Chapter 3 Magnetic Circuits and Transformer

7/25/2019 Chapter 3 Magnetic Circuits and Transformer

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CHAPTER 3

MAGNETIC CIRCUITS


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Introduction

In the general sense, a magnetic circuit is anypath taken by magnetic flux. More specifically, it isassociated with the magnetic flux within siliconsteel cores such as those found in transformer,

generators, motors, relays, etc.


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Magnetic field

The space around the poles of a magnet is called the magnetic field ,andis represented by magnetic lines of forces.


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Magnetic Flux density

The number of flux lines per unit area is called the flux

density, is denoted by the capital letter B, and is measured in

teslas. Its magnitude is determined by the following equation:

B _ teslas (T)

_ webers (Wb) (11.1)

A _ square meters (m2)


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Similarities between Magnetic and

Electric Circuits


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Energy stored in Magnetic Field

So far we have discussed the inductance in static forms.

In earlier chapter we discussed the fact that work is

required to be expended to assemble a group of charges

and this work is stated as electric energy. In the samemanner energy needs to be expended in sending

currents through coils and it is stored as magnetic

energy.

Let us consider a scenario where we consider a coil in

which the current is increased from 0 to a value I. As

mentioned earlier, the self inductance of a coil in

general can be written as


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Faraday's law's of Electromagnetic

induction

FIRST LAWFirst Law of Faraday's Electromagnetic Induction state that whenever a

conductor are placed in a varying magnetic field emf are induced whichis called induced emf, if the conductor circuit are closed current are alsoinduced which is called induced current.

Or

Whenever a conductor is rotated in magnetic field emf is induced whichare induced emf.

SECOND LAW

Second Law of Faraday's Electromagnetic Induction state that the induced

emf is equal to the rate of change of flux linkages (flux linkages is theproduct of turns, n of the coil and the flux associated with it).


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Let,

Initial flux linkages = N1

Final flux linkages = N2

Change in flux linkages= N2 N1=N((2-1)

If (2-1)=

Then change in flux linkages=N

Rate of change of flux linkages= N/t wb/sec

Taking derivative of right hand side we getRate of change of flux linkages= Nd/dt wb/sec

Rut according to Faraday's laws of electromagnetic induction, the rate ofchange of flux linkages equal to the induced emf, hence we can write

= Nd/dt volt

Generally Faraday's laws is written as

e = -Nd/dt volt


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Self and Mutual inductance

The changing magnetic field created by one circuit (the primary) can

induce a changing voltage and/or current in a second circuit (the

secondary).

The mutual inductance, M, of two circuits describes the size of the

voltage in the secondary induced by changes in the current of the

primary: change in I (primary) V(secondary) = - M * ----------------------

change in time

The units of mutual inductance are henry, abbreviated "H".

A circuit can create changing magnetic flux through itself, which can

induce an opposing voltage in itself. The size of that opposing voltageis change in I V(opposing) = - L * ------------- change in time where Lis

the self-inductanceof the circuit, again measured in henries.


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The mutual inductance that exists between the two coils can be greatly increased by

positioning them on a common soft iron core or by increasing the number of turns

of either coil as would be found in a transformer. If the two coils are tightly wound

one on top of the other over a common soft iron core unity coupling is said to exist

between them as any losses due to the leakage of flux will be extremely small


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Here the current flowing in coil one, L1sets up a

magnetic field around itself with some of these

magnetic field lines passing through coil two, L2

giving us mutual inductance. Coil one has a currentof I1and N1turns while, coil two has N2turns.

Therefore, the mutual inductance, M12of coil two

that exists with respect to coil one depends on their

position with respect to each other and is given as:


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Transformer A transformeris a power converter that transfers energy between two

electrical circuits by inductive coupling between two or more windings.A varying current in the primary winding creates a varying magneticflux in the transformer's core and thus a varying magnetic flux throughthe secondary winding. This varying magnetic flux induces a varyingelectromotive force (EMF) or "voltage", in the secondary winding. Thiseffect is called inductive coupling.

If a load is connected to the secondary winding, current will flow in thiswinding, and electrical energy will be transferred from the primarycircuit through the transformer to the load. Transformers may be usedfor AC-to-AC conversion of a single power frequency, or forconversion of signal power over a wide range of frequencies, such as

audio or radio frequencies. In an ideal transformer, the induced voltage in the secondary winding

(Vs) is in proportion to the primary voltage (Vp) and is given by theratio of the number of turns in the secondary (Ns) to the number ofturns in the primary (Np) as follows:


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Transformer Construction


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Transformer Action -- DC


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11

de N

dt

2 2

de N

dt

Opposes battery voltage Opposes flux buildup


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Transformer Action -- AC


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max4.44P PE N f max4.44S SE N f

Opposes VT Opposes M

max

max

4.44

4.44

PP P

S S S

N fE N

E N f N


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No-Load Condition


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No load condition continued

Io= Ife+ IM

Io= exciting current

Ioprovides the magnetizing flux and the core lossIfe= core-loss current Ife= VT/ Rfe

IM= magnetizing current IM= VT/ jXM


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O fe M

P O P fe P M

I I I

N I N I N I

No-Load Excitation mmf

No-Load Core Loss mmf

Magnetizing mmf

P MM

core

N I

R


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T P P P

T PP O

P

V I R E

V EI IR


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Close the load switch

P M S SM

core

N i N i

R

Secondary current will set up an mmf in OPPOSITION to the

primary mmf. The core flux will DECREASE to


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The decrease in flux causes a decrease in the counter-

emf EP, and the primary current will increase by an

amount known as IP,load

, the load component of the

primary current. Additional mmf due to this current adds

to the magnetizing flux.


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P= net flux in window of primary S= net flux in window of secondary

lp= leakage flux of primary ls= leakage flux of secondary

M= mutual flux

P= M+ lp

S= Mls


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EFFICIENCY OF TRANSFORMER


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Losses in a transformer

There are mainly two kinds of losses in a transformer,namely(1)core loss.

(2)Ohmic loss.

1.Core loss:

These core losses in transformer consists of two components hysteresisloss and eddy current loss

i.e. core loss=hysteresis loss+eddy current loss.

hysteresis losses depends on applied voltage and its frequency

eddy current loss is proportional to squre of the applied votage and is

independent of frequency f.

3.Ohmic loss:

when transformer is loadded ohmic losses(i^2*r)occurs in both the

primary and secondary winding resisrances.

In addition to core loss the follwing loss has to be taken into consideration

.


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Stray losses: Leakage fields present in the transformer

induce eddy currents in conductors, tanks, channels,

bolts etc. and these eddy currents give rise to stray losses CORE LOSSES:

I. the energy dissipated in the core due to hysterisis over

one cycle is the area enclosed by the hysterisis loop.


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are the magnetic field (flux density) and

magnetizing intensity (or auxilliary field orjust "H") present in the core.

Physically, this loss is understood as theenergy required to orient and reorient themagnetic domains in the ferromagneticmaterial, when the direction of the magneticfield changes due to the A.C. current.

II. Eddy current "flows" through the core ofthe transformer-- this flux is proportional tothe current, so it is also CHANGING in time.


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OHMIC LOSS:

This one is the easiest to understand-- The copper windings of

the primary and secondary of the transformer are (obviously)conductors, so some energy will be dissipated in them. The

copper wire of the primary and secondary will have total

resistances of RP

andRS

energy will dissipate in them at a rate of

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Chapter 3 Magnetic Circuits and Transformer