Transformers

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



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

2



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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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

4



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

5



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

6



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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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

8



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

9



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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Figure 5: Stepped Core Construction

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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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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

13



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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LV

HV

Core

HV LV

L

L

H

H

(a) (b)

LV HV

Core

(c)

Figure 7: Concentric and Sandwich Coils

15



cross over coilsDisc coilsHelix coils

Figure 8: Disc, Crossover and Helical Coil Construction

16



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

17



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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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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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



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.

21



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

22



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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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

24



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.

25



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.

26



+ +

+

-

-

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

27



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

28



+

-

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.

29



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

30



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

31



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

32



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

33



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



~+

+

+

-

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



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

.

36



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



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

38



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



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

40



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.

41



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.

42



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



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

44



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



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

46



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



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



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



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

50



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

51



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.

52



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.

53



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

54



(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

55



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)

56



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.

57



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

58



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

59



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

60



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

61



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.

62



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.

63



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

64



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

65



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

66



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



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

68



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.

69



V1

(I2 -I1)

I1+I2

V2

I2

I1

I1

I1I2

r1,xl1

r2,xl2

Figure 29: Kirchoff’s Law Application to auto transformer

70



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

V1 = E1 + I1(r1 + jxl1) − (I2 − I1)(r2 + jxl2) (83)

Note that the resistance r1 and leakage reactance xl1 refer to that part of the winding where

only the primary current flows. Similarly on the load side we have,

E2 = V2 + (I2 − I1)(r2 + jxl2) (84)

The voltage ratio V1 : V2 = E1 : E2 = T1 : T2 = a where T1 is the total turns

of the primary.

Then E1 = aE2 and I2 = aI1

multiplying equation(84) by ’a’ and substituting in (83) we have

V1 = aV2 + a(I2 − I1)(r2 + jxl2) + I1(r1 + jxl1) − (I2 − I1)(r2 + jxl2)

= aV2 + I1(r1 + jxl1 + r2 + jxl2 − ar2 − ajxl2) + I2(ar2 + jaxl2 − r2 − jxl2)

= aV2 + I1(r1 + jxl1 + r2 + jxl2 + a2r2 + ja2xl2 − ar2 − ajxl2 − ar2 − jaxl2

= aV2 + I1(r1 + r2(1 + a2 − 2a) + jxl1 + xl2(1 + a2 − 2a))

= aV2 + I1(r1 + (a− 1)2r2 + jxl1 + (a− 1)2xl2) (85)

Equation (85) yields the equivalent circuit of Fig. 30 where Re = r1 + (a − 1)2r2 and

Xe = xl1 + (a − 1)2xl2. The magnetization branch can now be hung across the mains for

completeness. The above equivalent circuit can now be compared with the approximate

equivalent circuit of a two winding case Re = r1 + a2r2 and Xe = xl1 + a2xl2. Thus in the

case of an auto transformer total value of the short circuit impedance is lower and so also the

71



Re jXe

Rc jXmV1

Ic ImIo

V’2=aV1

Re=r1+(a-1)2r2

Xl=xl1+(a-1)2xl2

Figure 30: Equivalent Circuit of auto transformers

percentage resistance and reactance. Thus the full load regulation is lower. Having a smaller

value of short circuit impedance is sometimes considered to be a disadvantage. That is be-

cause the short circuit currents become very large in those cases. The efficiency is higher

in auto transformers compared to their two winding counter part at the same load. The

phasor diagram of operation for the auto transformer drawing a load current at a lagging

power factor angle of θ2 is shown in Fig. 31. The magnetizing current is omitted here again

for simplicity.

From the foregoing study it is seen that there are several advantages in going in for the

autotransformer type of arrangement. The voltage/current transformation and impedance

conversion aspects of a two winding transformer are retained but with lesser material (and

hence lesser weight) used. The losses are reduced increasing the efficiency. Reactance is

reduced resulting in better regulation characteristics. All these benefits are enhanced as the

voltage ratio approaches unity. The price that is required to be paid is loss of electrical

72



φ

I1

-E1

I1r1I1x1

(I2-I1)x2V1

(I2-I1)r2

(I2-I1)x2

(I2-I1)r2

V2

E2

I2

Figure 31: Phasor Diagram of Operation of an autotransformer

73



isolation and a larger short circuit current (and larger short circuit forces on the winding).

Auto transformers are used in applications where electrical isolation is not a critical require-

ment. When the ratio V2 : V1 is 0.3 or more they are used with advantage. The normal

applications are motor starters, boosters or static balancers. Another wide spread applica-

Vin

M oving contact

Variable

a.c output

Figure 32: Variable Secondary Voltage Arrangement

tion of auto transformer type of arrangement is in obtaining a variable a.c. voltage from a

fixed a.c. voltage supply. Here only one winding is used as in the auto transformer. The sec-

ondary voltage is tapped by a brush whose position and hence the output voltage is variable.

The primary conductor is bared to facilitate electrical contact Fig. 32. Such arrangement

cannot exploit the savings in the copper as the output voltage is required right from zero

volts upwards.

The conductor is selected based on the maximum secondary current that could be drawn

74



as the output voltage varies in practically continuous manner. These are used in voltage

stabilizers, variable d.c. arrangements (with a diode bridge) in laboratories, motor starters,

dimmers etc.

12 Harmonics

In addition to the operation of transformers on the sinusoidal supplies, the harmonic behavior

becomes important as the size and rating of the transformer increases. The effects of the

harmonic currents are

1. Additional copper losses due to harmonic currents

2. Increased core losses

3. Increased electro magnetic interference with communication circuits.

On the other hand the harmonic voltages of the transformer cause

1. Increased dielectric stress on insulation

2. Electro static interference with communication circuits.

3. Resonance between winding reactance and feeder capacitance.

In the present times a greater awareness is generated by the problems of harmonic voltages

and currents produced by non-linear loads like the power electronic converters. These com-

bine with non-linear nature of transformer core and produce severe distortions in voltages

75



and currents and increase the power loss. Thus the study of harmonics is of great prac-

tical significance in the operation of transformers. The discussion here is confined to the

harmonics generated by transformers only.

12.1 Single phase transformers

e

t’ t’’ t iϕi’ϕi’’ϕiϕ

i’ϕ i’’ϕ

ϕ

ϕ’

ϕ’’

ϕ

ϕ’

ϕ’’

Figure 33: Harmonics Generated by Transformers

Modern transformers operate at increasing levels of saturation in order to reduce the weight

and cost of the core used in the same. Because of this and due to the hysteresis, the

transformer core behaves as a highly non-linear element and generates harmonic voltages

and currents. This is explained below. Fig. 33 shows the manner in which the shape of

the magnetizing current can be obtained and plotted. At any instant of the flux density

wave the ampere turns required to establish the same is read out and plotted, traversing the

hysteresis loop once per cycle. The sinusoidal flux density curve represents the sinusoidal

applied voltage to some other scale. The plot of the magnetizing current which is peaky is

76



analyzed using Fourier analysis. The harmonic current components are obtained from this

analysis. These harmonic currents produce harmonic fields in the core and harmonic voltages

in the windings. Relatively small value of harmonic fields generate considerable magnitude

of harmonic voltages. For example a 10% magnitude of 3rd harmonic flux produces 30%

magnitude of 3rd harmonic voltage. These effects get even more pronounced for higher

order harmonics. As these harmonic voltages get short circuited through the low impedance

of the supply they produce harmonic currents. These currents produce effects according to

Lenz’s law and tend to neutralize the harmonic flux and bring the flux wave to a sinusoid.

Normally third harmonic is the largest in its magnitude and hence the discussion is based on

it. The same can be told of other harmonics also. In the case of a single phase transformer

the harmonics are confined mostly to the primary side as the source impedance much smaller

compared to the load impedance. The understanding of the phenomenon becomes more clear

if the transformer is supplied with a sinusoidal current source. In this case current has to

be sinusoidal and the harmonic currents cannot be supplied by the source and hence the

induced emf will be peaky containing harmonic voltages. When the load is connected on

the secondary side the harmonic currents flow through the load and voltage tends to become

sinusoidal. The harmonic voltages induce electric stress on dielectrics and increased electro

static interference. The harmonic currents produce losses and electro magnetic interference

as already noted above.

12.2 Three phase banks of single phase transformers

In the case of single phase transformers connected to form three phase bank, each transformer

is magnetically decoupled from the other. The flow of harmonic currents are decided by the

type of the electrical connection used on the primary and secondary sides. Also, there

77



are three fundamental voltages in the present case each displaced from the other by 120

electrical degrees. Because of the symmetry of the a.c. wave about the time axis only

odd harmonics need to be considered. The harmonics which are triplen (multiples of three)

behave in a similar manner as they are co-phasal or in phase in the three phases. The

non-triplen harmonics behave in a similar manner to the fundamental and have ±120◦ phase

displacement between them. The harmonic behavior of poly-phase banks can be discussed

now.

Dd connection In three phase banks with mesh connection on both primary side and sec-

ondary side a closed path is available for the triplen harmonics to circulate currents.

Thus the supply current is nearly sinusoidal (but for the non-triplen harmonic cur-

rents). The triplen harmonic currents inside the closed mesh winding correct the flux

density wave to be nearly sinusoidal. The secondary voltages will be nearly sinusoidal.

Third harmonics currents flow both in the primary and the secondary and hence the

magnitudes of these currents, so also the drops due to them will be lower.

Dy and Yd connection (without neutral connection) Behavior of the bank with mesh

connection on one side is similar to the one discussed under Dd connection. The har-

monic currents and drops and the departure of the flux density from sinusoidal are

larger in the present case compared to Dd banks.

Yy connection without neutral wires With both primary and secondary connected in

star no closed path exists. As the triplen harmonics are always in phase, by virtue

of the Y connection they get canceled in the line voltages. Non-triplen harmonics

like fundamental, become√

3 times phase value and appear in the line voltages. Line

currents remain sinusoidal except for non-triplen harmonic currents. Flux wave in each

transformer will be flat topped and the phase voltages remain peaked. The potential

78



of the neutral is no longer steady. The star point oscillates due to the third harmonic

voltages. This is termed as ”oscillating neutral”.

Yy connection with neutral wires When a neutral wire is provided the triplen har-

monic current can flow and the condition is similar to the single phase case (with a

star connected 4 wire source or with the system earth). The neutral wire carries three

times the triplen harmonic current of one transformer as these currents are co-phasal.

Unloaded secondary neutral will not be operative. Other polyphase connections not

discussed above explicitly will fall under one type or the other of the cases discussed.

In a Yy connection, to obtain third harmonic suppression one may provide a third

winding which is connected in mesh, which can be an unloaded winding. It is called a

tertiary. This winding improves the single phase to earth fault detection also. Further,

this winding can be used to feed some permanent station loads also. Such transform-

ers are designated as Yyd transformers. If the neutral wires are provided and also

mesh connected winding is present, then triplen harmonics are ’shared’ between them

depending upon their impedances.

12.3 Three phase transformers units

As against a bank of three single phase transformers connected to three phase mains, a

three phase transformer generally has the three magnetic circuits that are interacting. The

exception to this rule is a 3-phase shell type transformer. In the shell type of construction,

even though the three cores are together, they are non-interacting. Three limb core type

3-phase transformer is the one in which the phases are magnetically also linked. Flux of

each limb uses the other two limbs for its return path. This is true for fundamental and

non-triplen harmonics. The triplen harmonics being co-phasal cannot use other limbs for

79



the return path (this holds good for zero sequence, unbalanced fundamental mmf also). The

flux path is completed through the air. So substantially large value of the mmf produces a

low value of third harmonic flux as the path of the flux is through the air and has a very

high reluctance. Thus the flux in the core remains nearly sinusoidal, so also the induced emf.

This happens irrespective of the type of connection used. The triplen order flux, sometimes

links the tank and produces loss in the same.

Other harmonics can be suppressed by connecting tuned filters at the terminals. Harmonic

current compensation using special magnetic circuit design is considered to be outside the

scope here.

13 Poly Phase connections and Poly phase Transform-

ers

The individual transformers are connected in a variety of ways in a power system. Due to

the advantages of polyphase power during generation, transmission and utilization polyphase

power handling is very important. As an engineering application is driven by techno-

economic considerations, no single connection or setup is satisfactory for all applications.

Thus transformers are deployed in many forms and connections. Star and mesh connections

are very commonly used. Apart from these, vee or open delta connections, zig zag connec-

tions , T connections, auto transformer connections, multi winding transformers etc. are a

few of the many possibilities. A few of the common connections and the technical and eco-

nomic considerations that govern their usage are discussed here. Literature abounds in the

description of many other. Apart from the characteristics and advantages of these, one must

also know their limitations and problems, to facilitate proper selection of a configuration for

80



an application. Many polyphase connections can be formed using single phase transformers.

In some cases it may be preferable to design, develop and deploy a polyphase transformer

itself. In a balanced two phase system we encounter two voltages that are equal in magnitude

differing in phase by 90◦. Similarly, in a three phase system there are three equal voltages

differing in phase 120 electrical degrees. Further there is an order in which they reach a

particular voltage magnitude. This is called the phase sequence. In some applications like

a.c. to d.c. conversion, six phases or more may be encountered. Transformers used in all

these applications must be connected properly for proper functioning. The basic relationship

between the primary and secondary voltages (brought about by a common mutual flux and

the number of turns), the polarity of the induced emf (decided by polarity test and used

with dot convention) and some understanding of the magnetic circuit are all necessary for

the same. To facilitate the manufacturer and users, international standards are also avail-

able. Each winding has two ends designated as 1 and 2. The HV winding is indicated by

capital letters and the LV winding by small letters. If more terminals are brought out from

a winding by way of taps there are numbered in the increasing numbers in accordance to

their distance from 1 (eg A1, A2, A3...). If the induced emf at an instant is from A1 to A2 on

the HV winding it will rise from a1 to a2 on the LV winding.

Out of the different polyphase connections three phase connections are mostly encountered

due to the wide spread use of three phase systems for generation, transmission and utiliza-

tion. Three balanced 3-phase voltages can be connected in star or mesh fashion to yield a

balanced 3-phase 3-wire system. The transformers that work on the 3-phase supply have

star, mesh or zig-zag connected windings on either primary secondary or both. In addition

to giving different voltage ratios, they introduce phase shifts between input and output sides.

These connections are broadly classified into 4 popular vector groups.

81



1. Group I: zero phase displacement between the primary and the secondary.

2. Group II: 180◦ phase displacement.

3. Group III: 30◦ lag phase displacement of the secondary with respect to the primary.

4. Group IV: 30◦ lead phase displacement of the secondary with respect to the primary.

A few examples of the physical connections and phasor diagrams are shown in Fig. 34 cor-

responding to each group. The capital letters indicates primary and the small letters the

secondary. D/d stand for mesh, Y/y - for star, Z/z for zig-zag. The angular displacement of

secondary with respect to the primary are shown as clock position, 0◦ referring to 12 o’clock

position. These vector groups are especially important when two or more transformers are

to be connected in parallel.

Star connection is normally cheaper as there are fewer turns and lesser cost of insulation. The

advantage becomes more with increase in voltage above 11kv. In a star connected winding

with earthed-neutral the maximum voltage to the earth is ( 1√3)of the line voltage. Also star

connection permits mixed loading due to the presence of the neutral. Mesh connections

are advantageous in low voltage transformers as insulation costs are insignificant and the

conductor size becomes ( 1√3) of that of star connection and permits ease of winding. The

common polyphase connections are briefly discussed now.

Star/star (Yy0, Yy6)connection This is the most economical one for small high voltage

transformers. Insulation cost is highly reduced. Neutral wire can permit mixed load-

ing. Triplen harmonics are absent in the lines. These triplen harmonic currents cannot

flow, unless there is a neutral wire. This connection produces oscillating neutral. Three

82



Group1 00 Phase shift

W indings & Term inals E.M .F Vector diagram s

A2 a2

B2 b2

C2 c2

N na1

b1

c1

A1

B1

C1

A2 a2

B2 b2

C2 c2

A1

B1

C1

a1

b1

c1

a4

b4

c4

A2

B2

C2

A1

B1

C1

n

a1

b1

c1

A2

C2 B2

a2

A2

C2

B2

a2

A2

C2 B2



A2

C2 B2

A2 a1

B2 b1

C2 c1

nA1

B1

C1

a1

b1

c1

N

a1

b1

c1

a1

b1

c1

a2

b2

c2

A2

B2

C2

A1

B1

C1

a3

b3

c3

n

a1

b1

c1

A2

B2

C2

A1

B1

C1

A2

C2 B2A2

C2 B2

(a) (b)



A2 a2

B2 b2

C2 c2

nA1

B1

C1

a1

b1

c1

A2 a2

B2 b2

C2 c2

NA1

B1

C1

a1

b1

c1

A2 a4

B2 b4

C2 c4

NA1

B1

C1

n

a1

b1

c1

A2

C2 B2

a2

A2

C2 B2

a2

A2

C2 B2

Group4 + 300 Phase shift


A2

B2

A1

B1

C1

a2

b2

c2

na1

b1

c1

A2

B2

C2

NA1

B1

C1

a1

b1

c1

a1

b1

c1

a2

b2

c2

A2

B2

C2

NA1

B1

C1

n

a1

b1

c1

a3

b3

c3

a4

b4

c4

A2

C2

B2

a2

C2 B2

C 2

B 2

A2

B2

C2

(c) (d)

Figure 34: Vector Groups for 3-phase Transformer Connections

83



phase shell type units have large triplen harmonic phase voltage. However three phase

core type transformers work satisfactorily. A tertiary mesh connected winding may be

required to stabilize the oscillating neutral due to third harmonics in three phase banks.

Mesh/mesh (Dd0, Dd6) This is an economical configuration for large low voltage trans-

formers. Large amount of unbalanced load can be met with ease. Mesh permits a cir-

culating path for triplen harmonics thus attenuates the same. It is possible to operate

with one transformer removed in open delta or Vee connection meeting 58 percent of

the balanced load. Three phase units cannot have this facility. Mixed single phase

loading is not possible due to the absence of neutral.

Star/mesh(Dy or Yd ) This arrangement is very common for power supply transform-

ers. The delta winding permits triplen harmonic currents to circulate in the closed

path and attenuates them.

Zig zag/ star (ZY1 or Zy11) Zigzag connection is obtained by inter connection of phases.

4-wire system is possible on both sides. Unbalanced loading is also possible. Oscillat-

ing neutral problem is absent in this connection. This connection requires 15% more

turns for the same voltage on the zigzag side and hence costs more.

Generally speaking a bank of three single phase transformers cost about 15% more

than their 3-phase counter part. Also, they occupy more space. But the spare capac-

ity cost will be less and single phase units are easier to transport.

Mesh connected three phase transformers resemble 3- single phase units but kept in

a common tank. In view of this single tank, the space occupied is less. Other than

that there is no big difference. The 3-phase core type transformer on the other hand

84



has a simple core arrangement. The three limbs are equal in cross section. Primary

and secondary of each phase are housed on the same limb. The flux setup in any limb

will return through the other two limbs as the mmf of those limbs are in the proper

directions so as to aid the same. Even though magnetically this is not a symmetri-

cal arrangement, as the reluctance to the flux setup by side limbs is different from

that of the central limb, it does not adversely affect the performance. This is due to

the fact that the magnetizing current itself forms a small fraction of the total phase

current drawn on load. The added advantage of 3-phase core is that it can tolerate

substantially large value of 3rd harmonic mmf without affecting the performance. The

3rd harmonic mmf of the three phases will be in phase and hence rise in all the limbs

together. The 3rd harmonic flux must therefore find its path through the air. Due to

the high reluctance of the air path even a substantially large value of third harmonic

mmf produces negligible value of third harmonic flux. Similarly unbalanced operation

of the transformer with large zero sequence fundamental mmf content also does not

affect its performance. Even with Yy type of poly phase connection without neutral

connection the oscillating neutral does not occur with these cores. Finally, three phase

cores themselves cost less than three single phase units due to compactness.

14 Parallel operation of one phase and two phase trans-

formers

By parallel operation we mean two or more transformers are connected to the same supply

bus bars on the primary side and to a common bus bar/load on the secondary side. Such

requirement is frequently encountered in practice. The reasons that necessitate parallel

85



operation are as follows.

1. Non-availability of a single large transformer to meet the total load requirement.

2. The power demand might have increased over a time necessitating augmentation of the

capacity. More transformers connected in parallel will then be pressed into service.

3. To ensure improved reliability. Even if one of the transformers gets into a fault or is

taken out for maintenance/repair the load can continued to be serviced.

4. To reduce the spare capacity. If many smaller size transformers are used one machine

can be used as spare. If only one large machine is feeding the load, a spare of similar

rating has to be available. The problem of spares becomes more acute with fewer

machines in service at a location.

5. When transportation problems limit installation of large transformers at site, it may

be easier to transport smaller ones to site and work them in parallel.

Fig. 35 shows the physical arrangement of two single phase transformers working in parallel

on the primary side. Transformer A and Transformer B are connected to input voltage bus

bars. After ascertaining the polarities they are connected to output/load bus bars. Certain

conditions have to be met before two or more transformers are connected in parallel and

share a common load satisfactorily. They are,

1. The voltage ratio must be the same.

2. The per unit impedance of each machine on its own base must be the same.

3. The polarity must be the same, so that there is no circulating current between the

transformers.

86



E1 E2

V1 V2IA

E1 E2

load

A

B

supply bus Load bus

IB

Figure 35: Parallel Operation of Two Single Phase Transformers - Physical

87



4. The phase sequence must be the same and no phase difference must exist between the

voltages of the two transformers.

These conditions are examined first with reference to single phase transformers and then the

three phase cases are discussed.

Same voltage ratio Generally the turns ratio and voltage ratio are taken to be the same.

If the ratio is large there can be considerable error in the voltages even if the turns ratios

are the same. When the primaries are connected to same bus bars, if the secondaries

do not show the same voltage, paralleling them would result in a circulating current

between the secondaries. Reflected circulating current will be there on the primary

side also. Thus even without connecting a load considerable current can be drawn

by the transformers and they produce copper losses. In two identical transformers

with percentage impedance of 5 percent, a no-load voltage difference of one percent

will result in a circulating current of 10 percent of full load current. This circulating

current gets added to the load current when the load is connected resulting in unequal

sharing of the load. In such cases the combined full load of the two transformers can

never be met without one transformer getting overloaded.

Per unit impedance Transformers of different ratings may be required to operate in par-

allel. If they have to share the total load in proportion to their ratings the larger

machine has to draw more current. The voltage drop across each machine has to be

the same by virtue of their connection at the input and the output ends. Thus the

larger machines have smaller impedance and smaller machines must have larger ohmic

impedance. Thus the impedances must be in the inverse ratios of the ratings. As the

voltage drops must be the same the per unit impedance of each transformer on its

own base, must be equal. In addition if active and reactive power are required to be

88



shared in proportion to the ratings the impedance angles also must be the same. Thus

we have the requirement that per unit resistance and per unit reactance of both the

transformers must be the same for proper load sharing.

Polarity of connection The polarity of connection in the case of single phase transform-

ers can be either same or opposite. Inside the loop formed by the two secondaries

the resulting voltage must be zero. If wrong polarity is chosen the two voltages get

added and short circuit results. In the case of polyphase banks it is possible to have

permanent phase error between the phases with substantial circulating current. Such

transformer banks must not be connected in parallel. The turns ratios in such groups

can be adjusted to give very close voltage ratios but phase errors cannot be compen-

sated. Phase error of 0.6 degree gives rise to one percent difference in voltage. Hence

poly phase transformers belonging to the same vector group alone must be taken for

paralleling.

Transformers having −30◦ angle can be paralleled to that having +30◦ angle by re-

versing the phase sequence of both primary and secondary terminals of one of the

transformers. This way one can overcome the problem of the phase angle error.

Phase sequence The phase sequence of operation becomes relevant only in the case of

poly phase systems. The poly phase banks belonging to same vector group can be

connected in parallel. A transformer with +30◦ phase angle however can be paralleled

with the one with −30◦ phase angle, the phase sequence is reversed for one of them

both at primary and secondary terminals. If the phase sequences are not the same

then the two transformers cannot be connected in parallel even if they belong to same

vector group. The phase sequence can be found out by the use of a phase sequence

indicator.

89



Performance of two or more single phase transformers working in parallel can be com-

puted using their equivalent circuit. In the case of poly phase banks also the approach

is identical and the single phase equivalent circuit of the same can be used. Basically

two cases arise in these problems. Case A: when the voltage ratio of the two trans-

formers is the same and Case B: when the voltage ratios are not the same. These are

discussed now in sequence.

14.1 CASE A: Equal voltage ratios

Always two transformers of equal voltage ratios are selected for working in parallel. This

way one can avoid a circulating current between the transformers. Load can be switched on

subsequently to these bus bars. Neglecting the parallel branch of the equivalent circuit the

above connection can be shown as in fig37. The equivalent circuit is drawn in terms of the

secondary parameters. This may be further simplified as shown under Fig. 36. The voltage

drop across the two transformers must be the same by virtue of common connection at input

as well as output ends. By inspection the voltage equation for the drop can be written as

IAZA = IBZB = IZ = v (say) (86)

HereI = IA + IB (87)

And Z is the equivalent impedance of the two transformers given by,

Z =ZAZB

ZA + ZB

(88)

Thus IA =v

ZA

=IZ

ZA

= I.ZB

ZA + ZB

and (89)

IB =v

ZB

=IZ

ZB

= I.ZA

ZA + ZB

(90)

90



V2’

V1ZA

ZB

IA

IB

RA

RB

jXA

jXB

V2’IV1

IA

IB

RA

RB

jXA

jXB

(a) (b)

V’L

IA

IB

ZL

I

ZA

ZB

V Load

VL

(c)

Figure 36: Equivalent Circuit for Transformers working in Parallel -Simplified circuit and

Further simplification for identical voltage ratio

91



If the terminal voltage is V = IZL then the active and reactive power supplied by each of

the two transformers is given by

PA = Real(V I∗A)andQA = Imag(V I∗A)and (91)

PB = Real(V I∗B)andQB = Imag(V I∗B) (92)

(93)

From the above it is seen that the transformer with higher impedance supplies lesser load

current and vice versa. If transformers of dissimilar ratings are paralleled the transformer

with larger rating shall have smaller impedance as it has to produce the same drop as the

other transformer, at a larger current. Thus the ohmic values of the impedances must be in

the inverse ratio of the ratings of the transformers. IAZA = IBZB ∴IA

IB= ZB

ZA. Expressing the

voltage drops in p.u basis, we aim at the same per unit drops at any load for the transformers.

The per unit impedances must therefore be the same on their respective bases. Fig. 37 shows

the phasor diagram of operation for these conditions. The drops are magnified and shown

to improve clarity. It is seen that the total voltage drop inside the transformers is v but the

currents IA and IB are forced to have a different phase angle due to the difference in the

internal power factor angles θA and θB. This forces the active and reactive components of

the currents drawn by each transformer to be different ( even in the case when current in

each transformer is the same). If we want them to share the load current in proportion to

their ratings, their percentage ( or p.u) impedances must be the same. In order to avoid

any divergence and to share active and reactive powers also properly, θA = θB. Thus the

condition for satisfactory parallel operation is that the p.u resistances and p.u reactance must

be the same on their respective bases for the two transformers. To determine the sharing of

currents and power either p.u parameters or ohmic values can be used.

This is illustrated with the help of an example below.

92



φBφA

E

θBθA

IB

IA

φ

IL

V2

V

IARAIR IBRB

IBXBIXIAXA

Figure 37: Phasor Diagram of Operation for two Transformers working in Parallel

14.2 Case B :unequal voltage ratios

One may not be able to get two transformers of identical voltage ratio in spite of ones best

efforts. Due to manufacturing differences, even in transformers built as per the same design,

the voltage ratios may not be the same. In such cases the circuit representation for parallel

operation will be different as shown in Fig. 38. In this case the two input voltages cannot

be merged to one, as they are different. The load brings about a common connection at the

output side. EA and EB are the no-load secondary emf. ZL is the load impedance at the

secondary terminals. By inspection the voltage equation can be written as below:

EA = IAZA + (IA + IB)ZL = V + IAZA · (94)

EB = IBZB + (IA + IB)ZL = V + IBZB· (95)

93



VL

IA

IB

ZLEA EB

I

RAjXA

RB jXB

Figure 38: Equivalent Circuit for unequal Voltage Ratio

Solving the two equations the expression for IA and IB can be obtained as

IA =EAZB + (EA − EB)ZL

ZAZB + ZL(ZA + ZB)and (96)

IB =EBZA + (EB − EA)ZL

ZAZB + ZL(ZA + ZB)

ZA and ZB are phasors and hence there can be angular difference also in addition to the

difference in magnitude. When load is not connected there will be a circulating current

between the transformers. The currents in that case can be obtained by putting ZL = ∞ (

after dividing the numerator and the denominator by ZL ). Then,

IA = −IB =(EA − EB)

(ZA − ZB)(97)

If the load impedance becomes zero as in the case of a short circuit, we have,

IA =EA

ZA

and IB =EB

ZB

(98)

94



Instead of the value of ZL if the value of V is known , the currents can be easily determined

( from eq 76 and eq 77) as

IA =EA − V

ZA

and IB =EB − V

ZB

(99)

If more than two transformers are connected across a load then the calculation of load cur-

V

IA

IB

IC

ZL

EA

EB

EC

I

RA

RB

jXA

jXB

RC jXC

Figure 39: Parallel Generator Theorem

rents following the method suggested above involves considerable amount of computational

labor. A simpler and more elegant method for the case depicted in Fig. 39 is given below.

It is known by the name parallel generator theorem.

IL = IA + IB + IC + ...... (100)

But IA =EA − V

ZA

, IB =EB − V

ZB

, IC =EC − V

ZC

(101)

V = IL.ZL (102)

Combining these equations

V

ZL

=EA − V

ZA

+EB − V

ZB

+EC − V

ZC

+ ... (103)

95



Grouping the terms together

V (1

ZL

+1

ZA

+1

ZB

+1

ZC

+ ...) =EA

ZA

+EB

ZB

+EC

ZC

+ ...

= ISCA + ISCB + ISCC + .... (104)

(1

ZL

+1

ZA

+1

ZB

+1

ZC

+ ...) =1

Z(105)

V = Z(ISCA + ISCB + ISCC + ....) (106)

From this V can be obtained. Substituting V in eqn 83, IA, IB etc can be obtained. Knowing

the individual current phasor, the load shared by each transformer can be computed. To

illustrate the method, the previous problem is solved again using this method.

Example.

15 Transformer voltage control and Tap changing

Regulating the voltage of a transformer is a requirement that often arises in a power appli-

cation or power system.

In an application it may be needed

1. To supply a desired voltage to the load.

2. To counter the voltage drops due to loads.

3. To counter the input supply voltage changes on load.

On a power system the transformers are additionally required to perform the task of regu-

lation of active and reactive power flows. The voltage control is performed by changing the

turns ratio. This is done by provision of taps in the winding. The volts per turn available in

96



Reverser

Booster transformer

Regulation transform

Booster transformer

Reverser

Main transform

tertiary

Figure 40: Tap changing and Buck Boost arrangement

large transformers is quite high and hence a change of even one turn on the LV side repre-

sents a large percentage change in the voltage. Also the LV currents are normally too large

to take out the tapping from the windings. LV winding being the inner winding in a core

type transformer adds to the difficulty of taking out of the taps. Hence irrespective of the

end use for which tapping is put to, taps are provided on the HV winding. Provision of taps

to control voltage is called tap changing. In the case of power systems, voltage levels are

some times changed by injecting a suitable voltage in series with the line. This may be called

buck-boost arrangement. In addition to the magnitude, phase of the injected voltage may

be varied in power systems. The tap changing arrangement and buck boost arrangement

with phase shift are shown in Fig. 40.

Tap changing can be effected when a) the transformers is on no- load and b) the load is still

remains connected to the transformer. These are called off load tap changing and on load

tap changing. The Off load tap changing relatively costs less. The tap positions are changed

when the transformer is taken out of the circuit and reconnected. The on-load tap changer

on the other hand tries to change the taps without the interruption of the load current. In

view of this requirement it normally costs more. A few schemes of on-load tap changing are

now discussed.

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HV LV

Reactor

s

1

23

4

5

Figure 41: Reactor Method of Tap Changer ( with table of switching)

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Reactor method The diagram of connections is shown in Fig. 41. This method employs

an auxiliary reactor to assist tap changing. The switches for the taps and that across

the reactor(S) are connected as shown. The reactor has a center tapped winding on

a magnetic core. The two ends of the reactor are connected to the two bus bars to

which tapping switches of odd/even numbered taps are connected. When only one tap

is connected to the reactor the shorting switch S is closed minimizing the drop in the

reactor. The reactor can also be worked with both ends connected to two successive

taps. In that case the switch ’S’ must be kept open. The reactor limits the circulating

current between the taps in such a situation. Thus a four step tapped winding can be

used for getting seven step voltage on the secondary(see the table of switching). The

advantage of this type of tap changer are

1. Load need not be switched off.

2. More steps than taps are obtained.

3. Switches need not interrupt load current as a alternate path is always provided.

The major objection to this scheme seems to be that the reactor is in the circuit always

generating extra loss.

Parallel winding, transformer method In order to maintain the continuity of supply

the primary winding is split into two parallel circuits each circuit having the taps as

shown in Fig. 42. Two circuit breakers A and B are used in the two circuits. Initially

tap 1a and 1b are closed and the transformer is energized with full primary voltage.

To change the tap the circuit breaker A is opened momentarily and tap is moved from

1a to 2a. Then circuit breaker A is closed. When the circuit A is opened whole of

the primary current of the transformer flows through the circuit B. A small difference

in the number of turns between the two circuit exists. This produces a circulating

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current between them. Next, circuit breaker B is opened, the tap is changed from 1b

to 2b and breaker is closed. In this position the two circuits are similar and there is

no circulating current. The circulating current is controlled by careful selection of the

leakage reactance. Generally, parallel circuits are needed in primary and secondary

to carry the large current in a big transformer. Provision of taps switches and circuit

breakers are to be additionally provided to achieve tap changing in these machines.

A B

HVLV

a1a2a3a4

b1 b2 b3 b4

Figure 42: Parallel Primary Winding Tap Changing

Series booster method In this case a separate transformer is used to buck/boost the

voltage of the main transformer. The main transformer need not be having a tapped

arrangement. This arrangement can be added to an existing system also. Fig. 42

shows the booster arrangement for a single phase supply. The reverser switch reverses

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the polarity of the injected voltage and hence a boost is converted into a buck and

vice versa. The power rating of this transformer need be a small fraction of the main

transformer as it is required to handle only the power associated with the injected

voltage. One precaution to be taken with this arrangement is that the winding must

not be open circuited. If it gets open circuited the core (B in fig) gets highly saturated.

In spite of the small ratings and low voltages and flexibility, this method of voltage

control costs more mainly due to the additional floor space it needs. The methods of

voltage regulation discussed so far basically use the principle of tap changing and hence

the voltage change takes place in steps. Applications like a.c. and d.c. motor speed

control, illumination control by dimmers, electro-chemistry and voltage stabilizers need

continuous control of voltage. This can be obtained with the help of moving coil voltage

regulators.

moving coil voltage regulators Fig. 43a shows the physical arrangement of one such

transformer. a, b are the two primary windings wound on a long core, wound in

the opposite sense. Thus the flux produced by each winding takes a path through

the air to link the winding. These fluxes links their secondaries a2 and b2. A short

circuited moving coil s is wound on the same limb and is capable of being held at any

desired position. This moving coil alters the inductances of the two primaries. The

sharing of the total applied voltage thus becomes different and also the induced emf

in the secondaries a2 and b2. The total secondary voltage in the present case varies

from 10 percent to 20 percent of the input in a continuous manner. The turns ratios

of a1 : a2 and b1 : b2 are 4.86 and 10.6 respectively. 5

4.86+ 95

10.6= 10% when s is in the

top position. In the bottom position it becomes 95

4.86+ 5

10.6= 20%. By selecting proper

ratios for the secondaries a2 and b2 one can get the desired voltage variation.

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Vin

5%

95%

a1

b1

S

a2

b2

Vout

Figure 43: Moving Coil Voltage Regulator

V1

sliding contact

Variable secondary

a.c voltage

V2

a) without electrical isolation

V1

b) with electrical isolation

(a) (b)

Figure 44: Sliding Contact Regulator a) without isolation b) with isolation

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Sliding contact regulators These have two winding or auto transformer like construction.

The winding from which the output is taken is bared and a sliding contact taps the

voltage. The minimum step size of voltage change obtainable is the voltage across a

single turn. The conductor is chosen on the basis of the maximum load current on the

output side. In smaller ratings this is highly cost effective. Two winding arrangements

are also possible. The two winding arrangement provides electrical isolation also.These

are shown in Fig. 44.

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Transformers

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