Machine1 Handout Abebe PDF

Course Title: Introduction to Electrical Machines Instructor: Abebe W. ECE 3303

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CHAPTER ONE

ELECTROMAGNETIC PRINCIPLES

1.1. INTRODUCTION

Nature of Magnetism Magnets can be found in a natural state in the form of a magnetic ore (permanent magnet), with

the two main types being Magnetite also called "iron oxide", (FE3O4) and Lodestone, also

called "leading stone". For most practical applications these natural occurring magnets can be

disregarded as their magnetism is very low and because nowadays, man-made artificial magnets

can be produced in many different shapes, sizes and magnetic strengths.

Electromagnetism is produced when an electrical current flows through a simple conductor

such as a piece of wire or cable. A small magnetic field is created around the conductor with the

direction of this magnetic field with regards to its "North" and "South" poles being determined

by the direction of the current flowing through the conductor.

Magnetism plays an important role in Electrical and Electronic Engineering because without it

components such as relays, solenoids, inductors, chokes, coils, loudspeakers, motors, generators,

transformers, and electricity meters etc, would not work if magnetism did not exist.

Then every coil of wire uses the effect of electromagnetism when an electrical current flows

through it.

All electromagnetic devices make use of magnetic fields in their operation. These magnetic

fields may be produced by permanent magnets or electromagnets. Magnetic fields are created

by alternating- and direct-current sources to provide the necessary medium for developing

generator action and motor action. Throughout this course we will be studying the application of

magnetic fields to electromechanical energy conversion processes as demonstrated in rotating

electric machinery. Also, transformers provide energy transfer from one electric circuit to

another via the changing magnetic field. It will become apparent that there is both transfer and

storage of energy in the magnetic fields of the various electromagnetic devices. Hence all

electromagnetic devices are constructed with appropriate magnetic circuits.


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1.2. MAGNETIC FIELDS 1.2.1. Properties of Magnetic Lines of Force

Some important properties of magnetic lines of force are ascribed bellow:

1. Magnetic lines of force are directed from north to south outside a magnet. The

direction is determined by the north pole of a small magnet held in the field.

Figure 1.1 Magnetic field pattern near a magnet

Figure 1.2 Magnetic field distortion

2. Magnetic lines of force are continious.

3. Magnetic lines of force enter or leave a magnetic surface at right angles.

4. Magnetic lines of force cannot cross each other.

5. Magnetic lines of force in the same direction tend to repel each other.

6. Magnetic lines of force tend to be as short as possible.

7. Magnetic lines of force occupy three-dimensional space extending (theoretically) to

infinity.


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1.2.2. Magnetic Field Produced by Current-Carrying Conductor(Electromagnetism):

A magnetic field is always associated with a current-carrying conductor, as illustrated in Figure

1.3. Exploring the magnetic field by means of a compass, we observe the following:

The magnetic field is strongest perpendicular to the current direction.

Figure 1.3 Direction of magnetic field around a currcnt-carrying conductor.

As we traverse a path around the conductor, we find that the magnetic field is always tangent to

the direction of current flow. We can trace a path around the conductor so that continuous

magnetic lines of force surround the conductor.

If we reverse the direction of current flow, the direction of the magnetic field also changes.

The field is strongest near the wire and decreases as we move farther from it. (We can obtain a

measure of field strength by trying to deflect the magnet needle from the position it has assumed

in the field. At a point where the field is strong, it will be more difficult to deflect it than at a

point where it is weak.)

If we look at a single current-carrying conductor end on, and draw it as in Figure 1.3, where the

symbol indicates current flowing into the page, it is easier to draw the magnetic field. If we

reverse the current, we have the symbol for current coming out of the page, and we have the

situation depicted in Figure 1.3. The dot and cross symbols, respectively, represent the head and

tail of an arrow.

If we grasp the conductor with our right hand, the thumb pointing in the direction of the current,

our fingers will point in the same direction as the north pole of the compass. This method of


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determining the directions of current flow in a conductor and the surrounding lines of force is

called Ampere's right-hand rule as illustrated in Figure 1.4.

Current-carryingconductor

Field or fluxline

Figure 1.4 Amperes right hand rule showing the direction of field

For a single current carrying conductor; if the thumb of our right hand indicates to the direction

of current flow, then the other fingers indicates the direction of magnetic field line.

If we construct a coil of many turns, we can increase the magnetic field strength very greatly, as

shown in Figure 1.5. We can also increase the magnetic field strength by increasing the

magnitude of current in the coil. A cylindrical coil closely wound with a large number of turns of

insulated wire is called solenoid . Thus we see that the magnetic field strength is proportional to

both the number of turns and the current.

We can determine the direction of the magnetic field in a cylindrical coil of many turns of

insulated wire by using our right hand. If we grasp the coil with our right hand with the fingers

pointing in the direction of the current, the thumb will point in the direction of the north pole.

This method of determining directions of current flow in a coil and magnetic fields of force is

another form of Ampere's right-hand rule. Andre Marie Ampere (1775-1836), pursuant to the

experimental work of Oersted, developed extensively the foundations of electromagnetic theory.

Refer to Figure 1.5.


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Figure 1.5 Magnetic field direction of solenoid by right hand rule.

1.3. ELECTROMAGNETIC RELATIONSHIPS

1.3.1. Magnetic Lines of Force

The "quantity of magnetism" which exists in a magnetic field is the magnetic line of force, or

more simply, the magnetic flux. In the SI system magnetic flux is measured in units called

webers, abbreviated Wb, and its symbol is ( (the Greek lowercase letter phi). Although there is

no actual flow of magnetic flux, we will consider flux to be analogous to current in electric

circuits.

3.3.2. Magnetic Flux Density

The total magnetic flux that comes out of the magnet is not uniformly distributed, as can be seen

in Figure 1.2. A more useful measure of the magnetic effect is the magnetic flux density, which

is the magnetic flux per unit cross-sectional area. We will consider two equal areas through

which the magnetic flux penetrates at right angles near one end of the permanent magnet along

its centerline. From the illustration it becomes apparent that there is a greater amount of magnetic

flux passing through an area that is nearer the magnet pole. In other words, the magnetic flux

density increases as we approach closer to the end of the magnet. However, it must be noted that

the magnetic flux density inside the magnet is uniformly constant. Magnetic flux density is

measured in units of tesla (T) and is given the symbol B. One tesla is equal to 1 weber of

magnetic flux per square meter of area.We can state that

A

B = 1.1


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where B = magnetic flux density, T

= magnetic flux, Wb

A = area through which penetrates perpendirularly, m2

1.3.3. Magnetomotive Force We have seen that an increase in the magnitude of current in a coil or a single conductor results

in an increase in the magnetic flux. If the number of turns in a coil are increased (with the current

remaining constant), there is an increase in magnetic flux. Therefore, the magnetic flux is

proportional to the products of amperes and turns. This ability of a coil to produce magnetic flux

is called the magnetomotive force. Magnetomotive force is abbreviated MMF and has the units

of ampere-turns (At). The magnetomotive force is given the symbol Fm.

We may write

NIm =F 1.2 where Fm = magnetomotive force (MMF), At

N = number of turns of coil

I = excitation current in coil, A

Magnetomotive force in the magnetic circuit is analogous to electromotive force in an electric

circuit.

1.3.4. Magnetic Reluctance

Doubling the driving force (MMF) in the circuit results in a doubling of the output quantity

(magnetic flux). We consider this ratio of MMF to magnetic flux:

mm =F 1.3

where NIm =F , the MMF, At

= magnetic flux, Wb

m= reluctance of the magnetic circuit. At/Wb

Transposing, we have

= mmF


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which shows us that the magnetic flux is directly proportional to the magnetomotive force. This

equation represents Ohm's law of magnetic circuits. The proportionality factor m, is called the

reluctance of the magnetic circuit and is obviouslv, analogous to resistance in an electric circuit.

A much larger amount of flux can be produced in an iron-core coil than in an air-core coil. Thus

we see that the reluctance of the magnetic circuit depends on the material properties of the mag-

netic circuit. For our purposes, the materials are classified as either magnetic or nonmagnetic.

Only the ferrous (irons and steels) group of metals, including cobalt and nickel, are magnetic

materials. All other materials, such as air, insulators, wood, paper, plastic, brass, and bronze,

including vacuum, are nonmagnetic materials.

The reluctance of a homogeneous magnetic circuit may be expressed in terms of its physical

dimensions and magnetic property as follows:

A

lm = 1.4

where m = reluctance of the magnetic circuit, At/Wb

l = average or mean length of the magnetic path, m

A = cross-sectional area of the magnetic path, m2

= 0r , absolute (or total) permeability of the magnetic path, H/m

Reluctance is in essence magnetic resistance, that is, the property of a magnetic circuit which is

reluctant or unwilling to set up magnetic flux. The reciprocal of reluctance is termed as

permeance, which is anologous to conductance in electric circuits.

1.3.5. Magnetic Field Intensity

One other important magnetic quantity is the magnetomotive force gradient per unit length of

magnetic circuit, or more commonly, the magnetic field intensity. Its symbol is H and from the

definition,

l

=H mF 1.5

the unit is ampere-turns per meter (At/m). The former name for magnetic field intensity was

magnetizing force. We have seen that more ampere-turns (MMF) are required to set up the same


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magnetic flux in magnetic circuits of air than in iron of similar configuration. Hence the

magnetic field intensity for the air path is much larger than for the iron path.

1.3.6. Magnetization (B-H) Curve

Typical magnetization or B-H curves for sheet steel, cast iron, and air are plotted in Figure 1.6.

The nonlinear relationship between magnetic flux density B (teslas) and magnetic field intensity

H (ampere-turns per meter) is illustrated. It is observed that the magnetic flux density increases

almost linearly with an increase in the magnetic field intensity up to the knee of the

magnetization curve. Beyond the knee, a continued increase in the magnetic field intensity

results in a relatively small increase in the magnetic flux density. When ferromagnetic materials

experience only a slight increase in magnetic flux density for a relatively large increase in

magnetic field intensity, the materials are said to be saturated. Magnetic saturation occurs

beyond the knee of the magnetization curve.

Figure 1.6 Typical Magnitizations curves.

The characteristic of saturation is present only in ferromagnetic materials. An explanation of

magnetic saturation is based on the theory that magnetic materials are composed of very many

tiny magnets (magnetic domains) that are randomly positioned when the material is totally

demagnetized. Upon application of a magnetizing force (H), the tiny magnets will tend to align

themselves in the direction of this force. In the lower part of the magnetizing curve, the


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alignment of the randomly positioned tiny magnets increases proportionately to the magnetic

field intensity until the knee of the curve is reached. Beyond the knee of the curve, fewer tiny

magnets remain to be aligned, and therefore large increases in the magnetic field intensity result

in only small increases in magnetic flux density. When there are no more tiny magnets to be

aligned, the ferromagnetic material is completely saturated. In the saturation region of the curve,

the magnetic flux density increases linearly with magnetic field intensity, just as it does for free

space or nonmagnetic materials. From the origin of the B-H curve there is a slight concave

curvature beyond which is the essentially linear region. We shall see that the nonlinear

characteristics of the magnetization curve have practical implications in the operation of

electrical machines.

1.3.7. Hysteresis

Hysteresis is the name given to the "lagging" of flux density B behind the magnetizing force H.

when a specimen of ferromagnetic material is taken through a cycle of magnetization.If the

specimen has been completely demagnetized and the magnetizing force H is increased in steps

from zero, the relationship between flux density B and H is represented by the curve oa (Figure

1.7) which is the normal magnetization curve. If the value of H is now decreased, the trace of B

is higher than oa and follows the curve ab until H is reduced to zero. Thus when H reaches

zero, there is a residual flux density referred to as remnant flux density ob. In order to reduce B

to zero, a negative field strength oc must be applied. The magnetic field intensity OE required to

wipe out the residual magnetism ob is called coercive force. As H is further increased in the

negative direction, the specimen becomes magnetized with the opposite polarity as shown by

the curve cd. If H is varied backwards from -H to +H, the flux density curve follows a path

defa, which is similar to the curve abcd. The closed loop abcdef thus traced out is called the

hysteresis loop of the specimen. The term remnant flux density Br is also called retentivity and

the term coercive force is often called coercivity.

The shape of the hysteresis loop will depend upon the nature of magnetic material. Steel alloyed

with 4 % silicon has a very narrow hysteresis loop.

Hysteresis in magnetic materials results in dissipation of energy, which is proportional to the

area of the hysteresis loop. Hence the following conclusions can be drawn:

1. Flux density B always lags with respect to the magnetizing force H.


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2. An expenditure of energy is essential to carry the specimen through a complete cycle of

magnetization.

3. Energy loss is proportional to the area of hysteresis loop and depends upon the quality of

the magnetic material.

Figure 1.7 Hysteresis loop

1.3.8. Permeability

Permeability is the magnetic property that determines the characteristics of magnetic materials

and nonmagnetic materials. The permeability of free space and nonmagnetic materials has the

following symbol and constant value in SI units:

m/H104 70=

As we can see, the reluctance of magnetic materials is much lower than that of air or

nonmagnetic material. From the inverse relationship of reluctance and permeability, we

determine that the total permeability of magnetic materials is much greater than that of air.

However, the value of permeability varies with the degree of magnetization of the magnetic

material and, of course, the type of material. Since the permeability of magnetic materials r is

variable, we must employ magnetic saturation (B-H) curves to perform magnetic circuit

calculations. Permeability in magnetic circuits is somewhat analogous to conductivity in electric

circuits.


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1.3.8.1. Relative Permeability Transposition of Eq. (1.6) gives the absolute permeability as the ratio of the magnetic flux

density to the corresponding magnetic field intensity:

HB

= 1.6

Thus we can obtain the values of absolute permeability of ferromagnetic materials from the

magnetization (B-H) curves. Another method of obtaining the absolute permeability would be to

take the slope (differential) of the curve at various points.

If we wish to compare the permeability of magnetic materials with that of air, we may use the

relative permeability r, which is defined by the equation

0

=r 1.7

Where = absolute permeability of the material. H/m

0 = 410-7H/m = permeability of free space

r = relative permeability

From the typical magnetization curves of Figure 1.6, we can calculate the value of absolute and

relative permeabilities for any magnetic operating condition. When we do this we observe that

the value of relative permeability is not a constant but obtains a maximum value at about the

knee of the B-H curve.

2. MAGNETIC CIRCUITS

A toroid of homogeneous magnetic material, such as iron or steel, is wound with a fixed number

of turns of insulated wire as shown in Figure 1.8. The magnetic flux () and the excitation

current (I) are related by Eq. (1.6):


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Figure 1.8: Toroid coil.

A

HB ==

Thus

lNI

A=

that is, = (constant) I

where the constant is NA/l. At the outset, the sample of ferromagnetic material in the toroid was

totally demagnetized. In experimental measurements, the excitation current is varied and the

corresponding values of magnetic flux recorded. Then the calculated values of B and H are

plotted on linear scales as illustrated in Figure 1.7.

2.1. Electric Circuit Analogs

In our discussion so far, we note the following analogous relationships between magnetic

quantities and electric quantities:

Electric circuit Magnetic circuit

E (volts)

I (amperes)

R (ohms)

)(1 yconductvit

=

Fm (NI ampere-turns)

(webers)

m (ampere-turns/weber)

(permeability)(henries/meter)

We can draw useful electrical analogs for the solution of magnetic circuit problems. In an

electrical circuit the driving force is the voltage, the output is the current, and the opposition to


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establishing current is the resistance. In the same way, the driving force in the magnetic circuit is

the magnetomotive force, the output is the magnetic flux, and opposition to establishing the flux

is the reluctance.

2.2. Series and Parallel Magnet Circuits

By definition, a series magnetic circuit contains magnetic flux, which is common throughout the

series magnetic elements. These series magnetic elements may consist of composite sectors of

ferromagnetic materials of different lengths and cross-sectional areas, and of air gaps. The

simplest series magnetic circuit would be of a toroid of homogeneous material and the steel core

of a transformer. More complex series circuits which contain air gaps .

Parallel magnetic circuits are defined by the number of paths that the magnetic flux may follow.

Any of these paths or branches may consist of composite sectors of magnetic materials,

including air gaps.

2.2.1. Series Magnet Circuit For the magnetic circuit of Figure 1.9a the analogous electric circuit and the analogous magnetic

circuit are in Figure 1.9b and c, respectively. The iron and air portions of the magnetic circuit are

analogous to the two series resistors of the electric circuit. Analogous to the electric circuit, the

magnetomotive force must overcome the magnetic potential drops of the two series reluctances

in accordance with Kirchhoff's voltage law applied to magnetic circuits.

Therefore, += magmironmF 1.8

is the equivalent magnetic-potential-drop equation. Since the permeability of ferromagnetic

materials (iron) is a variable depending on the state of magnetization, we must use the B-H

curves to obtain the magnetic field intensity if the magnetic flux density is available. Hence we

can calculate the MMF drop for the iron from Eq. (1.5) as follows:

ironironmiron lH=F [At(Amper-turns)] 1.9

Finally, the general MMF-drop equation for series magnetic circuits is modified for calculation

purposes to the following form:

ag0

agironironm A

llH

+=F 1.10


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+-

Rl

Rag

I

E

+

-NI

(a) Magnetic circuit; (b) analogous electric circuit (c) analogous magnetic circuit

Figure 1.9 Iron-core toroid with air gap

Given the physical parameters of the series magnetic circuit and the value of magnetic flux or

magnetic flux density, the required magnetomotive force can be calculated in a straightforward

manner using Eq. (1.10).

The general principles of electric circuits embodied in Ohm's and Kirchhotf's laws are applied as

analogous equivalents to parallel magnetic circuits. With the presence of air gaps, most complex

magnetic circuits are solved using the series-parallel equivalent analogs.In analogous

equivalents, Kirchhoff's current law for magnetic circuits states that the sum of magnetic fluxes

entering a junction or node is equal to the sum of magnetic fluxes leaving the junction or node.

2.2.2. Parallel Magnetic Circuit

Figure 1.10a shows a parallel magnetic circuit.There are NI ampere-turns on the center leg.The

flux that is produced by the MMF in the center leg exists in the center leg and then divides into

two parts, one going in the path afe and the other in the path bcd. If we assume for simplicitv that

afe = bcd, the flux is distributed evenly between the two paths. Now

g = afe + bcd 1.11

Where g = flux in portion g

afe = flux in portion afe

bcd = flux in portion bcd


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Equation (1.11) is actually the analog of Kirchhoff's current law, but now we can say that the

amount of flux entering a junction is equal to the amount of flux leaving the junction.

Another observation that we may make on this circuit is that the MMF drops around a circuit are

the same no matter what path we take. Thus the MMF drop around afe must be equal to the

MMF drop around bcd. This can be stated more precisely as

Hala + Hflf + Hele = Hblb + Hclc + Hdld 1.12

+

-

+

-

RgRcRf

Re Rd

Iafe Ibcd

Ig

(a) Magnetic circuit (b) equivalent magnetic circuit (c) analogous electric circuit

Figure 1-10 Magnetic circuit with center leg:

The drop in MMF around either path afe or bcd must also be equal to the MMF drop along path

g. But g also has an "active source," the NI ampere-turns of the coil. The actual MMF existing

between X and Y is the driving force NI minus the drop Hglg in path g. Then we can write

(NI - Hglg) = Hala + Hflf + Hele 1.13

= Hblb + Hclc + Hdld

Again we can draw analogous magnetic and electrical circuits as in Figure 1.10b and c. For

Figure 1.l0b we may write

NI - mgg = bcd (mb + mc + md ) 1.14

= afe (ma + mf + me )

and in Figure 1.l0c we may write

E - RgIg = Ibcd (Rb + Rc + Rd ) 1.15

= Iafe (Ra + Rf + Re )

In the analogous magnetic circuit, note that NI is drawn in series with Rmg, although physically

the coil surrounds the central magnetic path.


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CHAPTER TWO

2. TRANSFORMERS

2.1. INTRODUCTION The transformer is a static device that transfers electrical energy from one electrical circuit to

another electrical circuit through the medium of magnetic field and without a change in the

frequency. The electric circuit which receives energy from the supply mains is called primary

winding and the other circuit which delivers electrical energy to the load is called secondary

winding.

Actually the transformer is an electric energy conversion device, since the energy received by the

primary is converted to useful electrical energy in the other circuits (secondary winding circuit).

If the secondary winding has more turns than the primary winding, then the secondary voltage is

higher than the primary voltage and the transformer is called a step-up transformer. When the

secondary winding has less turns than the primary windings then the secondary voltage is lower

than the primary voltage and the transformer is called step down transformer.

Note that a step-up transformer can be used as a step-down transformer, in which the secondary

of step-up transformer becomes the primary of the step-down transformer. Actually a

transformer can be termed a step-up or step-down transformer only after it has been put into

service.

The most important tasks performed by transformers are:-

i. Changing voltage and current levels in electrical power systems

ii. Matching source and load impedances for maximum power transfer in electronic and

control circuit and

iii. Electrical isolation (isolating one circuit from another )

Transformers are used extensively in ac power systems. AC electrical power can be generated at

one central location, its voltage stepped up for transmission over long distances at very low

losses and its voltage stepped down again for final use.

2.2. Transformer Construction

The construction of a simple two-winding transformer consists of each winding being wound on

a separate limb or core of the soft iron form which provides the necessary magnetic circuit. This


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magnetic circuit, know more commonly as the "transformer core" is designed to provide a path

for the magnetic field to flow around, which is necessary for induction of the voltage between

the two windings.

The two most common and basic designs of transformer construction are the core-type Transformer and the Shell-core Transformer.

In the core form transformer, the primary and secondary windings are wound outside and surround the core ring.

In the shell type transformer, the primary and secondary windings pass inside the steel magnetic circuit (core) which forms a shell around the windings.

2.2.1. Transformer Construction of the Core & Shell-Type

A transformer is dependent upon how the primary and secondary windings are wound around

the central laminated steel core. The two most common and basic designs of transformer

construction are the Closed-core Transformer and the Shell-core Transformer. In the "closed-

core" type (core form) transformer, the primary and secondary windings are wound outside and

surround the core ring. In the "shell type" (shell form) transformer, the primary and secondary

windings pass inside the steel magnetic circuit (core) which forms a shell around the windings as

shown below.

a. Transformer Core Construction

Figure 2.1: Transformer core and Shell type core constructions


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In both types of transformer core design, the magnetic flux linking the primary and secondary

windings travels entirely within the core with no loss of magnetic flux through air. In the core

type transformer construction, one half of each winding is wrapped around each leg (or limb) of

the transformers magnetic circuit as shown above.

The coils are not arranged with the primary winding on one leg and the secondary on the other

but instead half of the primary winding and half of the secondary winding are placed one over

the other concentrically on each leg in order to increase magnetic coupling allowing practically

all of the magnetic lines of force go through both the primary and secondary windings at the

same time. However, with this type of transformer construction, a small percentage of the

magnetic lines of force flow outside of the core, and this is called "leakage flux".

Shell type transformer cores overcome this leakage flux as both the primary and secondary

windings are wound on the same centre leg or limb which has twice the cross-sectional area of

the two outer limbs. The advantage here is that the magnetic flux has two closed magnetic paths

to flow around external to the coils on both left and right hand sides before returning back to the

central coils. This means that the magnetic flux circulating around the outer limbs of this type of

transfrmer construction is equal to /2. As the magnetic flux has a closed path around the coils,

this has the advantage of decreasing core losses and increasing overall efficiency.

b. Transformer Laminations

But you may be wondering as to how the primary and secondary windings are wound around

these laminated iron or steel cores for this types of transformer constuctions. The coils are firstly

wound on a former which has a cylindrical, rectangular or oval type cross section to suit the

construction of the laminated core. In both the shell and core type transformer constructions, in

order to mount the coil windings, the individual laminations are stamped or punched out from

larger steel sheets and formed into strips of thin steel resembling the letters "E's", "L's", "U's"

and "I's" as shown in figure 2.2.


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Figure 2.2:Transformer lamination types

These lamination stampings when connected together form the required core shape. For example,

two "E" stampings plus two end closing "I" stampings to give an E-I core forming one element

of a standard shell-type transformer core. These individual laminations are tightly butted together

during the transformers construction to reduce the reluctance of the air gap at the joints

producing a highly saturated magnetic flux density.

Transformer core laminations are usually stacked alternately to each other to produce an

overlapping joint with more lamination pairs being added to make up the correct core thickness.

This alternate stacking of the laminations also gives the transformer the advantage of reduced

flux leakage and iron losses. E-I core laminated transformer construction is mostly used in

isolation transformers, step-up and step-down transformers as well as auto transformers.

2.3. PRINCIPLE OF TRANSFORMER ACTION

V1P SN1 N2

Figure 2.3 Schematic diagram of a two-winding transformer

The primary winding P is connected to an alternating voltage source, therefore, an alternating

current Im starts flowing through N1 turns. The alternating mmf N1Im sets up an alternating flux

which is confined to the high permeability iron path as indicated in Figure 2.3. The alternating


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flux induces voltage E1 in the primary P and E2 in secondary S. If a load is connected across the

secondary, load current starts flowing.

2.4. IDEAL TWO-WINDING TRANSFORMER For a transformer to be an ideal one, the various assumptions are as follows

1. Winding resistances are negligible (no copper losses).

2. All the flux set up by the primary links the secondary windings i.e. all of the flux is

confined to the magnetic core.

3. An infinitely permeable core with no core losses (hysteresis and eddy current losses are

negligible).

4. The core has constant permeability, i.e. the magnetization curve for the core is linear.

2.4.1. EMF Equation of A Transformer Let the voltage V1 applied voltage primary be sinusoidal (or sine wave). Then the current Im and,

therefore, the flux will flow with the variations of Im . That is, the flux is in time phase with

the current Im and varies sinusoidally. Let sinusoidal variation of flux be expressed as

tSinm =

Where m is maximum of the magnetic flux in Weber and = 2f is the angular frequency in

rad/sec and f is the supply frequency in Hz. The emf e1 in volt, induced in the primary of N1 turns by the alternating flux is given by

)tsin(N

tCosNdtdNe

m

m

21

1

11

=

=

=

Its maximum value, E1max occurs when

2tSin is equal to 1.

mm NE = 11

and

=

211tsinmEe

The RMS value of the induced emf E1 in the primary winding is


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mm

mm

fN.fN

fNEE

==

==

11

11

1

4442

22

2 (2.1)

Since the primary winding resistance is negligible hence e1, at every instant, must be equal and opposite of V1. That is,

dtdNev == 111

or 11 EV =

The emf induced in the secondary is

)t(sinE

)t(insN

tcosNdtdNe

m

m

2

2

2

2

222

=

=

=

=

Rms value of emf E2 induced in secondary winding is given by

m

mm

fN.

fNE

E

=

==

2

22

2

444

22 (2.2)

2.4.2. Voltage Transformation Ratio From Eqs. (2.1) and (2.2), we get

kNN

EE

==2

1

2

1 (2.3)

The ratio is known as voltage transformation ratio.

i. If N2 > N1 i.e., K1, then the transformer is known as a step-down transformer.

Again in an ideal transformer

2211 IVIVVAOutputVAInput

==

and k1

VV

II

1

2

2

1 ==

Hence, the currents are in the inverse ratio of the (voltage) transformation ratio of Eq. (2.3).

Also, the ratio of mfNE

NE

== 22

2

1

1 and this shows that the emf per turn in each of the

windings is the same.

If impedance Z2 is connected to the secondary, the impedance Z1 seen at the primary satisfies:


22

=K2

2.5. EQUIVALENT CIRCUIT OF NONIDEAL TRANSFORMER A nonideal transformer differs from an ideal transformer in that the former has hysteresis and

eddy current (or core) losses, and has resistive (i2R) losses in its primary and secondary

windings. Furthermore, the core of a nonideal transformer is not perfectly permeable, and the

transformer core requires a finite mmf for its magnetization. Also, not all fluxes link with the

primary and secondary windings simultaneously because of leakages. Referring to Fig. 2-4, we

observe that R1 and R2 are the respective resistances of the primary and secondary windings. The

flux c which replaces the flux of Fig. 2.3, is called the core flux or mutual flux, as it links both the primary and secondary windings. The primary and secondary leakages fluxes are shown

as l1 and l2 respectively. Thus in Fig. 2.4 we have accounted for all the imperfections listed above, except the core losses. We will include the core losses as well as the rest of the

imperfections in the equivalent circuit of a nonideal transformer. This circuit is also known as the

exact equivalent circuit, as it differs from the idealized equivalent circuit and the various

approximate equivalent circuits. We now proceed to derive these circuits.

Figure 2.4: Circuit diagram of two port single phase transformer

An equivalent circuit of an ideal transformer is shown in Fig. 2.5(a). When the nonideal effects

of winding resistances, leakage reactance, magnetizing reactance, and core losses are included,

the circuit of Fig. 2.5(a) is modified to that of Fig. 2.5(b), where the primary and the secondary

are coupled by an ideal transformer. By using , and

, the ideal transformer may be removed from Fig. 2.5(b) and the entire


23

equivalent circuit may be referred either to the primary, as shown in Fig. 2.6(a), or to the

secondary, as shown in Fig. 2.6(b).

Figure 2.5: Exact equivalent circuit of ideal & nonideal transformers

Figure 2.6: Equivalent circuits of a nonideal transformer referred to primary or secondary.

A phasor diagram for the circuit Fig. 2.6(a), for lagging power factor, is shown in Fig. 2.7.

Fig. 2-7: Phasor diagram corresponding to Fig. 2.6(a).

In Figs.2.5, 2.6, and 2.7 the various symbols are:

a=k= turns ratio (> 1 ) E1 = primary induced voltage E2 = secondary induced voltage


24

V1 = primary terminal voltage V2 = secondary terminal voltage I1 = primary current I2 = secondary current I0 = no-load (primary) current R1 = resistance of the primary winding R2 = resistance of the secondary winding X1 =primary leakage reactance X2 = secondary leakage reactance Im, X m = magnetizing current and reactance Ie, Rc = current and resistance accounting for the core losses

Approximate Equivalent circuits of a Transformer:

The derived equivalent circuit is detailed but it is considered to be too complex for practical

engineering applications. The main problem in calculations will be the excitation and the eddy

current and hysteresis loss representation adds an extra branch in the calculations.

In practical situations, the excitation current will be relatively small as compared to the load

current, which makes the resultant voltage drop across Rp and Xp to be very small, hence Rp and

Xp may be lumped together with the secondary referred impedances to form and equivalent

impedance. In some cases, the excitation current is neglected entirely due to its small magnitude.

(a) Referred to the primary with no excitation (b) Referred to the secondary with no excitation

Figure 2.8: Approximate equivalent circuits of a transformer

2.6. OPEN-CIRCUIT AND SHORT-CIRCUIT TESTS These two tests on a transformer help to determine

i. The parameters of the equivalent circuit of Figure 2.8


25

ii. the voltage regulation and

iii. efficiency

The equivalent circuit parameters can also be obtained from the physical dimensions of the

transformer core and its winding details. Complete analysis of the transformer can be carried out,

once its equivalent circuit parameters are known. The power required during these two tests is

equal to the appropriate power loss occurring in the transformer.

2.7. Transformer Efficiency A transformer does not require any moving parts to transfer energy. This means that there are no

friction or windage losses associated with other electrical machines. However, transformers do

suffer from other types of losses called "copper losses" and "iron losses" but generally these are

quite small. The resulting efficiency of a transformer is equal to the ratio of the power output of

the secondary winding (PS) to the power input of the primary winding (PP).

Thus

powerInputpowerOutput

=Efficiency

RI+P+cosIV

cosIV=

22

c222

222

2.4

Where Pc = total core loss

I22R = total ohmic losses V2I2 = output VA Cos 2 = load power factor

Since the efficiencies of power and distribution transformers are usually very high, it is therefore,

more accurate to determine the efficiency from measurement of losses than from the

measurement of output.

An ideal transformer is 100% efficient because it delivers all the energy it receives. Real

transformers on the other hand are not 100% efficient and at full load, the efficiency of a

transformer is between 94% to 96% which is quiet good. For a transformer operating with a

constant voltage and frequency with a very high capacity, the efficiency may be as high as 98%.

2.8. VOLTAGE REGULATION OF A TRANSFORMER

Constant voltage is the characteristics of most domestic, commercial and industrial loads. It is

therefore, necessary that the output voltage of a transformer must remain within narrow limits as


26

the load and its power factor vary. This requirement is more stringent in distribution transformers

as these directly feed the load centers. The voltage drop in a transformer on load is chiefly

determined by its leakage reactance which must be kept as low as design and manufacturing

techniques would permit.

The voltage regulation is defined as voltage in secondary terminal voltage, expressed as a

percentage (per unit) of secondary rated voltage i.e.

u.pinvoltageratedondarysec

VEregulationVoltage 22 =

where E2 = Secondary terminal voltage at no load

V2 = Secondary terminal voltage at any load

It is stipulated that the secondary rated voltage of a transformer is equal to the secondary

terminal voltage at no load, i.e. E2.

percentageinE

VEu.pinE

VEregulationVoltage 1002

22

2

22

=

=

At no-load, the primary leakage impedance drop is almost negligible, therefore, the secondary

no-load voltage1

212 N

NVE = . The expression for voltage regulation can also be written as

percentageinV

NNVV

percentagein

NNV

VNNV

1001001

2

121

1

21

21

21

=

Here V1 is the primary applied voltage.

The change in secondary terminal voltage with load current is due to the primary and secondary

leakage impedances of the transformer. The magnitude of this change depends on the load power

factor, load current, total resistance and leakage reactance of a transformer.

2.9. THREE-PHASE TRANSFORMERS

Generation, transmission and distribution of electric energy is invariably done through the use of

three-phase systems because of its several advantages over single-phase systems. As such, a

large number of three-phase transformers are inducted in a 3-phase energy system for stepping-

up or stepping down the voltage as required. For 3-phase up or down transformation, three

units of 1-phase transformers or one unit of 3-phase transformer may be used. When three


27

identical units of 1-phase transformers are used as shown in Figure 2.9(a), the arrangement is

usually called a bank of three transformers or a 3-phase transformer bank. A single 3-phase

transformer unit may employ 3phase core-type construction Figure 2.9(b) or three phase shell

type construction.

P S P S P S

A B C

a b c

Input

Output

I IIIII

P

S

P

S

P

S

(a) (b)

Figure 2.9 (a) 3-phase transformer both windings in star; (b) three-phase core-type transformer

A single-unit 3-phase core-type transformer uses a three-limbed core, one limb for each phase

winding as shown in Figure 2.9(b). Actually, each limb has the L.V. winding placed adjacent to

the laminated steel core and then H.V. winding is placed over the 1.v. winding. Appropriate

insulation is placed in between the core and 1.v. winding and also in between the two windings.

A 3-phase core-type transformer costs about 15% less than a bank of three 1-phase transformers.

Also, a single unit occupies less floor space than a bank.

2.9.1. Three-Phase Transformer Connections

Three-phase transformers may have four standard connections

(a) Star-Delta ( Y-) (b) Delta-Star (-Y)

(c) Delta-Delta (-) (d) Star-Star (Y-Y)

These connections are shown in Figures 2.10 and 2.11, where V and I are taken as input line

voltage and line current respectively. Primary and secondary windings of one phase are drawn

parallel to each other. With phase turns ratio from primary to secondary as


28

N1/N2= a, the voltages and current in the windings and lines are shown in Figures 2.10 and 2.11.

The various connections are now described briefly.

A. Star-delta (Y-) Connection This connection is commonly used for stepping down the voltage

from a high level to a medium or low level. The insulation on the h.v. side of the transformer is

stressed only to 57.74% voltagelinetolineof100x3

1

=

For per-phase m.m.f. balance, I2N2 =I1N1

Here primary phase current, I1 = primary line current I

aI.3I3currentlineSecondary

aIINNI,currentphaseSecondary

2

12

12

==

==

Also, voltage per turn on primary = voltage per turn on secondary

2

2

1

1.3 N

VN

V=

Secondary phase voltage, 3.3

.1

22 a

VVNNV ==

Secondary line voltage = secondary phase voltage = 3.a

V

Input VA = 3 .3

V I = output VA = 3. VIaIa

V 3.3.

=

Phase and line values for voltages and currents on both primary and secondary sides of star-delta

transformer are shown in Figure 2.10(a)


29

VV VIV Figure 2.10 (a) Star-delta connection and (b) delta-star connection of 3-phase transformers (b) Delta-Star (-Y) connection:- This type of connection is used for stepping up the voltage to a high level. For example, these are used in the beginning of h.v. transmission lines so that insulation is stressed to about 57.74% of line voltage Delta-star transformers are also generally used as distribution transformers for providing mixed line to line voltage to high-power equipment and line to neutral voltage to 1-phase low-power equipment. For example, 11kV/400V, delta-star distribution transformer is used to distribute power to consumers by 3-phase four-wire system. Three-phase highpower equipment is connected to 400V, three line wires, whereas 1-phase low-power equipment is energized from 231 V line to neutral circuits. VI33Ia.aV.3VAOutputVI33I.V.3VAInput aV.3voltagelineSecondary aVV.NNV,voltagephaseSecondary NVNVAlso 31aINNI,currentphaseSecondary )Icurrentlineprimary(31I,currentphaseprimaryHere NINI,balance.f.m.mphaseperFor 1122 1122 12121 1122 ===== = === === = Phase and line values for voltages and currents on primary as well as secondary sides of a 3-phase delta-star transformer are shown in Figure 2.10(b).


30

(c) Delta-Delta (-) Connection

This scheme of connections is used for large 1.v transformers. It is because a delta-connected

winding handles line voltage, so it requires more turns per phase but of smaller cross-sectional

area. The absence of star point may be a disadvantage in some applications.

In case a bank of three transformers is used, then one transformer can be removed for

maintenance purposes while the remaining two transformers (called an open-delta or V-

connection) can still deliver 58% of the power delivered by the original 3-phase transformer

bank.

For per phase mmf balance, I2N2 = I1N1:

aI3

aI3,currentlineSecondary

3

aIINNI,currentphaseSecondary

)Icurrentlineprimary(3

1I,currentphaseprimary

12

12

1

=

=

==

=

VI33

aI,aV.3VAoutput

3I.V3VAInput

aVVvoltagelineSecondary

)VVHere(aVV

NNV,voltagephaseSecondary

N

VNVAlso

2

111

22

1

1

2

2

====

==

===

==

Phase and line value for voltages and currents on both primary and secondary sides of a 3-phase

delta-delta transformer are shown in Figure 2.11 (a).


31

IVIkkV VVIVV Figure 2.11 (a) Delta-delta connection and (b) Star-star connection of three-phase transformers. (d) Star-Star (Y-Y) Connection This connection is used for small h.v transformers . As stated before, with star connection, turns per phase are minimum and the winding insulation is stressed to 57.74% of line voltage. Star-star connection is rarely used in practice because of oscillatory neutral problems. VI3aI.a.3V.3VAoutputI3V.3VAInput aVa.3V.3V3voltagelineSecondary a3VVNNV,voltagephaseondarysec currentlineSecondaryaIINNI,currentphaseSecondary I,currentlineprimaryI,CurrentPhaseimaryPr NINI,balance.f.m.mphaseperFor 2 1122 1212 1 1122 ==== === === == = = Phase and line values of voltages and currents on both sides of a star-star transformer are shown in Figure 2.11(b).


32

Three-phase Voltage and Current

Connection Phase Voltage Line Voltage Phase Current Line Current

Star VP = VL 3 VL = 3 VP IP = IL IL = IP

Delta VP = VL VL = VP IP = IL 3 IL = 3 IP

Other possible connections for three phase transformers are star-delta Yd, where the primary

winding is star-connected and the secondary is delta-connected or delta-star Dy with a delta-

connected primary and a star-connected secondary.

Three-phase Transformer Line Voltage and Current

Primary-Secondary

Configuration Line Voltage Line Current

Delta Delta

Delta Star

Star Delta

Star Star

Nature of Magnetism2.2. Transformer Construction2.2.1. Transformer Construction of the Core & Shell-TypeA transformer is dependent upon how the primary and secondary windings are wound around the central laminated steel core. The two most common and basic designs of transformer construction are the Closed-core Transformer and the Shell-core Transformer...a. Transformer Core Construction

b. Transformer Laminations2.7. Transformer EfficiencyFigure 2.9 (a) 3-phase transformer both windings in star; (b) three-phase core-type transformer

Three-phase transformers may have four standard connectionsPhase and line values of voltages and currents on both sides of a star-star transformer are shown in Figure 2.11(b).Three-phase Voltage and CurrentThree-phase Transformer Line Voltage and Current

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