## What is Lissajous Figure?

The lissajou figure is constructed using a cathode ray oscilloscope. The lissajou figure is also known as the Bowditch curve, named after the inventor. The lissajou figure or the lissajou 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 lissajou figure was initially studied by the American mathematician Nathaniel Bowditch in the year 1815. They investigated the curves independently using a cathode ray oscilloscope.

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In this article, we will learn about the lissajou figure, Lissajous figures definition, Lissajous figures uses, etc with suitable examples.

### Lissajous Figures Definition

The definition of a Lissajou 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 figures use 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 is also known as knots in general.

Lissajous Figures Uses:

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 Lissajou 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 & vertical deflection plates of CRO. Therefore, supply the sinusoidal signal, whose frequency is known to the horizontal deflection probe of CRO. Similarly, supply the sinusoidal signal of unknown frequency to the vertical deflection probe of CRO.

Let, f_{H} and f_{V} are the frequencies of sinusoidal signals, which are implemented to the horizontal and vertical deflection probes of CRO respectively. The relationship between known and unknown frequencies i.e., f_{H} and f_{V }can be mathematically represented as below.

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

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

Where,

n_{H }- The number of horizontal squares.

n_{V }- The number of vertical squares.

The values of n_{H} and n_{V }can be calculated by measuring the number of horizontal and vertical squares covered by the Lissajou figure. From equation (1) we can determine the applied unknown sinusoidal frequency.

### 2. Measurement of the Phase Difference

We know that the Lissajous figure is constructed and displayed on the screen of a cathode ray oscilloscope (CRO) when sinusoidal signals are employed to both horizontal and vertical deflection probes of the CRO. Therefore, supply the sinusoidal signals with the identical amplitude and frequency to both horizontal and vertical deflection channels of the CRO.

For a few Lissajous figures based on their shape, we can directly 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

^{0}with the positive x-axis of the CRO display, then the phase difference between the two sinusoidal signals used will be 0^{0}. This implies that there is no phase difference between those two sinusoidal signals used.If the Lissajous figure resulted in a straight line that is inclined at an angle of 135

^{0 }with the positive x-axis of the CRO display, then the phase difference between the two sinusoidal signals used will be around 180^{0}. 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

^{0}or 270^{0}.

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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 Lissajou 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:

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

Where,

x_{1 }- The total distance measured from the origin to the point on the x-axis of the display, where the intersection of elliptical Lissajous figure can be noticed

x_{2 }- The total distance measured from the origin to the vertical tangent of the elliptical Lissajous pattern

If the major axis of an elliptical Lissajou 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,

y_{1 }- The total distance measured from the origin to the point on the x-axis of the display, where the intersection of elliptical Lissajous figure can be noticed

y_{2 }- The total distance measured from the origin to the horizontal tangent of the elliptical Lissajous pattern (figure de lissajous)

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### 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 summarized. 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 analyze the characteristics of any pair of simple oscillators that are perpendicular to each other.

1. What is the Use of the Lissajou Figure?

Answer: Before the advent of digital frequency meters and synchronization loops, Lissajous numbers were used to determine the frequency of sound or radio signals. The signal of known frequency is applied to the horizontal axis of the oscilloscope, and the signal to be measured is applied to the vertical axis.

2. What is the Lissajou Wave?

Answer: When two identical sinusoidal signals intersect they construct a pattern that results in a loop or knot. Such waves which are used to construct the lissajou figure are called the lissajou wave.

3. How Do We Read the Lissajou Figures?

Answer: With the help of lissajou figures we can determine the unknown frequency of the sinusoidal wave. It can be calculated by measuring the number of horizontal and vertical divisions enclosed in a particular lissajou figure.