==================================================== Performance characteristics of measuring instruments ==================================================== ------------------------------------- David Baird, 2006-02-01, HW 1, EE 521 ------------------------------------- .. contents:: 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 .. Confidence interval .. ? 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 |mgr|\ m - Correct: the precision of the measurement results, expressed as the standard deviation obtained under repeatability conditions is 2 |mgr|\ m .. _nistwebsite: http://physics.nist.gov/Pubs/guidelines/appd.1.html#d12 __ _nistwebsite .. Tolerance .. --------- .. ? .. Dynamic range (range of span) .. ----------------------------- .. ? 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 .. .. |Mgr| unicode:: U+0039C .. GREEK CAPITAL LETTER MU .. .. |mgr| unicode:: U+003BC .. GREEK SMALL LETTER MU .. |mgr| replace:: u