# How do precision differential levels achieve such high accuracy

As you do, I was watching a video about levelling granite surface plates used in precision machine shops and labs. They're ground and lapped to a high level of flatness (I think AA plates are flat to within around 0.005mm) and a key tool for this process is a set of electronic differential levels like these.

These devices are used to measure the relative inclination of different parts of the plate and the paths are integrated to give a map of global roughness across the plate. The manufacturers of these levels claim that they can measure .1 arcseconds repeatably. If I did the maths right, across a 1m surface plate that corresponds to vertical displacement of 4.8e-7m.

My question is, how do these things work? I'm not electronic engineer but I assume each unit contains a (hinged or flexible) pendulum ~0.1m in length with capacitive plates on both sides charged from an AC source. As the pendulum flexes or rotates around the hinge, the relative change in capacitance between the plates on either side would result in a voltage difference between the two capacitors that you could then feed into an amplifier to measure the change in pendulum displacement and thus the change in the orientation of the level. Combine this with another level as shown in the video and you can measure the relative change in orientation between the units.

I've left some of the maths in here for you lot to enjoy working out for yourselves but if anyone has an idea of what the circuitry looks like inside these things, I'd like to know whether my assumption was even close. I've been playing around with a circuit sim to try and build something that can measure very small changes in relative capacitance but alas, I've got nothing that really works.