Transcript 6. Electromagnetic Waves

Electromagnetic
Waves
EEL 3472
Electromagnetic Waves
Spherical
Wavefront
Direction of
Propagation
Plane-wave
approximatio
n
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Electromagnetic Waves
Electromagnetic Waves
 2 E  j ( E  j ) E
In the case of E   (fields inside a good insulator such
as air or vacuum) we have
vector Helmholtz
 2 E   2  E  0
equation
This equation has a rich variety of solutions.
Let us assume that E has only an x component and varies
only in the z direction. In this case the vector Helmholtz
equation simplifies to
scalar Helmholtz
 2 Ex
2



E

0
x
z 2
equation
The latter equation is similar to the voltage wave equation
for a lossless transmission line.
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The solution of the scalar Helmholtz equation is
E x  E o e  j
 z
k
 E0e z
j o
where E 0  A0e
, to be found from boundary conditions;
the minus and the plus correspond to waves moving in the
+z and –z directions, respectively.
Ex ( z, t )  Re[ E o e  j
 z
e jt ]  Ao cos(t    z  o )
In terms of propagation constant k
k   
Ex ( z, t )  Ao cos(t  kz  o )
The position of a field maximum is given by
t   zmax  o  0  zmax 
Up 
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dzmax
1

dt

t
o

  
If   o ,   o
Up  c  1 / o o  2.99793x108 m / s
EEL 3472
Electromagnetic Waves
Characteristics of Plane Waves
For a plane wave which propagates in the +z direction and
has an electric field directed in the x direction
E  E oe jkz ex
where k     2 / 
The time-varying electric field of the wave must, according
to Faraday’s law, be accompanied by a magnetic field.
Thus,
Ex
 jH    E 
ey   jkEoe jkz ey
z
dB
E 0  jkz

H
e ey
dt

wave impedance (intrinsic


impedance of the medium

when   o ,   o   377 and is called the characteristic
impedance of vacuum.
 
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If Eo  A oe j
o
E  A o cos(t  kz  o )ex
H
Ao
cos(t  kz  o )ey

1. E , H , and the direction of propagation are all
perpendicular to each other.
2. E and H are in phase
3. The wave appears to move in the +z direction as though
it were a rigid structure moving toward the right
(in exactly the same way as the transmission-line waves).
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0


E x ( z, t )  1
.5 cos(t  kz  )
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Ao
j

E x  1
.5
e 4 e  jkz
E0
(t  0)
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

 t  
2

j0j  jkz
E xEx AA
e
4  jkz
0 e e
e




o



plane
wave
plane
wave
0  kz  const defines the equiphase surfaces
0
o
zz
o
kk
= const - planes
If the amplitude is the same throughout any transverse plane, a
plane wave is called uniform.
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In general,
ex  e y  ez
eE  eH  ek

unit vector along the
direction of propagation
In the x-z plane
t  t0
In the y-z plane
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t  t0 

2
When the electric field of a wave is always directed along
the same line, the wave is said to be linearly polarized.
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z 0
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
E  Eoe jkz (ex  jey )


E(t)  Re Eo (ex  jey )e jt  E0 ex cost - ey sin t

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Poynting’s Theorem
Poynting’s theorem is an identity based on Maxwell’s
equations, which can often be used as an energy-balance
equation
B
Faraday’s law
t
D
Ampere’s law
H  J
t
  (E  H)  H  (  E )  E  (  H)
E  
vector identity
B
D
  (E  H)   H 
E
EJ
t
t
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If ,  and E do not change with time,
Magnetic
energy
density



B
( H) 1 ( H  H)   1 2 
H
 H


 H 
t
t
2
t
t  2

E
D
( E ) 1 ( E  E )   1 2 
E


 E 
t
t
2
t
t  2



E  J  E  (E E)  EE2
  (E  H)  
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Electric
energy
density
 1 2 1 2 
2
 E  H   EE
t  2
2

EEL 3472
Electromagnetic Waves
Integrating over the volume of concern and using the
divergence theorem to convert the volume integral of   (E  H)
to the closed surface integral of (E  H) we have equation
referred to as Poynting’s theorem:
ò (E ´ H ) × ds = S
S
Total power
leaving the
volume
S = (E ´ H) (W/m2 )
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¶
¶t
æ1 2 1
2ö
2
e
E
+
m
H
ç
÷dv - ò v s E E dv
ò è2
ø
2
v
Rate of decrease in
energy stored in electric
and magnetic fields
Ohmic power
dissipated as
heat
Poynting vector (represents the power
flow per unit area)
EEL 3472
Electromagnetic Waves
S = (E ´ H) (W/m2 )
Net power
entering the
volume
- ò (E ´ H ) × ds =
S
S
rate of
increase in
stored We
¶we
+
¶t
+
¶wm
+
¶t
rate of
increase in
stored wm
+ ò s E E 2 dv
v
ohmic losses
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Electromagnetic Waves
E(z, t)  Eoe z cos(t  z)ax
H(z, t) =
Eo
h
e-a z cos(w t - b z - jh )ay
Eo2 -2a z é
S (z, t) =
e ëcosQh + cos(2wt - 2 b z - jh )ùû az
2h
1


 cos A cos B  cos(A  B)  cos(A  B)
2


The time-average Poynting vector:
1
Save (z) =
T
1
Eo2 -2a z é
*
ò 0 S (z, t) = 2 Re ( E ´ H ) = 2 h e ëcosQh ùûaz
The total time-average power crossing a given surface is
obtained by integrating Save over that surface.
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2

E
dv 
e
v

v

v
E  J dv
E  J dv  E J LA cos 
(
= E L cosq
) ( J A) = VI
V=LA
where V is the voltage across the volume and I is the
current flowing through it.
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Example. Using Poynting’s theorem
to find the power dissipated in the
wire carrying direct current I
J=
J
E=
sE
I
ez
p b2
=
H=
I
ez
p b2s E
I
2p b
ej
æ I
ö æ I
ö
S = E ´ H = ç 2 ez ÷ ´ ç
ej ÷
è p b s E ø è 2p b ø
I2
= - 2 3 ep
2p b s E
P=
ò
v
ez  e   e
E × J dv = - ò ( E ´ H ) × ds
S
I
I 2l
= 2 3 (2p bl) = 2 = I 2 R
2p b s E
pb sE
2
R=
19
l
p b 2s E
EEL 3472
Electromagnetic Waves
Reflection and Transmission at Normal Incidence
When a plane wave from one medium meets a different
medium, it is partly reflected and partly transmitted.
If the boundary between the two media is planar and
perpendicular to the wave’s direction of propagation, the
wave is said to be normally incident on the boundary.
Let us consider a wave linearly polarized in the x direction
and propagating in the +z direction, that strikes a perfectly
conducting wall at normal incidence. The wall is located at
z=0
 jkz
E i  Eo e
ex
E r  E1e jkz ex
Hi 
Eo

Hr  
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incident wave
reflected wave
e  jkz e y
E1

e jkz e y
e x  ( e y )  e z
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Electromagnetic Waves
At the interface z=0, the boundary conditions require that
the tangential component of E must vanish (since  E   for
z>0)
(Eo  E1 )ex  0  E1  Eo
E  Ei  Er  Eo(e  jkz  e jkz )ex  2 jEo sin kzex
(This result is similar to the expression for the voltage on a
short-circuited transmission line)
H  Hi  Hr 
Eo

(e  jkz  e jkz )e y 
The wave impedance is defined as
Z(z) 
2 Eo

cos kze y
E x(z)
H y(z)
This impedance is analogous to the input impedance
Z(z)=V(z)/i(z) on a transmission line. For a single
x-polarized wave moving in the +z direction Z(z)   ,
intrinsic impedance of the medium.
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o 
Er (z  0)
 1
Ei(z  0)

reflection coefficient
Et (z  0)
0
Ei(z  0)
transmission coefficient
The incident wave is totally reflected with a phase reversal,
and no power is transmitted across a perfectly conducting
boundary. The conducting boundary acts as a short circuit.
SWR 
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E max
E min

1  o
1  o

standing wave ratio
EEL 3472
Electromagnetic Waves
Let us next consider the case of a plane wave normally
incident on a dielectric discontinuity
E i(z  0 )  E r(z  0 )  E t(z  0 )
H i(z  0 )  H r(z  0 )  H t(z  0 )
E i  Eo e  jkz ex
E r  E1e jkz ex
E t  E2 e  jkz ex
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Electromagnetic Waves
z=0
Eo  E1  E2

Eo E1 E2
    
 1
1
2
E1
 
 0  2 1
E0
 2  1
E2
2 2
 
E0
 2  1
1   0 / 1
2   0 / 2
reflection coefficient
transmission coefficient
  1  0
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Electromagnetic Waves
Z(z)

S av
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S av
EEL 3472