A Hall-effect sensor with internal resistance at is used to measure a weak

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

Electronics for Measurement, Ch Noise and the signal-to-noise ratio

Problem

A Hall-effect sensor with internal resistance

R_s = 5.0\,\mathrm{k\Omega}

295\,\mathrm{K}

is used to measure a weak, modulated magnetic field. Two front-end amplifiers are available: amplifier C (bipolar) with

e_n = 3.0\,\mathrm{nV}/\sqrt{\mathrm{Hz}}

i_n = 4.0\,\mathrm{pA}/\sqrt{\mathrm{Hz}}

; amplifier D (CMOS) with

e_n = 25\,\mathrm{nV}/\sqrt{\mathrm{Hz}}

i_n = 0.020\,\mathrm{pA}/\sqrt{\mathrm{Hz}}

. The field is modulated at

875\,\mathrm{Hz}

, chosen because the nearest harmonics of

50\,\mathrm{Hz}

mains (

850

and

900\,\mathrm{Hz}

) and of

60\,\mathrm{Hz}

mains (

840

and

900\,\mathrm{Hz}

) are all at least

25\,\mathrm{Hz}

away. Detection is by a lock-in with a first-order output time constant

\tau = 2.0\,\mathrm{s}

. The sensor's sensitivity, referred to the amplifier's input node, is

5.0\,\mathrm{mV\,mT^{-1}}

. ((a)) By comparing each amplifier's optimum source resistance

e_n/i_n

with

R_s

, make a first choice of amplifier, then confirm it by computing each amplifier's total input-referred density at

R_s

. ((b)) Using that amplifier, find the total noise voltage density referred to the input, combining the sensor's Johnson noise with the amplifier's

e_n

and

i_nR_s

, and state which term dominates. ((c)) Find the effective noise bandwidth of the lock-in for this time constant. ((d)) Find the rms noise voltage referred to the input and the smallest magnetic field detectable at a signal-to-noise ratio of one.

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