ELCT564 Spring /9/20151ELCT564 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 6/9/20152ELCT564 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 6/9/20153ELCT564 Kirchhoff’s voltage law Kirchhoff’s current law Telegrapher equations

Wave Propagation on a Transmission Line 6/9/20154ELCT564

Wave Propagation on a Lossless Line 6/9/20155ELCT564

Field Analysis of Transmission Lines 6/9/20156ELCT564 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 6/9/20157ELCT564 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 Input impedance

Short Terminated Lossless Transmission Line 6/9/20158ELCT564 Voltage Current Impedance Г=-1

Open Terminated Lossless Transmission Line 6/9/20159ELCT564 Voltage Current Impedance Г=1

Two Transmission Lines 6/9/201510ELCT564 Decibels and Nepers Insertion Loss Ratio of power levels dBm

The Smith Chart 6/9/201511ELCT564

The Smith Chart: Resistance Circle 6/9/201512ELCT564 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 6/9/201513ELCT564 If Zo is 50 Ohm, indicate the position of j50, j10, -j25 in the plot

The Smith Chart 6/9/201514ELCT564 If Zo is 50 Ohm, indicate the position of 25+j50, 50+j100, 10-j25 in the plot

The Smith Chart: SWR Circles 6/9/201515ELCT564

The Smith Chart: Example 1 6/9/201516ELCT564 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 6/9/201517ELCT564 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 6/9/201518ELCT564 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 6/9/201519ELCT564 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.

6/9/2015ELCT56420 Transform of a complex load impedance into a real impedance?

The Multiple-Reflection Viewpoint 6/9/201521ELCT564 Zo Z1Z1

6/9/2015ELCT56422 The Quarter-Wave Transformer: Bandwidth Performance l=λ/4 at frequency f 0 Bandwidth

6/9/2015ELCT56423 The Quarter-Wave Transformer: Bandwidth Performance Zo Z1Z1 Z2Z2 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.

6/9/2015ELCT56424 Generator and Load Mismatches

6/9/2015ELCT56425 Generator and Load Mismatches Load matched to line Generator matched to loaded line Conjugate matching

6/9/2015ELCT56426 Lossy Transmission Line The low-loss line

6/9/2015ELCT56427 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

6/9/2015ELCT56428 The Terminated Lossy Line

6/9/2015ELCT56429 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 1.0 or 0.5 λ λ λ λ