-
1
-
LOGO
[email protected]
Chapter 10
Wave Propagation
Part One
-
TYPES OF WAVES
Transverse:
Longitudinal:
-
An electromagnetic wave is a transverse wave because the
electric and magnetic
fields are both perpendicular to the direction in which the wave
travels.
an Electromagnetic wave, unlike a wave on a string or a sound
wave, does not
require a medium in which to propagate.
Electromagnetic waves can travel through a vacuum or a material
substance.
All electromagnetic waves move through a vacuum at the same
speed, speed of
light in a vacuum and is 3 x 108 m/s
-
5
Electromagnetic Spectrum:
-
6
-
7
The existence of EM waves, predicted by Maxwell's equations,
was
first investigated by Heinrich Hertz.
Waves are means of transporting energy or information.
In this chapter, our major goal is to solve Maxwell's equations
and
derive EM wave motion in the following media:
1-Free space ( 0 , , )
2- Lossless dielectrics ( 0 , , , or )
3-Lossy dielectrics ( 0 , , )
4-Good conductors ( , , ,or )
o o
o r o r
o r o r
o o r
-
8
characteristics of waves
A wave is a function of both space and time.
Characteristics of the wave:
1- It is time harmonic
2- A is called the amplitude of the wave
3- (wt- Bz) is the phase (in radians) of the wave;
it depends on time t and space variable z.
4- w is the angular frequency (rad/sec); B is the phase constant
or wave
number (in rad/meters).
( , ) sin( )E t z A t z
-
9
Wave repeat itself every λ
λ is the wavelength (in meters).
Wave repeat itself every T
T is the period (in seconds).
( , ) sin( )E t z A t z
-
10
The speed:
T = l/f,
f: is the frequency (the number of cycles per second) of the
wave in Hertz (Hz).
sec
sec
2 2
rad
rad
mu m
f
f
for every wavelength of distance traveled, a wave undergoes
a
phase change of 2pi radians.
Phase constant:
u fT
-
11
The phase constant (B) or wave number (rad/meters).
Consider a wave is traveling with
a velocity u in the +z direction.
Consider a fixed point P on the wave.
Sketch E(t,z) at times t = 0,T/4, T/2
Point P is a point of constant phase
( , ) sin( )E t z A t z
-
12
In free space, H = 0.1 cos (2 X 10^8 t - kx) ay A/m.
Calculate
(a) k, λ, and T
(b) The time t1 it takes the wave to travel λ/8
(c) Sketch the wave at time t1.
x
8
8
8
8
8
8
8
8
1
1
propgation direction is +a
(a)
ω=2×10 rad/sec
u=c=3×10 m/sec (free space)
ω 2×10k=β= = =0.67 rad/m
u 3×10
1 2πω=2πf=2×10 , T= = =31.42ns
f 2×10
λu , λ (3×10 )(31.42ns) 9.426m
(b)
λu
(λ / 8) (9.42 / 83×10
wave
T
T
tt
8
)3.9
3×10ns
-
13
8 9 yy
1
(c)
H=0.1cos(2 10 3.9 10 0.67 )
=0.1cos(0.78 0.67 )
for t=0 H=0.1cos(0 ) 0.1cos(0 0.67 )
for t=t H=0.1cos(0.78 ) 0.1cos(0.78 0.67 )
figure at t1 is the same as at t but shifted by 0.78=pi/4
x
x
x x
x x
a
a
-
14
WAVE PROPAGATION IN LOSSY DIELECTRICS
-
15
WAVE PROPAGATION IN LOSSY DIELECTRICS
A lossy dielectric is a medium in which an EM wave loses power
as it
propagates due to poor conduction
Consider a linear, isotropic, homogeneous, lossy dielectric
medium
Linear: ε doesn't change with the applied field.
isotropic: ε doesn't change with the direction of field.
Homogenous : ε doesn't change from point to point
-
16
E has only an x-component and it is traveling along the +z
direction
: is called the propagation constant in per meter
: attenuation costant (Np/m)or (dB/m)
: phase costant (rad/m)
2
1 12
2
1 12
-
17
l
1
Anattenuation of 1 neper denotes a reduction to e of the
original value whereas an increase
of 1 neper indicates an increase by a facto
neper
r
/
e
s
of
m
- E and H are out of phase by θη.
- E leads H (or H lags E) by θη.
-
18
The intrinsic impedance (in ohms) of the medium
(1 )j
tan(2 )
1/42/
| |
1
-
19
tan θ : is known as the loss tangent and θ is the loss angle
good conductor ,tan is very large
good insulator (lossless), tan is very small
-
20
-The characteristic behavior of a medium depends not only on
its
constitutive parameters ε,σ, and µ but also on the frequency
of
operation.
- A medium that is regarded as a good conductor at low
frequencies
may be a good dielectric at high frequencies.
Complex permittivity
' , '' =c j
-
21
PLANE WAVES IN Lossless Dielectrics
thus E and H are in time phase with each other.
(1 )j
=0 , , 0 , 0 o o
2
1 12
2
1 12
LOSSLESS DIELECTRICS
2
tan(2 )
-
22
PLANE WAVES IN free space
tan(2 )
thus E and H are in time phase with each other.
(1 )j
8 =0 , = = , =120 =377 , 0
3 10
oo o
oc
2
1 12
2
1 12
Free space
3
-
23
PLANE WAVES IN good conductor
thus E leads H by 45o.
(1 )j
= , 45 , 45 2
o o
2
1 12
2
1 12
Free space
4
tan(2 ) 2 90
1/4 1/22
/ /| |
1
-
24
2
2
8
8
2
2
8
8
1 12
8 2110 1 1
3 2 10 8
0.00364
1 12
8 2 0.0036410 1 1 1.37 rad/m
2 10 8
o o
o
o o
o
-
25
8
8
1/42
0.00364(b) loss tangent: tan = 0.515
10 8
2(c) intrinsic impedance: 177.72 13.6
0.00364(1 ) 8 (1 )
8 10
/: | | 177.72 , tan(2 ) 13.6
1
2( ) =
o
oo
o
o
o
j j
or
d
8710 7.278 10 m/s
1.37u
u
-
26
3oo
k E H
E H
3 /3 8
E 0.5(e) H 2.8 10
| | 177.72
=
,
2.8 10 sin(10 1.37 13.6 ) A/m
note:E leads H by 13.6
or H lags E by 13.6
x k z y
z o
y
o
o
H e t z
a a a
a a a a a a
a
-
27
8
2
8
88
8
(a)
(b) =0 , =0 , non-magnetic material 1
2=
23.142 m
2
2 10 16 MHz
2 1 1 102
10 1 10 6 36
3 10 3
k z
r
o r o
o o r r r r r
u
f f
a a
-
28
oo
8
(c) intrinsic impedance: 62.8
(1 )
| | 62.8 and 0
or tan(2 ) 0 0
E 50 H 0.8
| | 62.8
0.8 sin(10 2 ) A/m
r o
r o
o
x
j
H t z
a
-
29
-
30
1- Magnetic field lines must be closed on itself.vD
B 0
dBE
dt
dDH J
dt
Only, the first equation confirm the presence of static
electric
field only, and there is no magnetic field as the right
side the fourth equation equals zero.
B 0
From M(2nd equation) the magnetic charge is not exists so the
field lines
must be closed on itself (divergenless)
Explain the following statements using Maxwell’s equation
2- Only a static electric charges exists.
vD
H 0
-
31
the fourth equation confirm the presence of static magnetic
field only in the absence of the electric field as the right
side
of the first equation equal to zero.
3- Only constant electric current (J) exists.
D 0
H J
4- Only a varying electric charges .
From first equation:there are a time varying electric field
which
will produce a time varying magnetic field as illustrated in
4th
equation.This varying magnetic field will produce electric
field
As in 3rd
equation.
v
dD dBD H J E
dt dt
vD
B 0
dBE
dt
dDH J
dt
