8/19/2019 Generation of Sinusoidal Waveforms
http://slidepdf.com/reader/full/generation-of-sinusoidal-waveforms 1/10
Generation of Sinusoidal Waveforms
Review Electromagnetism
1. An electric current flowing through a conductor can be used to generate a magneticfield around itself,
2. If a single wire conductor is moved or rotated within a stationary magnetic field, an
“EM!, "Electro#Motive orce$ will be induced within the conductor due to this
movement.
We learnt that a relationship exists between Electricity
and Magnetism giving us, as Michael Faraday
discovered the effect of “Electromagnetic nduction!
and it is this basic principal that is used to generate
a %inusoidal &aveform"
If a conductor moves in 'arallel with the magnetic
field as in the case of 'oints A and (, no lines of flu)
are cut and no EM is induced into the conductor, but
if the conductor moves at right angles to the magnetic
field as in the case of 'oints * and +, the ma)imum
amount of magnetic flu) is cut 'roducing the
ma)imum amount of induced EM"
#lso, as the conductor cuts the magnetic field at different
angles between points # and $, % and &%o the amount of
induced EMF will lie somewhere between this 'ero and
maximum value" (hen the amount of emf inducedwithin a conductor depends on the angle between the
conductor and the magnetic flux as well as the strength
of the magnetic field"
#n #$ generator uses the principal of Faraday)s electromagnetic induction to convert a
mechanical energy such as rotation, into electrical energy, a %inusoidal &aveform" # simple
generator consists of a pair of permanent magnets producing a fixed magnetic field between a
north and a south pole" nside this magnetic field is a single rectangular loop of wire that can be
rotated around a fixed axis allowing it to cut the magnetic flux at various angles as shown below"
8/19/2019 Generation of Sinusoidal Waveforms
http://slidepdf.com/reader/full/generation-of-sinusoidal-waveforms 2/10
*asic Single $oil #$ Generator
#s the coil rotates anticloc+wise around the central axis which is perpendicular to the magnetic
field, the wire loop cuts the lines of magnetic force set up between the north and south poles at
different angles as the loop rotates" (he amount of induced EMF in the loop at any instant of
time is proportional to the angle of rotation of the wire loop"
#s this wire loop rotates, electrons in the wire flow in one direction around the loop" ow whenthe wire loop has rotated past the -.%o point and moves across the magnetic lines of force in the
opposite direction, the electrons in the wire loop change and flow in the opposite direction" (hen
the direction of the electron movement determines the polarity of the induced voltage"
So we can see that when the loop or coil physically rotates one complete revolution, or /0% o, one
full sinusoidal waveform is produced with one cycle of the waveform being produced for each
revolution of the coil" #s the coil rotates within the magnetic field, the electrical connections are
made to the coil by means of carbon brushes and slip1rings which are used to transfer the
electrical current induced in the coil"
(he amount of EMF induced into a coil cutting the magnetic lines of force is determined by the
following three factors"
• 2 Speed 3 the speed at which the coil rotates inside the magnetic field"
• 2 Strength 3 the strength of the magnetic field"
• 2 4ength 3 the length of the coil or conductor passing through the magnetic field"
8/19/2019 Generation of Sinusoidal Waveforms
http://slidepdf.com/reader/full/generation-of-sinusoidal-waveforms 3/10
We +now that the fre5uency of a supply is the number of times a cycle appears in one second and
that fre5uency is measured in 6ert'" #s one cycle of induced emf is produced each full
revolution of the coil through a magnetic field comprising of a north and south pole as shown
above, if the coil rotates at a constant speed a constant number of cycles will be produced per
second giving a constant fre5uency" So by increasing the speed of rotation of the coil the
fre5uency will also be increased" (herefore, fre5uency is proportional to the speed of rotation,7 8 ∝ 9 : where 9 ; r"p"m"
#lso, our simple single coil generator above only has two poles, one north and one south pole,
giving <ust one pair of poles" f we add more magnetic poles to the generator above so that it now
has four poles in total, two north and two south, then for each revolution of the coil two cycles
will be produced for the same rotational speed" (herefore, fre5uency is proportional to thenumber of pairs of magnetic poles, 7 8 ∝ = : of the generator where = ; is the number of “pairs
of poles!"
(hen from these two facts we can say that the fre5uency output from an #$ generator is>
Where> 9 is the speed of rotation in r"p"m" = is the number of “pairs of poles! and 0% converts it
into seconds"
nstantaneous ?oltage
(he EMF induced in the coil at any instant of time depends upon the rate or speed at which the
coil cuts the lines of magnetic flux between the poles and this is dependant upon the angle of
rotation, (heta 7 @ : of the generating device" *ecause an #$ waveform is constantly changing its
value or amplitude, the waveform at any instant in time will have a different value from its next
instant in time"
For example, the value at -ms will be different to the value at -"Ams and so on" (hese values are
+nown generally as the Instantaneous alues, or ?i (hen the instantaneous value of the
waveform and also its direction will vary according to the position of the coil within the
magnetic field as shown below"
8/19/2019 Generation of Sinusoidal Waveforms
http://slidepdf.com/reader/full/generation-of-sinusoidal-waveforms 4/10
Bisplacement of a $oil within a Magnetic Field
(he instantaneous values of a sinusoidal waveform is given as the “nstantaneous value ;
Maximum value x sin @ ! and this is generali'ed by the formula"
Where, ?max is the maximum voltage induced in the coil and @ ; Ct, is the angle of coil rotation"
f we +now the maximum or pea+ value of the waveform, by using the formula above the
instantaneous values at various points along the waveform can be calculated" *y plotting these
values out onto graph paper, a sinusoidal waveform shape can be constructed" n order to +eep
things simple we will plot the instantaneous values for the sinusoidal waveform at every Do andassume a maximum value of -%%?" =lotting the instantaneous values at shorter intervals, for
example at every /%o would result in a more accurate waveform construction"
Sinusoidal Waveform $onstruction
*oil Angle " - $ % D &% -/ -.% AA A% /- /0%
e ma).sin- % %"- -%% %"- % 1%"- 1-%% 1%"- 1%
8/19/2019 Generation of Sinusoidal Waveforms
http://slidepdf.com/reader/full/generation-of-sinusoidal-waveforms 5/10
(he points on the sinusoidal waveform are obtained by pro<ecting across from the various
positions of rotation between %o and /0%o to the ordinate of the waveform that corresponds to the
angle, @ and when the wire loop or coil rotates one complete revolution, or /0% o, one full
waveform is produced"
From the plot of the sinusoidal waveform we can see that when @ is e5ual to %o, -.%o or /0%o, the
generated EMF is 'ero as the coil cuts the minimum amount of lines of flux" *ut when @ is e5ual
to &%o and A%o the generated EMF is at its maximum value as the maximum amount of flux is
cut"
(herefore a sinusoidal waveform has a positive pea+ at &%
o
and a negative pea+ at A%
o
"=ositions *, B, F and 6 generate a value of EMF corresponding to the formula e ; ?max"sin@"
(hen the waveform shape produced by our simple single loop generator is commonly referred to
as a %ine &ave as it is said to be sinusoidal in its shape" (his type of waveform is called a sine
wave because it is based on the trigonometric sine function used in mathematics,
7 x7t: ; #max"sin@ :"
When dealing with sine waves in the time domain and especially current related sine waves the
unit of measurement used along the hori'ontal axis of the waveform can be either time, degrees
or radians" n electrical engineering it is more common to use theRadian as the angular
measurement of the angle along the hori'ontal axis rather than degrees" For
example, C ; -%% rads, or %% rads"
Hadians
(he Radian, 7rad: is defined mathematically as a 5uadrant of a circle where the distance
subtended on the circumference e5uals the radius 7r : of the circle" Since the circumference of a
circle is e5ual to AI x radius, there must be AI radians around a /0%o circle, so - radian ;
/0%oAI ; /0.o" n electrical engineering the use of radians is very common so it is important to
remember the following formula"
8/19/2019 Generation of Sinusoidal Waveforms
http://slidepdf.com/reader/full/generation-of-sinusoidal-waveforms 6/10
Befinition of a Hadian
Jsing radians as the unit of measurement for a sinusoidal waveform would give AI radians for
one full cycle of /0%o" (hen half a sinusoidal waveform must be e5ual to -I radians or <ust I 7pi:"
(hen +nowing that pi, I is e5ual to /"-DA or AAK, the relationship between degrees and radians
for a sinusoidal waveform is given as"
Helationship between Begrees and Hadians
#pplying these two e5uations to various points along the waveform gives us"
8/19/2019 Generation of Sinusoidal Waveforms
http://slidepdf.com/reader/full/generation-of-sinusoidal-waveforms 7/10
(he conversion between degrees and radians for the more common e5uivalents used in
sinusoidal analysis are given in the following table"
Helationship between Begrees and Hadians
+egrees Radians +egrees Radians +egrees Radians
%o % -/o /ID A%o /IA
/%o I
0 -%o I
0 /%%o I
/
Do I
D -.%o I /-o I
D
0%o I
/ A-%o I
0 //%o --I
0
&%o
I
A AAo
I
D /0%o AI
-A%o AI
/ AD%o DI
/
(he velocity at which the generator rotates around its central axis determines the fre5uency of
the sinusoidal waveform" #s the fre5uency of the waveform is given as 8 6' or cycles per
8/19/2019 Generation of Sinusoidal Waveforms
http://slidepdf.com/reader/full/generation-of-sinusoidal-waveforms 8/10
second, the waveform has angular fre5uency, C, 7Gree+ letter omega:, in radians per second"
(hen the angular velocity of a sinusoidal waveform is given as"
#ngular ?elocity of a Sinusoidal Waveform
and in the Jnited Lingdom, the angular velocity or fre5uency of the mains supply is given as>
in the JS# as their mains supply fre5uency is 0%6' it is therefore> / rads
So we now +now that the velocity at which the generator rotates around its central axis
determines the fre5uency of the sinusoidal waveform and which can also be called its angular
velocity, C" *ut we should by now also +now that the time re5uired to complete one revolution
is e5ual to the periodic time, 7(: of the sinusoidal waveform"
#s fre5uency is inversely proportional to its time period, 8 ; -( we can therefore substitute the
fre5uency 5uantity in the above e5uation for the e5uivalent periodic time 5uantity and
substituting gives us"
(he above e5uation states that for a smaller periodic time of the sinusoidal waveform, the greater
must be the angular velocity of the waveform" 4i+ewise in the e5uation above for the fre5uency5uantity, the higher the fre5uency the higher the angular velocity"
Sinusoidal Waveform Example o-
# sinusoidal waveform is defined as> ?m ; -0&". sin7/t: volts" $alculate the HMS voltage of
the waveform, its fre5uency and the instantaneous value of the voltage after a time of 0ms"
We +now from above that the general expression given for a sinusoidal waveform is>
(hen comparing this to our given expression for a sinusoidal waveform above
of ?m ; -0&". sin7/t: will give us the pea+ voltage value of -0&". volts for the waveform"
(he waveforms HMS voltage is calculated as>
8/19/2019 Generation of Sinusoidal Waveforms
http://slidepdf.com/reader/full/generation-of-sinusoidal-waveforms 9/10
(he angular velocity 7C: is given as / rads" (hen AI8 ; /" So the fre5uency of the
waveform is calculated as>
(he instantaneous voltage ?i value after a time of 0mS is given as>
ote that the phase angle at time t ; 0mS is given in radians" We could 5uite easily convert this
to degrees if we wanted to and use this value instead to calculate the instantaneous voltage value"
(he angle in degrees will therefore be given as>
8/19/2019 Generation of Sinusoidal Waveforms
http://slidepdf.com/reader/full/generation-of-sinusoidal-waveforms 10/10
Sinusoidal Waveform
(hen the generalised format used for analysing and calculating the various values of
a %inusoidal &aveform is as follows>
Top Related