CHAPTER 1

SIGLE-PHASE TRASFORMER

1.0 DEFIITIO OF TRASFORMER

1.1 COSTRUCTIO

1.2 PRICIPLE OF OPERATIO

1.3 FARADAYS LAW AD LEZS LAW

1.3.1 FARADAYS LAW

1.3.2 LENZS LAW

1.4 EMF EQUATIO

1.5 TRASFORMER EQUIVALET CIRCUIT MODEL

1.5.1 IDEAL TRANSFORMER

1.5.2 PRACTICAL TRANSFORMER

1.5.3 IMPEDANCE TRANSFER

1.5.4 EXACT EQUIVALENT CIRCUIT

1.5.5 APPROXIMATE EQUIVALENT CIRCUIT

1.6 PHASOR DIAGRAMS

1.7 TRASFORMER TESTS

1.7.1 OPEN CIRCUIT TEST

1.7.2 SHORT CIRCUIT TEST

1.8 TYPES OF LOSSES

1.9 VOLTAGE REGULATIO

1.10 EFFICIECY AD MAXIMUM EFFICIECY

1.0 DEFIITIO OF TRASFORMER

A transformer is a device that transfers electrical energy from one circuit to

another through a shared magnetic field. A changing current in the first circuit

(the primary) creates a changing magnetic field; in turn, this magnetic field

induces a voltage in the second circuit (the secondary).

It can raise (step-up) or lower (step-down) the voltage in a circuit but with a

corresponding decrease or increase in current

1.1 COSTRUCTIO

A transformer is a static machine. Although it is not an energy conversion device, it is

indispensable in many energy conversion systems. It is a simple device, having two or

more electric circuits coupled by a common magnetic circuit.

A transformer essentially consists of two or more windings coupled by a mutual magnetic

field. Ferromagnetic cores are used to provide tight magnetic coupling and high flux

densities. Such transformers are known as iron core transformers. They are invariably

used in high-power applications. Air core transformers have poor magnetic coupling and

are sometimes used in low-power electronic circuits.

Two types of core constructions are normally used, as shown in Fig. 1.0. In the core type

(Fig. 1.0a), the windings are wound around two legs of a magnetic core of rectangular

shape. In the shell type (Fig. 1.0b), the windings are wound around the center leg of a

three-legged magnetic core. To reduce core losses, the magnetic core is formed of a stack

of thin laminations.

L-shaped laminations are used for core-type construction and E-shaped laminations are

used for shell-type construction. To avoid a continuous air gap (which would require a

large exciting current), laminations are stacked alternately as shown in Fig. l.0c and

Fig. 1.0d.

Fig. 1.0: Transformer core construction. (a) Core-type (b) Shell-type (c) L-shaped lamination

(d) E-shaped lamination

A schematic representation of a two-winding transformer is shown in Fig. 1.1. The two

vertical bars are used to signify tight magnetic coupling between the windings. One

winding is connected to an ac supply and is referred to as the primary winding. The other

winding is connected to an electrical load and is referred to as the secondary winding.

The winding with the higher number of turns will have a high voltage and is called the

high-voltage (HV) or high-tension (HT) winding. The winding with the lower number of

turns is called the low-voltage (LV) or low-tension (LT) winding. To achieve tighter

magnetic coupling between the windings, they may be formed of coils placed one on top

of another (Fig. l.0a) or side by side (Fig. l.0b) in a pancake coil formation where

primary and secondary coils are interleaved. Where the coils are placed one on top of

another, the low-voltage winding is placed nearer the core and the high-voltage winding

on top.

Fig. 1.1: Schematic representation of a two-winding transformer

Transformers have widespread use. Their primary function is to change voltage level. A

transformer is a device for converting electric energy at one voltage level to electric

energy at another voltage level through the action of a magnetic field. It plays an

extremely important role in modern life by making possible the economical long distance

transmission of electric power.

1.2 PRICIPLE OF OPERATIO

When a voltage is applied to the primary of a transformer, a flux is produced in the core

as given by Faradays law. The changing flux in the core then induced a voltage in the

secondary winding of the transformer. Because transformer cores have very high

permeability, the net magnetomotive force required in the core to produce its flux is very

small. Since the net magnetomotive force is very small, the primary circuits

magnetomotive force must be approximately equal and opposite to the secondary

circuits magnetomotive force. This fact yields the transformer current ratio.

The transformer is based on two principles:

First, that an electric current can produce a magnetic field (electromagnetism)

and,

Second, that a changing magnetic field within a coil of wire induces a voltage

across the ends of the coil (electromagnetic induction).

Transformer is only converting from AC signal into AC signal.

The primary is connected to source of alternating (AC) voltage

By changing the current in the primary coil, one changes the strength of its

magnetic field; since the secondary coil is wrapped around the same magnetic field,

which it produces mutually-induced e.m.f (electromotive force)

If the secondary coil is closed, a current flows in it and so a voltage is induced across

the secondary terminal. Therefore, electrical energy is transferred from primary to

the secondary terminal.

Example

Refer to Fig. 1.2

Fig. 1.2: Example of transformer structure

When the switch is closed:

Current in primary coil increases

Creates increasing magnetic field in primary coil

Induces current in secondary coil

Lamp lights up

When current in primary coil becomes steady:

No more changes in the magnetic field in primary coil

No more current induced in secondary coil

Lamp goes off

When switch is opened:

Current in primary coil decreases to zero

Creates decreasing magnetic field in primary coil

Induces current in opposite direction in secondary coil

Lamp lights up again

1.3 FARADAYS LAW AD LEZS LAW

1.3.1 FARADAYS LAW

Faradays law states that if a flux passes through a turn of a coil of wire, a voltage will be

induced in the turn of wire that is directly proportional to the rate of change in the flux

with respect to time. In equation form,

dt

de

ind

= (1.0)

where ind

e is the voltage induced in the turn of the coil and is the flux passing through

the turn. If a coil has turns and if the same flux passes through all of them, then the

voltage induced across the whole coil is given by

dt

de

ind

= (1.1)

where:

inde = voltage induced in the coil

= number of turns of wire in coil

= flux passing through coil

1.3.2 LEZS LAW

The minus sign in the Faradays Law equations (Eqn. 1.1) above is an expression of

Lenzs law. Lenzs law states that the direction of the voltage build up in the coil is such

that if the coil ends were short circuited, it would produce current that would cause a flux

opposing the original flux change. Since the induced voltage opposes the change that

causes it, a minus sign is included in Eqn. 1.1. To understand this concept clearly,

examine Fig. 1.3. If the flux shown in the figure is increasing in strength, then the voltage

built up in the coil will tend to establish a flux that will oppose the increase.

Fig.1.3: The meaning of Lenzs law: (a) A coil enclosing an increasing magnetic flux;

(b) Determining the resulting voltage polarity.

A current flowing as shown in Fig. 1.3(b) would produce a flux opposing the increase, so

the voltage on the coil must be built up with the polarity required to drive that current

through the external circuit. Therefore, the voltage must be built up with the polarity

shown in the figure. Since the polarity of the resulting voltage can be determined from

physical considerations, the minus sign in Eqn. 1.0 and Eqn. 1.1 is often left out.

There is one major difficulty involved in using Eqn. 1.1 in practical problems. That

equation assumes that exactly the same flux is present in each turn of the coil.

Unfortunately, the flux leaking out of the core into the surrounding air prevents this from

being true. If the windings are tightly coupled, so that the vast majority of the flux

passing through one turn of the coil does indeed pass through all of them, then Eqn. 1.1

will give valid answers. But if leakage is quite high or if extreme accuracy is required, a

different expression that does not make that assumption will be needed. The magnitude of

the voltage in the i th turn of the coil is always given by

( )dt

de i

ind

= . (1.2)

If there are turns in the coil of wire, the total voltage on the coil is

=

=

1i

1indee

( )

=

=

1i

i

dt

d

= =

1i

idt

d .. (1.3)

The term in parentheses in Eqn. 1.3 is called the flux linkage of the coil, and

Faradays law can be rewritten in terms of flux linkage as

dt

de

ind

= (1.4)

where

=

=

1i

i . (1.5)

The units of flux linkage are weber-turns.

Faradays law is the fundamental property of magnetic fields involved in transformer

operation. The effect of Lenzs law in transformers is to predict the polarity of the

voltages induced in transformer windings.

Faradays law also explains the eddy current losses. A time-changing flux induces

voltage within a ferromagnetic core in just the same manner as it would in a wire

wrapped around that core. These voltages cause swirls of current to flow within the core.

It is the shape of these currents that gives rise to the name eddy currents. These eddy

currents are flowing in a resistive material (the iron of the core), so energy is dissipated

by them. The lost energy goes into heating the iron core.

The amount of energy lost to eddy currents is proportional to the size of the paths they

follow within the core. For this reason, it is customary to break up any ferromagnetic core

that may be subject to alternating fluxes into many small strips, or laminations, and to

build the core up out of these strips. An insulating oxide or resin is used between the

strips, so that the current paths for eddy currents are limited to very small areas. Because

the insulating layers are extremely thin, this action reduces eddy current losses with very

little effect on the cores magnetic properties. Actual eddy current losses are proportional

to the square of the lamination thickness, so there is a strong incentive to make the

laminations as thin as economically possible.

Example

Figure below shows a coil of wire wrapped around an iron core. If the flux in the core is

given by the equation

t377sin05.0= Wb

If there are 100 turns on the core, what voltage is produced at the terminals of the coil?

Of what polarity is the voltage during the time when flux is increasing in the reference

direction shown in the figure? Assume that all the magnetic flux stays within the core

(i.e., assume that the flux leakage is zero).

Solution

The direction of the voltage while the flux is increasing in the reference direction must be

positive to negative, as shown in figure. The magnitude of the voltage is given by

dt

de

ind

=

( ) ( )t377sin05.0dt

dturns100=

t377cos1885= @ ( )V90t377sin1885 +

1.4 EMF EQUATIO

Let 1

= No. of turns in primary

2

= No. of turns in secondary

m = Maximum flux in core in Webers

= A*Bm

f = Frequency of AC input in Hz

The flux increases from its zero value to maximum value from in one quarter of the cycle

i.e. in 1/4 f second

Rate of change of flux = f4/1

m

Unit: wb/s or volt

Average e.m.f/ turns = m

f4

If flux ( )m

varies sinusoidally, then:

R.M.S value/turn = *11.1 average value

= m

f4*11.1

= m

f44.4 volts

E.M.F induced in primary = (r.m.s / turn) * (No. of primary turns)

= 1m

f44.4

= 1m

AfB44.4 (a)

E.M.F induced in secondary = (r.m.s / turn) * (No. of secondary turns)

= 2m

f44.4

= 2m

AfB44.4 (b)

From (a) and (b):

m

1

1 f44.4

E=

m

2

2 f44.4

E=

E.M.F per turns is same in both windings

From equation (a) and (b), we get:

a

E

E

2

1

2

1 == Where a = voltage transformation ratio

If a 1, then the transformer is called step- down transformer

1.5 TRASFORMER EQUIVALET CIRCUIT MODEL

1.5.1 IDEAL TRASFORMER

Fig. 1.4: Basic Transformer Fig. 1.5: Ideal Transformer Equivalent Circuit

Consider a transformer with two windings, a primary winding of 1

turns and a

secondary winding of 2

turns, as shown schematically in Fig.1.6. In a schematic

diagram it is a common practice to show the two windings in the two legs of the core,

although in an actual transformer the windings are interleaved. Let us consider an ideal

transformer that has the following properties:

Fig. 1.6(a): Ideal transformer

1. The winding resistances are negligible.

2. All fluxes are confined to the core and link both windings; that is, no leakage

fluxes are present. Core losses are assumed to be negligible.

3. Permeability of the core is infinite (i.e. ). Therefore, the exciting current

required to establish flux in the core is negligible; that is, the net mmf required to

establish a flux in the core is zero.

ss

pp

EV

EV

=

=

When the primary winding is connected to a time-varying voltage1

v , a time-varying flux

is established in the core. A voltage 1

e will be induced in the winding and will equal

the applied voltage if resistance of the winding is neglected:

dt

dev

111

== ..... (1.6)

The core flux also links the secondary winding and induces a voltage2

e , which is the

same as the terminal voltage2

v :

dt

dev

222

== .... (1.7)

From Eqn. 1.6 and Eqn. 1.7,

a

v

v

2

1

2

1 == .. (1.8)

where a is the turns ratio.

Eqn. 1.8 indicates that the voltages in the windings of an ideal transformer are directly

proportional to the turns of the windings.

Let us now connect a load (by closing the switch in Fig. 1.6(a)) to the secondary winding.

A current 2

i will flow in the secondary winding, and the secondary winding will provide

an mmf 22

i for the core. This will immediately make a primary winding current 1

i flow

so that a counter mmf 11

i can oppose22

i . Otherwise 22

i would make the core flux

change drastically and the balance between 1

v and 1

e would be disturbed. Note that in

Fig. 1.6 that the current directions are shown such that their mmfs oppose each other.

Because the net mmf required to establish a flux in the ideal core is zero,

=2211

ii net mmf 0=

2211ii = ... (1.9)

a

1

i

i

1

2

2

1 == .... (1.10)

The currents in the windings are inversely proportional to the turns of the windings. Also

note that if more current is drawn by the load, more current will flow from the supply. It

is this mmf-balancing requirement (Eqn. 1.9) that makes the primary know of the

presence of current in the secondary.

From Eqn. 1.8 and Eqn. 1.10

2211

iviv = (1.11)

That is, the instantaneous power input to the transformer equals the instantaneous power

output from the transformer. This is expected, because all power losses are neglected in

an ideal transformer. Note that although there is no physical connection between load and

supply, as soon as power is consumed by the load, the same power is drawn from the

supply. The transformer, therefore, provides a physical isolation between load and supply

while maintaining electrical continuity.

If the supply voltage 1

v is sinusoidal, then Eqn. 1.8, 1.10, and 1.11 can be written in

terms of rms values:

a

V

V

2

1

2

1 ==

a

1

I

I

1

2

2

1 ==

2211

IVIV = . (1.12)

1.5.2 PRACTICAL TRASFORMER

Fig. 1.6(b): Equivalent circuit of practical transformer

pV = Primary terminal voltage (input)

sV = Secondary terminal voltage (output)

spR,R = Leakage resistance on the primary and secondary respectively

spX,X = Leakage reactance on the primary and secondary respectively

cR = Core resistance

mX = Magnetizing reactance

pI = Primary current

sI = Secondary current

p = Primary winding turns

s = Secondary winding turns

1.5.3 IMPEDACE TRASFER

Consider the case of a sinusoidal applied voltage and secondary impedance2

Z , as shown

in Fig. 1.7a.

Fig. 1.7: Impedance transfer across an ideal transformer.

2

2

2I

VZ =

The input impedance is

2

22

2

2

1

1

1I

Va

a/I

aV

I

VZ ===

2

2 Za=

So

22

2

1'ZZaZ ==

Impedance 2

Z connected in the secondary will appear as impedance 2

'Z looking from

the primary. The circuit in Fig. 8a is therefore equivalent to the circuit in Fig. 1.7b.

Impedance can be transferred from secondary to primary if its value is multiplied by the

square of the turns ratio. Impedance from the primary side can also be transferred to the

secondary side, and in that case its value has to be divided by the square of the turns

ratio:

121Z

a

1'Z =

This impedance transfer is very useful because it eliminates a coupled circuit in an

electrical circuit and thereby simplifies the circuit.

1.5.4 EXACT EQUIVALET CIRCUIT

Fig. 1.8: The model of the real transformer

The figure above shows the accurate model of the transformer but it is quite

difficult to analyze practical circuits containing transformers

Thus the equivalent circuit normally has converted to the equivalent circuit at a

single voltage level.

Therefore, the equivalent circuit must be referred either to its primary side or to

its secondary side in problem solutions.

Fig. 1.9: Exact equivalent circuit (referred to the primary side)

Fig. 1.10: Exact equivalent circuit (referred to the secondary side)

1.5.5 APPROXIMATE EQUIVALET CIRCUIT

Fig. 1.11: The model of the real transformer

The excitation branch has a very small current compared to the load current of the

transformers.

Moreover, it is so small that under normal circumstances it causes a completely

negligible voltage drop in p

R andp

X . Thus, it can be simplified into approximate

equivalent circuits.

Fig. 1.12: Approximate equivalent circuit (referred to the primary side)

Fig. 1.13: Approximate equivalent circuit (referred to the secondary side)

1.6 PHASOR DIAGRAMS

It is easy to determine the effect of the impedances and the current phase on the

transformer voltage regulation by drawing the phasor diagram.

s

V is assumed to he at an angle of 0 and all other voltages and currents are

compared to that references.

A transformer phasor diagram is presented by applying Kirchhoffs \/voltage the

transformer equivalent circuit and an equation will be as follows.

Lagging Power Factor

Fig. 1.14: Lagging power factor

Unity Power Factor

Fig. 1.15: Unity power factor

++=seqseqs

pIjXIR0V

a

V

++= seqseqs

pIjXIR0V

a

V

++= 0IjX0IR0Va

Vseqseqs

p

Leading Power Factor

Fig. 1.16: Leading power factor

1.7 TRASFORMER TEST

1.7.1 OPE CIRCUIT TEST

Must be conducted at the rated terminal voltage

The values of c

R (core resistance) and m

X (magnetizing reactance) can be

determined by opening the output line at the secondary side of transformer as in

figure below

Fig. 1.17: Open-circuit test

ococ

oc1

OCIV

Pcos=

C

OC

cI

VR =

M

OC

MI

VX =

cosIIOCC

= sinIIOCM

=

++++= seqseqs

pIjXIR0V

a

V

CHECK

If ocp

VV = Therefore c

R and m

X (Primary side)

Else

ocsVV = Therefore

cR and

mX (Secondary side)

1.7.2 SHORT CIRCUIT TEST

Always conducted at the rated winding current

Current flowing in excitation branch is neglect since the input voltage is so low

during the short-circuit test.

Thus the entire voltage drop in the transformer can be attributed to the series

elements in the circuit:

The determined values of eq

R (winding resistance) and eq

X (winding reactance)

Fig. 1.18: Short-circuit test

(Note: minus sign inductance only)

scsc

SCSC

IV

P1cos=

scsc

sc

eqI

0VZ

=

eqeqeqjXRZ +=

CHECK

If scp

II =

Therefore pppeq

jXRZZ +==

Else

If scs

II =

Therefore ssseq

jXRZZ +==

p

rated

pV

SI =

s

rated

sV

SI =

1.8 TYPES OF LOSSES

Copper ( )RI 2 Losses : resistive heating losses in the secondary and primary windings

Eddy current losses : resistive heating losses in the core of the

transformer

Hysteresis losses : a heat loss caused by the magnetic properties of

the armature

Leakage flux : in turn causes losses due to frictional heating in

susceptible cores. Magnetic flux lines produced by

the primary winding that do not link the turns of the

secondary winding

1.9 VOLTAGE REGULATIO

Defined as the difference between the voltage magnitude at the load terminals of the

transformer at full load and at no load in percent of full load voltage.

a) Equivalent circuit referred to primary side

Fig. 1.19(a): Approximate equivalent circuit (referred to primary)

Voltage regulation %100V

VV

FLs

FLsLs

=

Since at no load, the voltage regulation can also be expressed as

Voltage regulation( )

( )

%100V

VV

FLs

FLsp

=

( ) pLs VV =

b) Equivalent circuit referred to secondary side

Fig. 1.19(b): Approximate equivalent circuit (referred to secondary)

Voltage regulation %100V

VV

FLs

FLsLs

=

Since at no load, the voltage regulation can also be expressed as

Voltage regulation( )

( )

%100V

Va/V

FLs

FLsp

=

( ) a/VV pLs =

1.10 EFFICIECY AD MAXIMUM EFFICIECY

To measure the efficient of the transformer

a) Equivalent circuit referred to primary side

Fig. 1.20(a): Approximate equivalent circuit (referred to primary side)

%100xPP

P

lossout

out

+=

%100xPPcosIV

cosIV

coreCuss

ss

++=

%100xPPncosInV

cosInV

coreCu

2

ss

ss

++=

c

2

p

coreR

VP =

1eq

2

pcuRIP =

b) Equivalent circuit referred to secondary side

Fig. 1.20(b): Approximate equivalent circuit (referred to secondary side)

When maximum efficiency occurs:

cu

c

P

Pn = and 1cos =

%100xPP

P

lossout

out

+=

%100xPPncosInV

cosInV

coreCu

2

ss

ss

++=

( )c

2

p

coreR

a/VP =

2eq

2

scuRIP =