Waveguides and Antennas
Rectangular waveguide TE and TM modes, cutoff frequency, dominant TE10 mode, guide wavelength and phase and group velocity, antenna parameters, dipoles, arrays and the Friis transmission equation.
At microwave frequencies, power travels in waveguides and is launched into space by antennas.
Rectangular waveguides
For inner dimensions a × b (a > b), a hollow guide supports TE_mn and TM_mn modes but not TEM.
- Cutoff frequency f_c = (c/2)√((m/a)² + (n/b)²).
- Dominant mode TE₁₀: f_c = c/(2a), λ_c = 2a. TM₁₁ is the lowest TM mode (TM₁₀ and TM₀₁ do not exist).
- Only frequencies above f_c propagate; below it, the mode decays (evanescent).
- Guide wavelength λ_g = λ/√(1 − (f_c/f)²) (longer than in free space).
- Phase velocity v_p = c/√(1 − (f_c/f)²) > c; group velocity v_g = c√(1 − (f_c/f)²) < c; v_p v_g = c².
- Wave impedance (TE): η/√(1 − (f_c/f)²).
Antenna parameters
- Radiation pattern, beamwidth (half-power), side lobes.
- Directivity D = maximum radiation intensity/average; gain G = η_rad D.
- Effective aperture A_e = Gλ²/(4π).
- Radiation resistance R_r: power radiated = ½ I² R_r.
- Polarisation, bandwidth, input impedance.
Common antennas
| Antenna | Key values |
|---|---|
| Hertzian (short) dipole, length dl | R_r = 80π²(dl/λ)²; D = 1.5 |
| Half-wave dipole | R_r ≈ 73 Ω; D = 1.64 (2.15 dBi) |
| Quarter-wave monopole on a ground plane | R_r ≈ 36.5 Ω |
| Small loop | Very low R_r |
| Horn, parabolic dish, patch, Yagi | High gain and directivity |
Arrays: the total pattern = element pattern × array factor. Broadside arrays radiate perpendicular to the line of elements (zero progressive phase); end-fire arrays along it.
Link budget
Friis transmission equation: P_r = P_t G_t G_r (λ/(4πd))²
Free-space path loss (dB) = 20 log(4πd/λ).
Worked example
A standard X-band guide has a = 2.286 cm. At 10 GHz:
- f_c = 3 × 10⁸/(2 × 0.02286) = 6.56 GHz.
- λ = 3 cm; λ_g = 3/√(1 − 0.656²) = 3.98 cm.
Practice
A rectangular guide has a = 3 cm. What is the TE₁₀ cutoff frequency in GHz?
Show answer and explanation
Answer: 5f_c = c/(2a) = 3 × 10⁸/0.06 = 5 GHz.
A waveguide carries a mode with phase velocity 5 × 10⁸ m/s. What is the group velocity in units of 10⁸ m/s?
Show answer and explanation
Answer: 1.8v_p v_g = c², so v_g = 9 × 10¹⁶/(5 × 10⁸) = 1.8 × 10⁸ m/s.
An antenna has directivity 10 and radiation efficiency 0.8. What is its gain in dB? (Give 2 decimal places.)
Show answer and explanation
Answer: 9.03G = 0.8 × 10 = 8, which is 10 log 8 = 9.03 dB.
Which mode cannot propagate in a hollow rectangular waveguide?
Show answer and explanation
Answer: AA TEM wave needs two separate conductors; a hollow guide has only one.