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Lissajous figures

Curricula: Sek II

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Aim

Obtain Lissajous figures and use them to determine the ratio of the frequencies of two signals.

The bench

Two sinusoidal voltage sources with an amplitude of 5 V: the right one — 100 Hz, connected to CH1 (X axis); the left one — 200 Hz, to CH2 (Y axis). The common terminals of the sources are connected to the oscilloscope's GND. X-Y mode, 2 V/div on both channels.

Procedure

  1. Look at the figure on the screen.
  2. Set 200 and 300 Hz on the sources.
  3. Set equal frequencies, for example 100 and 100 Hz.
  4. For each figure, count the touches of the horizontal and vertical sides of the frame.
Expected result

At 100 and 200 Hz you get a figure with two loops (ratio 1:2). At 200 and 300 Hz — a figure with a touch ratio of 2:3. At equal frequencies — an ellipse or a line segment: the shape depends on the phase shift. The frequency ratio equals the touch ratio: the frequency along Y to the frequency along X is as the number of touches of the horizontal side to the number of touches of the vertical side. The bench's sources hold the frequency exactly, so the figure stays still. With two real function generators the frequencies differ slightly and the figure slowly rotates, passing through all its forms; the slower the rotation, the more precisely the frequencies are matched.

Questions

  1. How can you find an unknown frequency from a Lissajous figure?
  2. Why is the figure stationary only when the frequency ratio is a ratio of integers?
Answers
  1. Feed a signal of known frequency to one channel and the unknown signal to the other, and adjust the known frequency until the figure stands still. Then \(f_Y / f_X = n_h / n_v\), where \(n_h\) is the number of touches of a horizontal side of the frame and \(n_v\) of a vertical side. For example, with 100 Hz on X and a figure that touches the top twice and the side once, the frequency on Y is 100 Hz · 2 / 1 = 200 Hz.
  2. The figure repeats itself only if the spot comes back to the same point after a whole number of periods of both signals: \(n \cdot T_X = m \cdot T_Y\), which means \(f_Y / f_X = m / n\). At 100 and 200 Hz this happens every 10 ms, after one period of X and two of Y. For any other ratio the phase between the signals keeps drifting, the path never closes, and the figure slowly changes its shape.