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ELCT564 Spring 2012

ELCT564 Spring 2012. 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. Voltage and current definitions of an incremental length of transmission line.

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ELCT564 Spring 2012

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  1. ELCT564 Spring 2012 Chapter 2: Transmission Line Theory ELCT564

  2. 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 ELCT564

  3. The Lumped-Element Circuit Model of T-Line Kirchhoff’s voltage law Kirchhoff’s current law Telegrapher equations ELCT564

  4. Wave Propagation on a Transmission Line ELCT564

  5. Wave Propagation on a Lossless Line ELCT564

  6. 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 ELCT564

  7. 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 ELCT564

  8. Short Terminated Lossless Transmission Line Г=-1 Impedance Current Voltage ELCT564

  9. Open Terminated Lossless Transmission Line Г=1 Impedance Current Voltage ELCT564

  10. Two Transmission Lines Insertion Loss Decibels and Nepers Ratio of power levels dBm ELCT564

  11. The Smith Chart ELCT564

  12. 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 ELCT564

  13. The Smith Chart: Reactance Curves If Zo is 50 Ohm, indicate the position of j50, j10, -j25 in the plot ELCT564

  14. The Smith Chart If Zo is 50 Ohm, indicate the position of 25+j50, 50+j100, 10-j25 in the plot ELCT564

  15. The Smith Chart: SWR Circles ELCT564

  16. 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 ELCT564

  17. 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? ELCT564

  18. 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. ELCT564

  19. 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. ELCT564

  20. Transform of a complex load impedance into a real impedance? ELCT564

  21. The Multiple-Reflection Viewpoint Z1 Zo ELCT564

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

  23. 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. Z1 Z2 Zo ELCT564

  24. Generator and Load Mismatches ELCT564

  25. Generator and Load Mismatches Load matched to line Generator matched to loaded line Conjugate matching ELCT564

  26. Lossy Transmission Line The low-loss line ELCT564

  27. 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 ELCT564

  28. The Terminated Lossy Line ELCT564

  29. 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 ELCT564

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