Quality FactorMicrowave EngineeringEE 172Dr. Ray KwokRay Kwok & Ji-Fuh LiangCharacterization of High-Q Resonators for Microwave-Filter ApplicationsIEEE Trans MTT vol.47, p111-114, 1999& references therein.Q factors - Dr. Ray KwokQuality Factor• Often referred to as the Q-factor.• It indicates how good a quality the “device” has. “Good” here means low loss.e.g. a capacitor is said to be high-Q when it’s low loss.• Unloaded-Q (Qu), loaded-Q (QL) and external-Q (Qe)of resonators are often quoted in literature.• Q-factor is often difficult to calculate precisely. Engineers measure it directly using either S12or S11.Q factors - Dr. Ray Kwok4 5 6 7Frequency (GHz)Filter1 Response-60-50-40-30-20-100DB(|S[1,1]|)Filter1DB(|S[2,1]|)Filter14 5 6 7Frequency (GHz)Filter Response-60-50-40-30-20-100DB(|S[1,1]|)FilterDB(|S[2,1]|)FilterEffect of Q in a bandpass filterQ=5000Q=100Q factors - Dr. Ray Kwok4.75 5.25 5.75 6.25Frequency (GH z)Insertion Loss-4-3-2-10DB(|S[1,2]|)FilterDB(|S[1,2]|)Filter1Qu– dictates design typesQu=5000Qu=100Q factors - Dr. Ray KwokUnloaded Q⋅⋅⋅++=≈dcuuQQQdissipatedenergystoredenergyQ111__conductionlossdielectriclossfor a rectangular waveguide. Rsis the surface resistance of the resonator.See Pozar Ch.6.adding series “resistance” !!!Q factors - Dr. Ray KwokLoaded QeuLQ1Q1Q1+=Input / output coupling “de-Q” resonators.The coupling is related to the external Q (Qe).QLis measured by fo/∆f at the 3 dB (usually in S12).Quis the desired parameters for any passive design.Qecan be calculated or evaluated for any given coupling structure.Q factors - Dr. Ray KwokTransmission MeasurementfoutputfoQ ≡ fo/∆ fQuality Factormany resonantsS213 dB bandwidthQ factors - Dr. Ray KwokS21requires weak coupling1.46 1.48 1.5 1.52 1.54Freq uency (GH z)Q 500-60-50-40-30D B (|S [1,2 ]|)S ch em atic 1∆f = 3.06 MHzfo= 1.5002 GHzQ = fo/∆f = 490< 30 dBweak coupling means 1/Qe~ 0. QL~ QuS21(dB)Q factors - Dr. Ray KwokTransmission vs. Reflection |S11|2+ |S12|2≈ 1 Easier, no additional parts to make. Use existing coupling feature…. Often just need to change coupling strength. Precise measurement. True Quof that structure.comblinefilterQ factors - Dr. Ray KwokReflection < 3 dB?!1.48 1.49 1.5 1.51 1.52Frequency (GHz)Q500-0.25-0.2-0.15-0.1-0.050DB (|S[1,1]|)Sc hematic 1∆ω∆ω∆ω∆ωxωωωωoρρρρo-x dBQL(x) ≡ ωo/∆ωS11(dB)Q factors - Dr. Ray KwokEquivalent Circuit - resonatorF ig . 1 (a ) E q u iv a le n t c irc u it o f a se rie s re so n a to r c o u p le d to a so u rce im p e d a n ce Zo. (b ) E q u iv a le n t c ircu it o f a p a ra lle l re so n a to r c o u p le d to a so u rce a d m itta n c e Yo.Z oK0 1rLCgY oJ0 1LCrAgA S1 1 S1 1A A(a )(b )gLCgArrALC ρA ρ ρ ρ ρAA ’A ’rAKZo=012Q factors - Dr. Ray KwokOne-port ReflectionCrALrΓin( )( )( )( )rrrLQjQ1jQ1rLjrr1rLjrr1LjrrLjrrrZrZLjrLjrLC11LjrCjLjrZAuoouuAAAAAinAininoooo2in≡βω≡ωω−ωω≡ΩΩ+β+Ω+β−≡Ωω++Ωω+−=Ωω++Ωω+−=+−=ΓΩω+≡ωω−ωωω+=ω−ω+=ω−ω+=coupling parameterLC1o=ωQ factors - Dr. Ray KwokAround ωωωωo()),x(Q1Loxooooβ=ωω∆=ωω−ω≈ωω−ωω≡Ω( )( )2Lu22Lu22x),x(QQ1),x(QQ1β+β+β+β−=ρ2x22x2Lu1)1()1(),x(F),x(QQ),x(Fρ−β−−ρβ+=ββ≡βmagnitude of Γingeneralized QLDefine a mapping functionNote: F & QLare functions of x and β, but Quis independent of both.Q factors - Dr. Ray KwokMapping Function F(x,ββββ)ooooolog20RL110ρ−=β+β−=ρ=ωω−ωω=ΩAt fosoandis the return loss at resonantoooo1111ρ−ρ+=βρ+ρ−=βNote: ρo= 0 if β = 1 (critically-coupled)if β < 1 (under-coupled)if β > 1 (over-coupled)2x2o2xo112),x(Fρ−ρ−ρρ=βmthenfor the over- / under-coupled casesQ factors - Dr. Ray KwokExample (not ideal, RL too low)1.48 1.49 1.5 1.51 1.52Frequency (GHz)Q500-0.25-0.2-0.15-0.1-0.050DB (|S[1,1]|)Sc hematic 1∆ω∆ω∆ω∆ωxωωωωoρρρρo=10-0.21/20= 0.976-x ~ -0.075 dBρx= 10-x/20= 0.991QL(x) ∼∼∼∼ 1.5/0.004 ~ 375S11(dB)F = 1.342Qu ~ 503Q factors - Dr. Ray KwokFor x = 3 dB 00.511.522.530 5 10 15 20 25 30 35 40Return Loss at resonant (dB)F(3,b)over-coupledunder-coupledρo= 0.70795Q factors - Dr. Ray Kwok1.497 1.498 1.499 1.5 1.501 1.502 1.503Frequency (GHz)Return Loss-30-20-100DB(|S[1,1]|)Schematic 1Critically-Coupled (ββββ = 1)01.01.0-1.010.010.0-10.05.05.0-5.02.02.0-2.03.03.0-3.04.04.0-4.00.20.2-0.20.40.4-0.40.60.6-0.60.80.8-0.8Smith ChartSwp Max2.5GHzSwp Min0.5GHzS[1,1]Schematic 1sharp & high return lossradius ~ 1 circleQ factors - Dr. Ray Kwok1.497 1.498 1.499 1.5 1.501 1.502 1.503Frequency (GHz)Return Loss-30-20-100DB(|S[1,1]|)Schematic 101.01.0-1.010.010.0-10.05.05.0-5.02.02.0-2.03.03.0-3.04.04.0-4.00.20.2-0.20.40.4-0.40.60.6-0.60.80.8-0.8Smith ChartSwp Max2.5GHzSwp Min0.5GHzS[1,1]Schematic 1Over-Coupled (ββββ > 1)lower return lossradius > 1Q factors - Dr. Ray KwokUnder-Coupled (ββββ < 1)lower return lossradius > 101.01.0-1.010.010.0-10.05.05.0-5.02.02.0-2.03.03.0-3.04.04.0-4.00.20.2-0.20.40.4-0.40.60.6-0.60.80.8-0.8Smith ChartSwp Max2.5GHzSwp Min0.5GHzS[1,1]Schematic 11.497 1.498 1.499 1.5 1.501 1.502 1.503Frequency (GHz)Return Loss-30-20-100DB(|S[1,1]|)Schematic 1Q factors - Dr. Ray KwokExternal Q – input coupling1.495 1.496 1.497 1.498 1.499 1.5 1.501 1.502 1.503 1.504 1.505Frequency (GHz)Graph 1-180-90090180Ang(S[1,1]) (Deg)Schematic 1RA∆f = ∆f/gog1oAAf/frR∆=Use delay instead for dc shiftSee paper for more