inductance of a solenoid with ferrite rod core

The power inductor gap may be realized in one of two fashions, discrete or distributed. The physics of soft magnetic materials result in the case that commericially useful materials range from about 0.3T to 1.8T in Bsat. One envelope constraint is that Bsat is not widely variable. Since μ = B/H, the lower the value of μ, the greater the value of H (or current) that is supported at a level of B that is less than the maximum value of flux density (Bsat) inherent in the magnetic material. Another way to express the function of the air gap is to say that it reduces and controls the effective permeability of the magnetic structure. The purpose of the gap is to store the energy, and to prevent the core from saturating under load.

Power inductors require the presence of an air gap within the core structure. By resisting change in current, the filter inductor essentially accumulates stored energy as an AC current crests each cycle, and releases that energy as it minimizes. FWIW, single turns on powdered iron toroids (mu >= 50, say) are about the right ballpark there.An inductor is a current filtering device. Easy enough to wind a few and measure saturation, at least make a scale model if necessary. Still very rough, but perhaps good enough for use. I think I would estimate by starting with the mu_eff estimate, and assume that mu_eff gives the average reduction in magnetic path length, and so we can assume the same ratio gives the increase in flux density over the air-cored case. But this wouldn't be very useful as the inductance gain is also tiny (equivalent to the given, that the core length is a small fraction of the total magnetic path length ). In the extreme, for a rod much shorter than the winding, the reduction of magnetic path length is very small indeed, so we might use the solenoid formula to find the flux density inside the winding and assume it saturates when B_air = B_sat. It will have a higher saturation current, but it's hard to say by how much.

The field will still be strongest in the middle, but the number of turns acting exactly upon the middle is fewer, and the coupling from more distant turns is less than one. For a distributed winding, significant leakage will span the winding itself, and the effective area will be larger. The above will work if the winding is confined to a small fraction of the length, so that leakage around the rod largely avoids the winding itself.

Inductance of a solenoid with ferrite rod core