DIGITAL KILOWATT HOUR METER RESOLUTION AND ACCURACY

Resolution
The following example illustrates the resolution and accuracy of the digital-ramp A/D converter. Assume the following values for the A/D converter of Figure 6-1: D/A converter has a 10-bit input and a full scale analog output of 10.23 volts; the comparator can detect a voltage difference of 1 millivolt or greater; Vin is 3.728 volts.

Since the D/A converter has a 10-bit input, the maximum number of steps possible is (210 1)  1023. With a full-scale output of 10.23 volts reached in 1023 steps, the step size is 10 millivolts. This means Vout increases in steps of 10 mV as the counter counts up from zero.

Since Vin  3.728 volts and the comparator threshold is 1 mV, then Vout has to reach 3.729 volts or greater before the comparator switches Low. At 10 mV per step, this requires 373 steps.

At the end of the conversion, the counter holds the binary equivalent of 373, which is 0101110101. This is the digital equivalent of the analog input of Vin  3.728 volts. The resolution of this A/D converter is equal to the step size of the D/A converter which is 10 mV, or approximately 0.1% (.010/10.23  100  0.1%).

The resolution of an A/D converter is equal to the resolution of the D/A converter that it contains. The D/A output voltage Vout is a staircase waveform (digital ramp) that goes up in discrete steps until it exceeds Vin. Thus, Vout approximates Vin.

When the resolution (step size) is 10 mV, the accuracy we can expect is that Vout is within 10 mV of Vin. The resolution of the D/A converter is an inherent error, often referred to as a quantizing error. This quantizing error can be reduced by increasing the number of bits in the counter and in the D/A converter. It is specified as an error ± 1 least significant bit (LSB), indicating that the result can vary by that much due to the step size.

From another point of view, the input voltage Vin can take on an infinite number of values, from 0 to full scale. However, the output voltage Vout has only a finite number of discrete values. This means that similar values of Vin within a small range could have the same digital representation.

For example, if the counter goes through 1,000 steps from zero to full scale, any value of Vin from 3.720 to 3.729 will require 373 steps, thus resulting in the same digital representation. In other words, Vin must change by 10 millivolts (the resolution) to produce a change in the digital output.


Accuracy
The D/A converter accuracy is not related to the resolution. It is related to the accuracy of the components in its circuit such as the resistors in the D/A network, comparator, level amplifiers, and the reference power supply. If a D/A has an accuracy of 0.01% full scale, the A/D converter may be off by 0.01% full scale owing to non-perfect components.

This error is in addition to the quantizing error due to resolution. These two sources of error are usually specified separately, and for a given A/D converter are usually of the same order of magnitude.


In addition to the inherent errors noted above, the accuracy of an electronic instrument depends on proper selection of the meter range. Normally, the uncertainty of measurements is expressed as a percent of the reading plus the number of counts of the least significant digit (LSD) displayed for that range.

If the 1,000 volt DC range is selected to measure a 2 volt signal for a three-and-a-half digit digital multimeter with a nameplate accuracy of 0.5% of input voltage 1 LSD, this setup would result in a meter accuracy of 50.5%, as shown below.

Given: Meter Range Accuracy (MRA) is 0.5% of input voltage 1 LSD
Meter range set to 1,000 volts DC
Input voltage is 2 volts DC
Then: Meter Accuracy  [(MRA  input V  LSD)/Input Voltage]  100
 [(0.5%  2  1)/2]  100
 50.5%

However, selecting a meter range of 2 volts DC on the same digital multimeter would result in an accuracy of 0.60%, nearly 100 times better, as shown below.

Given: Meter Range Accuracy is 0.5% of input 1 LSD
Meter range set to 2 volts DC
Input voltage is 2 volts DC
Then: Meter Accuracy  [(MRA  input V  LSD)/Input Voltage]  100
 [(0.5%  2  0.002)/2]  100
 0.6%


Digital Display Resolution and Accuracy
Typical handheld digital instruments display from 3 to 5 digits. Laboratory digital instruments often offer 7 or 8 digits. The number of digits directly affects the available resolution of the reading.

For example, a full 4-digit display is capable of presenting numbers from 0 to 9999 (with a decimal point somewhere in the display depending on the range setting of the instrument). This display can provide 10000 different readings for a particular range setting, so its resolution is limited to 1 part in 10000, or 0.01%. You may see this display referred to as a 10000-count display.

A 6-digit display can present numbers from 0 to 999999. This display resolution would be 1 part in 1,000,000 or 0.0001%. It may be called a 1,000,000-count display.

Examples in the previous section used LSD, Least Significant Digit, to adjust accuracy calculations to the characteristics of the display.

The design of a digital instrument often further limits the display. A 4-digit display, by design, may display numbers from 0 to 3999, rather than to 9999. That is, the left-most digit is programmed such that it only displays the numbers 0 to 3.

This display is described as a 31/2-digit display or as a 4000-count display. This design does not further affect the accuracy of calculations. The value of the LSD is the same for a 31/2-digit display as for a 4-digit display.