Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao Indian Institute of Technology Madras 1 Introduction Michael Faraday propounded the principle of electro-magnetic induction in 1831. It states that a voltage appears across the terminals of an electric coil when the flux linked with the same changes. The magnitude of the induced voltage is proportional to the rate of change of the flux linkages. This finding forms the basis for many magneto electric machines. The earliest use of this phenomenon was in the development of induction coils. These coils were used to generate high voltage pulses to ignite the explosive charges in the mines. As the d.c. power system was in use at that time, very little of transformer principle was made use of. In the d.c. supply system the generating station and the load center have to be necessarily close to each other due to the requirement of economic transmission of power. Also the d.c. generators cannot be scaled up due to the problem of the commutator. This made the world look for other efficient methods for bulk power generation and transmission. During the second half of the 19th century the alternators, transformers and induction motors were invented. These machines work on alternating power supply. The role of the transformers became obvious. The transformer which consisted of two electric circuits linked by a common magnetic circuit helped the voltage and current levels to be changed keeping the power invariant. The efficiency of such conversion was extremely high. Thus one could choose a moderate voltage for the generation of a.c. power, a high voltage for the transmission of this power over long distances and finally use a small and safe operating voltage at the user end. All these are made possible by transformers. The a.c. power systems thus got well established. Transformers can link two or more electric circuits. In its simple form two electric circuits can be linked by a magnetic circuit, one of the electric coils is used for the creation of a time varying magnetic filed. The second coil which is made to link this field has an induced voltage 1
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
1 Introduction
Michael Faraday propounded the principle of electro-magnetic induction in 1831. It states
that a voltage appears across the terminals of an electric coil when the flux linked with the
same changes. The magnitude of the induced voltage is proportional to the rate of change
of the flux linkages. This finding forms the basis for many magneto electric machines. The
earliest use of this phenomenon was in the development of induction coils. These coils were
used to generate high voltage pulses to ignite the explosive charges in the mines. As the d.c.
power system was in use at that time, very little of transformer principle was made use of.
In the d.c. supply system the generating station and the load center have to be necessarily
close to each other due to the requirement of economic transmission of power. Also the
d.c. generators cannot be scaled up due to the problem of the commutator. This made the
world look for other efficient methods for bulk power generation and transmission. During
the second half of the 19th century the alternators, transformers and induction motors were
invented. These machines work on alternating power supply. The role of the transformers
became obvious. The transformer which consisted of two electric circuits linked by a common
magnetic circuit helped the voltage and current levels to be changed keeping the power
invariant. The efficiency of such conversion was extremely high. Thus one could choose a
moderate voltage for the generation of a.c. power, a high voltage for the transmission of
this power over long distances and finally use a small and safe operating voltage at the user
end. All these are made possible by transformers. The a.c. power systems thus got well
established.
Transformers can link two or more electric circuits. In its simple form two electric circuits
can be linked by a magnetic circuit, one of the electric coils is used for the creation of a time
varying magnetic filed. The second coil which is made to link this field has an induced voltage
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
in the same. The magnitude of the induced emf is decided by the number of turns used in
each coil. Thus the voltage level can be increased or decreased. This excitation winding is
called a primary and the output winding is called a secondary. As a magnetic medium forms
the link between the primary and the secondary windings there is no conductive connection
between the two electric circuits. The transformer thus provides an electric isolation between
the two circuits. The frequency on the two sides will be the same. As there is no change in
the nature of the power, the resulting machine is called a ‘transformer’ and not a ‘converter’.
The electric power at one voltage/current level is only ‘transformed’ into electric power, at
the same frequency, to another voltage/current level.
Even though most of the large-power transformers can be found in the power systems,
the use of the transformers is not limited to the power systems. The use of the principle
of transformers is universal. Transformers can be found operating in the frequency range
starting from a few hertz going up to several mega hertz. Power ratings vary from a few
milliwatts to several hundreds of megawatts. The use of the transformers is so wide spread
that it is virtually impossible to think of a large power system without transformers. Demand
on electric power generation doubles every decade in a developing country. For every MVA
of generation the installed capacity of transformers grows by about 7MVA. These figures
show the indispensable nature of power transformers.
2 Basic Principles
As mentioned earlier the transformer is a static device working on the principle of Faraday’s
law of induction. Faraday’s law states that a voltage appears across the terminals of an
electric coil when the flux linkages associated with the same changes. This emf is proportional
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
to the rate of change of flux linkages. Putting mathematically,
e =dψ
dt(1)
Where, e is the induced emf in volt and ψ is the flux linkages in Weber turn. Fig. 1 shows a
Figure 1: Flux linkages of a coil
coil of N turns. All these N turns link flux lines of φ Weber resulting in the Nφ flux linkages.
In such a case,
ψ = Nφ (2)
and
e = Ndφ
dtvolt (3)
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.
• flux lines may be constant and not changing in time but the coil may move in space
linking different value of flux with time.
• both 1 and 2 above may take place. The flux lines may change in time with coil moving
in space.
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
These three cases are now elaborated in sequence below, with the help of a coil with a simple
geometry.
+-
Figure 2: Static coil
Fig. 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. A loop of one turn links part of this flux. The flux φ
linked by the turn is L ∗ B ∗X Weber. Here X is the length of overlap in meters as shown
in the figure. 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. Such a condition does not yield any
useful machine. 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. The magnitude of B is assumed
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
to be varying sinusoidally, and can be expressed as,
B = Bm sinωt (4)
where Bm is the peak amplitude of the flux density. ω is the angular rate of change with
time. Then, the instantaneous value of the flux linkage is given by,
ψ = Nφ = NLXBm sinωt (5)
The instantaneous value of the induced emf is given by,
e =dψ
dt= Nφm.ω cosωt = Nφm.ω. sin(ωt+
π
2) (6)
Here φm = Bm.L.X. The peak value of the induced emf is
em = Nφm.ω (7)
and the rms value is given by E = Nφm.ω√
2volt.
Further, this induced emf has a phase difference with respect to the flux linked by the turn.
This emf is termed as ‘transformer’ emf and this principle is used in a transformer. Polarity
of the emf is obtained by the application of Lenz’s law. Lenz’s law states that the reaction to
the change in the flux linkages would be such as to oppose the cause. The emf if permitted
to drive a current would produce a counter mmf to oppose this changing flux linkage. In the
present case, presented in Fig. 2 the flux linkages are assumed to be increasing. The polarity
of the emf is as indicated. The loop also experiences a compressive force.
Fig. 2) shows the same example as above but with a small difference. The flux density is held
constant at B Tesla. The flux linked by the coil at the current position is φ = B.L.X Weber.
The conductor is moved with a velocity v = dx/dt normal to the flux, cutting the flux lines
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
and changing the flux linkages. The induced emf as per the application of Faraday’s law of
induction is e = N.B.L.dx/dt = B.L.v volt.
Please note,the actual flux linked by the coil is immaterial. Only the change in the flux
linkages is needed to be known for the calculation of the voltage. The induced emf is in step
with the change in ψ and there is no phase shift. If the flux density B is distributed sinu-
soidally over the region in the horizontal direction, the emf induced also becomes sinusoidal.
This type of induced emf is termed as speed emf or rotational emf, as it arises out of the
motion of the conductor. The polarity of the induced emf is obtained by the application of
the Lenz’s law as before. Here the changes in flux linkages is produced by motion of the
conductor. The current in the conductor, when the coil ends are closed, makes the conductor
experience a force urging the same to the left. This is how the polarity of the emf shown in
fig.2b is arrived at. Also the mmf of the loop aids the field mmf to oppose change in flux
linkages. This principle is used in d.c machines and alternators.
The third case under the application of the Faraday’s law arises when the flux changes and
also the conductor moves. This is shown in Fig. 2.
The uniform flux density in space is assumed to be varying in magnitude in time as B =
Bm sinωt. The conductor is moved with a uniform velocity of dxdt
= v m/sec. The change in
the flux linkages and hence induced emf is given by
e = N.d(Bm. sinωt.L.X)
dt= N.L.X.Bm.ω. cosωt.+N.Bm. sinωt.L.
dx
dtV olt. (8)
The first term is due to the changing flux and hence is a transformer emf. The second term is
due to moving conductor or is a speed emf. When the terminals are closed such as to permit
a current the conductor experiences a force and also the mmf of the coil opposes the change
in flux linkages. This principle is used in a.c. machines where the field is time varying and
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
conductors are moving under the same.
The first case where there is a time varying field and a stationary coil resulting in a trans-
former emf is the subject matter in the present section. The case two will be revisited under
the study of the d.c machines and synchronous machines. Case three will be extensively used
under the study of a.c machines such as induction machines and also in a.c. commutator
machines.
Next in the study of the transformers comes the question of creating a time varying filed.
This is easily achieved by passing a time varying current through a coil. The winding which
establishes the field is called the primary. The other winding, which is kept in that field and
has a voltage induced in it, is called a secondary. 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. These aspects will be examined in the later sections. At first the common
constructional features of a transformer used in electric power supply system operating at
50 Hz are examined.
3 Constructional features
Transformers used in practice are of extremely large variety depending upon the end use. 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. Here the focus is on power transformers only. The
principle of operation of these transformers also is the same but the user requirements differ.
Power transformers of smaller sizes could be air cooled while the larger ones are oil cooled.
These machines are highly material intensive equipments and are designed to match the
applications for best operating conditions. Hence they are ‘tailor made’ to a job. This brings
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
in a very large variety in their constructional features. Here more common constructional
aspects alone are discussed. These can be broadly divided into
1. Core construction
2. Winding arrangements
3. Cooling aspects
3.1 Core construction
Transformer core for the power frequency application is made of highly permeable material.
The high value of permeability helps to give a low reluctance for the path of the flux and
the flux lines mostly confine themselves to the iron. µr well over 1000 are achieved by the
present day materials. Silicon steel in the form of thin laminations is used for the core
material. Over the years progressively better magnetic properties are obtained by going in
for Hot rolled non-oriented to Hot rolled grain oriented steel. Later better laminations in
the form of cold Rolled Grain Oriented (CRGO), -High B (HiB) grades became available.
The thickness of the laminations progressively got reduced from over 0.5mm to the present
0.25mm per lamination. These laminations are coated with a thin layer of insulating varnish,
oxide or phosphate. The magnetic material is required to have a high µ and a high saturation
flux density, a very low remanence and a small area under the B-H loop-to permit high flux
density of operation with low magnetizing current and low hysteresis loss. The resistivity
of the iron sheet itself is required to be high to reduce the eddy current losses. The eddy
current itself is highly reduced by making the laminations very thin. If the lamination is
made too thin then the production cost of steel laminations increases. The steel should not
have residual mechanical stresses which reduce their magnetic properties and hence must
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
be annealed after cutting and stacking. In the case of very small transformers (from a few
(a ) (b)
(c)
Figure 3: E and I,C and I and O Type Laminations
volt-amperes to a few kilo volt-amperes) hot rolled silicon steel laminations in the form of
E & I, C & I or O are used and the core cross section would be a square or a rectangle
as shown in Fig. 3. The percentage of silicon in the steel is about 3.5. Above this value
the steel becomes very brittle and also very hard to cut. The saturation flux density of
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
the present day steel lamination is about 2 Tesla. Broadly classifying, the core construction
HV LVHVLV
core
1.phase
3.phase
(a) (b)
Figure 4: Core and Shell Type Construction
can be separated into core type and shell type. In a core type construction the winding
surrounds the core. A few examples of single phase and three phase core type constructions
are shown in Fig. 4. In a shell type on the other hand the iron surrounds the winding. A few
examples are shown in Fig. 4. In the case of very small transformers the conductors are very
thin and round. These can be easily wound on a former with rectangular or square cross
section. Thus no special care is needed for the construction of the core. The cross section of
the core also would be square or rectangular. As the rating of the transformer increases the
conductor size also increases. Flat conductors are preferred to round ones. To wind such
conductor on a rectangular former is not only difficult but introduces stresses in the conduc-
tor, at the bends. From the short circuit force with stand capability point of view also this
is not desirable. Also, for a given area enclosed the length of the conductor becomes more.
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
Figure 5: Stepped Core Construction
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
Hence it results in more load losses. In order to avoid all these problems the coils are made
cylindrical and are wound on formers on heavy duty lathes. Thus the core construction is
required to be such as to fill the circular space inside the coil with steel laminations. Stepped
core construction thus becomes mandatory for the core of large transformers. Fig. 5 shows
a few typical stepped core constructions. When the core size increases it becomes extremely
difficult to cool the same (Even though the core losses are relatively very small). Cooling
ducts have to be provided in the core. The steel laminations are grain oriented exploiting
the simple geometry of the transformer to reduce the excitation losses. Another important
WindingsCore
Path of
flux
LV
HV
(a) (b)
Figure 6: Typical Core and wound core Constructional Features
aspect to be carefully checked and monitored is the air gaps in series in the path of the main
flux. 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.
Hence butt joints between laminations must be avoided. Lap joints are used to provide
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
alternate paths for flux lines thus reducing the reluctance of the flux paths. Some typical
constructional details are shown in Fig. 6. In some power transformers the core is built up
by threading a long strip of steel through the coil in the form of a toroid. This construction
is normally followed in instrument transformers to reduce the magnetizing current and hence
the errors.
Large cores made up of laminations must be rendered adequately stiff by the provision of
stiffening plates usually called as flitch plates. Punched through holes and bolts are progres-
sively being avoided to reduce heating and melting of the bolts. The whole stack is wrapped
up by strong epoxy tapes to give mechanical strength to the core which can stand in upright
position. Channels and angles are used for the frame and they hold the bottom yoke rigidly.
3.2 Windings
Windings form another important part of transformers. In a two winding transformer two
windings would be present. The one which is connected to a voltage source and creates the
flux is called as a primary winding. The second winding where the voltage is induced by
induction is called a secondary. If the secondary voltage is less than that of the primary
the transformer is called a step down transformer. If the secondary voltage is more then it
is a step up transformer. A step down transformer can be made a step up transformer by
making the low voltage winding its primary. Hence it may be more appropriate to designate
the windings as High Voltage (HV) and Low Voltage (LV) windings. The winding with more
number of turns will be a HV winding. The current on the HV side will be lower as V-I
product is a constant and given as the VA rating of the machines. Also the HV winding
needs to be insulated more to withstand the higher voltage across it. HV also needs more
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
clearance to the core, yoke or the body. These aspects influence the type of the winding
used for the HV or LV windings. Transformer coils can be broadly classified in to concentric
coils and sandwiched coils Fig. 7. The former are very common with core type transformers
while the latter one are common with shell type transformers. In the figure the letters L and
H indicate the low voltage and high voltage windings. In concentric arrangement, in view
of the lower insulation and clearance requirements, the LV winding is placed close to the
core which is at ground potential. The HV winding is placed around the LV winding. Also
taps are provided on HV winding when voltage change is required. This is also facilitated
by having the HV winding as the outer winding. Three most common types of coils viz.
helical, cross over and disc coils are shown in Fig. 8.
Helical Windings One very common cylindrical coil arrangement is the helical winding.
This is made up of large cross section rectangular conductor wound on its flat side.
The coil progresses as a helix. This is commonly used for LV windings. The insulation
requirement also is not too high. Between layers no insulation (other than conductor
insulation) is needed as the voltage between layers is low. The complexity of this
type of winding rapidly increases as the current to be handled becomes more. The
conductor cross section becomes too large and difficult to handle. The eddy current
losses in the conductor rapidly increases. Hence two or more conductors have to be
wound and connected in parallel. The parallel circuits bring in problems of current
sharing between the circuits. Transpositions of the parallel paths have to be adopted
to reduce unequal current distribution. The modern practice is to use continuously
transposed and bunched conductors.
Cross over coils The second popular winding type is the cross over coil. These are made
of circular conductors not exceeding 5 to 6 sq mm in cross section. These are used for
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
LV
HV
Core
HV LV
L
L
H
H
(a) (b)
LV HV
Core
(c)
Figure 7: Concentric and Sandwich Coils
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
cross over coilsDisc coilsHelix coils
Figure 8: Disc, Crossover and Helical Coil Construction
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
HV windings of relatively small transformers. These turns are wound in several layers.
The length and thickness of each block is made in line with cooling requirements. A
number of such blocks can be connected in series, leaving cooling ducts in between the
blocks, as required by total voltage requirement.
Disc coils Disc coils consist of flat conductors wound in a spiral form at the same place
spiralling outwards. Alternate discs are made to spiral from outside towards the center.
Sectional discs or continuous discs may be used. These have excellent thermal prop-
erties and the behavior of the winding is highly predictable. Winding of a continuous
disc winding needs specialized skills.
Sandwich coils Sandwich windings are more common with shell type core construction.
They permit easy control over the short circuit impedance of the transformer. By
bringing HV and LV coils close on the same magnetic axis the leakage is reduced
and the mutual flux is increased. By increasing the number of sandwiched coils the
reactance can be substantially reduced.
3.3 Insulation
The insulation used in the case of electrical conductors in a transformer is varnish or enamel in
dry type of transformers. 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. The transformer oil thus has dual
role. It is an insulator and also a coolant. The porous insulation around the conductor helps
the oil to reach the conductor surface and extract the heat. The conductor insulation may
be called the minor insulation as the voltage required to be withstood is not high. The major
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
insulation is between the windings. Annular bakelite cylinders serve this purpose. Oil ducts
are also used as part of insulation between windings. The oil used in the transformer tank
should be free from moisture or other contamination to be of any use as an insulator.
3.4 Cooling of transformers
Scaling advantages make the design of larger and larger unit sizes of transformers econom-
ically attractive. This can be explained as below. Consider a transformer of certain rating
designed with certain flux density and current density. If now the linear dimensions are made
larger by a factor of K keeping the current and flux densities the same the core and conductor
areas increase by a factor of K2. The losses in the machine, which are proportional to the
volume of the materials used, increase by a factor of K3.The rating of the machine increases
by a factor of K4.
The surface area however increases by a factor of K2 only. Thus the ratio of loss per surface
area goes on increasing by a factor of K. The substantial increase in the output is the major
attraction in going in for larger units. However cooling of the transformer becomes more
and more difficult. As the rating increases better cooling techniques are needed.
Simple air cooling of the transformers is adopted in dry type transformers. The limit for
this is reached by the time the rating is a few kVA. Hence air cooling is used in low voltage
machines. This method of cooling is termed as AN(Air Natural). Air Blast(AB) method
improves on the above by directing the blast of air at the core and windings. This permits
some improvement in the unit sizes.
Substantial improvement is obtained when the transformer is immersed in an oil tank. The
oil reaches the conductor surface and extracts the heat and transports the same to the sur-
face of the tank by convection. This is termed as ON (Oil Natural) type of cooling. This
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
method permits the increase in the surface available for the cooling further by the use of
ducts, radiators etc.
OB(Oil Blast) method is an improvement over the ON-type and it directs a blast of air on
the cooling surface. In the above two cases the flow of oil is by natural convective forces.
The rate of circulation of oil can be increased with the help of a pump, with the cooling at
the surface remaining natural cooling to air. This is termed as OFN (Oil Forced Natural).
If now a forced blast of air is also employed, the cooling method become OFB( Oil Forced
Blast). A forced circulation of oil through a radiator is done with a blast of air over the
radiator surface. Substantial amount of heat can be removed by employing a water cooling.
Here the hot oil going into the radiator is cooled by a water circuit. Due to the high specific
heat of water, heat can be evacuated effectively. 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. Some cooling arrangements are shown in Fig. 9.
In many large sized transformers the cooling method is matched with the amount of heat
that is required to be removed. As the load on the transformer changes the heat generated
within also changes. Suitable cooling method can be pressed into service at that time. This
gives rise to the concept of mixed cooling technique.
ON/OB Works as ON but with increased load additional air blast is adopted. This gives
the ratings to be in the ratio of 1:1.5
ON/OB/OFB Similarly gives the ratings in the ratio of 1:1.5:2
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
Tubes
main tankRadiator
tank
(a)
Radiator
water inlet
water outlet
oil pump
BushingConservator
& Breather
(b)
Conservator&
Breather
Radiator
Fan motorOil pump
for O.F.B
Bushing
(c)
Figure 9: Some Typical Cooling Arrangements20
Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
3.4.1 Properties of the transformer coil
Even though the basic functions of the oil used in transformers are a) heat conduction
and b) electrical insulation, there are many other properties which make a particular oil
eminently suitable. 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.
Mineral oils are suitable from the point of electrical properties but tend to form sludge.
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. However most of the oils naturally
fulfil this. Therefore deterioration in insulating property due to moisture or contami-
nation may be more relevant.
Viscosity It is important as it determines the rate of flow of the fluid. Highly viscous fluids
need much bigger clearances for adequate heat removal.
Purity The oil must not contain impurities which are corrosive. Sulphur or its compounds
as impurities cause formation of sludge and also attack metal parts.
Sludge formation Thickening of oil into a semisolid form is called a sludge. Sludge for-
mation properties have to be considered while choosing the oil as the oil slowly forms
semi-solid hydrocarbons. These impede flows and due to the acidic nature, corrode
metal parts. Heat in the presence of oxygen is seen to accelerate sludge formation. If
the hot oil is prevented from coming into contact with atmospheric air sludge formation
can be greatly reduced.
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
Acidity Oxidized oil normally produces CO2 and acids. The cellulose which is in the paper
insulation contains good amount of moisture. These form corrosive vapors. A good
breather can reduce the problems due to the formation of acids.
Flash point And Fire point Flash point of an oil is the temperature at which the oil
ignites spontaneously. This must be as high as possible (not less than 160◦C from the
point of safety). Fire point is the temperature at which the oil flashes and continuously
burns. This must be very high for the chosen oil (not less than 200◦C).
Inhibited oils and synthetic oils are therefore used in the transformers. Inhibited oils contain
additives which slow down the deterioration of properties under heat and moisture and hence
the degradation of oil. 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.
It is therefore necessary to check the quality of the oil periodically and take corrective steps
to avoid major break downs in the transformer.
There are several other structural and insulating parts in a large transformer. These are
considered to be outside the scope here.
4 Ideal Transformer
Earlier it is seen that a voltage is induced in a coil when the flux linkage associated with
the same changed. If one can generate a time varying magnetic field any coil placed in the
field of influence linking the same experiences an induced emf. A time varying field can
be created by passing an alternating current through an electric coil. This is called mutual
induction (see also fig 4.1). The medium can even be air. Such an arrangement is called air
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
cored transformer. Indeed such arrangements are used in very high frequency transformers.
Even though the principle of transformer action is not changed, the medium has considerable
influence on the working of such devices. These effects can be summarized as the followings.
1. The magnetizing current required to establish the field is very large, as the reluctance
of the medium is very high.
2. There is linear relationship between the mmf created and the flux produced.
3. The medium is non-lossy and hence no power is wasted in the medium.
4. Substantial amount of leakage flux exists.
5. It is very hard to direct the flux lines as we desire, as the whole medium is homogeneous.
If the secondary is not loaded the energy stored in the magnetic field finds its way back to
the source as the flux collapses. 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.
The medium does not absorb and lose any energy. Power is required to create the field and
not to maintain the same. As the winding losses can be made very small by proper choice
of material, the ideal efficiency of a transformer approaches 100%. The large magnetizing
current requirement is a major deterrent. However if now a piece of magnetic material is
introduced to form the magnetic circuit Fig. 10 the situation changes dramatically. These
can be enumerated as below.
1. Due to the large value for the permeance ( µr of the order of 1000 as compared to
air) the magnetizing current requirement decreases dramatically. This can also be
visualized as a dramatic increase in the flux produced for a given value of magnetizing
current.
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
x
Primary
Leakageflux
Mutual fluxSecondary
(a)
Leakage flux
Primary
Mutual flux
Secondary
Iron core
X
(b)
Figure 10: Mutual Induction a) air core b) iron core
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
2. The magnetic medium is linear for low values of induction and exhibits saturation type
of non-linearity at higher flux densities.
3. The iron also has hysteresis 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.
4. 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.
5. 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.
6. As the medium is made of a conducting material eddy currents are induced in the
same and produce losses. These are called ‘eddy current losses’. To minimize the
eddy current losses the steel core is required to be in the form of a stack of insulated
laminations.
From the above it is seen that the introduction of magnetic core to carry the flux introduced
two more losses. Fortunately the losses due to hysteresis and eddy current for the available
grade of steel is very small at power frequencies. Also the copper losses in the winding due
to magnetization current is reduced to an almost insignificant fraction of the full load losses.
Hence steel core is used in power transformers.
In order to have better understanding of the behavior of the transformer, initially certain
idealizations are made and the resulting ‘ideal’ transformer is studied. These idealizations
are as follows:
1. Magnetic circuit is linear and has infinite permeability. The consequence is that a van-
ishingly small current is enough to establish the given flux. Hysteresis loss is negligible.
As all the flux generated confines itself to the iron, there is no leakage flux.
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
2. Windings do not have resistance. This means that there are no copper losses, nor there
is any ohmic drop in the electric circuit.
In fact the practical transformers are very close to this model and hence no major departure
is made in making these assumptions.
Fig. ?? shows a two winding ideal transformer. The primary winding has T1 turns and is
connected to a voltage source of V1 volts. The secondary has T2 turns. Secondary can be
connected to a load impedance for loading the transformer. The primary and secondary are
shown on the same limb and separately for clarity.
As a current I0 amps is passed through the primary winding of T1 turns it sets up an mmf
of I0T1 ampere which is in turn sets up a flux φ through the core. 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. As I0 establishes the field inside the transformer it
is called the magnetizing current of the transformer.
F lux φ =mmf
Reluctance=I0T1
lµA
=I0T1Aµ
l. (9)
This current is the result of a sinusoidal voltage V applied to the primary. 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. Writing this in terms of instantaneous values we
have,
v1 − e1 = 0 (10)
where v1 is the instantaneous value of the applied voltage and e1 is the induced emf due to
Faradays principle. 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. Kirchoff’s law application to the loop will result in
the same thing.
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
+ +
+
-
-
T1
T2
e1
e2
φ µ
~
8v1=V1mcosωtio 0
+
e1
i1
+e2
i2
+ -
v1=V1sinωt
(a) (b)
N+ +
+
-
-
T1
T2
e1
e2
f µ 8
-
i1
i2
ZL
v1=V1cosωt
(c)
Figure 11: Two winding Ideal Transformer unloaded and loaded
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
This equation results in v1 = e1 or the induced emf must be same in magnitude to the applied
voltage at every instant of time. Let v1 = V1peak cosωt where V1peak is the peak value and
ω = 2πf t. f is the frequency of the supply. As v1 = e1; e1 = dψ1/dt but e1 = E1peak cosωt
∴ E1 = V1 . It can be easily seen that the variation of flux linkages can be obtained as
ψ1 = ψ1peak sinωt. Here ψ1peak is the peak value of the flux linkages of the primary. Thus
the RMS primary induced emf is
e1 =dψ1
dt=d(ψ1peak sinωt)
dt(11)
= ψ1peak.ω. cosωt or the rms value (12)
E1 =ψ1peak.ω√
2=
2πfT1φm√2
= 4.44fφmT1 volts
Here ψ1peak is the peak value of the flux linkages of the primary. The same mutual flux links
the secondary winding. However the magnitude of the flux linkages will be ψ2peak = T2.φm.
The induced emf in the secondary can be similarly obtained as ,
e2 =dψ2
dt=d(ψ2peak sinωt)
dt(13)
= ψ2peak.ω. cosωt or the rms value (14)
E2 =2πfT2φm√
2= 4.44fφmT2 volt
which yields the voltage ratio as
E1
E2
=T1
T2
(15)
The voltages E1 and E2 are obtained by the same mutual flux and hence they are in
phase. If the winding sense is opposite i.e., 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. Similar problem arises even when the sense of winding is kept the same, but the
two windings are on opposite limbs (due to the change in the direction of flux). Hence in the
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
+
-
E1
I1
V1
+
-
E2
I2
V2
Figure 12: Dot Convention
circuit representation of transformers a dot convention is adopted to indicate the terminals of
the windings that go high (or low) together. Fig. 12. This can be established experimentally
by means of a polarity test on the transformers. At a particular instant of time if the current
enters the terminal marked with a dot it magnetizes the core. Similarly a current leaving
the terminal with a dot demagnetizes the core.
So far, an unloaded ideal transformer is considered. If now a load impedance ZL is connected
across the terminals of the secondary winding a load current flows as marked in (figure 12b).
This load current produces a demagnetizing mmf and the flux tends to collapse. However
this is detected by the primary immediately as both E2 and E1 tend to collapse. The current
drawn from supply increases up to a point the flux in the core is restored back to its original
value. 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 the transformer operates under constant induced emf mode.
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
Thus,
i1T1 − i2T2 = i0T1 but i0 → 0 (16)
i2T2 = i1T1 and the rms value I2T2 = I1T1. (17)
If the reference directions for the two currents are chosen as in the fig13, then the above
equation can be written in phasor form as,
I1T1 = I2T2 or I1 =T2
T1
.I2 (18)
AlsoE1
E2
=T1
T2
=I2I1
E1I1 = E2I2 (19)
Thus voltage and current transformation ratio are inverse of one another. If an impedance
of ZL is connected across the secondary,
I2 =E2
ZL
or ZL =E2
I2(20)
The input impedance under such conditions is
Zi =E1
I1= (
T1
T2
)2.E2
I2= (
T1
T2
)2.ZL (21)
An impedance of ZL when viewed ‘through’ a transformer of turns ratio (T1
T2
) is seen as
(T1
T2
)2.ZL. Transformer thus acts as an impedance converter. The transformer can be inter-
posed in between a source and a load to ‘match’ the impedance. Finally, the phasor diagram
for the operation of the ideal transformer is shown in Fig. 13 in which θ1 and θ2 are power
factor angles on the primary and secondary sides. As the transformer itself does not absorb
any active or reactive power it is easy to see that θ1 = θ2.
Thus, from the study of the ideal transformer it is seen that the transformer provides elec-
trical isolation between two coupled electric circuits while maintaining power invariance at
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
E1
I1
θ2
f
V1
E2
I2
θ2
f
V2
Figure 13: Phasor diagram of Operation of an Ideal Transformer
its two ends. This can be used to step up or step down the voltage /current at constant
volt-ampere. Also, the transformer can be used for impedance matching. In the case of an
ideal transformer the efficiency is 100% as there are no losses inside the device.
5 Practical Transformer
An ideal transformer is useful in understanding the working of a transformer. But it can-
not be used for the computation of the performance of a practical transformer due to the
non-ideal nature of the practical transformer. In a working transformer the performance as-
pects like magnetizing current, losses, voltage regulation, efficiency etc are important. 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.
Conversely, if these effects are removed from a working transformer what is left behind is an
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
ideal 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. The cur-
rent and mmf required is proportional to the flux density B that is required to be established
in the core.
B = µH;B =φ
A(22)
where A is the area of cross section of the iron core m2. H is the magnetizing force which is
given by,
H = i.T1
l(23)
where l is the length of the magnetic path, m. or
φ = B.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. Once saturation sets in, the current has to vary in a
nonlinear manner to establish the flux of sinusoidal shape. This non-linear current can be
resolved into fundamental and harmonic currents. This is discussed to some extent under
harmonics. At present the effect of this non-linear behavior is neglected as a secondary
effect. Hence the current drawn from the mains is assumed to be purely sinusoidal and
directly proportional to the flux density of operation. 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. The energy absorbed when the current increases is returned to
the electric circuit when the current collapses to zero. This current is called the magnetizing
current of the transformer. The magnetizing current Im is given by Im = E1/Xm where Xm
is called the magnetizing reactance. The magnetic circuit being lossy absorbs and dissipates
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
the power depending upon the flux density of operation. These losses arise out of hysteresis,
eddy current inside the magnetic core. These are given by the following expressions:
Ph ∝ B1.6f (25)
Pe ∝ B2f 2t2 (26)
Ph -Hysteresis loss, Watts
B- Flux density of operation Tesla.
f - Frequency of operation, Hz
t - Thickness of the laminations of the core, m.
For a constant voltage, constant frequency operation B is constant and so are these losses.
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. Thus the no-load
current I0 may be made up of Il(loss component) and Im (magnetizing component as )
I0 = Il − jIm (27)
I2l Rc– gives the total core losses (i.e. hysteresis + eddy current loss)
I2mXm- Reactive volt amperes consumed for establishing the mutual flux.
Finite µ of the magnetic core makes a few lines of flux take to a path through the air. Thus
these flux lines do not link the secondary winding. It is called as leakage flux. As the path of
the leakage flux is mainly through the air the flux produced varies linearly with the primary
current I1. Even a large value of the current produces a small value of flux. This flux
produces a voltage drop opposing its cause, which is the current I1. Thus this effect of the
finite permeability of the magnetic core can be represented as a series element jxl1. This is
termed as the reactance due to the primary leakage flux. As this leakage flux varies linearly
with I1, the flux linkages per ampere and the primary leakage inductance are constant (This
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
is normally represented ll1 Henry). The primary leakage reactance therefore becomes
xl1 = 2πfll1 ohm (28)
A similar effect takes place on the secondary side when the transformer is loaded. The sec-
ondary leakage reactance jxl2 arising out of the secondary leakage inductance ll2 is given by
xl2 = 2πfll2 (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 small).
This is represented as a series circuit element, as the power lost and the drop produced in the
primary and secondary are proportional to the respective currents. These are represented
by r1 and r2 respectively on primary and secondary side. A practical transformer sans
these imperfections (taken out and represented explicitly in the electric circuits) is an ideal
transformer of turns ratio T1 : T2 (voltage ratio E1 : E2). This is seen in Fig. 14. I′
2 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 total primary current
vectorially is I1 = I′
2 + I0 (30)
Here I′
2T1 = I2T2 or I′
2 = I2T2
T1
(31)
Thus I1 = I2T2
T1
+ I0 (32)
By solving this circuit for any load impedance ZL one can find out the performance of the
loaded transformer.
The circuit shown in Fig. 14. However, it is not very convenient due to the presence of
the ideal transformer of turns ratio T1 : T2. If the turns ratio could be made unity by
34
Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
~+
+
+
-
Rm
I2
φ
-
ZL
I’2
V2
r1
r2
I1
jXmE1
-
V1
E2 T2
Io
jxl1
jxl2
T1
(a)
r1
Rc jXmV1
Ic ImIo
ZL V2E1 E2
I’2I1 I2r2jXl1
jXl2
(b)
Figure 14: A Practical Transformer a) Physical b) Equivalent Circuit
35
Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
some transformation the circuit becomes very simple to use. This is done here by replacing
the secondary by a ‘hypothetical’ secondary having T1 turns which is ‘equivalent ’ to the
physical secondary. The equivalence implies that the ampere turns, active and reactive power
associated with both the circuits must be the same. Then there is no change as far as their
effect on the primary is considered. Thus
V′
2 = aV2, I′
2 =I2a, r
′
2 = a2r2, x′
l2 = a2xl2 Z′
L = a2ZL.
where a -turns ratio T1
T2
This equivalent circuit is as shown in Fig. 14. 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. This particular equivalent circuit is
as seen from the primary side. 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. Such an equivalent circuit having all the parameters referred
to the secondary side is shown in Fig. 14
The equivalent circuit can be derived, with equal ease, analytically using the Kirchoff’s
equations applied to the primary and secondary. Referring to fig. 15. We have (by neglecting
the shunt branch)
V1 = E1 + I1(r1 + jxl1) (33)
E2 = V2 + I2(r2 + jxl2) (34)
T1I0 = T1I1 + T2I2 or I1 = −I2a
+ I0 (35)
= −I2a
+ Ic + Im
a =T1
T2
.
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
Multiply both sides of Eqn.34 by ‘a’ [This makes the turns ratio unity and retains the power
invariance].
aE2 = aV2 + aI2(r2 + jxl2) but aE2 = E1 (36)
Substituting in Eqn.33 we have
V1 = aV2 + aI2(r2 + jxl2) + I1(r1 + jxl1)
= V′
2 + I1(a2r2 + ja2xl2) + I1(r1 + jxl1)
= V′
2 + I1(r1 + r′
2 + jxl1 + x′
l2) (37)
A similar procedure can be used to refer all parameters to secondary side shown in fig. 15
r’1
R’c jX’mV’1
I’o
ZL V2
x’l1 xl2r2 I2I’1
I’l I’m
Figure 15: Equivalent Circuit Referred to the Secondary Side
37
Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
r1
Rc jXmV1
Io
Z’LV’2
xl1 x’l2r’2
Ic Im
I1
(a)
r1
Io
Z’L
jxl1 jx’l2r’2I’2I1
Ic Im
jxmRcV1
V1 V’2V1
I1 R XI’2
R=r1+r’2
x=xl1+xl2
I1=I’2
(b) (c)
Figure 16: Exact,approximate and simplified equivalent circuits
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
6 Phasor diagrams
The resulting equivalent circuit as shown in Fig. 16 is known as the exact equivalent circuit.
This circuit can be used for the analysis of the behavior of the transformers. As the no-
load current is less than 1% of the load current a simplified circuit known as ‘approximate’
equivalent circuit (see Fig. 16) is usually used, which may be further simplified to the one
shown in Fig. 16. On similar lines to the ideal transformer the phasor diagram of operation
can be drawn for a practical transformer also. The positions of the current and induced emf
phasor are not known uniquely if we start from the phasor V1. Hence it is assumed that the
phasor φ is known. The E1 and E2 phasor are then uniquely known. Now, the magnetizing
and loss components of the currents can be easily represented. Once I0 is known, the drop
that takes place in the primary resistance and series reactance can be obtained which when
added to E1 gives uniquely the position of V1 which satisfies all other parameters. This is
represented in Fig. 17 as phasor diagram on no-load.
Next we proceed to draw the phasor diagram corresponding to a loaded transformer. The
position of the E2 vector is known from the flux phasor. Magnitude of I2 and the load
power factor angle θ2 are assumed to be known. But the angle θ2 is defined with respect
to the terminal voltage V2 and not E2. By trial and error the position of I2 and V2 are
determined. V2 should also satisfy the Kirchoff’s equation for the secondary. Rest of the
construction of the phasor diagram then becomes routine. The equivalent primary current
I′
2 is added vectorially to I0 to yield I1. I1(r1 + jxl1) is added to E1 to yield I1. I1(r1 + jxl1)is
added to E1 to yield V1. This is shown in fig:trfr-prac-phasor as phasor diagram for a loaded
transformer.
39
Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
V1
E1
IoXl1
Ior1
Io
Im Ilφ φ
E2
(a)
V1
E1
I1Xl1
I1r1
Io
I’2 Il
φ φ
E2V2
I2I2x2 I2v2
(b)
Figure 17: Phasor Diagram of a Practical Transformer unloaded and loaded
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
7 Testing of Transformers
The structure of the circuit equivalent of a practical transformer is developed earlier. The
performance parameters of interest can be obtained by solving that circuit for any load
conditions. The equivalent circuit parameters are available to the designer of the transformers
from the various expressions that he uses for designing the transformers. But for a user
these are not available most of the times. Also when a transformer is rewound with different
primary and secondary windings the equivalent circuit also changes. In order to get the
equivalent circuit parameters test methods are heavily depended upon. From the analysis of
the equivalent circuit one can determine the electrical parameters. But if the temperature
rise of the transformer is required, then test method is the most dependable one. There are
several tests that can be done on the transformer; however a few common ones are discussed
here.
7.1 Winding resistance test
This is nothing but the resistance measurement of the windings by applying a small d.c
voltage to the winding and measuring the current through the same. The ratio gives the
winding resistance, more commonly feasible with high voltage windings. For low voltage
windings a resistance-bridge method can be used. From the d.c resistance one can get the
a.c. resistance by applying skin effect corrections.
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
V1
V2
V3
Vs
~V
S
+
-
+
(a) (b)
Figure 18: Polarity Test a) a.c. test b) d.c. test
7.2 Polarity Test
This is needed for identifying the primary and secondary phasor polarities. It is a must for
poly phase connections. Both a.c. and d.c methods can be used for detecting the polarities
of the induced emfs. The dot method discussed earlier is used to indicate the polarities. The
transformer is connected to a low voltage a.c. source with the connections made as shown
in the 18. A supply voltage Vs is applied to the primary and the readings of the voltmeters
V1, V2 and V3 are noted. V1 : V2 gives the turns ratio. If V3 reads V1 − V2 then assumed dot
locations are correct (for the connection shown). The beginning and end of the primary and
secondary may then be marked by A1 −A2 and a1 −a2 respectively. If the voltage rises from
A1 to A2 in the primary, at any instant it does so from a1 to a2 in the secondary. If more
secondary terminals are present due to taps taken from the windings they can be labeled
as a3, a4, a5, a6. It is the voltage rising from smaller number towards larger ones in each
winding. The same thing holds good if more secondaries are present. Fig. 18 shows the d.c.
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
method of testing the polarity. When the switch S is closed if the secondary voltage shows
a positive reading, with a moving coil meter, the assumed polarity is correct. If the meter
kicks back the assumed polarity is wrong.
7.3 Open Circuit Test
A
VV1
W
V2Rc
jXmV1
IcIm
Io
(a) (b)
Figure 19: No Load Test a) Physical Arrangement b) Equivalent Circuit
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.
In fig. 19 V1, A,W are the voltmeter, ammeter and wattmeter respectively. Let these meters
read V1, I0 and W0 respectively. Figure( 6.20) shows the equivalent circuit of the transformer.
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. The active component Ic of the no load current I0 represents the core losses
and reactive current Im is the current needed for the magnetization. Thus the wattmeter
43
Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
reading
W0 = V1Ic = Pcore (38)
∴ Ic =W0
V1
(39)
∴ Im =√
I20 − I2
c or (40)
Rc =V1
IcandXm =
V1
Im(41)
The parameters measured already are in terms of the primary. Sometimes the primary
Io
V1
Figure 20: Open Circuit Characteristics
voltage required may be in kilo-Volts and it may not be feasible to apply nominal voltage
to primary from the point of safety to personnel and equipment. If the secondary voltage is
low, one can perform the test with LV side energized keeping the HV side open circuited.
In this case the parameters that are obtained are in terms of LV . These have to be referred
to HV side if we need the equivalent circuit referred to HV side.
Sometimes the nominal value of high voltage itself may not be known, or in doubt, especially
in a rewound transformer. 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. This is a non
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
linear curve as shown in Fig. 20. This graph is obtained by noting the current drawn by
transformer at different applied voltage, keeping the secondary open circuited. The usual
operating point selected for operation lies at some standard voltage around the knee point
of the characteristic. After this value is chosen as the nominal value the parameters are
calculated as mentioned above.
7.4 Short Circuit Test
A
VVsc
(a)
V1Isc
Vsc
xl1 x’l2r’2
(b)
Figure 21: Short Circuit Test a) Physical Arrangement b) Equivalent Circuit
45
Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
The purpose of this test is to determine the series branch parameters of the equivalent circuit
of fig. 21. As the name suggests, in this test primary applied voltage, the current and power
input are measured keeping the secondary terminals short circuited. Let these values be
Vsc, Isc and Wsc respectively. The supply voltage required to circulate rated current through
the transformer is usually very small and is of the order of a few percent of the nominal
voltage. The excitation current which is only 1 percent or less even at rated voltage becomes
negligibly small during this test and hence is neglected. The shunt branch is thus assumed
to be absent. Also I1 = I′
2 as I0 ' 0. Therefore Wsc is the sum of the copper losses in
primary and secondary put together. The reactive power consumed is that absorbed by the
leakage reactance of the two windings.
Wsc = I2sc(r1 + r
′
2) (42)
Zsc =Vsc
Isc(43)
(xl1 + x′
l2) =√
Z2sc − (r1 + r
′
2)2 (44)
If the approximate equivalent circuit is required then there is no need to separate r1 and
r′
2 or xl1 and x′
l2. However if the exact equivalent circuit is needed then either r1 or r′
2 is
determined from the resistance measurement and the other separated from the total. As for
the separation of xl1 and x′
l2 is concerned, they are assumed to be equal. This is a fairly valid
assumption for many types of transformer windings as the leakage flux paths are through
air and are similar.
7.5 Load Test
Load Test helps to determine the total loss that takes place, when the transformer is loaded.
Unlike the tests described previously, in the present case nominal voltage is applied across
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
the primary and rated current is drown from the secondary. Load test is used mainly
1. to determine the rated load of the machine and the temperature rise
2. to determine the voltage regulation and efficiency of the transformer.
Rated load is determined by loading the transformer on a continuous basis and observing
the steady state temperature rise. The losses that are generated inside the transformer on
load appear as heat. This heats the transformer and the temperature of the transformer
increases. The insulation of the transformer is the one to get affected by this rise in the
temperature. Both paper and oil which are used for insulation in the transformer start get-
ting degenerated and get decomposed. If the flash point of the oil is reached the transformer
goes up in flames. Hence to have a reasonable life expectancy the loading of the transformer
must be limited to that value which gives the maximum temperature rise tolerated by the
insulation. This aspect of temperature rise cannot be guessed from the electrical equivalent
circuit. Further, the losses like dielectric losses and stray load losses are not modeled in the
equivalent circuit and the actual loss under load condition will be in error to that extent.
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 these permit
different levels of loading for the same transformer. Hence the only sure way of ascertaining
the rating is by conducting a load test.
It is rather easy to load a transformer of small ratings. As the rating increases it becomes
difficult to find a load that can absorb the requisite power and a source to feed the necessary
47
Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
current. As the transformers come in varied transformation ratios, in many cases it becomes
extremely difficult to get suitable load impedance.
Further, the temperature rise of the transformer is due to the losses that take place ‘inside’
the transformer. The efficiency of the transformer is above 99% even in modest sizes which
means 1 percent of power handled by the transformer actually goes to heat up the machine.
The remaining 99% of the power has to be dissipated in a load impedance external to
the machine. This is very wasteful in terms of energy also. Thus the actual loading of the
transformer is seldom resorted to. Equivalent loss methods of loading and ‘Phantom’ loading
are commonly used in the case of transformers. The load is applied and held constant till the
temperature rise of transformer reaches a steady value. If the final steady temperature rise
is lower than the maximum permissible value, then load can be increased else it is decreased.
That load which gives the maximum permissible temperature rise is declared as the nominal
or rated load.
In the equivalent loss method a short circuit test is done on the transformer. The short circuit
current is so chosen that the resulting loss taking place inside the transformer is equivalent
to the iron losses, full load copper losses and assumed stray load losses. By this method
even though one can pump in equivalent loss inside the transformer, the actual distribution
of this loss vastly differs from that taking place in reality. Therefore this test comes close to
a load test but does not replace one. In Phantom loading method two identical transformers
are needed. The windings are connected back to back as shown in Fig. 22. Suitable voltage
is injected into the loop formed by the two secondaries such that full load current passes
through them. An equivalent current then passes through the primary also. The voltage
source V1 supplies the magnetizing current and core losses for the two transformers. The
second source supplies the load component of the current and losses due to the same. There
is no power wasted in a load ( as a matter of fact there is no real load at all) and hence the
48
Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
A
V
Io
I2I’2
V
A
I2 I’2
Vs
W 2
W 1
2Io
V1
Io
Figure 22: Back to Back Test - Phantom Loading
49
Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
name Phantom or virtual loading. The power absorbed by the second transformer which
acts as a load is pushed back in to the mains. The two sources put together meet the core
and copper losses of the two transformers. The transformers work with full flux drawing full
load currents and hence are closest to the actual loading condition with a physical load.
8 Per Unit Calculations
As stated earlier, transformers of various sizes, ratings, voltage ratios can be seen being used
in a power system. The parameters of the equivalent circuits of these machines also vary over
a large range. Also the comparison of these machines are made simple if all the parameters
are normalized. If simple scaling of the parameters is done then one has to carry forward
the scaling factors in the calculations. Expressing in percent basis is one example of scaling.
However if the scaling is done on a logical basis one can have a simple representation of the
parameters without the bother of the scaling factors. Also different units of measurement are
in use in the different countries (FPS, CGS, MKS, etc;). These units also underwent several
revisions over the years. If the transformer parameter can be freed from the units then system
becomes very simple. The ‘per unit’ system is developed keeping these aspects in mind. The
parameters of the transformer are referred to some base values and thus get scaled. In the
case of power system a common base value is adopted in view of different ratings of the
equipments used. In the case of individual equipments, its own nominal parameters are used
as base values. Some base parameters can be chosen as independent base values while some
others become derived base parameters. Once the base values are identified the per unit
values are calculated for any parameter by dividing the same by its base value. The units
must be the same for both the parameters and their bases. Thus the per unit value is a
unit-less dimensionless number. Let us choose nominal voltage and nominal current on the
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
primary side of a transformer as the base values Vbase and Ibase. Other base values like volt
ampere Sbase, short circuit impedance Zbase can be calculated from those values.
Pbase, Qbase, Sbase = Vbase ∗ Ibase (45)
Rbase, Xbase, Zbase =Vbase
Ibase
(46)
Gbase, Bbase, Ybase =Ibase
Vbase
(47)
Normally Sbase and Vbase are known from name plate details. Other base values can be
derived from them.
Vp.u =V (volt)
Vbase(volt),
Ip.u =I(Amps)
Ibase(amps)=I(amps)
Sbase
Vbase
(48)
Zp.u =Z(ohm)
Zbase(ohm)= Z(ohm) ∗
Ibase
Vbase
= Z(ohm).Sbase
V 2base
(49)
Many times, when more transformers are involved in a circuit one is required to choose a
common base value for all of them. Parameters of all the machines are expressed on this
common base. This is a common problem encountered in the case of parallel operation of
two or more transformers. The conversion of the base values naturally lead to change in the
per unit values of their parameters. An impedance Zp.u.old on the old base of Sbaseold and
Vbaseold shall get modified on new base Sbasenew,Vbasenew as
Zp.u.new = (Zp.u.old.V 2
base old
Sbase old
)Sbase new
V 2base new
(50)
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.
If all the equivalent circuit parameters are referred to the secondary side and per unit values
of the new equivalent circuit parameters are computed with secondary voltage and current
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
as the base values, there is no change in the per unit values. This can be easily seen by,
Z′
p.u. = Z′
ohm.S
′
base
V′2base
but Z′
ohm =1
a2.Zohm (51)
Where a - is the turns ratio of primary to secondary
Z - impedance as seen by primary,
Z′
- impedance as seen by secondary.
S′
base = Sbase - as the transformer rating is unaltered.
V′
base = Vbase.1
a
From the above relationships it can be seen that Z′
p.u. = Zp.u..
This becomes obvious if we realize that the mmf of the core for establishing a given flux is
the same whether it is supplied through primary or the secondary. Also the active power and
reactive power absorbed inside the transformer are not dependant on the winding connected
to supply. This is further illustrated by taking the equivalent circuit of a transformer derived
earlier and expressing the same in per unit form.
Thus the per unit values help in dispensing away the scaling constants. The veracity of the
parameters can be readily checked. Comparison of the parameters of the machines with those
of similar ones throw in useful information about the machines. Comparing the efficiencies
of two transformers one can say that the transformer with a higher p.u.resistance has higher
copper losses without actually computing the same.
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.
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
9 Voltage Regulation
Modern power systems operate at some standard voltages. The equipments working on
these systems are therefore given input voltages at these standard values, within certain
agreed tolerance limits. In many applications this voltage itself may not be good enough
for obtaining the best operating condition for the loads. A transformer is interposed in
between the load and the supply terminals in such cases. There are additional drops inside
the transformer due to the load currents. 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. If
undue voltage drop is permitted to occur inside the transformer the load voltage becomes
too low and affects its performance. 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. This drop is termed as the voltage regulation and is expressed as a ratio of
the terminal voltage (the absolute value per se is not too important).
The voltage regulation can be defined in two ways - Regulation Down and Regulation up.
These two definitions differ only in the reference voltage as can be seen below.
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”.
Expressed in symbolic form we have,
Regulation =Vnl − Vl
Vnl
(52)
Vnl and Vl are no-load and load terminal voltages. This is the definition normally used
in the case of the transformers, the no-load voltage being the one given by the power
supply provider on which the user has no say. Hence no-load voltage is taken as the
reference.
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
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. This definition if expressed in symbolic form results in
Regulation =Vnl − Vl
Vl
(53)
Vnl is the no-load terminal voltage.
Vl is load voltage. Normally full load regulation is of interest as the part load regulation
is going to be lower.
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. He has to generate proper
no-load voltage at the generating station to provide the user the voltage he has asked for.
In the expressions for the regulation, only the numerical differences of the voltages are taken
and not vector differences.
In the case of transformers both definitions result in more or less the same value for the
regulation as the transformer impedance is very low and the power factor of operation is
quite high. The power factor of the load is defined with respect to the terminal voltage on
load. Hence a convenient starting point is the load voltage. Also the full load output voltage
is taken from the name plate. Hence regulation up has some advantage when it comes to its
application. Fig. 23 shows the phasor diagram of operation of the transformer under loaded
condition. The no-load current I0 is neglected in view of the large magnitude of I′
2. Then
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
(a)
O
V1
C
I2’
φ V’2θ B
D
I2’Re
I2’XeE
A
(b)
Figure 23: a) Equivalent Circuit and b) Phasor Diagram for Regulation of Transformer
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
I1= I′
2.
V1 = I′
2(Re + jXe) + V′
2 (54)
OD = V1 =√
[OA+ AB +BC]2 + [CD]2
=√
[V2 + I′
2Re cosφ+ I′
2Xe sinφ]2 + [I′
2Xe cosφ− I′
2Re sinφ]2 (55)
φ - power factor angle,
θ- internal impedance angle=tan−1 Xe
Re
Also
V1 = V′
2 + I′
2.(Re + jXe) (56)
= V′
2 + I′
2(cosφ− j sin φ)(Re + jXe)
∴ RegulationR =V1 − V
′
2
V′
2
=√
(1 + v1)2 + v22 − 1 (57)
(1 + v1)2 + v2
2 ' (1 + v1)2 + v2
2 .2(1 + v1)
2(1 + v1)+ [
v22
2(1 + v1)]2 = (1 + v1 +
v22
2(1 + v1))2 (58)
Taking the square root√
(1 + v1)2 + v22 = 1 + v1 +
v22
2(1 + v1)(59)
where v1 = er cos φ+ ex sin φ and v2 = ex cosφ− er sin φ
er =I
′
2Re
V′
2
=per unit resistance drop
ex =I
′
2Xe
V′
2
=per unit reactance drop
as v1 and v2 are small.
∴ R ' 1 + v1 +v22
2(1 + e1)− 1 ' v1 +
v22
2(60)
∴ regulationR = er cosφ± ex sin φ+(ex sin φ− er cosφ)2
2(61)
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
v22
2(1 + v1)'v22
2.(1 − v1)
(1 − v21)
'v22
2.(1 − v1) '
v22
2(62)
Powers higher than 2 for v1 and v2 are negligible as v1 and v2 are already small. 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 cosφ± ex sinφ
The negative sign is applicable when the power factor is leading. It can be seen from the
above expression, the full load regulation becomes zero when the power factor is leading and
er cosφ = ex sinφ or tanφ = er/ex
or the power factor angle φ = tan−1(er/ex) = tan−1(Re/Xe) leading.
Similarly, the value of the regulation is maximum at a power factor angle φ = tan−1(ex/er) =
tan−1(Xe/Re) lagging. An alternative expression for the regulation of a transformer can be
derived by the method shown in fig. 24. Here the phasor are resolved along the current axis
and normal to it.
We have,
OD2 = (OA+ AB)2 + (BC + CD)2 (63)
= (V ‘2 cosφ+ I ‘
2Re)2 + (V ‘
2 sinφ+ I ‘2Xe)
2(64)
∴ RegulationR =OD − V
′
2
V′
2
=OD
V′
2
− 1 (65)
√
(V ‘2 cosφ+ I ‘
2Re
V ‘2
)2 +(V ‘
2 sinφ+ I ‘2Xe
V ‘2
)2 − 1 (66)
=√
(cosφ+Rp.u)2 + (sinφ+Xp.u)2 − 1 (67)
Thus this expression may not be as convenient as the earlier one due to the square root
involved.
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
25
50
75
100
0.5 1x
Effic
iency
%
0
(a)
AB
O
D
Cθ
φ
I2’I2’Re
I2’Xl
V1
V2
(b)
Figure 24: An Alternate Method for the Calculation of Regulation
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
Fig. 24 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. It is seen from Fig. 25that the full load regulation at unity power factor is nothing
1
2
3
4
5
leading lagging
power factor
%Regulation
1.0 0.5 00 0.5
-1
-2
-3
-4
-5
Figure 25: Variation of Full Load Regulation with Power Factor
but the percentage resistance of the transformer. It is therefore very small and negligible.
Only with low power factor loads the drop in the series impedance of the transformer con-
tributes substantially to the regulation. In small transformers the designer tends to keep
the Xe very low (less than 5%) so that the regulation performance of the transformer is
satisfactory.
A low value of the short circuit impedance /reactance results in a large short circuit current
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
in case of a short circuit. This in turn results in large mechanical forces on the winding. So,
in large transformers the short circuit impedance is made high to give better short circuit
protection to the transformer which results in poorer regulation performance. 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. In other cases care has to be exercised in the
selection of the short circuit impedance as it affects the voltage regulation.
10 Efficiency
Efficiency of a power equipment is defined at any load as the ratio of the power output to
the power input. Putting in the form of an expression,
Efficiency η =output power
input power=Input power − losses inside the machine
Input power(68)
= 1 −losses inside the machine
inputpower= 1 − efficiency
=output power
output+ losses inside the machine
More conveniently the efficiency is expressed in percentage. %η = output power
input power∗ 100
The losses that take place inside the machine expressed as a fraction of the input is some
times termed as deficiency. Except in the case of an ideal machine, a certain fraction of
the input power gets lost inside the machine while handling the power. Thus the value for
the efficiency is always less than one. In the case of a.c. machines the rating is expressed
in terms of apparent power. It is nothing but the product of the applied voltage and the
current drawn. The actual power delivered is a function of the power factor at which this
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
current is drawn. As the reactive power shuttles between the source and the load and has a
zero average value over a cycle of the supply wave it does not have any direct effect on the
efficiency. The reactive power however increases the current handled by the machine and
the losses resulting from it. Therefore the losses that take place inside a transformer at any
given load play a vital role in determining the efficiency. The losses taking place inside a
transformer can be enumerated as below:
1. Primary copper loss
2. Secondary copper loss
3. Iron loss
4. Dielectric loss
5. Stray load loss
These are explained in sequence below.
Primary and secondary copper losses take place in the respective winding resistances due to
the flow of the current in them.
Pc = I21r1 + I2
2r2 = I′22 Re (69)
The primary and secondary resistances differ from their d.c. values due to skin effect and the
temperature rise of the windings. While the average temperature rise can be approximately
used, the skin effect is harder to get analytically. The short circuit test gives the value of Re
taking into account the skin effect.
The iron losses contain two components - Hysteresis loss and Eddy current loss. The Hys-
teresis loss is a function of the material used for the core.
Ph = KhB1.6f
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
For constant voltage and constant frequency operation this can be taken to be constant.
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. This again produces a power loss Pe in
the lamination.
Pe = KeB2f 2t2
where t is the thickness of the steel lamination used. As the lamination thickness is much
smaller than the depth of penetration of the field, the eddy current loss can be reduced by
reducing the thickness of the lamination. Present day laminations are of 0.25 mm thickness
and are capable of operation at 2 Tesla. These reduce the eddy current losses in the core.
This loss also remains constant due to constant voltage and frequency of operation. The
sum of hysteresis and eddy current losses can be obtained by the open circuit test.
The dielectric losses take place in the insulation of the transformer due to the large electric
stress. In the case of low voltage transformers this can be neglected. For constant voltage
operation this can be assumed to be a constant.
The stray load losses arise out of the leakage fluxes of the transformer. These leakage fluxes
link the metallic structural parts, tank etc. and produce eddy current losses in them. Thus
they take place ’all round’ the transformer instead of a definite place , hence the name ’stray’.
Also the leakage flux is directly proportional to the load current unlike the mutual flux which
is proportional to the applied voltage. Hence this loss is called ’stray load’ loss. This can also
be estimated experimentally. It can be modeled by another resistance in the series branch
in the equivalent circuit. The stray load losses are very low in air-cored transformers due to
the absence of the metallic tank.
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
Thus, the different losses fall in to two categories Constant losses (mainly voltage dependant)
and Variable losses (current dependant). 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 + x2Pvar
(70)
Here S in the volt ampere rating of the transformer (V′
2 I′
2 at full load), Pconst being constant
losses and Pvar the variable losses at full load. For a given power factor an expression for η in
terms of the variable x is thus obtained. By differentiating η with respect to x and equating
the same to zero, the condition for maximum efficiency is obtained. In the present case that
condition comes out to be
Pconst = x2Pvar or x =
√
Pconst
Pvar
(71)
That is, when constant losses equal the variable losses at any fractional load x the efficiency
reaches a maximum value. The maximum value of that efficiency at any given power factor
is given by,
ηmax =xS cos θ2
xS cos θ2 + 2Pconst
=xS cos θ2
xS cos θ2 + 2x2Pvar
(72)
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. It
may be considered a good practice to select the operating load point to be at the maximum
efficiency point. 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 losses.
However, in the modern transformers the iron losses are so low that it is practically impossible
to reduce the full load copper losses to that value. Such a design wastes lot of copper. This
point is illustrated with the help of an example below.
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
Two 100 kVA transformers A nd B are taken. Both transformers have total full load losses
to be 2 kW. The break up of this loss is chosen to be different for the two transformers.
Transformer A: iron loss 1 kW, and copper loss is 1 kW. The maximum efficiency of 98.04%
occurs at full load at unity power factor. Transformer B: Iron loss =0.3 kW and full load
copper loss =1.7 kW. This also has a full load η of 98.04%. Its maximum η occurs at
a fractional load of√
0.31.7
= 0.42. The maximum efficiency at unity power factor being
42
42+0.6∗ 100 = 98.59%. At the corresponding point the transformer A has an efficiency of
42
42+1.0+0.1764∗ 100 = 97.28%. 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.
10.1 All day efficiency
50
100
6 12 18 24
Load
% o
f ful
l loa
d
Time,hrs
s
P
50
100
12 24
Pow
er L
oss
%
(a) (b)
Figure 26: Calculation of Load Factor and Loss Factor
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
Large capacity transformers used in power systems are classified broadly into Power trans-
formers and Distribution transformers. The former variety is seen in generating stations and
large substations. Distribution transformers are seen at the distribution substations. The
basic difference between the two types arise from the fact that the power transformers are
switched in or out of the circuit depending upon the load to be handled by them. Thus at
50% load on the station only 50% of the transformers need to be connected in the circuit.
On the other hand a distribution transformer is never switched off. It has to remain in the
circuit irrespective of the load connected. In such cases the constant loss of the transformer
continues to be dissipated. Hence the concept of energy based efficiency is defined for such
transformers. It is called ’all day’ efficiency. The all day efficiency is thus the ratio of the
energy output of the transformer over a day to the corresponding energy input. One day
is taken as a duration of time over which the load pattern repeats itself. This assumption,
however, is far from being true. The power output varies from zero to full load depending
on the requirement of the user and the load losses vary as the square of the fractional loads.
The no-load losses or constant losses occur throughout the 24 hours. Thus, the comparison
of loads on different days becomes difficult. Even the load factor, which is given by the
ratio of the average load to rated load, does not give satisfactory results. The calculation
of the all day efficiency is illustrated below with an example. The graph of load on the
transformer, expressed as a fraction of the full load is plotted against time in Fig. 26. In an
actual situation the load on the transformer continuously changes. This has been presented
by a stepped curve for convenience. The average load can be calculated by
Average load over a day =
∑ni=1
Pi
24=Sn
∑ni=1
xiti cos θi
24(73)
where Pi is the load during an interval i. n intervals are assumed. xi is the fractional load.
Si = xiSn where Sn is nominal load. The average loss during the day is given by
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
Average loss = Pi +Pc
∑n
i=1x2
i ti24
(74)
This is a non-linear function. For the same load factor different average loss can be there
depending upon the values of xi and ti. Hence a better option would be to keep the constant
losses very low to keep the all day efficiency high. Variable losses are related to load and are
associated with revenue earned. The constant losses on the other hand has to be incurred
to make the service available. The concept of all day efficiency may therefore be more useful
for comparing two transformers subjected to the same load cycle.
The concept of minimizing the lost energy comes into effect right from the time of procure-
ment of the transformer. The constant losses and variable losses are capitalized and added
to the material cost of the transformer in order to select the most competitive one, which
gives minimum cost taking initial cost and running cost put together. Obviously the iron
losses are capitalized more in the process to give an effect to the maximization of energy
efficiency. If the load cycle is known at this stage, it can also be incorporated in computation
of the best transformer.
11 Auto Transformer
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. The currents drawn by these
two windings are out of phase by 180◦. This prompted the use of a part of the primary as
secondary. This is equivalent to fusing the secondary turns into primary turns. The fused
section need to have a cross sectional area of the conductor to carry (I2 − I1) ampere! This
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
V1
T1
T2
I1
V2 ZL
I1
I2
I2
Figure 27: Autotransformer - Physical Arrangement
ingenious thought led to the invention of an auto transformer. Fig. 27 shows the physical
arrangement of an auto transformer. Total number of turns between A and C are T1. At
point B a connection is taken. Section AB has T2 turns. 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 (75)
For simplifying analysis, the magnetizing current of the transformer is neglected. When the
secondary winding delivers a load current of I2 ampere the demagnetizing ampere turns is
I2T2 . This will be countered by a current I1 flowing from the source through the T1 turns
such that,
I1T1 = I2T2 (76)
A current of I1 ampere flows through the winding between B and C . The current in the
winding between A and B is (I2 − I1) ampere. The cross section of the wire to be selected
67
Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
for AB is proportional to this current assuming a constant current density for the whole
winding. Thus some amount of material saving can be achieved compared to a two winding
transformer. The magnetic circuit is assumed to be identical and hence there is no saving in
the same. To quantify the saving the total quantity of copper used in an auto transformer
is expressed as a fraction of that used in a two winding transformer as,
copper in autotransformer
copper in two winding transformer=
(T1 − T2)I1 + T2(I2 − I1)
T1I1 + T2I2(77)
= 1 −2T2I1
T1I1 + T2I2
ButT1I1 = T2I2 (78)
∴ The Ratio = 1 −2T2I12T1I1
= 1 −T2
T1
(79)
This means that an auto transformer requires the use of lesser quantity of copper given by
V1
I1+I2
V1+V2
V2
I2
I1
ZL
I1+I2 I1
I2
I2
φ
Figure 28: Two Winding Transformer used as auto transformer
the ratio of turns. This ratio therefore denotes the savings in copper. As the space for the
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
second winding need not be there, the window space can be less for an auto transformer,
giving some saving in the lamination weight also. The larger the ratio of the voltages,
smaller is the savings. As T2 approaches T1 the savings become significant. Thus auto
transformers become ideal choice for close ratio transformations. The savings in material is
obtained, however, at a price. The electrical isolation between primary and secondary has
to be sacrificed.
If we are not looking at the savings in the material, even then going in for the auto transformer
type of connection can be used with advantage, to obtain higher output. This can be
illustrated as follows. Fig. 28 shows a regular two winding transformer of a voltage ratio
V1 : V2, the volt ampere rating being V1I1 = V2I2 = S. If now the primary is connected
across a supply of V1 volt and the secondary is connected in series addition manner with the
primary winding, the output voltage becomes (V1 + V2) volt. The new output of this auto
transformer will now be
I2(V1 + V2) = I2V2(1 +V1
V2
) = S(1 +V1
V2
) (80)
= V1(I1 + I2) = S(1 +I2I1
) (81)
Thus an increased rating can be obtained compared to a two winding transformer with the
same material content. The windings can be connected in series opposition fashion also.
Then the new output rating will be
I2(V1 − V2) = I2V2(V1
V2
− 1) = S(V1
V2
− 1) (82)
The differential connection is not used as it is not advantageous as the cumulative connection.
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
V1
(I2 -I1)
I1+I2
V2
I2
I1
I1
I1I2
r1,xl1
r2,xl2
Figure 29: Kirchoff’s Law Application to auto transformer
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Elecrical Machines I Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao
Indian Institute of Technology Madras
11.1 Equivalent circuit
As mentioned earlier the magnetizing current can be neglected, for simplicity. Writing the
Kirchoff’s equation to the primary and secondary Fig. 29 we have