Prof. David R. Jackson Notes 13 Transmission Lines (Impedance Matching) ECE 3317 [Chapter 6]

Smith Chart Impedance matching is very important to avoid reflected power, which causes a loss of efficiency and interference. ZgZg z sinusoidal source ZLZL z = 0 Z0Z0 S We will discuss two methods:  Quarter-wave transformer  Single-stub matching

Quarter-Wave Transformer Quarter-Wave Transformer: First consider a real load. Z L = R L Z0Z0 Z 0T Z in Hence

Z L = R L Z0Z0 Z 0T Z in Set This gives us Hence Quarter-Wave Transformer (cont.)

Next, consider a general (complex) load impedance Z L. shunt (parallel) susceptance Y L = 1 / Z L Z0Z0 Z 0T Y L = G L Z0Z0 Z 0T New model: Z L = R L = 1 / G L

Quarter-Wave Transformer (cont.) Summary of quarter-wave transformer matching method Y L = G L + j B L Z0Z0 Z 0T Y s = jB s

Quarter-Wave Transformer (cont.) Realization using a shorted stub Y L = G L + j B L Z0Z0 Z 0T B s = - B L lsls Z0sZ0s (An open-circuited stub could also be used.)

Single-Stub Matching A susceptance is added at a distance d from the load. ZLZL Y 0 = 1 / Z 0 d 1) We choose the distance d so that at this distance from the load 2) We then choose the shunt susceptance so that (i.e., G in = Y 0 )

Single-Stub Matching (cont.) ZLZL Y 0 = 1 / Z 0 d The feeding transmission line on the left sees a perfect match.

Realization using a shorted stub ZLZL Z0Z0 lsls Z0sZ0s d (An open-circuited stub could also be used.) Single-Stub Matching (cont.)

ZLZL Z0Z0 lsls Z0sZ0s d We use the Smith chart as an admittance calculator to determine the distance d. 1)Convert the load impedance to a load admittance Y L. 2)Determine the distance d to make the normalized input conductance equal to )If desired, we can also use the Smith chart to find the stub length. Single-Stub Matching (cont.)

ZLZL Z0Z0 lsls Z0sZ0s d Example Single-Stub Matching (cont.)

X X X X wavelengths toward load wavelengths toward generator Smith chart scale: use this one Single-Stub Matching (cont.)

ZLZL z UNMATCHED z X Crank diagram Single-Stub Matching (cont.)

ZLZL z ZLZL z jB s MATCHED UNMATCHED z SWR = 1.0 z Single-Stub Matching (cont.) SWR = 4.26

0-j0.5 0-j1 0+j0.5 0+j1 0+j0 0+j2 0-j2 Next, we find the length of the short-circuited stub: Rotate clockwise from S/C to desired B s value. Assume Z 0s = Z 0 Otherwise, we have to be careful with the normalization! admittance chart Single-Stub Matching (cont.)

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