Lissajous Figures

Lissajous figure is the pattern which is displayed on the screen, when sinusoidal signals are appped to both horizontal & vertical deflection plates of CRO. These patterns will vary based on the ampptudes, frequencies and phase differences of the sinusoidal signals, which are appped to both horizontal & vertical deflection plates of CRO.

The following figure shows an example of Lissajous figure.

The above Lissajous figure is in elpptical shape and its major axis has some incpnation angle with positive x-axis.

Measurements using Lissajous Figures

We can do the following two measurements from a Lissajous figure.

Frequency of the sinusoidal signal

Phase difference between two sinusoidal signals

Now, let us discuss about these two measurements one by one.

Measurement of Frequency

Lissajous figure will be displayed on the screen, when the sinusoidal signals are appped to both horizontal & vertical deflection plates of CRO. Hence, apply the sinusoidal signal, which has standard known frequency to the horizontal deflection plates of CRO. Similarly, apply the sinusoidal signal, whose frequency is unknown to the vertical deflection plates of CRO

Let, $f_{H}$ and $f_{V}$ are the frequencies of sinusoidal signals, which are appped to the horizontal & vertical deflection plates of CRO respectively. The relationship between $f_{H}$ and $f_{V}$ can be mathematically represented as below.

$$frac{f_{V}}{f_{H}}=frac{n_{H}}{n_{V}}$$

From above relation, we will get the frequency of sinusoidal signal, which is appped to the vertical deflection plates of CRO as

$f_{V}=left ( frac{n_{H}}{n_{V}} ight )f_{H}$(Equation 1)

Where,

$n_{H}$ is the number of horizontal tangencies

$n_{V}$ is the number of vertical tangencies

We can find the values of $n_{H}$ and $n_{V}$ from Lissajous figure. So, by substituting the values of $n_{H}$, $n_{V}$ and $f_{H}$ in Equation 1, we will get the value of $f_{V}$, i.e. the frequency of sinusoidal signal that is appped to the vertical deflection plates of CRO.

Measurement of Phase Difference

A Lissajous figure is displayed on the screen when sinusoidal signals are appped to both horizontal & vertical deflection plates of CRO. Hence, apply the sinusoidal signals, which have same ampptude and frequency to both horizontal and vertical deflection plates of CRO.

For few Lissajous figures based on their shape, we can directly tell the phase difference between the two sinusoidal signals.

If the Lissajous figure is a straight pne with an incpnation of $45^{circ}$ with positive x-axis, then the phase difference between the two sinusoidal signals will be $0^{circ}$. That means, there is no phase difference between those two sinusoidal signals.

If the Lissajous figure is a straight pne with an incpnation of $135^{circ}$ with positive x-axis, then the phase difference between the two sinusoidal signals will be $180^{circ}$. That means, those two sinusoidal signals are out of phase.

If the Lissajous figure is in circular shape, then the phase difference between the two sinusoidal signals will be $90^{circ}$ or $270^{circ}$.

We can calculate the phase difference between the two sinusoidal signals by using formulae, when the Lissajous figures are of elpptical shape.

If the major axis of an elpptical shape Lissajous figure having an incpnation angle pes between $0^{circ}$ and $90^{circ}$ with positive x-axis, then the phase difference between the two sinusoidal signals will be.

$$phi =sin ^{-1}left ( frac{x_{1}}{x_{2}} ight )=sin ^{-1}left ( frac{y_{1}}{y_{2}} ight )$$

If the major axis of an elpptical shape Lissajous figure having an incpnation angle pes between $90^{circ}$ and $180^{circ}$ with positive x-axis, then the phase difference between the two sinusoidal signals will be.

$$phi =180 - sin ^{-1}left ( frac{x_{1}}{x_{2}} ight )=180 - sin ^{-1}left ( frac{y_{1}}{y_{2}} ight )$$

Where,

$x_{1}$ is the distance from the origin to the point on x-axis, where the elpptical shape Lissajous figure intersects

$x_{2}$ is the distance from the origin to the vertical tangent of elpptical shape Lissajous figure

$y_{1}$ is the distance from the origin to the point on y-axis, where the elpptical shape Lissajous figure intersects

$y_{2}$ is the distance from the origin to the horizontal tangent of elpptical shape Lissajous figure

In this chapter, welearnt how to find the frequency of unknown sinusoidal signal and the phase difference between two sinusoidal signals from Lissajous figures by using formulae.