When two sinusoidal voltages are applied to the horizontal (X-axis) and vertical (Y-axis) deflection plates of a Cathode Ray Oscilloscope (CRO) simultaneously, the resulting patterns observed on the screen are known as Lissajous figures. These figures are incredibly useful for determining the phase difference and frequency ratio between the two applied signals.

CRO Operation for Lissajous Figures

To display Lissajous figures, the CRO is typically operated in X-Y mode. In this mode, an external signal is applied to the horizontal input (X-input) instead of the internal sweep generator, and another external signal is applied to the vertical input (Y-input).

• Let the voltage applied to the X-input be: $$X = A \sin(\omega t)$$
• Let the voltage applied to the Y-input be: $$Y = B \sin(\omega t + \phi)$$

Here, $A$ and $B$ are the amplitudes of the X and Y signals, respectively, $\omega$ is their angular frequency (since the question states equal frequency), and $\phi$ is the phase difference between the Y and X signals.

Lissajous Patterns and Phase Shift

The shape of the Lissajous figure depends on several factors, including the frequency ratio of the two signals, their amplitude ratio, and their phase difference. The question specifies "equal frequencies" and "equal voltages" (implying equal amplitudes, $A=B$), but with a "phase shift (not equal to 0° or 90°)". Let's analyze the common patterns:

Ellipse Formation on CRO

The question states that the phase shift is "not equal to 0° or 90°". Since the voltages are equal (meaning equal amplitudes), and the frequencies are equal, we can consider the general case:

• If the phase shift is $$0^\circ$$ or $$180^\circ$$, a straight line is formed. A straight line can be thought of as a degenerate ellipse with zero eccentricity.
• If the phase shift is $$90^\circ$$ or $$270^\circ$$ and the amplitudes are equal, a perfect circle is formed. A circle is a special case of an ellipse where both axes are equal in length.
• If the phase shift is any value other than $$0^\circ$$, $$90^\circ$$, $$180^\circ$$, or $$270^\circ$$ (e.g., $$30^\circ$$, $$45^\circ$$, $$60^\circ$$), and the amplitudes are equal, the resulting figure will be an ellipse. This ellipse will be tilted relative to the axes.
• If the amplitudes are unequal, but the phase shift is $$90^\circ$$ or $$270^\circ$$, an ellipse (not a circle) is formed.

Given the condition that the phase shift is "not equal to 0° or 90°", it means the phase shift falls into the general category that produces an ellipse. The term "equal voltages" implies equal amplitudes ($A=B$). When $A=B$ and $\phi$ is neither $0^\circ$/$180^\circ$ nor $90^\circ$/$270^\circ$, the resulting Lissajous figure is an ellipse. This is the most general shape obtained when the frequencies are equal, and the phase shift is not one of the special values that produce a line or a perfect circle.

Therefore, when two equal voltages of equal frequency but with a phase shift (not equal to $$0^\circ$$ or $$90^\circ$$) are applied to a CRO, we obtain an ellipse.