1. Prof. Ji Chen Notes 13 Transmission Lines (Impedance Matching) ECE 3317 1 Spring 2014

2. 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 2

3. Quarter-Wave Transformer Quarter-Wave Transformer: First consider a real load. Z L = R L Z0Z0 Z 0T Z in Hence 3

4. Z L = R L Z0Z0 Z 0T Z in Set This gives us Hence Quarter-Wave Transformer (cont.) 4 Example:

5. 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 TOT = G L Z0Z0 Z 0T New model: Z L TOT = 1 / G L 5 (real) B s = -B L

6. 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 6

7. 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.) 7

8. Quarter-Wave Transformer with Extension 8 ZLZL Z0Z0 Z 0T Z in (-d) Z0Z0  We choose the length d to make the input impedance Z in (- d ) real.  We then use a quarter-wave transform to change the impedance to Z 0.

9. Quarter-Wave Transformer with Extension (cont.) 9 ZLZL Z0Z0 Z 0T Z0Z0 Example

10. 10 Quarter-Wave Transformer with Extension (cont.) Wavelengths towards generator

11. 11 Z 0T Z0Z0 Quarter-Wave Transformer with Extension (cont.)

12. 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 ) 12

13. Single-Stub Matching (cont.) ZLZL Y0Y0 d The feeding transmission line on the left sees a perfect match. 13

14. Realization using a shorted stub ZLZL Z0Z0 lsls Z0sZ0s d (An open-circuited stub could also be used.) Single-Stub Matching (cont.) 14

15. 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 1.0. 3)Determine the required value of B s to cancel B in. 4)If desired, we can also use the Smith chart to find the stub length l s. Single-Stub Matching (cont.) 15

16. ZLZL Z0Z0 lsls Z0sZ0s d Example Single-Stub Matching (cont.) 16

17. X X Wavelengths toward load Wavelengths toward generator Smith chart scale: Use this one Single-Stub Matching (cont.) 17 X X

18. Next, we find the length of the short-circuited stub: Rotate clockwise from S/C to desired B s value. 0-j0.5 0-j1 0+j0.5 0+j1 0+j0 0+j2 0-j2 Assume Z 0s = Z 0 Otherwise, we have to be careful with the normalization (see the note below). Admittance chart Single-Stub Matching (cont.) 18 Note: In general,

19. From the Smith chart: Analytically: Single-Stub Matching (cont.) 19 X Admittance chart

20. ZLZL z UNMATCHED Single-Stub Matching (cont.) z 1.62 1.55 0.38 1.0 0.78 20 X Crank diagram

21. ZLZL z ZLZL z jB s MATCHED UNMATCHED z 1.62 1.55 0.78 SWR = 1.0 z 1.62 1.55 0.38 1.0 0.78 Single-Stub Matching (cont.) SWR = 4.26 21