This document provides information about magnetic circuits and transformers: 1. It defines key magnetic concepts like ferromagnetic materials, magnetic flux, magnetic field intensity, hysteresis, and reluctance. 2. It describes how solenoids, toroids, and conductors carrying current produce magnetic fields according to mathematical relationships. 3. It explains the principles of electromagnetic induction, inductance, and how transformers work by magnetically coupling two coils to step up or step down voltages based on turns ratios.

1. BITS Pilani
Pilani Campus
EEE G529T MAGNETIC CIRCUITS
13.1-13.5
EEE Department
BITS Pilani, Pilani

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MAGNETIC FIELDS
 Ferromagnetic Material – that has the property of attracting other pieces of such materials
 A permanent magnet is a ferromagnetic material
 Just as we have the Electric Field associated with a stationary charge, we have magnetic
field associated with a moving charge (current)
 Magnetic Lines of Force – Imaginary closed loops
that origin from the north pole and end at
the south pole of the magnet
 The tangent at a point on these lines gives the direction of
the magnetic field.
Bar Magnet
Magnetic Lines of
Force

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MAGNETIC FLUX AND FLUX DENSITY
 Magnetic flux (denoted as Φ), is the amount of magnetic field passing through a surface
(such as a conducting coil).
 The SI unit of magnetic flux is the weber (Wb)
More
magnetic
lines pass
through the
circular ring

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LORENTZ FORCE
When a charge q moves with a velocity v in the presence of a Magnetic Field B, a force is
exerted on the charge known as the Lorentz Force.
The direction of this force is given by Right Hand Rule

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MAGNETIC FIELD DUE TO A CONDUCTOR
CARRYING CURRENT
A conductor carrying a current I produces a magnetic field B
around it in a cylindrically radial pattern as shown
The direction of the magnetic field is given by the right hand rule which
states that if you hold the conductor in your right hand with thumb
pointing in the direction of the current, then the direction in which the
fingers curl gives you the direction of the magnetic field.
𝑩 =
𝝁𝑰
𝟐𝝅𝒅
Magnetic Field B at a radial distance d from the conductor
carrying current I is given by
μ is permeability of the medium

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MAGNETIC FIELD DUE TO A SOLENOID
Solenoid : Cylindrical coil of wire wound on an air-core, an iron core or any other core
𝐵 =
𝜇𝑁
𝑙
I
Consider a cylindrical coil wound on a soft iron core
fed by a DC current of magnitude I. A magnetic field
will be setup in the core that will be almost constant
provided that the length of the cylinder (l) is much
larger than its radius.
The magnetic field inside the Iron Core is given by
The magnetic flux is given by B.A = B.(∏r2)
Permeability defines the easy with which magnetic field can be set up in the material

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MAGNETIC FIELD DUE TO A TOROID
 If we bend the iron core in the form of a ring, we get a toroid
 The magnetic field at the center of the toroid is given by
RELATIVE PERMEABILITY
 It is the ratio of permeability of the medium to the permeability of free space
 Ferromagnetic materials have high values of relative permeability (>1000)
𝐵 =
𝜇𝑁𝐼
2∏𝑟

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MAGNETIC CIRCUITS
 Magnetic circuits are analogue of electrical circuits.
 The magneto motive force of N-turn current carrying
coil is
 The reluctance R of a magnetic path depends on the
mean length l, the area A, and the permeability μ of
the material.
 Magnetic flux is analogous to current in electrical
circuit and is related to F and R in a similar way as
Ohm’s law
A
l

R
NiF 
RF

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ANALOGY BETWEEN ELECTRIC
AND MAGNETIC CKTS

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IRON CORES WITH AN AIR GAP
Core with an air gap Equivalent magnetic circuit The air gap will have some
reluctance that will be in series
with the reluctance of the iron
core.
 Fringing of the flux lines occur
when the air gap length is
somewhat large
 More flux is concentrated in
the inner portion of the core
than in the outer portion

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MAGNETIZATION CURVES
H
B
B
H
Linear
knee
saturation
 A plot of B v/s H is a magnetization curve
 As 𝑩 = ∅/𝑨 and ∅ =
𝑴𝑴𝑭
𝑹
we get 𝑩 = 𝝁𝑯
where H is the Magnetic Field Intensity given by
 𝑯 =
𝑴𝑴𝑭
𝒍
 H is independent of core material
Practical B-H Curve
Ideal Curve

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HYSTERESIS
Demagnetized Material : The Magnetic Flux Density (B) is 0 when no external Magnetizing
Field (H) is applied
Consider a demagnetized material and let us apply a magnetizing field to it to magnetize it
 For smaller values of H, B-H curve is almost linear
 For larger values of H, B-H curve saturates and B is almost constant
 If we increase the magnetizing field even more, no increase in B is observed (point b)
 The material is said to be saturated now
 If we reduce H now, B does not reduce the same way as it increased
 There is some amount of magnetism left over when H reduces to 0
 This is called Residual Magnetism (point c)
 Permanent Magnets are made of materials having high values of residual magnetism

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 If H is increased again but in the –ve direction, then
at a particular value of H, the Residual Magnetism
goes away and B becomes 0
 This value of H is called the coercive force
(point d)
 Permanent magnets have high coercivity
 If H is made –ve enough, material saturates
but in the opposite direction (point e)
 On increasing H from its max –ve value, we reach
point f that indicates negative residual magnetism
 The resulting loop is called a hysteresis loop and
the phenomenon Hysteresis
HYSTERESIS

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If the magnetizing force applied to the
demagnetized material is less than the required
to produce saturation, a hysteresis loop as
shown is produced.
HYSTERESIS
SQUARE LOOP MATERIALS
 Materials having hysteresis loop approximately
rectangular are called square loop materials
 Slopes of the sides of the hysteresis loop are quite large.
A small change in H can lead to a large change in B
 Small cores made from such materials are used as
binary memory devices in switching circuits and digital
computers

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HYSTERESIS LOSS
 When a magnetizing material is periodically magnetized and demagnetized using an
alternate current, Energy is absorbed by the material that gets converted to heat
 Energy lost per cycle – (Area under the B-H curve)
 Power loss due to Hysteresis –
 Kh and n depends on the core material
 loss is directly proportional to frequency
HdB
Kh(Bm)nf
Use cores made of materials
that have thin hysteresis loop

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ELECTROMAGNETIC INDUCTION
 EMF (Electro-Motive-Force) is induced in a multi-turn coil when the magnetic flux
passing through the coil varies with time
 First discovered by Michael Faraday in 1831
 The induced EMF depends on the rate of change
of total flux linkages with the coil
𝑒 = −𝑁 𝑑Φ
𝑑𝑡
Induced emf
Lenz law
The polarity of the voltage induced by a
changing flux tends to oppose the change in
flux that produced the induced voltage
Circuit is closed and thus
current flows

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ELECTROMAGNETIC INDUCTION
The principle of
electromagnetic induction is
also involved in power
generation when the armature
of a DC Machine is rotated
quite fast in the presence of a
radial magnetic field
POWER GENERATION
The current reverses its direction
whenever there is a change in
the direction of motion of the
magnet

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v= 𝐿
𝑑𝑖
𝑑𝑡
= 𝑁
𝑑∅
𝑑𝑡
Consider an inductor fed by a time varying current. An EMF is induced across the
inductor governed by the equation
On solving the above equation, we get a relation
𝐿 = 𝑁
∅
𝑖
As Φ=Ni/R we get
𝐿 =
𝑁2
𝑅 Reluctance
Number of flux linkages per ampere
1 Henry = 1 Weber per ampere
INDUCTANCE OF AN INDUCTOR

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 Consider an iron core that has a primary coil
and a secondary coil
 AC Sine wave is fed through the primary coil
 The current in the primary coil produces a
magnetic field and hence flux lines
 The magnetic flux has a sinusoidal nature
and is this variable flux travels through the
soft iron core
 This variable flux cuts the secondary coil
and induces and EMF in it that follows the
Lenz rule
 If a load is connected across the
secondary, time varying current
flows in the secondary coil
MAGNETICALLY COUPLED COILS
N1 N2

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 Suppose the primary winding having Inductance L1 has N1 turns and secondary winding
having Inductance L2 has N2 turns
 Since a time varying current in the primary induces a voltage across the secondary, we
say that the 2 coils are magnetically coupled
 The flux that is setup in the core on account of the current in the primary is given by
 Neglecting flux leakages, the same flux links the secondary coil inducing an EMF across it
Φ=N1i/R Reluctance
𝑒 = 𝑁2
𝑑Φ
𝑑𝑡
𝑒 =
𝑁1 𝑁2
𝑅
𝑑𝑖
𝑑𝑡
On solving
M=N1N2/R
Mutual Inductance
between the coils
MUTUAL INDUCTANCE

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DOT CONVENTIONS

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TRANSFORMERS
 A transformer is a magnetic circuit consisting of 2 coils wound on a common iron core
 More than 2 windings can also be used
 Used in efficient transfer of Electric Power from the Generating station to our homes
 2 types – Step Up and Step down
 Step Up : Steps up the voltage at lower currents ( v x I = constant ) ( Neglecting leakage flux)
 Step Down : Steps down the voltage but at a higher current
 Voltages are stepped up prior to transmission so that the Copper losses are minimal
 Used in Electronic, Control and Communication systems
 Used for isolating 2 circuits as there is a magnetic coupling between the two and no
physical contact
 Used for impedance matching to have maximum power transfer from source to load

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TRANSFORMERS
Circuit symbol
 There may be connections to both windings so i1 and i2 both can be non zero
 As a result, i1 that passes through L1 produces a voltage L1
𝑑𝑖1
𝑑𝑡 and i2 that passes
through L2 induces a voltage M12
𝑑𝑖2
𝑑𝑡 across the primary
 Total voltage across the primary :
v1 = 𝐋 𝟏
𝒅𝒊 𝟏
𝒅𝒕 + 𝐌 𝟏𝟐
𝒅𝒊 𝟐
𝒅𝒕
M12
i1
i2
L1 L2
M21

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v2 = 𝐋 𝟐
𝒅𝒊 𝟐
𝒅𝒕 + 𝐌 𝟐𝟏
𝒅𝒊 𝟏
𝒅𝒕 Similarly the voltage across the secondary winding is
 𝐌 𝟐𝟏 = 𝐌 𝟏𝟐 = M = N1N2 / R
 The Energy stored in the form of magnetic field in the transformer is given by
𝒘 𝒕 =
𝟏
𝟐
𝑳 𝟏 𝒊 𝟏
𝟐 𝒕 +
𝟏
𝟐
𝑳 𝟐 𝒊 𝟐
𝟐 𝒕 + 𝑴𝒊 𝟏 𝒕 𝒊 𝟐(𝒕)
 As a transformer works on AC, the currents and voltages are all phasors, we
can represent the transformer equations as follows
𝑽 𝟏 = 𝒋𝝎𝑳 𝟏 𝑰 𝟏 + 𝒋𝝎𝑴𝑰 𝟐
𝑽 𝟐 = 𝒋𝝎𝑳 𝟐 𝑰 𝟐 + 𝒋𝝎𝑴𝑰 𝟏
𝑣 = 𝐿
𝑑𝑖
𝑑𝑡
corresponds to 𝑽 = 𝑗𝜔𝐿1 in the frequency domain
TRANSFORMERS

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TRANSFORMER CIRCUIT REPRESENTATION
 The transformer can be represented by 3
uncoupled inductors as shown here
TRANSFORMER LOSSES
HYSTERESIS LOSS EDDY CURRENT LOSSES
Energy dissipation in the form of heat in
the core of the t/f on account of rapid
magnetization and demagnetization
As a core is a conductor and a time
varying magnetic flux will pass
through it, an EMF hence circulating
currents are generated in the core
that lead to I2R losses (core heating)

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COUPLING COEFFICIENT
 It is the measure of the magnetic coupling between the 2 coils
 Denoted by k
 0<k<1
 𝑘 =
𝑀
√𝐿1
𝐿2
 Coupling coefficient depends upon
 Permeability of the core material
 Number of turns in each coil
 Relative position and the dimensions of the 2 coils
 Loosely Coupled T/F -> k=0 (almost) (Air Core T/F)
 Tightly Coupled T/F -> k=1 (almost) (Iron Core T/F)

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IDEAL TRANSFORMER
k=1 perfect coupling
L1 , L2 = ∞
no losses
𝑽 𝟏 = 𝑗𝜔𝐿1 𝑰 𝟏 + 𝑗𝜔𝑀𝑰 𝟐
𝑽 𝟐 = 𝑗𝜔𝐿2 𝑰 𝟐 + 𝑗𝜔𝑀𝑰 𝟏
i1
i2
L1 L2
k=1
V1 V2
I1 I2
 Figure shows the circuit symbol for an ideal t/f (k=1)
 The phasor relationship is as follows
 As 𝑀 = √𝐿1 𝐿2 , we can write V2 in terms of V1 as
 Turns Ratio (N)
Ratio of secondary to primary turns
𝑁 = 𝐿2
𝐿1
= 𝑁2
2
/𝑅
𝑁1
2
/𝑅 =
𝑁2
𝑁1
𝑽2 = 𝐿2
𝐿1
𝑽1
N
 N>1 : Step Up T/F
 N<1 : Step Down T/F
 N=1 : Isolation T/F

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IDEAL TRANSFORMER MODEL
A transformer with perfect coupling is said to be ideal if L1 and L2 approach ∞ and the
turns ratio remains constant
For an ideal t/f 𝑽2 = 𝑁𝑽1
𝑰2 = −𝑰1/𝑁
+
+-
--
+
𝑰1
𝑰2
𝑽1 𝑽2
𝑽2/𝑁 𝑰1/𝑁+
-
--
++
𝑰1
𝑰2
𝑽1 𝑽2𝑁𝑰2 𝑁𝑽1
IDEAL TRANSFORMER MODEL ALTERNATE IDEAL TRANSFORMER MODEL

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IDEAL TRANSFORMER AS A LOSSLESS DEVICE
Instantaneous power absorbed by the primary winding : 𝒑 𝟏 = 𝒗 𝟏 𝒊 𝟏
Instantaneous power absorbed by the secondary winding : 𝒑 𝟐 = 𝒗 𝟐 𝒊 𝟐
Total Instantaneous power absorbed by the T/F: p = p1 + 𝑝2
𝑝 = 𝑣1 𝑖1 + 𝑣2 𝑖2 𝑝 = 𝑣1 𝑖1 + 𝑁𝑣1
−𝑖1
𝑁
𝐩 = 𝟎
Since the instantaneous power is 0, the average power and the energy stored = 0
IDEAL TRANSFORMER IS A LOSSLESS DEVICE