Circular Waveguide Calculator
Calculates the cutoff frequency for every TEmn and TMmn mode (using tabulated Bessel function zeros), the dominant TE11 guide wavelength and wave impedance, phase and group velocities, and conductor loss at the operating frequency.
Equations & Parameters ▸
\(f_{c,mn}^{\mathrm{TE}} = \dfrac{x'_{mn}\,c}{2\pi a\sqrt{\varepsilon_r}}, \quad f_{c,mn}^{\mathrm{TM}} = \dfrac{x_{mn}\,c}{2\pi a\sqrt{\varepsilon_r}}\)
\(\beta = \sqrt{k^2 - k_c^2}, \quad \lambda_g = \dfrac{2\pi}{\beta}, \quad Z_{\mathrm{TE}_{11}} = \dfrac{\omega\mu_0}{\beta}\)
\(\alpha_c^{\mathrm{TE}_{mn}} = \dfrac{R_s}{a\eta\sqrt{1-(f_c/f)^2}}\!\left[\!\left(\frac{f_c}{f}\right)^{\!2}\!+\dfrac{m^2}{x_{mn}^{\prime\,2}-m^2}\right]\)
| a | Inner radius of the waveguide |
| x'mn | n-th root of J'm(x) = 0 (TE modes); e.g. x'11 = 1.8412 |
| xmn | n-th root of Jm(x) = 0 (TM modes); e.g. x01 = 2.4048 |
| TE11 | Dominant mode. Single-mode bandwidth: fc,TE₁₁ to 1.307 × fc,TE₁₁ (TM₀₁ cutoff) |
| TE01 | m = 0 → no m²/(x'²−m²) term; conductor loss decreases with frequency. Used in mm-wave transmission. |
| vph·vgr | Phase × group velocity = c²/εr (always) |
Note: the TE11 mode is degenerate (two orthogonal polarisations). Circular waveguide is less common than rectangular due to its narrower single-mode bandwidth (~31% vs ~100% for standard WR), but is used for rotating joints, TE01 mm-wave links, and MRI bore liners.
Reference: M. Pozar, Microwave Engineering, 4th ed., §3.4 (Wiley, 2012)
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