22-Agric-A5 Principles of Instrumentation · May 2015
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. 04-Agric-A5 Principles of Instrumentation, National Exams May 2015 — 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); 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); 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) Johnson (thermal) noise originates from the random thermal agitation of charge carriers inside any resistive element — it is present in every resistor at any temperature above absolute zero even with zero current flowing, since it is a Brownian-motion-driven voltage fluctuation, not a current-dependent effect.
b) An ideal voltmeter should draw no current from the circuit it measures. A real voltmeter with input impedance $R_v$ in parallel with the source's own (often unknown) internal/Thevenin resistance $R_s$ forms a voltage divider, so the meter reads $V_{true}\cdot R_v/(R_v+R_s)$ — a value lower than the true open-circuit voltage. Making $R_v \gg R_s$ drives that loading error toward zero, so the meter reads the source's voltage without disturbing it.
c) Common mode rejection is a differential amplifier's ability to reject a signal that appears equally on both its inputs (the "common mode" signal, e.g. ground-loop offsets or 60 Hz mains pickup on both sensor leads) while still amplifying the difference between the two inputs (the wanted signal). It is important because sensor signals are frequently small differential voltages riding on a much larger common-mode interference; without adequate common mode rejection ratio (CMRR) that interference would swamp the genuine signal.
d) A 12-bit ADC has $2^{12}=4096$ discrete output codes across its full-scale range, so its resolution (one least-significant-bit step) is $$1/4096 \times 100\% \approx 0.0244\%\ \text{of full scale}.$$
e) The damping coefficient (damping ratio) of a sensor system governs the trade-off between speed of response and overshoot/ringing in its dynamic (transient) response. An under-damped system responds quickly but overshoots and oscillates before settling, delaying a trustworthy reading; an over-damped system settles smoothly but sluggishly. Since most measurements are taken from a system that has just been disturbed (a load applied, a step change), the damping coefficient directly sets how long must be waited before the output can be trusted, and how large a transient error would result from reading too soon.
f) An anti-aliasing filter is an analog low-pass filter placed before the sample-and-hold/ADC stage, with a cutoff below half the sampling rate. It removes signal and noise content above that cutoff so those high frequencies cannot fold back ("alias") into the sampled data as spurious low-frequency content that sampling alone cannot distinguish from real low-frequency signal.
g) The Nyquist criterion: the sampling rate must be at least twice the highest frequency component present in the (already anti-alias filtered) signal, $f_s \ge 2f_{max}$. Sampling slower than that cannot uniquely represent the original waveform — the missing information reappears as aliasing.
h) A "3½ digit" display has four digit positions, but the left-most (most significant) position can only show 0 or 1 (i.e. a partial, "half" digit) rather than the full 0-9 range of the other three. Its maximum reading is therefore ±1999, not the ±9999 a true 4-digit display would give, even though it occupies four physical digit positions.
i) Placing the pre-amplifier immediately at the sensor amplifies the weak raw signal to a robust level before it has to travel over any appreciable length of cable. Any noise picked up along that cable run, or voltage drop from cable resistance, then adds to an already-large signal and is a much smaller fraction of it (better SNR). If the raw, still-weak signal were sent over a long cable and only amplified at the far end, the same absolute noise pickup would corrupt a much smaller signal and be amplified along with it, giving a far worse signal-to-noise ratio.
j) Twisting the pair cancels inductively (magnetically) coupled noise: each successive twist reverses the loop area the interfering field acts on, so the induced common-mode voltages largely cancel when a differential measurement is taken across the pair. The outer shield, in turn, intercepts capacitively (electric-field) coupled interference and diverts it to ground (grounded at one end only, to avoid a ground-loop current in the shield itself) before it can couple onto the signal conductors. Together the two mechanisms protect against both the magnetic and electric components of interference typical of an industrial sensor wiring run.
