[femm] Re: Problems with axisymmetric problems

[Dcm3c@xxxxxxx]

> At the Vienna University of Technology we are
> making experiments with magnetic pulse
> fields. Essentially, the pulse has the shape
> of a damped cosine and we wanted to compare
> the experimental results with time harmonic
> solutions of Maxwell's equations as a first
> approximation. I have used FEMM to calculate
> the eddy currents in copper and iron samples.
> For cylindrical samples the results are
> comparable, but for spherical samples we have
> found rather large differences.
> 
> After some searching in the library I found
> an analytical solution for the eddy current
> density in a sphere in a homogeneous
> sinusoidally varying magnetic field. The
> results for "large" spheres (around 5 cm) are
> in excellent agreement. However, small
> spheres of a few millimeters in diameter,
> which have been used in our experiments, give
> unreliable results.

[...]

> I welcome any comments!

Thanks for the note--your writeup is pretty interesting. However, I think 
that femm is actually doing the correct thing in the case of the small 
spheres; something seems to be wrong with the analytical solution presented 
for this case.

To see the problem, the easiest case to consider is the one corresponding to 
Figure 31, located at

http://atp6000.tuwien.ac.at/MAGNET/WS/projects/proj3/node10.html

In this case, a copper sphere with a radius of 3.65 mm and a conductivity of 
sigma=56.82 MS/m is exposed to a source field of a Bsrc=1 Tesla amplitude 
varying at 109.89 Hz (omega=690.46 rad/sec). This apparently corresponds to 
the miniturnsphere.fem example problem.

This is a good case to consider because the radius and frequency are small 
enough that the reaction field from the eddy currents can be neglected (The 
skin depth at this frequency and conductivity is 6.4 mm. Since the skin 
depth is substantially greater than the radius of the sphere, neglecting the 
reaction currents for the purpose of estimating the induced current density 
is reasonable). When you can ignore the reaction currents, you can 
substitute directly into Faraday's law to get an expression for the eddy 
current density:

J=-j*omega*sigma*r*Bsrc/2

(see http://members.aol.com/gmagnetics/scholz.pdf for details of how I got 
this...)

This formula predicts an eddy current density of 71.6 MA/m^2 at the farthest 
radius of the sphere. Looking at the finite element solution, the amplitude 
of the induced current density at the point (r=3.649,z=0) is 71.63 MA/m^2, 
showing a good agreement.

Now, evaluating the sphere.nb Mathematica notebook under the above conditions 
yields a current 106.8 MA/m^2, which is substantially larger than one might 
expect.

So, what is the difference? I loaded sphere.nb into Mathematica and took the 
power series about omega=0 using the Series[] function and subsitituted in 
mu0 for mu, since we are considering the copper sphere. The result is:

J=-j*(3/4)*omega*sigma*r*Brc

Now, this doesn't match the low-frequency limiting case that that one can 
obtain from Faraday's law. There is an extra factor of 3/2 in there for this 
limiting case.

As a sanity check, I also ran this problem on the student version 3.4a of 
Quickfield. The resulting amplitude of the flux density for the same point 
was 71.5 MA/m^2 from this solution, which closely matches both the femm and 
simple Faraday solutions. (Check out 
http://members.aol.com/gmagnetics/msphere.zip for the qfield version of this 
problem. If you check out this model, make sure that you set the results to 
complex or peak to get the `right' values for comparison to femm--Quickfield 
defaults to RMS values, whereas femm only displays the currents as complex 
amplitudes)

> In addition, the new release 2.1a of femmview
> seems to display wrong eddy current densities.
> For example, the eddy current density does not
> vanish at r=0. However, if the results of fkern
> 2.1a are displayed with femmview 2.1,
> everything seems to be alright. But I have not
> tested the new version more thoroughly, yet.

Well, it looks like I really "fixed" this one good.... Anyhow, you are 
correct--I did screw up the eddy current densities in release 2.1a. Thanks 
for pointing this out to me. I have updated the version of the program 
available on the femm homepage. (http://members.aol.com/gmagnetics/setup.exe) 
Even though it still says 2.1a, it now has the fixed version of femmview in 
it. Rather than download the entire distribution again, you can download the 
file:

http://members.aol.com/gmagnetics/femmview.zip

This has the fixed version of femmview in it. Just replace the femmview.exe 
in you \program files\femm\bin directory with the new version in this zip 
file.

Dave.
--
http://members.aol.com/dcm3c