# Resonant Modes and the Lumped Element Model of Cylindrical Cavity Resonators

### Key Takeaways

• Cylindrical cavity resonators can be used to construct microwave transmitters, filters, amplifiers, band-pass filters, and oscillators.

• A cylindrical cavity resonator is usually described by the inner radius (Ro) and height (d) of the cylinder. As both the ends of the cylinder are short-circuited, the signals start oscillating inside the cavity with greater amplitudes.

• By moving the walls of a cylindrical cavity resonator inwards and outwards, the size of the cavity is changed, thereby tuning the frequency. Cavity resonator-based Magnetron tube

Cavity resonators are spaces enclosed by metal conductors that can confine electromagnetic waves. In microwave applications, oscillators, amplifiers, and filters are often constructed from cavity resonators. There are a variety of resonator types, but this article will focus on cylindrical cavity resonators. These types of resonators are found in applications such as microwave transmitters, band-pass filters, and receivers. Let’s take a closer look at how cylindrical cavity resonators work.

## How Do Cylindrical Cavity Resonators Work?

When a cavity is excited by signals from coaxial cables, those signals get reflected back and forth between the boundaries of the cavity. Currents in the walls build up standing electromagnetic waves that correspond to certain frequencies at specific resonant modes. Under various resonance conditions, electromagnetic energy is localized inside the cylindrical cavity.

When conducting walls are placed at both ends of a circular waveguide, it forms a hollow circular cylindrical cavity resonator. The cylindrical cavity resonator is usually described by the inner radius (Ro) and height (d) of the cylinder. As both ends of the cylinder are short-circuited, signals start oscillating inside the cavity with greater amplitudes. The cylindrical cavity ends are short-circuited by conductors

## The Lumped Element Model of a Cylindrical Cavity Resonator

Cavity resonators can be represented by equivalent lumped elements. In a cylindrical cavity resonator, the capacitance between the end walls carries the displacement current. The return current along the walls completes the circuit. The flow of real current induces inductance in the cavity.

The value of resonant frequency depends on the capacitance and inductance values in the cavity. Variations in cylindrical cavity geometry are responsible for the change in inductance and capacitance values within the cavity resonator.

By moving the walls of the cylindrical cavity resonator inwards and outwards, the size of the cavity is changed, thereby inducing frequency tuning, which varies the inductance and capacitance in the cavity. In summary, the geometric parameter variation results in the tuning of resonant frequency in a cylindrical cavity resonator. The displacement current and real current in the cavity walls of (a) one resonant LC circuit and (b) a cavity divided into 2 interacting resonant LC circuits

## Resonant Modes in Cylindrical Cavity Resonators

Resonant modes are natural electromagnetic oscillations in cavity resonators. The resonant mode is the localization of the energy density in the cylindrical cavity. The total energy density corresponding to each resonant mode is the contribution of the electric field and the magnetic field prevailing in the cylindrical cavity. Once a cavity is in resonance, it remains in that mode without any input energy. Features of electromagnetic oscillations in the cylindrical cavity can be determined from the following equations: E and B denote the electric field and magnetic flux density, respectively

### Electromagnetic Oscillations in the Cylindrical Cavity

In the cylindrical cavity, electromagnetic oscillations can be treated as two wave modes:

1. Transverse Magnetic (TM) - In this mode, magnetic fields are perpendicular to the direction of propagation in the cylindrical cavity.

2. Transverse Electric (TE) - In this mode, the electric field is perpendicular to the direction of propagation in the cylindrical cavity.

Each wave mode is associated with an azimuthal mode number, radial mode number, and longitudinal mode number, which are written as subscripts along with the mode. From the mode description, the distribution of electromagnetic fields E and H can be fully determined.

For example, in TM0n0, the azimuthal number and longitudinal number are both zero. A zero azimuthal number represents an azimuthally symmetrical mode and a zero value of the longitudinal number indicates that the electric field is constant in the longitudinal direction. The figure below indicates the various lowest order TE and TM resonant modes in a cylindrical cavity resonator. Lowest-order TE and TM resonant modes in a cylindrical cavity resonator

Cylindrical cavity resonators are widely used in microwave and RF circuits for the construction of transmitters, filters, amplifiers, and oscillators.  Depending upon the application requirements, the electromagnetic wave modes, either TM and TE, are selected in cavity resonators. By varying the geometry of the cylindrical cavity walls, the resonant frequency can be easily tuned. The geometric variation changes the values of capacitance and inductance in the cylindrical cavity and provides the required resonant frequency.