Understanding Lissajous Patterns and Frequency Ratios on a CRO

This question relates to Lissajous patterns, which are distinctive figures generated on a Cathode Ray Oscilloscope (CRO) screen. These visual representations arise when two sinusoidal voltage signals, applied perpendicularly to the electron beam's deflection plates (typically the X and Y inputs), are displayed simultaneously. The resulting pattern's shape is a direct indicator of the relationship between the frequencies and the phase difference of these two signals.

Key Factors Determining Lissajous Figure Shape

The appearance of a Lissajous pattern is determined by several parameters of the input signals:

• Frequency ($f_x$): The frequency of the signal applied to the horizontal deflection plates (X-input).
• Frequency ($f_y$): The frequency of the signal applied to the vertical deflection plates (Y-input).
• Phase Difference ($\delta$): The phase angle difference between the two sinusoidal signals.

The Straight Line Condition in Lissajous Figures

The question specifies that the observed pattern is a straight line. This specific, simple geometric form occurs only under a precise set of conditions. For a Lissajous figure to be a straight line, the frequencies of the two input signals must be identical ($f_x = f_y$). Furthermore, the phase difference ($\delta$) between these signals must be either $0$ radians ($0^\circ$) or $\pi$ radians ($180^\circ$).

Let's break this down:

• When $f_x = f_y$ and $\delta = 0^\circ$, the signals are perfectly in phase. The pattern traced is a straight line with a slope determined by the ratio of their amplitudes ($y = (A_y/A_x)x$).
• When $f_x = f_y$ and $\delta = 180^\circ$, the signals are exactly out of phase. The pattern traced is also a straight line passing through the origin, but with the opposite slope ($y = -(A_y/A_x)x$).

Therefore, the fundamental requirement for observing a straight line Lissajous pattern is that the two signals must possess the same frequency.

Evaluating Alternative Frequency Ratios

It is important to understand why other frequency relationships do not produce a straight line:

As illustrated, unequal frequencies (Option 2) or specific integer ratios like $1:2$ or $2:1$ (Options 3 and 4) lead to different, more complex patterns such as ellipses or figure-eights, particularly when the phase difference is $90^\circ$. A straight line is exclusively associated with the condition of equal frequencies ($f_x = f_y$).

Final Conclusion

Based on the principles of Lissajous figures, the observation of a straight line pattern on a CRO screen definitively implies that the frequencies of the two applied sinusoidal signals are equal.