Figure1: E-field phase animation inside a rectangular waveguide at 10 GHz.
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 TEmn and TMmn 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 TE10 and its E, H and J fields are shown in Fig. 2.
Figure 2: Fields pattern of the fundamental mode, TE10. The green lines represent the E-field, the purple lines the H-field and orange lines the J-field.
The electromagnetic analysis of a rectangular waveguide is well known, and can be easily found in the literature, like . Here we list only final results that can be used to verify the simulation results.
kcmn (Eq. 1)
)2 (Eq. 2)
βmn=√k2−kcmn2 (Eq. 3)
fcmn : cutoff frequency of the mode mn
βmn : propagation constant corresponding to the mode mn
αmn : attenuation constant corresponding to the mode mn
k : free-space wavenumber
kcmn : wavenumber corresponding to the mode mn
a and b : width and height of the waveguide respectively (see Fig. 3)
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.
Big edge dimension
Small edge dimension
Length of the waveguide
Figure 3: Section of a rectangular waveguide (WR-90) modeled in CST STUDIO SUITE, where α=22.86 mm, b =10.16 mm and l=40 mm.
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”.
Figure 4: Waveguide port mode information summarizing results for the first 3 modes of a WR90 at 10 GHz
In Table 2 we compare the results shown in Fig. 4 and compare with the analytical results given by the Eq. 1 – 3.
Table 2: Comparison between simulation and analytical results at 10 GHz
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 kcmn>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 TE01 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.
Figure 5: E-, H- and J-field patterns for the first 3 modes at 15 GHz.
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?
Figure 6: E-field pattern for the TE01 mode below cutoff, plotted with a dB scale and a lower limit of -40 dB.
 D.M. Pozar, Microwave Engineering, 4th Edition, John Wiley & Sons, pp. 110-119