Instruments - Static Characteristics

Desirable Characteristics

Accuracy

Accuracy is

the closeness of a measurement to the true value

Relative accuracy can be expressed as

a_r = (y_max - x) / x (1)

where

a_r = relative accuracy (unit/unit)

x = input true value (unit)

y = instrument output (unit)

Example - Accuracy

The true length of a steel beam is 6 m. Three repeated readings with a laser meter indicates a length of 6.01 m, 6.0095 and 6.015 m. The accuracy based on maximum difference can be calculated as

an absolute value like

a_a = (6.015 m) - (6 m)

= 0.015 m

or as a relative value

a_r = ((6.015 m) - (6 m)) / 6 m

= 0.0025 m/m

or as relative value in percentage

a_% = (((6.015 m) - (6 m)) / (6 m)) 100%

= 0.25 %

The accuracy of an instrument can be related to

maximum measured value possible for the instrument
maximum range for the instrument
actual output from the instrument

Two terms commonly used in connection with accuracy are precision, trueness and calibration.

Precision

Precision is the closeness of agreement among a set of results

Example - for the steel beam above all measurements are within ±0.01 m - and we could say that the precision is good.

Trueness

Trueness is the closeness of the mean of a set of measurement results to the actual (true) value

Example - for the steel beam above the mean value of the set of measurements is 6.01 m - and we could say that the trueness could have been better.

Calibration

The precision of the laser meter used in the example above is good and the accuracy of the meter can be improved with calibration as

calibration = (6.01 m) - (6 m)

= 0.01 m

Sensitivity

Sensitivity is the increment of the output signal (or response) to the increment of the input measured signal - and can be expressed as

s = dy / dx (2)

where

s = sensitivity (output unit / input unit)

dy = change instrument output value (output unit)

dx = change in input true value (input unit)

Example - Temperature measurement with a Pt100 Platinum Resistance Thermometer

When temperature is changed from 0 ^oC to 50 ^oC - the resistance in a Pt100 thermometer changes from 100 ohm to 119.4 ohm. The sensitivity for this range can be calculated as

s = ((119.4 ohm) - (100 ohm)) / ((50 ^oC) - (0 ^oC))

= 0.388 ohm/^oC

Repeatability

Repeatability is the variation in measurements taken on the same item under the same conditions.

Reproducibility

Reproducibility is the ability of a measurement to be duplicated, either by the same person or by someone else under changed conditions.

Undesirable Characteristics

Drift

Drift is the change in instrument output over time - when the true value is constant.

Dead zone

Dead zone errors are created by

Threshold

Threshold is when a minimum input is required to generate change in output.

Hystersis

Hysteris is when unloading applied input don't creates the same output.

An example can be a nut that is screwed a number of turns on a threaded rod. When turned back the same number of turns the nut will not be in the exact the same position as at the start. This is a typical a problem that must be adressed in applications like cnc machines and 3d printers.

Creep

Creep is caused by the time an instrument need to adapt to change in aplied input.

Resolution

Depending on the instrument - but minimum change in input may required for change in output.

Significant Figures

Uncertainty in measurements or calculated values is indicated by the number of significant figures.

Significant figures are the digits of a number known with certain certainty

The last digit in a number is taken as uncertain while the other digits are regarded as certain.

Example - Significant Figures

For the number 31.2 we regard the two first digits 30 as certain and the last digit 6 as uncertain. Unless otherwise stated an uncertainity of ±1 is assumed for the last digit.