22-Agric-A5 Principles of Instrumentation · December 2018
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. 04-Agric-A5 Principles of Instrumentation, National Exams December 2018 — a three-hour open-book exam; any non-communicating calculator is permitted. Questions 1 and 2 are compulsory (20 marks each); candidates then choose any three (3) of Questions 3-7 (20 marks each) for a 100-mark paper. All seven questions are worked here.
Reference texts. E.O. Doebelin, Measurement Systems: Application and Design, 5th ed. (calibration, standards, static/dynamic sensor characteristics, second-order step response, sampling and ADCs); J.P. Bentley, Principles of Measurement Systems, 4th ed. (accuracy vs. precision, error propagation, signal conditioning); P. Horowitz and W. Hill, The Art of Electronics, 3rd ed. (Johnson noise, CMRR, ADC architectures, anti-aliasing, op-amp signal conditioning); J. Fraden, Handbook of Modern Sensors: Physics, Designs, and Applications, 5th ed. (thermistors, capacitive sensors, photodetectors); D.A. Skoog, F.J. Holler and S.R. Crouch, Principles of Instrumental Analysis, 7th ed. (detection limits, selectivity, optical sensing).
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
a) Two broad classes of standard are used. Primary standards are traceable directly to a national metrology institute (e.g. NRC in Canada, NIST in the US) — certified mass pieces, Zener-diode voltage references, triple-point cells, gauge blocks — and are used sparingly to protect their certified accuracy. Working (secondary/transfer) standards are calibrated against a primary standard and used day-to-day on the shop floor; they must themselves be periodically re-checked against a primary standard to stay traceable. Reference materials (certified solutions, calibration gases) play the same role for chemical sensors.
b) A minimum of two points (zero and full-scale span) fixes only a straight line and cannot reveal curvature; in practice a calibration uses enough points (commonly five or more, spanning the full range) to both establish the working curve and check for non-linearity and hysteresis. The points are normally run in both the ascending and descending direction — an even number of directional passes — so that hysteresis (a direction-dependent difference) can be detected rather than averaged away.
c) The acceptable range is bounded at the low end by where the output signal becomes indistinguishable from the sensor's own noise/resolution floor, and at the high end by where the response departs from linearity, saturates, or risks damaging the sensing element. Between those two bounds the sensor's output stays within its specified accuracy tolerance of the true value; outside them the calibration curve is no longer trustworthy.
d) Hold the measurand at a fixed, known value while deliberately varying one candidate interferent (temperature, humidity, supply voltage, electromagnetic field, orientation) over its expected service range, and record the resulting shift in sensor output. The slope of output versus interferent level, with the true measurand held constant, is the interference sensitivity for that variable; repeating for each suspected interferent in turn characterizes the full cross-sensitivity of the sensor.
e) A calibration is normally repeated several times (a minimum of three full-range passes is typical) rather than run once, because a single pass gives a curve but no estimate of scatter. Repetition lets the standard deviation of the readings at each calibration point be estimated, which is what separates random precision error from the underlying calibration curve itself — a single run cannot distinguish a genuine curve feature from a one-off noise excursion.
f) Drift is detected by re-running the same calibration, against the same reference standards, at a later time (after a period of use, or at fixed intervals) and comparing the new curve to the original. A parallel shift at every point indicates zero (offset) drift; a change in slope indicates span (gain) drift. Because drift is defined relative to time rather than to input level, it can only be caught by repeating the calibration, not by any single calibration run.
g) Apply a step change to the sensor's input and record its transient output. For a typical underdamped (second-order) sensor the response overshoots and rings before settling; measuring the ratio of two successive overshoot peaks gives the logarithmic decrement, from which the damping ratio follows, $$\zeta=\dfrac{-\ln(\text{OS})}{\sqrt{\pi^2+\ln^2(\text{OS})}},$$ where OS is the fractional overshoot of the first peak above the final steady value. Counting how many oscillations occur before the response settles within a stated tolerance band is a quicker, cruder alternative.
h) Hysteresis is checked by driving the calibration through the full range in the increasing direction, then back down through the same nominal input values in the decreasing direction, and comparing the output recorded at each common input value for the two directions. Any systematic difference between the up-going and down-going curves at the same input is hysteresis; it shows up as a closed loop rather than a single curve on an output-versus-input plot.
i) A model (commonly a straight line, but a low-order polynomial for a known-curved response) is fitted by least squares: choosing the model's coefficients to minimize the sum of the squared differences (residuals) between the model's prediction and each measured calibration point. This is a closed-form linear-algebra solution for a polynomial model and gives the best-fit curve in the minimum-mean-square-error sense.
j) Plot the residuals (measured minus fitted) against the standard value and confirm they scatter randomly about zero with no systematic trend or curvature left over — a visible pattern means the model is the wrong shape. Quantify the fit with the standard error of estimate or $R^2$, and, where enough data exists, hold back some calibration points from the fit and confirm the model predicts them correctly (cross-validation) — a model that only fits the points it was trained on can still be the wrong shape.
