Waveguides

Waveguides are contrasted with solid conductors that only support TEMs and have an upper cap on the frequency due to skin effect and dielectric losses.

Waveguides support TEM/non-TEMs (transverse electromagnetic waves) above a certain very high cutoff frequency (in the GHz).

But Waveguides are hollow, inner-reflective things. Difficult to install.

Waveguide insides coated with Gold or Silver for max reflectance.

Types of waveguides

1. Rectangular Waveguide

Rectangular waveguides support TE and TM. They have a higher dominant mode bandwidth compared to rectangular waveguide.

If we assume a and b to be inner dimesions, the lowest cutoff will happen at TE₁₀ mode.

2. Circular Waveguide

Tubular, can be worked on basically like regular plumbing but limited dominant mode bandwidth. Easier to manufacture.

Only support TE and TM modes.

Modes

Four Modes: TE, TM, TEM, and HE.

1. TEM mode: Both $H_z$ and $E_z$ are 0. If the wave is moving in the $z$ direction, both the E field and the H field are transverse to $z$ direction. Nonexistent in waveguides.

2. TE mode: The Electric field is transverse to the propagation of the wave. $H_z$ is not 0, but $E_z$ is (for wave in z direction)

3. TM mode: contrast with TE mode. $H_z$ is zero.

4. Hybrid HE mode: both $E$ field and $H$ field not transverse to wave motion.

Dominant mode

Mode with the lowest cutoff frequency is dominant mode. What mode supports the lowest possible frequency in the waveguide?

Cutoff frequency

info

The frequency at which attenuation occur, and over which propagation takes place.

For rectangular waveguide ${TE}_{m,n}$ or ${TM}_{m,n}$

$$f_c = \frac{v}{2} \sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2}$$

Of course, the $v$ is the speed of em wave in a medium, and can be replaced with $\frac{c}{\sqrt{\varepsilon_r}}$.

$$f_c = \frac{c}{2\sqrt{\varepsilon_r}} \sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2}$$