chapter 2. transmission line theoryocw.sogang.ac.kr/rfile/2013/course21-microwave... ·...
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EMLAB
Chapter 2. Transmission line theory
1
EMLAB
Types of transmission lines
Microstrip line
Coaxial cable
Two-wire transmission line
전기 신호를 손실이 적게 전송하기 위한 구조
2
EMLAB
Signal propagation in tx-line
+
V
-
+
전기장의 진행 속도는 빛의 속도 c임
LZ
전기장의 방향
3
EMLAB
i (z, t)
v (z, t) +
-
z
L z C z
z
i (z+z, t)
v (z+ z,t) +
-
i (z, t)
v (z, t)
),(),(
),( tzzVt
tzIzLtzV
),(),(
),( tzzIt
tzzVzCtzI
Transmission line circuits modeling 4
EMLAB
Transmission line eq. solution
t
VC
z
I
t
IL
z
V
, 0,0
2
2
2
2
2
2
2
2
t
ILC
z
I
t
VLC
z
V
)()(),(
)()(),(
tzItzItzI
tzVtzVtzV
01
)(),(
,1
2
2
2
2
2
2
22
2
2
22
2
2
2
2
2
2
dX
Vd
dX
Vd
t
V
z
V
dX
Vd
t
X
t
V
Xt
V
dX
dV
t
X
dX
dV
t
V
dX
Vd
z
X
z
V
Xz
V
dX
dV
z
X
dX
dV
z
V
C
LZ0)()(),()( tI
C
LtVtI
C
LtV
LC
1
tzX
I
C
LV
dX
dIL
dX
dV
t
IL
z
V
5
EMLAB
Coax
a
W
a
bZ
ab
C
a
bL
ln2
1
ln
2
ln2
0
Parallel Plate
d
W
dZ
d
WC
W
dL
0
Transmission line parameter - examples
b
6
EMLAB
+ -
Parallel wire
Coplanar waveguide
a
DZ
aD
C
a
DL
2cosh
1
2cosh
2cosh
1
0
1
1
a
D
7
EMLAB
LZ+
V
-
Transmission line의 특징
H
E
진행 방향
H
1. 한 방향으로 진행하는 전파의 E+/H+의 비율이 일정.
2. 한 방향으로 진행하는 V+/I+ wave의 amplitude 비율도 일정. → 특성 임피던스 Z0
3. 비율이 맞지 않는 경우 반사파 생김.
0ZI
V
SZI
V
0ZI
V
I
V
8
EMLAB
LZ+
V
-
SZI
V
I
V
VVVL
IIIL
LLL IZV
)tcoefficien reflection;(, VV
0Z
II
VV
L
L
)1(
)1(
0
0
01
1
1
1
ZZ
ZZ
ZZI
V
I
V
L
L
L
L
L
Reflection coefficient 9
EMLAB
LZ+
V
-
SZ
20 40 60 800 100
0
1
-1
2
time, nsec
Vin
, V
Vo
ut, V
20 40 60 800 100
0
1
-1
2
time, nsec
Vin
, V
Vo
ut, V
20 40 60 800 100
0
1
-1
2
time, nsec
Vin
, V
Vo
ut,
V
20 40 60 800 100
0
1
-1
2
time, nsec
Vin
, V
Vo
ut, V
20 40 60 800 100
0
1
-1
2
time, nsec
Vin
, V
Vo
ut,
V
LZ+
V
-
SZ
LZ+
V
-
SZ
LZ+
V
-
SZ
LZ+
V
-
SZ
Zs = 20
Z0= 50
ZL= 1k
0.5m
Line길이에 따른 반사파 영향
rc
LT
/delay
[ns]250 T
[ns]6d T
[ns]3d T
[ns]5.1
[ns]375.0
[ns]75.0
Impedance mismatched
Vin Vout R
R2
R=1k Ohm
MLIN R
R1
R=20 Ohm
VtPulse
SRC1
t
Z0= 50
43.05020
50200
ZZ
ZZ
s
ss
9.05020
50200
ZZ
ZZ
L
LL
10
EMLAB
20 40 60 800 100
0
1
-1
2
time, nsecV
in,
VV
ou
t, V
20 40 60 800 100
0
1
-1
2
time, nsec
Vin
, V
Vo
ut,
V
20 40 60 800 100
0
1
-1
2
time, nsec
Vin
, V
Vo
ut, V
20 40 60 800 100
0
1
-1
2
time, nsec
Vin
, V
Vo
ut, V
20 40 60 800 100
0
1
-1
2
time, nsec
Vin
, V
Vo
ut,
V
LZ+
V
-
SZ
LZ+
V
-
SZ
LZ+
V
-
SZ
LZ+
V
-
SZ
Zs = 1
Z0= 50
ZL= 50
0.5m
rc
LT
/delay
[ns]6d T
[ns]3d T
[ns]5.1
[ns]375.0
[ns]75.0
LZ+
V
-
SZ
Impedance matched
Line 길이에 따른 수신 신호
Vin Vout R
R2
R=50 Ohm
MLIN R
R1
R=50 Ohm
VtPulse
SRC1
t
Z0= 50
11
EMLAB
Narrow band signal 12
EMLAB
0,02
2
2
2
2
2
2
2
t
iLC
z
i
tLC
z
Frequency domain solution
0,0 2
2
22
2
2
LCI
z
ILCV
z
V
tjtj ezItziezVtz ),(),(,),(),(
zjzj
zjzj
eIeIzI
eVeVzV
),(
),(
VCjz
IILj
z
V
,
)()(),()( IC
LVI
C
LV
LCLC p
p
1,
β : propagation constant, vp : speed of light
13
EMLAB
dezItzi
dezVtz
tj
tj
),(2
1),(
),(2
1),(
0,02
2
2
2
2
2
2
2
t
iLC
z
i
tLC
z
0),(),(
2
1
0),(),(
2
1
2
2
2
2
2
2
dezILCz
zI
dezLCVez
zV
tj
tjtj
0,0 2
2
22
2
2
ILC
z
IVLC
z
V
zjzj
zjzj
eIeIzI
eVeVzV
),(
),(
14
EMLAB
]}[Re{
}Re{
)/()/(),(
)/()/(
zjzjtj
cztjcztj
eVeVe
eVeV
cztVcztVtz
Phasor representation
zjzj eVeVzV ),(
}Re{)/( )/( cztjeVcztV
LZ+
V
-
SZI
V
I
V
0Z
tj
SS eVtlz ),(
}),(Re{),( tjezVtz lz 0z
)cos(),( ztVtzV
)cos(),( ztVtzV
15
EMLAB
Transmission line terminated with short, open
Zs = Zo
Vrefl V
inc
For reflection, a transmission line terminated in a short or open reflects all power
back to source
In phase (0 ) for open o
Out of phase (180 ) for short
Vrefl
o
16
EMLAB
Transmission Line Terminated with 25 Ω
Zs = Zo
ZL = 25
Vrefl V
inc
Standing wave pattern does not go to zero as with short or open
3
1
5025
5025
17
EMLAB
)(][1
),(
)(),(
00
ljljljlj
ljljljlj
eeZ
VeVeV
ZlI
eeVeVeVlV
0
0
ZZ
ZZ
L
L
ljZZ
ljZZZ
ee
eeZ
lI
lVZ
L
L
ljlj
ljlj
in
tan
tan
),(
),(
0
000
Equivalent input impedance 18
EMLAB
Input impedance of short
ljZZin tan0
19
EMLAB
Input impedance of open
ljZZin cot0
20
EMLAB
Input impedance of ¼ wavelength line
L
inZ
ZZ
2
0
4/
Quarter wavelength transformer
21
EMLAB
Reflection/Transmission
IV
IV TV
TI
TT
TT
IIVV
IIIVVV
)1(,)1(
,
101
1
)1(
)1(
ZZI
V
III
TVVV
T
T
TT
T
22
EMLAB
Reflection measurement – slotted line
||1
||1SWR
min
max
V
V
Standing wave ratio
23
EMLAB
Smith chart
j
L
L
L
L eZZ
ZZ
ZZ
ZZ||
1/
1/
0
0
0
0
각각의 반사계수에 해당하는 부하 임피던스를 표시한 그림
real
imag
0/ ZZz L
Normalized impedance
je||
반사계수 측정을 위해 사용된
transmission line의 특성 임피던스 = Z0
24
EMLAB
25
EMLAB
Network analyzer 26
EMLAB
Smith chart review
.
-90 o
0 o
180 o +
- .2
.4
.6
.8
1.0
90 o
0
0 +R
+jX
-jX
Smith Chart maps rectilinear
impedance plane onto polar plane
Rectilinear impedance plane
Polar plane
Z = Zo L
= 0
Constant X
Constant R
Z = L
= 0 O
1
Smith Chart
(open)
L
Z = 0
= ±180 O
1
(short)
0
0
ZZ
ZZ
Z-plane
Γ-plane Z-to-Γ transform
27
EMLAB
Smith chart :
• graphical representation in the reflection coefficient plan plane
• passive impedance plane Re(z)
• z one-to-one correspondance
)2()1(
2
)1()1(
1
)1(
})1}{()1{(
)1(
)1(
1
1
1
1
impedance normalized:/ 1
1
22
22
22
22
00
0
0
0
xr
xv
xr
xru
xr
jxrjxr
jxr
jxr
z
zivulet
z
Z
Xj
Z
Rjxrz
ZZzz
z
ZZ
ZZ
Constant resistance, reactance circles 28
EMLAB
22
2
2
2
2
2222
22
11)1( )2()6(
)1(
1
1 )1()5(
)6()1(
)5(1
1
222
2 )4()3(
)4(12)1(2
)2(
)3(12 ))2/()1((2
xxvu
ru
r
ru
v
vuxr
vu
rx
rv
x
v
ux
xrrxrv
x
xrv
uxx
29
EMLAB
x
R 0 0.5 1 2
r=2
r=1
r=0.5
r=0
x
R
2
1
0.5
0.5
1
2
x=2
x=0.5 x=1
x=-0.5
x=-1
x=-2
real
imag
real
imag
Constant resistance, reactance circles 30
EMLAB
Constant admittance circles
1/
1/
/1
/1
/1
/1
0
0
0
0
0
0
0
0
YY
YY
YY
YY
ZZ
ZZ
ZZ
ZZ
1
1
1
1
y
y
z
z
0
+jX
-jX
Z-plane
0
+jB
Y-plane
+jG 1
1)(
y
y
Impedance chart를 원점에 대칭이동하면 admittance chart가 됨.
31
EMLAB
Basic smith chart operation
ljzj
zj
zj
L
L
ezezeV
eV
lV
lVlz
ZZ
ZZ
V
Vz
22
0
0
)0()0()(
)()(
)0(
)0( z)( lz
ljezlz 2)0()(
Wavelength toward generator
32
EMLAB
Example 2.2
ZL = 40 + j 70Ω 의 부하가 특성 임피던스 100 Ω이고 길이가 0.3λ인
transmission line 에 연결되어 있다.
)0( z)( lz
부하단에서 반사계수, transmission line의 입력단에서 반사계수, 입력임피던스, SWR, return loss를 구하라.
7.04.0100
7040j
jzL
1045890.00.5714 + -0.1429
1
1j
z
z
L
LL
1125890.0
21611045890.02 lj
Lin e
dB6.4||log20RL
87.359.01
59.01SWR
33
EMLAB
34
EMLAB
Example 2.4
)2(2)( lj
L
lj
L eelz
)2(1 lje
)2(1 lje
0.2, 2.2, 4.2cm
0.72, 2.72, 4.72cm
특성 임피던스 50Ω 인 slotted
line을 이용한 측정 결과이다.
부하 임피던스를 구하라.
SWR =1.5
7.193.471
150
1996.00126.0*2.0
2.0
4
1.481.48cm0.52cm2cm
48.1*4/4
48.1*4/42
min
min
jZ
je
ee
l
L
LL
j
L
j
L
lj
L
35
EMLAB
Example 2.5
10050
71.7050*1001Z
36
EMLAB
2.7 Lossy transmission line 37
EMLAB
0I(z)dz
I(z)d
0V(z)dz
V(z)d
2
2
2
2
2
2
)()(dz
dI(z)
)()(dz
dV(z)
zVCjG
zILjR
C)jL)(Gj(R
j
C
Lossy transmission line 38
EMLAB
zz
zz
ee
ee
00
00
III(z)
VVV(z)
Wave Solution
CjG
LjRZ
LjR0
]VV[LjR
I(z) 00
zz ee
: Characteristic Impedance
0
00
0
00
I
VZ
I
VZ
2 fvp
For Lossless Line
LCv
LC
p
1
22
0
LC
LCjj
C
LZ 0
zjzj
zjzj
eZ
e
ee
0
0
0
0
00
I
Z
VI(z)
VVV(z)
39