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An introduction to electronics
An introduction to electronics

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The square wave challenge

The wave shown in Figure 26 approximates what is called a ‘square’ or ‘rectangular’ wave. The challenge is to design this wave using sine waves as components.

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Figure 26  Approximation to a square wave
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This is a screenshot from Interactive 3, showing just the bottom wave. As usual, the horizontal axis shows time from 0 to 20 milliseconds and the vertical axis shows voltage from minus 1 to 1 volt. There are 8 oscillations of a ‘rectangular’ shaped wave in which the ‘flat’ tops and bottoms of the wave have small ripples corresponding to the high-frequency components. The vertical sides of the rectangles slope left to right a bit as they go down from 0.6 volts to minus 0.6 volts, and slope the other way as the graph rises from minus 0.6 volts to 0.6 volts. This is quite a convincing approximation to a rectangular wave.

Figure 26  Approximation to a square wave

To make this wave using Interactive 3 (which you should still have open in a separate tab), start with two sine waves, the first having A = 0.7 and f = 400 Hz, and the second having A = 0.2 and f = 1200 Hz. This will give something similar to the wave shown in Figure 27, which is a good start.

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Figure 27  Combination of two sine waves
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This is another screenshot from Interactive 3, showing just the bottom wave. As usual, the horizontal axis shows time from 0 to 20 milliseconds and the vertical axis shows voltage from minus 1 to 1 volt. Figures 27 to 29 show how the ‘rectangular’ wave is made by adding higher-frequency sine waves.

Here there are two sine waves. The second, higher-frequency wave causes depressions in the tops and bottoms of the lower-frequency wave, effectively ‘chopping’ a bit off the bottom and top to make it more rectangle-like.

Figure 27  Combination of two sine waves

To make the top and bottom smoother requires sine waves with other frequencies. To get another sine wave, click on the ‘add another sine wave’ button below the second sine wave. Set the frequency of this to f = 2000 Hz.

SAQ 10

Suggest an appropriate value of A for the wave with frequency f = 2000 Hz.

Answer

Setting A = 0.1 gives the wave shown in Figure 28. This is closer to what is desired.

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Figure 28  Combination of three sine waves
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Following on from Figure 27, here there are three sine waves. The third wave chops a bit more off the tops and bottoms of the lower-frequency waves. The overall wave is getting flatter at the top and bottom.

Figure 28  Combination of three sine waves

To finish this challenge requires one more sine wave. Click again on the ‘add another sine wave’ button to get a fourth sine wave. Set its frequency to f = 2800 Hz.

SAQ 11

Suggest an appropriate value of A for the wave with frequency f = 2800 Hz.

Answer

Setting A = 0.05 gives the wave shown in Figure 29. This is even closer to the desired square wave.

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Figure 29  Combination of four sine waves
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Following on from Figure 28, here there are four sine waves. The fourth wave makes the tops and bottoms even flatter. This is now a good approximation to the rectangular wave.

Figure 29  Combination of four sine waves

You can now close the interactive.

In the square wave challenge, the shape of the wave was made closer to that required by adding higher frequencies with decreasing amplitudes. This is a general principle behind a very powerful theory for representing and processing signals.