Waveguides

Hollow metal pipes guide microwaves with very low loss and high power handling. Learn TE and TM modes, cutoff frequency, the dominant TE₁₀ mode in rectangular guides, standard WR sizes, guide wavelength, phase and group velocity, circular guides, and components (bends, couplers, cavities).

At microwave frequencies coax becomes lossy and power-limited. Radar transmitters, satellite ground stations and particle accelerators use waveguides to carry kilowatts or megawatts efficiently.

An analogy

A waveguide is like a hallway for sound: only wavelengths small enough fit the hallway's width — longer ones can't propagate (cutoff).

Modes

Waveguides carry field patterns called modes. TE (transverse electric) and TM (transverse magnetic); no TEM mode exists in a hollow guide. Below cutoff, a mode is evanescent and decays rapidly.

Operating band

Rectangular guides are used between the TE₁₀ cutoff and the next mode (TE₂₀ at 2f_c) — typically 1.25 f_c to 1.9 f_c. Standard sizes: WR-90 (X band, 8.2–12.4 GHz, a = 22.86 mm), WR-62, WR-28, etc.

Dispersion

Phase velocity exceeds c while group velocity (energy speed) is less than c; both depend on frequency, causing dispersion of wideband pulses.

Components

E- and H-plane bends, tees, magic tees, directional couplers, circulators, cavity resonators and filters, waveguide-to-coax transitions, and slotted-waveguide antennas. Substrate-integrated waveguides bring the concept to PCBs.

Formulas

• Cutoff (TE_mn): f_c = (c/2)√((m/a)² + (n/b)²)
• Dominant mode TE₁₀: f_c = c / (2a)
• Guide wavelength: λ_g = λ₀ / √(1 − (f_c/f)²)
• Velocities: v_p = c/√(1 − (f_c/f)²), v_g = c√(1 − (f_c/f)²), v_p v_g = c²

Worked example: WR-90 cutoff

Given: a = 22.86 mm

Find: TE₁₀ cutoff.

• f_c = 3 × 10⁸/(2 × 0.02286) = 6.56 GHz

Answer: 6.56 GHz (use 8.2–12.4 GHz)

Worked example: Guide wavelength at 10 GHz

Given: WR-90, f = 10 GHz

Find: λ_g.

• λ₀ = 30 mm
• λ_g = 30/√(1 − 0.656²) = 39.7 mm

Answer: 39.7 mm

Common mistakes

• Wrong: Operating below cutoff. — Right: Stay above ≈ 1.25 f_c.
• Wrong: Operating where higher modes propagate. — Right: Stay below the second mode cutoff.

Applications

• Radar: High-power feeds and rotary joints.
• Satellite ground stations: Low-loss feed networks.
• Accelerators & heating: Megawatt klystron outputs, industrial microwaves.

Key takeaways

• TE₁₀ f_c = c/2a.
• Use between 1.25 f_c and 1.9 f_c.
• λ_g = λ₀/√(1 − (f_c/f)²).
• Low loss, high power.

Going further

• Cavity resonators: High-Q microwave filters.
• Substrate-integrated waveguide: Waveguides on PCBs.
• Mode converters: TE₁₀ to circular modes.