Electronic – How much is the noise of a multiplexer for analog inputs

The question which noise sources need to be taken into consideration depends on how severe they are. Your question indicates that you are interested in noise generated at the op amp and not noise generated by interference from neighboring circuits (internal/external noise).

In order to make things comparable, all noise is referred to the op amp's input (RTI). In theory, I guess any point in your circuit might work as long as you refer all noise sources to that point, but it is common practice to act as if all noise sources were directly at the input pins. Sources include noise in the resistors, noise generated by current flowing into the op amp's input pins and noise that may be considered as a voltage between the inputs pins.

There is a very good discussion at this Q&A-style source and also in this nice article from 1969 (!), both authored by Analog Devices' staff.

Without re-typing everything in these sources, here are some rules of thumb:

Noise in the resistors becomes bad when the resistor values are high (some 100k or some 1M) and when the circuits are designed for high bandwidth since the noise is proportional to \$ \sqrt{4k \cdot T \cdot B \cdot R}.\$

You can try to minimize R, you can try to limit the bandwidth B if possible, you can put the circuit in liquid nitrogen (low temperature T), but you can't go for a low Boltzmann constant, because Boltzmann is dead (quote stolen at Analog Devices).

Current noise, i.e. noise generated by current flowing into the op amp inputs, will be converted to a noise voltage by the resistors around the input (\$R_f\$, \$R_g\$) and amplified by the circuit's gain. This is one of the reasons why one prefers op amps with very low input currents especially for high-ohmic circuits.

Voltage noise results from a real op amp's inability to completely null the voltage between the input pins.

All noise sources can be combined as the square root of the sum of their squares since they are independent of each other, which will work only if all sources are RTI.

ΔX assumes that as the signal gets bigger or smaller there will be a non-linearity. Let's look at that first: -

Let's say you are powering the 4052 from 5 volts (left graph). The change in resistance between -1 volt and +1 volt on your input is about 470 ohms - 350 ohms = 120 ohms. If you have a signal this large AND you are feeding the output of the multiplexer into a low impedance input (say 1 kohm) then yes, you will have a substantial ΔX misshaping of the output.

But, in all likelihood you are using an instrumentation amplifier than has input impedances approaching 1 giga ohm (or at least several Mohms). Let's say worst case 1 Mohm and let's say the nominal resistance of the analogue switch is 400 ohms.

What will the basic potential divider effect be: -

\$\dfrac{V_{OUT}}{V_{IN}}=\dfrac{1,000,000}{1,000,400}\$ = 0.9996 or an error of 0.04%

On top of this is the ΔX error of maybe +/-60 ohms - this will be a worst case error of about: -

\$\dfrac{V_{OUT}}{V_{IN}}=+/-\dfrac{1,000,000}{1,000,060}\$ = +/- 0.006%

However, your input aint +/-1V it's much, much smaller so I'd say forget it.

The error you DO need to be concerned about is leakage currents in and out of the input pins to the load cell - it has a 1 kohm resistance and the leakage quoted on page 5 is +/- 100nA (worst case).

This gives rise to an error voltage across the load-cell of 1 kohm * 100nA = 100 \$\mu\$V.

Can you live with this? It doesn't sound like you can.

FOOTNOTE

I use DG309 multiplexers for miniscule signals from thermocouples and have no problem with them - however, a thermocouple has a very low source impedance of maybe 1 ohm (or maybe 100 ohms if you include the cable). At ambient temperatures the DG309 exhibit a leakage current of +/-5nA max and this gives rise to a dc error of about 0.5 \$\mu\$V. These will work better than the CD4052 I reckon.