EXPERIMENT No. : 01
Aim: To construct a triangular wave with the
help of fundamental frequency and its harmonic component.
Apparatus
Required:
S.
No.

Component

Quantity

1.

Trainer kit ST2603

1

2.

CRO

1

3.

Connecting leads

2

Theory: Fourier synthesis is a method of
electronically constructing a signal with a specific, desired periodic
waveform. It works by combining a sine wave signal and sine wave or cosinewave
harmonics (signals at multiples of the lowest, or fundamental, frequency) in
certain proportions. The scheme gets its name from a French mathematician and
physicist named Jean Baptiste Joseph Baron de Fourier, who
lived during the 18th and 19th centuries.
A
mathematical theorem stating that a Periodic function f(x) which is reasonably
continuous may be expressed as the sum of a series of sine or cosine terms
(called the Fourier series), each of which has specific Amplitude and
Phase coefficients known as Fourier coefficients.Many waveforms
represent signal energy at a fundamental frequency and also at harmonic
frequencies (wholenumber multiples of the fundamental). The relative
proportions of energy concentrated at the fundamental and harmonic frequencies
determine the shape of the wave. The wave function (usually amplitude, frequency,
or
phase
versus time) can be expressed as of a sum of sine and cosine functions called a
Fourier series, uniquely defined by constants known as Fourier coefficients. If
these coefficients are represented by a0, a1, a2, a3, ..., an, ... and b1,
b2, b3, ..., bn ..., then the Fourier series F(x), where x is
an independent variable (usually time), has the following form :
F(x)
= a0/2 + a1 cos x + b1sin x + a2 cos
2x + b2 sin 2x + ... + an cos nx + bn sin
nx +...
In
Fourier synthesis, it is necessary to know, or to determine, the coefficients a0,
a1 a2, a3, ..., an ... and b1 b2, b3,
..., bn ... that will produce the waveform desired when "plugged
into" the generalized formula for the Fourier series. Then, sine and
cosine waves with the proper amplitudes (as defined by the coefficients)
must
be electronically generated and combined, up to the highest possible value of
n. The larger the value of n for which sinewave and cosinewave signals are
generated, the more nearly the synthesized waveform matches the desired
waveform.
Following
is the fourier series equation and wave form for the triangular wave.
i.e equation for triangular wave is
A*(
cos x + 1/9 cos 3x + 1/25 cos 5x + ………)
where
A = 8/()2
Example:
(0V DC + 3Cos(x) + 0.44 Cos (3x) + 0.304 Cos(5x) + 0.152 Cos(7x) + 0.092
Cos(9x))
Procedure
:
1.
Switch
‘On’ the Power Switch and LCD.
2.
Minimize
all the amplitudes of frequencies by potentiometer.
3.
Set
DC potentiometer to 0V DC by using Output BNC & Oscilloscope DC mode.
4.
Switch
each frequency for cosine and positive value.
5.
Use
“Harmonic Output” BNC to set and observe individual harmonic and use “Output”
BNC to observe the resultant waveform.
6.
Use
Harmonic select push button switch and Set the fundamental cosine Frequency (1
KHz) to 3V, observe it on oscilloscope.
7.
Use
Harmonic select push button switch and Set the 3rd harmonics i.e. cos 3x to
0.44V.
8.
Use
Harmonic select push button switch and Set the 5th harmonics i.e. cos 5x to
0.30V, 7th harmonic i.e. cos7x to 0.16V & 9th harmonic i.e. cos9x to 0.08V
Observation:
Result: Traced
the waveform from CRO & observed the effect of harmonics on the waveform.
Precautions:
·
Make sure that kit is powered off when
connections are made.
·
Handle the trainer kit properly.
Pre Expeiment Questions:
Q:1
What is Fourier Synthesis?
Q:2
What is the harmonics of the Fourier Series?
Q:3
The effect of various harmonics on the waveform?
Post Experiment Questions:
EXPERIMENT No. : 02
Aim: To construct a square wave with the help
of fundamental Frequency and its harmonic component.
Apparatus
Required:
S.
No.

Component

Quantity

1.

Trainer kit ST2603

1

2.

CRO

1

3.

Connecting leads

2

Theory: Fourier synthesis is a method of
electronically constructing a signal with a specific, desired periodic
waveform. It works by combining a sine wave signal and sine wave or cosinewave
harmonics (signals at multiples of the lowest, or fundamental, frequency) in
certain proportions. The scheme gets its name from a French mathematician and
physicist named Jean Baptiste Joseph Baron de Fourier, who
lived during the 18th and 19th centuries.
A
mathematical theorem stating that a Periodic function f(x) which is reasonably
continuous may be expressed as the sum of a series of sine or cosine terms
(called the Fourier series), each of which has specific Amplitude and
Phase coefficients known as Fourier coefficients.Many waveforms
represent signal energy at a fundamental frequency and also at harmonic
frequencies (wholenumber multiples of the fundamental). The relative
proportions of energy concentrated at the fundamental and harmonic frequencies
determine the shape of the wave. The wave function (usually amplitude, frequency,
or
phase
versus time) can be expressed as of a sum of sine and cosine functions called a
Fourier series, uniquely defined by constants known as Fourier coefficients. If
these coefficients are represented by a0, a1, a2, a3, ..., an, ... and b1,
b2, b3, ..., bn ..., then the Fourier series F(x), where x is
an independent variable (usually time), has the following form :
F(x)
= a0/2 + a1 cos x + b1sin x + a2 cos
2x + b2 sin 2x + ... + an cos nx + bn sin
nx +...
In
Fourier synthesis, it is necessary to know, or to determine, the coefficients a0,
a1 a2, a3, ..., an ... and b1 b2, b3,
..., bn ... that will produce the waveform desired when "plugged
into" the generalized formula for the Fourier series. Then, sine and
cosine waves with the proper amplitudes (as defined by the coefficients)
must
be electronically generated and combined, up to the highest possible value of
n. The larger the value of n for which sinewave and cosinewave signals are
generated, the more nearly the synthesized waveform matches the desired
waveform.
Following
is the fourier series equation and wave form for the square wave.
i.e.
equation for rectangular sawtooth wave is
A
*(sin x + 1/3 sin3x + 1/5sin5x)
where
A = 4/
Example:
0V DC + 5 Sin(x) + 1.72 Sin(3x) + 1.04 Sin(5x) + 0.74 Sin(7x) + 0.56 Sin(9x)
Procedure
:
1.
Switch ‘On’ the Power and LCD Switch.
2.
Minimize
all the amplitudes of frequencies by potentiometer.
3.
Set
DC potentiometer to 0V DC by using Output BNC & Oscilloscope DC mode.
4.
Switch
each odd frequency for sine and positive value.
5.
Use
“Harmonic Output” BNC to set and observe individual harmonic and use “Output”
BNC to observe the resultant waveform.
6.
Use
Harmonic select push button switch and set the fundamental sine Frequency H1 (1
KHz) to 5V observe it on oscilloscope.
7.
Use
Harmonic select push button switch and set the 3rd harmonics H3 i.e. sin3x to
1.72V output should be similar to following waveform.
8.
Use
Harmonic select push button switch and set the harmonic H5 (sin5x) to 1.04V and
observe the output.
9.
Use
Harmonic select push button switch and set the harmonic H7 (sin7x) to 0.74V and
observe the output.
10.
Use
Harmonic select push button switch and set the harmonic H9 (sin9x) to 0.56V and
observe the output.
Observation:
Result: Traced
the waveform from CRO & observed the effect of harmonics on the waveform.
Precautions:
·
Make sure that kit is powered off when
connections are made.
·
Handle the trainer kit properly.
Pre Expeiment Questions:
Q:1
What is Fourier Synthesis?
Q:2
What is the harmonics of the Fourier Series?
Q:3
The effect of various harmonics on the waveform?
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