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# Smith chart

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1. Smith chart

2. -z Zc ZL z = 0 impedance along the line Look at a distance z = - L toward the generator

3. -z Zc ZL z = 0 impedance along the line impedance is periodic every half-wavelength

4. -z Zc ZL z = 0 impedance along the line quarter wavelength Z <==> Y short ==> open open ==> short

5. -z quarter wave transformer Zc1 Zc2 ZL z = 0 matching frequency sensitive

6. -z Zc ZL z = 0 shorted line open line all values of reactance

7. L X = Zc tan kL

8. Should I add something in series or in parallel?

10. -z -z -z Zc Zc Zc ZL ZL ZL z = 0 z = 0 z = 0 Eureka! Use a stub.

11. -z impedance along the line Zc ZL z = 0 shorted line all values of susceptance

12.  2 1.5 1 My VSWR is 1.25! Zc ZL z = 0 -z

13. VSWR

14. -z -z -z Not matched! Zc Zc Zc ZL ZL ZL z = 0 z = 0 z = 0 Needs some -jB somewhere, but where? Eureka! Use a stub, somewhere. How long should it be?

15. An article appeared in the January, 1939 issue of Electronics that changed forever the way radio engineers think about transmission lines. Phil Smith [1907-1985] devised an extraordinarily clever circular chart that revealed graphically the complex impedance anywhere along a line. No math and minimum fuss. There's a marvelous symmetry in its design - everything fits together neatly. So ingenious was his invention that it has been the standard of the industry - for over sixty years.

16. -z Zc ZL z = 0 multiply numerator and denominator by the complex conjugate

17. -z Zc ZL z = 0 equate real and imaginary parts

18. -z Zc ZL z = 0 real

19. -z Zc ZL z = 0 real

20. -z Zc ZL z = 0 imaginary

21. r b a Equation of circle (x - a)2 + (y - b)2 = r2 y x

22. clf; clear; plot([-1 1], [0 0], ’y’) axis equal axis off hold on

23. for r = [0 .2 .5 1 2 5] rr = 1 / (r + 1); cr = 1 - rr; tr = 2 * pi * (0 : .01: 1); plot(cr + rr * cos (tr), rr * sin (tr), ‘y’); end

24. Ri Rr = 1 Rr

25. for x = [.2 .5 1 2 5] rx = 1 / x; cx = rx; tx = 2 * pi * (0 : .01: 1); plot(1 - rx * sin (tx), cx - rx * cos (tx), ‘y’); plot(1 - rx * sin (tx),- cx + rx * cos (tx), ‘y’); end

26. Ri Rr = 1 Rr

27. for x = [.2 .5 1 2 5] rx = 1 / x; cx = rx; tx = 2 * atan(x) * (0 : .01: 1); plot(1 - rx * sin (tx), cx - rx * cos (tx), ‘y’); plot(1 - rx * sin (tx),- cx + rx * cos (tx), ‘y’); end

28. Ri Rr = 1 Rr

29. Ri Rr = 1 Rr reflection coefficient = constant

31. 1 |R| 0

32. radius = 0 means no reflection!

33. z = 0.5 - j0.5

34. z=j1

35. Movie to illustrate the frequency dependence of the impedance of a series resonant circuit

36. Movie to illustrate the frequency dependence of the impedance of a parallel resonant circuit

37. Movie to illustrate the transformation of a load impedance at various locations on the transmission line l/2

38. y =  z = 0

39. Zin = ? Zc = 50 W l/8

40. z = j1 z = 0

41. Zin = ? Zc = 50 W l/8

42. Zin = ? Zc = 100 W 0.434 l

43. z = 0.7 + j1.2 zL = 2.6 + j1.8

44. Zin = ? Zc = 100 W 0.434 l

45. Single stub matching

46. y = 1 + j1 y = 1 - j1 yL = 2 + j1

47. y = + j1 y = - j1 yL = 