# Basic Rectangular Waveguide Theory

### Key Takeaways

• Usually, a basic waveguide can be constructed from a hollow conducting tube. If the conducting tube is rectangular in shape, then it forms a rectangular waveguide.

• Rectangular waveguides are extensively used in radars, couplers, isolators, and attenuators to transmit signals.

• The electromagnetic fields corresponding to (m,n) are called TEmn mode. The TE10 mode is the dominant waveguide in rectangular waveguides.

In satellite systems, waveguides are used to transmit electromagnetic signals

In satellite systems, high precision test applications, or high power microwave and radiofrequency systems, waveguides are used to transmit electromagnetic signals. There are various types of waveguiding structures available for signal transmissions, including metallic waveguides, dielectric waveguides, parallel-plate waveguides, and rectangular waveguides.

Rectangular waveguides are the earliest waveguiding structure utilized for transporting signals. This waveguide supports transverse electric mode (TE) and transverse magnetic mode (TM), but not transverse electromagnetic mode (TEM). This article discusses basic rectangular waveguide theory, starting with the geometry, modes of propagation, and cut-off frequency.

## Rectangular Waveguide Theory

In electromagnetics, a waveguide confines electromagnetic signals within the structure, preventing spreading, losses, and signal transmission from one point to another. Usually, a basic waveguide can be constructed from a hollow conducting tube. If the conducting tube has a rectangular cross-section, then it forms the rectangular waveguide. In the section below, we will discuss various aspects of rectangular waveguide theory.

### The Structure of a Rectangular Waveguide

Rectangular waveguides are the most commonly used waveguides. They consist of a hollow metallic structure with a rectangular cross-section. A rectangular waveguide is usually constructed with a length of a > b, where b is the breadth of the rectangle. A common trend for the dimension of a rectangular waveguide is a=2b.

The advantages of rectangular waveguides include:

• Wide frequency bandwidth for single-mode propagation

• Low attenuation

• Excellent mode stability for fundamental propagation modes

### Modes of Propagation

Rectangular waveguides are extensively used in radars, couplers, isolators, and attenuators for signal transmission. When electromagnetic waves are transmitted longitudinally through a rectangular waveguide, they are reflected from the conducting walls. The total reflection inside the rectangular waveguide results in either an electric field or magnetic field component in the direction of the propagation. There is no TEM mode in rectangular waveguides. The modes of propagation in a hollow rectangular waveguide with only one conductor are either TE or TM modes.

#### Transverse Electric Mode

In the TE mode of electromagnetic wave propagation, the electric field is transverse to the direction of propagation; however, in the magnetic field, it is not transverse. There is an infinite number of solutions for the magnetic fields corresponding to this mode from the wave equation. The solution of magnetic fields can be given by equation (1), where m=0,1,2 and n=0,1,2 but m≠n. The z component of the wave vector is kz. The other fields are determined from this solution using Maxwell's equations.

The electromagnetic fields corresponding to (m,n) are called TEmn mode. The TE10 mode is the dominant waveguide in the rectangular waveguide.

#### Transverse Magnetic Mode

In the TM mode of electromagnetic wave propagation, the magnetic field is transverse to the direction of propagation; however, the electric field is not transverse. The generalized solution of the electric field in the TM mode is given by equation (2), where m=0,1,2 and n=0,1,2.

The electromagnetic fields corresponding to (m,n) in this mode are called TMmn mode. There are infinite TEmn modes in rectangular waveguides.

### Cut-Off Frequency

A rectangular waveguide propagates signals only above a certain frequency, called the cut-off frequency. Below the cut-off frequency, there is no propagation in a rectangular waveguide. The cut-off frequency defines wave propagation modes in the rectangular waveguide, and this frequency is dependent on the dimensions of the waveguide.

The mode of propagation with the lowest cut-off frequency is called dominant mode and TE10 corresponds to the lowest cut-off frequency in the rectangular waveguide. In a rectangular waveguide, equation (3) gives the cut-off frequency for TEmn mode and TMmn mode. However, the condition m=0 or n=0 cannot be applied to TMmn mode cut-off frequency calculations.

A solid understanding of rectangular waveguide theory is essential to understanding other complex waveguides. Cadence’s software can help you design all types of waveguides, including rectangular waveguides.