Calculator
Example data table
| Geometry | Mode | Dimensions | Material | Estimated fc (GHz) | Notes |
|---|---|---|---|---|---|
| Rectangular | TE10 | a=22.86 mm, b=10.16 mm | εr=1, μr=1 | ~6.56 | WR-90 baseline |
| Rectangular | TE20 | a=22.86 mm, b=10.16 mm | εr=1, μr=1 | ~13.12 | Higher-order in a |
| Circular | TE11 | r=10 mm | εr=1, μr=1 | ~8.79 | Uses x≈1.8412 |
| Circular | TM01 | r=10 mm | εr=1, μr=1 | ~11.48 | Uses x≈2.4048 |
Formula used
- kc = π √[(m/a)² + (n/b)²]
- fc = (c/2) √[(m/a)² + (n/b)²]
- kc = x / r
- fc = (c·kc) / (2π)
- β = k √[1 − (fc/f)²], where k = 2πf/c
- λg = λ / √[1 − (fc/f)²], where λ = c/f
- vp = c / √[1 − (fc/f)²] , vg = c √[1 − (fc/f)²]
How to use this calculator
- Select the waveguide type and mode family.
- Enter dimensions in your preferred length unit.
- Set mode indices (m,n) and the material (εr, μr).
- Optionally add an operating frequency to evaluate propagation.
- Press Submit to view results above the form.
Insights
Cutoff frequency as a design gate
Waveguides behave like spatial filters: below cutoff, fields decay and power cannot propagate. This calculator turns geometry and mode choice into a clear pass or fail boundary. For a given filling medium, increasing the broad dimension lowers cutoff, while higher indices raise it. A practical rule is to target operating frequency at least 25% above cutoff to reduce dispersion and improve bandwidth planning. This supports fast sizing decisions during early concept studies today.
Rectangular modes and dimensional leverage
In rectangular guides, the broad wall a strongly controls the dominant TE10 cutoff, setting the usable band. The narrow wall b mainly affects modes with n>0 such as TE01. If your computed TE20 or TE11 cutoff is close to your operating frequency, multimode behavior becomes likely and repeatability can drop. The comparison chart helps spot those margins quickly.
Circular modes and Bessel roots
Circular waveguides rely on Bessel roots, so the dominant mode is typically TE11 rather than TE10. The calculator includes common Jm and J′m zeros for fast evaluation. When you work with higher-order modes, a custom root value keeps the method valid without changing the workflow. Because fc scales inversely with radius, machining tolerances can shift cutoff.
Material loading shifts the band
Relative permittivity and permeability slow wave speed in the medium, reducing cutoff frequency by the factor 1/√(εrμr). This is useful for compact components, but it also increases stored energy and can raise dielectric loss sensitivity. The output table reports wave speed and cutoff wavelength for quick material tradeoffs.
Propagation metrics above cutoff
When an operating frequency is provided, the calculator derives β, guide wavelength, phase velocity, and group velocity. Close to cutoff, β becomes small and λg grows, which often makes resonances and standing-wave sensitivity more pronounced. Farther above cutoff, dispersion reduces and vg approaches the medium wave speed. These metrics help timing and fixture length selection.
Exportable results for reviews and testing
Engineering reviews benefit from consistent evidence. The CSV export supports quick spreadsheet checks, while the PDF export provides a clean snapshot for design logs and lab notebooks. Pair exported margins with measured S-parameters to explain roll-off. If attenuation is unexpected, re-check nearby-mode cutoffs to rule out multimode excitation.
FAQs
What does “below cutoff” mean in practice?
Below cutoff, the selected mode cannot carry power down the guide. Fields decay exponentially, so transmission drops rapidly with length. Short sections may still show coupling, but it is not true propagation.
Which mode should I start with for rectangular guides?
Most designs start with TE10 because it is the dominant mode and usually has the lowest cutoff in a standard rectangular guide. Keeping other modes’ cutoffs comfortably higher helps avoid multimode behavior.
Why does circular waveguide use different constants than rectangular?
Circular solutions come from Bessel functions in cylindrical coordinates. The constants are zeros of Jm (TM) or J′m (TE). These roots replace the simple m/a and n/b terms used in rectangular guides.
How accurate are the built-in circular roots?
The included roots are standard numerical approximations for common low-order modes and are accurate enough for design screening. For unusual mode indices, use the custom root input from a trusted reference or solver.
What operating frequency margin is recommended?
A common guideline is to operate at least 20–30% above cutoff to reduce dispersion and sensitivity. Very close to cutoff, group velocity drops and guide wavelength grows, which can complicate tuning and measurements.
Does conductivity or wall loss affect cutoff?
Cutoff is primarily geometric and material-dependent through εr and μr. Wall loss affects attenuation above cutoff, not the cutoff boundary itself, although extreme surface conditions can alter effective dimensions slightly.