- Acknowledgments
- Definitions
- Static characteristics

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

- Measurand
- A physical parameter being quantified by measurement

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:

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

- See also the NIST website.
- 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

- 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

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)

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

- 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

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

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

- the non-coincidence between the loading (increasing) and the unloading (decreasing) measurement curves
- maximum input hysteresis, and maximum output hysteresis
- often seen in mechanical transducers or sensors with electrical windings formed around an iron core (transformers)

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

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

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

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)

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

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

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