An electret microphone capsule behaves

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

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

Problem

An electret microphone capsule behaves, for noise purposes, as a capacitance

C_{\rm cap}=50\,\mathrm{pF}

at the node where the transducer's signal appears. It is biased through a resistor

R_{\rm bias}=10\,\mathrm{M\Omega}

290\,\mathrm{K}

into a JFET-input amplifier with

e_n=15\,\mathrm{nV}/\sqrt{\mathrm{Hz}}

and

i_n=5.0\,\mathrm{fA}/\sqrt{\mathrm{Hz}}

, both referred to the amplifier's input. Downstream of this node the amplifier's own noise rolls off as a single first-order pole of corner frequency

f_c=20\,\mathrm{kHz}

; that roll-off does not act on anything at the bias node itself. The microphone's open-circuit sensitivity is

10\,\mathrm{mV\,Pa^{-1}}

, referred to the same node. Treat all three noise sources as independent. ((a)) Find the bias resistor's Johnson noise density

e_R

and the corner frequency

f_{RC}

of the low-pass that

R_{\rm bias}

forms with

C_{\rm cap}

at the node. Compare

f_{RC}

with

f_c

. ((b)) The resistor's noise reaching the node is shaped by that

R_{\rm bias}

–

C_{\rm cap}

low-pass, not by

f_c

: use the

k_BT/C

result to find its rms contribution at the node. The current noise

i_n

is injected at the same node and sees the same low-pass, but with a DC gain of

R_{\rm bias}

instead of one; extend the noise-equivalent-bandwidth argument to find its rms contribution. ((c)) The amplifier's own

e_n

is applied directly at its input, undiminished by the node's impedance, and is limited only by

f_c

: find its rms contribution over the amplifier's noise-equivalent bandwidth. Combine all three rms contributions in quadrature and identify the dominant one with its share of the noise power. ((d)) Find the smallest sound pressure detectable at a signal-to-noise ratio of one.

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