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Fig. 1. General view of the structure
Parallel-plate transmission line
In many practical cases, however, we may prefer to observe not only the spatial patterns, but also integral quantities, such as a line integral of the electric field along a pre-defined integration path.  If the field is potential, then this integral has the physical meaning of voltage. Knowing the voltage between two metal objects allows predicting the hazards of  arcing and electric breakdown.
The path of integration may be defined by the user with UDO objects from the contours library. Three of the available UDOs allow setting the path composed of 1, 3 or 7 segments (spanned between 2, 4 or 8 points, respectively). Another UDO facilitates reading the number of points and their coordinates from a text file. The points may be arbitrarily located in space but they will be snapped to the nearest mesh point (cell vertex), similarly as in the case of point sources. Each segment will be decomposed into a Manhattan-style route along cell edges. During the simulation, the electric field is integrated along the path, and this integral will be Fourier-transformed by FD-Probing postprocessing, analogously as voltages at lumped sources and probes.
Fig. 2. Reflection, transmission and characteristic impedance in extracted by S-parameter postprocessing
in extended mode
The parallel-plate transmission line is located along the x-axis and supports a TEM wave with Ez and Hy field components. The path of integration is marked in magenta, between two terminal points requested at (50,0,0) and (50,0,10) and effectively snapped to (50,0.5,0) and (50,0.5,10). Hence the integrated field will have the significance of voltage between the two metal plates.
Fig. 3. Voltage amplitude
Contours E, H and combination of them
Here we will present two more complicated and more practical examples. Namely, we will use integration of both electric and magnetic fields over specific contours and present the results of their ratio that can be interpreted as the characteristic impedance of a TEM transmission line.
Let us consider the example with five defined contours. Three of them CE, CE2 and CE3 are the E-field integration contours, and two of them: CH1 and CH2 are the H-field integration contours. They are all presented in Fig. 5. We can see that the problem concerns a TEM line (a shielded stripline) with the strip marked in yellow. Lower part of the line is filled with a substrate (light green). However, in this example we assume that the substrate has the same properties as air. This is in contrast to the next example where the substrate is made of FR4 of relative permittivity 3.8. The substrate has thickness of 20 mils.
The definition of the contours is specific for presentation of TEM and quasi-TEM line properties. CE1 and CE2 are different integration lines between the strip and ground. CE3 is a closed contour of E-field. One of H-field contours (CH1) makes a loop around the strip while the other provides the results of integration outside the strip.
Fig. 4. General view of the structure
Fig. 5. Contours considered in the examples
Fig. 6. Postprocessing list and results of integration over contours CE1 and CH1
Now please run the second example concerning a shielded microstrip on a dielectric substrate. Thus we consider a case of a quasi-TEM transmission line. The obtained results of simulation make it clear that this line can be considered as a TEM line only in a relatively narrow frequency band. In particular, the impedance calculated from contours CE1 and CH1 changes considerably as presented in Fig. 7.
Fig. 7. Results of integration over contours CE1 and CH1
discover accurate EM modelling