Consider the characteristics of patterns that appear on the screen of a CRT, when sinusoidal voltages are simultaneously applied to horizontal and vertical plates. These patterns are called 'Lissajous Patterns'. When two sinusoidal voltages of equal frequency which are in phase with each other are applied to the horizontal and vertical deflection plates, the pattern appearing on the screen is a straight line as is clear from fig.

Thus when two equal voltages of equal frequency but with 90 phase displacement are applied to a CRO, the trace on the screen is a circle. This is shown in fig.

      An ellipse is also obtained when unequal voltages of same frequency are applied to the CRO. A number of conclusions can be drawn from the above discussions. When two sinusoidal voltages of same frequency are applied:

(i)                               A straight line results when the two voltages are equal and are either in phase with each other or out of phase with each other. The angle formed with the horizontal is when the magnitudes of voltages are equal. An increase in the vertical deflection voltages causes the line to have an angle greater than with the horizontal. On the other hand, a greater horizontal voltage makes the angle less than with the horizontal.

(ii)                              Two sinusoidal waveforms of the same frequency produce a Lissajous pattern, which may be a straight line, a circle or an ellipse depending upon the phase and magnitude of the voltages.

              A circle can be formed only when the magnitude of the two signals are equal and the phase difference between them is either or however, if the two voltages are not equal and/or out of phase an ellipse with vertical major axis is formed while if the plate voltage has greater magnitude, the major axis of the ellipse lies along horizontal axis.

(iii) it is clear from fig. that for equal voltages of same frequency progressive variation of phase voltage causes the pattern to vary from a straight diagonal line to ellipses of different eccentricities and then to a circle, after that though another series of ellipses and finally a diagonal straight line again.

Regardless of the two amplitudes o the applied voltages the ellipse provides simple means of finding phase difference between two voltages. The sine of the phase angle between the voltages is given by:

 For convenience, the gains of the vertical and horizontal amplifiers are adjusted so that the ellipse fits exactly into a square marked by the lines on the graticule.

  If the major axis of the ellipse in the first and third quadrants the phase angle is either between or between. When the major axis of ellipse lies in second and fourth quadrants, when its slope is negative as in fig, the phase angle is either between and or between and 270.

     Lissajous patterns may be used for accurate measurement of frequency the signal, whose frequency is to be measured, is applied to the Y plates. An accurately calibrated standard variable frequency source is used to supply voltage to the X plates, with the internal sweep generator switched off. The standard frequency is adjusted unit the pattern appears as circle or an ellipse indicating that both signals are of the same frequency. Where it is not possible to adjust the standard signal frequency to the exact frequency of the unknown signal, the standard is adjusted to a multiple or a submultiples of the frequency of the unknown source so that the pattern appears stationary.

 Let us consider an example. Suppose sine waves are applied to X and Y plates as shown in fig.

 Let the frequency of wave applied to Y plates is twice that of the voltage applied to X plates. This means that the CRT spot travels two complete cycles in the vertical direction against one in the horizontal direction. The two waves start at the same instant. Lissajous pattern may be constructed in the usual way and a 8 shaped pattern with two loops is obtained. If the two waves do not start at the same instant we get different patterns for the same frequency ratio. The Lissajous patterns for other frequency ratios can be similarly drawn. Some of these patterns are shown in fig.

The simple rule mentioned above needs following modifications. Two lines are drawn, one horizontal and the other vertical so that they do not pass through any intersection of different parts of the Lissajous curve. The number of intersections of the horizontal and the vertical lines with the Lissajous are individually counted. The frequency ratio is given by:

   The modified rule is applicable in all cases whether the Lissajous pattern is open or closed. The ratio of frequencies when open ends as half tangencies as shown.

  There are some restrictions on the frequencies which can be applied to the deflection plates. One obviously is that the CRO must have the bandwidth required for these frequencies. The other restriction is that the ratio of the two frequencies should not be such as to make the pattern too complicated otherwise determination of frequency would become difficult. As a rule ratios as high as and as low as can be determined comfortably.