Microwave Electronics LabLHTL as the Dual of the Conventional TL 1 Conventional LCjZYj linearLC 0101LCddvLCvgpdistortion nocstevvgp LCjZYj nonlinear1LC 0022LCvLCvgpdistortion fctvvgp LjZ dzCjY ID: 515547
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Christophe Caloz, Hiroshi Okabe, TaisukeIwai and Tatsuo Itoh Microwave Electronics LabLH-TL as the Dual of the Conventional TL (1) Conventional LCjZYj linearLC 0101LCddvLCvgpdistortion nocstevvgp LCjZYj nonlinear1LC 0022LCvLCvgpdistortion)( fctvvgp LjZ)( dzCjY)( pass-low dzCjZ 1 dzLjY 1 pass-high LLjCj11 Microwave Electronics LabLH-TL as the Dual of the Conventional TL (2) Conventional (RH) TLLH-TL 1 LC slopegv slope pv p rop. z (energy) p rop. z (energy) z p rop. z v slope0 gv thefrom S21ABCD f p-S 21 matrix thefrom S21ABCD f as as m 10:lengthp Microwave Electronics LabDetermination of LH Material Parameters 1(, jCjYjLjZ TL / Fieldfor a plane wave in an (RH) medium: )2(1,1LjYCjZ LH-TL parameters: C21!01Equating (1) and (2) yields the dispersive Demonstration ofnegative (L. D. Landau, E. M. Lifshitzand L.P. Pitaevskii, Electrodynamics of Continuous Media, Pergamon, 1984) 0101CL Demonstration of negative indexof refraction cnncjjjZYLCjZYrrrr200!0 :Material1j :TL-LH Microwave Electronics LabLumped-Element Approx. of the Physical (RH) Line Physical LineLumped-el. Approx. Infinitesimal model Im 2) 1):dia g ram CjGLjRpGGpRRpCCpLLpSFH ,,,:length of line line coaxial :e.g. pRRroi,,tan,,, :parameters physical iRoRr tan ln2:parameters TL ioRRL ,ln'2ioRRC S 21S )2matrix scattering 1):diagram :circuitladder 12N:cellunit RuLuG ppu etc. ,NpLpLLuuH C dz :model TL mHL SG mR FC Microwave Electronics LabLumped-Element Approx. of the LH Line Physical LineLumped-el. Approx. Infinitesimal model Im 2) 1):dia g ram CjGLjR!!!naturally existingNot C,L,R,G :yarbitraril prescribed becan length)unit (per p S 21S )2matrix scattering 1):diagram :circuitladder 12N:cellunit RCG ppu etc. ,NpCpCCuuF L dz :model TL mFC SG mRHL :length of line p pGGpRRS , ,pCCpLLFH Microwave Electronics LabResults for the Physical Model of a RH-TL:line coaxial Physical (MHz)frequency parameters-S of Phase Cumulative v slope0 pv -S of Magnitude(MHz)frequency (some mismatch was intentionallyintroduced to create S11 peaks:Zin= 71 with ports Zin= 50 z phase p 21S 10,m 128pF, 470H, 56.2SGRFCHL parameters TL ingCorrespond 1/m,m 812pF/m, 47nH/m, 256 G.RCL 5.9 mm, 6.2eiRR 10 ,2.1pr Microwave Electronics Lab parameters-S of Magnitude(MHz)frequency approximat)(cutoffcf NNpeaksLumped-Element Approx. (RH-TL) approx.fNfc20N (MHz)frequency parameters-S of Phase Cumulative approximat)(cutoffcf (linear) LCPASS-LOW)(unmatched Phys. Model vsL.E. Approx. (RH-TL) / Matched TL -S of Magnitude(MHz)frequency (MHz)frequency parameters-S of Phase Cumulative parameters-S of Phase Cumulative(MHz)frequency TL-RHMatchedComparison parameters-S of Magnitude(MHz)frequency Microwave Electronics Lab (MHz)frequency parameters-S of Phase Cumulative approximat)(cutoffcf dependency 1 LC1Lumped-Element Realization of a LH-TL (1) 20NPASS-HIGH parameters-S of Magnitude(MHz)frequency NNpeaksionapproximat)(cutoffcf (matched) approx.1fNfc !! Microwave Electronics Lab parameters-S of Magnitude Lumped-Element Realization of a LH-TL (2)behaviorsfrequenciehigher LC1 (cutoffcf approx. el. lu. unlimited BW (MHz)frequency parameters-S of Phase as as � unlimited BW unwrappingphase (MHz)frequency s velocitiegroup and Phase(MHz)frequency vg2 LCvp2 Microwave Electronics LabPossible Distributed Realization of the LH Line shunt via toground unit cellFeatures:by Q-factorof C-L, but still broadstill very low lossesand moderate dispersion Microwave Electronics Lab 00 :star t p :star t p Conventional (RH) TLLH-TL jpjjeeSeSS 212121 :generalIn "negatively increases" pLCLC ... variationphase oppositeexpect might we,0 Since :Paradox LH idem! :"positively decreases"1LCpLC 122LCbecause !!!22LCbecauseIn a LH material, phase rotatesin the same senseas in a RH material. Microwave Electronics LabGeneral Considerationsexhibit a LH effect (e.g. periodic)low losses, broad bandwidth, moderate dispersionpotential interest for microwave applicationsof the distributed LH-TL to a2D/3D LH materialconstitutive parameters differentthan in other approaches:link with the split-rings / wiresstructure: 022221,1Fp L221,1 -LH H Microwave Electronics LabPlane Wave Propagation at Interface RH-LH (1)Normal Incidence MHz 20 nH/m,256pF/m,470fLC LCRH LH1 LH and RH for the diagram - LHRH offct a as amplitude field RHmedium 39.2LH LH Microwave Electronics LabPlane Wave Propagation at Interface RH-LH (2)Oblique Incidence ( law, s(Snell'18,45rrtransincn .201LCLH 65.00LCRHMHz) 30 nH/m,256 pF/m,47(0 fLCCLrr 11, )/(1),/(12121CLrr Microwave Electronics LabPlane Wave Propagation at Interface RH-LH (3)Oblique Incidence (| law, s(Snell'18,45rrtransincn .201LCLH 65.00LCRHMHz) 30 nH/m,256 pF/m,47(0 fLCCLrr 11, )/(1),/(12121CLrr