# 25 Electromagnetic Theorems 1

Electronics - Devices

Oct 8, 2013 (4 years and 9 months ago)

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Electromagnetic Theorems 1
UCF
Duality Theorem (1)
UCF
Duality Theorem (2)
UCF
Duality Theorem (3)
t


B
ME
t


D
JH
ED

B

t


B
ME
)(
t


D
JH
)(ED

B

'
4
dv
R
e
jkR
V


JA

'
4
dv
R
e
jkR
V


MF

')(
4
dv
R
e
jkR
V


JA

'
4
dv
R
e
jkR
V


MF

UCF
Duality Theorem (4)
UCF
Duality Theorem (5)
ztz
c
zt
c
z
ztz
c
zt
c
z
E
k
j
H
k
jk
H
k
j
E
k
jk






aH
aE
t
t
22
22


Example 1:
Example 2:
FAAE


1
)(
1
jj
AFFH


1
)(
1
jj
UCF
Duality Theorem (6)
')]'([
}'))]'(([
1
')]'([{)(
dsRG
dsRG
jk
dsRGjk
S
sS




)(rEn
)(rHn)(rHnrE

')]'([
}'))]'(([
1
')]'([{
1
)(
dsRG
dsRG
jk
dsRGjk
S
sS




)(rHn
)(rEn)(rEnrH

Example 3: Kirchhoff-Huygens Formula
UCF
Uniqueness Theorem (1)
UCF
Uniqueness Theorem (2)
UCF
Uniqueness Theorem (3)
UCF
Uniqueness Theorem (4)
UCF
Uniqueness Theorem (5)
UCF
Uniqueness Theorem (6)
UCF
Image Theory
UCF
Image Theory for Dielectric
Boundary (Electrostatics) (1)
1

1

1

2

2

2

q
h
0
1
2


0
2
2


1

2

h
h
q

qk
1
1
r
2
r
qk2
h
1
r
z
z
z
(a) Original problem
(b) Equivalent problem
for the upper region
(c) Equivalent problem
for the lower region
UCF
Image Theory for Dielectric
Boundary (Electrostatics) (2)
On the Boundary



0
2
20
1
1
0201
zz
zz
zz








222
1
222
1
21
1
11
1
)()(
1
4

44
hzyx
k
hzyx
q
r
qk
r
q



222
2
2
2
)(
4
hzyx
kq







21
2
2
1
1
1
)1(
1
kk
k
k


21
2
2
21
21
1
2




k
k
UCF
Reciprocity Theorem (1)
UCF
Reciprocity Theorem (2)
UCF
Reciprocity Theorem (3)
. and , when trueAlso
TT
μμεε
UCF
Reciprocity Theorem (4)
UCF
Reciprocity Theorem (5)
UCF
Reciprocity Theorem (6)
Two-port reciprocal network
Port 1
Port 2
Typically should be passive with no transistor, diodes, etc
Inside the network is the reciprocal media

In the following, we will discuss the relationship between Z
21
and Z12
for an reciprocal network

I
I
ZZ
ZZ
V
V
2
1
2221
1211
2
1
0
2
1
1
11
Z

I
I
V
0
1
2
1
12
Z

I
I
V
0
2
1
2
21
Z

I
I
V
0
1
2
2
22
Z

I
I
V

 1
V

2
V
1
I
2
I
UCF
Reciprocity Theorem (7)
z
x
1
zz
2
zz
A
E

A
V
1
3
zz
4
zz
A
I
2
1
zz
2
zz
B
E

B
V
2
3
zz
4
zz
B
I
1
OC at Port 1
OC at Port 2



elsewhere 0
for )()(
4322
zzzyxxI
z
A
A
a
J

1
x
2
x



elsewhere 0
for )()(
2111
zzzyxxI
z
B
B
a
J

UCF
Reciprocity Theorem (8)
No magnetic source



V
AB
V
BA
dvdv''JEJE
A
z
z
BB
z
z
A
IdzEIdzE
21
4
3
2
1


ABBA
IVIV
2211

2port at OC
1
2
1port at OC
2
1
B
B
A
A
I
V
I
V

2112
ZZ

UCF
Reciprocity Theorem (9)
It can be generalized to multi‐port reciprocal network

NNNN
N
N
I
I
ZZ
ZZZ
V
V
...
...
...
...
...
1
1
11211
1
with
n
m
mn
ZZ

or
T
ZZ

Since
1

ZY
YZZZY

111
)()(
TTT

NNNN
N
N
V
V
YY
YYY
I
I
...
...
...
...
...
1
1
11211
1
nmmn
YY

1
2
N
UCF
Volume Equivalence Theorem (1)
UCF
Volume Equivalence Theorem (2)
UCF
Volume Equivalence Theorem (3)
UCF
Volume Equivalence Theorem (4)
(a)
(                 can also be uniform
plane wave field)
00
&HE
Equivalent Problem of (b)
EJ)(
0

j
eq
HM)(
0

j
eq



s
s
HHH
EEE
0
0
(c)
(b)