# The Role of the Parallel Plate Waveguide Propagation Constant in TE Mode

### Key Takeaways

• The simplest type of waveguide is the parallel plate waveguide.

• Parallel plate waveguides consist of two parallel plates made from two conducting plates. Between these plates is a medium characterized by the refraction index or permittivity and permeability.

• The parallel plate waveguide solution is found by applying two boundary conditions to the general solution of a wave equation. Waveguides are a significant component in electromagnetic wave technology

Waveguides are a significant component in electromagnetic wave technology and are most often used for the low-loss transmission of electromagnetic power at high frequencies. These waveguides guide electromagnetic waves to their desired location. There are several kinds of waveguides including rectangular waveguides, circular waveguides, and parallel plate waveguides.

Parallel plate waveguides are promising electromagnetic components that support TE, TM, and TEM modes. Each of these modes satisfies Maxwell’s equation, and a solution is obtained by solving Maxwell’s equation with boundary conditions. The condition for a wave to propagate in a particular direction, say the z-direction, depends on the parallel plate waveguide propagation constant, frequency, permittivity, permeability, and mode of propagation.

The parallel plate propagation constant plays an essential role in designing the cut-off frequency and propagation mode in parallel plate waveguides. This constant is dependent on dielectric loss or conductor loss in the parallel plate waveguide. Let’s take a closer look at the influence of the parallel plate waveguide propagation constant.

## Defining the Parallel Plate Waveguide

A parallel plate waveguide is the simplest waveguide, consisting of two parallel plates made from two conducting plates. Between these plates, there is a medium characterized by the refraction index or permittivity and permeability. The geometry of a parallel plate waveguide is such that the width of the plate is greater than the separation between the plates. This helps prevent fringing fields and any other variations.

Parallel plate waveguides support three modes of electromagnetic wave propagation: transverse electric (TE) mode, transverse magnetic (TM), and transverse electromagnetic (TEM) mode. Understanding wave propagation through a parallel plate waveguide is essential to understanding how other waveguides function.

## How the Parallel Plate Waveguide Propagation Constant Influences TE Mode

Picture this: our parallel plates are infinite in the y-direction and the wave propagates in the z-direction. The electric field in the y-direction is Ey. The wave equation defines the electromagnetic wave propagation through the parallel plate waveguide: The partial derivative for y is zero and the derivatives for x and y are non-zero. So, the wave equation defining TE mode reduces to: The solution to equation (2) is given as the following equation, where ßx and ßz are the phase constant in x and z direction, respectively, and A=-B= E0 / 2j,  E0is an arbitrary constant: The parallel plate propagation constant ý is defined as à +jß, where à is the attenuation constant and ß is the phase constant. Assuming the parallel plates are conducting perfectly with an ideal medium sandwiched between the plates, there is no conductor loss or dielectric loss. This assumption reduces the propagation constant of the parallel plate waveguide into the phase constant. However, if the plates are lossy, replace  ß with ý.

## Applying Parallel Plate Waveguide Boundary Conditions

The following boundary conditions are applied to equation (3) to obtain the solution for the parallel plate waveguide:

1. x=0
2. x=a, where a is the distance between the two parallel plates.

The two solutions of Ey are obtained by applying the boundary conditions (a) and (b) to equation (3), which lead to equation (4): ## The Dispersion Relation and Conditions for Wave Propagation in a Parallel Plate Waveguide

According to equation (2), the dispersion relation can be given as equation (5): Equation (6) gives the phase constant in the z-direction: The wave propagation in a parallel plate waveguide follows z-direction when the phase constant is: The parallel plate waveguide propagation constant plays a crucial role in controlling the direction of wave propagation. It is critical to carefully design the cut-off frequency of a parallel plate waveguide and the different modes of propagation going through it. Cadence’s software can help design waveguides with specific cut-off frequencies and modes of propagation.