Module 11 − Microwave Principles
...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.
Figure 1-12 - E fields on a two-wire line with half-wave frames.
Figure 1-13 - E field of a voltage standing wave across a 1-wavelength
section of a waveguide.
Figure 1-14 - Magnetic field on a single wire.
Figure 1-15 - Magnetic field on a coil.
Figure 1-16A - Magnetic fields on a two-wire line with half-wave
Figure 1-16B - Magnetic fields on a two-wire line with half-wave
Figure 1-17 - Magnetic field pattern in a waveguide.
Figure 1-18 - Magnetic field in a waveguide three half-wavelengths
Figure 1-21 - The Poynting vector.
Figure 1-22 - Wavefronts in space.
Figure 1-23 - Combined wavefronts.
Figure 1-24 - Radiation from probe placed in a waveguide.
Figure 1-25 - Wavefronts in a waveguide.
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
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
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:
Figure 1-19A - E field boundary condition. MEETS BOUNDARY CONDITIONS.
Figure 1-19B - E field boundary condition. DOES NOT MEET BOUNDARY
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
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
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
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
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Posted July 28, 2021