommodate tapered roller bearings, capable of sustaining both one direction axial loads and large radial loads, as well as housing for them has been left in the design. The shaft will also have to be slightly modified to accommodate the bearing mountings. The lower bearing will 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, the exact design of the top and bottom composite boards will be dependent on the bearing housing for the central hole and holes for fastening the bearing housing to the board. In addition, mechanical tolerances of all parts have to be determined to ensure 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 the bearings as well as the design of the bearing housing will be needed before the design can be completely finalised. A final choice of bearings also needs to be 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 generator with a two-dimensional cross section of the geometry and then multiplying by machine length is a well established method, and while it is not perfect at modelling the effects at the end of the generator, it gives good results. The PMs were modelled as regions with constant magnetisation, and this should be a good approximation. There are other ways to model PMs, such as a region with near unity relative permeability and two out of plane current sheets along the sides along the magnetisation. The soft magnetic materials were not modelled with hysteresis loops but rather a one-to-one relationship, BH-curve, between B~ and H~ . Also, for the pole shoes, the magnetic properties of the steel were approximated with a generic soft iron material from COMSOL’s material library, since the exact steel grade had not yet been decided. This should not be a problem since most irons will have permeabilities that are orders of magnitude larger than that of free space. This is true unless the steel is saturated and magnetic flux densities high enough to saturate the steel occur, mostly in the stator for which a proper BH-curve was available. Due to the above simplifications, there are some uncertainties in the results of the electromagnetic simulations. It is, however, clear that the air gap flux density amplitude will be lower with the new design and, therefore, the output voltage will decrease. The rated power can be maintained by increasing the current density in the stator winding. A higher current density

40 will cause increased resistive losses, which in turn lowers the efficiency of the machine. The structural mechanics calculations had simplifications. The largest simplification was that parts joined by bolts were modelled as bonded. This should, however, be a valid approximation as long as the forces do not load the bolts outside their elastic domain. In the static simulations, the pole shoe holders were in places subjected to stresses larger than half of their yield strength (FoS less than two), which indicates that it would be a good idea to use thicker material. There are, however, few U-profiles of suitable dimensions with greater material thickness than those used. A possible way around this problem would be to change the design to use two L-profiles for making the pole shoe holder as there seems to be more L-profiles with thicker material available on the market. Another concern is the first natural mode of vibration, mode a of Figure 4.15 at a frequency of about 43 Hz. Even though it is not expected to be excited during normal operation, it could become so. Should the mode become a problem, there are ways to shift the frequency upward so much it will not be a problem by adding steel beams between the bearing housing and the bars holding the stator together. There are five other natural frequencies below the cogging frequency of 200 Hz. The lowest of these, the second natural frequency, is most likely an artifact of how the simulation was made and should not be a problem. The four remaining frequencies have such shape that it seems likely that they will not be excited during normal operation. The bolted joints in the rotor might also provide some dampening. However, the vibrations should be carefully monitored 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 holes could be fitted in the face of the plates facing the pole shoe holder. This was realised too late in the design process to be properly evaluated. This could possibly could simplify production of the pole shoes significantly by allowing the pole shoes to be cut from a plate. The pole shoes in the current design will require a lot of milling to shape from the raw material. Drawbacks with laminated poles are that holes going through the entire length of the pole shoe would be required, disrupting the magnetic flux, and that more bolts would be needed to fasten the pole shoe to the holder. The design has been strongly influenced by the requirement to leave the stator intact. The lack of space for the ferrite PMs has been a problem and has prevented the new design from achieving the same performance as the old design. A larger stator inner diameter or larger pole pitch angle would probably 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 shaped rotor 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 at the 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 in the armature windings. The overall performance of the new design will be similar to the performance of the old design, making the new design a viable substitute. Assembly of the new design has been investigated. The investigation indicated no major difficulty with the assembly process. The new design will also be able to withstand the mechanical loads it will be subjected to. The design has been constrained by the requirement to keep the stator design unchanged. If the whole machine had been designed at once, a different stator design would have been used and better performance achieved. The conclusion of this report is that the new rotor design suggested, once finalised, will be a viable substitute for the old rotor.

43 Bibliography

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46 Appendix A

Drawings

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

Table A.1: Table for translating the part names on the drawings to the part names used in the report. In order of appearance in Section 4.1. In report On drawing Bottom rotor end plate rotorbottenskiva Top rotor end plate rotortoppskiva Pole shoe holder polskohållare Pole shoe polsko_omparam Magnet holder bar ändstopp Inner support innerstöd_v3 Large support ring stödring Flange krage Small support ring liten_stödring Shaft axel2 Generator end board generatorlock

47 10 686 120 4,50

R222,14 R206,16 9

16 R330 16

16 97,50 ±0

R80

R174,75

R143,25 112

R111,75

UNLESS OTHERWISE SPECIFIED: FINISH: DEBUR AND DO NOT SCALE DRAWING REVISION DIMENSIONS ARE IN MILLIMETERS BREAK SHARP SURFACE FINISH: EDGES TOLERANCES: LINEAR: ANGULAR:

NAME SIGNATURE DATE TITLE: DRAWN

CHK'D

APPV'D

MFG

Q.A MATERIAL: DWG NO. rotorbottenskiva A4

WEIGHT: SCALE:1:5 SHEET 1 OF 1 484 10

4,50 9

484 R80 R111,75

R174,75

112 R143,25

484

R222,14 97,50

R206,16 120 143,25

UNLESS OTHERWISE SPECIFIED: FINISH: DEBUR AND DO NOT SCALE DRAWING REVISION DIMENSIONS ARE IN MILLIMETERS BREAK SHARP SURFACE FINISH: EDGES TOLERANCES: LINEAR: ANGULAR:

NAME SIGNATURE DATE TITLE: DRAWN

CHK'D

APPV'D

MFG

Q.A MATERIAL: DWG NO. rotortoppskiva A4

WEIGHT: SCALE:1:5 SHEET 1 OF 1 50

5 5 13 38,80 38,80 220 38,80 38,80 38,80

5 13 4,50 36,18 10,09 36,18

21 13

UNLESS OTHERWISE SPECIFIED: FINISH: DEBUR AND DO NOT SCALE DRAWING REVISION DIMENSIONS ARE IN MILLIMETERS BREAK SHARP

SURFACE FINISH: EDGES 10,09 TOLERANCES: LINEAR: ANGULAR:

NAME SIGNATURE DATE TITLE: DRAWN

CHK'D

APPV'D

MFG

Q.A MATERIAL: DWG NO. polskohållare A4

WEIGHT: SCALE:1:2 SHEET 1 OF 1 27 60

11,25° 9,68 40,72

34,23 11,25° 11,25° 125,20 3,30

17 38,80 38,80 224 38,80 38,80 38,80 13

27 60 3,30

129,98

UNLESS OTHERWISE SPECIFIED: FINISH: DEBUR AND DO NOT SCALE DRAWING REVISION DIMENSIONS ARE IN MILLIMETERS BREAK SHARP SURFACE FINISH: EDGES TOLERANCES: LINEAR: ANGULAR:

NAME SIGNATURE DATE TITLE: DRAWN

CHK'D

APPV'D

MFG

Q.A MATERIAL: DWG NO. polsko_omparam A4

WEIGHT: SCALE:1:2 SHEET 1 OF 2 4

4,50

R10

83,81

UNLESS OTHERWISE SPECIFIED: FINISH: DEBUR AND DO NOT SCALE DRAWING REVISION DIMENSIONS ARE IN MILLIMETERS BREAK SHARP SURFACE FINISH: EDGES TOLERANCES: LINEAR: ANGULAR:

NAME SIGNATURE DATE TITLE: DRAWN

CHK'D

APPV'D

MFG

Q.A MATERIAL: DWG NO. ändstopp A4

WEIGHT: SCALE:1:1 SHEET 1 OF 1 DRAWN APPV'D CHK'D MFG ANGULAR: LINEAR: TOLERANCES: SURFACE FINISH: DIMENSIONS AREINMILLIMETERS UNLESS OTHERWISESPECIFIED: Q.A 10 30 140 30 15 NAME

15,75 SIGNATURE 9 16 FINISH: 30 DATE WEIGHT: MATERIAL:

31,50 6,80 125 95 31,50 95 EDGES BREAK SHARP DEBUR AND 16 TITLE: SCALE:1:2 DWG NO.

R2 R2 30 innnerstöd_v3 DO NOTSCALEDRAWING 220 SHEET 1OF 1 REVISION A4 468 30

468

4,50

16 174,75 16 388

R206,16

9 R222,14

UNLESS OTHERWISE SPECIFIED: FINISH: DEBUR AND DO NOT SCALE DRAWING REVISION DIMENSIONS ARE IN MILLIMETERS BREAK SHARP SURFACE FINISH: EDGES TOLERANCES: LINEAR: ANGULAR:

NAME SIGNATURE DATE TITLE: DRAWN

CHK'D

APPV'D

MFG

Q.A MATERIAL: DWG NO. stödring A4

WEIGHT: SCALE:1:5 SHEET 1 OF 1 8

R0,50 R0,50 7,80

R47,50 R80 R0,25 R0,50

10 R57,50 R0,50

R92,50 11,50

TRUE R0,25 1

16 16 8 16 16 13

UNLESS OTHERWISE SPECIFIED: FINISH: DEBUR AND DO NOT SCALE DRAWING REVISION DIMENSIONS ARE IN MILLIMETERS BREAK SHARP SURFACE FINISH: EDGES TOLERANCES: LINEAR: ANGULAR:

NAME SIGNATURE DATE TITLE: DRAWN

CHK'D

APPV'D

MFG

Q.A MATERIAL: DWG NO. krage A4

WEIGHT: SCALE:1:2 SHEET 1 OF 1 DRAWN APPV'D CHK'D MFG ANGULAR: LINEAR: TOLERANCES: SURFACE FINISH: DIMENSIONS AREINMILLIMETERS UNLESS OTHERWISESPECIFIED: Q.A 7 185 NAME SIGNATURE FINISH:

DATE 122 WEIGHT: MATERIAL:

185 18° 18°

EDGES BREAK SHARP DEBUR AND 18° 18° TITLE: SCALE:1:2 DWG NO.

R80

DO NOTSCALEDRAWING

liten_stödring 9 SHEET 1OF 1 REVISION A4 DRAWN APPV'D CHK'D MFG ANGULAR: LINEAR: TOLERANCES: SURFACE FINISH: DIMENSIONS AREINMILLIMETERS UNLESS OTHERWISESPECIFIED: Q.A NAME SIGNATURE FINISH:

DATE 990 WEIGHT: MATERIAL:

EDGES BREAK SHARP DEBUR AND 145 425 53 53 TITLE: SCALE:1:10 DWG NO.

35 240 39 55 DO NOTSCALEDRAWING

110 97 95 95 97 95 axel2 SHEET 1OF 1 REVISION A4 DRAWN APPV'D CHK'D MFG ANGULAR: LINEAR: TOLERANCES: SURFACE FINISH: DIMENSIONS AREINMILLIMETERS UNLESS OTHERWISESPECIFIED: Q.A

NAME 395 395

11,50°

SIGNATURE 22°

FINISH: R16 R16 DATE WEIGHT: MATERIAL:

280 230 120

R34,01

1000

EDGES BREAK SHARP DEBUR AND

13,50 13,50 11 TITLE: SCALE:1:10 DWG NO. generatorlock DO NOTSCALEDRAWING

R455 R485 25 1000 SHEET 1OF 1 REVISION A4