Air-Filled Rectangular Cavity Resonator Calculator

Estimate the resonant frequency of an air-filled rectangular cavity resonator.

Using the formula:
\[ f_{mnl} = \frac{c}{2\pi}\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2 + \left(\frac{l}{d}\right)^2} \]
where \(c \approx 3 \times 10^8\) m/s.

* Enter the cavity dimensions (in meters) and mode numbers (m, n, l).
(For the common TE101 mode, use m=1, n=0, l=1.)

Step 1: Enter Cavity Parameters

Example: 0.1 m

Example: 0.05 m

Example: 0.2 m

Example: 1

Example: 0

Example: 1

Formula: \[ f_{mnl} = \frac{c}{2\pi}\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2 + \left(\frac{l}{d}\right)^2} \] with \( c \approx 3 \times 10^8 \) m/s.

Air-Filled Rectangular Cavity Resonator Calculator (In-Depth Explanation)

Air-Filled Rectangular Cavity Resonator Calculator (In-Depth Explanation)

Rectangular cavity resonators are widely used in microwave engineering for filters, oscillators, and various high-frequency components. When the cavity is air-filled, its resonant frequency depends purely on its physical dimensions and the electromagnetic mode of operation. This guide explains how to estimate the resonant frequency of an air-filled rectangular cavity resonator, focusing on a common mode.

Table of Contents

  1. Overview of Rectangular Cavity Resonators
  2. Understanding the TE₁₀₁ Mode
  3. Resonant Frequency Formula
  4. Step-by-Step Calculation Process
  5. Practical Examples
  6. Common Applications
  7. Conclusion

1. Overview of Rectangular Cavity Resonators

An air-filled rectangular cavity resonator is a hollow, metal box with rectangular cross-section that supports electromagnetic standing waves at specific resonant frequencies. The dimensions of the cavity (length, width, and height) determine which frequencies are sustained. These resonators are essential components in many microwave circuits.

2. Understanding the TE₁₀₁ Mode

In rectangular cavity resonators, various transverse electric (TE) and transverse magnetic (TM) modes can exist. A commonly used mode is the TE₁₀₁ mode, where:

  • \( m = 1 \): There is one half-wave variation along the width (\(a\)).
  • \( n = 0 \): No variation along the height (\(b\)).
  • \( l = 1 \): One half-wave variation along the length (\(d\)).

This mode is often chosen for its favorable field distribution and ease of excitation in practical applications.

3. Resonant Frequency Formula

For a rectangular cavity resonator operating in the TE₁₀₁ mode, the resonant frequency \( f_{101} \) is given by:

\( f_{101} = \frac{c}{2} \sqrt{\left(\frac{1}{a}\right)^2 + \left(\frac{1}{d}\right)^2} \)

Where:

  • \( c \) is the speed of light in air (\( \approx 3 \times 10^8\, \text{m/s} \)).
  • \( a \) is the width of the cavity (in meters).
  • \( d \) is the length (or depth) of the cavity (in meters).

Note that in this simplified formula for the TE₁₀₁ mode, the height \( b \) does not appear because there is no field variation in that dimension.

4. Step-by-Step Calculation Process

  1. Measure the Cavity Dimensions:
    • Determine the width \(a\) (in meters).
    • Determine the length (or depth) \(d\) (in meters).
  2. Substitute Known Constants:
    • Use \( c \approx 3 \times 10^8\, \text{m/s} \).
  3. Apply the Formula:
    \( f_{101} = \frac{3 \times 10^8}{2} \sqrt{\left(\frac{1}{a}\right)^2 + \left(\frac{1}{d}\right)^2} \)
  4. Compute the Result:

    Evaluate the terms inside the square root and multiply by the prefactor to obtain the resonant frequency in Hertz (Hz).

5. Practical Examples

Example 1: Typical Microwave Cavity

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

Calculation:

\( f_{101} = \frac{3 \times 10^8}{2} \sqrt{\left(\frac{1}{0.05}\right)^2 + \left(\frac{1}{0.10}\right)^2} \)

Compute the terms:

  • \( \frac{1}{0.05} = 20\,m^{-1} \) and \((20)^2 = 400 \)
  • \( \frac{1}{0.10} = 10\,m^{-1} \) and \((10)^2 = 100 \)
\( \sqrt{400 + 100} = \sqrt{500} \approx 22.36\,m^{-1} \)

Now, the prefactor:

\( \frac{3 \times 10^8}{2} = 1.5 \times 10^8\, m/s \)

Thus:

\( f_{101} \approx 1.5 \times 10^8 \times 22.36 \approx 3.35 \times 10^9\,Hz \)

The resonant frequency is approximately 3.35 GHz.

Example 2: Smaller Cavity

Given: A cavity with \(a = 0.03\,m\) and \(d = 0.06\,m\).

Calculation:

\( f_{101} = \frac{3 \times 10^8}{2} \sqrt{\left(\frac{1}{0.03}\right)^2 + \left(\frac{1}{0.06}\right)^2} \)
  • \( \frac{1}{0.03} \approx 33.33\,m^{-1} \) and \((33.33)^2 \approx 1111 \)
  • \( \frac{1}{0.06} \approx 16.67\,m^{-1} \) and \((16.67)^2 \approx 278 \)
\( \sqrt{1111 + 278} = \sqrt{1389} \approx 37.26\,m^{-1} \)
\( \frac{3 \times 10^8}{2} = 1.5 \times 10^8\,m/s \)
\( f_{101} \approx 1.5 \times 10^8 \times 37.26 \approx 5.59 \times 10^9\,Hz \)

The resonant frequency in this case is approximately 5.59 GHz.

6. Common Applications

  • Microwave Filters: Selectively pass or block specific frequency bands in communication systems.
  • Oscillators: Generate stable high-frequency signals for radar and RF applications.
  • Sensors: Used in resonant sensors for precise environmental and material measurements.
  • Scientific Research: Investigate electromagnetic wave propagation in controlled environments.

7. Conclusion

The Air-Filled Rectangular Cavity Resonator Calculator for the TE₁₀₁ mode provides an efficient method to estimate the resonant frequency of a rectangular cavity based on its dimensions. By applying the formula:

\( f_{101} = \frac{c}{2} \sqrt{\left(\frac{1}{a}\right)^2 + \left(\frac{1}{d}\right)^2} \)

where \(c\) is the speed of light, \(a\) is the cavity width, and \(d\) is the cavity length (or depth), engineers and researchers can accurately design resonant structures for a variety of microwave and RF applications. Mastery of these calculations is key to developing efficient, high-performance components in advanced electronics.