TRANSMISSION LINES By
Dr. Sikder Sunbeam Islam
EEE, IIUC.
INTRODUCTION
Transmission lines (TL) are interconnections that convey
electromagnetic energy from one point to another. Example
include two-wire, coaxial, strip, microstrip, waveguides, optical
fiber lines etc.
Transmission lines basically consists of two or more parallel
conductors used to connect a source to load. Source may be
generator, transmitter, oscillator and load may be factory,
antenna, oscilloscope, etc.
(c)
Fig.1. Example TL: (a)coaxial line, (b)two-wire line, (c) waveguide. 2
TRANSMISSION LINE: CIRCUIT THEORY
Transmission line (TL) can be described by circuit parameters
that are distributed throughout its length.
Consider a differential length ∆z of a TL that is described by the
following four parameters:
R= Series Resistance per unit length (both conductor),ohm/m
L= Series Inductance per unit length (both conductor),H/m
G=Shunt Conductance per unit length (both conductor),S/m
C=Shunt Capacitance per unit length (both conductor),H/m
Fig.2: Distributed parameters of two conductor TL
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TRANSMISSION LINE: GENERAL EQUATION
Fig.3 shows a equivalent electric circuit of such line segment.
The quantities v(z, t) and v( z+∆z, t) denotes the instantaneous
voltages at z and z+∆z, respectively. Similarly, i(z,t) and i(z+∆z, t)
denotes the instantaneous currents at z and z+∆z, respectively.
Applying Kirchhoff’s voltage law, we find
Fig.3:
Equivalent
circuit of
differential
length ∆z
of two wire
TL.
------------------(1) Or, 4
TRANSMISSION LINE: GENERAL EQUATION (CONTINUE.)
In the limit as ∆z→0, from equ.(1),
Now, applying Kirchhoff’s Current law to node N, we find
On dividing by ∆z and ∆z→0, from equ.(3),
Equation (2) and (4) are General Transmission Line Equations.
------------------(2)
------------------(3)
------------------(4)
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TRANSMISSION LINE: GENERAL EQUATION (CONTINUE.)
For simplifying the TL equations Assuming Harmonic Time
dependence,
Where, V(z) and I(z) are the phasors form of v(z, t) and i(z, t)
respectively. Therefore, equ. (2) and (4) becomes,
Equations (6.1) and (6.2) are Time harmonic TL-Equations. Equ.(6.1)
and (6.2) can be combined to solve V(z) and I(z) . Now taking second
derivative to equ. (6.1),
------------------(5.1)
------------------(5.2)
------------------(6.1)
------------------(6.2)
6 −𝑑2𝑉(𝑧)
𝑑𝑧2 = (R+j𝜔𝐿) 𝑑𝐼(𝑧)
𝑑𝑧
𝑑2𝑉(𝑧)
𝑑𝑧2 = (R+j𝜔𝐿) (G+j𝜔𝐶)V(z)
So,
Here, 𝛾 is propagation constant and α and β are attenuation
(Np/m) and phase (rad/m) constant respectively.
Solution of Equ. (7) and (8), are,
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------------------(7)
and similarly, ------------------(8)
TRANSMISSION LINE: GENERAL EQUATION (CONTINUE.)
Where,
------------------(9)
------------------(10.1)
------------------(10.2)
Here, +ve and –ve sign indicates the wave travelling in the +z and –z
direction respectively.
Thus we obtain instantaneous voltage expression,
The ratio of the voltage and the current at any (point of) z for an
infinitely long line is independent of z and is called the
characteristic impedance (Zo) of the line.
Lossless Line ( R=G=0): A TL is said to be lossless if
conductors of the lines are perfect (σ≈∞) and the dielectric
medium separating them is lossless (σ ≈ 0) .
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TRANSMISSION LINE: GENERAL EQUATION (CONTINUE.)
------------------(10.3)
------------------(11)
------------------(12.1)
------------------(12.2)
------------------(12.3)
Distortion less Line (R/L=G/C): Distortion less line is one in
which attenuation constant is frequency independent where
phase constant is linearly dependent on frequency. So in this
case,
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TRANSMISSION LINE: GENERAL EQUATION (CONTINUE.)
𝛽
------------------(12.4)
------------------(12.5)
TRANSMISSION LINE PARAMETERS
Comparison of Parameters of Two-wire and coaxial
cable:
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PROBLEM.1:
So,
Problem.2: It is found that attenuation on a 50 Ω distortion less
transmission line is 0.01 (dB/m). The line has a capacitance of 0.1
(nF/m).
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1Np=8.686 dB
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Voltage Standing Wave Ratio(VSWR): It is ratio of maximum to minimum
value of voltage is called VSWR. ------------------(13)
Voltage Reflection Coefficient (ГL): Voltage Reflection Coefficient at any
point on the line is the ratio of magnitude of reflected wave to that of incident
wave. ------------------(14)
Where, Zo= Characteristics Impedance; ZL= Load Impedance
INPUT IMPEDANCE OF TL
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------------------(15)
------------------(16)
Zin = Input impedance,
l= line length or
distance from the load.
ZL= Load Impedance
Zo=Characteristics Impedance
MICROSTRIP LINE
Microstrip line (ML) is a form of parallel plate TL consists of dielectric substrate sitting on a grounded conducting plane with a thin narrow metal strip on top of the substrate (Seen in Fig.5).
Few Equations for ML:
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Zo= Characteristics Impedance
(infinite long TL)
------(17)
Fig.5
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IMPEDANCE MATCHING: SMITH CHART
Smith Chart is a valuable graphical tool. It is a graphical
indications of the impedance of TL as one moves along the line.
Invented by Phillip H. Smith in 1939
Used to solve a variety of transmission line and waveguide problems.
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Fig.6:
SMITH CHART: BASIC USES
For evaluating the rectangular components, or the magnitude
and phase of an input impedance or admittance, voltage,
current, and related transmission functions at all points
along a transmission line, including:
Complex voltage and current reflections coefficients
Complex voltage and current transmission coefficients
Power reflection and transmission coefficients
Reflection Loss
Return Loss
Standing Wave Loss Factor
Maximum and minimum of voltage and current, and SWR
Shape, position, and phase distribution along voltage and
current standing waves. 18
Smith Chart constructed within a circle of unit radius (Г≤1). The chart is based on the equ.(14) , Where, Г𝑟and Г𝑖 are real and imaginary part of reflection coefficient (Г).
For a TL, the all impedance in the chart is normalized by a characterized impedance Zo. So, normalized load impedance ZL will be 𝑧𝐿,
The r and x are normalized resistance and reactance respectively. Different values of r yields circles having centers on Г𝑟-axis and centers of all x-circles lie on Г𝑟=1. Those, x >0 (inductive reactance)lie aboveГ𝑟-axis and x <0 (capacitivereactance)lie b𝐞𝐥𝐨𝐰 Г𝑟-axis. [See Fig.7]
Thex=0 circle becomes the Г𝑟-axis line.
All r-circles pass through (Г𝑟=1 , Г𝑖=0) point. Centers of all r-circles lie on Г𝑟-axis . The r=0 circle is the largest , centered at origin (of unity radius ).
At Psc on the chart r=0,x=0,(ZL=0+j0) represents short circuit on the TL. At Pocon the chart r=∞,x=∞,(ZL=∞+j∞)represents opencircuit on the TL.
The complete round-trip (3600) around the Smith Chart represents a
distance of 𝝀
𝟐 on the TL. The 𝛌 distance on the TL correspond to
𝟕𝟐𝟎𝟎movement on the chart.
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SMITH CHART: FEATURES
------(14) Or,
The clockwise movement on the chart regarded as moving towards
generator (away from the load).Similarly, The counter-clockwise
movement on the chart regarded as moving towards Load (away from
the source) [See Fig.8].
There are 3-scales around the periphery of the smith chart, seen in
Fig.8. The Outer most scale-determines the distance on the line
from Generator end in terms of 𝛌. The next one- determines the
distance on the line from Load end. The innermost scale is used for
determining 𝜽𝒓.
The𝑽𝒎𝒂𝒙occur where 𝒁𝒊𝒏,𝒎𝒂𝒙 is located on the chart; that is on the
positive Г𝑟-axis or on OPoc . The𝑽𝒎𝒊𝒏occur where 𝒁𝒊𝒏,𝒎𝒊𝒏 is located on the
chart; that is on the negative Г𝑟-axis or on OPsc . 𝑽𝒎𝒂𝒙 and 𝑽𝒎𝒊𝒏 are
𝟏𝟖𝟎𝟎apart .
Smith chart can be used both as impedance and admittance (Y=1/Z)
chart. In admittance chart (normalized impedance y= Y/Yo =G+jB), the
g and b correspond to r and x-circles respectively.
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SMITH CHART: FEATURES
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SMITH CHART: FEATURES
Fig.7:
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SMITH CHART: FEATURES
PROBLEM:4
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REFERENCE
Engineering Electromagnetics; William Hayt &
John Buck, 7th & 8th editions; 2012
Electromagnetics with Applications, Kraus and
Fleisch, 5th edition, 2010
Elements of Electromagnetics ; Matthew N.O.
Sadiku
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