A DC amplifier used to measure a slowly varying voltage has a noise spectral density

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Uncertainty in measurement · Designing and running experiments

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

Problem

e_n = 50\,\mathrm{nV/\sqrt{Hz}}

. Before its case has settled to a steady temperature its zero drifts at a rate

D = 0.20\,\mathrm{\mu V\,s^{-1}}

. Each measurement is the difference between two averages, each of readings taken for a time

t

, and the total standard uncertainty on that difference is

u(t) = \sqrt{\frac{e_n^2}{t} + (Dt)^2}.

((a)) Find the optimum averaging time

t_{\rm opt}

and the resulting minimum uncertainty

u(t_{\rm opt})

. ((b)) A thermally regulated enclosure is fitted, cutting the drift by a factor of

8

D' = 25\,\mathrm{nV\,s^{-1}}

. Find the new optimum averaging time

t_{\rm opt}'

and the new minimum uncertainty

u(t_{\rm opt}')

. ((c)) State, as a ratio, how much longer the new averaging takes and how much the minimum uncertainty improves, and comment on whether these two ratios are equal.

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