Prof. Ji Chen
Notes 15
Plane Waves
ECE 3317
1
Spring 2019
z
x
E
ocean

Introduction to Plane Waves
A plane wave is the simplest solution to Maxwell’s equations for a wave that
travels through free space.
The wave does not requires any conductors – it exists in free space.
A plane wave is a good model for radiation from an antenna, if we are far
enough away from the antenna.
x
z
E
H
S
2
S
E
H

The Electromagnetic Spectrum
3
0
/
c
f

Source
Frequency
Wavelength
U.S.
AC Power
60 Hz
5000 km
ELF Submarine
Communications
500
Hz
600 km
AM radio (KTRH)
740
kHz
405 m
TV ch. 2 (VHF)
60 MHz
5 m
FM radio (Sunny 99.1)
99.1
MHz
3 m
TV ch. 8 (VHF)
180
MHz
1.7 m
TV ch. 39 (UHF)
620
MHz
48 cm
Cell phone (PCS)
850
MHz
35 cm
Cell Phone (PCS 1900)
1.95 GHz
15 cm
μ-
wave oven
2.45 GHz
12 cm
Police radar (X-band)
10.5 GHz
2.85 cm
mm wave
100 GHz
3 mm
Light
5
10
14
[Hz]
0.60
m
X-ray
10
18
[Hz]
3
Å
The Electromagnetic Spectrum (cont.)
4
0
/
c
f

TV and Radio Spectrum
VHF TV:
55-216 MHz (channels 2-13)
Band I : 55-88 MHz (channels 2-6)
Band III: 175-216 MHz (channels 7-13)
FM Radio:
(Band II) 88-108 MHz
UHF TV:
470-806 MHz (channels 14-69)
AM Radio:
520-1610 kHz
Note: Digital TV broadcast takes place primarily in UHF and VHF Band III.
5

Vector Wave Equation
Start with Maxwell’s equations in the phasor domain:
Faraday’s law
Ampere’s law
Assume
free space
:
We then have
Ohm’s law:
6
E
H
H
J
E
j
j
J
E = 0
0
0
E
H
H
E
j
j
0
0
,

Vector Wave Equation (cont.)
Take the curl of the first equation and then substitute the
second equation into the first one:
Vector wave equation
Define:
Then
Wavenumber of free space [
rad/m
]
7
0
0
0
E
H
E
j
j
j
0
0
0
k
2
0
E
E
0
k
0
0
E
H
H
E
j
j

Vector Helmholtz Equation
Recall the vector Laplacian identity:
Hence we have
Also, from the divergence of the vector wave equation, we have:
8
Note:
There can be no charge density
in the sinusoidal steady state, in
free space (or actually in any
homogenous region of space).
2
0
E
E
0
k
2
V
V
V
2
2
0
E
E
E
0
k
2
0
E
E
0
k
E
0
0
0
1
1
E
D
0
v
2
0
E
E
0
k

Hence we have:
Vector Helmholtz equation
Recall the property of the vector Laplacian in rectangular coordinates:
Taking the
x
component of the vector Helmholtz equation, we have
Vector Helmholtz Equation (cont.)
Scalar Helmholtz equation
9
Reminder: This only works in
rectangular coordinates.

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