How do you calculate the TE10 cutoff frequency of a rectangular waveguide?

For the dominant TE10 mode the cutoff frequency is fc = c / (2a), where c is the speed of light in the dielectric filling the guide and a is the broad-wall (longer) interior dimension. For air-filled WR-90 with a = 22.86 mm, fc = 6.557 GHz, which is why WR-90 is rated for the 8.2 to 12.4 GHz X-band, comfortably above cutoff. If the guide is filled with a dielectric of relative permittivity er, the cutoff drops by a factor of the square root of er, since c becomes c0 divided by the square root of er. The general TEmn or TMmn cutoff is fc = (c/2) times the square root of (m/a) squared plus (n/b) squared, so every higher-order mode has its own, higher cutoff.

Why is a waveguide operated between 1.25 and 1.9 times its cutoff frequency?

The recommended band keeps operation in single-mode TE10 propagation while avoiding both edges. The lower edge, about 1.25 fc, stays clear of the high dispersion and high attenuation that occur near cutoff where the guide wavelength stretches toward infinity. The upper edge, about 1.9 fc, stays below the cutoff of the next mode (TE20, whose cutoff is 2 fc for the standard 2-to-1 aspect ratio), preventing multimode propagation and mode conversion. This roughly 1.5-to-1 usable bandwidth is why standard rectangular waveguide bands such as WR-90 X-band span only about an octave-and-a-half fraction of frequency.

Transmission Lines

Cutoff Frequency (TL)

Q: What happens to a signal below the waveguide cutoff frequency?

Below cutoff the propagation constant becomes purely imaginary, so the mode does not propagate; it is evanescent and its amplitude decays exponentially along the guide as exp(-alpha z). The attenuation is set by alpha = (2 pi / lambda_c) times the square root of (1 minus (f/fc) squared). A below-cutoff section therefore behaves like a reactive, near-lossless high-pass attenuator, which is exactly how piston (waveguide-below-cutoff) attenuators and cutoff-waveguide calibration standards work. At a frequency well below fc the attenuation can exceed 30 dB per centimeter, providing excellent isolation.

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Below a sharply defined threshold, a guided electromagnetic mode simply cannot carry power along a hollow-pipe transmission line: that threshold is the cutoff frequency, f_c. For each mode the propagation constant turns real only above f_c; below it the field is evanescent and decays exponentially with distance. In a rectangular waveguide the lowest cutoff belongs to the dominant mode TE₁₀, given by f_c = c / (2a) where a is the broad-wall dimension. Air-filled WR-90 has a = 22.86 mm, so its TE₁₀ cutoff is 6.56 GHz, setting the floor of the X-band 8.2 to 12.4 GHz operating range. The guide behaves as a high-pass structure: dispersion, attenuation, and guide wavelength all blow up as frequency approaches f_c from above.

Category: Transmission Lines

TE₁₀ Cutoff: f_c = c / (2a)

Usable Band: 1.25 to 1.9 × f_c

How Cutoff Defines the Waveguide Operating Band

A hollow metal waveguide does not support a true TEM mode the way a coaxial or stripline does; instead it supports a discrete set of transverse-electric (TE) and transverse-magnetic (TM) modes, each with its own cutoff frequency. The cutoff arises directly from the boundary conditions on the conducting walls: the transverse field pattern must fit an integer number of half-wavelengths across the guide cross-section. When the free-space wavelength grows too large to satisfy that pattern, the axial wavenumber becomes imaginary and the mode can no longer propagate. This is why a waveguide is fundamentally a high-pass device, in sharp contrast to a low-pass lumped network.

The practical consequence is a usable single-mode band bounded on both sides. Just above f_c, the phase velocity and guide wavelength tend toward infinity while attenuation and dispersion spike, so operation is kept above roughly 1.25 f_c. The upper edge is set by the next-higher mode: for the standard 2-to-1 broad-to-narrow aspect ratio, TE₂₀ cuts on at exactly 2 f_c, so designers stay below about 1.9 f_c to preserve clean TE₁₀ propagation and avoid mode conversion. The result is the familiar set of EIA WR bands, each about 1.5-to-1 in frequency span.

Filling the guide with a dielectric of relative permittivity ε_r scales every cutoff frequency down by the square root of ε_r, because the in-medium speed of light drops by the same factor. This is exploited in dielectric-loaded and ridged waveguides to shrink physical dimensions or widen the single-mode band. The same cutoff physics also governs below-cutoff (piston) attenuators and waveguide-beyond-cutoff filters, where a guide is deliberately operated where f < f_c to obtain a precise, near-lossless evanescent attenuation.

Cutoff and Propagation Equations

General TE_mn / TM_mn Cutoff (rectangular guide, broad-wall a, narrow-wall b):
f_c = (c / 2) × √((m/a)² + (n/b)²)

Dominant TE₁₀ Cutoff:
f_c = c / (2a) with cutoff wavelength λ_c = 2a

Guide Wavelength Above Cutoff:
λ_g = λ / √(1 − (f_c/f)²) → ∞ as f → f_c

Below-Cutoff Attenuation Constant:
α = (2π / λ_c) × √(1 − (f/f_c)²) nepers/m

c = speed of light in the guide medium (c₀ / √ε_r); m, n = mode indices; λ = free-space wavelength. Example: air-filled WR-90, a = 22.86 mm → f_c ≈ 6.56 GHz.

Cutoff Frequency of Common Standard Waveguides

Band / Guide	Broad-wall a (mm)	TE₁₀ f_c (GHz)	Recommended Band (GHz)	Dominant Mode
WR-284 (S-band)	72.14	2.078	2.60 to 3.95	TE₁₀
WR-137 (C-band)	34.85	4.301	5.85 to 8.20	TE₁₀
WR-90 (X-band)	22.86	6.557	8.20 to 12.40	TE₁₀
WR-42 (K-band)	10.67	14.05	18.00 to 26.50	TE₁₀
WR-28 (Ka-band)	7.112	21.08	26.50 to 40.00	TE₁₀
WR-15 (V-band)	3.759	39.88	50.00 to 75.00	TE₁₀

Common Questions

Frequently Asked Questions

How do you calculate the TE₁₀ cutoff frequency of a rectangular waveguide?

For the dominant TE₁₀ mode, f_c = c / (2a), where c is the speed of light in the guide medium and a is the broad-wall (longer) dimension. Air-filled WR-90 with a = 22.86 mm gives f_c = 6.557 GHz, which is why WR-90 serves the 8.2 to 12.4 GHz X-band. A dielectric filling of permittivity ε_r lowers f_c by √ε_r. The general result is f_c = (c/2) × √((m/a)² + (n/b)²).

What happens to a signal below the waveguide cutoff frequency?

The propagation constant becomes purely imaginary, so the mode is evanescent and its amplitude decays as exp(−αz) rather than propagating. The decay rate is α = (2π/λ_c) × √(1 − (f/f_c)²). A below-cutoff section acts as a reactive, near-lossless high-pass attenuator, which is exactly how piston attenuators and waveguide-beyond-cutoff standards achieve precise isolation, often exceeding 30 dB/cm well below f_c.

Why is a waveguide operated between roughly 1.25 and 1.9 times its cutoff frequency?

That window keeps single-mode TE₁₀ operation while staying clear of both edges. The lower bound, about 1.25 f_c, avoids the steep dispersion and attenuation near cutoff where λ_g stretches toward infinity. The upper bound, about 1.9 f_c, stays below the TE₂₀ cutoff at 2 f_c for the standard 2-to-1 aspect ratio, preventing multimode propagation and mode conversion.

Related Terms

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