Chapter II Resonators and Impedance Matching

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

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

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

Frequency response of series resonant circuit near resonance Fractional Bandwidth

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

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

Fractional Bandwidth Loaded Q Resonant Circuit Unloaded Q RL

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

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

Frequency responses of RLC resonant circuits near resonance

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

Equivalent series RLC resonant circuit

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

Yin Open-Circuited l/2 Line

Equivalent parallel RLC resonant circuit

Example: Find Q of Half-Wave Microstrip Line Resonators

Yin Short-Circuited l/4 Line

Equivalent parallel RLC resonant circuit

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

Q factor of Microstrip Components Attenuation due to conductor loss

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

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

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.

Radiation Loss Yin open-end

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.

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.

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

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

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

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

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

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.

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.

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

(1) (2) (3)

(4) (5) (6)

Estimation of Matching Bandwidth parallel series

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

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

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

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Three-Element (High Q) Matching X j 2b

+

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

+

+

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Cascaded L-Section (Low Q) Matching +

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

+

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Open or shorted stub Single-Stub Matching

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

H4

Double-Stub Matching

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

H4

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.

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