Two claims are made about two different apparatus

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Electronics, Instrumentation & Communication · Feedback, operational amplifiers and analogue signal processing

Measurement and Method, Ch Designing the measurement: noise, averaging, null and differential methods

Problem

Two claims are made about two different apparatus. Neither is correct as stated. ((a)) A demonstrator averages a step of size

0.50\,\mathrm{\mu V}

buried in noise that, for an averaging time

t

, falls as

0.080/\sqrt{t}\,\mathrm{\mu V}

(with

t

in seconds), while a slow drift moves the baseline by

D t

with

D = 0.020\,\mathrm{\mu V\,s^{-1}}

. A quick calculation gives an optimum averaging time of a few seconds, but the demonstrator argues: "averaging always helps, so I will average for

10\,\mathrm{s}

, five times longer than that estimate, to get the best possible signal-to-noise ratio." Find the true optimum averaging time

t_{\rm opt}

and the resulting

S/N

, and compare it with the

S/N

the demonstrator actually gets at

10\,\mathrm{s}

. ((b)) A strain-gauge bridge has four arms each of nominal resistance

R = 120\,\mathrm{\Omega}

, supplied at

V = 5.0\,\mathrm{V}

, and is balanced against a decade resistor before loading. A strain changes one arm's resistance by

\Delta R = 1.0\times10^{-3}\,\mathrm{\Omega}

, unbalancing the bridge by

\Delta V_{\rm out} = V\,\Delta R/(4R)

. Someone argues: "this measurement can only be trusted if the null detector is a voltmeter with a calibration accuracy of

1\,\%

, because accuracy is what determines how well we know the voltage." Compute

\Delta V_{\rm out}

and explain what is wrong with this argument, stating what property of the detector the measurement really needs.

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