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Module 11 - Microwave Principles: Waveguide Boundary Conditions

Navy Electricity and Electronics Training Series (NEETS)
Chapter 1:  Pages 1-11 through 1-20

Module 11 - Microwave Principles

Pages i, 1-1, 1-11, 1-21, 1-31, 1-41, 1-51, 1-61, 2-1, 2-11, 2-21, 2-31, 2-41, 2-51, 2-61, 3-1, 3-11, AI-1, Index-1, Assignment 1, Assignment 2


...Pages 1-1 through 1-10

The two-wire transmission line, illustrated in figure 1-11B, has an instantaneous standing wave of voltage applied to it by the generator. The line is short-circuited at one-wavelength, at the positive and negative voltage peaks, but the arrows, representing each field, point in opposite directions. The voltage across the line varies sinusoidally. Therefore, the density of the E-lines varies sinusoidally.

The development of the E field in a waveguide can be illustrated by a two-wire transmission line separated by several, double quarter-wave sections, called half-wave frames, as illustrated in figure 1-12. As shown, the voltage across the two-wire line varies in a sine-wave pattern and the density of the E field also varies in a sine-wave pattern. The half-wave frames located at high-voltage points (1) and (3) have a strong E field. The frames at the zero-voltage points (2) have no E fields present. Frame (4) has a weak E field and is located at a point between maximum and minimum voltage. This illustration is a buildup to the three-dimensional aspect of the full E field in a waveguide.

E fields on a two-wire line with half-wave frames - RF Cafe

Figure 1-12 - E fields on a two-wire line with half-wave frames.

E field of a voltage standing wave - RF Cafe 

Figure 1-13 - E field of a voltage standing wave across a 1-wavelength section of a waveguide.

Magnetic field on a single wire - RF Cafe 

Figure 1-14 - Magnetic field on a single wire. 

Magnetic field on a coil 

Figure 1-15 - Magnetic field on a coil.  

Magnetic fields on a two-wire line with half-wave frames - RF Cafe 

Figure 1-16A - Magnetic fields on a two-wire line with half-wave frames. 

Magnetic fields on a two-wire line - RF Cafe 

Figure 1-16B - Magnetic fields on a two-wire line with half-wave frames.  

Magnetic field pattern in a waveguide - RF Cafe 

Figure 1-17 - Magnetic field pattern in a waveguide. 

Magnetic field in a waveguide three half-wavelengths long - RF Cafe 

Figure 1-18 - Magnetic field in a waveguide three half-wavelengths long. 

The Poynting vector - RF Cafe 

Figure 1-21 - The Poynting vector. 

Wavefronts in space - RF Cafe 

Figure 1-22 - Wavefronts in space. 

Combined wavefronts - RF Cafe 

Figure 1-23 - Combined wavefronts.  

Radiation from probe placed in a waveguide - RF Cafe 

Figure 1-24 - Radiation from probe placed in a waveguide. 

Wavefronts in a waveguide - RF Cafe


Wavefronts in a waveguide - RF Cafe


Wavefronts in a waveguide - RF Cafe


Figure 1-25 - Wavefronts in a waveguide.

Reflection of a single wavefront - RF Cafe 

Figure 1-26. - Reflection of a single wavefront.

Figure 1-13, view (A), shows the E-field pattern created by a voltage sine wave applied to a one-wavelength section of waveguide shorted at one end. The electric fields are represented by the arrows shown in views (B) and (C). In the top view of view (A), the tip of each arrow is represented by a dot and the tail of each arrow is represented by an X. The E field varies in density at the same sine-wave rate as the applied voltage. This illustration represents the instant that the applied voltage wave is at its peak. At other times, the voltage and the E field in the waveguide vary continuously from zero to the peak value. Voltage and E-field polarity reverse with every reversal of the input. Note that the end view shown in view (B) shows the E field is maximum at the center and minimum near the walls of the waveguide. View (C) shows the arrangement of electromagnetic fields within a three-dimensional waveguide.

H Field

The magnetic field in a waveguide is made up of magnetic lines of force that are caused by current flow through the conductive material of the waveguide. Magnetic lines of force, called H lines, are continuous closed loops, as shown in figure 1-14. All of the H lines associated with current are collectively called a magnetic field or H field. The strength of the H field, indicated by the number of H lines in a given area, varies directly with the amount of current.

Although H lines encircle a single, straight wire, they behave differently when the wire is formed into a coil, as shown in figure 1-15. In a coil the individual H lines tend to form around each turn of wire. Since the H lines take opposite directions between adjacent turns, the field between the turns is cancelled. Inside and outside the coil, where the direction of each H field is the same, the fields join and form continuous H lines around the entire coil.

A similar action takes place in a waveguide. In figure 1-16A, a two-wire line with quarter-wave sections is shown. Currents flow in the main line and in the quarter-wave sections. The current direction produces the individual H lines around each conductor as shown. When a large number of sections exist, the fields cancel between the sections, but the directions are the same both inside and outside the waveguide. At half-wave intervals on the main line, current will flow in opposite directions. This produces H-line loops having opposite directions. In figure 1-16A, current at the left end is opposite to the current at the right end. The individual loops on the main line are opposite in direction. All around the framework they join so that the long loop shown in figure 1-16B is formed. Outside the waveguide the individual loops cannot join to form a continuous loop. Thus, no magnetic field exists outside a waveguide.

If the two-wire line and the half-wave frames are developed into a waveguide that is closed at both ends (as shown in figure 1-16B), the distribution of H lines will be as shown in figure 1-17. If the waveguide is extended to 1 1/2λ, these H lines form complete loops at half-wave intervals with each group reversed in direction. Again, no H lines can form outside the waveguide as long as it is completely enclosed.

Figure 1-18 shows a cross-sectional view of the magnetic field pattern illustrated in figure 1-17. Note in view (A) that the field is strongest at the edges of the waveguide where the current is highest. The minimum field strength occurs at the zero-current points. View (B) shows the field pattern as it appears λ/4 from the end view of the waveguide. As with the previously discussed E fields, the H fields shown in figures 1-17 and 1-18 represent a condition that exists at only one instant in time. During the peak of the next half cycle of the input current, all field directions are reversed and the field will continue to change with changes in the input.

Boundary Conditions in an Waveguide

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 BOUNDARY CONDITIONS, must be satisfied for energy to travel through a waveguide.

The first boundary condition (illustrated in figure 1-19A) can be stated as follows:

E field boundary condition. MEETS BOUNDARY CONDITIONS - RF Cafe 

Figure 1-19A - E field boundary condition. MEETS BOUNDARY CONDITIONS.  

E field boundary condition. DOES NOT MEET BOUNDARY CONDITIONS - RF Cafe 

Figure 1-19B - E field boundary condition. DOES NOT MEET BOUNDARY CONDITIONS.  

H field boundary condition - RF Cafe 

Figure 1-20 - H field boundary condition.

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

The opposite of this boundary condition, shown in figure 1-19B, is also true. An electric field CANNOT exist parallel to a perfect conductor.

The second boundary condition, which is illustrated in figure 1-20, 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.

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. You should briefly review the principles of electromagnetic propagation in free space (NEETS, Module 10, Introduction to Wave Propagation, Transmission Lines, and Antennas). This review will help you understand how a waveguide satisfies the two boundary conditions necessary for energy propagation in a waveguide.

Wavefronts Within a Waveguide

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 figure 1-21.

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 figure 1-22. Angle " is the direction of travel of the wave with respect to some reference axis.

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 figure 1-23, 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 figure 1-23, the lines labeled A, C, F, and H are addition points, and those labeled B, D, E, and G are cancellation points.

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 condition is, 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 figure 1-24. 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 figure 1-25. Notice that the wavefronts crisscross down the center of the waveguide and produce the same resultant field pattern that was shown in figure 1-23.

The reflection of a single wavefront off the "b" wall of a waveguide is shown in figure 1-26. The wavefront is shown in view (A) as small particles. In views (B) and (C) particle 1 strikes the wall and is bounced back from the wall without losing velocity. If the wall is perfectly flat, the angle at which it strikes the wall, known as the angle of incidence ("), is the same as the angle of reflection (ø) and are measured perpendicular to the waveguide surface. An instant after particle 1 strikes the wall, particle 2 strikes the wall, as shown in view (C), and reflects in the same manner. Because all the particles are traveling at the same velocity, particles 1 and 2 do not change their relative position with respect to each other. Therefore, the reflected wave has the same shape as the original. The remaining particles as shown in views (D), (E) and (F) reflect in the same manner. This process results in a reflected wavefront identical in shape, but opposite in polarity, to the incident wave.

Figures 1-27A and 1-27B, each illustrate the direction of propagation of two different electromagnetic wavefronts of different frequencies being radiated into a waveguide by a probe. Note that only the direction of propagation is indicated by the lines and arrowheads. The wavefronts are at right angles to the direction of propagation. The angle of incidence (") and the angle of reflection (ø) of the wavefronts vary in size with the frequency of the input energy, but the angles of reflection are equal to each other in a waveguide. The CUTofF Frequency in a waveguide is a frequency that would cause angles of incidence and reflection to be zero degrees. At any frequency below the cutoff frequency, the wavefronts will be reflected back and forth across the guide (setting up standing waves) and no energy will be conducted down the waveguide.

...Pages 1-21 through 1-30

NEETS Modules
- Matter, Energy, and Direct Current
- Alternating Current and Transformers
- Circuit Protection, Control, and Measurement
- Electrical Conductors, Wiring Techniques, and Schematic Reading
- Generators and Motors
- Electronic Emission, Tubes, and Power Supplies
- Solid-State Devices and Power Supplies
- Amplifiers
- Wave-Generation and Wave-Shaping Circuits
- Wave Propagation, Transmission Lines, and Antennas
- Microwave Principles
- Modulation Principles
- Introduction to Number Systems and Logic Circuits
- - Introduction to Microelectronics
- Principles of Synchros, Servos, and Gyros
- Introduction to Test Equipment
- Radio-Frequency Communications Principles
- Radar Principles
- The Technician's Handbook, Master Glossary
- Test Methods and Practices
- Introduction to Digital Computers
- Magnetic Recording
- Introduction to Fiber Optics
Note: Navy Electricity and Electronics Training Series (NEETS) content is U.S. Navy property in the public domain.

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