Aim: Study of field patterns of various modes inside a rectangular cavity
This gives the basic idea of the field pattern; that is, electric and magnetic field patterns for various
modes inside a rectangular cavity resonator. One can observe the field patterns of various modes in xy,
xz and yz planes for different frequency bands. Surface current density can also be observed on the walls
of a rectangular cavity resonator.
Field Pattern for dominant mode (TE101) along xy plane
About the experiment:
This experiment provides the field patterns of various modes inside a rectangular cavity. The conducting
walls of the cavity confine the electromagnetic fields inside the structure and hence the cavity acts as a
resonator. A number of distinct field configurations or modes can exist in cavities. In rectangular cavity,
modes are designated as TEmnp or TMmnp , where m, n, p are the number of half wave variations in x, y, z
directions respectively. In this experiment, you can get better understanding of how the field patterns
vary with the parameters m, n and p for Transverse Electric (TE) and Transverse Magnetic (TM) modes in
xy, yz and xz planes for different frequency bands. The surface current density plot for the TE and TM
modes can also be observed on the walls of the rectangular cavity. The figure below shows the planes of
a rectangular cavity.
In this experiment, a rectangular cavity of dimension x=a, y =b and z=d has been considered. The
dimensions of the cavity depend on the frequency band in which we are observing the field pattern. For
example, in X Band (8-12GHz), the U.S. standard waveguide WR-90 has inner width of 2.286 cm('a'), an
inner height of 1.106 cm ('b') and the dimension in z direction, d is usually an odd multiple of the guide
wavelength (g).
Theory
Electromagnetic waves propagating in open space travel out in all directions. As we know the waveguide
operates by confining the electromagnetic wave inside a metallic structure so that it does not spread
out, and losses resulting from this effect are eliminated. By definition, a resonant cavity is any space
completely enclosed by conducting walls that can contain oscillating electromagnetic fields and possess
resonant properties. Signals propagate within the confines of the metallic walls that act as boundaries.
The signal is confined by total internal reflection from the walls of the cavity. Resonant cavities have a
very high Q and can be built to handle relatively large amounts of power. They are used principally at
frequencies in the microwave range and can act as a resonator above a certain frequency, known as the
cut-off frequency. This cut-off frequency of the cavity depends upon its dimensions.
Rectangular Cavity
A rectangular cavity is a hollow metallic tube with a rectangular cross section. It can be simply described
as a rectangular waveguide which is shorted at both ends. The conducting walls of the waveguide
confine the electromagnetic fields and hence standing waves are created which leads to resonant
phenomenon. The rectangular cavity is basically characterized by its dimensions i.e., length d, breadth a
and height b.
Modes: Like waveguides, cavities are also analyzed by solving Maxwell's equations, or their reduced
form, the electromagnetic wave equation, with boundary conditions determined by the properties of
the materials and their interfaces. These equations have multiple solutions, or modes, which are eigenfunctions of the equation system. Each mode is therefore characterized by an eigen-value, which
corresponds to a cutoff frequency below which the mode cannot exist in the guide.
These resonant modes depend on the operating wavelength and the shape and size of the cavity. The
modes of the cavity are typically classified into following types:
TE modes (Transverse Electric) have no electric field component in the direction of propagation.
TM modes (Transverse Magnetic) have no magnetic field component in the direction of propagation.
Field Theory
As we know, an electromagnetic field is comprised of electric and magnetic fields which are
perpendicular to each other. These fields have different patterns for each mode. These patterns depend
upon the mode numbers (m, n and p) and the dimensions (a, b and d) of the cavity. The electric field and
magnetic field pattern are different for various modes in different cavities .The electric field component
of an EM wave is characterized by Ex, Ey and Ez components of the wave.
Similarly the magnetic field component of an EM wave is characterized by Hx, Hy and Hz components of
the wave.
For TEmnp mode, the field equations for a rectangular cavity are:
0 cos(
=
= 0
) sin(
) cos(
) cos(
= 2 0 cos(
= 0 cos(
) cos(
) sin(
) sin(
0 sin(
= ( 2 )
1
) sin(
0 sin(
) cos(
) sin
)
)
)
cos(
)
2
Where m=0,1,2, n=0,1,2., p=1,2,3 and m=n 0 and 2 =
For TMmnp mode, the field equations for a rectangular cavity are:
1
= ( 2 )
1
= 2 0 sin(
= 0 sin(
0
2
) sin(
) cos(
cos(
) sin(
) cos
) cos(
sin(
0
2
=
= 0
) sin(
cos(
) cos(
) sin(
) cos(
sin(
Where m=1,2, n=1,2., p=0,1,2,3 and 2 =
The resonant frequency for TEmnp and TMmnp modes is same and is given by
( ) =
+ +
Procedure
Please download the file shown on the left to perform the actual experiment.
Step 1: Select the frequency band in which you wish to see the field pattern.
Enter the frequency in GHz
Step 2: Select the type of mode, i.e., either Tranverse Electric (TE) or Transverse Magnetic (TM).
Select Mode (TE/TM)
Step 3: Select Pattern:
i) Electric Field: Select this to view the electric field pattern of the given mode.
ii) Magnetic Field : Select this to view the magnetic field pattern of the given mode.
iii) Surface Current: Select this option to view the surface current density for TE10 mode.
Select pattern
Step 4: Enter the values of m, n and p to obtain the field pattern, where m stands for no. of half waves
of electric or magnetic intensity in the X- direction, n stands for number of half waves in the y direction
and p stands for number of half waves in the z direction.
Step 5: Run the VI up to see the desired field pattern in XY, YZ and XZ planes. In case, you wish to see the
other field pattern then click stop and repeat steps 1-4 before running the program again.
You may see the following example for your reference, where appropriate buttons are selected in order
to observe the electric field pattern of TE101 mode in X-band:
View the pattern
XY plane
YZ plane
XZ plane
