In this example, the impedance of both odd and even mode of a coupled microstrip is calculated and compared with published results from Collin [1].

## Simulation Setup

The cross section of the coupled microstrip waveguide is illustrated in the image above. This structure supports both an even and an odd mode.

The structure is composed of 2D PEC microstrips on a 1 mm thick substrate. The substrate has a relative permittivity of 9.7, and a metal boundary condition is used to simulate the ground plane below the substrate. The microstrips have a width of 2 mm and a gap of 0.25 mm between them.

A mesh override region is placed over the microstrip structures to ensure that the width of microstrips and the width of the gap are accurately resolved.

No symmetry is enforced across the x=0 plane, however the even or odd modes of the coupled waveguide could be selectively found by the solver if symmetric or anti-symmetric boundaries are used for the x min boundary condition.

## Running Simulation and Result Analysis

Run the coupled_microstrip.lms simulation file and solve for the modes. Modes 1 and 2 in the mode list are the even and odd modes respectively. A simple way to determine whether the mode is even or odd is to visualize the E dataset from the mode as a vector plot.

### Even mode

### Odd mode

The following image shows how to generate a vector plot of the E fields from the visualizer.

Select the even or odd mode in the mode list and calculate the characteristic impedance of the mode by choosing the “Power and Impedance Integration” option and integrating the “current” term. For the even mode, the common mode impedance is obtained by setting the integration region to enclose both microstrips as shown below [2].

The even mode impedance is twice the common mode impedance, giving a value of approximately 40 ohms.

For the odd mode, the differential impedance can be obtained by setting the integration region to enclose only one of the microstrips, and the characteristic impedance of the odd mode is half the differential impedance, approximately 22 ohms.

The calculated characteristic impedance agrees with the reported characteristic impedance for the even mode and odd modes from figure 3.31 of the text by Collin [1].

### Related references

[1] Robert E. Collin, Foundations for Microwave Engineering, Second Edition. Wiley-Interscience (2001).

[2] Thierauf, Stephen C. High-speed circuit board signal integrity. Chapter 11, pp. 498-505 Artech House (2004).