Waveguide Mode Calculator

Example data table

Use these sample inputs to sanity-check the calculator.

Formula used

Let v = c / √(εr μr), ω = 2πf, and k = ω√(με). The cutoff wavenumber kc depends on guide geometry:

• Rectangular: kc = π √[(m/a)² + (n/b)²] and fc = (v/2) √[(m/a)² + (n/b)²]
• Circular: kc = xmn / a and fc = (v / 2π) kc, where xmn is a TE (derivative) or TM (Bessel) root

For propagation (f > fc), the phase constant is β = √(k² − kc²), guided wavelength λg = 2π/β, phase velocity vp = ω/β, and group velocity vg = v²/vp.

Wave impedance (lossless) is ZTE = ωμ/β and ZTM = β/(ωε). For f ≤ fc, fields are evanescent with attenuation α = √(kc² − k²).

How to use this calculator

1. Select guide type and choose TE or TM.
2. Enter dimensions: rectangular uses a and b; circular uses radius a.
3. Set frequency and units, then choose mode indices m and n.
4. Enter εr and μr for the filling material.
5. Submit to view results above, then export as CSV or PDF.

Technical article

1) Why waveguide modes matter

Metallic waveguides confine electromagnetic energy by boundary conditions, producing discrete transverse electric and transverse magnetic field patterns. Each pattern behaves like a separate channel with its own cutoff frequency and dispersion. Correct mode selection reduces loss, distortion, and unwanted coupling in microwave links, radar front ends, and laboratory test fixtures.

2) Rectangular guide reference numbers

In an air-filled rectangular guide, the dominant mode is typically TE10. For a common geometry of a = 22.86 mm and b = 10.16 mm (often used in X-band hardware), the TE10 cutoff is about 6.56 GHz. Operating at 10 GHz provides comfortable margin, improving mode purity and stability.

3) Higher-order modes and spacing

As frequency increases, additional modes become allowed when f > f_c. For the same example, TE20 cuts on near 13.1 GHz and TE01 near 14.8 GHz, so an operational band below 13 GHz helps prevent multimode propagation. The calculator highlights this by showing whether the chosen mode is propagating or evanescent.

4) Circular guide mode behavior

Circular waveguides use Bessel-function roots, producing families such as TE11 and TM01. A practical takeaway is that the first propagating mode is often TE11. For a 10 mm radius, TE11 cutoff is roughly 8.78 GHz in air, so a 12–18 GHz band can be feasible with careful transitions and alignment.

5) Material fill effects

Dielectric fill shifts cutoff downward by the factor 1/√(εr μr). For instance, increasing εr from 1.0 to 2.25 reduces cutoff by one third. This is useful for compact components, but dispersion increases and phase velocity changes, which can affect timing and matching.

6) Propagation constant and guided wavelength

Above cutoff, the guide phase constant is β = √(k² − k_c²). A smaller β near cutoff means longer guided wavelength λg, making structures electrically shorter. Designers exploit this for resonators, but must control reflections because impedance changes rapidly near cutoff.

7) Phase and group velocities

Waveguides are dispersive: v_p = ω/β can exceed the material wave speed, while energy transport follows v_g. In lossless guides, v_p v_g = v², so a slow group velocity near cutoff corresponds to high phase velocity. This impacts pulse broadening and modulation fidelity.

8) Using results in real designs

Use the cutoff check to pick a safe operating band and confirm single-mode margins. Then review impedance, β, and λg to size matching sections, irises, and resonant cavities. Export CSV/PDF for reports, and document geometry, material, and mode indices so results remain traceable during fabrication and testing.

FAQs

1) What does cutoff frequency mean?

Cutoff is the minimum frequency where a chosen mode can propagate. Below cutoff, fields decay exponentially and power does not travel down the guide effectively.

2) Why is TE10 called the dominant rectangular mode?

TE10 typically has the lowest cutoff in a standard rectangular guide, so it is the first to propagate as frequency increases, making it easier to keep single-mode operation.

3) Why does the calculator show evanescent behavior?

If the operating frequency is at or below cutoff, the propagation constant becomes imaginary. The tool reports attenuation alpha to quantify how quickly fields decay along the guide.

4) How do εr and μr change results?

They set the wave speed in the filling medium. Higher εr or μr lowers cutoff and changes beta, impedance, and velocities, which can alter bandwidth and matching.

5) Are the circular roots exact?

The calculator includes accurate standard roots for common low-order modes (m up to 3, n up to 3). For other modes, use a Bessel root solver and enter the corresponding root.

6) What units should I use for dimensions?

Select the length unit first, then enter a and b consistently. Rectangular uses a as the broad wall and b as the narrow wall; circular uses a as radius.

7) Can I export multiple runs at once?

Exports contain the latest computed result. For multiple cases, run each scenario and save the CSV or PDF each time, or copy results into your own comparison table.