# Acknowledgments

This document is copied almsot verbatim, with a few extra annotations, from Dr. Hai Xiao's lecture notes of Spring 2006.

# Definitions

Measurand
A physical parameter being quantified by measurement

# Static characteristics

## Accuracy/unaccuracy/measurement uncertainty

Accuracy:

• Accuracy is a measure of how close the measured value is to the true value
• Accuracy is a qualitative concept

Measurement uncertainty:

• Uncertainty: parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand. The parameter may be, for example, a standard deviation (or a given multiple of it), or the half-width of an interval having a stated level of confidence.
• Standard uncertainty: uncertainty of the result of a measurement expressed as a standard deviation
• Expanded uncertainty: quantity defining an interval about the result of a measurement that may be expected to encompass a large fraction of the distribution of values that could reasonably be attributed to the measurand.

## Precision/repeatability/reproducibility

Precision:

• The closeness of agreement between independent test results obtained under stipulated conditions
• Qualitative concept
• Precision should not be confused with accuracy

Repeatability:

• Closeness of the agreement between the results of successive measurements of the same measurand carried out under the same conditions of measurement

• Same (repeatability) conditions include:

• the same measurement procedure
• the same observer
• the same measuring instrument, used under the same conditions
• the same location
• repeition over a short period of time
• Precision under repeatability conditions

• Also a qualitative concept

Reproducibility:

• Closeness of agreement between the results of measurements of the same measurand carried out under changed conditions of measurement

• The changed conditions may include:

• principle of measurement
• method of measurement
• observer
• measuring instrument
• reference standard
• location
• conditions of use
• time
• Precision under reproducibility conditions

• Reproducibility is also a qualitative concept

### Qualitative v. quantitative

Qualitative terms should never have a number directly associated with the term:

• Wrong: the precision of the measurement results is 2 um
• Correct: the precision of the measurement results, expressed as the standard deviation obtained under repeatability conditions is 2 um

## Linearity

• A measure of how close is the output of an instrument to a straight line
• Use least-square method to do line-fitting ofthe output, the non-linearity is then defined as the maximum deviation of any of the output from the fitted straight line
• A quantitative number

## Sensitivity of measurement

• A measure of the change in instrument output that occurs when the measurand changes by a given amount

• It can be caluclated as the slow of (or a portion of) the fitted straight line:

Sensitivity = (Scale deflection) / (value of measurand producing the deflection)

• Note that the sensitivity might vary at different portion of measurement (e.g. sensitivity is zero at the top of a sinusoidal output)

## Resolution

Classic definition based on analog output instruments:

• The smallest change of the magnitude of the measurand that produces a minimum observable output of the instrument
• Can be expressed either as an absolute value or a percentage of the full scale deflection

What if the instrument's output is digital?

• More and more modern instruments have digital outputs because of the wide usage of computer
• The "resolution" of the digital output can be a very small number, but this is not the resolution of the instrument (e.g. what is the "resolution" of a 32-bit IEEE floating point number?)
• The resolution of an digital output instrument should be limited by the front end rather than the digital computation

## Threshold

• The minimum level of input that produces a large enough detectable output reading deflected from the initial states of the instrument, very often, the initial states of the instrument are at zero
• It can be expressed as either an absolute value or a percentage

• The range of input values over which there is no change in output values
• Example: rectifier circuits using diodes

## Bias

• A constant value that the instrument adds to its output even at zero input

## Hysteresis

• maximum input hysteresis, and maximum output hysteresis
• often seen in mechanical transducers or sensors with electrical windings formed around an iron core (transformers)

## Environmental effects

Survivability in harsh environment - Storage conditions: instrument is not required to operate - Operational but not to the full specifications

## Sensitivity to disturbance

• Temperature
• Humidity
• Ambient pressure (elevation, depth under water, etc.)
• Electromagnetic interference
• Acceleration
• Shock
• Vibration
• Duration exposed to harsh environment

## Mathematical model

Transfer function of the instrument

• Mathematic description of the entire measurement system (as for any other systems)
• Break the entire measurement system into small sybsystems (blocks) along the signal path through the system
• There will be nonlinear blocks, and approximations have to be made to linearize them so that transfer functions can be obtained
• Once the transfer function of the system is established, the static (or steady state) response of the system can be derived
• The dynamic characteristics can also be obtained based on the transfer function of the instrument

### Zero order instrument

Mathematical model

• qo = K qi
• qi is the input, qo is the output
• K is a constant (sensitivity of instrument)
• Theoretically, zero order instrument has infinite bandwidth (the output responses to the input instantaneously)

### First order instrument

Mathematical model

• a1 dqo/dt + a0 qo = b0 qi
• Qo(s)/Qi(s) = K / (1 + Ts)
• K = b0/a0 is the static sensitivity, T = a1/a0 is the time constant
• There will be a time lag (delay) between the change measurand and the update of the instrument reading

### Second order instrument

Mathematical model

• a1 d^2qo/dt^2 + a1 dqo/dt + a0 qo = b0 qi
• Qo(s)/Qi(s) = (K w0^2) / (s^2 + 2w0 xi s + w0^2)
• K = b0/a0 is the static sensitivity
• xi = a1/(2a0 a2) is the damping ratio
• w0 is the natural frequency
• optimal choice of xi: between 0.6 and 0.8

## Dynamic characteristics

• Frequency response/Bandwidth
• Warm up time
• Delay
• Stability/undershoot
• Jitter