waveforms 1 2003 - Course Materialscoursematerials.chamberlainradio.com/courses/wp-content/... ·...

WaveformsWaveformsWaveformsWaveforms

•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

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!!

OscilloscopeOscilloscope

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

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’

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

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

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)

y = sinx (1st harmonic)

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)

y = sinx + 1/3sin3x (2nd harmonic)

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)

y = sinx + 1/3sin3x + 1/5sin5x (3rd harmonic)

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)

y = sinx + 1/3sin3x + 1/5sin5x +1/7sin7x (4th harmonic)

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)

y = sinx + 1/3sin3x + 1/5sin5x +1/7sin7x +1/9sin9x (5th harmonic)

All together!!

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

y = sinx – sin3x + sin5x - sin7x + sin9x

9 25 49 81

How many harmonics shown? 5

Etc

Draw the graph of the 5th

Harmonic

TriangularTriangular

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)

y = sinx - sin3x/9 + sin5x/25 - sin7x/49 + sin9x/81 (5th harmonic)

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

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

y = sinx – sin2x + sin3x - sin4x + sin5x

2 3 4 5