Cathode Ray Oscilloscope (CRO) is very important electronic device. CRO is very useful to analyze the voltage wave form of different signals. The main part of CRO is CRT (**Cathode Ray Tube**). A simple CRT is shown in figure below-

When both pairs of the deflection plates (horizontal deflection plates and vertical deflection plates) of CRO (**Cathode Ray Oscilloscope**) are connected to two sinusoidal voltages, the patterns appear at CRO screen are called the **Lissajous pattern**.

Shape of these Lissajous pattern changes with changes of phase difference between signal and ration of frequencies applied to the deflection plates (traces) of **CRO**. Which makes these **Lissajous patterns** very useful to analysis the signals applied to deflection plated of CRO. These lissajous patterns have two Applications to analysis the signals. To calculate the phase difference between two sinusoidal signals having same frequency. To determine the ratio frequencies of sinusoidal signals applied to the vertical and horizontal deflecting plates.

#### Calculation of the phase difference between two Sinusoidal Signals having same frequency

When two sinusoidal signals of same frequency and magnitude are applied two both pairs of deflecting plates of **CRO**, the Lissajous pattern changes with change of phase difference between signals applied to the CRO.

For different value of phase differences, the shape of Lissajous patterns is shown in figure below,

SL No. | Phase angle difference ‘ø’ | Lissajous Pattern appeared at CRO Screen |

1 | 0^{o} & 360^{o} | |

2 | 30^{o} or 330^{o} | |

3 | 45^{o} or 315^{o} | |

4 | 60^{o} or 300^{o} | |

5 | 90^{o} or 270^{o} | |

6 | 120^{o} or 240^{o} | |

7 | 150^{o} or 210^{o} | |

8 | 180^{o} |

There are two cases to determine the phase difference ø between two signals applied to the horizontal and vertical plates,

Case – I: When, 0 o or 270^{o}o : –

As we studied above it clear that when the angle is in the range of 0 o or 270^{o}o, the Lissajous pattern is of the shape of Ellipse having major axis passing through origin from first quadrant to third quadrant:

Let’s consider an example for 0 o or 270^{o}o, as shown in figure below,

In this condition the phase difference will be,

Another possibility of phase difference,

From Above given Lissajous pattern

Another Possibility of Phase Difference,

Case – II: When 90^{o}o or 180^{o}o

As we studied above it Clear that when the angle is in the range of 0^{o}o or 270^{o}o, the Lissajous Pattern is of the shape of Ellipse having major axis passing through origin from second quadrant to fourth quadrant:

Let’s consider an example for When, 90^{o}o or 180^{o}o, as shown in figure below,

In this condition the phase difference will be,

Another possibility of phase difference,

From Above given Lissajous pattern

Another Possibility of Phase Difference,

To determine the ratio of frequencies of signal applied to the vertical and horizontal deflecting plates:

To determine the ratio of frequencies of signal by using the Lissajous pattern, simply draw arbitrary horizontal and vertical line on lissajous pattern intersecting the Lissajous pattern. Now count the number of horizontal and vertical tangencies by Lissajous pattern with these horizontal and vertical line.

Then the ratio of frequencies of signals applied to deflection plates,

Or,

Let consider some example to clear the concept in details:

##### Examples Lissajous Pattern

Sl No. | Lissajous Pattern | Ratio of Frequencies f_{y}/f_{x} |

1 | f_{y}/f_{x} = 4/2 = 2 | |

2 | f_{y}/f_{x} = 3/1 = 3 | |

3 | f_{y}/f_{x} = 6/4 = 3/2 | |

4 | f_{y}/f_{x} = 6/8 = 3/4 | |

5 | f_{y}/f_{x} = 4/3 |