# Electrical Engineering Laboratory EE391 Fourier Series Laboratory Supplement Procedure 2 Use a Harmonic Synthesizer to generate a

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```					                            Electrical Engineering Laboratory EE391
Fourier Series Laboratory - Supplement

Procedure:

2. Use a Harmonic Synthesizer to generate a square wave. These instruments are found in 2C74, and there
are only three of them available. Use the spectrum analyzer to measure/adjust the amplitude of
each harmonic, as it is much more precise than using a digitising oscilloscope to measure their peak-
to-peak voltages. Once the amplitude and phase of each component has been properly adjusted,
the composite waveform is present on the (sum) output. Your goal should be a 2 Vp-p square
wave, but you should be aware that there is a slight attenuation between the individual harmonic
outputs and the sum output. The shape of the wave is much more important than its amplitude.
The procedure 2 is replaced by the applet (a program written in the Java programming language that can
be included in an HTML page) available from http://homepages.gac.edu/%7Ehuber/fourier/index.html.
The Fourier series is given in the following two forms,
∞
a0
x(t)       =       +         (ak cos(kω0 t) + bk sin(kω0 t))               (1)
2
k=1
∞
=    c0 +         ck cos(kω0 t + θk )                           (2)
k=1

a0                                                    bk
c0 =      ,       ck =     a2 + b2 , and θk = tan−1 −
k    k
2                                                     ak
Any periodic wave can be expressed by the Fourier series which breaks down the periodic wave into a
series of harmonic waves. Each harmonic wave has amplitude ck and phase θk . The amplitude ck and
phase θk can be speciﬁed in terms of the cosine component ak and the sine component bk according to
the formula above. In this new procedure 2, we adjust ak and bk to construct a rectangular (square)
wave and a saw-tooth wave, instead of adjusting amplitude ck and phase θk .
1. Adjust the sliders in the applet to construct the following four waves. You can enter a number
followed by the "Enter" key, if you need a better precision.
• An anti-symmetric (odd function) square wave,
1             1            1
xR (t) = sin(ω0 t) +        sin(3ω0 t) + sin(5ω0 t) + cos(7ω0 t) + · · ·
3             5            7
• A symmetric (even function) square wave,
1             1            1
xR (t) = cos(ω0 t) −         cos(3ω0 t) + cos(5ω0 t) − cos(7ω0 t) + · · ·
3             5            7
• A saw-tooth wave of negative slope,
1             1            1
xS (t) = sin(ω0 t) +        sin(2ω0 t) + sin(3ω0 t) + sin(4ω0 t) + · · ·
2             3            4
• A saw-tooth wave pf positive slope,
1             1            1
xS (t) = sin(ω0 t) −        sin(2ω0 t) + sin(3ω0 t) − sin(4ω0 t) + · · ·
2             3            4

1
2. This applet can parse equations entered in the boxes provided above the sliders and text boxes for
ak and bk . Entering "0" clears all text boxes both for Cos and Sin column. Enter the following
formula to obtain a better approximated waveform with high precision numbers.
•   An anti-symmetric (odd function) square wave: ((-1)^(x-1)+1)/2/x
•   A symmetric (even function) square wave: You come up with a formula.
•   A saw-tooth wave of negative slope: 1/x
•   A saw-tooth wave pf positive slope: (-1^(x-1))/x
3. Construct a periodic pulse wave deﬁned by
A       for |t| < ∆/2
f (t) =
0       for ∆/2 < |t| < π
Its Fourier series is given in the mathematic handbook as follows:
∞
A∆ 1             sin(k∆/2)
f (t) =        +                     cos kt
π 2               (k∆/2)
k=1

Use ∆ = π/2 and A∆/π = 1, which make a pulse wave having a duty cycle of 25%

The screen captured from the Java applet looks like this.

Note: The web browser must have Java run time environment (Java plugin). Otherwise applets don’t
run. In the event that our lab computers do not have the Java run time environment installed, you may
do this part of the Fourier series laboratory on your own computer, this year only.
— The E N D —

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