The Simulation Corner |
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Modeling objects with arbitrary dimensions using MEFiSTo-2D
At first glance it might appear that MEFiSTo-2D allows you to enter only dimensions that are integer multiples of the mesh size. Nothing could be further from the truth. In fact there are two ways to finely adjust the position of boundaries and thus obtain highly accurate S-parameters and resonant frequencies. This note explains how it can be done.
The Variable Link Feature
This feature is described in detail in the MEFiSTo-2D operating manual. It allows you to finely adjust the position of an electric wall and even to make it move during the simulation. However, there are several restrictions:
- The wall must be a perfect electrical conductor
- The wall must be normal to the z-axis
- The wall must be situated on the right-hand side of the computational domain
- All Variable Links in a simulation must have the same specifications
Therefore, the Variable Link is mostly useful for specific demonstrations such as the Doppler effect in the time domain.
The Interface Wall Feature
This feature allows you to insert an ideal transformer into the link line between two neighboring cells or rows of cells. You only need to specify the square of the turns ratio. The interface wall will then act as an impedance transformer. You can use this property to finely tune the position of any boundary in the TLM mesh.
Consider the situation shown in Fig. 1. Two structures of width
are drawn into the TLM grid. While
structure 2 is exactly 8 long, structure 1 is
somewhat shorter. Without corrective measure, structure 1 could thus
not be modeled accurately by a 1 mm square mesh but would have to be
entered with the same dimensions as structure 2, and TLM results would
be identical for both.
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Two structures in a 2D-TLM mesh with square cells. |
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The corrective measure amounts to modifying structure 2 electrically
such that it behaves like structure 1. By inserting an ideal
transformer in the plane indicated by the dotted green line, we can
change the input impedance of the eighth cell of length
so that it equals the input impedance of
the shorter section of length 
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The input impedance of the short-circuited cell of length is
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(1) |
while the input impedance of the shorter cell of length
is
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(2) |
We can transform Zi into Z'i using a transformer with a turns ratio (n2n1)2 = Zi/Z'i.
For short stubs (
), a condition normally satisfied in
TLM simulations, the tangent is nearly equal to its argument and we can
write:
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(3) |
This relationship applies to the equivalent circuit shown in Fig. 2
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Ideal transformer used to adjust the electrical length of a structure. |
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Note that the turns ratio is independent of frequency. The transformer
is thus appropriate for time domain modeling. The correction can also
be used if is larger than , but should not exceed 1.5, to preserve accuracy.
Adjusting the position of a magnetic wall (ideal open circuit) calls
for a different expression. The input impedance of an open-circuited
stub of length is
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(4) |
and hence the turns ratio required for adjusting the position of a magnetic wall is:
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(5) |
Expressions (3) and (5) are valid even if both the walls and the TLM mesh are slightly lossy.
Example
Find the lowest resonant frequency of a parallel-plate waveguide resonator with the following dimensions:
| Width: | w = 7.0 mm |
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| Length: | l = 11.3 mm |
The resonator is short-circuited along one of its short sides and open at the three remaining sides.
If we discretize this structure into 1 mm square cells, only w can be accurately resolved, while l must be shortened to 11 mm or stretched to 12 mm. In both cases the resulting resonant frequency will be wrong.
The solution is to enter a resonator of 7 x 12 mm and electrically move either the electric or the magnetic wall by 0.7 mm. Fig. 3 shows the editor window of MEFiSTo-2D with three structures. The left resonator is either 11 mm or 12 mm long. The center resonator contains an interface wall (black line) in front of the electric boundary, and the right resonator contains an interface wall in front of the magnetic wall. The source and probe are located identically in all three structures.
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Three resonators used to verify the fine adjustment of boundaries. |
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When drawing the interface wall in front of the electric wall (center resonator) you must enter the following properties in the dialog box:
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(6) |
where n22 is the square of the number of turns on the side of the electric wall (above the interface).
The interface wall in front of the magnetic wall (right resonator) must have the following properties:
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(7) |
where n22 is the square of the number of turns on the side of the magnetic wall (below the interface).
The resonant frequencies of the three resonators are compared with each other in Fig. 4. We can clearly distinguish the resonant frequencies of the left resonator for both l = 12 mm and l = 11 mm. The responses of the center and right resonator perfectly overlap and represent the correct resonant frequency for l =11.3 mm.
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Resonant frequencies for different values of the resonator length.
The fine length adjustment using the Interface Wall feature indeed
permits accurate modeling of structures with arbitrary dimensions. This figure was produced by pasting screen images of MEFiSTo-2D into Microsoft® Paint (you must deselect Draw Opaque in the Image menu item of Paint). |
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Note that this boundary adjustment requires virtually no additional computer resources and is far superior to selecting a finer meshsize to accommodate all dimensions with a minimum of error.
Click here to download the zipped TLM file used in this example.
