Concept:
Phase velocity is given by:
\({v_p} = \frac{\omega }{\beta }\)
Where, ω = angular frequency
β = phase constant, calculated for a parallel waveguide as:
\(\beta = \sqrt {{\omega ^2}\mu - {{\left( {\frac{{m\pi }}{a}} \right)}^2}} \) [for parallel waveguide]
The Phase velocity can now be written as:
\({v_p} = \frac{\omega }{{\sqrt {{\omega ^2}\mu - {{\left( {\frac{{m\pi }}{a}} \right)}^2}} }}\)
\({v_p} = \frac{1}{{\sqrt {\mu - {{\left( {\frac{{m\pi }}{{a\omega }}} \right)}^2}} }}\)
\({v_p} = \frac{{\frac{1}{{\sqrt {\mu } }}}}{{\sqrt {1 - {{\left( {\frac{{m\pi }}{{a\omega \sqrt {\mu } }}} \right)}^2}} }}\)
With \(c = \frac{1}{{\sqrt {\mu } }}\)
\({v_p} = \frac{c}{{\sqrt {1 - {{\left( {\frac{{m\pi c}}{{a\omega }}} \right)}^2}} }}\;\)
Also, the cut-off frequency is defined as:
\({\omega _c} = \frac{{m{\pi _c}}}{a}\)
\({v_p} = \frac{c}{{\sqrt {1 - {{\left( {\frac{{{\omega _c}}}{\omega }} \right)}^2}} }}\)
\(\sin \theta = \frac{{{\omega _c}}}{\omega }\)
\({v_p} = \frac{c}{{\sqrt {1 - {{\sin }^2}\theta } }}\)
\({v_p} = \frac{C}{{\cos \theta }}\)
\({v_p} = \frac{c}{{\cos \theta }}\)
Note: This is valid for the Rectangular waveguide also.
Calculation:
Given
a = 8 cm
b = 4 cm
f = 3.4 GHz
The cut-off frequency will be:
\({f_c} = \frac{c}{{2a}} = \frac{{3 \times {{10}^{10}}}}{{2 \times 8}}\)
\(\sin \theta = \frac{{{f_c}}}{f}\)
\(\sin \theta = \frac{{1.875}}{{3.4}} = 0.5514\)
∴ cos θ = 0.83419
\(\frac{{{v_p}}}{c} = \frac{1}{{\cos \theta }}\)
\(\frac{{{v_p}}}{c} = 1.198\)
