Boundry Conditions

Boundry Conditions in Waveguides

The travel of energy down a waveguide is similar, but not identical, to the travel of electromagnetic waves in free space. The difference is that the energy in a waveguide is confined to the physical limits of the guide. Two conditions, known as BOUNDRY CONDITIONS , must be satisfied for energy to travel through waveguides.

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The first of boundary conditions (illustrated in the figure below view A) can be stated as follows:

For an electric field to exist at the surface of a conductor it must be perpendicular to the conductor

E field boundary conditions MEETS BOUNDARY CONDITIONS.

The opposite of these boundary conditions, shown in the figure below view B, is also true. An electric field CANNOT exist parallel to a perfect conductor.

E field boundary conditions DOES NOT MEET BOUNDARY CONDITIONS.

The second boundary conditions, which are illustrated in the next figure beow, can be stated as follows:

For a varying magnetic field to exist, it must form closed loops in parallel with the conductors and be perpendicular to the electric field.

H field boundary conditions.

Since an E field causes a current flow that in turn produces an H field, both fields always exist at the same time in a waveguide. If a system satisfies one of these boundary conditions, it must also satisfy the other since neither field can exist alone.


Electromagnetic energy transmitted into space consists of electric and magnetic fields that are at right angles (90 degrees) to each other and at right angles to the direction of propagation. A simple analogy to establish this relationship is by use of the right-hand rule for electromagnetic energy, based on the POYNTING VECTOR. It indicates that a screw (right-hand thread) with its axis perpendicular to the electric and magnetic fields will advance in the direction of propagation if the E field is rotated to the right (toward the H field). This rule is illustrated in the figure below.

The Poynting vector.

The combined electric and magnetic fields form a wavefront that can be represented by alternate negative and positive peaks at half-wavelength intervals, as illustrated in the next figure below. Angle " is the direction of travel of the wave with respect to some reference axis.

Wave fronts in space.

If a second wavefront, differing only in the direction of travel, is present at the same time, a resultant of the two is formed. The resultant is illustrated in the figure below, and a close inspection reveals important characteristics of combined wavefronts.

Both wavefronts add at all points on the reference axis and cancel at half-wavelength intervals from the reference axis. Therefore, alternate additions and cancellations of the two wavefronts occur at progressive half-wavelength increments from the reference axis. In the figure, the lines labeled A, C, F, and H are addition points, and those labeled B, D, E, and G are cancellation points

Combined wave fronts.

If two conductive plates are placed along cancellation lines D and E or cancellation lines B and G, the first boundary condition for waveguides will be satisfied; that is, the E fields will be zero at the surface of the conductive plates. The second boundary conditions are, therefore, automatically satisfied. Since these plates serve the same purpose as the "b" dimension walls of a waveguide, the "a" dimension walls can be added without affecting the magnetic or electric fields.

When a quarter-wavelength probe is inserted into a waveguide and supplied with microwave energy, it will act as a quarter-wave vertical antenna. Positive and negative wavefronts will be radiated, as shown in the figure below. Any portion of the wavefront traveling in the direction of arrow C will rapidly decrease to zero because it does not fulfill either of the required boundary conditions.

The parts of the wavefronts that travel in the directions of arrows A and B will reflect from the walls and form reverse-phase wavefronts. These two wavefronts, and those that follow, are illustrated in the last three figures below. Notice that the wavefronts crisscross down the center of the waveguide and produce the same resultant field pattern that was shown in the figure above.

Radiation from probe placed in a waveguide.

Wave fronts in a waveguide.

Wave fronts in a waveguide.

Wave fronts in a waveguide.

We will continue our discussion on the reflection of a single wavefront off the "b" wall of a waveguide in the next tutorial.

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