# Hollow Rectangular Waveguide Simulation

## Self-guided learning

This example of simulating a hollow rectangular waveguide contains a short description of the theory, detailed information on how to construct the model, a video showing how to construct the model, and the fully constructed model ready for you to download.

When working through the example, you may notice some small differences in your model compared to ours – this is usually simply due to the fact that you are using a different software version.

### The Physics

A hollow waveguide is a transmission line that looks like an empty metallic pipe. It supports the propagation of transverse electric (TE) and transverse magnetic (TM) modes, but not transverse electromagnetic (TEM) modes. There is an infinite number of modes that can propagate as long as the operating frequency is above the cutoff frequency of the mode. The notation TE_{mn} and TM_{mn} are commonly used to denote the type of wave and its mode, where m and n are the mode number in the horizontal and vertical directions respectively. The mode with the lowest cutoff frequency is called the fundamental mode or dominant mode. For a hollow rectangular waveguide the dominant mode is TE_{10} and its E, H and J fields are shown in Fig. 2.

The electromagnetic analysis of a rectangular waveguide is well known, and can be easily found in the literature, like [1]. Here we list only final results that can be used to verify the simulation results.

### The Model

A section of a rectangular waveguide is modeled in CST Studio Suite® and the first 3 modes are calculated and their field distributions analyzed. The dimensions used are the standard for WR-90 waveguide. Because the background is set to perfectly electrical conductor (PEC) material, we only need to model the vacuum inside the waveguide, with a waveguide port at each end. The boundary conditions are “electric” in all directions, and the model is simulated using the time domain solver. In this model the first 3 modes are calculated, and E- and H-field monitors are set-up at 10, 13.5 and 15 GHz

Parameter | Value | Description |
---|---|---|

a | 22.86 mm | Big edge dimension |

b | 10.16 mm | Small edge dimension |

l | 40 mm | Length of the waveguide |

### Discussion of Results

In Fig. 4 we show the dialog window of the Port Mode Information at 10 GHz. This can be found by right-clicking on the port in the dialog tree, and selecting “Object Information”.

In Table 2 we compare the results shown in Fig. 4 and compare with the analytical results given by the Eq. 1 – 3.

Note that for the higher modes, there is no real beta at 10 GHz. That is because those modes are below the cutoff frequency, and the propagation constant becomes entirely imaginary (alpha). As Eq. 2 demonstrates, this indicates that k_{cmn}>k . As these modes do not propagate, CST Studio Suite also calculates the distance of -40 dB of attenuation.

In the 2D/3D Results, we can see the field patterns for each mode. Fig. 5 shows the E-, H- and J-field patterns for each mode at 15 GHz. Fig 6 shows the field pattern for the TE_{01} mode at 10 GHz, below its cutoff frequency. As expected from the port mode calculation (Fig. 4), the mode does not propagate, and is attenuated below the -40 dB level around 20 mm into the waveguide.

## Additional Tasks

Model the WR-90 with a metallic wall made of copper (assume that the wall is infinitely thin, and use the “Copper (annealed)” from the CST material library) and estimate the waveguide attenuation at 10 GHz for the fundamental mode. Compare your results with the analytical results. You can also try comparing different mesh settings (for example, 10, 30 and 50 lines per wavelength) and different solvers. Which would you expect to give the closest fit to the analytical results?

Waveguides are sometimes “loaded” with dielectric in order to reduce their cutoff frequency, effectively reducing the size of waveguide required for a given mode to propagate. Replace the vacuum material with a dielectric, and perform a parameter sweep for the permittivity of this material. What permittivity would be required to allow the TE01 mode to propagate at 10 GHz?

## References

- [1] D.M. Pozar,
*Microwave Engineering, 4th Edition*, John Wiley & Sons, pp. 110-119