Validation of Eddy Current Loss
Models for Permanent Magnet Machines
with Concentrated Windings
Dong Liu
4123077
Supervised by
Dr. Ir. Henk Polinder
A thesis submitted in partial fulfillment of the requirement for the degree of
Master in Electrical Engineering
Electrical Power Processing
Faculty of Electrical Engineering, Mathematics and Computer Science
Delft University of Technology
Delft, the Netherlands
August 2012
Acknowledgement
Thanks to my parents. Without your spiritual and financial supports, I could not
have got the chance to study in the Netherlands and finish my master programme at
TU Delft. I feel very sorry that you must have been feeling alone without the only child
at home. Thank you all the time for your understanding.
Thanks to my supervisor Dr. Henk Polinder. You have given me great help and
advice and you are so kind that I cannot complaint anything about working with you.
I am very willing to go on studying electrical machines under your supervision.
Thanks to Anoop Jassal who is my friend and my tutor. Thank you for giving me
the opportunity to do the internship at XEMC Darwind B.V. and for offering me a very
challenging but interesting graduation project. I was really enjoying when we fought
together to fulfill this project. I was so lucky to meet you.
Thanks to Prof. Ferreira who leads a very competitive research group. I have been
growing rapidly in EPP group and I hope to learn more from you on the field of
electromagnetics.
Thanks to Domenico Lahaye for your guide of finite element methods which
benefited me in the thesis project and will also be beneficial in the future.
Thanks to Ren Yating and Venugopal Prasanth for studying together with me in
the EPP student room. I would have been feeling alone if you were not there.
Thanks to Kasper Zwetsloot and other laboratory staff who often helped me with
the setups and instruments.
Many thanks should also be given to my master classmates at TU Delft, my
colleagues at XEMC Darwind B.V and all people who supplied power to me and made
me going forward.
i
Abstract
Concentrated windings are considered to replace distributed windings for
permanent magnet electrical machines as they save manufacture time and costs and
even can be produced with automatic processes. Compared with distributed windings,
however, concentrated windings lead to much more eddy current losses in the rotor of
machines due to a great number of space harmonics of magnetic field. These losses
result in much heat that could do harm to the machines. Investigating eddy current
losses is therefore necessary in designing electrical machines with concentrated
windings.
Analytical and finite element models are proposed to predict these eddy current
losses in magnets and rotor back-iron. The accuracy of such models must be validated
by experimental tests. This thesis firstly carries out analytical calculations and finite
element simulations for the tested electrical machines with concentrated windings.
Then experimental tests are done for these machines by applying proper methodology.
The conclusions are drawn by comparing the results from models and from experiments,
and the validation can be evaluated in the end. The results show the models can to some
extent predict the eddy current losses in the rotor but further improvements such as a
three dimensional model are needed.
ii
List of Abbreviations
MMF - Mangetomotive Force
FEM - Finite Element Method
PMSM - Permanent Magnet Synchronous Machine
PDE - Partial Differential Equations
PM - Permanent Magnet
FDM - Finite Difference Method
FEA - Finite Element Analysis
AC - Alternating current
DC - Direct current
ALE - Arbitrary Lagranian-Eulerian
OS - Open-slot
SCS - Semi-closed-slot
EMF - Electromotive Force
PMSG - Permanent Magnet Synchronous Generator
RMS - Root Mean Square
IEEE - Institute of Electrical and Electronics Engineers
iii
Table of Contents
Acknowledgement ......................................................................................... i
Abstract .............................................................................................................. ii
List of Abbreviations ......................................................................................... iii
Chapter 1 Introduction ..................................................................................1
1.1 Background ..................................................................................................... 1
1.2 Problem Definition .......................................................................................... 3
1.3 Thesis Layout .................................................................................................. 4
Chapter 2 Review of Electromagnetics ........................................................ 5
2.1 Introduction ................................................................................................... 5
2.2 Electromagnetics ............................................................................................ 5
2.2.1 Maxwell’s Equations .............................................................................................5
2.2.2 Boundary Conditions .......................................................................................... 6
2.2.3 Vector Potential ................................................................................................... 6
2.3 Permanent Magnet Synchronous Machines ....................................................... 7
2.3.1 Principles ..............................................................................................................7
2.3.2 Magnetic Materials.............................................................................................. 8
2.4 Space Harmonics for Concentrated Windings .................................................. 11
2.5 Modeling Eddy Current Losses ....................................................................... 12
2.5.1 Introduction ........................................................................................................ 13
2.5.2 Assumptions ....................................................................................................... 13
2.5.3 Derivation of Partial Differential Equations (Pdes) .......................................... 14
2.5.4 Boundary Conditions ......................................................................................... 17
2.5.5 General Solution of the Partial Differential Equations ..................................... 18
2.5.6 Modeling the Armature Current ........................................................................18
2.5.7 Effect of Permanent Magnets Only ................................................................... 20
2.5.8 Combined Effect of Armature Currents and Permanent Magnets.................... 21
2.5.9 Derived Quantities from Az ............................................................................... 21
2.6 Finite Element Methods ................................................................................ 22
2.6.1 Why Finite Elements? ....................................................................................... 22
2.6.2 Using Finite Element Methods ......................................................................... 22
Chapter 3 Finite Element Simulations ...................................................... 29
3.1 Introduction ................................................................................................. 29
3.2 Simulation Tool ............................................................................................ 30
3.3 Description of Modeled Machines ................................................................... 30
3.4 Simulations and Results ................................................................................ 35
3.4.1 Static Simulation ................................................................................................35
3.4.2 Rotary Simulation ............................................................................................. 39
Chapter 4 Experimental Testing ................................................................ 49
4.1 Introduction ................................................................................................. 49
4.2 Experimental Test ......................................................................................... 49
4.2.1 Static Experiment .............................................................................................. 49
4.2.2 Rotary Experiment ............................................................................................ 56
Chapter 5 Validation .................................................................................. 69
5.1 Introduction ................................................................................................. 69
5.2 Static Test Validation..................................................................................... 69
5.2.1 Comparisons between Simulation and Experiment Results ............................ 69
5.2.2 Discussion on Possible Causes of The Deviations .............................................73
5.2.3 Summary ............................................................................................................ 77
5.3 Rotary Test Validation ................................................................................... 77
5.3.1 Separation of Stator and Rotor Iron Losses ....................................................... 77
5.3.2 Validating Finite Element Models by Experimental Measurements ............... 82
Chapter 6 Conclusions and Future Work ...................................................91
6.1 Conclusions .................................................................................................. 91
6.2 Future Work ................................................................................................. 92
Appendix A Stator Drawings ...................................................................... 93
Appendix B Solving a Problem with COMSOL .......................................... 95
Appendix C Estimated Stator Iron Losses in Rotary Tests ....................... 97
Appendix D Phase Variation of Armature Current in FEM ...................... 97
Bibliography .................................................................................................... 101
Chapter 1
Introduction
1.1 Background
The energy demand is growing very fast nowadays. Producing energy by using
fossil fuels is not a reliable solution any more as the fossil fuels tend to be exhausted in
some years. Moreover the emission of CO2 from the burning of fossil fuels can lead to the
green house effect and subsequently the global warming which does harm to the
ecological environment of the earth. Renewable energy is promising and considered as a
potential replacement for fossil fuels. It is clean and renewable, and regarded as a
fascinating solution for the construction of a sustainable world.
The renewable energy consists of wind, solar, geothermal, hydro and other possible
energy. Wind energy has been developing most rapidly among them. The growth of
global cumulative capacity is considerable as shown in Figure 1-1 and it will be
dominant in the development of renewable energy in future as shown in Figure 1-2.
Figure 1-1: Global cumulative capacity growth of wind power, showing top ten
countries 1990 – 2008 (GW)
Source: Technology Roadmap Wind Energy, IEA (2008a)
Wind energy is usually utilized in the form of electricity. The fast growth of wind
energy generation requires correspondingly fast manufacture of wind turbines which
convert wind energy to electrical power. In the production process of wind turbines, the
electrical generator takes a large amount of time and costs because the windings of the
generator is designed to be distributed as illustrated in Figure 1-3 (source: Internet).
The process of winding is very complex and needs manual production. Automatic
1
production by machines is not feasible although it can save time and human resources.
Alternative windings are therefore necessary.
Figure 1-2: Electricity from renewable energy sources up to 2050
in the ETP 2008 BLUE Map scenario
Source: Technology Roadmap Wind Energy, IEA (2008a)
The concentrated winding is considered to be a promising replacement of the
conventional distributed winding. As shown in Figure 1-4 we can see a very simple
arrangement of windings. This type of windings saves copper and production time, and
automatic manufacture can be achieved.
(a) Production process of distributed windings
(b) Copper usage of distributed windings
Figure 1-3: Manufacture and copper usage of distributed windings
2
The concentrated winding is proposed to be a promising replacement of the
conventional distributed winding. As shown in Figure 1-4 we can see a very simple
arrangement of windings. This type of windings saves copper and production time, and
automatic manufacture can be achieved with machines or robots.
Figure 1-4: Demonstration of a permanent magnet machine with concentrated windings
(8 pole rotor and 6 slot stator)
Source: Paul Nylander, bugman123.com
1.2 Problem Definition
However, machines with concentrated winding also have problems. One of the
critical problems is eddy current losses. Unlike the conventional permanent magnet
machines with distributed windings which hardly have eddy current losses in rotors due
to armature currents, concentrated windings introduce a relatively large number of
eddy current losses in rotors. This is due to the space harmonics. Figure 1-5(a)
illustrated the spatial distribution of the magnetomotive force (MMF) in the air gap of
a concentrated winding machine and Figure 1-5(b) shows the Fourier decomposition of
the air gap MMF in space. The high orders of space harmonics of MMF rotating relative
to the rotor can induce eddy current losses in the rotor. Such losses can heat up the
electrical machines and may demagnetize the magnets and lead to other damages.
(a) MMF distribution in space
(b) Fourier decomposition of MMF
Figure 1-5: Origin of space harmonics of MMF
3
In design of electrical machines, analytical models or finite element models are used
to predict eddy current losses. The accuracy of the results from these models must be
validated by experimental tests. This thesis aims at validating such models, especially
finite element models, for permanent magnet machines with concentrated windings.
1.3 Thesis Layout
The thesis starts with a review of electromagnetic theories in Chapter 2. Besides
the basic theory background, an analytical model is formulated in this chapter and the
finite element method is generally introduced.
Chapter 3 describes the machines used for this thesis and the finite element
simulations are operated to find predictions of eddy current losses in rotor of these
machines.
In Chapter 4 experimental tests based on power flow models are carried out and the
total iron loss in the machine is measured. The total iron loss is separated into the iron
loss in stator and the eddy current loss in rotor in Chapter 5. The obtained loss in rotor
is then compared with results from the analytical model and the finite element
simulations to validate the accuracy of the models.
The conclusions are drawn and shown in Chapter 6 and some useful work after this
thesis project is pointed out finally.
4
Chapter 2
Review of Electromagnetics
2.1 Introduction
This chapter presents a review of electromagnetic theory used for estimating eddy
current losses. Maxwell’s equations are used to describe electromagnetic phenomena,
accompanied with boundary conditions. A scalar or vector magnetic potential is used to
solve Maxwell’s equations. Non-linear magnetic materials and permanent magnets are
important for respectively characterizing the iron cores and magnets which are used in
permanent magnet electrical machines. In the case of concentrated windings, space
harmonics of magnetomotive force (MMF) due to armature currents are to be
investigated. Simple models for calculating eddy current losses are discussed as well.
Finally finite element method (FEM) is introduced.
2.2 Electromagnetics
2.2.1 Maxwell’s Equations
Starting from electromagnetics, we can understand the physics of an electrical
machine. The evaluation of magnetic field is critical in this study so we must
concentrate on the mathematics to solve for it.
The Scottish physicist James Clerk Maxwell put together the laws of
electromagnetism in the form of four equations for time-varying conditions [1]. The
differential form of the time-varying Maxwell’s equations are shown as
∇×Η = J +
∇×E = −
∂D
∂t
(2.1)
∂B
∂t
(2.2)
∇⋅B = 0
(2.3)
∇ ⋅ D = ρv
(2.4)
where H is the magnetic field intensity, J is the current density, D is the electric field
displacement, E is the electric field intensity, B is the magnetic flux density and ρv is the
electric charge in unit volume.
Equation (2.1) is known as the Ampere’s circuit law. Equation (2.2) is the
Faraday’s law. Equation (2.3) represents nonexistence of isolated magnetic charge,
which is also referred to as the Gauss’s law for magnetic fields. Equation (2.4) is the
original Gauss’s law which is for electric fields.
5
There are other equations that go hand in hand with Maxwell’s equations. One of
them to be emphasized is the equation of continuity:
∇⋅J = −
∂ρv
∂t
(2.5)
In linear, homogeneous and isotropic medium characterized by σ , ε and μ . σ is the
electrical conductivity, ε is the permittivity and μ is the magnetic permeability. The
constitutive equations are shown below to describe the properties of a medium.
J = σE
(2.6)
D = εE
(2.7)
B = μH
(2.8)
2.2.2 Boundary Conditions
The Maxwell’s equations require some or all of the four boundary conditions are the
keys to solve for the four variables E, H, D, B . These boundary conditions must be
satisfied by an electromagnetic field existing in two different media separated by an
interface. The final-form of boundary conditions are written as
(E1 − E2 ) × n = 0
(2.9)
(H1 − H2 ) × n = K
(2.10)
(D1 − D2 ) ⋅ n = ρs
(2.11)
(B1 − B2 ) ⋅ n = 0
(2.12)
where n is the unit normal vector to the boundary, K is the surface current density, and
ρs is the surface charge.
These boundary conditions can also be written in the following scalar form for
simplicity.
E1t − E 2t = 0
(2.13)
H 1t − H 2t = K
(2.14)
D1n − D2n = ρs
(2.15)
B1n − B2n = 0
(2.16)
where the subscript t means the tangential direction and n means the normal direction.
2.2.3 Vector Potential
In some electric field problems we relate the electric potential φ to the electric field
intensity E (E = −∇φ) . In this case, φ is a scalar potential. Similarly a scalar potential
φm is introduced for magnetic fields. This scalar potential is only available when the
current density is zero.
6
When non-zero currents exist, we cannot make use of the scalar potential, because
the vector H is no more able to be defined as the gradient of φm . In such a case, a vector
potential is introduced to replace the scalar potential to solve the Maxwell’s equations.
We define the vector potential A as:
B = ∇×A
(2.17)
Finally the Poisson’s equation in vector potential is yielded as
∇2A = −μJ
(2.18)
Equation (2.18) is to be solved for the vector potential A with boundary conditions.
The variables B and H can be then found out.
Note that the Poisson’s equation means a convenient approach to deal with
two-dimensional problems. For example, if the current density J only flows in the z
direction J z (x , y ) , then the flux density B does not have a z component at all.
In problems of 2-D geometry, only z component exists in the vector potential A .
We can write it as A = (0, 0, Az (x , y )) and Equation (2.17) becomes
Bx =
∂Az
∂Az
, By =
∂y
∂x
(2.19)
Here Equation (2.18) is simplified in a two-dimensional quasi-static magnetic field
as
∂ 2Az (x , y ) ∂ 2Az (x , y )
+
= −μJ z (x , y )
∂x 2
∂y 2
(2.20)
Deriving B and H can therefore be achieved by only solving Equation (2.20) for
Az (x , y ) . This is a great advantage for modeling magnetic fields of electrical machines in
two dimensions.
2.3 Permanent Magnet Synchronous Machines
2.3.1 Principles
Permanent magnet synchronous machines (PMSMs) are electromagnetically same
as conventional synchronous machines [2] [3] except the way of field excitation. PMSMs
use permanent magnets as field excitation to provide magnetic flux instead of electrical
excitation with coils in the conventional. The elimination of coils for excitation reduces
the volume and the mass of PMSMs, gets rid of losses in field windings and saves copper
and iron. Permanent magnets have a better exciting capability with a remanent flux
density of 1.2T (e.g. NdFeB) [4] and the energy density becomes higher [5]. However
PMSMs have a drawback that the excitation field is unable to control.
7
2.3.2 Magnetic Materials
2.3.2.1 Ferromagnetic materials
Ferromagnetic materials, e.g. electrical steels, are employed in electrical machines
to form the cores which act as the path for magnetic flux density and as the support
structures.
The permeability of these materials is nonlinear and multivalued. The flux density
through the material is not unique for a given field intensity. It is a function of the past
history of the field intensity. Therefore the magnetic properties of ferromagnetic
materials are often described graphically in terms of their B-H curve, hysteresis loop,
and core losses. Figure 2-1 illustrates the magnetization curves for several ferromagnetic
materials [6].
Figure 2-1: Magnetization curves for some ferromagnetic materials
Figure 2-2: Hysteresis B-H loops of a ferromagnetic material
When the magnetic field intensity H is back to zero, the magnetization disappears
but the curve cannot return to the origin due to hysteresis, which is illustrated in Figure
2-2 with the demagnetization curves as well excited by a negative H .
B-H loops lead to losses due to the fact that the area enclosed by the B-H loop
represents a power loss described by
8
Ph = fVcore ∫ HdB
(2.21)
where Ph is the hysteresis loss, f is the frequency of variation of the exciting current,Vcore
is the volume of the ferromagnetic material, and ∫ HdB is the area of the B-H loop.
A rapidly changing flux density though a magnetic material induces eddy currents
and results in heat due to the fact that magnetic materials normally have considerable
conductivities which allow induced currents to flow.
Eddy current losses drop by lamination that dramatically reduces the conductivity
of the magnetic material, as indicated in Figure 2-3. A laminate is a material that can
be constructed by uniting two or more layers of material together. The process of
creating a laminate is lamination, which in common parlance refers to the placing of
something between layers of plastic and gluing them with heat, pressure, and an
adhesive. There are also other approaches available to do the same [7].
ie
ie
B
B
Figure 2-3: Eddy currents and laminations to reduce eddy currents
There are also excess losses but the total core losses mainly consist of hysteresis
losses and eddy current losses. The typical core loss characteristics of ferromagnetic
material are illustrated in Figure 2-4 [8]. The core loss depends on both frequency and
flux density.
The characteristics of a ferromagnetic material can be simply described by the
relation:
B = μ0 μr H
(2.22)
where μ0 is the permeability of vacuum, and μr is the relative permeability.
Equation (2.22) applies in both linear and saturation regions where the relative
permeability μr is totally different. Normally the ferromagnetic material performs in the
linear region with a very high μr and a low value of H around zero. But sometimes the
operating point goes beyond this region and saturation occurs.
Saturation is an important feature of a ferromagnetic material. As the magnetic
field intensity H increases, the magnetic materials saturate more severely and
eventually all the B-H curves follow the B-H curve of vacuum with the relative
permeability μr = 1 . Saturation seriously limits the maximum magnetic field
9
achievable in a ferromagnetic material. This puts a restriction on the dimensioning of
structures built with such a material.
Core loss (Watts/Kg)
102
101
100
Increasing
Flux Density, B
10-1
101
102
103
Frequency (Hz)
104
Figure 2-4: Typical core loss characteristics of ferromagnetic material
2.3.2.2 Permanent magnets
Permanent magnets are also known as hard magnetic materials or hard iron. They
are active magnetic materials as they have a relatively high remanent flux density even
in absence of current excitation. Their operating point is determined by surrounding
magnetic circuits [9].
A permanent magnet is modeled by the permeability function μ throughout its
volume and the remanent flux density Br , which appears only in its surface if the
magnetization is uniform,
B = μ0μH + Br
(2.23)
where μ0 is the permeability of vacuum and μ is the vector relative permeability.
Generally, from its B-H curve, the magnetic characteristic of a permanent magnet
can be described in a simpler way, following the assumption that the magnet remains in
the linear region (the straight line of demagnetization, as shown in Figure 2-5):
B = μ0 μr H + Br
(2.24)
In Figure 2-5, Br is the scalar remanent flux density, H c is the coercive force,
and( μ0H m1, Bm1 ) and ( μ0H m 2 , Bm 2 ) are two operating points.
Permanent magnets generally operate along the demagnetization curve. In practice,
a large reverse excitation currents or a high temperature could demagnetize the
permanent magnets. Therefore favorable cooling and excitation current amplitudes are
necessarily required.
Like ferromagnetic materials, permanent magnets are not immune to iron losses
due to their considerable conductivity [10]. Both hysteresis and eddy current losses are
10
still present especially at high frequencies and they lead to heat that could harm the
magnets themselves. An effective way to reduce eddy current losses, for example, is to
have the magnets segmented [11], which applies the same principle as lamination.
B
Operating points
Br
Bm1
Bm2
-μ0Hc
μ0Hm2
μ0Hm1
μ0 H
Figure 2-5: Operating points along the demagnetization line
2.4 Space Harmonics for Concentrated Windings
Distributed winding machines are popular in the industry because of their good
performance. In distributed winding machines [12], the MMF is distributed sinusoidally
in space. The distribution is manipulated to maximize the fundamental space harmonic
and results in an advantage that little sub-harmonic content exists.
Concentrated winding machines are promising because they are cost-effective [13].
Unlike distributed winding machines, they have a large amount of space harmonics.
Only the torque producing harmonic (with a wavelength equal to twice the pole pitch)
is what we need for running the machine and other harmonics are not useful but
contributes to iron losses. Investigations have been performed to reveal the impact of
pole-slot combinations on PM machines with concentrated windings [14] [15], and they
are very useful in machine designing.
Solid back-iron is also not immune to eddy currents as its conductivity is in the
order of 106 S/m which is considerable for eddy current flow. However for the purpose of
lowering costs, it is still widely used for building the rotor of a PM machine. Therefore
eddy current losses could be very high in such part of a machine and the resultant heat
could demagnetize the permanent magnets on the rotor if not designed properly.
11
The origin of eddy currents comes from varying magnetic flux density whose speed
is not synchronized with the rotor speed. Taking a PM machine with 2 poles per 3 teeth
with 3 coils for example, we can derive the spatial distribution of flux density.
a
b
c
a
b
c
Figure 2-6: A fractional pitch concentrated winding with 2 poles per 3 teeth
A fractional pitch concentrated winding with 2 poles per 3 teeth is illustrates in
Figure 2-6. If the current flowing in phase a is iˆsa and the currents in the other two
phases are zero, the flux density can be calculated using Ampere’s law and the
continuity of flux density as [14]
Bsa
⎧
⎪
2μ0N siˆsa
⎪
⎪
⎪ 3geff
=⎪
⎨
⎪
μ0N siˆsa
⎪
⎪
−
⎪
3geff
⎪
⎪
⎩
for −
λ1
λ
< xs < 1
6
6
(2.25)
5λ
λ
for 1 < xs < 1
6
6
The amplitudes of the space harmonics of this flux density can be calculated as
2
k π μ N iˆ
sin( ) 0 s sa
Bˆsk =
3
kπ
geff
(2.26)
Similarly we can derive the amplitudes of space harmonics for other machine
topologies [16]. Because the machine with 2 poles per 3 teeth will be used as a prototype
for the research of this thesis, we do not need to consider other topologies here.
2.5 Modeling Eddy Current Losses
For a simple one-dimensional problem, [17] has derived expressions to calculate
eddy current losses in solid iron poles. A three layer model is used:
1. a layer with perfect iron (resistivity and permeability both infinite),
2. a layer of air (the air gap),
3. a layer of solid iron with a realistic resistivity and permeability.
The eddy current loss per square meter of surface area PA is given as
PA =
Bˆ02v 2
4ρFe Re[ γ ]
(2.27)
12
where B̂0 is the amplitude of the flux density wave, v is the speed of the flux density
wave, ρFe is the resistivity of iron and γ is the complex factor for skin effect.
Equation (2.27) can be simplified as
PA =
Bˆ02v 2δ
4ρFe
(2.28)
When the problem of magnetic field is expanded in two dimensions, the eddy
current loss is calculated with:
Peddy =
∫∫
s
J z2 (x , y )
dA
σ
(2.29)
where Peddy is the eddy current losses, J z is the induced current density, A is the area, and
σ is the conductivity.
To find the eddy current loss described in Equation (2.29) in 2-D, we introduce
another analytical model to solve for the current density J z .
2.5.1 Introduction
An analytical model is a basic approximation to calculate the magnetic field. There
are two magnetic fields active in a permanent magnet (PM) machine. One is the field
due to armature currents and the other is the field due to permanent magnets. Some
assumptions must be made as it is not easy to take into account all the problem details
in an analytical model. Derivation and solution of partial differential equations (PDEs)
starting with Maxwell equations and boundary conditions forms the basis of the model.
Only the field due to armature current is highlighted in detail. The mathematical
derivation of the field due to magnets in the analytical model is left for future studies.
2.5.2 Assumptions
The following assumptions are made to simplify analytical modeling [18]:
1. All materials are isotropic.
2. The stator is slotless and the teeth are replaced by an effective air gap using
Carter’s factor [19].
3. The stator windings are replaced by an equivalent current sheet traveling in
space and varying in time. This current sheet is located at the stator surface between
the stator and the effective air gap.
4. The stator iron conductivity is zero.
5. The relative permeability of stator iron is infinite.
6. The relative permeability of rotor back-iron is realistic.
7. The magnets occupy the whole surface of rotor without a boundary between each
other.
13
8. The magnets do not demagnetize.
9. 2-D modeling in Cartesian coordinates has been selected with the assumption
that every quantity remains constant in the z direction. This is valid because the axial
length is much larger than the air gap.
Figure 2-7 shows a real PM machine cross-section and Figure 2-8 shows the
simplified model according to the assumptions. The analytical model of a PM machine
consists of four layers, a current sheet and several boundaries.
Stator
Winding
Magnets
Rotor back-iron
Figure 2-7: Actual geometry of a PM machine
In this model, B1 to B7 are the boundaries where boundary conditions apply. B1
and B3 are the boundaries of magnetic insulation. On other boundaries, continuity
conditions are applied.
B1
Region 1: Iron with Permeability μ1
B4
b
Region 2: Air with Permeability μ2
B5
B6
B2
Region 3: Magnet Region with Permeability μ 3 and Remanent Flux Density of 1.2T
B7
Region 4: Iron with Permeability μ4
Y
B3
l
X
Z
Figure 2-8: Simplified geometry for analytical modeling
2.5.3 Derivation of Partial Differential Equations (PDEs)
The aim is to calculate eddy current losses in the magnet and the rotor back-iron
regions (regions 3 and 4 in the simplified geometry) due to the current sheet. We start
the derivation with Maxwell’s equations for quasi-static magnetic fields:
∇ × Η = J + Jext
(2.30)
14
∇×E +
∂B
=0
∂t
(2.31)
∇⋅B = 0
(2.32)
where H is the magnetic field intensity, J is the induced current density, Jext is the
external current density, E is the electric field intensity, B is the magnetic flux density
and ρv is the electric charge in unit volume.
The equation of continuity for external currents is
∇ ⋅ Jext = 0
(2.33)
Note that Equation (2.30) is somewhat different from Equation (2.1). The
∂D
is neglected since we are only dealing with quasi-static
displacement current
∂t
magnetic fields. Besides, induced currents and external currents need to be separately
treated in eddy current problems
The constitutive relations for the materials are:
J = σE
(2.34)
B = μΗ + Br
(2.35)
where σ is the conductivity and μ is the permeability,
Br is the remanent flux density of a magnet.
From Equation (2.35) we can write for H as:
1
H = (B − Br )
μ
Substituting this H in Equation (2.30) we have
(2.36)
1
∇ × [ (B − Br )] = J + Jext
μ
(2.37)
We rewrite Equation (2.37) in an elaborate from:
∇ × B = μ(J + Jext ) + ∇ × Br
(2.38)
The definition of the vector potential A has been introduced as
B = ∇×A
(2.39)
∇⋅A = 0
(2.40)
Substituting for B from Equation (2.39) in Equation (2.38), we get
∇ × (∇ × A) = μ(J + Jext ) + ∇ × Br
(2.41)
An important identity for vector calculus is
∇ × ∇ × A = ∇(∇ ⋅ A) − ∇ 2 A
(2.42)
Comparing Equation (2.41) and Equation (2.42), we can see
−∇ 2 A = μ J + μ Jext + ∇ × Br
(2.43)
15
where the condition Equation (2.40) has been applied.
From the Faraday’s law Equation (2.31), we can derive another relation
E=−
∂A
∂t
(2.44)
Combining this with the constitutive relation Equation (2.34), we have
J = −σ
∂A
∂t
(2.45)
Substituting for J in Equation (3.43), we achieve the Poisson’s equation considering
permanent magnets:
−∇2A + μσ
∂A
= μJext + ∇ × Br
∂t
(2.46)
The Poisson’s equation Equation (2.46) will be the key equation to evaluate eddy
current losses. It is valid for general cases and holds true for vectors in three dimensions.
For the case of electrical machines, however, analytical models are usually good enough
in two dimensions under the following assumptions:
1. The stator current Jext in Equation (2.46) flows only in z direction (perpendicular
to the plane of Figure 2-8).
2. Br has only a y component and its curl is only in z direction.
3. We can say from Equation (2.46) that only z component of the vector potential,
i.e. Az , is present. Because we assume 2-D geometry, every quantity is constant in z
direction. In other words, Az depends only on x and y coordinates as nothing changes in z
direction.
We can now rewrite Equation (2.46) in its component form as:
−(
∂Br ,y
∂ 2Az
∂ 2Az
∂Az
) + μσ
+
= μJ z ,ext +
z
2
2
∂t
∂x
∂x
∂y
(2.47)
where J z ,ext is the z component of external current density and By ,r is the y component of
remanent flux density pointing in z direction.
We use unit vectors x, y, z for x, y, z coordinates.
We have assumed that nothing varies in z direction and that Br only has a y
component. When we take curl of Br , there is thus a variation along x direction which
represents the change of magnet polarity. Note that the direction of the curl is along z
axis.
The magnetic field in the PM machine can be decomposed into two according to
their origins of excitation. One field is produced by the permanent magnets. The other
is excited by the stator current sheet which represents the stator current. In each region
of the machine, we can independently solve for the corresponding vector potentials in z
16
direction Az with one excitation and subsequently find out other magnetic quantities
from Az . Then superposition of fields is to be used to combine the effects due to both
excitations.
The Poisson’s equation (2.47) in 2-D of the machine can be simplified depending on
which excitation is present in each particular region. Table 2-1 shows the governing
equation for each region only with the excitation due to the stator current sheet. In this
case the magnets are inactive. Eddy currents, however, still exist in these magnets due
to their considerable conductivity. The equations for stator and air-gap are Laplace
equations while for magnets and rotor back-iron are Poisson’s equations.
Table 2-1: Governing PDEs for each region
Region
Equation
Stator
−∇ 2Az = 0
Air-gap
−∇ 2Az = 0
Magnets
∇2Az = μσ
∂By ,r
∂Az
−
z
∂t
∂x
∇2Az = μσ
Rotor back-iron
∂Az
∂t
2.5.4 Boundary Conditions
In Figure 2-8 there are boundaries represented by B1 to B7. We must apply
boundary conditions to achieve particular solutions for the PDEs. The first boundary
condition originates from Ampere’s law:
n × (H1 − H2 ) = K
(2.48)
where
H1 is the magnetic field intensity in region 1 on one side of boundary,
H 2 is the magnetic field intensity in region 2 on the other side of boundary,
K is the surface current density at the boundary and
n is the unit normal vector to the boundary.
Equation (2.48) indicates that the tangential components of magnetic field
intensity are equal on both sides of the boundary. We can translate Equation (2.48) in
the coordinate of Figure 2-8 to
H x1 − H x 2 = K
(2.49)
Using B = μH we can write
1
1
B −
B =K
μ1 x 1 μ2 x 2
(2.50)
17
Using B = ∇ × A , we can then write
1 ∂Az 1
1 ∂Az 2
−
=K
μ1 ∂y
μ2 ∂y
(2.51)
where Az 1 is the z component of magnetic vector potential in region 1, Az 2 is the z
component of magnetic vector potential in region 2.
The second boundary condition comes from magnetic flux continuity:
n ⋅ (B1 − B2 ) = 0
(2.52)
Similarly we can write for the vector potential in the coordinate of Figure 2-8 as
∂Az 1
∂Az 2
=
∂x
∂x
(2.53)
2.5.5 General Solution of the Partial Differential Equations
We can now solve the PDEs summarized in Table 2-1 for each region [26]. The
general solution for Laplace equation is given as
Az (x , y ) = (m cos kx + n sin kx )(ge ky + he −ky )
(2.54)
The general solution for Poisson’s equation is given as
Az (x , y) = (m cos αx + n sin αx )(gey
α2 + j β 2
+ he−y
α2 + j β 2
)
(2.55)
where
β =
ωσμ and α is a constant defining variation along x -axis,
m , n , g and h are constants to be found by applying boundary conditions,
σ is the conductivity, μ is the permeability and ω is the frequency.
2.5.6 Modeling the Armature Current
The armature current in a concentrated winding machine can be modeled as
rectangular columns, as illustrated in Figure 2-9.
The rectangular current waveforms can be Fourier-decomposed into several
sinusoidal harmonics along x-axis, which can be seen in Figure 2-9. As a result, the
MMF due to the current is distributed as sinusoidal harmonics in space. We solve for
the loss due to each harmonic and then add them up to obtain the total loss.
The excitation for the magnetic field in the model is a wave of travelling surface
current density. The rotor rotates at a constant speed. All harmonics of the current
density which rotate at a different speed with respect to the rotor produce eddy current
losses.
The relative motion between the harmonics and the rotor can be modeled by
changing the reference from stator to rotor.
18
xs = xr + vt = xr ±
ωτ p
ωλ1
t = xr ±
t
2pλ π
π
(2.56)
where v is the relative velocity with which the flux density leads the rotor, τ p is the pole
pitch (the distance between two adjacent rotor poles), pλ is the number of pole pairs
within the wave length λ1 , and ± is + if the frequency ω is in the same direction with v ,
and – if the frequency ω is in the opposite direction with v .
This transformation results in a modification of the velocity of the harmonic flux
density with respect to the rotor velocity, which can be summarized as
vn =
λ1 f
n
(2.57)
where vn is the velocity of nth space harmonic, λ1 is the fundamental wave length and
f is the electrical frequency.
The travelling current density is a function of both time and space. In general it can
be expresses as
K = Kˆ cos(kx − ωt )
(2.58)
where K is surface current density, ω is the electrical frequency in rotor frame of
reference, k is a constant for space distribution and x is a space coordinate.
Equation (2.58) is used in the boundary condition Equation (2.51) and then the
PDEs can be solved.
A
C
B
A
Flux density
x
Current/m
x
One wavelength λ
Figure 2-9: Flux density and surface current density (A/m) waveforms
for a concentrated winding machine
19
Figure 2-9: Surface current density decomposition for a concentrated winding machine
2.5.7 Effect of Permanent Magnets Only
If the magnetic field is only excited by permanent magnets (armature current is
zero), the cause of eddy currents is stator slotting. The magnets are mounted on the
rotor and this means the field due to magnets rotates at the same speed of rotor. This
implies that no eddy currents are supposed to be induced because of relative motion in
any part of rotor. Slotting due to armature slots, however, changes the distribution of
air-gap flux density. The flux density changes abruptly in vicinity of armature slot
edges, because the magnetic permeability suddenly changes in such regions, as simply
illustrated in Figure 2-10. More investigation on slotting effect can be found in [20] and
[21].
Stator
Magnet
Φ
Rotor back-iron
Figure 2-10: Slotting effect to change flux density
As the slotting effect has been neglected in the geometry of our model shown in
Figure 2-8 and the air-gap is simplified to be a smooth surface, no eddy currents are
20
induced in rotor back-iron by the magnets. Here we generally apply the equations in
Table 2-1 to see how the magnetic field is distributed by the magnets.
The analytical model for the field due to only magnets is left for future studies,
because
1. The modeling is complex.
2. Finite element method instead of analytical model is capable of actual
calculation for this field.
2.5.8 Combined Effect of Armature Currents and Permanent Magnets
The combined effect can be a superposition of the fields due to each excitation.
Without saturation in magnet and iron, the magnetization can been considered linear
even when the operating point deviates.
∇×Η = J
(2.59)
∇ × Η = J + Jext
(2.60)
Therefore the fields due to different excitations can be superimposed.
2.5.9 Derived Quantities from Az
After solving for Az , several quantities can be derived from it. What we need is the
induced current density J .
J = −σ
∂A
∂t
(2.61)
The vector potential Az is a function of time and space so we can express it in such
a way by utilizing the periodicity of sine and cosine functions:
Az (x , y , t ) = S (x , y )e j ωt
(2.62)
The induced current density can be then written as
J z (x , y ) = − j σω[S (x , y )e j ωt ]
(2.63)
At last the eddy current losses can be obtained from J z (x , y ) as
Peddy =
∫∫
s
J z2 (x , y )
dA
σ
(2.64)
This analytical model has been implemented already in [22]. Other approaches to
analytically solving for eddy current losses in magnets and in rotor back-iron are
presented in [23] and [24].
21
2.6 Finite Element Methods
2.6.1 Why Finite Elements?
Theoretical formulas are accurate to solve electromagnetic problems. When the
complexity of problems makes the analytic solutions limited, we resort to non-analytic
methods, which include graphical, experimental, analog and numerical methods. The
former three methods have limitations so that they are not applicable in many cases.
Numerical methods have become attractive and popular with the development of fast
computers. Three commonly used numerical methods are the moment method, the
finite difference method (FDM) and the finite element method (FEM). Partial
differential equations (PDEs) are usually solved by either FDM or FEM while integral
equations are solved by using the moment method. Although numerical methods give
approximate solutions, the solutions are sufficiently accurate for engineering purposes.
Frequently we have to encounter the following issues in an electromagnetic
problem:
- irregular geometric shapes of components
- non-linearity of magnetic or electric materials
- induced currents in non-regular components
- anisotropic materials or structures
- external circuits
- integration of thermal/mechanical effects
- non-sinusoidal time variation of currents and fields
Analytic techniques are limited while numerical methods are capable of modeling
and solving such problems. The most important advantage of FEM over the other two
numerical methods is that it is well suited to modeling non-linear materials [25]. Despite
large demands of computer memory and high software costs, the FEM is believed to be
best suited to the needs of designers.
2.6.2 Using Finite Element Methods
The finite element analysis of any problem involves basically four steps [26]:
- discretizing the solution domain into finite number of subdomains or elements
- deriving governing equations for a typical element,
- assembling all the elements in the solution domain
- solving the system of equations obtained
2.6.2.1 Finite element discretization
We divide the solution domain into a number of finite elements. The types of
element are diverse and triangles are most often used in two-dimension because a
22
triangle has minimum area of subdomain and lowest possible energy. With the
triangular elements illustrated in Figure 2-11, we seek an approximation for the
magnetic vector potential in z direction Az (x , y ) within an element e and then
inter-relate the potential distributions in various elements such that the magnetic
potential is continuous across inter-element boundaries. The approximate solution for
the whole domain is
N
Az (x , y ) ≈
∑ Az ,e (x , y )
(2.65)
e =1
where N is the number of triangular elements into which the solution domain is divided.
Polynomial approximation is used for the approximation of Az ,e within an element:
Az ,e (x , y ) = a + bx + cy
(2.66)
for a triangular element. we assume that the same linearity applies to the magnetic field
which is uniform within the element so that:
B = ∇ × A = cx − by
(2.67)
where the magnetic vector potential has only z component.
1
6
(1)
(2)
2
3
(5)
(3)
i
(4)
4
Node number
(j) Element number
5
Approximate boundary
Actual boundary
Figure 2-11: A typical finite element subdivision of an irregular domain
2.6.2.2 Element-governing equations
Consider a triangular element shown in Figure 2-12, whose numbering 1-2-3 must
be counterclockwise. The potential Az ,e1 , Az ,e 2 and Az ,e 3 at nodes 1, 2 and 3, respectively,
are derived from Equation (2.66) as:
⎛ Az ,e1 ⎞⎟ ⎛ 1 x1 y1 ⎞⎛a ⎞
⎜⎜
⎟⎜ ⎟
⎟ ⎜
⎜⎜ A ⎟⎟ = ⎜⎜ 1 x y ⎟⎟⎟ ⎜⎜b ⎟⎟⎟
z
,
e
2
2
2
⎟
⎜
⎟⎟ ⎜⎜ ⎟⎟
⎜⎜
⎟ ⎜
⎜⎝ Az ,e 3 ⎠⎟⎟ ⎜⎝⎜ 1 x 3 y3 ⎟⎠⎝⎜⎜c ⎟⎠
(2.68)
23
The coefficients can be thus found as
⎛a ⎞⎟ ⎛ 1 x1 y1 ⎞⎟−1 ⎛⎜ Az ,e1 ⎞⎟
⎜⎜ ⎟ ⎜⎜
⎟⎟
⎟⎟ ⎜⎜
⎟
⎜⎜⎜b ⎟⎟⎟ = ⎜⎜⎜ 1 x 2 y2 ⎟⎟⎟ ⎜⎜ Az ,e 2 ⎟⎟⎟
⎜⎜⎝c ⎟⎠ ⎜⎜⎝ 1 x y ⎟⎠ ⎜⎜ A ⎟⎟
⎝ z ,e 3 ⎠
3
3
(2.69)
Substituting this into Equation (2.66) gives
3
Az ,e =
∑ αi (x , y)Az ,ei
(2.70)
i =1
where
α1 =
1
[(x y − x 3y2 ) + (y2 − y3 )x + (x 3 − x 2 )y ]
2A 2 3
(2.71)
α2 =
1
[(x y − x1y3 ) + (y3 − y1 )x + (x1 − x 3 )y ]
2A 3 1
(2.72)
α3 =
1
[(x y − x 2y1 ) + (y1 − y2 )x + (x 2 − x1 )y ]
2A 1 2
(2.73)
and A is the area of the element:
A=
1
[(x − x1 )(y3 − y1 ) − (x 3 − x1 )(y2 − y1 )]
2 2
(2.74)
y
Az ,e 3 (x 3 , y3 )
3
1
Az ,e1 (x1, y1 )
2
Az ,e 2 (x 2 , y2 )
x
Figure 2-12: Typical triangular element
Equation (2.70) gives the potentials at any point within the element, provided that
the potentials at the nodes are known. Note that αi are linear interpolation functions
and αi are called element shape functions. They have the following properties:
⎧1, i = j
⎪
αi (x j , y j ) = ⎨
⎪
⎪
⎩ 0, i ≠ j
(2.75)
3
∑ αi (x , y) = 1
(2.76)
i =1
24
The magnetic energy per unit length associated with the element e is
We =
1
1 1
2
B ⋅ HdS = ∫
∇ × Az ,e dS
2∫
2 μ
S
S
(2.77)
From Equation (2.70) we have
3
∇ × Az ,e =
∑ Az ,ei (∇ × αi )
(2.78)
i =1
So Equation (2.77) can be written as
3
We =
3
1
1
A [ (∇ × αi )(∇ × αj )dS ]Az ,ej
∑
∑
2 i =1 j =1 μ z ,ei ∫
(2.79)
S
We define the term in brackets as
C ij(e ) =
∫ (∇ × αi )(∇ × αj )dS
(2.80)
S
The magnetic energy can then be described in matrix form:
We =
11
[A ]T [C (e ) ][Az ,e ]
2 μ z ,e
(2.81)
where the matrix [C (e ) ] is called the element coefficient matrix and this is
C (e )
⎛C (e ) C (e ) C (e ) ⎟⎞
12
13 ⎟
⎜⎜ 11
⎜⎜ (e )
(e )
(e ) ⎟
⎟
= ⎜C 21 C 22 C 23
⎟⎟⎟
⎜⎜
⎜⎜C (e ) C (e ) C (e ) ⎟⎟⎟
⎝ 31
32
33 ⎠
(2.82)
and
⎡ Az ,e1 ⎤
⎢
⎥
[Az ,e ] = ⎢⎢ Az ,e 2 ⎥⎥
⎢A ⎥
⎢⎣ z ,e 3 ⎥⎦
(2.83)
We can check our calculation by the relationship:
3
∑Cij(e) =
i =1
3
∑ Cij(e) = 0
(2.84)
j =1
2.6.2.3 Assembling elements
Having figuring out one element, the next is to assemble all elements in the solution
domain. The energy associated with the assemblage of all elements in the mesh is
N
W =
∑We
e =1
=
11
[A ]T [C ][Az ]
2μ z
(2.85)
where
25
⎡ Az 1 ⎤
⎢
⎥
⎢ Az 2 ⎥
⎥
Az = ⎢
⎢
⎥
⎢
⎥
⎢ Azn ⎥
⎣
⎦
(2.86)
n is the number of nodes, N is the number of elements, and [C ] is called global
coefficient matrix compared with the local coefficient matrix [C (e ) ] for a single element.
The global coefficient matrix has the form
⎛C11 … C1n ⎟⎞
⎜⎜
⎟⎟
[C ] = ⎜⎜
⎟⎟
⎜⎜
⎟
⎝⎜C n1
C nn ⎟⎠
(2.87)
which is a n × n symmetric matrix since n nodes are involved.
C ij represents the coupling between nodes i and j . If no coupling exists between
nodes i and j , the coefficientC ij is zero. We know the fact that the potential must be
continuous across inter-element boundaries. All the elements containing nodes i and j
determine C ij and we can obtain C ij by adding all the contributions in each element
containing nodes i and j . The example for demonstration in detail can be found in [26].
2.6.2.4 Solving equations
The principle of minimum total potential energy says that Laplace’s or Poisson’s
equation is satisfied when the total energy in the solution domain is the minimum. So
the partial derivatives ofW with respect to the potential Az of each node should be zero:
∂W
= 0,
∂Azk
k = 1, 2,..., n
(2.88)
From Equation (2.85), this results in
n
∑ AziCik
=0
i =1
(2.89)
where n is the number of nodes in the mesh.
We can obtain a set of equations by implementing Equation (2.89) for all nodes in
the mesh, from which the solution of [Az ]T = [Az 1 , Az 2 ,..., Azn ] can be found. This can be
done by the iteration method and band matrix method. The iteration process starts
with setting the potentials at the free nodes equal to zero or to the average value:
Az ,av =
1
(A
+ Az ,max )
2 z ,min
(2.90)
where Az ,min and Az ,max are the minimum and maximum values of the prescribed
potentials at the fixed nodes. With such initial values, the potentials at the free nodes
are calculated by using
26
Azk = −
1
C kk
n
∑
AziC ik
(2.91)
i =1,i ≠k
which is derived from Equation (2.89).
At the end of the first iteration, when the new values have been calculated for all
the free nodes, these values become the old values for the next iteration. The procedure
is repeated until the change between subsequent iterations is negligible.
If all free nodes are numbered first and the fixed nodes last, Equation (2.85) can be
written as
W =
⎛ C ff
1
ε[Vf Vp ] ⎜⎜
⎜⎝C pf
2
C fp ⎟⎞ ⎡Vf ⎤
⎟⎢ ⎥
C pp ⎠⎟ ⎢⎣Vp ⎥⎦
(2.92)
where subscripts f and p , respectively, refer to nodes with free and fixed (or prescribed)
potentials. Since isVp constant, we only differentiateVf with respect to so that applying
Equation (2.88) to (2.92) yields
[C ff ][Vf ] = −[C fp ][Vp ]
(2.93)
This equation can be written as
[V ] = [A]−1[B ]
(2.94)
where [V ] = [Vf ] , [A] = [C ff ] and B = −[C fp ][Vp ] Since [A] is, in general, non-singular, the
potential at the free nodes can be found using Equation (2.93). We can solve for [V ] in
Equation (2.94) using Gaussian elimination technique. We can also solve for [V ] in
Equation using matrix inversion if the size of the matrix to be inverted is not large.
The minimum potential energy is the basis of finite element methods. The
computer programs, e.g. COMSOL Multiphysics, using finite element methods to solve
PDEs, often implement the weak form for PDEs. The weak form is based on the
minimum potential energy and it is the statement of the potential energy method. The
approaches to apply the weak form in finite element problems can be found in [27].
27
28
Chapter 3
Finite Element Simulations
3.1 Introduction
Analytical models can be used to calculate eddy current losses in some cases. But
they have several important limitations:
1. Simplified geometry
2. Only linearity considered
3. Complexity of combining magnetic fields due to different excitations
These facts limit the application of analytical models. Finite element methods
(FEM) are therefore widely implemented to overcome these limitations. With the
assistance of computer technology, finite element models have many advantages over
analytical models, such as
1. Complicated geometry possible
2. Non-linear model possible
3. No need to separate field excitations
FEM simulation is now widely applied in electrical engineering and it is very
efficient for engineers to design electrical machines. We also use FEM simulations in
this thesis to do research on eddy current losses. COMSOL Multiphysics 3.5a is used as
the FEM simulation software.
In this and next two chapters, we do several tests to observe eddy current losses in
electrical machines with concentrated windings. The tests are carried out by means of
FEM simulations and laboratory experiments. The tests consist of two main parts:
1. Static test
Metal rings are used together with stators. The machine runs as a motor at static
mode. The FEM simulation result of eddy current losses is validated by experiments
and in the meantime used to validate the analytical model introduced in Chapter 2.
This test is referred to as static simulation in FEM simulations (Chapter 3) and static
experiment in laboratory experiments (Chapter 4).
2. Rotary test
An actual rotor is used instead of metal rings. The machine rotates as a generator
in no-load and on-load conditions. The FEM simulation result of eddy current losses is
validated by experiments. This test is referred to as rotary simulation in FEM
simulations (Chapter 3) and rotary experiment in laboratory experiments (Chapter 4).
29
The different machines to be modeled in FEM and to be measured in experiments
are described at the beginning of this chapter. Then the focus is on FEM simulations for
the static and the rotary tests. The method and procedure of doing these simulations
are stated. The static and the rotary simulations are executed independently and their
results are shown in figures and some important observations are discussed.
3.2 Simulation Tool
COMSOL Multiphysics is a finite element analysis, solver and Simulation software
/ FEA Software package for various physics and engineering applications, especially
coupled phenomena, or multiphysics. COMSOL Multiphysics also offers an extensive
interface to MATLAB and its toolboxes for a large variety of programming,
preprocessing and postprocessing possibilities. The packages are cross-platform
(Windows, Mac, Linux, Unix). In addition to conventional physics-based
user-interfaces, COMSOL Multiphysics also allows for entering coupled systems of
partial differential equations (PDEs). The PDEs can be entered directly or using the so
called weak form [28].
Figure 3-1: COMSOL Multiphysics 3.5a as the finite element simulation tool
In our case, only AC/DC Module of the COMSOL Multiphysics is used. This
module facilitates the modeling and simulation of electrostatics, direct current,
electro-quasistatic approximation, magneto-quasistatic approximation and rotating
electrical machines.
Normally most electrical machines are of rotating type except some linear machines.
To get the rotary motion effect, the mathematical technique of Arbitrary
Lagranian-Eulerian (ALE) method is implemented in COMSOL for mesh movement.
The ALE method is intermediate between the Lagranian and the Eulerian methods. It
combines the best features of both and allows moving boundaries without the need for
the mesh movement to follow the material [29].
3.3 Description of Modeled Machines
We have two stators, as shown in Figure 4-1, for both static and rotary tests. Both
of them have concentrated windings, laminated iron cores (M-19 steel, 0.35mm
30
thickness lamination) and 27 teeth (and 27 slots). They are designed to be inner stators
and coupled with outer rotors. There are two decisive differences in the stators:
1. Slot opening
One stator is with open slots and we call it open-slot (OS) stator (Figure 3-1(a)).
The other is with semi-closed slots and we call it semi-closed-slot (SCS) stator (Figure
3-1(b)).
2. Number of turns per coil per tooth
In the open-slot stator the number of turns per coil per tooth is 10 while in the
semi-closed-slot stator it is 9. Note that each turn consists of two identical wires, each
of which carries half current. This reduces the resistivity of the winding and in the
meantime enhances the current-carrying capability of the winding.
The neutral of the winding of each stator is connected internally but not accessible.
As a result, only line-to-line voltages are available to measure in the tests. The yoke of
each stator is designed to be mounted with bearings.
(a)
(b)
Figure 3-1: Concentrate winding stators with 27 teeth outwards
(a) Open-slot (OS) stator, (b) Semi-closed-slot (SCS) stator
31
We have three metal rings for the static test and one actual rotor for the rotary test.
The three rings are of different materials respectively: copper, aluminum and steel, as
shown in Figure 3-2(a), (b) and (c) respectively.
(a)
(b)
(c)
Figure 3-2: Three metal rings for the static test
(a) Copper ring, (b) Aluminum ring, (c) Steel ring
(a)
(b)
Figure 3-3: Actual rotor with magnets and back-iron for the rotary test
(a) Rotor, (b) Dimension of the magnets in mm
For the rotary test, we have an actual rotor with magnets and a solid iron back, as
shown in Figure 3-3(a). The outer rotor is a promising solution for flywheel generators
[30]. It is equipped with 18 magnets (poles). The combination of this rotor and either of
the stators is thus with 27 teeth and 18 poles, which can be described as a 3/2
combination. The magnets are made of Neodymium-Iron-Boron (NdFeB) with a
remanent flux density of 1.2T at 20℃ and their dimensions are shown in Figure 3-3(b).
So far we have had six combinations of stator and metal rings for the static test as
indicated in Table 3-1, and two combinations for the rotary test as indicated in Table
3-2. The specifications of the machine for each test are summarized in Table 3-3 and
Table 3-4.
32
Table 3-1: Combinations of stator and metal ring for the static test
No.
Code
Stator
Ring
1
OS_Cu
OS
Copper
2
OS_Al
OS
Aluminum
3
OS_St
OS
Steel
4
SCS_Cu
SCS
Copper
5
SCS_Al
SCS
Aluminum
6
SCS_St
SCS
Steel
Table 3-2: Combinations of stator and rotor for the rotary test
No.
Code
Stator
Rotor
1
OS_R
OS
Actual rotor
2
SCS_R
SCS
Actual rotor
In fact, the combinations for the rotary test are two permanent magnet machines.
They have been designed as 9kW machines.
Table 3-3: Specification of the metal rings for the static test
Copper
Aluminum
Steel
Inner radius [mm]
90.5
90.5
90.5
Outer radius [mm]
92.5
92.5
92.5
Conductivity [S/m]
5.6×107
3.7×107
5.0×106
Relative permeability
1
1
Variable*
Table 3-4: Specification of the 9kW machines for the rotary test
Parameter
OS
SCS
Rotor radius [mm]
97
97
Stator radius [mm]
90
90
Stack length [mm]
45
45
Air-gap length [mm]
2
2
Number of Phases
3
3
Number of Coils
27
27
Number of poles
18
18
Topology
Y*
Y*
Continued on the next page
33
Continued
Pole width [mm]
22.5
22.5
Pole pitch [mm]
32.1
32.1
Magnet thickness [mm]
5
5
Yoke thickness [mm]
18
18
Slot width [mm]
12.85
12.85
Slot inner width [mm]
5.9
5.9
Slot pitch [mm]
20.9
20.9
Stator core material
M-19
M-19
Thickness of lamination [mm]
0.35
0.35
Slot height [mm]
30
26.5
Dovetail height (tooth cap) [mm]
N/A
3.5
Dovetail width (tooth cap) [mm]
N/A
17
Turns per coil
10
9
Remanent flux density of magnet [T]
1.2
1.2
Conductivity of magnet [S/m]
0.76×106
0.76×106
Relative permeability of magnet
1.05
1.05
Conductivity of rotor back-iron [S/m]
5.0×106
5.0×106
Relative permeability of rotor back-iron
Variable**
Variable**
* The three phase windings are star-connected.
** The relative permeability is according to Figure 3-4.
Saturation of iron must be taken into account according to the B-H curve, as
illustrated in Figure 3-4, otherwise the flux density in the iron could be so high that the
iron losses could be overestimated.
(b) μr -B curve
(a) B-H curve
Figure 3-4: Magnetic properties of the steel
34
3.4 Simulations and Results
3.4.1 Static Simulation
3.4.1.1 Modeling the geometry
The static simulation uses three metal rings instead of rotor. Since there are no
magnets on the rings and the surface of the rings is smooth, the machine cannot rotate.
This reduces the computational work and therefore we can model the full geometry for
the machines in 2-D as shown in Figure 3-5. We use time-harmonic solver in COMSOL
for linear materials (copper and aluminum in our test) so that the computation can be
further simplified.
Surrounding air
Metal ring
Air gap
Stator winding
Stator iron
Figure 3-5: Full geometry of the open-slot stator with a rotor ring
Figure 3-6: Full geometry of the semi-closed-slot stator with a rotor ring
35
The surrounding air should be modeled here because of two reasons:
1. The ring is very thin so that a large amount of magnetic flux could go through it
and into the surrounding air. The amount of flux going to the surrounding air also
depends on the permeability of the ring. This portion of flux decreases with higher
permeability of the ring.
2. The magnetic flux crowds into the ring if we set the outer boundary of the ring
as the magnetic insulation. This could lead to higher eddy current losses in the ring.
Therefore we add the domain of surrounding air, the outer boundary of which is set as
the magnetic insulation.
Similarly we have the geometry for the semi-closed-slot stator with the ring as
shown in Figure 3-6.
3.4.1.2 Simulation procedure
We inject balanced three-phase sinusoidal currents into the winding and see how
much resistive heat exists in the ring. It is assumed that all resistive heat in the ring
comes from induced eddy currents.
The simulation would be better with higher currents. But we set the RMS value of
the three-phase currents to be 9A in order to facilitate the future laboratory
experiments in which the AC power supply can only provide at most 10A currents
(RMS value). The phase arrangement of stator windings is shown in Figure 3-7 and here
only the open-slot stator is illustrated as an example. The sequence of phase A, B and
C follows the counter-clockwise and repeats for all 27 teeth.
B
A
C
B
C
A
A
Figure 3-7: Phase sequence of the stator windings
The frequency of injected current is also manipulated. We set several frequencies
for analysis: 17Hz, 20Hz, 30Hz, 45Hz, 65Hz, 100Hz, 150Hz, 230Hz, 350Hz, 500Hz,
750Hz and 1000Hz.
These are the fundamental frequencies. Due to the nature of concentrated windings,
the MMF in space contains space harmonics. The range of these frequencies covers the
applications from low-speed to high-speed machines.
36
We have six combinations of stators and rings according to Table 3-1 so we seem to
have to complete 72 times transient simulations. But due to the fact that the copper
and the aluminum ring can be regarded as linear materials since they do not saturate,
we can make a simplification by using time-harmonic simulation. For the copper or the
aluminum ring, we only do the time-harmonic simulation once for each ring at all
frequencies. Transient simulations valid for non-linear material are executed only for
the cases with the steel ring. This reduces the workload from 72 times to 28 times.
3.4.1.3 Results
The simulations aim at the total resistive heating in the domains of magnets and
rotor back-iron. We separate the eddy current losses in magnets and rotor back-iron
here to show the effect of shielding. In Chapter 5, we will treat the eddy current losses
in rotor as a whole for the validation.
The total resistive heating in the rotor is obtained by integrating the simulated
resistive heating over the area of the magnets and the rotor back-iron, multiplying it by
nine (the number of symmetrical sections) and then by the axial length of the rotor. The
processing here is under the assumption that the machine over the whole axial length is
uniform and none of the variables change in the axial direction ( z -axis).
We need the mesh on the domain of ring as fine as possible because of two reasons:
1. The ring is very thin and needs plenty of mesh elements to get accurate solutions.
2. Due to the skin effect, the majority of eddy currents are supposed to be induced
close to the inner boundary of the ring.
However we cannot apply as fine meshing as we want, because excessive
computation could lead to memory problems for a computer. The mesh of the open-slot
machine as an example is illustrated in Figure 3-8. We can see approximately 4 mesh
elements on the domain of the ring in the radial direction and at the inner boundary of
the ring the mesh elements get smaller.
(a)
(b)
Figure 3-8: Finer meshing for the domain of ring and its inner boundary (open-slot machine)
37
In the simulations the conductivity of the stator iron is set to zero so there will be
not any iron losses in the stator iron. This is an assumption for simplification. In reality,
however, stator iron has non-zero conductivities even if it is laminated. But it is rather
difficult here to know the value of the stator iron conductivity.
Saturation is taken into account for the stator iron and we use the non-linearity
curves of B-H relation and permeability in Figure 3-4 for it. Otherwise the flux density
in the stator iron would be higher and this would lead to incorrect losses.
We obtain the resistive heating in the rings as the rotor eddy current loss. For
giving a reference of choosing a right stator topology in machine designing, we draw
together the curves of eddy current losses in the rotor of the open-slot stator and the
semi-closed-slot stator. The simulation results are drawn in Figure 3-9, 3-10 and 3-11
for the copper ring, the aluminum ring and the steel ring, respectively.
Figure 3-9: Eddy current loss curves of copper ring with each stator
Figure 3-10: Eddy current loss curves of aluminum ring with each stator
38
Figure 3-11: Eddy current loss curves of steel ring with each stator
In these figures we can observe that the eddy current loss is quite dependent on
frequency. The eddy current loss rises with an increasing frequency. But the pattern of
rise is different in the cases of copper and aluminum rings and the case of steel ring.
In Figure 3-9 and 3-10, we can see the curves are linear with a steep slope at low
frequencies and they become flat at high frequencies. In Figure 3-11, the curves are
almost linear at all frequencies. This implies:
1. At high frequencies, the magnetic field excited by induced eddy currents tends to
cancel the main field produced by the armature current.
2. The steel ring is able to reduce such cancellation but the other two rings are not.
The eddy current loss is higher with the semi-closed-slot stator. Although the
number of turns is slightly higher, the open-slot stator induces less eddy current losses
in the ring. All the results above in the static simulation will be validated in Chapter 5.
3.4.2 Rotary Simulation
3.4.2.1 Simulation procedure
The rotary simulation consists of two parts: no-load and on-load simulations. The
actual rotor rotates instead of stationary metal rings, resulting in rotary movement.
The same geometries respectively for the open-slot stator and the semi-closed-slot
stator are used over all the procedure. In the no-load simulation there are not any
currents flowing in the stator winding. This means to simulate the generator at no load.
In the on-load simulation we give three-phase sinusoidal currents to the stator winding
to simulate the generator outputting electrical power.
1. In the no-load simulation, no current is given to the stator windings. The rotor
rotates at various speeds and we observe the total resistive heating in magnets and rotor
back-iron.
39
The speeds are chosen to cover a wide range and they are from 100rpm to 1500rpm
with a step size of 100rpm.
2. In the on-load simulation, we observe the total resistive heating in magnets and
rotor back-iron as in the no-load simulation but now the resistive heating is induced not
only by the magnets but also the current we give to the stator winding. Both the effects
of speed and current can be simulated as a whole.
The speeds are chosen as:
600rpm, 800rpm, 1100rpm, 1300rpm and 1500rpm.
The values of current are chosen as:
4A, 7A, 10A, 13A, 16A and 18A.
The number of speed values is less than in no-load simulation. It is not necessary to
apply smaller step sizes for the speed since the consistence and stability of such
simulations in COMSOL is good enough. The values of current cover the small load (4A,
7A, 10A and 13A), the rated load (16A) and the overload (18A).
In the rotary simulations, the conductivity of stator iron is kept zero and therefore
the eddy current losses in the stator iron is neglected. It is assumed that eddy current
losses only occur in the magnets and rotor back-iron.
3.4.2.2 Modeling the geometry
From now on we use the symmetry shown in Figure 3-14 to model the machine
geometry because this can considerably reduce the computation workload when a very
fine mesh with motion is applied. The machine is divided into 9 identical sections with
their own corresponding angles in the full machine.
Tooth cap
Winding
Winding
③
③
④
⑤
②
①
④
⑤
③
(a)
Figure 3-12: One ninth section of the modeled machines
②
①
③
(b)
(a) Open-slot machine, (b) Semi-closed-slot machine
① Surrounding air, ② Rotor back-iron, ③ Magnets, ④ Air-gap, ⑤ Stator iron core
40
Boundary and symmetry conditions must be properly set up, especially for the
boundary between moving and stationary domains. The domain of surrounding air also
rotates with the rotor at the same speed and current can exist or not in the stator
windings.
3.4.2.3 No-load simulation and results
The no-load simulation used the geometry in Figure 3-12. We do not give any
current in the stator winding. The stator is fixed and the rotor rotates at a speed. In this
case, only the magnetic field due to magnets is present and the slotting effect induces
eddy current losses in magnets and rotor back-iron. In other words, the no-load
simulation investigates the slotting effect on eddy current losses. This effect can be
illustrated with finite element plotting in Figure 3-13. We can observe that the
magnetic field suddenly changes where slotting changes.
Figure 3-13: Finite element plotting showing the slotting effect
The meshing is fine enough as illustrated in Figure 3-14 (only the open-slot
machine as an example) because the mesh elements at the moving boundary are
sufficiently small and numerous. Furthermore the skin effect is also considered as the
meshing at the inner side of magnets is fine. This meshing will be used for all rotary
simulations.
Here the remanent flux density is set with temperature according to a real NdFeB
magnet. The remanent flux density is 1.2T at 20℃ and it follows the function of
temperature:
Br (T ) = Br ,20°C + αT (T − 20)
(3.1)
where Br (T ) is the remanent flux density at T ℃, Br ,20°C is the remanent flux
density at 20℃ and αT is the temperature coefficient of NdFeB which is -0.0012T/℃.
41
The temperature is assumed to be 70℃ and the remanent flux density used in the
simulation is 1.14T. The stator iron conductivity is set to be zero. The rotational speed
ranges from 100rpm to 1500rpm with a step size of 100rpm. We are concerned about the
total resistive heating in the magnets and the rotor back-iron.
Figure 3-14: Meshing used for all rotary simulations
Results
The result for the open-slot machine is plotted in Figure 3-15(a). We can see the
eddy current loss in rotor increases as the speed goes higher and the relation between
the eddy current loss and the speed is not linear but a little curved as an exponential
function. The curves of magnets and rotor back-iron are also shown in this figure. The
eddy current loss in magnets dominates here. The magnets act as a shield of the rotor
back-iron and reduce the eddy current loss in the rotor back-iron.
(a) Open-slot machine
42
(b) Semi-closed-slot machine
Figure 3-15: Eddy current losses in rotor for the no-load simulation
The result for the semi-closed-slot machine is illustrated in Figure 3-15(b). A
similar trend as in the open-slot machine can be observed. But the eddy current loss in
the rotor here is much lower and even negligible. The shielding effect of magnets also
reduces the eddy current losses in the rotor back-iron.
3.4.2.4 On-load simulation and results
To reveal the influence of both stator armature currents and permanent magnets
on the eddy current losses in rotor, we perform on-load simulations. We give
three-phase sinusoidal currents to the stator windings to simulate the generated
currents by the rotating machine. The sequence of the three-phase current should follow
the rotation direction of the rotor as indicated in Figure 3-16 (only the open-slot
machine is shown as an example).
The rotor is modeled to rotate in counter-clockwise. In reality, the electromotive
force (EMF) of phase A is induced first. The EMF of phase B is next and the EMF of
phase C is the last. There is a phase shift of 120° in time between adjacent phases.
It must be emphasized that the frequency of given current must agree with the
corresponding rotational speed. For example, if the rotational speed is 1500rpm, the
frequency should be 225Hz because of the relation:
f =
n
⋅p
60
(3.2)
where f is the frequency of armature current, n is the rotational speed and p is the
number of pole pairs.
43
The definition of current density in COMSOL is written as:
ja =
2N t I a
2
cos(ωt + π)
Ac
3
jb =
2N t I b
cos(ωt )
Ac
jc =
2N t I c
2
cos(ωt − π)
Ac
3
(3.3)
where ji , i = a, b, c is the current density of each phase,
I i , i = a, b, c is the RMS value of current of each phase,
N t is the number of turns per coil,
Ac is the cross-section area of one coil on one side of a tooth, and
ω = 2π f is the electrical frequency.
ω
C
B
A
Figure 3-16: Phase sequence and rotation direction for on-load simulation
Eddy current losses on phase current
At first we fix the speed and seek for the influence of the current on the eddy
current loss. The result of this case is plotted in Figure 3-17. Some of the no-load
simulation result is also included.
We can notice from Figure 3-17(a) that increasing phase current cannot change the
eddy current loss in rotor evidently in the open-slot PMSG. This suggests that
armature current has a rather small influence on the eddy current loss in rotor. But
there is still a slight rise when current increases.
Higher eddy current losses in rotor were expected with higher armature currents
because the armature current could produce a large amount of space harmonics of MMF.
44
In the finite element simulations, however, the eddy current loss in rotor does not show
so. This could be due to the following reason:
The machine runs as a generator so the armature current is induced by the rotation
of the magnetic field excited by magnets. According to Lenz’s law [31], the magnetic
field due to the induced current tends to reduce the magnetic field coming from magnets.
The total magnetic field therefore does not vary too much with the additional armature
currents. As a result, the eddy current loss in rotor depending on the magnetic field
could remain almost the same.
(a) Open-slot machine
(b) Semi-closed-slot machine
Figure 3-17: Dependence of the eddy current losses in rotor on current at difference speeds
45
The same phenomenon is also found in the magnets or the rotor back-iron
separately. Hence the curves for the eddy current loss in magnets and rotor back-iron
are omitted here.
The result from the semi-closed-slot PMSG is different from the other machine.
The losses are relatively rather low. The variations of losses with different currents are
much greater. The loss with 18A current is almost twice as high as the loss at no load.
This implies that the armature current significantly contributes to the eddy current loss
in the rotor.
Eddy current losses on speed
Then we fix the RMS value of armature current and look for the relation between
the eddy current loss and the speed. The eddy current loss with the phase current of
18A is shown here in Figure 3-18(a) and (b) respectively for the open-slot and the
semi-closed-slot PMSG. The eddy current losses in magnets and rotor back-iron are
illustrated together with the total rotor loss. Since the variation with different currents
is very small in the open-slot PMSG and the values of losses in the semi-closed-slot
PMSG is very low, showing all results with different currents is not necessary here.
The eddy current losses in magnets and rotor back-iron are quite dependent on
speed according to Figure 3-18. A higher speed results in a higher eddy current loss. The
magnets play an important role to protect the rotor back-iron from high eddy current
losses because the majority of loss occurs in the magnets rather than in the rotor
back-iron.
The variation of eddy current losses with different currents is considerably large in
the semi-closed-slot PMSG, so the dependence of the total eddy current losses in rotor
with all simulated current values is plotted in Figure 3-19.
(a) Open-slot machine
46
(b) Semi-closed-slot machine
Figure 3-18: Dependence of the eddy current losses in rotor on speed with phase current of 18A
Figure 3-19: Dependence of the eddy current losses in rotor on speed with different currents
An important observation is that the eddy current loss in rotor is much lower in
the semi-closed-slot machine than in the open-slot machine. This contributes to a key
difference between two types of stators. We can predict that the eddy current loss in
rotor is dominant in the open-slot machine but rather small in the semi-closed-slot
machine. This will be validated in Chapter 5.
47
48
Chapter 4
Experimental Testing
4.1 Introduction
It is necessary to validate the analytical model and the finite element simulation
with experimental tests to see how accurate the models are. Therefore experimental
tests have to be designed and executed for the purpose of validation. At the beginning
of this chapter, we describe the methods with which we design and perform the
experimental tests. An analysis is done prior to the main static experiment in order to
find out the influence of frequency and ring material on resistance and inductance. The
following experimental tests as well as tested machines have been described as in
Section 3.1and 3.3. In the formal static experiment, we inject three-phase current at
different frequencies to the stator and measure the eddy current losses in the metal ring
coupled with the stator. In the rotary experiment, the stator mounted with an actual
rotor acts as a generator. The power flow model for permanent magnet synchronous
generators (PMSGs) is thus introduced for the rotary test and we refer to this as power
balance method. Two power flow models are explained and applied in this method, and
a comparison is made to find which method is more reliable. Afterwards static and
rotary experiments are carried out, and we measure and calculate the total loss in stator
and rotor as a whole. In the end, comparisons between the open-slot machine and the
semi-closed-slot machine are presented to give clues for designing and testing a PM
machine with concentrated windings.
4.2 Experimental Test
4.2.1 Static Experiment
4.2.1.1 Analysis of resistance and inductance
It is known that resistance is dependent on frequency due to skin effect and
proximity effect [6]. In this analysis we intend to know how these effects play in our
tested machines. We are also interested in how different metal rings affect the resistance
and the inductance of the tested machines. This will help us understand more about the
reaction between armature current and the rotor material.
We use a digital impedance analyzer illustrated in Figure 4-1. After calibration we
connect two terminals of the stator winding (i.e. phase A and phase B) to the analyzer
and measure the line-to-line resistance and inductance at the frequencies of 20Hz, 40Hz,
100Hz, 200Hz, 300Hz, 400Hz, 500Hz, 600Hz, 800Hz and 1000Hz. We change the metal
ring for the open-slot stator and the semi-closed-slot stator (including the case of the
stator without any ring). So we do the measurement eight times in total and four times
for each stator.
49
Figure 4-1: Digital impedance analyzer
Results
The results are shown in Figure 4-2 and 4-3 for the open-slot stator and
semi-closed-slot stator respectively. The dependence of resistance or inductance on
frequency and ring material is plotted in the same figure for the purpose of comparison.
For the two stators without ring, the inductance hardly varies with frequency while
the resistance goes twice higher from 20Hz to 1000Hz. When the copper ring or the
aluminum ring is added, the inductance drops slightly by approximately100 μH while
the resistance increases to five times from 20Hz to 1000Hz. Then the pattern
considerably changes with the steel ring. The inductance decreases by around 900 μH
which is much greater than in the previous two cases. The resistance rises to nearly 30
times higher at 1000Hz than at 20Hz.
Comparing the resistance with different rings, we can see the resistance with the
steel ring is the much greater than with either of the other two rings when the frequency
is high, e.g. five times at 1000Hz. The same phenomenon can be also found in the
comparison of inductance with different rings, e.g. three times at 20Hz.
Resistance is quite dependent on frequency and ring material. Inductance is not as
sensitive to frequency as resistance only when the steel ring is added around the stator.
This suggests that the material of ring plays an important role in resistance and
inductance. Because the steel ring with a high permeability, say μr = 200 , much
enhances the magnetic field inside the machine and elevates the influence of frequency
on the magnetic field. On the contrary, too much magnetic flux goes through the ring
and into the surrounding air in the cases of copper ring and aluminum ring, because the
relative permeability of copper and aluminum is only equal to 1.
The difference in this test between the open-slot stator and the semi-closed-slot
stator lies in the inductance. The latter stator has higher inductance in any case due to
its structure of semi-closed slots. This structure can result in higher inductance. The
resistance is not quite different between these two stators.
50
(a) Resistance
(b) Inductance
Figure 4-2: Resistance and inductance at different frequencies with the open-slot stator
and different metal rings
(a) Resistance
51
(b) Inductance
Figure 4-3: Resistance and inductance at different frequencies with the semi-closed-slot stator
and different metal rings
We have seen the effect of frequency on resistance and the influence of ring
materials on both resistance and inductance. This will be helping us when the result of
static experiment is analyzed.
4.2.1.2 Static experiment
In this experiment we use a three-phase AC power supply (Figure 4-4) to inject
three-phase current into the stator winding. The frequencies of the current are 17Hz,
20Hz, 30Hz, 45Hz, 65Hz, 100Hz, 150Hz, 230Hz, 350Hz, 500Hz, 750Hz and 1000Hz.
The RMS value of the current is controlled to be about 9A. As the AC power
supply does not have current control mode so we can only adjust the voltage to get
suitable current values. This leads to slight deviation from 9A. Nevertheless the
accuracy of current output is qualified.
Because the neutral lines of both the stators are not accessible, we do not use the
neutral points of the AC power supply. Now the connection between the supply and the
stator winding is drawn in Figure 4-5.
Figure 4-4 Three-phase AC power supply
52
All combinations shown in Table 4-1 in Chapter 4 are to be tested and the stator
without any rotor ring is regarded as reference. We denote the references as OS_ref and
SCS_ref for the bare open-slot stator and the bare semi-closed-slot stator.
Figure 4-5: Connection between the power supply and the stator winding for the static test
According to the simple power balance, all power delivered by the power supply is
consumed in the tested machine. For the reference machines, we have the following
power balance:
Pin = PFe ,s + PCu ,s
(4.1)
where Pin is the power delivered from the power supply, PFe ,s is the stator iron losses and
PCu ,s is the copper loss in the stator winding.
For the other combinations, we have the power balance:
Pin' = PFe ,s + PCu ,s + Pr
(4.2)
where Pin' is the power delivered by the power supply and Pr is the eddy current loss in
the ring.
As the armature current is kept constant as 9A, the copper loss in the stator
winding PCu ,s is fixed for all tests. Furthermore we assume the stator iron loss PFe ,s is also
fixed for all static experiments. Now the difference between the delivered power from
the power supply Pin and Pin' is the eddy current loss in rotor ring Pr :
Pr = Pin' − Pin
(4.3)
Equation (4.3) gives us the method to measure the eddy current loss in the ring.
That is:
1. Measure the power consumed by the stator without ring,
2. Keep the stator fixed and change the rotor ring and measure the power given by
the power supply, and
3. Subtract the power measured in 1 from the power measured in 2. The difference
is the eddy current loss in the rotor ring.
It must be remembered that Equation (4.3) is only valid under the assumption that
the losses in stator do not vary in all measurements. Actually this assumption is not
true, because the iron losses in the stator also depend on frequency and ring material.
53
The dependence on frequency may not follow the pattern of no ring (reference case).
Changing the metal ring could also result in different losses in the stator iron. Now
Equation (4.2) turns to
'
Pin' = PFe
,s + PCu ,s + Pr
(4.4)
Comparing Equation (4.4) with Equation (4.1), we have the calculation for Pr as
'
Pr = (Pin' − Pin ) − (PFe
,s − PFe ,s )
(4.5)
We denote ΔPFe,s as the difference of stator iron losses:
'
ΔPFe,s = PFe
,s − PFe ,s
(4.6)
where ΔPFe ,s > 0 .
Therefore by measuring Pin and Pin' , we can only obtain a value which is larger than
the real eddy current loss in the ring:
Pin' − Pin = Pr + ΔPFe,s
(4.7)
The deviation of stator iron loss could lead to higher measured eddy current losses
in the ring than true values.
Results
The results are indicated in Figure 4-6(a), (b) and (c) for the machines with copper,
aluminum and steel ring respectively. We place the curves for the open-slot stator and
the semi-closed-slot stator in the same figure.
(a) Copper ring
54
(b) Aluminum ring
(c) Steel ring
Figure 4-6: Eddy current losses in the metal rings with the two stators
Under the assumption that changing rings does not affect the stator iron loss, we
can observe the follows:
1. The higher the frequency of the injected three-phase current, the more eddy
current induced in the ring. The dependence of eddy current losses on frequency is
almost linear in the steel ring but obviously curved to be flat at higher frequencies in the
copper or the aluminum ring.
2. The eddy current losses in the rings with the semi-closed-slot stator are the
greater in all the cases.
55
4.2.2 Rotary Experiment
4.2.2.1 Modeling the power flow for PMSG
We must know the power flow for the tested machines in order to apply the method
of power balance to find out rotor iron losses. Here we introduce a model of power flow
illustrated in Figure 4-7 which derives from synchronous generators [3].
Figure 4-7: Origin power flow model for PMSG
The input power drawn by the prime mover is finally converted into electrical
power output. In the intermediate process a portion of the input power is consumed as
losses. These losses mainly consist of prime mover and mechanical losses, stator winding
copper losses, stator iron losses and rotor iron and magnet losses. We need to calculate
the amount of rotor iron and magnet losses by using this power flow model as:
Pr = Pin − Ppm − PCu − PFe ,s − Pout
(4.8)
where Pr is the total loss of rotor iron and magnets, Pin is the input power to the prime
mover, Ppm is the sum of the prime mover and mechanical losses, PCu is the stator
winding copper loss, PFe ,s is the stator iron loss and Pout is the output power for load.
Figure 4-8: Modified model of power flow for PMSG by using electromagnetic power
An alternative method coming from the model above is derived in order to
eliminate the accuracy problem of measuring the prime mover and mechanical losses.
We can jump over the input power, the prime mover loss and the mechanical loss, and
directly measure the electromagnetic power within the tested PMSG. The bearing loss
consuming the electromagnetic power by means of friction between the stator and the
56
bearing is not considered as the friction is negligibly small. All other stray losses are
neglected as well.
The model of power flow is modified as shown in Figure 4-8. The electromagnetic
power can be believed to be the input power to the PMSG and now the relation in
Equation (4.8) is modified as:
Pr = Pem − PCu − PFe ,s − Pout
(4.9)
4.2.2.2 Setup
The following equipments are needed to test the rotating PMSG:
1. Prime mover
A separately excited DC motor is used as the prime mover (Figure 4-9(a)).
2. DC power supply to drive the DC motor
A DC power supply provides armature current (Figure 4-9(b)) and another one
excites the field winding (Figure 4-9(c)).
3. A spring balance measuring the electromagnetic force of the PMSG (Figure
4-9(d), (e))
We cannot directly measure the electromagnetic power so instead we measure the
force with which the rotor drags the stator, and calculate the electromagnetic power by
Tem = Fr
(4.10)
Pem = Tem ωm
(4.11)
whereTem is the electromagnetic torque, F is the dragging force imposed by the rotor to
the stator, r is the distance between the force F and the center of the machine, Pem is the
electromagnetic power and ωm is the mechanical rotating speed.
The quantity we can measure with the spring balance is mass rather than force.
Therefore we should use the following equation to transform mass to force:
F = mg
(4.12)
where m is the mass and g is the acceleration of gravity equal to 9.8m/s2.
The hook of spring balance is fixed with the stator such that the spring prevents
the stator from moving with the rotating rotor and it is thus able to measures the force.
The position of the spring balance is fixed such that the force F is perpendicular to the
distance r . The spring balance is selected properly such that the length of the spring is
hardly stretched by the force.
4. Resistive load
Three single-phase resistors are used as the load (Figure 4-9(f)). The resistances in
three phases can be manually adjusted independently.
57
5. Mechanical connections and bearings
6. An iron cage preventing the PMSG from flying off and in the meantime fixing
the spring balance
7. A tachometer measuring the rotational speed and outputting the corresponding
DC voltage signal, and multi-meters
(a) A DC motor as the prime mover driving the PMSG with connecting structures
(b) DC armature power supply
(c) DC supply for field excitation
(d) Spring balance and its position
(e) Three-phase resistor
Figure 4-9: Main equipments for rotary tests
58
The PMSG with both stator and rotor is shown in Figure 4-10 (only with the
open-slot stator for indication) when it is mounted on the shaft coming from the prime
mover. In Figure 4-10 we can clearly see the structure that fixes the stator with the
spring balance. The structure also provides the bearing that dissociates the stationary
stator and the rotating shaft.
Connected with the
spring balance
Figure 4-10: Fully mounted PMSG (with the open-slot stator)
The setup is demonstrated in Figure 4-11 The DC power supplyUda delivers power
to the armature of DC motor. V1 and A1 measure the armature voltageUda and current
Idc respectively. The tachometer gives DC voltage signals measured by V2 as the
indication of the rotational speed n of DC motor. A2 measures the current I f exciting
the field winding of DC motor. I f is kept constant for all tests and ensure sufficient
power that the DC motor can give to the shaft. The reading of m gives the value of mass
m that the spring balance measures.
Figure 4-11: Setup for rotary tests
The described setup so far is used for both the no-load test and the on-load test.
The part enclosed by the dashed line is not used in the no-load test but working in the
on-load test. When the PMSG is loaded, A3, A4 and A5 measure the true RMS values
of phase current Ia , Ib , and Ic while V3, V4 and V5 measure the RMS phase voltageUa ,
Ub , andUc . The reason to measure each phase is that the three-phase resistor is not
balanced and that the three-phase terminal voltage of the PMSG is not completely
59
symmetrical. These two facts could lead to a relatively large error for the load
measurement if conventional two-meter method is implemented.
4.2.2.3 Measuring the prime mover and mechanical losses
In the original model described in Equation (4.8), the input power is the power
absorbed from the DC power supply (connected with the armature) by the DC motor
which acts the prime mover. The prime mover and mechanical losses consist of the loss
consumed by the copper armature winding of the DC motor, the losses within the DC
motor (e.g. iron losses, bearing losses, friction losses), the mechanical losses of bearings
and air friction due to the PMSG. We can easily measure the input power Pin by reading
and multiplying the armature voltageUdc and current Idc of the DC motor.
Pin = Udc Idc
(4.13)
Then we deal with the prime mover and mechanical losses by considering the
following facts:
1. The copper loss in the armature winding of DC motor depends on the magnitude
the current which is variable in the test.
2. The amount of mechanical losses both inside and outside the DC motor is
constant at a fixed rotational speed.
3. The mechanical losses have a portion due to the mass of the PMSG. The mass is
constant in any case but it affects the bearing loss at different speeds.
Due to these facts, we simplify the prime mover and mechanical losses Ppm to
consist of two parts. One is the copper loss PCu ,dc in the armature winding of DC motor
and the other is denoted as the mechanical loss Pmech (of the driving system).
Ppm = PCu ,dc + Pmech
(4.14)
The resistance of the armature winding of DC motor Rdc is measured by applying
Ohm’s law. The excitation filed is set to zero by removing the excitation current so that
the DC motor cannot rotate. The voltage of several proper values is imposed to the
armature winding and corresponding currents will flow. The ratio between voltage and
current is the resistance.
Rdc =
Udc
Idc
(4.15)
Averaging the resistances calculated from the several pairs of voltage and current
gives an accurate value Rdc = 0.85Ω . All the measurements are conducted when the
machine has warmed up (temperature stable). From now on the copper loss in the
armature winding of the DC motor Pdc can be calculated with a variable current Idc .
2
PCu ,dc = Rdc I dc
(4.16)
To find the mechanical loss we conduct three measurements stated as follows:
60
1. Remove the PMSG from the shaft and measure the current and power of the DC
motor at different rotational speeds. The power balance here is
Pmech ,0 = Pin − PCu ,dc
(4.17)
where Pmech ,0 is the mechanical loss without the influence of the mass of rotor and stator.
2. Mount only the rotor onto the shaft and measure the current and power of the
DC motor at the same speeds in 1. The power balance here is
Pmech ,1 = Pin − PCu ,dc
(4.18)
where Pmech ,1 is the mechanical loss with only the influence of the mass of rotor.
3. Do the same in 2 using the stator instead of the rotor. The power balance can be
written here as
Pmech ,2 = Pin − PCu ,dc
(4.19)
where Pmech ,2 is the mechanical loss with only the influence of the mass of stator.
Afterwards the mechanical loss at a certain speed is calculated by
Pmech = Pmech ,1 + Pmech ,2 − Pmech ,0
(4.20)
The speeds to be used are from 100rpm to 1500rpm with a step size of 100rpm.
It is not possible to mount the rotor and the stator together to measure the
mechanical loss because this combination results in electromagnetic reaction between
the rotor and the stator. The resultant loss calculated from Equation (4.19) would be
incorrect as it contains a large portion of electromagnetic power.
Now we have got the amount of prime mover and mechanical losses at each speed.
This result is independent on the load so it can be used in both no-load and on-load
tests.
Results
Measuring the prime mover and mechanical losses is very important to apply the
original power flow model. We have separated the losses Ppm into copper losses in the
armature winding of DC motor PCu ,dc and the mechanical loss Pmech .
PCu ,dc is related to resistance and current and easily to calculate but the current
varies at different speeds. Only the mechanical loss Pmech is measured in this test. The
obtained data of Pmech at various speeds will be used in both no-load and on-load
experiments. The measurement result is illustrated in Figure 4-12.
We can see from the figure:
1. The mechanical loss measured with the semi-closed-slot machine is a little bit
higher than with the open-slot machine. This is mainly due to the slightly higher mass
of the semi-closed-slot stator.
61
2. Each curve shows an almost linear relationship. This is implies that most of the
mechanical loss comes from bearing losses, because bearing losses vary linearly with
speed while air friction losses have a quadratic or even cubic variation with speed.
Figure 4-12: Mechanical losses with the open-slot machine and the semi-closed-slot machine
4.2.2.4 No-load test
In this test the terminals of PMSG remain open and no current flows in the stator
winding. The rotor is driven by the prime mover and the stator is kept stationary by the
spring balance. The original power flow model is now simplified as illustrated in Figure
4-13. The copper loss in the stator winding and the output power become both zero due
to the absence of armature current.
Figure 4-13: Power flow model for the no-load test
This power flow can be formulated as:
62
Pr = Pin − Ppm − PFe ,s
(4.21)
The modified method using the electromagnetic power can be written as:
Pr = Pem − PFe ,s
(4.22)
In this case electromagnetic power is all converted into iron losses (both stator iron
and rotor iron) and magnet losses in the PMSG [32].
We measure the input power and the electromagnetic power simultaneously to
ensure the identical conditions for the both methods.
Besides we measure the armature current for the copper loss calculation in the DC
motor. We use the mechanical loss measured in Section 4.2.2.3 in the modified Equation
(4.20):
Pr = Pin − PCu ,dc − Pmech − PFe ,s
(4.23)
In both power flow models we have to know the stator iron loss, otherwise only the
total iron loss in the PMSG can be found out. If we treat the losses in the stator iron and
in the rotor as a whole as the total iron loss PFe , PFe will be much easier to measure.
PFe = Pin − PCu ,dc − Pmech
(4.24)
whereis the amount of stator iron loss and rotor loss.
Hence we choose to measure the total iron loss PFe only for the purpose of
measurement simplification. In fact the stator iron loss and the eddy current losses in
rotor have different characteristics. This approach will also be used in on-load
experiments.
The Equation (4.21) is also modified as
PFe = Pem
(4.25)
Now we can measure the quantities on the right side of Equations (4.23) and (4.24)
at different rotational speeds and calculate the corresponding total iron losses in the
PMSG. The speed is chosen ranging from 100rpm to 1500rpm with a step size of 100rpm
for both the original power flow model and the modified power flow model.
All experiments are executed for both the open-slot machine and the
semi-closed-slot machine.
Results
We measure the total iron loss within the PMSG without any load current. We do
not separate the rotor iron loss or stator iron loss from the total iron loss here but leave
the separation in Chapter 5.
First we compare the iron loss obtain by using the original and the modified power
flow model. The compared curves are plotted in Figure 4-14 for the open-slot PMSG as
well as for the semi-closed-slot PMSG. We can see a good agreement in both PMSGs.
63
This implied that the original and the modified power flow model are equivalent to
calculate the total iron loss.
Figure 4-14: Comparison between the open-slot PMSG and the semi-closed-slot PMSG
by using different power flow models
Then we compare the performance of the open-slot PMSG and the semi-closed-slot
PMSG. It is apparently indicated in the figure that the total iron loss in the open-slot
PMSG is much greater no matter which power flow model is applied. Taking the data
obtained by the modified power flow model, the total iron loss in the open-slot PMSG is
nearly twice of the loss in the semi-closed-slot PMSG at 1500rpm. This suggests that
the semi-closed-slot PMSG can save much more power by reducing the total iron loss,
compared with the open-slot PMSG.
We assume in the rotary test that the mechanical loss is consistent with the curves
in Figure 4-12. However those curves are obtained without the electromagnetic reaction
between rotor and stator. In case this reaction is present the mechanical loss must more
or less deviate from the curves in Figure 4-12. Relatively speaking, the modified model
is more reliable. Because it measures the power within the PMSG and it eliminates the
process of measuring the prime mover and mechanical losses which may introduce
measurement errors. We need a model with as low error as possible and therefore we
select the modified model for the on-load experiments afterwards and further for the
validations in Chapter 5.
4.2.2.5 On-load test
In order to find the influence of armature currents together with magnets on iron
losses, we connect the PMSG to a purely resistive load as described in Section 4.2.2.2.
Therefore, generated armature currents flow in the stator winding. We adjust the
64
resistor for different resistance, rotate the PMSG at different speeds, and measure the
total iron loss PFe by using power balance method from the modified power flow models.
This model has advantages over the original model. This test can reveal the combined
effect of fields due to magnet and armature currents.
The modified power flow model for this test turns to be complete as shown in
Figure 4-8. The equation used for calculating the total iron loss as a whole is
PFe = Pem − PCu − Pout
(4.26)
where PCu is the copper loss in the stator winding of PMSG and Pout is the output power
which means the load power in this test.
The copper loss in the PMSG is calculated as
PCu = (I a2 + I b2 + I c2 )Rs
(4.27)
where Rs is the resistance in each phase of the stator winding. Rs at different frequencies
is found by looking up the result of the resistance analysis described in Section 4.2.1.1.
The output power is calculated as the sum of all three-phase loads:
Pout = Ua Ia + UbIb + UcIc
(4.28)
We do not use the load resistance to calculate the copper loss here because
1. The load resistance can be measured only when the PMSG is offline while
voltages can be measured on line.
2. The load resistance is quite sensitive to temperature. Precise measuring is
difficult especially when the resistor is offline, because the temperature of the resistor
decreases fast.
The RMS values of current for the open-slot PMSG are selected as 4A, 7A, 10A,
13A, 16A and 18A. These values cover the low load (4A, 7A, 10A and 13A), rated load
(16A) and overload (18A) of the PMSG. The current values chosen for the
semi-closed-slot PMSG are only 4A, 7A, 10A and 13A, because in this case the higher
current (16A and 18A) leads to an improper values of total iron losses and the modified
power flow model is not available any more. The chosen speeds for both PMSGs are
600rpm, 800rpm, 1100rpm, 1200rpm, 1300rpm, 1400rpm and 1500rpm. Too low speeds
(e.g. 100 rpm, 300rpm) could lead to unacceptable errors in measurement so they are
not used. First we fix the speed and seek for the influence of the current on the iron loss.
Both the open-slot machine and the semi-closed-slot machine are tested. Then we fix
the RMS value of current and look for the relation between the total iron loss and the
speed.
Results
We measure the total iron loss within the PMSG with load current. We do not
separate the rotor or stator iron loss from the total iron loss here but leave the
separation in Chapter 5.
65
First we compare the total iron loss measured by applying different RMS phase
currents at a fixed speed and then compare the total iron loss measured at various
speeds with a fixed RMS phase current. The curves of the first comparison are
illustrated in Figure 4-15 for the open-slot PMSG and the semi-closed-slot PMSG. The
result for no load is also included.
(a) Open-slot PMSG
(b) Semi-closed-slot PMSG
Figure 4-15: Dependence of total iron loss on phase current measured
by using the modified model
66
At the same speed the total iron loss drops when the current rises. This implies that
the value of armature current affects the induction of iron losses. We can say that the
magnetic field due to armature current cannot be negligible but play an important role.
With a rising current, the induction of iron losses is more suppressed. This can be also
regarded as an extension of Lenz’s law since the magnetic field excited by the generated
current opposes the field due to the magnets.
(a) Open-slot PMSG
(b) Semi-closed-slot PMSG
Figure 4-16: Dependence of total iron loss on speed measured
by using the modified model
67
The curves of the second comparison are illustrated in Figure 4-16 for the open-slot
PMSG and the semi-closed-slot PMSG respectively. The total iron loss rises with an
increasing speed. The trend of the curves almost agrees with a linear function but slight
exponential content can be observed.
68
Chapter 5
Validation
5.1 Introduction
Finite element simulations and experimental tests have been completed
independently. It is necessary to validate the simulation results by comparing the
simulation results with the experiment results to conclude if the finite element model is
capable of simulating the actual machine. The compared quantity is the eddy current
loss in metal ring in the static test and the total eddy current loss in magnets and rotor
back-iron in the rotary test.
At first the static test results are validated. The agreement between simulations
and experiment results is shown and discussed. The reasons for the deviations are
presented and analyzed. An experiment of heating distribution is done to observe the
edge effect of magnetic flux in the tested machines. Then the validation of rotary tests
is carried out and the results are analyzed and discussed. In this validation, the iron
losses in stator and rotor measured in the experiments are separated by looking into the
stator iron loss. The stator iron loss is estimated by looking up the core loss as a
function of flux density and corrected by a factor to make the estimation more accurate.
All losses in rotor are regarded as eddy current losses since hysteresis losses in rotor are
neglected. The eddy current loss in the rotor is thus known and used to validate the
finite element simulations done in Chapter 3.
The focus in this chapter is on validating the accuracy of finite element models of
PMSGs from the point of view of eddy current losses. The difference between the open
slots and the semi-closed slots is no longer the emphasis.
5.2 Static Test Validation
5.2.1 Comparisons between Simulation and Experiment Results
The comparison of eddy current losses in all the three metal rings for the machine
with open-slot stator is illustrated in Figure 5-1. The simulations with the copper and
the aluminum rings are taken by using time-harmonic solutions instead of transient
solution used for the non-linear steel ring. The simulation with the steel ring uses
transient solutions. Time-harmonic solutions for copper and aluminum respectively are
nearly the same as transient solutions but cost much less computation time. The same
simulation setting applies to the semi-closed-slot machine whose results are shown in
Figure 5-2.
The eddy current losses calculated by using the analytical model in Section 2.5 are
also plotted in the figures. The analytical model is valid only for linear materials
69
( μr = constant ) and only when no magnets are active and only armature currents
excite the magnetic field. But it can also estimate eddy current losses for the non-linear
steel ring by assuming μr = constant for the steel ring.
From the comparisons between FEM and experiments in these figures, we can
observe large deviations in the cases of copper and aluminum rings. The values of eddy
current losses with respect to frequency are generally much lower in FEM but they
follow a similar trend as in experiment. At low frequencies less than 350Hz, the
agreement is fine and the FEM results are even slightly higher in the open-slot machine.
The two results are no longer close with the frequency going higher. At 1000Hz, the
relative deviations are 37% and 28.5% respectively for the copper and the aluminum
rings with the open-slot stator, and 26.5% and 22.4% with the semi-closed-slot stator.
We consider the experiment result as the reference for all comparisons here.
In the case of steel ring, however, the two curves match very well in the open-slot
machine. The experiment result is always slightly higher. The deviation over all
frequencies is below 5%. In the semi-closed-slot machine, the agreement between FEM
and experiment is not very bad although there is still an apparent difference in between.
The deviation at 1500Hz is 19.8% for example.
(a) Copper ring
(b) Aluminum ring
70
(c) Steel ring
Figure 5-1: Eddy current losses in metal rings with the open-slot stator at various frequencies
The results calculated from the analytical model are between the FEM and the
experiment results at high frequencies and close to the experiment results in the cases of
copper and aluminum rings. For the steel ring, the difference between linearity and
non-linearity looks obvious as the linear analytical model overestimates the eddy
current loss in the non-linear steel ring.
The analytical model appears to have better predictions over the finite element
model. However the difficulty in the analytical model in find a proper Carter’s factor
could lead to inaccuracy of such a model. The Carter’s factor used here is just an
approximation by estimating flux in the air gap. The accuracy of the Carter’s factor
determines the validity of the analytical model.
Hence we can say the capability of the analytical model mainly depends on:
- Carter’s factor
- Linearity
(a) Copper ring
71
(b) Aluminum ring
(c) Steel ring
Figure 5-2: Eddy current losses in metal rings with the semi-closed-slot stator
at various frequencies
The eddy current losses in the copper and the aluminum rings rise fast with
increasing frequency but tend to be flat at higher frequencies. The eddy current losses in
the steel ring do not follow the same pattern but show an almost linear characteristic
over the range of all frequencies. All the analytical, finite element and experiment
results can indicate this phenomenon.
This can be explained physically by means of the equivalent circuit of an electrical
machine. We notice that in the static test the machine acts as an induction motor but
the rotor (metal ring) is stalled. In the IEEE-recommended equivalent circuit shown in
Figure 5-3, the slip is one so there should be current flowing in the rotor. The mutual
reactance Xm increases with frequency (because of X m = 2 π fLm ), less current goes
through it and the current flowing in the rotor I 2' becomes more. This reaction gets more
severe if the frequency goes higher. But it cannot unlimitedly increase, because the
reactance of rotor X2' will also go higher due to the induction in the metal ring to limit
the current flowing in the ring
72
Figure 5-3: IEEE-recommended equivalent circuit of an induction machine
5.2.2 Discussion on Possible Causes of the Deviations
The analytical modeling and FEM results are basically always lower in the
foregoing comparisons. This implies that either the models underestimated the eddy
current loss in the ring, or some errors were introduced in the experimental
measurements or they both contributed. The factors stated below are likely to cause the
deviations:
- Edge effect
- End effect
- Variable stator iron losses
5.2.2.1 Edge effect
In the analytical and the finite element models, only the magnetic flux in parallel
with the 2-D plane is modeled as illustrated in Figure 5-4. The flux perpendicular to the
2-D plane cannot be involved. All the MMF is to build the flux density in parallel with
the 2-D plane.
Stator
Flux lines along the
main magnetic path
Y
Metal ring
Z
X
Figure 5-4: Cross-section view of the machine to observe flux lines along the main magnetic path
In the experimental tests, however, some of the flux density produced by the
armature current (the armature current in parallel with 2-D X-Y plane but not modeled
in FEM) strays away from the magnetic path shown in Figure 5-4. This portion of flux
density circulates along the path on the Y-Z plane indicated in Figure 5-5 which is
perpendicular to the X-Y plane. The stray flux density also links with the metal ring
73
but only at the edge of the ring. The variation of such stray flux density in time could
induce eddy currents at the edge of the ring.
Metal ring
Stray flux lines
Windings
Stator iron core
Y
Z
X
Figure 5-5: Side view of the machine to observe stray flux lines
This is called the edge effect. Eddy current losses due to this effect are not modeled
in the analytical and the finite element models because the models are only in 2-D. The
edge effect results in deviations between analytical or finite element simulations and
laboratory experiments. The effect is considerable especially when the ring is made of
copper or aluminum which has the same permeability of air, since the flux density
cannot be effectively confined in the main magnetic path on the X-Y plane. The steel
ring has a relatively high permeability (e.g. μr ≈ 200 ) so that the flux density is almost
confined in the main magnetic path on the X-Y plane and much less stray flux density
can circulate on the Y-Z plane.
Taking thermal photos with a thermal camera can be used to verify if the edge
effect really works. The stator with the ring is put into a box preventing wrong
measurements from the light outside. The light cannot be eliminated completely but
controlled to be as little as possible. The area where the machine is tested and the
thermal photos with different rings are shown in Figure 5-6. The rainbow color indicates
the temperature in the thermal photos. Red color means hot while blue means cold. The
red zone represents the copper windings which should have been hot due to copper
losses. Some yellow or red spots lie on the middle part of the ring but these are due to
the unexpected light leaking from outside or due to uniform material of the ring.
We can observe a very thin yellow strip at each edge of the ring. Beside the yellow
strips the area turns to green and finally blue. The transition of color means the
temperature at the edges is significantly higher. The heat, resulting in the temperature
74
rise at the edges, is likely to come from the edge effect. The edge effect can be now
considered to induce more eddy current losses at the edges.
Photo area
(a) Area of the thermal photos
(b) Cold state before test
(d) Loading state with aluminum ring
(c) Loading state with copper ring
(e) Loading state with steel ring
Figure 5-6: Thermal photos to investigate the edge effect
The edge effect is not modeled in FEM because the finite element model is only in
2-D. If this effect is required to be simulated, a 3-D model has to be built instead.
75
5.2.2.2 End effect
The end copper windings in parallel with the X-Y plane (Figure 5.4) are not
modeled in the analytical and finite element models but they do exist in the acutual
stators. This portion of winding contributes to deviations of resistance between the
models and the actual machine. If such windings are present, the resistance is higher
due to the extension of length. This means the resistance of copper winding in the
experiments is higher than in the analytical or finite element models.
With the help of the analysis in Section 4.2.1.1, the end effect can be explained as
follows:
At low frequencies, the electromagnetic reaction of the machine is small and the
loss of the machine mainly depends upon the resistance. In the models, the resistance is
lower so that the loss is higher according to
P=
U2
R
(5.1)
whereU is the terminal voltage of the machine, R is the resistance of the machine seen
from the winding terminals.
At higher frequencies, the resistance increases due to skin effects and proximity
effects. The electromagnetic reaction is very fierce and the total loss becomes a complex.
Other factors contribute to the imaginary part of the complex loss while the conductive
loss is the real part and can be calculated with current and resistance as
P = I 2R
(5.2)
The machine in the finite element model has a lower resistance without the end
winding than the machine in the experiments with the end winding. So the eddy current
losses in models are lower. The extent of this end effect depends on how much the end
winding is neglected in the finite element model. According to the specification of the
stators in Table 3-4, the length of the end winding could reach 30% of the length of the
winding along the axial direction. This could results in the deviations shown in Figure
5-1 and 5-2.
5.2.2.3 Variable stator iron losses
In the experiments we assume the stator iron loss is constant at a frequency no
matter which metal ring is coupled with the stator. The assumption allows us to use
Equation (4.1), (4.2) and (4.3) to measure the eddy current loss in the ring.
If the stator iron loss is not fixed at the same frequency, however, Equation (4.7)
has to be used. We cannot measure the difference of the stator iron loss ΔPFe ,s directly
and therefore this factor could lead to additional eddy current losses measured in the
experiments. The eddy current losses by measurement could be higher than the true
values. This can also be a possible explanation for the deviation in Figure 5-1 and 5-2.
The details have been presented in Section 4.2.1.2.
76
5.2.3 Summary
The finite element simulation can to some extent predict the performance on eddy
current losses of the machine when the rotor is replaced with a static metal ring. Several
factors may result in finite element predictions lower than experimental values. One of
the most decisive limitations of FEM is that the model is built in 2-D plane on which
the edge effect and the end effect are neglected. If a 3-D model taking these effects into
consideration, predictions made by finite element simulations can be much more
accurate.
Meanwhile the variation of stator iron losses at the same frequency may introduce
more eddy current losses measured in the metal ring. An approach to measure the
stator iron loss or keep the stator iron loss constant at the same frequency can address
this problem.
The analytical model described in Section 2.5 can also predict eddy current losses in
the rings. The accuracy is acceptable but it depends on the right estimation of the
Carter’s factor and the proper modeling of linearity.
5.3 Rotary Test Validation
5.3.1 Separation of Stator and Rotor Iron Losses
The iron losses measured in Chapter 4 in the rotary tests are the sum of stator and
rotor iron losses. In that chapter, we did not separate them, because it is rather difficult
to do so by measurement. Here in this chapter, a method combined with finite element
simulations can give an estimation of stator iron losses.
Figure 5-7: Iron loss as a function of flux density at different frequencies
Iron losses in ferromagnetic materials depend on the flux density and its frequency.
Eddy current loss per unit volume can be written as
77
pe = Ke f 2Bm2 W/Kg
(5.3)
if the flux density is a sinusoidal function as
B = Bm sin ωt
(5.4)
where Bm is the peak value of flux density, Ke is the eddy current coefficient.
The eddy current coefficient is usually hard to determine as it is not a constant.
This makes the application of Equation (5.3) rather difficult. Manufacturers of
ferromagnetic materials therefore provide the users with curves or look-up tables. The
curves or tables show the iron losses as a function of the peak value and the frequency
of flux density.
The ferromagnetic material used for the stators in this thesis is M-19-29-Ga fully
processed non-oriented silicon steel. Its functions with respect to peak flux density at
different frequencies are plotted in Figure 5-7. The unit for the flux density is Gauss and
for the iron loss is Watt/ Pound. The conversion of these units to SI units follows:
1 Gauss= 0.0001 Tesla
1 Watt/Pound=2.20462 Watt/Kg
Yoke
Tooth cap
Tooth
Figure 5-8: Indication of the regions where the flux density is tested (the straight thick red lines)
The work now is to find the peak values of flux density in the stator iron. Finite
element simulations are used again to solve for the flux density. With the same
geometry and the same parameters of the actual PMSG, the finite element simulation
can provide a relatively precise prediction of flux density in the stator teeth, the tooth
caps and the stator yoke. We run a simulation and check the flux density at the regions
with red lines across the teeth or the yoke, as shown in Figure 5-8. The test is taken
78
when the flux density in the top tooth is at its maximum value. The semi-closed-slot
machine is used here as an example. The same test lines also apply to the open-slot
machine except that we do not have to deal with tooth caps as well as their flux density.
Table 5-1: Peak value of flux density in different regions [T]
Tooth
Tooth cap
Yoke
Open-slot PMSG
1.50
N/A
0.90
Semi-closed-slot PMSG
1.65
1.40
0.95
The value flux density measured along the red lines in the finite simulation is
plotted in Figure 5-9 for the open-slot and the semi-closed-slot PMSGs respectively. We
can get the peak values of flux density in different regions from these figures as
indicated in Table 5-1.
(a) Open-slot PMSG
79
(b) Semi-closed-slot PMSG
Figure 5-9: Flux density tested in tooth, tooth cap and yoke of the semi-closed-slot PMSG
80
The flux density represents the rotational magnetic field and therefore the
frequency is obtained according to the number of pole pairs p and the rotational speed n
of machines. The frequency is calculated as:
f =
np
60
(5.5)
The frequency at different speeds is used in the functions in the form of curves in
Figure 5-7. With the peak values of flux density written in Table 5-1, the iron loss per
unit mass can be found.
The mass is not all mass of the stator but obtained by using the following
procedure:
1. The total mass of the stator is measured.
2. The total stator iron mass is obtained after the mass of copper windings is
deducted.
3. The mass of teeth, tooth caps or yoke is calculated by multiplying the proportion
of its own volume in the total volume with total stator iron mass, as the density of the
stator iron is uniform.
4. The part of stator yoke in which the flux density hardly varies is deducted. The
effective yoke mass contributing to iron losses is approximately 0.6 time of the total
mass of yoke.
The mass of each region is summarized in Table 5-2. Then the iron loss is calculated
in each region with its own mass and iron loss per unit mass. The total stator iron loss
is subsequently obtained by adding the losses in all regions of the stator iron.
Table 5-2: Mass of different regions of stator iron [Kg]
Tooth
Tooth cap
Effective yoke
Open-slot stator
2.55
N/A
0.99
Semi-closed-slot stator
2.02
0.56
0.89
It should be noticed that this stator iron loss is not absolutely accurate but is
estimation. The accuracy is dependent on several factors. One decisive factor is the
deterioration of the iron material during its production process. Some processes such as
punching can lead to higher iron losses in the iron. A correction factor c p is thus
introduced to take the increase of iron losses due to production process into account [33].
According to experiences, the correction factor for iron losses of the open-slot stator is
1.5 and for the semi-closed-slot stator is 1.55. Because the latter stator has tooth caps
which complicate its production and bring about relatively more iron losses than the
former one. The factor is used as:
81
'
'
PFe
,s = cp ⋅ PFe,s0
(5.6)
'
'
where PFe
,s is the stator iron loss before correction and PFe,s0 is after correction. As the
'
peak flux density does not vary very much when current changes, PFe
,s is fixed with all
currents for the open-slot and the semi-closed-slot PMSG respectively. The stator iron
losses at different speeds before and after correction are shown in Appendix C.
As the total iron loss in stator and rotor have already been measured in Chapter 4,
it is very easy now to have the iron losses in rotor by simply subtracting the stator iron
loss from the total iron loss. It can be written as
'
Pr = PFe − PFe
,s
(5.7)
'
where PFe
,s represents the estimated value.
The further separation of eddy current losses in magnets and rotor back-iron is not
an object of this thesis and therefore we keep the eddy current loss in both magnets and
rotor back-iron as a whole for the validation.
5.3.2 Validating Finite Element Models by Experimental Measurements
So far the total eddy current loss in rotor has been found. In Chapter 3 the total
eddy current loss in rotor was also simulated in FEM. The comparison between the two
results is feasible now and we can use the experimentally measured result to validate
the simulation to judge if the finite element model can predict the performance on eddy
current losses of the actual PMSGs.
The simulation results must be modified according to the actual temperature which
considerably influences the remanent flux density of the magnets. The temperatures
measured by a thermal meter (Figure 5-10) are summarized in Table 5-3.
Figure 5-10: Thermal meter to measure the temperature of magnets
Equation (3.1) is used again and we rewrite it here:
Br (T ) = Br ,20°C + αT (T − 20)
(5.8)
82
where Br (T ) is the remanent flux density atT ℃, Br ,20°C is the remanent flux density at
20℃ and αT is the temperature coefficient of NdFeB which is -0.0012T/℃.
Table 5-3: Temperature of magnets in different experiments [℃]
Open-slot PMSG
Semi-closed-slot PMSG
No load
70
70
4A
70
70
7A
75
75
10A
80
75
13A
90
80
16A
95
90
18A
95
95
Now the remanent flux density at each temperature is clear and we redo the same
simulations as in Chapter 3 with such values of remanent flux density.
5.3.2.1 No load
All results at 15 speeds (from 100rpm to 1500rpm) with no armature current are
compared between simulations and experimental measurements. The comparisons are
illustrated in Figure 5-11 for the open-slot and the semi-closed-slot PMSGs respectively.
The experiment results apply the modified power flow model.
We can observe in the open-slot PMSG the eddy current loss in rotor is
overestimated while in the semi-closed-slot PMSG it is underestimated. Both the curves
of FEM simulation and experiment match well in the former machine. There is a little
difference in between but it is not unacceptable (12.3% deviation at 1500rpm if the
experiment regarded as the reference). The deviation may come from the distinction
between the two dimensional distribution of magnetic field in FEM and the three
dimensional distribution in the actual machines.
The agreement for the semi-closed-slot machine is not that good especially when
the speed is high (more than 800rpm). The values of eddy current losses in FEM are
relatively rather small because the tooth caps of stators hamper the variation of flux
density in the magnets and the farther rotor back-iron. The experimental curves tell the
simulation that at higher speed the tooth caps cannot effectively do so as at lower speed,
and that considerable eddy current losses can actually be induced in rotor.
We can draw conclusions here:
1. For the open-slot PMSG running at no load, the finite element simulation can
well predict the amount of eddy current losses in rotor.
83
2. For the semi-closed-slot PMSG running at no load, the finite element simulation
underestimates the amount of eddy current losses in rotor.
(a) Open-slot PMSG
(a) Semi-closed-slot PMSG
Figure 5-11: Comparisons of eddy current losses in rotor between finite element simulation and
experimental measurement at no load
5.3.2.2 On load
The PMSGs are loaded with armature current. The load is purely resistive. The
values of current (RMS) are selected as: 4A, 7A, 10A, 13A, 16A and 18A for the
open-slot PMSG and 4A, 7A and 10A for the semi-closed-slot PMSG. The values of 13A,
16A and 18A are absent for the latter machine, because the modified power flow model
is not valid any more in these cases and the values of eddy current losses in rotor are
negative after calculation. Other factors must have lowered the electromagnetic power
measured by the spring balance.
84
The values of speed are selected as 600rpm, 800rpm, 1100rpm, 1300rpm and
1500rpm for the simulation and 600rpm, 800rpm, 1000rpm, 1100rpm, 1200rpm,
1300rpm, 1400rpm and 1500rpm.
Open-slot PMSG
The comparisons are illustrated in Figure 5-12 for the open-slot PMSG.
For the open-slot PMSG, the eddy current losses in rotor simulated by FEM hardly
vary with current changing. However, the experimental curves increasingly deviate
from the no load curve if the current rises. The accuracy of finite element modeling to
predict eddy current losses in rotor turn to drop as the armature current becomes higher.
Besides the measurement errors, this could be explained as follows:
1. The addition of armature current makes the air gap magnetic flux crowded and
some of the flux is pushed out of the air gap in the form of fringing flux as shown in
Figure 5-13. The fringing flux goes along a longer path and the variation of this portion
of flux induces less eddy current compared with the flux inside the air gap. Besides, the
flux goes through a smaller area and thus the flux density decreases consequently.
Higher armature current results in more fringing flux off the air gap and less the eddy
current loss induced in the rotor. The fringing flux is not modeled in the FEM
simulations on a two dimensional Y-Z plane. Hence all flux is concentrated inside the air
gap and more eddy current is likely to be induced.
2. The current in the end windings not modeled in the finite element simulations
could lead to extra leakage flux, but this portion of leakage flux is all included in the
flux linked with the rotor because the model is two dimensional. The higher linkage flux
in the simulations results in higher eddy current losses in the rotor. The higher current
results in more leakage flux due to end windings and the deviation therefore becomes
larger.
(a) 4A
85
(b) 7A
(c) 10A
(d) 13A
86
(e) 16A
(f) 18A
Figure 5-12: Comparisons of eddy current losses in rotor between finite element simulation and
experimental measurement on load for the open-slot PMSG
3. In the finite element simulations we assumed the phase current was in phase with
the no-load voltage, but actually the phase shift between them depends on the value of
phase current because the synchronous inductance may vary with armature current [34].
The equivalent circuit of a synchronous machine is illustrated in Figure 5-14. If the
synchronous reactance increases due to the variation of current, the phase shift between
E f and I s varies from the phase shift in the case of no load. Hence according to each
armature current, there should be a corresponding phase shift between E f and I s . We
should set the phase shift in the finite element model for each armature current level
instead of keeping it constant in any case. Changing the phase of armature current in
FEM could results in up to 19.8% reduction of eddy current losses in rotor (see
87
Appendix D). However, measuring the phase shift in experiments is also a difficulty.
Some appropriate methods are required.
Rotor
Fringing flux
Y
Stator
Z
X
Figure 5-13: Fringing flux at the edges of air gap
Figure 5-14: Equivalent circuit of a synchronous machine (generator mode)
We can also observe the current dependency of the measured eddy current losses in
rotor as shown in Figure 5-15. The loss at the same speed decreases considerably with
the current rise, which is beyond the expectation in the space harmonics point of view
that the armature current could induce much more eddy current losses. This may be an
effect due to Lenz’s law in two aspects:
1. The armature current cancels out the magnetic field which induces it. This effect
opposes the slotting effect and leads to lower flux density in the rotor. Lower eddy
current losses could therefore be possible.
2. The induced eddy current excites its own magnetic field which opposes the
original field which just produced it. This opposition overcomes the space harmonics
from armature current and thus the total eddy current loss appears to decrease.
Semi-closed-slot PMSG
The comparisons are illustrated in Figure 5-16 for the open-slot PMSG. The same
phenomenon as in the no-load validation can be observed. At relatively high speeds, the
finite element simulation underestimates the eddy current loss in rotor. At low speeds,
the values are too small for validation. In short, the finite element simulation is not
88
capable of predicting accurate eddy current losses in the rotor of the semi-closed-slot
PMSG even though the values are both very small compared with the open-slot PMSG.
The large deviation could be caused by the following reason:
The presence of tooth caps makes a shorter path for flux between adjacent teeth. A
large amount of flux does not go to the magnets and the rotor back-iron but circulates
within the stator iron core. The finite element model overestimates this portion of flux
staying in the stator, but in reality much flux is still linked with the rotor in the plane
perpendicular to the modeled 2-D plane in FEM. Hence less flux reaching the rotor is
modeled by FEM and lower eddy current losses are induced in the rotor in the
simulations as a consequence.
Figure 5-15: Current dependence of eddy current losses in rotor for the open-slot PMSG
The experiment results also indict the eddy current loss decreases when the
armature current goes higher, but the difference is not very obvious. Therefore the
curves of the dependence on current are omitted here.
(a) 4A
89
(b) 7A
(c) 10A
Figure 5-16: Comparisons of eddy current losses in rotor between finite element simulation and
experimental measurement on load for the semi-closed-slot PMSG
So far the validation has been completed for both the static and the rotary tests. The
phenomena observed by comparing finite element simulations and experimental
measurements have been interpreted with several possible reasons.
90
Chapter 6
Conclusions and Future Work
6.1 Conclusions
Simulations based on finite element methods were considered to predict eddy
current losses in the rotor for permanent magnet synchronous machines with
concentrated windings. Experimental measurements were executed to validate the
accuracy of the prediction from the simulations. We now draw conclusions as follows:
Drawn from the static test
- The predictions for the machine with non-linear steel ring are good and acceptable
but for the linear copper or aluminum ring have relatively large deviations.
- The predictions generally underestimate the eddy current loss in these rings. This
is caused by
a. The inability of 2-D models dealing with the edge effect and the end effect.
b. The variation of stator iron loss in the experiments brings about inaccurate
calculations.
- The analytical model based on the vector magnetic potential could be used to
predict for the machines with linear metal rings ( μr = constant ). They gave similar
results as in the finite element simulations. The analytical model can be used to replace
the finite element model if the tested object is of linear materials.
Drawn from the rotary test
- The modified power flow model using the electromagnetic power can be applied to
calculate the total iron loss.
- Separation of the eddy current loss in rotor from the total loss is feasible since the
stator iron loss can be estimated.
- The predictions of finite element simulations for the open-slot PMSG are
acceptable at no load and with small armature current. The accuracy gets increasingly
worse when the armature current goes higher. The simulations overestimate the eddy
current losses in rotor and this may mainly come from the incomprehensive 2-D
modeling.
- The predictions of finite element simulations for the semi-closed-slot PMSG are
not good both at no load and with armature current. They show negligible eddy current
losses in rotor but actually from the experiment some eddy current losses can still be
observed.
91
- As observed both in simulations and experiments, the semi-closed-slot PMSG has
the less total iron loss and the less eddy current loss in rotor compared with the
open-slot PMSG. This suggests that the semi-closed-slot PMSG is preferred over the
open-slot PMSG if eddy current losses are a big issue in machine designs.
6.2 Future Work
As concluded above, a few pieces of future work are need and could be very
contributive.
Rotary test with demagnetized magnets
We already investigated the eddy current losses due to the magnetic field excited
both by magnets and by armature currents in the rotary test. It is necessary to study
the influence of only armature currents on the eddy current losses when the machine is
rotating. In this case, the magnets are replaced by their demagnetized copies with the
same conductivity and permeability. The armature current is obtained by injecting
three-phase sinusoidal currents from an AC power supply.
This research on eddy current losses will be more complete with this test.
Improvement on the 2-D model
We can see many causes for the deviations given by simulations come from the 2-D
model itself. The 2-D model therefore requires improvements. If the results from 2-D
models are always not good enough, building 3-D models is necessary.
Separation of stator iron losses from the total losses
As directly estimating the eddy current loss in rotor is rather difficult, estimating
the stator iron loss is the best method to separate these two losses from the total. The
combination of using finite element simulations and manufacturer’s data sheets is a
feasible approach but may not be the most accurate one. Thus a better way to measure
the stator iron loss is an important challenge. This would be also helpful to deal with
the variation of stator iron losses in the static test.
Measuring phase shift between EMF and armature current in rotary test
We encountered a problem on the simulating the phase shift between EMF and
armature current in FEM for the rotary test. A proper approach is required to measure
this phase shift. The finite element model needs the actual phase of armature current to
do accurate simulations.
Improving the Analytical model
Even if finite element simulations can do the work of prediction, analytical models
are still useful in some occasions. The analytic model used in the static test is valid
when the magnetic field is excited only by armature currents. A more general model is
necessary when the magnetic field is produced by both currents and magnets.
92
Appendix A
Stator Drawings
93
94
Appendix B
Solving a Problem with COMSOL
There is a problem with COMSOL Multiphysics 3.5a. The average value with time
of the z component of total current density in each magnet is not zero but a DC
component exists. This is not true in reality because the oscillation of the total current
density in z direction should be alternating around zero. This is caused by the symmetry
used in the finite element model to allow finer meshing and to reduce computation.
We introduce compensation for the z component of total current density by
intentionally adding extra induced current into the magnet. The processing is
illustrated in the following figures.
(a) Creating variables with the values of original induced current density
(b) Creating variables with negative values of the original induced current density
95
(c) Adding the obtained variables as external current density in each magnet
The processing consists of three steps:
1. Creating variables with the values of original induced current density.
2. Creating variables with negative values of the original induced current density
3. Adding the obtained variables as external current density in each magnet
Now the average value with time of the z component of total current density in each
magnet becomes zero and this problem has been settled.
96
Appendix C
Estimated Stator Iron Losses in Rotary Tests
Table C-1: Estimated stator iron losses in the open-slot PMSG
Speed [rpm]
Stator iron loss [W]
100
3.36
200
5.52
300
9.14
400
12.75
500
17.09
600
21.41
700
26.12
800
31.52
900
36.93
1000
42.33
1100
48.62
1200
54.90
1300
61.20
1400
68.99
1500
76.89
Figure C-1: Curve of estimated stator iron losses in the open-slot PMSG
97
Table C-2: Estimated stator iron losses in the semi-closed-slot PMSG
Speed [rpm]
Stator iron loss [W]
100
2.62
200
6.74
300
11.28
400
16.24
500
22.17
600
27.95
700
34.08
800
40.94
900
47.80
1000
54.65
1100
62.71
1200
70.79
1300
78.83
1400
88.82
1500
99.23
Figure C-2: Curve of estimated stator iron losses in the semi-closed-slot PMSG
98
Appendix D
Phase Variation of Armature Current in FEM
In the finite element modeling, the phase of armature current affects the eddy
current loss in the rotor. We choose several values of the phase and do finite element
simulations as in the on-load rotary test. The phase sequence is indicated in Figure D-1.
ω
C
B
A
Figure D-1: Phase sequence and rotation direction for on-load simulation
We set the phase angles of phase A as 0, ± 20°, ± 40°, ± 80°, ± 100° and ± 120°.
The other two phases follow the manner of three-phase symmetry. The armature
current is set to 16A as this is the rated current value. The speed is 1500rpm. Only the
open-slot PMSG is simulated.
Figure D-2: Dependence of eddy current losses on the variation of current phase
99
The eddy current loss in rotor as a function of phase angle is plotted in Figure D-2.
We can observe that the eddy current loss in rotor reaches its maximum at 20° and
becomes lower towards both sides. The difference of eddy current loss is up to 33.8%
and the difference between ± 120° reaches 19.8%.
100
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