Circular Waveguide Calculator computes TE11 cutoff frequency, higher-order mode cutoffs, guide wavelength, wave impedance, and phase velocity for a circular waveguide of a given radius.
Circular Waveguide Calculator: TE11 Cutoff, Mode Chart, and Single-Mode Bandwidth
This calculator computes the dominant TE₁₁ cutoff frequency, the cutoff points of the next higher-order modes, and the guide wavelength and wave impedance for a circular waveguide of a given radius and frequency. Microwave and satellite-communication engineers use it to size circular waveguide runs, feed horns, and rotary joints so they operate cleanly in a single mode.
Circular Waveguide Calculator Inputs and Outputs
Enter the inner radius $r$ in cm, operating frequency $f$ in GHz, and dielectric constant $\varepsilon_r$ (1 for air). The calculator returns the TE₁₁, TM₀₁, TE₂₁, and TE₀₁ cutoff frequencies and wavelengths, the recommended single-mode band, guide wavelength, wave impedance, and phase velocity.
How the Circular Waveguide Cutoff Formula Works
Per C. A. Balanis’s “Circular Waveguides” reference article (Wiley Encyclopedia of RF and Microwave Engineering), the cutoff frequency of a TEmn mode in a circular waveguide of radius $r$ is set by the roots of the derivative of the Bessel function:
$$f_c = \dfrac{p’_{mn}\,c}{2\pi r\sqrt{\varepsilon_r}}$$
where $p’_{mn}$ is 1.8412 for the dominant TE₁₁ mode, 3.0542 for TE₂₁, and 3.8318 for TE₀₁. TM modes use the same formula with the plain (non-derivative) Bessel roots $p_{mn}$, where $p_{01} = 2.4049$ for TM₀₁. Cutoff wavelength is $\lambda_c = 2\pi r/p$. Above cutoff, guide wavelength, wave impedance, and phase velocity are:
$$\lambda_g = \dfrac{\lambda_0}{\sqrt{1-(f_c/f)^2}} \qquad Z_{TE} = \dfrac{\eta_0}{\sqrt{1-(f_c/f)^2}} \qquad v_p = \dfrac{c}{\sqrt{1-(f_c/f)^2}}$$
where $\lambda_0 = c/f$ and $\eta_0 = 376.73\ \Omega$ is the intrinsic impedance of free space. The most common input mistake is entering diameter where the formula calls for radius (or vice versa) — since $f_c \propto 1/r$, this halves or doubles the calculated cutoff frequency, which is easy to miss because the result still looks like a plausible waveguide number.
A detail not covered by generic cutoff-frequency tools: TE₁₁ is degenerate — two orthogonal polarizations of TE₁₁ share the exact same cutoff frequency, unlike rectangular waveguide’s TE₁₀, where geometry fixes the field orientation.
A perfectly round guide doesn’t prefer either orientation, so any small ellipticity, weld seam, or flange misalignment can rotate the polarization or split it into two slightly different modes — a real effect in long runs and rotary joints.
This degeneracy is also deliberately exploited in dual-polarized antenna feeds and rotary joints, where both orthogonal TE₁₁ orientations are excited on purpose.
Circular Waveguide Mode Order and Single-Mode Band
Bessel Root Values for Circular Waveguide Modes
The values below are the Bessel and Bessel-derivative roots from Balanis’s “Circular Waveguides” reference tables, used directly in the cutoff formula above.
| Mode | Type | Root Value | Cutoff Ratio to TE11 |
|---|---|---|---|
| TE11 | Derivative root $p’_{11}$ | 1.8412 | 1.000 |
| TM01 | Bessel root $p_{01}$ | 2.4049 | 1.306 |
| TE21 | Derivative root $p’_{21}$ | 3.0542 | 1.659 |
| TE01 / TM11 | Derivative/Bessel root | 3.8318 | 2.081 |
Circular Waveguide Cutoff and Mode Questions
Why is TE11 the dominant mode in circular waveguide?
TE₁₁ has the lowest Bessel-derivative root (1.8412) of any TE or TM mode, giving it the lowest cutoff frequency, so it’s the first mode to propagate as frequency rises from zero.
Why is circular waveguide’s single-mode bandwidth narrower than rectangular?
Rectangular waveguide can reach a 2:1 bandwidth between TE₁₀ and the next mode. Circular waveguide is fixed by geometry to about 1.306:1 between TE₁₁ and TM₀₁, and that ratio can’t be changed by resizing the guide.
What happens if the operating frequency exceeds the next mode’s cutoff?
Multiple modes propagate simultaneously. Since each mode travels at a different phase velocity, the signal disperses and distorts — this is the condition the calculator’s multi-moding warning is flagging.
Does the dielectric constant input matter for an air-filled waveguide?
For air, $\varepsilon_r \approx 1$, so it has virtually no effect. It matters when the guide is filled with a solid or gas dielectric, since cutoff frequency scales down by $1/\sqrt{\varepsilon_r}$.
Why does circular waveguide have no fixed polarization reference?
A round cross-section has no geometric axis to lock the TE₁₁ field to, unlike a rectangular guide’s fixed width and height. Both orthogonal TE₁₁ orientations share the same cutoff and can mix or rotate along the run.