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For a noise-free DMM, resolution is the smallest change in an input signal that produces, on average, a change in the output signal. Resolution can be expressed in terms of bits, digits, or absolute units, which are related to each other.
Resolution is often expressed in bits, which specifically refers to the performance of the analog-to-digital converter (ADC). Theoretically, a 12-bit ADC can convert an analog input signal into 212 (4,096) distinct values. 4,096 is the number of least significant bits (LSB). LSB can be translated into digits of resolution:
Digits of resolution = log10 (Number of LSB)
Using the previous equation, a DMM with a 12-bit ADC has a resolution of:
Log10 (4,096) = 3.61 digits
|Note If a 12-bit ADC is used to digitize signals in a DMM, it might be insufficient to call this DMM a 3½–digit DMM, as noise must also be considered. Noise may reduce the number of LSBs and, therefore, the number of digits.|
For traditional DMMs, 5½ digits refers to the number of digits displayed on the readout of a DMM. A traditional 5½–digit DMM has five full digits that only display values from 0 to 9 and one half digit that only displays 0 or 1. This DMM can show positive or negative values from 0 to 199,999.
For more sophisticated digital instruments, and particularly virtual instruments, digits of resolution does not directly apply to the digits displayed by the readout.
Counts for a DMM is analogous to LSBs for an ADC. A count represents a value that a signal can be digitized to and is equivalent to a step in a quantizer. The weight of a count, or the step size, is the absolute unit of resolution.
Absolute unit of resolution = total span/counts
Digits can be defined as:
Digits of resolution = log10 (total span/absolute unit of resolution)
For example, a noise–free DMM set to the 10 V range (20 V total span) with 200,000 available counts has an absolute unit of resolution of:
Absolute unit of resolution = 20.0 V/200,000 = 100 µV
The readout of this noise–free DMM would display six digits. A change of the last digit would indicate a change of 100 µV of the input signal.
An 18-bit ADC provides the minimum number of LSB. You can now calculate the digits of resolution:
(217 = 131,072, 218 = 262,144)
Digits of resolution = log10 (20.0 V/100 x 10-6 V)
Digits of resolution = 5.3
This noise–free DMM can be called a 5½ digit DMM.
The quantization process introduces into any converted signal an irremovable error, called the quantization noise. For input signals through a uniform quantizer (without overload distortion), the rms value of the quantization noise in a noise–free DMM can be expressed as:
rms of quantization noise = absolute units of resolution/√12
In reality, a noise-free DMM does not exist and you need to account for the noise level when calculating absolute units of resolution. You can define the effective absolute units of resolution of a noisy DMM as the step size of a noise–free DMM with a quantization noise equal to the total noise of the noisy DMM.
Effective absolute units of resolution = √12 * rms noise
You can define the Effective Number of Digits (ENOD) of this noisy DMM as:
ENOD = log10(total span/Effective absolute units of resolution)
If a DMM set on the 10 V range (20 V total span) shows readings with an rms noise level of 70 µV, its effective absolute units of resolution and the ENOD is:
Absolute units of resolution = √12 * 70 µV = 242.49 µV
ENOD = log10 (20.0 V/242.49*10-6 V) = 4.92 digits
This DMM can be called a 5 digit DMM.
The minimum number of counts needed for this DMM would be 20 V/242.49*10-6 V = 82,478. The minimum number of bits needed would be 17 (216 = 65,536, 217 = 131,072).
If the same DMM demonstrates an rms noise level of 20 µV:
Absolute units of resolution = √12 * 20 µV = 69.28 µV
ENOD = log10 (20 V/69.28*10-6 V) = 5.46 digits
This DMM can be considered a "5½" digit DMM.
The minimum number of counts needed for this DMM would be 20 V/69.28*10-6 V = 288,675. The minimum number of bits needed would be 19 (218 = 262,144, 219 = 524,288).
The following table relates bits, counts, and ENOD to conventional digits of resolution for DMMs. As evidenced by the table, bits, counts, and ENOD are deterministically related. A direct mathematical relationship between ENOD and digits does not exist because digits are used only as an approximation.