A microstrip is a popular transmission line typically used at RF frequencies that consists of a conducting metallic strip placed on top of a substrate which is in turn backed by a metallic ground plane. This example investigates the characteristic impedance (Z0) of the microstrip’s fundamental quasi-TEM mode using the FDE solver in MODE. We perform convergence testing of the mesh step size and extrapolate the data to get an accurate simulated value of the characteristic impedance which is compared to theory.

## Background

Using the FDE solver of MODE, we can find the eigenmodes supported by a microstrip transmission line. The characteristic impedance of a microstrip mode is calculated using the equation \(Z0=P/|I|^2\). P is the power carried by the mode which can be calculated by integrating the normal component of the Poynting vector over the area of the mode,

$$ P=\int_{S} P \cdot d S $$

and I is the current carried by the strip which can be calculated by integrating the H fields over a closed path around the conductor,

$$ I=\oint_{c} H \cdot d l$$

The characteristic impedance calculated using this method is returned as a result from the FDE solver. For comparison, an approximate characteristic impedance can be calculated for this structure using an equation from Pozar [1].

## Simulation setup

The microstrip transmission line in this example is composed of a copper strip with thickness of 34.1 um. Instead of modelling the volumetric strip’s deeply subwavelength thickness which can be computationally intensive, a 2D rectangle is used instead. The 2D rectangle is assigned a 2D conductive material model whose surface conductivity is calculated from the bulk conductivity of copper at RF frequencies (5.8e7 S/m) and the previously mentioned layer thickness. A lossless FR4 substrate is used with a relative permittivity 4.34 and thickness 0.5 mm that is backed by a volumetric copper ground plane of thickness 35um.

Since the fields of the quasi-TEM mode are highly confined to the copper strip, it is appropriate in this case to use a PEC boundary condition at the x max and y max boundaries provided they are placed a sufficiently distance away from the strip such that the fields have significantly decayed at the boundary. Even higher levels of accuracy can be achieved by placing the PEC boundaries farther away from the strip or replacing them with PML boundaries. A symmetry boundary condition is placed at x min to reduce the simulation domain.

To enable the calculation of the characteristic impedance using the FDE solver, the “calculate characteristic impedance” option under the “impedance” tab of the FDE solver object has been selected. The integration region x1, x2, y1 and y2 settings are defined to enclose the copper strip, whose values are shown in the figure below.

Since the fields are highly confined to the strip and vary sharply near the strip’s edges, a mesh override region is placed over the copper strip to resolve the width of the strip and the surrounding modal fields.

To perform convergence testing of the mesh, a parameter sweep task in the Optimizations and Sweeps window has been set up to collect the Z0 result from the mode list’s first mode (quasi-TEM mode) as the dx and dy mesh step size is reduced.

## Running Simulation and Analysis

Open the *microstrip.lms* simulation file. Run the simulation and calculate the modes to view the field profile of the microstrip’s quasi-TEM mode (mode #1 in Mode list). The mode profile is plotted below in linear and log scale.

Next, run the *microstrip_convergence.lsf* script file with the *microstrip.lms* simulation file. The script file will run the parameter sweep which records the characteristic impedance of the microstrip mode as the mesh is made finer. The ratio of the mesh step size in the x and y directions are constant (i.e., dx/dy=constant) in the sweep. The script produces a plot of impedance versus mesh size, which appears to linearly converge.

By fitting a line to the results using the “polyfit” script function, we can get the intercept position for a mesh step size of 0 in order to find an accurate final result for the characteristic impedance and loss. This technique of extrapolating the results is called Richardson extrapolation and gives a higher order accuracy.

Note that when there are singularities in the mode profile, the results may not always converge linearly.

Using this approach, the final characteristic impedance is found to be approximately 76.9 Ohms. By running the *microstrip_Z0.lsf* script file, a theoretical approximation of the characteristic impedance for the microstrip transmission line is calculated using analytically expressions in Pozar [1] which gives 77.58 Ohms. This approximate characteristic impedance is within 1% of the fitted simulated value.

### Related references

[1] D.M. Pozar, Microwave Engineering, Fourth Edition. John Wiley & Sons (2012).