- 2069 Bhadra (old course) · 8 marks
Rectangular waveguide modes and critical parameters
Short answer
Rectangular waveguide is a hollow metal pipe supporting TE and TM modes above cut-off, acting as a high-pass filter. The dominant TE10 mode has cut-off frequency fc=c/(2a), guide wavelength λg=λ0/√(1−(fc/f)²), phase velocity vp > c, and group velocity vg < c. Parameters depend on mode indices m and n and waveguide dimensions a (broad) and b (narrow).
Describe rectangular waveguide based on modes of propagation and other critical parameters.
A rectangular waveguide is a hollow metal pipe of inner width a and height b (a > b, often a ≈ 2b). Being a single conductor, it cannot carry TEM waves; it carries TE and TM modes above their cut-off frequencies, acting as a high-pass filter.
y
b +-----------------+
| E (TE10) ↑ | propagation along z
+-----------------+
0 a -> x
Modes of propagation
- TEmn (H modes): Ez = 0, Hz = H0 cos(mπx/a) cos(nπy/b). m, n = 0, 1, 2 … but not both zero. Lowest: TE10.
- TMmn (E modes): Hz = 0, Ez = E0 sin(mπx/a) sin(nπy/b). m, n ≥ 1. Lowest: TM11.
- m and n are the numbers of half-wave field variations along the broad (a) and narrow (b) walls.
- Dominant mode: TE10 (lowest fc). Degenerate modes: TEmn and TMmn with the same m, n share the same fc.
Critical parameters (air-filled)
| Parameter | Expression |
|---|---|
| Cut-off frequency | fc = (c/2)√((m/a)² + (n/b)²) |
| Cut-off wavelength | λc = 2/√((m/a)² + (n/b)²); TE10: 2a |
| Phase constant | β = (2π/λ0)√(1 − (fc/f)²) |
| Guide wavelength | λg = λ0/√(1 − (fc/f)²) |
| Phase velocity | vp = c/√(1 − (fc/f)²) > c |
| Group velocity | vg = c√(1 − (fc/f)²) < c |
| TE wave impedance | Z_TE = η/√(1 − (fc/f)²) |
| TM wave impedance | Z_TM = η√(1 − (fc/f)²) |
Also 1/λ0² = 1/λg² + 1/λc² and vp·vg = c².
Example (WR-90, a = 2.286 cm, b = 1.016 cm, f = 10 GHz, TE10): fc = 6.562 GHz, λg = 3.976 cm, vp = 3.976 × 10⁸ m/s, vg = 2.264 × 10⁸ m/s, Z_TE ≈ 500 Ω. TE20 cuts off at 13.12 GHz, so only TE10 propagates.
✓ Answer checked by Claude Opus 5.5 (high effort)
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