(I) Finding frequency ratio and phase difference (φ) from Lissajous figures

(a) 0^{o}, 90^{o}, 180^{o}
(Amp = Amplitude; assume same voltage sensitivity)

φ = 0^{o}

φ = 180^{o}



φ = 90^{o} (Amp_{x} = Amp_{y})

φ = 90^{o} (Amp_{x} > Amp_{y})

φ = 90^{o} (Amp_{x} < Amp_{y})




(b) General Case 










We do not know whether x leads y or y leads x unless the sense of
rotation is known.
Clockwise rotation: y leads x; counterclockwise rotation: x leads y.

(II) Frequencies are different, but in a simple ratio

The pattern generally depends on x and yfrequency and their initial phases.
The ratio of the two frequencies can be found easily by the following method:


Draw a vertical and a horizontal line to cut the pattern curve, but avoid passing through any selfintersection
points of the pattern curve.
Count the number of intersection points of the vertical line with the pattern
curve. Let this number be m. As well, count the number of intersection points of the horizontal line with
the pattern curve. Let this number be n.
Hence, xfrequency : yfrequency = m : n.
In this example, the ratio is 6 : 4 = 3 : 2.


One more example,
xfrequency : yfrequency = 2 : 3
