Air-Filled Circular Cavity Resonator Calculator

For the TE₁₁₁ mode, the resonant frequency is given by:
\[ f_{111} = \frac{c}{2\pi}\sqrt{\left(\frac{x’_{11}}{a}\right)^2+\left(\frac{\pi}{d}\right)^2} \] where \(x’_{11}\approx1.8412\), \(a\) is the cavity radius, and \(d\) is the cavity height.

* Enter the cavity radius (m) and height (m).

Step 1: Enter Cavity Dimensions

Cavity Radius, \( a \) (m):

Example: 0.05 m

Cavity Height, \( d \) (m):

Example: 0.1 m

Air-Filled Circular Cavity Resonator Calculator (TE₁₁₁ Mode)

The air-filled circular cavity resonator is a key component in microwave engineering, used to create precise frequency filters and oscillators. In the TE₁₁₁ mode, the resonant frequency is determined by the dimensions of the cavity. This guide explains how to calculate the resonant frequency for the TE₁₁₁ mode using a dedicated calculator, detailing the underlying principles, formula, and practical examples.

Overview of Circular Cavity Resonators
Understanding the TE₁₁₁ Mode
Resonant Frequency Formula
Step-by-Step Calculation Process
Practical Examples
Common Applications
Conclusion

1. Overview of Circular Cavity Resonators

A circular cavity resonator is a hollow, cylindrical structure designed to support electromagnetic waves at specific resonant frequencies. When the cavity is air-filled, its dielectric properties are essentially those of free space, simplifying the design and calculations.

Such resonators are used in filters, oscillators, and various microwave components. Their resonant frequency is strongly dependent on their physical dimensions—primarily the cavity’s radius and length.

2. Understanding the TE₁₁₁ Mode

The TE₁₁₁ mode (Transverse Electric mode) is one of the fundamental modes in a circular cavity resonator. In this mode:

Electric fields have no component in the direction of propagation (along the cavity’s axis).
The field distribution is described by Bessel functions, and the boundary conditions lead to discrete resonant frequencies.

For the TE₁₁₁ mode, the resonant frequency is mainly determined by the first zero of the derivative of the Bessel function \(J_1′(x)\) and the cavity dimensions.

3. Resonant Frequency Formula for TE₁₁₁ Mode

For an air-filled circular cavity resonator operating in the TE₁₁₁ mode, the resonant frequency \( f_{111} \) is given by:

\( f_{111} = \frac{c}{2\pi} \sqrt{\left(\frac{X’_{11}}{a}\right)^2 + \left(\frac{\pi}{d}\right)^2} \)

Where:

\( c \) is the speed of light in air (\( \approx 3 \times 10^8 \, \text{m/s} \)).
\( X’_{11} \) is the first zero of the derivative of the Bessel function \(J_1′(x)\), approximately \(1.84118\).
\( a \) is the radius of the cavity (in meters).
\( d \) is the length (or height) of the cavity (in meters).

This equation combines the contributions from the radial and axial dimensions of the cavity to determine the overall resonant frequency.

4. Step-by-Step Calculation Process

Input Cavity Dimensions: Determine the radius \( a \) and the length \( d \) of the resonator.
Use Known Constants: Use \( c \approx 3 \times 10^8 \, \text{m/s} \) and \( X’_{11} \approx 1.84118 \).
Apply the Formula: Substitute the values into the resonant frequency formula:
\( f_{111} = \frac{c}{2\pi} \sqrt{\left(\frac{1.84118}{a}\right)^2 + \left(\frac{\pi}{d}\right)^2} \)
Compute \( f_{111} \): Perform the arithmetic operations to obtain the resonant frequency in Hz.

5. Practical Examples

Example 1: Typical Microwave Cavity

Given: A circular cavity with a radius \(a = 0.05\,m\) and a length \(d = 0.10\,m\).

Calculation:

\( f_{111} = \frac{3 \times 10^8}{2\pi} \sqrt{\left(\frac{1.84118}{0.05}\right)^2 + \left(\frac{\pi}{0.10}\right)^2} \)

First, calculate the individual components:

\( \frac{1.84118}{0.05} = 36.8236 \, \text{m}^{-1} \), then \((36.8236)^2 \approx 1355.4 \)
\( \frac{\pi}{0.10} = 31.416 \, \text{m}^{-1} \), then \((31.416)^2 \approx 986.96 \)

Sum under the square root:

\( 1355.4 + 986.96 \approx 2342.36 \)

Taking the square root:

\( \sqrt{2342.36} \approx 48.4 \, \text{m}^{-1} \)

Now, calculate the prefactor:

\( \frac{3 \times 10^8}{2\pi} \approx \frac{3 \times 10^8}{6.2832} \approx 47.75 \times 10^6\, \text{m/s} \)

Finally, the resonant frequency is:

\( f_{111} \approx 47.75 \times 10^6 \times 48.4 \approx 2.312 \times 10^9 \, \text{Hz} \)

Thus, the resonant frequency is approximately 2.31 GHz.

Example 2: Smaller Cavity

Given: A cavity with \(a = 0.03\,m\) and \(d = 0.05\,m\).

Calculation: Substituting into the formula:

\( f_{111} = \frac{3 \times 10^8}{2\pi} \sqrt{\left(\frac{1.84118}{0.03}\right)^2 + \left(\frac{\pi}{0.05}\right)^2} \)

Calculate the terms:

\( \frac{1.84118}{0.03} \approx 61.3727 \, \text{m}^{-1} \) and \((61.3727)^2 \approx 3766.8 \)
\( \frac{\pi}{0.05} \approx 62.832 \, \text{m}^{-1} \) and \((62.832)^2 \approx 3947.8 \)

Sum:

\( 3766.8 + 3947.8 \approx 7714.6 \)

Square root:

\( \sqrt{7714.6} \approx 87.8 \, \text{m}^{-1} \)

Prefactor:

\( \frac{3 \times 10^8}{2\pi} \approx 47.75 \times 10^6\, \text{m/s} \)

Therefore:

\( f_{111} \approx 47.75 \times 10^6 \times 87.8 \approx 4.194 \times 10^9 \, \text{Hz} \)

The resonant frequency is approximately 4.19 GHz.

7. Common Applications

Microwave Filters: Used in telecommunications and radar systems for frequency selection.
Oscillators: Serve as key components in generating stable high-frequency signals.
Sensing and Measurement: Employed in resonant sensors for precise detection of environmental changes.
Scientific Research: Essential in experiments involving electromagnetic fields and wave propagation.

8. Conclusion

The Air-Filled Circular Cavity Resonator Calculator for the TE₁₁₁ mode provides an efficient way to determine the resonant frequency of a circular cavity based on its physical dimensions. By using the formula:

\( f_{111} = \frac{c}{2\pi} \sqrt{\left(\frac{1.84118}{a}\right)^2 + \left(\frac{\pi}{d}\right)^2} \)

and understanding the role of each parameter—cavity radius \(a\), cavity length \(d\), and the relevant constants—engineers and researchers can accurately design and analyze microwave components and resonant structures. This calculator is an invaluable tool in the field of high-frequency electronics and resonant circuit design.