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Lecture 4
1.5 The terminated lossless transmission line
What is a voltage reflection coefficient?
V(z) = V + (z)+V - (z)
I(z) = I + (z)+ I - (z)
 Assume an incident wave ( V0 e  jz ) 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.
What is a voltage reflection coefficient?
Total voltage and current on the line
(superposition of incident and reflected
waves):
V ( z )  V0 e  jz  V0 e jz
V0  jz V0 jz
I ( z) 
e

e
Z0
Z0
(V0+: incident; V0-: reflected)
The total voltage and current at the load are related by the load impedance, so
at z = 0, we must have
V (0) V0  V0
ZL 

Z0
I (0) V0  V0
Voltage reflection coefficient Γ:
V0 Z L  Z 0
0   
V0
Z L  Z0
(Phase
difference: π)
What is a voltage reflection coefficient?
The total voltage and current waves on the line :
V (z) = V0+e- jb z +V0-e jb z = V0+ [e- jb z + G 0 e jb z ]
V0+ - jb z V0- jb z V0+ - jb z
I(z) =
e
e =
[e
- G 0 e jb z ]
Z0
Z0
Z0
V0G0 = +
V0
Consider the time-average power flow along the line at the point z:
2
+
V
1
1
2
0
Pav = Re[V (z)I(z)* ] =
Re{1- G*0 e-2 jb z + G 0 e2 jb z - G 0 }
2
2 Z0
which can be simplified:
+ 2
1 V0
2
Pav =
(1- G 0 )
2 Z0
• 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
V (z, t) = V0+ [e- jb z + G 0 e jb z ]× e jwt
= V0+ [e jb z - e- jb z ]× e jwt
= 2iV sin(b z) × e
+
0
jw t
l
l = -z
Re[V(z, t)] = -2V0+ sin(b z)sin(wt)
Define Voltage standing wave ratio to measure the reflection intensity
at 0  < 1
V (z) = V0+ 1+ G 0 e2 jb z = V0+ 1+ G 0 e-2 jbl
SWR =
Vmax 1+ G 0
=
Vmin 1- G 0
(z = -l)
(1  SWR < ， where SWR=1 implied a match load.)
What is a voltage reflection coefficient?
(l)
The reflection coefficient at z = -l:
V0
(0)
Zin
ZL
Z0
V0 e  jl
(l )   jl  (0)e 2 jl
V0 e
V0
z
l
0
At a distance l = -z from the load, the input impedance seen looking
toward the load is

jl
 jl
1   e 2 jl
V (l ) V0 (e   e )
Z in 

Z0 
Z0
I (l ) V0 (e jl   e  jl )
1   e 2 jl
A more usable form of input impedance:
( Z L  Z 0 ) e j  l  ( Z L  Z 0 ) e  j l )
Z in  Z 0
( Z L  Z 0 ) e j  l  ( Z L  Z 0 ) e  j l )
 Z0
Z L cos l  jZ0 sin l
Z 0 cos l  jZ L sin l
 Z0
Z L  jZ0 tan l
Z 0  jZ L tan l
• Input impedance of oe portion of transmission
line with an arbitrary load impedance.
• Transmission line impedance Equation.
Special termination conditions
> Input impedance of lossless transmission line
Z L + jZ0 tan b l
Zin (l) = Z0
Z 0 + jZ L tan b l
V0
> Characteristic impedance
Zin
L
Z0 =
C
> Voltage and current along the line:
V ( z )  V0 [e  jz  e jz ]
V0  jz
I ( z) 
[e
 e jz ]
Z0
ZL
Z0
V0
z
l
0
(1). Short circuit transmission line (ZL = 0)
voltage
Zin
l =-z
0
current
Voltage:
Re[V(l)] = 2iV0+ sin(bl)
Current:
2V0+
Re[I(l)] =
cos(b l)
Z0
Input impedance:
Zin (l) = iZ0 tan(bl)
impedance
(2). Open circuit transmission line ZL = 
voltage
Zin
current
Voltage:
Re[V(l)] = 2V0+ cos(bl)
Current:
2iV0+
Re[I(l)] =
sin(b l)
Z0
Input impedance:
Zin (l) = -iZ0 cot(bl)
impedance
(3). Quarter-wave transmission line
(l   / 4  n / 2,
n  1,2,3,...)
(4). Interface of two transmission lines
Reflection coefficient:
Transmission coefficient:
Insertion loss:
RL  20 log 
(dB) (return loss)
1.6 Sourced and loaded transmission lines
Total input voltage:
 The incident input voltage:
Z in
1
Vin  Vg
Z in  Z g 1 in
Eliminate Zin
Here the input impedance is
1+ G 0 e-2 jbl
1+ G in
Zin = Z 0
=
Z
0
1- G 0 e-2 jbl
1- G in
and the reflection coefficient seen
looking into the generator is
s 
Z g Z 0
Z g Z 0
Vin  Vg
Z0
1
Z 0  Z g 1  s in
Power consideration:
Vin+ = Vg
Z0
1
Z0 + Z g 1- G s G in
G in = G 0 e-2 jbl
Homework
3. 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.
l = 0.3 
Zin
Z0= 75 
ZL
ZL = 30 –j20 
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?