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EMLAB
1
Network parameter transformation
EMLAB
S-matrix conversion from Z-matrix
)(1
0
nnnnn
nnn
VVZ
III
VVV
][][V
]][[]][[Z
1
]][[]][[]][[][
0
V
VZVZ
IZIZIZV
022021
012011
1
11
1111
//
//][
) ][][ () ][][ (][
1
1Zport -one
ZzZz
ZzZzZ
SUSUZ
ZS
][ ][][Z
1 ][ ][][
Z
1
00
VUZVUZ
][][Z
1 ][][
Z
1][
0
1
0
UZUZS
10
10
001
][
U
2
EMLAB
][][ ][][ Re
2
1
][][ Re2
1
**
0
*
VVVVZ
IVP
tt
tav
**
0
****
0
][][ ][][ Re2
1
][][ ][][ ][][ ][][ Re2
1
VVVVZ
VVVVVVVVZ
tt
tttt
incident power reflected power
Lossless network, incident power = reflected power
]][[][
][][ ][][ **
VSV
VVVV tt
][ ][
][][
][][][][ ][][
1*
*
***
t
t
ttt
SS
USS
VSSVVV
Unitary matrix
S-matrix property for Lossless network
N
kijkjkiSS
1
*
3
EMLAB
Example 4.5 S-parameter 계산
00.2450.85
45-0.8500.15[S]
1) Port 2 가 matched load 로 termination 된 경우 port 1 의 return loss2) Port 2 가 short 되었을 때 port 1 의 return loss.
111
1
2
2
1
2221
1211
2
1 0,)1
SV
V
VV
V
SS
SS
V
V
122
2112111
22
12122221212
2121111
22
2
1
2221
1211
2
1
1
1
,)2
VS
SSSV
S
VSVVSVSV
VSVSV
VVV
V
SS
SS
V
V
4
EMLAB
Reciprocal network
IZV
I
I
I
ZZ
ZZZ
V
V
V
NNNN
N
N
2
1
1
11211
2
1
xNMy
xMNy
IZV
IZV
xI
yV
port N
port M
yV
xI
jiij ZZ Port N 에 전류원 Ix 를 연결했을 때 port M 에 전압 Vy 가 측정되면 →
Port M 에 전류원 Ix 를 연결했을 때 port N 에 전압 Vy 가 측정된다 .
5
EMLAB
1
00
1
00
0
1
0
][][Z
1][][
Z
1
][][Z
1][][
Z
1
][][Z
1 ][][
Z
1][
UZUZ
UZUZ
UZUZS
tttt
t
t
S-matrix property for Reciprocal network
]])[[]([2
1][],])[[]([
2
1][
)(2
1),(
2
1
)(1
00
00
0
IUZZVIUZZV
IZVVIZVV
VVZ
III
VVV
nnnnnn
nnnnn
nnn
1
00
][][Z
1 ][][
Z
1][
UZUZS
tSS ][][
6
EMLAB
A shift in reference planes
jkzeVzV 0)(
nV
Ljn
jnn eVeVV n
nV
Ljn
jnn eVeVV n
]][[][ VSV
]][[][ VSV
][][2
1
n
j
j
j
n V
e
e
e
V
N
][][2
1
n
j
j
j
n V
e
e
e
V
N
NN j
j
j
j
j
j
e
e
e
S
e
e
e
S
2
1
2
1
][][
7
EMLAB
Generalized Scattering parameters
nnn
nnn
ZVb
ZVa
0
0
/
/
Port 별로 연결된 transmission line 의 임피던스가 다른 경우
)(1
)(1
)(
)(
00
0
0
nn
n
nnn
n
nnnnn
nnnnnn
baZ
VVZ
I
baZVV
baZVVV
jkVij
ji
jkaj
iij
kk
ZV
ZV
a
bS
for0
0
0
for0
8
EMLAB
Vector network analyzer9
EMLAB
4.4 Transmission (ABCD) parameter
2
2
1
1
221
221
I
V
DC
BA
I
V
DICVI
BIAVV
3
3
22
22
11
11
1
1
3
3
22
22
2
2
2
2
11
11
1
1 ,
I
V
DC
BA
DC
BA
I
V
I
V
DC
BA
I
V
I
V
DC
BA
I
V
10
EMLAB
Z-to-ABCD transform
2221212
2121111
IZIZV
IZIZV
2221212
2121111
VYVYI
VYVYI
Reciprocal network 인 경우 jiij ZZ
2221212
2121111
IZIZV
IZIZV
2
2
22
1221221111
21
2
2
22
121121
211
1
2
2
22
12
1
1
21
11
1
1
1
0
10
1
1
0
0
1
I
V
Z
ZZZZZ
Z
I
V
Z
ZZZ
ZI
V
I
V
Z
Z
I
V
Z
Z
11
EMLAB
12
EMLAB
Example 4.6
221
221
DICVI
BIAVV
1
0
/
1
02
1
02
1
1
1
02
1
02
1
2
2
2
2
I
I
V
I
I
ID
V
IC
ZZV
V
I
VB
V
VA
13
EMLAB
Figure 4.12 (p. 188) A coax-to-microstrip transition and equivalent circuit representations. (a) Geometry of the transition. (b) Representation of the transition by a “black box.” (c) A possible equivalent circuit for the transition [6].
14
EMLAB
Figure 4.23 (p. 199)Some common microstrip discontinuities. (a) Open-ended microstrip. (b) Gap in microstrip. (c) Change in width. (d) T-junction. (e) Coax-to-microstrip junction.
15
EMLAB
Properties of Z(ω) and Γ(ω)
deVt tj)(
2
1)( )()( * tt
deVdeVdeV tjtjtj )()()( **
)()( * VV
)()()()()()()()( **** IZVIZIZV
)()(),()(
)()()()(
)()(*
XXRR
jXRjXR
ZZ
)()()(
)()()(
)()(
)()(
)(
)()(
0
0*
0
0
0
0
jXZR
jXZR
jXZR
jXZR
ZZ
ZZ
)()(* ZZ
)()(*
2*2)()()()()()(
2
211111*1111
2
11 )(1)()()()()( SSSSSS
(A signal is a Real number function)
16
EMLAB
rE dV
CdI rH
Equivalent voltage and current
TEM 인 경우 unique
17
EMLAB
Figure 4.2 (p. 163)Electric field lines for the TE10 mode of a rectangular waveguide.
22
),(sin
),(sin
ak
eyxAhea
xA
ajH
eyxAeea
xA
ajE
zjx
zjx
zjy
zjy
k
H
EZ
x
yTE
TEZ
yxzyx
),(ˆˆ),(ˆ e
h
Rectangular waveguide Z018
EMLAB
k
ZTE GHz10f 2.54,ε
1.016cmb 2.286cm,a
r
10
12
20
12
20
4.2092
304
158
mc
fk
ma
k
ma
k
rd
a
316.0
6.259
500158
3774.209
00
00
0
000
ad
ad
dd
aa
ZZ
ZZ
Z
kZ
Example 4.2
Figure 4.3 (p. 167)Geometry of a partially filled waveguide and its transmission line equivalent for Example 4.2.
19
EMLAB
Electromagnetic energy and power flow
dW
dW
Vm
Ve
*
*
Re4
1
Re4
1
HB
ED • Time averaged electric field energy stored in volume V
• Time averaged magnetic field energy stored in volume V
daPS
)(Re2
1 *HE
Lem
VS
PWWj
djda
)(2
)(2
1)(
2
1 **
EEEDHBHE **
Poynting theorem
VLem
*2
***
)(2
2
1
2
1
2
1
1
PWWj
RIIC
IIILIjωIZIP
CjLjRjXRZ
Circuit analogy
20
EMLAB
Example
01Z 02Z
0102
0102
1
1
011
01111
/
/
ZZ
ZZ
V
V
ZV
ZVS
nnn
nnn
ZVb
ZVa
0
0
/
/
0102
0201
0102
01
01
02
2
1
01
02
022
01112
22
/
/
ZZ
ZZ
ZZ
Z
Z
Z
V
V
Z
Z
ZV
ZVS
0102
0201
0102
02
02
01
1
2
02
01
011
02221
22
/
/
ZZ
ZZ
ZZ
Z
Z
Z
V
V
Z
Z
ZV
ZVS
0201
0201
2
2
022
02222
/
/
ZZ
ZZ
V
V
ZV
ZVS
21