How build coil for respiratory magnetometer or find a better option (hall effect)?
The general principles are made clear enough in the patent. See "Summary of the invention".

This is an "easy enough" task to do by a number of means.
How well the method per se works is unknowable, but what the equipment needs to do is track the motion of 2 points on the rib cage relative to points on the patients back. Larger coils are liable to be no great problem. 

Linear hall sensors are usable - field strengths gets small with increasing distance but some extremely capable active magnetic detectors can be obtained. 

A near field inductive power transfer system would work and the ranges are within what is available commercially. Resonant transmit and receive coils and suitably high Q will give adequate range.

If this is of interest then more specific questions are welcome.



I've been looking at the specs of various linear high sensitivity hall sensors - and there are integrated modules that do vastly better at not vastly more. You can buy linear hall sensors with a 6 mG full scale range so lets say you can deal with fields down to 1 mG to start.

A modern rare earth magnet gives ABOUT 1/2 to 1 G at 1/2 the magnet length from the pole face. So if you had a say 10mm thick magnet you'd get 500 mG at 5mm (very very roughly). 1 mG is 1/500th of that. If field falls as d^2 you get to about 110mm and if field falls as d^3 you get only 40mm. At large distances field fall off is cube law but nearer its square law. (Ignore why for now). Realistically this means that you are near the edge of what is easily done off the shelf.

I measure ABOUT 250mm from back to chest front so I'd expect somewhere under 1 mG at my back with the above magnet. Larger magnets are one answer BUT a pulsed coild may do better AND you can use GMR effect sensors (don't even ask) that measure magnetic angle and these may work here and you could probably make a simple front reference harness that would greatly reduce distance

There are MANY possible solutions and I have little doubt that a few days of playing my a competent engineer would produce something usable.

HOWEVER THE big problem here is that we don't know enough and there is a great danger of dying the death of 1000 cuts / answerlets / questionlets here as we slowly edge towards understanding what you really want. If you are willing and able to provide a decent overview of what you are trying to do and why it would help heaps.

If you are trying to produce a commercial solution and do not want to say too much then you could look for competent people who seem liable to be able to help and take this offline. If you just want an answer and do not care who hears the details then this is an excellent place to ask.

There are various other methods that would allow you to track a say 3mm dia sphere in a space several meters on a side with mm's of resolution - but you don't need that. Lower res versions of that would do. Cost effort budget aim market ... need to be known. [Longish ago researchers managed to not only track a flying blow fly BUT were able to see which way it turned its head.! What you want is trivial compared.].