22-Agric-A5 Principles of Instrumentation · May 2018
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. 04-Agric-A5 Principles of Instrumentation, National Exams May 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, thermocouples, capacitive and photo sensors).
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) Standard deviations combine as variances, not linearly, because independent random errors add in quadrature: for independent sources $\sigma_{total}=\sqrt{\sigma_1^2+\sigma_2^2+\cdots}$. Simply summing $\sigma_1+\sigma_2+\cdots$ treats the errors as if they were perfectly correlated (always at their worst simultaneously), which overstates the combined uncertainty — the correct root-sum-square rule reflects that independent errors partially cancel on average.
b) A two-point calibration (typically zero and full-scale/span) can only fix a straight line, so it implicitly assumes the instrument's true response is linear across the entire working range between those two points. If the real response has any curvature, a two-point calibration will silently misrepresent the reading everywhere except at the two calibration points themselves.
c) Statistical methods (repeatability, standard deviation, RMS scatter of repeated readings) only characterize precision — how tightly the instrument's readings cluster around whatever value it happens to report. Accuracy is how close that cluster is to the true value, which requires comparison against an independent, traceable reference/standard; a sensor can be extremely precise (very small spread) while being consistently offset from the truth by a systematic (bias) error that repeated measurements of the same instrument can never reveal on their own.
d) Plotting the calibration error (reading minus true/standard value) as a function of the standard value used, rather than just quoting one aggregate number, reveals the structure of the error — whether it is a constant offset (zero error), a slope error (span/gain error), or a curvature/non-linearity that is worse in some part of the range than others. An aggregate RMS or maximum-error figure hides exactly where in the measurement range the instrument is least trustworthy, which the error-vs-standard-value plot exposes directly.
e) A blank is a sample prepared and measured in exactly the same way as a real sample but containing none of the analyte/quantity being measured (e.g. pure solvent, or a control specimen with zero of the property under test). It establishes the instrument's own zero-level response and baseline noise, which must be subtracted from every subsequent real reading.
f) An internal standard is a known, fixed amount of a reference substance (or reference signal) added directly into every sample — including the unknowns — before or during measurement. Because the internal standard experiences the same handling, drift, and instrument-response variations as the analyte in that same run, the ratio of the analyte's response to the internal standard's response cancels out run-to-run variability (injection volume, detector drift, sample-to-sample loss) that a simple external calibration curve would not correct for.
g) The lowest quantity a sensor can measure — its detection limit — is defined by the point where the signal can no longer be reliably distinguished from the instrument's own baseline noise, conventionally taken as the input that produces a response some fixed multiple (commonly 3×) of the standard deviation of the noise/blank measurement, not by the sensor's nominal resolution or display digits alone.
h) Sensitivity is the local slope of the calibration curve ($dV_{out}/dX_{input}$), not a single number for a non-linear instrument. The instrument is most sensitive wherever that curve is steepest — graphically, wherever the tangent line to the calibration curve has the largest slope — which for a typical saturating (concave-down) curve is near the low end of the range, and for a typical threshold/rising (concave-up) curve is near the high end; it must be read off the actual curve's slope rather than assumed.
i) A primary standard is a reference whose value is fixed directly by a physical law, constant, or defined phenomenon (e.g. the triple point of water for temperature, a fundamental physical constant), established without calibration against any other physical standard. It sits at the top of the traceability chain — every other (secondary, working) standard is ultimately calibrated back to a primary standard.
j) For a set of errors distributed about a mean (with a nonzero mean/bias offset removed), the RMS error and the standard deviation are numerically the same quantity, $\text{RMS}=\sqrt{\frac{1}{n}\sum e_i^2}$ versus $\sigma=\sqrt{\frac{1}{n}\sum(e_i-\bar e)^2}$ — they coincide exactly when the mean error $\bar e$ is zero (i.e. no systematic bias), and diverge ($\text{RMS}^2=\sigma^2+ \bar e^2$) whenever a bias is present, since the RMS error also captures that offset while the standard deviation only captures scatter about the (possibly biased) mean.
