Lissajous Figure

The Lissajous figure is constructed using a cathode ray oscilloscope (CRO). The Lissajous figure is also known as the Bowditch curve, named after the inventor. The Lissajous figure or the Lissajous pattern is constructed by the intersection of two sinusoidal curved axes of which are perpendicular to each other or maintained at right angles to each other. The Lissajous figure was initially studied by the American mathematician Nathaniel Bowditch in the year 1815. Later in 1857, Jules Antoine Lissajous investigated it in more detail. They investigated the curves independently using a cathode ray. 

In this article, we will learn about the Lissajous figure, Lissajous figures definition, Lissajous figures uses, measurement using Lissajous figures with suitable examples so that you can have a better understanding of the concept.

Let’s start with the definition of the Lissajous figure. 

Lissajous Figures Definition

The definition of a Lissajous figure can be defined as one of an infinite number of curves formed by combining two simple oscillations that are perpendicular to each other. This is usually viewed by an oscilloscope and is used to study the frequency, amplitude, and phase relationships of harmonic variables.

The Lissajous figures oscilloscope allows you to plot one sine wave along the x-axis against another sine wave along the y-axis. The result is the number of measurements. The Lissajous figure shows the phase difference between the two signals and the relationship between their frequencies.

Lissajous is the pattern that appears on the screen when a sinusoidal signal is applied to the horizontal and vertical caps of the CRO. These patterns depend on the amplitude, frequency, and phase difference of the sinusoidal signal applied to the horizontal and vertical baffles of the CRO.

Lissajous Figures Uses

The Lissajous figure’s uses mainly consist of measurement of the frequency and measurement of the phase difference. The Lissajous figure is of high importance in physics in order to study the sinusoidal waves. The Lissajous figures are mainly used in analogue electronics to analyse the intersection of two or more sinusoidal wave constructing loops which are also known as Lissajous knots in general.

Some of the usage of Lissajous figure are as follows:

• The Lissajou figures are used to determine the unknown frequency by comparing it with the known frequency.

• Verifying audio oscillator with a known-frequency signal.

• Monitoring audio amplifiers and feedback networks for phase shift.

Measurements Using the Lissajous Figure

As previously mentioned in the Lissajous figure, using the Lissajous figure, we can measure the frequency of the sinusoidal waves and the phase difference between the two sinusoidal waves. Let us describe the measurement of frequency and phase difference with the help of the Lissajous figure definition or Lissajous pattern definition.

1. Measurement of Frequency

When two sinusoidal frequencies are applied, a Lissajous figure or pattern will be formed. The Lissajous figure constructed will be illustrated on the CRO screen. To obtain the Lissajou figure, we must apply the sinusoidal signals to both horizontal and vertical deflection plates of CRO. Therefore, apply the sinusoidal signal, frequency of which is known to the horizontal deflection plates of CRO. Similarly, apply the sinusoidal signal, frequency of which is unknown to the vertical deflection plates of CRO.

Suppose fH and fV are the frequencies of sinusoidal signals, which are implemented to the horizontal and vertical deflection plates of CRO, respectively. The relationship between known and unknown frequencies, i.e., fH and fV can be mathematically expressed as below:

\[\Rightarrow \frac{f_{V}}{f_{H}}=\frac{n_{H}}{n_{V}}\]

From the above equation, the frequency of sinusoidal signal applied to the vertical deflection plates of CRO can be drawn which is given below:

\[\Rightarrow f_{V}=\left ( \frac{n_{H}}{n_{V}} \right )f_{H}\] ....(1)

where, nH is the number of horizontal squares and nV is the number of vertical squares.

The values of nH and nV also can be obtained from the Lissajous figure. Now, if we substitute the value of nH and nV in equation (1), we can determine the unknown frequency of sinusoidal signal applied to the vertical deflection plates of CRO.

2. Measurement of the Phase Difference

We know that the Lissajous figure is constructed and displayed on the screen of a CRO when sinusoidal signals are employed to both horizontal and vertical deflection plates of the CRO. Therefore, apply the sinusoidal signals with the identical amplitude and frequency to both horizontal and vertical deflection plates of the CRO.

Based on the shape of a few Lissajous figures, we can simply recognise the phase difference between the two sinusoidal signals used.

• If the Lissajous figure resulted in a straight line that is inclined at an angle of 45° with the positive x-axis of the CRO display, then the phase difference between the two sinusoidal signals used will be 0°. This implies that no phase difference is present between those two sinusoidal signals used.

• If the Lissajous figure resulted in a straight line that is inclined at an angle of 135° with the positive x-axis of the CRO display, then the phase difference between the two sinusoidal signals used will be around 180°. This implies that the sinusoidal signals used are out of phase.

• If the Lissajous figure resulted in a circular shape, then the phase difference between the two sinusoidal signals used will be either 90° or 270°.

We can estimate the phase difference between the two sinusoidal signals (Lissajous figures phase difference) by applying formulae when the constructed Lissajous figures are of elliptical shape, i.e., Lissajous ellipse.

• If the major axis of an elliptical Lissajous figure is inclined at angles between 0° and 90° with the positive x-axis of the CRO display screen, then the phase difference between the two sinusoidal signals can be determined using the following equation:

\[\Rightarrow \varphi =sin^{-1}\left ( \frac{x_{1}}{x_{2}} \right )=sin^{-1}\left ( \frac{y_{1}}{y_{2}} \right )\]

• If the major axis of an elliptical Lissajous figure is inclined at angles between 90° and 180° with the positive x-axis of the CRO screen, then the phase difference between the two sinusoidal signals can be calculated using the following equation:

\[\Rightarrow \varphi =180-sin^{-1}\left (\frac{x_{1}}{x_{2}}  \right )=180-sin^{-1}\left ( \frac{y_{1}}{y_{2}} \right )\]

where, 

x1 is the total distance measured from the origin to the point on the x-axis of the display, where the intersection of elliptical Lissajous figures can be noticed.

x2 is the total distance measured from the origin to the vertical tangent of the elliptical Lissajous pattern.

y1 is the total distance measured from the origin to the point on the x-axis of the display, where the intersection of the elliptical Lissajous figure can be noticed.

y2 is the total distance measured from the origin to the horizontal tangent of the elliptical Lissajous pattern (figure de Lissajous)

Did You Know?

• Lissajous figures have been widely studied since their discovery over 100 years ago. Especially in the last 30 years, the shape, evolutionary trend, periodicity, and symmetry of Lissajous figures have been studied and a series of important characteristics and laws have been summarised. However, the analysis of Lissajous figures from the point of view of 3D curves has not yet been carried out. 

• Curves of a particular value in electronics can be displayed on an oscilloscope, and their shape helps determine the characteristics of unknown electrical signals. Therefore, one of the two curves is a signal with known characteristics. 

• In general, curves can be used to analyse the characteristics of any pair of simple oscillators that are perpendicular to each other.

Conclusion

In this article, we have learnt about the Lissajous Figures and how to find the frequency of unknown sinusoidal signals and the phase difference between two sinusoidal signals from Lissajous figures by using formulae.