Introduction
The area enclosed in an X-Y plot can be calculated as:
$$Area = {\int y(x)dx} = {\int x(y)xy}$$
The oscilloscope has the data for both traces as a function of time, t. The variables can be changed in the integral to calculate the area based on the acquired traces:
$$Area = {\int y(t) {{dx(t)}\over{dt}} dt}$$ $$Area = {\int x(t) {{dy(t)}\over{dt}} dt}$$
To implement this on a scope we have to differentiate one of the traces then multiply it by the other trace and integrate the result. The integral, evaluated over 1 cycle of the periodic waveform, equals the area contained within the X-Y plot.
Figure 1 shows an X-Y plot enclosing a circular area. Based on the geometry of the display measured using the X-Y cursors we can determine the enclosed area as a test of the process outlined above. The relative amplitude cursors measure the diameter of the circle as 756 mV.
In Figure 2 the actual calculation is performed with closely matching results. The area parameter of the product waveform (F2) over the same region (one cycle of the input waveform) produces the same result.
Some hints to maximize accuracy:
- The derivative should be calculated with the minimum number of points to minimize noise. In the following examples the math operations were performed using 500 points.
- Maximize the input signals in the dynamic range by using the variable attenuator.
Figures 3 and 4 contain another example using an easily verified figure, a square. The geometric area is determined using the X-Y cursor readouts as 0.284 V2 . Calculation of the area yields a result of 0.285 V2 as read from the cursor readout field for math trace F3 (integral).
Teledyne LeCroy oscilloscopes offer a high level of functional integration which is evident in this application. X-Y displays include both Cartesian and polar coordinate cursors making reference measurements easy. Math functions can be chained permitting up to 16 simultaneous math operations (model dependent) at one time yielding an answer on a single screen display.
