Measuring nanoamps and below like a Ninja

[Kleinstein]

The resistor noise from the input AC coupling mainly effects the frequency range at lower end and below the transition. With higher frequencies the capacitor and source impedance reduce the noise. Similar the 1 GOhms resistor in the capacitive feedback version does contribute in the same low frequency range. The 1 Gohms resistor noise is relative to the output and thus less relevant.

One can get around the resistor noise with an even lower cross over frequency for the input and thus larger resistor and than setting the lower frequency limit in a later stage (could be software).
An alternative viewpoint is having the resistor as a source of current noise that than sees the input capacitor as "load" impedance. A higher resistor gives less noise.
Trying to go with a relative low resistance and large capacitor is more causing problems when connecting to the DUT and the lower impedance loads down the source.
The inverting configuration also has more loading of the DUT. The extra series resistor to set the upper frequency limit also contributes to the noise - this time for the full band and not just at the lower transition range.


It absolutely makes sense to have battery power for a low noise amplifier, but for some uses mains powered operation also helps.

[Kleinstein]

The somewhat jagged pattern of the noise in the time domain could be due to DNL errors of the DAC. It looks like there is repetitive part with some 14 seconds period, that can be seen as some 16 steps. For comparison it may help to know at which time the DAC values are changed. 

I don't see much effect of interference or an indication of a firmware problem. It is more a lack if low pass filtering and a limitation of the DAC.

[Kleinstein]

It is well possible that this are the DNL limitations:
The DAC11001 data-sheet gives a typical (with respect to the codes used) DNL error of some 0.1 LSB or 0.1 ppm (with 20 bit resolution). With a 10 V full scale range this would be around 1 µV.
So the observed steps are about what is to be expected.

The interface of the high resolution DACs is seriell and the digital interference would thus be faster. In addition the main part with digital interference would be capacitive coupling and thus more very fast spikes well beyound the BW of the test. If not carefulll with the software, the INL correction could result in some steps getting twice as large, using 2 LSB jumps. The current observed steps however are smaller, more like 0.1 or 0.15 LSB.

Some lag / rounding is expected from some analog low pass filtering after the DAC. Not sure how much filtering is present in the test.

A ramp of 1 mV/s is relatively slow. With a 10 V full scale range this would allow a 3 hours ramp and this a quite long averaging time. If one does not need the full 3 hours time window, one could divide down the voltage signal and run the DAC with a faster ramp. For testing the slow ramp is of cause a good idea.

The instruments to measure pA range currents are usually not very fast and would average over much of the modulation on top. So the jagged curve is not a real problem, more a sign that the amplifier to test it is good.

[Kleinstein]

Doing the correction via the DDS part is a good idea, as it should give plenty of resolution. The DAC seems to be quite good and quantization there already an issue.

It looks like one can measure the DNL steps. So in principle one could use a correction also for the DNL steps. So a correction value for the 20 bits, or at least most of them - the higher bits are the more critical ones not so much the lowest ones, though they are more visible. There are still limits to this method as the bit values may not be all that stable. Especially the MSB can be drift limited.

[3roomlab]

The somewhat jagged pattern of the noise in the time domain could be due to DNL errors of the DAC. It looks like there is repetitive part with some 14 seconds period, that can be seen as some 16 steps. For comparison it may help to know at which time the DAC values are changed. 

I don't see much effect of interference or an indication of a firmware problem. It is more a lack if low pass filtering and a limitation of the DAC.

is that the similar error as fig2 in this link?
https://www.analog.com/en/technical-articles/16bit-parallel-dac-has-1lsb-linearity-ultralow-glitch-and-accurate-4quadrant-resistors.html

[Kleinstein]

The curve measured over time corresponds to the DNL over code from the link.  Fron the curve it looks like 16 (maybe 32 with a very good LSB) code steps every 14 seconds. The part in between is just noise and there is also some extra noise from the amplifier.

For the correction one could use the same measurement as before: just correlate the steps with the bits changing. So it is only some 20 constants for the correction, not some 15000 points for the center part.
It still makes sense to measure over a certain range to get some averaging also for the higher bits.  For the start one could likely use only the lowest 2 to 4 bits and see how much the "noise" improves.

[Kleinstein]

The popcorn noise of the LM399 often mainly effects really low frequencies and may not show up very much in the frequency range of the preamplifier. The curve beyound some 20 seconds is to a large part effected by the frequency response of the amplifier. So the curves going down beyond some 10 s is no from the references.

For the ramp, there is a part from the DAC and amplifiers and a part from the reference. The ref. part depends on the voltage and not just the ramp speed.

[Kleinstein]

I would somewhat doubt, that the BW really goes down to 0.02 Hz. At least the curve for just the amplifier starts to go down from 20 s on. With a normal circuit the expected shape ot the Allan deviation curve is to finally go up on the long run. There is essentially no reasonable way (other than the frequency response) to let the curve go down to very long times (corresponds to low frequencies). White noise would give a horizontal curve and usually the noise is more going up to low frequencies, but not down. Getting the lower frequency limit from a resistor in the 10s of Gohm range is somewhat prone to errors (e.g. from leakage).