Electromagnetic Waves in a Parallel-Plate Waveguide

​
mode
TM
TE
mode number n
0
1
2
3
4
5
channel width b (m)
0.25
0.5
0.75
1.
1.25
frequency f (GHz)
0.2
time in ω t (degree)
0
f >
f
c
= 0.15 (GHz) ⟶ wave passes through.
displayed fields
electric field
magnetic field
Poynting vector
(power density)
values
1000 V/m
2.654 A/m
2000 W/
2
m
displayed
0.2 m length
in red arrow
0.2 m length
in blue arrow
1 m length in
horizontal graph
Electromagnetic waves can propagate through a parallel-plate waveguide under appropriate conditions. This Demonstration determines the corresponding fields, energy distributions, and energy transport. The parallel plates support transverse magnetic (TM) and transverse electric (TE) waves. Specifying one of those modes, the mode number
n
, channel width
b
, and frequency
f
, the instantaneous fields and energy density distribution are determined for the designated time or phase
ωt
. The maximum electric field is fixed at 1000 V/m in all cases. The frequency has to be higher than the cut-off frequency
f
c
, which is determined by
b
and
n
. Taking the wave propagation direction as the
z
axis, the fields are function of
y
,
z
, and
t
. The energy flows along the channel (in the positive
z
direction).
The electric and magnetic fields are shown by red and blue arrows, respectively. The energy density is represented by color variation. The energy transport or power density equals the averaged Poynting vector, whose magnitude depends on the
y
position, as indicated by the curve on the right. The magnitudes of the fields are shown, as described in the table.

Details

Snapshot 1: the energy density has a maximum at the conductor surface in
TM
1
mode (TM mode with
n=1
)
Snapshot 2: the energy density oscillates three times along the
y
axis in
TM
3
mode
Snapshot 3: the directions of the electrical and magnetic fields mode are interchanged in
TE
3
mode
Periodic solutions of the wave equation satisfying the boundary conditions
E
z
(y=0)=
E
z
(y=b)=0
take the following forms, for
TM
n
modes:
E
y
=-
γb
nπ
A
n
cos
nπy
b
jωt-γz
e
,
E
z
=
A
n
sin
nπy
b
jωt-γz
e
,
H
x
=
jω
ϵ
0
b
nπ
A
n
cos
nπy
b
jωt-γz
e
,
where
ω
is the angular frequency,
ϵ
0
is the permittivity of air (approximately equal to its vacuum value), and
γ
is the propagation constant given by
γ=
2
nπ
b
-
2
ω
c
. The constant
γ
becomes pure imaginary if the frequency is higher than a certain value, the cut-off frequency. A similar solution is derived for TE modes. The propagation constant and the cut-off frequency are the same.
The energy density can be calculated by
W=
ϵ
0
2
2
E
+
μ
0
2
2
H
, where
E
and
H
are the instantaneous field values. The average Poynting vector is given by
S
ave
=
1
2
E
H
, which is always in the
z
direction.
In the case
n=0
,
TM
0
mode is constructed by
E
y
(≠0)
and
H
x
(≠0)
with
E
z
=0
. This means the transverse electromagnetic (TEM). However,
TE
0
mode does not exist. If this is selected, no fields are displayed.
It is possible to select the frequency below the cut-off frequency without error. If selected, it will be noted that the energy density is no longer periodic along the
z
axis, but decays with distance.
The electromagnetic field in parallel plate wave guides can be represented as the superposition of two plane waves reflected at the upper and lower conductor surfaces.

References

[1] D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Reading, MA: Addison-Wesley, 1989.

External Links

Electromagnetic Wave Incident on a Perfect Conductor

Permanent Citation

Y. Shibuya
​
​"Electromagnetic Waves in a Parallel-Plate Waveguide"​
​http://demonstrations.wolfram.com/ElectromagneticWavesInAParallelPlateWaveguide/​
​Wolfram Demonstrations Project​
​Published: December 2, 2012