Notes 10 Transmission Lines (Reflection and Impedance) ECE 3317 Applied Electromagnetic Waves Prof. David R. Jackson Fall 2018 Notes 10 Transmission Lines (Reflection and Impedance)

Consider a transmission line that is terminated with a load: Reflection Consider a transmission line that is terminated with a load: + - Voltage and current on the line:

Reflection (cont.) Important point: The forward-traveling and backward-traveling wave amplitudes are the amplitudes that describe the two waves in sinusoidal steady-state (after all bounces have died down and we are in steady state). + - A = amplitude of net forward-traveling wave B = amplitude of net backward-traveling wave

Reflection (cont.) + - At the load (z = 0): Hence we have or

Reflection Coefficient We define the load reflection coefficient: Hence We then have, form the last slide, or

Voltage and Current We can then write: Note: The generator (source) will determine the unknown (complex) constant A.

Impedance We define the input impedance at any point z on the line: + - The input impedance is the impedance “seen” looking to the right.

Impedance (cont.) + - + - Note: The input impedance does not care what is to the left of the point z. We can remove everything to the left if we wish. (What is to the left of the point z only affects the amplitude level of the voltage and current.)

Impedance (cont.) We then have Canceling the constant A, we have or

Impedance (cont.) Now let z = - d : + -

Impedance (cont.) At the beginning of the line we have: l = length of line + -

Impedance (cont.) Substituting for the load reflection coefficient, we have:

Impedance (cont.) Rearranging the expression, we have:

Impedance (cont.) Now let z = - d : + - d = distance from load

Impedance (cont.) At the beginning of the line we have: l = length of line + -

These limiting cases agree with circuit theory. Impedance (cont.) Limiting cases 1) General (lossy) line: 2) Lossless line: These limiting cases agree with circuit theory.

Impedance (cont.) Limiting cases (cont.) 3) Lossy infinite line: Proof:

Impedance (cont.) Lossless Case Use We then have where

Summary of final formula for a lossless line: Impedance (cont.) Summary of final formula for a lossless line:

Impedance (cont.) For a lossless line: The input impedance repeats every one-half wavelength. The voltage and current repeat every wavelength. The magnitude of the voltage and current repeat every one-half wavelength. The voltage and current become their negatives after one-half wavelength.

Special Cases of Lossless Line Impedance (cont.) Special Cases of Lossless Line Short-circuit line or

Impedance (cont.) Short-circuit line or π/2 3π/2 inductive capacitive 5π/2  Short circuit The line is one-half of a wavelength long.

Impedance (cont.) Low frequency: Short-circuit line

Special Cases of Lossless Line (cont.) Impedance (cont.) Special Cases of Lossless Line (cont.) Open-circuit line or

Impedance (cont.) Open-circuit line or π 2π 3π inductive capacitive The line is one-half of a wavelength long.

Impedance (cont.) Low frequency: Open-circuit line

Microstrip filter (application of open-circuited “stubs”)

Appendix: Summary of Formulas