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) The applied bias voltage electrochemically consumes CO at the sensing electrode as fast as it arrives, holding the CO concentration inside the cell essentially at zero at all times. The rate at which CO reaches the electrode is therefore controlled entirely by diffusion through the external barrier, which (by Fick's first law) is directly proportional to the concentration difference across it — and since the inside concentration is pinned near zero, that difference is essentially just the outside (ambient) CO concentration. Each CO molecule that diffuses in and reacts releases a fixed number of electrons (2, per the given reaction), so the resulting current is a direct, linear count of CO molecules arriving per unit time — i.e. directly proportional to the ambient CO concentration, with no saturating or non-linear chemical step in the measurement path.
b) Fouling is the gradual buildup of contaminants, reaction by-products, or other deposits on the sensing electrode surface or within the diffusion barrier (or electrode poisoning by an interfering species), which reduces the active electrode area available for the CO reaction and/or partially blocks the diffusion pathway. For this sensor, fouling reduces the effective reaction rate and/or the diffusion coefficient, which lowers the output current produced for a given ambient CO concentration — the sensor's calibrated sensitivity drifts downward over time (a dangerous failure mode for a safety sensor, since it under-reports true CO levels) and its response time can also slow, so periodic recalibration or barrier/electrode replacement is required.
c) Given. Transimpedance (current-to-voltage) op-amp circuit: sensing-electrode current $I_{in}=-1.2\,\mu\text{A}$ feeding, via a small $10\,\Omega$ series resistor, into the op-amp's inverting input; non-inverting input grounded; feedback resistor $R_f=10\,\text{k}\Omega$ from the output back to the inverting input.
Find. $V_{out}$.
Approach. Apply the standard virtual-ground transimpedance relation: with an ideal op-amp and negative feedback, all of the input current is forced through $R_f$ (none enters the op-amp), so the stage inverts: $V_{out}=-I_{in}\,R_f$.
The printed figure draws the $I_{in}$ arrow pointing from the sensing electrode into the $10\,\Omega$ resistor and on to the summing junction, so $I_{in}$ is the current entering that node and the stage is inverting: $V_{out}=-I_{in}R_f$. The sign of the result is also confirmed by the chemistry: the half-reaction $CO+H_2O\rightarrow CO_2+2H^++2e^-$ is an oxidation, releasing electrons at the sensing electrode, so conventional current flows out of the summing junction and into the cell — i.e. in the direction opposite to the drawn arrow, which is exactly why the question states $I_{in}$ as a negative $-1.2\,\mu\text{A}$. The op-amp must therefore source that $1.2\,\mu\text{A}$ through $R_f$, which requires its output to sit above the virtual ground: a rising CO concentration drives $V_{out}$ positive, as a single-supply detector front end should. The magnitude is $|V_{out}|=|I_{in}|R_f=12\,\text{mV}$.
d) A company selling this type of life-safety sensor faces product liability exposure primarily around failure-to-warn: if a unit's response drifts low or fails silently due to fouling (part b) or self-calibration lapse and a user is injured or killed by undetected CO, this is a strong negligence/product-defect claim, particularly if the failure mode was known and not adequately disclosed or mitigated (e.g. via a mandated end-of-life or self-test/fault indicator). There is a duty to properly specify and communicate calibration/replacement intervals and to design in self-diagnostics that flag degraded sensitivity before it becomes dangerous; the product should meet applicable life-safety certification standards (e.g. UL/CSA CO-detector standards) as evidence of due diligence, and warranty/negligence exposure is significantly higher for a device whose entire purpose is to prevent death from an undetectable, odourless gas — any known failure mode not addressed in the design or disclosed to users is a direct liability risk.
| Item | Result |
|---|---|
| Sensor output current $I_{in}$ | $-1.2\,\mu\text{A}$ |
| Transimpedance gain | $R_f=10\,\text{k}\Omega$ |
| Output voltage | $V_{out}=-I_{in}R_f=+12\,\text{mV}$ |
