MODULE 2 EE100 Basics of Electrical Engineering
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MAGNETIC CIRCUITS
Magnet is a material which has the property of attracting iron pieces. This property is called
magnetism.
When a magnet is suspended freely, it will align itself in the north-south direction of earth. The
tip of the magnet pointing north is called north pole and the tip pointing the south is called the
south pole of the magnet.
Magnetic Field : It is the region around a magnet within which the influence of the
magnet can be experienced.
Magnetic Lines of Force : These are the imaginary lines surrounding the magnet which
represent the magnetic field. The lines of force always flown from North pole to South
pole of the magnet.
Magnetic Flux (ฯ) : The total number of magnetic lines of force existing in a particulat
magnetic field is called magnetic flux. The unit of flux is Weber (Wb).
1 Weber = 108 lines of force.
Magnetic Flux Density (B) : It is defined as the flux per unit area.
B = ฯ
๐ ; Unit : Wb/m2 or Tesla (T)
Magnetic Field Strength (H) : The force experienced by a unit north pole when placed
at any point in magnetic field is known as Magnetic field strength at that point.
H = Ampere Turns
Length=
N I
๐ ; Unit = AT/m
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Where N = number of turns in the coil
I = current through the coil
l = length of magnetic field
Permeability : The ease with which the magnetic material forces the magnetic flux
through a medium.
Absolute Permeability (ยต) : It is the ratio of magnetic field density (B) in a particular
medium to the magnetic strength (H) producing this flux.
ยต = ๐ต
๐ป ; Unit = H/m.
Permeability of free space or vaccum (ยต0) : It is the ratio of magnetic field density
(B) to magnetic field strength (H) in vaccum.
ยต0 = 4ฯ ร 10-7 H/m
Relative permeability (ยตr) : It is the ration of magnetic flux density in a particular
medium to magnetic flux density in free space
ยตr = ๐ต
๐ต0
ยตr of air = 1 and
ยตr = ยต0 ยตr
Magneto Motive Force (mmf) : It is the driving force behind the magnetic circuit
mmf = N I; Unit = Ampere Turns (AT).
Reluctance (S) = Opposition to flow of flux is called reluctance (S).
S ฮฑ ๐
a ;
where l = length of magnetic field
a = area of crosssection of wire
S = K ๐
a ; but K =
1
ยต ;
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Therefore, S = ๐
ยต a
Or, S = ๐
ยต0ยต๐ a ; Unit = AT/Wb
Reluctance S is also defined as S = mmf
flux =
N I
ฯ AT/Wb
Permeance : It is the reciprocal of reluctance
Permeance = 1
Reluctance
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ELECTROMAGNETIC INDUCTION
FARADAYS LAWS OF ELECTROMAGNETIC INDUCTION
Michael Faraday, a British Scientist as stated 2 laws related to electromagnetic induction.
Faradayโs First Law of Electromagnetic Induction:
โWhenever the flux linking with a coil changes, an EMF is induced in the coilโ
Faradayโs Second Law of Electromagnetic Induction :
โThe magnitude of induced EMF is directly proportional to rate of change of flux
linkagesโ
e = dฯ
dt
Where, e = induced EMF in the coil
ฮฆ = flux linking with the coil.
If the coil has N turns, then
e = N dฯ
dt
Lenzโs Law :
โThe direction of induced EMF is such that it always opposes the causeโ
By considering the effect due to Lenzโs Law, we can write induced EMF as
e = -N dฯ
dt
NATURE OF INDUCED EMF
Depending upon the nature of inducing EMF, the induced EMF is classified into two :
Dynamically Induced EMF and Statically Induced EMF.
Dynamically Induced EMF
Here the EMF is induced by moving the flux with respect to the coil or by moving the
coil with respect to flux. In either way there involves the physical movement of coil or
the flux.
Magnitude of Dynamically Induced EMF
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Consider a conductor of length l moving perpendicular to a magnetic field with flux density
B with a velocity of v m/s.
Let the conductor moves a very small distance of dx in a small time dt.
The area, A covered by the conductor = l . dx
Flux dฯ in the area A = flux density ร area
i.e dฯ = B.l.dx
Induced EMF, e = dฯ
dt =
B.๐.dx
dt but
dx
dt = velocity, v
Therefore, e = B.l.v Unit = volts V
If the motion is not perpendicular i.e the motion is at an angle ฮธ from the vertical plane, the
induced EMF is given by
e = B.l.v sin ฮธ Volts
Statically Induced EMF
If the EMF is induced without the physical movement of coil or magnet, then it is called
statically induced EMF.
Consider an electromagnet as shown below.
If the current in the coil is reversed, the direction of magnetic field also gets reversed.
Thus by periodically changing the direction of current, the flux can also be changed.
This periodically changing flux is linked with the same coil and generates an EMF in
the coil. This is called self induced EMF.
In other words โEMF induced in a coil due to change in its own flux linked with it is
called self indued EMFโ.
Self Inductance
The direction of self induced EMF is such that it always oppose the cause producing it.
Here the cause is change in direction of current. So self inductance can be defined as
โthe property of the coil to oppose any change in current through itโ.
Self induced EMF is given by e = -N dฯ
dt
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ฯ = ฯ
I ร I
therefore, dฯ
dt =
ฯ
I ร
dI
dt
So, self induced EMF, e = -N ฯ
I dI
dt
Here N ฯ
I = L;
Hence we can write self induced EMF as
e = -L dI
dt Volts
Where L is the co efficient of self inductance.
Co-efficient of Self Inductance
L = N ฯ
I; But flux, ฯ =
mmf
relucatnce =
N I
S
So we can write L = N2 I
I S =
N2
S; But reluctance, S =
๐
ยต0ยต๐ a
So L = N2ยต0ยต๐ a
๐
Where N = number of turns
a = area of cross section
l = length of magnetic circuit.
Mutually Induced EMF
If flux produced by one coil is linking with other coil and the change in this flux induces
an EMF in the second coil , such an EMF is called mutually induced EMF.
Consider the figure below.
Let N1 = Number of turns in coil 1;
ฯ1 = flux produced in coil 1;
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I1 = current flowing through coil 1
N2 = Number of turns in coil 2;
ฮฆ2 = flux linking with coil 2;
I2 = current induced in coil 2
The EMF induced in coil 2 id given by
e2 = -N2 dฯ2
dt
ฯ2 = ๐2
I1 ร I1
Therefore, e2 = - ๐2๐2
I1 dI1
dt
Here, ๐2๐2
I1 = M ; M is the co-efficient of mutual inductance
Only a portion of the flux ฯ1 is linking with coil 2. So we can write ฯ2 = K. ฯ1
So M can be written as M = ๐2๐พ๐1
I1 ; ๐1 =
N1 I1S
Therefore, M = ๐2๐พN1 I1
I1S =
๐2๐พN1
S ; S =
๐
ยต0ยต๐ a
So, M = ๐2๐พN1ยต0ยต๐ a
๐ ; if K = 1
We can write M = ๐2๐1ยต0ยต๐ a
๐
Co-efficient of Coupling
M can be written as M = ๐2๐พ๐1
I1 ------------ (1)
M can also be written as M = ๐1๐พ๐2
I2 ------------ (2)
Multiplying equations 1 and 2, we get
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M2 =
๐2๐พ๐1 ๐1๐พ๐2
I1I2
i.e M2 =
๐พ๐1๐1 . ๐พ๐2๐2
I1 . I2
in the above equation put N1 ฯ1
I1 = L1 and
N2 ฯ2
I2 = L2
i.e M2 = K2 L1.L2
or M = ๐พโL1L2
i.e K = M
โL1L2
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AC FUNDAMENTALS
INTRODUCTION
In Direct Current (DC), the magnitude is constant and direction remains same with
respect to time. Whereas in an Alternating Current (AC), the current changes both its
magnitude and direction periodically.
GENERATION OF AC VOLTAGE
The machine used to produce electricity is called generators. The generator which produce ac
voltages is called alternators. The working principle behind working of alternators is Faradays
Laws of Electromagnetic Induction. The figure below shows the simple diagram of an
alternator.
It consist of two permanent magnets, north and south. A single turn coil is placed in between
the magnets. The coil is connected to a resistor through slip rings and brushes. As the coil is
MODULE 2 EE100 Basics of Electrical Engineering
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rotated, coil cuts the magnetic field lines. This results in generation of alternating EMF as stated
by faradays law.
The induced EMF is given by e = B l v sinฮธ
Different instants during the generation of alternating EMF is shown below.
INSTANT โ 1 (ฮธ = 0)
At this instant the coil is along the horizontal axis. The velocity of the coil is parallel to the field
as shown in figure. So the angle ฮธ = 0O and sin 0 = 0. So during this instant the induced EMF is zero.
INSTANT โ 2 (ฮธ < 90)
At this instant the coil is rotated by an angle ฮธ. During this instant the induced EMF generated
in the coil whose magnitude is proportional to v sin ฮธ which is increasing. That is during this instant
the induced EMF gradually increases from zero to maximum value.
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INSTANT โ 3 (ฮธ = 90)
At this instant the coil is rotated by an angle 90. Since sin 90 = 1, the magnitude of the EMF
generated is maximum.
Induced EMF, e = B l v
INSTANT โ 4 (ฮธ > 90)
At this instant the coil is rotated by an angle ฮธ which is greater than 90. During this instant
the magnitude of induced EMF is proportional to the value v sinฮธ which is decreasing. That is during
this instant the induced EMF gradually decreases from maximum value to zero.
INSTANT โ 5 (ฮธ =180)
At this instant the coil has rotated over an angle of 180o and is along the horizontal axis. The
velocity of the coil is parallel to the field. Here sin 180 = 0. So during this instant the induced EMF is
zero.
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INSTANT โ 6 (ฮธ > 180)
At this instant the coil is rotated by an angle ฮธ greater than 180. During this instant the induced
EMF generated in the coil whose magnitude is proportional to v sin ฮธ which is negative and
increasing in negative direction. That is during this instant the induced EMF increasing in
negative direction.
The waveform for all the 6 instants is shown below.
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STANDARD TERMS RELATED TO ALTERNATING QUANTITY
Before further analysis of alternating quantity, it is necessary to be familiar with the
different terms and different parameters of the alternating quantity. For this, consider the
waveform of one cycle of alternating EMF shown in figure 2d.
Instantaneous Value
Since the magnitude of an alternating current varies with time as shown in figure 3, the
value of the current at a particular instant is called instantaneous value. In the figure, e1 and e2
are the instantaneous value of alternating emf at instants t1 and t2 respectively.
Waveform
The graph of instantaneous values of an alternating quantity plotted against time is
called its waveform.
Cycle
A set of positive and negative half cycles of an alternating quantity is called a cycle.
One cycle = 360o or 2ฯ radians.
Time Period
Time taken by an alternating quantity to complete on cycle is called its time period. It
is denoted by T seconds.
Frequency
It is the number of cycles of alternating quantity in one second. It is denoted by f and it
is measured in cycles/second, which is also known as Hertz (Hz). Frequency can be also stated
as reciprocal of time period.
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f = ๐
๐ป Hz
Amplitude
The maximum value attained by the alternating quantity during positive or negative
half cycle is called amplitude. The amplitude of voltage is denoted by Em and amplitude of
current is denoted by Im.
Angular Frequency
It is the frequency expressed in electrical radians per second. Since one cycle of
alternating quantity corresponds to 2ฯ radians, the angular frequency can be expressed as (2ฯร
cycles/sec). It is denoted by ฯ and its unit is radians/second.
ฯ = 2ฯf rad/sec
or
ฯ = ๐๐
๐ป rad/sec
Equation For Instantaneous Voltage And Current
If the amplitude of voltage and amplitude of current of an AC supply with angular
frequency ฯ are Em and Im, then the instantaneous values of voltage and current can be given
as
e = Em sin(ฯt)
i = Im sin(ฯt)
RMS Value or Effective Value
The rms value or effective value of an alternating current is given by that steady
current (DC) which, when flowing through a given circuit for a given time, produces same
amount of heat as produced by alternating current, which when flowing through the same
circuit for the same time.
All the values of magnitudes of alternating quantities are expressed in rms values. The
standard value of 230V and 415V are rms values of alternating voltage.
All the ammeters and voltmeters measure the rms value of the current and voltage.
RMS value = ๐
โ๐ [Maximum Value]
Irms = ๐ฐ๐
โ๐
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Vrms = ๐ฝ๐
โ๐
Average value
The average value of an alternating quantity is defined as that value which is obtained
by averaging all the instantaneous values over a period of half cycle. For a complete cycle, the
average value is zero.
Average value = 0.637 [Maximum Value]
Iavg = 0.637 Im
Vavg = 0.637 Vm
Form Factor
Form factor of the alternating quantity is the ratio of rms value to average value.
Kf = ๐ฐ๐๐๐
๐ฐ๐๐๐ =
๐.๐๐๐ ๐ฐ๐
๐.๐๐๐ ๐ฐ๐ = 1.11
or
Kf = ๐ฝ๐๐๐
๐ฝ๐๐๐ =
๐.๐๐๐ ๐ฝ๐
๐.๐๐๐ ๐ฝ๐ = 1.11
Peak Factor
Peak factor of the alternating quantity is the ratio of maximum value to rms value.
KP= ๐ฐ๐
๐ฐ๐๐๐ =
๐ฐ๐
๐.๐๐๐ ๐ฐ๐ = 1.414
or
KP = ๐ฝ๐
๐ฝ๐๐๐ =
๐ฝ๐
๐.๐๐๐ ๐ฝ๐
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