Transmission Lines
We have obtained the following solutions for the steady-state
voltage and current phasors in a transmission line:
Loss-less line
V (z) V e jz V e j z
1
I (z) V e jz V e j z
Z0
Lossy line
V (z) V ez V ez
1
I (z) V ez V ez
Z0
Since V (z) and I (z) are the solutions of second order differential
(wave) equations, we must determine two unknowns, V+ and V ,
which represent the amplitudes of steady-state voltage waves,
travelling in the positive and in the negative direction, respectively.
Therefore, we need two boundary conditions to determine these
unknowns, by considering the effect of the load and of the
generator connected to the transmission line.
© Amanogawa, 2000 – Digital Maestro Series
65
Transmission Lines
Before we consider the boundary conditions, it is very convenient
to shift the reference of the space coordinate so that the zero
reference is at the location of the load instead of the generator.
Since the analysis of the transmission line normally starts from the
load itself, this will simplify considerably the problem later.
ZR
New Space Coordinate
z
d
0
We will also change the positive direction of the space coordinate,
so that it increases when moving from load to generator along the
transmission line.
© Amanogawa, 2000 – Digital Maestro Series
66
Transmission Lines
We adopt a new coordinate d = z, with zero reference at the load
location. The new equations for voltage and current along the lossy
transmission line are
Loss-less line
jd
jd
V (d) V e
Lossy line
d
d
V e
1
I (d) V e jd V e jd
Z0
V (d) V e
V e
1
I (d) V ed V e d
Z0
At the load (d = 0) we have, for both cases,
V (0) V V 1
I (0) V V
Z0
© Amanogawa, 2000 – Digital Maestro Series
67
Transmission Lines
For a given load impedance ZR , the load boundary condition is
V (0) ZR I (0)
Therefore, we have
ZR
V V V V
Z0
from which we obtain the voltage load reflection coefficient
V
ZR Z0
R Z Z
V
R
0
© Amanogawa, 2000 – Digital Maestro Series
68
Transmission Lines
We can introduce this result into the transmission line equations as
Loss-less line
Lossy line
V e jd
2 j d
I (d) e
1
R Z0
V (d) V e jd 1 R e2 j d
V e d
2 d
I (d) e
1
R Z0
V (d) V ed 1 R e2 d
At each line location we define a Generalized Reflection Coefficient
(d) R e2 j d
(d) R e2 d
and the line equations become
V (d) V e jd 1 (d) V (d) V ed 1 (d) V e jd
I (d) 1 (d) Z0
V ed
I (d) 1 (d) Z0
© Amanogawa, 2000 – Digital Maestro Series
69
Transmission Lines
We define the line impedance as
1 (d )
V (d )
Z0
Z( d ) 1 (d )
I (d )
A simple circuit diagram can illustrate the significance of line
impedance and generalized reflection coefficient:
Req = (d)
d
© Amanogawa, 2000 – Digital Maestro Series
ZR
Zeq=Z(d)
0
70
Transmission Lines
If you imagine to cut the line at location d, the input impedance of
the portion of line terminated by the load is the same as the line
impedance at that location “before the cut”. The behavior of the line
on the left of location d is the same if an equivalent impedance with
value Z(d) replaces the cut out portion. The reflection coefficient of
the new load is equal to (d)
Req ( d ) ZReq Z0
ZReq Z0
If the total length of the line is L, the input impedance is obtained
from the formula for the line impedance as
1 ( L)
Vin V ( L)
Z0
Zin 1 ( L)
Iin
I ( L)
The input impedance is the equivalent impedance representing the
entire line terminated by the load.
© Amanogawa, 2000 – Digital Maestro Series
71
Transmission Lines
An important practical case is the low-loss transmission line, where
the reactive elements still dominate but R and G cannot be
neglected as in a loss-less line. We have the following conditions:
L R
C G
so that
( j L R)( j C G)
R G 1
j L j C 1 j
L
j
C
R
G
RG
j LC 1 j L j C 2 LC
The last term under the square root can be neglected, because it is
the product of two very small quantities.
© Amanogawa, 2000 – Digital Maestro Series
72
Transmission Lines
What remains of the square root can be expanded into a truncated
Taylor series
1 R
G j LC 1 2
j
L
j
C
1
C
L
R
G
j LC
2
L
C
so that
1
C
L
R
G
2
L
C
© Amanogawa, 2000 – Digital Maestro Series
LC
73
Transmission Lines
The characteristic impedance of the low-loss line is a real quantity
for all practical purposes and it is approximately the same as in a
corresponding loss-less line
R j L
L
Z0 G j C
C
and the phase velocity associated to the wave propagation is
1
vp LC
BUT NOTE:
In the case of the low-loss line, the equations for voltage and
current retain the same form obtained for general lossy lines.
© Amanogawa, 2000 – Digital Maestro Series
74
Transmission Lines
Again, we obtain the loss-less transmission line if we assume
R0
G0
This is often acceptable in relatively short transmission lines, where
the overall attenuation is small.
As shown earlier, the characteristic impedance in a loss-less line is
exactly real
L
Z0 C
while the propagation constant has no attenuation term
( j L)( j C ) j LC j
The loss-less line does not dissipate power, because = 0.
© Amanogawa, 2000 – Digital Maestro Series
75
Transmission Lines
For all cases, the line impedance was defined as
1 (d)
V (d)
Z0
Z(d) 1 (d)
I (d)
By including the appropriate generalized reflection coefficient, we
can derive alternative expressions of the line impedance:
A) Loss-less line
1 Re2 jd
ZR jZ0 tan( d)
Z0
Z(d) Z0
2 jd
jZR tan( d) Z0
1 Re
B) Lossy line (including low-loss)
1 R e2 d
ZR Z0 tanh( d)
Z0
Z(d) Z0
2 d
ZR tanh( d) Z0
1 Re
© Amanogawa, 2000 – Digital Maestro Series
76
Transmission Lines
Let’s now consider power flow in a transmission line, limiting the
discussion to the time-average power, which accounts for the
active power dissipated by the resistive elements in the circuit.
The time-average power at any transmission line location is
1
P(d , t ) Re V (d) I * (d)
2
This quantity indicates the time-average power that flows through
the line cross-section at location d. In other words, this is the
power that, given a certain input, is able to reach location d and
then flows into the remaining portion of the line beyond this point.
It is a common mistake to think that the quantity above is the power
dissipated at location d !
© Amanogawa, 2000 – Digital Maestro Series
77
Transmission Lines
The generator, the input impedance, the input voltage and the input
current determine the power injected at the transmission line input.
Iin
Zin
Vin VG
ZG Zin
ZG
VG
Vin
Generator
© Amanogawa, 2000 – Digital Maestro Series
Zin
Line
1
Iin VG
ZG Zin
1
*
Pin Re Vin Iin
2
78
Transmission Lines
The time-average power reaching the load of the transmission line
is given by
1
P(d=0 , t ) Re V (0) I * (0)
2
*
1
1
Re V 1 R V 1 R 2
Z0*
This represents the power dissipated by the load.
The time-average power absorbed by the line is simply the
difference between the input power and the power absorbed by the
load
P line Pin P(d 0 , t ) Remember that the internal impedance of the generator dissipates
part of the total power generated.
© Amanogawa, 2000 – Digital Maestro Series
79
Transmission Lines
The time-average power injected into the input of the transmission
line is maximized when the input impedance of the transmission
line and the internal generator impedance are complex conjugate of
each other.
Zin
Load
Generator
ZG
VG
ZG Z*in
© Amanogawa, 2000 – Digital Maestro Series
Transmission line
ZR
for maximum power transfer
80
Transmission Lines
In a loss-less transmission line no power is absorbed by the line, so
the input time-average power is the same as the time-average
power absorbed by the load. The characteristic impedance of the
loss-less line is real and we can express the power flow as
1
P(d , t ) Re V (d) I* (d)
2
1
Re V e j d 1 Re j 2 d
2
1
* j d
j 2 d * (V ) e
1 Re
Z0
1
1
2
2
2
V
V
R
2 Z0
2 Z0
Incident wave
© Amanogawa, 2000 – Digital Maestro Series
Reflected wave
81
Transmission Lines
In the case of low-loss lines, the characteristic impedance is again
real, and the time-average power flow along the line is given by
1
P(d , t ) Re V (d) I * (d)
2
1
Re V e d e j d 1 Re2 d
2
1
* d j d
2 d * (V ) e e
1 Re
Z0
1
1
2
2 2 d
2 2 d
V
e
V
e
R
2 Z0
2 Z0
Incident wave
© Amanogawa, 2000 – Digital Maestro Series
Reflected wave
82
Transmission Lines
Note that in a lossy line the reference for the amplitude of the
incident voltage wave is at the load and that the amplitude grows
exponentially moving towards the input. The amplitude of the
incident wave behaves in the following way
V e L
V e d
V
input
inside the line
load
The reflected voltage wave has maximum amplitude at the load, and
it decays exponentially moving back towards the generator. The
amplitude of the reflected wave behaves in the following way
V R e L
V R e d
V R
input
inside the line
load
© Amanogawa, 2000 – Digital Maestro Series
83
Transmission Lines
For a general lossy line the characteristic impedance is complex,
and the time-average power is
1
P(d , t ) Re V (d) I * (d)
2
1
Re V e d e j d 1 (d) 2
*
* * d j d
Y0 (V ) e e
1 (d) G0 2 2 d G0 2 2 d
2
R
V
e
V
e
2
2
B0 V
© Amanogawa, 2000 – Digital Maestro Series
2
e2 d Im((d))
84
Transmission Lines
We have introduced for convenience the characteristic admittance
of the line
Y0 1
G0 jB0
Z0
since a complex characteristic impedance would appear at
denominator in the expression for the power.
Note that for a low-loss transmission line the characteristic
impedance is approximately real and
B0 0
The previous result for the low-loss line can be readily recovered
from the time-average power for the general lossy line.
© Amanogawa, 2000 – Digital Maestro Series
85
Transmission Lines
To completely specify the transmission line problem, we still have
to determine the value of V+ from the input boundary condition.
The load boundary condition imposes the shape of the
interference pattern of voltage and current along the line.
The input boundary condition, linked to the generator, imposes
the scaling for the interference patterns.
We have
Zin
Vin V ( L) VG
ZG Zin
or
with
1 (L)
Zin Z0
1 (L)
ZR jZ0 tan( L)
Zin Z0
jZR tan( L) Z0
loss - less line
ZR Z0 tanh( L)
Zin Z0
ZR tanh( L) Z0
lossy line
© Amanogawa, 2000 – Digital Maestro Series
86
Transmission Lines
For a loss-less transmission line:
V (L) V e j L 1 (L) V e j L (1 Re j 2 L )
!
1
Zin
V VG
ZG Zin e j L (1 Re j 2 L )
For a lossy transmission line:
V (L) V e L 1 (L) V e L 1 Re2 d
!
1
Zin
V VG
ZG Zin e L (1 Re2 L )
© Amanogawa, 2000 – Digital Maestro Series
87
Transmission Lines
In order to have good control on the behavior of a high frequency
circuit, it is very important to realize transmission lines as uniform
as possible along their length, so that the impedance behavior of
the line does not vary and can be easily characterized.
A change in transmission line properties, wanted or unwanted,
entails a change in the characteristic impedance, which causes a
reflection. Example:
Z01
Z02
ZR
1
Z01
© Amanogawa, 2000 – Digital Maestro Series
Zin
Zin Z01
1 Zin Z01
88
Transmission Lines
Special Cases
ZR 0 (SHORT CIRCUIT)
ZR = 0
Z0
The load boundary condition due to the short circuit is V (0) = 0
V (d 0) V e j 0 (1 R e j 2 0 )
V (1 R ) 0
© Amanogawa, 2000 – Digital Maestro Series
R 1
89
Transmission Lines
Since
R V
V
V V We can write the line voltage phasor as
V (d) V e j d V e j d
V e j d V e j d
V ( e j d e j d )
2 jV sin( d)
© Amanogawa, 2000 – Digital Maestro Series
90
Transmission Lines
For the line current phasor we have
1
(V e j d V e j d )
I (d) Z0
1
(V e j d V e j d )
Z0
V j d
(e
e j d )
Z0
2V cos( d)
Z0
The line impedance is given by
2 jV sin( d)
V (d)
jZ0 tan( d)
Z(d) I (d) 2V cos( d) / Z0
© Amanogawa, 2000 – Digital Maestro Series
91
Transmission Lines
The time-dependent values of voltage and current are obtained as
V (d, t ) Re[V (d) e j t ] Re[2 j | V | e j sin( d) e j t ]
2| V |sin( d) Re[ j e j( t ) ]
2| V |sin( d) Re[ j cos(t ) sin(t )]
2| V |sin( d) sin(t )
I (d, t ) Re[ I (d) e j t ] Re[2 | V | e j cos( d) e j t ] / Z0
2| V |cos( d) Re[ e j ( t ) ] / Z0
2| V |cos( d) Re[ (cos(t ) j sin(t )] / Z0
| V |
cos( d) cos(t )
2
Z0
© Amanogawa, 2000 – Digital Maestro Series
92
Transmission Lines
The time-dependent power is given by
P(d, t ) V (d, t ) I (d, t )
| V |2
sin( d) cos( d) sin(t ) cos(t )
4
Z0
| V |2
sin(2 d) sin (2t 2)
Z0
and the corresponding time-average power is
1 T
P(d, t ) P(d, t ) dt
T 0
| V |2
1 T
sin(2 d) sin (2t 2) 0
Z0
T 0
© Amanogawa, 2000 – Digital Maestro Series
93
Transmission Lines
ZR (OPEN CIRCUIT)
ZR Z0
The load boundary condition due to the open circuit is I (0) = 0
V j 0
I (d 0) e (1 R e j 2 0 )
Z0
V
(1 R ) 0
Z0
© Amanogawa, 2000 – Digital Maestro Series
R 1
94
Transmission Lines
Since
R V
V
V V
We can write the line current phasor as
1
(V e j d V e j d )
I (d) Z0
1
(V e j d V e j d )
Z0
2 jV V j d j d
(e
)
sin( d)
e
Z0
Z0
© Amanogawa, 2000 – Digital Maestro Series
95
Transmission Lines
For the line voltage phasor we have
V (d) (V e j d V e j d )
(V e j d V e j d )
V ( e j d e j d )
2V cos( d)
The line impedance is given by
2V cos( d)
V (d)
Z0
j
Z(d) tan( d)
I (d) 2 jV sin( d) / Z0
© Amanogawa, 2000 – Digital Maestro Series
96
Transmission Lines
The time-dependent values of voltage and current are obtained as
V (d, t ) Re[V (d) e j t ] Re[2 | V | e j cos( d) e j t ]
2| V |cos( d) Re[ e j( t ) ]
2| V |cos( d) Re[ (cos(t ) j sin(t )]
2| V |cos( d) cos( t )
I (d, t ) Re[ I (d) e j t ] Re[2 j | V | e j sin( d) e j t ] / Z0
2| V |sin( d) Re[ j e j ( t ) ] / Z0
2| V |sin( d) Re[ j cos( t ) sin(t )] / Z0
| V |
sin( d) sin(t )
2
Z0
© Amanogawa, 2000 – Digital Maestro Series
97
Transmission Lines
The time-dependent power is given by
P(d, t ) V (d, t ) I (d, t ) | V |2
cos( d) sin( d) cos(t ) sin(t )
4
Z0
| V |2
sin(2 d) sin (2t 2)
Z0
and the corresponding time-average power is
1 T
P(d, t ) P(d, t ) dt
T 0
| V |2
1 T
sin(2 d) sin (2t 2) 0
Z0
T 0
© Amanogawa, 2000 – Digital Maestro Series
98
Transmission Lines
ZR Z0 (MATCHED LOAD)
Z0
ZR = Z0
The reflection coefficient for a matched load is
ZR Z0 Z0 Z0
R 0
ZR Z0 Z0 Z0
no reflection!
The line voltage and line current phasors are
V (d) V e j d (1 R e2 j d ) V e j d
V j d
V
(1 R e2 j d ) I( d ) e
e j d
Z0
Z0
© Amanogawa, 2000 – Digital Maestro Series
99
Transmission Lines
The line impedance is independent of position and equal to the
characteristic impedance of the line
V (d) V e j d
Z0
Z(d) I (d) V j d
e
Z0
The time-dependent voltage and current are
V (d, t ) Re[| V | e j e j d e j t ]
| V | Re[e j( t d ) ] | V | cos(t d )
I (d, t ) Re[| V | e j e j d e j t ] / Z0
| V |
|
|
V
j ( t d )
]
cos(t d )
Re[e
Z0
Z0
© Amanogawa, 2000 – Digital Maestro Series
100
Transmission Lines
The time-dependent power is
|
|
V
cos(t d )
P(d, t ) | V | cos(t d )
Z0
| V |2
cos2 (t d )
Z0
and the time average power absorbed by the load is
1 t | V |2
cos2 (t d ) dt
P(d) T 0 Z0
| V |2
2 Z0
© Amanogawa, 2000 – Digital Maestro Series
101
Transmission Lines
ZR jX (PURE REACTANCE)
ZR = j X
Z0
The reflection coefficient for a purely reactive load is
ZR Z0 jX Z0
R ZR Z0 jX Z0
( jX Z0 )( jX Z0 )
( jX Z0 )( jX Z0 )
© Amanogawa, 2000 – Digital Maestro Series
X 2 Z02
Z02 X 2
2j
XZ0
Z02 X 2
102
Transmission Lines
In polar form
R R exp( j)
where
R 2 2
X Z0
4 X 2 Z02
2
2
Z02 X 2
Z02 X 2
2
2
2 2
Z0 X
1
2
Z02 X 2
1 2 XZ0 tan X 2 Z2 0
The reflection coefficient has unitary magnitude, as in the case of
short and open circuit load, with zero time average power absorbed
by the load. Both voltage and current are finite at the load, and the
time-dependent power oscillates between positive and negative
values. This means that the load periodically stores and returns
powers to the line without dissipation.
© Amanogawa, 2000 – Digital Maestro Series
103
Transmission Lines
Reactive impedances can be realized with transmission lines
terminated by a short or by an open circuit. The input impedance of
a loss-less transmission line of length L terminated by a short
circuit is purely imaginary
2 f 2 L
Zin j Z0 tan L j Z0 tan L j Z0 tan vp At a specified frequency f, any reactance value can be obtained by
changing the length of the line from 0 to /2. Since the tangent
function is periodic, the behavior of the impedance will repeat
identically for each line increment of length /2.
Note that a similar periodic behavior is obtained when the length of
the transmission line is fixed and the frequency of operation is
changed.
© Amanogawa, 2000 – Digital Maestro Series
104
Transmission Lines
Shorted transmission line – Fixed frequency
L
L 0
0 L
L
4
4
L
4
2
L
2
3
L
2
4
3
L
4
3
L 4
Z in 0
s h o rt c irc u it
Im Z in 0
in d u c ta n c e
Z in o p e n c irc u it
c a p a c ita n c e
Im Z in 0
Z in 0
s h o rt c irc u it
Im Z in 0
in d u c ta n c e
Z in o p e n c irc u it
Im Z in 0
c a p a c ita n c e
© Amanogawa, 2000 – Digital Maestro Series
105
Z(L)/Zo = j tan( L)
Transmission Lines
Impedance of a short circuited transmission line
(fixed frequency, variable length)
40
30
20
inductive
Normalized Input Impedance
10
inductive
inductive
0
-10
capacitive
cap.
-20
-30
-40
0
100
200
300
400
500
L
[deg]
Line Length L
© Amanogawa, 2000 – Digital Maestro Series
106
Z(L)/Zo = j tan( L)
Transmission Lines
Impedance of a short circuited transmission line
(fixed length, variable frequency)
40
30
20
inductive
Normalized Input Impedance
10
inductive
inductive
0
-10
capacitive
cap.
-20
-30
-40
0
100
vp / (4L)
200
vp / (2L)
300
3vp / (4L)
400
vp / L
500
5vp / (4L)
[deg]
f
Frequency of operation
© Amanogawa, 2000 – Digital Maestro Series
107
Transmission Lines
For a transmission line of length L terminated by an open circuit,
the input impedance is again purely imaginary
Z0
j
Zin j
tan L Z0
Z0
j
2 2 f tan L tan L
v
p We can use the open circuited line to realize any reactance, but
starting from a capacitive value when the line length is very short.
The reactance varies linearly with frequency in the case of lumped
elements since
X L (inductance)
or
1
(capacitance)
X
C
This is not the case when the reactance is realized with a section of
transmission line.
© Amanogawa, 2000 – Digital Maestro Series
108
Transmission Lines
Open transmission line – Fixed frequency
L
L 0
0 L
L
4
4
L
4
2
L
2
3
L
2
4
3
L
4
3
L 4
Z in o p e n c irc u it
Im Z in 0
c a p a c ita n c e
Z in 0
s h o rt c irc u it
Im Z in 0
in d u c ta n c e
Z in o p e n c irc u it
Im Z in 0
c a p a c ita n c e
Z in 0
s h o rt c irc u it
Im Z in 0
in d u c ta n c e
© Amanogawa, 2000 – Digital Maestro Series
109
Normalized Input Impedance
Z(L)/ Zo = - j cotan( L)
Transmission Lines
Impedance of an open circuited transmission line
(fixed frequency, variable length)
40
30
20
inductive
inductive
inductive
10
0
-10
capacitive
capacitive
capacitive
-20
-30
-40
0
100
200
300
400
500
[deg]
L
Line Length L
© Amanogawa, 2000 – Digital Maestro Series
110
Normalized Input Impedance
Z(L)/ Zo = - j cotan( L)
Transmission Lines
Impedance of an open circuited transmission line
(fixed length, variable frequency)
40
30
20
inductive
inductive
inductive
10
0
-10
capacitive
capacitive
capacitive
-20
-30
-40
0
100
vp / (4L)
200
vp / (2L)
300
3vp / (4L)
400
vp / L
500
5vp / (4L)
[deg]
f
Frequency of operation
© Amanogawa, 2000 – Digital Maestro Series
111
Transmission Lines
It is possible to realize resonant circuits by using transmission
lines as reactive elements. For instance, consider the circuit below
realized with lines having the same characteristic impedance:
I
L1
short
circuit
Z0
L2
V
Zin1
Zin1 j Z0 tan L 1
© Amanogawa, 2000 – Digital Maestro Series
Z0
short
circuit
Zin2
Zin2 j Z0 tan L 2
112
Transmission Lines
The circuit is resonant if L1 and L2 are chosen such that an
inductance and a capacitance are realized.
A resonance condition is established when the total input
impedance of the parallel circuit is infinite (or, equivalently, when
the input admittance of the parallel circuit is zero)
1
1
0
j Z0 tan r L1 j Z0 tan r L 2 or
r
r
tan L1 tan L 2 vp
vp
with
2 r
r r vp
Since the tangent is a periodic function, there is a multiplicity of
possible resonant angular frequencies r that satisfy the condition
above. The solutions can be found by using a numerical procedure.
© Amanogawa, 2000 – Digital Maestro Series
113