Transformer

T.C

SÜLEMAN DEMİREL UNIVERSITY

FEN BİLİMLERİ ENSTİTÜSÜ/MÜHENDİSLİK FAKÜLTESİ

ELEKTRONİK VE HABERLEŞME MÜHENDİSLİĞİ

COURSE SUBJECT

ELECTROMAGNETIC WAVES THEORY OFFERED BY

Prof. Dr. MUSTAFA MERDAN

HOME WORK:

TRANSFORMER

SUBMITTED BY:

KHALID SAEED LATEEF AL-BADRI

1

Table of content:

no Subject Page

1 Introduction 2

2 Electromagnetic induction 3

3 History 3

4 Magnetic units 4

5 Definition of transformer 6

6 Basic principles 7

7 Behavior of magnetic materials 10

8 Diamagnetic 10

9 Paramagnetism 11

10 Ferromagnetism 12

11 Magnetic core 14

12 Cores structures 19

13 Transformer losses 22

14 Heat transfer effects 29

15 The magnetic circuit 30

16 Appendix Maxwell’s equations 32

17 References 33

2

INTRODACTION

There are many kinds of transformers, and all share the

same set of fundamental operating principles. Since this

project focuses on principle of transformers, and answer the

first equation , ‘‘What exactly is a transformer?’’ . and

describing the underlying physics behind transformer

operation. A theoretical foundation is absolutely necessary

in order to understand what is going on inside a transformer

and why. The magnetic properties of materials, a review of

magnetic units, and analysis of magnetic circuits to gain full

comprehension of the physics involved.

Then we will show that (i) a time varying field will cause

eddy currents to be induced in the core causing power loss

and (ii) hysteresis effect of the material also causes

additional power loss called hysteresis loss. The effect of

both the losses will make the core hotter. We must see that

these two losses, (together called core loss) are kept to a

minimum in order to increase efficiency of the apparatus

such as transformers & rotating machines, where the core

of the magnetic circuit is subjected to time varying field. If

we want to minimize something we must know the origin

and factors on which that something depends. In the

following sections we first discuss eddy current

phenomenon and then the phenomenon of hysteresis.

3

ELECTROMAGNETIC INDUCTION

Is the production of an electromotive force across a conductor when it is

exposed to a varying magnetic field. It is described mathematically

by Faraday's law of induction, named after Michael Faraday who is

generally credited with the discovery of induction in 1831.

HISTORY

Electromagnetic induction was discovered independently by Michael

Faraday in 1831 and Joseph Henry in 1832. Faraday was the first to

publish the results of his experiments. In Faraday's first experimental

demonstration (August 29, 1831), he wrapped two wires around

opposite sides of an iron ring or "torus" (an arrangement similar to a

modern toroidal transformer). Based on his assessment of recently

discovered properties of electromagnets, he expected that when current

started to flow in one wire, a sort of wave would travel through the ring

and cause some electrical effect on the opposite side. He plugged one

wire into a galvanometer, and watched it as he connected the other wire

to a battery. Indeed, he saw a transient current (which he called a "wave

of electricity") when he connected the wire to the battery, and another

when he disconnected it. This induction was due to the change in

magnetic flux that occurred when the battery was connected and

disconnected. Within two months, Faraday found several other

manifestations of electromagnetic induction. For example, he saw

transient currents when he quickly slid a bar magnet in and out of a coil

of wires, and he generated a steady (DC) current by rotating a copper

disk near the bar magnet with a sliding electrical lead ("Faraday's disk").

4

Faraday explained electromagnetic induction using a concept he

called lines of force. However, scientists at the time widely rejected his

theoretical ideas, mainly because they were not formulated

mathematically. An exception was Maxwell, who used Faraday's ideas

as the basis of his quantitative electromagnetic theory. In Maxwell's

model, the time varying aspect of electromagnetic induction is expressed

as a differential equation which Oliver Heaviside referred to as Faraday's

law even though it is slightly different from Faraday's original formulation

and does not describe motional EMF. Heaviside's version (see Maxwell–

Faraday equation below) is the form recognized today in the group of

equations known as Maxwell's equations.

Heinrich Lenz formulated the law named after him in 1834, to describe

the "flux through the circuit". Lenz's law gives the direction of the

induced EMF and current resulting from electromagnetic induction

(elaborated upon in the examples below).

Following the understanding brought by these laws, many kinds of

device employing magnetic induction have been invented.

MAGNETIC UNITS[1]

The basic operation of all transformers is deeply rooted in

electromagnetics, whether or not the transformer has a magnetic iron

core. There are three basic systems of measurement used in

engineering: English, MKS (meter-kilogram-second), and cgs

(centimeter-gram-second) also called SI unite. To make matters worse,

some transformer textbooks even mix English units with cgs or MKS

units. For consistency and ease of understanding, we will use MKS

units. The fist magnetic quantity is the magnetomotive force (MMF). In

electrical terms, MMF is roughly equivalent to the electromotive force

(EMF), that causes current to flow in an electrical circuit. The units and

conversion factors for MMF are

MKS: ampere-turn

cgs: gilbert

1 Gb = 0.4π amp-turn

The next magnetic quantity is flux, represented by the Greek letter φ.

Since a magnetic field can be visualized as a bundle of lines flowing

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from a north magnetic pole to a south magnetic pole, flux is the total

number of ‘‘lines.’’ The units and conversion factors of φ are

MKS: weber

cgs: maxwell

1 Ma =1 line = 10-8

Wb

The magnetic flux density B is the concentration of magnetic of lines

across an area. The units and conversion factors for B are

MKS: tesla

cgs: gauss

1 G = 10-4

T

1 T = 1 Wb/m2

The magnetic field intensity H is the distribution of MMF along a

magnetic path. If the flux density is constant, H is merely the total MMF

divided by the length of the magnetic path. The units and conversion

factors for H are

MKS: amp-turns/meter

cgs: oersted

1 Oe =(250/π) amp-turns/m

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DEFINITION OF TRANSFORMER[2]

A transformer is a static device which transforms electrical energy from

one circuit to another without any direct electrical connection and with

the help of mutual induction between two windings. It transforms power

from one circuit to another

without changing its frequency

but may be in different

voltage level. This is a very

short and simple definition of

transformer. Or a transformer

is an electrical device that

transfers energy between two

or more circuits through

electromagnetic induction.

A varying current in the

transformer's primary winding

creates a varying magnetic flux in the core and a varying magnetic field

impinging on the secondary winding. This varying magnetic field at the

secondary induces a varying electromotive force (EMF) or voltage in the

secondary winding. Making use of Faraday's Law in conjunction with

high magnetic permeability core properties, transformers can thus be

designed to efficiently change AC voltages from one voltage level to

another within power networks.

Transformers range in size from RF transformers less than a cubic

centimeter in volume to units interconnecting the power grid weighing

7

hundreds of tons. A wide range of transformer designs are encountered

in electronic and electric power applications. Since the invention in 1885

of the first constant potential transformer, transformers have become

essential for the AC transmission, distribution, and utilization of electrical

energy.

BASIC PRINCIPLES[1]

It is very common, for simplification or approximation purposes, to

analyze the transformer as an ideal transformer model as represented in

the two images. An ideal

transformer is a theoretical

, linear transformer that is

lossless and perfectly coupled;

that is, there are no energy

losses and flux is completely

confined within the magnetic

core. Perfect coupling implies

infinitely high core magnetic

permeability and winding inductances and zero net magneto motive

force.

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A varying current in the transformer's primary winding creates a varying

magnetic flux in the core and a varying magnetic field impinging on the

secondary winding. This varying magnetic field at the secondary induces

a varying electromotive force(EMF) or voltage in the secondary winding.

The primary and secondary windings are wrapped around a core of

infinitely high magnetic permeability[d] so that all of the magnetic flux

passes through both the primary and secondary windings with a voltage

connected to the primary winding and load impedance connected to the

secondary winding the transformer currents flow in the indicated

directions. (See also Polarity.)

According to Faraday's law of induction, since the same magnetic flux

passes through both the primary and secondary windings in an ideal

transformer, a voltage is induced in each winding[3]

, according to eq. (1)

in the secondary winding case, according to eq. (2) in the primary

winding case.

The primary EMF is sometimes termed counter EMF This is in

accordance with Lenz's law, which states that induction of EMF always

opposes development of any such change in magnetic field.

The transformer winding voltage ratio is thus shown to be directly

proportional to the winding turns ratio according to eq. (3).

According to the law of Conservation of Energy(In physics, the law of

conservation of energy states that the total energy of an isolated system

(1)

(2)

(3)

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cannot change—it is said to be conserved over time. Energy can be

neither created nor destroyed, but can change form, for instance

chemical energy can be converted to kinetic energy in the explosion of a

stick of dynamite.

A consequence of the law of conservation of energy is that a perpetual

motion machine of the first kind cannot exist. That is to say, no system

without an external energy supply can deliver an unlimited amount of

energy to its surroundings.) any load impedance connected to the ideal

transformer's secondary winding results in conservation of apparent, real

and reactive power consistent with eq. (4).

The ideal transformer identity shown in eq. (5) is a reasonable

approximation for the typical commercial transformer, with voltage ratio

and winding turns ratio both being inversely proportional to the

corresponding current ratio.

By Ohm's Law and the ideal transformer identity:

the secondary circuit load impedance can be expressed as eq. (6)

the apparent load impedance referred to the primary circuit is derived in

eq. (7) to be equal to the turns ratio squared times the secondary circuit

load impedance

(4)

(5)

(6)

(7)

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BEHAVIOR OF MAGNETIC MATERIALS[7]

In Eq. ( M=Xm H), we describe the macroscopic magnetic property of a

linear. isotropic medium) defining the magnetic susceptibility Xm. which is

unit less. The magnetic susceptibility and the relative permeability are

related as follows:

Magnetic material can be roughly classified into three maim group in

accordance with the μr values.

Diamagnetic, if μ r≤ 1 (Xm is a very small negative number).

Paramagnetic. if μ r≥ 1 (Xm is a very small positive number).

Ferromagnetic, if μr >> 1(Xm is a large positive number).

DIAMAGNETISM

Diamagnetic materials create an

induced magnetic field in a direction

opposite to an externally applied

magnetic field, and are repelled by

the applied magnetic field. In

contrast, the opposite behavior is

exhibited by paramagnetic

materials. Diamagnetism is a

quantum mechanical effect that

occurs in all materials; when it is the

only contribution to the magnetism

the material is called a diamagnet.

Unlike a ferromagnet, a diamagnet

is not a permanent magnet. Its

magnetic permeability is less than μ0

(the permeability of free space). In most materials diamagnetism is a

weak effect, but a superconductor repels the magnetic field entirely,

apart from a thin layer at the surface. Diamagnetic materials, like water,

or water based materials, have a relative magnetic permeability that is

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less than or equal to 1, and therefore a magnetic susceptibility less than

or equal to 0, since susceptibility is defined as

χm = μr − 1.

This means that diamagnetic materials are repelled by magnetic fields.

However, since diamagnetism is such a weak property its effects are not

observable in everyday life. For example, the magnetic susceptibility of

diamagnets such as water is χm = −9.05×10−6

. The most strongly

diamagnetic material is bismuth, χm = −1.66×10−4

, although pyrolytic

carbon may have a susceptibility of χm = −4.00×10−4

in one plane.

Nevertheless, these values are orders of magnitude smaller than the

magnetism exhibited by paramagnets and ferromagnets. Note that

because χm is derived from the ratio of the internal magnetic field to the

applied field, it is a dimensionless value.

All conductors exhibit an effective diamagnetism when they experience a

changing magnetic field. The Lorentz force on electrons causes them to

circulate around forming eddy currents. The eddy currents then produce

an induced magnetic field opposite the applied field, resisting the

conductor's motion.

PARAMAGNETISM

is a form of magnetism whereby certain materials are attracted by an

externally applied magnetic field, and form internal, induced magnetic

fields in the direction of the applied magnetic field. In contrast with this

behavior, diamagnetic materials are repelled by magnetic fields and form

induced magnetic fields in the direction opposite to that of the applied

magnetic field. Paramagnetic materials include most chemical elements

and some compounds; they have a relative magnetic permeability

greater than or equal to 1 ( μ r≥ 1 ) (i.e., a positive magnetic

susceptibility) and hence are attracted to magnetic fields. The magnetic

moment induced by the applied field is linear in the field strength and

rather weak. It typically requires a sensitive analytical balance to detect

the effect and modern measurements on paramagnetic materials are

often conducted with a SQUID magnetometer.

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Paramagnetic materials have a small, positive susceptibility to magnetic

fields. These materials are slightly attracted by a magnetic field and the

material does not retain the magnetic properties when the external field

is removed. Paramagnetic properties are due to the presence of some

unpaired electrons, and from the realignment of the electron paths

caused by the external magnetic field. Paramagnetic materials include

magnesium, molybdenum, lithium, and tantalum.

Unlike ferromagnets, paramagnets do not retain any magnetization in

the absence of an externally applied magnetic field because thermal

motion randomizes the spin orientations. Some paramagnetic materials

retain spin disorder at absolute zero, meaning they are paramagnetic in

the ground state. Thus the total magnetization drops to zero when the

applied field is removed. Even in the presence of the field there is only a

small induced magnetization because only a small fraction of the spins

will be oriented by the field. This fraction is proportional to the field

strength and this explains the linear dependency. The attraction

experienced by ferromagnetic materials is non-linear and much stronger,

so that it is easily observed, for instance, by the attraction between a

refrigerator magnet and the iron of the refrigerator itself.

FERROMAGNETISM

Is the basic mechanism by

which certain materials (such as

iron) form permanent magnets,

or are attracted to magnets. In

physics, several different types

of magnetism are distinguished.

Ferromagnetism (including

ferrimagnetism) is the strongest

type: it is the only one that

typically creates forces strong

enough to be felt, and is

responsible for the common

phenomena of magnetism

encountered in everyday life.

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Substances respond weakly to magnetic fields with three other types of

magnetism, paramagnetism, diamagnetism, and antiferromagnetism, but

the forces are usually so weak that they can only be detected by

sensitive instruments in a laboratory. An everyday example of

ferromagnetism is a refrigerator magnet used to hold notes on a

refrigerator door. The attraction between a magnet and ferromagnetic

material is "the quality of magnetism first apparent to the ancient world,

and to us today".

Permanent magnets (materials that can be magnetized by an external

magnetic field and remain magnetized after the external field is

removed) are either ferromagnetic or ferrimagnetic, as are other

materials that are noticeably attracted to them. Only a few substances

are ferromagnetic. The common ones are iron, nickel, cobalt and most of

their alloys, some compounds of rare earth metals, and a few naturally-

occurring minerals such as lodestone.

Ferromagnetism is very important in industry and modern technology,

and is the basis for many electrical and electromechanical devices such

as electromagnets, electric motors, generators, transformers, and

magnetic storage such as tape recorders, and hard disks.

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MAGNETIC CORE

A magnetic core is a piece of magnetic material with a high

permeability(permeability[4]

is the measure of the ability of a material to

support the formation of a magnetic field within itself. In other words, it is

the degree of magnetization that a material obtains in response to an

applied magnetic field. Magnetic permeability is typically represented by

the Greek letter μ. The term was coined in September 1885 by Oliver

Heaviside. The reciprocal of magnetic permeability is magnetic

reluctivity. In SI units, permeability is measured in henries per meter

(H·m−1

), or newtons per ampere squared (N·A−2

). The permeability

constant (μ0), also known as the magnetic constant or the permeability of

free space, is a measure of the amount of resistance encountered when

forming a magnetic field in a classical vacuum. The magnetic constant

has the exact (defined) value µ0 = 4π×10−7

H·m−1

≈

1.2566370614…×10−6

H·m−1

or N·A−2

).) used to confine and guide

magnetic fields in electrical, electromechanical and magnetic devices

such as electromagnets, transformers, electric motors, generators,

inductors, magnetic recording heads, and magnetic assemblies. It is

made of ferromagnetic metal such as iron, or ferromagnetic compounds

such as ferrites. The high permeability, relative to the surrounding air,

causes the magnetic field lines to be concentrated in the core material.

The magnetic field is often created by a coil of wire around the core that

carries a current. The presence of the core can increase the magnetic

field of a coil by a factor of several thousand over what it would be

without the core.

The use of a magnetic core can enormously concentrate the strength

and increase the effect of magnetic fields produced by electric currents

and permanent magnets. The properties of a device will depend crucially

on the following factors:

the geometry of the magnetic core.

the amount of air gap in the magnetic circuit.

the properties of the core material (especially permeability and

hysteresis).

the operating temperature of the core.

whether the core is laminated to reduce eddy currents.

15

In many applications it is undesirable for the core to retain magnetization

when the applied field is removed. This property, called hysteresis can

cause energy losses in applications such as transformers. Therefore

'soft' magnetic materials with low hysteresis, such as silicon steel, rather

than the 'hard' magnetic materials used for permanent magnets, are

usually used in cores.

SOFT IRON

"Soft" (annealed) iron is used

in magnetic assemblies,

electromagnets and in some

electric motors; and it can

create a concentrated field

that is as much as 50,000

times more intense than an

air core. Iron is desirable to

make magnetic cores, as it

can withstand high levels of

magnetic field without

saturating (up to 2.16 teslas )

It is also used because, unlike

"hard" iron, it does not remain

magnetised when the field is

removed, which is often

important in applications

where the magnetic field is

required to be repeatedly

switched.Unfortunately, due

to the electrical conductivity of

the metal, at AC frequencies

a bulk block or rod of soft iron

can often suffer from large

eddy currents circulating

within it that waste energy

and cause undesirable

heating of the iron.

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LAMINATED SILICON STEEL

Because iron is a relatively good conductor, it cannot be used in bulk

form with a rapidly changing field, such as in a transformer, as intense

eddy currents would appear due to the magnetic field, resulting in huge

losses (this is used in induction heating).Two techniques are commonly

used together to increase the resistivity of iron: lamination and alloying

of the iron with silicon.

CARBONYL IRON

Powdered cores made

of carbonyl iron, a

highly pure iron, have

high stability of

parameters across a

wide range of

temperatures and

magnetic flux levels,

with excellent Q factors

between 50 kHz and

200 MHz. Carbonyl iron

powders are basically

constituted of micrometer-size spheres of iron coated in a thin layer of

electrical insulation. This is equivalent to a microscopic laminated

magnetic circuit (see silicon steel, above), hence reducing the eddy

currents, particularly at very high frequencies. A popular application of

carbonyl iron-based magnetic cores is in high-frequency and broadband

inductors and transformers.

IRON POWDER

Powdered cores made of hydrogen

reduced iron have higher permeability

but lower Q. They are used mostly for

electromagnetic interference filters

and low-frequency chokes, mainly in

switched-mode power supplies.

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FERRITE

A ferrite is a type of

ceramic compound

composed of iron oxide

(Fe2O3) combined

chemically with one or

more additional metallic

elements. They are

ferrimagnetic, meaning

they can be magnetized

or attracted to a magnet,

and are electrically

nonconductive, one of

the few substances that combine these two properties. Ferrites can be

divided into two families based on their magnetic coercivity, their

resistance to being demagnetized. Hard ferrites have high coercivity;

they are difficult to demagnetize. They are used to make magnets, for

devices such as refrigerator magnets, loudspeakers and small electric

motors. Soft ferrites have low coercivity. They are used in the electronics

industry to make ferrite

cores for inductors and

transformers, and in

various microwave

components. Yogoro

Kato and Takeshi Takei

of the Tokyo Institute of

Technology invented

ferrite in 1930.Ferrite

ceramics are used for

high-frequency

applications. The ferrite

materials can be

engineered with a wide

range of parameters. As ceramics, they are essentially insulators, which

prevents eddy currents, although losses such as hysteresis losses can

still occur.

18

VITREOUS METAL

Amorphous metal (also known metallic glass or glassy metal) is a solid

metallic material, usually an alloy, with a disordered atomic-scale

structure. Most metals are crystalline in their solid state, which means

they have a highly ordered arrangement of atoms. Amorphous metals

are non-crystalline, and have a glass-like structure. But unlike common

glasses, such as window-glass, which are typically insulators,

amorphous metals have good electrical conductivity. There are several

ways in which amorphous metals can be produced, including extremely

rapid cooling, physical vapor deposition, solid-state reaction, ion

irradiation, and mechanical alloying.

More recently, batches of amorphous steel have been produced that

demonstrate strengths much greater than conventional steel alloys is a

variety of alloys that are non-crystalline or glassy. These are being used

to create high-efficiency transformers. The materials can be highly

responsive to magnetic fields for low hysteresis losses, and they can

also have lower conductivity to reduce eddy current losses. China is

currently making widespread industrial and power grid usage of these

transformers for new installations. Currently the most important

application is due to the special magnetic properties of some

ferromagnetic metallic glasses. The low magnetization loss is used in

high efficiency transformers (amorphous metal transformer) at line

frequency and some higher frequency transformers. Amorphous steel is

a very brittle material which makes it difficult to punch into motor

laminations. Also electronic article surveillance (such as theft control

passive ID tags,) often uses metallic glasses because of these magnetic

properties.

AIR

A coil not containing a magnetic core

is called an air core. This includes coils

wound on a plastic or ceramic form in

addition to those made of stiff wire that

are self-supporting and have air inside

them. Air core coils generally have a

much lower inductance than similarly

19

sized ferromagnetic core

coils, but are used in radio

frequency circuits to prevent

energy losses called core

losses that occur in magnetic

cores. The absence of

normal core losses permits a

higher Q factor, so air core

coils are used in high

frequency resonant circuits,

such as up to a few

megahertz. However,

losses such as proximity

effect and dielectric losses

are still present. Air cores

are also used when field

strengths above around 2

Tesla are required as they are not subject to saturation.

CORES STRUCTURES

STRAIGHT CYLINDRICAL ROD

Most commonly made of ferrite or

a similar material, and used in

radios especially for tuning an

inductor. The rod sits in the

middle of the coil, and small

adjustments of the rod's position

will fine tune the inductance.

Often the rod is threaded to allow

adjustment with a screwdriver. In

radio circuits, a blob of wax or

resin is used once the inductor

has been tuned to prevent the core from moving.

20

The presence of the high

permeability core increases the

inductance but the field must still

spread into the air at the ends of the

rod. The path through the air ensures

that the inductor remains linear. In

this type of inductor radiation occurs

at the end of the rod and

electromagnetic interference may be

a problem in some circumstances.

"C" OR "U" CORE

U and C-shaped cores are used with I or another C or U core to make a

square closed core, the simplest closed core shape. Windings may be

put on one or both legs of the core.

"E" CORE

E-shaped core are more symmetric solutions to form a closed magnetic

system. Most of the time, the electric circuit is wound around the center

leg, whose section area is twice that of each individual outer leg. In 3-

phase transformer cores, the legs are of equal size, and all three legs

are wound.

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"E" AND "I" CORE

Sheets of suitable iron stamped out in

shapes like the (sans-serif) letters "E"

and "I", are stacked with the "I"

against the open end of the "E" to

form a 3-legged structure. Coils can

be wound around any leg, but usually

the center leg is used. This type of

core is frequently used for power

transformers, autotransformers, and

inductors.

RING OR BEAD

The ring is essentially identical in shape and performance to the toroid,

except that inductors commonly pass only through the center of the

core, without wrapping around the core multiple times.

The ring core may also be composed of two separate C-shaped

hemispheres secured together within a plastic shell, permitting it to be

placed on finished cables with large connectors already installed, that

would prevent threading the cable through the small inner diameter of a

solid ring.

22

TRANSFORMER LOSSES

When the core is subjected to a changing magnetic field, as it is in

devices that use AC current such as transformers, inductors, and AC

motors and alternators, some of the power that would ideally be

transferred through the device is lost in the core, dissipated as heat and

sometimes noise.[2]

EDDY CURRENTS AND WINDING STRAY LOSSES

The load loss of a transformer consists of losses due to ohmic resistance

of windings (I2R losses) and some additional losses. These additional

losses are generally known as stray losses, which occur due to leakage

field of windings and field of high current carrying leads/bus-bars. The

stray losses in the windings are further classified as eddy loss and

circulating current loss. The other stray losses occur in structural steel

parts. There is always some amount of leakage field in all types of

transformers, and in large power transformers (limited in size due to

transport and space restrictions) the stray field strength increases with

growing rating much faster than in smaller transformers. The stray flux

impinging on conducting parts (winding conductors and structural

components) gives rise to eddy currents in them. The stray losses in

windings can be substantially high in large transformers if conductor

dimensions and transposition methods are not chosen properly.

Today’s designer faces challenges like higher loss capitalization and

optimum performance requirements. In addition, there could be

constraints on dimensions and weight of the transformer which is to be

designed. If the designer lowers current density to reduce the DC

resistance copper loss (I2R loss), the eddy loss in windings increases

due to increase in conductor dimensions. Hence, the winding conductor

is usually subdivided with a proper transposition method to minimize the

stray losses in windings.

In order to accurately estimate and control the stray losses in windings

and structural parts, in-depth understanding of the fundamentals of eddy

currents starting from basics of electromagnetic fields is desirable. The

fundamentals are described in first few sections of this chapter.

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EDDY CURRENTS

Eddy currents (also called Foucault currents[5]

) are circular electric

currents induced within conductors by a changing magnetic field in the

conductor, due to Faraday's law of induction. Eddy currents flow in

closed loops within conductors, in planes perpendicular to the magnetic

field. They can be induced within nearby stationary conductors by a

time-varying magnetic field created by an AC electromagnet or

transformer, for example, or by relative motion between a magnet and a

nearby conductor. The magnitude

of the current in a given loop is

proportional to the strength of the

magnetic field, the area of the loop,

and the rate of change of flux, and

inversely proportional to the

resistivity of the material.

By Lenz's law, an eddy current

creates a magnetic field that

opposes the magnetic field that

created it, and thus eddy currents

react back on the source of the

magnetic field. For example, a

nearby conductive surface will

exert a drag force on a moving

magnet that opposes its motion, due to eddy currents induced in the

surface by the moving magnetic field. This effect is employed in eddy

current brakes which are used to stop rotating power tools quickly when

they are turned off. The current flowing through the resistance of the

conductor also dissipates energy as heat in the material. Thus eddy

currents are a source of energy loss in alternating current (AC)

inductors, transformers, electric motors and generators, and other AC

machinery, requiring special construction such as laminated magnetic

cores to minimize them. Eddy currents are also used to heat objects in

induction heating furnaces and equipment, and to detect cracks and

flaws in metal parts using eddy-current testing instruments.Under certain

assumptions (uniform material, uniform magnetic field, no skin effect,

24

etc.) the power lost due to eddy currents per unit mass for a thin sheet or

wire can be calculated from the following equation:[6]

where

P is the power lost per unit mass (W/kg),

Bp is the peak magnetic field (T),

d is the thickness of the sheet or diameter of the wire (m),

f is the frequency (Hz),

k is a constant equal to 1 for a thin sheet and 2 for a thin wire,

ρ is the resistivity of the material (Ω m), and

D is the density of the material (kg/m3).

This equation is valid only under the so-called quasi-static conditions,

where the frequency of magnetisation does not result in the skin effect;

that is, the electromagnetic wave fully penetrates the material.

SKIN EFFECT

In very fast-changing fields, the magnetic field does not penetrate

completely into the interior of the material. This skin effect renders the

above equation invalid. However, in any case increased frequency of the

same value of field will always increase eddy currents, even with non-

uniform field penetration

The penetration depth for a good conductor can be calculated from the

following equation

25

USE OF THIN PLATES OR LAMINATIONS FOR CORE

We must see that the power loss due to eddy current is minimized so

that heating of the core is reduced and efficiency of the machine or the

apparatus is increased. It is obvious if the cross sectional area of the

eddy path is reduced then eddy voltage induced too will be reduced

(Eeddy ∞ area), hence eddy loss will be less. This can be achieved by

using several thin electrically insulated plates (called laminations)

stacked together to form the core instead a solid block of iron. The idea

is depicted in the Figure bellow where the plates have been shown for

clarity, rather separated from each other. While assembling the core the

laminations are kept closely pact. Conclusion is that solid block of iron

should not be used to construct the core when exciting current will be ac.

However, if exciting current is dc, the core need not be laminated.

EDDY CURRENT AND HYSTERESIS LOSSES HYSTERESIS

LOSS

The material used for structural components in transformers is usually

magnetic steel (mild steel), which is a ferromagnetic material having a

much larger value of relative permeability (μr) as compared to the free

26

space (for which μr=1). The material has non-linear B-H characteristics

and the permeability itself is a function of H. Moreover, the

characteristics also exhibit hysteresis property. Equation below

μh=μe-jθ

(B=μH)has to be suitably modified to reflect the non-linear characteristics

and hysteresis behavior.Hysteresis introduces a time phase difference

between B and H; B lags H by an angle (θ) known as the hysteresis

angle. One of the ways in which the characteristics can be

mathematically expressed is by complex or elliptical permeability,

HYSTERESIS LOSS

Consider a magnetic circuit with constant (d.c) excitation current I0. Flux

established will have fixed value with a fixed direction. Suppose this final

current I0 has been attained from zero current slowly by energizing the

coil from a potential divider arrangement as. Let us also assume that

initially the core was not magnetized. The exciting current therefore

becomes a function of time till it reached the desired current I and we

stopped further increasing it. The flux too naturally will be function of

time and cause induced voltage e12 in the coil with a polarity to oppose

the increase of inflow of current as shown. The coil becomes a source of

emf with terminal-1, +ve and terminal-2, -ve. Recall that a source in

which current enters through its +ve terminal absorbs power or energy

while it delivers power or energy when current comes out of the +ve

terminal. Therefore during the interval when i(t) is increasing the coil

absorbs energy. Is it possible to know how much energy does the coil

absorb when current is increased from 0 to I0? This is possible if we

have the B-H curve of the material with us.

27

HYSTERESIS LOOP WITH ALTERNATING EXCITING

CURRENT

let us see how the operating point is traced out if the exciting current is i

= Imax sin ωt. The nature of the current variation in a complete cycle can

be enumerated as follows:

Let the core had no residual field when the coil is excited by i = Imax sin

ωt. In the interval 0<ωt<π/ 2 will rise

along the path OGP. Operating point at

P corresponds to +Imax or +Hmax. For the

interval 2π<ωt<π operating moves

along the path PRT. At point T, current

is zero. However, due to sinusoidal

current, i starts increasing in the –ve

direction as shown in the bellow and

operating point moves along TSEQ. It

may be noted that a –ve H of value OS

is necessary to bring the residual field

to zero at S. OS is called the coercivity

of the material. At the end of the

interval π<ωt<3π/2, current reaches –

Imax or field –Hmax. In the next internal,

3/2π<ωt<π, current changes from –Imax

to zero and operating point moves from M to N along the path MN. After

this a new cycle of current variation begins and the operating point now

never enters into the path OGP. The movement of the operating point

can be described by two paths namely: (i) QFMNKP for increasing

current from –Imax to +Imax and (ii) from +Imax to –Imax along PRTSEQ.

28

In other words the operating point trace the perimeter of the closed area

QFMNKPRTSEQ. This area is called the B-H loop of the material. We

will now show that the area enclosed by the loop is the hysteresis loss

per unit volume per cycle variation of the current. In the interval 0<ωt<π/

2 is +ve and di/dt also +ve, moving the operating point from M to P

along the path MNKP. Energy absorbed during this interval is given by

the shaded area MNKPLTM shown in Figure (i). In the interval 2π≤ωt≤π,

i is +ve but di/dt is –ve, moving the operating point from P to T along the

path PRT. Energy returned during this interval is given by the shaded

area PLTRP shown in Figure (ii). Thus during the +ve half cycle of

current variation net amount of energy absorbed is given by the shaded

area MNKPRTM which is nothing but half the area of the loop.

In the interval 3/2π≤ωt≤π i is –ve and di/dt is also –ve, moving the

operating point from T to Q along the path TSEQ. Energy absorbed

during this interval is given by the shaded area QJMTSEQ shown in

Figure (iii).

In the interval 3/2π≤ωt≤2π i is –ve but di/dt is + ve, moving the operating

point from Q to M along the path QEM. Energy returned during this

interval is given by the shaded area QJMFQ shown in Figure (iv).

Thus during the –ve half cycle of current variation net amount of energy

absorbed is given by the shaded area QFMTSEQ which is nothing but

the other half the loop area.

Therefore total area enclosed by the B-H loop is the measure of the

hysteresis loss per unit volume per unit cycle. To reduce hysteresis loss

one has to use a core material for which area enclosed will be as small

as possible.

29

LOAD LOSSES

The load losses in a power transformer are due to the electric resistance

of windings and stray losses. The resistive action of the winding

conductor to the current flow will be lost in the form of heat and will be

dissipated in the surrounding area inside the transformer. The

magnitude of that loss increases by the square of the current . Stray

losses occur due to the leakage field of winding and due to high currents

seen in internal structural parts such as bus bars. Stray losses can affect

the overall rating of the transformer because they can create hot spots

when the current leads become excessive, affecting the overall life of the

transformer .

HEAT TRANSFER EFFECTS

A load serving transformer not only experiences an electrical process but

also goes through a thermal process that is driven by heat. The heat

generated by the no-load and load losses is the main source of

temperature rise in the transformer. However, the losses of the windings

and stray losses seen from the structural parts are the main factors of

heat generation within the transformer. The thermal energy produced by

the windings is transferred to the winding insulation and consequently to

the oil and transformer walls. This process will continue until an

equilibrium state is reached when the heat generated by the windings

equals the heat taken away by some form of coolant or cooling system .

This heat transfer mechanism must not allow the core, windings, or any

structural parts to reach critical temperatures that could possibly

deteriorate the credibility of the winding insulation. The dielectric

30

insulating properties of the insulation can be weakened if temperatures

above the limiting values are permitted . As a result, the insulation ages

more rapidly, reducing its normal life. Due to the temperature

requirements of the insulation, transformers utilize cooling systems to

control the temperature rise. The best method of absorbing heat from

the windings, core, and structural parts in larger power transformers is to

use oil .For smaller oil-field transformers, the tank surface is used to

dissipate heat to the atmosphere. For larger transformers, heat

exchangers, such as radiators, usually mounted beside the tank, are

employed to cool the oil. The standard identifies the type of cooling

system according to Table 1.

THE MAGNETIC CIRCUIT

The magnetic core has been introduced, an understanding of the

magnetic circuit is necessary to quantify the relationships between

voltage, current, flux, and field density.

31

Consider the magnetic circuit shown in Figure above consisting of a coil

of wire wound around a magnetic yoke. The coil has N turns and carries

a current i. The current in the coil causes a magnetic flux to flow along

the path a-b-c-d-a. For the time being, let us assume that the flux

density is small so that the permeability of the yoke is a constant. The

magnitude of the flux is given by

where( N X i ) is the magnetomotive force (MMF) in ampere-turns and is

the reluctance of the magnetic circuit a-b-c-d-a.

As the name implies, reluctance is a property that resists magnetic flux

when MMF is applied to a magnetic circuit. Reluctance is roughly

equivalent to the resistance in an electrical circuit.

For a homogeneous material where the mean length of the flux path isl

and the cross-sectional area is A, the reluctance is calculated in the

MKS system of measurement as follows:

The coil’s inductance L is equal to N 2 (μ X A)/l. Therefore, the coil’s

inductance is inversely proportional to reluctance of the magnetic circuit.

For series elements in the magnetic path, the total reluctance is found by

adding the values of reluctance of the individual segments along the

magnetic path. The reluctance values of parallel elements in a magnetic

circuit are combined in a manner similar to combining parallel

resistances in an electrical circuit.

32

APPENDIX

MAXWELL’S EQUATIONS

33

REFERENCES

1- Power Transformers Principles and Applications, John J. Winders,

Jr. , Copyright © 2002 by Marcel Dekker, Inc.

2- Transformer Engineering Design and Practice, S.V.Kulkarni.,

S.A.Khaparde., Indian Institute of Technology, Bombay, Mumbai,

India., MARCEL DEKKER, INC.

3- http://en.wikipedia.org/wiki/Transformer

4- http://en.wikipedia.org/wiki/Permeability_(electromagnetism)

5- Electronics engineers' handbook, Donald G. Fink, Donald

Christiansen ,McGraw-Hill, Jan 1, 1989 - Technology &

Engineering

6- Measurement and characterization of magnetic materials, F.

Fiorillo, Elsevier Academic Press, 2004

7- Fundamentals of Engineering Electromagnetics, Cheng. David K.

Copyright 1993 by Addison-Weceley Publishing Company, In c.