For estimating and education only. Results come from the simplified models described below, are provided as is with no warranty, and are not a substitute for engineering analysis, simulation or measurement. Check them before relying on them in a design, purchase or safety decision. See the terms of use.
Equations
Rectangular
f_c(m,n) = \frac{c}{2\sqrt{\varepsilon_r}}\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2}Circular
f_c = \frac{\chi\,c}{\pi d\sqrt{\varepsilon_r}}, \qquad \chi_{\mathrm{TE}_{11}} = 1.8412, \quad \chi_{\mathrm{TM}_{01}} = 2.4049\lambda_g = \frac{\lambda}{\sqrt{1 - (f_c/f)^2}}v_p = \frac{v}{\sqrt{1 - (f_c/f)^2}}, \qquad v_g = v\sqrt{1 - (f_c/f)^2}Z_{\mathrm{TE}_{10}} = \frac{\eta}{\sqrt{1 - (f_c/f)^2}}ΩRecommended band
1.25\,f_c(\mathrm{TE}_{10}) \le f \le 1.89\,f_c(\mathrm{TE}_{10})Variables
| Symbol | Meaning | Unit |
|---|---|---|
a,\ b | Inner width and height of rectangular guide (a ≥ b) | m |
d | Inner diameter of circular guide | m |
m,\ n | Mode indices | — |
v | c / √εr | m/s |
\eta | Wave impedance of the fill medium | Ω |
Worked example
| Input | Value |
|---|---|
| Guide | WR-90, 22.86 × 10.16 mm, air-filled |
| Frequency | 10 GHz |
| Result | Value |
|---|---|
| Dominant mode | TE10 |
| Cutoff of dominant mode | 6.5571 GHz |
| Single-mode range | 6.557 – 13.11 GHz |
| Recommended band (1.25–1.89 × fc) | 8.196 – 12.39 GHz |
| At 10 GHz | Single mode (TE10) |
| Guide wavelength | 39.71 mm |
| Phase velocity | 1.324 c |
| Group velocity | 0.755 c |
| TE10 wave impedance | 499 Ω |
WR-90 cuts off at 6.557 GHz and the next mode starts at 13.114 GHz; the standard 8.2–12.4 GHz band sits inside that single-mode window.
Load this example in the calculatorValidity
- Perfect conductor walls and a uniform fill. Wall loss is not modeled.
- Dimensions are inner dimensions. Use the manufacturer’s tolerance if you are near cutoff.
- Below cutoff the field decays exponentially: nothing propagates.
Sources
- D. M. Pozar, Microwave Engineering, 4th ed., Wiley, 2012. Chapter 3.
- IEC 60153-2, Hollow metallic waveguides: relevant specifications for ordinary rectangular waveguides.