Chapter 2: Transmission Line Theory ELCT564 Spring 2013 Chapter 2: Transmission Line Theory 4/16/2017 ELCT564

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 4/16/2017 ELCT564

The Lumped-Element Circuit Model of T-Line Kirchhoff’s voltage law Kirchhoff’s current law Telegrapher equations 4/16/2017 ELCT564

Wave Propagation on a Transmission Line 4/16/2017 ELCT564

Wave Propagation on a Lossless Line 4/16/2017 ELCT564

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 4/16/2017 ELCT564

Transmission Lines Parameters 4/16/2017 ELCT564

Terminated Lossless Transmission Line A transmission line terminated in a load impedance ZL A superposition of an incident and a reflected wave: standing waves Voltage reflection coefficient Return loss Standing Wave Ratio Input impedance 4/16/2017 ELCT564

Short Terminated Lossless Transmission Line Г=-1 Impedance Current Voltage 4/16/2017 ELCT564

Open Terminated Lossless Transmission Line   Г=1 Impedance Current Voltage 4/16/2017 ELCT564

Two Transmission Lines Insertion Loss Decibels and Nepers Ratio of power levels dBm 4/16/2017 ELCT564

The Smith Chart                 4/16/2017 ELCT564

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 4/16/2017 ELCT564

The Smith Chart: Reactance Curves If Zo is 50 Ohm, indicate the position of j50, j10, -j25 in the plot 4/16/2017 ELCT564

The Smith Chart If Zo is 50 Ohm, indicate the position of 25+j50, 50+j100, 10-j25 in the plot 4/16/2017 ELCT564

The Smith Chart: SWR Circles 4/16/2017 ELCT564

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 4/16/2017 ELCT564

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? 4/16/2017 ELCT564

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. 4/16/2017 ELCT564

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. 4/16/2017 ELCT564

Transform of a complex load impedance into a real impedance? 4/16/2017 ELCT564

The Multiple-Reflection Viewpoint             Zo Z1                       4/16/2017 ELCT564

The Quarter-Wave Transformer: Bandwidth Performance   l=λ/4 at frequency f0 Bandwidth   4/16/2017 ELCT564

The Quarter-Wave Transformer: Bandwidth Performance Design a single-section quarter-wave matching transformer to match a 10Ω load to a 50Ω transmission line at f0=3GHz. Determine the percent bandwidth for which the SWR≤1.5. Zo Z1 Z2 4/16/2017 ELCT564

Generator and Load Mismatches 4/16/2017 ELCT564

Generator and Load Mismatches Load matched to line Generator matched to loaded line Conjugate matching 4/16/2017 ELCT564

Lossy Transmission Line The low-loss line 4/16/2017 ELCT564

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 4/16/2017 ELCT564

The Terminated Lossy Line 4/16/2017 ELCT564

Bounce Diagram 4/16/2017 ELCT564

Bounce Diagram 4/16/2017 ELCT564

Additional Examples Use the Smith Chart to find the shortest lengths of a short-circuited 75Ω line to give the following input impedance: Zin = 0 Zin = infinity Zin = j75 Ω Zin = -j50 Ω 0 or 0.5 λ 0.25 λ 0.125 λ 0.406 λ 4/16/2017 ELCT564