Notes 13 Transmission Lines (Impedance Matching) ECE 3317 Applied Electromagnetic Waves Prof. David R. Jackson Fall 2018 Notes 13 Transmission Lines (Impedance Matching)

Impedance Matching Impedance matching is very important to avoid reflected power, which causes a loss of efficiency and interference. + - We will discuss two methods: Quarter-wave transformer Single-stub matching

Quarter-Wave Transformer Quarter-Wave Transformer: First consider a real load on a lossless line. Hence

Quarter-Wave Transformer (cont.) Set Example: Hence This gives us

Quarter-Wave Transformer with Shunt Susceptance Next, consider a general (complex) load impedance ZL. Goal: determine Bs, Z0T New model:

Quarter-Wave Transformer with Shunt Susceptance (cont.) Summary of quarter-wave transformer matching method

Quarter-Wave Transformer with Shunt Susceptance (cont.) Realization using a shorted stub: (An open-circuited stub could also be used.) Hence we have:

Quarter-Wave Transformer with Line Extension In this method we use a line extension “d” instead of a shunt susceptance. Goal: determine d, Z0T We choose the length d to make the input impedance Zin (-d) real. We then use a quarter-wave transformer to change the impedance to Z0.

Quarter-Wave Transformer with Line Extension (cont.) Example

Quarter-Wave Transformer with Line Extension (cont.) Wavelengths towards generator

Quarter-Wave Transformer with Line Extension (cont.)

Quarter-Wave Transformer with Line Extension (cont.) Summary of Design

A parallel (shunt) susceptance is added at a distance d from the load. Single-Stub Matching A parallel (shunt) susceptance is added at a distance d from the load. Goal: determine Bs, Z0T 1) We choose the distance d so that at this distance from the load 2) We then choose the shunt susceptance so that

Single-Stub Matching (cont.) The feeding transmission line on the left sees a perfect match!

Single-Stub Matching (cont.) Realization using a shorted stub (An open-circuited stub could also be used.) Goal: Find d and ls.

Single-Stub Matching (cont.) We use the Smith chart as an admittance calculator to determine the distance d. Convert the load impedance ZL to a load admittance YL. Determine the distance d to make the normalized input conductance equal to 1.0. Determine the required value of Bs to cancel Bin (Bs = - Bin). Determine the stub length ls from the value of Bs. Note: If desired, we can use the Smith chart to also find the stub length ls.

Single-Stub Matching (cont.) Example

Single-Stub Matching (cont.) Use this one Smith chart scale: Wavelengths toward load Wavelengths toward generator

Single-Stub Matching (cont.) Next, we find the length of the short-circuited stub: Rotate clockwise from S/C to desired Bs value. Note: Here we have Z0s = Z0. Otherwise, we have to be careful with the normalization (see the note below). 0-j0.5 0-j1 0+j0.5 0+j1 0+j0 0+j2 0-j2 Note: In general, Admittance chart

Single-Stub Matching (cont.) From the Smith chart: X Admittance chart Analytically:

Single-Stub Matching (cont.) Final Design

Single-Stub Matching (cont.) z Unmatched Crank Diagram Recall: The stub is located at d = 0.219 

Single-Stub Matching (cont.) z Unmatched Matched z