Dr. Vahid Nayyeri. Microwave Circuits Design

Transcription

1 Lect. 8: Microwave Resonators Various applications: including filters, oscillators, frequency meters, and tuned amplifiers, etc. microwave resonators of all types can be modelled in terms of equivalent RLC resonators (either in series or in parallel.. Series Resonant Circuit Z in R jl j C

2 The complex power delivered to the resonator is P in * Zin I I VI ( R R: dissipate power I R 4 P loss L: store magnetic energy I L C: store electric energy So, we have P in P loss W m W jl j C e VC C I 4 4 j( Wm We C

3 Then Z in can be written as Z in P I in P loss j( W I m W e At resonance, W m = W e, then we have Z in P loss I R and LC Where is the resonant frequency.

4 (average energy stored Quality factor: Q Q (energy loss/second Lower loss implies higher Q. Wm W P loss e For the series resonant circuit and at resonance Q W L P R RC m loss Q increases as R decreases.

5 Bandwidth consideration: near resonance When = o +and is small, Z in Since R jl R jl LC L Q we have R Zin R j L R( j Q Can be used to identify equivalent circuits for distributed element resonators.

6 Modeling a resonator with loss as a lossless resonator Replace the resonant frequency with a complex effective resonant frequency j Q

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8 Z R jrq ( BW R in BW Q

9 . Parallel Resonant Circuit Z in P in P R loss jl jc j( Wm We LC Q R RC L BW Q

10 When loss is small, we can take it into account by having j Q

11 Loaded and Unloaded Q Unloaded Q (Q : The resonant circuit is not connected to any external circuitry, no loading effect. A resonant circuit connected to an external load, R L. L RL where Qe RL L Qe: external Q Loaded Q (Q L : With an external load, R L which will always lower the overall or loaded Q (Q L. for series circuits for parallel circuits Q Q Q L e

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13 Transmission Line Resonators At microwave frequencies, distributed elements, such as transmission lines are more commonly used as the resonators. Z in Short-Circuited / Line Z Z tanh( j l tanhl j tan l j tan l tanhl l = n/ Under the assumption of small loss and l = /. Z in Z ( l j

14 Compare with the expression for the series resonator circuit We have The equivalent capacitance is R At resonance, l = n/ (n=,, Q Z l C L R l Z in R jl L L Z Z Z in See Example 6. for practice R Z l Only true for TEM or quasi-tem lines!

15 Short-Circuited /4 Line Under the assumption of small loss and l = /4. Z in l Z j / Compare with the expression for Z in the parallel resonator circuit / R jc We have Z R l C 4 Z L C 4Z Q R L 4l

16 Open-Circuited / Line Under the assumption of small loss and l = /. Z in Z l j / ( Compare with the expression for Z in the parallel resonator circuit / R jc We have Z R l C Z L C Z Q R L l

17 Excitation of Resonators Common coupling techniques:. Gap coupling. Aperture coupling Gap coupling feed coupling aperture coupling Microstrip feedline coupling

18 Critical Coupling - a resonator matched to a feedline at the resonance frequency: to obtain maximum power transfer between a resonator and a feedline Consider this: a series resonant circuit coupled to a feedline Zin R j L R( j Q The unloaded Q is L Q R At resonance, we need to have Zin Z R R Then Q e L Q Z L Q Z the external Q

19 Coupling Coefficient: g g / Q for series resonator Q e Z R / Z Three different coupling situation: R for parallel resonator ( g <, undercoupled; ( g =, critically coupled; (3 g >, overcoupled A Gap-Coupled Microstrip Resonator A / open-circuited microstrip resonator is coupled to a microstrip feedline.

20 Equivalent circuit of the gap-coupled microstrip resonator The normalized input impedance seen by the feedline is z Z Z [(/ C Z j Z cot l] j tan l b b tan l Where b c =Z C is the normalized susceptance of the coupling capacitor, C. c c

21 When does the resonance occur? z In practice, b c The first intersection is close to where l= tan l b c (can be solved by numerical method Effect of the coupling: lower the resonant frequency.

22 Using a series RLC circuit to represent the gap-coupled microstrip resonator ( ( ( ( d dz z z First, expand z( in a Taylor series about the resonant frequency ( ( tan sec ( c p c p c c c b j v l b j v l b b j d l d l b l j d dz Since b c << and l v p /, we have ( ( b c j z then Microwave Circuits Design

23 What about the losses? Replace with (+j/q, then z( j ( j ( z( b c Qb c bc The capacitive-coupled / resonator looks like a series RLC resonant circuit near resonance. But an open-circuited / line resonator is described as a parallel RLC circuit. So what happened? The coupling capacitor serves as an impedance inverter (will see details in the discussion on filters. At resonance, For critical coupling, we have The coupling coefficient is R Z /Qbc R Z or b c Q g Z / R Q b / c