Digital Multimeter Measurement Fundamentals

This tutorial recommends tips and techniques for using a National Instruments digital multimter (DMM) to build the most accurate measurement system possible. In this tutorial, you learn how the NI 4070 can operate as both a 6½ digit digital multimeter and a fully isolated, high-voltage digitizer, capable of acquiring waveforms at sample rates up to 1.8 MS/s at ±300 V input. This section of the tutorial covers the topics below.

For more information return to the Complete Digital Multimeter Measurement Tutorial.

Accuracy

Accuracy essentially represents the uncertainty of a given measurement because a reading from a digital multimeter can differ from the actual input. Accuracy is often expressed as:

(% Reading) + Offset
or
(% Reading) + (% Range)
or
±(ppm of reading + ppm of range)
Note Refer to the digital multimeter specifications included with your digital multimeter to determine which method is used.

For example, assume a digital multimeter set to the 10 V range is operating 90 days after calibration at 23ºC ±5 ºC, and is expecting a 7 V signal. The accuracy specifications for these conditions state ±(20 ppm of reading + 6 ppm of range). To determine accuracy of the digital multimeter under these conditions, use the following formula:

Accuracy = ±(ppm of reading + ppm of range)

Accuracy = ±(20 ppm of 7 V + 6 ppm of 10 V)

Accuracy = ±((7 V(20/1,000,000) + (10 V(6/1,000,000))

Accuracy = 200 µV

Therefore, the reading should be within 200 µV of the actual input voltage.
Accuracy can also be defined in terms of the deviation from an ideal transfer function as follows:

y = mx + b
where x is the input
m is the ideal gain of a system
b is the offset

Applying this example to a digital multimeter signal measurement, y is the reading obtained from the digital multimeter with x as the input, and b is an offset error that you may be able to null before the measurement is performed. If m is 1, the output measurement is equal to the input. If m is 1.000001, then the error from the ideal is 1 ppm or 0.0001%.

ppm to Percent Conversions

ppm	Percent
1	0.0001
10	0.001
100	0.01
1,000	0.1
10,000	1

High-resolution, high-accuracy digital multimeters describe accuracy in units of ppm and are specified as ±(ppm of reading + ppm of range). The ppm of reading is the deviation from the ideal m; ppm of range is the deviation from the ideal b, which is zero. The b errors are most commonly referred to as offset errors.

Temperature can have a significant effect on the accuracy of a digital multimeter and is a common problem for precision measurements. Temperature coefficient, or tempco, expresses the error caused by temperature. Errors are calculated as ±(ppm of reading + ppm of range)/ºC. Therefore the gain and offset in the digital multimeter transfer function vary with temperature, but are not worse than those specified by the tempco specification.

Learn more about NI Digital Multimeters.

Sensitivity

Sensitivity is the smallest unit of a given parameter that can be meaningfully detected with the instrument when used under reasonable conditions. For example, assume the sensitivity of a digital multimeter in the volts function is 100 nV. With this sensitivity, the digital multimeter can detect a 100 nV change in the input voltage.

Learn more about NI Digital Multimeters.

Resolution

For a noise-free digital multimeter, 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 can be related to each other.

Bits
The resolution of general-purpose digitizers are often expressed in bits. Bits specifically refer 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)  (1)

Using the above equation, a digital multimeter 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 digital multimeter, it would be insufficient to call this digital multimeter a 3½ digit digital multimeter as noise must also be considered. Noise may reduce the number of LSBs, therefore reducing the number of digits.

Digital Multimeter Absolute Units and Digits of Resolution
Traditionally, 5½ digits refers to the number of digits displayed on the readout of a digital multimeter. A 5½ digit digital multimeter would have five full digits that display values from 0 to 9 and one half digit that could only display 0 or 1. This digital multimeter could 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. Therefore, care must be taken when specifying the number of digits for these measurement devices.

Absolute Units
Counts for a digital multimeter 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 called the absolute unit of resolution.

Absolute unit of resolution = total span/counts  (2)

Digits
Digits can be defined as:

Digits of resolution = log₁₀ (total span/absolute unit of resolution)   (3)

For example, a noise-free digital multimeter 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 digital multimeter 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 digital multimeter could be called a 5½ digit digital multimeter.

The quantization process introduces into any converted signal an irremovable error, the quantization noise. For input signals through a uniform quantizer (without overload distortion), the rms value of the quantization noise in a noise-free digital multimeter can be expressed as:

rms of quantization noise = absolute units of resolution /   (4)

In reality, a noise-free digital multimeter does not exist and you need to account for the noise level when calculating its absolute units of resolution. Using formula 4, you can define the effective absolute units of resolution of a noisy digital multimeter as the step size of a noise-free digital multimeter with a quantization noise equal to the total noise of the noisy digital multimeter.

Effective absolute units of resolution = * rms noise   (5)

From formula 3, you can define the Effective Number of Digits (ENOD) of this noisy digital multimeter as:

ENOD = log₁₀(total span / Effective absolute units of resolution)  (6)

For example, if a digital multimeter 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 = * 70 µV = 242.49 µV

ENOD = log₁₀ (20.0 V/242.49*10^-6 V) = 4.92 digits

This digital multimeter can be called a 5 digit digital multimeter.
The minimum number of counts needed for this digital multimeter 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).

As another example, if the same digital multimeter demonstrates an rms noise level of 20 µV:

Absolute units of resolution = * 20 µV = 69.28 µV

ENOD = log₁₀ (20 V/69.28*10^-6 V) = 5.46 digits

This digital multimeter would be considered a "5½" digit digital multimeter.

The minimum number of counts needed for this digital multimeter 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, ENOD, to conventional digits of resolution for digital multimeters. 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 is used only as an approximation.



Learn more about NI Digital Multimeters.

Noise

Noise in a measurement can originate from the instrument taking the measurement or from an interfering signal passing through the instrument and causing measurement instability. When considering noise, you need to know the measurement bandwidth because it sets the bounds for how you can manage the noise. You can decrease the measurement bandwidth by increasing the aperture of the measurement or by averaging the measurement.

Noise in the system is a common and problematic challenge in designing measurement systems. Noise sources in the environment can be electrostatically or inductively coupled in from the powerline; therefore, most digital multimeters specify noise rejection at line frequencies of 50 Hz or 60 Hz. The rejection at 400 Hz is, at a minimum, as good as the rejection at 50 Hz because the aperture time for 50 Hz also eliminates 400 Hz components. For more information about how to configure the NI 4070 Digital Multimeter for optimum NMRR, refer to DC Noise Rejection.

A commonly overlooked source of noise in precision instrumentation is the source noise resistance (ohms), as shown in the following figure.

This noise is present in every resistor at common laboratory temperatures and is caused by random thermal motion of electrically charged carriers within the device. The noise is a function of temperature, the value of the resistance (ohms), and the bandwidth of the measurement. The noise is defined as:

You can convert this equation to:

R =resistance (ohms) being measured
f = noise bandwidth of the measurement

This equation assumes ideal resistor elements exhibiting white noise that is Gaussian in distribution. Some resistors, such as certain carbon film resistors, can generate noise from other mechanisms when current (amps) is passed through them. Metal foil and wire wound resistors approach this theoretical limit.

As a point of reference to simplify the calculation, a 1 kΩ resistor has rms noise density (1 Hz bandwidth). You can scale this value to get to any noise level given any resistor or frequency by multiplying it by . For example, a 100 kΩ resistor in a 100 Hz bandwidth has a noise of:

e_n = 400 nVrms

If the digital multimeter is digitizing at a 1 kS/s, the measurement bandwidth is 1 kHz and the effective noise is:

e_n = 4 nV x 316

e_n = 1.26 µVrms or 8.3 µV_p-p

Thus, the source resistance (ohms) limits the noise floor of the measurement over a 1 kHz bandwidth to 8.3 µVp-p.

Learn more about NI Digital Multimeters.

Precision

Precision is a measure of the stability of the digital multimeter and its capability of resulting in the same measurement over and over again for the same input signal. Precision is given by:

Precision = 1 – |X_n – Av(X_n)|/|Av(X_n)|

where

X_n = the value of the nth measurement

Av(X_n) = the average value of the set of n measurement

For instance, if you are monitoring a constant voltage of 1 V, and you notice that your measured value changes by 20 µV between measurements, then your measurement precision is:

Precision (1 – 20 µV/1 V) x 100 = 99.998%

Precision is most valuable when you are using the digital multimeter to calibrate a device or performing relative measurements.

Learn more about NI Digital Multimeters.

Reader Comments | Submit a comment »

Another ref to digits
The NI definition of digits is correct. Check another source at http://en.wikipedia.org/wiki/Multimeter
- Gurdas Singh. gurdas@qagetech.com - Oct 5, 2006

Digital Multimeter Absolute Units and Digits of Rsolution
Check, please, the sentence "This digital multimeter (5 1/2 digits) could show positive or negative values from 0 to 199 999" Is it right? Wouldn´t it to be 0 to 100 000 ? See the HP 34401A "User´s Guide", pg. 226, item "Number of Digits and Overranging". Thanks for reply.
- itl@mbox.trilog.cz - Apr 6, 2006