Transcript No Slide Title

Fields and Waves
Lesson 5.2
PLANE WAVE PROPAGATION
Lossless Media
Lale T. Ergene
Time Harmonic Fields
EM wave propagation involves electric and magnetic fields
having more than one component, each dependent on all
three coordinates, in addition to time.
e.g. Electric field

~
E( x, y, z, t )  Re E( x, y, z)e jt

instantaneous field
Valid for the other fields

vector phasor

  
D, H , B and their sources J , v
Maxwell’s Equations in Phasor Domain
time domain

  E  ρ /ε
v

dB
E  
dt

 H  0
~ ~
  E  v / 
~
~
  E   jH
~
 H  0
~ ~
~
  H  J  jE
remember


D  E


B  H

  D
 H  J 
t


J  E
Complex Permittivity
 ~
~
~
  H  (  j ) E  j (  j ) E

c
complex permittivity
 c   ' j ' '
For lossless medium σ=0 ε’’=0
εc =ε’=ε
Wave Equations (charge free)
~
2~
 E  E  0
2
 2   2 c
~
2 ~
 H  H  0
2
Homogenous wave equation for E~

propagation constant
Plane Wave Propagation in Lossless Media
There are three constitutive parameters of the medium: σ, ε, μ
If the medium is nonconducting σ=0 α=0 LOSSLESS
 2   2 
εc =ε’=ε
Wavenumber k
  k
2
2
k   
~
2 ~
 Ek E 0
~
~
2 H  k 2 H  0
2
(for a lossless medium)
Transverse Electromagnetic Wave
•Electric and magnetic fields that are perpendicular to each other
and to the direction of propagation
•They are uniform in planes perpendicular to the direction of
propagation
x

•At large distances from
E
physical antennas and
Direction of
ground, the waves can be
propagation
approximated as uniform
plane waves
z


H
y
E  Ex ( z, t )a x

H  H y ( z, t )a y
Transverse Electromagnetic Wave
Spatial
variation


of E and H at
t=0
Traveling waves
The Electric Field in phasor form (only x component)
~
d Ex
2 ~
 k Ex  0
2
dz
2
General solution of the differential equation
~
~
~
Ex ( z )  Ex ( z )  Ex ( z )  Ex0 e jkz  Ex0e jkz
Amplitudes (constant)
Uniform Plane waves
In general, a uniform plane wave traveling in the +z direction,
may have x and y components
~
~
~
E ( z )  Ex ( z )a x  E y ( z )a y
~
~
~
H ( z )  H x ( z )a x  H y ( z )a y
The relationship between them
~ 1
~

H  az  E

~
~
E  az  H
Do Problem 1
Intrinsic impedance (η) of a lossless medium
• Similar to the characteristic impedance (Z0) of a transmission line
• Defines the connection between electric and magnetic fields of
an EM wave

Phase velocity
wavelength

k


1

k  

2 u p
[m]


k
f
up 





 

[Ω]
[m/s]
If the medium is vacuum : up=3x108 [m/s], ηc=377 [Ω]
Do Problem 2
Electromagnetic Power Density

• Poynting Vector S , is defined
  
S  EH
[W/unit area]

S is along the propagation direction of the wave
Total power

P  A S  an dA

OR P  S A cos 
[m/s]
[W]
[W]
Average power density of the wave S av 

1
~ ~
Re E  H 
2

[W/m2]
Plane wave in a Lossless Medium
~
~
~
E ( z )  Ex ( z )a x  E y ( z )a y
~
E ( z )  ( Ex 0 a x  E y 0 a y )e  jkz
~
1
~ 1

H ( z )  a Z  E  (  E y 0 a x  Ex 0 a y )e  jkz


2
1
2
S av  a z
( Ex 0  E y 0 )
2
~2
E
[W/m2]
S av  a z
2
Do Problem 3