Electrical Machines in the Power
Engineering and Automatics
Section II
Electrical Machines in the Power Engineering
1. Power Transformers.........................................................................................................2
-
Introduction – part in the power system;
Construction and technology - core, windings, housing, cooling;
Energy losses;
Circulation and magnetization current;
Group of winding connections;
Parallel operation;
Exciting current inrush.
2. Turbogenerators and Hydrogenerators.............................................................................23
-
Parameters and characteristics;
Construction and technology - core, windings, housing, cooling;
Selected problems of exploitation;
Surge short circuit;
Circuit parameters.
3. Induction Motors and Generators.....................................................................................41
-
Construction;
Parameters and characteristics;
Condition of operation of Induction Generator.
4. Generators for wind power plants....................................................................................55
-
Wind power plants;
5. Permanent Magnet Generator...........................................................................................67
-
Magnets.
6. References.........................................................................................................................83
Laboratory
1. Determination of selected parameters of synchronous generator.
2. Testing the magnetization conditions of the 3-phase transformer.
3. Measurements of induction generator.
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1. Power Transformers
1.1. Transformer types:
Air core transformer (transformator bezrdzeniowy) – special transformer, seen often
in radio-frequency circuits (Fig.1.1). The degree of coupling (mutual inductance) between
windings is many times less than that of an equivalent iron – core transformer, but the
undesirable characteristics of a ferromagnetic core (eddy current losses, hysteresis, saturation,
etc.) are completely eliminated.
Core type transformer (transformator rdzeniowy) – (Fig.1.2).
Fig.1.1. Air core tramsformers of cylindrical
(a) or toroidal (b) forms.
Fig.1.2. Single phase core type transformer
Column transformer (transformator kolumnowy) – Fig.1.10. Three phase column
transformer is topologically derived from the set of three 1-phase transformers (Fig.1.3.).
Three limb column transformer has unsymmetrical magnetic circuit, wile five limb column
transformer could be treated as symmetrical one (Fig.1.6.).
Fig.1.3. Origin of three limb 3-phase column transformer.
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Shell-core transformer (rdzeń złożony z blach kształtu ―E‖ i ―I‖ ).The most popular
and efficient transformer core. Each layer of the core consists of E- and I-shaped sections of
metal (Fig.1.4).
Fig.1.4. Shell-type core construction.
Shell core transformers (and core type transformers) are burdensome in production. For
easy assemble purpose, distributed gap cores are proposed (Fig.1.5). Due to reduction of
assembled lamination elements (what results in reduction of air gap) significant increase of
efficiency is achieved. However high assemble precision must be kept.
Fig.1.5. Easy assemble distributed gap core of transformer.
Shell type transformer (transformator płaszczowy) – core encircles most part of the
winding (Fig.1.6). Construction used for very high voltage transformers.
Fig.1.6 Difference in construction between shell-type and core-type transformer.
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Unit transformer (transformator blokowy). The unit transformer is tweed-off from
the main connectons of the generator to the generator transformer. It is energized when
generator is in service and supplies loads which are essential to the operation of the unit.
Booster transformer (transformator dodawczy). Booster transformers are used to
eliminate the stray currents and the disturbances, obliging the return current to flow to the
return conductor. The use of booster transformers is growing due to new EMC regulations and
the increasing amount of sensitive electronic appliances. Booster transformers are often used
to improve old railway feeder systems.
Buck-Boost transformer is a small, single phase, dry type distribution transformers
designed and shipped as insulating/isolating transformers. It has a dual voltage primary and a
dual voltage secondary. This transformer can be connected for a wide range of voltage
combinations. The most common use is to buck (lower) or boost (raise) the supply voltage a
small amount, usually 5 to 27%.
Step down transformer (transformator obniżający napięcie).
Step up transformer (transformator podwyższający napięcie).
Regulating transformer (transformator regulacyjny). Is usually implemented as
autotransformer - with this no galvanic separation between in and output voltage exists.
Shielded transformer (transformator ekranowany). Electrostatically (faraday)
sheilded transformers are designed to protect sensitive electrical and electronic devices and
systems from high frequency voltages (electrical noise) or transients that occur due to
switching and loading on distribution lines.
Forced cooling transformer (transformator o chłodzeniu wymuszonym);
Air-blast transformer (transformator o wymuszonym chłodzeniu powietrznym), rely
on the forced air-blast provided by electrical fan.
Surge-proof transformer (transformator odporny na napięcia udarowe), this
transformer would withstand severe surges incident to a direct stroke of lightning to the
transmission line close to the transformers.
Oil immersed transformer (transformator olejowy).
Fig.1.7. Oil immersed transformers. Transformer core
and windings are enclosed in the vat tube (housing)
filled with oil.
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Some power transformers are immersed in transformer oil that both cools and insulates the
windings. The oil is a highly refined mineral oil that remains stable at transformer operating
temperature. Indoor liquid-filled transformers must use a non-flammable liquid, or must be
located in fire resistant rooms
Isolating transformer (transformator izolujący). Primary and secondary windings have the
same number of turns.
1.2. Transformer winding types:
Sandwich winding (uzwojenie krążkowe);
Concentric winding (uzwojenie współosiowe);
Fig.1.8. Sandwich winding
Fig.1.9. Concentric winding.
Coil winding (uzwojenie cewkowe) – Fig.1.10;
Compensating winding (uzwojenie wyrównawcze), is a set of isolated coils arranged
as delta connection, created for zero sequence current compensation.
Taped winding (uzwojenie z zaczepami). Made for voltage adjustement purpose.
a)
b)
.
Fig.1.10. Three phase medium power column transformer with taped coil winding. a) Dry transformer – voltage
adjustment made by changing keeper-connector. b) Oil immersed transformer – voltage adjustment made by
external connector.
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1.3. High power transformer
A major application of transformers is to increase voltage before transmitting electrical
energy over long distances through energy system wires. Wires have resistance and so
dissipate electrical energy at a rate proportional to the square of the current through the wire.
By transforming electrical power to a high-voltage (and therefore low-current) form for
transmission and back again afterward, transformers enable economic transmission of power
over long distances. Consequently, transformers have shaped the electricity supply industry,
permitting generation to be located remotely from points of demand. All but a tiny fraction of
the world's electrical power has passed through a series of transformers by the time it reaches
the consumer.
Fig.1.11. Typical high power transformer.
1.4. Transformer energy losses
When transformers transfer power, they should do so with a minimum of loss. Modern
power transformer designs typically exceed 98% efficiency. Transformer losses are
divided into losses in the windings, termed copper loss, and those in the magnetic circuit,
termed ―iron loss‖ or ―core loss‖. Losses in the transformer arise from:
Power lost due to resistance of the wire windings. Unless superconducting wires are
used, there will always be power dissipated in the form of heat through the resistance of
current-carrying conductors. Because transformers require such long lengths of wire, this loss
can be a significant factor. Increasing the gauge (cross section) of the winding wire is one
way to minimize this loss, but only with substantial increases in cost, size, and weight.
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The most significant of “core losses” is eddy-current loss, which is resistive power
dissipation due to the passage of induced currents through the iron of the core. Because iron
is a conductor of electricity as well as being an excellent ―conductor‖ of magnetic flux, there
will be currents induced in the iron just as there are currents induced in the secondary
windings from the alternating magnetic field. These induced currents - as described by the
perpendicularity clause of Faraday's Law - tend to circulate through the cross-section of the
core parallel to the primary winding turns. Their circular motion gives them their unusual
name: like eddies in a stream of water that circulates rather than move in straight lines.
Fig.1.12. Explanation of limitation of eddy currents by using iron sheets.
The main strategy in mitigating these wasteful eddy currents in transformer cores is to
form the iron core in sheets, each sheet covered with an insulating varnish so that the core is
divided up into thin slices. The result is very little width in the core for eddy currents to
circulate in.
Another ―core loss‖ is that of magnetic hysteresis.
Fig.1.13. Core iron histeresis loop of 3,5 kVA column 3-phase transformer.
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All ferromagnetic materials tend to retain some degree of magnetization after exposure to
an external magnetic field. This tendency to stay magnetized is called ―hysteresis‖, and it
takes a certain investment in energy to overcome this opposition to change every time the
magnetic field produced by the primary winding changes polarity (twice per AC cycle).
This type of loss can be mitigated through good core material selection (choosing a core
alloy with low hysteresis, as evidenced by a ―thin‖ B/H hysteresis curve), and designing the
core for minimum flux density (large cross-sectional area). However, high saturation
polarisation of modern magnetic materials allows reducing core cross-sectional area (Fig1.14).
Modern magnetic materials as Cobalt-Iron are also characterized by lover hysteresis losses
because of lower coercivity forces and lower eddy current losses because of higher electrical
resistivity.
Fig.1.14. Cobalt-Iron Alloys have the highst saturation polarisation and surpass all known soft magnetic
materials.
Transformer energy losses tend to worsen with increasing frequency. Lets discuss
behavior of single conductor. Self induced eddy currents that appear in the conductor oppose
main current changes in the centre of the conductor and support changes in the surface area as
it is explained in Fig.1.15.
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Fig.1.15. Self induced eddy currents in the conductor.
AC current density is highest close to the surface and lowest along conductor axe. In case
of high frequency currents, non-zero current density exists only within the thin surface region.
Changes of conductor resistance evoked by this phenomenon describe equation:
R
R0
1 k4 /3
0,997k 0,277
k 1 / 4 3 / 64k
k
dla
1,5
k
1
k
10
(1.1)
10
where: Rω – RMS conductor resistance of radius r for AC current of pulsation ω;
R0 – conductor resistance for DC current;
r
r
;
k
r
rsk
2
γ – characteristic electric conductivity;
2
rsk
- efficacious depth of AC current penetration (distance from conductor
r
surface in which current density changes e - times).
In a conductor carrying current, if currents are flowing through one or more other nearby
conductors, such as within a closely wound coil of wire, the distribution of current within the
first conductor will be constrained to smaller regions (Fig.1.16).
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Fig.1.16. Transformer current distribution.
The resulting current crowding is termed the proximity effect or skin effect.The skin
effect within winding conductors reduces the available cross-sectional area for electron
flow, thereby increasing effective AC resistance as the frequency goes up and creating more
power lost through resistive dissipation (the AC resistance of a conductor can easily exceed
ten times its DC resistance).
Magnetic core losses are also exaggerated with higher frequencies, eddy currents and
hysteresis effects becoming more severe. For this reason, transformers of significant size are
designed to operate efficiently in a limited range of frequencies. In most power distribution
systems where the line frequency is very stable, one would think excessive frequency would
never pose a problem.
Magnetostriction. Magnetic flux in a ferromagnetic material, such as the core, causes it
to physically expand and contract slightly with each cycle of the magnetic field, an effect
known as magnetostriction. This produces the buzzing sound commonly associated with
transformers, and in turn causes losses due to frictional heating in susceptible cores.
Mechanical losses. In addition to magnetostriction, the alternating magnetic field causes
fluctuating electromagnetic forces between the primary and secondary windings. These incite
vibrations within nearby metalwork, adding to the buzzing noise, and consuming a small
amount of power.
Stray losses. Leakage inductance is by itself lossless, since energy supplied to its
magnetic fields is returned to the supply with the next half-cycle (Fig.1.17). However, any
leakage flux that intercepts nearby conductive materials such as the transformer's support
structure will give rise to eddy currents and be converted to heat.
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Fig.1.17. Explanation of the flux leakage phenomena.
1.5. Tramsformer principles
Induction law. The voltage induced across the secondary coil may be calculated from
Faraday's law of induction, which states that:
dΦ12
dΦ
,
e2
z2 12 ,
dt
dt
where: e1, e2 – primary and secondary side Emf;
Φ12 – main flux.
e1
(1.2)
z1
For sinusoidal voltage:
2 E1 sin t .
e1
(1.3)
Primary voltage U1 is balanced by Emf e1 and drop of voltage over winding impedance
forced by flux exciting current (which waveform is different from sinusoidal due to iron core
non linear characteristic – Fig.1.18).
Flux exciting current varays from fraction of % to few % of nominal current. In this case
one can simplify:
dΦ
1
2 E1 sin tdt
z1
1
z1
2 E1 sin( t
2
)
Φm sin( t
2
) . (1.4)
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Fig.1.18. Primary voltage U1, Emf e1, exciting current i and main flux Φ12 waveforms.
Flux advances Emf e1 by π/2 (Fig.1.18.). Finaly:
E1
2
z1Φm
Winding turn ratio –
4,44 fz1Φm .
E1
E2
(1.5)
z1
.
z2
Voltage ratio(transformer voltage ratio) –
(1.6)
u
U1
.
U 20
(1.7)
1.6. Problem of circulating currents
In the windings with large current ratings, conductor has to be sub-divided into a number
of parallel conductors to reduce the eddy loss to an acceptable value. Without proper
transposition, the leakage flux linked by these parallel conductors is different, inducing
different voltages in them. Since the conductors are paralleled at the ends, there will be net
induced voltages in the loops formed by these conductors resulting into circulating currents
that do not flow outside the windings. The appearing value of circulating currents adds to the
other stray loss components resulting into higher load loss. If not controlled circulating
currents may cause hot spots in the windings of large power transformers. Equivalent circuit
for calculation of circulating currents is shown on Fig.1.19. This loss can be avoided if
location of each conductor in the layer and from layer to layer is such that all conductors are
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linked with an equal amount of leakage flux. This can be achieved by doing suitable
transpositions between the parallel conductors (Fig.1.20.).
Fig.1.19. Equivalent circuit for calculation of circulating currents.
V1…n – voltage induced by leakage flux, R1…n – resistance of the parallel conductors.
Fig.1.20. Transposition schemes in a layer winding.
1.7. Magnetization current
In addition to the current entering the primary because of the secondary load, there is the
core exciting current I0 which flows in the primary whether the secondary load is connected or
not. This current is drawn by the primary core reactance Xf and equivalent core-loss resistance
RFe. It has two components: If, the magnetizing component which flows 90° lagging behind
induced voltage e1; and Iow, the core-loss current which is in phase with e1. Ordinarily this
current is small and produces negligible voltage drop in the winding. Core-loss current (as
stated before) is often divided into two components: eddy current and hysteresis. Eddy-current
loss is caused by current circulating in the core laminations. Hysteresis loss is the power
required to magnetize the core first in one direction and then in the other on alternating halfcycles. Hysteresis loss and magnetization are intimately connected, as can be seen from
Fig.1.21. and Fig.1.13.
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Fig.1.21. Transformer voltage, flux, and exciting current.
Induced voltage e1 is plotted against time, and core flux Φ lags e1 by 90°. This flux is also
plotted against magnetizing current in the loop at the right. This loop has the same shape as
the B-H loop for the grade of iron used in the core, but the scales are changed:
BAc
,
i Hlc / 0,4 N
where: B – core flux density in gauss,
Ac – core cross-sectional area in cm2,
H – core magnetizing force in oersteds,
lc – core flux path length in cm,
N – number of wire turns.
(1.8)
Current is projected from the Φ-i loop to obtain the alternating current i at the bottom of
Fig.1.20. This current contains both the magnetizing and the hysteresis loss components of
current. In core-material research it is important to separate these components, for it is mainly
through reduction of the B-H loop area (and hence hysteresis loss) that core materials have
been improved. Techniques have been developed to separate the exciting current components,
but it is evident that these components cannot be separated by current measurement only. It is
nevertheless convenient for analysis of measurements to add the loss components and call
their sum Iow, and to regard the magnetizing component If as a separate lagging current
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(Fig.1.22). As long as the core reactance is large, the vector sum I0 of If and I0w is small,
however the shape of I0 is highly non-sinusoidal. Core flux reactance may be found by
measuring the magnetizing current, i.e., the current component which lags the applied voltage
90° with the secondary circuit open. Because of the method of measurement, this is often
called the open-circuit reactance, and this reactance divided by the angular frequency is called
the open-circuit inductance.
Fig.1.22. Equivalent open circuit of the transformer.
Although the applied voltage to a transformer is sinusoidal, the magnetization current
related to the flux through the magnetization curve is non-sinusoidal, as shown in Fig.1.21.
The non-sinusoidal current is symmetrical around its peak. Such a waveform is mainly
composed of odd harmonics. In particular, the 3rd harmonic, as well as the 5th and the 7th
harmonics have a significant contribution. This is because their magnitudes are large, usually
between 5%-10% of the magnetizing current.
Fig.1.23. Emf shape in case of: a) absence of higher harmonics in the magnetiztion current, b) presence of
higher harmonics in the magnetiztion current.
When the source cannot provide higher harmonics of current to flow (for example third
order harmonics in the case of Wye without zero line connection) the flux (flux density)
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waveform contains higher odd harmonics and in consequence phase voltage (emf) also
contains higher order harmonics (Fig.1.23a). However higher harmonics do not exist in the
line voltages. Distorted phase voltage has higher peak value that can be dangerous for load. If
the higher current harmonics can appear like in the case of delta winding connection (higher
current harmonics flow in delta circuit), the flux waveforms don’t contain higher odd
harmonics and in consequence phase voltage (emf) also doesn’t contain higher order
harmonics (Fig.1.23b). Higher current harmonics do not exist in the line current.
1.8. Winding connections
Three phase primary and secondary winding can be arranged in many ways. In practice
only the only 4 possible transformer combinations connections that secure symmetrical
secondary side voltages are used. The most common are Wye to Wye (Y-Y) – and Delta to
Wye (D-Y) or Wye to Delta (Y-D) connections:
1.
Delta to Delta - use: industrial applications.
2.
Wye to Delta - use : high voltage transmissions.
3.
Delta to Wye - use : most common; commercial and industrial.
4.
Wye to Wye - use : rare, causes harmonics and balancing problems.
Special kind of connection is Wye to Zigzag (Y-Z) connection. We rarely use zigzag
configurations for typical industrial or commercial use, because they are more expensive to
construct than conventional wye-connected transformers. But zigzag connections are useful in
special applications where conventional transformer connections aren't effective.
Each of described above winding arrangement has unique features. The most important is
ability of secure transformer operation under the load asymmetry. Load asymmetry leads to
phase current asymmetry that can be transformed in to current positive, negative and zero
sequences:
IA
IP
IB
IC
C IN ,
I0
were: C
h
h2
(1.9)
1
1
1
h2
h
h 1,
h2 1
1
C
1
h
1 h2
1 1
h2
h ;
1
1
3
;
j
2
2
1
3
j
.
2
2
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Positive and negative current sequences can flow in any transformer winding
arrangements, and each creates transformer symmetrical load. Negative sequence currents in
secondary windings forced by asymmetrical load will also flow in the primary winding.
Ampere-turns of these currents are to be compensated and voltage and magnetic symmetry of
the transformer is kept. Zero sequence currents can be forced by the asymmetrical load in the
transformer with zero line. Zero sequence currents can also appear in delta connection
primary winding. In this case zero sequence primary and secondary ampere-turns are to be
compensated as well.
Wye to Wye winding connection is the simplest one. In comparison with delta
arrangement wining requires 3 less winding turns but wire cross section has to be 3
greater (Fig.1.24).
Fig.1.24. Wye to Wye winding connecton. a) Schematic winding arrangement (ABC – three phase input
voltage). b) Primary side voltages vector representation. c) Secondary side 0 o voltages shift in respect to input
voltage. d) Secondary side 180o voltages shift in respect to input voltage.
Transformer requires less sandwich winding, less insulation, more simple cooling system.
In consequence the copper filling factor is better. This kind of the transformer is used as a
small unit with symmetrical load as well as big energy transmission unit. Because of
enormous costs there is no zero line in high (primary) voltage side of transformer. Zero
sequence currents can’t flow and voltage and magnetic asymmetry appears. That’s why
usually transmission unit is equipped with an additional compensating winding that
compensate zero sequence current and additionally secures third order magnetization current
sequence. Secondary side voltage phase shift (time shift) could be 0o (0h) or 180o (2h) as
explained in Fig.1.24c,d.
Delta-Wye winding connection admits all kind of unsymmetrical transformer load.
Delta connection allows the current of zero (that appears in case of load asymmetry) and third
order to flow. This kind of transformer is used for high voltage up to 35 kV. Secondary side
voltage phase shift (time shift) could be 330o (11h) or 150o (5h) as explained in Fig.1.25.
Primary phase voltage is 3 times higher than transformer input voltage (in this case
assumed as a Wye voltage system).
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Fig.1.25. Delta-Wye winding connection. a) Schematic winding arrangement (ABC – three phase input voltage).
b) Primary side voltages vector representation. c) Secondary side 330 o voltages shift in respect to input voltage.
d) Secondary side 150o voltages shift in respect to input voltage
Star-Zigzag winding connection (Fig.1.26a) admits unsymmetrical transformer load.
The main important feature of zigzag connection is that zero order ampere-turn selfcompensate within this kind of winding connection. Output voltage in case of zigzag winding
connection is 2 / 3 1.15 times lower than typical wye connection (compare Fig.26c and
Fig.26d).
Fig.1.26. Star-zigzag winding connection. a) Schematic winding arrangement (ABC – three phase input
voltage). b) Primary side voltage vectors representation. c) Example of secondary side 0o voltages shift in respect
to input voltage – winding arranged in typical wye connection. d) Example of secondary side 330o voltages shift
in respect to input voltage – winding arranged in one of the possible zigzag connection.
All the transformers connections can be classified in to distinct vector groups. Each vector
group notation consists of first an uppercase letter denoting HV (primary) connection, a
second lover case letter denoting LV (secondary) connection, followed by a clock number
representing LV winding’s phase displacement with respect o HV winding (at twelve o’clock).
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There are four groups into which all possible tree phase connections can be classified:
Group 1: zero phase displacement (Yy0, Dd0, Dz0);
Group 2: 180o phase displacement (Yy6, Dd6, Dz6);
Group 3: -30o (330o) phase displacement (Yd11, Dy11, Yz11);
Group 4: 30o phase displacement (Yd1, Dy1, Yz1);
1.9. Parallel operation of transformers
For supplying a load in excess of the rating of the existing transformer, one or more
transformers may be connected in parallel with the existing transformer. It is usually
economical to install another transformer in parallel instead of replacing the exiting
transformer by a single larger unit. The cost of a spare unit in the case of two parallel
transformers (of equal ratings) is also lower than a single larger unit. In addition it is
preferable to have a parallel transformer for the reason of reliability. With this, at least half
the load can be supplied with one transformer out of service. For parallel connections of
transformers, primary winding of the transformers are connected to source bus-bar and
secondary windings are connected to the load bus-bars. There are various conditions that must
be fulfilled for the successful parallel operation of the transformers:
1- The line voltage ratio of the transformers should be equal. If the transformers
connected in parallel have slightly different voltage ratios, then due to the inequality of
induced emfs in the secondary windings the circulating current will flow in the loop formed
by the secondary windings under the no load conditions which may be much greater then the
normal no load current. The current will be quite high as the leakage impedance is quite low.
When the secondary windings are loaded, this circulating current will tend to produce unequal
loading on the two transformers, and it may be impossible to take he full load from this group
of two parallel (ore more) transformers. One transformer may get overloaded.
2- The transformers should have equal per unit leakage impedances and the same ratio
of equivalent leakage reactance to the equivalent resistance (X/R). If the ratings of both
the transformer are equal, their per unit leakage impedances should be equal in order to have
equal loading of both the transformers:
I1r Zl1r
I 2r Zl 2r , since I1r
I 2r
Zl1r
Zl 2 r .
(1.10)
If the ratings are unequal, their per unit leakage impedances based on their own ratings should
be equal so that the currents carried by them will be proportional to their ratings. In other
words, for unequal ratings the numerical (ohmic) values of their impedances should be in the
inverse proportion to their ratings, to have current in them in line with their ratings.
A difference in the ratio of impedance value to resistance value of the per unit
impedance results in a different phase angle of the currents carried by the two parallel
transformers:
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cos( 1 )
cos( 2 ) if
X l1
R1
Xl2
.
R2
(1.11)
One transformer will be working with the higher power factor and the other with the lower
power factor than that of the combined output. Hence the real power will not be
proportionally shared by the transformers.
3- The transformers should have the same polarity. The transformers should be properly
connected with regard of their polarity. If they are connected with incorrect polarities then the
two emfs, induced in the secondary windings which are in parallel, will act together in the
local secondary circuit and produce a short circuit.
The previous three conditions are applicable to both single-phase as well as three-phase
transformers. In addition to these three conditions, two more conditions are essential for the
parallel operation of three-phase transformers:
4- Transformers should have the same phase sequence. The phase sequence of line
voltages of both the transformers must be identical for parallel operation of three-phase
transformers. If the phase sequence is an incorrect, in every cycle each pair of phases will get
short-circuited.
5- The transformers should have the zero relative phase displacement between the
secondary line voltages. The transformer windings can be connected in a variety of ways
which produce different magnitudes and phase displacements of the secondary voltage.
In order to have zero relative phase displacements of secondary side line voltages the
transformers belonging to the same group can be paralleled. For example two transformers
with Yd1 and Dy1 connections can be paralleled. The transformers of groups 1 and 2 can only
be paralleled with transformers from their own group. The transformers of groups 3 and 4 can
be paralleled by reversing the phase sequence of one of them.
1.10. Exciting current inrush
When a transformer is initially energized, there is a phenomenon known as exciting
current inrush. When initially energizing a single-phase transformer the flux in the core is
equal to the integral of the excitation voltage. If the circuit is closed when the voltage is
passing through zero and the initial flux is zero, the sinusoidal flux will be fully offset from
zero. The full-offset flux has a peak value that is twice the peak value of a symmetrical
sinusoidal flux and this is generally sufficient to drive the transformer core into saturation
(Fig.27b).
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Fig.1.27. Flux, steady state flux, voltage and induced voltage of transformer in different initial conditions of
closing the circuit. a) initial energizing when the voltage reaches maximum. b) initial energizing when the
voltage is passing through zero. c) initial energizing when the voltage is passing through zero when there is
residual flux.
Flux is related to induced voltage e:
e
U m sin( t
)
z
d
dt
ut 0
U m sin ,
1
U m cos( t
) C
2 fz
where: Um – voltage amplitude,
ψ – phase angle,
z – number of the winding turns.
Constant C can be determined as follow:
et
(1.12)
0
Um
cos
Φ0 Φm cos ,
2 fz
where: Φ0 – residual flux.
C
Φ0
(1.13)
In consquence:
Φm cos( t
Φ0 Φm [cos
) Φ0 Φm cos
cos( t
)]
,
(1.14)
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and:
Φ0
peak
2Φm .
(1.15)
At this point, the only thing that limits exciting current is the air-core impedance of the
winding, which is several orders of magnitude smaller than the normal magnetizing
impedance. Therefore, the exciting current is much greater than the normal exciting current
during the half cycle when the core is saturated. During the opposite half cycle, the core is no
longer saturated and the exciting current is approximately equal to the normal exciting current.
The situation is even more extreme when there is residual flux in the core and the direction of
the residual flux is in the same direction as the offset in the sinusoidal flux wave (Fig.27c,
Fig.28).
Peak inrush current is limited only by the air-core reactance XL [μH] which is nothing else
but transformer winding without core reactance. XL can be determined on the base of
calculations:
A
,
l
where: A – winding coil surface,
l – length of the flux path (axial length of the coil),
A
z2 0
L – inductance of the winding.
l
2 fz 2
XL
0
(1.16)
Generated flux υL is equal to the residual flux plus 2 times the normal flux change minus
the saturation flux. Inductance is related to the flux υL and the current as follow:
z
L
L
I
.
(1.17)
Therefore, the peak inrush current is expressed as:
2 u
s)
,
L
where: φ0 – residual flux, φu – normal flux change, φ0 – saturation flux.
I peak
z(
0
(1.18)
Without resistance in the circuit, each successive peak would have the same value and the
current inrush would go on indefinitely. With resistance in the circuit, however, there is a
significant voltage drop across the resistance and the flux does not have to rise quite as high
as the previous cycle. The integral of the voltage drop represents a net decrease in the flux
required to support the applied voltage. Since the i * R voltage drop is always in the same
direction, each cycle decreases the amount of flux required.
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Fig.28. Voltage, flux and current after energizing of a power transformer with residual flux.
When the peak value of flux falls below the saturation value of the core the inrush current
disappears. The rate of decay is not exponential although it resembles an exponentially
decaying current. For large power transformers, the inrush current can persist for several
seconds before it finally dies off. The line reactance has the effect of reducing the peak inrush
current by simply adding inductance to the air-core inductance of the winding. There is a
definite relationship between inrush current and short-circuit current because both are related
to the air-core inductance of the windings. (Remember that short circuits tend to exclude flux
from the core.) Typically, a rule of thumb is that peak inrush currents are a little over 90% of
peak short-circuit currents. The magnetic forces caused by inrush currents are generally much
smaller than short-circuit forces, however. Because only one winding per phase is involved,
there is no magnetic repulsion between windings. The whole problem of analyzing exciting
current inrush gets much more difficult when 3-phase transformers are involved. This is
because the phase angles of the exciting voltages are 120° apart, there are interactions of
currents and voltages between phases and the three poles of the switching device do not close
at exactly the same time. Nevertheless, it is safe to say that the peak magnitude of inrush
current for three-phase transformers approaches the short-circuit current levels. One of the
interesting features of exciting current inrush is that since the current is fully offset, there are
large percentages of even harmonics present. Even harmonics are otherwise rarely
encountered in power circuits.
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The duration of exciting current inrush is on the order of seconds (as compared to on
the order of cycles with fault/short-circuit currents). Exciting current inrush conditions also
occur much more frequently than short circuits.
There is also a phenomenon known as sympathetic inrush, where a transformer that is
previously energized will exhibit a sudden change in current when a nearby transformer is
switched on. Sympathetic inrush is caused by changes in line voltages from the inrush
currents of the second transformer.
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2. Synchronous generator
2.1. Salient pole and non-salient pole construction:
Synchronous AC generators are considered either brush or brushless, based on the method
used to transfer DC exciting current to the generator field. In addition AC generators are
classified as salient-pole and non-salient-pole depending on the configuration of the field
poles. Projecting poles are salient poles units, and turbo type (slotted) are non-salient units.
Fig.2.1. Schematic illustration of synchronous machines of
(a) salient rotor structures, (b) round or cylindrical rotor.
Salient pole (medium power synchronous machines, hydrogenerators).
Fig.2.2. Flux density distribution in air gap and induced emf in the phase winding of a (a) two pole and (b) four
pole synchronous machine.
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Fig.2.3. Air gap magnetic field distribution in the
cross-section of salient pole synchronous machine.
Fig.2.4. Flux density for x air gap position.
By changing the shape of pole shoe (air gap thickness) the sinusoidal distribution of flux
density (magnetic induction) is obtained.
Field ampere-turn:
where:
where:
nf
Θf
If ,
p
nf – number of wire turns
p – number of pole pairs.
(2.1)
Θf
(2.2)
U
U
,
x
H xlx ,
x
Hx – magnetic field strength,
lx – flux path length.
Magnetomotive force:
Ff
Air gap field ampere-turn:
U
where:
1
Θf
2
f
H
1
2
fx
l
U
x
H
x
.
(2.3)
k
fx c
x
,
(2.4)
kc – Carter coefficient (1,05 – 1,2) that represents tooth-grove
structure of the stator.
Neglecting the drop of magnetic potential in the magnet core and yoke results in:
U
f
Ff ,
(2.5)
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So flux density in the air gap:
Bx
H
x
B
nf
fx
2 pkc
If .
(2.6)
x
In Fig.2.5 the 2p rotor of small power synchronous machine and in Fig.2.6 shape of rotor
lamination is presented. Pole shoe holes are prepared for rotor rotational speed stabilization
cage.
Fig.2.5. Four pole small synchronous machine.
Fig.2.6. Four pole rotor lamination.
Non-salient pole generators.
Fig.2.7. The mmf of a distributed wondong on the rotor of non-salient pole generator.
Due to the winding distribution and/or air gap shape (Fig.2.8, Fig.2.9) resultant mmf is
very close to the space fundamental mmf that is stationary in respect to the rotor:
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Ff 1
Ff t
(2.7)
In practice higher tooth harmonics of induced stator winding voltage are dumped and
output voltage is sinusoidal.
Fig.2.8. Non-uniform rotor groves distribution.
Fig.2.9. Non-uniform air gap distribution.
Turbogenerator - generator consisting of a steam turbine coupled to an electric generator for
the production of electric power. Normally 2p = 2 so p = 1 for 3000 rpm turbogenerators.
Fig.2.10. Turbogenerator inside coal power plant.
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Data of generator
GTHW 360:
Power - 426 MVA
Real power - 370 MW
Output voltage - 22 kV
Armature current 11180A
Power factor - 0,85
Field current - 2800 A
Field voltage - 533 V
H2 pressure - 0,3 MPa
Fig.2.11. Construction of typical turbogenerator.
In turbogenerators as well as in hydrogenerators oil bearings are used. Oil (fluid) bearings
are frequently used in high load, high speed or high precision applications where ordinary ball
bearings have short life or high noise and vibration. They are also used increasingly to reduce
cost.
Fig.2.12. Stator with winding.
Fig.2.13. Head part of stator winding.
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Hydrogenerator - generator consisting of a water turbine coupled to an electric generator for
the production of electric power. Normally multi-salient-pole constructions, for slow rotor
speed. Vertical position of the water turbine shaft determine also vertical electric generator
position.
Fig.2.14. Hydrogenerator inside water power plant.
Fig.2.15. Cross-section of hydrogenerator.
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Fig.2.16. Synchronous generator widing.
Due to the high flux density in the air gap stator windings are one turn ones. For the
purpose of minimalization of additional copper losses due to the non-uniform current density,
winding bar is made of several parallel branches with transposition.
Fig.2.17. Winding bar parallel branches transposition.
Fig.2.18. Head part of the winding has to be
protected against the centrifugal forces.
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Fig.2.19. Salient pole
rotor with exciter.
The exciter is the "backbone" of the generator control system.
Two basic kinds of excitors:
rotating and static.
Rotating exciters:
brush exciter – require sliprings, commutators, brushes and
require periodic maintenance,
brushless exciter– do not require
slip-rings, commutators, brushes
and are practically maintenance
free.
Static exciters (static excitation
means no moving parts. It
provides faster transient
response than rotary exciters):
Shunt Type – operating field
power is delivered from
generator output voltage,
Series Type – operating field
power is delivered from
generator output voltage &
current.
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It is the power source that supplies the dc magnetizing current to the field windings of a
synchronous generator thereby ultimately inducing ac voltage and current in the generator
armature.
2.2. Rotor and stator cooling problems.
Depending upon rating and design, the generator stator core and windings may be cooled
by air, oil, hydrogen or water. For direct cooled generators, the coolant is in direct contact
with the heat producing members such as the stator winding. For indirect cooled generators,
the coolant cools the generator by relying on heat transfer through the insulation. For any
generator, a failure of the cooling system can result in rapid deterioration of the stator core
lamination insulation and/or stator winding conductors and insulation.
Turbogenerators up to 50 MW are equipped with closed circulation air cooling system.
Positioned at the end of rotor barrel, fan blades force air to circulate through cooling canals in
the rotor as well as in the stator. Cooling system with air-to air heat exchanger is illustrated in
Fig.2.20 (radial stator cooling canals are visible).
Increasing of generator power output (up to 1500 MW and higher) is possible by
increasing cooling efficiency. For this purpose hydrogen is used because of better ability of
heat acceptance (under the pressure of 5 atmospheres). In some constructions also deionized
water (flowing through canals in the winding bars) is used.
Fig.2.20. Radial stator and rotor cooling canals of machine cooling system.
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Fig.2.21. Canals in stator winding for direct water/hydrogen cooling.
Air cooled generators
Air cooled generators are produced in two basic configurations: open ventilated (OV) and
totally enclosed water-to-air-cooled (TEWAC). In the OV design, air is drawn from outside
the unit through filters, passes through the generator and is discharged outside the generator.
In the TEWAC design, air is circulated within the generator passing through frame mounted
air-to-water heat exchangers.
Hydrogen cooled generators
Hydrogen has attractive characteristics as a fluid to bathe the windings of the generator,
and to remove heat from the windings and deliver that heat to the cooling water. Hydrogen is
nearly the perfect cooling gas, except for being flamable:
- lowest density gas yields lowest drag (drag is proportional to fluid density, and
related to the square of the velocity),
- highest heat conductivity of any gas,
- controlled atmosphere to maintain Clean & Dry,
- inexpensive,
- easy to detect,
- excellent electrical properties,
- easy to manage – not readily miscible with CO2 purge gas.
Hydrogen has a wide flammability range (4% to 75% hydrogen in air is hydrogen
flammability range). Unlike most applications involved with flammable gases, where the
effort is to keep the gas below the LFL (4%), the safety of hydrogen generator cooling is
based on staying above the UFL (75%). Operating above the UFL require experience in
operating a system. Every leak is a source of concern – because the leak will pass through the
flammability envelope. Leak detection and mitigation is critical.
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2.22. Hydrogen cooled generator – sealed cavity – recirculating pressurized hydrogen atmosphere.
2.3. Surge short-circuit of synchronous generator.
Surge short-circuit of synchronous generator is instantaneous, direct short circuit terminals
of rotating, exciting synchronous machine. Such a short-circuit results in huge currents in the
machine circuits dangerous because of high dynamic forces proportional to the square of
current affecting the head part of windings. Surge short-circuit current depends on the flux
linkages of all possible electrical circuits (i.e. field flux value, armature flux value, rotor
position). Expression for surge short-circuit current has a form:
iz izp iza i i izu iza
(2.8)
where: izp – periodic short-circuit current component,
iza – aperiodic short-circuit current component.
i‖ – short-circuit current subtransient component (i‖ represents difference between
short-circuit current periodic component of generator with damping cage and shortcircuit current periodic component of generator without damping cage,
i’ – short-circuit current transient component (i’ represents difference between shortcircuit current periodic component of generator without damping cage and steady state
short-circuit current component,
izu – steady state short-circuit current component,
iz
2U if
1
Xd
1
e
Xd
t
Td
1
Xd
1
e
Xd
t
Td
1
cos( t
Xd
0
)
1
cos( 0 )e
Xd
t
Ts
, (2.9)
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where: Uif – stator induced voltage
X‖d – direct axis subtransient reactance nearly equal to leakage reactance Xl (X‖d is the
reactance with which synchronous machine opposes the current in the first instance of
time of surge short-circuit),
X’d – direct axis transient reactance (reactance with which synchronous machine
without damping cage opposes the periodic current component in the first instance of
time of surge short-circuit),
T‖d – time constant of current decaying in the damping cage with short-circuited stator
and excitation circuits,
Xd – direct axis reactance equal to the sum of leakage reactance Xl and armature
reaction reactance Xad,
T’d – time constant of excitation circuit for short-circuited stator windings (T’d >> T‖d),
γ0 – angle between phase axis and direct rotor axis
Ts – time constant of short-circuited stator phase winding (Ts > T’d).
Surge short-circuited current waveforms of synchronous generator are presented in
Fig.2.23. Current waveforms presents short-circuit phenomena for general generator
conditions, that means that rotor position in time instant of short-circuit keeps position
between direct and quadrature axis.
Fig.2.23. Surge short-circuited current waveform of synchronous generator.
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2.3.1 Circuit parameters of synchronous machine (definition and evaluation).
-
Machine impedance –
where:
-
U phN
,
I phN
UphN – nominal phase voltage,
IphN – nominal phase current.
Zn
(2.10)
Resistance of phase winding (temperature 75 oC) –
Rt75,
Resistance of excitation winding (temperature 75 oC) – Rf75,
Synchronous reactance Xd,
Synchronous reactance Xq,
Poter reactance Xp (Xp is an equivalent to leakage reactance Xl),
Symmetrical components reactance (zero sequence reactance X0, positive sequence
reactance X1, negative sequence reactance X2),
Synchronous subtransient reactance X‖d,
Synchronous subtransient reactance X‖q,
Synchronous reactance Xd.
Direct axis synchronous reactance is a reactance that represents magnetic conductivity of
all armature flax: Xad (direct axis armature reaction reactance) and Xal (leakage reactance):
(2.11)
X d X ad X al .
Fig.2.24. Equivalent circuit of synchronous machine.
Fig.2.25. Direct axis flux
distribution
Direct axis synchronous reactance Xd can be defined as quotient of fundamental induced
(by total direct armature flux) voltage wave and fundamental periodical current component in
steady state of generator operation in direct rotor position:
Xd
U if
I
.
(2.12)
d position
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Synchronous reactance Xq.
Similarly quadrature axis synchronous reactance is a reactance that represents magnetic
conductivity of all armature flax: Xaq (direct axis armature reaction reactance) and Xal
(leakage reactance):
Xd
X aq
X al .
(2.13)
Quadrature axis synchronous reactance Xq can be defined as quotient of fundamental
induced (by total direct armature flux) voltage wave and fundamental periodical current
component in steady state of generator operation in quadrature rotor position:
Xd
U if
I
.
(2.14)
q position
Above definitions leads to measurement method of Xd and Xq determination. The method
is called small slip method. Generator is supplying with symmetrical voltage equal 0,1 ÷ 0,2
U. Excitation winding terminals are connected to voltmeter. Generator rotor should rotate in
the same direction as armature resultant flux with velocity ωr that is close to the synchronous
speed ω (slip value around 0,01). Rotor rotation is secured by coupling DC motor.
Fig.2.26. Measurement of Xd and Xq parameters.
Very small slip cause slow change of direct axis position in respect to the rotating magnetic
field. That means that periodically with slip frequency permaence of flux path changes
respectively from value Λad to Λaq. Permeance change is equivalent to the armature reaction
reactance change from Xad to Xaq and in consequence synchronous reactance from Xd to Xq. As
a result of above changes also the armature current Ia changes from minimum to the
maximum value as well as indispensable - supply voltage U and voltage induced in the field
winding Uf. Zero value of Uf means direct axis rotor position, Uf = Ufmax means quadrature
rotor position. According to the above equation:
Xd
U max
,
3I min
Xq
U min
.
3I max
(2.15)
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Fig.2.27. Quadrature axis flux distribution.
Synchronous subtransient reactance X”d and X”q.
From synchronous subtransient reactance X‖d and X‖q definition it is clear that the best
way to determine them is to perform surge short-circuit of synchronous generator. Since the
method is difficult to perform the static method is proposed. For direct and quadrature rotor
position the oscillatory ampere-turn is forced (supply voltage of value equel to (0,2 ÷ 0,7)U is
applied to two phase generator terminals). Excitation winding is short-circuited with amperemeter – the induced current appears. Flux produced changes with high frequency and the
phenomena is similar to the surge short-circuit. The corresponding formulas are as follow:
Xd
U max
,
2I min
Xq
U min
.
2I max
(2.17)
Fig.2.28. Measurement of X‖d and X‖q parameters.
Zero sequence reactance X0.
Zero sequence reactance X0 is the parameter that characterizes unsymmetrical one-phase
state of generator operation. Zero sequence reactance X0 is the reactance that opposes the zero
current sequence. This definitions leads to measurement method in which all three phase
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windings are supplied from the same voltage source and fixed rotor excitation winding is
short-circuited.
Fig.2.29. Measurement of zero sequence reactance.
The corresponding formula is as follow:
3U
.
I
X0
(2.18)
2.4. Electromagnetic torque in respect to the load angle.
Dependence of power of synchronous machine on load angle can be evoked from
simplified synchronous machine vector representation (Fig.2.30).
Fig.2.30. Vector representation of synchronous machine.
Ui
U s cos
U sin
X q Iq
X d Id
.
(2.19)
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Since synchronous machine power is equal:
(2.20)
P mUs I cos(
),
expression describing electromagnetic torque in respect to the load angle results:
Tem
U
3 s
2
UU
3 s i sin
Xd
Synchronous torque
2
1
Xq
1
Xd
sin 2
.
(2.21)
Reluctance torque
Fig.2.30. Electromagnetic torque in respect to the load angle.
From expression 2.21 it is evident than if Xd ≠ Xq reluctance torque appears.
2.5. Equivalent circuit of synchronous machine
On the base of equivalent circuit of synchronous machine,
Fig.2.31. Equivalent circuit of synchronous motor.
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with assumptions concerning rotational speed:
n = nN = const,
following parameters of synchronous machine can be evaluated:
Induced voltage:
Ui
Nominal induced voltage:
U iN
U iN
If
I fN
,
(U phN
X s I N sin
N
)2 ( X s I N cos
N
)2 ,
Phazor`s diagram:
Real power:
Pin
operating angle:
Stator current: I s
30
nN T 3I phs Rs ,
ZsIs
arcsin
PX s
,
3U phNU iN
1
U i2 U s2 2U sU i cos
Xs
Electromagnetic torque:
Tem
cΦ sin ,
Us
Ui
,
Is
Glossary of Synchronous Machine Terms
AC: Alternating current.
Armature: The part of an electrical machine in which emf is generated and the load current flows.
Armature reaction: Effect of armature current on the resultant magnetic field distribution in an
electrical machine.
Automatic voltage regulator (AVR): A device that senses the terminal voltage and adjusts the
field excitation so that the terminal voltage is maintained at the specified value. For a gridconnected synchronous machine, the AVR is used for reactive power control and to improve
stability.
Busbars: The locations to which synchronous generators and loads are connected in a power
system.
DC: Direct current.
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Field: The part of an electrical machine producing the magnetic flux.
Load angle: This is the electrical angle between the no-load voltage E and the terminal voltage V
f
of a synchronous machine.
Phasor diagram: Diagram showing the relationship between electrical quantities, the concise
phasor representation being used for each quantity.
Synchronous speed: This is the speed of the armature rotating mmf. If the frequency of the
armature current is f Hz and the number of pole pairs is p, then the synchronous speed is equal to
f/p revolutions per second.
Synchronous reactance: This is a hypothetical internal reactance used in the per-phase
equivalent circuit model of a synchronous machine. The synchronous reactance is the sum of the
armature reaction reactance and leakage reactance of the armature winding. The typical value of
synchronous reactance is 1.5–2.0 p.u.
Voltage regulation: For a synchronous generator, the voltage regulation is the voltage rise at the
terminals when a given load is removed, both the speed and field excitation remaining unchanged.
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3. Asynchronous Machines (Induction Motors)
Induction motors are the simplest and most rugged of all electric motors. They consist of
two basic electrical assemblies: the wound stator and the rotor assembly.
Fig.3.1. Cross-section of low power induction motor.
Fig.3.2. Placement of distributed stator winding in the stator slots.
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The rotor consists of laminated, cylindrical iron cores with slots for receiving the
conductors. On early motors, the conductors were copper bars with ends welded to copper
rings known as end rings.
Fig.3.3. Squirrel cage made of copper bars with ends welded to copper rings.
Viewed from the end, the rotor assembly resembles a squirrel cage, hence the name
squirrel- cage motor is used to refer to induction motors. In modern induction motors, the
most common type of rotor has cast-aluminum conductors and short-circuiting end rings.
Fig.3.4. Cast-aluminum squirrel cage.
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The rotor turns when the moving magnetic field induces a current in the shorted
conductors. The speed at which the magnetic field rotates is the synchronous speed of the
motor and is determined by the number of pair of poles in the stator and the frequency of the
power supply:
60 f1
, [min-1],
(3.1)
ns
p
where: ns – synchronous speed,
f1 – frequency,
p – number of pair of poles.
Fig.3.5. Induction motor power in respect of number of pole pairs.
Synchronous speed is the absolute upper limit of motor speed. At synchronous speed,
there is no difference between rotor speed and rotating field speed, so no voltage is induced in
the rotor bars, hence no torque is developed. Therefore, when running, the rotor must rotate
slower than the magnetic field. The rotor speed is just slow enough to cause the proper
amount of rotor current to flow, so that the resulting torque is sufficient to overcome windage
and friction losses, and drive the load. This speed difference between the rotor and magnetic
field, called slip, is normally referred to synchronous speed:
s
ns
nr
ns
s 1
break
1 s 0
motor
s 0
generator ,
s 1
transforme r
s 0
coil
(3.2)
where: s = slip,
ns – synchronous speed,
nr – rotor speed.
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In consequence rotor speed can be expressed as:
nr
ns (1 s) , or nr
60 f s
(1 s) ,
p
(3.3)
while rotor frequency as:
(3.4)
f r sf s .
Stator induced voltage is proportional to the synchronous speed and resultant flux:
Ui 4,44kB N s f s kwsΦ ,
where: kB – flux distribution factor,
Ns – number of stator phase winding turns,
kws – stator winding factor,
and rotating rotor induced voltage is proportional to the rotor speed:
(3.5)
Uir 4,44kB Nr sf s kwrΦ ,
where: Nr – number of rotor phase winding turns,
kwr – rotor winding factor,
(3.6)
Fig.3.6. Equivalent circuit of induction motor.
Induction motor equivalent circuit (Fig.3.6) that represents also mechanical power produced
in the machine Pm + Pout that is proportional to the rotor real resistance R2:
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1 s
R2 ,
s
where: Pm – mechanical losses,
Pout – power on the motor shaft,
Pm
(3.7)
Pout
allows to formulate power balance equations.
Input power:
Copper losses in the stator windings:
Pin msU phs I phs cos ,
where: ms – number of stator phases,
Uphs – stator phase voltage,
Iphs – stator phase current,
φ – power factor,
Pus
2
ms Rs I phs
,
(3.8)
(3.9)
where: Rs – stator phase winding resistance,
Rotor and stator iron losses depends on performance of used magnetic materials. New
technologies allow producing soft magnetic laminations with high saturation level and low
losses. Additionally new glued lamination technologies (combination of isolation and
oxidation layers) allow reaching maximum package densities for highest induction values. In
Fig 3.7 the comparison of iron losses for standard (Silicon-Iron) magnetic materials and
Cobalt-Iron magnetic materials with thinnest isolation layers is presented.
Fig.3.7. Reduction of iron losses due to better magnetic material and improved stacking technology.
Electromagnetic power:
Pem
Pin Pus PFe Pad ,
(3.10)
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where: PFe – core losses,
Pad – additional loses,
Copper losses in rotor windings:
Output power:
Pur
sPem , (for s=1 Pur
Pem ),
Pout Pem Pur Pm , and Pout
where: ω = 2πfs – rotor speed,
T – torque.
Torque on the shaft:
T
Efficiency:
Pout
60 Pout
2 n
Pout
Pin
Pin
9,55
Plosses
Pin
(3.11)
T , (3.12)
Pout
,
n
(3.13)
,
(3.14)
Motor efficiency depends on the motor nominal power as it is shown in Fig.3.8
Fig.3.8. Dependence of induction motor efficiency on motor nominal power.
Assuming that RFe >> Rs, Xf >> Xs and R’r/s >> Rs, following parameters can be derived:
Electromagnetic torque:
ms Rr' U s2
Tem
s
s Xs
X
' 2
r
' 2
2
(R )
s
,→ T
U2,
(3.15)
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Critical torque (maximal torque):
Critical slip:
Tkr
2
msU phs
2
s
X r' )
(Xs
,
Rr'
,
X s X r'
(if R’r = Xs + X’r → skr = 1 and Tem = Tkr).
skr
(3.16)
(3.17)
From equations 3.15 and 3.16,
Kloss equation can be formulated:
Motor overload:
Tem
Tkr
2
s
skr
skr
s
.
Tkr
,
Tld
(typically u > 2).
u
(3.18)
(3.19)
Polyphase induction motors — are classified according to locked rotor torque and current,
breakdown torque, pull up torque, and percent slip.
Fig.3.9. Speed – torque characteristic.
- Locked rotor torque is the minimum torque that the motor develops at rest for all angular
positions of the rotor at rated voltage and frequency.
- Locked rotor current is the steady state current from the line at rated voltage and
frequency with the rotor locked.
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- Breakdown torque is the maximum torque that the motor develops at rated voltage and
frequency, without an abrupt drop in speed.
- Pull up torque is the minimum torque developed during the period of acceleration from
rest to the speed that breakdown torque occurs. Various speed-torque characteristics are
presented in Fig.10.
Fig.3.10. Typical speed-torque characteristics of induction motors.
- Characteristic A - motors have a higher breakdown torque than characteristic B motors
(most common characteristic) and are usually designed for a specific use. Slip is 5%, or
less. Better torque –speed characteristics are achieved by developing copper die-cast rotor
conductor (Fig.3.11).
Fig.3.11. Copper die-cast rotor conductor.
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Increase of motor efficiency is the result of a lower rotor resistance resulting in lower
rotor losses and longer bearing life. Motor with die-cast-copper rotor efficiency in
comparison to die-cast aluminium rotor is presented in Fig.3.12.
Fig.3.12. Calculated efficiency of 3 kW, 4 pole induction motor.
- Characteristic B motors account for most of the induction motors sold. Often referred to
as general purpose motors, slip is 5% or less.
-Characteristic C motors have high starting torque with normal starting current and low
slip. This design is normally used where breakaway loads are high at starting, but
normally run at rated full load, and are not subject to high overload demands after running
speed has been reached. Slip is 5% or less.
-Characteristic D motors exhibit high slip (5 to 13%), very high starting torque, low
starting current, and low full load speed. Because of high slip, speed can drop when
fluctuating loads are encountered. This design is subdivided into several groups that vary
according to slip or the shape of the speed-torque curve. These motors are usually
available only on a special order basis.
Wound-rotor motors — Although the squirrel-cage induction motor is relatively inflexible
with regard to speed and torque characteristics, a special wound-rotor version has controllable
speed and torque. Application of wound-rotor motors is markedly different from squirrel-cage
motors because of the accessibility of the rotor circuit. Various performance characteristics
can be obtained by inserting different values of resistance in the rotor circuit.
Wound rotor motors are generally started with secondary resistance in the rotor circuit.
This resistance is sequentially reduced to permit the motor to come up to speed. Thus the
motor can develop substantial torque while limiting locked rotor current. The secondary
resistance can be designed for continuous service to dissipate heat produced by continuous
operation at reduced speed, frequent acceleration, or acceleration with a large inertia load.
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External resistance gives the motor a characteristic that results in a large drop in rpm for a
fairly small change in load (characteristic D). Reduced speed is provided down to about 50%,
rated speed, but efficiency is low.
Single-phase motors — These motors are commonly fractional-horsepower types, though
integral sizes are generally available to 10 hp. The most common single phase motor types are
shaded pole, split phase, capacitor-start, and permanent split capacitor.
- Shaded pole motors have a continuous copper loop wound around a small portion of
each pole (Fig.3.13).
Fig.3.13. Shaded pole induction motor.
The loop causes the magnetic field through the ringed portion to lag behind the field in the
unringed portion. This produces a slightly rotating field in each pole face sufficient to turn the
rotor. As the rotor accelerates, its torque increases and rated speed is reached. Shaded pole
motors have low starting torque and are available only in fractional and subfractional
horsepower sizes. Slip is about 10%, or more at rated load.
- Split phase motors, use both a starting and running winding (Fig.3.14).
Fig.3.14. Split phase induction motor.
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The starting winding is displaced 90 electrical degrees from the running winding. The running
winding has many turns of large diameter wire wound in the bottom of the stator slots to get
high reactance. Therefore, the current in the starting winding leads the current in the running
winding, causing a rotating field. During startup, both windings are connected to the line.
Fig.3.15. Split phase induction motor windings connection.
As the motor comes up to speed (at about 25% of full-load speed), a centrifugal switch
actuated by the rotor, or an electronic switch, disconnects the starting winding (Fig.3.15).
Split phase motors are considered low or moderate starting torque motors and are limited to
about 200 W.
- Capacitor-start motors are similar to split phase motors. The main difference is that
a capacitor is placed in series with the auxiliary winding (Fig.3.15). This type of motor
produces greater locked rotor and accelerating torque per ampere than does the split phase
motor. Sizes range from fractional to 5 kW at 750 to 3000 rpm.
- Split-capacitor motors also have an auxiliary winding with a capacitor, but they
remain continuously energized and aid in producing a higher power factor than other
capacitor designs. This makes them well suited to variable speed applications.
3.1 Condition of operation of Induction Generator.
Induction generator can work being integrated with the energy system or independently
(autonomously). For the over-synchronous speed (n > nn, s < 0), induction machine enters
generator state of operation with changed direction of induced voltage Uir, changed direction
of power flow (delivered to the rotor mechanical power is converted into electrical power Pe –
equivalent of Tem, that is transferred to the stator), changed direction of active current, but
unchanged direction of reactive (magnetizing) current (Fig.3.16).
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Fig.3.16. Torque-speed characteristic with generating operation area.
That means that induction generator delivers active power to the grid but (similarly like
induction motor) absorbs reactive inductive power from grid. This is the main disadvantage of
induction generator. Another disadvantage is that induction generator can’t individually force
voltage frequency.
Fig.3.17. Operation of induction motor as induction generator.
Using external drive with regulated rotational speed, it is possible to force synchronous
state of operation of induction motor by increasing motor rotational speed up to synchronous
speed (s=0). Drive system delivers power to cover all mechanical losses in induction motor
(bearings and ventilation), and balancing parasitic electromagnetic torques. On the other side
grid delivers active power to cover winding and core losses as well as reactive power
necessary to build up flux in the motor (Fig.3.17).
After increasing shaft speed over synchronous speed (n > ns) slip became negative (s < 0)
and internal motor power became negative:
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Pi
ms
Rr 2
Ir ,
s
(3.20)
so machine starts to operate as a generator.
While operating autonomously there is necessity to deliver reactive power, source of
reactive power could be external block of capacitors connected to the motor phases. External
capacitors capacity is determined by two border states of load:
- Minimum value of connected capacitor Cmin is determined by it’s ability to deliver
reactive magnetizing power to the motor in the no load state. This could be analyzing with the
help of simplified induction generator equivalent circuit (Fig.3.18),
Fig.3.18. Simplified induction generator equivalent circuit.
With assumptions of negligible value of stator resistance Rs in comparison to stator reactance
Xf and Xs:
Rs
Xs
Xf ,
(3.21)
voltage-current equations has the form:
U
Ui
Ui
jX s I f
jX f I f
,
(3.22)
Uc
jX c I c
where: Xc – capacitor reactance.
Capacitor current became magnetizing current:
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Ic
U
Xc
If
U
Xs
Xf
,
(3.23)
so reactance of connected capacitors should be equal to the total motor reactance:
Xc
Xf
1
C
Xs
Ls
Lf
Cmin
1
2
( Ls
Lf )
.
(3.24)
- Maximum value of connected capacitor Cmax is determined by maximum load state
of induction generator and reactive value of load current:
Ic I f I L ,
where: IL – load reactive current.
(3.25)
Assuming generator overload up to 1,2 nominal current for determined power factor cosφ of
induction character current equation has form:
Ic
If
I load cos ,
(3.26)
so
If
Cmax
I load cos
U
,
(3.27)
and finally:
1
2
( Ls
Lf )
C
If
I load cos
U
.
(3.28)
Border values of capacitance C delivering necessary reactive power to the induction generator
together with magnetizing characteristic of induction motor/generator are presented in
Fig.3.19). Crossing points A1, A2, A3 of three values of capacitances C1, C2, C3 characteristics
with magnetizing characteristic determine generator operational voltage in no load state. For
capacitance value C2 and generator load IL voltage on the generator terminals is described by
point A. Voltage point A of generator operation is consequence of decreasing magnetizing
current (part of magnetizing current – reactive current produced by capacitor C is absorbed by
load). Necessary condition to build up voltage of autonomously working induction generator
is preexistence of remanence magnetization that is representing in Fig.3.19 by remanence
voltage UR.
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Fig.3.19. Determination of operating point of induction generator using magnetization and capacitor
characteristics.
Exemplary external characteristic of induction generator are presented in Fig. 3.20.
External characteristic is the function of generator terminals voltage on load current for
constant value of connected capacitors.
Fig.3.20. External characteristics of induction generator.
External characteristics of induction generator working autonomously characterize low
stiffness – or in another words big voltage change between no load and full load state of
operation. Low characteristic stiffness is determined by the drop of voltage on internal
generator impedance and decrement of magnetizing current with increase of load current of
inductive character.
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4. Wind Turbines
4.1 Wind Power
Wind power is the conversion of wind energy into a useful form of energy, such as using
wind turbines to make electricity, wind mills for mechanical power, wind pumps for pumping
water or drainage, or sails to propel ships.
Large-scale wind farms (Fig.4.1) are connected to the electric power transmission network,
smaller facilities are used to provide electricity to isolated locations. Utility companies
increasingly buy back surplus electricity produced by small domestic turbines. However, wind
power is non-dispatchable, meaning that for economic operation, all of the available output
must be taken when it is available. Other resources, such as hydropower, and standard load
management techniques must be used to match supply with demand. The intermittency of
wind seldom creates problems when using wind power to supply a low proportion of total
demand.
Even though the cost of wind power has decreased dramatically in the past 10 years, the
technology requires a higher initial investment than fossil-fueled generators. Roughly 80% of
the cost is the machinery, with the balance being the site preparation and installation. If windgenerating systems are compared with fossil-fueled systems on a "life-cycle" cost basis
(counting fuel and operating expenses for the life of the generator), wind power costs are
much more competitive with other generating technologies because there is no fuel to
purchase and minimal operating expenses.
Fig.4.1. Large-scale coast wind farm.
Wind Power Density (WPD) is a calculation relating to the effective force of the wind at a
particular location, frequently expressed in terms of the elevation above ground level over a
period of time. It takes into account wind velocity and mass. Color coded maps are prepared
for a particular area described, for example, as "Mean Annual Power Density at 50 Meters."
The larger the WPD calculation, the higher it is rated by class. In Fig.4.3 mean wind velocity
map in Europe is presented.
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Fig.4.2. Map of the effective force of the wind due to the direction.
Fig.4.3. Color map presenting mean wind velocity in Europe.
Wind power is nonlinear function of wind velocity as it is presented in Fig.4.4. Wind turbines
locations with constantly high wind speeds bring best return on investment. With a wind
resource assessment it is possible to estimate the amount of energy the wind turbine will
produce.
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Fig.4.4. Wind power as a function of wind velocity.
Fig4.5. Wind speed over Polish terrytory (autumn).
The first known electricity generating windmill operated was a battery charging machine
installed in 1887 by James Blyth in Scotland. A forerunner of modern horizontal-axis wind
generators was in service at Yalta, USSR in 1931. This was a 100 kW generator on a 30 m
tower, connected to the local 6.3 kV distribution system. It was reported to have an annual
capacity factor of 32 per cent, not much different from current wind machines. The modern
wind power industry began in 1979 with the serial production of wind turbines by Danish
manufacturers Kuriant, Vestas, Nordtank, and Bonus. These early turbines were small by
today's standards, with capacities of 20–30 kW each. Since then, they have increased greatly
in size (Fig.4.5, Fig.4.6), while wind turbine production has expanded to many countries.
Fig.4.5. 20 years of development of wind turbines (capacitances grew from 20 kW to 4,5 MW).
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Fig.4.6a. Wind Generator – rated power 5 MW, rotor (propeller) diameter 116 m, weight 290 t.
Fig.4.6b. Multi-pole generator construction.
Wind energy as a power source is attractive as an alternative to fossil fuels, because it is
plentiful, renewable, widely distributed, clean, and produces no greenhouse gas emissions. In
spite of the fact that the construction of wind farms is not universally welcomed because of
their visual impact and other effects on the environment installed wind power grows
expotentially (Fig.4.7).
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Fig.4.7. Installed wind power in the years 1990 – 2002.
At the end of 2008, worldwide nameplate
capacity (installed power) of wind-powered
generators was 121.2 gigawatts (GW), which is
about 1.5% of worldwide electricity usage, and
is growing rapidly, having doubled in the three
years between 2005 and 2008. Several countries
have achieved relatively high levels of wind
power penetration, such as 19% of stationary
electricity production in Denmark, 11% in
Spain and Portugal, and 7% in Germany and the
Republic of Ireland in 2008. As of May 2009,
eighty countries around the world are using
wind power on a commercial basis. The total
amount of economically extractable power
available from the wind is considerably more
than present human power use from all sources.
An estimated 72 terawatt (TW) of wind power
on the Earth potentially can be commercially
viable, compared to about 15 TW average
global power consumption from all sources in
2005. Not all the energy of the wind flowing
past a given point can be recovered. Because so
much power is generated by higher wind speed,
much of the energy comes in short bursts – half
of the energy available arrived in just 15% of
the operating time. The consequence is that
wind energy from a particular turbine or wind
farm does not have as consistent an output as
fuel-fired power plants.
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4.2 Wind Turbines and the Wind Energy
A wind turbine converts the kinetic energy in wind into the mechanical energy of a
rotating shaft. In conformity with the first law of thermodynamics the ―energy out‖ of a wind
turbine has to equal the ―energy in‖. The ―energy in‖ is the kinetic energy from the wind's
velocity and air density. It is not possible to convert all of the wind's kinetic energy into
mechanical energy. Some energy must remain in the wind (otherwise wind turbine can’t
operate). The "energy out" is the energy converted by the turbine blades into mechanical
energy then to the electrical energy, plus energy is left in the air after it passes through the
turbine rotors.
Fig.4.8. Schematic of fluid flow through wind turbine.
Wind turbine power Pw (assuming very thin propeller, incompressible and axial flow) can be
expressed as:
1
D2 V 3 C g b ,
2
4
where: ρ – air density,
D – diameter of turbine blades,
V – velocity of the wind passing through the turbine blades,
C – constant – Betz' Law limiting factor,
ηg – electric generator efficiency,
ηb – gearbox efficiency.
Pw
(4.1)
Constant C shows the maximum possible that may be derived by means of an infinitely
thin rotor from a fluid flowing at a certain speed. According to the mass conservation
principle, the mass flow rate is given by:
m
A1v1
Sv
A2 v2 ,
(4.2)
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where: A1 – area in front of the turbine
v1 – speed in the front of the turbine,
S – area of the turbine,
v – speed at the fluid power device,
A2 – area downstream of the turbine,
v2 – speed downstream of the turbine.
The force exerted on the wind by the rotor may be written as:
F
ma
m
dv
dt
m v
Sv(v1 v2 ) .
(4.3)
Incremental work done by the force may be written as:
dE
F dx ,
(4.4)
so the power content in the wind is:
P
dE
dt
F
dx
dt
Fv .
(4.5)
Substituting the force writen in the form 4.3 yelds:
P
Sv 2 (v1 v2 ) .
(4.6)
Power can also be computed usin gthe kinetic energy:
P
dE
dt
1
m (v12 v22 ) .
2
(4.7)
Taking into account continuity equation 4.2 a substitution for the mass flow rate yields the
following:
P
1
Sv(v12 v22 ) .
2
(4.8)
Equating expressions 4.6 and 4.8 yields:
1 2
(v1 v22 )
2
1
(v1 v2 )( v1 v2 )
2
v(v1 v2 )
v
1
(v1 v2 ) .
2
(4.9)
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So the wind velocity at the rotor may be taken as the average of the upstream and downstream
velocities. Power based on kinetic energy can be expressed as follow:
E
1
Sv(v12 v22 )
2
1
Sv13 1
4
v2
v1
2
1
S (v1 v2 )(v12 v22 )
4
v2
v1
v2
v1
3
.
(4.10)
By differentiating E with respect to v2/v1 for a given fluid speed v1 and a given area S one
finds the maximum or minimum value for E (Fig.4.9). E reaches maximum value when
v2/v1=1/3.
Fig.4.9. Dependence of the Betz coefficient on the ratio v1/v2.
Substituting v1/v2 by 1/3 results in:
Pmax
16 1
Sv13 .
27 2
(4.11)
In consequence work rate obtainable from a cylinder of fluid with cross sectional area S and
velocity v1 is:
P
1
Sv13C ,
2
(4.12)
The Betz coefficient (coefficient of efficience C(=P/Pmax) has a maximum value of
Cmax=16/27=0.593 (or 59.3%) and shows a limiting factor for this form of renewable energy.
Rotor (propeller) losses are the most significant energy losses in a wind generator. So it is,
essential to reduce these as much as possible. Modern rotors achieve values for C in the range
of 0.4 to 0.5, which is 70 to 80% of the theoretically possible.
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4.3 Constructional constrains.
On the world energy market the most important thing is to get the most electric power for
the least money. This may require design trade-offs that limit efficiency in order to get the
best overall system for the money. For example, it might not be possible to make the most
efficient blade shape strong enough to hold together during a strong wind. As you can see
from the formula 4.1, being able to get power from a stronger wind is probably worth more
than efficiency. On a very windy site, stronger less efficient blades might end up getting us
more power. Engineers have to deal with conflicting design trade-offs all the time.
One should also remember that turbine efficiency varies with wind speed. A given wind
turbine has a "design point" that generally defines its peak efficiency at the wind speed for
which the system is designed. At wind speeds above and below the design speed the
efficiency is the same or less. So wind turbine will generally operate at lower than its best
efficiency, because wind speeds are never constant or average.
Generator efficiency, engineering constraints regarding proppeller, energy storage and
transmission losses and other factors mean that even the best modern turbines operate at
efficiencies substantially below the Betz Limit.
4.4 Wind Turbines.
Turbines used in wind farms for commercial production of electric power are usually
three-bladed and pointed into the wind by computer-controlled motors. These have high tip
speeds of over 320 km/h, high efficiency, and low torque ripple, which contribute to good
reliability. The blades are usually colored light gray to blend in with the clouds and range in
length from 20 to 40 metres or more. The tubular steel towers range from 60 to 90 metres tall.
The blades rotate at 10-22 revolutions per minute. At 22 rotations per minute the tip speed
exceeds 40 metres per second. A gear box is commonly used to step up the speed of the
generator, although designs may also use direct drive of an annular generator. Some models
operate at constant speed, but more energy can be collected by variable-speed turbines which
use a solid-state power converter to interface to the transmission system. All turbines are
equipped with shut-down features to avoid damage at high wind speeds.
4.4.1 Horizontal-axis wind turbines.
Horizontal-axis wind turbines (HAWT) have the main rotor shaft and electrical generator
at the top of a tower (Fig.4.5, Fig.4.11), and must be pointed into the wind. Small turbines are
pointed by a simple wind vane, while large turbines generally use a wind sensor coupled with
a servo motor. Most have a gearbox, which turns the slow rotation of the blades into a quicker
rotation that is more suitable to drive an electrical generator.
Since a tower produces turbulence behind it, the turbine is usually pointed upwind of the
tower. Turbine blades are made stiff to prevent the blades from being pushed into the tower
by high winds (Fig.4.10). Additionally, the blades are placed a considerable distance in front
of the tower and are sometimes tilted up a small amount.
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Fig.4.10. Turbine blades pushed into the tower by high winds.
Fig.4.11. Construction of the HAWT.
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Horizontal axis construction advantages:
- Variable blade pitch, which gives the turbine blades the optimum angle of attack.
Allowing the angle of attack to be remotely adjusted gives greater control, so the turbine
collects the maximum amount of wind energy for the time of day and season.
- The tall tower base allows access to stronger wind in sites with wind shear. In some
wind shear sites, every ten meters up, the wind speed can increase by 20% and the power
output by 34%.
- High efficiency, since the blades always move perpendicularly to the wind, receiving
power through the whole rotation. In contrast, all vertical axis wind turbines, and most
proposed airborne wind turbine designs, involve various types of reciprocating actions,
requiring airfoil surfaces to backtrack against the wind for part of the cycle. Backtracking
against the wind leads to inherently lower efficiency.
- The face of a horizontal axis blade is struck by the wind at a consistent angle
regardless of the position in its rotation. This results in a consistent lateral wind loading over
the course of a rotation, reducing vibration and audible noise coupled to the tower or mount.
4.4.2 Vertical-axis wind turbines.
Vertical-axis wind turbines (VAWT) have the main rotor shaft arranged vertically
(Fig.4.12). Key advantages of this arrangement are that the turbine does not need to be
pointed into the wind to be effective. This is an advantage on sites where the wind direction is
highly variable. With a vertical axis, the generator and gearbox can be placed near the ground,
so the tower doesn't need to support it, and it is more accessible for maintenance. Drawbacks
are that some designs produce pulsating torque.
Fig.4.12. A helical twisted VAWT.
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It is difficult to mount vertical-axis turbines on towers, meaning they are often installed
nearer to the base on which they rest, such as the ground or a building rooftop. The wind
speed is slower at a lower altitude, so less wind energy is available for a given size turbine.
Air flow near the ground and other objects can create turbulent flow, which can introduce
issues of vibration, including noise and bearing wear which may increase the maintenance or
shorten the service life. However, when a turbine is mounted on a rooftop, the building
generally redirects wind over the roof and this can double the wind speed at the turbine. If the
height of the rooftop mounted turbine tower is approximately 50% of the building height, this
is near the optimum for maximum wind energy and minimum wind turbulence.
Vertical axis construction advantages:
- A massive tower structure is less frequently used, as VAWTs are more frequently
mounted with the lower bearing mounted near the ground.
- Designs without yaw mechanisms are possible with fixed pitch rotor designs.
- The generator of a VAWT can be located nearer the ground, making it easier to
maintain the moving parts.
- VAWTs have lower wind startup speeds than HAWTs. Typically, they start creating
electricity at 10 km/h.
- VAWTs may be built at locations where taller structures are prohibited.
- VAWTs situated close to the ground can take advantage of locations where mesas,
hilltops, ridgelines, and passes funnel the wind and increase wind velocity.
- VAWTs may have a lower noise signature.
4.5 Wind Turbines conversion systems.
The major components of a typical wind energy conversion system include a wind turbine,
a generator, interconnection apparatus, and control systems. At the present time and for the
near future, generators for wind turbines will be synchronous generators, permanent magnet
synchronous generators, and induction generators, including the squirrel-cage type and wound
rotor type (Fig.4.13). For small to medium power wind turbines, permanent magnet
generators and squirrel-cage induction generators are often used because of their reliability
and cost advantages. Induction generators, permanent magnet synchronous generators, and
wound field synchronous generators are currently used in various high power wind turbines.
Interconnection apparatuses are devices to achieve power control, soft start, and
interconnection functions. Very often, power electronic converters are used as such devices.
Most modern turbine inverters are forced commutated PWM inverters to provide a fixed
voltage and fixed frequency output with a high power quality (Fig.4.14). Both voltage source
voltage controlled inverters and voltage source current controlled inverters have been applied
in wind turbines. For certain high power wind turbines, effective power control can be
achieved with double PWM (pulse-width modulation) converters which provide a
bidirectional power flow between the turbine generator and the utility grid.
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Fig.4.13. Wind energy conversion systems.
Fig.4.14. Turbine forced commutated PWM inverter.
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5. Permanent Magnet Generator
Permanent magnet generators are regarded as high performance at low RPM and long
lasting solution for both horizontal and vertical wind turbines since there is no need of
external excitation and conductor losses are removed from the rotor. PMG come in a wide
range of powers and are designed for wind turbines application but can also be adapted for
small hydro systems. Basically, PM generators can be divided into radial-flux and axial-flux
machines, according to the flux direction in the air gap. The availability of modern high
energy density magnet materials, such as NdFeB, has made it possible to design special
topologies such as toothless stators with air gap windings In Fig.5.1 the slot and slot-less PM
generator constructions are presented.
a)
b)
Fig.5.1. a) Slotted and b) slot-less radial flux PM generator.
a)
b)
Fig.5.2. Flux distribution in a) slotted and b) slotless PM generator/motor. Flux distribution in slot-less PM
generator reveals lack of cogging torque.
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In Fig.5.3 the different rotor constructions for radial flux PM generator are presented.
Fig.5.3. Types of rotors for radial flux PM generators.
Slotted axial-flux machine (Fig.5.4), the shape of the stator as well as the rotor resembles
a pancake and these machines are commonly referred to as pancake machines. Double stator
slotted axial flux machine consists of two external stators and one inner rotor. The permanent
magnets are axially magnetized and they are surface mounted or inset into a cut window on
the rotor disc. In all axial flux machines, the rotor rotates relative to the stator with the flux
crossing the air-gap in the axial direction. The stator iron core is laminated in the radial
direction and resembles concentric rings that have a constant slot width and tapered teeth.
Fig.5.4. Axial-flux construction with double stators.
Generally axial-flux slotted machines have a smaller volume for a given power rating, making
the power density very high. However, it should be mentioned that as the power rating
increases and the outer radius becomes larger, the mechanical dynamic balance must be taken
into consideration for axial flux machines.
The low speed constructions (multi-pole) are superior to the high speed constructions,
which means that multi-pole PM generators are preferred in the application of small, gearless,
low speed wind system.
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5.1 Permanent Magnets
A permanent magnet is an object made from a material that is magnetized and creates its
own persistent magnetic field. Materials that can be magnetized, which are also the ones that
are strongly attracted to a magnet, are called ferromagnetic (or ferrimagnetic). These include
iron, nickel, cobalt, some rare earth metals and some of their alloys (e.g., Alnico), and some
naturally occurring minerals such as lodestone.
Fig.5.5. Permanent magnets development in last century. (BH max – magnet energy)
Although ferromagnetic (and ferrimagnetic) materials are the only ones attracted to a
magnet strongly enough to be commonly considered magnetic, all other substances respond
weakly to a magnetic field, by one of several other types of magnetism. Some ferromagnetic
materials can be magnetised by a magnetic field but do not tend to remain magnetised when
the field is removed; these are termed soft. Permanent magnets are made from magnetically
hard ferromagnetic materials that stay magnetized.
Characteristics of magnetic materials are prezented on B and H plane and are named
histeresis loop (Fig.5.6).
Fig.5.6. Histeresis loop with magnetization curve.
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The BH plane is divided into quadrants. The first quadrant is often referred to as the
magnetizing quadrant, and the second is often referred to as the demagnetizing quadrant.
The B axis represents magnetic induction field (given in SI units of tesla (T)) or in another
words the magnetic output either of the magnet or the combination of the magnet and the
applied field. The tesla is equivalent to the magnetic flux (in webers) per unit area (in meters
squared), thus giving B the unit of a flux density. In CGS, the unit of B is the gauss (G). One
tesla equals 104 G. The H axis represents the magnetic field H (given in SI units of ampereturns per meter (A-turn/m)) or in another words the magnitude of an externally applied field
to the magnet material. The A-turns appears because when H is produced by a currentcarrying wire, its value is proportional to the number of turns of that wire. In CGS, the unit of
H is the oersted (Oe). One A-turn/m equals 4π×10-3 Oe.
Note that in air, one G = one Oe. Note also the distinction that oersted refers to applied field
and gauss to the measured, resultant (induced) field.
Magnetization M that describes property of magnet is defined as the quantity of magnetic
moment per unit volume, V:
N
m nm ,
V
where: N – number of magnetic moments in the sample of magnetic material,
N/V = n – density of magnetic moments.
M
(5.1)
The origin of the magnetic moments responsible for magnetization can be either microscopic
electric currents resulting from the motion of electrons in atoms, or the spin of the electrons or
the nuclei. Net magnetization results from the response of a material to an external magnetic
field, together with any unbalanced magnetic dipole moments that may be inherent in the
material itself; for example, in ferromagnets.
The magnetization M is given in SI units of ampere per meter (A/m). In CGS, the unit of
M is the oersted (Oe). One A/m equals 10-3 emu. A good permanent magnet can have a
magnetization as large as a million ampere per meter.
In SI units, oblige the relation:
B
M) ,
0 (H
where: μ0 – permeability of space equals 4π×10-7 T·m/A.
(5.2)
In Gaussian units the above equation takes form:
B
H
4 M,
(5.3)
which is frequently considered as much easier for calculations and commonly used.
Materials that are not permanent magnets usually satisfy the linear relation:
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M
H,
(5.4)
where: χ – dimensionless magnetic susceptibility.
In CGS, M = χH, but χSI = 4πχCGS, and μ = μr.
Most non-magnetic materials have a relatively small χ (on the order of a millionth), but
soft magnets can have χ on the order of hundreds or thousands. For materials satisfying
equation 5.4 the relation:
B
)H
H,
0 (1
0 rH
where: μr = 1 + χ – dimensionless relative permeability,
μ = μ0μr is the magnetic permeability,
(5.5)
is also valid. Both hard and soft magnets have a more complex (nonlinear), history-dependent,
relation between B and H, described by hysteresis loops (Fig.5.6, Fig.5.7).
Fig.5.7. Soft and hard magnetic material histeresis loop.
From the point of view of magnets applied in electrical machine the demagnetization
characteristic is most important one. There are two very important points here. First is
Br(H=0). As the H field is reduced, B also falls. The value of B where H=0 is called
remenance, Br. The second is Hc(B=0), or coercive force. When the magnet reaches this
condition, it has no observable (external) field. This is because the applied field H=Hc is
balanced out by the flux M of the magnet material. Because they are in the opposite direction,
the net observable induction B is equal to zero.
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a)
b)
Fig.5.8. a) Dependence of demagnetization characteristic on magnet material:
1 – NdFeB, 2 – SmCo, 3 – Alnico, 4 – Ferrite.
b) Dependence of demagnetization characteristic on magnet temperature.
Demagnetization characteristic depends on magnet material (Fig.5.8a) and temperature
(Fig.5.8b).
5.2 Determination of the operating point of the magnet (calculation of
permanent magnets excitation)
Equivalent circuit of magnetic circuit of PM motors (presented in Fig.5.1) is presented in
Fig.5.9.
Fig.5.9. Equivalent circuit of magnetic circuit consists of permanent magnet, ferromagnetic core and air gap.
Magnetic induction (flux density) insides the magnet according to 5.2 is equal:
Bm
Mm ,
0Hm
where: Hm – magnetic field strength inside magnet,
Mm – magnetization of magnet.
(5.6)
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Magnet magnetization depends on demagnetization curve shape and magnet operating point.
According to Ampere’s law magnetic circuit is described by formula:
lm H m
H
lFe H Fe ,
(5.7)
where: lm – magnet length,
Hm – magnetic field strength determined by magnet operating point,
Hδ – magnetic field strength in the air-gap,
δ – air-gap width.
Magnetic field strength in the air gap and magnetic field strength in the magnetic circuit core
(made of ferromagnetic material) have the same direction, so magnetic field strength inside
magnet has to have opposite direction. Accepting very high permeability of ferromagnetic
lFe H Fe can be neglected:
core μFe=∞ equation 5.7 component
H .
lm H m
(5.8)
On the base of magnetic field strength in the air gap the flux density in the air gap can de
determined:
B
0
H .
(5.9)
Flux density in the air gap and magnet (neglecting flux leakage) is the same:
B
B.
Bm
(5.10)
Taking in to consideration 5.8, 5.9, 5.10 yields:
B
lm
0
Hm .
(5.11)
In consequence assuming the same cross-sections of magnet and air-gap (which is normally
fulfilled), magnet operating point Bd on the demagnetization curve is determined by common
point of characteristic B=f(H) and characteristic of flux density in the air gap (magnet load
line) described by dependence 5.11. For non-operating machine this point is called open
circuit operating point (Fig.5.10).
Any change of the width of air-gap of premanent magnet circuit moves the magnetic
operating point on the demagnetization curve. Especially with steep demagnetization curve
the effect can cause strong dependence of the air gap induction from the air gap width.
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Fig.5.10. Determination of the magnet operating point.
Magnet operating point can be also determined by approximation of demagnetization
curve by the straight line:
H
,
(5.12)
H ca
where: Hca – common point of approximated demagnetization characteristic with axe H.
B
Br 1
Finally magnet operating point Bd describes equation:
H ca Br
0lm H ca Br
.
Bd
, Hd
Br
Br
0lm H ca
0lm H ca
(5.13)
Inclination of magnet load line is described by permeance coefficient P.C. Permanent magnet
flux in no-load state can be determined using formula (Fig.5.11):
d re
m,
2
where: dre – diameter of rotor with permanent magnet poles,
αm – magnet angle.
Φ
B p lm
(5.14)
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Fig.5.11. Magnet pole diameters.
The flux needed for excitation is set by the high, length and angle of the magnet. Stray
factor σ gives the ratio of the magnets total flux to usable flux in the air-gap and usually lies
between 1,5 and 5. In another words stray factor gives information of usable efficiency of
system. The field decay along the flux conducting parts and inner air-gaps of the magnetic
circuit and is counted fir by factor τ. Its value lies between 1 and 1,5 and normally mean value
of 1,2 is used. Having above in mind one can calculate magnets length:
lm
B
0
Hp
,
(5.16)
and magnets cross-section Sm:
Sm
BS
Bp
.
(5.17)
The magnets volume results as product of Sm and lm:
Vm
B 2V
0 H p Bp
.
(5.18)
When the operating point is known, the energy produced by the magnet Em may be
calculated. This is calculated by multiplying the Bd value by the Hd value:
Em
Bd H d .
(5.19)
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This calculates the rectangular area under the normal curve between the origin and the
operating point. Magnet energy is calculated in Mega Gauss Oersteds (MGOe) in cgs unit
system and Joules/cubic meter in SI unit system. Maximum Energy is found by locating a
single point on the demagnetization curve where the product of BdHd is greater than at any
other point (Fig.5.12).
Fig.5.12. Magnet energy characteristic.
For ―Straight Line demagnetization curve‖ materials this point is at approximately Bd=1.1 Hd
(it is suitable to choose the operating point a little above BHmax – the point of maximum BH
product). BHmax is commonly used to rate materials for maximum energy output per unit
volume. However, this does not mean that material with greater BHmax will always perform
better.
5.3 Armature reaction
The armature current produces magnetic flux which is opposite to the excitation flux. The
mmf of armature current describes equation:
Θa m
,
(5.20)
U ma
2p
where: a – ampere-turns of one pole pair.
As a consequence of appearance of external demagnetization field magnet load line moves
along H axe to the new position (Fig.5.10). The mmf of armature can’t exceed value
determined by equation:
hm H k
H k U ma ,
where: Hk – critical strength of magnetic field in the magnet.
(5.21)
Exciding maximum allowable value results in permanent magnet demagnetization
(irreversible loss in magnetic output due to ―exceeding the knee‖ of the demagnetization
curve). Magnet operating point belongs to the new demagnetization (recoil) curve (Fig.5.10).
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Glossary of Magnet Terms
Air gap, is a low permeability gap in the flux path of a magnetic circuit. Often air, but
inclusive of other materials such as paint, aluminum, etc.
Anisotropy. Literally means having different properties depending on the inspected
direction. Magnets which are anisotropic, or have an easy axis of magnetization, have their
anisotropy developed by two methods: Shape and Magnetocrystalline.
As - Area of the air gap, or the cross sectional area of the air gap perpendicular to the flux
path, is the average cross sectional area of that portion of the air gap within which the
application interaction occurs. Area is measured in sq. cm. in a plane normal to the central
flux line of the air gap.
Am - Area of the magnet, is the cross sectional area of the magnet perpendicular to the
central flux line, measured in sq. cm. at any point along its length. In design, Am is usually
considered the area at the neutral section of the magnet.
B - Magnetic induction, is the magnetic field induced by a field strength, H, at a given point.
It is the vector sum, at each point within the substance, of the magnetic field strength and
resultant intrinsic induction. Magnetic induction is the flux per unit area normal to the
direction of the magnetic path.
Bd - Remnant induction, is any magnetic induction that remains in a magnetic material after
removal of an applied saturating magnetic field, Hs. (Bd is the magnetic induction at any
point on the demagnetization curve: measured in gauss.)
Bd/Hd. Slope of the operating line, is the ratio of the remnant induction, Bd, to a
demagnetizing force, Hd. It is also referred to as the permeance coefficient, shear line, load
line and unit permeance.
BdHd. Energy product indicates the energy that a magnetic material can supply to an external
magnetic circuit when operating at any point on its demagnetization curve; measured in
megagauss-oersteds.
(BH)max - Maximum energy product, is the maximum product of (BdHd) which can be
obtained on the demagnetization curve.
Bis (or J) - Saturation intrinsic induction, is the maximum intrinsic induction possible in a
material.
Bg - Magnetic induction in the air gap, is the average value of magnetic induction over the
area of the air gap, A; or it is the magnetic induction measured at a specific point within the
air gap; measured in gauss.
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Bi (or J) - Intrinsic induction, is the contribution of the magnetic material to the total
magnetic induction, B. It is the vector difference between the magnetic induction in the
material and the magnetic induction that would exist in a vacuum under the same field
strength, H. This relation is expressed by the equation:
Bi=B-H,
where: Bi = intrinsic induction in gauss; B = magnetic induction in gauss; H = field strength
in oersteds
Bm - Recoil induction, is the magnetic induction that remains in a magnetic material after
magnetizing and conditioning for final use; measured in gauss.
Bo - Magnetic induction, at the point of the maximum energy product (BH)max; measured in
gauss.
Br - Residual induction (or flux density), is the magnetic induction corresponding to zero
magnetizing force in a magnetic material after saturation in a closed circuit; measured in
gauss.
A closed circuit condition exists when the external flux path of a permanent magnet is
confined with high permeability material.
Curie Temperature. The transition temperature above which the alloy loses its magnetic
properties. This is not the maximum serviceable temperature, which is usually much lower.
The demagnetization curve is the second (or fourth) quadrant of a major hysteresis loop.
Points on this curve are designated by the coordinates Bd and Hd.
Domains. Areas in a magnetic alloy which have the same orientation. The magnetic domains
are regions where the atomic moments of atoms cooperate and allow for a common magnetic
moment. It is the domains which are rotated and manipulated by an external magnetizing field
to create a useful magnet which has a net magnetic moment. In un-magnetized material the
domains are un-oriented and cancel each other out. In this condition there is no net external
field.
Eddy currents, are circulating electrical currents that are induced in electrically conductive
elements when exposed to changing magnetic fields, creating an opposing force to the
magnetic flux. Eddy currents can be harnessed to perform useful work (such as dampening of
movement), or may be unwanted consequences of certain designs, which should be accounted
for of minimized.
Electromagnet, is a magnet, consisting of solenoid with an iron core, which has a magnetic
field existing only during the time of current flow through the coil.
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f - Reluctance factor, accounts for the apparent magnetic circuit reluctance. This factor is
required due to the treatment of H, and H, as constants.
F - Leakage factor, accounts for flux leakage from the magnetic circuit. It is the ratio
between the magnetic flux at the magnet neutral section and the average flux present in the air
gap:
F=(Bm Am)/(B Ag).
F - Magneto motive force, (magnetic potential difference), is the line integral of the field
strength, H, between any two points, p1 and p2:
p2
F=∫ H dl,
p1
where: F = magneto motive force in gilberts, H = field strength in oersteds, dl = an element of
length between the two points, in centimeters.
Ferromagnetic material, is a material whose permeability is very much larger than 1 (from
60 to several thousands times 1), and which exhibits hysteresis phenomena.
Flux is the condition existing in a medium subjected to a magnetizing force. This quantity is
characterized by the fact that an electromotive force is induced in a conductor surrounding the
flux at any time the flux changes in magnitude. The cgs unit of flux is the Maxwell.
A flux meter is an instrument that measures the change of flux linkage with a search coil.
Fringing fields are leakage flux particularly associated with edge effects in a magnetic circuit.
The gauss is the unit of magnetic induction, B, in the cgs electromagnetic system. One gauss
is equal to one maxwell per square centimeter.
A gauss meter is an instrument that measures the instantaneous value of magnetic induction,
B. Its principle of operation is usually based on one of the following: the Hall effect, nuclear
magnetic resonance (NMR), or the rotating coil principle.
The gilbert is the unit of magneto motive force F, in the cgs electromagnetic system.
H - Magnetic field strength (magnetizing or demagnetizing force), is the measure of the
vector magnetic quantity that determines the ability of an electric current, or a magnetic body,
to induce a magnetic field at a given point; measured in oersteds.
Hc - Coercive force of a material, is equal to the demagnetizing force required to reduce
residual induction, Br to zero in a magnetic field after magnetizing to saturation; measured in
oersteds.
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Hci - Intrinsic coercive force of a material, indicates its resistance to demagnetization. It is
equal to the demagnetizing force which reduces the intrinsic induction Bi, in the material to
zero after magnetizing to saturation; measured in oersteds.
Hd is that value of H corresponding to the remnant induction, Bd; on the demagnetization
curve, measured in oersteds.
Hmv is that value of H corresponding to the recoil induction, B,; measured in oersteds.
Ho is the magnetic field strength at the point of the maximum energy product (BH)max;
measured in oersteds.
Hs Net effective magnetizing force, is the magnetizing force required in the material, to
magnetize to saturation measured in oersteds.
A hysteresis loop is a closed curve obtained for a material by plotting (usually to rectangular
coordinates) corresponding values of magnetic induction B, for ordinates and magnetizing
force H, for abscissa when the material is passing through a complete cycle between definite
limits of either magnetizing force H, or magnetic induction B.
Irreversible losses are defined as partial demagnetization of the magnet, caused by exposure
to high or low temperatures external fields or other factors. These losses are recoverable by
remagnetization. Magnets can be stabilized against irreversible losses by partial
demagnetization induced by temperature cycles or by external magnetic fields.
J, see Bi Intrinsic induction.
Js, see Bis, Saturation intrinsic induction.
A keeper is a piece (or pieces) of soft iron that is placed on or between the pole faces of a
permanent magnet to decrease the reluctance of the air gap and thereby reduce the flux
leakage from the magnet. It also makes the magnet less susceptible to demagnetizing
influences.
Keepers. A keeper is a high permeability material, typically mild steel, which is installed on a
magnet or magnetic assembly to reduce the reluctance of the magnetic circuit. This reduces
the overall leakage fields generated by the magnet or magnetic assembly. Keepers are
typically installed to help the magnet or magnetic assembly resist demagnetization during
handling, transportation, or storage. Keepers are typically found on Alnico magnets and
Alnico magnetic assemblies.
Knee of the demagnetization curve is the point at which the B-H curve ceases to be linear.
All magnet materials, even if their second quadrant curves are straight line at room
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temperature, develop a knee at some temperature. Alnico 5 exhibits a knee at room
temperature. If the operating point of a magnet falls below the knee, small changes in H
produce large changes in B, and the magnet will not be able to recover its original flux output
without re-magnetization.
Leakage flux is flux Φ, whose path is outside the useful or intended magnetic circuit;
measured in maxwells.
lg - Length of the air gap, is the length of the path of the central flux line of the air gap;
measured in centimeters.
lm - Length of the magnet, is the total length of magnet material traversed in one complete
revolution of the centerline of the magnetic circuit; measured in centimeters.
lm/D - Dimension ratio, is the ratio of the length of a magnet to its diameter, or the diameter
of a circle of equivalent cross-sectional area. For simple geometries, such as bars and rods, the
dimension ratio is related to the slope of the operating line of the magnet BdHa.
Load line is a line drawn from the origin of the demagnetization curve with a slope of B/H,
the intersection of which with the B-H curve represents the operating point of the magnet.
Also see permeance coefficient.
Magnetic Assemblies. A combination of materials, magnetic and non-magnetic, which form
a particular solution. Incorporates a permanent magnet as the flux generator and usually relies
on mild steel to conduct the flux to the workface. Allows for better means of mounting-tapped
holes, threads, press fits, etc.
Magnetic circuit, an assembly consisting of some or all of the following: permanent magnets,
ferromagnetic conduction elements, air gaps, and electrical currents.
Magnetic Length. The physical length of the magnet dimension which corresponds to the
direction the magnet is magnetized. This may or may not be the magnet's orientation direction.
The major hysteresis loop of a material is the closed loop obtained when the material is
cycled between positive and negative saturation.
The maxwell is the unit of magnetic flux in the cgs electromagnetic system. One maxwell is
one line of magnetic flux.
The neutral section of a permanent magnet is defined by a plane passing through the magnet
perpendicular to its central flux line at the point of maximum flux.
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North pole, is the pole of a magnet which, when freely suspended, would point to the north
magnetic pole of the earth. The definition of polarity can be a confusing issue, and it is often
the best to clarify by using ―north seeking pole‖ instead of ―north pole‖ in specifications.
The oersted is the unit of magnetic field strength, H, in the cgs electromagnetic system. One
oersted equals a magneto motive force of one gilbert per centimeter of flux path.
An open circuit condition exists when a magnetized magnet is by itself with no external flux
path of high permeability material.
The operating line for a given permanent magnet circuit is a straight line passing through the
origin of the demagnetization curve with a slope of negative Bd/Hd. (Also known as
permeance coefficient line).
The operating point of a permanent magnet is that point on a demagnetization curve defined
by the coordinates (BdHd) or that point within the demagnetization curve defined by the
coordinates (BmHm).
Orientation direction, is the direction in which an anisotropic magnet should be magnetized
in order to achieve optimum magnetic properties. Also known as the ―axis,‖ ―easy axis,‖ or
―angle of inclination‖.
An oriented (anisotropic) material is one that has better magnetic properties in a given
direction.
Paramagnetic material, is a material having a permeability slightly greater than 1.
A permeameter is an instrument that can measure, and often record, the magnetic
characteristics of a specimen.
P - Permeance, is the reciprocal of the reluctance, R, measured in maxwells per gilbert.
Pole pieces, are ferromagnetic materials placed on magnetic poles used to shape and alter the
effect of lines of flux.
R - Reluctance, is somewhat analogous to electrical resistance. It is the quantity that
determines the magnetic flux, Φ, resulting from a given magneto motive force F:
R=F/Φ,
where: R=reluctance, in gilberts per maxwell, F=magneto motive force, in gilberts, Φ=flux, in
maxwells.
Return path, are conduction elements in a magnetic circuit, which provide a low reluctance
path for the magnetic flux.
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Saturation. A condition where the increase in applied external field yields no increase in
induction. When this condition is met, all of the elementary magnetic moments have the same
alignment. This condition is important in permanent magnet alloys and in Ferromagnetic
alloys. Magnet alloys must always be magnetized to saturation. The magnet may not be used
at this level, but before conditioning and stabilization the magnet must always first be
magnetized to saturation. Usually saturation should not be exceeded in Ferromagnetic alloys
which comprise the yoke or return path elements of a magnetic circuit. If ferromagnetic
elements are saturated there will be flux leakage in the system and a redesign should be
considered.
A search coil is a coiled conductor, usually of known area and number of turns that is used
with a flux meter to measure the change of flux linkage with the coil.
Sintered. A sintered magnet is comprised of a compacted powder which is then subjected to a
heat treat operation where the full density and magnetic orientation is achieved.
Stabilization, is exposure of a magnet to demagnetizing influences expected to be
encountered in use in order to prevent irreversible losses during actual operation.
Demagnetizing influences can be caused by high or low temperatures, or by external magnetic
fields.
Tc - Curie temperature, is the transition temperature above which a material loses its magnet
properties.
Tmax - Maximum service temperature, is the maximum temperature to which the magnet
may be exposed with no significant long-range instability or structural changes.
Reversible temperature coefficients are changes in flux which occur with temperature
change. These are spontaneously regained when the temperature is returned to its original
point.Magnetic saturation of a material exists when an increase in magnetizing force produces
no increase in intrinsic induction.
The temperature coefficient is a factor which describes the reversible change in a magnetic
property with a change in temperature. The magnetic property spontaneously returns when the
temperature is cycled to its original point. It usually is expressed as the percentage change per
unit of temperature.
An unoriented (isotropic) material has equal magnetic properties in all directions.
Vg - Air gap volume, is the useful volume of air or nonmagnetic material between magnetic
poles; measured in cubic centimeters.
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Weber is the practical unit of magnetic flux. It is the amount of magnetic flux which, when
linked at a uniform rate with a single-turn electric current during an interval of 1 second, will
induce in this circuit an electromotive of force of 1 volt.
µ - permeability, is the general term used to express various relationships between magnetic
induction, B, and the field strength, H.
µre - recoil permeability, is the average slope of the recoil hysteresis loop. Also known as the
minor loop.
Φ - magnetic flux, is a contrived but measurable concept that has evolved in an attempt to
describe the ―flow‖ of a magnetic field. Mathematically, it is the surface integral of the
normal component of the magnetic induction B, over an area A.
Φ = ∫∫B • dA,
where: Φ = magnetic flux, in maxwells, B = magnetic induction, in gauss, dA = an element of
area, in square centimeters.
When the magnetic induction, B, is uniformly distributed and is normal to the area, A, the
flux, Φ = BA.
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References
John J. Winders – Power transformers. Principles and Applocations
Ion Boldea, Syed A. Nasar - The induction machine handbook
Anuszczyk Jan - Maszyny elektryczne w energetyce : zagadnienia wybrane
Przybysz Jerzy - Turbogeneratory
Bytnar Andrzej - Wybrane zagadnienia z konstrukcji i eksploatacji generatorów
Śliwiński Tadeusz – Parametry rozruchowe silników indukcyjnych
Kamiński G., Przyborowski W., Kosk J. - Laboratorium maszyn elektrycznych
Internet data
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