### An Introduction to Lissajous Patterns

Lissajous patterns created on the scope using 2 function generators

Purpose
This application note describes the functionality of Lissajous patterns and how they are used to calibrate the frequency of waveform.

Background
Lissajous (pronounced LEE-suh-zhoo) figures were discovered by the French physicist Jules Antoine Lissajous. He would use sounds of different frequencies to vibrate a mirror. A beam of light reflected from the mirror would trace patterns which depended on the frequencies of the sounds. Lissajous' setup was similar to the apparatus which is used today to project laser light shows.

Before the days of digital frequency meters and phase-locked loops, Lissajous figures were used to determine the frequencies of sounds or radio signals. A signal of known frequency was applied to the horizontal axis of an oscilloscope, and the signal to be measured was applied to the vertical axis. The resulting pattern was a function of the ratio of the two frequencies.Lissajous figures are useful in the calibration of frequencies in tuning forks. With these properly calibrated tuning forks one is able to verify the functionality of police radar, or the tuning of musical instruments.

A Lissajous pattern is a graph of one frequency plotted on the y axis combined with a second frequency plotted on the x axis. Y and X are both periodic functions of time t given by equations such as x = sin (w*n*t + c) and y = sin w*t. Different patterns may be generated for different values of n(period shift) and c (phase shift). The simplest patterns are formed when n is a ratio of small whole numbers such as 1/2, 2/3, or 1/3. The value of c is usually taken as 0 or 1.57 (which is actually p /2).

Lissajous Demonstration
The simplest Lissajous pattern is the circle which is formed when both the x and y frequency are the same. For this example both frequencies are set to 1.000kHz.

X Frequency= 1.000KHz and Y Frequency =1.000kHZ

The Lissajous pattern will change if the X and Y frequencies differ slightly. In the following picture the X frequency is set to 1.001kHz and the Y frequency is set to 1.000kHz. The pattern is only a circle when the phase differs by 90 degrees. The next two picture show when the frequencies are out of phase by 90 degrees. The green line corresponds to the x axis frequency and the purple line coresponds to the y axis frequency. These lines are included to illustrate how the Lissajous pattern is affected by phase changes. The Lissajous pattern that they form is represented by the yellow line.

The purple line leads the green line by a 90 degree phase shift.
The purple lags the green line by a 90 degree phase shift.

The Lissajous pattern changes to a line when the X and Y frequencies are in phase by either 180 or 0 degrees. The next two slides demonstrate this.

The frequencies are almost in phase.
The frequencies are out of phase by almost 180 degrees

There are many examples of lissajous patterns. These are dependant on what the frequency ratio is. A figure eight is formed when one frequency is two times the other. For example 2.000kHz for the Y axis frequency versus 1.000kHz for the X axis frequency as shown below.

In this case the x axis frequency is twice the y axis frequency.

If one frequency is three times the other frequency, an image similar to the one below will appear.

In this case the Y axis frequency is three times the X axis frequency with a phase that is not equal to zero.

Tools and Resources

The waveforms in this application note were created using two waveform function generators and an oscilloscope (setup pictured below). One wave form generator was set to the x-axis and the other waveform function generator was connected to the y-axis. It is possible to create lissajous patterns by using one wave form function generator, an oscilloscope, a tuning fork and a microphone. In fact this is the current setup used for calibrating tuning forks. If the tuning fork frequency equals the frequency of the wave form function generator then a circle is formed. Similiarly other patterns can be formed such as the figure eight. Technicians observe these patterns to determine the accuracy of tuning forks.

A figure eight Lissajous pattern

MATLAB Program:
Matlab was used to simulate Lissajous patterns. The function has seven input parameters and the code is provided below.

function lissajous(amp1, z, phase, amp2, points, cycles, k, p)
clf;
inc=cycles/(z*points);
t=0:inc:cycles/z;
for n=1:k;
x=amp1*sin(2*pi*z.*t+(pi/180)*phase);
y=amp2*sin(2*pi.*t);
pause(1/5);
if p == 'r';
plot(x, y, 'k-');
elseif p == 'p';
r=(x.^2+y.^2).^0.5;
theta=atan2(y, x);
polar(theta, r, 'k-');
end;
t=t(length(t)):inc:t(length(t))+points*inc;
end;

The amp1 and amp2 are the amplitudes of the x and y sinusoids. The ratio n(period shift) is imputed with the parameter z. The phase between the sunusoids is inputed as phase and the number of points plotted is inputed as points. The number of cycles can be inputed whith cycles and repeated whith the k input. Reapeted cycles gives the impression of a signal varying with time. The last input p is an 'r' for a rectangular plot and a 'p' for a polar plot.

Ex1. lissajous(1,1/2,90,1,100,1,1,'r') produces a figure eight
Ex2. lissajous(1,2/3,90,1,100,1,1,'p')
Ex3. lissajous(1,4/5,0,1,100,4,1,'r')