22-Agric-A5 Principles of Instrumentation · May 2016
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. 04-Agric-A5 Principles of Instrumentation, National Exams May 2016 — 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) A signal is decomposed into its frequency components mathematically by the Fourier transform (for a general, non-periodic or finite-duration signal) or the Fourier series (for a strictly periodic signal), which represent the time-domain signal $x(t)$ as a sum (or integral) of sinusoids of different frequencies, amplitudes and phases, $$x(t)=\int_{-\infty}^{\infty}X(f)\,e^{j2\pi ft}\,df,$$ where $X(f)$ — the frequency-domain spectrum — gives the amplitude and phase of the component at each frequency $f$.
b) The low-pass filter is the most used filter type in instrument applications — nearly every measurement chain needs to reject high-frequency noise/interference above the signal's own bandwidth, whether as a dedicated signal-conditioning stage or, essentially universally, as the anti-aliasing filter placed immediately before analog-to-digital sampling (part c).
c) An anti-aliasing filter is a low-pass filter placed before an analog-to-digital converter's sampling stage, with its cutoff set at or below half the sampling frequency (the Nyquist frequency, $f_s/2$). Its purpose is to attenuate any signal content above $f_s/2$ before sampling occurs, because any such content would otherwise be misrepresented (aliased) as a false lower-frequency component once sampled — an error that cannot be detected or removed after the fact, since the aliased and genuine low-frequency content become indistinguishable in the sampled data.
d) Given. Second-order low-pass filter, cutoff frequency $f_c=30$ Hz; interference frequency $f=120$ Hz.
Find. The reduction (attenuation) of the 120 Hz signal.
Approach. A generic second-order (two-pole) low-pass filter rolls off at $-40$ dB/decade ($-12$ dB/octave) well above its cutoff; evaluate the standard maximally-flat 2nd-order magnitude response at the given frequency ratio and cross-check against the straight-line Bode asymptote.
| Quantity | Value |
|---|---|
| Frequency ratio $f/f_c$ | 4 (two octaves above cutoff) |
| Amplitude ratio $|H(f)/H(0)|$ | ≈ 0.0624 (≈ 6.2% remains) |
| Attenuation | ≈ 24.1 dB (≈ 16× reduction) |
e) A running average is a linear operation, but a non-linear sensor's output is a non-linear function $y=f(x)$ of the true measurand $x$. Averaging the sensor's raw output readings and then converting the average through the (inverse) calibration curve is not the same as averaging the true underlying measurand and converting that single value — in general $\overline{f(x)}\ne f(\overline{x})$ for a non-linear $f$ (Jensen's inequality guarantees a systematic bias whenever $f$ is curved, not just random scatter). The running-average (low-pass) filter therefore introduces a systematic bias in the recovered measurand whenever the raw signal fluctuates over a range where the sensor's calibration curve is significantly non-linear — the size and sign of the bias depend on the curve's local concavity and the amplitude of the fluctuations being averaged over.
f) A low-pass filter can be built as a purely mechanical/pneumatic analog of the electrical RC filter: connect the air-flow sensing line to a small enclosed chamber (an "air capacitor," analogous to a capacitance storing charge) through a narrow restriction such as a small-bore capillary tube or orifice (a flow resistance, analogous to an electrical resistor). Fast pressure/flow fluctuations are damped because the restriction limits how quickly the chamber's pressure can respond, exactly as an electrical $R$-$C$ pair limits how fast a capacitor's voltage can follow a changing input — giving a first-order low-pass response with time constant $\tau\approx R_{pneumatic}C_{chamber}$ (analogous to $\tau=RC$), while the slow, genuine average flow trend passes through with negligible attenuation. (A mechanically damped diaphragm/dashpot, or simply a sensor with deliberately large thermal/mechanical inertia, achieves the same low-pass effect by the same physical principle — a restriction that only lets slow changes propagate.)
