Jensen, Thomas

Krozer, Viktor

Meincke, Peter

Zhurbenko, Vitaliy

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Analytical model of planar double split ring resonator

This paper focuses on accurate modelling of microstrip double split ring resonators. The impedance matrix representation for coupled lines is applied for the first time to model the SRR, resulting in excellent model accuracy over a wide frequency range. Phase compensation is implemented to take into account the curved shape of the resonators. The presented results show that the accuracy better than 0.5% for the first four resonances. Analysis of a periodical structure with frequency selective properties is presented

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Downloaded from orbit.dtu.dk on: Dec 17, 2017Analytical Model of Planar Double Split Ring ResonatorZhurbenko, Vitaliy; Jensen, Thomas; Krozer, Viktor; Meincke, PeterPublished in:Proceedings of the International Microwave &amp; Optoelectronics ConferenceLink to article, DOI:10.1109/IMOC.2007.4404369Publication date:2007Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Zhurbenko, V., Jensen, T., Krozer, V., & Meincke, P. (2007). Analytical Model of Planar Double Split RingResonator. In Proceedings of the International Microwave & Optoelectronics Conference (pp. 753-756). IEEE.DOI: 10.1109/IMOC.2007.4404369Analytical Model of Planar Double Split Ring ResonatorVitaliy Zhurbenko, Thomas Jensen, Viktor Krozer, Peter MeinckeTechnical University of Denmark, Ørsted•DTU, ElectroScience,Ørsteds Plads, building 348, 2800 Kgs. Lyngby, Denmark, Phone:+45-45253861, Fax: +45-45931634E-mail: tje@oersted.dtu.dkAbstract — This paper focuses on accurate modelling ofmicrostrip double split ring resonators. The impedance matrixrepresentation for coupled lines is applied for the first time tomodel the SRR, resulting in excellent model accuracy over a widefrequency range. Phase compensation is implemented to take intoaccount the curved shape of the resonators. The presented resultsshow that the accuracy better than 0.5% for the first fourresonances. Analysis of a periodical structure with frequencyselective properties is presented.Index Terms — Split ring resonator, metamaterial, microstripcircuit, coupled lines, impedance matrix.I. INTRODUCTIONecently microstrip filters based on electromagneticbandgap (EBG) structures and metamaterials haveattracted a great deal of interest because of improvedcharacteristics in comparison to traditional filter design [1-3].In planar filter design the synthesis of metamaterials issuccessfully done with double split ring resonators, originallyproposed by Pendry et al. [4]. The main obstacle for filterdesign with the SRR technology is the lack of accurateanalytical models and the designer usually has to use full-wave electromagnetic simulations in order to obtain asufficiently accurate prediction of the response. This approachis inefficient because it is extremely time consuming.Analytical models based on lumped elements where theelement values have been extracted from electromagneticsimulations or measurements have been presented [5], butthey are restricted to a narrow frequency range around theresonant frequency. The model presented in [6] provides awideband analysis, but, unfortunately, doesn’t take intoaccount the presence of a ground plane in planar structures.Here it is shown that the response of microstrip double SRRcan be accurately predicted using sections of coupledtransmission lines. The proposed analytical representationgreatly reduces the simulation time with good correlationbetween model and full-wave electromagnetic simulations. Insection II an analytical model based on impedance matricesfor coupled lines, transmission lines and microstrip gaps isderived for microstrip double SRR. Section III showsinvestigations of the models accuracy and section IV providesan example of a stopband structure analysis based on theproposed model.II. MODEL FOR MICROSTRIP DOUBLE SRRThe double SRR structure consists of a pair of concentricrings with splits etched on opposite sides. The double SRRlayout and proposed model representation are shown in Fig.1.Fig. 1. Microstrip SRR geometry and corresponding modelrepresentation.wslclw, l=gwslclw, l=g w, lw, lw, l w, lC1C1C2C1C1C2rgsw(a)(b)R7531-4244-0661-7/07/$20.00 c© 2007 IEEEThe model is based on two coupled line sections with alength equal togwrl 	2cl . (1)However, this length only corresponds to the length of theinner semi-ring.It is obvious that the lengths of the two resonators are notequal. In most practical cases this difference between theelectrical lengths at the first resonant frequency reaches a fewtens of degrees and even higher. It is not sufficient to describea double SRR by just two coupled line sections with a lengthequal to the average resonators length. To compensate for thisdifference four transmission lines with a length equal to		 cl2324 lgswrl  (2)have to be introduced as shown in Fig. 1(b). Such a phasecompensation significantly improves the model accuracyresulting in good agreement between simulated and calculatedresults, as shown later.An equivalent circuit for the microstrip gap g of theresonators is a symmetric 2-port -network, as shown inFig. 1(b). The capacitance C2 is due to the charge build-upbetween the two microstrip lines. The capacitance C1 is due tothe fringing fields at the open circuits. The calculation ofvalues for these capacitors is well described in literature [7-9]and is not considered here.The transmission line with a length equal to g unites thetwo semi-rings, which are represented by coupled lines.The circuit in Fig. 1(b) is transformed to an impedancematrix network with the corresponding matrix representationof the circuit elements. This network is shown in Fig. 2, where[ZCL] stands for the impedance matrix of the coupled linesfour-port.A total of four quantities is required to describe symmetriccoupled lines [10]: Z0o and Z0e, which are, respectively, thecharacteristic impedances for odd and even modes ofpropagation; o and e, the propagation constants of odd andeven modes. The elements of the impedance matrix [ZCL] aregiven by   cl0cl044332211 cotcot2lZlZjZZZZ ooee  	  , (3a)   cl0cl043342112 cotcot2lZlZjZZZZ ooee    , (3b)   cl0cl042243113 csccsc2lZlZjZZZZ ooee    , (3c)   cl0cl032234114 csccsc2lZlZjZZZZ ooee  	  , (3d)where lcl is the length of the coupled line section, as it isshown in Fig. 1(b).[ZCL] [ZCL][ZC][ZC][Z] [Z][Z] [Z][Zg][Zg]Fig. 2. Double SRR model based on an impedance matrixrepresentation.The matrices for the transmission lines of the model(Fig. 1.(b)) are given to be       gZgZgZgZZ gcothcschcschcoth0000 , (4)and        lZlZlZlZZcothcschcschcoth0000 . (5)Here Z0 is the characteristic impedance of the microstrip line,and  its propagation constant.The calculation of electrical parameters for the microstriplines and coupled lines can be performed using empiricalexpressions [11] or any free or commercially availablesoftware transmission line calculators.The impedance matrix for the microstrip gap is 						)CC2(CCC)CC2(CC)CC2(CC)CC2(CCC121211212121212121jjjjZC . (6)Using this model representation the response of theresonator can be predicted in a wide frequency range.754 2007 SBMO/IEEE MTT-S International Microwave & Optoelectronics Conference (IMOC 2007)III. MODEL VALIDATIONTo estimate the proposed model accuracy the calculatedresponse of the microstrip double SRR is compared to full-wave simulations using method of moments (MoM). Theconsidered configuration is based on a substrate with adielectric constant r = 3.38 and thickness h = 0.8 mm. TheSRR parameters and the relative difference betweencalculated and simulated resonant frequency fi are given inTable 1 for the first four resonant frequencies, respectively.TABLE 1RESONATOR PARAMETERS AND ESTIMATION OF MODEL ACCURACYParameter r(mm)w(mm)s(mm)g(mm)f1(%)f2(%)f3(%)f4(%)Value 12.7 0.2 0.17 0.17 0.02 0.06 0.17 0.3The SRR structure exhibits resonant frequencies within awide range of frequencies. The first resonant frequency ispredicted with an accuracy better than 0.02% and all resonantfrequencies are predicted with an accuracy better than 0.5%.The main contribution to the inaccuracy originates from theelectrical model parameter relations for coupled lines,transmission lines, and capacitances C1, C2.As expected, the resonant frequency calculation inaccuracyincreases for higher order resonances, but is significantly lessthan 1%. Such an accuracy is deemed sufficient for filterdesign.IV. MODELLING OF STOPBAND STRUCTUREA double SRR is often used as a unit cell in electromagneticmetamaterials. Such materials are formed from repeated unitcells containing scattering elements. Therefore it is interestingto investigate the reliability of the proposed modelconsidering a structure, which contains a number of doubleSRRs.Double SRRs are a well known alternative to EBGstructures for the rejection of frequency bands in planartransmission lines. A behaviour similar to EBG structures canbe achieved by implementing periodical loading of a planartransmission line with double SRRs.The microstrip double SRR was implemented to form astopband structure in a microstrip transmission line. Twentydouble SRRs with varying resonance frequency are coupled toa transmission line using short stubs. In such a type oftransmission line the signal propagation is inhibited in acertain frequency band. This property is used to build amicrostrip stopband structure in a form as shown in Fig. 3.Fig. 3. Periodically loaded 50  microstrip transmission line.The considered structure is electrically large resulting incomplicated, time consuming full-wave analysis. The varyingresonant frequency of individual cells prohibits the reuse offull-wave single cell simulation results for the overallstructure.The model developed here is used to design the structurewith a frequency stopband around 2 GHz, using a substratewith r = 3.38 and height h = 0.8 mm. The transmission linewidth is equal to 1.86 mm. The connecting stubs length andwidth are 0.4 mm and 0.2 mm, respectively. The distancebetween two adjacent double SRRs is equal to 5 mm. Thedimensions for the SRR, as illustrated in Fig. 1(a), ares = g = 0.2 mm, w = 0.4 mm. The radiuses r of the resonatorsare 6.55 mm, 6.45 mm, 6.3 mm, 6.15 mm, 5.95 mm, 5.75 mm,5.55 mm, 5.35 mm, 5.15 mm, and 5.00 mm.The calculated response of the microstrip SRR is comparedto MoM simulations. The microstrip lines metal is assumed tobe lossless with zero thickness to improve MoMcomputational efficiency.The calculated and simulated responses are shown in Fig. 4.It can be seen from the presented results that the agreementbetween calculated and simulated is very good at lowerfrequencies and the stopband is slightly moved towards lowerfrequencies (Fig. 4(a)). Discrepancy between calculated andsimulated transmission and reflection magnitudes increases inthe high frequency region. The simulated S11 bandwidth isnarrower, than predicted by the model, as can be seen inFig. 4(b).The discrepancy may be attributed to the parasiticinteractions between the SRRs and the transmission line,which are not considered in the model. This interaction maycause the excitation of undesirable modes in the SRRs.Although the simulation and modelling results agreereasonably well, more work is needed to include mutualcoupling affects and parasitic mode excitation in SRRs.IN50OUT502007 SBMO/IEEE MTT-S International Microwave & Optoelectronics Conference (IMOC 2007) 755Fig. 4: Calculated and simulated responses of the structureshown in Fig. 3.V. CONCLUSIONIn this work, an analytical model for the microstrip doubleSRR has been developed based on the impedance matrixrepresentations of coupled lines, transmission lines andmicrostrip gaps.The developed model has been validated with MoMsimulations. The presented results demonstrate an accurateprediction of the first four resonant frequencies of an SRRwith a relative error below 0.5%. The proposed model isreliable in a wide frequency range and provides prediction ofhigher order resonances, which has not been demonstrated yetwith circuit based models.The proposed analytical model makes it simple andeffective to analyze microstrip double SRR structures andelectrically large circuits based on it.As an example the developed model has been used foreffective design and analysis of a microstrip stopbandstructure based on a number of double SRRs. The predictedtransmission and reflection performance of the structurecompares well with full-wave simulations.The developed model is intended to be used in a design ofcompact and high performance components for a microwavecamera.ACKNOWLEDGEMENTThe authors wish to acknowledge the partial financialsupport of the Danish Ministry of Research and Education.REFERENCES[1] J. García-García, et al., “Miniaturized Microstrip and CPWFilters Using Coupled Metamaterial Resonators,” IEEETransactions on Microwave Theory and Techniques, vol.54, no.6, pp. 2628–2634, June 2006.[2] A. Garcia-Lamperez, M. Salazar-Palma, “Dual Band Filter withSplit-Ring Resonators,” IEEE MTT-S International MicrowaveSymposium Digest, pp. 519-522, 2006.[3] J. García-García, et al., “Microwave Filters with ImprovedStopband Based on Sub-Wavelength Resonators,” IEEETransactions on Microwave Theory and Techniques, vol.53, no.6, pp. 1997–2006, June 2005.[4] Pendry, J. B., A. J. Holden, D. J. Robbins, and W. J. Stewart,“Magnetism from conductors and enhanced nonlinearphenomena,” IEEE Transactions on Microwave Theory andTechniques, vol.47, pp. 2075–2084, 1999.[5] J. D. Baena, et al., ”Equivalent-Circuit Models for Split-RingResonators and Complementary Split-Ring Resonators Coupledto planar Transmission Lines,” Transactions on MicrowaveTheory and Techniques, vol. 53, no. 4, April 2005.[6] M. Shamonin, E. Shamonina, V. Kalinin, and L. Solymar,“Resonant Frequencies of A Split-Ring Resonator: AnalyticalSolutions and Numerical Simulations,” Microwave and OpticalTechnology Letters, Vol. 44, No. 2, pp. 133-136, January 202005.[7] M. Kirschning, R. H. Jansen, N. H. L. Koster, “MeasurementAnd Computer-Aided Modelling Of Microstrip DiscontinuitiesBy An Improved Resonator Method,” IEEE MTT-SInternational Microwave Symposium Digest, pp. 495-497, 1983.[8] E. Hammerstad, “Computer-Aided Design Of MicrostripCouplers with Accurate Discontinuity Models,” IEEE MTT-SInternational Microwave Symposium Digest, pp. 54-56, 1981.[9] N. H. L. Koster, R. H. Jansen, ”The Equivalent Circuit of theAsymmetrical Series Gap in Microstrip and SuspendedSubstrate Lines,” Transactions on Microwave Theory andTechniques, vol. MTT-30, no. 8, August 1982.[10] G. I. Zysman, A. K. Johnson, “Coupled Transmission LineNetworks in Inhomogeneous Dielectric Medium” Transactionson Microwave Theory and Techniques, vol. MTT-17, no. 10,October 1969.[11] R. Mongia, I. Bahl, P. Bhartia, RF and Microwave CoupledLine Circuits. Norwood: Artech House microwave library, 1999MoMModel0.5 1.0 1.5 2.0 2.5 3.0 3.50.0 4.0-30-20-10-400Frequency, GHzMagnitudeofS11,dB(a)(b)-70-60-50-40-30-20-10-800xMoMModelx0.5 1.0 1.5 2.0 2.5 3.0 3.50.0 4.0Frequency, GHzMagnitudeofS21,dB(dB)( )(dB)( )756 2007 SBMO/IEEE MTT-S International Microwave & Optoelectronics Conference (IMOC 2007)

Analytical Model of Planar Double Split Ring Resonator

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Analytical Model of Planar Double Split Ring Resonator

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