Calculator Inputs
Example Data Table
| Geometry | Mode | Frequency | Dimensions | εr | μr | Expected behavior |
|---|---|---|---|---|---|---|
| Rectangular | TE10 | 10 GHz | a = 22.86 mm, b = 10.16 mm | 1.0 | 1.0 | Common X-band guide, propagating above cutoff |
| Rectangular | TE20 | 15 GHz | a = 22.86 mm, b = 10.16 mm | 1.0 | 1.0 | Higher mode with higher cutoff frequency |
| Circular | TE11 | 8 GHz | Radius = 12.5 mm | 1.0 | 1.0 | Dominant circular mode in many round guides |
| Circular | TM01 | 12 GHz | Radius = 8 mm | 2.1 | 1.0 | Dielectric-filled guide with shifted velocity |
Formula Used
1) Velocity in the filling medium
v = c / √(εrμr)
2) Rectangular waveguide cutoff frequency
fc = (v / 2) × √[(m / a)² + (n / b)²]
3) Circular waveguide cutoff frequency
fc = (v × X) / (2πr)
Here, X is the Bessel-function root for the selected circular mode.
4) Phase velocity above cutoff
vp = v / √[1 - (fc / f)²]
5) Group velocity above cutoff
vg = v × √[1 - (fc / f)²]
6) Propagation constant and guide wavelength
β = (2π / λ) × √[1 - (fc / f)²]
λg = λ / √[1 - (fc / f)²]
Below cutoff, propagation becomes evanescent. In that region, a real phase velocity for power transmission is not reported, because the field does not propagate normally down the guide.
How to Use This Calculator
Step 1: Choose the waveguide geometry
Select rectangular for broad-wall and narrow-wall dimensions, or circular for a round guide using radius and a preset circular mode.
Step 2: Enter frequency and units
Input the operating frequency and choose Hz, kHz, MHz, or GHz. The calculator converts the value internally to hertz.
Step 3: Enter geometry details
For rectangular guides, enter a, b, m, and n. For circular guides, enter radius and choose a supported TE or TM mode.
Step 4: Set material properties
Enter relative permittivity and relative permeability to model air-filled or dielectric-filled guides accurately.
Step 5: Review the results and export
After calculation, inspect cutoff frequency, phase velocity, group velocity, wavelength values, and the graph. Then download results as CSV or PDF.
Frequently Asked Questions
1) Why can phase velocity exceed the speed of light?
Phase velocity can be greater than light speed in a waveguide without violating relativity. It does not represent information or energy transport. Group velocity remains below the medium wave speed for normal propagation.
2) What happens at frequencies below cutoff?
Below cutoff, the mode becomes evanescent. Fields decay along the guide instead of carrying power efficiently. That is why the calculator marks the state as below cutoff and does not show a real propagating phase velocity.
3) Why do TE and TM modes matter?
TE and TM modes have different field structures, cutoff behavior, and impedance relationships. Selecting the correct mode is essential because phase velocity depends strongly on the cutoff frequency for that exact mode.
4) How do dielectric materials affect the result?
Higher relative permittivity lowers wave speed in the filling medium, which also changes cutoff frequency, guide wavelength, group velocity, and phase velocity. This is important in loaded or partially filled waveguide designs.
5) Which rectangular mode is usually dominant?
For standard rectangular waveguides, TE10 is usually the dominant mode. It has the lowest cutoff frequency among valid rectangular modes, so it is commonly selected for efficient single-mode operation.
6) Why does phase velocity grow near cutoff?
As operating frequency approaches cutoff from above, the square-root term in the denominator becomes very small. That makes phase velocity rise sharply, while group velocity drops toward zero.
7) Can I use inches or millimeters for dimensions?
Yes. The calculator supports millimeters, centimeters, meters, and inches. All dimensions are converted internally to meters before applying the engineering formulas.
8) What does the graph show?
The graph plots phase velocity above cutoff across a frequency sweep. It helps you see how strongly the selected geometry and mode respond as operating frequency moves away from cutoff.