Lecture 5

Fourier

James Uren

BTC108Electronics

Lecture 5

Fourier

Joseph Fourier was a brilliant mathematician of the 18th/19th centuries. His work is used throughout broadcasting, primarily in analysis of the spectrum (frequency response) of electronic signals.

He proved the mathematical connection between the time domain and the frequency domain. For example, a simple sine wave:

will have a spectrum of a single spike (called a delta function) in the frequency domain:

Lecture 5: FourierBTC108: Electronics – James Uren

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Time

Amplitude

Frequency

Amplitude

Fourier showed that cyclic waves like the square wave below could be built from a sum of different frequency sine and cosine waves. For example a square wave:

So the harmonic frequencies of the original sine wave, when added together, produce a square wave.

In the square wave case above, it is the ‘odd’ harmonic frequencies, i.e. x1, x3, x5, x7 etc, that are added together to produce the square.

So the frequency response, or spectrum of an ideal square wave will look like this:

Note that the amplitude of the higher harmonics gets smaller.

Lecture 5: FourierBTC108: Electronics – James Uren

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Frequency

Amplitude

Fourier went on to show that this is true for any repeated wave. The following applet is a live and interactive demonstration of this

http://www.falstad.com/fourier/

Redraw the wave by dragging over the top waveform and reduce the number of terms to see how any repeated wave is produced from a sum of sines and cosines.

Lecture 5: FourierBTC108: Electronics – James Uren

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Practical:

Please ensure you have thoroughly read the Health and Safety information at the back of this handout.

Using a simple RC filter with an appropriate -3db frequency and a square-wave source. Observe the effects of increasing frequency of the source. Repeat this for other repeated waveforms.

Lecture 5: FourierBTC108: Electronics – James Uren

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Health & Safety Considerations

Soldering and de-soldering:

Solder melts at between 180 and 200°C. Soldering irons will heat up to between 250 and 400°C. Be extremely careful when soldering and take the following precautions:

Switch off the soldering iron at the mains when not in use Always keep the iron in its stand Make sure your workspace is clear, well lit and well ventilated Never solder while your circuit is powered up Never solder without tutor supervision Only apply the soldering iron for the minimum amount of time Keep your soldering tidy and use the minimum amount of solder Avoid breathing in solder fumes You must only use the lead-free solder provided You must use tools e.g. pliers to support components that are

being soldered and ensure the board is secure.

Switching it on:

Powering up a circuit that is incorrectly connected can cause components or equipment to get extremely hot or even ‘blow’. A short circuit (where unintended electrical connections are made) for example may damage equipment or blow components causing them to behave in an unpredictable way.

Before powering up your circuit you MUST have it checked by the tutor

Have your neighbour physically inspect your work before powering on

If your circuit does not behave as you expect, switch it off immediately

Use your nose! A faulty circuit with hot components will often smell or smoke

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If your circuit does not behave as you expect:

With the power off, confirm by eye that your circuit is connected correctly and that you are using all the correct components and mounted with the correct polarities

Inspect your circuit closely for short circuits, soldering faults and dry joints:

Do all the testing on your circuit that you can with it powered off. Be extremely careful when probing your circuit live as the probe

itself can cause short circuits When probing with an oscilloscope ensure the earth connection is

applied safely

Lecture 5: FourierBTC108: Electronics – James Uren

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