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Ionization Chamber: Resistor Noise Calculations

Given the ionization chamber’s tiny currents and the huge resistors required to turn them into voltages, reviewing the thermal noise I generally ignore seems in order…

The RMS noise voltage of an ordinary resistor:

vn = √ (4 kT R Δf)

The constants:

  • kB – Boltzman’s Constant = 1.38×10-23 J/K
  • T – temperature in kelvin = 300 K (close enough)

Mashing them together:

vn = √ (16.6x10-21 R Δf)

vn = 129x10-12 √ (R Δf)

For a (generous) pulse current of 20 fA, a 10 GΩ resistor produces a mere 200 μV, so wrap a gain of 100 around the op amp to get 20 mV. An LMC6081 has a GBW just over 1 MHz, giving a 10 kHz bandwidth:

vn = 129x10-12 √ (10x109 10x103) = 1.3 mV

Which says the noise will be loud, but not deafening.

A 100 GΩ resistor increases the voltage by a factor of 10, so you can decrease the gain by a factor of ten for the same 20 mV output, which increases the bandwidth by a factor of ten, which increases the noise by a factor of … ten.

Ouch.

With the same gain of 100 (and therefore 10 kHz bandwidth) after the 100 GΩ resistor, the output increases by a factor of ten to 200 mV, but the noise increases by only √10 to 4 mV.

The LMC6081 has 22 nV/√Hz and 0.2 fA/√Hz input-referred noise, neither of which will rise above the grass from the resistor.

With 10 kHz bandwidth, the pulse rise time is:

tr = 0.34 / BW = 0.34 / 10 kHz = 34 μs

The LMC6081 has a 1 V/μs slew rate that poses no limitation at all for these tiddly signals.

That’s significantly better than the stacked Darlingtons and might be Good Enough for my simple needs.

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