By coincidence, I had two inquiries this week on eddy-current simulations. The first concerned a pulsed valve using a solenoid and iron holding plate. How quickly could the magnetic force be extinguished? The second inquiry was whether Aether could be used to determine the inductance of a 3D magnetic sensor.
I’ll concentrate on the second issue because it is an important consideration for Aether development. The code solves the full Maxwell equations. Ordinarily, we would consider such a code to be an inefficient tool for eddy-current simulations because the propagation time for light across the simulation volume is much less than the time-scale for changes of current and magnetic field. (If L is the system size and c is the speed of light, the the electromagnetic time scale is L/c.) In other words, the time step to satisfy the Courant stability condition would be extremely small compared to the time scale for system changes.
As I have shown throughout these articles, scaling laws can come to the rescue. In the eddy-current limit, displacement currents in the system are negligible compared to real currents. The thing to recognize is that 1/1000 is almost as negligible as 1/100000000. Suppose we multiply all values of relative dielectric constant in the system by a factor ?^2. The speed of light would changed according to c’ = c/? and the Courant time step would increase by a factor of ?. As long as L/c‘ is short compared to the time scale for pulsed currents or the period of harmonic currents, the solution will be accurate.
I prepared a 2D benchmark example to verify the method and to make sure there were no lurking physical problems. I performed the same calculation with Pulse (designed specifically for magnetic diffusion calculations) and EMP (an electromagnetic code) with adjusted dielectric constants. The z-r plot in the first figure below shows a detail of the geometry. I used a relatively large solution volume (-7.5 cm ? z ? 7.5 cm, 0.0 cm ? r ? 10.0 cm) to reduce the effects of the boundaries. The pulsed coil produced a field that diffused into the metal slug. The coil current increased smoothly to 1.0 A with a risetime tr = 2.5 ?s. The drive current waveform was:
I(t) = 0.5*[1 - cos(?*t/2.5E-6)] A, t < 2.5 ?s (1)
I(t) = 1.0, t > 2.5 ?s
The coil had length Lc = 4.0 cm and radial thickness ?Rc = 0.5 cm. The slug had length Ls = 4.0 cm, radius Rs = 2.0 cm, relative magnetic permeability ?r = 50.0 and conductivity ? = 1500.0 S/m. The air and coil regions had ?r = 1.0 and a small conductivity 0.001 S/m required for the diffusion calculation. The approximate time for field penetration was roughly
?t ? ?r*?0*?*(Rs/2)^2/2 ? 5 ?s. (2)
The calculation extended to 11 ?s. The figure below show lines of magnetic flux density B at t = 2.5 us. The top of the second figure shows the magnetic flux density at a point on the midplane between the coil and slug. The initial high value drops as flux penetrates into the slug. A probe on axis inside the slug shows B slowly approaching equilibrium at 11 us.
The boundary in the EMP calculation was an ideal absorber rather than a flux excluder. I set the value of ?r in air regions to 90,000 (? = 300) and in the slug to 90000/50 to preserve the speed of light. With c‘ = 1.0E6 m/s, the electromagnetic transit time across the system was about 0.25 ?s. Because of the peculiarities of 2D electromagnetic solutions, it was necessary to supply the value of dj?/dt in the coil region. For the coil area and the drive waveform in Eq. (1), the value was
dj?/dt = (?/2*tr) sin(?*t/tr), t ? tr (3)
dj?/dt = 0.0. t > tr
I used the linear interpolation option in EMP because the drive function had discontinuous derivatives. The bottom graph in the second figure below shows the variation of Hz determined by EMP. The relative signals from the two codes were almost identical at all positions inside and near the slug. The absolute agreement was good to about 10%, an acceptable result considering the wide differences in the computational approaches and boundary conditions. There was no detectable difference in the EMP signals when I lowered ? to 150. The runtime for the EMP calculation was about 50% longer than that for the Pulse solution.
The implication is that Aether can be used effectively for eddy current calculations in both time and frequency domain solutions. The 3D code will combine the function of four of our 2D codes (Pulse, EMP, Nelson and WaveSim). To ensure that the user doesn’t have address all the details I worried about in the benchmark test, we will add the following entry to the Aether command set:
EDDYCURRENT [Xi]
In response, the code adjusts ?r for all material regions by a factor ? before computing conductivities for absorbing layers. If the user doesn’t supply a value for ?, the code will make an estimate based on the system dimensions and the material properties. The Aerial post-processor will break out values of magnetic, electrical and electromagnetic energy for inductance calculations. There will also be an option to plot B instead of H.
Here are links to input files if you want to try the Pulse and EMP calculations:
http://www.fieldp.com/myblog/examples/bdiffusion_pulse.min
http://www.fieldp.com/myblog/examples/bdiffusion_pulse.pin
http://www.fieldp.com/myblog/examples/smooth.cur
http://www.fieldp.com/myblog/examples/bdiffusion_emp.min
http://www.fieldp.com/myblog/examples/bdiffusion_emp.ein
http://www.fieldp.com/myblog/examples/dsmooth.cur
