Transcript lect2-03.ppt

Plane Wave in a Good Conductor
  
    j  j 
s 

1

j




 (1  j)
2
j
2

1

 (1  j )
 (1  j)
 s
2
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Boundary Conditions at the Surface
of a Good Conductor
The field amplitude decays exponentially from its surface
According to e-u/s where u is the normal distance into the
Conductor, s is the skin depth
s 
2

,
J  E
The surface Impedance :
1 j
Zm 
,
Et  Z m J s  Z m n̂  H
 s
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Reflection From A Dielectric Interface
Parallel Polarization
Er
0

x
n2
q2
n3
q3
q1
Et
z
n1
E ijk n r
Ei  E1e 0 1 1
, H i  Y0n1  Ei
Er  E2e  jk0n2 r1
, H r  Y0n2  Er
k0   0
0
, Y0 
0
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E t  E 3e
 jk n3  r1
Y  nY0
, H t  Y n3  Et
, k  k0 , n  
k0n1 x  k0n2 x  kn3 x  nk0n3 x


n1  a x sin q1  a z cos q1


n2  a x sin q 2  a z cosq 2


n3  a x sin q 3  a z cos q 3
q1  q 2
sin q1  n sin q 3
E1 x  E1 cos q1
, E1z   E1 sin q1
E2 x  E2 cos q 2
, E1z  E2 sin q 2
E3 x  E3 cos q 3
, E1z   E3 sin q 3
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Energy and Power
Under steady-state sinusoidal time-varying
Conditions, the time-average energy stored in the
Electric field is
1
1
*
We  Re  E  D dV    E  E * dV
4V
4V
If  is constant and real, then
We 

E E

4
*
dV
V
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Time average energy stored in the magnetic field is :
1
*
Wm  Re  H  B dV
4V


4
*
H

H
dV if  is real and constant

V
The time average power tran smitted across a closed
surface S is given by :
1
P  Re  E  H *  dS
2
S
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Poynting Theorem
  E  H *  (  E )  H *  (  H * )  E
 ( jB  M )  H *  jD *  E  E  J *
J  J s  E

1
1
*
*


E

H
d
V


E

H
 dS


2V
2S
 
1
 j  ( B  H *  E  D * )dV   ( E  J *  H *  M s )dV
2V
2V
 B  H * E  D* 
1
*
*


 2 j  

dV

(
E

J

H
 M s )dV


4
4 
2V
V
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If the medium is characteri zed by :
    - j  and conductivi ty 
     j ,
1
1
*
-  ( E  J Ss  H  M S )dV   E  H *  dS 
2S
2V

1
*
*
*




E  E dV   (  H  H   E  E )dV 

2V
2V
j

*
*


)dV
E

E


H

H

(

2V
1
P0   E  H *  dS
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P 

*
*




(

H

H


E

E
)dV 

2V
1
*

E

E
dV

2V
 Time average power loss
1
*
Ps    ( E  J s  H s  M s )dV
2V
*
*


1
H

H
E

E
*



dV
 Im  E  H  dS  2   

2S
4
4 
V
 2 (Wm  We )
Ps  P0  P  2 j (Wm  We )
The power delivered by the sources (Ps ) is equal to the sum of
the power tran smitted through t he surface P0 , the power lost to heat
in the volume ( P ) and 2 times the reactive energy stored in the
volume.
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Circuit Analogy
I
L
R
C
V
1 * 1
1
j
VI  ZII *  II * ( R  jL 
)
2
2
2
C
*
1
1
1
II
 RII *  2 j ( LII * 
)
2
2
4
4 C
 P  2 j (Wm  We )
P  2 j (Wm  We )
Z
1 *
II
2
General Definitio n of the impedance of a network
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