Rectangular Waveguides

Waveguides are just rectangular tubes full of air for transmission of power waves at high frequencies. Waveguides with a single conductor support either TE or TM waves, but not TEM waves.

• Modes define the properties of how a wave propagates through a guide
• Modes are defined by $m$ and $n$
  • Obtained through solving the wave equations for different boundary conditions
• Mode with lowest cutoff frequency is the dominant mode
  • Dominant TM mode is $T{M}_{11}$
  • Dominant TE mode is $T{E}_{10}$

TM Modes

Phase Constant

A wave is travelling inside the guide along the z-direction. It's phase factor is $e^{-j\beta z}$ with:

$$\beta =\sqrt{k^2-k_c^2}=\sqrt{{\omega }^2\mu \epsilon -{\left(\frac{m\pi }{a}\right)}^2-{\left(\frac{n\pi }{a}\right)}^2}$$

Cutoff Frequency

Corresponding to each mode there is a cutoff frequency $f_{cmn}$ at which $\beta =0$. A mode can only propagate if $f>f_{cmn}$, as only then is $\beta$ real.

$$f_{cmn}=\frac{u_{p0}}{2}\sqrt{{\left(\frac{m}{a}\right)}^2+{\left(\frac{n}{b}\right)}^2}$$

$u_{p0}=1/\sqrt{\mu \epsilon }$ is the phase velocity of a TEM wave in an unbounded medium with parameters $\mu$ and $\epsilon$.

Phase Velocity

$$u_p=\frac{\omega }{\beta }=\frac{u_{p0}}{\sqrt{1-(f_{cmn}/f{)}^2}}$$

Wave Impedance

$$Z_{TM}=\frac{\beta \eta }{k}=\eta \sqrt{1-{\left(\frac{f_{cmn}}{f}\right)}^2}$$

$\eta =\sqrt{\mu /\epsilon }$ is the intrinsic impedance of the lossless medium.

TE Mode

All the parameters are the same as for TM mode, except for wave impedance

$$Z_{TE}=\frac{\eta }{\sqrt{1-{\left(\frac{f_{cmn}}{f}\right)}^2}}$$

The TE dominant mode, assuming $a>b$ where $a$ and $b$ are the width and height of the waveguide ,is $T{E}_{10}$ with

$$f_{c10}=\frac{1}{2a\sqrt{\mu \epsilon }}$$

Zigzag Reflections

For the $T{E}_{10}$ mode, the field component can be expressed as the sum of two TEM plane waves, both travelling in the $+z$ direction, but zigzagging between opposite walls of the waveguide. The phase velocity of these waves is $u_{p0}$ and their direction is at angles. The phase velocity of their combination $u_p$ is that of the $T{E}_{10}$ mode.

$${\theta }^{\prime }=\arctan \frac{\pi }{\beta a}{\theta }^{\prime \prime }=-\arctan \frac{\pi }{\beta a}$$

$${\theta }^{\prime }=\arctan \frac{1}{\sqrt{(f/f_{c10}{)}^2-1}}$$

Table