Lissajous patterns obtained on a Cathode Ray Oscilloscope (CRO) are used to determine the phase shift and frequency of the applied signal.

Lissajous patterns are formed when two periodic signals, typically with different frequencies, are applied to the X and Y inputs of a CRO. These signals can be sinusoidal, square waves, or any other periodic waveforms. The resulting pattern on the CRO screen depends on the phase relationship between the two signals.

Here is how Lissajous patterns can be used to determine the phase shift and frequency of the applied signal:

1. Phase Shift:
- When the two signals have the same frequency and are in phase, a straight line is formed on the CRO screen, either vertically, horizontally, or diagonally.
- If there is a phase shift between the two signals, the resulting pattern will be an ellipse.
- The orientation and shape of the ellipse provide information about the phase shift. For example, if the ellipse is tilted to the right, it indicates a positive phase shift, while a tilt to the left indicates a negative phase shift.
- By analyzing the Lissajous pattern, the phase shift between the two signals can be determined accurately.

2. Frequency:
- The Lissajous pattern also provides information about the frequency of the applied signal.
- The number of lobes in the pattern indicates the ratio of the frequencies of the two signals.
- For example, if the pattern has two lobes, it means that the frequency of the Y signal is twice that of the X signal.
- By counting the number of lobes, the frequency ratio and hence the frequency of the applied signal can be determined.

It is important to note that Lissajous patterns are accurate only when the frequencies of the two signals are known and stable. Additionally, the amplitude and distortion of the signals do not affect the formation of the Lissajous pattern.

In conclusion, Lissajous patterns obtained on a CRO are useful for determining the phase shift and frequency of an applied signal. These patterns provide visual representations that help in accurately analyzing and measuring the phase relationship between two signals.