Download Transmission Line Theory

ELCT564
Spring 2012
Chapter 2: Transmission Line
Theory
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The Lumped-Element Circuit Model of T-Line
Transmission line theory bridges the gap between field analysis and basic circuit theory
Voltage and current definitions of an incremental length of transmission line
R: Series resistance per unit length (Ω/m)
L: Series inductance per unit length (H/m)
G: Shunt conductance per unit length (S/m)
C: Shunt capacitance per unit length (F/m)
Lumped-element equivalent circuit of an incremental length of transmission line
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The Lumped-Element Circuit Model of T-Line
Kirchhoff’s voltage law
Kirchhoff’s current law
Telegrapher equations
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Wave Propagation on a Transmission Line
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Wave Propagation on a Lossless Line
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Field Analysis of Transmission Lines
Field lines on an arbitrary TEM transmission line
Time-average stored magnetic energy
Time-average stored electric energy
Power loss per unit length in
lossy dielectric
Power loss per unit length due to conductor
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Terminated Lossless Transmission Line
A transmission line terminated in a load impedance ZL
A superposition of an
incident and a reflected
wave: standing waves
Return loss
Standing Wave Ratio
Input impedance
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Short Terminated Lossless Transmission Line
Г=-1
Impedance
Voltage
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Current
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Open Terminated Lossless Transmission Line
Г=1
Impedance
Voltage
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Current
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Two Transmission Lines
Insertion Loss
Decibels and Nepers
Ratio of power
levels
dBm
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The Smith Chart
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The Smith Chart: Resistance Circle
If Zo is 50 Ohm, indicate the
position of 10, 25, 50 and
250 Ohm in the plot
If Zo is 100 Ohm, indicate
the position of 10, 25, 50
and 250 Ohm in the plot
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The Smith Chart: Reactance Curves
If Zo is 50 Ohm, indicate the
position of j50, j10, -j25 in
the plot
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The Smith Chart
If Zo is 50 Ohm, indicate the
position of 25+j50, 50+j100,
10-j25 in the plot
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The Smith Chart: SWR Circles
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The Smith Chart: Example 1
Suppose we have a transmission line with a characteristic impedance of 50Ω
and an electrical length of 0.3λ. The line is terminated with an impedance
having a resistive component of 25Ω and an inductive reactance of 25Ω.
What is the input impedance to the line?
Basic Steps using Smith Chart:
• Normalize and plot a line input/load impedance and construct a constant
SWR circle
• Apply the line length to the wavelengths scales
• Read normalized load/input impedance, and convert to impedance in ohms
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The Smith Chart: Example 2
Suppose we have a measured input impedance to a 50Ω of 70-j25 Ω. The line
is 2.35λ long, and is terminated in an antenna. What is the antenna feed
impedance?
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The Slotted Line
The following two step procedure has been carried out with a 50 Ω coaxial
slotted line to determine an unknown load impedance:
•
•
A short circuit is placed at the load plane, resulting in a standing wave on the line with infinite
SWR, and sharply defined voltage minima recorded at z=0.2 cm, 2.2cm, 4.2cm
The short circuit is removed, and replaced with the unknown load. The SWR is measured as 1.5,
and voltage minima are recorded at z=0.72cm, 2.72cm, 4.72cm.
Find the load impedance.
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The Quarter-Wave Transformer
Consider a load resistance RL=100Ω to be matched to a 50Ω line with a quarter-wave
transformer. Find the characteristic impedance of the matching line section and plot the
magnitude of the reflection coefficient versus normalized frequency, f/fo, where fo is the
frequency at which the line is λ/4 long.
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Transform of a complex
load impedance into a
real impedance?
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The Multiple-Reflection Viewpoint
Zo
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Z1
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The Quarter-Wave Transformer:
Bandwidth Performance
l=λ/4 at frequency f0
Bandwidth
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The Quarter-Wave Transformer:
Bandwidth Performance
Design a single-section quarter-wave matching
transformer to match a 10Ω load to a 50Ω ;ome. At
f0=3GJz/ Determine the percent bandwidth for which
the SWR≤1.5.
Zo
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Z1
Z2
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Generator and Load Mismatches
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Generator and Load Mismatches
Load matched to line
Generator matched to loaded line
Conjugate matching
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Lossy Transmission Line
The low-loss line
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The Distorionless Line
When the phase term is not a linear function of frequency, the various frequency components
of a wideband signal will travel with different phase velocities and arrive the receiver end of the
transmission line at slight different times. This will lead to dispersion.
Distortionless line
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The Terminated Lossy Line
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Additional Examples
Use the Smith Chart to find the shortest lengths of a short-circuited 75Ω line to give the
following input impedance:
1. Zin = 0
2. Zin = infinity
3. Zin = j75 Ω
4. Zin = -j50 Ω
1.
2.
3.
4.
0 or 0.5 λ
0.25 λ
0.125 λ
0.406 λ
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