Synthesis of Split-Rings-Based Artificial Transmission Lines Through a New Two-Step, Fast...

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 61, NO. 6, JUNE 2013 2295

Synthesis of Split-Rings-Based ArtificialTransmission Lines Through a New Two-Step,Fast Converging, and Robust AggressiveSpace Mapping (ASM) Algorithm

Jordi Selga, Student Member, IEEE, Ana Rodríguez, Student Member, IEEE, Vicente E. Boria, Senior Member, IEEE,and Ferran Martín, Fellow, IEEE

Abstract—This paper is focused on the synthesis of artificialtransmission lines based on complementary split ring resonators(CSRRs). The considered structures are microstrip lines withCSRRs etched in the ground plane and microstrip lines loadedwith both CSRRs and series capacitive gaps. An aggressive spacemapping (ASM) optimization algorithm, able to automaticallygenerate the layout of these artificial lines, has been developed.The tool has been optimized in order to achieve fast convergenceand to provide accurate results. The main relevant aspects of theproposed algorithm (based on a novel two-step ASM optimizationapproach) are: 1) the capability to provide the implementablecircuit elements of the equivalent circuit model of the consideredartificial lines and 2) the ability to converge in a few (unprece-dented) iteration steps, due to a new procedure to generate theinitial layouts (which are very close to the final ones). First, thesoftware is tested through the synthesis of several CSRR-basedmicrostrip lines, and then some practical application examples ofsuch artificial lines are reported to illustrate the potential of theproposed synthesis tool.

Index Terms—Artificial transmission lines, complementarysplit ring resonator (CSRR), metamaterial transmission lines,microstrip, space mapping.

I. INTRODUCTION

T HE topic of artificial transmission lines has attractedmuch attention in the last decade and opened a path to a

new type of artificial transmission lines based on metamaterialconcepts [1]–[6]. Metamaterial-based (or inspired) transmis-sion lines consist of a host line loaded with reactive elements(e.g., inductors, capacitors, resonators, or a combination of

Manuscript received November 28, 2012; revised March 22, 2013; acceptedMarch 26, 2013. Date of publication May 13, 2013; date of current versionMay 31, 2013. This work was supported in part by MICIIN-Spain under ProjectTEC2010-17512METATRANSFER, Project TEC2010-21520-C04-01, ProjectCONSOLIDER EMET CSD2008-00066, and Grant AP2008-04707, General-itat de Catalunya under Project 2009SGR-421, and MITyC-Spain under ProjectTSI-020100-2010-169 METASINTESIS.J. Selga and F. Martín are with GEMMA/CIMITEC, Departament d’Enginy-

eria Electrònica, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain(e-mail: [email protected]).A. Rodríguez and V. E. Boria are with Departamento de Comunicaciones-

iTEAM, Universidad Politécnica de Valencia, 46022 Valencia, Spain (e-mail:[email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TMTT.2013.2259254

them). Such lines exhibit controllable dispersion and char-acteristic impedance, this being the main relevant advantageas compared with conventional lines or with other artificialtransmission lines. In conventional lines, the line impedancecan be tailored to some extent through the geometry; however,the phase constant (for a given substrate) is barely dependenton the geometry. In certain artificial transmission lines, such asslow-wave transmission lines, a host line is periodically loadedwith capacitors, inductors, or by means of semi-lumped reac-tive elements, which effectively increase the line capacitanceor inductance and, hence, reduce the phase velocity [7]–[9].In metamaterial transmission lines, the presence of reactive

elements increases the design flexibility of the lines. Thus, thecharacteristic impedance and the phase constant (as well as thephase and group velocities) can be engineered in order to fulfillline specifications or functionalities not achieved before. Sizereduction [10], [11], broadband [10], [12]–[16] and multibandoperation [17]–[21], leaky wave structures with steerable radi-ation beams [22], [23], or novel broadband filters and diplexerswith spurious suppression [24]–[26], (including tunable filters[27]) are some of the applications of metamaterial transmis-sion lines. In these applications, device design is based on bothimpedance and dispersion engineering.The early metamaterial transmission lines were implemented

by loading a host line with series capacitors and shunt in-ductors (CL-loaded lines) [2]–[4]. At low frequencies, theloading elements are dominant, and the structure behaves asa backward wave transmission line. In this regime, the phaseand group velocities are anti-parallel and the structure supportsbackward waves, or, in other words, it exhibits a left-handedbehavior. At higher frequencies, the line inductance andcapacitance are dominant, and the propagation is forward(or right-handed). Therefore, these CL-loaded lines exhibitcomposite right-/left-handed (CRLH) wave propagation [28],[29]. Such CRLH behavior can also be achieved by meansof metamaterial transmission lines based on electrically smallresonators, such as split ring resonators (SRRs) [30] or comple-mentary split ring resonators (CSRRs) [31]. In the latter case,which is the relevant one for the purposes of the present paper,a host line, typically (although not exclusively) a microstripline, is loaded with CSRRs (etched in the ground plane beneaththe conductor strip) and series capacitive gaps on top of theCSRRs. These CSRR-based lines do also exhibit a CRLH

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2296 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 61, NO. 6, JUNE 2013

Fig. 1. Typical topologies and circuit models (unit cell) of a CSRR-loaded mi-crostrip line (a) with and (b) without series gap. The ground plane, where theCSRR are etched, is depicted in gray.

behavior. However, the circuit model, depicted in Fig. 1(a),reveals the presence of a coupling capacitance, , between theCSRR (modeled by and ) and the line, responsible forthe presence of a transmission zero. is the line inductanceand is related to the gap capacitance. In fact, assumingthat the series gap can be modeled by a -network with seriescapacitance and shunt (fringing) capacitance , it follows[32] that

(1)

(2)

where and is the line capacitance.The frequency response of CL-loaded and CSRR-based

CRLH lines exhibits a passband in the left-handed and theright-handed regions, and these two bands merge if the CRLHline is balanced, that is, if the series and shunt resonancefrequencies are identical. By removing the gap, the passbandbehavior of CSRR-based lines switches to a stopband, causedby the presence of the transmission zero [the circuit model forthese structures is also depicted in Fig. 1(b)]. This stopbandfunctionality is of interest for the implementation of notchfilters and bandstop filters [33].Despite the numerous applications and advantages of CSRR-

based artificial lines, a weak point towards the further pene-tration of these lines into the market concerns their synthesis.In this regard, recent efforts have been made by the authorsin order to implement efficient algorithms for layout genera-tion. Specifically, in [34] and [35], the potential of space map-ping optimization for the synthesis of CSRR-based lines actingas stop band structures (i.e., without the presence of the gaps)was demonstrated. In the present paper, we extend the spacemapping optimization to the synthesis of CSRR and gap loadedlines.Moreover, as compared to [34] and [35], we introduce sev-eral improvements that provide faster convergence and majorrobustness to the algorithm, and a method able to give the im-plementable element values of the equivalent circuit model.

This paper is organized as follows. In Section II, the gen-eral formulation of ASM is provided for completeness and forbetter comprehension of the paper. Section III is focused onthe application of ASM optimization to the synthesis of CSRR-based lines, where the new two-step ASM algorithm (based on anovel method to determine the initial geometry) is presented. InSection IV, the new two-step ASM-based tool is applied to thesynthesis of practical microwave circuits, including a bandstopfilter and a dual-band power divider. Finally, the main conclu-sions are highlighted in Section V.

II. GENERAL FORMULATION OF AGGRESSIVESPACE MAPPING (ASM)

Space mapping is a technique extensively used in the op-timized design process of microwave devices, which makesproper use of two simulation spaces [36]–[38]. In the opti-mization space, , the variables are linked to a coarse model,which is simple and computationally efficient, although notaccurate. On the other hand, the variables corresponding to thevalidation space, , are linked to a fine model, typically morecomplex and CPU intensive, but significantly more precise.In each space, we can define a vector containing the differentmodel parameters. Let us denote such vectors as andfor the fine model and coarse model parameters, respectively.Using the same nomenclature, and will denotethe responses of the fine and coarse models, respectively. Formicrowave applications, the model response is related to theevaluation of the device behavior, e.g., a scattering parameter,such as or , computed in a certain frequency range.The key idea behind the space mapping algorithm is to gen-

erate an appropriate parameter transformation

(3)

mapping the fine model parameter space to the coarse modelparameter space such that

(4)

in some predefined region, where is a certain suitable normand is a small positive number close to zero. If is invertible,then the inverse transformation

(5)

is used to find the fine model solution, which is the image of thecoarse model solution that gives the target response .The determination of according to the procedure reported

in [36] follows an iterative process that is rather inefficient.However, the efficiency of the method can be improved by in-troducing a quasi-Newton type iteration [37]. This method ag-gressively exploits each fine model EM analysis with the resultof a faster convergence. Hence, the new approach was calledaggressive space mapping (ASM) [37], and it is the optimiza-tion procedure considered in this work. Essentially, the goal inASM is to solve the following set of nonlinear equations:

(6)

SELGA et al.: SYNTHESIS OF SPLIT-RINGS-BASED ARTIFICIAL TRANSMISSION LINES THROUGH A FAST CONVERGING ROBUST ASM ALGORITHM 2297

For a better understanding of the iterative optimization process,a superscript is added to the notation that actually indicatesthe iteration number. Hence, let us assume that is theth approximation to the solution of (6) and is the errorfunction corresponding to . The next vector of theiterative process is obtained by a quasi-Newton iterationaccording to

(7)

where is given by

(8)

and is an approach to the Broyden matrix [37]

(9)

which is also updated at each iterative step. In (9), isobtained by evaluating (6) using a certain parameter extractionmethod providing the coarse model parameters from the finemodel parameters, and the super-index stands for transpose.The implementation of the ASM algorithm is well reported

in [37]. If the fine and coarse models involve the same spaceparameters, then the first vector in the fine space is typically setequal to the target vector in the coarse space, and the Broydenmatrix is initialized by forcing it to be the identity. However,since this is not the case we are considering in this paper, wehave followed a different approach (Section III-C).

III. SYNTHESIS OF CSRR-BASED ARTIFICIALTRANSMISSION LINES BY MEANS OF THENOVEL TWO-STEP ASM APPROACH

From now on, and for clear distinction, microstrip linesloaded only with CSRRs will be called CSRR-loaded met-alines, and those lines loaded with both CSRRs and seriesgaps will be designated as CSRR-gap-loaded metalines. Letus first describe the two spaces involved in the proposed ASMalgorithm for both line types. The optimization space is thecircuit model, and the space variables are the circuit elementsappearing in the equivalent circuits of Fig. 1(a) and (b) forthe CSRR-gap- and CSRR-loaded lines, respectively. Thevalidation space is the electromagnetic (EM) model ofthe physical structure that provides the frequency responsefrom the geometry and substrate parameters. The substrateparameters will be set to certain values and therefore theseparameters are not considered as variables of the fine model.The chosen substrate is the Rogers RO3010 with dielectricconstant and thickness either 1.27 mm or0.635 mm (losses are not considered in the EM model). Someconstraints will be applied to the geometry parameters in orderto reduce the degrees of freedom and thus work with the samenumber of variables in both spaces, as it was done in [34] (thematrix is thus square and invertible, and the computationaleffort is minimized).The most relevant difference between the ASM algorithm re-

ported in this paper, with regard to the one reported in [34], is

Fig. 2. Schematic of the new proposed two-step ASM algorithm (see the textfor better comprehension).

the determination of the initial layout. With the new proposedstrategy, the initial estimated layout (i.e., ) is already veryclose to the optimum final solution , leading to a fast and verysoft convergence rate of the whole synthesis procedure. More-over, our previous implementations of ASM algorithms [34],[35], [39] are not able to guarantee if there is a mapping betweenthe optimal coarse model solution (i.e., a set of circuit parame-ters corresponding to the target response) and a fine model pointproviding a physically implementable layout. In this paper, weovercome this practical limitation through the determination ofa convergence region in the coarse model space. Hence, wewill first present this novel strategy for the determination of theconvergence regions, which involves a pre-optimization ASMscheme. For this reason, our proposed synthesis tool is based ona two-step ASM approach (see Fig. 2).

A. Determination of the Convergence Region:Pre-Optimization ASM Algorithm

The implementable layouts of CSRR based lines are lim-ited to certain combinations of the element values of the circuitmodels. Despite some efforts having made by means of para-metric analysis [32], there is not yet a systematic procedure todetermine whether a given combination of circuit model ele-ments can be translated to an implementable geometry or not.This is a fundamental problem to solve in order to implement apractical synthesis tool, and it is the subject of this section.Let us first consider the case of CSRR-loaded metalines.

The geometrical variables are the microstrip line length andwidth and the width and separation of the slot rings [seeFig. 1(b)]. In order to deal with the same number of variablesin the coarse and fine model spaces, we consider and

, where is the external radius of the CSRR. Thischoice is justified to avoid distributed effects that may appearif and are not accounted for by the circuit model andto ensure that the ports of the structure are accessible.Given a set of circuit (target) variables and ,

a procedure to determine if this set of variables has a physi-cally implementable layout is needed. According to the circuitmodel, and are physically realizable if these values arenot too extreme and a microstrip line with reasonable widthand length results. The important aspect is, thus, given a pairof implementable values of and , to determine the con-vergence region for the circuit values modeling the CSRR (


and ). Thus, the strategy consists of calculating the line ge-ometry that provides the element values and under fourdifferent scenarios corresponding to the extreme values ofand , namely, , and

. The parameters and are the minimumachievable slot and strip widths, respectively, with the tech-nology in use (we have set 0.15 mm in thiswork, unless otherwise specified). On the other hand, and

are set to a reasonable (maximum) value that guaranteesthe validity of the model in the frequency region of interest. Ithas been found that, for values exceeding 0.4–0.5 mm, the cou-pling between the slot rings is very limited, and hence we haveset 0.4-0.5 mm (larger values expand the con-vergence region, but at the expense of less accuracy in the finalsolutions).For each case, the single geometrical variables are and

( and are fixed and ). These variables must beoptimized with the goal of recovering the and values cor-responding to the target ( and ). The extraction of the ele-ments of the circuit model from the EM simulation of the CSRR-loaded line layout (which is a straightforward technique) hasbeen reported before [35], but it is reproduced in Appendix A forcompleteness (parameter extraction is a fundamental buildingblock of the proposed ASM algorithm).For the determination of and , we do also consider an

ASM optimization scheme, which is very simple since onlytwo variables in each space are involved. The initial values ofand are obtained from the characteristic impedance andphase constant resulting from and , with the help of atransmission line calculator. With the resulting line geometry,we extract the parameters of the circuit model and also ini-tialize the Broyden matrix (the initiation of the Broyden ma-trix follows a similar approach to that corresponding to the coreASM algorithm, which is reported in Section III-C). Then, theprocess iterates until convergence is achieved. In this pre-opti-mization ASM process, the stopping criterion is usually tighterthan the one considered for the full-ASM optimization, i.e.,smaller values of the error functions are forced. This more re-strictive criterion is chosen not only to accurately determine thevertices of the convergence region, but also to obtain a better es-timate of the initial layout of the core ASM algorithm (howeverthis does not require much computational effort due to the verysmall number of terms—coarse model parameters—involved inthe error calculation).Once the and values corresponding to the target andfor a given and combination (for instance )

are found, the whole geometry is known, and the element valuesand can be obtained by means of the parameter extractor.

These element values ( and ) correspond to the consid-ered CSRR geometry , and actually define the firstvertex, , of the polygon which defines the convergenceregion; see Fig. 3. Notice that the nomenclature used for identi-fying the vertices indicates (subscript) the values of and inmicrometers.Then, the same process is repeated for obtaining the next

points, i.e., the and pairs corresponding to, and with the target and (for

these cases the initial values of and are set to the solutions of

Fig. 3. (a) Convergence region for a CSRR-loaded line model with4.86 nH and 1.88 pF, defined by four points in the subspace,and considering 0.15 mm and 0.4 mm.(b) Various regions defined with different constraints for the same values ofand . The thickness of the substrate is 1.27 mm.

the previous vertex, since it provides a much better approachesto the optimal final solution). As a result, we obtain a set ofpoints in the coarse model sub-space of and that definesa polygon (see Fig. 3). This polygon is a rough estimate for theregion of convergence in the subspace, for the targetvalues and . The criterion to decide if the target elementvalues can be physically implemented is the pertinence or not ofthe point (i.e., ) to the region enclosed by the polygon. Itis possible to refine the convergence region by calculating morepoints, meaning further values/combinations of and (for in-stance, those corresponding to a combination of a extreme andan intermediate value).As a representative example of a CSRR-loaded line model,

we have chosen 4.86 nH and 1.88 pF. The conver-gence region for the subspace calculated following theprocedure explained above is depicted in Fig. 3(a). Obviously,decreasing the maximum allowable value of and hasthe effect of reducing the area of the polygon, as Fig. 3(b) illus-trates. However, 0.4 mm is a reasonable valuethat represents a tradeoff between accuracy (related to the fre-quency responses of the fine and coarse models) and size of theconvergence region.The other case under study, i.e., CSRR-gap-loaded metalines,

presents a coarse model which depends on five variables, thoseof the line without gap plus [Fig. 1(a)]. The additional ge-ometrical variable is simply the gap space . The convergenceregion can be obtained following a similar procedure, but nowthe target response is given by the electrical parameters thatcharacterize the microstrip line with gap ( and ). Thepre-optimization ASM algorithm is thus more complex sinceit involves three variables, which are the line length , width, and gap space , with the unknown dimension parameters

to be calculated. As in the previous case, the initial layout forthe first vertex is found analytically, as detailed in Appendix B.Once the geometry of the line with gap is estimated, the elec-trical parameters are extracted according to the procedure givenin Appendix A, the Broyden matrix is initiated, and the processis iterated until convergence is achieved. The result is a set ofcircuit model element values corresponding to the target ele-ments of and , and and values that depend onthe considered CSRR geometry.


Fig. 4. CSRR-based microstrip line with T-shaped gap.

Fig. 5. Convergence region for a CSRR-gap-loaded line model with9.45 nH, 17.93 pF, and 1.01 pF, defined by eight points in the

subspace, and considering 0.15 mm and0.5 mm. The thickness of the substrate is 0.635 mm.

One differential aspect of CSRR-gap-loaded lines, as com-pared with CSRR-loaded lines, is the number of necessarypoints in the subspace to accurately determine theconvergence region. It has been found that four points do notsuffice in this case. Hence, the pre-optimization ASM algorithmhas been applied eight times: four of them by considering theCSRR geometries with extreme values of and and theother four cases by considering the following combinations:

, and ,where and .In some occasions, where a large gap capacitance is required,

it is convenient to use a gap with T-shaped geometry (Fig. 4). Inthis case, in order to preserve the number of variables in the vali-dation space, the gap distance is set to a small value (this helpsto increase the series capacitance of the gap), and the gap width

is considered as the geometry variable associated to thepresence of the gap. Thus, the gap distance is set to 0.2 mm(notice that —see Fig. 4—is also set to 0.2 mm to avoidan excess of variables). However, the pre-optimizer and coreASM algorithms do not experience any change by consideringT-shaped gaps.Fig. 5 depicts the convergence region in the subspace

that results by considering 9.45 nH and 17.93 pF

and 1.01 pF for a CSRR-gap-loaded line with T-shapedgap. In view of this figure, it is clear that four points are notsufficient to determine the convergence region (notice that the

points corresponding to the CSRR geometries given by, and

are significantly misaligned with the lines forming the polygonresulting by considering the four cases with extreme values ofand ).

B. Determination of the Initial Layout

Let us consider the synthesis of a given set of circuit param-eters of the circuit model (target coarse model solution), fora CSRR-loaded metaline, and let us assume that the previousanalysis reveals that such target parameters in the coarse modelspace have an implementable fine model solution. The next stepis the determination of the initial layout, unless and co-incide with any of the vertices of the converging polygon in the

subspace (in this case, the layout is already known, and,hence, no further optimization is necessary). From the previousanalysis, it is expected that the dimensions of the CSRR afterASM optimization depend on the position of the pointin the convergence region. Namely, if the point is closeto a vertex, it is expected that and are similar to the valuescorresponding to that vertex.In this paper, a new procedure to determine the initial layout,

which is necessary to start up the core ASM algorithm, is pro-posed. First, the determination of the initial layout for CSRR-loaded lines is considered. The aim is to express any of the geo-metrical variables ( or ) as a function of and . Toobtain the initial value of each geometrical dimension involvedin the optimization process, we will assume it has a linear de-pendence with and . For instance, the initial value of(for the other variables we will use identical expressions) willbe estimated according to

(10)

The previous expression can be alternatively written as

(11)

where the constants determine the functional dependence ofthe initial value of with and . To determine the constants, four conditions are needed. Let us consider the following

error function:

(12)

where the subscript is used to differentiate between the dif-ferent vertices, and hence is the value of in the vertex ,and are the corresponding values of and for thatvertex.We have considered a number of vertices equal to 4 (cor-responding to CSRR-loaded lines), but the formulation can be


generalized to a higher number of vertices. Expression (12) canthen be written as

(13)

To find the values of the constants , we obtain the partialderivatives of the previous error function with regard to andforce them to be equal to zero [40] as

(14)

for . Following this least-squares approach, fourindependent equations for the constants are obtained. Suchequations can be written in matrix form as given by (15), shownat the bottom of this page. Once the constants are obtained(solving the previous equations), the initial value of is inferredfrom (11). The process is repeated for and , and the initialgeometry necessary for the initiation of the ASM algorithm isthus obtained.For the CSRR-gap-loaded lines, a similar approach is fol-

lowed, although the number of vertices is eight, rather than four.As compared to the algorithm reported in [34], [35], the initialgeometry derived from the procedure explained in this sub-sec-tion is very close to the final geometry, and this is fundamental toreduce the number of steps towards ASM convergence, as it willbe shown later. As compared to previous works by the authors,the main relevant contributions of the present paper are the de-termination of the initial layout from the least squares approach,and the pre-optimizer ASM algorithm to determine the conver-gence region in the space. These are fundamental as-pects to make the synthesis approach practical and competitive.

C. Core ASM Algorithm

Once the initial geometry is calculated, the response of thefine model is obtained through EM analysis, and the circuit

Fig. 6. Flow diagram of the ASM algorithm. The stopping criterion applied inthis work is .

parameters are extracted, from which the error function (6) canbe obtained. To initiate the Broyden matrix, we slightly per-turb each geometrical variable from the value corresponding tothe initial layout, and we obtain the circuit parameters resultingfrom each geometry variation. The relative changes can be ex-pressed in a matrix form as follows:

(16)

which corresponds to the initial Broyden matrix values. Oncethe Broyden matrix is known, the geometry of the followingiteration can be derived from (7), and the process is iterateduntil convergence is obtained. To avoid that, the variables inthe fine model space exceed the limits of the implementablerange of values with the available technology, geometrical con-straints and a shrinking factor are introduced. As can be seenin Fig. 6, the vector is tested, and if some of the geometricalvariables exceed the geometrical constraints, the algorithm usesthe shrinking factor in order to obtain a new vector inside thelimits.The iterative algorithm is controlled by theMATLAB commer-

cial software [41], which is the core tool of the implementedASM algorithm. The three main building blocks of the MATLABprogram are the initial geometry calculator (that previously de-termines the convergence regions in the subspace as ex-plained before), the EM solver, and the parameter extractor. Forthe EM simulation, we have considered two alternative options:

(15)


TABLE IOPTIMAL COARSE SOLUTION

the Agilent Momentum simulator (incorporated in the AgilentADS commercial software) [42], and Ansoft Designer [43]. TheEM solver creates the geometry of the structure, which is thenexported to either Agilent Momentum or Ansoft Designer tocarry out the EM simulation. The results of this simulation (the-parameters) are then imported back toMATLAB and processedby the parameter extraction module, which allows us to ex-tract the circuit parameters. These values are compared with thetarget parameters, and the error function is obtained accordingto

(17)

From this, the next geometry of the iterative process is inferredfollowing the update procedure described before. The flow dia-gram of the complete ASM algorithm (including constraints toavoid unwanted solutions) is depicted in Fig. 6.

D. Effects of the New ASM Algorithm on Convergence Speed

The main novelty of the proposed ASM algorithm is the cal-culation of a very good initial geometry. First, we have appliedthe new ASM algorithm to the CSRR-loaded metalines whoseoptimal coarse parameters are detailed in Table I, using AnsoftDesigner as the EM solver. In Fig. 3(a), the location of thesethree different cases with respect to the shared convergence re-gion (they have the same and values) is shown. Noticethat, in the case, that the point is out of the convergence regionbut close to it, we let the algorithm to continue and change thelimiting constrains of and .The evolution of the error function with the iteration number

is plotted in Fig. 7, and compared to that of our previous ASMalgorithm [34], [35]. Convergence is much faster with the newASM approach since the initial layout is closer to the final solu-tion. It is noticeable that there are points, such as , that do notconverge to a final solution with our previous ASM algorithm,whereas now convergence is obtained with the new approach.The number of iterations that needs the ASM algorithm to con-verge to the final layout dimensions and its corresponding eval-uated error are summarized in Table II, for the three differentexamples considered ( has been set to 0.01).The final layout obtained for the first example (S) is slightly

out of the lower boundary ( mm), as it was expectedsince the target S was placed out of the convergence region [see

Fig. 7. Evolution of the error function of the ASM algorithm for the pointsof Table I. Dashed lines correspond to the evolution of the error function byconsidering the initial layout resulting from analytical expressions [34], [35];solid lines give the error function for the initial layout calculated from the least-squares approach proposed in this paper.M (blue), G (green) and S (pink) points.

TABLE IIFINAL LAYOUT

Fig. 3(a)]. However, we have let the algorithm to continue to-wards convergence by relaxing the limiting values of and .Thus, the fact that the target is out of the convergence regiondoes not necessarily mean that convergence is not possible, butthat the resulting geometric valuesmight be beyond or below theconsidering limits of and . For the S point, the close proximityto the line of the polygon corresponding to mm (Fig. 3)explains that the final value of is 0.41 mm. The agreementbetween the frequency response obtained by EM simulation ofthe final layout (obtained after a single iteration) and the circuitsimulation of the target parameters is excellent, as it can be seenin Fig. 8. This is reasonable since the value of is not far fromthe considering limiting value that guarantees that the CSRRsare accurately described by the models depicted in Fig. 1.Fig. 9 depicts the comparison of the frequency responses cor-

responding to point G, where, again, very good agreement be-tween circuit simulation (target parameters) and EM simulationof the final layout results.We have also obtained the error function with the iteration

number for CSRR-gap (T-shaped) loaded lines with the targetsindicated in Fig. 5 and given in Table III (the target T corre-sponds to a dual-band impedance inverter, which is the artificialline used in one of the examples discussed in Section IV). Theresulting geometrical values and number of required iterationsare given in Table IV (in this case, has been set to 0.04, stillproviding very accurate results). Fig. 10 depicts the frequency


Fig. 8. (a) Magnitude and (b) phase of the scattering parameters andat initial solution , and circuital simulation of the for point S.

Fig. 9. (a) Magnitude and (b) phase of the scattering parameters andat initial solution , and circuital simulation of for point G.

response for the initial and final layout, as well as the circuitsimulation of the target parameters for the T case (the consid-ered electromagnetic solver for these CSRR-gap-loaded lines isthe Agilent Momentum).It should be noted that the initial layout (inferred from the

pre-optimization algorithm) is already very close to the final

Fig. 10. (a) Magnitude and (b) phase of the scattering parameters andat initial solution , and circuital simulation of for point T. Devia-tions between the circuit simulation and the electromagnetic response of the finallayout are due to the fect that the circuit model is not valid at high frequencies.

TABLE IIIOPTIMAL COARSE SOLUTION

TABLE IVFINAL LAYOUT

solution. However, it can be observed a progressive mismatchbetween the electromagnetic response of the final layout andthe circuit simulation as frequency increases due to the inac-curacy of the circuit model at high frequencies. We do not pro-vide the convergence curves that lead to the number of iterations


TABLE VOPTIMAL COARSE SOLUTIONS OF UNIT CELLS

TABLE VISYNTHESIZED LAYOUTS FOR THE DIFFERENT UNIT CELLS

and error functions indicated in Table IV, since we cannot com-pare such curves with those curves that result by using the pre-vious version of the ASM algorthm (i.e., that reported in [39]).The reason is that our old ASM algorithm was not adapted forT-shaped gaps. Nevertheless, convergence speed is good (fewiterations suffice to obtain a small error function), although notso fast, as compared to the structures without gap (Fig. 7).

IV. APPLICATION EXAMPLES

A. Stopband Filter

A practical application of CSRR-loaded lines without gapsis the implementation of stopband filters with controllable re-sponse (bandwidth and rejection level). Compact planar stop-band filters can be designed by cascading several CSRR-loadedline unit cells. Let us illustrate this possibility through the de-sign procedure and experimental verification of a three-unit cellstopband filter in microstrip technology.The first step consists of the synthesis of the central unit

cell, whose transmission zero frequency is chosen to bethe central frequency of the stop-band, i.e., 2.45 GHz. Sincethe other cells involved in the design will have similar targetresponses and very close transmission zero frequencies (i.e.,2.36 and 2.53 GHz), the optimal coarse solutions for all of theinvolved cells were forced to be placed in the same conver-gence region (i.e., with the same and values, as seen inTable V) in order to speed up the design process. By cascadingthe three cells with the circuit elements indicated in Table V,a satisfactory stopband response at the circuit simulation levelresults. By using the ASM algorithm described in Section IIIand considering the Rogers RO3010 substrate with thickness

1.27 mm and dielectric constant , the threelayouts summarized in Table VI have been obtained.However, it is expected that, by directly cascading the three

designed cells, coupling effects between the CSRRs appear(this has been verified through EM simulation, not shown).

Fig. 11. Equivalent circuit and layout of the implemented stopband filter. Thesmall blue rectangles correspond to lines of physical length equal to 2.22 mmand width according to the value for each unit cell (see Table VI).

Fig. 12. Photograph of the fabricated CSRR-based stopband filter.

Therefore, it is necessary to insert transmission-line sectionsbetween the synthesized unit cells, as depicted in Figs. 11and 12. The length of the transmission-line sections betweenadjacent CSRRs has been set to 3/4 the CSRR radius (the widthbeing identical to that of the microstrip on top of the CSRR,indicated in Table VI). With these transmission-line lengths,coupling effects between adjacent CSRRs are not present, whilethe circuit response does not substantially change as comparedwith the case with direct connection between unit cells at thecircuit level.The completed manufactured stop-band filter can be seen in

Fig. 12, where tapers between the input/output 50- transmis-sion lines (needed to perform the measurements) and the de-signed structure are included. It was fabricated by using the cir-cuit board plotter LPKF ProMat S103.The measured filter response, displayed in Fig. 13, is com-

pared with the equivalent circuit and full-wave electromagneticsimulated responses. The measured rejection is better than20 dB within a 345-MHz frequency band. The agreementbetween circuit and electromagnetic simulation is very good, asexpected on account of the small errors that appear in Table VI.


Fig. 13. Frequency response of the stopband filter: the target response (i.e.,that corresponding to the equivalent circuit simulation) is depicted by the greendotted line, the full-wave EM simulation by blue dashed line, and the measure-ments by black solid lines.

The small discrepancies in measurement and the additionalreflection zero are attributed to fabrication related tolerances.

B. Dual-Band CSRR-Based Power Divider

Let us now consider the application of the proposed two-stepASM algorithm to the synthesis of a dual-band power splitterbased on a dual-band impedance inverter implemented bymeans of the CSRR-gap-loaded line. To achieve the dual-bandfunctionality, the inverter must provide a phase shift ofand at the operating frequencies and , respectively.We are thus exploiting the composite right-/left-handed be-havior of the structure, with and located in the left-handedand right-handed regions, respectively (see [19]). The inverterimpedance is set to 35.35 at both frequencies in order toguarantee a good matching when the inverter output port isloaded with a pair of matched loads (in practice two 50access lines) to implement the divider. Notice that these fourconditions do not univocally determine the five circuit elementsof the model of the CSRR-gap loaded line [Fig. 1(a)]. However,we can set the transmission zero (see (A2) in Appendix A)to a certain value, and the five parameters thus have a uniquesolution. Specifically, the design frequencies of the inverterhave been set to 0.9 GHz and 1.8 GHz and thetransmission zero to 0.5 GHz.With the previous specifications, the element values can be

easily found from (A2), (A3), and the characteristic impedanceof the symmetric T-network, given by

(18)

where and are the series and shunt impedance, respec-tively, of the T-circuit model of Fig. 1(a). The resulting elementvalues are 12.5 nH, 24.9 pF, 1.25 pF,

3.38 nH and 5.10 pF. With the element valuesof , and , we have obtained the convergence regionin the – plane according to the method reported be-fore, and the target values of and do not belong tosuch region. This means that it is not possible to implementthe dual-band impedance inverter by merely considering theCSRR-gap-loaded line (some element values are too extreme).

Fig. 14. Dependence of the element values of the CSRR-gap loaded line withthe phase of each cascaded transmission-line section at .

However, it is expected that, by cascading transmission-linesections at both sides of the CSRR-gap loaded line, the ele-ment values of the cell are relaxed, and a solution within theconvergence region arises. Therefore, we have cascaded twoidentical 35.35- transmission-line sections at both sides ofour CSRR-gap loaded line. The width of these line sectionsis 1.127 mm, corresponding to the indicated characteristicimpedance in the considered substrate (the Rogers RO3010,with thickness 0.635 mm and dielectric constant 10.2).Notice that, by cascading such 35.35- lines, the electricallength at the operating frequencies is the sum of the electricallengths of the lines and the CSRR-based cell. Thus, the phasecondition that must satisfy the CSRR-gap loaded line can beexpressed as and ,where and are the electrical lengths of the CSRR-basedcell at the design frequencies and , and is the phaseintroduced by the line at the indicated frequency.We have made a parametric analysis consisting on obtaining

the element values of the CSRR-based cell for different valuesof the length of the cascaded transmission-line sections (andhence and ). The results are depicted in Fig. 14.It can be observed that, for small values of and aretoo large to be implemented. Large means a small value of, but this is not compatible with a large value. On the other

hand, the values of and without cascaded line sections,i.e., , give extreme values of and , that is, alarge value of and a small value of . However, by increasing

(or the length of the cascaded lines), the variation ofthe elements of the central CSRR-gap-based cell goes in thecorrect direction for their implementation. Specifically, we haveconsidered a pair of transmission-line sections with, which means that the required electrical lengths for the

CSRR-based cell at the operating frequencies areand . The element values corresponding to thesephases are 9.45 nH and 17.9 pF, 1.01 pF,

4.85 nH, and 2.95 pF, and these values lead usto an implementable layout. The reason is that this phase shiftgives the minimum value of (see Fig. 14) and a reasonablysmall value of , with not so small and not too large.Notice that does not experience significant variations with

.


Fig. 15. Comparison between the electromagnetic and circuit simulation cor-responding to the characteristic impedance and electrical length of the designeddual-band impedance inverter.

Fig. 16. Schematic and photograph of the fabricated dual-band power divider.

We have applied the proposed two-step ASM algorithm to theprevious element values, and we have synthesized the layoutof the CSRR-gap-loaded line, considering a T-shaped geom-etry for the gap. The geometrical parameters of the synthesizedstructure are (in reference to Fig. 4) 14.42 mm,0.87 mm, 0.34 mm, 0.40 mm, 7.13 mm, andconvergence (with an error of 0.012) has been obtained after sixiterations.The comparison of the electrical length and characteristic

impedance inferred from EM simulation of the synthesizedimpedance inverter (the CSRR-based cell plus the cascaded35.35- transmission-line sections) and the ones inferred fromcircuit simulation are shown in Fig. 15. The agreement isexcellent in the left-handed region, where the model describesthe structure to a very good approximation and progressivelydegrades as frequency increases, as it is well known andexpected. Nevertheless, the phase shift and the characteristicimpedance at are reasonably close to the nominal values,

Fig. 17. Frequency response (circuit and electromagnetic simulation and mea-surement) of the designed and fabricated dual-band power divider.

and hence we do expect that the functionality of the powerdivider at is preserved.We have cascaded two output 50- access lines for connector

soldering, and the structure has been fabricated by means of aphoto/mask etching process (Fig. 16). Fig. 17 shows the fre-quency response of the divider, where it can be appreciated thatoptimum matching occurs at and slightly below , for thereasons explained. Nevertheless, the functionality of the powerdivider covers both design frequencies. The discrepancy be-tween themeasured response and the target is not due to a failureof the ASM algorithm, but to the fact that the circuit modelof the CSRR-gap-loaded line does not accurately describe thestructure at high frequencies, including part of the right-handedband. A more accurate model increases the complexity of theASM algorithm.

V. CONCLUSION

A new tool for the automated synthesis of CSRR-based trans-mission lines, based on aggressive space mapping (ASM), hasbeen proposed and developed. As compared with previous re-ported tools, the proposed algorithm is able to provide the con-vergence region in the coarse space; namely, it is able to predictwhether a given set of element values of the equivalent circuitmodel of the unit cell is physically able to be implemented ornot. Moreover, the proposed tool is a two-step ASM algorithm,where the first (and fast) ASM optimizer is used not only to ob-tain the convergence region, but also to provide the necessaryinputs (geometrical variables) for the determination of the ini-tial layout from a novel developed least-squares approach. Thus,the initial layout is very close to the target, and the convergencetime of the core (second step) ASM algorithm is dramaticallydecreased as compared to previous proposals (in some cases,two iterations have been enough to achieve convergence).To illustrate the potential of the new two-step ASM algorithm

to the synthesis of CSRR-based devices, the tool has been usedfor the automated synthesis of a stopband filter and a dual-bandpower divider. In the former case, the filter is achieved by cas-cading three unit cells. In order to avoid coupling effects thatthe equivalent circuit does not predict, small transmission-line


sections between adjacent cells have been added to the struc-ture. Good agreement between the circuit simulation and theelectromagnetic simulation with the layout of the CSRR-basedcells obtained with the proposed algorithm has been achieved.The measured results are also in good agreement with the circuitand electromagnetic simulations, hence validating the designingprocess in the region of interest. For the dual-band power di-vider, the pre-optimizer ASM has revealed that the target ele-ment values (inferred from the considered specifications) are notphysically able to be implemented. However, it has been foundthat, by cascading two transmission-line sections to the inputand output ports, the element values of the unit cell are not soextreme, resulting in an implementable CSRR-gap-loaded unitcell that has been synthesized by means of the proposed tool.The power divider has been fabricated and demonstrated to ex-hibit its functionality at the design frequencies.The considered electromagnetic solvers of the new proposed

ASM tool have been the Agilent Momentum and the Ansoft De-signer commercial software packages (i.e., two versions of thetwo-step ASM algorithm have been developed). The tool can beextended to other available electromagnetic solvers. The resultsof this work open the path to the integration of this new ASMalgorithm in such commercial tools for the fast and efficient op-timization of CSRR-based circuits.

APPENDIXPARAMETER EXTRACTION

The parameters of the circuit of Fig. 1(a) can be extractedfrom the EM simulation of the CSRR-loaded line according tothe following procedure. First, the reflection coefficient isrepresented in the Smith Chart. At the intercept of with theunit resistance circle, the shunt branch opens and hence we candetermine the resonance frequency of the CSRR

(A1)

as well as the value of the line inductance (from the simu-lated series reactance at ). To unequivocally determine thethree circuit elements of the shunt branch, we need two addi-tional conditions, apart from (A1). One of them is the transmis-sion zero frequency , which can be easily determined fromthe representation of the magnitude of with frequency. Thetransmission zero obeys the expression

(A2)

The third required condition can be derived from the followingexpression providing the electrical length of the structure:

(A3)

where and are the series and shunt impedance ofthe T-circuit model. Forcing , it follows that

(A4)

where is the angular frequency where , whichmeans that the phase of the transmission coefficient is also

and can be easily computed. Thus, from (A1),(A2), and (A4) we can determine the three reactive elementvalues that contribute to the shunt impedance.The circuit of Fig. 1(b) has an additional parameter ;

therefore, we need an additional condition to fully determinethe circuit parameters. This condition can be the resonance fre-quency of the series branch

(A5)

which is given by the intercept of with the unit conductancecircle, where the impedance of the series branch nulls.

APPENDIXINITIAL LINE AND GAP GEOMETRY

Inversion of (1) and (2) gives

(B1)

(B2)

If we assume that the fringing capacitance is small as comparedwith the line capacitance, it follows that , and fromthe value of and , we obtain the line impedance accordingto

(B3)

We use this impedance value to determine the line width ac-cording to well-known expressions (the line width is inferredby means of a transmission-line calculator). From the substrateparameters and the line width, the gap space necessary to ob-tain the value of given by (B1) can be inferred accordingto well-known expressions [44]. Finally, the line length is in-ferred by inverting the phase velocity given by

(B4)

where is the speed of light in vacuum and is the effectivedielectric constant (that can be easily computed from a trans-mission-line calculator).

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Jordi Selga (S’11) was born in Barcelona, Spain,in 1982. He received the Telecommunications En-gineering Diploma, Electronics Engineering degree,and M.S. degree in micro and nano electronicsengineering from the Universitat Autònoma deBarcelona, Barcelona, Spain, in 2006, 2008, and2009, respectively, where he is currently workingtoward the Ph.D. degree.In 2008 he joined CIMITEC Universitat

Autònoma de Barcelona, Barcelona, Spain, aresearch centre on Metamaterials supported by

TECNIO (Catalan Government). He held a national research fellowship fromthe Formación de Profesorado Universitario Program of the Education andScience Ministry (Reference AP2008-4707). His research interests are meta-materials, computer-aided design of microwave devices, and electromagneticoptimization methods.

Ana Rodriguez (S’10) was born in Lugo, Spain.She received the Telecommunications Engineeringdegree from the Universidade de Vigo, Vigo, Spain,in 2008, and the “Master en Tecnología, Sistemas yRedes de Comunicaciones” degree from UniversitatPolitècnica de València (UPV), València, Spain, in2010, where she is currently working toward thePh.D. degree.As a student, she participated in the Erasmus

exchange program, developing the Master Thesisat the University of Oulu, Finland. Since the end of

2008, she has been with the Institute of Telecommunications and MultimediaApplications (iTEAM), which is part of the scientific park at the UniversitatPolitècnica de València (UPV), València, Spain. Her main research interestsinclude computer-aided design of microwave devices, electromagnetic opti-mization methods, and metamaterials.

Vicente E. Boria (S’91–A’99–SM’02) was bornin Valencia, Spain, on May 18, 1970. He receivedthe “Ingeniero de Telecomunicación” degree(with first-class honors) and “Doctor Ingeniero deTelecomunicación” degree from the UniversidadPolitécnica de Valencia, Valencia, Spain, in 1993and 1997, respectively.In 1993, he joined the Departamento de Comu-

nicaciones, Universidad Politécnica de Valencia,Valencia, Spain, where he has been a Full Professorsince 2003. In 1995 and 1996, he held a Spanish

Trainee position with the European Space Research and Technology Centre,European Space Agency (ESTEC-ESA), Noordwijk, The Netherlands, where

he was involved in the area of electromagnetic analysis and design of passivewaveguide devices. He has authored or coauthored seven chapters in technicaltextbooks, 75 papers in refereed international technical journals, and over 150papers in international conference proceedings. he is a member of the EditorialBoards of Proceeding of the IET (Microwaves, Antennas and Propagation),IET Electronics Letters, and Radio Science. His current research interestsare focused on the analysis and automated design of passive components,left-handed and periodic structures, as well as on the simulation and measure-ment of power effects in passive waveguide systems.Dr. Boria has been a member of the IEEE Microwave Theory and Tech-

niques Society (IEEE MTT-S) and the IEEE Antennas and Propagation So-ciety (IEEE AP-S) since 1992. He is member of the Editorial Boards of theIEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and the IEEEMICROWAVE ANDWIRELESS COMPONENTS LETTERS. He is also a member of theTechnical Committees of the IEEE-MTT International Microwave Symposiumand of the European Microwave Conference.

Ferran Martín (M’04–SM’08–F’12) was born inBarakaldo, Spain in 1965. He received the B.S.degree in physics and Ph.D. degree from the Univer-sitat Autònoma de Barcelona, Barcelona, Spain, in1988 and 1992, respectively.From 1994 up to 2006, he was an Associate

Professor of electronics with the Departamentd’Enginyeria Electrònica, Universitat Autònomade Barcelona (UAB), Barcelona, Spain, and, since2007, a Full Professor of electronics. In recent years,he has been involved in different research activities,

including modelling and simulation of electron devices for high frequencyapplications, millimeter-wave and terahertz-generation systems, and the appli-cation of electromagnetic bandgaps to microwave and millimeter-wave circuits.He is now very active in the field of metamaterials and their application to theminiaturization and optimization of microwave circuits and antennas. He is thehead of the Microwave and Millimeter Wave Engineering Group (GEMMAGroup) at UAB and Director of CIMITEC, a research center on Metamaterialssupported by TECNIO (Generalitat de Catalunya). He has authored andcoauthored over 350 technical conference, letter, and journal papers, and heis coauthor of the monograph on metamaterials entitled Metamaterials withNegative Parameters: Theory, Design and Microwave Applications (Wiley,2008). He has filed several patents on metamaterials and has headed severalDevelopment Contracts.Dr. Martín was the recipient of the 2006 Duran Farell Prize for Technological

Research, he holds the Parc de Recerca UAB—Santander Technology TransferChair, and he was been the recipient of an ICREA ACADEMIA Award. Hehas organized several international events related to metamaterials, includingWorkshops at the IEEE International Microwave Symposium (years 2005 and2007) and EuropeanMicrowave Conference (2009). He has acted as guest editorfor three special issues on metamaterials in three international journals.

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