Performance characteristics of measuring instruments

David Baird, 2006-02-01, HW 1, EE 521

Table of Contents

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:

  • 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

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

Dead space

  • Dead zone
  • 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

  • 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)

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
  • Radiation
  • 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