Journal of Magnetism and Magnetic Materials 383 (2015) 144–151

Contents lists available at ScienceDirect

Journal of Magnetism and Magnetic Materials

http://d0304-88

n CorrE-m

Ferran.M

journal homepage: www.elsevier.com/locate/jmmm

Analysis of transmission lines loaded with pairs of coupled resonantelements and application to sensors

J. Naqui, L. Su n, J. Mata, F. MartínGEMMA/CIMITEC, Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain

a r t i c l e i n f o

Article history:Received 13 June 2014Received in revised form4 December 2014Accepted 11 December 2014Available online 12 December 2014

Keywords:MetamaterialsSplit ring resonators (SRR)Stepped impedance resonators (SIR)Microstrip lineCoplanar waveguides (CPW)Microwave sensorsMagnetic coupling

x.doi.org/10.1016/j.jmmm.2014.12.01153/& 2014 Elsevier B.V. All rights reserved.

esponding author.ail addresses: lijuan.suri.su@gmail.com (L. Su)artin@uab.es (F. Martín).

a b s t r a c t

This paper is focused on the analysis of transmission lines loaded with pairs of magnetically coupledresonators. We have considered two different structures: (i) a microstrip line loaded with pairs ofstepped impedance resonators (SIRs), and (ii) a coplanar waveguide (CPW) transmission line loaded withpairs of split ring resonators (SRRs). In both cases, the line exhibits a single resonance frequency(transmission zero) if the resonators are identical (symmetric structure with regard to the line axis), andthis resonance is different to the one of the line loaded with a single resonator due to inter-resonatorcoupling. If the structures are asymmetric, inter-resonator coupling enhances the distance between thetwo split resonance frequencies that arise. In spite that the considered lines and loading resonators arevery different and are described by different lumped element equivalent circuit models, the phenom-enology associated to the effects of coupling is exactly the same, and the resonance frequencies are givenby identical expressions. The reported lumped element circuit models of both structures are validated bycomparing the circuit simulations with extracted parameters with both electromagnetic simulations andexperimental data. These structures can be useful for the implementation of microwave sensors based onsymmetry properties.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Artificial transmission lines inspired by metamaterial concepts,also called metamaterial transmission lines, have been the subjectof an intensive research in the last decade. These lines are im-plemented by loading a host line (e.g., a microstrip line or a co-planar waveguide – CPW) with reactive elements, namely in-ductors, capacitors, and/or resonant elements. Thanks to the pre-sence of these reactive loading elements, there are more degreesof freedom for design purposes (as compared to ordinary trans-mission lines), and the implementation of microwave componentswith reduced size and high performance, or the design of devicesexhibiting novel functionalities, is possible. There are many re-ported examples of microwave components based on metama-terial transmission lines, such as enhanced bandwidth compo-nents [1–4], multiband components [5–8], leaky-wave antennaswith end-fire to back-fire scanning capability [9–12], compact andmultiband filters [13–16], couplers [17,18], etc. In most of theseimplementations, the key aspect to achieve the required deviceperformance or functionality is the controllability of both the

,

characteristic impedance and the dispersion in the constitutivemetamaterial transmission lines. Thus, the design of metamater-ial-inspired planar microwave components is mainly based onimpedance and dispersion engineering [19].

Many of the previous metamaterial based components havebeen implemented by means of the so-called resonant-type ap-proach, where a host line is loaded with resonant elements, ty-pically split ring resonators (SRR) [20], complementary split ringresonators (CSRRs) [21], open SRRs and CSRRs [22], etc., and,eventually, other reactive elements (inductors or capacitors). Theformer resonant-type metamaterial transmission line was im-plemented by loading a CPW transmission line with split ringresonators [20]. This line exhibits a stop band functionality thatmay be interpreted as due to the extreme values of the effectivepermeability of the line in the vicinity of the first SRR resonance(positive below it, and negative above it). However, by addingshunt connected inductive elements (strips), it was found that thestop band switches to a pass band with backward (left-handed)wave propagation at low frequencies and forward wave propaga-tion at high frequencies (i.e., the SRR/strip loaded CPW exhibits acomposite right/left handed – CRLH – behavior, similar to trans-mission lines loaded with series capacitances and shunt in-ductances [23]).

In spite that the behavior of SRR or SRR/strip loaded CPWtransmission lines can be interpreted from the theory of

www.sciencedirect.com/science/journal/03048853www.elsevier.com/locate/jmmmhttp://dx.doi.org/10.1016/j.jmmm.2014.12.011http://dx.doi.org/10.1016/j.jmmm.2014.12.011http://dx.doi.org/10.1016/j.jmmm.2014.12.011http://crossmark.crossref.org/dialog/?doi=10.1016/j.jmmm.2014.12.011&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.jmmm.2014.12.011&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.jmmm.2014.12.011&domain=pdfmailto:lijuan.suri.su@gmail.commailto:Ferran.Martin@uab.eshttp://dx.doi.org/10.1016/j.jmmm.2014.12.011

J. Naqui et al. / Journal of Magnetism and Magnetic Materials 383 (2015) 144–151 145

metamaterials, it is useful to analyze these lines, and other me-tamaterial transmission lines, from their lumped element equiva-lent circuit model, valid as long as the unit cell of these lines iselectrically small. In this regard, the former model of SRR loadedlines was reported in [20], and later revised in [24] to account forthe effects of the inductive strips on the transmission zero fre-quency, present below the first transmission band. In these mod-els, coupling between SRRs of adjacent cells was not included, andthe orientation of the SRRs was considered to be with their sym-metry plane orthogonal to the line axis. With such orientation, thesingle coupling mechanism between the line and the SRRs ismagnetic. However, by rotating the SRRs, electric coupling mustalso be accounted for in the model (giving rise to mixed coupling),and it explains the asymmetry in the reflection coefficientsmeasured from the two ports, as demonstrated in [25] (an ex-tensive study on the coupling mechanisms between SRRs and theeffects of relative orientation is reported in [26,27]). Couplingbetween adjacent SRRs was considered in [28,29]. In [29] it wasdemonstrated that stop band bandwidth enhancement in SRR-loaded CPW transmission lines is due to the presence of complexmodes [30], which appear as conjugate pairs and do not carry netpower.

In the present paper, we study the effects of coupling betweenthe SRRs forming the pair in a CPW transmission line, neglectingthe coupling between SRRs of adjacent cells. Moreover, orthogonalorientation between the symmetry plane of the CPW and thesymmetry plane of the pair of SRRs is considered. Although for theimplementation of metamaterial transmission lines the pairs ofSRRs are implemented with identical resonators, disruption ofsymmetry may be of interest in certain applications, such as sen-sors and comparators based on symmetry properties [31]. There-fore, we will consider the more general case of CPWs withasymmetric SRR loading (i.e., with different SRRs forming the pair,including different inductive coupling between the line and theresonators). Moreover, since microstrip lines loaded with pairs ofshunt connected stepped impedance resonators (SIRs) have beenfound to be useful for sensing purposes [32], we will also study indetail these lines, where coupling between resonators is alsopresent. It is important to highlight that, in spite that SRR-loadedCPWs and SIR-loaded microstrip lines are described by distinctlumped element circuits, the phenomenology associated to inter-resonator coupling is roughly the same in both structures, and theresonance frequencies predicted by the respective models are gi-ven by identical expressions.

a b

Fig. 1. Typical topology of a CPW loaded with a pair of SRRs (a), and lumped element equblack) are etched in the back substrate side of the CPW (depicted in gray). The model c

The paper is organized as follows: in Section 2, the models ofthe SRR-loaded and SIR-loaded lines are presented, and the re-sonance frequencies predicted by these models are obtained forthe general (asymmetric) case, and for specific cases (asymmetricstructure but considering identical resonance frequencies for thetwo resonators, and symmetric structure). In Section 3, the modelsare validated by comparing circuit simulations with electro-magnetic simulations and experimental responses obtained infabricated structures. The potentiality of these structures for sen-sing is discussed in Section 4. Finally, the main conclusions arepointed out in Section 5.

2. Circuit model and analysis

In this section, the circuit models of the considered structuresare reported, and the resonance frequencies, given by the trans-mission zero frequencies, are inferred. We will first study the CPWloaded with a pair of magnetically coupled SRRs, and then themicrostrip line loaded with a pair of coupled (also magnetically)SIRs.

2.1. CPW loaded with a pair of coupled SRRs

Fig. 1(a) depicts the typical topology (and relevant dimensions)of a CPW loaded with a pair of SRRs, etched in the back substrateside, underneath the slots. The symmetry plane of the SRRs isorthogonal to the line axis, so that electric coupling between theline and the SRRs can be neglected [25], the main coupling me-chanism thus being magnetic. Moreover, the SRRs are orientedwith the gap of the outer ring located on the opposite side fromthe interaction region between SRRs (i.e., the vicinity of the lineaxis). By this means, we ensure that the dominant coupling me-chanism between SRRs is magnetic. Moreover, we have consideredrectangular shaped SRRs to enhance such inter-resonator coupling.Although the structure depicted in Fig. 1 is symmetric, the lumpedelement equivalent circuit model, depicted in Fig. 1(b), corres-ponds to the general case of an asymmetric structure with dif-ferent SRRs and line-to-resonators couplings (the model is iden-tical to the one reported in [20], but considering asymmetry andinter-resonator coupling). In the model, L and C are the line in-ductance and capacitance, respectively; the SRRs are described bythe resonant tanks L1–C1 and L2–C2, and their coupling to the lineis accounted for by the mutual inductances M1 and M2; finally,

ivalent circuit model, considering magnetic coupling between SRRs (b). The SRRs (inonsiders the general case of different SRRs.

J. Naqui et al. / Journal of Magnetism and Magnetic Materials 383 (2015) 144–151146

M′ is the mutual inductance between SRRs. This model is valid inthe frequency region of interest (i.e., the vicinity of the funda-mental resonance of the SRR) as long as the SRRs are electricallysmall resonant particles (a necessary condition for this is that thetwo metallic loops forming the SRR are tightly coupled, and hencevery close [33,34]).

The circuit of Fig. 1(b) is a π-circuit where the shunt branchesare purely capacitive. Thus, the transmission zeros (or resonancefrequencies of the coupled resonators) are given by the poles ofthe series reactance. From Kirchhoff's equations, including thedifferent mutual couplings, applied to the series branch of thecircuit of Fig. 1(b), the impedance of this branch is found to be

⎧⎨⎪⎩⎪⎫⎬⎪⎭⎪

( ) ( )( )( )

Z j L

j

C M C M M M M C C

M C C

( )

1 / 1 / 2

1 / 1 /

(1)

s

3

1 12 2

22

2 22 2

12 2

1 2 1 2

212 2

22 4 2

1 2

ω ω

ω

ω ω ω ω ω

ω ω ω ω ω

= +

×

− + − + ′

− − − ′

where ω1¼(L1C1)�1/2 and ω2¼(L2C2)�1/2 are the resonance fre-quencies of the isolated resonators. By forcing the denominator ofthe last term to be zero, the notch (or transmission zero) fre-quencies can be derived, namely

M C C

M C C

( ) 4

2[1 ] (2)2 1

222

12

22 2 2

14

24

1 22

12

22

1 2

ωω ω ω ω ω ω

ω ω=

+ ± − + ′

− ′±

Note that the mutual coupling M′ between resonators enhancesthe distance between transmission zeros, i.e.,

M C C

M C C

( ) 4

[1 ] (3)2 2 1

222 2 2

14

24

1 22

12

22

1 212

22ω ω

ω ω ω ωω ω

ω ω− =− + ′

− ′> −+ −

However, the resonance frequencies, ωþ and ω� , do not de-pend on the mutual coupling between the line and the SRRs (M1and M2).

Let us now consider two particular cases of interest. The firstone is the symmetric SRR loaded CPW, namely M1¼M2¼M, L1¼L2¼Lr, and C1¼C2¼Cr (and hence ω1¼ω2¼ω0). Under these con-ditions, the two solutions of (2) are

M L k1 / 1 (4)r M

0 0ωω ω

=∓ ′

=∓±

Fig. 2. Typical topology of a microstrip line loaded with a pair of SIRs (a), and lumped elemodel considers the general case of different SIRs.

where kM is the magnetic coupling coefficient [36]. However, oneof the solutions, the one with the (�) sign in the radicand of (4)(ωþ), is not actually a transmission zero frequency. The reason isthat this frequency also nulls the numerator of the last term in (1).By applying l’Hôpital’s rule, it follows that the series impedance isfinite at the frequency ωþ; hence, we can conclude that only onetransmission zero, to the left of ω0 (since M′40), appears in thetransmission coefficient for symmetric structures with magneti-cally coupled SRRs.

The second case of interest is the asymmetric structure (dif-ferent coupled SRRs), but considering that their fundamental re-sonance frequencies are identical, that is, L1≠L2, C1≠C2, ω1¼ω2¼ω0. In this case, expression (2) gives:

M L L k1 / 1 (5)M

0

1 2

0ωω ω

=∓ ′

=∓±

Note that (5) is formally identical to (4). The two frequenciesgiven by (5) null the denominator of the last term in (1), but, ingeneral, none of them null the numerator. However, if the fol-lowing condition is satisfied

C M C M M M C C2 (6)1 12

2 22

1 2 1 2+ =

then the numerator of the last term in (1) is also null at ωþ , andthe impedance of the series branch is finite at this frequency (asthe application of l’Hôpital’s rule reveals). Expression (6) can besimplified to:

MM

LL (7)

1

2

1

2=

Thus, if condition (7) is satisfied, then only one transmissionzero (at ω�) is expected (note that (7) represents a balance thatforces the structure to behave similarly to the symmetric one).Otherwise, two transmission zeros at the frequencies given by (5),i.e., one above and the other below ω0, are expected.

2.2. Microstrip line loaded with a pair of coupled SIRs

The typical topology, and lumped element equivalent circuitmodel, of a microstrip line loaded with a pair of SIRs (a resonatorused in [35] for the implementation of band pass filters) are de-picted in Fig. 2. Even for wide (low impedance) host microstriplines, the coupling between SIRs cannot be neglected and it hasbeen found that such coupling is magnetic and negative [32], asindicated in Fig. 2(b). The negative sign is consequence of thecurrents flowing in opposite directions in both resonant elements

ment equivalent circuit model, considering magnetic coupling between SIRs (b). The

Fig. 3. Two-port with magnetically coupled inductors and its equivalent T-circuit model (a), and model of the SIR loaded line that results by applying the previousequivalence (b).

J. Naqui et al. / Journal of Magnetism and Magnetic Materials 383 (2015) 144–151 147

for the symmetric case. We have opted to designate such couplingas –|M′| to avoid any confusion related to the sign. The circuitparameters L and C are the inductance and capacitance of the line,respectively, and the SIRs, in general of different dimensions, aredescribed by the series resonators L1–C1 and L2–C2. Using the well-known transformation indicated in Fig. 3(a) [36], the lumpedelement equivalent circuit model can be depicted as shown inFig. 3(b), where the mutual coupling between inductive elementsis suppressed. The circuit of Fig. 3(b) is a T-model where the seriesbranches are purely inductive. Thus, the transmission zeros (orresonance frequencies of the coupled resonators) are given by thezeros of the shunt reactance. Excluding the capacitance C (whichdoes not affect the zeros of the shunt reactance), the impedance ofthe shunt branch is given by:

{ }{ }

ZC C M L L L C L C

j C C L L M C C( )

{ } 1

( 2 ) ( ) (8)p

1 24 2

1 22

1 1 2 2

1 22

1 2 1 2

ωω ω

ω ω=

′ − + + −

+ + ′ − +

Forcing the numerator to be zero, the transmission zero fre-quencies are given by

M C C

M C C

( ) 4

2[1 ] (9)2 1

222

12

22 2 2

14

24

1 22

12

22

1 2

ωω ω ω ω ω ω

ω ω=

+ ± − + ′− ′±

where ω1¼(L1C1)�1/2 and ω2¼(L2C2)�1/2. Note that M′2¼ |M′|2,and hence expressions (9) and (2), providing the transmission zerofrequencies in significantly different structures, are identical.

Obviously, for the symmetric structure (L1¼L2¼Lr, C1¼C2¼Crand ω1¼ω2¼ω0), expression (9) simplifies to (4). However, thereis an important difference: in this case (SIR-loaded line), the so-lution ω� nulls the denominator of (8), which means that thevalid solution (i.e., the one with physical meaning) is ωþ . In otherwords, for the symmetric case, like in SRR loaded lines, SIR loadedlines exhibit a single transmission zero. However, unlike in SRRs, itappears above the resonance frequency, ω0, of the isolated SIR.

For the asymmetric structure (different coupled SIRs), butconsidering that their fundamental resonance frequencies areidentical (L1≠L2, C1≠C2, ω1¼ω2¼ω0), the solutions are given byexpression (5). However, unlike in SRR loaded lines, in SIR loadedlines there is not a balance condition (similar to the one of ex-pression (7)) able to null the denominator of (8) for one of thesolutions of (5). In other words, there are always two transmissionzeros in SIR loaded lines satisfying L1≠L2, C1≠C2, ω1¼ω2¼ω0. In-deed the analysis carried out by introducing (5) in the denomi-nator of (8) reveals that this denominator nulls for ω� if the fol-lowing condition is satisfied:

C CC C

2 (10)1 2

1 2+

=

which corresponds to the pure symmetric case. This is an im-portant difference between SRR and SIR loaded lines with coupledresonators. Notice that the balance condition in (7) depends on themutual coupling M1 and M2 between the line and the resonators.

M1 and M2 add extra parameters in the model of Fig. 1(b), ascompared to the model of Figs. 2(b) or 3(b), and for this reason abalance condition able to give only a single resonance for theconsidered case (L1≠L2, C1≠C2, ω1¼ω2¼ω0) in SIR loaded lines isnot possible.

3. Validation

The validation of the models of the previous structures is car-ried out by extracting circuit parameters and comparing the circuitsimulations to electromagnetic simulations and, in some cases, tomeasurements on fabricated structures.

3.1. SRR loaded CPWs

Let us first consider the symmetric case, where SRR and CPWdimensions are set to l1¼4.8 mm, l2¼3.8 mm, c¼d¼0.2 mm,l¼5.6 mm, W¼3 mm, and G¼1.01 mm. The lossless electro-magnetic simulations (inferred from Agilent Momentum) of thestructure, considering different values of inter-resonator distance,d′, are depicted in Fig. 4 (the considered substrate is the RogersRO3010 with dielectric constant εr¼11.2 and thicknessh¼1.27 mm). The circuit parameters for the four considered cases(extracted from the method reported in [37]) are shown in Table 1.As d′ increases, the mutual coupling M′ decreases, and the re-sonance frequency increases, in agreement with the ω� solutionof (4). Note that the other circuit parameters do not significantlyvary, and the agreement between circuit and electromagnetic si-mulation in the region of interest is very good, pointing out thevalidity of the model. We have fabricated these structures, and wehave obtained the measured response, also included in Fig. 4 (thedetails of the SRRs in one of the samples are shown in the inset ofFig. 4a). The agreement is reasonable, taking into account the ty-pical tolerances in the fabrication process (a milling machine hasbeen used) and in the dielectric constant as well. Note also that theelectromagnetic and circuit simulations are lossless. Although byexcluding losses in electromagnetic and circuit simulation, weobtain less accuracy as compared to measurement, parameterextraction is preferred by excluding losses. Otherwise, too manyelements appear in the model. However, the resistive elementsmodeling losses at the circuit level (i.e., parasitics), which may beincluded in the resonant elements and eventually in the linemodel, can be introduced and determined by curve fitting once thereactive parameters are known. Notice, however, that in asym-metric structures this involves two resistances in the resonantelements plus the resistance/s modeling losses in the lines. For thisreason, we do not have considered such curve fitting in this work.

Let us now consider the asymmetric SRR-loaded line withidentical resonance frequency for both SRRs (ω1¼ω2¼ω0). Theconsidered geometrical parameters are: l¼5.6 mm, W¼3 mm,G¼1.01 mm; upper SRR: l1¼4 mm, l2¼3 mm, c¼0.2 mm,

Fig. 4. Frequency response (symmetric case) for different values of d′. (a) Magnitude transmission coefficient, (b) magnitude reflection coefficient, (c) phase transmissioncoefficient, (d) phase reflection coefficient. The inset of (a) shows a picture of the SRRs (for one of the samples), etched in the back side of the CPW.

Table 1Extracted circuit parameters (symmetric case) for different values of d′.

d′(mm) L (nH) C (pF) Cr (pF) Lr (nH) M (nH) M′(nH)

0.105 1.82 1.58 0.44 6.85 0.82 1.740.305 1.86 1.58 0.44 6.85 0.82 1.290.505 1.84 1.57 0.43 6.85 0.81 1.020.755 1.85 1.55 0.43 6.85 0.80 0.80

J. Naqui et al. / Journal of Magnetism and Magnetic Materials 383 (2015) 144–151148

d¼0.1 mm, d′¼0.305 mm; lower SRR: l1¼4.5 mm, l2¼4.1 mm,c¼0.2 mm, d¼0.725 mm, d′¼0.155 mm. The extracted parametersare: L¼1.77 nH, C¼1.60 pF, L1¼5.0 nH, C1¼0.42 pF, L2¼7.20 nH,C2¼0.29 pF, M1¼0.51 nH, M2¼0.77 nH, M′¼0.96 nH. As expected,two notches (one above and the other below ω0) are present. Theagreement between the circuit and electromagnetic simulation isvery good, according to Fig. 5. This figure also includes the simu-lated and circuit response of the CPW loaded with a single re-sonator, to verify that the resonance frequency is identical in both

Fig. 5. Frequency response for the considered asymmetric case with identical SRRs reso¼0.774 nH M1¼0.645 nH, i.e., satisfying (7) (a). The responses of the CPW loaded with

resonant elements. Moreover, we have verified through circuitsimulation that when condition (7) is satisfied, only one notchappears (see Fig. 5). However the verification of this throughelectromagnetic simulation and experiment is difficult since it isnot so simple to find out the layout whose extracted parametersexactly satisfy (7).

Finally, we have considered an asymmetric structure with dif-ferent SRR resonance frequencies. The geometry is as follows:l¼5.6 mm, W¼3 mm, G¼1.01 mm; upper SRR: l1¼4.8 mm, l2¼4.6 mm, c¼d¼0.2 mm; lower SRRs: l1¼4.8 mm, l2¼3.8 mm,c¼d¼0.2 mm. We have obtained the frequency response and thecircuit parameters for four different values of d′ (see Table 2). Thecomparison between the circuit and electromagnetic simulationsis depicted in Fig. 6, where good agreement is again obtained,further validating the proposed model. The measured responses(also included in the figure for comparison purposes) reveal thatthe agreement with circuit and electromagnetic simulations isreasonable. The pair of notches (at f�¼ω�/2π and fþ¼ωþ/2π)

nance frequencies and circuit simulation corresponding to L1¼5 nH, L2¼7.2 nH, M2only the larger or smaller SRR are depicted in (b).

Table 2Extracted circuit parameters (asymmetric case) for different values of d′.

d′(mm) L (nH) C (pF) C1 (pF) L1 (nH) M1 (nH) C2 (pF) L2 (nH) M2 (nH) M′ (nH)

0.105 1.87 1.58 0.37 9.85 1.00 0.44 6.85 0.83 1.910.305 1.84 1.58 0.37 9.85 0.98 0.44 6.85 0.82 1.430.505 1.83 1.57 0.37 9.85 0.97 0.43 6.85 0.81 1.140.755 1.86 1.55 0.37 9.85 0.93 0.43 6.85 0.80 0.90

Fig. 7. Variation of the notch frequencies as a function of d′ for the asymmetricSRR-loaded CPW.

J. Naqui et al. / Journal of Magnetism and Magnetic Materials 383 (2015) 144–151 149

are depicted in Fig. 7, and verify that their distance increases as d′decreases, in agreement to (3).

3.2. SIR loaded microstrip lines

In this case, we have validated the proposed model by con-sidering three different structures: one symmetric and twoasymmetric. These structures have been fabricated on the RogersRO4003C substrate with dielectric constant εr¼3.1 and thicknessh¼812.8 mm, and are depicted in Fig. 8. The difference betweenthese SIR loaded lines is simply the size of the capacitive patch ofthe upper SIR. The dimensions for the symmetric structure are:W¼1.83 mm, l¼15.9 mm, l1¼ l2¼2.6 mm, w1¼5.5 mm, w2¼250 mm. For the asymmetric structures, we have simply varied l1with Δl1¼70.5 mm. Fig. 9(a) depicts the frequency responses(circuit and electromagnetic simulations) of the microstrip lineloaded with only the upper SIR, whereas Fig. 9(b) shows the re-sponses (in this case including also measurement) of the struc-tures of Fig. 8(i.e., with the coupled SIRs). For the three consideredcases the following circuit values have been extracted: L¼1.80 nH,C¼0.57 pF, L1¼L2¼2.46 nH, C1¼0.65 pF, and |M|¼0.31 nH. Thecapacitance of the upper patch is C2¼0.65 pF (Fig. 8a), C2¼0.80 pF(Fig. 8b), and C2¼0.50 pF (Fig. 8c). The agreement is good in allcases pointing out the validity of the proposed model. For the

Fig. 6. Frequency response (asymmetric case) for different values of d′. (a) Magnitude trcoefficient, (d) phase reflection coefficient. The inset of (a) shows a picture of the SRRs (fthe ground plane regions are connected through vias and metallic strips, also etched in

symmetric structure (Fig. 8a), only one resonance appears, and it islocated to the right of the one of the isolated resonator, as ex-pected according to the results of the previous section. For theasymmetric structures, inter-resonator coupling enhances thedistance between resonances, as compared to the distance be-tween the resonances of the individual SIRs, also in accordance tothe results of the previous section.

ansmission coefficient, (b) magnitude reflection coefficient, (c) phase transmissionor one of the samples), etched in the back side of the CPW. To avoid the slot mode,the back side of the substrate (at both sides of the SRRs).

Fig. 8. Photograph of the fabricated SIR loaded lines. (a) Symmetric, (b) asymmetric with larger capacitance of the upper SIR, and (c) asymmetric with smaller capacitance ofthe upper SIR.

Fig. 9. Transmission coefficient corresponding to the lossless electromagnetic andcircuit simulations for a microstrip line loaded (a) with a single SIR (not fabricated),and (b) with a pair of SIRs (fabricated and shown in Fig. 8). In (b), measurementsare included. The frequencies indicated in the right correspond to theoreticalvalues.

J. Naqui et al. / Journal of Magnetism and Magnetic Materials 383 (2015) 144–151150

4. Potential application to microwave sensors

The previous structures can be useful for sensing purposes,especially for sensors and comparators based on dielectric loading.The idea is to disrupt the symmetry of a SIR or a SRR loadedtransmission line by means of a dielectric load, located on top or inthe vicinity of one of the resonant elements. This causes the ap-pearance of two notches in the transmission coefficient and thiscan be useful for sensing. Note that these structures are of specialinterest, for instance, for the determination of defects or ab-normalities in a substance by comparison to a reference. Indeed,the asymmetric SIR loaded transmission lines of Fig. 8 exhibitdifferent patch capacitance for the upper SIR in order to emulatedifferent dielectric loads between the two resonators forming thepair. According to Fig. 9, this asymmetry produces two notches inthe transmission coefficient, which can be easily detected. Theexperimental validation of this sensing principle was reported in[32], where a conductor backed CPW transmission line loadedwith a pair of symmetric SIRs was tested with and without thepresence of a piece of glass on the top of one of the SIRs. As ex-pected, a frequency response with two notches in the transmissioncoefficient, similar to those depicted in Fig. 9 for the asymmetriccases, was obtained. Work is in progress for the implementation ofsensors and comparators based on SRR loaded lines.

5. Conclusions

In summary, we have studied the transmission properties oftransmission lines loaded with pairs of magnetically coupled re-sonant elements. Specifically, microstrip lines loaded with coupledSIRs and CPWs loaded with coupled SRRs have been considered. Inboth cases, the lumped element equivalent circuit model of thestructure has been proposed. From these models, the transmission

zero frequencies (or resonance frequencies of the coupled re-sonators) have been inferred, and it has been found that thegeneral analytical expressions providing such resonances areidentical in both structures. The main difference concerns thesymmetric case. If the resonators are identical, then only one re-sonance appears, and this resonance is below and above ω0, theresonance of the isolated resonator, for the SRR and SIR loadedlines, respectively. The second difference concerns the case ofdistinct resonators but exhibiting the same resonance frequency.In both cases (SIR and SRR loaded lines), two resonance fre-quencies (one below and the other above ω0) appear. However, forSRR loaded lines there is a kind of balance condition that nulls oneof these frequencies, whereas such condition does not exist in SIRloaded lines, and the two resonances are always present. Themodels have been validated by comparing circuit simulations withextracted parameters with electromagnetic simulations and ex-perimental responses obtained in fabricated prototypes. The po-tentiality of these structures for microwave sensors has been alsopointed out.

Acknowledgments

This work has been supported by MINECO (Spain) under Pro-jects TEC2010-17512, CSD2008-00066, TEC2013-40600-R andAGAUR-Generalitat de Catalunya under Project 2009SGR-421. JordiNaqui is in debt to MECD (Spain) for supporting his work throughthe FPU Grant AP2010-0431. Ferran Martín is in debt to ICREA forsupporting his work.

References

[1] H. Okabe, C. Caloz, T. Itoh, A compact enhanced bandwidth hybrid ring usingan artificial lumped element left handed transmission line section, IEEE Trans.Microw. Theory Tech. 52 (2004) 798–804.

[2] M.A. Antoniades, G.V. Eleftheriades, A broadband series power divider usingzero-degree metamaterial phase shifting lines, IEEE Microw. Wirel. Comp. Lett.15 (2005) 808–810.

[3] M.A. Antoniades, G.V. Eleftheriades, A broadband Wilkinson balun using mi-crostrip metamaterial lines, IEEE Antenna Wirel. Propag. Lett. 4 (2005)209–212.

[4] G. Sisó, J. Bonache, M. Gil, F. Martín, Application of resonant-type metamaterialtransmission lines to the design of enhanced bandwidth components withcompact dimensions, Microw. Opt. Tech. Lett. 50 (2008) 127–134.

[5] I.-H. Lin, M. DeVincentis, C. Caloz, T. Itoh, Arbitrary dual-band componentsusing composite right/left-handed transmission lines, IEEE Trans. Microw.Theory Tech. 52 (2004) 1142–1149.

[6] A.C. Papanastasiou, G.E. Georghiou, G.V. Eleftheriades, A quad-band wilkinsonpower divider using generalized NRI transmission lines, IEEE Microw. Wirel.Comp. Lett. 18 (2008) 521–523.

[7] M. Durán-Sindreu, G. Sisó, J. Bonache, F. Martín, Planar multi-band microwavecomponents based on the generalized composite right/left handed transmis-sion line concept, IEEE Trans. Microw. Theory Tech. 58 (12) (2010) 3882–3891.

[8] M. Durán-Sindreu, J. Bonache, F. Martín, T. Itoh, Single-layer fully-planar ex-tended-composite right/left handed transmission lines based on substrateintegrated waveguides for dual-band and quad-band applications, Int. J. Mi-crow. Wirel. Technol. 5 (2013) 213–229.

[9] S. Lim, C. Caloz, T. Itoh, Electronically scanned composite right/left handedmicrostrip leaky-wave antenna, IEEE Microw. Wirel. Comp. Lett. 14 (2004)

http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref1http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref1http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref1http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref1http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref2http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref2http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref2http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref2http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref3http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref3http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref3http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref3http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref4http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref4http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref4http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref4http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref5http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref5http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref5http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref5http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref6http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref6http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref6http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref6http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref7http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref7http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref7http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref7http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref8http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref8http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref8http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref8http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref8http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref9http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref9

J. Naqui et al. / Journal of Magnetism and Magnetic Materials 383 (2015) 144–151 151

277–279.[10] M.A. Antoniades, G.V. Eleftheriades, A CPS leaky-wave antenna with reduced

beam squinting using NRI-TL metamaterials, IEEE Trans. Antennas Propag. 56(3) (2008) 708–721.

[11] M. Durán-Sindreu, J. Choi, J. Bonache, F. Martín, T. Itoh Dual-band leaky waveantenna with filtering capability based on extended-composite right/left-handed transmission lines, IEEE MTT-S International Microwave Symposium,2013, Seatle (USA).

[12] G. Zamora, S. Zuffanelli, F. Paredes, F. Javier Herraiz-Martínez, F. Martín,J. Bonache, Fundamental mode leaky-wave-antenna (LWA) using slot line andsplit-ring-resonator (SRR) based metamaterials, IEEE Antenna Wirel. Prop.Lett. 12 (2013) 1424–1427.

[13] J. Bonache, F. Martín, J. García-García, I. Gil, R. Marqués, M. Sorolla, Ultra wideband pass filtres (UWBPF) based on complementary split rings resonators,Microw. Opt. Technol. Lett. 46 (2005) 283–286.

[14] J. Bonache, I. Gil, J. García-García, F. Martín, Novel microstrip band pass filtersbased on complementary split rings resonators, IEEE Trans. Microw. TheoryTech. 54 (2006) 265–271.

[15] M. Gil, J. Bonache, J. García-García, J. Martel, F. Martín, Composite right/lefthanded (crlh) metamaterial transmission lines based on complementary splitrings resonators (csrrs) and their applications to very wide band and compactfilter design, IEEE Trans. Microw. Theory Tech. 55 (2007) 1296–1304.

[16] M. Gil, J. Bonache, F. Martín, Metamaterial filters: a review, Metamaterials 2(2008) 186–197.

[17] R. Islam, F. Elek, G.V. Eleftheriades, Coupled line metamaterial coupler havingco-directional phase but contra-directional power flow, Electron. Lett. 40 (5)(2004) 315–317.

[18] C. Caloz, A. Sanada, T. Itoh, A novel composite right/left handed coupled-linedirectional coupler with arbitrary coupling level and broad bandwidth, IEEETrans. Microw. Theory Tech. 52 (2004) 980–992.

[19] M. Durán-Sindreu, A. Vélez, G. Sisó, J. Selga, P. Vélez, J. Bonache, F. Martín,Recent advances in metamaterial transmission lines based on split rings, Proc.IEEE 99 (2011) 1701–1710.

[20] F. Martín, F. Falcone, J. Bonache, R. Marqués, M. Sorolla, Split ring resonatorbased left handed coplanar waveguide, Appl. Phys. Lett. 83 (2003) 4652–4654.

[21] F. Falcone, T. Lopetegi, J.D. Baena, R. Marqués, F. Martín, M. Sorolla, Effectivenegative-ε stop-band microstrip lines based on complementary split ringresonators, IEEE Microw. Wirel. Comp. Lett 14 (2004) 280–282.

[22] M. Durán-Sindreu, A. Vélez, F. Aznar, G. Sisó, J. Bonache, F. Martín, Applicationof open split ring resonators and open complementary split ring resonators tothe synthesis of artificial transmission lines and microwave passive compo-nents, IEEE Trans. Microw. Theory Tech. 57 (2009) 3395–3403.

[23] C. Caloz and T. Itoh, Novel microwave devices and structures based on thetransmission line approach of metamaterials, IEEE-MTT International Micro-wave Symposium, 1 Philadelphia, PA, 2003, pp. 195–198.

[24] F. Aznar, J. Bonache, F. Martín, Improved circuit model for left handed linesloaded with split ring resonators, Appl. Phys. Lett. 92 (2008) (paper 043512).

[25] J. Naqui, M. Durán-Sindreu, F. Martín, Modeling split ring resonator (SRR) andcomplementary split ring resonator (CSRR) loaded transmission lines ex-hibiting cross polarization effects, IEEE Antenna Wirel. Propag. Lett 12 (2013)178–181.

[26] F. Hesmer, E. Tatartschuk, O. Zhuromskyy, et al., Coupling mechanisms for splitring resonators: theory and experiment, Phys. Status Solidi B 244 (2007)1170–1175.

[27] E. Tatartschuk, N. Gneiding, F. Hesmer, A. Radkovskaya, E. Shamonina, Map-ping inter-element coupling in metamaterials: scaling down to infrared, J.Appl. Phys. 111 (2012) (paper 094904).

[28] R.R.A. Syms, L. Solymar, Effective permeability of a metamaterial: againstconventional wisdom, App. Phys. Lett. 100 (2012) (paper 124103).

[29] J. Naqui, A. Fernández-Prieto, F. Mesa, F. Medina, F. Martín, Effects of inter-resonator coupling in split ring resonator (SRR) loaded metamaterial trans-mission lines, J. Appl. Phys. 115 (2014) (paper 194903).

[30] T. Tamir, A.A. Oliner, Guided complex waves, Proc. Inst. Electr. Eng 110 (1963)310–334.

[31] J. Naqui, M. Durán-Sindreu, F. Martín, Novel sensors based on the symmetryproperties of split ring resonators (SRRs), Sensors 11 (2011) 7545–7553.

[32] J. Naqui, C. Damm, A. Wiens, R. Jakoby, L. Su, F. Martín, Transmission linesloaded with pairs of magnetically coupled stepped impedance resonators:modeling and application to microwave sensors, IEEE MTT-S InternationalMicrowave Symposium, Tampa, FL (USA), 2014.

[33] J.B. Pendry, A.J. Holden, D.J. Robbins, W.J. Stewart, Magnetism from conductorsand enhanced nonlinear phenomena, IEEE Trans. Microw. Theory Tech. 47(1999) 2075–2084.

[34] J. García-García, F. Martín, J.D. Baena, R. Marques, L. Jelinek, On the resonancesand polarizabilities of split rings resonators, J. Appl. Phys. 98 (2005) (pp.033103-1-9).

[35] M. Makimoto, S. Yamashita, Compact bandpass filters using stepped im-pedance resonators, Proc. IEEE 67 (1) (1979) 16–19.

[36] J.-S. Hong, M.J. Lancaster, Microstrip Filters for RF/Microwave Applications,John Wiley, New Jersey, 2001.

[37] F. Aznar, M. Gil, J. Bonache, J.D. Baena, L. Jelinek, R. Marqués, F. Martín,Characterization of miniaturized metamaterial resonators coupled to planartransmission lines, J. Appl. Phys. 104 (2008) (paper 114501-1-8).

http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref9http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref9http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref10http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref10http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref10http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref10http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref11http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref11http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref11http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref11http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref11http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref12http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref12http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref12http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref12http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref13http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref13http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref13http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref13http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref14http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref14http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref14http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref14http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref14http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref15http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref15http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref15http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref16http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref16http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref16http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref16http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref17http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref17http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref17http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref17http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref18http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref18http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref18http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref18http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref19http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref19http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref19http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref20http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref20http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref20http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref20http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref20http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref20http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref21http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref21http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref21http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref21http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref21http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref22http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref22http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref23http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref23http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref23http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref23http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref23http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref24http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref24http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref24http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref24http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref25http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref25http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref25http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref26http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref26http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref27http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref27http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref27http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref28http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref28http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref28http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref29http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref29http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref29http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref30http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref30http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref30http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref30http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref31http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref31http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref31http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref32http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref32http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref32http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref33http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref33http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref34http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref34http://refhub.elsevier.com/S0304-8853(14)01213-X/sbref34

Analysis of transmission lines loaded with pairs of coupled resonant elements and application to sensorsIntroductionCircuit model and analysisCPW loaded with a pair of coupled SRRsMicrostrip line loaded with a pair of coupled SIRs

ValidationSRR loaded CPWsSIR loaded microstrip lines

Potential application to microwave sensorsConclusionsAcknowledgmentsReferences