electrical machines ii compiled by mumtaz buriro
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Transformers
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 limitations of the commutator. This
made the world look for other efficient methods for bulk power generation and transmis-
sion. 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 transmis-
sion 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.
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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 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 by changing
the number of turns. 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 onthe two sides will be the same. As there is no change in the nature of the power, the re-
sulting 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 an-
other 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.
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2 Basic Principles
As mentioned earlier the transformer is a static device working on the principle of
Faradays law of induction. Faradays 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 to the rate of change of flux linkages. Putting mathematically,
e =d
dt(1)
Where, e is the induced emf in volt and is the flux linkages in Weber turn. Fig. 1 shows a
Figure 1: Flux linkages of a coil
coil ofN turns. All these N turns link flux lines of Weber resulting in the N flux linkages.
In such a case,
= N (2)
and
e = Nd
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.
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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.
These three cases are now elaborated in sequence below, with the help of a coil with a simple
geometry.
L
B
X
+-
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 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 notyield 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
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is assumed to be varying sinusoidally, and can be expressed as,
B = Bm sin t (4)
where Bm is the peak amplitude of the flux density. is the angular rate of change withtime. Then, the instantaneous value of the flux linkage is given by,
= N = NLXBm sin t (5)
The instantaneous value of the induced emf is given by,
e =d
dt= Nm. cos t = Nm.. sin(t +
2) (6)
Here m = Bm.L.X. The peak value of the induced emf is
em = Nm. (7)
and the rms value is given by
E =Nm.
2volt.
Further, this induced emf has a phase difference of /2 radian 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 Lenzs law. Lenzs
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(b) 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
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= B.L.X Weber. The conductor is moved with a velocity v = dx/dt normal to the flux,
cutting the flux lines and changing the flux linkages. The induced emf as per the application
of Faradays law of induction is e = N.B.L.dx/dt = B.L.v volt.(Here N=1)
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 and there is no phase shift. If the flux density B is distributed
sinusoidally over the region in the horizontal direction, the emf induced also becomes sinu-
soidal. 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 applicationof the Lenzs 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 Faradays law arises when the flux changes
and also the conductor moves. This is shown in Fig. 2(c).
The uniform flux density in space is assumed to be varying in magnitude in time as
B = Bm sin t. The conductor is moved with a uniform velocity ofdx
dt= v m/sec. The
change in the flux linkages and hence induced emf is given by
e = N.d(Bm. sin t.L.X)
dt= N.L.X.Bm.. cos t. + N.Bm. sin t.L.
dx
dtV olt. (8)
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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 conductors are moving under the same.
The first case where there is a time varying field and a stationary coil resulting in
a transformer emf is the subject matter in the present section. The case two will be re-
visited 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.
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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 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. Relative permeability r
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 prop-
erties are obtained by going in for Hot rolled non-oriented to Hot rolled grain oriented steel.
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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 permeability and a high saturation flux density, a very low remanence Br and a
small area under the B-H loop-to permit high flux density of operation with low magnetizing
current and low hysteresis loss. 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 be annealed after cutting and stacking.
In the case of very small transformers (from a few volt-amperes to a few kilo volt-
amperes) hot rolled silicon steel laminations in the form of E & I, C & I or O as shown in
Fig. 3 are used and the core cross section would be a square or a rectangle. 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 the present day steel lamination is about 2
Tesla.
Broadly classifying, the core 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.
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
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(a ) (b)
(c)
Figure 3: E and I,C and I and O Type Laminations
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HV LVHVLV
coreHVLV
Single phase Three phase
1.phase
3.phase
(a)Core type (b) Shell type
Figure 4: Core and Shell Type Construction
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 conductor, 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. 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 thetransformer to reduce the excitation losses. The iron losses in the lamination, when the flux
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d d dduct
duct
0.71D
0.160.16
0.1 0.1
0.53
0.14 0.14
0.42
0.12 0.120.09 0.09
0.07 0.07
0.3
Figure 5: Stepped Core Construction
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is oriented in the direction of grain orientation, is about 30% of that in the normal direction.
Another important aspect to be carefully checked and monitored is the air gaps in
WindingsCore
Path of
flux
LV
HV
(a) (b)
Figure 6: Typical stacked Core and wound core Construction
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 pro-
vide 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.
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Large cores made up of laminations must be rendered adequately stiff by the provi-
sion of stiffening plates usually called as flitch plates. Punched through holes and bolts are
progressively being avoided to reduce heating and melting of the through 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 trans-
former 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 desig-nate 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
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
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LV
HV
Core
HV LV
(a)Concentric coil
LVHV
Core
(b) Sandwich coil
Figure 7: Concentric and Sandwich Coils
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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.
cross over coilsDisc coils
Helical coils
Figure 8: Disc, Crossover and Helical Coil Construction
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
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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
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 placespiralling 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
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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 trans-
former 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 with-
stood is not high. The major 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
economically 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 ofK2. The losses in the machine, which are proportional
to the volume of the materials used, increase by a factor ofK3.The rating of the machine
increases by a factor ofK4.
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The surface area however increases by a factor ofK2 only. Thus the ratio of loss per
surface area goes on increasing by a factor ofK. 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
surface of the tank by convection. This is termed as ON (Oil Natural) type of cooling. This
method permits the increase in the surface available for the cooling further by the use ofducts, 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
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Tubes
Radiator
Main tank
(a)
Radiator
water inlet
water outlet
oil pump
Bushing
Conservator
& Breather
(b)
Conservator&
Breather
Radiator
Fan motorOil pump
for O.F.B
Bushing
(c)
Figure 9: Some Typical Cooling Arrangements20
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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
The temperature rise permitted in the British standard specification for power transformers
are tabulated below.
Type winding oil core
Class A Class B
C C C
AN,AB 55 75 - As
ON,OB,OW 60 - 50 for
OFN,OFB 65 - 50 adjacent
OFW 70 - 50 winding
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3.4.1 Properties of the transformer coil
Even though the basic functions of the oil used in transformers are a) heat conduc-
tion and b) electrical insulation, there are many other properties which make a particular oileminently 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
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can be greatly reduced.
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 160C 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 200C).
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 correctivesteps 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.
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4 Ideal Transformer
Earlier it is seen that a voltage is induced in a coil when the flux linkage associated
with the same changed. 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. The medium can even be air. Such an arrangement is called air 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
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x
PrimaryLeakage
flux
Mutual flux
Secondary
(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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current requirement is a major deterrent. However if now a piece of magnetic material is
introduced to form the magnetic circuit Fig. 10(b) the situation changes dramatically. These
can be enumerated as below.
1. Due to the large value for the permeance ( r 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.
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 type of non-linearity due to which certain amount of power
is lost in the iron (in the form of hysteresis loss), as the B-H characteristic is traversed.
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 fluxintroduced two more losses. Fortunately the losses due to hysteresis and eddy current for
the available grades of steel is very small at power frequencies. Also the copper losses in the
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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 ideal-
izations 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.
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. 11 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 ofI0T1 ampere which is in turn sets up a flux through the core. Since the reluctance
of the iron path given by R = l/Ais zero as , a vanishingly small value of currentI0 is enough to setup a flux which is finite. As I0 establishes the field inside the transformer
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++
+
-
-
T1
T2
e1
e2
~
8v1=V1mcost
io 0
+
e1
i1
+
e2
i2
+ -
v1=V1sint
(a)Unloaded machine (b) Circuit
form
N
++
+
-
-
T1
T2
e1
e2
8
-
i1
i2
ZL
v1=V1cost
(c)Loaded machine
Figure 11: Two winding Ideal Transformer unloaded and loaded
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it is called the magnetizing current of the transformer.
F lux =mmf
Reluctance=
I0T1lA
=I0T1A
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 Lenzs law and shows
that it is in the form of a voltage drop. Kirchoffs law application to the loop will result in
the same thing.
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 cos t where V1peak is the peak
value and = 2f t. f is the frequency of the supply. As v1 = e1; e1 = d1/dt bute1 = E1peak cos t E1 = V1 . It can be easily seen that the variation of flux linkages can
be obtained as 1 = 1peak sin t. Here 1peak is the peak value of the flux linkages of the
primary.
Thus the RMS primary induced emf is
e1 =d1dt
=d(1peak sin t)
dt(11)
= 1peak.. cos t or the rms value (12)
E1 =1peak.
2=
2f T1m2
= 4.44f mT1 volts
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Here 1peak 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 2peak =
T2.m. The induced emf in the secondary can be similarly obtained as ,
e2 =d2dt
=d(2peak sin t)
dt(13)
= 2peak.. cos t or the rms value (14)
E2 =2f T2m
2= 4.44f mT2 volt
which yields the voltage ratio asE1E2
=T1T2
(15)
+
-
E1
I1
V1
+
-
E2
I2
V2
Figure 12: Dot Convention
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
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two windings are on opposite limbs (due to the change in the direction of flux). Hence in
the circuit representation of transformers a dot convention is adopted to indicate the ter-
minals 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
Fig. 11(c). 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.
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 Fig. 12, then the above
equation can be written in phasor form as,
I1T1 = I2T2 or I1 =T2T1
.I2 (18)
AlsoE1E2
=T1T2
=I2I1
E1I1 = E2I2 (19)
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Thus voltage and current transformation ratio are inverse of one another. If an impedance
of ZL is connected across the secondary,
I2 =E2
ZLor ZL =
E2
I2(20)
The input impedance under such conditions is
Zi =E1I1
= (T1T2
)2.E2I2
= (T1T2
)2.ZL (21)
An impedance of ZL when viewed through a transformer of turns ratio (T1T2
) is seen
as (T1T2
)2.ZL. Transformer thus acts as an impedance converter. The transformer can be
interposed in between a source and a load to match the impedance.
E1
I1
1
V1
E2
I2
2
V2
Figure 13: Phasor diagram of Operation of an Ideal Transformer
Finally, the phasor diagram for the operation of the ideal transformer is shown in
Fig. 13 in which 1 and 2 are power factor angles on the primary and secondary sides. As
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the transformer itself does not absorb any active or reactive power it is easy to see that
1 = 2.
Thus, from the study of the ideal transformer it is seen that the transformer provides
electrical isolation between two coupled electric circuits while maintaining power invariance
at its two ends. However, grounding of loads and one terminal of the transformer on the
secondary/primary side are followed with the provision of leakage current detection devices
to safe guard the persons working with the devices. Even though the isolation aspect is
a desirable one its utility cannot be over emphasized. It can be used to step up or step
down the voltage/current at constant volt-ampere. Also, the transformer can be used forimpedance matching. In the case of an ideal transformer the efficiency is 100% as there are
no losses inside the device.
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5 Practical Transformer
An ideal transformer is useful in understanding the working of a transformer. But it
cannot be used for the computation of the performance of a practical transformer due to the
non-ideal nature of the practical transformer. 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
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 current and mmf required is proportional to the flux density B that is required to be
established in the core.
B = H; B =
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.T1l
(23)
where l is the length of the magnetic path, m. or
= B.A =A(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
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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
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 B2f2t2 (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 theno-load current I0 may be made up of Ic(loss component) and Im (magnetizing component
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as )
I0 = Ic jIm (27)
I2c
Rc gives the total core losses (i.e. hysteresis + eddy current loss)
I2mXm- Reactive volt amperes consumed for establishing the mutual flux.
Finite 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 inductive 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 is normally represented by ll1 Henry). The primary leakage reactance
therefore becomes
xl1 = 2f ll1 ohm (28)
A similar effect takes place on the secondary side when the transformer is loaded.
The secondary leakage reactance jxl2 arising out of the secondary leakage inductance ll2 is
given by
xl2 = 2f ll2 (29)
Finally, the primary and secondary windings are wound with copper (sometimes alu-
minium in small transformers) conductors; thus the windings have a finite resistance (though
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~
+
+
+
-
Rc
I2
-
ZL
I2
V2
r1
r2
I1
jXmE1
-
V1
E2 T2
Io
jxl1
jxl2
T1
(a)Physical arrangement
r1
RcjXmV1
Ic ImIo
ZL V2E1 E2
I2I1I2
r2jXl1
jXl2
(b)Equivalent circuit
Figure 14: A Practical Transformer
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small). This is represented as a series circuit element, as the power lost and the drop pro-
duced 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 = I2T2T1
+ 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(b). However, it is not very convenient for use due to
the presence of the ideal transformer of turns ratio T1 : T2. If the turns ratio could be made
unity by 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= I2
a, r
2= a2r2, x
l2= a2xl2 Z
L= a2ZL.
where a -turns ratio T1T2
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This equivalent circuit is as shown in Fig. ??(a). 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 pri-
mary 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. 15.
The equivalent circuit can be derived, with equal ease, analytically using the Kir-
choffs equations applied to the primary and secondary. Referring to fig. 14(a), 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 =T1T2
.
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
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V1 = aV2 + aI2(r2 + jxl2) + I1(r1 + jxl1)
= V
2 + I1(a2
r2 + ja2
xl2) + 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.)
r1
Rc jXmV1
Io
ZLV2
jxl1 jxl2r2I2I1
Ic Im
Figure 15: Equivalent Circuit Referred to the Secondary Side
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6 Phasor diagrams
r1
Rc jXmV1
Io
ZLV2
jxl1 jxl2r2
Ic Im
I1
(a)
r1
Io
ZL
jxl1 jxl2r
2
I2I1
Ic Im
jxmRcV1
V2V2V1
I1R jX
I2
R=r1+r2
x=xl1+xl2
I1=I2
(b) (c)
Figure 16: Exact,approximate and simplified equivalent circuits
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 transform-ers. As the no-load current is less than 1% of the load current a simplified circuit known
as approximate equivalent circuit (see Fig. 16(b)) is usually used, which may be further
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simplified to the one shown in Fig. 16(c).
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
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 ofV1 which satisfies all other parameters. This is represented
in Fig. 17(a) 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 2 are assumed to be known. But the angle 2 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 Kirchoffs equation for the secondary. Rest of the
construction of the phasor diagram then becomes routine. The equivalent primary current
I
2is added vectorially to I0 to yield I1. I1(r1 +jxl1)is added to E1 to yield V1. This is shown
in fig. 17(b) as phasor diagram for a loaded transformer.
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V1
E1
IoXl1
Ior1
Io
Im Il
E2
(a)No-load
V1
E1
I1Xl1
I1r1
Io
I2
Il
E2
V2
I2
I2x2I2r2
(b)On-load
Figure 17: Phasor Diagram of a Practical Transformer
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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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V1
V2
V3
Vs
~
A2
A1
a2
a1
V
S
+
-
+
A2
A1
a2
a1
(a)A.C.test (b)D.C.test
Figure 18: Polarity 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 fig. 18(a). 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. IfV3 reads
V1V2 then assumed dot locations are correct (for the connection shown). The beginning and
end of the primary and secondary may then be marked by A1A2 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.
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Fig. 18(b) shows the d.c. 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
V2
RcjXmV1
IcIm
Io
(a)Physical Arrangement
(b)Equivalent Circuit
Figure 19: No Load Test
As the name suggests, the secondary is kept open circuited and nominal value
of the input voltage is applied to the primary winding and the input current and power are
measured. In Fig. 19(a) V,A,W are the voltmeter, ammeter and wattmeter respectively.
Let these meters read V1, I0 and W0 respectively.Fig. 19(b) shows the equivalent circuit of
the transformer under this test. 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
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represents the core losses and reactive current Im is the current needed for the magnetization.
Thus the watt meter reading
W0 = V1Ic = Pcore (38)
Ic =W0
V1(39)
Im =I20 I2
cor (40)
Rc =V1
IcandXm =
V1
Im(41)
Io
V1
Figure 20: Open Circuit Characteristics
The parameters measured already are in terms of the primary. Sometimes the pri-
mary 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 ofLV
. These have tobe referred to HV side if we need the equivalent circuit referred to HV side.
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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 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
The purpose of this test is to determine the series branch parameters of the equiv-
alent circuit of Fig. 21(b). 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 ratedvoltage becomes negligibly small during this test and hence is neglected. The shunt branch
is thus assumed to be absent. Also I1 = I
2as 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 = I2
sc(r1 + r
2) (42)
Zsc =
Vsc
Isc (43)
(xl1 + x
l2) =
Z2sc (r1 + r
2)2 (44)
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A
VVsc
W
(a)Physical Arrangement
r1Isc
Vsc
jxl1 jxl2r2
(b)Equivalent Circuit
Figure 21: Short Circuit Test
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If the approximate equivalent circuit is required then there is no need to separate r1
and r
2or xl1 and x
l2. However if the exact equivalent circuit is needed then either r1 or
r
2is 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 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 observ-
ing 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 theinsulation. 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
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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 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. ( If the load is of unity power
factor) 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 current which gives the
maximum permissible temperature rise is declared as the nominal or rated load current and
the volt amperes are computed using the same.
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alent 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 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.
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8 Per Unit Calculations
As stated earlier, transformers of various sizes, ratings, voltage ratios can be seen
being used in a power system. 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 the 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 thebase 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 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
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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)
SbaseVbase
(48)
Zp.u =Z(ohm)
Zbase(ohm)= Z(ohm)
Ibase
Vbase= Z(ohm).
Sbase
V2base(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.V2base oldSbase old
)Sbase new
V2base ne w(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
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current 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
V2
base
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 trans-
former 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 at any load 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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9 Voltage Regulation
Modern power systems operate at some standard voltages. The equipments work-
ing 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
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supply provider on which the user has no say. Hence no-load voltage is taken as the
reference.
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 itsapplication. 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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Re jXe
V1
I2
V2
(a) Equivalent Circuit
O
V1
C
I2
V2
B
D
I2Re
I2Xe
E
A
(b)Phasor Diagram
Figure 23: Regulation of Transformer
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I1= I
2.
V1 = I
2(Re + jXe) + V
2 (54)
OD = V1 =
[OA + AB + BC]2
+ [CD]2
=
[V
2 + I
2Re cos + I
2Xe sin]2 + [I
2Xe cos I
2Re sin]2 (55)
- power factor angle,
- internal impedance angle=tan1 XeRe
Also,
V1 = V
2 + I
2.(Re + jXe) (56)
= V
2 + I
2(cosj sin )(Re + jXe)
RegulationR =|V1| |V
2 |
|V
2 |=
(1 + v1)2 + v22 1 (57)
(1 + v1)2 + v22 (1 + v1)
2 + v22 .2(1 + v1)
2(1 + v1)+ [
v222(1 + v1)
]2 = (1 + v1 +v22
2(1 + v1))2 (58)
Taking the square root (59)
(1 + v1)2 + v22 = 1 + v1 + v22
2(1 + v1)
(60)
where v1 = er cos + ex sin and v2 = ex cos er sin
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 + v
22
2(61)
regulation R = er cos ex sin +(ex sin er cos)
2
2(62)
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v222(1 + v1)
v222.(1 v1)
(1 v21)
v222.(1 v1)
v222
(63)
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 cos ex sin
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 cos = ex sin or tan = er/ex
or the power factor angle = tan1(er/ex) = tan1(Re/Xe) leading.
Similarly, the value of the regulation is maximum at a power factor angle =
tan1(ex/er) = tan1(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 (64)
= (V
2 cos + I
2Re)2 + (V
2 sin + I
2Xe)2 (65)
RegulationR =OD V
2
V
2
=OD
V
2
1 (66)
(V
2 cos + I
2Re)V
2
2
+ (V
2 sin + I
2Xe)V
2
2
1 (67)
=
(cos + Rp.u)2 + (sin + X2p.u) 1 (68)
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A
B
O
D
C
I2
I2Re
I2Xe
V1
V2
Figure 24: An Alternate Method for the Calculation of Regulation
Thus this expression may not be as convenient as the earlier one due to the square root
involved.
Fig. 25 shows the variation of full load regulation of a typical transformer as the
power factor is varied from zero power factor leading, through unity power factor, to zero
power factor lagging.
It is seen from Fig. 25 that the full load regulation at unity power factor is nothing 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 contributes
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.
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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
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A low value of the short circuit impedance /reactance results in a large short circuit
current 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.
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D.C Machines
1 Introduction
The steam age signalled the beginning of an industrial revolution. The advantages
of machines and gadgets in helping mass production and in improving the services spurred
the industrial research. Thus a search for new sources of energy and novel gadgets received
great attention. By the end of the 18th century the research on electric charges received
a great boost with the invention of storage batteries. This enabled the research work on
moving charges or currents. It was soon discovered ( in 1820 ) that, these electric currents
are also associated with magnetic field like a load stone. This led to the invention of an
electromagnet. Hardly a year later the force exerted on a current carrying conductor placed
in the magnetic field was invented. This can be termed as the birth of a motor. A better
understanding of the inter relationship between electric and magnetic circuits was obtained
with the enumeration of laws of induction by Faraday in 1831. Parallel research was contem-
porarily being done to invent a source of energy to recharge the batteries in the form of a
d.c. source of constant amplitude (or d.c. generator). For about three decades the research
on d.c. motors and d.c. generators proceeded on independent paths. During the second half
of the 19th century these two paths merged. The invention of a commutator paved the way
for the birth of d.c. generators and motors. These inventions generated great interest in the
generation and use of electrical energy. Other useful machines like alternators, transformers
and induction motors came into existence almost contemporarily. The evolution of these
machines was very quick. They rapidly attained the physical configurations that are being
used even today. The d.c. power system was poised for a predominant place as a preferred
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system for use, with the availability of batteries for storage, d.c. generators for conversion of
mechanical energy into electrical form and d.c. motors for getting mechanical outputs from
electrical energy.
The limitations of the d.c. system however became more and more apparent
as the power demand increased. In the case of d.c. systems the generating stations and the
load centers have to be near to each other for efficient transmission of energy. The invention
of induction machines in the 1880s tilted the scale in favor of a.c. systems mainly due to
the advantage offered by transformers, which could step up or step down the a.c.voltage
levels at constant power at extremely high efficiency. Thus a.c. system took over as thepreferred system for the generation transmission and utilization of electrical energy. The
d.c. system, however could not be obliterated due to the able support of batteries. Further,
d.c. motors have excellent control characteristics. Even today the d.c. motor remains an
industry standard as far as the control aspects are concerned. In the lower power levels and
also in regenerative systems the d.c. machines still have a major say.
In spite of the apparent diversity in the characteristics, the underlying princi-
ples of both a.c. and d.c. machines are the same. They use the electromagnetic principles
which can be further simplified at the low frequency levels at which these machines are used.
These basic p