waveforms 1 2003 - course materialscoursematerials.chamberlainradio.com/courses/wp-content/... ·...
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WaveformsWaveformsWaveformsWaveforms
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•Identify the different waveforms produced
Outcomes from this sessionOutcomes from this session
At the end of this session you should be able to
•Plot at least two types of waveform
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Basic Sinusoidal waveformsBasic Sinusoidal waveforms
A basic sinusoidal wave can be described with the mathematical formula y = sinx
What happens if we change the values, For example
Y = Sin2x or y = 2Sinx or y = 2Sin2x
Twice the waveform in the periodic
time
Twice the height of the waveform in the
periodic time
Both!!
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OscilloscopeOscilloscope
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Draw the Graph of Y = sin 2x between 0 - 3600
Y = Sin 2x
2x 0 60 120 180 240 300 360 420 480 540 600 660 720
X 0 30 60 90 120 150 180 210 240 270 300 330 360
Sin2x 0 0.866 0.866 0 -0.866 -0.866 0 0.866 0.866 0 -0.866 -0.866 0
0 30 60 90 120 150 180 210 240 270 300 330 360
1
-1
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HarmonicsIf the frequency remains fundamental, then the waveform should be
sinusoidal, however if we start interfering with the frequency then complex waveforms are evident such as:
Changing these fundamental frequencies is simply termed ‘Harmonics’
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Generating Harmonics
Harmonics are generated when a sinusoidal signal is passed through a non linear amplifier:
When a signal is amplified through a linear amplifier the signal remains sinusoidal
When a signal is amplified through a Non linear amplifier the signal becomes synthesised
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Producing WaveformsSquare waveform
As with the sinusoidal wave there is a formula in order to produce the wave. With a sinusoidal wave
it is based upon y = sinx
y = sinx + 1/3sin3x (2nd harmonic)
y = sinx + 1/3sin3x + 1/5sin5x (3rd harmonic)
y = sinx + 1/3sin3x + 1/5sin5x +1/7sin7x (4th harmonic)
y = sinx (1st harmonic)
And so on….. The higher up the harmonics the closer to a square wave
To get to ‘square’ further harmonics are introduced
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X 0 30 60 90 120 150 180 210 240 270 300 330 360
Sinx 0 0.5 0.866 1 0.866 0.5 0 -0.5 -0.866 - 1 -0.866 -0.5 0
y = sinx (1st harmonic)
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y = sinx (1st harmonic)
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X 0 30 60 90 120 150 180 210 240 270 300 330 360
Sinx 0 0.5 0.866 1 0.866 0.5 0 -0.5 -0.866 - 1 -0.866 -0.5 0
1/3Sin3x 0 0.33 0 -0.33 0 0.33 0 -0.33 0 0.33 0 -0.33 0
Total 1 0 0.83 0.866 0.67 0.866 0.83 0 -0.83 -0.866 -0.67 -0.866 -0.83 0
y = sinx + 1/3sin3x (2nd harmonic)
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y = sinx + 1/3sin3x (2nd harmonic)
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X 0 30 60 90 120 150 180 210 240 270 300 330 360
Sinx 0 0.5 0.866 1 0.866 0.5 0 -0.5 -0.866 - 1 -0.866 -0.5 0
1/3Sin3x 0 0.33 0 -0.33 0 0.33 0 -0.33 0 0.33 0 -0.33 0
1/5Sin5x 0 0.1 -0.173 0.2 -0.173 0.1 0 -0.1 0.173 -0.2 0.173 -0.1 0
Total 1 0 0.83 0.866 0.67 0.866 0.83 0 -0.83 -0.866 -0.67 -0.866 -0.83 0
Total 2 0 0.93 0.693 0.87 0.693 0.93 0 -0.93 -0.693 -0.87 -0.693 -0.93 0
y = sinx + 1/3sin3x + 1/5sin5x (3rd harmonic)
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y = sinx + 1/3sin3x + 1/5sin5x (3rd harmonic)
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X 0 30 60 90 120 150 180 210 240 270 300 330 360
Sinx 0 0.5 0.866 1 0.866 0.5 0 -0.5 -0.866 - 1 -0.866 -0.5 0
1/3Sin3x 0 0.33 0 -0.33 0 0.33 0 -0.33 0 0.33 0 -0.33 0
1/5Sin5x 0 0.1 -0.173 0.2 -0.173 0.1 0 -0.1 0.173 -0.2 0.173 -0.1 0
Total 1 0 0.83 0.866 0.67 0.866 0.83 0 -0.83 -0.866 -0.67 -0.866 -0.83 0
Total 2 0 0.93 0.693 0.87 0.693 0.93 0 -0.93 -0.693 -0.87 -0.693 -0.93 0
Total 3 0 0.859 0.817 0.727 0.817 0.859 0 -0.859 -0.817 -0.727 -0.817 -0.859 0
1/7Sin7x 0 -0.07 0.124 - 0.143 0.124 -0.07 0 0.07 -0.124 0.143 -0.124 0.07 0
y = sinx + 1/3sin3x + 1/5sin5x +1/7sin7x (4th harmonic)
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y = sinx + 1/3sin3x + 1/5sin5x +1/7sin7x (4th harmonic)
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X 0 30 60 90 120 150 180 210 240 270 300 330 360
Sinx 0 0.5 0.866 1 0.866 0.5 0 -0.5 -0.866 - 1 -0.866 -0.5 0
1/3Sin3x 0 0.33 0 -0.33 0 0.33 0 -0.33 0 0.33 0 -0.33 0
1/5Sin5x 0 0.1 -0.173 0.2 -0.173 0.1 0 -0.1 0.173 -0.2 0.173 -0.1 0
Total 1 0 0.83 0.866 0.67 0.866 0.83 0 -0.83 -0.866 -0.67 -0.866 -0.83 0
Total 2 0 0.93 0.693 0.87 0.693 0.93 0 -0.93 -0.693 -0.87 -0.693 -0.93 0
Total 3 0 0.859 0.817 0.727 0.817 0.859 0 -0.859 -0.817 -0.727 -0.817 -0.859 0
Total 4 0 0.748 0.817 0.838 0.817 0.83 0 -0.748 -0.817 -0.838 -0.817 -0.83 0
1/7Sin7x 0 -0.07 0.124 - 0.143 0.124 -0.07 0 0.07 -0.124 0.143 -0.124 0.07 0
1/9Sin9x 0 -0.11 0 0.111 0 -0.111 0 0.111 0 -0.111 0 0.111 0
y = sinx + 1/3sin3x + 1/5sin5x +1/7sin7x +1/9sin9x (5th harmonic)
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y = sinx + 1/3sin3x + 1/5sin5x +1/7sin7x +1/9sin9x (5th harmonic)
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All together!!
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Here on a sawtooth waveform the ‘Fourier’ series is used, and as can be seen it takes at least 25 harmonics
to create the form!
SawtoothSawtooth
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y = sinx – sin3x + sin5x - sin7x + sin9x
9 25 49 81
How many harmonics shown? 5
Etc
Draw the graph of the 5th
Harmonic
TriangularTriangular
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X 0 30 60 90 120 150 180 210 240 270 300 330 360
-Sin3x/9 0 -0.11 0 0.11 0 -0.11 0 0.11 0 -0.11 0 0.11 0
Sin5x /25 0 0.02 -0.035 0.04 -0.035 0.02 0 0.02 -0.035 0.04 -0.035 0.02 0
Sin9x /81 0 -0.012 0 0.012 0 -0.012 0 -0.012 0 0.012 0 -0.012 0
Sinx 0 0.5 0.866 1 0.866 0.5 0 -0.5 -0.866 - 1 -0.866 -0.5 0
-Sin7x /49 0 0.01 -0.018 0.02 -0.018 0.01 0 0.01 -0.018 0.02 -0.018 0.01 0
Total 0 0.408 0.813 1.183 0.813 0.407 0 -0.408 -0.813 -1.183 -0.813 -0.407 0
y = sinx - sin3x/9 + sin5x/25 - sin7x/49 + sin9x/81 (5th harmonic)
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y = sinx - sin3x/9 + sin5x/25 - sin7x/49 + sin9x/81 (5th harmonic)
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y = sinx – sin2x + sin3x - sin4x + sin5x
2 3 4 5
Whats the next harmonic? - sin6x
6
Etc
Draw the graph of the 5th
Harmonic
SawtoothSawtooth
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X 0 30 60 90 120 150 180 210 240 270 300 330 360
-Sin2x/2 0 -0.43 - 0.43 0 0.43 0.43 0 -0.43 -0.43 0 0.43 0.43 0
Sin3x /3 0 0.33 0 -0.33 0 0.33 0 -0.33 0 0.33 0 -0.33 0
Sin5x /5 0 0.1 -0.17 0.2 -0.17 0.1 0 -0.1 0.17 -0.2 0.17 -0.1 0
Sinx 0 0.5 0.866 1 0.866 0.5 0 -0.5 -0.866 - 1 -0.866 -0.5 0
-Sin4x /4 0 -0.22 0.22 0 -0.22 0.22 0 -0.22 0.22 0 -0.22 0.22 0
Total 0 0.28 0.486 0.87 0.906 1.58 0 -1.58 -0.906 -0.87 -0.486 -0.28 0
y = sinx – sin2x + sin3x - sin4x + sin5x
2 3 4 5
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y = sinx – sin2x + sin3x - sin4x + sin5x
2 3 4 5
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