Calculating Noise Free Resolution of ADC

Other Hardware Sensors
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Hello,

I am attempting to figure out how to calculate the noise free resolution of my load cell and HX711 ADC system.

In this post , a user made some experimental estimates of the HX711 noise free bit resolution. However I've had a hard time understanding how he calculates the noise free resolution in grams (see comment #5), which is further complicated by the fact that other sources calculate it in different ways. His equation is as follows:

NFR (g) = (load cell full scale load (g))/(2^(noise free bits) * gain * (load cell sensitivity (mV/V)))

(Shouldnt the load cell sensitivity be multiplied by the load cell voltage output at full scale?)

Whereas this source calculates it as follows:

NFR (g) = (load cell full scale load (g)) * (ADC full-scale voltage range (V))/((load cell full scale output (V)) * 2^(noise free bits) )

I am working on a university project involving measuring rocket thrust from a load cell, and it would be very helpful to be able to form an estimate on what my measurement resolution is. Can anyone offer insight into how NFR in units of your measurement mass should be calculated (or whether I am completely misunderstanding the calculations)? Or any useful resources to read into?

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Furthermore, this source (page 15) doesn't list how they calculated NFR, but I do not get the same final values when using the equations in my original post.

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Good discussion here: ADC Input Noise: The Good, The Bad, and The Ugly. Is No Noise Good Noise? | Analog Devices

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Thanks for the link. Unfortunately they dont go further into calculating the final noise free resolution. Useful resource though!

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Another good discussion here: A Reference Design for High-Performance, Low-Cost Weigh Scales | Analog Devices

Excerpt from the above shows how you can calculate noise-free resolution in bits, for your particular application:
image

Regarding the math:

If your setup uses the full "bandwidth" of a 24-bit ADC, then in a perfect, noiseless world, the best possible resolution you can get is...

full scale load / 2^24

But if, because of the load cell sensitivity or a one-sided application, you can only use half of the "bandwidth," then the best possible resolution is...

full scale load / (0.5 * 2^24)

For the HX711 in a single-sided application, with 128 gain and sensitivity of "S" mV/V (less than or equal to approx 3.9), the best possible resolution is...

full scale load / (0.128 * S * 2^24)

And note that if S=3.9 mV/V, you will get, at best...

full scale load / (0.5 * 2^24)

(If S > 3.9, then the ADC will clip.)

Noise will increase those numbers, as will a load cell with less than optimal S.

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Thank you Dave for the excellent response and resource!

Would the magnitude of the excitation voltage of the load cell not affect this calculation, since the output signal of the load cell is determined by the sensitivity and the excitation voltage (as well as the applied load)?

(Referring not to the equation in the picture but the equation you wrote out).

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With a HX711 you haven't got a choice regarding excitation voltage.
It must be the same as the HX711 board provides, to keep the ratiometric A/D happy,
otherwise you're trading noise for (much worse) instability.
The only choice is shielding (metal enclosure), a board with split analogue/digital supply (so you can decouple the analogue part more), and a load cell that closely matches the load.
Leo..

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