Resonant frequency of rectangular resonator¶

A rectangular resonator consists of metal walls and a material filling it.

Notation:

\(c\) (c) is speed_of_light.

Links:

Mahatma Gandhi Central University, formula 17 on page 12 (PDF file).

resonant_frequency¶: Resonant temporal_frequency of the resonator.

Symbol:: f_r
Latex:: \(f_\text{r}\)
Dimension:: frequency

first_index¶: Index that changes along the width of the resonator, see positive_number.

Symbol:: m
Latex:: \(m\)
Dimension:: dimensionless

second_index¶: Index that changes along the height of the resonator, see positive_number.

Symbol:: n
Latex:: \(n\)
Dimension:: dimensionless

third_index¶: Index that changes along the length of the resonator, see positive_number.

Symbol:: p
Latex:: \(p\)
Dimension:: dimensionless

width¶: length of the resonator along the axis perpendicular to the axis of wave propagation and to height.

Symbol:: l_1
Latex:: \(l_{1}\)
Dimension:: length

height¶: length of the resonator along the axis perpendicular to the axis of wave propagation and to width.

Symbol:: l_2
Latex:: \(l_{2}\)
Dimension:: length

length¶: length of the resonator along the axis of wave propagation.

Symbol:: l_3
Latex:: \(l_{3}\)
Dimension:: length

relative_permittivity¶: relative_permittivity of the medium filling the resonator.

Symbol:: epsilon_r
Latex:: \(\varepsilon_\text{r}\)
Dimension:: dimensionless

relative_permeability¶: relative_permeability of the medium filling the resonator.

Symbol:: mu_r
Latex:: \(\mu_\text{r}\)
Dimension:: dimensionless

law¶

f_r = c / (2 * sqrt(epsilon_r * mu_r)) * sqrt((m / l_1)^2 + (n / l_2)^2 + (p / l_3)^2)

Latex:: \[f_\text{r} = \frac{c}{2 \sqrt{\varepsilon_\text{r} \mu_\text{r}}} \sqrt{\left(\frac{m}{l_{1}}\right)^{2} + \left(\frac{n}{l_{2}}\right)^{2} + \left(\frac{p}{l_{3}}\right)^{2}}\]