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Title: Electrical Machines
Description: its has contain of basic work of an machines and there properties with their principles and the working procudires
Description: its has contain of basic work of an machines and there properties with their principles and the working procudires
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Transformers
1
Introduction
Michael Faraday propounded the principle of electro-magnetic induction in 1831
...
The magnitude of the induced voltage is proportional to the rate of
change of the flux linkages
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The earliest use of this phenomenon was in the development of induction coils
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As
the d
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power system was in use at that time, very little of transformer principle was made
use of
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c
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Also the d
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generators cannot be scaled up due to the limitations of the commutator
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During the second half of the 19th century the alternators, transformers and induction
motors were invented
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The role of the
transformers became obvious
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The efficiency of such conversion was extremely high
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c
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All these are made possible by transformers
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c
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Transformers can link two or more electric circuits
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The second coil which is made to link this field has an
induced voltage in the same
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Thus the voltage level can be increased or decreased by changing
the number of turns
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As a magnetic medium forms the link between the primary and the
secondary windings there is no conductive connection between the two electric circuits
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The frequency on
the two sides will be the same
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The electric power at one
voltage/current level is only ‘transformed’ into electric power, at the same frequency, to another voltage/current level
...
The use of the principle
of transformers is universal
...
Power ratings vary from a few
milliwatts to several hundreds of megawatts
...
Demand
on electric power generation doubles every decade in a developing country
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These figures
show the indispensable nature of power transformers
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Faraday’s law states that a voltage appears across the terminals
of an electric coil when the flux linkages associated with the same changes
...
Putting mathematically,
e=
dψ
dt
(1)
Where, e is the induced emf in volt and ψ is the flux linkages in Weber turn
...
1 shows a
Figure 1: Flux linkages of a coil
coil of N turns
...
In such a case,
ψ = Nφ
(2)
dφ
dt
(3)
and
e=N
volt
The change in the flux linkage can be brought about in a variety of ways
• coil may be static and unmoving but the flux linking the same may change with time
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• both 1 and 2 above may take place
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These three cases are now elaborated in sequence below, with the help of a coil with a simple
geometry
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2 shows a region of length L m, of uniform flux density B Tesla, the
flux lines being normal to the plane of the paper
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The flux φ linked by the turn is L ∗ B ∗ X Weber
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If now B does not change with time and the loop is unmoving then
no emf is induced in the coil as the flux linkages do not change
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On the other hand if the value of B varies with time a voltage is
induced in the coil linking the same coil even if the coil does not move
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ω is the angular rate of change with
time
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ω cos ωt = Nφm
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L
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The peak value of the induced emf is
em = Nφm
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ω
√
2
volt
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This emf is termed as ‘transformer’ emf and this principle is used
in a transformer
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Lenz’s
law states that the reaction to the change in the flux linkages would be such as to oppose
the cause
...
In the present case, presented in Fig
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The polarity of the emf is as indicated
...
Fig
...
The flux
density is held constant at B Tesla
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L
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The conductor is moved with a velocity v = dx/dt normal to the flux,
cutting the flux lines and changing the flux linkages
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B
...
dx/dt = B
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v volt
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Only the change in the
flux linkages is needed to be known for the calculation of the voltage
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If the flux density B is distributed
sinusoidally over the region in the horizontal direction, the emf induced also becomes sinusoidal
...
The polarity of the induced emf is obtained by the application
of the Lenz’s law as before
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The current in the conductor, when the coil ends are closed, makes the conductor
experience a force urging the same to the left
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2b is arrived at
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This principle is used in d
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The third case under the application of the Faraday’s law arises when the flux changes
and also the conductor moves
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2(c)
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The conductor is moved with a uniform velocity of
dx
dt
= v m/sec
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d(Bm
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L
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L
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Bm
...
+ N
...
sin ωt
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V olt
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The second
term is due to moving conductor or is a speed emf
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This principle is used in a
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machines where the field is time
varying and conductors are moving under the same
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The case two will be revisited under the study of the d
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Case three will be
extensively used under the study of a
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c
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Next in the study of the transformers comes the question of creating a time varying
filed
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The winding
which establishes the field is called the primary
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It should not be forgotten that
the primary also sees the same time varying field set up by it linking its turns and has an
induced emf in the same
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At first
the common constructional features of a transformer used in electric power supply system
operating at 50 Hz are examined
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In addition to the transformers used in power systems, in power transmission and
distribution, a large number of special transformers are in use in applications like electronic
supplies, rectification, furnaces, traction etc
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The principle of operation of these transformers also is the same but the user requirements
differ
...
These machines are highly material intensive equipments and are designed to
match the applications for best operating conditions
...
This brings in a very large variety in their constructional features
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These can be broadly divided into
1
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Winding arrangements
3
...
1
Core construction
Transformer core for the power frequency application is made of highly permeable
material
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Relative permeability µr
well over 1000 are achieved by the present day materials
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Over the years progressively better magnetic properties are obtained by going in for Hot rolled non-oriented to Hot rolled grain oriented steel
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The thickness of the laminations progressively got reduced from
over 0
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25mm per lamination
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The magnetic material is required to have
a high permeability µ and a high saturation flux density, a very low remanence Br and a
small area under the B-H loop-to permit high flux density of operation with low magnetizing
current and low hysteresis loss
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The eddy current itself is highly reduced by making the
laminations very thin
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The steel should not have residual mechanical stresses which reduce
their magnetic properties and hence must be annealed after cutting and stacking
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3 are used and the core cross section would be a square or a rectangle
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5
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The saturation flux density of the present day steel lamination is about 2
Tesla
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In a core type construction the winding surrounds the core
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4
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In the case of very small transformers the conductors are very thin and round
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Thus no
special care is needed for the construction of the core
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As the rating of the transformer increases the conductor size
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(a )
(b)
(c)
Figure 3: E and I,C and I and O Type Laminations
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1
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phase
LV HV
HV LV core
Single phase
LV HV
Three phase
(a)Core type
(b) Shell type
Figure 4: Core and Shell Type Construction
also increases
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To wind such conductor on a
rectangular former is not only difficult but introduces stresses in the conductor, at the bends
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Also, for a given area enclosed the length of the conductor becomes more
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In order to avoid all these problems the coils are made cylindrical and are
wound on formers on heavy duty lathes
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Stepped core construction thus
becomes mandatory for the core of large transformers
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5 shows a few typical stepped core
constructions
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Cooling ducts have to be provided
in the core
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The iron losses in the lamination, when the flux
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0
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1
0
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14
0
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42
0
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1 0
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12
0
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3
0
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12
0
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71D
d
d
d
duct
Figure 5: Stepped Core Construction
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duct
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is oriented in the direction of grain orientation, is about 30% of that in the normal direction
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As the reluctance of air path is about 1000 times more
than that of the steel, an air path of 1mm will require a mmf needed by a 1 meter path in iron
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Lap joints are used to provide alternate paths for flux lines thus reducing the reluctance of the flux paths
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6
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This construction
is normally followed in instrument transformers to reduce the magnetizing current and hence
the errors
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Punched through holes and bolts are
progressively being avoided to reduce heating and melting of the through bolts
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Channels and angles are used for the frame and they hold the
bottom yoke rigidly
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2
Windings
Windings form another important part of transformers
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The one which is connected to a voltage source and
creates the flux is called as a primary winding
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If the secondary voltage is less than that of the
primary the transformer is called a step down transformer
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A step down transformer can be made a step up transformer
by making the low voltage winding its primary
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The winding with
more number of turns will be a HV winding
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Also the HV winding
needs to be insulated more to withstand the higher voltage across it
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These aspects influence the type of the winding
used for the HV or LV windings
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In the figure the letters L and H indicate the low
voltage and high voltage windings
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The HV winding is placed around the LV winding
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This is also facilitated by having the HV winding
as the outer winding
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helical, cross over and disc coils are shown in Fig
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Helical coils
Disc coils
cross over coils
Figure 8: Disc, Crossover and Helical Coil Construction
Helical Windings One very common cylindrical coil arrangement is the helical winding
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The coil progresses as a helix
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The insulation
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requirement also is not too high
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The complexity of this
type of winding rapidly increases as the current to be handled becomes more
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The eddy current
losses in the conductor rapidly increases
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The parallel circuits bring in problems of current
sharing between the circuits
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The modern practice is to use continuously
transposed and bunched conductors
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These are made
of circular conductors not exceeding 5 to 6 sq mm in cross section
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These turns are wound in several layers
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A
number of such blocks can be connected in series, leaving cooling ducts in between the
blocks, as required by total voltage requirement
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Alternate discs are made to spiral from outside towards the center
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These have excellent thermal properties and the behavior of the winding is highly predictable
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Sandwich coils Sandwich windings are more common with shell type core construction
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By
bringing HV and LV coils close on the same magnetic axis the leakage is reduced
and the mutual flux is increased
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3
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In larger transformers to improve the heat transfer
characteristics the conductors are insulated using un-impregnated paper or cloth and the
whole core-winding assembly is immersed in a tank containing transformer oil
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It is an insulator and also a coolant
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The
conductor insulation may be called the minor insulation as the voltage required to be withstood is not high
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Annular bakelite cylinders
serve this purpose
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The oil
used in the transformer tank should be free from moisture or other contamination to be of
any use as an insulator
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4
Cooling of transformers
Scaling advantages make the design of larger and larger unit sizes of transformers
economically attractive
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Consider a transformer of certain
rating designed with certain flux density and current density
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The losses in the machine, which are proportional
to the volume of the materials used, increase by a factor of K 3
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The surface area however increases by a factor of K 2 only
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The substantial increase in the output is
the major attraction in going in for larger units
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As the rating increases better cooling techniques are needed
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The limit
for this is reached by the time the rating is a few kVA
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This method of cooling is termed as AN(Air Natural)
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This
permits some improvement in the unit sizes
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The oil reaches the conductor surface and extracts the heat and transports the same to the
surface of the tank by convection
...
This
method permits the increase in the surface available for the cooling further by the use of
ducts, radiators etc
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In the above two cases the flow of oil is by natural convective
forces
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This is termed as OFN (Oil Forced
Natural)
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A forced circulation of oil through a radiator is done with a blast of air
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Main tank
Radiator
Tubes
(a)
Conservator
Bushing
& Breather
water outlet
Radiator
oil pump
water inlet
(b)
Conservator&
Breather
Bushing
Radiator
Oil pump
Fan motor
for O
...
B
(c)
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Figure 9: Some Typical Cooling Arrangements
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over the radiator surface
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Here the hot oil going into the radiator is cooled by a water circuit
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Next in hierarchy comes OFW
which is similar to OFB except that instead of blast of air a forced circulation of cool water
in the radiator is used in this
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9
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As the load on the transformer changes the heat
generated within also changes
...
This gives rise to the concept of mixed cooling technique
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This gives
the ratings to be in the ratio of 1:1
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5:2
The temperature rise permitted in the British standard specification for power transformers
are tabulated below
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4
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Organic oils of vegetative or animal origin are good insulators but tend
to decompose giving rise to acidic by-products which attack the paper or cloth insulation
around the conductors
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The properties that are required to be looked into before selecting an oil for transformer
application are as follows:
Insulting property This is a very important property
...
Therefore deterioration in insulating property due to moisture or contamination may be more relevant
...
Highly viscous fluids
need much bigger clearances for adequate heat removal
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Sulphur or its compounds
as impurities cause formation of sludge and also attack metal parts
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Sludge formation properties have to be considered while choosing the oil as the oil slowly forms
semi-solid hydrocarbons
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Heat in the presence of oxygen is seen to accelerate sludge formation
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Acidity Oxidized oil normally produces CO2 and acids
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These form corrosive vapors
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Flash point And Fire point Flash point of an oil is the temperature at which the oil
ignites spontaneously
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Fire point is the temperature at which the oil flashes and continuously
burns
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Inhibited oils and synthetic oils are therefore used in the transformers
...
Synthetic transformer oil like chlorinated diphenyl has
excellent properties like chemical stability, non-oxidizing, good dielectric strength, moisture
repellant, reduced risk due fire and explosion
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There are several other structural and insulating parts in a large transformer
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4
Ideal Transformer
Earlier it is seen that a voltage is induced in a coil when the flux linkage associated
with the same changed
...
A time varying field can
be created by passing an alternating current through an electric coil
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The medium can even be air
...
Indeed such arrangements are used in very high frequency transformers
...
These effects can be summarized as the followings
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The magnetizing current required to establish the field is very large, as the reluctance
of the medium is very high
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There is linear relationship between the mmf created and the flux produced
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The medium is non-lossy and hence no power is wasted in the medium
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Substantial amount of leakage flux exists
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It is very hard to direct the flux lines as we desire, as the whole medium is homogeneous
...
If the secondary winding is connected to a load then
part of the power from the source is delivered to the load through the magnetic field as a link
...
Power is required to create the field and
not to maintain the same
...
The large magnetizing
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Primary
Leakage
flux
x
Secondary
Mutual flux
(a)
Leakage flux
X
Primary
Mutual flux
Secondary
Iron core
(b)
Figure 10: Mutual Induction a) air core b) iron core
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current requirement is a major deterrent
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10(b) the situation changes dramatically
...
1
...
This can also be
visualized as a dramatic increase in the flux produced for a given value of magnetizing
current
...
The magnetic medium is linear for low values of induction and exhibits saturation type
of non-linearity at higher flux densities
...
The iron also has hysteresis type of non-linearity due to which certain amount of power
is lost in the iron (in the form of hysteresis loss), as the B-H characteristic is traversed
...
Most of the flux lines are confined to iron path and hence the mutual flux is increased
very much and leakage flux is greatly reduced
...
The flux can be easily ‘directed’ as it takes the path through steel which gives great
freedom for the designer in physical arrangement of the excitation and output windings
...
As the medium is made of a conducting material eddy currents are induced in the
same and produce losses
...
To minimize the
eddy current losses the steel core is required to be in the form of a stack of insulated
laminations
...
Fortunately the losses due to hysteresis and eddy current for
the available grades of steel is very small at power frequencies
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Hence steel core is used in power transformers
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These idealizations are as follows:
1
...
The consequence is that a vanishingly small current is enough to establish the given flux
...
As all the flux generated confines itself to the iron, there is no leakage flux
...
Windings do not have resistance
...
In fact the practical transformers are very close to this model and hence no major
departure is made in making these assumptions
...
11 shows a two winding ideal transformer
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The secondary has T2 turns
...
The primary and secondary are
shown on the same limb and separately for clarity
...
Since the reluctance
of the iron path given by R = l/µAis zero as µ −→ ∞, a vanishingly small value of current
I0 is enough to setup a flux which is finite
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F lux φ =
mmf
I0 T1
I0 T1 Aµ
= l =
...
As the
current through the loop is zero (or vanishingly small), at every instant of time, the sum of
the voltages must be zero inside the same
...
The negative sign is due to the application of the Lenz’s law and shows
that it is in the form of a voltage drop
...
This equation results in v1 = e1 or the induced emf must be same in magnitude to
the applied voltage at every instant of time
...
f is the frequency of the supply
...
It can be easily seen that the variation of flux linkages can
be obtained as ψ1 = ψ1peak sin ωt
...
Thus the RMS primary induced emf is
dψ1
d(ψ1peak sin ωt)
=
dt
dt
= ψ1peak
...
ω
2πf T1 φm
√
√
=
= 4
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The same mutual
flux links the secondary winding
...
φm
...
cos ωt or the rms value
e2 =
2πf T2 φm
√
= 4
...
If the winding sense is opposite i
...
, if the primary is wound in clockwise sense
and the secondary counter clockwise sense then if the top terminal of the first winding is
at maximum potential the bottom terminal of the second winding would be at the peak
potential
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Hence in
the circuit representation of transformers a dot convention is adopted to indicate the terminals of the windings that go high (or low) together
...
12)
...
At a particular instant of
time if the current enters the terminal marked with a dot it magnetizes the core
...
So far, an unloaded ideal transformer is considered
...
11(c)
...
However this is detected by the primary immediately as both E2 and E1 tend to collapse
...
The demagnetizing mmf produced by the secondary is neutralized by
additional magnetizing mmf produces by the primary leaving the mmf and flux in the core
as in the case of no-load
...
Thus,
i1 T1 − i2 T2 = i0 T1
i2 T2 = i1 T1
but i0 → 0
(16)
and the rms value I2 T2 = I1 T1
...
12, then the above
equation can be written in phasor form as,
T2 ¯
¯
or I1 =
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If an impedance
of ZL is connected across the secondary,
¯
E2
¯
I2 = ¯
ZL
¯
E2
¯
or ZL = ¯
I2
(20)
The input impedance under such conditions is
¯
¯
E1
T1 E2
T1
¯
¯
Zi = ¯ = ( )2
...
ZL
T2 I2
T2
I1
(21)
An impedance of ZL when viewed ‘through’ a transformer of turns ratio ( T1 ) is seen
T2
T1
as ( T2 )2
...
Transformer thus acts as an impedance converter
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V1
E1
I2
E2
V2
I1
θ2
θ1
φ
φ
Figure 13: Phasor diagram of Operation of an Ideal Transformer
Finally, the phasor diagram for the operation of the ideal transformer is shown in
Fig
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As
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the transformer itself does not absorb any active or reactive power it is easy to see that
θ1 = θ2
...
However, grounding of loads and one terminal of the transformer on the
secondary/primary side are followed with the provision of leakage current detection devices
to safe guard the persons working with the devices
...
It can be used to step up or step
down the voltage/current at constant volt-ampere
...
In the case of an ideal transformer the efficiency is 100% as there are
no losses inside the device
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But it
cannot be used for the computation of the performance of a practical transformer due to the
non-ideal nature of the practical transformer
...
Hence
the effects of the non-idealization like finite permeability, saturation, hysteresis and winding
resistances have to be added to an ideal transformer to make it a practical transformer
...
Finite permeability of the magnetic circuit necessitates a finite value of the current
to be drawn from the mains to produce the mmf required to establish the necessary flux
...
B = µH;
B=
φ
A
(22)
where A is the area of cross section of the iron core m2
...
T1
l
(23)
where l is the length of the magnetic path, m
...
A =
Aµ(iT1 )
l
=
permeance ∗ mmf (here that of
primary)
(24)
The magnetizing force and the current vary linearly with the applied voltage as long
as the magnetic circuit is not saturated
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This non-linear current can be
resolved into fundamental and harmonic currents
...
At present the effect of this non-linear behavior is neglected as a secondary
effect
...
This current can be represented by a
current drawn by an inductive reactance in the circuit as the net energy associated with the
same over a cycle is zero
...
This current is called the magnetizing
current of the transformer
...
The magnetic circuit being lossy absorbs and dissipates
the power depending upon the flux density of operation
...
These are given by the following expressions:
Ph ∝ B 1
...
f - Frequency of operation, Hz
t - Thickness of the laminations of the core, m
...
An active power consumption by the no-load current can be represented in the input
circuit as a resistance Rc connected in parallel to the magnetizing reactance Xm
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e
...
Finite µ of the magnetic core makes a few lines of flux take to a path through the air
...
It is called as leakage flux
...
Even a large value of the current produces a small value of flux
...
Thus this effect of
the finite permeability of the magnetic core can be represented as a series inductive element
jxl1
...
As this leakage flux
varies linearly with I1 , the flux linkages per ampere and the primary leakage inductance
are constant (This is normally represented by ll1 Henry)
...
The secondary leakage reactance jxl2 arising out of the secondary leakage inductance ll2 is
given by
xl2 = 2πf ll2
(29)
Finally, the primary and secondary windings are wound with copper (sometimes aluminium in small transformers) conductors; thus the windings have a finite resistance (though
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I1
V1
r1
jxl1
φ
I’2
Io
+
~
+
Rc
-
jXm
E1
T1
-
r2
jxl2
I2
+
ZL
V2
T2
E2
-
(a)Physical arrangement
I1
r1
jXl1
I’2
Ic
V1
Rc
r2 jXl2
I2
Io
Im
E1
jXm
E2
(b)Equivalent circuit
Figure 14: A Practical Transformer
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ZL V2
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small)
...
These are
represented by r1 and r2 respectively on primary and secondary side
...
This is seen in Fig
...
I2
in the circuit represents the primary current component that is required to flow from the
mains in the primary T1 turns to neutralize the demagnetizing secondary current I2 due to
the load in the secondary turns
...
The circuit shown in Fig
...
However, it is not very convenient for use due to
the presence of the ideal transformer of turns ratio T1 : T2
...
This is done here by
replacing the secondary by a ‘hypothetical’ secondary having T1 turns which is ‘equivalent ’
to the physical secondary
...
Then there is no change as far
as their effect on the primary is considered
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This equivalent circuit is as shown in Fig
...
As the ideal transformer in this
case has a turns ratio of unity the potentials on either side are the same and hence they
may be conductively connected dispensing away with the ideal transformer
...
It is also possible to refer all the primary parameters to secondary by making the hypothetical equivalent primary winding on
the input side having the number of turns to be T2
...
15
...
Referring to fig
...
T2
Multiply both sides of Eqn
...
aE2 = aV2 + aI2 (r2 + jxl2 )
but aE2 = E1
Substituting in Eqn
...
jntuworld
...
jntuworld
...
(Shown in fig
...
)
r’1
jx’l1
r2
I’1
jxl2
I2
I’o
I’c
R’c
V’1
I’m
jX’m
ZL
V2
Figure 15: Equivalent Circuit Referred to the Secondary Side
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6
Phasor diagrams
r1
I1
r’2
jxl1
Io
Ic
Im
Rc
V1
jx’l2
V’2
jXm
Z’L
R
jX
(a)
I1
Ic
V1 Rc
r1
I’2
Io
jxl1
r’2
jx’l2
I1
I’2
R=r1+r’2
Im
Z’L
jxm
V’2 x=xl1+x’l2
V’2
V1
I1=I’2
(b)
(c)
Figure 16: Exact,approximate and simplified equivalent circuits
The resulting equivalent circuit as shown in Fig
...
This circuit can be used for the analysis of the behavior of the transformers
...
16(b)) is usually used, which may be further
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simplified to the one shown in Fig
...
On similar lines to the ideal transformer the phasor diagram of operation can be
drawn for a practical transformer also
...
Hence it is assumed that the phasor
φ is known
...
Now, the magnetizing and loss
components of the currents can be easily represented
...
This is represented
in Fig
...
Next we proceed to draw the phasor diagram corresponding to a loaded transformer
...
Magnitude of I2 and the load
power factor angle θ2 are assumed to be known
...
By trial and error the position of I2 and V2 are
determined
...
Rest of the
construction of the phasor diagram then becomes routine
...
I1 (r1 + jxl1 )is added to E1 to yield V1
...
17(b) as phasor diagram for a loaded transformer
...
jntuworld
...
jntuworld
...
jntuworld
...
jntuworld
...
The performance parameters of interest can be obtained by solving that circuit for any load
conditions
...
But for a user
these are not available most of the times
...
In order to get the
equivalent circuit parameters test methods are heavily depended upon
...
But if the temperature
rise of the transformer is required, then test method is the most dependable one
...
7
...
c voltage to the winding and measuring the current through the same
...
For low voltage
windings a resistance-bridge method can be used
...
c resistance one can get the
a
...
resistance by applying skin effect corrections
...
jntuworld
...
jntuworld
...
C
...
C
...
2
Polarity Test
This is needed for identifying the primary and secondary phasor polarities
...
Both a
...
and d
...
The dot method discussed earlier is used to indicate the
polarities
...
c
...
18(a)
...
V1 : V2 gives the turns ratio
...
The beginning and
end of the primary and secondary may then be marked by A1 − A2 and a1 − a2 respectively
...
If more secondary terminals are present due to taps taken from the windings
they can be labeled as a3 , a4 , a5 , a6
...
The same thing holds good if more secondaries are present
...
jntuworld
...
jntuworld
...
18(b) shows the d
...
method of testing the polarity
...
If the meter kicks back the assumed polarity is wrong
...
3
Open Circuit Test
W
A
V1
Io
V2
V
Im
V1
jXm
Ic
Rc
(a)Physical Arrangement
(b)Equivalent Circuit
Figure 19: No Load Test
As the name suggests, the secondary is kept open circuited and nominal value
of the input voltage is applied to the primary winding and the input current and power are
measured
...
19(a) V, A, W are the voltmeter, ammeter and wattmeter respectively
...
Fig
...
The no load current at rated voltage is less than 1 percent of
nominal current and hence the loss and drop that take place in primary impedance r1 + jxl1
due to the no load current I0 is negligible
...
jntuworld
...
jntuworld
...
Thus the watt meter reading
W0 = V1 Ic = Pcore
(38)
W0
V1
(39)
∴ Ic =
∴ Im =
Rc =
2
2
I0 − Ic
V1
Ic
or
andXm =
(40)
V1
Im
(41)
V1
Io
Figure 20: Open Circuit Characteristics
The parameters measured already are in terms of the primary
...
If the secondary
voltage is low, one can perform the test with LV side energized keeping the HV side open
circuited
...
These have to
be referred to HV side if we need the equivalent circuit referred to HV side
...
jntuworld
...
jntuworld
...
In such cases an open circuit characteristics is first
obtained, which is a graph showing the applied voltage as a function of the no load current
...
20
...
The
usual operating point selected for operation lies at some standard voltage around the knee
point of the characteristic
...
7
...
21(b)
...
Let
these values be Vsc , Isc and Wsc respectively
...
The excitation current which is only 1 percent or less even at rated
voltage becomes negligibly small during this test and hence is neglected
...
Also I1 = I2 as I0 ≃ 0
...
The reactive power consumed is that
absorbed by the leakage reactance of the two windings
...
jntuworld
...
jntuworld
...
jntuworld
...
jntuworld
...
However if the exact equivalent circuit is needed then either r1 or
′
r2 is determined from the resistance measurement and the other separated from the total
...
This is a
fairly valid assumption for many types of transformer windings as the leakage flux paths are
through air and are similar
...
5
Load Test
Load Test helps to determine the total loss that takes place, when the transformer
is loaded
...
Load test is used mainly
1
...
to determine the voltage regulation and efficiency of the transformer
...
The losses that are generated inside the transformer
on load appear as heat
...
The insulation of the transformer is the one to get affected by this rise in the
temperature
...
If the flash point of the oil is reached the transformer
goes up in flames
...
This aspect of temperature rise cannot be guessed from the electrical equivalent
circuit
...
jntuworld
...
jntuworld
...
Many external means of removal of heat from the transformer in the form of different cooling
methods give rise to different values for temperature rise of insulation
...
Hence the only sure way of ascertaining
the rating is by conducting a load test
...
As the rating increases it
becomes difficult to find a load that can absorb the requisite power and a source to feed the
necessary current
...
Further, the temperature rise of the transformer is due to the losses that take place
‘inside’ the transformer
...
The remaining 99% of the power has to be dissipated in a load impedance external
to the machine
...
( If the load is of unity power
factor) Thus the actual loading of the transformer is seldom resorted to
...
The load is applied and held constant till the temperature rise of transformer reaches a
steady value
...
That load current which gives the
maximum permissible temperature rise is declared as the nominal or rated load current and
the volt amperes are computed using the same
...
jntuworld
...
jntuworld
...
The
short circuit current is so chosen that the resulting loss taking place inside the transformer
is equivalent to the sum of the iron losses, full load copper losses and assumed stray load
losses
...
Therefore
this test comes close to a load test but does not replace one
...
The windings
are connected back to back as shown in Fig
...
Suitable voltage is injected into the loop
formed by the two secondaries such that full load current passes through them
...
jntuworld
...
jntuworld
...
The voltage source V1 supplies the
magnetizing current and core losses for the two transformers
...
There is no power wasted
in a load ( as a matter of fact there is no real load at all) and hence the name Phantom
or virtual loading
...
The two sources put together meet the core and copper losses
of the two transformers
...
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8
Per Unit Calculations
As stated earlier, transformers of various sizes, ratings, voltage ratios can be seen
being used in a power system
...
Also the comparison of these machines are made simple if all
the parameters are normalized
...
Expressing in percent basis is one
example of scaling
...
Also different
units of measurement are in use in the different countries (FPS, CGS, MKS, etc;)
...
If the transformer parameter can be
freed from the units then the system becomes very simple
...
The parameters of the transformer are referred to some base
values and thus get scaled
...
In the case of individual equipments,
its own nominal parameters are used as base values
...
Once the
base values are identified the per unit values are calculated for any parameter by dividing
the same by its base value
...
Thus the per unit value is a unit-less dimensionless number
...
Other base values like volt ampere Sbase , short circuit impedance Zbase can be
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calculated from those values
...
Other base values can be
derived from them
...
u =
Ip
...
u
Z(ohm)
Ibase
Sbase
=
= Z(ohm) ∗
= Z(ohm)
...
Parameters of all the machines are expressed on this
common base
...
The conversion of the base values naturally lead to change in the
per unit values of their parameters
...
u
...
u
...
u
...
2
Vbase
Sbase
old
)
old
Sbase
2
Vbase
new
(50)
new
The term inside the bracket is nothing but the ohmic value of the impedance and this gets
converted into the new per unit value by the new Sbase and Vbase
...
jntuworld
...
jntuworld
...
This can be easily seen by,
′
′
Zp
...
S
= Zohm
...
Zohm
a2
(51)
Where a - is the turns ratio of primary to secondary
Z - impedance as seen by primary,
′
Z - impedance as seen by secondary
...
′
1
Vbase = Vbase
...
u
...
u
...
Also the active
power and reactive power absorbed inside the transformer are not dependant on the winding
connected to supply
...
Thus the per unit values help in dispensing away the scaling constants
...
Comparison of the parameters of the machines
with those of similar ones throw in useful information about the machines
...
u
...
Application of per unit values for the calculation of voltage regulation, efficiency and
load sharing of parallel connected transformers will be discussed later at appropriate places
...
jntuworld
...
jntuworld
...
The equipments work-
ing on these systems are therefore given input voltages at these standard values, within
certain agreed tolerance limits
...
A transformer is interposed
in between the load and the supply terminals in such cases
...
While input voltage is the responsibility of
the supply provider, the voltage at the load is the one which the user has to worry about
...
It is therefore necessary to quantify the drop that takes
place inside a transformer when certain load current, at any power factor, is drawn from its
output leads
...
The voltage regulation can be defined in two ways - Regulation Down and Regulation
up
...
Regulation down: This is defined as ” the change in terminal voltage when a load current
at any power factor is applied, expressed as a fraction of the no-load terminal voltage”
...
This is the definition normally used
in the case of the transformers, the no-load voltage being the one given by the power
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supply provider on which the user has no say
...
Regulation up: Here again the regulation is expressed as the ratio of the change in the
terminal voltage when a load at a given power factor is thrown off, and the on load
voltage
...
Vl is load voltage
...
This definition is more commonly used in the case of alternators and power systems
as the user-end voltage is guaranteed by the power supply provider
...
In the expressions for the regulation, only the numerical differences of the voltages are
taken and not vector differences
...
The power factor of the load is defined with respect to the terminal voltage on
load
...
Also the full load output voltage
is taken from the name plate
...
Fig
...
The no-load current I0 is neglected in view of the large magnitude of I2
...
jntuworld
...
jntuworld
...
jntuworld
...
jntuworld
...
′
′
V1 = I2 (Re + jXe ) + V2
OD = V1 =
(54)
[OA + AB + BC]2 + [CD]2
′
′
′
′
′
[V2 + I2 Re cos φ + I2 Xe sin φ]2 + [I2 Xe cos φ − I2 Re sin φ]2
=
(55)
φ - power factor angle,
e
θ- internal impedance angle=tan−1 Xe
R
Also,
′
′
′
′
V1 = V2 + I2
...
2
(1 + v1 )2 + v2 − 1
(57)
2
2
2(1 + v1 )
v2
v2
+[
]2 = (1 + v1 +
)2 (58)
2(1 + v1 )
2(1 + v1 )
2(1 + v1 )
Taking the square root
2
(1 + v1 )2 + v2 = 1 + v1 +
(59)
2
v2
2(1 + v1 )
(60)
where v1 = er cos φ + ex sin φ and v2 = ex cos φ − er sin φ
′
er =
I2 Re
′ =per
V2
unit resistance drop
′
ex =
I2 Xe
′ =per
V2
unit reactance drop
as v1 and v2 are small
...
jntuworld
...
jntuworld
...
≃
...
As
v2 is small its second power may be neglected as a further approximation and the expression
for the regulation of the transform boils down to
regulation R = er cos φ ± ex sin φ
The negative sign is applicable when the power factor is leading
...
Similarly, the value of the regulation is maximum at a power factor angle φ =
tan−1 (ex /er ) = tan−1 (Xe /Re ) lagging
...
24
...
We have,
OD 2 = (OA + AB)2 + (BC + CD)2
′
′
′
(64)
′
= (V2 cos φ + I2 Re )2 + (V2 sin φ + I2 Xe )2 (65)
′
OD
OD − V2
= ′ −1
∴ RegulationR =
′
V2
V2
′
′
2
′
′
2
(V sin φ + I2 Xe )
(V2 cos φ + I2 Re )
+ 2
−1
′
′
V2
V2
=
(66)
2
(cos φ + Rp
...
u) − 1
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(67)
(68)
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V1
I2’Xe
V2
O
D
θ
φ
I2’Re
I2’
C
A
B
Figure 24: An Alternate Method for the Calculation of Regulation
Thus this expression may not be as convenient as the earlier one due to the square root
involved
...
25 shows the variation of full load regulation of a typical transformer as the
power factor is varied from zero power factor leading, through unity power factor, to zero
power factor lagging
...
25 that the full load regulation at unity power factor is nothing but
the percentage resistance of the transformer
...
Only
with low power factor loads the drop in the series impedance of the transformer contributes
substantially to the regulation
...
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5
4
3
%Regulation
2
power factor
1
leading
0
1
...
5
0
...
jntuworld
...
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...
This in turn results in large mechanical forces on the
winding
...
In the case of transformers provided with taps on windings, so that the turns ratio can be
changed, the voltage regulation is not a serious issue
...
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10
Efficiency
Transformers which are connected to the power supplies and loads and are in op-
eration are required to handle load current and power as per the requirements of the load
...
As the load is increased the primary and secondary currents
increase as per the load requirements
...
Due to the presence of no load losses and I 2 R losses in the windings
certain amount of electrical energy gets dissipated as heat inside the transformer
...
Efficiency of a power equipment is defined at any load as the ratio of the power output to
the power input
...
%η =
output power
input power
(69)
∗ 100
While the efficiency tells us the fraction of the input power delivered to the load, the
deficiency focuses our attention on losses taking place inside transformer
...
The temperature rise decides the rating of the equipment
...
The peak temperature
attained directly affects the life of the insulations of the machine for any class of insulation
...
5 on load test
...
jntuworld
...
jntuworld
...
5
x
1
Figure 26: Efficiency
A typical curve for the variation of efficiency as a function of output is given in Fig
...
Except in the case of an ideal machine, a certain fraction of
the input power gets lost inside the machine while handling the power
...
In the case of a
...
machines the rating is expressed
in terms of apparent power
...
The actual power delivered is a function of the power factor at which this
current is drawn
...
The reactive power however increases the current handled by the machine and
the losses resulting from it
...
jntuworld
...
jntuworld
...
The losses taking place inside a
transformer can be enumerated as below:
1
...
Secondary copper loss
3
...
Dielectric loss
5
...
Primary and secondary copper losses take place in the respective winding resistances
due to the flow of the current in them
...
c
...
While the average temperature rise can be approximately
used, the skin effect is harder to get analytically
...
The iron losses contain two components - Hysteresis loss and Eddy current loss
...
Ph = Kh B 1
...
jntuworld
...
jntuworld
...
The eddy current loss in the core arises because of the induced emf in the steel
lamination sheets and the eddies of current formed due to it
...
Pe = Ke B 2 f 2 t2
where t is the thickness of the steel lamination used
...
Present day laminations are of 0
...
These reduce the eddy current losses in the core
...
The
sum of hysteresis and eddy current losses can be obtained by the open circuit test
...
In the case of low voltage transformers this can be neglected
...
The stray load losses arise out of the leakage fluxes of the transformer
...
and produce eddy current losses in them
...
Also the leakage flux is directly proportional to the load current unlike the mutual
flux which is proportional to the applied voltage
...
This can also be estimated experimentally
...
The stray load losses are very low in air-cored transformers due to the absence of the metallic tank
...
jntuworld
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jntuworld
...
The expression for the efficiency of the
transformer operating at a fractional load x of its rating, at a load power factor of θ2 , can
be written as
η=
xS cos θ2
xS cos θ2 + Pconst + x2 Pvar
′
(71)
′
Here S in the volt ampere rating of the transformer (V2 I2 at full load), Pconst being constant
losses and Pvar the variable losses at full load
...
By differentiating η with respect to x and equating the same to zero, the condition for
maximum efficiency is obtained
...
The maximum value of that efficiency at any given
power factor is given by,
ηmax =
xS cos θ2
xS cos θ2
=
xS cos θ2 + 2Pconst
xS cos θ2 + 2x2 Pvar
(73)
From the expression for the maximum efficiency it can be easily deduced that this
maximum value increases with increase in power factor and is zero at zero power factor of
the load
...
Thus if a transformer is on full load, for most part of the time
then the ηmax can be made to occur at full load by proper selection of constant and variable
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losses
...
Such a design wastes lot of
copper
...
Two 100 kVA transformers A nd B are taken
...
The break up of this loss is chosen to be different for the two transformers
...
The maximum efficiency of 98
...
Transformer B: Iron loss =0
...
7 kW
...
04%
...
6
0
...
7
= 0
...
The maximum efficiency at unity power factor being
∗ 100 = 98
...
At the corresponding point the transformer A has an efficiency of
42
42+1
...
1764
∗ 100 = 97
...
Transformer A uses iron of more loss per kg at a given flux
density, but transformer B uses lesser quantity of copper and works at higher current density
...
1
All day efficiency
Large capacity transformers used in power systems are classified broadly into Power transformers and Distribution transformers
...
Distribution transformers are seen at the distribution substations
...
Thus at
50% load on the station only 50% of the transformers need to be connected in the circuit
...
It has to remain in the
circuit irrespective of the load connected
...
Hence the concept of energy based efficiency is defined for such
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50
50
Power Loss %
100
Load % of full load
100
s
P
6
12
Time,hrs
18
24
24
12
(a)Load factor
(b) Loss factor
Figure 27: Calculation of Load Factor and Loss Factor
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transformers
...
The all day efficiency is thus the ratio of the
energy output of the transformer over a day to the corresponding energy input
...
This assumption,
however, is far from being true
...
The no-load losses or constant losses occur throughout the 24 hours
...
Even the load factor, which is given by the
ratio of the average load to rated load, does not give satisfactory results
...
The graph of load on the
transformer, expressed as a fraction of the full load is plotted against time in Fig
...
In an
actual situation the load on the transformer continuously changes
...
The average load can be calculated by
n
i=1
Average load over a day =
Pi
24
=
Sn
n
i=1
xi ti cos θi
24
(74)
where Pi is the load during an interval i
...
xi is the fractional load
...
The average loss during the day is given by
Average loss = Pi +
Pc
n
i=1
x2 ti
i
24
(75)
This is a non-linear function
...
Hence a better option would be to keep the
constant losses very low to keep the all day efficiency high
...
The constant losses on the other hand has to
be incurred to make the service available
...
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The concept of minimizing the lost energy comes into effect right from the time of
procurement of the transformer
...
Obviously the
iron losses are capitalized more in the process to give an effect to the maximization of energy
efficiency
...
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11
Auto Transformer
I1
C
T1
V1
B
I2
T2
V2
ZL
I2
A
I1
Figure 28: Autotransformer - Physical Arrangement
The primary and secondary windings of a two winding transformer have
induced emf in them due to a common mutual flux and hence are in phase
...
This prompted the use of a part
of the primary as secondary
...
The fused section need to have a cross sectional area of the conductor to carry (I2 −I1 )
ampere! This ingenious thought led to the invention of an auto transformer
...
28 shows
the physical arrangement of an auto transformer
...
At point B a connection is taken
...
As the volts per turn,
which is proportional to the flux in the machine, is the same for the whole winding,
V1 : V2 = T1 : T2
(76)
For simplifying analysis, the magnetizing current of the transformer is neglected
...
jntuworld
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...
This will be countered by a current I1 flowing from the source through the
T1 turns such that,
I1 T1 = I2 T2
(77)
A current of I1 ampere flows through the winding between B and C
...
The cross section of the wire to be
selected for AB is proportional to this current assuming a constant current density for the
whole winding
...
The magnetic circuit is assumed to be identical and hence there is
no saving in the same
...
This ratio therefore denotes the savings in copper
...
The larger the ratio of the
voltages, smaller is the savings
...
Thus
auto transformers become ideal choice for close ratio transformations
...
The electrical isolation between primary and secondary
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I1+I2
I1
φ
I2
I2
V2
V1
V1+V2
I1+I2
ZL
I1
I2
Figure 29: Two Winding Transformer used as auto transformer
has to be sacrificed
...
This
can be illustrated as follows
...
29 shows a regular two winding transformer of a voltage
ratio V1 : V2 , the volt ampere rating being V1 I1 = V2 I2 = S
...
The new output of this auto
transformer will now be
V1
V1
) = S(1 + )
V2
V2
I2
= V1 (I1 + I2 ) = S(1 + )
I1
I2 (V1 + V2 ) = I2 V2 (1 +
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(81)
(82)
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Thus an increased rating can be obtained compared to a two winding transformer
with the same material content
...
Then the new output rating will be
I2 (V1 − V2 ) = I2 V2 (
V1
V1
− 1) = S( − 1)
V2
V2
(83)
The differential connection is not used as it is not advantageous as the cumulative
connection
...
1
Equivalent circuit
I1
I1
r1,xl1
I2
V1
r2,xl2
V2
(I2 -I1)
I1
I2
Figure 30: Kirchoff’s Law Application to auto transformer
As mentioned earlier the magnetizing current can be neglected, for simplicity
...
jntuworld
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...
30 we have
V1 = E1 + I1 (r1 + jxl1 ) − (I2 − I1 )(r2 + jxl2 )
(84)
Note that the resistance r1 and leakage reactance xl1 refer to that part of the winding where
only the primary current flows
...
Then E1 = aE2 and I2 = aI1
multiplying equation(84) by ’a’ and substituting in (83) we have
V1 = aV2 + a(I2 − I1 )(r2 + jxl2 ) + I1 (r1 + jxl1 ) − (I2 − I1 )(r2 + jxl2 )
= aV2 + I1 (r1 + jxl1 + r2 + jxl2 − ar2 − ajxl 2) + I2 (ar2 + jaxl2 − r2 − jxl2 )
= aV2 + I1 (r1 + jxl1 + r2 + jxl2 + a2 r2 + ja2 xl2 − ar2 − ajxl2 − ar2 − jaxl2 )
= aV2 + I1 (r1 + r2 (1 + a2 − 2a) + jxl1 + xl2 (1 + a2 − 2a))
= aV2 + I1 (r1 + (a − 1)2 r2 + jxl1 + (a − 1)2 xl2 )
(86)
Equation (85) yields the equivalent circuit of Fig
...
The magnetization branch can now be hung across the mains for completeness
...
Thus in the case of an auto
transformer total value of the short circuit impedance is lower and so also the percentage
resistance and reactance
...
Having a smaller value
of short circuit impedance is sometimes considered to be a disadvantage
...
jntuworld
...
jntuworld
...
The efficiency is higher in auto
transformers compared to their two winding counter part at the same load
...
32
...
From the foregoing study it is seen that there are several advantages in going in for the
autotransformer type of arrangement
...
The losses are reduced increasing the efficiency
...
All these benefits are enhanced as
the voltage ratio approaches unity
...
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I1x1
(I2-I1)r2
I1r1
(I2-I1)x2
V1
I2
E1
E2
I1
(I2-I1)x2
(I2-I1)r2
V2
θ2
θ1
I2
φ
Figure 32: Phasor Diagram of Operation of an autotransformer
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Auto transformers are used in applications where electrical isolation is not a critical
requirement
...
3 or more they are used with advantage
...
Vari e
abl
a
...
c
...
c
...
Here only one winding is used
as in the auto transformer
...
The primary conductor is bared to facilitate electrical
contact Fig
...
Such arrangement cannot exploit the savings in the copper as the output
voltage is required right from zero volts upwards
...
These are used in
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voltage stabilizers, variable d
...
arrangements (with a diode bridge) in laboratories, motor
starters, dimmers etc
...
jntuworld
...
jntuworld
...
The effects
of the harmonic currents are
1
...
Increased core losses
3
...
On the other hand the harmonic voltages of the transformer cause
1
...
Electro static interference with communication circuits
...
Resonance between winding reactance and feeder capacitance
...
These combine with non-linear nature of transformer core and produce severe distortions in
voltages and currents and increase the power loss
...
The discussion here is confined to the
harmonics generated by transformers only
...
jntuworld
...
jntuworld
...
1
Single phase transformers
Modern transformers operate at increasing levels of saturation in order to reduce
the weight and cost of the core used in the same
...
This is explained below
...
34 shows the manner in which the shape of the
magnetizing current can be obtained and plotted
...
The sinusoidal flux density curve represents the sinusoidal
applied voltage to some other scale
...
The harmonic current components are obtained from this
analysis
...
Relatively small value of harmonic fields generate considerable magnitude
of harmonic voltages
...
jntuworld
...
jntuworld
...
These effects get even more pronounced for higher
order harmonics
...
These currents produce effects according to
Lenz’s law and tend to neutralize the harmonic flux and bring the flux wave to a sinusoid
...
The same can be told of other harmonics also
...
The understanding of the phenomenon becomes
more clear if the transformer is supplied with a sinusoidal current source
...
When the load is connected on
the secondary side the harmonic currents flow through the load and voltage tends to become
sinusoidal
...
The harmonic currents produce losses and electro magnetic interference
as already noted above
...
2
Three phase banks of single phase transformers
In the case of single phase transformers connected to form three phase bank, each
transformer is magnetically decoupled from the other
...
Also, there are three fundamental voltages in the present case each displaced from the other
by 120 electrical degrees
...
c
...
The harmonics which are triplen (multiples of
three) behave in a similar manner as they are co-phasal or in phase in the three phases
...
jntuworld
...
jntuworld
...
The harmonic behavior of poly-phase banks can be discussed
now
...
Thus the supply current is nearly sinusoidal (but for the non-triplen harmonic currents)
...
The secondary voltages will be nearly sinusoidal
...
Dy and Yd connection (without neutral connection) Behavior of the bank with mesh
connection on one side is similar to the one discussed under Dd connection
...
Yy connection without neutral wires With both primary and secondary connected in
star no closed path exists
...
Non-triplen harmonics
√
like fundamental, become 3 times phase value and appear in the line voltages
...
Flux wave in each
transformer will be flat topped and the phase voltages remain peaked
...
The star point oscillates due to the third harmonic
voltages
...
Yy connection with neutral wires When a neutral wire is provided the triplen har86
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monic current can flow and the condition is similar to the single phase case (with a
star connected 4 wire source or with the system earth)
...
Unloaded secondary neutral will not be operative
...
In a Yy connection, to obtain third harmonic suppression one may provide a third
winding which is connected in mesh, which can be an unloaded winding
...
This winding improves the single phase to earth fault detection also
...
Such transformers are designated as Yyd transformers
...
12
...
The exception to this rule is a 3-phase shell type transformer
...
Three limb
core type 3-phase transformer is the one in which the phases are magnetically also linked
...
This is true for fundamental
and non-triplen harmonics
...
The flux path is completed through the air
...
jntuworld
...
jntuworld
...
Thus the flux in the core remains nearly sinusoidal, so also the induced emf
...
The triplen order flux, sometimes
links the tank and produces loss in the same
...
Harmonic current compensation using special magnetic circuit design is considered to be
outside the scope here
...
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...
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...
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...
Sridhara Rao, Prof
...
Sasidhara Rao
Poly Phase connections and Poly phase Transformers
The individual transformers are connected in a variety of ways in a power system
...
As an engineering application is driven by
techno-economic considerations, no single connection or setup is satisfactory for all applications
...
Star and mesh
connections are very commonly used
...
are a few of the many possibilities
...
Literature
abounds in the description of many other
...
Many polyphase connections can be formed using single
phase transformers
...
In a balanced two phase system we encounter two voltages
that are equal in magnitude differing in phase by 90◦
...
Further there is an
order in which they reach a particular voltage magnitude
...
In some applications like a
...
to d
...
conversion, six phases or more may be encountered
...
The basic relationship between the primary and secondary voltages (brought about by
a common mutual flux and the number of turns), the polarity of the induced emf (decided
by polarity test and used with dot convention) and some understanding of the magnetic
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...
P
...
To facilitate the manufacturer and users, international standards are also available
...
The
HV winding is indicated by capital letters and the LV winding by small letters
...
If the induced emf
at an instant is from A1 to A2 on the HV winding it will rise from a1 to a2 on the LV winding
...
Three balanced 3-phase voltages can be connected in star or mesh fashion
to yield a balanced 3-phase 3-wire system
...
In
addition to giving different voltage ratios, they introduce phase shifts between input and
output sides
...
1
...
2
...
3
...
4
...
A few examples of the physical connections and phasor diagrams are shown in Fig
...
36 corresponding to each group
...
D/d stand for mesh, Y/y - for star, Z/z for zig-zag
...
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...
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...
Sasidhara Rao
G roup1 0 0 Phase shit
f
E
...
Vector di
M F
agram s
W i ngs & Term i s
ndi
nal
N n
A2
A1
A 2 a1
a1
B1
B 2 b1
b1
C1
C 2 c1
c1
C2
A2
A1
B1
C1
B2
a2
A 2a1
B 2b1
C 2c1
a1
b2
b1
c2
c1
C2
n
A1
B 2b 3
C1
C 2 c3
B2
A 2 a3
B1
A2
a1
b1
C2
c1
B2
(b)
Figure 35: Vector Groups for 3-phase Transformer Connections
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Vector di
M F
agram s
W i ngs & Term i s
ndi
nal
A2
n
A1
A 2 a2
B1
B 2 b2
b1
C 2 c2
a2
a1
c1
C1
N
C2
A1
A 2 a2
B1
B 2 b2
C1
C 2 c2
A2 B2
a2
a1
b1
c1
A1
B 2 b4
C1
A2
A 2 a4
B1
B2
C2
N n
C 2 c4
a1
b1
c1
B2
C2
(a)
G roup4 + 30 0 Phase shit
f
E
...
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...
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...
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...
Sasidhara Rao
referring to 12 o’clock position
...
Star connection is normally cheaper as there are fewer turns and lesser cost of insulation
...
In a star connected
1
winding with earthed-neutral the maximum voltage to the earth is ( √3 )of the line voltage
...
Mesh connections are advantageous in low voltage transformers as insulation costs are insignificant
1
and the conductor size becomes ( √3 ) of that of star connection and permits ease of winding
...
Star/star (Yy0, Yy6)connection This is the most economical one for small high voltage
transformers
...
Neutral wire can permit mixed loading
...
These triplen harmonic currents cannot
flow, unless there is a neutral wire
...
Three
phase shell type units have large triplen harmonic phase voltage
...
A tertiary mesh connected winding may be
required to stabilize the oscillating neutral due to third harmonics in three phase banks
...
Large amount of unbalanced load can be met with ease
...
It is possible to operate
with one transformer removed in open delta or Vee connection meeting 58 percent of
the balanced load
...
Mixed single phase
loading is not possible due to the absence of neutral
...
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...
Sasidhara Rao
Star/mesh(Dy or Yd ) This arrangement is very common for power supply transformers
...
Zig zag/ star (ZY1 or Zy11) Zigzag connection is obtained by inter connection of phases
...
Unbalanced loading is also possible
...
This connection requires 15% more
turns for the same voltage on the zigzag side and hence costs more
...
Also, they occupy more space
...
Mesh connected three phase transformers resemble 3- single phase units but kept in
a common tank
...
Other than
that there is no big difference
...
The three limbs are equal in cross section
...
The flux setup in any limb
will return through the other two limbs as the mmf of those limbs are in the proper
directions so as to aid the same
...
This is due to the fact
that the magnetizing current itself forms a small fraction of the total phase current
drawn on load
...
jntuworld
...
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Krishna Vasudevan, Prof
...
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...
Sasidhara Rao
large value of 3rd harmonic mmf without affecting the performance
...
The 3rd harmonic flux must therefore find its path through the air
...
Similarly unbalanced operation of the
transformer with large zero sequence fundamental mmf content also does not affect its
performance
...
Finally, three phase cores
themselves cost less than three single phase units due to compactness
...
jntuworld
...
jntuworld
...
Krishna Vasudevan, Prof
...
Sridhara Rao, Prof
...
Sasidhara Rao
Parallel operation of one phase and two phase transformers
By parallel operation we mean two or more transformers are connected to the same
supply bus bars on the primary side and to a common bus bar/load on the secondary side
...
The reasons that necessitate parallel
operation are as follows
...
Non-availability of a single large transformer to meet the total load requirement
...
The power demand might have increased over a time necessitating augmentation of the
capacity
...
3
...
Even if one of the transformers gets into a fault or is
taken out for maintenance/repair the load can continued to be serviced
...
To reduce the spare capacity
...
If only one large machine is feeding the load, a spare of similar
rating has to be available
...
5
...
Fig
...
Transformer A and Transformer B are connected to input
voltage bus bars
...
jntuworld
...
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Krishna Vasudevan, Prof
...
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...
Sasidhara Rao
V1
A
E1
IA
V2
E2
load
IB
E1
E2
B
supply bus
Load bus
Figure 37: Parallel Operation of Two Single Phase Transformers - Physical
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Certain conditions have to be met before two or more transformers are connected in
parallel and share a common load satisfactorily
...
The voltage ratio must be the same
...
The per unit impedance of each machine on its own base must be the same
...
The polarity must be the same, so that there is no circulating current between the
transformers
...
The phase sequence must be the same and no phase difference must exist between the
voltages of the two transformers
...
Same voltage ratio Generally the turns ratio and voltage ratio are taken to be the same
...
When the primaries are connected to same bus bars, if the secondaries
do not show the same voltage, paralleling them would result in a circulating current
between the secondaries
...
Thus even without connecting a load considerable current can be drawn
by the transformers and they produce copper losses
...
This circulating
current gets added to the load current when the load is connected resulting in unequal
sharing of the load
...
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If they have to share the total load in proportion to their ratings the larger
machine has to draw more current
...
Thus the
larger machines have smaller impedance and smaller machines must have larger ohmic
impedance
...
As the
voltage drops must be the same the per unit impedance of each transformer on its
own base, must be equal
...
Thus
we have the requirement that per unit resistance and per unit reactance of both the
transformers must be the same for proper load sharing
...
Inside the loop formed by the two secondaries
the resulting voltage must be zero
...
In the case of polyphase banks it is possible to have
permanent phase error between the phases with substantial circulating current
...
The turns ratios in such groups
can be adjusted to give very close voltage ratios but phase errors cannot be compensated
...
6 degree gives rise to one percent difference in voltage
...
Transformers having −30◦ angle can be paralleled to that having +30◦ angle by reversing the phase sequence of both primary and secondary terminals of one of the
transformers
...
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The poly phase banks belonging to same vector group can be
connected in parallel
...
If the phase sequences are not the same
then the two transformers cannot be connected in parallel even if they belong to same
vector group
...
Performance of two or more single phase transformers working in parallel can be computed using their equivalent circuit
...
Basically
two cases arise in these problems
...
These are
discussed now in sequence
...
1
Case A: Equal voltage ratios
Always two transformers of equal voltage ratios are selected for working in parallel
...
Load can be switched
on subsequently to these bus bars
...
38(a),(b)
...
This may be further simplified as shown under Fig
...
The
voltage drop across the two transformers must be the same by virtue of common connection
at input as well as output ends
...
jntuworld
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Krishna Vasudevan, Prof
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Sasidhara Rao
j A
X
RA
V1
IA
RA
V 2’
jX A
IA
ZA
V1
RB
j B
X
RB
I
jX B
V 2’
IB
IB
ZB
(a)
(b)
ZA
IA
I
ZB
IB
VL
V ’L
V
Load
ZL
(c)
Figure 38: Equivalent Circuit for Transformers working in Parallel -Simplified circuit and
Further simplification for identical voltage ratio
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ZA
ZA
ZA + ZB
v
IZ
ZA
=
=
= I
...
If transformers of dissimilar ratings are paralleled the transformer with larger rating shall have smaller impedance as it has to produce the same drop
as the other transformer, at a larger current
...
IA ZA = IB ZB , therefore
IA
IB
=
ZB
...
u basis, we aim at the same per unit drops at any load for
the transformers
...
Fig
...
The drops are
magnified and shown to improve clarity
...
jntuworld
...
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Krishna Vasudevan, Prof
...
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...
Sasidhara Rao
E
V
IAXA
V2
φ
φB
IX
θA θB
φA
IBXB
IB
IA
IL
IARA
IR
IBRB
Figure 39: Phasor Diagram of Operation for two Transformers working in Parallel
transformers is v but the currents IA and IB are forced to have a different phase angle due
to the difference in the internal power factor angles θA and θB
...
If we want them to share the load
current in proportion to their ratings, their percentage ( or p
...
In order to avoid any divergence and to share active and reactive powers also properly,
θA = θB
...
u resistances
and p
...
To
determine the sharing of currents and power either p
...
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Case B :Unequal voltage ratios
RA
RB
j A
X
IA
j B
X
I
IB
ZL
EB
EA
VL
Figure 40: Equivalent Circuit for unequal Voltage Ratio
One may not be able to get two transformers of identical voltage ratio in
spite of ones best efforts
...
In such cases the circuit
representation for parallel operation will be different as shown in Fig
...
In this case the
two input voltages cannot be merged to one, as they are different
...
EA and EB are the no-load secondary emf
...
By inspection the voltage equation can be
written as below:
EA = IA ZA + (IA + IB )ZL = V + IA ZA ·
EB = IB ZB + (IA + IB )ZL = V + IB ZB ·
Solving the two equations the expression for IA and IB can be obtained as
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...
When load is not connected there will be a circulating current
between the transformers
...
Then,
IA = −IB =
(EA − EB )
(ZA + ZB )
(95)
If the load impedance becomes zero as in the case of a short circuit, we have,
IA =
EA
ZA
and IB =
EB
ZB
(96)
Instead of the value of ZL if the value of V is known , the currents can be easily determined
( from Eqns
...
A simpler and more elegant method for the case depicted in Fig
...
It is known by the name parallel generator theorem
...
EA − V
,
ZA
= IL
...
jntuworld
...
jntuworld
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Krishna Vasudevan, Prof
...
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...
Sasidhara Rao
RA
jXA
RB
jXB
RC
jXC
EA
IA
IB
I
EB
IC
EC
ZL
V
Figure 41: Parallel Generator Theorem
Combining these equations
V
EA − V
EB − V
EC − V
=
+
+
+
...
) =
+
+
+
...
1
1
1
1
1
+
+
+
+
...
)
(
(100)
(101)
(102)
From this V can be obtained
...
98, IA , IB etc can be obtained
...
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15
Transformer voltage control and Tap changing
Regulating the voltage of a transformer is a requirement that often arises in a power
application or power system
...
To supply a desired voltage to the load
...
To counter the voltage drops due to loads
...
To counter the input supply voltage changes on load
...
Booster transformer
Regulation
transformer
B
Main
transformer
1
2
Booster transformer
B
1
tertiary2
Reverser
Reverser
Figure 42: Tap changing and Buck Boost arrangement
The voltage control is performed by changing the turns ratio
...
The volts per turn available in large transformers is quite high
and hence a change of even one turn on the LV side represents a large percentage change
in the voltage
...
LV winding being the inner winding in a core type transformer adds to the
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difficulty of taking out of the taps
...
Provision of taps to control voltage is called tap
changing
...
This may be called buck-boost arrangement
...
The
tap changing arrangement and buck boost arrangement with phase shift are shown in Fig
...
Tap changing can be effected when a) the transformers is on no- load and b) the load
is still remains connected to the transformer
...
The Off load tap changing relatively costs less
...
The on-load
tap changer on the other hand tries to change the taps without the interruption of the load
current
...
A few schemes of on-load tap
changing are now discussed
...
jntuworld
...
jntuworld
...
jntuworld
...
jntuworld
...
43
...
The switches for the taps and that across
the reactor(S) are connected as shown
...
The two ends of the reactor are connected to the two bus bars to
which tapping switches of odd/even numbered taps are connected
...
The reactor can also be worked with both ends connected to two successive
taps
...
The reactor limits the circulating
current between the taps in such a situation
...
The
advantage of this type of tap changer are
1
...
2
...
3
...
110
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The major objection to this scheme seems to be that the reactor is in the circuit always
generating extra loss
...
44
...
Initially
tap 1a and 1b are closed and the transformer is energized with full primary voltage
...
Then circuit breaker A is closed
...
A small difference in
the number of turns between the two circuit exists
...
Next, circuit breaker B is opened momentarily, the tap is changed from
1b to 2b and the breaker is closed
...
The circulating current is controlled by careful selection of
the leakage reactance
...
Provision of taps switches and circuit
breakers are to be additionally provided to achieve tap changing in these machines
...
The main transformer need not be having a tapped
arrangement
...
Fig
...
The reverser switch reverses the
polarity of the injected voltage and hence a boost is converted into a buck and vice
versa
...
One precaution to be taken with this arrangement is that the winding must
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B
A
a2
b1
a1
b2 b3
b4
a4 a3
LV
HV
Figure 44: Parallel Primary Winding Tap Changing
112
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not be open circuited
...
In spite of the small ratings and low voltages and flexibility, this method of voltage
control costs more mainly due to the additional floor space it needs
...
Applications like a
...
and d
...
motor speed
control, illumination control by dimmers, electro-chemistry and voltage stabilizers need
continuous control of voltage
...
5%
a1
a2
V out
Vi
n
S
95%
b1
b2
Figure 45: Moving Coil Voltage Regulator
moving coil voltage regulators Fig
...
a, b are the two primary windings wound on a long core, wound in the opposite
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sense
...
These fluxes link their secondaries a2 and b2
...
This moving coil alters the inductances of the two primaries
...
The total secondary voltage in the present case varies from 10 percent to 20
percent of the input in a continuous manner
...
86 and 10
...
5
4
...
6
= 10% when s is in the top position
...
86
+
5
10
...
By selecting proper ratios for the
secondaries a2 and b2 one can get the desired voltage variation
...
c voltage
V1
V1
V2
sliding contact
a) without electrical isolation
b) with electrical isolation
Figure 46: Sliding Contact Regulator
Sliding contact regulators These have two winding or auto transformer like construction
...
The minimum step size of voltage change obtainable is the voltage across a
single turn
...
jntuworld
...
jntuworld
...
In smaller ratings this is highly cost effective
...
The two winding arrangement provides electrical isolation also
...
46
...
jntuworld
...
jntuworld
...
C Machines
1
Introduction
The steam age signalled the beginning of an industrial revolution
...
Thus a search for new sources of energy and novel gadgets received
great attention
...
This enabled the research work on
moving charges or currents
...
This led to the invention of an
electromagnet
...
This can be termed as the birth of a motor
...
Parallel research was contemporarily being done to invent a source of energy to recharge the batteries in the form of a
d
...
source of constant amplitude (or d
...
generator)
...
c
...
c
...
During the second half
of the 19th century these two paths merged
...
c
...
These inventions generated great interest in the
generation and use of electrical energy
...
The evolution of these
machines was very quick
...
The d
...
power system was poised for a predominant place as a preferred
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system for use, with the availability of batteries for storage, d
...
generators for conversion of
mechanical energy into electrical form and d
...
motors for getting mechanical outputs from
electrical energy
...
c
...
In the case of d
...
systems the generating stations and the
load centers have to be near to each other for efficient transmission of energy
...
c
...
c
...
Thus a
...
system took over as the
preferred system for the generation transmission and utilization of electrical energy
...
c
...
Further,
d
...
motors have excellent control characteristics
...
c
...
In the lower power levels and
also in regenerative systems the d
...
machines still have a major say
...
c
...
c
...
They use the electromagnetic principles
which can be further simplified at the low frequency levels at which these machines are used
...
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1
...
The electrostatic principles do not yield practical machines for commercial electric power generation
...
Though one sees a variety of electrical machines in the market, the basic underlying principles of all these are the same
...
1
...
Magnetic circuit law -
Ampere′ s Law
3
...
Law of electromagnetic interaction -BiotSavart′ s Law
Most of the present day machines have one or two electric circuits linking a common
magnetic circuit
...
The attention is focused on the Faraday’s law and Biot Savart’s law in the
present study of the electrical machines
...
1
...
It states that if the magnetic
flux lines linking a closed electric coil changes, then an emf is induced in the coil
...
This can be expressed
mathematically,
e∝
dψ
dt
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(1)
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where ψ is the flux linkages given by the product of flux lines in weber that are linked
and N the number of turns of the coil
...
The direction of the induced emf can be determined by the application of
Lenz’s law
...
It is analogous to the inertia in the mechanical
systems
...
This is now explained with the help of a simple geometry
...
1 shows a rectangular loop of one turn (or N=1)
...
The flux lines, the conductor and the motion are in mutually perpendicular directions
...
If the flux is unchanging
and conductor stationary, no emf will be seen at the terminals of the loop
...
cos ωt, as in Case(i), an emf given by
e=
d
(Bm
...
N cos ωt) = −(Bm
...
Nω)
...
dt
= −jBm
...
Nω
...
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...
This is termed as a ”transformer” emf
...
L
...
L
...
L
...
v
volts
(5)
The emf induced in the loop is directly proportional to the uniform flux density under which
it is moving with a velocity v
...
The Case(iii) refers to the situation where B is changing with time and so also is X
...
L
...
N
...
cos ωt
...
N
...
L
...
N
...
dt
dt
dt
In this case both transformer emf and speed emf are present
...
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...
This is the principle
used in transformers
...
The induced emf
of this variety is often termed as the transformer emf
...
c
...
A
static magnetic field is produced by a permanent magnet or by a coil carrying a d
...
current
...
In order to produce the emf on a continuous manner a cylindrical geometry
is chosen for the machines
...
In the example shown above, only one conductor is taken and the flux ’cut’ by
the same in the normal direction is used for the computation of the emf
...
This greatly simplifies the
computation of the induced voltage as the determination of flux linkages and finding its rate
of change are dispensed with
...
If the conductor, field and motion are not normal to each other then
the mutually normal components are to be taken for the computation of the voltage
...
Application of Faradays law according to Case(iii) above for electro mechani6
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cal energy conversion results in the generation of both transformer and rotational emf to be
present in the coil moving under a changing field
...
c
...
The direction of the induced emf is
emf and
current
Force
Motion
F
B
(a)
(b)
Figure 2: Law of induction-Generator action
decided next
...
This is illustrated in Fig
...
In Case(i), the induced emf will be in such a direction as to cause a opposing
mmf if the circuit is closed
...
Also the coil experiences a compressive force when the flux tries to increase and
a tensile force when the flux decays
...
In Case(ii), the direction of the induced emf is as shown
...
The changes in the flux linkages is
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emf
current
F
Motion,Force
B
(a)
(b)
Figure 3: Law of interaction- Motor action
brought about by the sweep or movement of the conductor
...
If one looks
closely at the field around the conductor under these conditions it is as shown in Fig
...
The flux lines are more on one side of the conductor than the other
...
As F opposes v and the applied force,
mechanical energy gets absorbed in this case and the machine works as a generator
...
Fig
...
c
...
The current carrying conductor reacts with the field to develop a force which urges the conductor to the right
...
In Case (iii) also the direction of the induced emf can be determined in a
similar manner
...
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...
Putting mathematically, in the present study of d
...
machines,
F = B
...
I
Newton
When the generated voltage drives a current, it produces a reaction force on the
mechanical system which absorbs the mechanical energy
...
The converse happens in the case of the motor
...
Thus, it is seen that the motoring and generating
actions are easily changeable with the help of the terminal conditions
...
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c
...
C
...
c
...
c
...
These
machines can be broadly classified into two types, on the basis of their magnetic structure
...
Homopolar machines
2
...
These are discussed in sequence below
...
1
Homopolar machines
Homopolar generators
Even though the magnetic poles occur in pairs, in a homopolar generator the conductors
are arranged in such a manner that they always move under one polarity
...
Since the conductor encounters the magnetic
flux of the same polarity every where it is called a homopolar generator
...
The conductor can be situated on the surface of the rotor
with one slip-ring at each end of the conductor
...
4
...
The conductor thus sees only one pole polarity or the flux directed in one sense
...
The polarity
of the induced voltage can be reversed by reversing either the excitation or the direction of
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+
Flux
N
-
A
Brush
B
S
S
Field
coil
+
A
N
Figure 4: Homopolar Generator
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-
B
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rotation but not both
...
Such sources are used in some applications
like pulse-current and MHD generators, liquid metal pumps or plasma rockets
...
If higher voltages are required one is forced to connect many conductors in series
...
Many conductors must be situated on the
rotating structure each connected to a pair of slip rings
...
The magnitude
of the induced emf in a conductor 10 cm long kept on a rotor of 10 cm radius rotating at
3000 rpm, with the field flux density being 1 Tesla every where in the air gap, is given by
e = BLv
= 1 ∗ 0
...
1 ∗
3000
= 3
...
Even though homopolar machines are d
...
generators in a
strict sense that they ’generate’ steady voltages, they are not quite useful for day to day use
...
c
...
2
...
c
...
The
induced emf in the conductor therefore is not a constant but alternates in magnitude
...
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If the flux density variation is sinusoidal in space, then a sine
wave voltage is generated
...
c generators
...
c
...
c
...
This is achieved by employing an external element,
which is called a commutator, with the winding
...
5 shows an elementary hetero-polar, 2-pole machine and one-coil armature
...
As
the polarity of the induced voltages changes the connection to the brush also gets switched
so that the voltage seen at the brushes has a unidirectional polarity
...
The brushes are placed
on the commutator
...
The
idea of a commutator is an ingenious one
...
Similarly the sum of a set of coils also remains a constant
...
The coils connected between the two brushes must be
”similarly located” with respect to the poles irrespective of the actual position of the rotor
...
If a winding satisfies this condition then
it is suitable for use as an armature winding of a d
...
machine
...
It is easy to follow the action of the d
...
machine using a ring winding,
hence it is taken up here for explanation
...
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6 shows a 2-pole, 12 coil, ring wound armature of a machine
...
The junction of each coil with its neighbor
is connected to a commutator segment
...
Two brushes A and B are placed on the commutator which
looks like a cylinder
...
In each path a set of voltages get added up
...
The constancy of this magnitude is altered by a small value corresponding
to the coil short circuited by the brush
...
If
the armature is turned by a distance of one slot pitch the sum of emfs is seen to be constant
even though a different set of coils participate in the addition
...
This variation in the output voltage is called the ’ripple’
...
Another important observation from the working principle of a heterogeneous
generator is that the actual shape of the flux density curve does not matter as long as the
integral of the flux entering the rotor is held constant; which means that for a given flux
per pole the voltage will be constant even if the shape of this flux density curve changes
(speed and other conditions remaining unaltered)
...
A rectangular flux density wave form has some advantages in the derivation
of the voltage between the brushes
...
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...
With this back
ground the emf induced between the brushes can be derived
...
L
...
L- Length of the ’active’ conductor, m
...
If there are Z conductors on the armature and they form b pairs of parallel circuits between
the brushes by virtue of their connections, then number of conductors in a series path is
Z/2b
...
Z
2b
E = Bav
...
v
...
Y
...
Also Bav can be written in terms of pole pitch Y , core length L, and flux per pole φ as
Bav =
φ
(L
...
9,
E=
φ
Z
...
(2p
...
n)
...
Y )
2b
=
φpZn
b
volts
(11)
The number of pairs of parallel paths is a function of the type of the winding chosen
...
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2
...
1
Torque production
When the armature is loaded, the armature conductors carry currents
...
This
force is in such a direction as to oppose their cause which in the present case is the relative
movement between the conductors and the field
...
Hence it absorbs mechanical energy
...
The electrical power generated by an armature delivering a
current of Ia to the load at an induced emf of E is EIa Watts
...
Substituting for E from Eqn
...
Z
...
Ia
b
(13)
which gives torque T as
T =
1
Ia
...
( )Z Nm
2π
b
(14)
This shows that the torque generated is not a function of the speed
...
The expression for the torque generated
can also be derived from the first principles by the application of the law of interaction
...
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...
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...
Force on a single conductor Fc is given by,
Fc = B
...
Ic
Newton
(15)
The total work done by an armature with Z conductors in one revolution is given by,
Wa = Bav
...
Ic
...
(2p
...
L
...
Z
...
Y
L
...
2p
...
Ic
...
p
...
n
...
The above principles can easily be extended to the case of motoring mode
of operation also
...
c
...
2
...
2
Motoring operation of a d
...
machine
In the motoring operation the d
...
machine is made to work from a d
...
source and
absorb electrical power
...
This is briefly
discussed here
...
c
...
c
...
If the field is already excited then
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these current carrying conductors experience a force as per the law of interaction discussed
above and the armature experiences a torque
...
The conductors now move under the
field and cut the magnetic flux and hence an induced emf appears in them
...
Thus a ’back emf’ appears and tries to reduce the current
...
This absorbed electrical power gets converted into mechanical
form
...
The absorbed power
gets converted into electrical or mechanical power
...
These aspects would be discussed in detail at a later stage
...
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c
...
c
...
The initial pace of development work was phenomenal
...
Less effective options were
discarded
...
c
...
To appreciate the working and the characteristics
of these machines, it is necessary to know about the different parts of the machine - both
electrical and non-electrical
...
An exploded view of a small d
...
Figure 7: Exploded view of D
...
Machine
machine is shown in Fig
...
Click here to see the assembling of the parts
...
Body
2
...
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Armature
4
...
Commutating poles
6
...
Other mechanical parts
The constructional aspects relating to these parts are now discussed briefly in sequence
...
This will be closed at both the ends by two end covers which also support the bearings
required to facilitate the rotation of the rotor and the shaft
...
c
...
Hence the shell or frame supports
the poles and yoke of the magnetic system
...
Cast steel is used as a material for the frame and yoke as the
flux does not vary in these parts
...
Those are called as fabricated frames
...
In small special machines these could be
made of stack of laminations suitably fastened together to form a solid structure
...
Pole shoes are generally laminated
...
Stiffeners are used on both
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sides of the laminations
...
The pole
shoes are shaped so as to have a slightly increased air gap at the tips
...
These can
be solid, or laminated just as the main poles
...
Sometimes the yoke may be slotted to receive these poles
...
These are also called as commutating
poles or compoles
...
Armature The armature is where the moving conductors are located
...
Thickness of these lamination
is kept low to reduce eddy current losses
...
The core is divided into packets to facilitate ventilation
...
Open parallel sided equally spaced slots are normally punched in
the rotor laminations
...
Large sized machines
employ a spider on which the laminations are stacked in segments
...
Armature construction process
must ensure provision of sufficient axial and radial ducts to facilitate easy removal of
heat from the armature winding
...
These carry the excitation current and produce the main field in the machine
...
Two types of windings are generally
employed
...
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The resistance of such winding would be an order of magnitude larger than the
armature winding resistance
...
The resistance of such windings is low and is comparable
to armature resistance
...
The total ampere turns required to establish the necessary flux under the poles is
calculated from the magnetic circuit calculations
...
The mmf
required to be shared between shunt and series windings are apportioned as per the
design requirements
...
Mmf ’per pole’ is normally used in these calculations
...
The conductors may fly
away when the armature starts rotating
...
In the case of
small machines, they can be hand wound
...
In the case of large machines slot wedges are additionally used
to restrain the coils from flying away
...
The armature must be dynamically balanced to reduce the centrifugal forces at
the operating speeds
...
This is mostly found in d
...
machines of very large rating
...
In smaller machines, they may be absent
...
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4
3
2
2
1
1
...
Insulating cups
3
...
Riser
5
...
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...
It consists of copper segments tightly fastened together with mica/micanite
insulating separators on an insulated base
...
Each commutator segment is provided with a ’riser’ where the ends of the armature coils get
connected
...
Under-cutting the
mica insulators that are between these commutator segments has to be done periodically to avoid fouling of the surface of the commutator by mica when the commutator
gets worn out
...
8
...
Normally
electro-graphite is used as brush material
...
The hardness of
the graphite brush is selected to be lower than that of the commutator
...
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More number of relatively smaller width brushes are preferred in place of large broad
brushes
...
The connection
Pigtail
Pressure
spring
Brush
Brush holder box
(a)
Radial
Trailing
Reaction
Motion of commutator
(b)
Figure 9: Brush holder with a Brush and Positioning of the brush on the commutator
from the brush is taken out by means of flexible pigtail
...
This is to ensure proper contact between
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the brushes and the commutator even under high speeds of operation
...
Fig
...
Radial positioning of the
brushes helps in providing similar current collection conditions for both direction of
rotation
...
9-(b) Reaction arrangement is preferred as it results in zero side
thrust on brush box and the brush can slide down or up freely
...
This is especially true if the machine is operating in a dusty
environment like the one found in cement plants
...
End covers are completely solid or have opening for ventilation
...
Proper machining is to be ensured for
easy assembly
...
In most machines the fan is on the
non-commutator end sucking the air from the commutator end and throwing the same
out
...
Bearings Small machines employ ball bearings at both ends
...
The bearings are mounted press-fit
on the shaft
...
The bearings must be
kept in closed housing with suitable lubricant keeping dust and other foreign materials
away
...
Care must be taken to see that there are no bearing currents or axial forces on
the shaft both of which destroy the bearings
...
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10 gives the cross sectional view of a modern d
...
machine showing all the salient parts
...
c
...
This is where the emf is induced and hence its effective deployment
enhances the output of the machine
...
11(a) shows one coil of an armature of Gramme
ring arrangement and Fig
...
Earlier,
a simple form of this winding in the form of Gramme ring winding was presented for easy
understanding
...
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The ring winding has only one conductor in
a turn working as an active conductor
...
Thus the effectiveness of the electric circuit is only 50 percent
...
Also, the
return conductor has to be wound inside the bore of the rotor, and hence the rotor diameter
is larger and mounting of the rotor on the shaft is made difficult
...
Both the conductors have emf induced in them
...
The rotor is mechanically robust with more area being available for carrying
the flux
...
The rotor diameters are smaller
...
The coils
could be made of single conductors (single turn coils) or more number of conductors in series
(multi turn coils)
...
The two sides
of the coil lie under two poles one north and the other south, so that the induced emf in
them are always additive by virtue of the end connection
...
Thus there is no circulating
current when the armature is not loaded
...
Hence slots are made on the surface and the conductors
are placed in these slots and fastened by steel wires to keep them in position
...
The winding is called as
a double layer winding
...
The
distance, measured along the periphery of the armature from any point under a pole to a
similar point under the neighboring pole is termed as a pole pitch
...
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The junction of two coils is terminated
on a commutator segment
...
In a double layer winding in S slots there are 2S layers
...
The S junctions of these S coils are terminated on S
commutator segments
...
While the number of parallel circuits in the case of ring winding is
equal to the number of poles, in the case of drum winding a wide variety of windings are
possible
...
The physical
arrangement of a single coil is shown in Fig
...
Fig
...
13-(b) shows the cut and spread view
of the machine
...
As seen earlier the sum of the instantaneous emfs appears across the brushes
...
As this coil in general lies in the magnetic neutral
axis it has a small value of voltage induced in it
...
In order to reduce the ripple,
one can increase the number of coils coming in series between the brushes
...
However increasing the slot number makes the tooth
width too narrow and makes them mechanically weak
...
This is shown in Fig
...
In the figure, the conductors a, b and c belong to a
coil
...
In the present case the number
of coils in the armature is 2S/3S
...
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Such coils are called ’full pitched’ coils
...
If the
span is more than or less than the full pitch then the coil is said to be ’chorded’
...
Slight short chording of the coil reduces overhang length
and saves copper and also improves commutation
...
Similar to the pitch of a coil one can define the winding pitch and commutator
pitch
...
c
...
Winding pitch provides a means of indicating this
...
Commutator pitch is the number of ’micas’ between the ends of
a coil
...
One simple method is to number only the
top coil side of the coils in sequence
...
As a
double layer is being used the bottom coil side is placed in a slot displaced by one coil span
′
′
from the top coil side
...
indicating
′
′
the second sides by 1 , 2 etc
...
The commutator segment connected to top coil side of coil 1 is numbered 1
...
It should be noted that changing of the pitch
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of a coil slightly changes the induced emf in the same
...
The armature windings are classified into two families based on this
...
They can be simply stated in terms of the commutator
pitch used for the winding
...
1
Lap winding
The commutator pitch for the lap windings is given by
yc = ±m,
m = 1, 2, 3
...
For m = 1 we get a simple lap winding, m = 2 gives duplex lap winding etc
...
The sign refers to the direction of progression
of the winding
...
Fig
...
Fig
...
The connections of the coils to the commutator segments are also
shown
...
The position and polarity of the brushes are also indicated
...
The number of parallel paths formed by the winding
equals the number of poles
...
Thus the lap winding is well suited for high
current generators
...
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The increase in the number of parallel paths in the armature winding brings
about a problem of circulating current
...
This will be more with the
increase in the number of poles in the machine
...
This circulating current wastes power, produces heat
and over loads the brushes under loaded conditions
...
c
...
As the name suggests
these connections identify similar potential points of the different parallel paths and connect
them together to equalize the potentials
...
Also, the circulating current does not
flow through the brushes loading them
...
An
example of equalizer connection is discussed now with the help of a 6-pole armature having
150 commutator segments
...
There are 50 groups like that
...
Then
1,11,21,31, and 41 are the coils under the first pair of poles
...
There are 10 coils
connected in series between any two successive links
...
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1
S
N
S
N
20 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
A2
+
A1
+
- B2
- B1
Motion
(a)Winding layout
_
Full pitch: 21/4=5
...
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2
Wave windings
In wave windings the coils carrying emf in the same direction at a time are all
grouped together and connected in series
...
To implement a wave winding one should select the commutator pitch
as
yc =
C±1
p
(21)
where C is the total segments on the commutator
...
Here also the positive sign refers to the progressive
winding and the negative sign yields a retrogressive winding
...
A simple wave winding for 4 poles in 21 slots is illustrated in
Fig
...
As could be seen from the figure, the connection to the next (or previous) adjacent
coil is reached after p coils are connected in series
...
The position for the brushes is indicated in the diagram
...
The number of commutator segments can only be
one more or one less than some multiple of pole pairs
...
As mentioned earlier the simple wave winding forms two parallel paths, duplex
wave winding has 2*2=4 etc
...
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Two
brush sets are therefore adequate
...
This arrangement does not increase the number of parallel circuits but reduces
the current to be collected by each brush set
...
A
4-pole wave connected winding with 21 commutator segments is taken
...
A retrogressive wave winding results
...
If coil number 1 is assumed to be in the neutral axis then other neutral axis
coils are a pole pitch apart i
...
coils 6, 11, 16
...
The polarity of the brushes are positive and negative
alternately
...
By
having four brushes at 1, 6, 11 and 16 and connecting 1,11 and 6,16 still only two parallel
circuits are obtained
...
This method permits the use
of commutator of shorter length as lesser current is to be collected by each brush and thus
saving on the cost of the commutator
...
17(b) illustrates this brush arrangement with
respect to a 21 slot 4 pole machine
...
Multiplex windings of order m have m times the circuits compared to a simplex
winding and so also more restriction on the choice of the slots, coil sides, commutator
and brushes
...
The duplex windings are used under very special circumstances when the number
of parallel paths had to be doubled
...
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3
Dummy coils and dummy commutator segments
Due to the restrictions posed by lap and wave windings on the choice of number
of slots and commutator segments a practical difficulty arises
...
Thus each machine may be tailor made for a given
specification
...
Sometimes due to the non-availability of a suitable slot number or commutator, one is forced to design the winding in an armature readily available in stock
...
If one is satisfied with approximate solutions then the designer can
omit the surplus coil or surplus commutator segment and complete the design
...
All the coils are placed in the armature slots
...
It serves to provide mechanical balance against centrifugal
forces
...
These are called dummy coils
and dummy commutator segments
...
Slightly un-symmetric
winding may be tolerable in machines of smaller rating with very few poles
...
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Armature reaction
Earlier, an expression was derived for the induced emf at the terminals of the
armature winding under the influence of motion of the conductors under the field established
by field poles
...
In such a case the armature conductors also carry currents and produce a field of their
own
...
As the magnetic structure is complex
and as we are interested in the flux cut by the conductors, we primarily focus our attention
on the surface of the armature
...
The convention used here is that
the mmf acting across the air gap and the flux density in the air gap are shown as positive
when they act in a direction from the field system to the armature
...
As the magnetic circuit is non-linear,
the field mmf and armature mmf are separately computed and added at each point on the
surface of the armature
...
The flux produced by field and that produced by armature could be added to
get the total flux only in the case of a linear magnetic circuit
...
5
...
1
MMF distribution due to the field coils acting alone
Fig
...
It
is a step curve with the width being equal to the pole arc
...
18 assuming the air gap under the pole to be uniform and neglecting
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the slotting of the armature
...
Allowing for the fringing of the flux, the actual flux density curve
would be as shown under Fig
...
5
...
2
MMF distribution due to armature conductors alone carrying currents
N
N-Pole
S
A
S-Pole
Generator
Flux
mmf
Figure 19: Mmf and flux distribution under the action of armature alone carrying current
The armature has a distributed winding, as against the field coils which
are concentrated and concentric
...
For a full pitched coil, each coil produces a rectangular mmf distribution
...
If we neglect
slotting and have uniformly spaced coils on the surface, then the mmf distribution due to the
armature working alone would be a triangular distribution in space since all the conductors
carry equal currents
...
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19
...
Z
Fa =
...
This
peak value of the mmf occurs at the inter polar area, shifted from the main pole axis by half
the pole pitch when the brushes are kept in the magnetic neutral axis of the main poles
...
0
...
20
...
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This shows the decrease in the mmf at one tip of a pole and a substantial rise
at the other tip
...
7 then 70% of the
armature reaction mmf gets added at this tip leading to considerable amount of saturation
under full load conditions
...
20
...
Actual flux distribution
differs from this slightly due to fringing
...
The air gaps under
the pole tips are also increased in practice to reduce excessive saturation of this part
...
It greatly mitigates the
effect of the armature reaction
...
Even though the armature
reaction produced a cross magnetizing effect, the net flux per pole gets slightly reduced,
on load, due to the saturation under one tip of the pole
...
c
...
5
...
4
Effect of brush shift
In some small d
...
machines the brushes are shifted from the position of the mag-
netic neutral axis in order to improve the commutation
...
Such a shift in the direction of rotation is termed ‘lead’ (or forward lead)
...
This lead is expressed
in terms of the number of commutator segments or in terms of the electrical angle
...
Fig
...
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S
N
Rotation
a
c
Field
flux
b
Armature flux
Total flux
(a)Armature reaction with brush shift
N
Rotation
b’
a’
a
θ
b
S
(b)Calculation of demagnetizing mmf per pole
Figure 21: Effect of brush shift on armature reaction
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The magnetization action due to the armature is no
longer entirely cross magnetizing
...
This may be seen as the price we pay for
improving the commutation
...
Fmin = Ff − α
...
Fa
where Ff is the field mmf, Fa is armature reaction mmf per pole, and α is the pole arc to
pole pitch ratio
...
Ic
...
This may
be to the tune of 20 percent in the modern d
...
machines
...
This can be resolved into two components, one in the quadrature axis and second along the pole axis as shown in Fig
...
(b) The demagnetizing and cross magnetizing
component of the armature ampere turn per pole can be written as
2θ
...
Fa
π
Fd =
(24)
(25)
where θ is the angle of lead
...
C 4p
4p
or
Cl
...
Z
C
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5
...
5
Armature reaction in motors
As discussed earlier, for a given polarity of the field and sense of rotation, the
motoring and generating modes differ only in the direction of the armature current
...
The leading and trailing edges of the poles change positions if direction
of rotation is made opposite
...
Hence it is highly desirable, even in the case of non-reversing drives,
to keep the brush position at the geometrical neutral axis if the machine goes through both
motoring and generating modes
...
This increases the stress on the ‘mica’ (micanite) insulation
used for the commutator, thus resulting in increased chance of breakdown of these insulating
sheets
...
The third effect of the armature reaction mmf distorting the flux density is
that the armature teeth experience a heavy degree of saturation in this region
...
The saturation of the teeth may
be too great as to have some flux lines to link the thick end plates used for strengthening
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The increase in iron loss could be as high as 50 percent more at full load
compared to its no-load value
...
This is provided by having a compensating
winding housed on the pole shoe which carries currents that are directly proportional to the
armature current
...
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+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
S
N
Rotation
mmf of
compensating
winding
Resultant
mmf
compole mmf
Armature
mmf
Main field
mmf
Figure 23: Armature reaction with Compensating winding
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The sign of the ampere conductors is made opposite to the armature
...
22 and Fig
...
Since the compensating winding is connected in series with
the armature, the relationship between armature mmf and the mmf due to compensating
winding remains proper for all modes of working of the machine
...
Z polearc
...
The axis of the mmf due to armature and the compensating winding being
the same and the signs of mmf being opposite to each other the flux density in the region
of geometric neutral axis gets reduced thus improving the conditions for commutation
...
Such a design results in overcompensation under the poles
...
It is better not to depend on the compensating winding for improving commutation
...
From the analysis of the phenomenon of armature reaction that takes place
in a d
...
machine it can be inferred that the equivalent circuit of the machine need not be
modified to include the armature reaction
...
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source of internal resistance equal to the armature circuit resistance and a series voltage drop
equal to the brush contact drop, under steady state
...
c
...
5
...
c
...
The
frequency of this induced emf is given by the product of the pole-pairs and the speed in
revolutions per second
...
In order to get maximum d
...
voltage in the external circuit the coil
should be shifted to the negative group
...
During a short interval when the two adjacent commutator segments get bridged by the
brush the coils connected in series between these two segments get short circuited
...
In a simple wave
winding in a 2p pole machine 2 coils get short circuited
...
Thus brush
and commutator play an important role in commutation
...
c
...
c
...
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Brushes
Brush forms an important component in the process of commutation
...
Further this
brush contact resistance is not a constant
...
Brush contact drop is influenced by the major factors like speed of operation,
pressure on the brushes, and to a smaller extent the direction of current flow
...
Soft graphite brushes working at a current density of about 10A/cm2 produce a
drop of 1
...
3V
...
12 and 0
...
The attention is focussed next on the process of
commutation
...
1
...
For simplicity, the brush
thickness is made equal to thickness of one commutator segment
...
24(b), the brush
is initially solely resting on segment number 1
...
As the commutator moves relative to the brush position, the brush
position starts to overlap with that of segment 2
...
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two segments
...
The currents I1 and I2 in brush
segments 1 and 2 are given by
I1 = 2Ia (1 −
x
) and
tb
I2 = 2Ia
x
tb
(28)
giving I1 + I2 to be 2 Ia
...
The process of commutation
would be over when the current through segment number 1 becomes zero
...
This type of linear commutation is very close
to the ideal method of commutation
...
25
...
Fig
...
(b) also shows the timing diagram for the
currents I1 and I2 and the current densities in entering edge αe , leaving edge αl and also the
mean current density αm in the brush
...
In general commutation will not be linear due to the presence of emf of self
induction and induced rotational emf in the coil
...
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1
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Here
the current transfer from 1 to 2 gets retarded as the name suggests
...
26
...
The variation of i is the change in
′
the current of the coil undergoing commutation, while i is that during linear commutation
...
26(b) shows the variation of I1 and current density in the brush at the leaving edge and
Fig
...
(c) shows the same phenomenon with respect to I2 at entering edge
...
αm is the mean current density in
the brush given by total current divided by brush area of cross section
...
As before,
I1 = Ia + i
and
I2 = Ia − i
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The variation of densities at leaving and entering edges are given as
αl =
AB
...
αm
DE
(33)
At the very end of commutation, the current density
′
αe
di di
= αm
...
/
dt Tc
(34)
If at this point di/dt = 0 the possibility of sudden breaking of the current and
hence the creation of an arc is removed
...
27
...
αe = αm
...
Hence it is very advantageous to have retarded commutation at the
entry time and accelerated commutation in the second half
...
27
...
It is termed as sinusoidal commutation
...
To obtain an accelerated commutation, the coil undergoing commutation must have in it an induced emf of such a polarity as that under the pole towards
which it is moving
...
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α’ =AB/AC
α" =DB/DC
α"
Ia
D
αm
C
Ia
α’
B
0
0
i
i
Tc
i’
i’
B
Tc
A
-Ia
-Ia
(a1 )
(a2 )
α’
Ia
C
Ia
S
α"
A
B
αm
R
α’ =PR/PQ
α" =SR/SQ
Q
i
Leaving edge
Entering
edge
Tc
0
Tc
0
time
time
i’
i
-Ia
P
-Ia
(b1 )
(b2 )
Figure 27: Accelerated and Sinusoidal commutation
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In a non-inter pole machine
the brush shift must be changed from forward lead to backward lead depending upon generating or motoring operation
...
The condition of commutation will be worse if commutating poles
are provided and not excited or they are excited but wrongly
...
It does not disturb the main field distribution
...
The commutating pole windings are connected in series with the armature of a d
...
machine to get
a load dependent compensation of armature reaction mmf
...
The air gap
under compole is made large and the width of compole small
...
This would ensure straight line commutation
...
The compole mmf in the presence of a compensating winding on the poles
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This could have been predicted as the axis of
the compensating winding and armature winding is one and the same
...
5
...
c
...
Also, the axis
of compensating mmf and mmf of commutating poles act in quadrature to the main field
...
The relative connections of armature, compole and compensating winding are
unaltered whether the machine is working as a generator or as a motor; whether the load
is on the machine or not
...
The terminals reflect only the additional ohmic drops due to the compole and compensating
windings
...
The armature circuit
can be simply modelled by a voltage source of internal resistance equal to the armature
resistance + compole resistance + compensating winding resistance
...
5
...
1
Excitation circuit
The excitation for establishing the required field can be of two types a) Permanent
magnet excitation(PM) b) Electro magnetic excitation
...
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Yoke
ly
lt lg
lg lt
lp
lp
Pole
Field
coil
la
Armature
da
Figure 28: Magnetization of a DC machine
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Also, permanent magnet excited fields cannot be varied for control purposes
...
However, an advantage
of permanent magnet is that there are no losses associated with the establishment of the field
...
Even though certain amount
of energy is lost in establishing the field it has the advantages like lesser cost, ease of control
...
MMF required for the poles, air gap,
armature teeth, armature core and stator yoke are computed and added
...
28 shows two
poles of a 4-pole machine with the flux paths marked on it
...
µ/l
Where P- permeance
A- Area of cross section of the part
mu- permeability of the medium
l- Length of the part
A flux loop traverses a stator yoke, armature yoke, and two numbers each
of poles, air gap, armature teeth in its path
...
The mmf requirement for
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permeance
= B
...
This value of mmf is required to establish two poles
...
In the case of small machines all this mmf
is produced by a coil wound around one pole
...
This procedure saves cost as only one coil need be wound for getting a pair of poles
...
In large machines, half of total mmf is assigned to each pole as the mmf per pole
...
Such winding has a high value of resistance and hence a large ohmic drop
...
Such method of
excitation is termed as shunt excitation
...
These
windings have extremely small resistance and can be connected in series with a large current
path such as an armature
...
A d
...
machine can have either of these or both these types of
excitation
...
29
...
The mmf of the two windings could be arranged to
aid each other or oppose each other
...
If the shunt winding is excited by a separate voltage source
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C generator connections
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If the excitation power comes from the same machine,
then it is called self excitation
...
The characteristics of these generators are discussed now in sequence
...
2
...
m
...
m
...
due to Residual Magnetism
F1
-
Exciting Current
(a)
(b)
Figure 30: Magnetization characteristics
Fig
...
The current drawn by
the field winding can be regulated from zero to the maximum value
...
In all these cases the presence of a prime mover rotating the armature is assumed
...
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separate excitation is normally used for testing of d
...
generators to determine their open
circuit or magnetization characteristic
...
The load current is kept zero
...
The graph showing the nature of variation of the induced emf as a
function of the excitation current is called as open circuit characteristic (occ), or no-load
magnetization curve or no-load saturation characteristic
...
30(b)
...
The
magnetization characteristic exhibits saturation at large values of excitation current
...
This is because of the remnant
field in the iron
...
Separate
excitation is advantageous as the exciting current is independent of the terminal voltage
and load current and satisfactory operation is possible over the entire voltage range of the
machine starting from zero
...
2
...
The shunt field is connected across the armature
...
The series field continues to be in series with the armature
...
31
...
When the
armature is rotated a feeble induced emf of 2 to 5 percent appears across the brushes depending upon the speed of rotation and the residual magnetism that is present
...
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1500 rev/min
oh
m
s
25
0
210
A2
F2
n
Prime
mover
17
0
ohm
s
500 rev/min
60
30
0
(a)Physical connection
1
...
m
...
m
...
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D
Open circuit
characteristic
C
Q’
Voltage
R’
QL
Q
P’
R
RL
P
O’
0
PL
P1
P"
Q1
Q"
A’
Armature drop
characteristic
A
Excitation current If
Armature current Ia
Figure 32: External characteristics of a self excited of a shunt generator
gets applied across the shunt field winding and produces a small mmf
...
It is like a positive feed back
...
Under
such situation there is no further increase in the field mmf and the build up of emf also stops
...
Fig
...
The field resistance
line is also shown by a straight line OC
...
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teristic (OCC) with the field resistance line, in this case C, represents the voltage build up
on self excitation
...
This value of the resistance is called the critical resistance
...
See Fig
...
Instead of increasing the field resistance if the speed of the machine is reduced then the same
resistance line becomes a critical resistance at a new speed and the self excitation collapses
at that speed
...
In the linear portion of the OCC the ordinates are proportional to the speed
of operation, hence the critical resistance increases as a function of speed Fig
...
(b) and (d)
...
1
...
2
...
3
...
4
...
5
...
Remedial measures to be taken if the machine fails to self excite are briefly
discussed below
...
The residual field will be absent in a brand new, unexcited, machine
...
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The polarity of connections have to be set right
...
If the generator had
been working with armature rotating in clockwise direction before stopping and if one
tries to self excite the same with counter clockwise direction then the induced emf
opposes residual field, changing the polarity of connections of the field with respect to
armature is normally sufficient for this problem
...
Field circuit resistance implies all the resistances coming in series with the field winding
like regulating resistance, contact resistance, drop at the brushes, and the armature
resistance
...
The dirt on
the commutator due to dust or worn out mica insulator can increase the total circuit
resistance enormously
...
So ensuring good speed,
clean commutator and good connections should normally be sufficient to overcome this
problem
...
Speed must be increased sufficiently to a high value to be above the critical speed
...
The load switch must be opened or the load resistance is made very high
...
2
...
In this case the field circuit resistance is the same as the load circuit
resistance and hence it must be made very low to help self excitation
...
To help
in the creation of maximum mmf during self excitation any field diverter if present must be
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In a series generator load current being the field current of the machine the
self excitation characteristic or one and the same
...
33
5
...
5
Self excitation of compound generators
Most of the compound machines are basically shunt machines with the series wind-
ing doing the act of strengthening/weakening the field on load, depending up on the connections
...
Due to the presence of
the shunt winding, the self excitation can proceed as in a shunt machine
...
It
can be a short shunt connection or a long shunt connection
...
But it does not affect the process
of self excitation as the mmf contribution from the series field is negligible
...
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Both series field winding and shunt field winding are wound around the main
poles
...
The designed
field windings must cater to the full range of operation of the machine at nominal armature
current
...
The d
...
machines give rise to a variety of external characteristics with considerable ease
...
The external characteristics, also known
as load characteristics, of these machines are discussed next
...
3
Load characteristics of d
...
generators
Load characteristics are also known as the external characteristics
...
If
they are not held constant then there is further change in the terminal voltage
...
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Open circuit e
...
f C
Induced e
...
f
B
Volts
Terminal voltage V
Ohmic Drop IaRa
0
A
Load current,Ia
Figure 34: External characteristics of a separately excited shunt generator
pensating winding resistance
...
Experimentally the external characteristics
can be determined by conducting a load test
...
This departure is due to the armature reaction which
causes a saturation at one tip of each pole
...
Hence the reduction in the flux per pole with
load is obvious
...
Thus there is some slight droop
in the external characteristics, which is good for parallel operation of the generators
...
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One could easily guess that the self excited machines have slightly higher droop
in the external characteristic as the induced emf E drops also due to the reduction in the
applied voltage to the field
...
The former is preferred due to the ease with which it
can be implemented
...
The value of this mmf magnetizes/demagnetizes the field depending on
whether the lead is backward or forward
...
4
External characteristics of a shunt generator
For a given no-load voltage a self excited machine will have more voltage drop at
the terminals than a separately excited machine, as the load is increased
...
After certain load
current the terminal voltage decreases rapidly along with the terminal current, even when
load impedance is reduced
...
Also, in a
self excited generator the no-load terminal voltage itself is very sensitive to the point of intersection of the magnetizing characteristics and field resistance line
...
If one determines
the load magnetization curves at different load currents then the external characteristics
can be easily determined
...
If such curves are determined for different load currents then by determining
the intersection points of these curves with field resistance line one can get the external
characteristics of a shunt generator
...
Thus
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P
...
Determining their intercepts with the field resistance line gives us the requisite result
...
These ordinates are then plotted on the original plot
...
32
...
The points of operation up to
the maximum current represent a stable region of operation
...
The decrease in the load resistance decreases the terminal voltage in this region
...
4
...
From the occ obtained by running the machine as a separately excited one, the armature drops are deducted to yield external /load characteristics
...
Fig
...
The first half of
the curve is unstable for constant resistance load
...
The load characteristics
in the first half however is useful for operating the series generator as a booster
...
This is shown in Fig
...
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-
S1
A1
A2
F2
S2
+
E2
A2
Booster
Generator
E1
F1
E=E1 + E2
-
A1
Main generator
Figure 35: Series generator used as Booster
5
...
2
Load characteristics of compound generators
In the case of compound generators the external characteristics resemble those of
shunt generators at low loads
...
This increases /decreases the flux per pole and the induced emf E
...
If this increased emf
cancels out the armature drop the terminal voltage remains practically same between no
load and full load
...
Any cumulative compounding below
this value is called under compounding and those above are termed over- compounding
...
36
...
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External characteristics for other voltages of operation can be similarly derived
by changing the speed or the field excitation or both
...
5
Parallel operation of generators
D
...
generators are required to operate in parallel supplying a common load when
the load is larger than the capacity of any one machine
...
This approach reduces the spare capacity requirement and its
cost
...
In all these cases two or more machines are connected to
operate in parallel
...
5
...
In the case of generators we can alter the external characteristics easily
while it is not possible with batteries
...
This avoids a circulating current between the machines
...
In the case of the loaded machine the
difference in the induced emf makes the load sharing unequal
...
37 shows two generators connected in parallel
...
As the load
is gradually applied a total load current of I ampere is drawn by the load
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s1
v
A2
Prime
mover
s2
A2
Load
G2
G1
A1
A1
F2
F2
Vf1
Vf2
F1
F1
Figure 37: Connection of two shunt generators in Parallel
E1
V2
k
V0
j
Total char
a
Terminal
Voltage
E2
V
A
cteristic
B
C
V2
V1
I1
I2
I=I1+I2
O
Load current
D
Figure 38: Characteristics of two shunt generators in Parallel
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Each machines will share this total current by delivering
currents of I1 and I2 ampere such that I1 + I2 = I
...
This is dictated
by the internal drop in each machine given by equations
V = E1 − I1 Ra1 = E2 − I2 Ra2
(37)
where Ra1 and Ra2 are the armature circuit resistances
...
If RL is not known then an iterative procedure has to be
adopted
...
This is based on the fact that the two machines have a
common terminal voltage when connected in parallel
...
38 the external characteristics
of the two machines are first drawn as I and II
...
Here OC = OA + OB
...
This can be called as the resultant or total external characteristics of the two machines put
together
...
The above procedure can be used even when the two voltages of the machines
at no load are different
...
Machine I gives out electrical power and machine II receives the same
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As can be seen larger the values of Ra1 and Ra2 larger is the tolerance for the error
between the voltages E1 and E2
...
When Ra1 and Ra2 are nearly zero
implying an almost flat external characteristic, the parallel operation is extremely difficult
...
5
...
Some applications
like electric braking may employ them and operate two or more series generates in parallel
...
39 shows two series generators connected in parallel supplying load current of I1 and I2
...
As the total current is I the current supplied
by machine II reduces, so also its excitation and induced emf
...
Ultimately the
current of machine II becomes negative and it also loads the first machine
...
One remedy is
for a problem as this is to make the two fields immune to the circulating current between the
machines
...
39-a
...
A better solution for such problems is to cross connect the two fields as shown in
Fig
...
A tendency to supply a larger current by a machine strengthens the field of the
next machine and increases its induced emf
...
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F2
+
-
F1
A2
A1
F2
F1
+
A2
A1
-
Load
V
Load
V
Equalizer
S2
S2
S1
S1
(a)Equalizer connection
F2
+
-
F1
A2
A1
F1
S2
S1
F2
+
A2
A1
-
S2
S1
(b)Cross connection of series fields
Figure 40: Compound generators operating in parallel
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Dif-
ferential compounding would produce a drooping external characteristics and satisfactory
parallel operation is made easy
...
In such cases the external characteristics will be nearly flat making the
parallel operation more difficult
...
Fig
...
5
...
C
...
C
...
The
simplicity, and linearity of the control method makes them highly preferred machines in
precision drives
...
c
...
Apart from high precision application they are preferred in
stand alone systems working on batteries and high speed drives off constant voltage mains
...
The direction of the torque can be readily obtained from the
law of interaction
...
This absorbed power appears as mechanical power
...
The connections to the supply of a d
...
shunt motor are given in Fig
...
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Commonly used connection is where in both the field and the armature are
energized simultaneously Fig
...
As the field has higher inductance and time constant
torque takes some time to reach the full value corresponding to a given armature current
...
41
...
By then the field
current would have reached the steady value
...
The only difference in the second connection Fig
...
(a) is that the shunt field
winding is connected to a separate source
...
c
...
The field voltage is
kept low in such cases for the sake of control purposes
...
Suitable interlock should be provided to prevent the
armature switch being closed prior to / without closing of field circuit as the armature
currents reach very large values still not producing any torque or rotation
...
Z
...
n where Ke =
b
b
1 pZ
1 p
...
= Kt φIa where Kt =
...
w - angular velocity = 2π
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The first one is an electrical equation, the second and the third are electro
mechanical in nature and the last equation is the mechanical equation of motion
...
Under
steady speed of operation the fourth equation is not required
...
These characteristics are similar to the external characteristics for a generator
...
This is termed as the torque speed characteristic of the motor
...
7
Torque speed characteristics of a shunt motor
A constant applied voltage V is assumed across the armature
...
The mmf of the field is assumed to be constant
...
The variation of these parameters are shown in Fig
...
Knowing the value of E and flux one can determine the value of the speed
...
This
procedure is repeated for different values of the assumed armature currents and the values
are plotted as in Fig
...
From these graphs, a graph indicating speed as a function of
torque or the torque-speed characteristics is plotted Fig
...
As seen from the figure the fall in the flux due to load increases the speed due
to the fact that the induced emf depends on the product of speed and flux
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Flux, Speed and Torque
A
No load speed
Line voltage
B
Speed
C
Back emf
E
F
Flux
Torque
G
Armature current
0
(a)Load characteristics
(ii)
Speed
(i)
0
Torque
(b)Torque speed curve
Figure 42: DC Shunt motor characteristics
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With highly saturated machines
the on-load speed may even slightly increase at over load conditions
...
This type of external characteristics introduces instability
during operation Fig
...
This may be simply achieved by
providing a series stability winding which aids the shunt field mmf
...
8
Load characteristics of a series motor
Following the procedure described earlier under shunt motor, the torque speed
characteristics of a series motor can also be determined
...
At large value of the armature currents the useful flux
would be less than the no-load magnetization curve for the machine
...
As the magnetic circuit becomes more
and more saturated the torque becomes proportional to Ia as flux variation becomes small
...
43(a) shows the variation of E1 , flux , torque and speed following the above procedure
from which the torque-speed characteristics of the series motor for a given applied voltage
V can be plotted as shown in Fig
...
(b) The initial portion of this torque-speed curve is
seen to be a rectangular hyperbola and the final portion is nearly a straight line
...
Such
high speeds are unsafe, as the centrifugal forces acting on the armature and commutator
can destroy them giving rise to a catastrophic break down
...
In order to
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safeguard the motor and personnel, in the modern machines, a ‘weak’ shunt field is provided
on series motors to ensure a definite, though small, value of flux even when the armature
current is nearly zero
...
It is
needless to say, this field should be connected so as to aid the series field
...
9
Load characteristics of a compound motor
Two situations arise in the case of compound motors
...
The first configuration
is called differential compounding and is rarely used
...
This
mode may sometimes result due to the motoring operation of a level-compounded generator,
say by the failure of the prime mover
...
In motors intended for constant speed operation the level of compounding is very
low as not to cause any problem
...
High degree of compounding will make the machine approach a series machine like characteristics but with a safe no-load speed
...
Thus the torque per ampere of the armature current is made
high
...
Due to the large speed variation between light load and peak load conditions, a
fly wheel can be used with such motors with advantage
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seen that all modern machines are compound machines
...
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6
Parallel operation of d
...
motors
As in the case of generators motors may also be required to operate in parallel
driving a common load
...
As the two machines are coupled to a common load the speed of the load is the common
parameter in the torque speed plane
...
If the torque speed lines are drooping the point
of intersection remains reasonably unaltered for small changes in the characteristics due to
temperature and excitation effects
...
The machine with flatter curve shares a larger portion
of the torque demand
...
The operation of
level compounded generators is much more difficult compared to the same machines working as cumulative compounded motor
...
Series motors are, with their highly falling
speed with the load torque, are ideal as far as the parallel operation is considered
...
One application where several series motors operate in parallel is in electric locomotives
...
The torque developed by each machine remains close to the other and there is no tendency
for derailment
...
44
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Speed
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A
0
C
B
D
Motors I and II
I
II
in parallel
Torque
Figure 44: Parallel operation of Series motors
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This forces a common
armature current through both the machines and the torques developed by the machines
are proportional to the flux in each machine
...
This method is ideally suited for shunt
and compound machines with nearly flat torque speed characteristics
...
This configuration
is used in steel mills
...
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8
Application of d
...
motors
Some elementary principles of application alone are dealt with here
...
TM − TL = J
dw
dt
(43)
Here TM and TL are the motor torque and the load torques respectively which are expressed
as functions of ω
...
The application of
motors mainly looks at three aspects of operation
...
Starting
2
...
Braking
The speed of the machine has to be increased from zero and brought to the operating speed
...
The operating speed itself should be
varied as per the requirements of the load
...
Finally, the running
machine has to be brought to rest, by decelerating the same
...
The
torque speed characteristics of the machine is modified to achieve these as it is assumed
that the variation in the characteristics of the load is either not feasible or desirable
...
When
more than one method is available for achieving the same objective then other criteria like,
initial cost, running cost, efficiency and ease operation are also applied for the evaluation of
the methods
...
c
...
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c
...
The relevant expressions may be written as,
E
V − Ia Ra − Vb
=
Ke φ
pZφ/b
1 p
...
...
Ia =
2π b
dω
= J
dt
n =
TM
TM − TL
(44)
(45)
(46)
As can be seen, speed is a function of E and φ and T is a function of φ and Ia
...
8
...
c
...
Then TM − TL will be positive so also is dω/dt, and the
machine accelerates
...
Normally the armature resistance of a d
...
machine is such as to cause 1 to 5 percent drop
at full load current
...
The same can be told of the torque if full flux is already established
...
As the speed increases the induced emf appears across the
terminals opposing the applied voltage
...
This continues till the load torque and the motor torque are equal to
each other
...
The starting is now discussed with respect to specific machines
...
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1
...
c
...
To
improve the torque per ampere of line current drawn it is advisable to energize the field
first
...
Variable voltage V can be obtained from a motor generator set
...
A schematic diagram of Ward-Leonard arrangement is shown
in Fig
...
By controlling the field of the Ward-Leonard generator one can get a variable
voltage at its terminals which is used for starting the motor
...
The
current and the torque get reduced
...
46(a)
...
Also,
the starting resistance wastes large amount of power
...
This has to be done progressively,
making sure that the current does not jump up to large values
...
Better starting torques are obtained for
compound motors as the torque per ampere is more
...
47
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1
...
To ascertain that at no step does
the current jump to a large value non-uniform reduction of resistances must be assorted to
...
The calculations
for a starter resistance of a shunt motor are shown below with the help of Fig
...
In
the figure an n element or n+1 step starter is shown
...
The total armature circuit resistance at step 1 is
R1 = (r1 + r2 +
...
The field winding is connected across the supply
...
One resistance element
is cut from the circuit when the current falls down to Im in
...
Thus during the starting the current changes between two limits Im ax and
Im in
...
=
=
= k(say)
=
Im in
R2
R3
Rn
Rn+1
R1 R2
Rn
R1
R1
kn =
∗
∗
...
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Then the number of steps required can
be calculated as
1
log Ra
R1
r
n log k = log
,n =
ra
log k
log R1 − log Rn
log k
(50)
Also,
R=
n
V
=
I1 ra
n
V
=
RI2 ra
n+1
V
I2 ra
(51)
From these expressions it is seen that to have the ratio k to be unity, the number
of steps should be infinity
...
Also, it is not possible to choose n and k independently
...
Im in
is found from the minimum torque against which the starting is required to be performed
...
In these cases the
ratio of currents and the ratio of fluxes are needed
...
Resistance method of starting is cheaper and simple and hence is used universally
...
Hence this method is not advised when
frequent starting of the motor is required
...
With the help of a auto transformer and rectifier set one can get variable
voltage d
...
supply from a constant voltage a
...
This is some times called a
static Ward-Leonard arrangement
...
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...
c
...
The voltage control can be from a variable voltage source like Ward-Leonard
arrangement or by the use of series armature resistance
...
That means
considerable energy is lost in these resistors
...
The variable voltage source on the other hand gives the
motor the voltage just needed by it and the losses in the control gear is a minimum
...
Field control or flux control is also used for speed control purposes
...
This causes operation at higher speeds than the nominal speed
...
Even though
flux weakening gives higher speeds of operation it reduces the torque produced by the machine for a given armature current and hence the power delivered does not increase at any
armature current
...
Above the nominal speed of operation, constant flux mode with increased
applied voltage can be used; but this is never done as the stress on the commutator insulation
increases
...
Above the
nominal speed field weakening is adopted
...
In the case of series motors however field weakening
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is done by the use of ’diverters’
...
8
...
c
...
Braking is employed when rapid stopping is required
...
The electric braking may be done for various reasons such as those mentioned
below:
1
...
2
...
3
...
4
...
5
...
In many cases electric braking makes more brake power available to the braking
process where mechanical brakes are applied
...
By recovering
the mechanical energy stored in the rotating parts and pumping it into the supply lines
the overall energy efficiency is improved
...
Where the safety of
the personnel or the equipment is at stake the machine may be required to stop instantly
...
Electric braking can help
in these situations also
...
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The reduction of the
process time improves the throughput
...
The electric motor can
be made to work as a generator by suitable terminal conditions and absorb mechanical energy
...
Braking can be broadly classified into:
1
...
Regenerative
3
...
8
...
1
Dynamic braking
• Shunt machine
In dynamic braking the motor is disconnected from the supply and connected to a
dynamic braking resistance RDB
...
49 this is done by changing the switch
from position 1 to 2
...
Due to the
rotation of the armature during motoring mode and due to the inertia, the armature
continues to rotate
...
This voltage drives a current through the braking resistance
...
Therefore, torque developed also gets reversed
...
The
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torque speed characteristics separate by excited shunt of the machine under dynamic
braking mode is as shown in Fig
...
The positive
torque corresponds to the motoring operation
...
50 shows the dynamic braking of
a shunt excited motor and the corresponding torque-speed curve
...
Below a certain speed the self-excitation collapses and the braking action becomes
Zero
...
In
order to achieve dynamic braking the series field must be isolated and connected to
a low voltage high current source to provide the field
...
When several machines are available at any
spot, as in railway locomotives, dynamic braking is feasible
...
• Compound generators
In the case of compound machine, the situation is like in a shunt machine
...
A cumulatively connected motor becomes differentially compounded generator and the
braking torque generated comes down
...
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2
F2 +
A2
Vf
F1
+
1
RDB
E
-
A1
2
1
(a)Connections
Speed
RDB increasing
Torque
(b)Characteristics
Figure 49: Dynamic Braking of a shunt motor
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2
F2 +
A2
Vf
F1
+
1
RDB
E
-
A1
2
1
(a)Connections
Speed
RDB increasing
Torque
0
(b)Characteristics
Figure 50: Dynamic braking of shunt excited shunt machine
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8
...
2
Regenerative braking
In regenerative braking as the name suggests the energy recovered from the rotating
masses is fed back into the d
...
power source
...
The armature current can be made to reverse for a constant voltage
operation by increase in speed/excitation only
...
Hence the best method for obtaining the regenerative braking is to operate, the
machine on a variable voltage supply
...
The field current is held constant by means
of separate excitation
...
c
...
51
...
In modern times static Ward-Leonard scheme is used for getting the variable
d
...
voltage
...
Static set
is compact, has higher efficiency, requires lesser space, and silent in operation; however it
suffers from drawbacks like large ripple at low voltage levels, unidirectional power flow and
low over load capacity
...
Series motors cannot be regeneratively braked as the characteristics do not extend
to the second quadrant
...
3
...
Fig
...
Initially the machine is connected to the supply with the
switch S in position number 1
...
The induced armature voltage E and supply voltage
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+
If
F2
Vf
Variable
votage
V source
A2
E
F1
A1
(a)Physical connection
Speed
A
B
V1
C
V2
V1 > V2
Torque
0
(b)Characteristics
Figure 51: Regenerative braking of a shunt machine
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RB
2
-
+
F2
1
+
A2
V
Vf
F1
E
2
-
A1
1
(a)Physical connection
A
Speed
B
C
0
T
orque
(b)Characteristics
Figure 52: Plugging or reverse voltage braking of a shunt motor
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V aid each other and a large reverse current flows through the armature
...
Hence plugging is also termed as reverse voltage
braking
...
If the motor is not switched off at this instant
the direction of rotation reverses and the motor starts rotating the reverse direction
...
1) plug to reverse and 2) plug to stop
...
If nothing is done it is plug to reverse mode
...
Just as in starting, during
plugging also it is necessary to limit the current and thus the torque, to reduce the stress on
the mechanical system and the commutator
...
• Series motors
In the case of series motors plugging cannot be employed as the field current too gets
reversed when reverse voltage is applied across the machine
...
This fact is used with advantage, in operating a
d
...
series motor on d
...
or a
...
supply
...
• Compound motors
Plugging of compound motors proceeds on similar lines as the shunt motors
...
A
cumulatively compounded motor becomes differentially compounded on plugging
...
This decreases the braking torque, and increases the
duration of the large braking current
...
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In such cases the
braking proceeds just as in a shunt motor
...
Unlike dynamic braking and regenerative
braking where the motor is made to work as a generator during braking period, plugging
makes the motor work on reverse motoring mode
...
4
Application of d
...
c
...
The need for generating
these characteristics will be clear only when they are seen along with the characteristics of
the loads that they operate with
...
Loads are broadly divided into,
(a) Passive loads
(b) Active loads
They may be unidirectional in operation or work in either direction (Reversible loads)
...
The
direction of rotation may be taken to be clockwise/counter clockwise rotation
...
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Any torque which accelerates the motor load system in the positive direction of
rotation is termed as a positive toque
...
The torque being taken
as an independent variable, is represented along the x-axis
...
Quadrants
...
53(a) represent ‘forward motoring’ and ‘reverse motoring’ operation respectively
...
Fig
...
The characteristics a, b,and c correspond to frictional torque, cutting torque and fan torque
respectively
...
These can only
absorb mechanical power and hence are represented in quadrantal II for positive direction
of rotation
...
Fig
...
Here an elevator is taken as an example
...
As seen from the Fig
...
The characteristics of the load exists in all the four quadrants
and is capable of delivering as well as absorbing mechanical power
...
The governing equation when the motor and a load are connected together is
TM (w) − TL (w) = J
dw
dt
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(52)
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Speed
II
I
Torque
III
IV
(a)
b
a
Speed
c
Torque
c
a
b
(b)
Figure 53: Typical load characteristics on a four quadrantal diagram
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W
T
speed
Hoisting an
empty cage
W
T
Hoisting a
loaded cage
Torque
o
T
W
W
T
Lowering a
loaded cage
Lowering an
empty cage
Figure 54: Four quadrantal diagram
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where TM (w) and TM (w) are motor and load torques respectively
...
dw
dt
is made positive when the
speed has to be increased in the positive direction and negative when reducing the speed
...
Both motor and load torques are expressed as
functions of the speed
...
By varying the characteristics of the motor (or the load),
this speed can be changed to suit our requirements
...
Thus most speed control methods adopt, varying the
motor characteristics to achieve speed control
...
d
...
shunt motor
lathes,fans,pumps disc and band saw drive requiring moderate torques
...
c
...
c
...
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9
Testing of d
...
machines
A d
...
machine has to be tested for proper fabrication and trouble free operation
...
Also, one can find the efficiency, rating and temperature rise of the machine
...
9
...
Field is not excited during this test
...
The armature circuit
consists of two resistances in series
...
The brush contact drop behaves like a non-linear
resistance
...
An equation of V = Vb + IRa form is fitted through
these test points shown graphically in Fig
...
For large values of I the equivalent armature
resistance is taken to be V /I ohm
...
9
...
It is a graph showing the variation
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v
A
A2
+
DC Supply
V
A1
-
(a)Physical connection
I
...
Characteristic
Ra =
dv
di
dv
di
V
Vb
0
I
(b)Characteristics
Figure 55: Measurement of Armature resistance and Brush drop
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of the induced emf as a function of excitation current, when the speed is held constant,
with the load current being zero
...
This is experimentally determined by running the machine
as a separately excited generator on no-load at a constant speed and noting the terminal
voltage as a function of the excitation current
...
9
...
The machine is demagnetized and an extremely small field current is passed
through the field
...
The speed is to be held constant during this test also
...
9
...
When the
machine is loaded, certain fraction of the input is lost inside the machine and appears as
heat, increasing the temperature of the machine
...
The load test gives the information about the efficiency of a given machine at any
load condition
...
If the temperature rise
is below the permissible value for the insulation then the machine can be safely operated
at that load, else the load has to be reduced
...
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...
Thus the load test alone can give us the
proper information of the rating and also can help in the direct measurement of the efficiency
...
5
Measurement of rotor inertia
The moment of inertia value is very important for the selection of a proper motor
for drives involving many starts and stops or requiring very good speed control characteristics
...
The test works on the principle that when a motor is switched off from the
mains it decelerates and comes to rest
...
The torque lost at
any speed is calculated by running the motor at that speed steadily on no load and noting
the power input
...
All this power is spent in over
coming the mechanical losses at that speed
...
In a retardation
test the motor speed is taken to some high value and the power to the motor is switched off
...
The
lost torque at any speed can be written as
PL = Tlost
...
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From these values the moment of inertia can be computed as
J=
9
...
dw
dt
(54)
Efficiency of a d
...
machine
A machine when loaded yields an output
...
The the efficiency in per unit is given as the ratio of output power
to input power
...
The other
two definitions are known as determination of efficiency using the loss segregation
...
c
...
The
losses that take place inside a d
...
machine can be listed as below
...
Armature copper loss
...
Brush and brush contact loss
...
Shunt field loss
4
...
Commutating pole loss
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6
...
Mechanical losses
8
...
Stray load losses
Out of these items 1,2,7,8 and 9 will be present in all the d
...
machines
...
These
losses change with temperature of operation
...
Iron losses change with the degree of saturation and distortion of the shape of the field flux
distribution under the poles
...
c
...
The input requirement also increases along with the output
...
The efficiency of power conversion is given by the ratio of output power to input power
...
The load dependant losses form the variable losses
...
Even though the loss that takes
place in a field winding is proportional to the square of the current through that winding, it
is classified under constant losses as the excitation current is held constant during loading
...
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c
...
The term inside the brackets is sometimes referred to as the deficiency
...
c
...
56(a) as a function of the load current
...
The curve d is the efficiency curve with all three components taken together
...
This can be easily seen from the graph that
this maximum occurs when constant losses equal the variable losses
...
I
Fig
...
c
...
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b
c
Efficiency
a
d
current
Efficiency
(a)Efficiency Vs Load current
Output
(b)Output Vs Efficiency
Figure 56: Efficiency of a D
...
machine
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Title: Electrical Machines
Description: its has contain of basic work of an machines and there properties with their principles and the working procudires
Description: its has contain of basic work of an machines and there properties with their principles and the working procudires