# Common Error Sources Affecting DC and RMS Measurements

Some common error sources for DC measurements are single-frequency components (or tones), multiple tones, or random noise. These same error signals can interfere with RMS measurements so in many cases the approach taken to improve RMS measurements is the same as for DC measurements.

## DC Overlapped with Single Tone

Consider the case where the signal you measure is composed of a DC signal and a single sine tone. The average of a single period of the sine tone is ideally zero because the positive half-period of the tone cancels the negative half-period.

Any remaining partial period, shown in the previous figure with vertical hatching, introduces an error in the average value and therefore in the DC measurement. Increasing the averaging time reduces this error because the integration is always divided by the measurement time t2 – t1. If you know the period of the sine tone, you can take a more accurate measurement of the DC value by using a measurement period equal to an integer number of periods of the sine tone. The most severe error occurs when the measurement time is a half-period different from an integer number of periods of the sine tone because this is the maximum area under or over the signal curve.

## Defining the Equivalent Number of Digits

Defining the Equivalent Number of Digits (ENOD) makes it easier to relate a measurement error to a number of digits, similar to digits of precision. ENOD translates measurement accuracy into a number of digits.

ENOD = log10(Relative Error)

A 1% error corresponds to two digits of accuracy, and a one part per million error corresponds to six digits of accuracy (log10(0.000001) = 6).

ENOD is only an approximation that tells you what order of magnitude of accuracy you can achieve under specific measurement conditions. This accuracy does not take into account any error introduced by the measurement instrument or data acquisition hardware itself. ENOD is only a tool for computing the reliability of a specific measurement technique.

Thus, the ENOD should at least match the accuracy of the measurement instrument or measurement requirements. For example, it is not necessary to use a measurement technique with an ENOD of six digits if the instrument has an accuracy of only 0.1% (three digits). Similarly, you do not get the six digits of accuracy from a six-digit accurate measurement instrument if the measurement technique is limited to an ENOD of only three digits.

## DC Plus Sine Tone

The following figure shows that for a 1.0 VDC signal overlapped with a 0.5 V single sine tone, the worst ENOD increases with measurement time— x-axis shown in periods of the additive sine tone—at a rate of approximately one additional digit for 10 times more measurement time. To achieve 10 times more accuracy, you need to increase the measurement time by a factor of 10. In other words, accuracy and measurement time are related through a first-order function.

## Windowing to Improve DC Measurements

The worst ENOD for a DC signal plus a sine tone occurs when the measurement time is at half-periods of the sine tone. You can greatly reduce these errors due to noninteger number of cycles by using a weighting function before integrating to measure the desired DC value. The most common weighting or window function is the Hann window, commonly known as the Hanning window.

The following figure shows a dramatic increase in accuracy from the use of the Hann window. The accuracy as a function of the number of sine tone periods is improved from a first-order function to a third-order function. In other words, you can achieve one additional digit of accuracy for every 101/3 = 2.15 times more measurement time using the Hann window instead of one digit for every 10 times more measurement time without using a window. As in the non-windowing case, the DC level is 1.0 V and the single tone peak amplitude is 0.5 V.

You can use other types of window functions to further reduce the necessary measurement time or greatly increase the resulting accuracy. The following figure shows that the Low Sidelobe (LSL) window can achieve more than six ENOD of worst accuracy when averaging a DC signal over only five periods of the sine tone (same test signal).

## RMS Measurements Using Windows

Like DC measurements, the worst ENOD for measuring the RMS level of signals sometimes can be improved significantly by applying a window to the signal before RMS integration. For example, if you measure the RMS level of the DC signal plus a single sine tone, the most accurate measurements are made when the measurement time is an integer number of periods of the sine tone. The following figure shows that the worst ENOD varies with measurement time (in periods of the sine tone) for various window functions. Here, the test signal contains 0.707 VDC with 1.0 V peak sine tone.

Applying the window to the signal increases RMS measurement accuracy significantly, but the improvement is not as large as in DC measurements. For this example, the LSL window achieves six digits of accuracy when the measurement time reaches eight periods of the sine tone.

## Using Windows with Care

Window functions can be very useful to improve the speed of a measurement, but you must be careful. The Hann window is a general window recommended in most cases. Use more advanced windows such as the LSL window only if you know the window will improve the measurement. For example, you can reduce significantly RMS measurement accuracy if the signal you want to measure is composed of many frequency components close to each other in the frequency domain.

You also must make sure that the window is scaled correctly or that you update scaling after applying the window. The most useful window functions are pre-scaled by their coherent gain—the mean value of the window function—so that the resulting mean value of the scaled window function is always 1.00. DC measurements do not need to be scaled when using a properly scaled window function. For RMS measurements, each window has a specific equivalent noise bandwidth that you must use to scale integrated RMS measurements. You must scale RMS measurements using windows by the reciprocal of the square root of the equivalent noise bandwidth.