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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