Download Construction, Modeling and Evaluation of a Low Loss Motor

Construction, Modeling and Evaluation of a Low
Loss Motor/Generator for Flywheels
Abstract
The application of flywheels (FWs) for energy storage requires generators with low losses,
converting electrical energy to rotational energy and vice versa. In order to keep losses low
and achieve a high power density, axial-flux generators with air-wound stator are surveyed.
A small-scale axial-flux permanent-magnet (AFPM) generator with ironless stator has been
designed and constructed. Magnetic field, voltage and frequency have been measured and
analyzed. The different losses, occurring in the experimental set-up have been investigated. A
model of the magnetic field of the rotor configuration has been created in COMSOL
MultiphysicsTM and the simulation results have been compared to measurements of the flux
density in the air gap.
Content
1
2
3
4
5
6
7
Introduction ........................................................................................................................ 4
Flywheels in General.......................................................................................................... 5
2.1
Historical Applications of Flywheels......................................................................... 5
2.2
Principles of Flywheel Technology............................................................................ 6
2.3
Improvements of Technology .................................................................................... 8
2.4
Applications for Flywheel Energy Storages (FES) .................................................... 8
2.4.1
FES for Power Supply........................................................................................ 9
2.3.2
FES for Vehicles .............................................................................................. 10
Theory .............................................................................................................................. 12
3.1
Magnetic Fields ........................................................................................................ 12
3.2
Field Calculations Using Numerical Methods ......................................................... 18
3.3
Principle of Synchronous Generators (SG).............................................................. 18
3.4
PM Machines............................................................................................................ 25
3.5
Axial-Flux Machines................................................................................................ 25
3.6
Axial-Flux PM Machines with Air-Wound Stator................................................... 27
3.7
Losses in an AFPM Machine with Air-Gap Winding.............................................. 27
The Experimental Set-Up................................................................................................. 29
4.1
Basic Properties of the Experimental FW Generator ............................................... 29
4.2
Simulation with FEM Simulation Tool.................................................................... 30
4.3
The Rotors ................................................................................................................ 33
4.4
The Air-Gap Winding Stator.................................................................................... 34
4.5
The Complete Small-Scale FW Generator............................................................... 36
Simulation of the Magnetic Field of the Rotor ................................................................ 38
5.1
The Modeling Software COMSOL Multiphysics™ ................................................ 38
5.2
Geometry Modeling ................................................................................................. 38
5.3
Physics Modeling ..................................................................................................... 39
5.4
Simulation Results.................................................................................................... 40
Measurements and Results ............................................................................................... 43
6.1
Measurement of the Magnetic Field of the Rotor .................................................... 43
6.2
Measurement of the Stator Resistance per Phase..................................................... 45
6.3
Measurement of Phase Voltage at Open-Circuit Operation..................................... 45
6.4
Connection of a Load ............................................................................................... 45
6.5
Voltage Harmonics................................................................................................... 46
6.6
Determination of Losses.......................................................................................... 48
6.6.1
Standby Losses................................................................................................. 48
6.6.2
Total Losses...................................................................................................... 52
Summary of Results and Discussion................................................................................ 57
7.1
Properties of the Experimental Generator................................................................ 57
7.1.1
Length of Cable in the Winding ....................................................................... 57
7.1.2
Mechanical Eigenfrequency ............................................................................. 57
7.1.3
Voltage Harmonics........................................................................................... 57
7.1.4
Generator Efficiency ........................................................................................ 58
7.1.5
Stored Kinetic Energy ...................................................................................... 58
7.2
Comparison of Measurement and Simulation Results ............................................. 58
7.2.1
Magnetic Flux Density ..................................................................................... 58
7.2.2
Voltage Output ................................................................................................. 61
7.3
Discussion of the Measurement and Simulation Results ......................................... 61
8
Conclusion and Suggestions for Future Work ................................................................. 63
8.1
Conclusion................................................................................................................ 63
8.2
Suggestions for Future Work ................................................................................... 63
9
Acknowledgements .......................................................................................................... 65
References ................................................................................................................................ 66
Appendix .................................................................................................................................. 68
A Abbreviations ............................................................................................................... 68
B Symbols........................................................................................................................ 69
C Properties of the Sintered NdFeB PM.......................................................................... 71
D Calculation of Polar Mass Moment of Inertia (m-file) ................................................ 72
E Cross Section of the Experimental Generator .............................................................. 74
F Pictures of the Experimental FW Generator ................................................................ 75
G Calculation of Losses of the Experimental Set-Up (m-file)......................................... 76
1 Introduction
4
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1
Introduction
The storage of energy can help to increase the efficiency and quality of electrical power
supply and traction units e.g. as part of a hybrid propulsion system for vehicles. Due to
improvements in material, magnetic bearings and power electronics, flywheels become a
reasonable alternative for energy storage. A motor/generator with both low no-load losses and
low load losses is needed, as electrical energy has to be converted to mechanical energy and
vice versa.
Excitation losses are avoidable by using permanent magnets (PMs) instead of electromagnetic
poles. Mechanical losses can be kept low by using magnetic bearings and operating the FW in
a vacuum vessel. Under low-pressure conditions axial-flux machines have an advantage over
radial-flux machines because of their easier cooling arrangements. In electrical machines
without ferromagnetic material in the stator, no core losses occur. Thus, coreless stator AFPM
generators seem quite suitable for energy storage with flywheels [1].
Connected to the shaft of a flywheel a motor/generator rotates even during standby operation
when no electrical energy is absorbed or supplied. The rotating magnetic field of the PMs
induces eddy-currents in the stator winding resulting in Joule heating, and consequently losses
[2].
During load operation, eddy currents in the PMs and the rotor discs can appear due to the
harmonic content of the air-gap flux-density distribution. It is important to estimate the eddycurrent losses in the rotor, since the heat generated in the rotor can demagnetize the PMs and
cooling is not easy under low-pressure conditions [3], [4], [5].
A motor/generator, which produces high voltage and low current, would be advantageous
concerning copper losses. Instead of rectangular conductors with large areas insulated circular
conductors (e.g. cables) have to be applied for the stator winding of such a machine [1].
For this thesis a small-scale flywheel motor/generator using cables for the air-gap winding has
been designed and constructed. The magnetic field of the rotor configuration has been
simulated by modeling the rotor configuration in COMSOL Multiphysics™.
Moreover, measurement results from the experimental set-up have been analyzed. In
particular it has been examined whether it was possible to determine eddy-current losses in
the magnets, the rotor discs and the stator cables by measurements on the experimental set-up.
2 Flywheels in General
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2
Flywheels in General
A flywheel (FW) is a rotating disc used to store energy in the form of kinetic energy. Modern
flywheels are used for storing electric energy and require a motor/generator for energy
conversion.
2.1 Historical Applications of Flywheels
The flywheel is considered to be one of the earliest inventions of human kind. Simple
flywheels have been used as bases for potter´s wheels and grindstones. With the increased
utilization of machines, more advanced flywheels have been adopted to achieve smooth
operation of machines. For example, they smoothed out the power of steam engines, which
were vital for the industrial revolution [1], [6].
In the 1950s the company Oerlikon developed a “gyrobus”. A flywheel of 1,500 kg generated
electricity for the electrical motor in the bus. The bus could go approximately 6 km before the
flywheel had to be turned up to 3,000 rpm again, taking about 3 minutes. Several busses were
in use in Yverdon (Switzerland), Belgium and in the Democratic Republic Congo [7], [8].
a)
b)
c)
Figure 2.1 Gyrobus [7], [8]
a) A gyrobus getting recharged
b) Schematic configuration of the electrical propulsion system when FW getting recharged
c) Schematic configuration of the electrical propulsion system with FW as energy source
As early as the late 1800s, a torpedo with flywheel propulsion was developed and built. A
flywheel was accelerated up to 12,000 rpm. The power was transmitted to turn propellers,
giving the weapon a speed of 30 knots [9], [10].
2 Flywheels in General
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2.2 Principles of Flywheel Technology
A flywheel stores kinetic energy of rotation, where the stored energy depends on the polar
mass moment of inertia Im and the rotational speed according to:
Ek =
where
1
⋅ Im ⋅ω 2 ,
2
Ek:
Im :
ω:
(2.1)
kinetic energy stored
polar mass moment of inertia
angular speed.
Connected to an electric motor/generator, a flywheel can be accelerated by the motor when it
is supplied with electric energy. Inversely, the generator can provide electrical energy by
slowing down the flywheel. In the first case, the flywheel unit absorbs electrical energy, while
it delivers electrical energy in the second case. Thus, such a unit can be utilized as storage for
electric energy.
Equation (2.1) shows that the energy stored in a flywheel is proportional to the polar mass
moment of inertia (i.e. the mass, if equal shapes for different materials are assumed) and to
the square of the angular speed. Thus, stored energy can be increased either by choosing a
material with higher mass density that leads to high polar mass moment of inertia, or by
speeding up the flywheel. Obviously, the latter is the more promising option. Fibre
composites are lighter than steel but allow higher rim speeds. Therefore, a good trade-off is
gained by using fibre composite instead of steel for the flywheel disc.
The simplest shape of a flywheel is that of a solid cylinder (e.g. a steel rotor). A common
shape is a rim attached to a shaft. This shape is often applied if composite materials are used.
Equations (2.2) and (2.3) specify how the polar mass moments of inertia of bodies of these
shapes are calculated. It depends on the geometry and the mass.
Im =
where
r:
m:
a:
ρ m:
Im =
where
1 2
1
⋅ r ⋅ m = ⋅ r 4 ⋅π ⋅ a ⋅ ρm ,
2
2
(
radius of solid cylinder
mass of the cylinder
length of the cylinder
density of the cylinder material
)
1
⋅ m ⋅ ro2 + ri 2 ,
4
ro:
r i:
(2.2)
(2.3)
outer radius of a hollow circular cylinder
inner radius of a hollow circular cylinder.
The rotational speed is limited by the tensile strength σ t , i.e. the stress developed within the
wheel due to inertial loads. At a given speed, lighter materials develop lower inertial loads.
2 Flywheels in General
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Since composite materials have a low density and high tensile strength, they enable higher rim
speeds for flywheels than steel rotors allow.
The maximum energy density of a flywheel with respect to the mass is given by:
em = K
where
σt
,
ρm
em:
K:
σt :
ρm :
(2.4)
kinetic energy per unit mass
shape factor
tensile strength
mass density.
The shape factor K that links the maximum energy density to the ratio between maximum
tensile strength and the density of the mass depends on the geometry of the flywheel rotor [1],
[11]. The values for K vary between 0 and 1. For a flat unpierced disc for example K amounts
to 0.606. Table 2.1 shows some examples for other shapes [6].
Table 2.1
Flywheel Geometry
Constant stress disc
Cross section
Shape Factor K
1.000
Modified constant stress disc
0.931
Truncated conical disc
0.806
Flat unpierced disc
0.606
Thin rim
0.500
Rim with web
0.400
Single filament bar
0.333
Flat pierced disc
0.305
So far, energy densities of about 286 kJ/kg have been achieved. It is probable that even much
higher energy densities will be possible due to further increase of graphite fiber strength [12].
Especially the high power density compared to chemical batteries makes flywheels useful for
applications where high peak power is demanded. Batteries can store a lot of energy, but they
cannot be discharged within a very short time without being destroyed. Furthermore, the
efficiency decreases considerably for high charge and discharge powers. In contrast,
flywheels can easily deliver the whole energy stored within a few seconds, without lowered
efficiency or loss of functionality. The number of load cycles is not limited as it is for
batteries. A flywheel can also be constructed in different sizes [1], [13].
2 Flywheels in General
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Obviously flywheels have to be operated with high revolution speed, in order to achieve high
energy and power density. By designing the FW in a way that results in a low eigenfrequency,
it can be operated above the corresponding revolution speed. This means that the FW is never
slowed down to standstill. But by slowing down a flywheel to e.g. half of the maximum
revolution speed, 75% of the total energy is available.
One of the disadvantages is the relatively short storage time. While batteries can store energy
for years, windage losses, bearing and other losses cause the flywheel to be slowed down,
even during standby time. Windage losses can be decreased by operating the flywheel under
low-pressure conditions in a vacuum vessel.
2.3
Improvements of Technology
The in- and output power is limited by the motor/generator and the power electronics
connecting the flywheel unit to a surrounding power system. Due to recent progresses in highpower insulated-gate bipolar transistors (IGBTs) and field-effect transistors (FETs), flywheels
can operate at high power with power electronics units comparable in size to the flywheel
itself. Thus, flywheels can be used for applications were high peak powers are necessary [1].
As the energy stored in a rotating mass is proportional to the polar mass moment of inertia
and to the square of the rotational speed, it is advantageous to have a rotor of strong materials
that allow high rotational speed. Instead of steel, modern flywheels consist of carbon fibres.
The decrease of stored energy due to lower mass density is overcompensated by much higher
rotational speed. Rim speeds up to 1.2 km/s are possible [12].
Magnetic bearings that started to appear in the 1980s enable to reduce the no-load losses
considerably. Since magnetic bearing control has been improved, they can replace roller
bearings and further more be used to monitor the dynamic behavior of the flywheel. In case of
occurrence of abnormal conditions it can be shut down safely [1], [14].
2.4
Applications for Flywheel Energy Storages (FES)
There are two main application areas for flywheels: Grid-connecting applications, e.g.
improving power quality, or uninterruptible power supply (UPS) and power supply on several
vehicles.
Figure 2.2 shows the basic layout of a FES with a motor/generator on the same shaft as the
flywheel and enclosed by a vacuum vessel.
2 Flywheels in General
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Bearing
Containment
Flywheel rotor
Motor/generator rotor
Motor/generator stator
Vacuum or very low
pressure
Bearing
Figure 2.2
2.4.1
Basic layout of a FES [1]
FES for Power Supply
Storage of electric energy by flywheels competes with chemical batteries and superconducting
magnetic energy storage (SMES). In SMES, energy is stored in the magnetic field
surrounding a coil of superconducting wire. The current can circulate in the coil without
losses as long as the temperature of the superconducting material is cooled to cryogenetic
temperatures. As the cooling consumes energy, long time storage is inconvenient. The
practical time to hold energy is in the magnitude of days.
For flywheels the practical storage time is even in the range of hours, although FWs rotating
for many days have been developed [15]. But the temperature range is much less limited, and
the relative size for equivalent power and energy is smaller compared to SMES. Currently, the
prices per kilowatt supplied by FWs or SMES are considerably higher than for chemical
batteries. On the other hand, further technical development will most likely lower the prices
for FWs and SMES, while costs for chemical batteries will probably not become significantly
lower than they are at present.
Moreover, chemical batteries have disadvantages of possible environmental hazards, lifetime
limited to a few years and their state-of-charge being difficult to determine. Taking into
consideration that the lifetime of FWs is expected to be more than 20 years, the life-cycle cost
of some FW applications are comparable to those of lead-acid batteries.
Since FWs can store energy for periods of several hours and can handle high power levels,
they are suitable for the use in electrical grids. Due to their short response time, they can be
applied to balance the grid frequency. Increasing distributed production of energy by utilizing
renewable energy sources raises the demand for short time storages that can compensate for
variations in energy production. FWs can contribute to generally improving the power quality
in public power supplies.
The currently most technically mature commercial application of FWs is the supply of highly
reliable electric power for seconds or minutes. For many companies, a short power breakdown
can result in huge economical losses, as their production suffers. In some cases, whole
production lines might be destroyed, when they are suddenly stopped.
One example is the FW energy storage in the combined heat and power station that supplies a
semiconductor fabrication facility of Advanced Micro Devices (AMD) in Dresden. While the
2 Flywheels in General
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overall power rating of this plant is 30 MW, the FW subsystem can supply or absorb 5 MW
for 5 seconds. Thus, it can store up to about 7 kWh. The 5-second storage interval is sufficient
for the plant to switch between the utility grid and local generators as power sources.
Most power line disturbances last for less than one second. Consequently, uninterruptible
power supply (UPS) is possible by using FWs, as they can deliver energy fast and long
enough to compensate for short voltage drops or in order to carry the whole load for about 15
seconds, while a standby generator is brought on line [1], [14].
2.3.2
FES for Vehicles
In hybrid electric vehicles, a combustion engine constantly supplies the average power needed
to drive the vehicle. Additionally required energy for acceleration and climbing hills is
supplied by an electric motor, which takes energy from a temporary store. Thus, the
combustion engine can operate at an almost constant optimum speed. Thereby fuel
consumption is reduced as well as air and noise pollution. Moreover, the lifetime of the
combustion engine is extended.
The electric drive motor can be used for regenerative braking. By acting as a generator it can
turn the kinetic energy of a vehicle into electric energy charging the store. Consequently,
waste heat, as produced by friction brakes, can be decreased. Instead, the energy can be used
for subsequent accelerations.
So far, nickel metal-hydride batteries are the most common energy storage device in vehicles.
But as the energy only has to be stored for relatively short time intervals, they could be
replaced by flywheels. Flywheels have longer lifetimes and a higher power and energy
density than batteries. While it is difficult to optimize the design of chemical batteries if
frequent shallow discharges are mixed with very deep discharges, flywheels operate
independently of the depth of discharge.
As the current costs for flywheels are higher than the costs for chemical batteries, they are
considered preferable in larger vehicles like busses or trains. Hybrid gas turbine/electrical
trains are an option to extend high-speed operation to reasonable prices. Trains that are solely
diesel-powered are too heavy for speeds higher than about 180 km/h. In contrast, electric
trains are quite convenient for operation at high-speed. However, a big drawback is the high
costs for electrification. In a hybrid train, the advantages of both propulsion alternatives could
be combined. A gas turbine driving a high-speed generator supplies the average power, while
a flywheel battery balances the difference between produced electrical energy and the one that
is needed for propulsion at a time [14].
Also, in diesel-electric trains for lower speeds, FWs can be applied as energy storage in order
to reduce fuel consumption and the environmental impact. Since peak demands on the diesel
engine are decreased, it can be sized smaller and thus weight is saved. Further advantages are
that the combustion engine can be stopped, although auxiliary devices have to be supplied and
emission-free operation on short sections of line e.g. tunnels and stations are possible. Figure
2.3 shows the principle of diesel-electric propulsion.
For the Alstom Light Innovative Regional Express (LIREX), a version with flywheels as
energy storage has been constructed. The Research Center of Deutsche BAHN AG carried out
simulations of the operation of such a version. According to these simulations, about 11% of
the energy that a LIREX without flywheels needs can be saved. The shorter the distance
between stops, the higher is the saving effect [16], [17].
2 Flywheels in General
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If an electrified railroad line exists, the voltage drop at the substations can be reduced by
installing FWs in the substations. Additionally, a part of the braking energy can be recovered
[14].
Figure 2.3
Principle of propulsion system of a diesel-electric train with energy storage
Furthermore, FES can be used for certain military applications. Future combat vehicles will
need a lot of electrical power for propulsion, suspension, communication, weapons and
defensive systems. One possibility for a hybrid system is an engine providing the power
demanded in average combined with fuel cells, supercapacitors and flywheels to supply
continous as well as pulsed power. Flywheels designed for applications where power has to be
provided within µs would charge a bank of supercapacitors, which then would supply the
high-speed systems. The peak power would be several MW. Even higher peak power is
required for an electromagnetic aircraft launcher, where 5 to 10 GW are necessary [14].
Moreover, the performance of space vehicles like satellites, planetary rovers or space stations
could be improved by equipping them with FES. In earth orbit the sun is the prime energy
source, i.e. stored energy is necessary while the space vehicle is in darkness. The initial design
for the energy storage on the International Space Station (ISS) uses batteries. It would be
advantageous to replace the chemical batteries by flywheels, since they have about the same
volume and mass as the batteries but provide energy twice as long and have much longer
lifetime. Additionally, the state-of-charge can easily be determined by measuring the
rotational velocity. The National Aeronautics and Space Administration (NASA) is
developing flywheel sets that can store 15 MJ and deliver a peak power of more than 4 kW. In
total, 48 flywheels would be needed to replace all batteries. By doing so, about 100 million
Euros could be saved, according to an estimation made by NASA. Taking into account the
motor, generator and flywheel losses, the net efficiency (chare-discharge) amounts to 93.7
percent, while the current batteries have an efficiency of about 80 percent [14], [18].
In vehicles like cars, trains and in the ISS, flywheel batteries are controlled as pairs and
situated to rotate in opposite direction. If their rotational speed is always changed equally, no
net torque will be produced. But in some cases, a net torque might be wanted e.g. in order to
supply attitude control. The aim of one program supported by the U.S. Air Force is to develop
flywheel-based systems that can store energy and provide torque to a spacecraft [1], [14].
3 Theory
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3
Theory
3.1
Magnetic Fields
Electrical charge in motion causes a magnetic field. This moving electrical charge can be a
current flowing in a conductor or orbital motions and spins of electrons that are called
Ampèrian currents. Such Ampèrian currents exist in permanent-magnet material leading to a
magnetization within the material and a magnetic field outside.
Ampère´s law relates the magnetic field intensity H to its source the current flowing in a
conductor. According to this law the line integral of the magnetic field intensity over any
closed loop C spanning the enclosed area A is proportional to the current flow penetrating
through this area. If the loop C encloses one conductor (N = 1) with the current I as shown in
Figure 3.1 a) this current is equal to the line integral of H along the curve C:
∫H ⋅ds = N ⋅ I ,
(3.1)
C
where
a)
H:
I:
N:
magnetic field intensity
electric current
number of conductors.
b)
Figure 3.1
Magnetic field density along a loop enclosing current flowing in conductors
a) One conductor carrying current I
b) Two conductors carrying current I each
In case of two conductors (N = 2) with identical current I (Figure 3.1 b), two times this current
is equal to the line integral. These both conductors could be part of the same coil. The sum of
the currents in all conductors penetrating A is called ampere turns or magnetomotive force
(mmf) Θ. In case of a number of N conductors carrying the same current I, the magnetomotive
force can be calculated according to (3.2).
3 Theory
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Θ = N⋅I ,
Θ:
where
N:
(3.2)
ampere turns
number of conductors
and Ampère´s law can be written as following:
∫H ⋅ds = Θ .
(3.3)
C
If a magnetic field intensity H is generated in a medium by a current I, the response of the
medium is its magnetic induction B, also called magnetic flux density. The relation between
magnetic flux density B and the magnetic field intensity H is a property of the medium, called
permeability µ.
The product of the average magnetic flux density B and the perpendicular area that is
penetrated by the flux density is called the magnetic flux Φ (see Figure 3.2). In many media
the flux density B is a linear function of the magnetic intensity H. In free space, B is the
product of H and the magnetic permeability of free space µ0:
B = μ0 ⋅ H ,
(3.4)
where
B:
µ0:
magnetic flux density
permeability of free space.
Figure 3.2
Magnetic flux density B penetrating a certain area
For other media than vacuum the general equation (3.5) has to be used:
B = μ ⋅ H = μr ⋅ μ0 ⋅ H ,
where
µ:
µr:
µ0:
(3.5)
permeability
relative permeability
permeability of free space.
While in paramagnetic and diamagnetic materials B is a linear function of H, the correlation
in ferromagnetic materials is a more complicated function.
3 Theory
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The effect that a magnetic material has on the flux density when a field passes through it is
represented by the magnetization M. While diamagnets make the flux density smaller, paraand ferromagnets make it larger. How the magnetic flux density is changed by the presence of
material is indicated by the relative permeability µr of the material. The magnetization M has
the same unit as the magnetic field intensity H:
Φ
B
,
=
μ0 ⋅ A μ0
M =
where
M:
Φ:
(3.6)
magnetization
magnetic flux.
If no external electric currents are present to generate an external magnetic field, the flux
density in a magnetic material is simply the magnetization times the permeability.
Since magnetization M and magnetic field intensity H contribute to the magnetic flux density
in a similar way, their contributions can be summed, when both magnetization and magnetic
field are present:
B = μ 0 (H + M ) .
(3.7)
The product of magnetization and µ0 is also called magnetic polarization JM. By rearranging
(3.5), it is obvious that the permeability is defined as:
μ=
B
.
H
(3.8)
Correspondingly, the susceptibility χ can be defined as:
χ=
M
.
H
(3.9)
Moreover, the relative permeability is defined as:
μr =
μ
.
μ0
(3.10)
Since the relative permeability is closely related to the susceptibility, the following equation is
always true:
μ r = χ + 1.
(3.11)
The permeability of free space is μ 0 = 4 ⋅ π ⋅ 10 −7 H/m and µr of free space is 1.
While the susceptibility of diamagnetic materials, e.g. copper, is negative, the susceptibility of
paramagnetic materials, e.g. aluminium, is small and positive. Typical values are χ =10-3 –
10-5. Ferromagnetic materials have a susceptibility being much greater than one, typically
3 Theory
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χ = 50 to 10 000. Examples are iron, cobalt, nickel and some rare earth metals as well as their
alloys.
The most important property of the ferromagnetic materials is their high relative permeability,
which is not constant as a function of the magnetic field intensity. As the relative permeability
changes with varying magnetic intensity, it is necessary to measure the flux density B as a
function of H over a continuous range to obtain a hysteresis loop.
Figure 3.3
Typical hysteresis loop of a ferromagnetic material [20]
Once exposed to a magnetic field, ferromagnetic materials retain their magnetization even
when the field is removed. Paramagnetic materials are also magnetized in an applied magnetic
field. However, they cannot maintain it after the field is removed.
Instead of plotting B for various H, plots of M as a function of H or JM as a function of H can
be used. Figure 3.3 shows the magnetic flux density and the magnetic polarization JM over the
magnetic field intensity H. The difference of both loops is the linear function B = μ 0 ⋅ H . In
the second and the forth quadrant, the loop of B depending on H is almost linear. The virgin
curve of the B(H)-loop describes how the flux density increases due to applied field, if the
ferromagnetic material has not been magnetized in its initial state.
In case of indefinitely increased magnetic field intensity, the magnetization reaches saturation
at a value M0. This value depends only on the material. The magnetization will not increase
beyond M0, even though the magnetic field intensity is increased even more.
When the external magnetic field is reduced to zero after magnetizing a magnetic material, a
certain magnetic flux remains. The corresponding values of the flux density are called the
remanent flux density BR and the remanent magnetization MR respectively. The algebraic sign
of the remanent flux density depends on the algebraic sign of the magnetic field intensity,
before it has been reduced to zero. By applying a reverse magnetic field intensity the flux
density can be reduced further. The field intensity that is needed for reducing the flux density
to zero, is known as coercive field intensity HCB, the one needed for reducing the
magnetization (and thus the polarization) to zero is called the intrinsic coercive field intensity
HCJ.
B
3 Theory
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The area enclosed by the hysteresis loop represents the energy expended during one cycle of
the hysteresis loop. This energy is also refered to as hysteresis loss. It depends on the
coercivity.
When a ferromagnetic material is heated it becomes paramagnetic at a certain transition
temperature, called the Curie temperature. As the permeability suddenly drops at this
temperature, both coercivity and remanence become zero.
Ferromagnetic materials can be classified on the basis of their coercivity. Magnetic materials
with high coercivity are designated hard magnetic materials and the other soft magnetic
materials.
An interesting class of magnetic materials are permanent magnets, finding applications in
electrical motors and generators. Important properties of these materials are represented by
the so-called demagnetization curve. This is the portion of the hysteresis curve in the second
quadrant, in which the magnetization is reduced from saturation. The properties depend on the
metallurgical treatment and processing of the material and the chemical composition.
Besides the coercivity and the remanence, the maximum energy product BHmax is given by the
demagnetizing curve. It is obtained by finding the maximum value of the product |BH| in the
demagnetizing quadrant of the hysteresis loop. BHmax represents the magnetic energy stored in
a permanent-magnet material.
However, in most applications, the stability of the permanent magnets is most important.
Therefore, the magnets have to be operated sufficiently far below the Curie temperature, as
the spontaneous magnetization decreases rapidly with temperature above about 75% of the
Curie temperature [19], [20].
Common permanent-magnet materials are Sm2Co17, SmCo5, AlNiCo and Nd2Fe14B [19], [20].
If a conductor carries a current I while a magnetic flux density B is present, a force is exerted
on this conductor. The force per meter on the conductor caused by B is given by:
F = I ⋅l × B ,
F:
l:
where
(3.12)
force on a current-carrying conductor caused by magnetic flux density
unit vector pointing in the direction of the current.
In order to visualize the strength and direction of a magnetic field, so-called lines of magnetic
induction are used. They are a geometrical abstraction and always form a closed path. Around
a linear current-carrying conductor these lines form loops, being coaxial with the conductor
and follow the right-hand rule (see Figure 3.1). In a solenoid the lines are uniform within the
solenoid and form a closed return path outside the solenoid. Around a bare magnet, the lines
of flux density are similar to those around a solenoid. Both act as magnetic dipoles. Since
lines of flux density always form a closed path, the amount of flux entering through any
closed surface is equal to the amount of flux leaving it. Thus, the divergence of B is always
zero as described by Gauss´ law:
∫B⋅d A = 0,
(3.13)
A
where
A:
closed area, where flux density penetrates.
3 Theory
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While a current flowing in a conductor causes a magnetic field, a change of magnetic flux
linking an electric circuit causes the induction of an electromotive force (emf). This
phenomenon is called electromagnetic induction. According to Faraday´s law, the voltage
induced in an electrical circuit is proportional to the rate of change of magnetic flux linking
the circuit. Another important law, namely the Lenz´s law states that the induced voltage is in
a direction, which opposes the flux change producing it. Equation (3.14) describes how the
voltage induced in a coil depends on the change of magnetic flux and the change of magnetic
flux density respectively.
U = −N ⋅
where
dΦ
dB
,
= −N ⋅ A ⋅
dt
dt
U:
N:
Φ:
A:
(3.14)
voltage induced in a coil
number of turns of a coil
magnetic flux passing through a coil
area of a coil.
Maxwell formulated four differential equations describing electromagnetic fields on a
macroscopic level. Written in integral form, these equations are quite similar to the laws
mentioned above. Table 3.1 shows the equations in differential and integral form and
associates them to one of the laws that described the electromagnetic field earlier.
Table 3.1
Maxwell´s equations
Differential form
∇× H = J +
∂D
∂t
∂B
∂t
Integral form
d
∫ H ⋅ d l = ∫ J ⋅ d A + dt ∫ D ⋅ d A
C
A
A
d
Name of law
Ampère´s law
(plus Maxwell´s
extension)
∫ E ⋅ d l = − dt ∫ B ⋅ d A
Faraday´s law
∇⋅B = 0
∫B⋅d A = 0
Gauss´ law
∇⋅D = ρ
∫ D ⋅ d A = ∫ ρ ⋅ dV
Gauss´ law
∇× E = −
C
A
A
A
V
Since the Maxwell´s equations apply whether the fields are steady or time dependent, their
solution provides a general result. As a special case, the first Maxwell equation reduces to the
steady-state Ampère´s law, if the displacement current term ∂D ∂t is small due to low
frequency (< 1014 Hz). That means that when examining electrical generators, ∇ × H is
determined by the current density J.
Another special case arises if magnetic materials are present. The properties of these materials
can alter the penetration of magnetic fields into the material considerably. When a timedependent field penetrates into an electrically conducting material, eddy currents are induced
due to Maxwell´s second equation (alternatively Faraday´s law of induction). This causes
alteration of the field amplitude with depth into the material. The attenuation of the exciting
field increases with increasing frequency of the field, resulting in a penetration depth, which
decreases with increasing frequency. Often the name skin depth is used for the penetration
depth [19]:
3 Theory
18
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1
δ skin ≈
π ⋅ μr ⋅ μ0 ⋅σ ⋅ f
δskin:
where
σ:
3.2
,
(3.15)
skin depth
electrical conductivity.
Field Calculations Using Numerical Methods
Numerical techniques have to be applied to determine the electric field in complicated
configurations. In most cases, Maxwell´s equations have to be solved for a finite region of
space, either two- or three-dimensional. Such a finite region of space is called the spatial
domain. For solving Maxwell´s equations, an appropriate set of boundary conditions is
necessary. Often the magnetic field in the air gap of an electrical machine is to be calculated.
Since there are no field sources in the gap the following equations hold:
∇× H = 0
∇⋅B = 0.
If field sources occur in the region of interest, the source distribution must be known and
included in the calculation. Among others, the finite-difference, the boundary-element method
and the finite-element method can be used. In the latter method, the spatial domain is divided
into triangle-shaped elements (also other polygonal shapes are possible) and the field values
are computed on the nodes of each element. As the size of the elements can be varied over the
region of interest, more elements can be included in regions where the field gradient is large
[19].
3.3
Principle of Synchronous Generators (SG)
An ordinary three-phase synchronous generator consists of a stator with three-phase current
winding (also called armature) and a rotor with a DC field winding that forms a number of
poles. This field winding is fed by means of slip rings. The stator winding connected to a
three-phase power system excites a rotary current field in the air gap between stator and rotor.
Since all SG have an even number of poles, the number of poles divided by two is called the
number of pole pairs. Rotor and stator have the same number of pole pairs. Equation (3.16)
describes how the rotational speed depends on the electrical frequency and the number of pole
pairs.
n=
where
f
,
p
(3.16)
f:
n:
p:
electrical frequency
revolution speed
number of pole pairs.
3 Theory
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If the rotor is driven mechanically, the rotor poles induce a three-phase voltage system in the
stator winding. When a load is connected current flows in the stator exciting a magnetic field
rotating synchronously with the rotor.
In case of motor-operation, the three-phase current system in the stator, which excites a
rotating field in the air gap, is supplied by the grid. The rotating field attracts the magnetic
field of the rotor. Consequently, the rotor rotates with exactly the same speed as the magnetic
air-gap field.
a)
b)
Figure 3.4
Synchronous machine with one phase of the stator winding, salient pole rotor and
magnetic field excited by the rotor winding [21], [22]
a) Number of pole pairs p = 1
b) Number of pole pairs p = 2
Figure 3.4 shows the principal construction of a synchronous generator with salient pole rotor,
i.e. the rotor poles are formed by a concentrated DC field winding. Alternatively, the field
winding can be distributed in many rotor slots. Such a machine with round rotor has a
constant air gap.
In either case, the stator winding is laid in slots. The magnetic field excited by current flowing
in this winding is not exactly sinusoidal, as the winding is not equally distributed but
concentrated in the slots. Besides the fundamental wave, the resulting air gap field contains
harmonics. There are two common types of stator windings, the lap winding and the wave
winding (see Figure 3.5).
a)
b)
Figure 3.5
Winding schemes for lap and wave winding for a SG with four poles [23]
a) Winding scheme for a lap winding in a stator with 12 slots (3 phases are shown)
b) Winding scheme for a wave winding in a stator with 12 slots (1 phase is shown)
3 Theory
20
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a)
b)
Figure 3.6
SG with one slot per pole and phase [23]
a) SG with 12 slots in the stator and a rotor with four salient poles
b) Voltage phasor diagram for SG with p = 2
As described in (3.16), the electric frequency depends on the revolution speed n as well as on
the number of pole pairs p. Assuming the same revolution speed, the electric frequency is
twice as high in a SG with p = 2 as in a SG with p = 1. For a 2 pole-pair generator, four poles
pass a certain stator slot while the rotor performs one turn, the induced voltage comprises two
periods of the fundamental wave. While the mechanical angle between the slots is only 30°,
the electrical angle between the voltages in two adjacent slots is 60°.
An arrow always pointing on the positive maximum of the fundamental wave of the voltage
and initially pointing towards slot 1, points towards slot 7 when the rotor has rotated half a
turn. But in slot 1, there is already a positive maximum again, i.e. a whole fundamental wave
has been induced in the conductor lying in this slot. Thus, another arrow can be introduced
that is now directed to slot 1, primarily pointing towards slot 7. In the slots 1 and 7, another
fundamental wave is induced, until the rotor has finished one whole turn. Of course the same
applies for all the other slots as well. This is illustrated by the voltage phasor diagram shown
in Figure 3.6 b).
a)
b)
c)
Figure 3.7
SG with 2 slots per pole and phase [23]
a) Winding scheme for a lap winding with q = 2 slots per pole and phase
b) SG with 12 stator slots and one pole pair
c) Voltage phasor diagram for a SG with p = 1 and q = 2
3 Theory
21
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Instead of using one slot per pole and phase (like in Figure 3.5 a)), two or more slots can be
used per pole and phase. Applying several slots per pole and phase, results in a more
distributed winding. Consequently, the magnetic field becomes more sinusoidal, i.e. its
content of harmonics becomes reduced. A winding scheme for a lap winding with q = 2 is
shown in Figure 3.7 a). Although the associated SG has only two poles instead of 4, as the one
shown in Figure 3.5, the stator has 12 slots, since in total 6 slots per pole are needed.
The total number of slots in the stator can be calculated by the following equation:
Q = 2 ⋅ p ⋅ q ⋅ me ,
where
Q:
q:
me:
(3.17)
total number of stator slots
number of slots per pole and phase
number of phases.
As shown in Figure 3.7 c), the voltage phasor diagram looks different for a machine with
distributed winding. The distances between the tops of the arrows are smaller corresponding
to the smaller differences in voltage of two adjacent slots. In Figure 3.8 the arrows connecting
the arrowheads of voltage phasors for q = 1 and q = 3 are compared.
Figure 3.8
Voltage phasors [23]
The electrical angle between two adjacent slots
α=
2 ⋅π
⋅p
Q
(3.18)
corresponds to the angle α between the voltage phasors in case of q = 3 in Figure 3.8. The
length of phasor AB can be calculated by
⎛α ⎞
AB = 2 ⋅ R ⋅ sin ⎜ ⎟ .
⎝2⎠
(3.19)
While the arithmetic sum of AB , BC and CD can be calculated by q times AB , the
geometric sum is
⎛ α⎞
AD = 2 ⋅ R ⋅ sin ⎜ q ⋅ ⎟ .
⎝ 2⎠
(3.20)
3 Theory
22
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By dividing the geometric sum by the arithmetic sum, the so-called distribution factor kd is
obtained:
⎛
⎛ π ⎞
π ⎞
Q
⎛ α⎞
⋅ ⋅ p ⎟⎟
⎟⎟
sin⎜⎜
sin ⎜ q ⋅ ⎟ sin⎜⎜
me ⋅ 2 ⋅ p Q ⎠
2
⋅
m
2⎠
e
⎝
⎝
⎝
⎠ .
=
=
kd =
⎛ π
⎞
⎛π
⎞
⎛α ⎞
q ⋅ sin⎜ ⎟
⎟⎟
q ⋅ sin⎜⎜ ⋅ p ⎟⎟
q ⋅ sin⎜⎜
⎝2⎠
2
⋅
⋅
m
q
⎝Q ⎠
e
⎝
⎠
(3.21)
The fundamental wave of the induced voltage is reduced by this factor but more importantly,
the harmonics are reduced to greater extent according to (3.22). As harmonics in principle are
disturbing in SG, they should be as low as possible.
k dν
where
⎛
sin⎜⎜ν
⎝
=
⎛
q ⋅ sin⎜⎜ν
⎝
ν:
q:
me:
⋅
⎞
⎟
2 ⋅ me ⎟⎠
π
⎞
⎟
⋅
2 ⋅ me ⋅ q ⎟⎠
π
,
(3.22)
number of harmonic
number of slots per slot and phase
number of phases.
Another option to reduce harmonics is to design the winding with a coil pitch w smaller than
the pole pitch τp. While the latter describes the distance between coil groups of the same
phase, the coil pitch (or coil span) describes the distance between conductors of the same coil,
like shown in Figure 3.9 a).
a)
b)
Figure 3.9
2/3-pitched winding [23]
a) Illustration of difference between coil pitch and pole pitch
b) Influence of pitch on the third harmonic
Having a coil span shorter than the pole pitch means that a coil encloses a smaller magnetic
flux compared to a full-pitch winding. In a single-layer winding, the coil pitch is always
identical to the pole pitch, since the length of a pole is determined by the length of the
corresponding coils. To enable a shortening of the coil pitch, at least two-layer winding is
required (see Figure 3.10).
3 Theory
23
_________________________________________________________________________________________________________________
a)
b)
c)
Figure 3.10
Single and two-layer winding distribution [20], [24]
a) Single-layer winding (q = 2) with one coil in each slot
b) Two-layer winding (q = 2) with two coils in each slot
c) Two-layer winding (q = 3) with two coils in each slot (one phase shown)
Instead of laying one coil in each slot, a second layer of coils with an identical winding
scheme can be added. This second layer can be shifted as illustrated in Figure 3.10 b). For the
full-pitch winding shown in Figure 3.10 a), the distance between conductor “A” in slot 1 and
the closest conductor “A´” (belonging to the same coil) amounts to 6 slot steps. In Figure 3.10
b), the distance between the conductor “A” (upper layer in slot 1) to the closest conductor
“A´”(lower layer in slot 6) is only 5 slot steps. Thus, the coil pitch of the winding in Figure
3.10 b) is only 5/6 of the coil pitch of the full-pitch winding. But the pole pitch remains the
same. For a SG with two poles, the pole pitch is always half the inner circumference of the
stator, i.e. it comprises half the slots.
As the ratio between coil pitch and pole pitch in Figure 3.10 b) is 5/6, this winding is called
5/6-pitched winding.
If the upper layer was shifted by two slot steps instead of one, it would be a 2/3 pitch. Figure
3.9 b) shows the effect of a 2/3 pitch winding on the 3rd harmonic. With full-pitch winding,
e.g. two positive half-waves and one negative half-wave of 3rd harmonic of the magnetic flux
are linked with the coil that consists of conductor “a” and “a-“. Since the sum is not zero,
voltage with three times the frequency of the fundamental wave is induced. In contrast the
sum of flux (ν = 3) linked with a coil of a 2/3-pitched winding is zero. Thus, no voltage is
induced due to the third harmonic of the flux.
3 Theory
24
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In general, the effect of pitching of coils is described by the pitch factor kpν according to:
⎛ w π⎞
k pν = sin ⎜ν ⋅ ⋅ ⎟ ,
⎜ τ 2⎟
p
⎝
⎠
where
w:
(3.23)
coil span
pole pitch.
τp:
Since the distribution factor and the pitch factor both describe the reduction of induced
voltage for the fundamental wave and each harmonic, they are combined to the winding factor
kw:
k wν = k pν ⋅ k dν
⎛
sin ⎜⎜ν
⎛ w π⎞
⎝
= sin ⎜ν ⋅ ⋅ ⎟ ⋅
⎜ τ 2⎟
p
⎠ q ⋅ sin ⎛⎜ν
⎝
⎜
⎝
⋅
⎞
⎟
2 ⋅ me ⎟⎠
π
⎞
⎟
⋅
2 ⋅ me ⋅ q ⎟⎠
π
.
(3.24)
Applying Faraday´s law, the voltage induced in one phase of the stator winding by the
magnetic field in the air gap can be calculated according to:
Up =
ω ⋅ N S ⋅ kw ⋅ Φ
2
NS:
l:
where
= 2 ⋅π ⋅ f ⋅ N S ⋅ kw ⋅
2
π
^
⋅ l ⋅τ p ⋅ B p ,
(3.25)
number of coil turns per phase
length of the machine
^
Bp :
amplitude of the fundamental wave of the flux density in the air gap.
The number of coil turns per phase for a single-layer winding is given by (3.26) and for a twolayer winding it is given by (3.27).
where
N=
p ⋅ q ⋅ Nc
a
(3.26)
N=
2 ⋅ p ⋅ q ⋅ Nc
,
a
(3.27)
Nc:
a:
number of turns per coil
number of coils connected parallel.
The equivalent circuit for a synchronous machine with round rotor is shown in Figure 3.11. RS
represents the resistance of the stator winding. US is the voltage between the terminal of one
phase and the neutral point and Up is the induced electromotive force due to the fundamental
air-gap flux linkage. The stator main reactance Xh and the stator leakage flux reactance Xs can
be combined to the synchronous reactance Xd that describes the effect of the total stator
magnetic field [20], [23], [24].
3 Theory
25
_________________________________________________________________________________________________________________
Figure 3.11
3.4
Equivalent circuit of a synchronous machine with round rotor
PM Machines
Instead of a field winding, permanent magnets can be applied to generate a magnetic field by
the rotor. As the rotor does not need to be supplied by any exciting current, no slip rings are
necessary. Consequently, there are neither friction losses due to slip rings nor excitation losses
in the rotor that would occur even at no-load operation. The relative permeability of the
permanent-magnet material is very similar to the one of air, i.e. µr_PM ≅ 1. Therefore,
concerning the behavior of the magnetic flux, the air gap of a PM machine with surface
mounted magnets can be considered to be constant, like in a round rotor machine, although its
geometry looks more like the one of a salient pole machine.
Consequently, the same equivalent circuit may be applied as for a round rotor machine [20].
3.5
Axial-Flux Machines
Radial-flux machines, as introduced above, have rotors that rotate within (or in some cases
outside) the stator. In contrast, the rotor of an axial-flux machine rotates in a plane, parallel to
the stator. In the simplest case, a disc-shaped stator and a rotor disc with PMs are used (see
Figure 3.12 a)). In such a single-rotor – single-stator design a strong force of attraction occurs,
attracting the rotor towards the stator. Strong bearings would be required to keep the air gap
constant [25].
By adding a second stator on the other side of the rotor, the forces in axial direction become
balanced (Figure 3.12 b)). Alternatively, a second rotor can be applied.
In case of a rotor-stator-rotor structure, the stator has two working surfaces. Consequently, the
active conductor length is the sum of the two radial portions that face the magnetic poles of
the rotors [26]. Since axial-flux machines have two working surfaces, higher power output
can be obtained, compared to radial-flux machines.
3 Theory
26
_________________________________________________________________________________________________________________
a)
b)
c)
Figure 3.12
Axial-flux machines [3]
a) single-rotor – single-stator configuration
b) two-stators – one rotor configuration
c) two-rotors – one stator configuration
The rotor field is provided by permanent magnets, which either are mounted on the surface of
the rotor disc or embedded inside the rotor [27]. The necessary cooling arrangements for an
axial-flux machine are simpler than the one for a radial-flux machine, what is of special
advantage, when operating under low-pressure conditions [1]. Furthermore, axial-flux
machines have a planar adjustable air gap. This can be used for widening the air gap e.g.
during the no-load operation of a FW, resulting in reduced losses, while waiting for discharge
of energy.
As an axial-flux machine has a flat shape, the shaft can be shorter than that of a radial-flux
machine. At a high-speed rotation, there is advantage of lower shaft vibrations due to the short
length of the shaft. In case of occurring eccentricity, aligning force is generated, reducing
radial load [28].
A draw back is the high rotation speed at the outer circumference, since common PM
materials have a low tensile strength [1]. For some applications, a reasonable tradeoff can be
obtained by applying multi-stage design as shown in Figure 3.13.
a)
b)
Figure 3.13
Multi-stage AFPM machine [29]
a) Principal configuration of a multi-stage AFPM machine with three stators
b) Path of the main flux in a multi-stage AFPM machine
3 Theory
27
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3.6
Axial-Flux PM Machines with Air-Wound Stator
Because of the varying magnetic field, hysteresis and eddy-current losses occur in the stator
iron. Instead of a slotted stator, an air-gap winding can be applied in order to eliminate the
iron losses. For instance, an ironless (or coreless) stator can consist of coils that are held
together and in position by a composite material of epoxy resin and hardener.
According to [30], the fundamental per phase equivalent circuit of a coreless AFPM machine
looks like the one shown in Figure 3.14. In this circuit, RS is the stator resistance, LS is the
stator inductance and Em represents the induced electromotive force due to the fundamental
air-gap PM flux linkage. Ua and Ia are the fundamental instantaneous phase voltage and
current respectively. The shunt resistance Re represents the eddy-current loss resistance of the
stator.
Figure 3.14
3.7
Per-phase equivalent circuit of an AFPM machine
Losses in an AFPM Machine with Air-Gap Winding
While excitation and iron losses are eliminated in an AFPM machine with air-gap winding,
there are still some losses both in the rotor and in the stator. Dominating are the resistive
copper losses in the stator winding. Further losses are windage losses, bearing losses and
losses due to eddy currents.
Eddy currents are induced in conducting material when it is subjected to a varying magnetic
field. Such currents result in reduction of magnetic flux and power loss. The eddy currents
generate Joule heating in the material and extract the necessary energy from the magnetic
field [31], [32].
In AFPM a relatively large part of the copper losses is generated in the end windings. Thus, it
is advantageous to reduce the length of the end windings by applying a pitched winding [4].
The resistive copper losses in the stator depend only on the current and the resistance of the
winding conductors:
Pcopper _ resistive = I S2 ⋅ RS ,
where
Pcopper_resistive: resistive copper losses in the stator winding
phase current of the stator
IS:
stator phase resistance.
RS:
(3.28)
3 Theory
28
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The rotation of the PMs mounted on the rotor causes an alternating field in the stator
conductors. According to (3.29) [33], [34], the eddy-current losses for round conductors
depend on the square of the frequency. Consequently, serious losses due to eddy currents
appear in the stator when operating with high revolution speed and using a high number of
pole pairs since the flux density varies with high frequency.
Peddy ∝
where
2
l ⋅ d 4 ⋅ ω 2 ⋅ B pk
ρ
Peddy:
l:
d:
ρ:
ω:
B pk :
,
(3.29)
eddy-current losses
length of radial portion of the conductor of the stator winding
diameter of the conductor strands in the stator winding
resistivity of the stator winding
electrical angular speed
peak value of the flux density.
High revolution speed is required in order to obtain a high energy density and high voltage.
Instead of limiting the revolution speed, the eddy-current losses in the stator can be effectively
reduced by applying individual conductors with small area, since they depend on the 4th
power of the diameter.
Usually the fundamental air-gap field rotates with the rotor. However, the winding
distribution causes an incremental rotation of the magnetic field. Thus, the waveform of the
flux density contains harmonics, which circulate faster than the rotor. The waveform of the
flux density in the air gap of a coreless AFPM machine is often close to a trapezoidal
waveform with a noticeable third and fifth harmonic content [2]. Since NdFeB magnets are
highly conductive, eddy currents are induced and increase the temperature in the magnets.
Also a solid rotor disc might experience eddy-current losses as magnetic steel is conductive.
For high-speed machines the rotor losses due to eddy currents may become a major part of the
total losses. An increased temperature caused by eddy-current losses in the rotor can
demagnetize the PM material and weaken the adhesive, which attaches the magnets on the
rotor disc.
In order to minimize the high-frequency related eddy currents, laminated rotor discs and
segmented magnets can be used. Furthermore, these eddy currents can be reduced by
increasing the magnet thickness and the air gap [2], [3], [4], [5].
4 The Experimental Set-Up
29
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4
The Experimental Set-Up
A small-scale generator had to be constructed, satisfying the main requirements for a
motor/generator for flywheel energy storage: high power output and low losses. Losses
appearing during no-load operation reduce the storage time or rather the energy available, at a
certain time after charging the FES. Therefore, the design of the constructed generator should
aim at minimizing standby-losses. Moreover, the losses of this set-up had to be determined by
the means of measurements.
A partially constructed generator model existed with some quadratic PM of dimension 20 mm
x 20 mm x 10 mm. Instead of completing this one, a new and somewhat larger set-up was
constructed. However, the existing model gave suggestions concerning the arrangement of the
set-up. The shaft and two wooden columns with the same height as the shaft have been used
from the old construction.
Figure 4.1
4.1
Pictures of the old set-up
Basic Properties of the Experimental FW Generator
An AFPM machine is applied in order to meet the demand of high power density. A tworotors – single-stator configuration has been chosen, i.e. one stator is located between two
rotor discs with permanent magnets. In order to eliminate hysteresis losses in the stator iron,
an ironless stator is used. This means the conductors cannot be laid in slots, properly fixing
their position. In such a configuration the magnetic flux passes from one pole on a rotor disc
across the air gap with the air-wound stator and through a magnet on the opposite disc. It then
it passes along the ferromagnetic rotor disc to the next magnet and returns through the air gap,
crossing the stator winding again (see Figure 4.2).
Figure 4.2
Simplified path for the main flux in an AFPM machine with air-gap winding
4 The Experimental Set-Up
30
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Further losses that have to be considered are resistive copper losses, i.e. losses because of the
ohmic resistance of the stator winding. Since these losses depend on the current and the
resistance of the winding, according to (3.28), there are two possibilities to reduce the losses.
Either a conductor with large cross section area is chosen, thus decreasing the resistance, or a
cable that enables high voltage, resulting in lower current, is used. The eddy-current losses in
the stator winding, which also occur during no-load operation, depend on the 4th power of the
diameter of a round conductor (3.29). In addition, increasing the amount of conducting
material in the stator increases the weight and the costs of the machine. Therefore, conductors
with large diameter are not a reasonable option.
The magnetic field generated by the PMs rotates with the rotor discs and the magnets
mounted on their surface. However, there will be flux changing in the rotors and the PMs,
because the air-gap field, to which the stator also contributes, contains harmonics. Since these
harmonics revolve with higher speed than the rotors, magnetic flux changes relative to the
rotor discs with the PMs. Consequently, eddy currents are induced in the ferromagnetic
material of the rotor discs and the PMs, in case of PMs having high conductivity.
Especially at high revolution speeds, theses losses become important, as they depend on the
square of the frequency. Reducing the content of harmonics in the air-gap field, results in
lower eddy-current losses in the rotor discs and the PMs.
4.2
Simulation with FEM Simulation Tool
In order to get a rough idea about the order of magnitude of electric characteristics that can be
achieved on the small-scale FW generator, a FEM simulation tool was used. This program is
programmed for the simulation of radial-flux PM machines. The geometry of the rotor and the
stator can be modelled. For example different PMs (shape, material) and pole shoes (different
shapes or no at all) can be chosen for the rotor. The design of the stator slots and the stator
cables can be modelled (e.g. area, outer diameter, number of strands). Different pitches and
numbers of slots per pole and phase can be chosen. Moreover, the dimensions of rotor, stator
and air gap have to be specified. Values for revolution speed, frequency, voltage and apparent
power have to be determined.
The program executes the calculation only for the smallest section that is symmetrically
repeated in the geometry of the machine. It provides values for e.g. the current and length of
the machine and shows the magnetic field under no-load conditions as well as under load
conditions.
Since it is not possible to choose the geometric configuration of an axial-flux machine, a
radial-flux machine, as similar as possible to the designated design of the AFPM machine was
simulated.
4 The Experimental Set-Up
31
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a)
b)
Figure 4.3
Radial- and axial-flux machines with different dimensions but same pole pitch
a) Machines with 4 poles of pole pitch τp and diameter d
b) Machines with 8 poles of the same pole pitch τp and diameter d´ = 2·d
Figure 4.3 a) shows the simplified flux density in the air gap for one pole of a radial-flux and
an axial-flux machine respectively. Figure 4.3 b) shows both machines with twice the
diameter, against which the width of one pole is the same, resulting in twice the amount of
poles. The air-gap fields of these two enlarged machines look more similar than the air-gap
fields of the smaller machines.
The larger both machines are dimensioned, the more similar become the circumstances in the
air gap. This has been utilized for the simulation in the FEM simulation tool, by modeling a
machine with ten times the dimension of the experimental set-up, as an approach of its
properties. Of course, several electrical characteristics are scaled as well. This has to be taken
into account, when evaluating the simulation results. Table 4.1 compares the dimensions and
characteristics of the design used in the experimental set-up to the one used in the simulation.
As the number of pole pairs was not part of the input data, it is not mentioned in the table.
However, it depends on the given frequency and revolution speed. For the experimental setup, the number of pole pairs p was 14, resulting in p = 140 for the in FEM model.
4 The Experimental Set-Up
32
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Table 4.1:
Input data for the FEM simulation tool model compared with the dimensions and
characteristics of the experimental set-up
Size
Rotor diameter d (m)
Electrical frequency f (Hz)
Revolution speed n (1/min)
Power P (kW)
Armature voltage U (V)
Air-gap length (mm)
Number of slots per pole and phase q
Outer diameter of cable (mm)
Cross section area of the cable (mm2)
Number of strands
Pitch (w/τp)
BR of the PM (T)
Tangential breadth of the PM lPM (m)
Hight of the PM hPM (m)
B
Experimental set-up
0.3
467.7
2,000
1
42
10
2
3.35
2.5
50
5.5/6
1.3
0.02
0.01
Model for the FEM
simulation tool
3
467.7
200
10
420
10
2
3.35
2.5
50
5/6
1.3
0.02
0.01
The length of a PM machine corresponds to the length of the magnets. In case of a radial-flux
machine, this length is the same as the axial length of the rotor. In case of an axial-flux
machine this length corresponds to the radial length of the magnets.
For a given power and voltage, the FEM simulation tool calculates the necessary length of the
machine and the occurring current density. Thus, different combinations of voltage and power
have been tried in order to find out which radial length the PMs should have. For the input
data as presented in table 4.1 a length of 40 mm was obtained. Thus, e.g. two of the available
magnets could be used in a row.
A power of 1 kW and an armature voltage of 42 V, resulting in a current density of 5.5
A/mm2 seemed to be reasonable characteristics. Therefore, this set of dimensions and
characteristics has been taken as the basis for the design of the small-scale FW generator.
According to the FEM simulation, the value for the air-gap flux density amounts to 0.47 T. As
only one rotor was considered, the flux density in the air gap of the experimental set-up was
expected to be about the same, although the air gap would be larger than 10 mm.
4 The Experimental Set-Up
33
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Figure 4.4
4.3
Plot of the field lines and the flux density (in T) of one pole, simulated with the FEM
simulation tool (rotor with PM on the left side and stator slots with cables on the right
side)
The Rotors
Both rotors consist of identical discs of ferromagnetic steel. Their outer diameter amounts to
31.5 cm and they are 10 mm thick. Along their outer circumference, they have a rim, 3 mm
high and 7.5 mm broad, in order to counteract the centrifugal force that accelerates the PMs
outwards.
a)
b)
c)
Figure 4.5
Rotor disc and PMs
a) Rotor disc, how it would look with rectangular magnets
b) Rotor disc with conical magnets and almost constant gap between them
a) Shape of one single PM with dimensioning
Placing two of the quadratic PMs in a row would result in a relatively large gap between the
magnets at the outside of the machine, while the gap would become quite narrow at the inside.
A gap between the magnets - much smaller than the air gap - means that a considerable part of
the magnetic flux would not pass across the air gap but pass to an adjacent magnet. Since the
air-wound stator has to fit between the rotors, the air gap will be quite large. Hence the gap
between the magnets must not be smaller than inevitable. This is only possible if conical
4 The Experimental Set-Up
34
_________________________________________________________________________________________________________________
magnets are used. Therefore, custom-made conical NdFeB-magnets with an average width of
20 mm and a height of 10 mm have been applied. On each rotor disc, 28 (two times the
number of pole pairs) PMs have been sticked with a two-component adhesive.
As PM material, sintered Neodymium-Iron-Boron (NdFeB) with a remanent flux density BR =
1.3 T is used. Although the Curie temperature of this material is higher than 300 °C, the
maximum working temperature amounts to only 80 °C. The value of energy product |BH|max
is 320 kJ/m3. Further properties of the used PM are given in appendix C.
As illustrated in Figure 4.2, approximately half of the flux passing a magnet, continues
through the disc by turning to the left and half of it continues in opposite direction. The
thickness of the rotor discs amounts to half the average width of the magnets. Thus half the
cross section area is available for half the flux, resulting in a nearly constant flux density.
B
4.4
The Air-Gap Winding Stator
Since an ironless stator was required, a material with high electrical resistivity was needed as
a base on which the cables of the winding could be laid. Moreover, it should be quite stiff in
order to avoid deflection, which could result in demolition of the winding in case of being
touched by the fast spinning lower rotor disc. As bakelite meets these two demands fairly
well, a 3 mm thick bakelite disc was chosen as base.
Further requirements on the stator were arrangements to reduce harmonics in the air-gap field
that means a distributed and pitched winding was necessary. Thus, the winding had to be a
two-layer winding with at least two slots per pole and phase. While usually coils incorporated
in resin (see Figure 4.6) are used for the ironless stator of an AFPM machine, cables with
polyolefine insulation have been applied as stator conductors for the experimental FW
generator. The eddy-current losses in the winding can be kept low by using a cable with
several thin strands.
a)
b)
Figure 4.6
Air-gap winding [35]
a) Stator coils incorporated in resin
b) Trial winding with cables
The number of poles had been chosen in a way so that the formation of cables would be closepacked with hardly any space between them if the number of slots per pole and phase is 2.
4 The Experimental Set-Up
35
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A higher value of q is not possible, while maintaining a reasonable number of poles. The pole
pitch at the middle of the radial length of the magnets is somewhat less than 30 mm. At the
inner edge of the magnets it is only 24.7 mm. Thus, 4.1 mm per cable are available per cable,
if 6 cables per pole are used. The applied cable has an outer diameter of 3.35 mm and consists
of fifty strands. In order to fix them on the bakelite disc, a glue gun and hot-melt adhesive was
used.
Any deviation from radial and horizontal alignment of the cables in the area between the
magnets would influence the content of harmonics. By means of a CNC-miller, 168 very
shallow notches have been milled into the bakelite disc. This should ease straight and radial
alignment of the cables of the lower layer.
a)
b)
c)
Figure 4.7
Winding scheme of the air-gap winding
a) Winding scheme of one layer
b) Cross section of the winding
c) Winding scheme of both layers
In contrast to lap windings, wave windings enable to wind the first layer completely before
starting to wind the second layer. This is why the air-gap winding has been designed as a
wave winding. Figure 4.7 shows the winding scheme with two cables per pole and phase.
4 The Experimental Set-Up
36
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Since the upper winding has been laid in the flute between the cables of the lower layer, it is
shifted by half a slot step compared to the lower layer. This results in a pitch of 5.5/6 = 11/12,
as illustrated in Figure 4.7 b). To obtain an air-gap winding, being as thin as possible, the
cables had to be pressed fairly strong against the base, alternatively against the lower layer, in
order to avoid having too much adhesive between the cables. In addition the cables had to be
laid in their positions properly in order to prevent them becoming bended in the area situated
between the magnets. Another challenge was the end-winding part inside the magnets, which
is quite narrow. As the bows of several cables cross each other in that area, there was a risk
that this part would become very thick, thus increasing the minimum air gap. Figure 4.8
shows the winding on the bakelite disk. Serial connection is applied for all cables belonging
to the same phase.
Figure 4.8
4.5
Picture of the two-layer air-gap winding on a bakelite disc
The Complete Small-Scale FW Generator
The lower rotor disc is attached to a shaft, which is supported by two ball bearings at its lower
end and its upper part. The second rotor disc is fixed by four thread rods, which are inserted in
tapped holes in the lower disc. The bakelite with the air-gap winding is bond on a plastic tube.
A second plastic tube enclosing both rotor discs and the stator, serves as protection in case of
hurtling parts. An allen screw is fixed on the top of the shaft. Thus, a drilling machine with
inserted allen key can be used for driving the generator.
In Table 4.2 the polar mass moment of inertia and the mass of the rotating parts of the FW are
shown. For the calculations, see Appendix D. Figure 4.8 shows the outline and pictures of the
whole experimental set-up. In Appendix E the outline with dimensioning is shown.
Table 4.2
Polar mass moment of inertia and mass of rotating parts
Part
Rotor disc (one)
PMs (all)
Other individual rotating parts (e.g.
shaft, thread rods)
All rotating parts
Mass (kg)
Polar mass moment
of inertia (kg·m2)
0.0788
0.0589
0.0024
6.21
3.36
0.77
0.2189
16.53
4 The Experimental Set-Up
37
_________________________________________________________________________________________________________________
a)
b)
c)
Figure 4.8
The experimental set-up
a) Outline of the experimental FW generator
b) Picture of the stator mounted on a plastic tube and the lower rotor disc with inserted thread
rods for holding the upper rotor disc in place
c) Picture of the whole experimental set-up with outer plastic tube as protection
More pictures of the experimental AFPM generator with air-gap winding are shown in
Appendix F.
5 Simulation of the Magnetic Field of the Rotor
38
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5
Simulation of the Magnetic Field of the Rotor
5.1
The Modeling Software COMSOL Multiphysics™
COMSOL Multiphysics™ is a software for modeling and solving scientific and engineering
problems based on partial differential equations (PDEs). Instead of having models for only
one type of physics, it enables models that can solve coupled physics phenomena without
requiring in-depth knowledge of numerical analysis.
Basically, a model can be built by creating the geometry and defining the relevant physical
quantities such as material properties or sources, while COMSOL Multiphsycis™ provides
modules for different application areas (e.g. heat transfer, structural mechanics) with
respective underlying equations and specialized solvers for the models. For simulations in this
thesis the electromagnetic module was used. This module is suitable for simulating
electromagnetic wave and field propagation as well as AC-DC electromagnetics.
A geometry model contains so-called manifolds, which are defined as mathematical functions
describing a surface, curve, or point in a geometry model of any dimension [36]. A domain is
a topological entity describing bounded parts of such manifolds as well as relations between
manifolds. A domain of dimension one less than the space dimension for the respective
model, is termed boundary. For example a face, i.e. a bounded part of a surface, in a 3D
geometry is a boundary. Consequently, a subdomain is a topological part of the modelling
space. Dependent on the dimension of the modelling space the geometrical representation of a
subdomain is either a line segment (in 1D), an area (2D), or a volume (3D). For more detailed
definitions see [36].
Since COMSOL Multiphysics™ in principle is unitless, any consistent set of units can be
used. But for the Maxwell equations that use the constants for permittivity and permeability in
a vacuum ε0 and μ0 the use of SI units is assumed. This has to be taken into account if another
unit system than the SI units is chosen. In the simulations with COMSOL Muliphysics™ for
this thesis SI units have been used.
5.2
Geometry Modeling
In order to reduce the calculation work, only a forth, i.e. seven pole pairs, of the generator was
simulated. As the magnetic flux that goes through one specific pole pair returns partly via
poles close by, it is necessary to consider more than one pole pair. By taking into account
seven pole pairs, there is one pole in the middle having three vicinal pole pairs on each sides.
The flux returning via even more distant poles is expected to be negligible compared to the
total flux through one pole.
The relevant parts of the experimental set-up for the simulation of the magnetic field are the
permanent magnets and the rotor discs. All other parts can be ignored as they do not
significantly influence the magnetic field. The bakelite disk and the insulation of the cable as
well as the copper conductor have a magnetic permeability close to the one of free space µ0.
Therefore, the whole stator is considered to have the same magnetic properties as free space.
5 Simulation of the Magnetic Field of the Rotor
39
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Thus, the geometry exists of segments of the steel discs with a rim at the outside and the
conical PMs placed on the discs. They are enclosed by a cuboid, representing a section of the
surrounding space filled with air. The shaft and the rods that connect the two discs are not
taken into consideration due to the long distance to the PMs.
The magnetostatics application mode allowing the modeling of conducting and magnetic
materials has been chosen. The geometry was created using the CAD tools that are provided
by COMSOL Multiphysics™.
Figure 5.1
5.3
Geometry of the modeled section of the rotors without surrounding cuboid
Physics Modeling
After having created the model geometry, the settings for the physics and the equations have
been entered into the model by specifying the subdomain settings and the boundary
conditions. They describe the material properties that have to be provided. For the
ferromagnetic steel discs, the relative permeability has been set to µr = 1000, thus having a
value between the one of iron and common construction steel.
COMSOL MultiphysicsTM provides sets of equations in order to solve the modeled problems.
For the constitutive relationship between the magnetic flux density and the magnetic field
intensity, three alternatives are available, using the relative permeability, magnetic
magnetization and the remanent flux density respectively:
B = μ0μr H ,
B = μ 0 (H + M ) ,
B = μ 0 μ r H + BR .
(5.1)
(5.2)
(5.3)
Concerning the permanent magnets equation (5.3) containing BR has been chosen as
constitutive relationship because it enables to consider that all PMs have a remanent magnetic
flux density in either z or –z direction (perpendicular to the rotor discs). While the remanent
flux density of a PM on the upper rotor disc has the same orientation as the associated PM on
the lower rotor disc, the orientation alternates along the circumference of one rotor. Remanent
magnetic flux density in other directions than perpendicular to the rotor discs is neglected.
Besides the remanent flux density BR = ± 1.3T, the relative permeability of the NdFeBB
B
5 Simulation of the Magnetic Field of the Rotor
40
_________________________________________________________________________________________________________________
magnets has to be specified for using the equation. According to the manufacturer of the
magnets, their relative permeability is µr = 1.05. For the steel disc and the air, equation (5.1)
has been chosen.
The basis for PDE formulation is Ampére´s law:
∇× H = J .
(5.4)
By using the definition of magnetic potential:
B = ∇× A
(5.5)
and the constitutive equation (5.3),
Ampère´s law becomes
(
(
))
∇ × μ 0−1 ⋅ μ r−1 ∇ × A − BR = J e ,
where
Je :
(5.6)
externally generated current density
or by using constitutive equation (5.1)
(
)
∇ × μ 0−1 ⋅ μ r−1 ⋅ ∇ × A = J e .
5.4
(5.7)
Simulation Results
Figure 5.2 a) shows the magnetic flux density along a slice through the middle of the two
magnets in the middle of the modeled rotor section. The highest flux density in the middle of
the air gap amounts to 0.56 T and the flux density is about 0.5 T in a larger range in the
middle of the magnets, the area where the stator is situated when inserted. In Figure 5.2 b) the
flux density in the plane parallel to the rotor discs and in the middle of them is shown.
Tangential and radial slices are shown in Figure 5.3, giving an impression of the threedimensional geometry.
5 Simulation of the Magnetic Field of the Rotor
41
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a)
b)
Figure 5.2
Magnetic flux density in the air gap (in T)
a) Along a radial slice through the middle of the modeled rotor configuration
b) In the plane parallel to and in the middle to the rotor discs
5 Simulation of the Magnetic Field of the Rotor
42
_________________________________________________________________________________________________________________
a)
b)
Figure 5.3
Magnetic flux density in the air gap (in T)
a) Along a tangential slice
b) Along tangential and radial slices
6 Measurements and Results
43
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6
Measurements and Results
6.1
Measurement of the Magnetic Field of the Rotor
By means of a gaussmeter that measures flux density perpendicular to a probe, the flux
density of both rotors without inserted stator has been investigated for two different air gaps.
Initially, the flux density along a line in radial direction in the middle of the air gap has been
measured. The probe was inserted from outside the rotor towards the center of it, following
the path shown in Figure 6.2 a). Clearly to perceive, the variation of the flux density is very
similar in the 16 mm and 18 mm thick air gaps, but of course higher values of flux density are
obtained when the air gap is smaller.
Magnetic field of the rotor
600
Magnetic flux density (mT)
500
400
18 mm air gap
16 mm air gap
300
200
100
0
-20
-10
0
10
20
30
40
50
Radial position (mm)
Figure 6.1
a) radial
Figure 6.2
Magnetic flux density in the middle of the air gap, measured from outside to inside
(the radial position zero corresponds to the outer edge of the magnets)
b) tangential
c) axial
Different directions, in which the probe has been moved for measurements
6 Measurements and Results
44
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The next measurement has been applied in tangential direction over the distance of one pole
pitch, again in the middle of the two rotor discs. Despite of the few measurement points, the
shape of the curve in Figure 6.3 can be said to be fairly sinusoidal.
500
400
Magnetic flux density (mT)
300
200
100
16 mm air gap
0
0
5
10
15
20
25
30
35
-100
-200
-300
-400
-500
Tangential position (mm)
Figure 6.3
Variation of the magnetic flux density in tangential direction (measured over one pole
pitch, starting in the middle of a magnet)
700
Magnetic flux density (mT)
600
500
400
16 mm air gap
300
200
100
0
0
1
2
3
4
5
6
7
8
9
Distance from surface of PM (mm)
Figure 6.4
Variation of the magnetic flux density from the middle of the air gap to the surface of
one of the permanent magnets
6 Measurements and Results
45
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Finally, a measurement was made by moving the probe in axial direction, more exactly
towards the magnet above the probe. Figure 6.4 shows how the flux density increases, when
the distance to the surface of the magnet is reduced to zero in steps of 1 mm.
6.2
Measurement of the Stator Resistance per Phase
By means of a milliohmmeter, the resistance of each phase of the stator winding has been
measured to be 134 mΩ.
6.3
Measurement of Phase Voltage at Open-Circuit Operation
Figure 6.5 shows the measured voltage of one phase at around 850 rpm, corresponding to an
electric frequency of somewhat less than 200 Hz. The amplitude amounts to about 24 V and
the shape of the curve is quite sinusoidal. Obviously the content of harmonics is low. The
maximum revolution speed that could be reached with the drilling machine was about 900
rpm. If the generator is spinning at this revolution when the drilling machine is removed, it
takes about 3 minutes until the revolution speed is reduced to zero due to no-load losses.
Phase voltage
30
20
Voltage (V)
10
Phase 1
0
-0,006
-0,004
-0,002
0
0,002
0,004
0,006
-10
-20
-30
Time (s)
Figure 6.5
6.4
Measured open-circuit voltage of one phase
Connection of a Load
A circuit with loads of 1 Ω in each phase was arranged (Y connection). While accelerating the
generator, the loads were disconnected. By switching on a circuit breaker, the loads were
connected when a rotation speed of approximately 850 rpm was reached. Figure 6.6 shows the
voltage drop due to the resistance of the stator winding when the load was connected.
6 Measurements and Results
46
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Furthermore, it shows how the generator was slowed down due to load losses. After
connecting the 1 Ω loads in each phase, it took about 6 seconds until the generator had
stopped to rotate.
30
20
Voltage (V)
10
0
Phase 1
0
5000
10000
15000
20000
25000
30000
35000
-10
-20
-30
Samples
Figure 6.6
Measured voltage of one phase before and after the connection of a load
Before the load was connected, the amplitude of the voltage amounted to 25.2 V. Directly
after the connection of the load, the phase voltage was reduced to 22.2 V. As the stator
resistance was 0.134 Ω, the voltage drop over the stator winding was:
U R _ S = U0 ⋅
0.134Ω
RS
= 25.2V ⋅
≈ 3V
1Ω + 0.134Ω
Rload + Rs
Since the load in all three phases is 1 Ω and the rms-value of the voltage directly after the
connection of the load was 22.2V / 2 = 15.7V , the phase current amounted to
IS =
15.7 V
US
=
= 15.7 A
1Ω
Rload
Thus, the initial power in all three phases together is:
P = 3 ⋅ PS = 3 ⋅ U S ⋅ I S = 3 ⋅ 15.7V ⋅ 15.7A = 3 ⋅ 246.5W = 739.5W
6.5
Voltage Harmonics
Using the FFT function of an oscilloscope, the content of harmonics in the voltage has been
examined. Again a load of 1 Ω was connected to every phase. A certain minimum frequency
was necessary for the oscilloscope to be able to execute FFT analysis. On the other hand, the
6 Measurements and Results
47
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drilling machine has not enough power to accelerate the FW generator to high revolution
speeds as long as a load of 1 Ω is connected. Consequently, the range of possible frequencies
concerning the fundamental wave was fairly limited. FFT analyses at rotational speeds of
about 400 rpm and 500 rpm have been accomplished.
Fraction of the amplitude of the fundamental wave (%)
0,5
0,45
0,4
0,35
0,3
408 rpm
0,25
0,2
0,15
0,1
0,05
0
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
47
49
Number of harmonic
Figure 6.7
Content of voltage harmonics at n = 408 rpm
Fraction of the amplitude of the fundamental wave (%)
0,7
0,6
0,5
0,4
522 rpm
0,3
0,2
0,1
0
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
Number of harmonic
Figure 6.8
Content of voltage harmonics at n = 522 rpm
31
33
35
37
39
41
43
45
47
49
6 Measurements and Results
48
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Figure 6.7 and 6.8 show the content of harmonics as fraction of the amplitude of the
fundamental wave. In both cases, the third harmonic is by far the largest one, and the fifth
harmonic is the second largest.
However, even the amplitude of the third harmonic amounts only to 0.6 percent of the
amplitude of the fundamental wave. Generally the total content of harmonics is low,
according to these measurements.
6.6
Determination of Losses
One of the main purposes of the project was to investigate the losses that occur in the
experimental generator. Besides the mechanical losses, the eddy-current and cooper losses in
the winding, some other losses may occur. Due to harmonics, the induction of eddy currents
in the rotor discs and the permanent magnets is probable. Since it is difficult to determine
such losses directly, it was tried to calculate the difference between the total losses and the
losses that were easier to determine. This method is applied for example to calculate core
losses, which are also difficult to obtain [37].
All appearing losses slow down the generator. When the rotation speed changes, the energy
stored in the rotor changes as well. Further on, the reduction of stored energy within a certain
time period can be considered as power loss. In addition, the electrical frequency decreases
linear with the revolution speed of the generator. The decrease of frequency can be described
dependent on the time if the voltage is measured over a certain time.
6.6.1
Standby Losses
A spin-down measurement was accomplished. The FW generator was accelerated by a
drilling machine to more than 850 rpm. From this time on, when the drilling machine was
removed, the rotational speed - and thus the electrical frequency - was decreasing until both
were zero. During the time from some seconds before the drilling machine was removed until
standstill of the generator the voltage was measured with 10,000 samples per second. As it
took about 3 minutes until the rotor had lost its rotational energy, a total of 2,000,000 samples
were recorded. The voltage was measured with relatively high resolution over a long period of
time. The period time of the voltage was determined by taking the average of some time
periods in a short time interval around certain points of time after the removal of the drilling
machine. Thus, the frequency could be plotted over the time (see Figure 6.9).
6 Measurements and Results
49
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Frequency over time
250
Frequency (Hz)
200
150
100
50
no load
0
0
20
40
60
80
100
120
Time (s)
Figure 6.9
Calculated frequency over time obtained from measurement results
As no load was connected to the generator, it was basically slowed down by friction in the
bearings and air friction. In addition, the moving magnetic field of the rotor induces eddy
currents in the stator cables. As these three losses occur during no-load operation, they can be
combined to standby losses.
It is assumed that the bearing losses depend linearly on the frequency, whereas the air friction
losses and the eddy-current losses in the stator cables increase with the square of the
frequency [3].
Equation (6.1) includes the losses increasing with the square of the frequency, which slow
down the rotation of the flywheel. In contrast equation (6.2) considers only the losses
depending linearly on the frequency.
J⋅
dω
+ K 1 ⋅ ω = J ⋅ ω& + K 1 ⋅ ω = 0
dt
(6.1)
J⋅
dω
+ K 2 = J ⋅ ω& + K 2 = 0
dt
(6.2)
The sum of 6.1 and 6.2 results in:
2 ⋅ J ⋅ ω& + K1 ⋅ ω + K 2 = 0
(6.3)
6 Measurements and Results
50
_________________________________________________________________________________________________________________
With
A=
K1
J
(6.4)
B=
K2
J
(6.5)
and
the equation of motion can be written as:
2 ⋅ ω& + A ⋅ ω + B = 0 ,
where
(6.6)
A > 0: coefficient describing the part of the losses depending on the square of
the frequency
B > 0: coefficient describing the constant part of the losses
The solution of this differential equation is given by equation (6.7)
ω (t ) = ω 0 ⋅ e
A
− ⋅t
2
A
⎞
B ⎛ − 2 ⋅t
⎜
+ ⎜e
− 1⎟⎟
A⎝
⎠
(6.7)
which is only valid for values of the time t resulting in a frequency ω (t ) ≥ 0 , i.e. for 0 ≤ t ≤ t *
with
t* = −
⎞
B
2 ⎛
⎟.
⋅ ln⎜⎜
A ⎝ B + A ⋅ ω 0 ⎟⎠
(6.8)
Equation (6.7) contains two coefficients, A and B, which can be determined by approximating
(6.7) to the values obtained from the measured at no load.
At the starting time t0 = 0, the value for the starting frequency ω0 amounted to 204.9 Hz (see
Figure 6.9).
By applying a curve fitting tool and using the measured values of frequency dependent on the
time, the coefficients A and B were determined to A = 0.01006 and B = 1.09438. Thus, (6.7)
was approximated to the measured values. Figure 6.10 shows the measured frequency
compared to the approximated curve, which is almost identical with the points obtained by
measurement. That means, the equation (6.7) with the determined coefficients A and B,
describes quite well how the frequency depends on the time.
6 Measurements and Results
51
_________________________________________________________________________________________________________________
Frequency over time
250
Frequency (Hz)
200
150
100
50
Frequency from
measurement
Approximationcurve
0
0
20
40
60
80
100
120
Time (s)
Figure 6.10
The approximated values of frequency compared to the values obtained by
measurement
The power at certain time can be approximated by the change of energy within a short time
interval. According to (2.1), the change of energy depends on the change of the angular speed.
With (6.7) and the determined values for A and B, the change of energy can be described
dependent on the time:
1
1
2
2
⋅ I m ⋅ (ω (t1 )) − ⋅ I m ⋅ (ω (t2 ))
⎛ t +t ⎞
2
.
Pstandby ⎜ t = 1 2 ⎟ = P(ω (t )) = 2
2 ⎠
t1 − t2
⎝
(6.9)
In a next step, a curve is approximated to the values calculated by the means of (6.9).
Initially it was assumed that bearing friction losses depend linearly on the frequency and
losses due to air friction and eddy-currents losses in the stator cables are proportional to the
square of the frequency. Consequently, equation (6.10) should be a suitable equation for being
approximated to the values obtained from (6.9), which links frequency to losses.
Pstandby (ω ) = A1 ⋅ ω + A2 ⋅ ω 2
(6.10)
Again the curve fitting tool was applied for determining the coefficients, resulting in the
values A1 = 0.054 and A2 = 0.0012 respectively. Thus, equation (6.10) was approximated to
the values calculated by using (6.9) and (6.7).
6 Measurements and Results
52
_________________________________________________________________________________________________________________
Finally, an expression had been developed, that can be applied for calculating the standby
losses depending on the frequency. No information about time was needed anymore. From the
curve in Figure 6.11, it can be seen that the losses for high frequencies increase much faster
than linearly. If the losses would only rise linearly, the standby losses at 200 Hz would be:
Plinear = A1 ⋅ ω = A1 ⋅
2 ⋅π ⋅ f
2 ⋅ π ⋅ 200Hz
= 0.054J ⋅
= 4.85W .
14
p
According to Figure 6.11, standby losses at 200 Hz amount to 14.3 W, i.e. about three times
higher than from the bearings.
Figure 6.11
6.6.2
Calculated standby losses dependent on the frequency
Total Losses
The same experiment has been accomplished with an electrical load of 1 Ω in each phase (Y
connection was applied for the generator and the loads). Again, frequency was determined
depending on time, by examining the period time at certain points of time. Figure 6.12 shows
the plot of the frequency over time.
6 Measurements and Results
53
_________________________________________________________________________________________________________________
Frequency over time
250
Frequency (Hz)
200
150
With load
100
50
0
0
1
2
3
4
5
6
Time (s)
Figure 6.12
Calculated frequency depending on time obtained from measurement results when
operating the flywheel with load
The dependence of the frequency on the time has been approximated to the measured values
by determining the coefficients f00 and A3 in:
f total (t ) = f 00 ⋅ e − A3 ⋅t .
(6.11)
The coefficients were determined to be A3 = 0.466381 and f 00 = 200.8. By using (6.11) and
(6.12), the total power losses occurring under load operation can be calculated depending on
the time.
1
⎛ 2 ⋅π ⋅ f
(t1 )⎞⎟ − 1 ⋅ I m ⋅ ⎛⎜ 2 ⋅ π ⋅ f (t 2 )⎞⎟
⋅ Im ⋅⎜
t +t ⎞
2
2
⎛
⎠ .
⎝ 14
⎠
⎝ 14
Ptotal ⎜ t = 1 2 ⎟ = P( f total (t )) =
2 ⎠
t1 − t 2
⎝
2
2
(6.12)
In order to obtain an expression, dependent on frequency, an equation containing only ω as
variable was needed. Again, linear dependence and dependence on the square of the
frequency was presumed. Thus, the equation will look similar to (6.10), but with coefficients
A4 and A5 instead of A1 and A2:
Ptotal (ω ) = A 4 ⋅ ω + A 5 ⋅ ω 2 .
(6.13)
6 Measurements and Results
54
_________________________________________________________________________________________________________________
With A4 = 0.003 and A5 = 0.1026 which again have been determined by the means of the
curve fitting tool (equation (6.13) was approximated to the values calculated by applying
(6.11) and (6.12)), the total losses can be calculated depending on the frequency for any value
of ω (see Figure 6.13).
Figure 6.13
Calculated sum of all losses (including power in the load) dependent on the frequency
The largest part of the energy initially stored by the FW generator is turned into heat due to
ohmic losses in the load and the winding. These powers of the load and the copper losses of
the winding can easily be calculated, since the voltage depends linearly on the frequency and
the constant kUf = f/U can be obtained by approximating a linear equation to the values of
voltage and frequency, measured during the experiment with electric load. This resulted in
obtaining kUf = 0.11.
Thus the copper losses in all three phases are given by:
PCopper
⎛ f ⋅ k fU
= 3 ⋅ I ⋅ RS = 3 ⋅ ⎜⎜
⎝ 2 ⋅ Rload
2
2
⎞
⎟ ⋅ RS .
⎟
⎠
(6.14)
The power in the load, regarding change of energy in the FW generator also a power loss, can
be calculated according to:
Pload = 3 ⋅ U 2 / Rload = 3 ⋅ ( f ⋅ k fU ) / Rload .
2
(6.15)
As seen from these equations, the copper losses and the power in the load depend on the
square of the frequency, what is apparent in Figure 6.14. The eddy-current losses in the stator
have the same behaviour. However, they are included in the standby losses.
6 Measurements and Results
55
_________________________________________________________________________________________________________________
a)
b)
Figure 6.14
Measured resistive copper losses and power in the load (voltage over load and
resistances of load and resistance of winding have been measured)
a) Resistive copper losses in the stator winding
b) The power in the ohmic load over frequency
6 Measurements and Results
56
_________________________________________________________________________________________________________________
The difference between the total losses and the sum of power in the load, copper losses in the
winding and the standby losses, might reveal if there were other losses. Figure 6.15 shows
Ptotal - Pstandby - Pload - Pcopper.
Figure 6.15
Calculated difference between total losses and sum of standby losses, copper losses in
the stator winding and power in load
The sum is negative and amounts to about 1% of the total power losses. Because of
inaccuracy of the measurement and arising differences by applying approximation of a curve
to a selection of values, the error of this method to determine losses presumably has an error
above 1%.
7 Summary of Results and Discussion
57
_________________________________________________________________________________________________________________
7
Summary of Results and Discussion
7.1
Properties of the Experimental Generator
7.1.1
Length of Cable in the Winding
Using the measured value for the resistance of the stator, the known cross section area and
resistivity of the cable, the cable length per phase in the air-gap winding can be calculated:
l = Rs ⋅
q
ρ copper
= 0.134 Ω ⋅
2.5mm 2
= 18.8m .
1.78 ⋅ 10 − 8 Ωm
About 3.8 m are used for connection of the stator. Thus, the winding itself comprises about
15 m of cable per phase.
7.1.2
Mechanical Eigenfrequency
At revolution speeds between 540 rpm and 570 rpm, the experimental generator vibrates
observably. This indicates that the experimental set-up has an eigenfrequency of about 550 Hz
and should preferably be operated above this range of revolution speed.
7.1.3
Voltage Harmonics
The voltage induced in the winding has a low content of harmonics. The pitched winding and
the large air gap reduce the harmonics. The third harmonic has clearly the highest share of the
harmonics and the fifth harmonic has the second largest share. This matches the statement,
that the content of these both harmonics usually is high for air-gap wound AFPM machines
[2].
Table 7.1 shows the winding factors of the air gap winding with q = 2 and 11/12 pitch for the
fundamental wave and the harmonics up to ν = 20. The winding factor for the third harmonic,
is among the highest. This is in accordance with the results of the FFT measurement.
7 Summary of Results and Discussion
58
_________________________________________________________________________________________________________________
Table 7.1
Winding factors depending on the number of harmonics
Number of
harmonic ν
1
2
3
4
5
6
7
8
9
10
7.1.4
Winding
factor kw
0.9577
0.2241
-0.6533
-0.2500
0.2053
0.0000
0.1576
0.4330
-0.2706
-0.8365
Number of
harmonic ν
11
12
13
14
15
16
17
18
18
20
Winding
factor kw
0.1261
1.0000
0.1261
-0.8365
-0.2706
0.4330
0.1576
-0.0000
0.2053
-0.2500
Generator Efficiency
Applying the curves, which have been approximated to the measurement values of the voltage
and the frequency, total losses of 813 W (including power in the load) and a power in the load
of 710 W are obtained for n = 850 rpm. Thus the overall efficiency at this revolution speed
amounts to
η=
Pload
Ptotal _ losses
=
710
≈ 87% .
813
The efficiency is quite low, since the connected load (1 Ω) was quite small, compared to the
resistance of the winding (0.134 Ω), resulting in high copper losses in the stator winding.
7.1.5
Stored Kinetic Energy
At a revolution speed of 850 rpm, the energy stored in the rotating part of the FW generator
amounts to:
1
1
⎛ 2 ⋅ π ⋅ 850 ⎞
Ek = ⋅ I m ⋅ ω 2 = ⋅ 0.22kg ⋅ m 2 ⋅ ⎜
⎟ = 872J .
2
2
⎝ 60s ⎠
2
7.2
Comparison of Measurement and Simulation Results
7.2.1
Magnetic Flux Density
The magnetic flux density B in the middle of the air gap has been measured and simulated
(see chapter 5 and 6). In Figure 7.1 the result from the simulation with COMSOL
MultiphysicsTM is compared to the measurement result for an air-gap length of 16 mm.
According to the simulation for the 16 mm large air gap, the highest value of B amounts to
0.56 T, which is to be found in an area situated between 16 and 18 mm distance from the
7 Summary of Results and Discussion
59
_________________________________________________________________________________________________________________
outer edge of the magnets. The highest value for B obtained by measurement was 0.509 T,
occurring at the positions 14 mm and 16 mm. Thus the highest value obtained by
measurement is about 10% lower than the highest value, obtained by simulation.
For the most other positions, the measurement results are up to 17% (at positions inside the
outer edges of the magnets) lower than the corresponding simulation results. The overall
shape of both curves is similar.
In Figure 7.2 the results of the simulation and the measurement of the flux density for an air
gap length of 18 mm, the size of the actual air gap of the experimental set-up, are compared.
Again, the measured values are lower than the one obtained from the simulation.
Obviously, the simulation is not very accurate, as the curve of the simulated flux density has
quite large steps, and in some regions the flux density increases with decreasing distance from
the outer edges of the magnets. The mesh for the simulation with COMSOL MultiphysicsTM
was rather coarse, since the calculation complexity of a FEM simulation for solids is high
anyway.
600
500
Flux density (mT)
400
Measurement result
Simulation result
300
200
100
0
-10
0
10
20
30
40
50
Radial position (mm)
Figure 7.1
Comparison of simulation result and measurement result of the magnetic flux density
in the air gap (16 mm)
7 Summary of Results and Discussion
60
_________________________________________________________________________________________________________________
600
Magnetic flux density (mT)
500
400
Measurement result
Simulation Result
300
200
100
0
-10
0
10
20
30
40
50
Radial position (mm)
Figure 7.2
Comparison of simulation result and measurement result of the magnetic flux density
in the air gap (18 mm)
On the other hand, the results of the measurement might be somewhat lower than the real flux
density, as the probe only measures flux density perpendicular to it. Consequently, the
measured value becomes too low, if the orientation of the probe is not exactly orthogonal to
the flux. Probably there has always been a small angle during the measurement, resulting in
reduced values.
This means that the actual flux density in the air gap of the rotor configuration has probably
values between those obtained by measurement and simulation. A flux density of 0.47 T
seems to be a reasonable value.
The voltage of the generator operated at 850 rpm can be calculated by applying (3.25) with
this value of the flux density:
Up =
ω ⋅ N S ⋅ kw ⋅ Φ
= 2 ⋅π ⋅ f ⋅ N S ⋅ kw ⋅
2
^
⋅ l ⋅τ p ⋅ B p =
π
2
850 ⋅ 14
2
= 2 ⋅π
Hz ⋅ 2 ⋅14 ⋅ 2 ⋅ 0.96 ⋅ ⋅ 0.04 m ⋅ 0.03 m ⋅ 0.47 T ≈ 17 V .
60
π
This calculated value is fairly close to the measured value of the voltage, suggesting that the
assumed value of the flux density is possible.
7 Summary of Results and Discussion
61
_________________________________________________________________________________________________________________
7.2.2
Voltage Output
With the used drilling machine, the generator could be accelerated to a maximum revolution
speed of about 900 rpm. At 850 rpm, the measured off-load voltage amounts to Û = 25 V, i.e.
the rms-value of the generated voltage is 17.7 V.
For the initial simulation with the FEM simulation tool applied for designing the experimental
generator, a revolution speed of 2,000 rpm was assumed (see Table 4.1). The generated
voltage according to the simulation results was 42 V.
Since it was not possible to achieve such a high revolution speed, the simulation was repeated
for 850 rpm, resulting in a no-load voltage of U = 14.4 V. This value is about 20% lower than
the value obtained by measurement on the experimental set-up. However, both values have
the same order of magnitude.
7.3
Discussion of the Measurement and Simulation Results
In order to obtain a flux density of 0.47 T in the air gap, a value of 10 mm has to be chosen
for the size of the air gap, when doing simulations in the FEM simulation tool. No larger air
gap was assumed in the initial simulation, since 10 mm already seemed large compared to the
pole pitch of about 3 cm.
In Figure 4.4, it is obvious that a large part of the field does not cross the air gap but returns
via one of the adjacent permanent magnets. A larger air gap would result in an even smaller
portion of field that crosses the air gap and returns via the stator iron.
However, as the air-gap wound stator itself is 10 mm thick and some safety margin below and
above is applied, the actual air gap of the experimental set-up is 18 mm. In contrast to the
stator modeled in the FEM simulation tool, the air-gap wound stator is ironless. Thus, no
material with a relative permeability being considerably higher than 1, exists in the air gap.
Consequently, the air gap of 18 mm could result in a quite low flux density in the middle of
the air gap. On the other hand, there is another magnetic pole on the other side of the air gap.
The measurements and calculations show that, by simulating a radial-flux machine with an air
gap of 10 mm in the FEM simulation tool, the same flux density is obtained as existing in the
18 mm large air gap of this axial-flux machine with two rotors.
First of all, this means that an axial-flux machine can be operated with a large air-gap, as a
reasonable flux density still can be achieved. Obviously, the magnetic field is straightened by
the PMs of the two rotors. Otherwise a lower fraction of the flux would cross the air gap due
to the absence of magnetically conducting material.
This statement is supported by the results of the simulation with COMSOL MultiphysicsTM.
Figure 5.3 shows that the magnetic flux density between two PMs of the same pole is
considerably higher than the flux density between two adjacent PMs.
Furthermore, it has been shown that the available FEM simulation tool can be applied as a
tool for designing an axial-flux machine, although the value for the air gap used for the
simulation differs from the actual air gap. However, it provides only coarse values, which
have to be specified by means of other tools.
7 Summary of Results and Discussion
62
_________________________________________________________________________________________________________________
The sum of standby losses, copper losses and the power in the load was slightly larger than
the determined total losses, indicating inaccuracy concerning measurement and approximation
of equations to the measurement values. This method is obviously not adequate for obtaining
information about further losses from the measured data recorded at low revolution speed.
However, other losses do obviously not contribute considerably. Had they done so, the
difference should be positive and not in the magnitude of the error of measurement. This error
is assumed to be reasonably low, since the measurements applied for the determination of
losses have been accomplished with high sample rate.
It is probable that the eddy-current losses in the PMs and the rotor discs are negligibly small
at the low revolution speeds, the generator has been operated on, since these losses increase
with the square of the frequency. The low content of harmonics and thus the low occurrence
of high frequencies support the assumption that other eddy-current losses than those in the
stator winding are very small.
To get a considerable contribution of these losses, the generator has to be operated on higher
revolution speeds and if possible with smaller air gap in order to increase the content of
harmonics in the voltage.
8 Conclusion and Suggestions for Future Work
63
_________________________________________________________________________________________________________________
8
Conclusion and Suggestions for Future Work
8.1
Conclusion
A working ironless AFPM air-gap wound generator with cables as stator conductors has been
designed and constructed. The resistance of all three phases has the same value RS = 0.134 Ω.
At a revolution speed of 850 rpm, an off-load peak voltage of Û = 25 V is generated. A power
output of 740 W was achieved by connecting a load of 1 Ω to every phase at a revolution
speed of 850 rpm, without any perceivable warming of the cables. Thus, a multiple of this
value is achievable by accelerating the generator to higher revolution speed before connecting
the load to it.
The generated voltage is almost sinusoidal, i.e. its content of harmonics is quite low at the
investigated ranges of revolution.
The standby losses, i.e. the sum of mechanical losses and eddy-current losses in the stator, and
the copper losses have been determined depending on the revolution speed. The contribution
of other losses than standby and copper losses could not be determined with the accomplished
method. However, it can be assumed that they are very low at the revolution speeds on which
the generator has been operated.
More accurate measurements at higher revolution speed are necessary in order to determine
the eddy-current losses in the PMs and the rotor discs.
The simulation results differ up to 20% from the measurement results. Thus, the simulation
tools can be applied for roughly designing AFPM generators with air-wound stator. It has
been shown that a reasonable high flux density can be obtained, despite a relatively large air
gap. The magnets on the two rotor discs straighten the magnetic field compared to a radial
flux machine.
The generator efficiency of the experimental set-up connected to loads of 1 Ω per phase was
about 87% at 850 rpm.
8.2
Suggestions for Future Work
A considerably higher contribution of eddy-current losses in the PMs and the rotor discs to the
total losses would be achieved by accelerating the experimental FW generator to much higher
revolution speeds. The sum of those eddy-current losses could then be determined by
applying the same method as in this thesis or by measuring the losses more directly, e.g. by
measuring heating of the PMs and the rotor discs.
Another way of increasing such losses would be to reduce the air gap if possible without
damaging the stator. A smaller air gap would be possible using smaller cables for the air-gap
winding or reducing the thickness of the bakelite disc. If a very stiff adhesive could be
applied, the disc might be entirely dispensable.
While the contribution of the air friction losses to the standby losses could be specified by
repeating the spin-down measurement in a vacuum vessel, the contribution of the eddy-
8 Conclusion and Suggestions for Future Work
64
_________________________________________________________________________________________________________________
current losses in the winding could be determined by another spin-down measurement with
removed stator.
A large reduction of the bearing losses could be achieved by applying magnetic bearings
instead of ball bearings.
The content of harmonics in the air-gap flux density could be reduced by applying skewed
magnets.
Connecting a flywheel disc to the generator would increase the stored energy.
When accelerating the experimental FW generator to very high revolution speed, the
capability of the winding to handle higher voltage could be investigated.
Instead of accelerating the set-up mechanically, it could be fed via an IGBT converter and
operated as both a motor and a generator, thus becoming an applicable flywheel energy
storage, which can absorb, store and supply energy.
9 Acknowledgements
65
_________________________________________________________________________________________________________________
9
Acknowledgements
The author would like to thank everybody at the Division for Electricity and Lightning
Research at the Uppsala University for their help during my work. A special thank is
addressed to my tutor Björn Bolund, to my supervisor Hans Bernhoff and the technician in the
tool shop Ulf Ring for assisting with help. Furthermore, I want to thank Csaba Zsolt Deák and
Prof. Dr.-Ing. habil. A. Binder at the Institute for Electrical Energy Conversion at the
Darmstadt University of Technology for their support.
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66
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Appendix
68
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Appendix
A
Abbreviations
Abbreviation
AFPM
emf
FEM
FET
FES
FFT
FW
IGBT
ISS
LIREX
mmf
PDE
PM
rpm
rms
SMES
NASA
ISS
DC
SG
SM
CNC
NdFeB
SmCo
AlNiCo
Meaning
Axial flux permanent-magnet
Electromotive force
Finite element method
Field-effect transistor
Flywheel energy storage
Fast Fourier Transformation
Flywheel
Insulated-gate bipolar transistor
International Space Station
Light Innovative Regional Express
Magnetomotive force
Partial differential equation
Permanent magnet
Revolutions per minute
Route mean square
Superconducting magnetic energy storage
National Aeronautics and Space Administration
International Space Station
Direct current
Synchronous generator
Synchronous machine
Computerized numeric control
Neodymium-Iron-Boron
Samarium-cobalt
Aluminium-Nickel-Cobalt
Appendix
69
_________________________________________________________________________________________________________________
B
Symbol
a
A
B
d
Ek
f
F
H
I
Im
J
JM
kp
kd
kw
K
l
L
m
M
me
n
Nc
Ns
p
P
Q
q
r
R
U
w
X
δskin
Θ
μ
μr
ρ
ρm
σ
σt
τα
τp
ν
Φ
Symbols
Quantity
Number of coils connected parallel
area
Magnetic flux density
Diameter
Kinetic energy
Electrical frequency
Force
Magnetic field intensity
Current
Polar mass moment of inertia
Current density
Magnetic polarization
Pitch factor
Distribution factor
Winding factor
Shape factor
Length
Inductance
Mass
Magnetization
Number of phases
Revolution speed
Number of turns per coil
Number of coil turns per phase
Number of pole pairs
Power
Total number of stator slots
Number of slots per pole and phase
Radius
Resistance
Voltage
Coil pitch
Reactance
Skin depth
Ampere turns
Permeability
Relative permeability
Resistivity
Mass density
Electrical conductivity
Tensile strength
Number of slot steps
Pole pitch
Number of harmonic
Magnetic flux
Unit
1
m2
T
m
J
Hz
N
A/m
A
kg·m2
A/mm2
T
1
1
1
1
m
H
kg
A/m
1
min-1
1
1
1
W
1
1
m
Ω
V
m
H/s
m
A
Vs/Am
1
Ωm
kg/m3
S/m
MPa
1
m
1
Wb
Appendix
70
_________________________________________________________________________________________________________________
χ
ω&
ω
Susceptibility
Angular acceleration
Angular speed
1
rad/s2
rad/s
Appendix
71
_________________________________________________________________________________________________________________
C
Properties of the Sintered NdFeB PM
Table A1: Properties of the sintered NdFeB magnets applied for the small-scale generator
Property
Remanence BR
Coercitivity HCB
Intrinsic coercitivity HCJ
Energy product |BH|max
Max. working temperature
Reversible temperature
coefficient of BR (20-100 °C)
Reversible temperature
coefficient of HCJ (20-100 °C)
Density
Vickers hardness
Tensile strength
Specific heat
Young´s modulus
Poisson´s ratio
Curie temperature
Electrical resistivity
Flexural strength
Coeff. Of thermal expansion
Thermal Conductivity
Rigidity
Compressibility
B
Unit
T
kA/m
kA/m
kJ/m3
°C
%/°C
Value
1.3
875
955
320
80
0.11
%/°C
-0.6
g/cm3
Hv
kg/mm2
kCal/(kg °C)
N/m2
7.6
570
8
0.12
1.6·1011
0.24
310
144
25
4·10-6
7.7
0.64
9.8·10-12
B
°C
µΩcm
kg/mm
/°C
kCal/(mh°C)
N/m2
m2/N
Appendix
72
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D
Calculation of Polar Mass Moment of Inertia (m-file)
Matlab m-file, used for calculating the polar moment of inertia
%% Densities
ro_iron = 7.84 * 1000; % kg/m^3
ro_NdFeB = 7.6 * 1000; % kg/m^3
%% Geometry ***********************************************
% Rotor
r_rotordisc_o = 0.1575; % m
r_rotordisc_i = 0.011; % m
h_rotordisc = 0.01; % m
r_rim_o = 0.1575; % m
r_rim_i = 0.150; % m
h_rim = 0.003; % m
% Shaft
r_shaft_upper_part = 0.0055; % m
h_shaft_upper_part = 0.06; % m
r_shaft_middlepart = 0.011; % m
h_shaft_middlepart = 0.095; % m
r_shaftring = 0.03; % m
h_shaftring = 0.005; % m
r_shaft_lower_part = 0.0055; % m
h_shaft_lower_part = 0.013; % m
% Rod
r_rod = 0.0045; % m
h_rod = 0.125; % m
% Nut
r_nut_o = 0.0085; % m
r_nut_i = 0.0045; % m
% Magnet
l_rectangle = 0.04; % m
h_rectangle = 0.01; % m
b_rectangle = 0.017; % m
a_triangel = 0.003
b_triangel = 0; % m
c_triangel = 0.04; % m
%% Masses ******************************************************
mass_hole = 0.005^2 * pi * 0.01 * ro_iron %kg
mass_shaft_upper_part = r_shaft_upper_part^2 * h_shaft_upper_part * pi * ro_iron %kg
mass_shaft_middlepart = r_shaft_middlepart^2 * h_shaft_middlepart * pi * ro_iron %kg
mass_shaft_lower_part = r_shaft_lower_part^2 * h_shaft_lower_part * pi * ro_iron %kg
mass_shaftring = r_shaftring^2 * h_shaftring * pi * ro_iron %kg
mass_rotordisc = (r_rotordisc_o^2 - r_rotordisc_i^2)* pi * h_rotordisc * ro_iron - 8 * mass_hole %kg
mass_rim = (r_rim_o^2 - r_rim_i^2)* pi * h_rim * ro_iron %kg
mass_shaft = mass_shaft_upper_part + mass_shaft_middlepart + mass_shaft_lower_part + mass_shaftring %kg
mass_rod = r_rod^2 * pi * h_rod * ro_iron %kg
Appendix
73
_________________________________________________________________________________________________________________
mass_nut = 0.01 %kg
mass_magnet = 0.02 * 0.04 * 0.01 * ro_NdFeB %kg
mass_rectangle = 0.017 * 0.04 * 0.01 * ro_NdFeB %kg
mass_triangel = a_triangel*c_triangel/2*h_rectangle*ro_NdFeB %kg
total_mass = 2 * mass_rotordisc + mass_rim + mass_shaft + 4 * mass_rod + 56 * mass_magnet + 20 * mass_nut
%kg
%% Polar mass moments of inertia
% Rotor:
I_disc = 0.5 * mass_rotordisc * r_rotordisc_y^2; % kg *m^2
I_rim = 0.5 * mass_rim * (r_rim_o^2 + r_rim_i^2); % kg * m^2
I_rotor = I_disc + I_rim % kg * m^2
% Shaft
I_shaft_ovre_del = 0.5 * massa_shaft_ovre_del * r_shaft_ovre_del^2; % kg *m^2
I_shaft_medeldel = 0.5 * massa_shaft_medeldel * r_shaft_medeldel^2; % kg *m^2
I_shaft_nedre_del = 0.5 * massa_shaft_nedre_del * r_shaft_nedre_del^2; % kg *m^2
I_shaftring = 0.5 * mass_shaftring * r_shaftring^2; % kg *m^2
I_shaft = I_shaft_ovre_del + I_shaft_medeldel + I_shaft_nedre_del + I_shaftring % kg * m^2
% Rod
I_rod = 0.5 * mass_rod * r_rod^2 + mass_rod * 0.07^2 % kg * m^2
% Nut
I_nut = 0.5 * mass_nut * (r_nut_o^2 + r_nut_i^2) + mass_nut * 0.07^2 % kg * m^2
% I_magnet
I_rectangle = 1/12 * mass_rectangle * (l_rectangle^2+b_rectangle^2) + mass_rectangle*0.13^2 % kg * m^2
I_triangel = 2*(1/6*a_triangel*c_triangel*c_triangel*c_triangel*massa_triangel + massa_triangel * (0.11 +
0.04*2/3)*(0.11 + 0.04*2/3)) % kg * m^2
I_magnet = I_rectangle + I_triangel % kg * m^2
I_other = 0.5 * 0.02 * 0.02 + 0.02 * 0.07^2 % kg * m^2 % grommets
%TOTALT
I_total = 2 * I_rotor + I_shaft + 4 * I_rod + 56 * I_magnet + 22 * I_nut % two nuts for welded part
Appendix
74
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E
Cross Section of the Experimental Generator
Cross section of the experimental FW generator with dimensioning
Appendix
75
_________________________________________________________________________________________________________________
F
Pictures of the Experimental FW Generator
Lower rotor disc with PMs
Both rotor discs without stator
Both rotor discs with inserted stator
Lower rotor disc and stator inside the tube
Completely assembled experimental AFPM generator with air-gap winding
Appendix
76
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G
Calculation of Losses of the Experimental Set-Up (m-file)
Matlab m-file, used for calculating and plotting the losses dependent on the frequency
clear
konst_f_U = 0.11
% determined by approximating a curve to the values for f
% and U of the "load measurement"
load_resistance = 1.006 % total resistance of load and connecting cable,
% solder and mechanical connections
winding_resistance = 0.134
freq = [200.8 188.68 176.06 159.24 148.37 138.12 125.94 112.11 99.6 88.5
78.62 61.88 48.73 37.94 29.45 23.58 18.2 14.04]; %frequencies measured by
%"load measurement"
%%%%%%%%%%%%%%%%%%%%%%%with no-load
f0=204.9
% value determined by approximating curve f_noload = f0.*exp(% A./2.*t)+B./A.*(exp(-A./2.*t)-1) to measured values;
A=0.01066
% value determined by approximating curve f_noload = f0.*exp(% A./2.*t)+B./A.*(exp(-A./2.*t)-1) to measured values;
B=1.094
% value determined by approximating to curve f_noload f0.*exp(% A./2.*t)+B./A.*(exp(-A./2.*t)-1) to measured values;
I = 0.22 % polar mass moment of inertia
t=[0:0.01:200];
f_noload = f0.*exp(-A./2.*t)+B./A.*(exp(-A./2.*t)-1); % freq. at certain
% times
plot (t, f_noload);
t1 = t - 0.005; % small time intervals
t2 = t + 0.005;
delta_P1 = 0.5 .*I.*(2.*pi./14).^2 .* ((f0.*exp(-A./2.*t1)+B./A.*(exp(A./2.*t1)-1)).^2-(f0.*exp(-A./2.*t2)+B./A.*(exp(-A./2.*t2)-1)).^2)./(t2t1);
%Power at certain times
%plot (t, delta_P1);
%plot (f_noload, delta_P1);
a0 = 0.000236342
a1 = 0.0242169
%
%
%
%
value determined by approximating curve P_mech =
f_noload.*a1 + f_noload.^2.*a0
value determined by approximating curve P_mech =
f_noload.*a1 + f_noload.^2.*a0
P_mech = f_noload.*a1 + f_noload.^2.*a0; % losses at certain frequency
%plot (f_noload, P_mech)
%plot (f_noload, P_mech);
%%%%%%%%%%%%%%%%%%% with load
f00=200.8
% value determined by approximating curve f_load =
% f00.*exp(- a3.*t) to measured values
a3 = 0.466381
% value determined by approximating curve f_load =
% f00.*exp(- a3.*t) to measured values
f_load = f00.*exp(-a3.*t); % frequency at certain time
P_loss = 0.5 .*I.*(2.*pi./14).^2 .*((f00.*exp(-a3.*t1)).^2
a3.*t2)).^2 )./(t2-t1);
%Power at certain times
%plot (f_load, P_loss);
a4= 0.00135552
a5= 0.0206586
%
%
%
%
%
- (f00.*exp(-
value determined by approximating curve
P_loss_total = a4.*f_load + a5.*f_load.^2
to
value approximated to curve P_loss_total =
a4.*f_load + a5.*f_load.^2
Appendix
77
_________________________________________________________________________________________________________________
%Figure(8)
%P_loss_total = a4.*f_load + a5.*f_load.^2;
%plot (f_load, P_loss_total);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
f = [0:0.01:200];
w2 = 2.*pi.*f;
voltage2 = f .* konst_f_U./sqrt(2) % voltage drop over load
Figure (1)
plot (f, voltage2)
title ('Voltage depending on frequency')
xlabel ('Frequency (f)')
ylabel ('Voltage (V)')
p_load2 = 3.*((voltage2).^2)./load_resistance; % power in load
Figure (2)
plot (f, p_load2)
title ('Power in load')
xlabel ('Frequency (f)')
ylabel ('Power (W)')
% copper losses
p_winding2 = 3.*(voltage2./(load_resistance)).^2.*winding_resistance;
Figure (3)
plot (f, p_winding2)
title ('Losses in winding')
xlabel ('Frequency (f)')
ylabel ('Power (W)')
p_standby = f.*a1 + f.^2.*a0; % standby losses
Figure (4)
plot (f, p_standby)
title ('Standby losses')
xlabel ('Frequency (f)')
ylabel ('Power (W)')
p_loss_total = a4.*f + a5.*f.^2; % total losses
Figure(6)
plot (f, p_loss_total)
title ('Total losses')
xlabel ('Frequency (f)')
ylabel ('Power (W)')
p_rest = p_loss_total - p_winding2 - p_load2 - p_standby
Figure (7)
plot (f, p_rest)
title ('Difference between total losses and electrical plus mechanical
losses')
xlabel ('Frequency (f)')
ylabel ('Power (W)')