COMSOL 6.4 - Thin Conductive Layer Using the Transition Boundary Condition

Thin Conductive Layer Using the Transition Boundary Condition

In electromagnetic simulations, such as transformers and converter stations, one often needs to include geometrically thin conductive layers. Explicitly meshing these thin layers can be computationally expensive and numerically challenging, especially when the layer thickness is much smaller than the wavelength of electromagnetic waves or the characteristic mesh size.

In COMSOL Multiphysics, you can approximate conductive layers by using the Transition Boundary Condition (TBC) or the Impedance Boundary Condition (IBC) features. Both the TBC and IBC features support not only frequency domain analysis but also time domain analysis. This example focuses on the TBC and demonstrates that it can produce accurate results in both time- and frequency-domain simulations. The model geometry used in this example was first proposed in Ref. 1, where a frequency-domain analysis is performed.

Model Definition

The 2D model shown in Figure 1 is a conducting layer consisting of two apertures. The aperture on the left is a conductor with a conductivity σ, whereas the right one is a small open aperture. The boundaries in blue are magnetically insulated. The excitation of the magnetic field is an oscillating line current, pointing out of the plane.

Figure 1: The geometry of the full-fidelity 2D model.

A layer being geometrically thin does not necessarily mean it is electrically thin under the external magnetic field. One needs to compare the layer’s geometrical size with the characteristic skin depth to determine the layer’s electrical type. There are 3 types of layers in EM simulations, depending on the ratio of the skin depth

and the layer’s geometric thickness ds. One can classify the electrical thickness of the layer according to the following definition: electrically very thin layer (ds/δ < 0.5), electrically thin layer (0.5 ≤ ds/δ ≤ 20), and electrically thick layer (20 < ds/δ).

To benchmark the simulation results using the TBC, one can compare them with the results from a model in which the interior of the layer is explicitly meshed and the skin effect is fully resolved, which will be referred to as the full-fidelity model.

First, a frequency-domain simulation is performed to calculate the magnetic field H at a point slightly above the layer, as indicated by the red dot in Figure 1. Next, a corresponding time-domain simulation is carried out by exciting the model with a time-dependent oscillating current. As will be demonstrated by this example, in both scenarios, the TBC produces accurate and reliable results.

Results (Frequency Domain)

The electrical thickness of the layer can be tuned by varying its conductivity σ. A parameter sweep over σ at a fixed frequency fc=100 Hz is performed and for each σ, the norm of the magnetic field is calculated at a point located 2 mm above the layer.

As shown in Figure 2, for small values of ds/δ the results from both the electrically very thin and electrically thin layer models agree well with those from the full-fidelity model. As the conductivity increases, the layer transitions from electrically very thin regime to electrically thin regime. In this transition region, one can observe a drop in the magnitude of H.

When the conductivity is increased further, the layer effectively becomes electrically thick, thereby preventing the magnetic field from penetrating through it. This occurs because the skin depth becomes small for a highly conductive layer. In the thick-layer regime, the magnetic field in the shielding region is mainly due to leakage through the small open aperture. This is also illustrated in Figure 3: for small conductivities, the magnetic field in the region between the conducting layer and the line source has a nonzero component perpendicular to the layer. However, for large conductivities, the field lines near the conductor are almost entirely parallel to the layer, and the magnetic field in the shielding region originates solely from the open aperture, as expected.

Figure 2: Magnetic field norm measured above the conductive layer as a function of ds/δ.

Figure 3: Magnetic field distribution (contour plots) for the thin- and thick-layer cases obtained with the TBC model, shown alongside their full-fidelity counterparts.

Results (Time Domain)

In the time-domain simulation, the model is excited with a time-dependent current I(t) = 1000*sin(2*pi*fc*t)*exp(-t*fc), which is plotted in as a function of t in Figure 4.

Figure 4: Current as a function of time.

Note that, to use the electrically thick layer in the TBC feature in the time domain, the button must first be clicked before solving the model. This step computes a partial-fraction fit for the real and imaginary parts of the surface admittance Ys for a wide range of frequencies, which is required for time-domain analysis. The accuracy of the partial-fraction fit (High, Normal, or Low) can be adjusted according to the user’s needs. For electrically thin or very thin layers, the partial-fraction fit is unnecessary, as the approximation is precalculated automatically. For time-domain studies, a characteristic or center frequency (fc) must be specified when choosing the electrically thick layer.

In Figure 5, the magnetic field H(t) on the shielding side of the thick layer is probed at different points in time. The parameter used for Figure 5 is chosen such that ds/δ = 48 > 20. Good agreement between the TBC (thick layer) and the full-fidelity model is found, whereas the very thin layer (magenta) and the thin layer (black) options show clear deviations, as expected. This shows that choosing the correct type of layer is important.

Figure 5: The magnitude of H(t) probed above the electrically thick conductive layer.

Reference

1. G. Eriksson, “Efficient 3D simulation of thin conducting layers of arbitrary thickness,” in 2007 IEEE International Symposium on Electromagnetic Compatibility, pp. 1–6, IEEE, 2007.

ACDC_Module/Verifications/thin_conductive_layer

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > > .

Click .

Click

In the tree, select > .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
conductivity	6*10^16[S/m]	6E16 S/m
current_input	10000exp(2ifcpi)	10000
delta	sqrt(2/(murmu0_constconductivityfc2*pi))	2.0547E-7 m
fc	0.0000001[GHz]	100 Hz
height_conductor	5[mm]	0.005 m
height_upperplane	10[mm]	0.01 m
mur	1	1
period	1/fc	0.01 s
ratio	height_layer/delta	48.669
width_aperture	0.2[mm]	2E-4 m
width_aperture_big	2[mm]	0.002 m
width_shift	2[mm]	0.002 m
width_whole	8[mm]	0.008 m
height_layer	10[um]	1E-5 m