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Dung Trinh, PhD
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1
Trinh Xuan Dung, PhD
Department of Telecommunications
Faculty of Electrical and Electronics Engineering
Ho Chi Minh city University of Technology
Chapter 1
Theory and Applications of
Transmission Lines
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Contents
2
1.
Introduction
2.
Lumped-Element Circuit Model for Transmission Lines
3.
Transmission Line Equations and Solutions
4.
Characteristic Impedance of Transmission Line
5.
Propagation constant and velocity
6.
Lossless and Lossy Transmission Lines
7.
Reflection Coefficient
8.
Transmission Line Impedance and Admittance
9.
Power Transmission on Transmission Lines
10. Standing Wave and Standing Wave Ratio
11. Impedance Matching
Problems
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1. Introduction
3
The previous class provided the analysis of EM field and wave traveling in the
free space.
This chapter
provides the analysis of
wave propagations in the
guided mediums : transmission lines.
For efficient point-to-point transmission of power and information, the source
energy must be directed or guided.
The key difference between circuit theory and Transmission Line is electrical
size.
At
low frequencies,
an electrical
circuit
is completely characterized by the
electrical
parameters
like
resistance,
inductance,
capacitance
etc.
and the
physical size of the electrical components plays no role in the circuit analysis.
As the frequency increases however,
the size of
the components becomes
important. The voltage and currents exist in the form of waves. Even a change
in the length of a simple connecting wire may alter the behavior of the circuit.
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1. Introduction
4
The circuit approach then has to be re-investigated with inclusion of the space
into
the
analysis.
This
approach
is
then
called
the
Transmission Line
approach.
Although
the
primary
objective
of
a
transmission
line
is
to
carry
electromagnetic energy efficiently from one location to other,
they find wide
applications in high frequency circuit design.
Also at high frequencies, the transmit time of the signals can not be ignored. In
the era of
high speed computers,
where data rates are approaching to few
Gb/sec,
the phenomena related to the electromagnetic waves,
like the bit
distortion, signal reflection, impedance matching play a vital role in high speed
communication networks.
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1. Introduction
5
At a given location along the line, find:
Current, voltage and power
Reflection coefficient, impedance, VSWR
Design real TLs, such as micro-strip lines, CPW lines
General problems of the chapter
Z
L
Z
S
V
S
I(l)
V(l)
z
l
0
Characteristic Impedance Z
0
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2. Lumped-Element Circuit Model for Transmission Lines
6
Examples of Transmission Lines:
Two-wire TL
Coaxial TL
Microstrip TL
Two-wire Transmission Line: consists of a pair of parallel conducting wires
separated by a uniform distance. Examples: telephone line, cable connecting
from roof-top antenna to TV receiver.
Coaxial
Transmission Line:
consists of inner conductor and and a coaxial
outer separated by a dielectric medium. Examples: TV Cable, etc.
Microstrip Transmission Line:
consists of
two parallel
conducting plates
separated by a dielectric slab. It can be fabricated inexpensively on PCB.
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2. Lumped-Element Circuit Model for Transmission Lines
7
Z
L
Z
S
V
S
I(l)
V(l)
z
l
0
Current i and voltage v
are
a
function
of
position
z
because
a
wire
is
never
a
“perfect” conductor.
It
will have:
Inductance (G)
Resistance (R)
Capacitance (C)
Conductance (L)
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2. Lumped-Element Circuit Model for Transmission Lines
8
R, L, G, and C are per-unit-length quantities defined as follows:
R = series resistance per unit length, for both conductors, in
/m.
L = series inductance per unit length, for both conductors, in H/m.
G = shunt conductance per unit length, in S/m.
C = shunt capacitance per unit length, in F/m.
o
Series
inductance
L represents
the
total
self-inductance
of
the
two
conductors.
o
Shunt capacitance C is due to the close proximity of the two conductors.
o
Series resistance R represents the resistance due to the finite conductivity
of the individual conductors.
o
Shunt conductance G is due to dielectric loss in the material between the
conductors.
o
R and G, therefore, represent loss.
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2. Lumped-Element Circuit Model for Transmission Lines
9
Table: Transmission Line Parameters of some common lines:
Further reading: Kỹ thuật SCT, p.25-p.33
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3. Transmission Line Equations and Solution
10
Applying Kirchoff’s Voltage Law (KVL):
Applying Kirchoff’s Current Law (KCL):
𝒗 𝒛 + 𝚫𝒛, 𝒕
= 𝒗 𝒛, 𝒕
− 𝑹𝚫𝒛𝒊 𝒛, 𝒕
− 𝑳𝚫𝒛
𝝏𝒊 𝒛, 𝒕
𝝏𝒕
𝒊 𝒛 + 𝚫𝒛, 𝒕
= 𝒊 𝒛, 𝒕
− 𝑮𝚫𝒛𝒗 𝒛, 𝒕
− 𝑪𝚫𝒛
𝝏𝒗 𝒛, 𝒕
𝝏𝒕
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3. Transmission Line Equations and Solution
11
Then:
𝒗 𝒛 + 𝚫𝒛, 𝒕
− 𝒗 𝒛, 𝒕
𝚫𝒛
= −𝑹𝒊 𝒛, 𝒕
− 𝑳
𝝏𝒊 𝒛, 𝒕
𝝏𝒕
𝒊 𝒛 + 𝚫𝒛, 𝒕
− 𝒊 𝒛, 𝒕
𝚫𝒛
= −𝑮𝒗 𝒛, 𝒕
− 𝑪
𝝏𝒗 𝒛, 𝒕
𝝏𝒕
When
∆𝑧 → 0
:
𝝏𝒗(𝒛, 𝒕)
𝝏𝒛
= −𝑹𝒊 𝒛, 𝒕
− 𝑳
𝝏𝒊 𝒛, 𝒕
𝝏𝒕
𝝏𝒊(𝒛, 𝒕)
𝝏𝒛
= −𝑮𝒗 𝒛, 𝒕
− 𝑪
𝝏𝒗 𝒛, 𝒕
𝝏𝒕
These equations
are “telegrapher’s
equations”.
There are infinite number
of
solutions
𝒗 𝒛, 𝒕
and
𝒗 𝒛, 𝒕
for the “telegrapher’s equations”. The problem can
be
simplified by assuming that
the
function of
time
is
“time
harmonic”
(sinusoidal).
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3. Transmission Line Equations and Solution
12
If a sinusoidal
voltage source with frequency
𝜔
is used to excite a linear,
time-invariant
circuit
then the voltage at
every point
with the circuit
will
likewise vary sinusoidal.
The voltage along a transmission line when excited by a sinusoidal
source
must have the form:
The time harmonic voltage at every location z along a transmission line:
where:
and
There is no reason to explicitly write the complex function
𝒆
𝒋𝝎𝒕
since the only
unknown is the complex function
𝑽 𝒛
.
Once we determine
𝑽 𝒛
,
we can
always recover the real function
𝒗 𝒛, 𝒕
:
𝒗 𝒛, 𝒕
= 𝒗 𝒛 𝒄𝒐𝒔 𝝎𝒕 + 𝝋 𝒛
= 𝕽𝒆 𝒗 𝒛 𝒆
𝒋𝝎𝒕
𝒆
𝒋𝝋 𝒛
𝑽 𝒛
= 𝒗 𝒛 𝒆
𝒋𝝋 𝒛
𝒗 𝒛
= 𝑽 𝒛
𝝋 𝒛
= 𝒂𝒓𝒈 𝑽 𝒛
𝒗 𝒛, 𝒕
= 𝕽𝒆 𝑽 𝒛 𝒆
𝒋𝝎𝒕
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3. Transmission Line Equations and Solution
13
Let’s assume that
𝒗 𝒛, 𝒕
and
𝒊 𝒛, 𝒕
each have the time harmonic form:
Then time derivative of these functions are:
The telegrapher’s equations thus become:
𝒗 𝒛, 𝒕
= 𝕽𝒆 𝑽 𝒛 𝒆
𝒋𝝎𝒕
𝒊 𝒛, 𝒕
= 𝕽𝒆 𝑰 𝒛 𝒆
𝒋𝝎𝒕
𝝏𝒗(𝒛, 𝒕)
𝝏𝒛
= 𝕽𝒆 𝒋𝝎𝑽 𝒛 𝒆
𝒋𝝎𝒕
𝝏𝒊(𝒛, 𝒕)
𝝏𝒛
= 𝕽𝒆 𝒋𝝎𝑰 𝒛 𝒆
𝒋𝝎𝒕
𝕽𝒆
𝝏𝑽 𝒛
𝝏𝒛
𝒆
𝒋𝝎𝒕
= 𝕽𝒆 − 𝑹 + 𝒋𝝎𝑳 𝑰 𝒛 𝒆
𝒋𝝎𝒕
𝕽𝒆
𝝏𝑰 𝒛
𝝏𝒛
𝒆
𝒋𝝎𝒕
= 𝕽𝒆 − 𝑮 + 𝒋𝝎𝑪 𝑽 𝒛 𝒆
𝒋𝝎𝒕
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3. Transmission Line Equations and Solution
14
𝝏𝑽 𝒛
𝝏𝒛
= − 𝑹 + 𝒋𝝎𝑳 𝑰 𝒛
𝝏𝑰 𝒛
𝝏𝒛
= − 𝑮 + 𝒋𝝎𝑪 𝑽 𝒛
Then the complex form of telegrapher’s equations are:
Note that these functions are not a function of time t.
Take the derivative with respect to z of the telegrapher’s equations, lead to:
Complex Value:
𝒗 𝒛 𝒆
𝒋𝝋 𝒛
𝝏
𝟐
𝑽 𝒛
𝝏𝒛
𝟐
= 𝑹 + 𝒋𝝎𝑳
𝑮 + 𝒋𝝎𝑪 𝑽 𝒛
𝝏
𝟐
𝑰 𝒛
𝝏𝒛
𝟐
= 𝑹 + 𝒋𝝎𝑳
𝑮 + 𝒋𝝎𝑪 𝑰 𝒛
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3. Transmission Line Equations and Solution
15
These equations can be written as:
where
𝜸 𝝎 =
𝑹 + 𝒋𝝎𝑳
𝑮 + 𝒋𝝎𝑪
is propagation constant.
Only
special
equations
satisfy
these
equations.
The
solution
of
these
equations can be found as:
where
γ = 𝛼 + 𝑗𝛽
.
𝝏
𝟐
𝑽 𝒛
𝝏𝒛
𝟐
= 𝜸
𝟐
𝝎 𝑽 𝒛
𝝏
𝟐
𝑰 𝒛
𝝏𝒛
𝟐
= 𝜸
𝟐
𝝎 𝑰 𝒛
𝑽 𝒛
= 𝑽
𝟎
+
𝒆
−𝜸𝒛
+ 𝑽
𝟎
−
𝒆
+𝜸𝒛
𝑰 𝒛
= 𝑰
𝟎
+
𝒆
−𝜸𝒛
+ 𝑰
𝟎
−
𝒆
+𝜸𝒛
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3. Transmission Line Equations and Solution
16
The current and voltage at a given point must have the form:
𝑽 𝒛
= 𝑽
𝟎
+
𝒆
−𝜶𝒛
𝒆
−𝒋𝜷𝒛
+ 𝑽
𝟎
−
𝒆
+𝜶𝒛
𝒆
+𝒋𝜷𝒛
𝑰 𝒛
= 𝑰
𝟎
+
𝒆
−𝜶𝒛
𝒆
−𝒋𝜷𝒛
+ 𝑰
𝟎
−
𝒆
+𝜶𝒛
𝒆
+𝒋𝜷𝒛
Z
L
Z
S
V
S
I(l)
V(l)
z
0
Incident wave
Reflected wave
𝑽
𝟎
+
𝑰
𝟎
+
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4. Characteristic Impedance of Transmission Line
17
The
terms
in
each
equation
describe
two
waves
propagating
in
the
transmission line,
one propagating in one direction (+z) and the other wave
propagating in the opposite direction (-z):
Then:
After re-arranging,
𝑰 𝒛
must be:
For the equations to be true for all z, I
0
and V
0
must be related as:
𝑽 𝒛
= 𝑽
𝟎
+
𝒆
−𝜸𝒛
+ 𝑽
𝟎
−
𝒆
+𝜸𝒛
𝑰 𝒛
= 𝑰
𝟎
+
𝒆
−𝜸𝒛
+ 𝑰
𝟎
−
𝒆
+𝜸𝒛
𝝏𝑽 𝒛
𝝏𝒛
= −𝜸𝑽
𝟎
+
𝒆
−𝜸𝒛
+ 𝜸𝑽
𝟎
−
𝒆
+𝜸𝒛
= − 𝑹 + 𝒋𝝎𝑳 𝑰 𝒛
𝑰 𝒛
=
𝜸
𝑹 + 𝒋𝝎𝑳
𝑽
𝟎
+
𝒆
−𝜸𝒛
−
𝜸
𝑹 + 𝒋𝝎𝑳
𝑽
𝟎
−
𝒆
+𝜸𝒛
= 𝑰
𝟎
+
𝒆
−𝜸𝒛
+ 𝑰
𝟎
−
𝒆
−𝜸𝒛
𝑰
𝟎
+
=
𝑽
𝟎
+
𝒁
𝟎
𝑰
𝟎
−
=
𝑽
𝟎
−
𝒁
and
where:
𝒁
𝟎
=
𝑹 + 𝒋𝝎𝑳
𝜸
=
𝑹 + 𝒋𝝎𝑳
𝑮 + 𝒋𝝎𝑪
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4. Characteristic Impedance of Transmission Line
18
𝑽
𝟎
+
and
𝑰
𝟎
+
are determined by the “boundary condition” (what is connected to
either
end of
the transmission line)
but
the ratio
𝑽
𝟎
+
𝑰
𝟎
+
is determined by the
parameters of the transmission line only.
Set
𝑍 = 𝑅 + 𝑗𝜔𝐿
and
𝑌 = 𝐺 + 𝑗𝜔𝐶
. Then:
Lossless transmission line:
In practice:
𝒁
𝟎
is always real.
In communications system:
𝒁
𝟎
= 𝟓𝟎𝛀
. In telecommunications: :
𝒁
𝟎
= 𝟕𝟓𝛀
.
𝒁
𝟎
= 𝑍Δ𝑥 +
1
𝑌Δ𝑥
∥ 𝑍
0
𝑥→0
𝑍
𝑌
=
𝑅 + 𝑗𝜔𝐿
𝐺 + 𝑗𝜔𝐶
𝒁
𝟎
=
𝐿
𝐶
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5a. Propagation Constant and Velocity
19
Propagation constant:
𝛼
: attenuation constant [Np/m] or [dB/m].
𝛽
: phase constant [rad/s].
The “wave velocity” is described by its “phase velocity”.
Since velocity is
change
in distance
with respect
to time,
we
need to first
express
the
propagation wave in its real form:
Let’s set the absolute phase to some arbitrary value:
𝝎𝒕 − 𝜷𝒛 = 𝝓
𝒄
.
Then:
and
𝜸 𝝎 = 𝜶 𝝎 + 𝒋𝜷 𝝎 =
𝑹 + 𝒋𝝎𝑳
𝑮 + 𝒋𝝎𝑪
𝑽
+
𝒛, 𝒕
= 𝕽𝒆 𝑽
+
𝒛 𝒆
−𝒋𝝎𝒕
= 𝑽
𝟎
+
𝒄𝒐𝒔 𝝎𝒕 − 𝜷𝒛
𝒛 =
𝝎𝒕 − 𝝓
𝒄
𝜷
𝒗
𝒑
=
𝝏𝒛
𝝏𝒕
=
𝝎
𝜷
𝛼 𝑑𝐵/𝑚 = 20𝑙𝑜𝑔
10
𝑒
𝛼 𝑁𝑝/𝑚
= 8.68𝛼 𝑁𝑝/𝑚
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20
5b. Line Impedance
The Line Impedance is NOT the T.L Impedance
𝒁
𝟎
. Recall that:
Therefore, the Line Impedance can be written as:
Or more specifically:
𝑽 𝒛
= 𝑽
+
𝒛 + 𝑽
−
𝒛
𝑰 𝒛
=
𝑽
+
𝒛 − 𝑽
−
𝒛
𝒁
𝟎
𝒁 𝒛
=
𝑽 𝒛
𝑰 𝒛
= 𝒁
𝟎
𝑽
+
𝒛 + 𝑽
−
𝒛
𝑽
+
𝒛 − 𝑽
−
𝒛
𝒁 𝒛
= 𝒁
𝟎
𝑽
𝟎
+
𝒆
−𝜸𝒛
+ 𝑽
𝟎
−
𝒆
𝜸𝒛
𝑽
𝟎
+
𝒆
−𝜸𝒛
− 𝑽
𝟎
−
𝒆
𝜸𝒛
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6. Lossless and Low-loss Transmission Line
21
In practice,
transmission lines have losses due to finite conductivity and/or
lossy dielectric but these losses are usually small.
In most practical microwave:
Losses may be neglected
Lossless Transmission Line.
Losses may be assumed to be very small
Low-loss Transmission Line.
Lossless Transmission Line
:
𝑹 = 𝟎, 𝑮 = 𝟎
Low-loss Transmission Line:
both conductor
and dielectric loss will
be
small,
and we can assume that
𝑅 ≪ 𝜔𝐿
and
𝐺 ≪ 𝜔𝐶
.
Then:
𝑅𝐺 ≪ 𝜔
2
𝐿𝐶
.
Then:
𝜸 𝝎 =
𝑹 + 𝒋𝝎𝑳
𝑮 + 𝒋𝝎𝑪 = 𝒋𝝎 𝑳𝑪
𝜶 𝝎 = 𝟎
𝜷 𝝎 = 𝝎 𝑳𝑪
𝜸 𝝎 ≃ 𝒋𝝎 𝑳𝑪 𝟏 − 𝒋
𝑹
𝝎𝑳
+
𝑮
𝝎𝑪
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6. Lossless and Low-loss Transmission Line
22
Using the Taylor series expansion* for:
Then:
Hence:
where:
𝑍
0
=
𝑅+𝑗𝜔𝐿
𝐺+𝑗𝜔𝐶
≃
𝐿
𝐶
* https://en.wikipedia.org/wiki/Taylor_series
𝜸 𝝎 ≃ 𝒋𝝎 𝑳𝑪 𝟏 − 𝒋
𝑹
𝝎𝑳
+
𝑮
𝝎𝑪
≃ 𝒋𝝎 𝑳𝑪 𝟏 −
𝒋
𝟐
𝑹
𝝎𝑳
+
𝑮
𝝎𝑪
𝟏 + 𝒙 ≃ 𝟏 +
𝒙
𝟐
−
𝒙
𝟐
𝟖
+
𝒙
𝟑
𝟏𝟔
+ ⋯
𝜶 ≃
𝟏
𝟐
𝑹
𝑪
𝑳
+ 𝑮
𝑳
𝑪
=
𝟏
𝟐
𝑹
𝒁
𝟎
+ 𝑮𝒁
𝟎
𝜷 ≃ 𝝎 𝑳𝑪
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7. Reflection Coefficient
23
Voltage Reflection Coefficient is defined as:
Current Reflection Coefficient is defined as:
𝚪
𝑽
𝒛
=
𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒆𝒅 𝑽𝒐𝒍𝒕𝒂𝒈𝒆
𝑰𝒏𝒄𝒊𝒅𝒆𝒏𝒕 𝑽𝒐𝒍𝒕𝒂𝒈𝒆
=
𝑽
𝟎
−
𝒆
+𝜸𝒛
𝑽
𝟎
+
𝒆
−𝜸𝒛
=
𝑽
𝟎
−
𝑽
𝟎
+
𝒆
𝟐𝜸𝒛
𝒁
𝑳
𝒁
𝑺
𝑽
𝑺
𝑰(𝒍
)
𝑽(𝒍)
𝒛
𝒍
𝟎
Incident wave
Reflected wave
𝑽 𝒛
= 𝑽
𝟎
+
𝒆
−𝜸𝒛
+ 𝑽
𝟎
−
𝒆
+𝜸𝒛
𝑰 𝒛
= 𝑰
𝟎
+
𝒆
−𝜸𝒛
+ 𝑰
𝟎
−
𝒆
+𝜸𝒛
𝚪
𝑰
𝒛
=
𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒆𝒅 𝑪𝒖𝒓𝒓𝒆𝒏𝒕
𝑰𝒏𝒄𝒊𝒅𝒆𝒏𝒕 𝑪𝒖𝒓𝒓𝒆𝒏𝒕
=
𝑰
𝟎
−
𝒆
+𝜸𝒛
𝑰
𝟎
+
𝒆
−𝜸𝒛
=
−
𝑽
𝟎
−
𝒁
𝟎
𝑽
𝟎
+
𝒁
𝟎
𝒆
𝟐𝜸𝒛
= −𝚪
𝑽
𝒛
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7. Reflection Coefficient
24
At load:
Note that:
Then:
At location z:
𝚪
𝑳
=
𝑽
𝟎
−
𝑽
𝟎
+
𝒆
𝟐𝜸𝒍
𝑽 𝒛
= 𝑽
𝟎
+
𝒆
−𝜸𝒛
+ 𝑽
𝟎
−
𝒆
+𝜸𝒛
𝑰 𝒛
= 𝑰
𝟎
+
𝒆
−𝜸𝒛
+ 𝑰
𝟎
−
𝒆
+𝜸𝒛
𝒁
𝑳
=
𝑽(𝒍)
𝑰(𝒍)
= 𝒁
𝟎
𝑽
𝟎
+
𝒆
−𝒋𝜷𝒍
+ 𝑽
𝟎
−
𝒆
𝒋𝜷𝒍
𝑽
𝟎
+
𝒆
−𝒋𝜷𝒍
− 𝑽
𝟎
−
𝒆
𝒋𝜷𝒍
= 𝒁
𝟎
𝟏 + 𝚪
𝑳
𝟏 − 𝚪
𝑳
𝚪
𝑳
=
𝒁
𝑳
− 𝒁
𝟎
𝒁
𝑳
+ 𝒁
𝟎
𝒁
𝑳
𝒁
𝑺
𝑽
𝑺
𝑰(𝒍
)
𝑽(𝒍)
𝒛
𝒍
0
Incident wave
Reflected wave
𝒛 = 𝒍 − 𝒅
𝚪 𝒛 = 𝒍 − 𝒅 =
𝑽
𝟎
−
𝑽
𝟎
+
𝒆
𝟐𝜸𝒛
=
𝑽
𝟎
−
𝑽
𝟎
+
𝒆
𝟐𝜸 𝒍−𝒅
=
𝑽
𝟎
−
𝑽
𝟎
+
𝒆
𝟐𝜸𝒍
𝒆
−𝟐𝜸𝒅
= 𝚪
𝑳
𝒆
−𝟐𝜸𝒅
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7. Reflection Coefficient - Representation on a complex plane
25
Reflection Coefficient at
𝑧 = 𝑙 − 𝑑
:
where:
𝛾 = 𝛼 + 𝑗𝛽
.
Then:
𝚪 𝒛 = 𝒍 − 𝒅 = 𝚪
𝑳
𝒆
−𝟐𝜸𝒅
𝚪 𝒛 = 𝒍 − 𝒅 = 𝚪
𝑳
𝒆
−𝟐𝜶𝒅
𝒆
−𝟐𝒋𝜷𝒅
/ 2
2
2
2
2
2
2
2
d
d
d
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26
Quiz 1: A 6-m section of 150
Ω
lossless line is driven by a source with
𝑣
𝑔
𝑡
= 5 cos 8𝜋 × 10
7
𝑡 − 30
0
(𝑉)
And
𝑍
𝑔
= 150Ω
. If the line, which has a relative permittivity
𝜀
𝑟
= 2.25
is terminated
in a load
𝑍
𝐿
= (150 − 𝑗50)Ω
, find:
a.
𝜆
on the line. Note that:
𝜆 =
𝑣
𝑃
𝑓
where
𝑣
𝑃
=
𝑐
𝜀
𝑟
.
b. The reflection coefficient at the load.
c. The input impedance.
d. The input voltage V
i
and time-domain voltage v
i
(t).
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27
8. Transmission Line Impedance and Admittance
𝒁(𝒙)
The line impedance at
𝑧 = 𝑙 − 𝑑
:
Note that:
Then the line impedance can be specified:
More specifically:
𝒁 𝒛
= 𝒁
𝟎
𝑽
𝟎
+
𝒆
−𝒋𝜸𝒛
+ 𝑽
𝟎
−
𝒆
𝒋𝜸𝒛
𝑽
𝟎
+
𝒆
−𝒋𝜸𝒛
− 𝑽
𝟎
−
𝒆
𝒋𝜸𝒛
𝚪 𝒛 = 𝒍 − 𝒅 =
𝑽
𝟎
−
𝑽
𝟎
+
𝒆
𝟐𝜸𝒍
𝒆
−𝟐𝜸𝒅
= 𝚪
𝑳
𝒆
−𝟐𝜸𝒅
𝒁 𝒛
= 𝒁
𝟎
𝟏 + 𝚪 𝒛
𝟏 − 𝚪 𝒛
𝒁 𝒛
= 𝒁
𝟎
𝒁
𝑳
+ 𝒁
𝟎
𝒆
𝜸𝒅
+ 𝒁
𝑳
− 𝒁
𝟎
𝒆
−𝜸𝒅
𝒁
𝑳
+ 𝒁
𝟎
𝒆
𝜸𝒅
− 𝒁
𝑳
− 𝒁
𝟎
𝒆
−𝜸𝒅
= 𝒁
𝟎
𝒁
𝑳
𝒆
𝜸𝒅
+ 𝒆
−𝜸𝒅
+ 𝒁
𝟎
𝒆
𝜸𝒅
− 𝒆
−𝜸𝒅
𝒁
𝑳
𝒆
𝜸𝒅
− 𝒆
−𝜸𝒅
+ 𝒁
𝟎
𝒆
𝜸𝒅
+ 𝒆
−𝜸𝒅
= 𝒁
𝟎
𝒁
𝑳
𝒄𝒐𝒔𝒉 𝜸𝒅 + 𝒁
𝟎
𝒔𝒊𝒏𝒉 𝜸𝒅
𝒁
𝑳
𝒔𝒊𝒏𝒉 𝜸𝒅 + 𝒁
𝟎
𝒄𝒐𝒔𝒉 𝜸𝒅
= 𝒁
𝟎
𝒁
𝑳
+ 𝒁
𝟎
𝒕𝒂𝒏𝒉 𝜸𝒅
𝒁
𝟎
+ 𝒁
𝑳
𝒕𝒂𝒏𝒉 𝜸𝒅
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28
8. Transmission Line Impedance and Admittance
Lossless T.L (
𝛼 = 0
):
𝑍
𝐿
= 𝑍
0
:
𝑍
𝐿
= 𝑗𝑋
𝐿
:
𝑍
𝐿
= 0
:
𝑍
𝐿
= ∞
:
𝒁(𝒙)
𝒁 𝒛
= 𝒁
𝟎
𝒁
𝑳
+ 𝒋𝒁
𝟎
𝒕𝒂𝒏 𝜷𝒅
𝒁
𝟎
+ 𝒋𝒁
𝑳
𝒕𝒂𝒏 𝜷𝒅
𝒁 𝒛
= 𝒁
𝟎
𝒁
𝑳
+ 𝒋𝒁
𝟎
𝒕𝒂𝒏 𝜷𝒅
𝒁
𝟎
+ 𝒋𝒁
𝑳
𝒕𝒂𝒏 𝜷𝒅
= 𝒁
𝟎
𝒁 𝒛
= 𝒁
𝟎
𝒋𝑿
𝑳
+ 𝒋𝒁
𝟎
𝒕𝒂𝒏 𝜷𝒅
𝒁
𝟎
− 𝑿
𝑳
𝒕𝒂𝒏 𝜷𝒅
has imaginary part only
𝒁 𝒛
= 𝒋𝒁
𝟎
𝒕𝒂𝒏 𝜷𝒅
pure reactance
𝒁 𝒛
=
𝒁
𝟎
𝒋𝒕𝒂𝒏 𝜷𝒅
= −𝒋𝒁
𝟎
𝒄𝒐𝒕𝒂𝒏 𝜷𝒅
pure reactance
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29
8. Transmission Line Impedance and Admittance
Inductance
Capacitance
At load
Shorted-circuit
T.L can be used to realize inductors
or
capacitors
at
specific
frequencies
Distributed Components.
𝑍
𝐿
= 0
:
𝒁 𝒛
= 𝒋𝒁
𝟎
𝒕𝒂𝒏 𝜷𝒅 = 𝒋𝑿(𝒅)
Pure reactance
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30
8. Transmission Line Impedance and Admittance
Open-circuit
T.L can be
used to realize
inductors
or
capacitors
at
specific
frequencies
Distributed Components.
𝑍
𝐿
= ∞
:
𝒁 𝒛
= −𝒋𝒁
𝟎
𝒄𝒐𝒕𝒂𝒏 𝜷𝒅 = 𝒋𝑿(𝒅)
Pure reactance
Inductance
Capacitance
At load
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31
8. Transmission Line Impedance and Admittance
If
𝑍
𝐿
→ ∞:
𝒁
𝒊𝒏
= 𝟎
.
If
𝑍
𝐿
= 0:
𝒁
𝒊𝒏
→ ∞
.
A quarter wavelength TL:
𝒁
𝑳
𝒍 =
𝝀
𝟒
𝒁
𝟎
𝒁
𝒊𝒏
𝒁
𝒊𝒏
= 𝒁
𝟎
𝒁
𝑳
+ 𝒋𝒁
𝟎
𝒕𝒂𝒏 𝜷𝒅
𝒁
𝟎
+ 𝒋𝒁
𝑳
𝒕𝒂𝒏 𝜷𝒅
=
𝒁
𝟎
𝟐
𝒁
𝑳
Application for impedance transformation:
𝒁
𝒊𝒏
=
𝒁
𝟎
𝟐
𝒁
𝑳
→ 𝒁
𝟎
=
𝒁
𝒊𝒏
𝒁
𝑳
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Dept. of Telecoms Engineering
32
8. Transmission Line Impedance and Admittance
Example 1:
The open-circuit
and short-circuit
impedances
measured at
the input
terminal of a very low-loss TL of length 1.5m which is less than a quarter wavelength,
are respectively -54.6j
(
Ω) and 103j
(
Ω)
a.
Find Z
0
and
𝛾
of the line.
b.
Without changing the frequency, find the input impedance of a short-circuited TL
that is twice the given length.
c.
How long should the short-circuited TL be in order to appear as an open circuit at
the input terminals?
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33
8. Transmission Line Impedance and Admittance
Quiz 2: A voltage generator with
𝑣
𝑔
𝑡
= 5 cos 2𝜋 × 10
9
𝑡
(𝑉)
and internal impedance is
𝑍
𝑔
= 50
Ω is connected to a
50
Ω lossless T.L. The line
length is 5cm and the line is terminated in a load with impedance
𝑍
𝐿
= 100 − 𝑗100
Ω.
Determine:
a.
Reflection coefficient at load
Γ
𝐿
?
b.
𝑍
𝑖𝑛
at the input of the T.L.
c.
The input voltage
𝑣
𝑖
𝑡
and input current
𝑖
𝑖
𝑡
?
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34
9. Power Transmission on Transmission Lines
Steps to find
𝑽
𝟎
+
and
𝑽
𝟎
−
:
1.
𝚪
𝑳
=
𝒁
𝑳
−𝒁
𝟎
𝒁
𝑳
+𝒁
𝟎
2.
𝚪
𝒊𝒏
= 𝚪
𝑳
𝒆
−𝟐𝜸𝒍
3.
𝒁
𝒊𝒏
= 𝒁
𝟎
𝟏+𝚪
𝒊𝒏
𝟏−𝚪
𝒊𝒏
𝒁
𝑳
𝒍 =
𝝀
𝟒
𝒁
𝟎
𝒁
𝒊𝒏
𝑽 𝒛
= 𝑽
𝟎
+
𝒆
−𝜸𝒛
+ 𝑽
𝟎
−
𝒆
+𝜸𝒛
𝑰 𝒛
= 𝑰
𝟎
+
𝒆
−𝜸𝒛
+ 𝑰
𝟎
−
𝒆
+𝜸𝒛
4.
𝑽
𝒊𝒏
= 𝑽
𝑺
𝒁
𝒊𝒏
𝒁
𝒊𝒏
+𝒁
𝑺
5.
𝑽
𝒊𝒏
= 𝑽
𝟎
+
+ 𝑽
𝟎
−
= 𝑽
𝟎
+
𝟏 + 𝚪
𝒊𝒏
6.
𝑽
𝟎
+
=
𝑽
𝒊𝒏
𝟏+𝚪
𝒊𝒏
𝑽
𝟎
−
= 𝚪
𝒊𝒏
𝑽
𝟎
+
If
𝒁
𝑳
= 𝒁
𝟎
:
𝑽
𝟎
+
=
𝑽
𝑺
𝟐
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Dept. of Telecoms Engineering
35
9. Power Transmission on Transmission Lines
𝒁
𝑳
𝒁
𝑺
𝑽
𝑺
𝑰(𝒍
)
𝑽(𝒍)
𝒛
𝒍
𝟎
𝑃
𝑖𝑛𝑐
𝑃
𝑡
𝑃
𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑
The time average power flows along a transmission line:
𝑷
𝒕
=
𝟏
𝟐
𝕽𝒆 𝑽 𝒛 𝑰
∗
(𝒛)
=
𝟏
𝟐𝒁
𝟎
𝕽𝒆
𝑽
𝟎
+
𝒆
−𝜶𝒛
𝒆
−𝒋𝜷𝒛
+ 𝑽
𝟎
−
𝒆
𝜶𝒛
𝒆
𝒋𝜷𝒛
𝑽
𝟎
+∗
𝒆
−𝜶𝒛
𝒆
𝒋𝜷𝒛
− 𝑽
𝟎
−∗
𝒆
𝜶𝒛
𝒆
−𝒋𝜷𝒛
=
𝟏
𝟐𝒁
𝟎
𝕽𝒆
𝑽
𝟎
+
𝟐
𝒆
−𝟐𝜶𝒛
− 𝑽
𝟎
+
𝑽
𝟎
−∗
𝒆
−𝒋𝟐𝜷𝒛
+ 𝑽
𝟎
+∗
𝑽
𝟎
−
𝒆
𝒋𝟐𝜷𝒛
− 𝑽
𝟎
−
𝟐
𝒆
𝟐𝜶𝒛
=
𝟏
𝟐𝒁
𝟎
𝑽
𝟎
+
𝟐
𝒆
−𝟐𝜶𝒛
− 𝑽
𝟎
−
𝟐
𝒆
𝟐𝜶𝒛
=
𝑽
𝟎
+
𝟐
𝟐𝒁
𝟎
𝒆
−𝟐𝜶𝒛
𝟏 − 𝚪
𝒛
𝟐
= 𝑷
𝒊𝒏𝒄
− 𝑷
𝒓𝒆𝒇𝒍
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36
9. Power Transmission on Transmission Lines
𝒁
𝑳
𝒛
𝒍
𝟎
𝑃
𝑖𝑛𝑐
𝑃
𝑡
𝑃
𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑
The time average absorbed by load:
𝑷
𝒕
=
𝟏
𝟐
𝕽𝒆 𝑽
𝑳
𝑰
𝑳
∗
=
𝑽
𝟎
+
𝟐
𝟐𝒁
𝟎
𝒆
−𝟐𝜶𝒍
𝟏 − 𝚪
𝑳
𝟐
= 𝑷
𝒊𝒏𝒄
− 𝑷
𝒓𝒆𝒇𝒍
𝒁
𝑳
𝒛
𝒍
𝟎
𝑷
𝒊𝒏𝒄,𝟎
=
𝑽
𝟎
+
𝟐
𝟐𝒁
𝟎
𝑷
𝒊𝒏𝒄,𝒍
=
𝑽
𝟎
+
𝟐
𝟐𝒁
𝟎
𝒆
−𝟐𝜶𝒍
𝑷
𝒓𝒆𝒇𝒍,𝒍
=
𝑽
𝟎
+
𝟐
𝟐𝒁
𝟎
𝒆
−𝟐𝜶𝒍
𝚪
𝑳
𝟐
𝑷
𝒓𝒆𝒇𝒍,𝟎
=
𝑽
𝟎
+
𝟐
𝟐𝒁
𝟎
𝒆
−𝟒𝜶𝒍
𝚪
𝑳
𝟐
Power Flow:
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37
10. Standing Wave and Standing Wave Ratio
𝒁(𝒙)
𝑽 𝒛
= 𝑽
𝟎
+
𝒆
−𝜸𝒛
+ 𝑽
𝟎
−
𝒆
+𝜸𝒛
= 𝑽
𝟎
+
𝒆
−𝜸𝒛
(𝟏 +
𝑽
𝟎
−
𝑽
𝟎
+
𝒆
+𝟐𝜸𝒛
)
= 𝑽
𝟎
+
𝒆
−𝜸𝒛
(𝟏 + 𝚪 𝒛 )
If
𝛼 = 0
:
𝑽 𝒛
= 𝑽
𝟎
+
𝒆
−𝒋𝜷𝒛
𝟏 + 𝚪 𝒛
→
𝑽 𝒛
= 𝑽
𝟎
+
𝟏 + 𝚪 𝒛
Then:
𝑽 𝒛
𝒎𝒂𝒙
= 𝑽
𝟎
+
𝟏 + 𝚪
𝑳
𝒘𝒉𝒆𝒏 𝚪 𝒛
= 𝚪
𝑳
𝑽 𝒛
𝒎𝒊𝒏
= 𝑽
𝟎
+
𝟏 − 𝚪
𝑳
𝒘𝒉𝒆𝒏 𝚪 𝒛
= − 𝚪
𝑳
𝚪 𝒛 = 𝒍 − 𝒅 = 𝚪
𝑳
𝒆
−𝟐𝜸𝒅
𝑽𝑺𝑾𝑹 =
𝑽 𝒛
𝒎𝒂𝒙
𝑽 𝒛
𝒎𝒊𝒏
=
𝟏 + 𝚪
𝑳
𝟏 − 𝚪
𝑳
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38
10. Standing Wave and Standing Wave Ratio
𝒁(𝒙)
We have:
𝑽 𝒛
= 𝑽
𝟎
+
𝒆
−𝒋𝜷𝒛
𝟏 + 𝚪 𝒛
𝚪 𝒛 = 𝒍 − 𝒅 = 𝚪
𝑳
𝒆
−𝟐𝜷𝒅
= 𝚪
𝑳
𝒆
𝒋𝜽
𝒓
𝒆
−𝟐𝒋𝜷𝒅
where:
Then:
𝑽 𝒛
= 𝑽
𝟎
+
𝟏 + 𝚪 𝒛
= 𝑽
𝟎
+
𝟏 + 𝚪
𝑳
𝟐
+ 𝟐 𝚪
𝑳
𝒄𝒐𝒔 𝟐𝜷𝒅 − 𝜽
𝒓
𝟏
𝟐
Matched TL:
𝑍
𝐿
= 𝑍
0
→
𝚪 = 𝟎
Short circuit TL:
𝑍
𝐿
= 0 →
𝚪 = −𝟏
Open circuit TL:
𝑍
𝐿
= ∞ →
𝚪 = 𝟏
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39
10. Standing Wave and Standing Wave Ratio
𝑽 𝒛
= 𝑽
𝟎
+
𝟏 + 𝚪
𝑳
𝟐
+ 𝟐 𝚪
𝑳
𝒄𝒐𝒔 𝟐𝜷𝒅 − 𝜽
𝒓
𝟏
𝟐
𝑽 𝒛
= 𝑽 𝒛
𝒎𝒊𝒏
= 𝑽
𝟎
+
𝟏 − 𝚪
𝑳
𝑤ℎ𝑒𝑛:
𝑐𝑜𝑠 2𝛽𝑑 − 𝜃
𝑟
= −1 ↔ 2𝛽𝑑 − 𝜃
𝑟
= 2𝑛 + 1 𝜋
𝑽 𝒛
= 𝑽 𝒛
𝒎𝒂𝒙
= 𝑽
𝟎
+
𝟏 + 𝚪
𝑳
𝑤ℎ𝑒𝑛:
𝑐𝑜𝑠 2𝛽𝑑 − 𝜃
𝑟
= 1 ↔ 2𝛽𝑑 − 𝜃
𝑟
= 2𝑛𝜋
https://www.youtube.com/watch?v=yCZ1zFPvrIc
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10. Standing Wave and Standing Wave Ratio
Example 2:
in an unknown load impedance is found to be 3.0.
The
distance between successive voltage minima is 30cm and the first minimum is
located at 12cm from the load. Determine:
a.
The reflection coefficient
Γ
.
b.
The load impedance
𝑍
𝐿
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Exercises
Exercise
1:
Two half-wave
dipole
antennas,
each with impedance
of
75Ω are
connected in parallel through a pair of T.L. and the combination is connected to a feed
T.L. as shown in the following figure. All lines are 50Ω lossless.
a.
Calculate
𝑍
𝑖𝑛1
b.
Calculate
𝑍
𝑖𝑛
of the feed line.
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Exercises
Exercise 2:
A 50Ω lossless line of length
𝑙 = 0.15𝜆
connects a 300MHz generator
with
𝑉
𝑔
= 300𝑉
and
𝑍
𝑔
= 50
Ω to a load
𝑍
𝐿
= 75
Ω.
a.
Compute
𝑍
𝑖𝑛
b.
Compute
𝑉
𝑖
and
𝐼
𝑖
.
c.
Compute the time-average power delivered to the line,
𝑃
𝑖𝑛
=
1
2
ℝ𝑒 𝑉
𝑖
𝐼
𝑖
.
d.
Compute
𝑉
𝐿
,
𝐼
𝐿
and the
time-average
power
delivered to the
load,
𝑃
𝐿
=
1
2
ℝ𝑒 𝑉
𝐿
𝐼
𝑙
.
e.
Compute the time-average power
delivered by the generator
and time-average
power dissipated by in
𝑍
𝑔
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Exercises
Exercise 3:
In addition to not
dissipating power,
a lossless line has two important
features:
(1) It is dispersionless (v
p
is independent of frequency).
(2) Its characteristic impedance Z
0
is real.
Sometimes it is not possible to design a T.L. such that
𝑅′ ≪ 𝜔𝐿′
and
𝐺′ ≪ 𝜔𝐶′
but it
is possible to choose the dimensions of the line and its material
properties so as to
satisfy the condition
𝑅’𝐶’ = 𝐿’𝐺’
(distortionless line).
Such a line is called a distortionless line because despite the fact that it is not lossless,
it
nonetheless possesses the previous mentioned features of the lossless line.
Show
that for a distortionless line:
𝛼 = 𝑅′
𝐶′
𝐿′
𝛽 = 𝜔 𝐿
′
𝐶′
𝑍
0
=
𝐿′
𝐶′
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Exercises
Exercise 4:
A 300Ω lossless line is connected to a complex load composed of
a
resistor
𝑅 = 600
Ω and an inductor with
𝐿 = 0.02𝑚𝐻
. At 10MHz, determine:
a.
Reflection coefficient at load
Γ
𝐿
?
b.
Voltage Standing Wave Ratio (VSWR).
c.
Location of voltage maximum nearest the load.
d.
Location of current maximum nearest the load.
Exercise 5: On a 150Ω lossless line, the following observations were noted: distance
of first voltage minimum from load is 3cm,
distance of first voltage maximum from
load is 9cm and VSWR=3. Find
𝑍
𝐿
?
Exercise 6: A load with impedance
𝑍
𝐿
= 25 − 𝑗50
Ω is to be connected to a lossless
T.L.
with characteristic impedance
𝑍
0
with chosen
𝑍
0
such that
the VSWR is the
smallest possible. What should
𝑍
0
be?
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Exercises
Exercise
7:
A 100MHz
FM broadcast
station uses
a
300Ω T.L.
between the
transmitter and a tower-mounted half-wave dipole antenna. The antenna impedance is
73Ω. You are asked to design a quarter-wavelength transformer to match the antenna
to the line.
a.
Determine the length and characteristic impedance of the quarter-wavelength
section?
b.
If the quarter-wavelength is a two-wire line with
𝐷 = 2.5𝑐𝑚
and the wires are
embedded in polystyrene with
𝜀
𝑟
= 2.6.
Determine the physical length of the
quarter-wave section and the radius of the two wire conductor.
Note that the characteristic parameters of T.Ls are given in the following table:
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Exercises
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Exercises
Exercise 8: Consider the circuit below.
A generator with
𝑅
0
= 75Ω
is connected to a
complex of
𝑍
𝐿
= 100 + 𝑗100
Ω through a T.L. of arbitrary length with
𝑍
0
= 75
Ω and
𝑣
𝑃
= 0.8𝑐
. Using the Smith Chart, evaluate the line for stub matching. The generator is
operating at 100MHz. Find
a.
The electrical length of
𝜆
of the T.L.
b.
The normalized load impedance.
c.
The closest stub location as measured from the load.
d.
The length of the stub at the closest location.
e.
The lumped load element value that could take the place of the stub at the nearest
location.
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Exercises
Exercise 9: A Vector Network Analyzer (VNA) is attached to the end of a lossless, 15m
long T.L. (
50Ω
,
𝜖
𝑟
= 2.3
) operating at 220MHz. The VNA shows an input impedance
of
𝑍
𝑖𝑛
= 75 − 𝑗35
Ω. Using the Smith Chart:
a.
Find the VSWR on the line.
b.
Find the normalized, denormalized and equivalent circuit of the load impedance
𝑍
𝐿
at the far end of the line.
The equivalent
circuit
must
show the correct
schematic
symbols (L and/or R and/or C) and the values of each symbol.
c.
Find the normalized load admittance Y
L
at the far end of the line. The length of the
stub at the closest location.
d.
Find the distance in meters from the load to the first matching point.
e.
What is the normalized admittance at the first match point?
f.
Find the shortest stub to match the susceptance found at the first match point. Give
the length of the stub in meters.
g.
If fabrication of a coaxial stub was not feasible but a lumped matching element was
necessary, draw the component schematic symbol and give its value.
h.
After the matching network is connected, where do standing waves exist and where
do they not exist in this system? What is the SWR at the input to the line?
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Q&A
49
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