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# How to Read Lissajous Curves on Oscilloscopes Oscilloscopes are full of fun features that can be used to conveniently visualize signal behavior, as well as compare two signals when working on multiple channels. One feature that can be found on multichannel oscilloscopes used in analog or RF domains is a Lissajous curve display. The feature provides a summative comparison of two waves, and the particular shape of the curve gives information about the relative phase, amplitude, and frequency of the two signals used to construct the curve.

Typically, Lissajous curves are used with two sine wave inputs, but in general the technique could be used to compare any pair of waveforms. To see how the measurement applies to more complex waveforms, keep reading the following sections and give it a try on your lab oscilloscope.

## The Math Behind Lissajous Curves

Named after Jules Antoine Lissajous, who studied them in 1857, Lissajous curves provide a parametric display of two waves overlaid on the same graph. In the case of a waveform measurement, the parameter being used to plot a curve in an x-y graph is the time variable. By measuring two waves over time, and setting the measured voltage value of each wave on the x and y axis respectively, a 2D curve is drawn out with a particular shape.

In terms of two waves with different possible amplitudes and frequencies, we can write the following for the x and y coordinates used to display the Lissajous figure:

• x(t) = Asin(⍵t)
• y(t) = Bsin(Ωt + δ)

Here there are 5 possible values that characterize the shape of the curve. The exact shape of the curve depends on the relative amplitudes, the two frequencies of the waves, and the phase difference between the waves. Of these 5 parameters, the 3 parameters that broadly determine the shape of a Lissajous figure are the two frequencies and the phase difference. When the frequencies are related to each other by an integer ratio (such as 3:1).

To see the possible range of Lissajous figures that can be viewed on an oscilloscope, the table below shows the set of Lissajous figures with integer ratio relationship for the waveform frequencies. The top row lists the phase difference between the two input waves; the figures will exhibit rotation and skew as the phase difference is changed between the particular values in the top row. Source: Flemming, J., and A. Hornes. "Lissajous-like figures with triangular and square waves." Revista Brasileira de Ensino de Física 35 (2013): 3702.

The above figures assume the two waves have the same amplitudes (essentially being normalized). If the amplitudes are different, the curve will be stretched along the X or Y axis (or both) depending on the difference in the two amplitudes.

## How Lissajous Curves Are Used

Lissajous curves are used in a range of fields beyond electronics, such as vibratometry and interferometry. In general, whenever a measured waveform needs to be compared with some reference, either in terms of phase or frequency (or both), a Lissajous curve is a very useful way to visualize this comparative measurement. Based on the shape of the curve and the tilt, the relative frequencies and phase difference can be quickly extracted.

Whenever testing the phase response of a DUT that involves two oscillators, a Lissajous curve can be used by sweeping across a range of frequencies. This kind of measurement might be used in an RF system that uses a selectable/programmable reference oscillator or waveform generator. While sweeping through frequencies for the test signal and matching to the reference oscillator, the phase response between the two signals can be determined by reading off the Lissajous curve on the oscilloscope at each frequency.

Finally, we should address the potential for two curves to be broadband. The above description assumes the two waveforms being compared are sinusoidal waves, but comparative measurements may be needed between a pair of multi-tone waveforms, or between broadband waveforms (square wave, triangle wave, etc.). There are Lissajous-like curves that can be generated with these alternative waveforms. As an example, the table below shows Lissajous-like curves generated from triangle waves and square waves with integer ratio frequencies. Lissajous-like curves for triangle waves (left) and square waves (right).

Source: Flemming, J., and A. Hornes. "Lissajous-like figures with triangular and square waves." Revista Brasileira de Ensino de Física 35 (2013): 3702.

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