1. Chapter II Resonators and Impedance Matching

2. Outline Lumped Resonators Transmission-Line Resonators
Excitation of Resonators
Impedance Matching Methods

3. - 1. Lumped Resonators Series Resonant Circuit R L C + Vc Zin(w) I
Resonance occurs when Wm=We

4. R L R C Quality factor At resonance (w=w0):
Q and D factors for series two-element circuits R L R C

5. Frequency response of series resonant circuit near resonance
Fractional Bandwidth

6. - Parallel Resonant Circuit + IL V R L C Yin(w)
Resonance occurs when Wm=We
Quality factor at resonance

7. L R C R Q and D factors for parallel two-element circuits
Frequency response of parallel resonant circuit near resonance

8. Fractional Bandwidth
Loaded Q
Resonant Circuit
Unloaded Q RL

9. External Q
Relations among loaded Q, unloaded Q, and external Q
Complex angular resonant frequency

10. Summary Relation between impedance and admittance
Q factor for two-element LR or CR circuit
Q factor for three-element L R C circuits at resonance

11. Frequency responses of RLC resonant circuits near resonance

12. 2. Transmission-Line Resonators
Short-Circuited l/2 Line
Zin
Frequency response near resonance

13. Equivalent series RLC resonant circuit

14. Example: Find Q of Half-Wave Coaxial Line Resonators
Short- circuited at the end
RG-401 b a

15. Yin
Open-Circuited l/2 Line

16. Equivalent parallel RLC resonant circuit

17. Example: Find Q of Half-Wave Microstrip Line Resonators

18. Yin
Short-Circuited l/4 Line

19. Equivalent parallel RLC resonant circuit

20. Zin Open-Circuited l/4 Line R L C
Equivalent series RLC resonant circuit R L C

21. Q factor of Microstrip Components
Attenuation due to conductor loss

22. Considering the conductor surface roughness
(provided by E.O. Hammerstad, et. al.)
Attenuation due to dielectric loss

23. For the most microstrip passive component design, the component length is proportional to the guided wavelength, ie, l = mlg. The Q factor can be then expressed as
However, the Q factor will be limited by radiation and surface-wave losses because both losses tend to increase with frequency.
Also, for the most microstrip passive component design, the ratio of substrate thickness to microstrip width is fixed to maintain the same characteristic impedance. From the planar waveguide model

24. Radiation and Surface-Wave Losses
However, the Q factor will be limited by radiation and surface-wave losses because both losses tend to increase with substrate thickness.
Radiation and Surface-Wave Losses
Radiated space waves and surface waves are generated at discontinuities like open-ends. Radiated space waves propagate upward into free space. Surface waves propagate along the substrate surface.

25. Radiation Loss
Yin
open-end

26. There are three directions to achieve high isolation factor preferred in circuit applications:
Use relatively high permittivity substrates.
Use fairly thin substrates.
Employ high impedance stubs in matching circuits.
For antenna applications, low isolation factor or high radiation efficiency is preferred.
Surface-Wave Loss
At high frequencies, say, above 10 GHz, surface-wave loss may surpass the radiation loss and becomes the dominant loss.
provided by T.S. Horng, et. al.

27. Summary
A short-circuited l/2 line is equivalent to a series RLC resonant circuit.
An open-circuited l/2 line is equivalent to a parallel RLC resonant circuit.
A short-circuited l/4 line is equivalent to a parallel RLC resonant circuit.
An open-circuited l/4 line is equivalent to a series RLC resonant circuit.
The above four transmission-line resonators have the same quality factor equal to b/(2a).
The Q factors of microstrip components increase with frequency but have upper limits due to radiation and surface-wave losses.
The Q factors of microstrips components increase with substrate thickness but have upper limits due to radiation and surface-wave losses.
The radiation and surface-wave losses increase with frequency, and also substrate thickness.

28. 3. Excitation of Resonators
Lumped Resonators Fed by Transmission Lines
Zin
Yin

29. H3 Series RLC case Parallel RLC case g>1 g>1 g=1 g=1 g<1
g<1 The resonator is said to be undercoupled to the feedline.
g=1 The resonator is critically coupled to the feedline.
g>1 The resonator is said to be overcoupled to the feedline.
Series RLC case
Parallel RLC case
g>1
g=1
g<1 w
g>1
g=1
g<1 w

30. Example: Find Coupling Coefficient of Half-Wave Microstrip Line Resonators
Assume that the same microstrip line resonator (patch) as the previous example in finding the Q factor was given.
50-W Coaxial feedline
patch

31. Z Zin Gap-Coupled Microstrip Resonators Open-Circuit l/2 Resonator
Gap Cap
Open-Circuit l/2 Resonator
Feed Line

33. H3 R L C Equivalent series RLC resonant circuit feedline feedline
patch
patch
Gap(Edge) coupled
Overlay coupled

34. Example: Find gap capacitance of a Gap-Coupled Microstrip Line Resonator in Critically Coupled Condition
Assume that the same microstrip line resonator (patch) as the previous example in finding the Q factor was given. The Z0 of feedline is 50 W.
feedline
patch
Gap coupled
Now that C is determined, the exact resonant frequency can be found by solving the transcendental equation bc+ tanq = 0 , which results in a value of about 4.8 GHz, slightly lower than the unloaded resonant frequency of 5 GHz.

35. Summary
To obtain maximum power transfer between a resonator and a feedline, the resonator must be matched to the feed at the resonant frequency. The resonator is then said to be critically coupled to the feed.
A gap-coupled microstrip resonator can reach critically coupled condition by choosing adequate value of gap capacitance. Its resonant frequency and Q factor are close to the values of the unloaded microstrip resonator.

36. 4. Impedance Matching Methods
Impedance matching circuit
Lumped - element matching
Transmission - line element matching
Two-Element (L-Section) Matching

37. (1)
(2)
(3)

38. (4)
(5)
(6)

39. Estimation of Matching Bandwidth
parallel
series

40. (series-to-parallel transformation)
Case (a)

41. Find 3-dB bandwidth of | G |
For low Q case,

42. Example: Find Element Values and 3-dB Fractional Bandwidth of the Two-Element Matching Network
Impedance matching circuit

43. H4

44. Three-Element (High Q) MatchingX j 2b

45. +

46. Example: Design of the Three-Element Matching Network+
Example: Design of the Three-Element Matching Network
Impedance matching circuit

47. +

48. +

49. H4

51. Cascaded L-Section (Low Q) Matching+

52. Example: Design of the Cascaded L-Section Matching Network+
Example: Design of the Cascaded L-Section Matching Network
Impedance matching circuit

53. +

54. H4

55. H4

56. Open or shorted stub
Single-Stub Matching

58. Example: Design of Single-Stub Shunt Matching
Impedance matching circuit

59. H4

60. Double-Stub Matching

61. Example: Design of Double-Stub Shunt Matching
Impedance matching circuit
(Use short-circuited shunt stubs)

63. H4

64. Summary
The lumped-element matching networks have the advantages of smaller size and wider matching bandwidth than the transmission-line matching networks. However, due to the self resonant frequencies of the lumped elements, the lumped-element matching networks can’t be applied as high frequencies as the transmission-line matching networks.
The lossless lumped element matching networks that are commonly used include L-section, P-section, T-section, and cascaded L-section. L-section has medium but not adjustable matching bandwidth. P and T sections have rather small and adjustable matching bandwidth. Cascaded L-section can increase bandwidth as the number of stages increases.
The lossless transmission-line matching networks that are commonly used include single-stub, double-stub, triple-stub, etc. Every stub is a section of transmission line that is open or short circuited at the end. Single-stub is not good for tuner applications. Double and triple stubs are commonly used in tuners. In principle, the more stubs a tuner has, the larger impedance-matching range it can achieve.

65. Considering the trade off between the size and upper limit of operating frequency, we can combine the lumped elements and transmission lines skillfully in the matching networks.
Bibliography
D.M. Pozar, Microwave Engineering, 4th Ed., §5(1-3), §6(1,2,6)
T.C. Edwards, Foundations of Interconnect and Microstrip Design, 3rd Ed., §(10)
C. Bowick, RF Circuit Design, §2, §4
Quiz 2
Covering: Lecture note §2, Textbook §4(1-5), §5(1-3), §6(1,2,6)
Closed book w/ 1 A4 paper