- CRO Definition: A Cathode Ray Oscilloscope (CRO) is defined as an electronic device used to analyze the voltage waveforms of different signals.
- Lissajous Patterns Definition: Lissajous patterns in CRO are the shapes formed on the CRO screen when sinusoidal voltages are applied to its deflection plates.
- Phase Difference Analysis: The phase difference between two sinusoidal signals of the same frequency is determined using the shape of Lissajous patterns.
- Frequency Ratio Calculation: The ratio of frequencies of signals applied to the CRO’s deflection plates is found by counting the tangencies of the Lissajous pattern.
- Ellipse Patterns: Lissajous patterns form ellipses that change shape depending on the phase difference and the quadrant through which the major axis passes.
A Lissajous pattern on a Cathode Ray Oscilloscope (CRO) is the X-Y trace of two sinusoids, one on the horizontal plates and one on the vertical plates. The display is a CRT (Cathode Ray Tube). A simple CRT is shown below.
Those screen figures are Lissajous patterns. Shape depends on phase difference and on the frequency ratio of the two sinusoids on the CRO plates.
Lissajous patterns have two lab uses on a Cathode Ray Oscilloscope: phase difference when both frequencies are equal, and the frequency ratio fy/fx when the ratio is a simple rational number and the figure is stationary.
Calculation of the phase difference between two Sinusoidal Signals having same frequency
Equal-frequency sinusoids of equal amplitude on both plate pairs of a CRO give a line, ellipse or circle as phase changes. The table below is that equal-amplitude case on a CRO.
Unequal amplitudes still give an ellipse at 90°, not a circle. The shapes for common phase angles are:
| SL No. | Phase angle difference ‘ø’ | Lissajous Pattern appeared at CRO Screen |
| 1 | 0o & 360o |  |
| 2 | 30o or 330o |  |
| 3 | 45o or 315o |  |
| 4 | 60o or 300o |  |
| 5 | 90o or 270o |  |
| 6 | 120o or 240o |  |
| 7 | 150o or 210o |  |
| 8 | 180o |  |
Phase from an ellipse uses sin φ = (intercept on one axis) / (maximum deflection on that axis). Two slope cases:
Case – I: When, 0 < ø < 90o or 270o < ø < 360o : –
For 0 < ø < 90o or 270o < ø < 360o the ellipse tilts through the first and third quadrants (positive slope).
For example, 0 < ø < 90o or 270o < ø < 360o gives the 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 90o < ø < 180o or 180o < ø < 270o
For 90o < ø < 180o or 180o < ø < 270o the ellipse tilts through the second and fourth quadrants (negative slope).
For example, 90o < ø < 180o or 180o < ø < 270o gives the 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, 
Frequency ratio from a stationary figure is found by counting tangencies to horizontal and vertical lines.
Draw one line in each direction so that it cuts the loops, then count the tangencies.
The ratio of frequencies of signals applied to deflection plates,
Or,
The figure is stable only when fy/fx is rational. Examples:
Examples Lissajous Pattern
| Sl No. | Lissajous Pattern | Ratio of Frequencies fy/fx |
| 1 |  | fy/fx = 4/2 = 2 |
| 2 |  | fy/fx = 3/1 = 3 |
| 3 |  | fy/fx = 6/4 = 3/2 |
| 4 |  | fy/fx = 6/8 = 3/4 |
| 5 |  | fy/fx = 4/3 |
