Modeling magnetic field shielding

Question

I am wondering what is the best way to model magnetic and electric field shielding from a magnetic dipole in the near field?

For example, let's say you have a coil of current carrying wire in the x-y plane which creates an AC magnetic field along the z-axis. I am interested in a model in which I could put in material parameters (such as $\sigma$, $\epsilon$, $\mu$) for a shielding plate located distance $r$ from the source.

I tried using section 5.4 of an online electromagnetic book which talks about reflection, and transmission of of EM waves. I would estimate wave impedance as $Z=\omega\mu_0r$ which is the equation for near field magnetic source. I determine the intrinsic impedance of the shielding material as $\eta=\sqrt {\frac {j\omega\mu} {\sigma + j\omega\epsilon}}$. I want to try good conductors as well as ferrite so I'm not making any assumptions such as good conductor or loss-less medium.

I implemented this but the result was not what I expected. I would expect highest energy reflection with the ferrite because it has low reluctance to magnetic fields and would guide the flux through it and return to source. However, a good conductor (e.g aluminum) had almost 100% energy reflection, but I was expecting the shielding mechanism of conductors would be all energy lost as absorption from eddy currents.

Let me know your thoughts of whether this is a legit model; I can give more info about parameters/equations used if that's helpful.

Answer 1

In general, the eddy equation for "low" frequency is voltage based (dB/dt). I've recently been working on this issue from this side of the spectrum, where "shielding" is assumed to be negligible. In extending this assumption to higher orders, I found this assumption to break down. The shielding equation in my case is what I'm calling "shield factor" $ S=\frac{B_{cur}}{B_{applied}}=k_{geom}\frac{\tau^2\mu f}{\rho}$ where $B_{cur}$ is flux density from eddy currents, $B_{applied}$ is flux density from external source, and $\tau$ is material thickness. The assumption that $\tau$ is small compared to other dimensions is also made. I'm not sure I kept all of the assumptions to extend this to $B_{cur} ~=B_{applied}$ where shielding is very effective, but for my case it seems to still be valid.

Selecting materials: One would want to select the highest conductivity first, then add some cheap permeability second. You'll see a lot of copper shielding for this reason, but also a lot of nickel shielding. Pure nickel is only 6x lower conductivity than copper, but you might get 10-100 for relative permeability in the MHz range. There is an added benefit of nickel that it can shield low frequency fields as well, so that can offset the cost. Copper will tend to function relatively better as the frequency increases, and nickel will tend to lose advantage at higher frequencies due to reduced permeability. This should give the first intuition into material selection. One might want to increase resistivity to reduce power loss in the shielding material, but $E^2/\sigma$ assumes a poor shield, or another way of saying $\int I\mu dA<<B$ at the frequency in question. For a shield, this is wrong, so you need to apply the field from the eddy currents back into the system, resolving the mesh for the given geometry (assuming this is FEA). http://www.public.iastate.edu/~nbowler/pdf%20final%20versions/conferences/QNDE2005Bowler.pdf

To get to the top equation, I used the assumptions and equations from NPTEL for eddy loss , which happened to be independent of permeability: $\frac{E^2}{\sigma} ->P_{eddy}=\frac{\pi^2f^2B^2_{max}\tau^2}{(6\rho)}(hL\tau)$ and $I=k_{geom}\frac{fB_{max}}{\rho}$

http://nptel.ac.in/courses/108105053/pdf/L-22(TB)(ET)%20((EE)NPTEL).pdf