Circular Waveguide Calculator — Cutoff Frequency, TE11 & TM01 Modes

A circular waveguide is a hollow metal tube that carries microwaves above a cutoff frequency set by its radius. Its fields form modes, transverse electric (TE) or transverse magnetic (TM), each shaped by a Bessel function, and each mode propagates only above its own cutoff:

Here $a$ is the inner radius, $\varepsilon_r$ the relative permittivity of the filling, and $\chi_{nm}$ a zero of a Bessel function: for TE modes the $m$-th zero $p'_{nm}$ of $J'_n$, for TM modes the $m$-th zero $p_{nm}$ of $J_n$ (Pozar, Microwave Engineering, 4th ed., section 3.4). The index $n$ counts the field's variations around the circumference and $m$ those across the radius. The calculator holds the roots to ten decimals, as tabulated in the NIST Digital Library of Mathematical Functions, section 10.21.

The single-mode band

The dominant mode is TE₁₁, with $p'_{11} = 1.8412$, and the next is TM₀₁, with $p_{01} = 2.4048$. A guide therefore carries a single mode only between their two cutoffs, a band of 2.4048/1.8412 = 1.306 to 1 whatever the radius or filling.

Above cutoff

With $k = 2\pi f\sqrt{\varepsilon_r}/c$, the phase constant is $\beta = \sqrt{k^2 - (\chi_{nm}/a)^2}$, the guide wavelength $2\pi/\beta$, the phase velocity $\omega/\beta$, and the group velocity $c^2/(\varepsilon_r v_p)$. The wave impedance is $k\eta/\beta$ for TE modes and $\beta\eta/k$ for TM modes, with $\eta = \mu_0 c/\sqrt{\varepsilon_r}$.

The dielectric attenuation is $\alpha_d = k^2\tan\delta/(2\beta)$. The wall loss, with $R_s = \sqrt{\omega\mu_0/2\sigma}$ and $x = f_c/f$, is

for TE modes, which is Pozar's equation 3.133 for TE₁₁, and $R_s/(a\eta\sqrt{1-x^2})$ for TM modes. With $n = 0$ the TE term in brackets is $x^2$ alone, so the wall loss of TE₀₁ keeps falling as the frequency rises.

Below cutoff

Below its cutoff a mode does not propagate. Its fields decay as $e^{-\alpha z}$ with $\alpha = \sqrt{(\chi_{nm}/a)^2 - k^2}$, which the calculator reports in dB/m. A short length of guide below cutoff is how a vent blocks low frequencies.

Validity

The guide is a perfect circular cylinder filled with one homogeneous material, and the mode indices run from n = 0 to 5 and m = 1 to 5. The loss formulas come from the perturbation method: they assume a good conductor, with a skin depth far smaller than the radius, and a small loss tangent, and they lose accuracy very close to cutoff, where β tends to zero. A loss tangent above 0.1 or a skin depth above 1% of the radius is shown with a warning.