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Measure vector and area with scope X-Y display

23 Feb 2015  | Arthur Pini

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X-Y displays are also used to study hysteresis effects. Magnetic hysteresis in inductive components is a common example. Hysteresis in an X-Y plot manifests itself as a closed loop, the area of which is a measure of the energy loss per cycle. So being able to measure the area within a closed curve on the X-Y plot is a useful function. Math functions in an oscilloscope can only be applied to time based waveforms, so computing the area within the X-Y plot has to be accomplished based on the component waveforms of the X-Y plot. The area enclosed in an X-Y plot can be calculated as

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:

To implement this on an oscilloscope, you have to differentiate one of the traces, multiply it by the other trace, and integrate the result over time. The integral, evaluated over a single cycle of the periodic waveform, equals the area contained within the X-Y plot.

Figure 3 shows an X-Y plot enclosing a circular area (chosen because it is easy to confirm the area measurement). 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 horizontal cursors measure the diameter of the circle as 756mV.

Area of circle = π(0.7558/2)2 = 0.4486V2

Figure 3: Area of circle = π(.7558/2)2 = 0.4486V2

In figure 4, the actual calculation is performed with closely matching results. The input signals are shown in the channel 1 (C1, yellow) and channel 2 (C2, pink) traces these appear in the left hand grids. C1 is differentiated in the math trace F1 (upper right, blue). The derivative should be calculated with the minimum number of points to minimise noise. The sparse math operator is used perform the calculations. In the following examples, the math operations were performed using 500 points.

The differentiated trace F1 is multiplied by the C2 trace in math trace F2 (right side second from the top). This function is integrated in math trace F3 (right side third from the top). Two different measurements are made. The first uses the relative time cursors to read the value of the integral over a single cycle of the input waveforms. This result is read in the trace descriptor for trace F3 as Δy = 449.4mV2.

The second method uses the area parameter (P1) applied to the product waveform in F2. This measurement is gated over the same single cycle of the input waveform. The resultant measurement is 449.48mV2. Both results are within 0.2% of the estimated value based on the geometrical estimate.

Figure 4: Calculated area = 0.449V2 using both cursors on the integral function (F3) and the area parameter on the product function (F2) both are gated to include only a single cycle of the source waveforms.

We have seen two different types of measurements applied to X-Y displays. The first showed how to make vector measurements on base band quadrature communications signals, the second demonstrated computing the area enclosed by an X-Y plot. Both are extremely useful in their respective fields. If you've used vector or area measurements, tell us how.

About the author
Arthur Pini is a technical support specialist and electrical engineer with over 50 years experience in the electronics test and measurement industry. He has supported oscilloscopes, real-time spectrum analysers, frequency synthesisers, digitizers and arbitrary waveform generators for leading manufacturers.

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