To observe Lissajous patterns, certain conditions need to be met. Here are the necessary conditions to observe Lissajous patterns:
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Frequency ratio: The simplest Lissajous patterns appear in the oscilloscope display when the frequencies of the signals are the same, i.e. their ratio is 1:1. The pattern closes if the frequencies are whole number ratios, i.e. a/b is rational.
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Phase shift: When the phase shift is 0° or 360°, the display consists of a straight line sloping upward from the left side of the screen to the right side. When the phase shift is 90° or 270°, and both signals are the same frequency, a perfect circle displays. When the phase shift is 45°, the Lissajous pattern is an ellipse whose centerline slopes upward from left to right. When the phase shift is 180°, the Lissajous pattern is a straight line sloping down from left to right. The Lissajous pattern indicates the phase difference by the shape of the X-Y plot.
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Amplitude: The angle of the line depends on the difference in amplitude between the two signals, a line at 45º to the horizontal means the amplitudes are equal. While a circle indicates a 90º difference. It will only be a true circle if the amplitudes are equal.
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Different frequency ratios: Increasingly complex but easily recognizable Lissajous patterns are obtained for phase shifts of two signals having different frequency ratios, and when the amplitudes are not the same, still more Lissajous patterns are generated.
Lissajous patterns are named after Jules Antoine Lissajous, who investigated, wrote, and lectured about them in 1857. They can be generated using an oscilloscope, and they are used to study the frequency, amplitude, and phase relationships of harmonic variables. They can also be used to determine the unknown frequency of the sinusoidal wave.
