1. 5. Impedance Matching and Tuning
Apply the theory and techniques of the previous chapters to practical problems in microwave engineering.
Impedance matching is the 1st topic.
Figure 5.1 (p. 223) A lossless network matching an arbitrary load impedance to a transmission line.

2. Impedance matching or tuning is important since
Maximum power is delivered when the load is matched to the line, and power loss in the feed line is minimized.
Impedance matching sensitive receiver components improves the signal-to-noise ratio of the system.
Impedance matching in a power distribution network will reduce the amplitude and phase errors.

3. Important factors in the selection of matching network.
Complexity
Bandwidth
Implementation
Ajdustability

4. 5.1 Matching with Lumped Elements
L-section is the simplest type of matching network.
2 possible configurations
Figure 5.2 (p. 223) L-section matching networks. (a) Network for zL inside the 1 + jx circle. (b) Network for zL outside the 1 + jx circle.

5. Analytic Solution
For Fig. 5. 2a, let ZL=RL+jXL. For zL to be inside the 1+jx circle, RL>Z0. For a match,
Removing X 

6. For Fig.5.2b, RL<Z0.

7. Smith Chart Solutions
Ex 5.1

8. Figure 5. 3b (p. 227) (b) The two possible L-section matching circuits
Figure 5.3b (p. 227) (b) The two possible L-section matching circuits. (c) Reflection coefficient magnitudes versus frequency for the matching circuits of (b).

9. Figure on page 228.

10. 5.2 Single Stub Tuning
Figure 5.4 (p. 229) Single-stub tuning circuits. (a) Shunt stub. (b) Series stub.

11. 2 adjustable parameters
d: from the load to the stub position.
B or X provided by the shunt or series stub.
For the shunt-stub case,
Select d so that Y seen looking into the line at d from the load is Y0+jB
Then the stub susceptance is chosen as –jB.
For the series-stub case,
Select d so that Z seen looking into the line at d from the load is Z0+jX
Then the stub reactance is chosen as –jX.

12. Shunt Stubs Ex 5.2 Single-Stub Shunt Tuning ZL=60-j80
Figure 5.5a (p. 230) Solution to Example (a) Smith chart for the shunt-stub tuners.

13. Figure 5. 5b (p. 231) (b) The two shunt-stub tuning solutions
Figure 5.5b (p. 231) (b) The two shunt-stub tuning solutions. (c) Reflection coefficient magnitudes versus frequency for the tuning circuits of (b).

14. To derive formulas for d and l, let ZL= 1/YL= RL+ jXL.
Now d is chosen so that G = Y0=1/Z0,

15. If RL = Z0, then tanβd = -XL/2Z0. 2 principal solutions are
To find the required stub length, BS = -B.
for open stub
for short stub

16. Series Stubs Ex 5.3 Single Stub Series Tuning ZL = 100+j80
Figure 5.6a (p. 233) Solution to Example (a) Smith chart for the series-stub tuners.

17. Figure 5. 6b (p. 232) (b) The two series-stub tuning solutions
Figure 5.6b (p. 232) (b) The two series-stub tuning solutions. (c) Reflection coefficient magnitudes versus frequency for the tuning circuits of (b).

18. To derive formulas for d and l, let YL= 1/ZL= GL+ jBL.
Now d is chosen so that R = Z0=1/Y0,

19. If GL = Y0, then tanβd = -BL/2Y0. 2 principal solutions are
To find the required stub length, XS = -X.
for short stub
for open stub

20. 5.3 Double-Stub Tuning
If an adjustable tuner was desired, single-tuner would probably pose some difficulty.
Smith Chart Solution
yL  add jb1 (on the rotated 1+jb circle)  rotate by d thru SWR circle (WTG)  y1  add jb2  Matched
Avoid the forbidden region.

21. Figure 5. 7 (p. 236) Double-stub tuning
Figure 5.7 (p. 236) Double-stub tuning. (a) Original circuit with the load an arbitrary distance from the first stub. (b) Equivalent-circuit with load at the first stub.

22. Figure 5.8 (p. 236) Smith chart diagram for the operation of a double-stub tuner.

23. Ex. 5.4 ZL = 60-j80
Open stubs, d = λ/8
Figure 5.9a (p. 238) Solution to Example (a) Smith chart for the double-stub tuners.

24. Figure 5. 9b (p. 239) (b) The two double-stub tuning solutions
Figure 5.9b (p. 239) (b) The two double-stub tuning solutions. (c) Reflection coefficient magnitudes versus frequency for the tuning circuits of (b).

25. Analytic Solution To the left of the first stub in Fig. 5.7b,
Y1 = GL + j(BL+B1) where YL = GL + jBL
To the right of the 2nd stub,
At this point, Re{Y2} = Y0

26. Since GL is real,
After d has been fixed, the 1st stub susceptance can be determined as
The 2nd stub susceptance can be found from the negative of the imaginary part of (5.18)

27. B2 =
The open-circuited stub length is
The short-circuited stub length is

28. 5.4 The Quarter-Wave Transformer
Single-section transformer for narrow band impedance match.
Multisection quarter-wave transformer designs for a desired frequency band.
One drawback is that this can only match a real load impedance.
For single-section,

29. Figure 5.10 (p. 241) A single-section quarter-wave matching transformer. at the design frequency f0.

30. The input impedance seen looking into the matching section is
where t = tanβl = tanθ, θ = π/2 at f0.
The reflection coefficient
Since Z12 = Z0ZL,

31. The reflection coefficient magnitude is

32. Now assume f ≈ f0, then l ≈ λ0/4 and θ ≈ π/2. Then sec2 θ >> 1. 

33. We can define the bandwidth of the matching transformer as
For TEM line,
At θ = θm,

34. The fractional bandwidth is
Ex. 5.5 Quarter-Wave Transformer Bandwidth
ZL = 10, Z0 = 50, f0= 3 GHz, SWR ≤ 1.5

35. Figure (p. 243) Reflection coefficient magnitude versus frequency for a single-section quarter-wave matching transformer with various load mismatches.

36. 5.5 The Theory of Small Reflection
Single-Section Transformer

37. Figure 5.13 (p. 244) Partial reflections and transmissions on a single-section matching transformer.

38. Multisection Transformer
Assume the transformer is symmetrical,

39. If N is odd, the last term is
while N is even,

40. 5.6 Binomial Multisection Matching Transformer
The response is as flat as possible near the design frequency.  maximally flat
This type of response is designed, for an N-section transformer, by setting the first N-1 derivatives of |Γ(θ)| to 0 at f0.
Such a response can be obtained if we let

41. Note that |Γ(θ)| = 0 for θ=π/2, (dn |Γ(θ)|/dθn ) = 0 at θ=π/2 for n = 1, 2, …, N-1.
By letting f  0,

42. Γn must be chosen as
Since we assumed that Γn are small, ln x ≈ 2(x-1)/(x+1),
Numerically solve for the characteristic impedance  Table 5.1

43. The bandwidth of the binomial transformer
Ex. 5.6 Binomial Transformer Design

44. Figure (p. 250) Reflection coefficient magnitude versus frequency for multisection binomial matching transformers of Example 5.6 ZL = 50Ω and Z0 = 100Ω.

45. 5.7 Chebyshev Multisection Matching Transformer
Chebyshev Polynomial
The first 4 polynomials are
Higher-order polynomials can be found using

46. Figure 5.16 (p. 251) The first four Chebyshev polynomials Tn(x).

47. Properties
For -1≤x ≤1, |Tn(x)|≤1  Oscillate between ±1  Equal ripple property.
For |x| > 1, |Tn(x)|>1  Outside the passband
For |x| > 1, |Tn(x)| increases faster with x as n increases.
Now let x = cosθ for |x| < 1. The Chebyshev polynomials can be expressed as
More generally,

48. We need to map θm to x=1 and π- θm to x = -1. For this,
Therefore,

49. Design of Chebyshev Transformers
Using (5.46)
Letting θ = 0,

50. If the maximum allowable reflection coefficient magnitude in the passband is Γm,

51. Once θm is known,
Ex 5.7 Chebyshev Transformer Design
Γm = 0.05, Z0 = 50, ZL = 100
Use Table 5.2

52. Figure (p. 255) Reflection coefficient magnitude versus frequency for the multisection matching transformers of Example 5.7.

53. Figure (p. 256) A tapered transmission line matching section and the model for an incremental length of tapered line. (a) The tapered transmission line matching section. (b) Model for an incremental step change in impedance of the tapered line.

54. Figure (p. 257) A matching section with an exponential impedance taper. (a) Variation of impedance. (b) Resulting reflection coefficient magnitude response.

55. Figure (p. 258) A matching section with a triangular taper for d(In Z/Z0/dz. (a) Variation of impedance. (b) Resulting reflection coefficient magnitude response.

56. Figure 5. 21 (p. 260) Solution to Example 5. 8
Figure (p. 260) Solution to Example (a) Impedance variations for the triangular, exponential, and Klopfenstein tapers. (b) Resulting reflection coefficient magnitude versus frequency for the tapers of (a).

57. Figure (p. 262) The Bode-Fano limits for RC and RL loads matched with passive and lossless networks (ω0 is the center frequency of the matching bandwidth). (a) Parallel RC. (b) Series RC. (c) Parallel RL. (d) Series RL.

58. Figure 5. 23 (p. 263) Illustrating the Bode-Fano criterion
Figure (p. 263) Illustrating the Bode-Fano criterion. (a) A possible reflection coefficient response. (b) Nonrealizable and realizable reflection coefficient responses.