MAGNETIC CIRCUITS

Q1. Define terms magnetic flux, flux density,magnetic field strength, permeability, MMF, Reluctance

Magnetic flux: Amount of magnetic field lines produced by the magnetic source is called magnetic flux (Ф). In other words it is the number of lines of force produced by Magnetic source. The symbol for magnetic flux is Φ.The unit of magnetic flux is weber  (Wb).

Magnetic flux density: it is the flux per unit area at right angle to the magnetic flux. OR It is the amount of flux passing through a defined area that is perpendicular to the direction of the flux: The symbol for magnetic flux density is B. Magnetic flux density = (magnetic flux) / (area) = B = Φ / AThe unit of magnetic flux density is the tesla (T) 1 T = 1Wb/m2

B=ФAwb/m2( tesla)

Ф−flux∈wb A−area∈m2

Magnetic field strength: Force experienced by unit north pole at that point in magnetic field.

OR mgneto motive force per length.

H= ATlengt h

=¿lATm

Permeability: The ability of material to carry the flux lines.

So when the permeability is high we can use that material as conductor in magnetic ckt (µo).

Absolute permeability (µ) : The ratio of magnetic flux density B in particular medium to magnetic field strength(H), which produces flux density is defined as absolute permeability of the material.

µ= BH

−−−H /m(µ=µ0µr)

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Relative permeability (µr): This is defined as ratio of flux density in particular medium produced by magnet to flux density in air (or) vacuum by same magnet under ideal conditions.

µr=BB0

(wit h same H )

B-flux density in medium B0- flux density in air (or) vacuum.

Permeability of free space (µ0) - 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

Magneto motive force: It drives flux through magnetic ckt same as emf in electric ckt.

MMF=N*I (AT) N-number of turns I-current

Reluctance : It is the property of material which opposes creation of magnetic flux in it. (or) opposition offered to passage of magnetic flux through material analogues to resistance in electric ckt.

s= lµ A

=mmfФ

At /wb

magnetic ckt: close path followed by magnetic flux is called magnetic ckt.

Permeance : it is the reciprocal of reluctance readiness with which magnetic flux is developed. Same as conductance in electric ckt

P= 1S= Ф

MMFwb /At (henry)

Comparison between electric ckt and magnet ckt

Electric Circuit Magnetic Circuit

Path traced by the current is known as electric

current. I= V/R (Amp)

Path traced by the magnetic flux is called as

magnetic circuit. Ф = mmf/ S (wb)

EMF is the driving force in the electric circuit. The

unit is Volts. EMF= IR (volts)

MMF is the driving force in the magnetic circuit.

The unit is ampere turns. MMF= ФS ( AT)

The current densityJ = I/a (A/m2) The flux density B = φ/a (wb/m2) OR Tesla

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Resistance (R) oppose the flow of the current.

The unit is Ohm R = V/I

Reluctance (S) is opposed by magnetic path to the

flux. S= mmf/ Ф The Unit is ampere turn/weber.

R = ρ. l/a. (units= ohms)

Directly proportional to l.Inversely proportional to a.

Depends on nature of material.

S = l / (μ0μra) (units = AT /wb)

Directly proportional to l. Inversely proportional

to μ = μ0μr.

Inversely proportional to a

resistivity reluctivity

Conductivity= 1/ resistivity Permeability = 1/resuctivity

Total emf=IR1 + IR2 _ _ Total mmf= ФS1 + ФS2 _ _ _

Ohms law, Kirchhoff current law and voltage law is

applicable to the electric circuit.

Ohms law, Kirchhoff mmf law and flux law is

applicable to the magnetic flux.

Dissimilarities between the two circuits which are listed below:

Electric Circuit Magnetic Circuit

 In the electric circuit, the current is actually flows.

i.e. there is movement of electrons.

Due to mmf flux gets established and does not

flow in the sense in which current flows.

There are many materials which can be used as insulators

(air, PVC, synthetic resins etc) which current can not

pass

There is no magnetic insulator as flux can

pass through all the materials, even through

the air as well.

Energy must be supplied to the electric circuit to

maintain the flow of current.

Energy is required to create the magnetic flux,

but is not required to maintain it.

The resistance and conductivity are independent of

current density under constant temperature. But may

change due to the temperature.

The reluctance, permanence and permeability

are dependent on the flux density.

Electric lines of flux are not closed. They start from

positive charge and end on negative charge.

Magnetic lines of flux are closed lines. They

flow from N pole to S pole externally while S

pole to N pole internally.

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What do you mean by series magnetic circuit?

Therefore, total ampere turns required in a series magnetic circuit is nothing but the arithmetic sum of

individual ampere turns each part of the circuit connected in series.

Leakage Flux And Fringing : Leakage flux is defined as the magnetic flux which does not follow the particular intended path in a magnetic circuit. Taking an example of solenoid you can explain the leakage flux and the fringing both.

When a current is passed through a solenoid, magnetic flux is produced by it. Most of the flux is set up in the core of the solenoid and passes through the particular path that is through the air gap and is utilized in the magnetic circuit. This flux is known as Useful flux φu.

As practically it is not possible that all the flux in the circuit follows a particular intended path and sets up in the magnetic core and thus some of the flux also sets up around the coil or surrounds the core of the coil, and is not utilized for any work in the magnetic circuit. This type of flux which is not used for any work is called Leakage Flux and is denoted by φl.

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A magnetic circuit consists of numbers of magnetic paths connected in series is referred as series magnetic circuit. In series magnetic circuit, the total reluctance of the circuit is the arithmetic sum of individual reluctance of each magnetic path connected in series. Total ampere turns required to establish a particular flux in the circuit is the arithmetic sum of ampere turns required for individual parts of the magnetic circuit for establishing same flux.Suppose magnetic circuit consists of different magnetic material of mean lengths L 1, L 2, L 3…… L n and cross-sectional areas of A1, A2, A3,…… A n respectively. The relative permeabilities are μ1, μ2, μ3…… μn, respectively.

Therefore, the total flux Φ produced by the solenoid in the magnetic circuit is the sum of the leakage

flux and the useful flux and is given by the equation shown below

Leakage coefficient : The ratio of the total flux produced to the useful flux set up in the air gap of the magnetic circuit is called leakage coefficient or leakage factor. It is denoted by (λ).

Fringing: The useful flux when sets up in the air gap, it tends to bulge outward at (b and b’) as show in above figureabove, because of this bulging the effective area of the air gap increases and the flux density

of the air gap decreases. This effect is known as Fringing.

Fringing is directly proportional to the length of the air gap that means if the length increases the fringing effect will also be more and vice versa.

B- H CURVE : Relation between B and H

For a magnetic material B=µH ; B=µ0 µr H=4π X10−7 X µr H

Where µrvaries with value of H

Magnetising curve: To draw the magnetising curve

1. One specimen ring of uniform cross sectional area wound uniformly with a coil “ p ” which is connected to a battery “B”

2. To reverse the flux direction a reversible switch "RS “ , variable resistance R1 and an ammeter are connected to the primeary winding.

3. To mesure the current and emf generated across secondary a resistance and a ballistic galvanometer is connected as shown in figure.

4. N2 are the number of turns which are connected to the secondary as shown in figure

5. To observe the beaviour of the flux density in the core, change the direction of current6. When the direction of the current is changed direction of the flux in the core is also changed.7. Suppose the initial flux through the core is ‘Ф’ and after changing the current direction of the current

flux through the core is ‘-Ф’.resultant flux=Ф−(−Ф )=2Ф total flux cut by t he secondery coil=N2 2Ф

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B

8. Suppose to change the flux direction takes a time of ‘ t ’

9. Then E.M.F generated in the secondary = E . M . F=N2 2Ф

t

current∈secondary coil= E .M .FR s

=N 22Фt Rs

θ : First deflection of the galvanometer when the current I is received by primary. K : Ballistic constant of galvanometer change per unit deflection. Let Ф is the flux in Wb produced by primary current I T :time of reversal of flux Charge passing through BG = k θ

For a particular instant current in secondary I=N22ФR s

−−−−−−1

When the change in flux present in the core it causes to deflection in the galvanometer which is connected in secondary soc hange∈Фαθ (θ ist hedeflection∈t h egalvonometer )

H in AT/m

BH curve for different material

Hysteresis loop:-- The lag or delay of a magnetic material known commonly as Magnetic Hysteresis, relates to the magnetisation properties of a material by which it firstly becomes magnetised and then de-magnetised.

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I=k θ−−−−−−2

From the above equations 1 & 2 N 22ФRS

=K θ

Ф=Kθ Rs

2N2−−−−3

magnetic fluxdensity= fluxarea

=ФA

−−−−4

Substitute the equation 3 in 4

Magnetic flux density B=KθRS

2 NSWb /m2

fluxdensity B wb /m2

Bmax

Starting with an unmagnetised core both B and H will be at zero, point 0 on the magnetisation curve.

1. If the magnetisation current, i is increased in a positive direction to some value the magnetic field strength H increases linearly with i and the flux density B will also increase as shown by the curve from point 0 to point a as it heads towards saturation.

2. Now if the magnetising current in the coil is reduced to zero, the magnetic field circulating around the core also reduces to zero. However, the coils magnetic flux will not reach zero due to the residual magnetism (remanance of flux) present within the core and this is shown on the curve from point a to point b. Ob is called retentitivity.

3. To reduce the flux density at point b to zero we need to reverse the current flowing through the coil. The magnetising force which must be applied to null the residual flux density is called a “Coercive Force”.or Coercivity. Now core becomes unmagnetised at point c.

4. An increase in this reverse current causes the core to be magnetised in the opposite direction and increasing this current further will cause the core to reach its saturation point but in the opposite direction, point d on the curve.

5. If the magnetising current is reduced again to zero the residual magnetism present in the core will be equal to the previous value but in reverse at point e.

6. Again reversing the magnetising current flowing through the coil this time into a positive direction will cause the magnetic flux to reach zero, point f on the curve and as before increasing the magnetisation current further in a positive direction will cause the core to reach saturation at point a.

Then the B-H curve follows the path of a-b-c-d-e-f-a as the magnetising current flowing through the coil alternates between a positive and negative value such as the cycle of an AC voltage. This path is called a Magnetic Hysteresis Loop.

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FARADAYS LAW OF ELECTROMAGNETISM

First law : whenever the magnetic flux cut by the conductor an emf is induced in the conductor.Or flux linked with conductor if changes emf is induced in te conductor.

Second law: The magnitude of the induced E.M.F is proportional to rate of charge of flux linkages.

E .M . F=NФ2−NФ1

twb /s e¿

N (Ф2−Ф1 )t

In differential from, we get

e= ddt

(NФ )=−N . dФdt

Lenz’s law: These law stats that the direction of induced emf is always tend to establish a current which opposes the change of flux responsible for inducing that emf.

Or electromagnetically induced current always flows in such direction that the action of the magnetic field set up by it tends to oppose the very cause which produces it.

Self inductance-- this is the property of coil due to which it opposes any increase or decrease or current or flux through it .

Coefficienct of Self inductance: “the Weber-turns per ampere in the coil” means when current flow

in the coil ‘flux linked with coil’L=NФI

henry

Or a coil has self inductance or one henry if one volt is induced in it when current through it changes at the rate of one ampere/sec.

1. e=−L didt

=−ddt

(LI )

e=−N dФdt

=−L dIdt

eL

eL=−L dIdt

=self induced emf

if w henweknow t heE .M .F∧change∈current :L= edi /dt

3. if we know t he phisical paremererst hen

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L=N dФdt

=NФI

but we knowФ= mmfreluctance

=¿s=

¿l

µ0µra=

¿ µ0µral

dФd I

=N µ0µra

lso L=N

N µ0µral

= N2

s

This property is analogous to inertia in material body. Initially it is a very difficult to set a heavy body into motion, but once in motion, it is equally difficult to stop it.

Mutual inductance: It is the ability of one coil to produce an emf in a nearby coil by induction when the current in the first coil changes.

Coefficient of mutual inductance (M):

1. The Weber turns in one coil due to one ampere current in other. M=N2Ф1

I 1

2. Flux in the first coil 1Ф=N2

s 1M=

N1N2

s

3. Two coils are said to have a mutual inductance of one henry if current changing at the rate of 1 amp/sec in one coil induces and emf of one volt in the other.

Coefficient of coupling: Consider two magnetically coupled coils A&B having N1and N2 turns respect.

inductanceof coil1 L1=N 1

2

lµ0µr A

inductanceof coil2 L2=N 2

2

lµ0µr A

Suppose a fraction K1 of flux (due to primary coil A) K1Ф1 is linked with B

M=K1Ф1N2

I 1−−−−−1

We already know the flux due to coil A

Ф=N1 I 1

lµ0µr A

=N1 I 1

S−−−2

So sub --2 in 1

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M 1=K1

N 1N 2

lµ0µr A

−−−3

M 2=K2

N1N2

lµ0µr A

−−−4

Multiplying 3 and 4 (when the current passing through both the coils are equal)

Fleming Right Hand Rule This rule states "Hold out the right hand with the first finger, second finger and thumb at right angle to each other. If forefinger represents the direction of the line of force, the thumb points in the direction of motion or applied force, then second finger points in the direction of the induced current.

Fleming’s left hand rule is used to determine the direction of force acting on the armature conductors of dc motor. Fleming’s left hand rule says that if we extend the index finger, middle finger and thumb of our left hand in such a way that the current carrying conductor is placed in a magnetic field (represented by the index finger) is perpendicular to the direction of current (represented by the middle finger), then the conductor experiences a force in the direction (represented by the thumb) mutually perpendicular to both the direction of field and the current in the conductor.

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K= M√L1L2

M 2=K 1 K 2

N12

lµ0µr A

N22

lµ0µr A

−−−5

M 2=K1 K2L1L2

M=√K1K 2L1L2