generation of sinusoidal waveforms

8/19/2019 Generation of Sinusoidal Waveforms

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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"



*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"



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"



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%



(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"



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"



(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



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>



(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>



Sinusoidal Waveform

(hen the generalised format used for analysing and calculating the various values of

a %inusoidal &aveform is as follows>

generation of sinusoidal waveforms

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