beee unit ii

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Basic Electrical & Electronics Engineering

Prepared By: Deependr Singh 1

Unit IITRANSFORMERS

Introduction:

Before really starting, let us look at some magnetic circuits shown in the following figures.

All of them have a magnetic material of regular geometric shape called core. A coil having a

number of turns (= N) of conducting material (say copper) are wound over the core. This coil is

called the exciting coil. When no current flows through the coil, we don’t expect any magnetic field

or lines of forces to be present inside the core. However in presence of current in the coil, magnetic

flux φ will be produced within the core. The strength of the flux, it will be shown, depends on the

product of number of turns (N) of the coil and the current (I) it carries. The quantity Ni called mmf

(magnetomotive force) can be thought as the cause in order to produce an effect in the form of flux φ

within the core. Is it not somewhat similar to an electrical circuit problem where a voltage (e.m.f.) is

applied (cause) and a current is produced (effect) in the circuit? Hence the term magnetic circuit is

used in relation to producing flux in the core by applying mmf (= NI). We shall see more similarities

between an electrical circuit and a magnetic circuit in due course as we go along further. At this

point you may just note that a magnetic circuit may be as simple as shown in figure 21.1 with a

single core and a single coil or as complex as having different core materials, air gap and multiple

exciting coils as in figure 21.2.

REVIEW OF LAWS OF ELECTROMAGNETISM

Biot-Savart’s Law: We know that any current carrying conductor produces a magnetic field. A

magnetic field is characterized either by H , the magnetic field intensity or by B, the magnetic flux

density vector. These two vectors are connected by a rather simple relation: B = μ0 μr H ; where μ0 =

4π×10-7 H/m is called the absolute permeability of free space and μr , a dimensionless quantity called

the relative permeability of a medium (or a material). For example the value of μr is 1 for free space

or could be several thousands in case of ferromagnetic materials.

Biot-Savart law is of fundamental in nature and tells us how to calculate dB or dH at a given

point with position vector r , due to an elemental current and is given by:

3

0

4 r

r l id Bd r





If the shape and dimensions of the conductor carrying the current is known, then the field at given

point can be calculated by integrating the RHS of the above equation.

length

r

r

r l id B

3

0

4

Where, length indicates that the integration is to be carried out over the length of the conductor.

AMPERE’S CIRCUITAL LAW: This law states that line integral of the vector H along any

arbitrary closed path is equal to the current enclosed by the path. Mathematically:

I l d H

For certain problems particularly in magnetic circuit problems Ampere’s circuital law is used

to calculate field instead of the more fundamental Biot-Savart law for reasons going to be explained

below. Consider an infinite straight conductor carrying current i and we want to calculate field at a

point situated at a distance d from the conductor. Now take the closed path to be a circle of radius d .

At any point on the circle the magnitude of field strength will be constant and direction of the field

will be tangential. Thus LHS of the above equation simply becomes H × 2πd. So field strength is:

m Ad

I H /

2

MAGNETIC CIRCUIT:

MAGNETIC FIELD: The region around the magnet where its poles exhibit a force of attraction or

repulsion is called Magnetic filed

MAGNETIC FLUX: The quantity of magnetic lines of force produced by a magnet is called

magnetic flux.

It is measured in weber.

1Wb = 108 lines of force or Maxwell

MAGNETIC FLUX DENSITY: The flux per unit area of the surface at right angles to the magnetic

field is known as flux density.

A B

MAGNETOMOTIVE FORCE: The force behind the flow of flux or production of flux in

magnetic circuit is called magnetomotive force.

The product of the number of turns of the magnetizing coil & the current passing through it.

Mmf = NI ampere turns

RELUCTANCE (S): In an electric circuit, current flow is opposed by the resistance of the material.

Similarly there is opposition by the material to the flow of flux which is called reluctance.

A

l

A

l S

r 0

A/Wb





Reluctance flux

mmf AT/Wb or A/Wb

PERMEANCE: The permeance of the magnetic circuit is defined as the reciprocal of the reluctance.

It is defined as the property of the magnetic circuit due to which it allows flow of the magnetic flux

through it.

MAGNETIC LEAKAGE AND FRINGING: The magnetic flux which does not follow the

intended path in a magnetic circuit is called leakage flux.

The leakage coefficient is defined as the ration of total flux (t) to the useful flux (u)

u

t

The flux is setup in the magnetic core and passes through the air gap. This flux is known as useful

flux u.

FRINGING: The useful flux passing across the gap tends to bulge outwards. This increases the

effective area in the air gap and decreases the flux density. This effect is known as fringing.

COMPARISON BETWEEN ELECTRICAL AND MAGNETIC CIRCUIT

SIMILARITIESMAGNETIC CIRCUIT ELECTRIC CIRCUIT

1. Path traced by the current is called

electric circuit

1. Path traced by magnetic flux is called

magnetic circuit

2. M.M.F., in AT 2. E.M.F., in Volts

3. Flux, in Wb 3. Current, I in Ampere

4. Reluctance, A

l

A

l S

r 0

AT/Wb 4. Resistance, R = A

l =

A

l

Ω

5. 5. 6. Permeability, µ 6. Conductivity,

1

7. Flux density, A

B 7. Current density,

A

I J

8. Magnetic intensity,

l

NI H 8. Electric field intensity,

d

V E

DISSIMILARITIES1. The magnetic flux does not flow but is

setup in the magnetic circuit.

1. The electric current actually flows in an

electric circuit.

2. For magnetic flux there is no perfect

insulator

2. For electric current there are large no of

perfect insulators.

3. The reluctance of magnetic circuit is not

constant rather it varies with the value of

flux density.

3. The resistance of an electric circuit is

almost constant as its value depends upon

the value of which is almost constant.

4. Energy is required to create the magnetic

flux but is not required to maintain it.

4. Energy must be supplied to the electrical

circuit to maintain the flow of current.

M.M.F., FLUX, AND THEIR RELATION





Let us now try to derive a relationship between flux produced φ and mmf NI applied for

linear case.

A B

A H r 0

Al

NI r

0

A

l

NI

r

0

1

Now defining NI = mmf and S =

A

l

r 0

1 , the above equation can be written as

A

l

NI

S

NI

r

0

1

This equation resembles the familiar current voltage relationship of an electric circuit which is

produced below for immediate comparison between the two:

The expression in the denominator is called resistance which impedes the flow of the current.

S =

A

l

r 0

1 , is known as reluctance of the magnetic circuit and permeance (similar to

admittance in electric circuit) is defined as the reciprocal of reluctance i.e. ρ =S

l .

ANALYSIS OF MAGNETIC CIRCUITSANALYSIS OF SERIES MAGNETIC CIRCUIT: Consider first a simple magnetic circuit, shown

in Figure 21.10 with a single core material having uniform cross sectional area A and mean length of

flux path l. Reluctance offered to the flow of flux is S . The corresponding electrical representation is

rather simple. Due to the fact that NI =Hl = φS the equivalent electrical circuit is also drawn beside

the magnetic circuit. Polarity of mmf is decided on the basis of the direction of the flux which is

clockwise inside the core in this case. Although in the actual magnetic circuit there is no physical

connection of the winding and the core, in the electrical circuit representation mmf and reluctance

are shown to be connected. One should not feel disturbed by this as because the relationship between





= g i

r

l Al A

00

= A

l

A

l g i

r 00

11

NI = g i S S

NI = )( g i S S

Or,)( g i S S

NI

As we can see these two reluctances are connected in series. Similarly, for series magnetic

circuits having different reluctance segments, total reluctance will always be the sum of individual

reluctances.

ANALYSIS OF SERIES-PARALLEL MAGNETIC CIRCUIT: We now take up the following

magnetic circuit (Figure 21.12). As a first step to solve this circuit, we would like to draw its

equivalent electrical representation. Vertical links of the core are called limbs and the horizontal

links are called yoke of the magnetic circuit. In the figure PU, QT and RS are the limbs whereas PQ,

QR, UT and TS are the yokes. We fix up the corner points P, Q, R etc from the given physical

dimensions, joining of which will give us the mean length of the flux paths.

If the coil carries a current I in the direction shown, flux φ, produced in the first limb will be

in the upward direction. Same φ is constrained to move along the yoke PQ. At point Q, two parallel

paths are available to φ for its path through namely (i) the central limb QT and (ii) the yoke QR. In

other words, φ will be divided into two components φ1 and φ2 as shown with the obvious

condition 21 . The relative values of these components will be decided by the respective

reluctances of the paths. φ1 and φ2 are combined together at point T and hence complete the path.

Now in the path TUPQ flux φ is same, it is made of same material and has same cross-sectional area





A, then its reluctance A

l S

TUPQ

TUPQ . In the central limb, flux is same (φ1), however it encounters two

materials, one is iron (QM and WT) and the other is a small air gap (MW). The reluctance of the air

gap A

l S

g

g

0

. The two reluctances S QM and S WT of the magnetic material may however be

combined into a single reluctance as = S QM + S WT . The portion of the magnetic circuit which

carries flux φ2 can be represented by a single reluctance A

l S

QRST

QRST . Instead of carrying on with

long suffixes let us call S QRST to be S 2. To write down the basic equations let us redraw the electrical

equivalence of the above magnetic circuit below (Figure 21.13):

IMPORTANT EQUATIONS:

The various equations which will hold good are written below:

Φ = φ1 + φ2

NI = Hl + H 1l 1 + H g l g = S φ + ( S 1+S g )φ1 balance in loop1

( S 1+S g )φ1 = S 2φ2 mmf balance in loop2

H 1l 1 + H g l g = H 2l 2 mmf balance in loop2

NI = Hl + H 2l 2 mmf balance in the outer loop.





SINGLE-PHASE TRANSFORMER

INTRODUCTION: Transformers are one of the most important components of any power system.

It basically changes the level of voltages from one value to the other at constant frequency. Being a

static machine the efficiency of a transformer could be as high as 99%. Big generating stations are

located at hundreds or more km away from the load center (where the power will be actually

consumed). Long transmission lines carry the power to the load centre from the generating stations.

Generator is a rotating machines and the level of voltage at which it generates power is limited to

several kilo volts only a typical value is 11 kV. To transmit large amount of power (several

thousands of mega watts) at this voltage level means large amount of current has to flow through the

transmission lines. The cross-sectional area of the conductor of the lines accordingly should be large.

Hence cost involved in transmitting a given amount of power rises many times. Also the

transmission lines have their own resistances. This huge amount of current will cause tremendous

amount of power loss or I2r loss in the lines. This loss will simply heat the lines and becomes a

wasteful energy. In other words, efficiency of transmission becomes poor and cost involved is high.

The above problems may be suppressed if we could transmit power at a very high voltage

say, at 200 kV or 400 kV or even higher at 800 kV. But as pointed out earlier, a generator is

incapable of generating voltage at these levels due to its own practical limitation. The solution to this

problem is to use an appropriate step-up transformer at the generating station to bring the

transmission voltage level at the desired value as depicted in figure 23.1 where for simplicity single

phase system is shown to understand the basic idea. Obviously when power reaches the load centre,

one has to step down the voltage to suitable and safe values by using transformers. Thus transformers

are an integral part in any modern power system. Transformers are located in places called

substations. In cities or towns you must have noticed transformers are installed on poles – these are

called pole mounted distribution transformers. These transformers change voltage level typically

from 3-phase, 6 kV to 3-phase 440 V line to line.

In this and the following lessons we shall study the basic principle of operation and

performance evaluation based on equivalent circuit.





BASIC CONCEPTS AND CONSTRUCTION FEATURES

BASIC CONSTRUCTION AND WORKING PRINCIPLE OF TRANSFORMER

An elementary transformer consists of a soft iron or silicon steel core and two windings,

placed on it. The windings are insulated from both the core and each other. The core is built up of

thin soft iron or low reluctance to the magnetic flux. The winding connected to the magnetic flux.

The winding connected to the supply main is called the primary and the winding connected to the

load circuit is called the secondary. Although in the actual construction the two windings are usually

wound one over the other, for the sake of simplicity, the figures for analyzing transformer theory

show the windings on opposite sides of the core, as shown below

SIMPLE TRANSFORMER:

When the primary winding is connected to an ac supply mains, current flows through

it. Since this winding links with an iron core, so current flowing through this winding produces an

alternating flux in the core. Since this flux is alternating and links with the secondary winding also,

so induces an emf in the secondary winding. The frequency of induced e.m.f. in secondary winding

is the same as that of the flux or that of the s supply voltage. The induced e.m.f in the secondary

winding enables it to deliver current to an external load connected across it. Thus the energy is

transformed from primary winding to the secondary winding by means of electro-magnetic induction

without any change in frequency. The flux Ø of the iron core links not only with the secondary

winding but also with the primary winding, so produces self-induced e.m.f in the primary winding:

This induced in the primary winding opposes the applied voltage and therefore sometimes it is

known as back e.m.f of the primary. In fact the induced e.m.f in the primary winding limits the

primary current in much the same way that the back e.m.f in a dc motor limits the armature

current.





CONSTRUCTION OF TRANSFORMER:

Assembly of a power transformer is shown in fig below:

The transformer is very simple in construction and consists of magnetic circuit linking with

two windings known as primary and secondary windings. Besides magnetic circuit and windings it

consists of a suitable container for the assembled core and windings, such as a tank, a suitable

medium for insulating the core and windings from its container such as transformer oil, suitable

bushings (either of porcelain, oil filled or condenser type) for insulating and bringing out terminals of

the windings from the container, temperature gauge for measurement of temperature of hot oil or

hottest spot temperature, and oil gauge to indicate the oil level inside the tank. Some transformers are

also provided with conservator tank in order to slow down deterioration of oil and keep the main

tank full of oil, emergency vent to relieve the pressure inside the tank incase the pressure inside the

transformer rises to danger point and gas operated relay (Buchholz relay) in order to give alarm to

indicate the presence of gas in case of some minor fault and take out the transformer out of circuit in

case of serious fault.

Magnetic circuit consists of an iron core. Since core is magnetic link between the two

systems connected to the transformer and it itself contains a lot of energy, therefore, it is not by any

means the passive component it would appear first. The core is made of thin laminations (0.35 to 0.5

mm thick) of cold rolled grain oriented steel (CRGO Steel ) in order to keep iron loss small in

rectangular x-section in case of small size transformers and stepped for large size one to reduce the

quantity of copper required. The laminations are insulated from each other by a thin varnish coating.

As already mentioned, the electric circuit consists essentially of two windings, each being

split up into two equal number of coils and the coils of the two windings are arranged on each limb





of iron core to reduce the magnetic leakage. In order to minimize the amount of high voltage

insulation, low voltage coils are placed adjacent to the iron.

CAUTION: - Transformer must not be connected to a direct source. If the primary winding of a

transformer is connected to a dc supply mains, the flux produced will not vary but remain constant in

magnitude and therefore no e.m.f. will be induced in the secondary winding except at the moment of

switching on. Thus the transformer can not be employed for raising or lowering the dc voltage. Also

there will be no back induced e.m.f. in the primary winding and therefore a heavy current will be

drawn from the supply mains which may result in the burning out of the winding.

A transformer in its simplest form will consist of a rectangular laminated magnetic structure

on which two coils of different number of turns are wound as shown in Figure 23.2.

The winding to which A.C. voltage is impressed is called the primary of the transformer andthe winding across which the load is connected is called the secondary of the transformer.

IDEAL TRANSFORMER

To understand the working of a transformer it is always instructive, to begin with the concept

of an ideal transformer with the following properties.

1. Primary and secondary windings have no resistance.

2. All the flux produced by the primary links the secondary winding i.e., there is no leakage

flux.3. Permeability μr of the core is infinitely large. In other words, to establish flux in the core

vanishingly small (or zero) current is required.

4. Core loss comprising of eddy current and hysteresis losses are neglected.

ANALYSIS OF IDEAL TRANSFORMER

Let us assume a sinusoidally varying voltage is supplied across the primary with secondary

winding open circuited. Although the current drawn I m will be practically zero, but its position will

be 90° lagging with respect to the supply voltage. The flux produced will obviously be in phase with





I m. In other words the supply voltage will lead the flux phasor by 90°. Since flux is common for both

the primary and secondary coils, it is customary to take flux phasor as the reference.

Let, t t sin)( max

Then,

2

sinmax1

t V v (1)

The time varying flux )(t will link both the primary and secondary turns inducing in

voltages e1 and e2 respectively.

Instantaneous induced voltage in primary

2

sinmax11

t N

dt

d N

2

sin2 max1

t fN (2)

Instantaneous induced voltage in primary

2sinmax22

t N dt d N

2

sin2 max2

t fN (3)

Magnitudes of the rms induced voltages will therefore be:

max1max11 44.42 fN fN E (4)

max2max22 44.42 fN fN E (5)

VOLTAGE, CURRENT AND IMPEDANCE TRANSFORMATION

The time phase relationship between the applied voltage v1 and e1 and e2 will be same. The

180° phase relationship obtained in the mathematical expressions of the two merely indicates that the

induced voltage opposes the applied voltage as per Lenz’s law. In other words if e1 were allowed to

act alone it would have delivered power in a direction opposite to that of v1. By applying Kirchhoff’s

law in the primary one can easily say that V 1 = E 1 as there is no other drop existing in this ideal

transformer. Thus under no load condition:

k N

N

E

E

V

V

1

2

1

2

1

2 , where k is the transformation ratio

We know that for an ideal transformer power transferred remains same, so we have

P 1 = P 2

i.e. 2211 I V I V

2

1

1

2

I I

V V





Or we can say, k I

I

N

N

E

E

V

V

2

1

1

2

1

2

1

2

Also, we know that for an ideal transformer power consumed also remains same, so we have

2

2

21

2

1 Z I Z I

i.e.1

2

2

2

21

Z

Z

I

I

1

2

2

1

Z

Z

I

I

Hence, k R

R

Z

Z

I

I

N

N

E

E

V

V

1

2

1

2

2

1

1

2

1

2

1

2

TRANSFORMER ON NO-LOAD (5.7 Sahdev & Chaturvedi)

A transformer is said to be on no-load when secondary winding is open circuited and the

secondary current I 2 is zero. Neither the secondary winding has any effect on the magnetic flux in the

core nor does it have any effect on primary current.

In actual transformer, the losses can not be neglected. Therefore if transformer is on no load,

a small current I 0 (usually 2 to 10% of the rated value) called exciting current or no-load current is

drawn by the primary. This current has to supply in iron losses (hysteresis and eddy current losses) in

the core and a very small amount of copper losses in primary (the primary copper losses are so small

as compared to core losses that hey are generally neglected and secondary copper losses are zero as

I 2 is zero). Therefore, current I 0 lags behind voltage V 1 by an angle φ0 which is less than 900, as

shown in fig. The angle of lag depends upon the losses in the transformer.

The no-load current I 0 has two components:

a) I w in phase with the applied voltage V 1, called active or working component. It

supplies the iron losses and a small quantity of primary copper losses.

b) I m in quadrature with the applied voltage V 1, called reactive or magnetizing

component. It produces flux in the core and does not consume any power.





From Phasor diagram shown in fig.:

Working component, 00 cos I I w

Magnetizing component, 00 sin I I m

No-Load current, 22

0 mw I I I

No-Load Power Input, 0010 cos I V P

TRANSFORMER ON NO-LOAD (neglecting winding resistance and leakage

flux ) (5.8 Sahdev & Chaturvedi)

When a certain load is connected across the secondary, a current I 2 flows through it as shown

in fig. 5.12. The magnitude of current I 2 depends upon terminal voltage V 2 and impedance of the

load. The phase angle of secondary current I 2 with respect to V 2 depends upon the nature of load i.e.

whether the load is resistive, inductive or capacitive.

The operation of the transformer on load is explained below with the aid of the diagram;

(i) When the transformer is on no load as shown in fig. 5.13(a) it draws no load current I 0

from the supply mains. The no load current I 0 produces an m.m.f. N 1 I 0 which sets up

flux φ in the core.

(ii) When the transformer is loaded, current I 2 flows in the secondary winding. This

secondary current I 2 produces an m.m.f. N 2 I 2 which sets up flux φ2 in the core. This

flux φ2 opposes the flux φ which is set up by the current I0, as shown in fig. 5.13(b)

according to Lenz’s Law.

(iii) Since φ2 opposes the flux φ, hence the resultant flux tends to decrease and causes the

reduction of self induced e.m.f. E 1 momentarily. Thus, V 1 predominates over E 1

causing additional primary current I’ 1 drawn from the supply. The amount of this

additional current is such that the original conditions i.e., flux in the core must be

restored to the original value φ





PHASOR DIAGRAM

EQUIVALENT CIRCUITS

VOLTAGE REGULATION

LOSSES AND EFFICIENCY

OC AND SC TEST

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