ELCT564 Spring 2012

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Chapter 2: Transmission Line Theory

The Lumped-Element Circuit Model of T-Line

Transmission line theory bridges the gap between field analysis and basic circuit theory

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Voltage and current definitions of an incremental length of transmission line

Lumped-element equivalent circuit 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)

The Lumped-Element Circuit Model of T-Line

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Kirchhoff’s voltage law

Kirchhoff’s current law

Telegrapher equations

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

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Field lines on an arbitrary TEM transmission line

Time-average stored magnetic energy

Time-average stored electric energy

Power loss per unit length due to conductor

Power loss per unit length in lossy dielectric

Terminated Lossless Transmission Line

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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

Short Terminated Lossless Transmission Line

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VoltageCurrent

Impedance

Г=-1

Open Terminated Lossless Transmission Line

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VoltageCurrent

Impedance

Г=1

Two Transmission Lines

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Decibels and Nepers

Insertion Loss

Ratio of power levels

dBm

The Smith Chart

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The Smith Chart: Resistance Circle

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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

The Smith Chart: Reactance Curves

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If Zo is 50 Ohm, indicate the position of j50, j10, -j25 in the plot

The Smith Chart

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If Zo is 50 Ohm, indicate the position of 25+j50, 50+j100, 10-j25 in the plot

The Smith Chart: SWR Circles

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The Smith Chart: Example 1

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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

The Smith Chart: Example 2

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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?

The Slotted Line

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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.

The Quarter-Wave Transformer

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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?

The Multiple-Reflection Viewpoint

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Zo 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

Zo Z1 Z2

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.

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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 = 02.Zin = infinity3.Zin = j75 Ω4.Zin = -j50 Ω

1. 0 or 0.5 λ2. 0.25 λ3. 0.125 λ4. 0.406 λ