22-Agric-A5 Principles of Instrumentation · December 2015
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. 04-Agric-A5 Principles of Instrumentation, National Exams December 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. (bridge circuits, instrumentation amplifiers, CMRR, ADC architectures); J. Fraden, Handbook of Modern Sensors: Physics, Designs, and Applications, 5th ed. (thermistors, strain gages, photodetectors); R.W. Fox, A.T. McDonald and P.J. Pritchard, Introduction to Fluid Mechanics, 7th ed. (orifice and venturi metering).
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) Given. Op-amp $A_1$'s non-inverting input is tied to $V_1$, $A_2$'s to $V_2$; their inverting inputs are joined through the series chain $R_1$-$R_g$-$R_2$ (feedback resistor $R_1$ from $A_1$'s own output to its inverting input, likewise $R_2$ for $A_2$); the outputs $V_{o1}$, $V_{o2}$ feed a unity-gain difference amplifier built from four matched resistors $R$. Find. $V_{out}$ as a function of $V_1$, $V_2$, $R_1$, $R_2$, $R_g$.
Approach. Use the virtual-short/no-input-current op-amp idealization to find the single current through $R_1$-$R_g$-$R_2$, express $V_{o1}$ and $V_{o2}$ from it, then apply the difference stage's unity gain.
| Quantity | Result |
|---|---|
| Current through $R_1$-$R_g$-$R_2$ | $I=(V_1-V_2)/R_g$ |
| $V_{o1}$, $V_{o2}$ | $V_1+I R_1$, $\;V_2-I R_2$ |
| $V_{out}$ | $(V_2-V_1)\left(1+\dfrac{R_1+R_2}{R_g}\right)$ |
b) Two advantages over a simple single-op-amp difference amplifier: (1) VERY HIGH input impedance at BOTH inputs — $V_1$ and $V_2$ connect directly to op-amp non-inverting inputs, presenting near-infinite impedance, unlike a single-op-amp difference amp whose input impedance is comparatively low and set (asymmetrically) by its own input resistors — critical for not loading a high-impedance sensor (Q2h). (2) Gain is adjustable with a SINGLE resistor $R_g$ that never touches the precisely matched 4-resistor network setting the output stage's CMRR — so gain can be changed (even made a trim potentiometer) without ever degrading common-mode rejection, which a single-op-amp difference amplifier cannot offer, since there changing gain means changing the very resistors whose matching sets CMRR.
c) A mismatch among the four nominally equal output-stage resistors degrades the difference amplifier's common-mode rejection ratio (CMRR) — the fraction of any common-mode signal present equally on $V_{o1}$ and $V_{o2}$ (e.g. residual common-mode voltage that survived stage 1, ground noise) that leaks through to the output instead of cancelling grows directly with the fractional mismatch (even a tight 0.1% resistor tolerance caps the achievable CMRR at roughly 60 dB regardless of how good the op-amps' own intrinsic CMRR is). It also introduces a small differential-gain error and DC offset, but degraded CMRR is the dominant, most damaging consequence.
d) In the derived result $V_{out}=(V_2-V_1)(1+(R_1+R_2)/R_g)$, $R_1$ and $R_2$ appear ONLY through their SUM — the individual values of $R_1$ and $R_2$ never appear separately. Any mismatch between $R_1$ and $R_2$ is therefore invisible to the overall transfer function as long as their sum is known and set accurately; only that sum needs to be precise, unlike the four output-stage resistors in (c), whose INDIVIDUAL matching (not merely their sum) is what sets CMRR.
e) The first stage typically carries most of the amplifier's total gain (often 100-1000$\times$ for small sensor signals), and that gain amplifies EVERYTHING present at the first stage's own input — including its own intrinsic input-referred noise and DC offset voltage — by that same large factor before it ever reaches the second stage. Errors introduced further downstream (stage 2, typically unity or modest gain) are NOT multiplied by the large first-stage gain when referred to the output, and when referred back to the INPUT they are effectively divided by that same first-stage gain. By this standard cascaded-stage error-budget rule, a poorly chosen first-stage op-amp corrupts the whole measurement in a way no later stage can undo, while a noisy second-stage op-amp barely matters.
