Magnetic Circuit

Magnetomotive Force (MMF) Reluctance
Magnetic Circuit
S UBHASH PATEL
Electronics & Communication
Year 2010
Outline
1 Magnetomotive Force (MMF)

Outline
1 Magnetomotive Force (MMF)
2 Reluctance
Magnetomotive Force (MMF)

The current flowing in the electrical circuit due to the existence of electromotive
force(emf) across the circuit

To drive magnetic flux through magnetic circuit, a magneto motive force (mmf)
is necessary.

is necessary.
Magnetomotive force can be produced when current flows in a coil of one or

more turns.

is necessary.

more turns.
The magnitude of mmf is directly proportional to the current I and number of

turns of the coil N,

is necessary.

more turns.

mmf = NI

is necessary.

more turns.

mmf = NI
If the magnetic circuit of magnetic material is homogeneous and of uniform

cross sectional area, the magnetomotive force per meter length of the magnetic
circuit (path of magnetic flux) is called the magnetic field strength (H).

is necessary.

more turns.

mmf = NI
If the magnetic circuit of magnetic material is homogeneous and of uniform

cross sectional area, the magnetomotive force per meter length of the magnetic
circuit (path of magnetic flux) is called the magnetic field strength (H).
mmf
H=
l
Reluctance
Reluctance
Let the ring have a mean circumference of

l meter, a cross-sectional area of a m2 and
N turns carrying a current of I ampere,
then the total flux flowing in the dotted
path is given by,
φ = flux density × cross sectional area

Reluctance

path is given by,
φ = B×a
Reluctance

path is given by,
φ = B×a
Reluctance
I
Also, mmf = magnetic field strength ×
length of flux path
mmf = Hl

path is given by,
φ = B×a
Reluctance
I
length of flux path
mmf = Hl
Taking ratio of φ and mmf,
φ Ba
=
mmf Hl
B
But, = µ = µ0 µr
H

path is given by,
φ = B×a
Reluctance
I
length of flux path
mmf = Hl
φ Ba
=
mmf Hl
B
But, = µ = µ0 µr
H
φ a mmf l
= µ0 µr or =
mmf l φ µ0 µr a

path is given by,
φ = B×a
Reluctance
I
length of flux path
mmf = Hl
φ Ba
=
mmf Hl
B
But, = µ = µ0 µr
H
φ a mmf l
= µ0 µr or =
mmf l φ µ0 µr a
For the electric circuits,
emf ρl
Let the ring have a mean circumference of =R=
current a
path is given by,
φ = B×a
Reluctance
I
length of flux path
mmf = Hl
φ Ba
=
mmf Hl
B
But, = µ = µ0 µr
H
φ a mmf l
= µ0 µr or =
mmf l φ µ0 µr a
emf ρl
current a
N turns carrying a current of I ampere, As emf/current is called resistance in
then the total flux flowing in the dotted electric circuits, mmf/flux can be termed
path is given by, as reluctance for magnetic circuits.
φ = B×a
Reluctance
I
length of flux path
mmf = Hl
φ Ba
=
mmf Hl
B
But, = µ = µ0 µr
H
φ a mmf l
= µ0 µr or =
mmf l φ µ0 µr a
emf ρl
current a
Reluctance is the property of magnetic
φ = flux density × cross sectional area material which opposes the flow of flux
through it.
φ = B×a
Reluctance
I
length of flux path
mmf = Hl
φ Ba
=
mmf Hl
B
But, = µ = µ0 µr
H
φ a mmf l
= µ0 µr or =
mmf l φ µ0 µr a
emf ρl
current a
Reluctance is the property of magnetic
φ = flux density × cross sectional area material which opposes the flow of flux
through it.
φ = B×a l
Reluctance R = ampere/weber
µ0 µr a
Laws of Magnetic Circuits

Ohm’s law for electric circuit is,

emf = current × resistance or E = I × R
Ohm’s law for magnetic circuit is,

mmf = flux × reluctance or mmf = φ × S


1 length l
For electric circuits, Resistance = × =ρ
conductivity area a
1 length 1 l
For magnetic circuits, Reluctance = × = ×
permeability area µ0 µr a


1 length l
For electric circuits, Resistance = × =ρ
conductivity area a
1 length 1 l
For magnetic circuits, Reluctance = × = ×
permeability area µ0 µr a
In series electric circuits, the total resistance of the circuit is equal to the sum of all the
resistance in series.
Similarly, when the flux has to permeate a number of portions of a magnetic circuit in
series, the total reluctance of the complete magnetic circuit will be equal to the sum of the
reluctances of various portions, i.e. S = S1 + S2 + S3 + · · ·


In a series electric circuit the total voltage drop is equal to the sum of the
voltage drop in various elements of the circuit.

The total mmf required to establish a given flux in the magnetic circuit is equal
to the sum of the mmfs necessary to establish the flux through the various parts
of the circuit. Thus total mmf of the complete magnetic circuit consisting of a
number of homogeneous parts is given by,
Total mmf F = F1 + F2 + F3 + · · ·
Total mmf F = H1 l1 + H2 l2 + H3 l3 + · · ·
B1 B B
Total mmf F = l + 2 l + 3 l +···
µ1 1 µ2 2 µ3 3

The total mmf required to establish a given flux in the magnetic circuit is equal
to the sum of the mmfs necessary to establish the flux through the various parts
of the circuit. Thus total mmf of the complete magnetic circuit consisting of a
number of homogeneous parts is given by,
Total mmf F = F1 + F2 + F3 + · · ·
Total mmf F = H1 l1 + H2 l2 + H3 l3 + · · ·
B1 B B
Total mmf F = l + 2 l + 3 l +···
µ1 1 µ2 2 µ3 3
The, mmf acting around a complete magnetic circuit is equal to the total ampere
turns required to force the given flux through the magnetic circuit.
Comparison between the Electric and Magnetic Circuit
Similarities in Electric and Magnetic Circuit
Current flows in the circuit. Flux is assumed to flow

The path of current is called electric circuit Path of flux is called magnetic circuit.
Current flows due to emf Flux flows due to mmf
Flow of current is restricted by resistance of Flow of flux is restricted by reluctance of
the circuit the circuit
Current = emf/Resistance Flux = mmf/reluctance
l l
R=ρ S=
a µA
Comparison between the Electric and Magnetic Circuit
Similarities in Electric and Magnetic Circuit
Current flows in the circuit. Flux is assumed to flow

The path of current is called electric circuit Path of flux is called magnetic circuit.
Current flows due to emf Flux flows due to mmf
Flow of current is restricted by resistance of Flow of flux is restricted by reluctance of
the circuit the circuit
Current = emf/Resistance Flux = mmf/reluctance
l l
R=ρ S=
a µA
Dissimilarities in Electrical and Magnetic circuit
Current is actually flows in the circuit. Flux does not flow, it is only assumed to
flow for finding out certain magnetic ef-
fects.
Energy is needed till the current flows. Energy is needed only to create the mag-
netic flux
Resistance of the circuit is independent of Reluctance of the circuit changes with the
the current. magnetic flux
Calculation of Ampere Turns for the Air Gap
Total ampere turns for the air gap is given by,
ATg = flux × reluctance
lg
ATg = φ ×
µ0 Ag
φ lg
ATg =
Ag µ0
1
ATg = Bg × lg
µ0
1
ATg = Bg lg
4π × 10−7
ATg = 0.796Bg lg × 106

Example 1
The magnetic frame shown in figure is built up of iron of square cross-section, 3 cm

side. Each air gap is 2 mm wide. Each of the coil is wound with 1000 turns and the
exciting current is 1.0 A. The relative permeability of Part A and Part B may be taken
as 1000 and 1200 respectively. Calculate the following. (i) reluctance of part A (ii)
reluctance of part B (iii) reluctance of two air gaps (iv) total reluctance of complete
magnetic circuit (v) the mmf (vi) total flux (vii)flux density.
Example 1
The magnetic frame shown in figure is built up of iron of square cross-section, 3 cm

side. Each air gap is 2 mm wide. Each of the coil is wound with 1000 turns and the
exciting current is 1.0 A. The relative permeability of Part A and Part B may be taken
as 1000 and 1200 respectively. Calculate the following. (i) reluctance of part A (ii)
reluctance of part B (iii) reluctance of two air gaps (iv) total reluctance of complete
magnetic circuit (v) the mmf (vi) total flux (vii)flux density.
17 cm
Part A
2 mm
10 cm
Part B
20 cm
Example 1
lA
Reluctance for part A, SA =
µ0 µr a
Cross sectional area of part
A = 0.03 × 0.03 = 9 × 10−4 m2
0.17
SA = Total Reluctance S = SA + SB + Sg
4π × 10−7 × 1000 × 9 × 10−4
SA = 15.03 × 104 AT/wb S = (15.03 + 25.04 + 353.5) × 104
S = 393.57 × 104 AT/wb
lb
Reluctance of part B, SB = mmf = 2 × 1000 × 1.0 = 2000AT
µ0 µr a
Length of part B,
lb = 17 + 8.5 + 8.5 = 34cm = 0.34m mmf 2000
0.34 flux = =
SB = reluctance 393.57 × 104
4π × 10−7 × 1000 × 9 × 10−4 flux = 5.08 × 10−4 wb
SB = 25.04 × 104 AT/wb
flux
lg Flux Density =
cross sectional area
Reluctance of air gap, Sg =
µ0 a 5.08 × 10−4
B= = 0.564Wb/m2
Length of air gap is 2+2=4mm 9 × 10−4
4 × 10−3
Sg =
4π × 10−7 × 9 × 10−4
Sg = 353.5 × 104 AT/wb
Example 2
A cast steel electromagnet has an air gap length of 3 mm and an iron path of length
40 cm. Find the number of ampere turn necessary to produced a flux density of 0.7
wb/m2 in the gap.
Example 2
wb/m2 in the gap.
Ampere turns for the air gap is given by,

Example 2
wb/m2 in the gap.
ATg = 0.796Bg lg × 106

Example 2
wb/m2 in the gap.
ATg = 0.796Bg lg × 106
ATg = 0.796 × 0.7 × 0.4 × 106

Example 2
wb/m2 in the gap.
ATg = 0.796Bg lg × 106
ATg = 0.796 × 0.7 × 0.4 × 106
ATg = 1671AT
Example 3
A steel ring of 25 cm mean diameter and of circular section 3 cm in diameter has an
air gap of 1.5 mm length. It wound uniformly with 700 turns of wire carrying a
current of 2A. Calculate (i) magnetomotive force (ii) flux density (iii)magnetic flux
(iv) reluctance (v) relative permeability of steel ring. Neglect magnetic leakage and
assume that the iron path takes about 35 percent of total megnetomotive force.
Example 3
A steel ring of 25 cm mean diameter and of circular section 3 cm in diameter has an
air gap of 1.5 mm length. It wound uniformly with 700 turns of wire carrying a
current of 2A. Calculate (i) magnetomotive force (ii) flux density (iii)magnetic flux
(iv) reluctance (v) relative permeability of steel ring. Neglect magnetic leakage and
assume that the iron path takes about 35 percent of total megnetomotive force.
Calculation Magnetomotive force
mmf = total turns × current
mmf = 700 × 2
mmf = 1400 AT
Calculation of Reluctance
mmf
Reluctance =
Calculation of flux density φ
Total AT = 1400 1400
Iron part takes about 35% of total mmf, thus air Reluctance =
0.538m = 2.6 × 106 AT/wb
part takes 65% of total mmf
ATg = 0.65 × 1400
Calculation of µr for steel ring,
ATg = 910
mmf for steel ring is 0.35 x 1400 = 490
910 = 0.796 × Bg lg × 106 B
910 mmf = Hl = l
Bg = = 0.762 wb/m2 µ0 µr
0.796 × 1.5 × 10−3 × 106 Bl 0.762 × π × 0.25
µr = =
mmf × µ0 490 × 4π × 10−7
Calculation of flux φ µr = 971.9
φ = B×A
φ = 0.762 × π × 0.0152 = 0.538 mWb
Example 4
A total flux of 0.0006 Wb is required in the air gap of an iron ring of cross-section 5.5
cm2 and mean length 2.7 m with an air gap of 4.5 mm. Find the number of ampere
turns required. Points on the B-H curve for the material of the ring are as follows:
H(AT/m): 200 400 500 600 800 1000
B(Wb/m2 ): 0.4 0.8 1.0 1.09 1.17 1.19
Example 4
A total flux of 0.0006 Wb is required in the air gap of an iron ring of cross-section 5.5
cm2 and mean length 2.7 m with an air gap of 4.5 mm. Find the number of ampere
turns required. Points on the B-H curve for the material of the ring are as follows:
H(AT/m): 200 400 500 600 800 1000
B(Wb/m2 ): 0.4 0.8 1.0 1.09 1.17 1.19
Flux φ = 0.0006 Wb
φ 0.0006
Flux density B = = = 1.09Wb/m2
A 5.5 × 10−4
ATg = 0.796 × Bg lg × 106

ATg = 0.796 × 1.09 × 4.5 × 10−3 × 106
ATg = 3904
Ampere turns calculation for iron ring

AT = Hl
AT = 600 × (2.7 − 4.5 × 10−3 )
AT = 1617
Total ampere turns = 3904 + 1617 = 5521

Leakage Flux and Fringing
The total magnetic flux, φt produced by the coil due to current I flowing in it is
divided into following two components.
1 Useful flux φ , which flows throughout the magnetic circuit and is utilized for
the useful purpose. In case of transformer, the flux which links both the primary
and secondary winding is called useful flux and in case of rotating machines the
flux which crosses the air gap is called the useful flux.
2 Leakage flux φl , links partly the magnetic circuit and complete its path through
air as shown in figure. In case of transformer the flux which links with one
winding only and complete its path trough air is called leakage flux. The value
of leakage flux depends on the load current.
2 Leakage flux φl , links partly the magnetic circuit and complete its path through
air as shown in figure. In case of transformer the flux which links with one
winding only and complete its path trough air is called leakage flux. The value
of leakage flux depends on the load current.
Thus, the total flux produced, φt is equal to the sum of the useful flux phi and the
leakage flux φl .
φT = φ + φl
The ratio of the total flux produced φT to the useful flux phi is called the leakage
factor or leakage coefficient, that is,
φT
leakage factor =
φ
The value of the leakage factor is always grater than unity and varies between 1.15 to
1.25 depending upon the type of magnetic circuit
φT
leakage factor =
φ
The value of the leakage factor is always grater than unity and varies between 1.15 to
1.25 depending upon the type of magnetic circuit
When the flux crosses the air gap, it tends to bulge out across the edges of the air gap.
This effect of bulging is called fringing. The effect of fringing is to increase the
effective gap area, which in turn reduces the flux density in the air gap. Fringing
depends upon the length of the air gap, that is higher the air gap greater is the
fringing.
Magnetic Hysteresis
Magnetic Hysteresis
If a magnetic material is magnetized in a strong magnetic field, it retains a
considerable portion of magnetism even after the removal of the magnetic force.
This phenomenon of lagging of magnetization of flux density B behind the
magnetizing force H is known as magnetic hysteresis.
Magnetic Hysteresis
Magnetic Hysteresis
Residual magnetism or remanent flux density is defined as the magnetic flux

density which still remains in magnetic material even when the magnetizing
force is completely removed.
Magnetic Hysteresis
Residual magnetism or remanent flux density is defined as the magnetic flux

density which still remains in magnetic material even when the magnetizing
force is completely removed.
The magnetic field intensity required to wipe out the residual magnetism is
called coercive force.
Magnetic Hysteresis
Retentivity is the power of retaining magnetism in the magnetic materials even after the
magnetizing force is completely removed.
Coercive force is the demagnetizing force which is necessary to neutralize completely the
magnetism in the magnetic material.
The shape of the hysteresis loop will depend upon the nature of magnetic material.
Hysteresis loop for hard steel is quite wide and hence possess high retentivity power and
large coercive force. This type of material is well suited for permanent magnets and not
suitable for rapid reversals of magnetization as in transformer core and choke cores.
Steel, silicon alloys have a very narrow hysteresis loop. Since they have very high
permeability and very low hysteresis losses, these materials are more suitable for
transformer core and armature cores which are subjected to rapid reversal of
magnetization.
When a magnetic material is subjected to cyclic changes of magnetization, the domains
changes the direction of their orientation according to the way the applied magnetizing
force H changes its direction. Work is done in changing the direction of the domains
which leads to the production of heat within the material. The energy required in taking
the material through one complete cycle of magnetization is proportional to the area
enclosed by the hysteresis loop.
Hysteresis loss
If the magnetization is carried through a complete cycle, the energy wasted is
proportional to the area of the hysteresis loop and the shape of hysteresis loop
depends upon the nature of the ferromagnetic material.
Hysteresis loss is equal to the energy consumed in magnetizing and
demagnetizing a magnetic material .
It is proportional to ,
Area enclosed in the hysteresis loop
Frequency of magnetic flux reversal
Volume of the magnetic material
Hysteresis loss, Ph = ηVf (Bmax )1.6
V = Volume of material in m3
η is a constant, steinmetz’s coefficient
f is frequency of magnetic flux reversal
Eddy Currents
When a magnetic material is linked with a variable or alternating flux, an emf is

induced in the magnetic material itself according to Faraday’s law of
electromagnetic induction. This induced emf circulates a current in the body of
the magnetic material. These circulating currents are called Eddy currents and
power losses due to the flow of this current are called Eddy current losses.
Eddy currents always tend to flow in planes perpendicular to the magnetic flux
as they are induced due to variation of this flux through the circuit.
Applications of eddy current include eddy current braking in induction energy

meters and eddy current damping in permanent magnet moving coil
instruments.
Eddy current loss
Power loss due to flow of eddy currents in the magnetic material is called Eddy
current loss. Heat is generated due to flow of eddy current. Basically, eddy current
loss depends upon the value of the emf induced and the resistance offered by the
magnetic material to the flow of eddy currents. The resistance can greatly increased
by laminating the material, thereby reducing the magnitude of the eddy currents to an
appreciable value.
The eddy current losses depends upon the following things,
1 Thickness of the material

2 Frequency of flux reversal
3 Maximum value of flux density
4 Volume of the material
5 Quality of material
Classification of Magnetic Materials
1 Ferromagnetic Materials
The materials whose relative permeability is very high (µr =100 to 10000) are
called the ferromagnetic materials, e.g. iron, steel, nickel, cobalt etc. Such
materials can be used to make strong magnets as they are strongly attracted by
magnets.
magnets.
2 Paramagnetic Materials
The materials having relative permeability equal to one or slightly more than
one are called the paramagnetic materials. e.g. platinum, manganese, aluminum
etc. These materials are attracted by a magnet but not very strongly.
magnets.
2 Paramagnetic Materials
The materials having relative permeability equal to one or slightly more than
one are called the paramagnetic materials. e.g. platinum, manganese, aluminum
etc. These materials are attracted by a magnet but not very strongly.
3 Diamagnetic Materials
The materials having their relative permeability less than one are called the
diamagnetic materials, e.g. zinc, bismuth, sulfur, phosphorous, tin, lead etc.
Such materials are actually repelled by a magnet.

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What are the different types of magnetic materials discussed?

What are the different types of magnetic materials discussed?

What is magnetomotive force and how is it related to current and number of turns?

What is magnetomotive force and how is it related to current and number of turns?

Magnetomotive Force (MMF) Reluctance

1 Magnetomotive Force (MMF)

1 Magnetomotive Force (MMF)

Magnetomotive Force (MMF)

Magnetomotive Force (MMF)

Magnetomotive Force (MMF)

Magnetomotive force can be produced when current flows in a coil of one or

Magnetomotive Force (MMF)

Magnetomotive force can be produced when current flows in a coil of one or

The magnitude of mmf is directly proportional to the current I and number of

Magnetomotive Force (MMF)

Magnetomotive force can be produced when current flows in a coil of one or

The magnitude of mmf is directly proportional to the current I and number of

Magnetomotive Force (MMF)

Magnetomotive force can be produced when current flows in a coil of one or

The magnitude of mmf is directly proportional to the current I and number of

If the magnetic circuit of magnetic material is homogeneous and of uniform

Magnetomotive Force (MMF)

Magnetomotive force can be produced when current flows in a coil of one or

The magnitude of mmf is directly proportional to the current I and number of

If the magnetic circuit of magnetic material is homogeneous and of uniform

Let the ring have a mean circumference of

φ = flux density × cross sectional area

Let the ring have a mean circumference of

φ = flux density × cross sectional area

Let the ring have a mean circumference of

φ = flux density × cross sectional area

Let the ring have a mean circumference of

φ = flux density × cross sectional area

Let the ring have a mean circumference of

φ = flux density × cross sectional area

Let the ring have a mean circumference of

φ = flux density × cross sectional area

φ = flux density × cross sectional area

φ = flux density × cross sectional area

Laws of Magnetic Circuits

Laws of Magnetic Circuits

Ohm’s law for electric circuit is,

Ohm’s law for magnetic circuit is,

Laws of Magnetic Circuits

Ohm’s law for electric circuit is,

Ohm’s law for magnetic circuit is,

Laws of Magnetic Circuits

Ohm’s law for electric circuit is,

Ohm’s law for magnetic circuit is,

Laws of Magnetic Circuits

Laws of Magnetic Circuits

Laws of Magnetic Circuits

Laws of Magnetic Circuits

Comparison between the Electric and Magnetic Circuit

Similarities in Electric and Magnetic Circuit

Current flows in the circuit. Flux is assumed to flow

Comparison between the Electric and Magnetic Circuit

Similarities in Electric and Magnetic Circuit

Current flows in the circuit. Flux is assumed to flow