Notes 11 Transmission Lines ECE 3317 Applied Electromagnetic Waves Prof. David R. Jackson Fall 2018 Notes 11 Transmission Lines (Standing Wave Ratio (SWR) and Generalized Reflection Coefficient)

Consider a lossless transmission line that is terminated with a load: Standing Wave Ratio Consider a lossless transmission line that is terminated with a load: + -

Standing Wave Ratio (cont.) Consider a lossless transmission line that is terminated with a load: + -

Standing Wave Ratio (cont.) Denote Then we have The magnitude is Maximum voltage: Maximum voltage:

Standing Wave Ratio (cont.) The voltage standing wave ratio is the ratio of Vmax to Vmin . We then have: Perfect match: L = 0

Standing Wave Ratio (cont.) For the current we have Hence we have: The current standing wave ratio is thus Hence

Note: V+ is the net wave going in the +z direction. Standing Wave Pattern Note: V+ is the net wave going in the +z direction.

Standing Wave Ratio: Real Load Special case of a real load impedance Case a:

Standing Wave Ratio: Real Load (cont.) Hence Case b: Hence

Standing Wave Ratio: Real Load (cont.) Hence, for a real load impedance we have:

Example (6.6, Shen and Kong) Given: Find: + -

Example (6.6, Shen and Kong) (cont.)

Example (6.6, Shen and Kong) (cont.) This problem has practical significance: often we are interested in figuring out what an unknown load is. Reverse problem: Given: What is the unknown load impedance? so (Any multiple of 2 can be added to .) Solve for .

Example (6.6, Shen and Kong) (cont.) Hence, we have The calculation yields:

Generalized Reflection Coefficient Define the “generalized reflection coefficient” at a point z0 on the line: + -

Generalized Reflection Coefficient Rearranging, we have: Solve for L We can then write where

Generalized Reflection Coefficient (cont.) where + - We identify (z0) as the reflection coefficient at the point z0, with Zin acting as the load impedance. Hence

Generalized Reflection Coefficient (cont.) Hence

Generalized Reflection Coefficient (cont.) Define a normalized input impedance at point z0: We then have Note: This can be used as an alternative formula to the tangent formula for calculating input impedance at z0 = -d. (This is the starting point for the Smith chart discussion.)

Crank Diagram where Note: 1 Moving from load (angle change 2 z) Note: We go all the way around the crank diagram when z changes by  / 2.

Example Given: Calculate the reflection coefficient and the input impedance at z0 = -0.125 so so

Appendix: Summary of Formulas