Lissajous Patterns on a CRO

Lissajous patterns are visual representations generated on a Cathode Ray Oscilloscope (CRO) when two sinusoidal signals are applied simultaneously to its horizontal (X) and vertical (Y) deflection plates. The shape of the resulting Lissajous pattern directly depends on the frequency ratio, amplitude ratio, and the phase shift between the two input signals. These patterns are a fundamental tool in electronics for comparing and characterizing signals without needing a time base.

The CRO's electron beam traces a path determined by the instantaneous voltage values of both signals, producing a unique stationary or slowly moving pattern on the screen. By analyzing the characteristics of this Lissajous pattern, engineers and technicians can accurately determine specific properties of the applied signals.

Determining Phase Shift with Lissajous Patterns

One of the primary uses of Lissajous patterns is to accurately determine the phase shift between two sinusoidal signals, especially when they have the same frequency. When the frequencies are identical, the Lissajous pattern formed on the CRO screen will be a stable, closed curve—typically an ellipse, a straight line, or a circle.

• If the phase shift is 0 degrees or 180 degrees, the Lissajous pattern will be a straight line passing through the origin.
• If the phase shift is 90 degrees or 270 degrees, and the amplitudes of both signals are equal, the pattern will be a perfect circle.
• For any other phase shift (between 0 and 360 degrees, excluding 0, 90, 180, 270), the pattern will be an ellipse.

The exact phase shift ($\phi$) for an elliptical Lissajous pattern can be calculated using the following formula:

$$\phi = \arcsin\left(\frac{\text{Y-intercept}}{\text{Maximum Y-deflection}}\right)$$

Where:

• Y-intercept: This is the vertical distance from the center of the screen to the point where the ellipse crosses the vertical (Y) axis.
• Maximum Y-deflection: This is the maximum vertical amplitude of the ellipse from the center of the screen to its highest or lowest point.

Determining Frequency Using Lissajous Patterns

Another crucial application of Lissajous patterns is to determine the frequency ratio between two signals, particularly when one frequency is known and the other is unknown. This method is highly effective for measuring an unknown frequency by comparing it to a precisely known reference frequency.

The ratio of the vertical frequency ($f_V$) to the horizontal frequency ($f_H$) can be found by counting the number of times the Lissajous pattern touches (becomes tangent to) the horizontal and vertical axes of the CRO display.

The formula to determine the frequency ratio is:

$$\frac{f_V}{f_H} = \frac{\text{Number of horizontal tangents}}{\text{Number of vertical tangents}}$$

• Number of horizontal tangents: Count how many times the Lissajous pattern touches or crosses the extreme top or bottom edges of the pattern.
• Number of vertical tangents: Count how many times the Lissajous pattern touches or crosses the extreme left or right edges of the pattern.

For example, if a Lissajous pattern shows 3 horizontal tangencies and 2 vertical tangencies, it means the vertical signal's frequency is 1.5 times the horizontal signal's frequency ($f_V/f_H = 3/2$).

Lissajous Patterns: Not for Amplitude, Current, Distortion

While the overall size of a Lissajous pattern is related to the amplitudes of the input signals, it is not the most precise or common method to measure their absolute amplitude. A CRO in its standard time-base mode, where the vertical deflection directly represents voltage and the horizontal axis represents time, is typically used for accurate amplitude measurements.

Similarly, Lissajous patterns do not directly determine current in a circuit. A CRO primarily measures voltage. To measure current, a current-to-voltage conversion (e.g., using a current probe or measuring voltage across a shunt resistor) is required before applying the signal to the CRO.

Although signal distortion can affect the purity of the sinusoidal waveform and thus the appearance of a Lissajous pattern, these patterns are not the primary tool for quantifying distortion. Specialized instruments like distortion analyzers or spectrum analyzers, or a detailed analysis of the waveform in the time domain on a CRO, are more suitable for measuring and characterizing signal distortion.

Therefore, the core strength and primary utility of Lissajous patterns obtained on a CRO lie in their ability to accurately determine the relative phase shift and frequency ratios between two applied signals.