Waveguide Mode Calculator
Waveguide Mode Calculator computes cutoff frequencies for TE/TM modes in a rectangular waveguide using dimensions and mode indices. (141 chars)
Formulas Used in Waveguide Mode Calculator
The calculator computes the cutoff frequency for TE and TM modes in a rectangular waveguide using the following formula:
Cutoff Frequency for TE/TM Modes:
\\[ f_{c,mn} = \frac{c}{2\pi\sqrt{\mu_r \varepsilon_r}} \sqrt{\left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2} \\]Speed of Light in Medium:
\\[ c = \frac{c_0}{\sqrt{\mu_r \varepsilon_r}} \\]Where:
- \\( f_{c,mn} \\): Cutoff frequency for mode indices \\( m, n \\) (Hz)
- \\( c_0 \\): Speed of light in vacuum (\\( 3 \times 10^8 \, \text{m/s} \\))
- \\( c \\): Speed of light in the medium (m/s)
- \\( a \\): Waveguide width (m)
- \\( b \\): Waveguide height (m)
- \\( m, n \\): Mode indices (non-negative integers)
- \\( \varepsilon_r \\): Relative permittivity of the medium
- \\( \mu_r \\): Relative permeability of the medium
Note: At least one of \\( m \\) or \\( n \\) must be non-zero for valid TE/TM modes. The TE\\(_{00}\\) mode does not exist in rectangular waveguides.
Example Calculations
Example: Waveguide with \\( a = 0.1 \, \text{m}, b = 0.05 \, \text{m}, m = 1, n = 0, \varepsilon_r = 1.0, \mu_r = 1.0 \\)
\\[
c = \frac{3 \times 10^8}{\sqrt{1.0 \cdot 1.0}} = 3 \times 10^8 \, \text{m/s}
\\]
\\[
f_{c,10} = \frac{3 \times 10^8}{2\pi \sqrt{1.0 \cdot 1.0}} \sqrt{\left(\frac{1 \cdot \pi}{0.1}\right)^2 + \left(\frac{0 \cdot \pi}{0.05}\right)^2}
\\]
\\[
f_{c,10} = \frac{3 \times 10^8}{2\pi} \sqrt{\frac{\pi^2}{0.01}} \approx 1.499 \times 10^9 \, \text{Hz} \approx 1.50 \, \text{GHz}
\\]
Result: \\( f_{c,10} \approx 1.50 \, \text{GHz} \\)