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Lecture 4
1.5 The terminated lossless transmission line
What is a voltage reflection coefficient?
Assume an incident wave ( ) generated from a source at z < 0. We have seen that the ratio of voltage to current for such a traveling wave is Z0, the characteristic impedance. But when the line is terminated in an arbitrary load ZL Z0, the ratio of voltage to current at the load must be ZL. Thus, a reflected wave must be excited with the appropriate amplitude to satisfy this condition.
zjeV 0
What is a voltage reflection coefficient?
Total voltage and current on the line (superposition of incident and reflected waves):
Voltage reflection coefficient Γ:
0
0
0
00 ZZ
ZZ
V
V
L
L
zjzj
zjzj
eZ
Ve
Z
VzI
eVeVzV
0
0
0
0
00
)(
)(
The total voltage and current at the load are related by the load impedance, so at z = 0, we must have
000
00
)0(
)0(Z
VV
VV
I
VZL
(V0+: incident; V0
-: reflected)
(Phase difference: π)
What is a voltage reflection coefficient?
The total voltage and current waves on the line :
Consider the time-average power flow along the line at the point z:
which can be simplified:
• Constant average power flow at any point on the line;• Total power delivered to the load = incident power – reflected power
To understand the reflection coefficient, eg, standing wave (Zl = 0, 0 = -1), we have
Define Voltage standing wave ratio to measure the reflection intensity at 0 < 1
(z = -l)
(1 SWR < ， where SWR=1 implied a match load.)
l = -z
l
The reflection coefficient at z = -l:
At a distance l = -z from the load, the input impedance seen looking toward the load is
02
2
00
0
1
1
)(
)(
)(
)(Z
e
eZ
eeV
eeV
lI
lVZ
lj
lj
ljlj
ljlj
in
ljlj
lj
eeV
eVl
2
0
0 )0()(
A more usable form of input impedance:
ljZZ
ljZZZ
ljZlZ
ljZlZZ
eZZeZZ
eZZeZZZZ
L
L
L
L
ljL
ljL
ljL
ljL
in
tan
tan
sincos
sincos
))()(
))()(
0
00
0
00
00
000
ZL
Zin Z0
0V
0V
lz
0
(l) (0)
• Input impedance of oe portion of transmission line with an arbitrary load impedance. • Transmission line impedance Equation.
What is a voltage reflection coefficient?
Special termination conditions
> Voltage and current along the line:
][)(
][)(
0
0
0
zjzj
zjzj
eeZ
VzI
eeVzV
> Input impedance of lossless transmission line
ZL
Zin Z0
0V
0V
lz
0
> Characteristic impedance
(1). Short circuit transmission line (ZL = 0)
voltage
current
impedance
0l =-z
Zin
Voltage:
Current:
Input impedance:
voltage
current
impedance
(2). Open circuit transmission line ZL =
Zin
Voltage:
Current:
Input impedance:
(3). Quarter-wave transmission line
),...3,2,1,2/4/( nnl
(4). Interface of two transmission lines
Reflection coefficient:
Transmission coefficient:
Insertion loss:
(return loss))(log20 dBRL
1.6 Sourced and loaded transmission lines
Total input voltage:
The incident input voltage:
ingin
ingin ZZ
ZVV
1
1
Here the input impedance is
0
0
ZZ
ZZ
g
gs
and the reflection coefficient seen looking into the generator is
insggin ZZ
ZVV
1
1
0
0
Eliminate Zin
Power consideration:
Homework3. A lossless transmission line of electrical length l = 0.3, is terminated with a complex load impedance as shown below. Find the reflection coefficient at the load, the SWR on the line, the reflection coefficient at the input of the line, and the input impedance to the line.
ZL = 30 –j20 ZL
l = 0.3
Zin Z0= 75
4. A radio transmitter is connected to an antenna having an impedance 80 + j40 with a 50 coaxial cable. If the 50 transmitter can deliver 30 W when connected to a 50 load. How much power is delivered to the antenna?
