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IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 33, NO. 1, FEBRUARY 2010 169

Wideband Circuit Model for Planar EBG StructuresBaharak Mohajer-Iravani, Member, IEEE, and Omar M. Ramahi, Fellow, IEEE

Abstract—In this paper, we present a comprehensive equivalentcircuit model to accurately characterize an important class ofelectromagnetic bandgap (EBG) structures over a wide range offrequencies. The model is developed based on a combination oflumped elements and transmission lines. The model presentedhere predicts with high degree of accuracy the dispersion diagramover a wide band of frequencies. Since the circuit model can besimulated using SPICE-like simulation tools, optimization of EBGstructures to meet specific engineering criteria can be performedwith high efficiency, thus saving significant computation time andmemory resources. The model was validated by comparison tofull-wave simulation results.

Index Terms—Electromagnetic bandgap (EBG) structures,metamaterials, noise suppression, switching noise.

I. INTRODUCTION

I N THE LAST few years, different methods have been usedto characterize engineered materials including metamate-

rials, electromagnetic bandgap (EBG) structures, and frequencyselective surfaces. Engineered metamaterials are electricallysmall resonators and their electrical size is typically smaller(and sometime much smaller) than over the range of oper-ating frequencies. These materials may lead to different typesof propagation such as left handed (LH), right handed (RH),or stopband (bandgap). While metamaterials have traditionallybeen considered as any engineered material that gives rise toleft-handed propagation, it also includes the class of engineeredmaterial that is referred to as electromagnetic bandgap (EBG)structures.

EBG structures have the primary characteristic that propaga-tion through the structures is inhibited. Characterization of EBGstructures is performed using one of the following methods:

• measurements [1];• full-wave numerical simulation of the entire structure

based on various methods such as the finite elementmethod, the finite-difference time-domain method, thefinite integration method, etc., [2], [3];

Manuscript received November 04, 2008; revised March 20, 2009. First pub-lished October 30, 2009; current version published February 26, 2010. Thiswork was supported in part by Research in Motion and in part by the NationalScience and Engineering Research Council of Canada under the NSERC/RIMIndustrial Research Associate Chair Program and the Discovery Program. Thiswork was recommended for publication by Associate Editor L.-T. Hwang uponevaluation of the reviewers comments.

B. Mohajer-Iravani is with the Electrical and Computer Engineering Depart-ment, University of Maryland, College Park, MD 20742 USA (e-mail: [email protected]).

O. M. Ramahi is with the Electrical and Computer Engineering Department,University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TADVP.2009.2021156

• dispersion diagram extraction using full-wave numericalsimulation of a single cell [4], [5];

• equivalent circuit modeling based on lumped elements[6]–[10];

• equivalent circuit modeling based on lumped elements andtransmission lines [11]–[13].

Among these characterization methods, experimental measure-ments give high accuracy; however, they are costly and timeconsuming because of the time it takes to build EBG struc-tures (either printed circuit boards or chip packages need to bemanufactured first). Full-wave numerical analysis provides ac-curate results, however, for structures with high variance in theirtopological dimensions, simulation time and computer memoryneeds can be excessive to render optimization impractical. Otherindirect and efficient numerical analysis techniques were devel-oped based on the assumption that the actual finite-size struc-tures behave in a manner similar to an infinite extension of thesame structure [14], [15]. While such assumption is reasonablewhen the structure is composed of many cells, it falls short of ac-curate prediction when only few cells are employed as in caseswhere the circuit real-estate is limited [3].

Circuit-based models were developed as an alternative totime-consuming 3-D full-wave based models. In additionto their high-efficiency, circuit-based models help in under-standing how the different operating regions (LH, RH, andbandgap) are related to the topology and composition of theengineered EBG structures. However, deriving a model whichpredicts accurately the dispersive effects through a wide rangeof frequencies is not viable. This is because electromagnetic ef-fects at higher frequencies are stronger and are more difficult tomodel. Therefore, to finalize a specific design, fine tunings andadjustments based on full-wave analysis and/or experimentalmethods become necessary.

In [6], a simple model based on lumped elements for themushroom EBG structures was proposed. This model works fornormal wave incidence only at low frequencies where the di-mensions of the EBG are much smaller than the wavelengthin the host media. Later, a model based on both transmissionline theory and circuit elements was presented in [13]. Thismodel partially overcoming previous limitation may predict thebandgap with higher accuracy. In [7], a method was developedfor extracting the parameters of the model for a mushroom-typeEBG embedded in parallel plate waveguide. The method in [7]uses simple formulas to derive initial values of the lumped el-ements in the model followed by a numerical algorithm basedon curve fitting the analytically extracted S-parameters to thoseextracted using full wave numerical simulations. The techniquereported in [7], while amongst the first efforts for extracting acircuit model for EBG structures employed for noise mitigation,was not efficient and robust enough for designing EBG struc-tures with complex topologies. In [8], a physical-based equiv-

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170 IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 33, NO. 1, FEBRUARY 2010

alent circuit model for mushroom EBGs embedded in parallelplate waveguides was derived. This model gives good predic-tion of the center frequency of bandgap but fails to predict theedges of the bandgap. In [11] and [12], transmission line seg-ments were added to the lumped elements model, leading tomore accurate prediction of the edges of the bandgap as wellas the center frequency. In [9] and [10], basic lumped-elementsmodels were developed for metamaterials. Those models givethe first LH and RH modes and the existing bandgap in be-tween. These models give good prediction of the bandgap if thecoupling and the parasitic effects (due to higher order modes)appear at much higher frequencies than the frequency band ofinterest.

In this work, we present a model for an important class ofplanar EBG structures that is applicable over a wide range offrequencies. The structures considered here were recently pro-posed in [3] and [16] as a robust, highly versatile, and tunablesolution in reducing electromagnetic interference (EMI) in mul-tilayer printed circuit board (PCB) boards and integrated cir-cuit (IC) packages. These are simple to construct as they arecomposed of square patches connected by meander lines. Theparameters of such topology are easily quantified and whencontrolled, can lead to designs that meet specific criteria. Inthis work, we develop an advanced equivalent circuit modelbased on lumped elements and transmission lines under TEMmode assumption. More specifically, we show that by consid-ering enough circuit elements in the modeling of the unit cell ofengineered material, it is possible to predict the performance ofstructure at higher frequencies up to the point where the TEMmode assumption becomes invalid. Therefore, the model pre-sented here eliminates the drawbacks of earlier circuit-basedmodels.

This paper is organized as follows. In Section II, we presentthe method used to predict the dispersive behavior of infinitearray of EBG structures using the circuit model of a unit cell. InSections III–VI, we present the modeling of the different com-ponents comprising the class of planar EBG structures consid-ered in this paper. In Section VII, the complete advanced circuitmodel is presented along with performance validation. Conclu-sions are presented in Section VIII.

II. OUTLINE OF MODEL FORMULATION

In this section, we present the method used to predict the dis-persive behavior of an infinite array of EBG structures usingthe circuit model of unit cells, along with the correspondingfull-wave numerical procedure. The actual extraction of the cir-cuit model will be presented in the next sections.

A 3-D view of the planar EBG structure under study con-sisting of square patches connected by meander lines is shownin Fig. 1. Without loss of generality, we develop our model inthe first region of Brillouin zone [17] ( region) where thewave experiences phase variation in -direction through prop-agation along an infinite array of EBGs ( ; where

is the propagation constant in the direction and is thecell size). The phase variation in -direction is ( ;where is the propagation constant in the direction) where

is an integer number. Notice that the structure is isotropic in

Fig. 1. Planar EBG structure based on patches connected by meander lines.

Fig. 2. Infinite 1-D array of the planar EBG structure with PMC boundariesused to model the 2-D infinite array in �� region of Brillouin zone (a) withand (b) without connecting meanders in �-direction.

the and directions. The model developed in this work can beextended to other regions of Brillouin zone.

By considering the region (in the Brillouin diagram),we reduce the analysis of the infinite 2-D array to the analysisof 1-D infinite array in -direction. The effects of the period-icity in the -direction is accounted for through the use of per-fect magnetic conducting (PMC) boundaries on both sides of the1-D array, as shown in Fig. 2(a). The PMC boundary is the onlycondition which provides zero phase shifts for all frequencies ofthe wave propagating in the -direction. By incorporating thisboundary, we have effectively ignored the side coupling effectsin the -direction. The top view of the unit cell of EBG is shownin Fig. 3(a). Our objective here is to develop a circuit model forthis type of structure that is valid over a wideband of frequen-cies. Therefore, the unit cell of the EBG array is reduced to ei-ther the structure shown in Fig. 3(b) or (c) which corresponds toan infinite array of 1-D EBGs shown in Fig. 2(b) (the results ofthe analysis of these two structures are identical). In this study,we will provide complete model for Fig. 3(c).

The dispersion diagram of the EBG structure is numericallyextracted using the procedure discussed in [4]. The computa-tional domain and the boundary setup for extracting the disper-sion diagram of the unit cell of EBG array are shown in Fig. 3(d).The pair of periodic boundary condition (PBC) models the pe-riodicity in the array structure.


MOHAJER-IRAVANI AND RAMAHI: WIDEBAND CIRCUIT MODEL FOR PLANAR EBG STRUCTURES 171

Fig. 3. Unit cell of the planar EBG structure with meander line as a connectingbridge. (a) Top view of unit cell of array in Fig. 2(a). (b) and (c) Top view ofunit cell of array in Fig. 2(b). (d) Computational setup used for extracting thedispersion diagram of EBG structure.

Next, the dispersive behavior of the EBG structure is pre-dicted through the circuit model of a unit cell and its effectiveequivalent transmission line using the ABCD matrix parame-ters. The infinite array of EBG structure cells is considered asa transmission line made of engineered materials representingquasi homogeneous media where the effective propagation con-stant is and the effective characteristic impedance is .The ABCD parameters for this transmission line correspondingto a unit cell (see Fig. 3) are given as

where is the length of the unit cell. The ABCD matrix of aunitcell of the EBG structure obtained from the equivalent circuitmodel is represented as

The frequency independent lumped elements to be used inthe circuit model represent the meander line, the patch, the mu-tual coupling between the meander line and the patch, and thestep discontinuity (junction) between the patch and the meanderline. The lumped elements will be connected through transmis-sion lines, which are critical to model the phase shift varia-tion across the structure. By equating those two ABCD matricesshown above, the dispersion diagram of the EBG structure canbe extracted.

In the following sections, we introduce the relationship be-tween the physical parameters of the structure (topological pa-rameters) and the equivalent lumped elements, and compute

Fig. 4. (a) Top view of patch connected to meander lines from both sides.(b) Equivalent circuit based on lumped elements.

the parameters of the transfer matrix , , , and .Throughout this work, we will assume that the metallizationthickness and both metallic and dielectric losses are neglected.

III. MODELS OF PATCH AND PATCH-MEANDER

LINE DISCONTINUITY

The patch and the meander line are implemented usingmicrostrip technology. Their characteristic impedances aredenoted by and , respectively. is much smaller than

as the width of patch microstrip is much wider than thewidth of the meander microstrip line. At low frequencies, thepatch behaves as an electrically short line compared to adjacentmeanders. Therefore, the patch is modeled by a -equivalentcircuit as shown in Fig. 4. The series reactance, , and shuntsusceptance, , in this model are approximated as [18]

(1)

(2)

leading to the capacitance and inductance (seeFig. 4)

where is the length of square patch, is the guided wave-length in dielectric substrate, is the speed of light in free space,and is the effective relative permittivity of dielectric sub-strate.

The asymmetric step discontinuity between the patch andthe meander as shown in Fig. 5(a) can be modeled through theT-equivalent circuit shown Fig. 5(b). is the width of patchand is the width of meander line. If we look to the struc-ture through the symmetry line of the meander-microstrip junc-tion, the width of patch is divided into two unequal parts withwidths and (where ). Two new sym-metric discontinuities are constructed such that the patch widthin the first one has a width as shown in Fig. 5(c) and in thesecond discontinuity the patch has the width as shown inFig. 5(d). The values of the capacitance and the inductances inthe model of asymmetric discontinuity can be approximated as



Fig. 5. (a) Asymmetrical discontinuity between the patch and the meanderedbridge. (b) Equivalent lumped-element circuit model. (c) and (d) are two newsymmetrical discontinuities introduced for modeling purpose.

the geometric mean of related values for two symmetric casesas

(3)

where , , andpresent the elements modeling

the step discontinuities at the connections shown inFig. 5(a), (c), and (d), respectively. The elements modeling thesymmetrical discontinuity in Fig. 5(c) where the width of firststrip is and the width of second strip is aregiven by [18], [19]

where the inductance per unit length of the microstrip line( , 2) is given by and

[18], [19].and are the characteristic impedance and effective relativepermittivity of the microstrip line with width , respectively,and is the dielectric thickness in . The capacitance is givenby [18], [19]

is in pF, , , and are in nH. The values of theequivalent circuit for the configuration in Fig. 5(d) are foundsimilarly.

Fig. 6. (a) Three-dimensional view of the meander line inductor. Design pa-rameters are included in the figure. (b) Equivalent circuit model.

For the special asymmetric case where(meander line is connected to the patch at the edge) same as thecase shown in Fig. 3, the value of the lumped elements can beapproximated by ignoring the effect of the fringing field alongthe side (where we have assumed -symmetry). Therefore, theasymmetric discontinuity can be approximated by a symmet-rical one where the width of the microstrip line at the junctionis twice that of the original structures ( , ).

IV. MODEL OF MEANDER LINE

A 3-D view of a planar meander line and its design parame-ters is shown in Fig. 6(a). The meander line which is placed ontop of a metal backed dielectric substrate consists of metallicarms and metallic bridges connecting arms. The con-cept of modeling to be discussed in this section is general andcan be extended to any meander line configuration. Without anyloss of generality, we assume that all traces in the meander havethe same parameters of width, , separation gap between ad-jacent arms, , arm length, , input and output connectingbridges’ length, , and input and outputarms’ length, . The thickness and rel-ative permittivity of dielectric substrate are and , respec-tively. The -equivalent circuit model of meander line is shownin Fig. 6(b). The inductor is the dominant element in this modelup to the first self resonant frequency. The capacitors model theparasitic effects. It should be noted that in this model, we havenot taken into account the effect of right angled bends in the me-ander line. The effect of these bends becomes more pronouncedat frequencies much higher than the range of interest consideredin this work.



Fig. 7. Magnetically coupled traces in the meander line. The mutual couplingbetween pairs of arms and bridges is marked with solid and dashed lines, re-spectively.

A. Computation of Inductance

Using Greenhouse formulas [20] and image theory [21], thetotal meander line inductance is given by

(4)

where is the inductance of meander pattern. is the mu-tual inductance between the meander line and its ground image.Here, it is assumed that the meander line is electrically short.The elements in circuit model are frequency independent. Thephase shift across the structure which is function of frequencywill be considered in section VI.

The top view of a meander line is shown in Fig. 7. The mag-netically-coupled arms are marked by the solid lines shown inFig. 7. The mutual inductance between one arm and all the mag-netically coupled arms are sequentially alternating between neg-ative and positive signs due to direction of current and magneticflux lines surrounding the arms. The mutual inductance betweenone bridge and all the magnetically coupled bridges are posi-tive. As a first-order approximation, we have considered onlythe coupling between adjacent bridges indicated by dashed lineas shown in Fig. 7. Coupling between other pairs of lines is as-sumed negligible as there is no overlap along the traces as wellas the increased distance between them. is given by

(5)

where the first sum is due to the self inductance of all sectionsin the inductor pattern (total of arms and bridges),the second sum gives the total mutual inductance between cou-pled arms, and the third sum gives the total mutual inductancebetween adjacent bridges. The self inductance of th section isgiven by [22], [23]

Fig. 8. Image of meander line by ground plane. Direction of currents in actualand image patterns are shown in the figure.

where is the length of the line. All dimensions are in andis in nH.

The mutual inductance between two parallel arms which arecompletely overlapping, as shown in Fig. 7, with effective length

and center to center average distanceis computed by (6) and (7) [20], [23],

[24]

(6)

(7)

All dimensions are in m and is in nH. is the geomet-rical mean distance between parallel strips.

The mutual coupling between adjacent bridges where the ef-fective length of strips is equal to and

is approximated using (6) and (7) by [24]

For other configurations of parallel strips with arbitrary shapes,approximate formulas to compute magnetic mutual couplingcan be found in [24] and [25].

In order to model the effect of the ground plane, we showin Fig. 8 a meander line and its image. The distance betweenthe actual meander and its image pattern is equal to twice thesubstrate-thickness. The current flowing through the image lineis in opposite direction leading to the total negative mutual in-ductance represented by , as shown in (4). The mutual in-ductance between magnetically coupled sections in the physicalline and the image line are computed through similar formulas



Fig. 9. Parasitic capacitances in meander lines.

stated earlier. In those formulas, the geometric mean distance isapproximated by the distance between the centers of the lines.

is computed by

if j is odd

if j is even(8)

It is to be noted that the mutual inductance between the imageand the physical line is not doubled as the image pattern is notpart of the actual structure.

B. Computation of Parasitic Capacitances

There are two types of parasitic capacitances in the meanderline model, and , as shown in Fig. 6(b). representsthe parasitic capacitance in the meander pattern, which consistsof the different interline capacitances such as , , and

, between adjacent arms, and , andbetween adjacent bridges, as shown in Fig. 9. Due to our as-sumption of zero metallization thickness, , andare negligible. Also, and between the arms aremuch higher than and between the bridges in ourmeander designs where we intentionally want to enlarge thevalue of inductance per unit area [3], [16]. Therefore, the para-sitic interline capacitance between two coupled adjacent armsand is given by

or

where , representing the effective coupled length betweentwo arms, is approximated by

The total fringing per-unit-length capacitance in the gap be-tween two arms through air and the dielectric is given by

and , respectively.In this modeling, we are approximating the parasitic capac-

itance by neglecting coupling between nonadjacent arms andbetween the patch and the nonadjacent arms (i.e., only the cou-pling effect between adjacent strips is considered). Any two

Fig. 10. (a) Uncoupled microstrip line. (b) Symmetric coupled microstrip lines.(c) Symmetric coupled striplines. The widths of strips and the gaps in betweenare equal for all cases. The parasitic capacitances per unit length between thestrips and the ground planes are shown.

adjacent arms in the meander pattern are considered as sym-metric coupled lines because of the constant strip width. In ad-dition, the equal gap between the arms gives and

where . Below, we will provideformulas to compute and . which is the equivalentcapacitance resulting from series of gap capacitances inthe -1 consecutive inter-arm gaps is computed as

(9)

In our case, the expression for is reduced toconsidering that all arms are of equal length.

To compute [refer to Fig. 6(b)], we have to considerbridge-sections and arm-sections separately due to differentcoupling mechanism. The per-unit-length capacitance betweenthe strip and the ground for bridge-sections, which are uncou-pled microstrip lines, is given by [23].

is result of parallel combination of the three capacitancesshown in Fig. 10(a) [18]

where is the parallel plate per-unit-length capaci-tance between the strip and the ground plane, given by

. The fringing per-unit-length capaci-tance between the uncoupled strip and the ground is givenby . and are the



relative permittivity and the effective relative permittivity ofthe meander microstrip, respectively.

For two symmetrically coupled microstrips shown inFig. 10(b), the per-unit-length capacitance between the stripand the ground is . The even mode capacitance perunit length is [18], [19] where is thefringing capacitance between the meander-strip and the groundin presence of another similar strip. It is given by [19]

where .The interline mutual capacitance between symmetric lines

discussed earlier within the context of is defined in termsof the odd and the even mode capacitances. The gap capaci-tance is given by where is theodd mode capacitance per unit length. It is defined as

[18], [19] [refer to Fig. 10(b)]. isthe fringing per-unit-length capacitance between two symmet-rical coupled microstrips through the dielectric gap and is ob-tained by using a corresponding structure of symmetrically cou-pled striplines with similar dimensions and twice the substrateheight as shown in Fig. 10(c). is equal towhere and are the per-unit-length even and the oddmode capacitances of the stripline setup, respectively. In the me-ander line, these odd and even capacitances for symmetricallycoupled arms are given by [19]where . and are the elliptic function andits complement [19], [26], respectively. is [19]

ifif

(10)

where . is the fringing per-unit-length ca-pacitance between two symmetrical coupled microstrip linesthrough the air gap, and is given by [19] and [27]

where represents the fringing per-unit-length capaci-tance between the symmetric coupled coplanar lines whenthe dielectric material is air. The width of strips and the sep-aration between them in coplanar configuration are similar tothe arm-strips in the original meander structure and they areequal to and , respectively. is given by [19]

where and. is the fringing per-unit-length capaci-

tance in the surrounding air (not in the air gap between the twosymmetric coupled arm-strips) and is approximated by [27]

where , , , and are the fringing per-unit-lengthcapacitances in the symmetric coupled microstrips andstriplines with air as a dielectric material .

Fig. 11. (a) Side view of patch and meander-arm microstrips as a pair of asym-metric coupled microstrip lines. The capacitances per unit length modeling thecoupled lines are shown. (b) Corresponding equivalent circuit model.

The formulas of fringing capacitances have been providedearlier. The fringing capacitances in the symmetric coupledstriplines are summed as

where .By using the above mentioned approximation, in three sym-

metric coupled microstrip configuration the parasitic per-unit-length capacitance between the middle strip and the ground isgiven by .

In the next section, the coupling effects between the meanderline and adjacent patches are computed.

V. MODEL OF COUPLING EFFECT BETWEEN THE PATCH

AND THE MEANDER LINE

In the planar EBG structures under study, the configurationof a patch and an adjacent arm represents asymmetric coupledmicrostrip lines, as shown in Fig. 11(a). All per-unit-length ca-pacitances characterizing that configuration including the par-allel plate capacitances and the fringing capacitances are shownin Fig. 11(a). Fig. 11(b) shows the -equivalent circuit modelfor this configuration, where is representing the capac-itance between the first arm and the ground plane, is thecapacitance between the patch and the ground, and isthe mutual coupling between the patch and the arm. Generally,the unit cell of this type of designs for EBG structure (patch +meander line) includes several parallel microstrip lines whichare symmetrically or asymmetrically coupled to each other. Tomodel these structures accurately appropriate fringing capaci-tances have to be included in the model.

To this end, we will use the lumped elements as defined inthis section to improve the magnitude value of those elements



introduced in earlier sections. The capacitance is in-cluded in (refer to Section IV-B) to complete the total valueof the parasitic capacitance between the meander line and theground. We can replace in Section III with . Then,the model approximates the capacitance between the patch andthe ground with higher accuracy due to the consideration of thefringing capacitances in presence of coupled lines. The lumpedelements in Fig. 11(b) may be approximated by

(11)

(12)

(13)

where is the effective overlapping length between thepatch and the meander arm. For the EBG structures understudy, is equal to . and rep-resent the parallel plate per-unit-length capacitance for thepatch and the meander-strip, respectively. anddemonstrate the fringing per-unit-length capacitances for thetwo coupled symmetric patch-microstrips [28]. Similarly,

and represent fringing per-unit-length capac-itances for the two coupled symmetric meander-microstrips[28]. Separation between the two symmetric strips is iden-tical to the original setup, [Fig. 6(a)]. According to theexpressions provided for meander line model in Section IV,

and . is obtainedfrom the formula for by replacing with the properseparation . If in the provided expressions, we replace thespecifications of the meander line with the specifications of thepatch including width ( is replaced by ), characteristicimpedance ( is replaced by ), and relative effectivepermittivity ( is replaced by ) then the values of

, , and are easily obtained. , theper-unit-length capacitance through the dielectric gap is givenby [28] whereis the total fringing per-unit-length capacitance between the twosymmetric coupled meander-strips through the dielectric gap.That capacitance is computed through the formula provided for

in Section IV by replacing the proper separation gap (is replaced by ). Similarly, , the total fringingper-unit-length capacitance between the two symmetric cou-pled patch-strips through the dielectric gap is computed. Thefringing per-unit-length capacitance through air gap is givenby [28] . is thetotal fringing per-unit-length capacitance between the asym-metric coplanar strips suspended in air. The widths of stripsand the separation between them in coplanar are similar to thepatch-arm microstrips and they are equal to , , and

. is given by [29]where and is given by

represents the fringing per-unit-length capacitance inthe surrounding air. This capacitance can be approximated by

(14)

Fig. 12. (a) A meander line between two patches. The bridge connecting thefirst arm to the patch is shown. (b) Equivalent circuit model of: i) the mutual cou-pling between the patch and the adjacent meander-arm and ii) the connectingbridge-trace. (c) Equivalent circuit model of the entire meander line. In thismodel, the effect of mutual coupling between the patch and the meander is con-sidered as interline parasitic capacitance.

where

, , , , , ,, and represent the fringing per-unit-length

capacitances between the strips and the ground in the symmetriccoupled microstrips and striplines. In these configurations, thedielectric material is air. The subscripts and referto meander-meander and patch-patch coupling, respectively.These capacitances are computed through the expressionsprovided in Section IV by substituting .

The bridge connecting the patch to adjacent arm shown inFig. 12(a) is modeled as follows. If the bridge-length, , is aconsiderable portion of the total distance separating the two con-secutive patches, then we can complete the modeling throughthe -equivalent circuit shown in Fig. 12(b). Where, is theinductance of the bridge-trace and is the capacitance be-tween the trace and the ground. is the mutual couplingcapacitance between the patch and the meander-arm. However,if is very small then and may be included in thetotal inductance and capacitance modeling the meander line (and ) discussed in Section IV. In this case, is mod-eled as part of the interline parasitic capacitances in the meanderpattern as shown in Fig. 12(c).



Fig. 13. Advanced equivalent circuit model made of lumped elements and transmission lines modeling the unit cell shown in Fig. 3(c).

VI. MODEL OF ELECTRICAL LENGTH OF PLANAR

EBG STRUCTURE

Due to distributive nature of the inductances and capaci-tances realized by EBG structures, it is important to includeboth lumped elements and transmission lines in the circuitmodel. This combination insures the inclusion of the phaseshift due to delay of the field propagating across the structure.

To consider the phase delay across the unit cell of the EBGstructure under study, we include two different types of mi-crostrip transmission lines in our model consisting of patch-mi-crostrip and meander-microstrip with total length of and ,respectively. Where represents the periodicity-length (or the length of a unit cell of EBG structure), rep-resents the length of a patch, and represents the length ofseparation between the two consecutive patches. Therefore, thepatch-microstrip models the phase difference across the patchand the meander-microstrip models the phase difference acrossthe meander line [30] (not along the meander line).

In this study, we adopted similar model as [30] in which themeander-antenna was modeled as a linear dipole antenna withinductive loading. For the application involving noise mitiga-tion in PCB and IC package, the proposed model gives sufficientlevels of accuracy, however, it is possible to make the modelmore accurate for higher frequencies by considering the cou-pling effects vectorially between the arms in the meander lineas in [21]. The ABCD parameters for the microstrip line spec-ified by the characteristic impedance, the relative effective per-mittivity, and the line length are available in microwave bookssuch as [15].

VII. FINAL CIRCUIT MODEL AND MODEL VALIDATION

In Fig. 13, the complete circuit model of a unit cell of theplanar EBG structure under study [Fig. 3(c)] is presented. Theelements along the -direction between the two input and outputports model the propagation in this direction in an infinite arrayof periodic structures. The propagation in -direction is bound

between two PMC boundaries resulting in zero phase shifts (be-tween the side PMC boundaries) for all frequencies, as dis-cussed in Section II.

The unit cell in the -direction is consisting of two open-ended microstrips. The widths of these microstrips are half ofthe width of patch and they are located at the ends of the unit cell.Each of these microstrip lines (from one side up to the middleof the patch) is modeled by the following elements: 1) the openend; 2) the microstrip line representing phase delay for wavepropagating in -direction; 3) the inductance of corre-sponding to the microstrip line with both width and length of

; 4) the capacitance of (which is common be-tween both the and directions) models the capacitive valueof half of the patch. In Fig. 13, in -direction the characteristicimpedance and the propagation constant of patch-microstrip arelabeled as and , respectively. is the length of patch.The characteristic impedance and the propagation constant ofmeander-microstrip are denoted as and , respectively.

is the distance between two consecutive patches. In the -di-rection, the characteristic impedance and the propagation con-stant of half-patch-microstrips are labeled as and .

The final ABCD matrix for this model is simply given by cas-cading the ABCD matrices of all the subcircuits. The disper-sion diagram of the structure is finally extracted by equating theABCD matrices and , as discussed in Section II.

To validate the circuit model, we consider two separate ge-ometries and obtain the dispersion diagram using our circuitmodel and using the full-wave finite element based simulationsoftware HFSS [31]. The design parameters of the EBG struc-tures are: , , ,

, , . The sub-strate in the first sample has a relative permittivity ofand loss tangent of . For the second example, wehave and . These two examples usedfor validation of our circuit model were considered earlier fornoise mitigation in packages [3], [16]. The dispersion/attenua-tion diagrams are shown in Fig. 14, where . Thecorresponding dispersion diagrams extracted by HFSS are alsoincluded. From the results presented in Fig. 14, we observe that



Fig. 14. Dispersion/attenuation diagrams for two planar EBG structures ex-tracted by two methods: 1) advanced equivalent circuit model and 2) full-wavenumerical simulator HFSS. The parameters of EBGs are as follows: � �

�� , � � �� , � � �� , � � �� , � �

�� , � � �� . The dielectric is specified as: (a) � � �� and� � � �� in the first sample and (b) � � �� and � � � �� inthe second sample.

the circuit model predicted the first three propagating modes ofthe EBG samples with high accuracy. In fact, strong agreementbetween the circuit model and the full-wave simulation resultsare observed over a frequency bandwidth extending from verylow frequency up to 5 GHz. What is a strong feature of the cir-cuit model developed here is that it predicts the edges of thebandgap with high degree of accuracy, something that earliercircuit models were not able to achieve.

VIII. CONCLUSION

In this study, we introduced, for the first time, a circuit modelfor planar EBG structure, with validity extending over a widerrange of frequencies. The circuit model is implemented aspatches and meander lines as connecting bridges. The circuitmodel was developed based on rigorous analysis of the prop-agation effects of each segment of the structure. As a result,the developed model predicts the dispersion diagram of EBGs

over a wide frequency range with high accuracy. We validatedthe circuit model by obtaining the dispersion diagram for twodifferent EBG structures reported earlier in [3] and comparedthe results to those obtained using full-wave analysis.

EBG structures are electrically small resonators and theirbandgap behavior is typically not readily predictable unlessfull-wave based extraction is performed. In the work presentedhere, a circuit model was developed for a wide class of EBGtopologies, thus facilitating expedient and efficient extraction ofthe bandgap behavior. For example, for the two EBG structuresconsidered here, it took few seconds to obtain the dispersion di-agram using the circuit model whereas it took over 4 h to extractthe dispersion diagram using full-wave simulation performedon a PC with an Intel Pentium 4 processor and 2 GB of RAM.It is important to note that the circuit model that was developedhere did not require any a priori full-wave based extraction aswas done in previous works (see, for example, [7]).

Once a circuit model is obtained, the effect of differentsources with varying time waveforms can be studied with highefficiency. Furthermore, the effect of varying the length ofthe meander line bridges and/or the effect of varying othertopological parameters can be obtained without resorting tofull-wave simulations.

Finally, we note that the circuit model developed in this workconsidered open planar EBG structure. However, the circuitmodel can easily be extended to include shielded EBG struc-tures (i.e., three-layer EBG structures). To include shieldingeffects, the circuit model would be developed based on striplineinstead of microstrip line propagation behavior.

REFERENCES

[1] S. Shahparnia and O. M. Ramahi, “Miniaturized electromagneticbandgap structures for broadband switching noise suppression inPCBs,” Electron. Lett., vol. 41, no. 9, pp. 519–520, Apr. 2005.

[2] F. Yang and Y. Rahmat-Samii, “Microstrip antennas integrated withelectromagnetic band-gap (EBG) structures: A low mutual couplingdesign for array applications,” IEEE. Trans. Antennas Propagat., vol.51, no. 10, pp. 2936–2946, Oct. 2003.

[3] B. Mohajer-Iravani and O. M. Ramahi, “Suppression of EMI and elec-tromagnetic noise in packages using embedded capacitance and minia-turized electromagnetic bandgap structures with high-k dielectrics,”IEEE Trans. Adv. Packag., vol. 30, no. 4, pp. 776–788, Nov. 2007.

[4] R. Remski, “Analysis of photonic bandgap surfaces using AnsoftHFSS,” Microwave J., vol. 43, no. 9, pp. 190–198, Sep. 2000.

[5] B. Mohajer-Iravani, S. Shahparnia, and O. M. Ramahi, “Couplingreduction in enclosures and cavities using electromagnetic band gapstructures,” IEEE Trans. Electromagn. Compatibil., vol. 48, no. 2, pp.292–303, May 2006.

[6] D. F. Sievenpiper, “High-impedance electromagnetic surface,” Ph.D.dissertation, Dept. Electrical Eng., Univ. California, Los Angeles, CA,1999.

[7] T. Kamgaing and O. M. Ramahi, “Design and modeling ofhigh-impedance electromagnetic surfaces for switching noise suppres-sion in power planes,” IEEE Trans. Electromagn. Compat., vol. 47,no. 3, pp. 479–489, Aug. 2005.

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[31] High Frequency Structure Simulator (HFSS). Ansoft Corp., Pittsburgh,PA.

Baharak Mohajer-Iravani (S’03–M’08) receivedthe B.Sc. degree in electrical engineering from SharifUniversity of Technology, Tehran, Iran, in 1998, theM.Sc. degree (with honors) in electrical engineeringfrom Amir-Kabir University of Technology (TehranPolytechnics), Iran, in 2001, and the M.Sc. and Ph.D.degrees in electrical and computer engineering fromthe University of Maryland, College Park, in 2004and 2007, respectively.

From 1997 to 2001, she worked on design andimplementation of the circuits and systems. She

was also involved in developing algorithms for speech processing. She wasvisiting scholar at University of Waterloo, Waterloo, ON, Canada, from 2006to 2007. Her research interests include electromagnetic band-gap structures(EBG), analysis and modeling of microwave and RF devices, high-speed pack-aging and signal integrity, and electromagnetic compatibility and interference(EMC/EMI).

Dr. Mohajer-Iravani has been selected for inclusion in 2009 Marquis Who’sWho in America.

Omar M. Ramahi (F’09) received the B.S. degree inmathematics and electrical and computer engineering(summa cum laude) from Oregon State University,Corvallis, the M.S. and Ph.D. degree in electrical andcomputer engineering from the University of Illinois,Urbana-Champaign.

From 1990 to 1993, he held a visiting fellowshipposition at the University of Illinois, Urbana-Cham-paign. From 1993 to 2000, he worked at DigitalEquipment Corporation (presently, HP), where hewas a member of the Alpha Server Product Devel-

opment Group. In 2000, he joined the faculty of the James Clark School ofEngineering at the University of Maryland at College Park as an AssistantProfessor and later as a tenured Associate Professor. At the University ofMaryland, he was also a faculty member of the CALCE Electronic Prod-ucts and Systems Center. Presently, he is a Professor in the Electrical andComputer Engineering Department, University of Waterloo, Waterloo, ON,Canada. He holds cross appointments with the Department of Mechanical andMechatronics Engineering and the Department of Physics and Astronomy. Hehas authored and co-authored over 190 journal and conference paper. He isa co-author of the book EMI/EMC Computational Modeling Handbook, 2ndEd. (Springer-Verlag, 2001). He served as a consultant to several companiesand was a co-founder of EMS-PLUS, LLC and Applied ElectromagneticTechnology, LLC.


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