## Lissajous Figures

Starting with the simplest comparison, if a voltage from a frequency standard is applied to one pair of the deflecting plates (say, the horizontal) of the cathode-ray oscillograph and a voltage from a source whose frequency is to be adjusted in terms of the standard to the other pair, patterns of the type illustrated in Figure 1 will be obtained.

These are the well-known Lissajous figures. For simple frequency ratios, expressible by small whole numbers, the patterns are not too complicated, and identification of the frequency ratio is possible, even when the pattern is rotating slowly.

If the pattern can be made to be nearly stationary, by adjustment of the frequency to be checked, then the fre quency ratio is found as follows: Count the horizontal tangent points (such as A, B, C, Figure 1); count the vertical tangent points (such as D). The frequency ratio is the ratio of the number of horizontal points to the number of vertical points, which, for the example of Figure 1, is 3 : 1. If the unknown frequency is on the vertical plates, then the unknown is three limes (as illustrated in Figure 1) the standard frequency.

As indicated by the successive parts of Figure 1, the appearance of the pattern changes progressively if there is a slight difference in frequency between the unknown and standard frequencies. Under such conditions the tangent points can be counted only for simple frequency ratios. If a frequency ratio of 7 : 5 is obtained, for example, the pattern appears almost as a network covering the area and, unless the pattern is steady, it is very difficult to count the tangent points.

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