Waveguides and modes
A waveguide is a pipe that does not let low frequencies through: a WR-90 blocks everything below 6.56 GHz. The holes in the door of a microwave oven work the same way.
01A pipe that carries waves
Above a few gigahertz, a coaxial cable starts to lose a lot: the center conductor is thin, the skin effect concentrates the current on its surface and the dielectric absorbs. To carry microwave power with low losses, a hollow metal tube is used, the waveguide. It has no center conductor: the wave travels through the air inside, bouncing between the walls.
Unlike a cable, a waveguide does not let every frequency through. It only propagates the waves whose field configuration satisfies the boundary conditions —zero tangential electric field on the metal— and that is only possible above a minimum frequency.
02The modes of a rectangular waveguide
Each field configuration that propagates is called a mode. In a rectangular waveguide of sides \( a \times b \) there are transverse electric (TE) modes, with no electric field in the direction of propagation, and transverse magnetic (TM) modes, identified by two indices that count how many half waves fit along each side. Each mode has its cutoff frequency:
The waveguide is used in the band where only the TE₁₀ propagates: above its cutoff and below the next mode. That way the energy travels in a single known configuration, without being split among modes that travel at different velocities. Standard waveguides have \( a \approx 2b \) so that this band is as wide as possible:
| Waveguide | Inside dimensions (mm) | TE₁₀ cutoff | Operating band |
|---|---|---|---|
| WR-284 | 72.14 × 34.04 | 2.08 GHz | 2.6 to 3.95 GHz (S band) |
| WR-90 | 22.86 × 10.16 | 6.56 GHz | 8.2 to 12.4 GHz (X band) |
| WR-28 | 7.11 × 3.56 | 21.1 GHz | 26.5 to 40 GHz (Ka band) |
03How the wave travels in the waveguide
Above cutoff, the mode advances with a wavelength longer than that of free space and with two different velocities:
Since the group velocity depends on frequency, the waveguide is dispersive: a pulse is distorted as it travels through it, the more so the closer to cutoff you work. That is why the operating band starts well above cutoff. The wave impedance of the TE₁₀ is \( \eta_0 / \sqrt{1 - (f_c/f)^2} \), greater than that of vacuum.
A waveguide below cutoff also has uses: as an attenuator, or like the holes in the door of a microwave oven, which are waveguides too small to let 2.45 GHz out.
The cutoff frequencies of the first modes of three standard waveguides, on a frequency axis. You choose the waveguide and the operating frequency: the modes whose cutoff lies to the left propagate.
04Other guided structures
| Structure | Dominant mode | Limit |
|---|---|---|
| Circular waveguide | TE₁₁, with \( f_c = 1.841\, c / 2\pi a \) | The polarization can rotate; it is used in rotary joints |
| Coaxial cable | TEM, with no cutoff frequency | Above a certain frequency the first TE mode appears and the cable stops being single-mode |
| Microstrip on a circuit board | Quasi-TEM | Radiation and surface modes at high frequencies |
| Optical fiber | Modes guided by total internal reflection | Covered in optical fibers |
05Where it appears in electronics
| Case | Why a waveguide |
|---|---|
| Airport or ship radar | Peak powers of kilowatts in S or X band with low losses |
| Satellite dish feed | The feed horn is the end of a waveguide that flares open |
| Microwave oven | A short waveguide carries the power from the magnetron to the cavity |
| Point-to-point cellular links | Waveguide transitions in the radio units at tens of GHz |
06In the lab
With an X-band waveguide bench, a slotted line and a detector, locate two consecutive minima of the standing wave with the waveguide short-circuited and calculate \( \lambda_g \). Compare it with the formula.
Sweep the generator frequency from 6 to 9 GHz with a section of WR-90 and measure the transmitted power. Locate the cutoff of the TE₁₀.
Measure the holes in the mesh of a microwave oven door and calculate the cutoff frequency of each hole as a circular waveguide.
07Common mistakes
- Working too close to cutoff. Dispersion and losses grow a lot.
- Using the waveguide above the second mode. The energy is split among modes and the signal degrades.
- Believing that a phase velocity greater than c violates relativity. Information travels at the group velocity.
- Confusing \( \lambda_g \) with \( \lambda_0 \). Quarter-wave sections in the waveguide are measured with \( \lambda_g \).
08Self-assessment
What is the TE₁₀ cutoff of a waveguide 30 mm wide?
\( c / 2a = 3 \times 10^8 / 0.06 = 5 \) GHz.
What is \( \lambda_g \) in a WR-90 at 10 GHz?
\( 30 / \sqrt{1 - (6.557/10)^2} \approx 39.7 \) mm.
Why are standard waveguides twice as wide as they are tall?
So that the band where only the TE₁₀ propagates is as wide as possible: with \( a = 2b \), the TE₂₀ and the TE₀₁ cut off at the same frequency, twice that of the TE₁₀.
What is the relationship between phase velocity and group velocity?
Their product is \( c^2 \).
09Further reading
- David M. Pozar. Microwave Engineering. 4th ed., Wiley, 2011. The reference text on microwaves: rectangular and circular waveguides with all their modes.
- Robert E. Collin. Foundations for Microwave Engineering. 2nd ed., Wiley-IEEE Press, 2001. The classic, rigorous treatment of guided structures.
- David K. Cheng. Field and Wave Electromagnetics. 2nd ed., Addison-Wesley, 1989. Waveguides derived from Maxwell's equations, with the continuity of this course.