Transcript No Slide Title
Fields and Waves Lesson 5.3
PLANE WAVE PROPAGATION Lossy Media Lale T. Ergene
Wave Equations for a Conducting Medium 2 ~
E
2 ~
E
0 2 ~
H
2 ~
H
0 Homogenous wave equation for ~
E
Homogenous wave equation for ~
H
; propagation constant is complex
2
c
' '
( '
j
' ' )
Propagation Constant
j
Attenuation constant Phase constant 2 ' 1
' ' ' 2 1
[Np/m]
(for a lossy medium) 2 ' 1
' ' ' 2 1
[rad/m]
Solution of the Wave Equation The Electric Field in phasor form (only x component) 2 ~
d E x dz
2
2 ~
E x
0 General solution of the differential equation for a lossy medium ~ ( )
x
~
E x
~
E x
E e x
0 ( )
z
E e x
0 ( )
z
forward traveling in +z direction backward traveling in -z direction
Intrinsic Impedance,
η c
The relationship between electric and magnetic field phasors is the same but the intrinsic impedance of lossy medium,
η c
is different If +z is the direction of the propagation ~
E
c a
z
intrinsic impedance
~
H
c
~
H
1
c a
z
~
E
c
' 1
j
' ' ' /
Skin Depth, δ s shows how well an electromagnetic wave can penetrate into a conducting medium Skin Depth
s
1 [m] Perfect dielectric: σ=0 α=0 δ s =∞ Perfect Conductor: σ=∞ α=∞ δ s =0
Low-Loss Dielectric defined when ε’’/ε’<<1 practically if ε’’/ε’<10 -2, dielectric the medium can be considered as a low-loss 2 ' '
'
2
[Np/m] [rad/m]
'
c
[ Ω]
Good Conductor defined when ε’’/ε’>>1 practically if ε’’/ε’>100 , the medium can be considered as a good conductor 2 ' ' 2
[Np/m] [rad/m]
c
j
' '
j
)
j
)
[ Ω]
•When 10-2≤ ε’’/ε’ ≤100, the medium is considered as a “Quasi-Conductor”.
Do Problem 1
Average Power Density ~ ( ) ~
E x
~ ( ) (
E a x
0
x
x
~
E y
( )
y y
0
y
) )
z
~ ( ) 1
c a
Z
~
E
1
c
(
E a
y
0
x
x
0
y
)
z e
Average power density
S av
1 2 Re ~
E
~
H
a
z
1 2 (
E x
0 2
E y
0 2 )
e
2
z
Re 1
c
[W/m 2 ]
Average Power Density If η c is written in polar form
c
c e j
Average power density where
S av
a
z E
0 2
c
2
e
2
z
cos
E
0
E x
0 2
E y
0 2
Do Problem 2
[W/m 2 ]