Nuts & Volts - January 2005

In The Trenches

at 100 samples per cycle). If we measure from zero crossing to zero crossing point (the proper method), the basic A/D error is 0.224 degrees (see above). So, our total error can be a maximum of 3. 82 degrees or 1.06%. The 1,000 Hz signal might be measured as 1,060 Hz. Again, this is well in excess of the basic 0.4% eight-bit A/D error.

You can improve things by sampling faster. You will need to sample every 1.44 degrees (0.4% of 360) or 250 samples per cycle to match the A/D error of 0.4%. For our 1,000 Hz signal, this comes to a 250,000 Hz sample rate.

Another way to increase the accuracy of your measurements is to measure multiple cycles. This is usually easy to do. Count 10 cycles instead of one for a 10-fold increase in accuracy. This spreads the error over 10 cycles and reduces the per-cycle error by 10. (Many hardware frequency counters do this. Some count 1,000s of cycles for very high accuracy.)

relatively long time. This is the opposite of the frequency error above. The amplitude error can be 1/256 of full-scale. This corresponds to 0.996 instead of 1.000 (full-scale equals 1.000). The inverse sine of 0.996 is about 85 degrees. This is a difference of 5 degrees from the proper value of 90. Add this 5 degree error to the sampling error of 3. 6 degrees and there is a whopping 8. 6 degree potential phase error (or 9.5%) at 90 degrees. This is nearly 25 times worse than the basic A/D error of 0.4%.

How fast do we need to sample to get the phase error down to our basic 0.4% A/D amplitude error? Quite simply, you can't get there from here. We just saw that there was an inherent 5 degree error due to the step-size of our eight-bit A/D. An eight-bit A/D is just not good enough to resolve 1.44 degrees (0.4% of 360

degrees). We will need a step-size of 0.03%, which corresponds to about 3,000 steps — which means a 12-bit A/D converter.

Here's how those values were determined. Our desired resolution is 1.44 degrees. The worst case point is at 90 degrees. So, we have to be able to resolve the difference in amplitude between 90 degrees and 88. 56 degrees (90-1.44). The sine of 90 degrees is 1.000 and the sine of 88. 56 degrees is 0.99968. They differ by 0.00032. This defines the minimum step-size necessary (with a full-scale of 1.000). There are 3,125 steps of 0.00032 in a full-scale value of 1.000 (or 1/3125). A 12-bit A/D has 4,096 steps.

We still have to sample faster, too. We saw that 100 samples per cycle yielded a 3. 6 degree error. It's easy to calculate our sampling speed for 1.44 degrees per sample. It's just 360/1.44 or 250 samples per cycle.

Phase Errors

The last basic error is phase. (Amplitude, frequency, and phase define any signal.) Basically, this is a delay error from the sampling and conversion process. This is probably the most obvious error of the three. Generally, it is usually the least significant. However, there are times when phase error can be really nasty.

First, how much error is there? With 100 samples per cycle, the error is 1% or 3. 6 degrees. However, this is only due to the sampling error. There is also the acquisition error. This depends on how long the sampling process takes. Usually, this is much less than the sampling error, but it is important to verify that. (We will ignore that factor because it is A/D dependent.)

The worst case phase error from the A/D comes at the peak and trough of the sine wave where the amplitude changes take a

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