In order to test the contact resistance at nominal current, which is 10mA here, the modulation of test current was chosen to be about 10mA +-2.5mA. This modulation is done by the help of BC337. Concretely spoken it makes the potential seen at the upper pin of 68R resistor (Ux) to toggle between about 0.51V and 0.85V. The actual signal at this point also depends on Rcont, of course, but is corrected with the methode shown here.

Rcont is determined as follows:

Rcont = Ureed / Ireed = Ureed / ((Ux - Ureed)/68 Ohm) =>

Rcont = 68 Ohm x Ureed / (Ux - Ureed)

Where Ureed is the voltage drop across reed contact and Ireed is the current flowing through it.

Because we have two states of Ureed and Ux, a higher and a lower one due to the modulation, we get:

Rcont = 68 Ohm x Ureed,h / (Ux,h - Ureed,h)

Rcont = 68 Ohm x Ureed,l / (Ux,l - Ureed,l)

Linear algebra teaches us, that this relation is linear and that it is also valid, when we take Ureed,ss = (Ureed,h - Ureed,l) and Ux,ss = (Ux,h - Ux,l):

Rcont = 68 Ohm x Ureed,ss / (Ux,ss - Ureed,ss)

Ureed,ss and Ux,ss is just the AC components of this modulation.

Because signals are very small, an amplification is useful. So, Ureed,ss is amplified by a factor of 256 and becomes the "Ucont" signal shown in the schematic, while Ux,ss is amplified by a factor of 8 and becomes the "Usig" signal.

So, we just have to give Ucont and Usig to the input of an ADC, measure the upper and lower values of both, calculate the difference Ucont,ss and Usig,ss and get Rcont after running the following formula:

Rcont = 68R x (Ucont,ss/256) / (Usig,ss/8 - Ucont,ss/256)

Why changing Ux every 900µsec?
There are a lot of high pass and low pass filters in the signal path. The involved time constants are matched for a change of Ux every 900µsec. The result is a suare wave with rounded knees, just without any relevant differentiations of high pass filters.

Change the state of BC337, wait for a suited amount of time, read Usig and Ucont by the ADC and immediately after that change the state of BC337 again, in such a way, that Ux is changed exactly every 900µsec.

900µsec was chosen to avoid interference with the harmonics of mains.

Take the average of several succeeding readings to cancel noise and hum interference.

"10xUcont" and "100xUcont" signals allow you to work with a simple 8bit ADC, even when Rcont becomes smaller and smaller. Otherwise you would need a 14bit ADC.

Connect a zero resistance at input and adjust the 10k trimmer, so that the gain of this stage is exactly "8". Then, connect a typically to be expected and well known resistance to the input and adjust the 1k trimmer, so that the reading of your LCD equals this resistance.

A lower modulation rate is not useful, because then much bigger electrolytics would be needed. If your ADC is fast enough, then you should even use a higher modulation rate.

Use this circuit here as a starting point for own experiments and developments. When using a bipolar supply voltage and precision OPamps, for instance, this circuit might look totally different. Or, you can also add analog precision rectifiers to fabricate Ucont,ss and Usig,ss. Etc. etc.

Kai