How to increase the "dynamic range" of shielding problems in Feko.
Introduction
For shielding applications using the default solver in Feko (MoM) a general rule of thumb is that shielding levels of around 40 dB are achievable. This is due to the formulation of the MoM and of course the fact that the model is discretised into meshed triangles. This how-to will show an easy way to achieve much higher shielding levels with essentially the same mesh.
Background
The total electric field is given by the sum of the incident electric field and the scattered electric field:
E_total = E_incident + E_scattered
For example, in a typical shielding scenario the source is an incident plane wave. The amplitude is typically set to E_incident = 1 V/m (plane wave). If the geometry provides, for example 40 dB of shielding, then inside the shielded area of the model, we must have E_total = 0.01 V/m. This means that E_scattered = -0.99 V/m.
Therefore minor errors in E_scattered (even as small as.1%) could result in huge errors in the shielding factor. This effect is denoted error amplification. E_scattered depends directly on the integration of the surface currents. Since a mesh of the geometry is being solved, and not the geometry itself, the finite sized meshed triangles containing linear basis functions introduces errors in E_scattered. Consequently, the error can be reduced by meshing finer, but meshing too fine introduces other non-idealities into the solution, such as low-frequency breakdown. In addition, meshing finer increases the computational resources.
Remedy: Employing the SEP
The Surface Equivalence Principle (SEP) for modelling dielectrics is exploited by creating two separate regions, the inside region where the shielding is to be computed and the outside or free space region (from where the incident field originates). The SEP decomposes the model into the two regions. More specifically, in the inside region, where the sensitive shielding calculation is required, E_incident is zero, and thus E_total = E_scattered. In this way the error amplification due to cancellation is removed. The following example demonstrates how these principles are applied.
Example 1: Shielding of a PEC cuboid with a small aperture
Consider a cuboid of equal side dimensions (volume 1.5m^3) and with a circular aperture of diameter 0.07 m on the top face. The shielding inside the cuboid is computed by calculating the near field inside the cuboid caused by an incident plane wave (at theta=0). The frequency range is 120 to 180 MHz.
Two versions of the model are created.
In the first version, a PEC cuboid is created and a small circle (ellipse) is subtracted from the top face to create the hole.
In the second version the small circle is retained - looking at the mesh, there is no hole in the cuboid. In addition, a dielectric is created with permittivity of 1 and label “air”. The inside region of the cuboid is changed from the default Free Space to “air”. All the faces of the cuboid, except that of the small ellipse, is set to PEC. The face of the ellipse is set to a dielectric face.
For both models, local meshing is applied on/around the hole. A single near field request in the middle of the cuboid is requested.
Results
The shielding is given by the ratio between the incident field (in this case 1 V/m) and the field inside the box. Conveniently then we only need to plot the E-field in dB to directly obtain the shielding. The graph shows that a much lower shielding factor is obtained for the free space case (incorrect) compared to the SEP case (correct)
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Example 2: Finite conductivity spherical shell with high permeability
Consider a fully closed sphere with conductivity 1e7 and relative permeability 112. The inside region of the sphere is free space. Shielding is calculated from 60 Hz to 100 kHz.
Reference: R. Jobava et.al. “Interaction of Low Frequency Magnetic Fields with Thin 3D Sheets of Combined Resistive and Magnetic Properties”, Proceedings of the 40th European Microwave Conference
Similar to example 1, the inside of the sphere is set to “air” dielectric. The surface (Face) of the sphere is set to a metallic medium with conductivity and permeability properties as stated.
Feko compares very well with the analytical result obtained by the reference.
Conclusions
The SEP is clearly the method of choice to model shielding problems where high levels of shielding is expected. Note that memory consumption will be approximately four times higher compared to the free space only method. The reason is the addition of magnetic basis functions on the surface of the air dielectric.
But note that for electrically very small problems, the SEP suffers from low frequency breakdown. A general rule of thumb for being cautious of low frequency breakdown is where the model size is of the order of 1/1000 of a wavelength.
In general, switching to double precision could help improve the shielding dynamic range and it can lower the minimum frequency where reliable results can be obtained.