Calculators › Transmission Lines

Rectangular Waveguide Calculator

Cutoff frequency, cutoff wavelength, guide wavelength, wave impedance, phase velocity and group velocity for any TEmn or TMmn mode. Includes presets for standard WR sizes from WR-284 (S-band) to WR-10 (W-band).

Parameters
a (broad wall) b TE₁₀ dominant mode
Broad Wall a
mm
Narrow Wall b
mm
Fill εr
—
Frequency
Mode
m / n
/
⚠ TM: m≥1, n≥1. TE₀₀ invalid.
WR Presets
Cutoff & Mode
Mode—
Cutoff Freq (fc)—GHz
Cutoff λ (λc)—mm
fc / f ratio—
Wave Impedance Zw—Ω
Cutoff
fc=(c/2√εr)·√((m/a)²+(n/b)²)
λc=2/√((m/a)²+(n/b)²) (geometric)
Z_TE=η/√(1−(fc/f)²)
Z_TM=η·√(1−(fc/f)²), η=377/√εr
Propagation
Free-Space λ₀—mm
Guide Wavelength λg—mm
λg/4 (resonator)—mm
Phase Velocity vp—× c
Group Velocity vg—× c
vp × vg—=c²/εr ✓
Propagation
λg=λ₀/√(1−(fc/f)²)
vp=c/(√εr·√(1−(fc/f)²))
vg=c·√(1−(fc/f)²)/√εr
vp·vg=c²/εr (always)

About the Rectangular Waveguide Calculator

Rectangular waveguide is a hollow metal tube used to guide microwave energy at frequencies typically above 1 GHz. Unlike coaxial cable or microstrip, waveguide has no centre conductor — energy propagates as electromagnetic field modes inside the tube. Waveguide offers extremely low loss and very high power handling, making it the transmission line of choice for radar systems, satellite ground stations and high-power microwave applications.

Dominant Mode — TE₁₀

The dominant mode is TE₁₀, with the lowest cutoff frequency: fc = c/(2a√εr). The standard WR waveguide operating band runs from 1.25×fc to 1.9×fc to ensure single-mode operation above cutoff and below the next mode (TE₂₀ or TE₀₁).

Cutoff Wavelength

The cutoff wavelength λc = 2/√((m/a)²+(n/b)²) is a purely geometric quantity — it depends only on the waveguide dimensions and the mode indices, not on the fill dielectric. For TE₁₀ in an air-filled WR-90: λc = 2a = 45.72 mm.

Guide Wavelength and Wave Impedance

Inside a waveguide, signals travel at a phase velocity greater than the speed of light and a group velocity less than the speed of light. The product vp × vg = c²/εr always, regardless of frequency or mode. Wave impedance for TE modes is always greater than η = 377/√εr Ω.