ELCT564
Spring 2013
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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
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Telegrapher equations
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Kirchhoff’s voltage law
Kirchhoff’s current law
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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
Time-average stored magnetic energy
Time-average stored electric energy
Field lines on an arbitrary TEM transmission line
Power loss per unit length in lossy dielectric
Power loss per unit length due to conductor
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Transmission Lines Parameters
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Terminated Lossless Transmission Line
Voltage reflection coefficient
A transmission line terminated in a load impedance Z
L
A superposition of an incident and a reflected wave: standing waves
Return loss Standing Wave Ratio
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Input impedance
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Short Terminated Lossless Transmission Line
Г=-1
Voltage
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Current
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Impedance
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Open Terminated Lossless Transmission Line
Г=1
Voltage
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Current
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Impedance
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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
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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 50 Ω and an electrical length of 0.3
λ. The line is terminated with an impedance having a resistive component of 25 Ω 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 50 Ω of 70-j25 Ω. The line is 2.35
λ 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=100
Ω 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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Zo
The Multiple-Reflection Viewpoint
Z
1
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The Quarter-Wave Transformer:
Bandwidth Performance l= λ/4 at frequency f
0
Bandwidth
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The Quarter-Wave Transformer:
Bandwidth Performance
Design a single-section quarter-wave matching transformer to match a 10 Ω load to a 50Ω transmission line at f0=3GHz. Determine the percent bandwidth for which the SWR≤1.5.
Zo Z
1
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Z
2
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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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The low-loss line
Lossy Transmission 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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Bounce Diagram
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Bounce Diagram
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Additional Examples
Use the Smith Chart to find the shortest lengths of a short-circuited 75
Ω line to give the following input impedance:
1. Zin = 0
2. Zin = infinity
3. Zin = j75
Ω
4. Zin = -j50
Ω
1. 0 or 0.5
λ
2. 0.25
λ
3. 0.125 λ
4. 0.406 λ
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