Transcript File

Magnetism and
Electromagnetism
Engr. Faheemullah Shaikh
Wire Coil
• Notice that a
current carrying
coil of wire will
produce
a
perpendicular field
Magnetic Field: Coil
• A series of coils produces a field similar
to a bar magnet – but weaker!
Magnetic Field: Coil

Magnetic Field
Flux Ф can be increased by
increasing the current I,
Ф
I
I

Magnetic Field
Flux Ф can be increased by
increasing the number of turns N,
Ф
N
I
N

Magnetic Field
Flux Ф can be increased by
increasing the cross-section area
of coil A,
Ф
N
A
I
A

Magnetic Field
Flux Ф can be increased by
increasing the cross-section area
of coil A,
Ф
N
I
A
A

Magnetic Field
Flux Ф is decreased by increasing
the length of coil l,
Ф
l
N
I
A
1
l

Magnetic Field
Therefore we can write
equation for flux Ф as,
Ф
NIA
l
l
N
or
I
μ0 NIA
Ф =
l
A
an
Magnetic Field
μ0 NIA
Ф =
l
Where μ0 is vacuum or non-magnetic material permeability
μ0 = 4π x 10-7 H/m
Solenoid
If a coil is wound on a steel rod and connected
to a battery, the steel becomes magnetized
and behaves like a permanent magnet.
Magnetic Field: Coil
• Placing a ferrous material inside
the coil increases the magnetic
field
• Acts to concentrate the field also
notice field lines are parallel
inside ferrous element
• ‘flux density’ has increased
Magnetic Field

By placing a magnetic material
inside the coil,
μ NIA
Ф =
l
N
I
A
l
Where
μ
is
the
permeability
of
the
magnetic material (core).
Magnetic Field

By placing a magnetic material inside the coil,
μ NIA
Ф =
l
N
l
Where μ is the permeability of the
magnetic material (core).
I
A
Flux Density
Permeability
• Permeability μ is a measure of the ease by which a
magnetic flux can pass through a material
(Wb/Am)
• Permeability of free space μo = 4π x 10-7 (Wb/Am)
• Relative permeability:
Reluctance
• Reluctance: “resistance” to
flow of magnetic flux
Associated with “magnetic
circuit” – flux equivalent to
current
• What’s equivalent of voltage?
Magnetomotive Force, F
• Coil generates magnetic
field in ferrous torroid
• Driving force F needed to
overcome
torroid
reluctance
• Magnetic equivalent of
ohms law
Circuit Analogy
Magnetomotive Force
• The MMF is generated by the coil
• Strength related to number of turns and
current, measured in Ampere turns (At)
Magnetic Field Intensity
• The longer the magnetic path the greater the
MMF required to drive the flux
• Magnetomotive force per unit length is known
as the “magnetizing force” H
• Magnetizing force and flux density related by:
Electric circuit:
Emf = V = I x R
Magnetic circuit:
mmf = F = Φ x
=Hxl
= (B x A) x
= (B x A) x
=Bx
l
μ
l
μA
=Hxl
Magnetic Force On A Current –
Carrying Conductor
Magnetic Force On A Current – Carrying
Conductor
• The magnetic force (F) the conductor
experiences is equal to the product of its
length (L) within the field, the current I in the
conductor, the external magnetic field B and
the sine of the angle between the conductor
and the magnetic field. In short
F= BIL (sin)
The force on a current-carrying conductor
in a magnetic field :
• When a current-carrying conductor is placed in a magnetic
field, there is an interaction between the magnetic field
produced by the current and the permanent field, which
leads to a force being experienced by the conductor:
• The magnitude of the force on the conductor depends
on the magnitude of the current which it carries. The
force is a maximum when the current flows
perpendicular to the field (as shown in diagram A on
the left below), and it is zero when it flows parallel to
the field (as in diagram B, on the right):
Fleming's left hand rule (for electric motors)
Fleming's left hand rule shows the direction of
the thrust on a conductor carrying a current in
a magnetic field.
The left hand is held with the thumb, index
finger and middle finger mutually at right
angles.
The First finger represents the direction of the Field.
The Second finger represents the direction of the Current (in the classical
direction, from positive to negative).
The Thumb represents the direction of the Thrust or resultant Motion.
Fleming’s left-hand rule
• The directional relationship of
I in the conductor, the external
magnetic field and the force
the conductor experiences
I
B
F
Faraday’s Law
Faraday’s Law
Magnetic Field can produce an electric current in a closed
loop, if the magnetic flux linking the surface area of the loop
changes with time.
This mechanism is called “Electromagnetic Induction”
The electric Current Produced  Induced Current
Faraday’s Law
First Experiments
Conducting
loop
Sensitive
current meter
Since there is no battery or
other source of emf included,
there is no current in the circuit
Move a bar magnet toward
the loop, a current suddenly
appears in the circuit
The current disappears when
the bar magnet stops
If we then move the bar magnet away, a
current again suddenly appears, but now in
the opposite direction
Faraday’s Law
Discovering of the First Experiments
1. A current appears only if there is relative
motion between the loop and the magnet
2. Faster motion produces a greater current
3. If moving the magnet’s N-pole towards the
loop causes clockwise current, then moving
the N-pole away causes counterclockwise.
Faraday’s Law
An Experiment - Situation A
Constant flux
though the loop
Current in the coil
produces a
magnetic field B
DC current I, in coil produces a constant magnetic field, in turn
produces a constant flux though the loop
Constant flux, no current is induced in the loop.
No current detected by Galvanometer
Faraday’s Law
An Experiment - Situation B: Disconnect battery suddenly
Magnetic field
drops to zero
Deflection of
Galvanometer
needle
Sudden change of magnetic flux to zero causes a
momentarily deflection of Galvanometer needle.
Faraday’s Law
An Experiment - Situation C: Reconnect Battery
Sudden change of
magnetic flux
through the loop
Deflection of
Galvanometer needle in
the opposite direction
Magnetic field
becomes non-zero
Current in the coil
produces a
magnetic field B
Link: http://micro.magnet.fsu.edu/electromag/java/faraday/index.html
Faraday’s Law
Conclusions from the experiment
• Current induced in the closed loop when magnetic flux changes,
and direction of current depends on whether flux is increasing or
decreasing
• If the loop is turned or moved closer or away from the coil, the
physical movement changes the magnetic flux linking its surface,
produces a current in the loop, even though B has not changed
In Technical Terms
Time-varying magnetic field produces an electromotive
force (emf) which establish a current in the closed
circuit
Faraday’s Law
Electromotive force (emf) can be obtained through the
following ways:
1. A time-varying flux linking a stationary closed path. (i.e.
Transformer)
2. Relative motion between a steady flux and a close path.
(i.e. D.C. Generator)
3. A combination of the two above, both flux changing and
conductor moving simultaneously. A closed path may
consists of a conductor, a capacitor or an imaginary line in
space, etc.
Faraday’s Law
Faraday summarized this electromagnetic phenomenon
into two laws ,which are called the Faraday’s law
Faraday’s First Law
When the flux magnet linked to a circuit
changes, an electromotive force (emf) will
be induced.
Faraday’s Law
Faraday’s Second Law
The magnetic of emf induced is equal to
the time rate of change of the linked
magnetic flux .
(volts)
Minus Sign  Lenz’s Law
Indicates that the emf induced is in such a direction as to
produces a current whose flux, if added to the original
flux, would reduce the magnitude of the emf
Faraday’s Law
Minus Sign  Lenz’s Law
The induced voltage acts to produce an opposing flux
Faraday’s Law
Minus Sign  Lenz’s Law
The induced voltage acts to produce an opposing flux
Faraday’s Law
Minus Sign  Lenz’s Law
The induced voltage acts to produce an opposing flux
Heinrich F.E. Lenz
•
•
•
•
Russian physicist
(1804-1865)
1834 Lenz’s Law
There is an induced current in
a closed conducting loop if
and only if the magnetic flux
through the loop is changing.
• Indicates that the emf induced
is in such a direction as to
produces a current whose flux,
if added to the original flux,
would reduce the magnitude
of the emf
There is an induced current in a closed conducting loop if and only if
the magnetic flux through the loop is changing. The direction of the
induced current is such that the induced magnetic field always opposes
the change in the flux.
Right Hand Rule
• If you wrap your
fingers around the
coil in the direction
of the current, your
thumb points
north.
2
Direction of induced current
b
Lenz's law
In both cases, magnet
moves against a
force.
Work is done during
the motion & it is
transferred as
electrical energy.
Induced I always flows to oppose the
movement which started it.
Applications of Magnetic Induction
• Magnetic Levitation (Maglev) Trains
– Induced surface (“eddy”) currents produce field in
opposite direction
 Repels magnet
 Levitates train
S
N
“eddy” current
rails
– Maglev trains today can travel up to 310 mph
 Twice the speed of Amtrak’s fastest conventional
train!
Liner induction
0-70 mph in 3 sec
FALLING MAGNET
• The copper tube "sees" a
changing magnetic field
from the falling magnet.
This changing magnetic
field induces a current in
the copper tube.
• The induced current in
the copper tube creates
its own magnetic field
that
opposes
the
magnetic
field
that
created it.
Faraday’s Law
Fleming Right Hand Rule
Direction of Induced e.m.f, Magnetic Flux, Conductor Motion
Fore Finger
Direction of
Field Flux
Middle Finger
Direction of Induced
emf or Current Flow
Thumb
Direction of
Conductor Motion
Fleming's right hand rule (for generators)
Fleming's right hand rule shows the direction
of induced current flow when a conductor
moves in a magnetic field.
The right hand is held with the thumb, first
finger and second finger mutually at right
angles, as shown in the diagram
The Thumb represents the direction of Motion of the conductor.
The First finger represents the direction of the Field.
The Second finger represents the direction of the induced or generated Current
(in the classical direction, from positive to negative).
Leakage Flux and Fringing
fringing
Leakage flux
Leakage Flux
It is found that it is impossible to confine all the flux
to the iron path only. Some of the flux leaks through
air surrounding the iron ring.
Leakage coefficient λ =
Total flux produced
Useful flux available
Fringing
Spreading of lines of flux at the edges of the
air-gap. Reduces the flux density in the airgap.
Hysteresis loss
Materials before applying m.m.f (H), the polarity of the
molecules or structures are in random.
After applying m.m.f (H) , the polarity of the molecules or
structures are in one direction, thus the materials become
magnetized. The more H applied the more magnetic flux (B
)will be produced
When we plot the mmf (H) versus the magnetic flux (B) will produce a curve so
called Hysteresis loop
1. OAC – when more H applied, B
increased until saturated. At this
point no increment of B when we
increase the H.
2. CD- when we reduce the H the B
also reduce but will not go to zero.
3. DE- a negative value of H has to
applied in order to reduce B to zero.
4. EF – when applying more H in the
negative direction will increase B in
the reverse direction.
5. FGC- when reduce H will reduce B
but it will not go to zero. Then by
increasing positively the also
decrease and certain point it again
change the polarity to negative until
it reach C.
Hysteresis Loss
• Empirical equation
Summary : Hysteresis loss is proportional to f and
ABH
Eddy current
When a sinusoidal current
enter the coil, the flux  also
varies sinusoidally according
to I. The induced current will
flow in the magnetic core.
This current is called eddy
current.
This
current
introduce the eddy current
loss. The losses due to
hysteresis and eddy-core
totally called core loss. To
reduce eddy current we use
laminated core
metal
insulator
Eddy Current Loss
Empirical equation
Core Loss
• Core Loss
Pc  Ph  Pe
where Ph  hysteresisloss
Pe  eddy current loss
Inductance
• A changing magnetic flux induces an e.m.f. in
any conductor within it
• Faraday’s law:
The magnitude of the e.m.f. induced in a circuit is
proportional to the rate of change of magnetic flux
linking the circuit
• Lenz’s law:
The direction of the e.m.f. is such that it tends to
produce a current that opposes the change of flux
responsible for inducing the e.m.f.
• When a circuit forms a single loop, the e.m.f.
induced is given by the rate of change of the flux
• When a circuit contains many loops the resulting
e.m.f. is the sum of those produced by each loop
• Therefore, if a coil contains N loops, the induced
voltage V is given by
where d/dt is the rate of change of flux in Wb/s
V  N dΦ
dt
• This property, whereby an e.m.f. is induced as a
result of changes in magnetic flux, is known as
inductance
TYPES OF INDUCED EMF
• Statically induced emf
– Conductor remains stationary and flux linked with it is
changed (the current which creates the flux changes i.e
increases or decreases)
TYPES
– Self induced
– Mutually induced
TYPES OF INDUCED EMF
• Dynamically induced emf
– Field is stationary and conductors cut across it
– Either the coil or the magnet moves.
Self-Inductance
Consider a coil connected to resistance R and voltage V.
When switch is closed, the rising current I increases flux,
producing an internal back emf in the coil.
Increasing I
Lenz’s Law:
The back emf (red
arrow)  must oppose
change in flux:
R
Decreasing I
R
Inductance
The back emf E induced in a coil is proportional to the
rate of change of the current DI/Dt.
Di
E  L ;
Dt
L  inductance
An inductance of one henry (H)
means that current changing at the
rate of one ampere per second will
induce a back emf of one volt.
Increasing Di/ Dt
R
1V
1 H
1 A/s
Example 1: A coil having 20 turns has an induced emf of
4 mV when the current is changing at the rate of 2 A/s.
What is the inductance?
Di/ Dt = 2 A/s
4 mV
R
Di
E  L ;
Dt
(0.004 V)
L
2 A/s
E
L
Di / Dt
L = 2.00 mH
Note: We are following the practice of using lower
case i for transient or changing current and upper
case I for steady current.
Calculating the Inductance
Recall two ways of finding E:
D
E  N
Dt
Di
E  L
Dt
Increasing Di/ Dt
R
Setting these terms equal gives:
D
Di
N
L
Dt
Dt
Thus, the inductance L can
be found from:
Inductance L
N
L
I
Inductance of a Solenoid
The B-field created by a current I for
length l is:
Solenoid
l
B
B
 0 NI
and  = BA
R
Inductance L
Combining the last two equations
gives:

0 NIA
L
N
L
I
0 N 2 A
Example 2: A solenoid of area 0.002 m2 and length 30
cm, has 100 turns. If the current increases from 0 to 2 A
in 0.1 s, what is the inductance of the solenoid?
First we find the inductance of the solenoid:
0 N 2 A (4 x 10-7 TAm )(100)2 (0.002 m2 )
L

0.300 m
l
L = 8.38 x 10-5 H
A
R
Note: L does NOT depend on
current,
but
on
physical
parameters of the coil.
Example 2 (Cont.): If the current in the 83.8-H
solenoid increased from 0 to 2 A in 0.1 s, what is
the induced emf?
l
L = 8.38 x 10-5 H
A
R
Di
E  L
Dt
(8.38 x 10-5 H)(2 A - 0)
E
0.100 s
E  1.68 mV
Energy Stored in an Inductor
At an instant when the current is changing at
Di/Dt, we have:
Di
EL ;
Dt
Di
P  Ei  Li
Dt
R
Since the power P = Work/t, Work = P Dt. Also the
average value of Li is Li/2 during rise to the final current
I. Thus, the total energy stored is:
Potential energy
stored in inductor:
U  12 Li 2
Example 3: What is the potential energy stored in a 0.3
H inductor if the current rises from 0 to a final value of
2 A?
U  12 Li 2
L = 0.3 H
R
U  12 (0.3 H)(2 A)2  0.600 J
U = 0.600 J
I=2A
This energy is equal to the work done in reaching
the final current I; it is returned when the current
decreases to zero.
The R-L Circuit
An inductor L and resistor R are
connected in series and switch 1 is
closed:
V – E = iR
Di
EL
Dt
Di
V  L  iR
Dt
V
S1
S2
i
R
L
E
Initially, Di/Dt is large, making the back emf large and the
current i small. The current rises to its maximum value I
when rate of change is zero.
The Rise of Current in L
V
i  (1  e  ( R / L ) t )
R
i
I
At t = 0, I = 0
At t = , I = V/R
The time constant
t:
L
t 
R
0.63 I
Current Rise
t
Time, t
In an inductor, the current will rise to 63% of its
maximum value in one time constant t = L/R.
The R-L Decay
Now suppose we close S2 after energy is
in inductor:
E = iR
For current decay
in L:
Di
EL
Dt
Di
L  iR
Dt
V
S1
S2
i
R
L
E
Initially, Di/Dt is large and the emf E driving the current is at
its maximum value I. The current decays to zero when the
emf plays out.
The Decay of Current in L
V  ( R / L )t
i e
R
i
I
At t = 0, i = V/R
At t = , i = 0
The time constant
t:
L
t 
R
Current Decay
0.37 I
t
Time, t
In an inductor, the current will decay to 37% of its
maximum value in one time constant t.
Example 5: The circuit below has a 40-mH inductor
connected to a 5-W resistor and a 16-V battery. What
is the time constant and what is the current after one
time constant?
L 0.040 H
t 
R
5W
16 V
5W
R
Time constant:
L = 0.04 H
After time
t:
i = 0.63(V/R)
t = 8 ms
V
i  (1  e  ( R / L ) t )
R
 16V 
i  0.63 

 5W 
i = 2.02 A
Inductors in Series and Parallel
• When several inductors are connected
together their effective inductance can be
calculated in the same way as for resistors –
provided that they are not linked magnetically
• Inductors in Series
• Inductors in Parallel
Mutual Inductance
• When two coils are linked magnetically then a
changing current in one will produce a changing
magnetic field which will induce a voltage in the other
– this is mutual inductance
• When a current I1 in one circuit, induces a voltage V2
in another circuit, then
dI
V M 1
dt
2
where M is the mutual inductance between the circuits. The
unit of mutual inductance is the Henry (as for self-inductance)
• The coupling between the coils can be
increased by wrapping the two coils around a
core
– the fraction of the magnetic field that is coupled is
referred to as the coupling coefficient
• Coupling is
transformers
particularly
important
– the arrangements below give
coefficient that is very close to 1
a
in
coupling