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>I'm looking for a simple closed form (or approximation) for the
inductance of a wire, surrounded by a magnetic material.
Sort of like having a wire go through a toroid or long tube of material.
If you look into e.g. the wikipedia entries about inductance, you'll find, that wire inductances depend on some current return path, such as either a parallel wire or a coaxial cable tube.
A wire bent into a loop or a rectangle will provide its own return path.
A typical equation of wire inductance is that of a coaxial cable. It is
L = mu*l/(2*pi)*ln(rout/rin)
where l is the length of the cable and rin the radius of the inner conductor and rout the (inner) radius of the outer one and mu the permeability of the material between the conductors. It is derived here:
https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electro-Optics/Book%3A_Electromagnetics_I_(Ellingson)/07%3A_Magnetostatics/7.14%3A_Inductance_of_a_Coaxial_Structure
In the case of a coaxial cable with a concentric tubular magnetic medium
of inner radius r1 and outer radius r2, such that rin<r1<r2<rout, the
inductance becomes:
L = l/(2*pi)*(mu*ln(r1/rin)+mu2*ln(r2/r1)+mu*ln(rout/r2))
where mu2 is the permeability of the magnetic material. All of the materials between the conductors are assumed to be insulating.
The derivation is practically the same as in the link, only the region between inner and outer conductor has been split into 3 parts.
>In part, I assume that the magnetic flux would be contained mostly
within the material - but is there a thing comparable to skin effect?
How would one calculate the flux in the "core"?
Yes, the flux will be mostly in the material if mu2>>mu. The flux inside the core is multiplied by mur, the relative permeability.
There will be a skin effect in the inner conductor, but I don't believe it to be different form that without the magnetics. The inductance will rise with a magnetic surrounding, which implies,
that there is a stronger electric field induced, which opposes the field wich causes the current in the wire. This opposing field is a bit larger in the center of the wire than at its surface. That will
drive the current to the surface of the wire. But important for the skin effect is the difference between center and surface field, not its absolute magnitude. And the difference
is not affected by the magnetic material.
I'm a bit rusty in this, so please correct me if necessary.
Udo
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