L14-1RLC RESONATORSResonators trap energy:RLCISeries RLC resonatorGCLV+-Parallel RLC resonatorAlso:terminated TEM lines,waveguides Circuit equations, series resonator:Idi 1L + Ri + i dt = 0 j L I + R I + = 0dt C j C⇒ωω∫→21,2 2 1 R1R 1s = - ± j - ( ) Note: s = s * ω' = for R 02L LC 2LLCω’2ii2(-b± b -4ac )Let j = s ; recall: as + bs + c = 0 s = 2aω*= 0 ⇒ (jω –s1)(jω –s2) = 0*⎡⎤⎢⎥⎣⎦2R1(jω)+(jω)+ ILLC R- tjω't2Le oi(t) = R { I e } eL14-2RLC RESONATOR WAVEFORMSSeries resonator current i(t):Let φ = 0wT=wToe-(R/L)t= wm(t) + we(t)wm(t)Stored energyt0wTowTo/e~Q radians,Q/ω’ secondswe(t)Energy w(t):e-2αt ≅ e-ω’t/Q= e-(R/L)tQ ≅ω’/2α =Lω’/Rwemax=wmmax⇒ vmax=imax0ti(t)Io e-(R/2L)t= Ioe-αtL/C /R≅(series resonance)Q ≅ω’/2α =Cω’/GC/L / G≅(parallel)LC φRR- t - tjω't2L 2Le o oi(t) = R { I e } e = I cos(ω't + )e ∝R-t22Lm12w(t)= Li cos(ω't) e ∝R-t22Le12w(t)= Cv sin(ω't) eL14-3COUPLING TO RLC RESONATORSThevenin and Norton Equivalent Sources:Thevenin equivalent sourceRLLCI(ω)VThRThPower dissipated Pd(f) in R = RL+RTh:Half-power frequencies: ω ≅ ωο± R/2L = ωο±α,where ωο= 1/ so: Δω = 2α = ωo/Q and Q = ωo/ΔωINoNorton equivalentGCLV+-GNoPd(ω)ωoωΔω11/2Resonance22Th Thd2222Th22|V | |V |11P( ) R R22 1|Z||R j L |jC|V | R1R 1R ( j )( j )2L 2LLC LC2L−ω= =+ω +ωω=ω−−ω+−LCDominant factor near ωoL14-4RESONATOR QGeneral derivation of Q (all resonators):Pd(ω)ωoωΔω11/2wT≅ wToe-ω’t/Q(total stored energy [J])Pd= -dwT/dt ≅ (ω’/Q)wT(power dissipated [W])Q ≅ω’wT/Pd* (resonator Q [“radians” is dimensionless])Internal, external, and loaded Q (QI, QE, QL):QI = ω’wT/PdI(PdIis power dissipated internally, in R)QE = ω’wT/PdE(PdEis power dissipated externally, in RTh)QL = ω’wT/PdT(PdTis the total power dissipated, in R and RTh)RLCVThRThGCL+-VThRThPdT= PdI+ PdE⇒ QL-1= QI-1+QE-1QL≈ωo/Δω Perfect Match: QI=QE*IEEE definition: Q = ωowT/PdL14-5MATCHING TO RESONATORSTransmission line feed:RLCVThZoZoΓVThZoZoΓCL+-RAt ωo: |Γ|2= = 0 if matched, R = Zo= 1/9 if R = Zo/2 or 2ZoPd(ω)ωΔω11/20Behavior away from resonance:Series resonance: Open circuitParallel resonance: Short circuit2ooRZRZ−+o 1L/Cω=L14-6EXAMPLE #1 – CELL PHONE FILTERBandpass filter specifications:VThZoZoΓCL+-RLooks like a short circuit far from ωoAt ωo: reflect 1/9 of the incident power and let Γ < 0ωo= 5 × 109and Δω = 5 × 107Zo= 100-ohm lineFilter solution:Parallel resonators look like short circuits far from ωo|Γ|2= 1/9 and Γ < 0 ⇒Γ= -1/3 at ωo. ⇒ R = 50Ω= 1/ωo= (5x109)-1QL= R’/ (parallel) ⇒ = R’/QL= 33/100 = 0.33 (R’ = R // Zo)L = = (5x109)-1 x 0.33 = 6.67 x 10-12[Hy]C = = (5x109)-1/0.33 = 6 x 10-10[F]Small, hard to build, use TEM?o1ZZ1+Γ=−ΓLCL/CL/CLC / L/CLC L/CL14-7EXAMPLE #2 – BAND-STOP FILTERFilter specifications:Far from ωothe load is matched (signal goes to amplifier R)At ωoreflect all incident power; let Γ = -1 (short circuit) ωo= 5 × 106Δω = 5 × 104(rejected band, notch filter) ⇒ Q = 100Zo= 100-ohm lineFilter solution:Lossless series resonators look like short circuits at ωoR = Zo= 100Ω⇒|Γ|2= 0 at ω far from ωo= 1/ωo= (5x106)-1QL= /RL(series) ⇒ = RLQL= 50x100 = 5000L = = (5x106)-1 x 5000 = 10-3[Hy]C = = (5x106)-1/5000 = 4 x 10-11[F]RLCVThZoZoLCL/CLC / L/CLC L/CL/CMIT OpenCourseWare http://ocw.mit.edu 6.013 Electromagnetics and Applications Spring 2009 For information about citing these materials or our Terms of Use, visit:
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