Quantised Conductance Resistance Standard

Putting together the tiny about of hardware for this project I've been thinking about what I really want to do with it. Really, measuring resistors using a micro isn't that exciting as a project. Measuring physical constants using common components is a lot more interesting. Actually, the conductance quantum is related to one of the most accurately measured constants of physics, the fine structure constant (https://en.wikipedia.org/wiki/Fine-structure_constant). The relation is:

The magnetic permeability is a defined constant (so, you know, pretty high accuracy) so I suppose you could even try to get a good measure for the speed of light.

All in all, I think it'd be a lot more fun to make a circuit I can sit downstairs that doesn't do anything but try to determine the conductance quantum.

I want to reiterate before going over the parts I want to use that part of the point of this project is to use cheap and cheerful components. At least partially because I enjoy the idea of seeing obviously quantum phenomena without spending a lot of money.

The first important thing to consider is the choice of contact mechanism. When you're only doing a few trials a couple of wires or a switch is fine, but I want to get reliability from a huge number of trials so a relay is the only choice. The choice of contact is a little bit more involving. The small amount of literature on measuring contact breaking QC shows that copper and silver work but aren't ideal, there's nothing I could find on silver alloys, and from my small experience and the literature gold works the best. Gold plated relays aren't the most common but cheap on the few websites I checked.

Measuring the current is a little interesting. Normally you might think to hook up a transimpedance amplifier into a proper ADC however the capacitance of the entire system should be tiny. I'm not sure if the trace capacitance, some coupling with the relay coil, or the ADC sample capacitance are the largest but they're all tiny. With such a small time constant I'll just be measuring voltage across a resistor.

I should clarify that there's no requirement for any sort of high accuracy resistor. In fact, because I'm lazy and can't think of any other easy way to rescale for testing a nice wide range of resistors I'm going to use a digital pot to vary range. It might also help during contact breaking events to reduce the effect of constant error in the ADC on ever smaller measurements.

Seeing as this is going to be a prototype, and a simple one at that, and I might want to iterate lots I'll just be doing the entire thing on veroboard with a Teensy and its delicious 13 ENOB ADC.