# Applying Waveguide Boundary Conditions in Waveguide Analysis

### Key Takeaways

• For a wave to propagate through the waveguide, it needs to satisfy Maxwell’s equation and waveguide boundary conditions.

• Waveguide boundary conditions are:

• The tangential components of the electric field should be equal to zero.

• The normal derivative of the tangential component of the magnetic field should be equal to zero.

• There are six field components in rectangular waveguides: three electric fields and three magnetic fields. Waveguides are used in RF and microwave systems

Waveguides are the structures that carry electromagnetic waves between their endpoints. Waveguides can be rectangular, circular, or made of parallel plates, but, typically, they come in the form of a hollow metal pipe used to transport radiation of a single frequency. In waveguides, there are two modes of propagation: transverse electric (TE) mode and transverse magnetic (TM) mode. For a wave to propagate through a waveguide, it needs to satisfy Maxwell’s equation and waveguide boundary conditions.

## Waveguide Boundary Conditions

When electromagnetic waves travel in free space, they are attenuated, and a fair part of the energy transmitted gets lost while in propagation. By using waveguides, the energy loss during propagation can be avoided. Waves traveling through a waveguide are confined to the physical limits of the waveguide structure. There are two waveguide boundary conditions that need to be satisfied for waves to travel through waveguides

1. The tangential components of the electric field should be equal to zero.

2. The normal derivative of the tangential component of the magnetic field should be equal to zero.

## How to Conduct Waveguide Analysis

Waveguide analysis involves the calculation of electric and magnetic fields associated with the structure. Let's use a rectangular waveguide as an example of how to conduct waveguide analysis.

### Waveguide Analysis in a Rectangular Waveguide

Imagine waves are propagating through a rectangular waveguide and the direction of propagation is in the z-direction, or longitudinal direction. There are six field components in a rectangular waveguide—three electric fields (Ex, Ey, and Ez) and three magnetic fields (Hx, Hy, and Hz) along the x, y, and z-axis, respectively. Out of the six components, two are independent components and the other four are dependent components. The four dependent components can be obtained using Maxwell’s equation and waveguide boundary conditions on the independent components.

The longitudinal components Ez and Hz are the independent components. The transverse components of electric and magnetic fields in the terms of Ez and Hz can be given as the following, where h is the transverse propagation constant, 𝜇 is the permeability of the waveguide medium, 𝜀 is the permittivity of the waveguide medium, and β is the phase constant of the wave propagating through the waveguide in the z-direction. The first step of waveguide analysis is to obtain the longitudinal components—either electric field Ez or magnetic field Hz—depending on the mode of propagation inside the waveguide (transverse magnetic or transverse electric). Irrespective of the propagation mode, the longitudinal component should be consistent with the aforementioned boundary conditions. Once the independent component is obtained, the transverse components are calculated using equations 1-4, above. In a transverse magnetic field, Hz=0 and Ez≠0, therefore, the transverse components are calculated from Ez. In a transverse electric field, Ez=0 and Hz≠0, so the dependent components are calculated using the Hz field.

Let’s assume for a moment that a finite length rectangular waveguide is supporting transverse magnetic mode. To obtain Ez, the wave equation represented by equation 5 must be solved by the method of separation of variables, where Ez can be given in terms of the variables in equation 6. Substituting equation 6 into equation 5 gives us the final equation shown below. In equation 7, the constants C1, C2, C3, C4, and C5 are arbitrary constants that are to be evaluated using the boundary conditions. In this solution, it is assumed that the fields are sinusoidal with angular velocity 𝜔.

In the transverse magnetic mode of a rectangular waveguide, Ez is parallel to all four walls of the waveguide. Applying the boundary condition of the tangential component of the electric field is equal to zero to the rectangular waveguide structure. The boundary condition for Ez is obtained in equation 8.  After substituting the other boundary conditions, equations 9 and 10, the solution for Ez is obtained in the equation below, where C gives the amplitude of Ez. To obtain the transverse components of the electric and magnetic fields present in the transverse magnetic mode of propagation, where Hz=0, substitute equation 11 in equations 1-4. The dependent fields are obtained in equations 12-15, where m and n represent the order of the mode in TMmn. By applying waveguide boundary conditions, engineers can properly conduct waveguide analysis. Cadence’s software provides tools that technically support the design of waveguides in RF and microwave circuits

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