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I have two solenoid coils, one is wounded on ferrite rod and other is wounded in larger diameter in proxmity of ferrite rod, see picture. It is used as wireless comm (simplex) between RX and TX where I use resonance on TX to maximise signal gains

After bit of reading and now familar with mutual inductance between two air solenoid, impact of cross section area and so on.

(1) How to calculate or estimate mutual inductance as well as coupling coeff under this diagram. (2) If possible tip how to calculate/estimate the reluctance for this circuit.

My gut feeling that not all flux path will goes to ferrites as flux concreator and need to estimate out how much fraction of flux goes to ferrite rod.

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The main complexity is the fact that the magnetic medium is mixed, and without obvious symmetry (apart from cylindrical).

If the coils were entirely in air, then you could do it exactly with Biot Savart, and there are two-coil approximations that are quite easy to find via your favourite search engine.

If the ferrite core was closed, then as its permeability is so much higher than 1, it could be treated as a simple inductor with magnetic length, area, and ignore the air.

As the ferrite rod is open, as there is a large air path to be negotiated by lines of flux that return round the RX coil, its influence on the inductance of that coil will be small, a factor of just a few, notwithstanding that its ur is 1000. Its effect on the TX coil will be even smaller.

There are no simple approximations. Your options are
a) a full mixed media finite element simulation
b) empirical measurement
c) neglect the ferrite, use air as a lower bound, and say the inductances will be 'a bit more'

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