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2614IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 8, AUGUST 2010

Artificial Magnetic Materials Using Fractal Hilbert

Curves

Leila Yousefi, Member, IEEE, and Omar M. Ramahi, Fellow, IEEE

Abstract—Novel configurations based on Fractal Hilbert curves

are proposed for realizing artificial magnetic materials. It is

shown that the proposed configuration gives significant rise to

miniaturization of artificial unit cells which in turn results in

higher homogeneity in the material, and reduction in the profile

of the artificial substrate. Analytical formulas are proposed for

design and optimization of the presented structures, and are

verified through full wave numerical characterization. The elec-

tromagnetic properties of the proposed structures are studied

in detail and compared to square spiral from the point of view

of size reduction, maximum value of the resultant permeability,

magnetic loss, and frequency dispersion. To validate the analytical

model and the numerical simulation results, an artificial substrate

containing second-order Fractal Hilbert curve is fabricated and

experimentally characterized using a microstrip-based character-

ization method.

Index Terms—Artificial magnetic materials, Fractal Hilbert

curves, metamaterials, microstrip-line based characterization,

permeability.

I. INTRODUCTION

W

andmagnetically,sotheyexhibitenhancedrelativepermeability

and permittivity.

Recently it has been shown that utilizing magneto-dielec-

tric materials instead of dielectrics with high permittivity offers

many advantages in an important class of applications [1]–[8].

In [1], it was shown that using materials with high permeability

for antenna miniaturization can increase the bandwidth while

materials with high permittivity would shrink the bandwidth.

Furthermore, when materials with only high permittivity are

used for antenna miniaturization, the high impedance mismatch

between the substrate and the air decreases the efficiency of the

system, while in the case of using magneto-dielectric materials,

the impedance mismatch is smaller leading to the higher effi-

ciency for the miniaturized antenna. It was shown in [2] that

magneto-dielectric resonator antennas have wider impedance

bandwidth than dielectric resonator antennas. In [3], a meander

HEN exposed to an applied electromagnetic field, mag-

neto-dielectric materials are polarized both electrically

Manuscript received August 27, 2009; revised November 21, 2009; accepted

February 01, 2010. Date of publication May 18, 2010; date of current version

August 05, 2010. This work was supported by Research in Motion and the

National Science and Engineering Research Council of Canada under the

NSERC/RIM Industrial Research Chair Program and the NSERC Discovery

Grant Program.

The authors are with the Electrical and Computer Engineering Depart-

ment, University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail:

lyousefi@uwaterloo.ca; oramahi@ece.uwaterloo.ca).

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/TAP.2010.2050438

line type antenna using a magneto-dielectric material as a sub-

strate was proposed for using in RFID systems. In [4] mag-

neto-dielectric materials were used as a superstrate to increase

the gain of a patch antenna leading to lower profile in compar-

ison to antennas with dielectric superstrates. In [7] magneto-di-

electric materials were used as the substrate of a mushroom-

type electromagnetic band gap structure (EBG). The results in

[7] showed that using magneto-dielectrics could increase the

in-phase reflection bandwidth of EBGs when they are used as

artificial magnetic ground planes.

For low-loss applications in the microwave region, natural

material choices are limited to nonmagnetic dielectrics. When

requiring relatively high permeability, the choices are limited to

ferrite composites which provide high levels of magnetic loss

[9]–[12]. Therefore, artificial magnetic materials are designed

to provide desirable permeability and permittivity with man-

ageable loss at these frequencies [13]–[19]. In spite of double

negative metamaterials which are designed to provide negative

permeability and permittivity, the goal of designing artificial

magnetic materials is to provide an enhanced positive value for

permeability.

The idea of using the split-ring as an artificial magnetic par-

ticle was introduced first in [20]. The works on realizing such

a media started in the late 1990’s [13]–[15]. Since then, engi-

neers have proposed numerous types of inclusions [16]–[19].

The single and coupled split ring resonators (SRR), modified

ring resonators, paired ping resonators, metasolenoid [16], split

square spiral configuration [18], are some of the most popular

configurations used in previous works. Each proposed structure

provides its own advantages and disadvantages in terms of re-

sultant permeability and dissipation. For example, in [16] it was

shown that the metasolenoid configuration provides higher per-

meability in comparison to SRR configurations, or using split

square spiral configuration results in artificial magnetic mate-

rials with smaller unit cells when compared to SRR and meta-

solenoid [17], [18].

One of the most important applications of artificial magnetic

materials is implementing miniaturized planar structures, spe-

cially miniaturized microstrip antennas [18], [19], [21]–[24].

Although it was shown that by using these materials, significant

miniaturization factors could be achieved in the planar sizes of

the antennas, the height of the substrate is limited by the size of

theunit cell of theartificial structures. Thesize of thedeveloped

artificial unit cells are typically much smaller than the wave-

length(smallerthan

),yettheyyieldalargeantennaprofile

(for example, for a microstrip antenna operating at 200 MHz,

the smallest profile achieved is 2 cm [18]). Therefore, miniatur-

izing the unit cell of artificial materials not only provides better

homogeneity, but also decreases the antenna profile.

0018-926X/$26.00 © 2010 IEEE

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YOUSEFI AND RAMAHI: ARTIFICIAL MAGNETIC MATERIALS USING FRACTAL HILBERT CURVES2615

With the aim of further miniaturization, other sub-wave-

length particles have been recently proposed such as spiral

resonators [17], [18], capacitively loaded embedded-circuit

particles [25]–[27], as well as lumped-element based metama-

terials [28]. In [17], it was shown that using spiral resonators

decreases the size of the inclusion by a factor of 2 when

compared to SRRs (split ring resonators). In [25]–[27], using

parallel-plate capacitors, inclusions as small as

oped. (Note that in this paper, by

the host dielectric at the resonance frequency. Since by using

a high-k dielectric, the size of the inclusion decreases, and

therefore, it is not useful to compare different inclusions based

on wavelength in the air.) However; the structures in [25]–[27]

are three dimensional and cannot be realized using printed

circuit board technology. Using lumped elements within the

inclusions, unit cells as small as

The solution proposed in [28] is promising, however, lumped

elements need to be soldered into place for each unit cell which

makes the fabrication process difficult and time-consuming.

In [18], using square spirals unit cells as small as

developed and used for antenna miniaturization. The structure

introduced in [17], can be realized by simple printed tech-

nology. In this paper, we introduce fractal curves as inclusions

for artificial magnetic material to further increase the miniatur-

ization potential.

Previous works on using fractal geometries in developing

artificial structures included frequency selective surfaces [29],

high-impedance surfaces [30]–[35], left-handed metamaterials

[36], and complementary split-ring resonators [37]. An ex-

tended class of space-filling wire structures based on grid-graph

Hamiltonian paths and cycles has also been investigated in [38].

In this work, the use of Fractal curves to miniaturize artificial

magnetic materials is investigated. This idea was proposed for

the first time as a conference paper in our pervious work [39].

Combining the square spiral loop configuration with fractal

Hilbert curves, a new configuration is proposed to realize fur-

ther miniaturization for artificial particles. It is shown that by

using fourth-order fractal Hilbert curves, inclusions as small as

can be realized. Using higher order Hilbert curves results

in even smaller unit cells. Analytical formulas for design and

analysis of the proposed structures are presented. Full-wave

numerical characterization and experimental testing is carried

out to validate the analytical results. The electromagnetic

behavior of the proposed structures are investigated from the

point of view of size reduction, maximum value of the resultant

permeability, magnetic loss, and frequency dispersion.

The organization of this paper is as follows: In Section II, the

new inclusions based on fractal Hilbert curves are introduced

and analyzed. Analytical formulas for design and analysis

of the proposed structures are presented, and full-wave nu-

merical characterization is performed to verify the analytical

results. Furthermore, using the analytical model and numerical

simulation results, the proposed inclusions are compared to

spiral inclusions. In Section III, second-order Fractal Hilbert

inclusions were fabricated and characterized. For experimental

characterization, we have used a new method reported in our

prior work [40]. Summary and conclusion are provided in

Section IV.

are devel-

we mean the wavelength in

were developed in [28].

were

Fig. 1. (a) SRR. (b) Square spiral. (c) Second-order Fractal Hilbert inclusion.

(d) Third-order Fractal Hilbert inclusion. (e) Fourth-order Fractal Hilbert in-

clusion. Note that as the order of Hilbert curve increases, the size of inclusion

decreases.

II. ARTIFICIAL MAGNETIC MATERIALS BASED ON FRACTAL

HILBERT CURVES

Fig. 1 shows the proposed structures for metamaterial inclu-

sions based on Hilbert curves, along with the split ring res-

onators(SRR)andspiralstructures.Usingcircuitmodels,itwas

shown in [17] that the effective capacitance of the spiral config-

uration is four times of that of the SRR. Therefore, spirals can

reduce the resonancefrequency of the inclusion by a factor of 2.

This claim was verified numerically and experimentally in [17].

The structures proposed in this work combine the idea of using

spiralconfiguration (proposed in[17])and FractalHilbert curve

to provide inclusions with smaller size.

The dimensions of the inclusions as a fraction of the wave-

length in the dielectric at the resonance frequency are shown in

Fig.1.Clearlyobservedisthatwhenusingafourth-orderFractal

Hilbert curve makes it possible to realize inclusion as small as

. This size is 63% of the size of spiral inclusion and 32%

of the size of SRR.

A. Analytical Model

Fig. 2 shows a unit cell of a third-order Hilbert inclusion. In

what follows, a general formulation, which can be used for any

order of Hilbert inclusions, is derived for the effective perme-

ability. The unit cell in Fig. 2 has dimensions of

inthex,y,andzdirections,respectively.Anappliedexternal

magnetic field

in the y direction induces an electromotive

,, and

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2616IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 8, AUGUST 2010

Fig. 2. A unit cell of engineered magnetic substrate composed of inclusions

with 3rd order Hilbert Curve.

force

the induced

in the ring. Applying Faraday’s law, one can obtain

as

(1)

where

duced magnetic field,

enclosed by the inclusion. Since the dimensions of the unit cell

inartificialmaterials aresmallincomparisontothewavelength,

weassumeauniformmagneticfieldovertheareaoftheunitcell.

is the frequency of the external field,

is the induced current, and

is the in-

is the area

The generated

and the induced current as

is related to the impedance of the rings

(2)

where

Equating (1) and (2) yields

is the impedance of the metallic inclusion.

(3)

From (3), it is observed that the effective inductance,

inclusions can be defined as

of the

(4)

The magnetic polarization,

induced magnetic dipole moments can be expressed as

defined as the average of the

(5)

Using (3) and (5), the relative permeability

is obtained as

(6)

It should be noted that the substrate whose unit cell is shown

in Fig. 2 can provide magnetic moment vectors only in the di-

rection perpendicular to the inclusion surface (i.e., the effective

permeability formulated in (6) represents the yy component of

thepermeabilitytensor).Forx-directedandz-directedmagnetic

fields, the effective permeability will be equal to that of the host

media which is unity for nonmagnetic substrates. Therefore, the

artificial substrate will be anisotropic with permeability tensor

of

(7)

To achieve isotropic artificial substrates with the same effec-

tive permeability in all the three directions of x, y, and z, two

inclusions with the surfaces perpendicular to the x, and z direc-

tions should be added to the unit cell shown in Fig. 2. If these

two inclusions have the same structure as the inclusion perpen-

dicular to the y direction, the same effective permeability for-

mulated in (6) will be achieved in the x, and z directions.

Theformulagivenin(6)isgeneralandcanbeusedforHilbert

inclusions of any order. The only difference between various

orders would be in the value of the inclusion impedance,

and the effective inductance,

The effective inductance,

area

, enclosed by the inclusion; and since this area varies by

the order,

, of Fractal Hilbert curve, the effective inductance

would be dependent on .

,

.

given in (4) is a function of the

is given as

(8)

where

(see Fig. 2). Equation (8) can be proved using mathematical

induction principle [41].

The inclusion impedance,

which models the ohmic loss of the metallic inclusions due to

thefiniteconductivityofthestrips,andtheotherpartmodelsthe

mutual impedance between external and internal loops,

, can be calculated using the Ohm’s law

is the dimension of the inclusion in the x direction

, consists of two parts:,

.

(9)

where

is the width of metallic strips and

the metallic strips.

andare the conductivity and skin depth, respectively,

is the total length of

is given as

(10)

Equation (10) can be proved using mathematical induction

principle [41]. In the proposed inclusions, the internal loop

follows the shape of the external loop, therefore the mutual

impedance between external and internal loops

modeled as the per-unit-length mutual impedance of the strips,

, times the average length of the strips,

can be

(11)

where

(12)

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YOUSEFI AND RAMAHI: ARTIFICIAL MAGNETIC MATERIALS USING FRACTAL HILBERT CURVES2617

and

impedance,

nique for the coplanar strip lines [42]

is the gap between the metallic strips. The per-unit-length

is calculated using conformal mapping tech-

(13)

where

gent of the host substrate, respectively. The impedance given

in (13) contains a parallel combination of a capacitance,

and a conductance,to model the capacitance between the

metallic strips and the loss due to the conductivity of the host

substrate.

It should be noted that this model considers the capacitance

betweentwoadjacentstripswhileitneglectsthecapacitancebe-

tweennon-adjacentstrips.Therefore,theaccuracyofthismodel

increases when the space between non-adjacent strips is much

larger than the space between adjacent strips. However, for the

compact structures or for high order fractal curves, where the

space between non-adjacent strips is comparable to the space

betweenadjacentstrips,thismodelgivesonlyanapproximation

of the relative permeability that can be used to initiate a design.

To extract exact relative permeability, one needs to use the full

wave simulation method as explained below in Section II-B.

The proposed analytical formula developed in (6) fits the

Lorentz model [43]

andare the relative permittivity and the loss tan-

(14)

if the following substitutions are made:

(15)

Here, the

typical host substrate, several orders of magnitude smaller than

one.

The approach introduced in this section for deriving relative

permeability of Hilbert inclusions can be used for square spi-

rals too, and results in the same formula derived in (6). The

only difference would be in the value of the effective lumped

elements,

, , and. Equation (16) illustrates the pa-

rametersthatshouldbereplacedsothat(6)canbeusedforspiral

inclusions.

term in (13) has been ignored since it is, for a

(16)

TABLE I

COMPARISON OF THE FRACTAL SPIRAL INCLUSIONS WITH NON-FRACTAL

SPIRAL INCLUSION

Using the above formulas, Table I compares the effective in-

ductance, capacitance, resistance, and the resonance frequency

of Hilbert inclusions with those parameters of the square spiral.

The electrical parameters of the host substrate and the geomet-

rical parameters of the unit cell and metallic strips are assumed

to be the same for all the Hilbert inclusions and spiral.

From Table I, using Fractal Hilbert inclusions instead of

square spirals reduces the effective inductance while increasing

the effective capacitance in such a way that the resonance

frequency would be smaller for Fractal Hilbert curves with

. Therefore, using structures with Hilbert curves of order

3 or higher results in miniaturization of inclusions. We also

observe that the miniaturization factor increases with the order

of Hilbert curves.

Table I also gives higher value of effective resistance for

Hilbert inclusions when compared to spirals. Increasing the

effective resistance in the circuit model results in lower

factor, which in turn results in lower rate of change in the

resultant permeability with respect to frequency. Therefore,

lower frequency dispersion can be achieved for the artificial

medium. On the other hand, any increase in the effective

resistance results in an increase in the magnetic loss of the

artificial medium. Furthermore, since the effective inductance

is reduced, it is expected that the maximum value of the relative

permeability would be smaller for Hilbert inclusions when

compared to spiral.

Asaconclusion,theanalyticalmodelsdevelopedherepredict

thatusingFractalHilbertcurveswithorder3orhigherresultsin

miniaturization of the inclusions, and, furthermore, in a reduc-

tion of the frequency dispersion of the artificial medium. The

drawbacks would be lower resultant permeability and higher

magnetic loss. This conclusion will be verified in part C using

full wave numerical simulation.

B. Numerical Full Wave Analysis

In order to verify the accuracy of the analytical model devel-

opedabove,afull-wavenumericalanalysissetupisdevelopedin

this section to characterize the proposed engineered materials.

The simulation setup is shown in Fig. 3.

In the simulation setup, periodic boundary conditions are

used in the z and y directions around a unit cell to model an

infinite slab, and perfect matched layers absorbing boundaries

are used in the x direction to prevent reflections from the com-

putational domain walls. For characterization, the well known

plane wave analysis method is used [44]–[46]. In this method,

the effective parameters,

and, are extracted from the

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2618IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 8, AUGUST 2010

Fig. 3. Simulation setup used for numerically characterization of metamate-

rials.

reflection and transmission coefficients. Equations (17)–(20)

give the relationship between the effective parameters and the

transmission and reflection coefficients [44]–[46].

(17)

(18)

(19)

(20)

where

number, respectively;

characteristic impedance, refractive index, permittivity, and

permeability of the engineered material, respectively;

are the reflection and transmission coefficients, and d is the

thickness of the engineered material. The sign in (17), and

(18) is determined by the requirements that

for passive media. The numerical process was

carried out based on the finite element method using Ansoft

HFSS10 full-wave simulation tool.

andare the air characteristic impedance and wave

,,, and are the effective

and

and

C. The Resultant Permeability

Using the aforementioned analytical and numerical methods,

the relative permeability of the proposed structures in Fig. 1

was derived and the results are shown in Figs. 4, 5. In these

figures, the resultant permeability of Hilbert curve structures

are compared to that of the square split spiral proposed in [17],

[18].Theparametersand dimensionsofallthestructuresare the

same and are given as:

, host dielectric of

Fig. 4. Real part of resultant permeability for engineered materials with inclu-

sions shown in Fig. 1. Analytical results (solid line) are compared with numer-

ical results (dash line).

Fig. 5. Imaginary part of resultant permeability for engineered materials with

inclusions shown in Fig. 1. Analytical results (solid line) are compared with

numerical results (dash line).

(,),,,

, and the strips are assumed copper to

model the loss.

Figs.4and5showtherealandimaginarypartsoftheresultant

permeability,computedanalytically andnumerically.As shown

in these figures, for inclusions with 3rd or 4th order Hilbert

curves,which resonate at lower frequencies, a strong agreement

is observed between the analytical and numerical results. How-

ever; for the inclusions with spiral or 2nd order Hilbert curves,

which resonate at higher frequencies, a frequencyshift reaching

a maximum value of 8.8% for the case of 2nd order Hilbert is

observedbetweenthenumericalandanalyticalresults.Sincethe

analytical model has been derived based on the assumption of

field uniformity throughout the unit cell, the smaller the inclu-

sionincomparisontothewavelength,themoreaccuratetheana-

lytical model is. Therefore, the numerical simulations for inclu-

sions based on the 3rd or 4th order Hilbert curves are expected

to give closer agreement with the analytical model in compar-

ison to inclusions based on lower order Hilbert curves.

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YOUSEFI AND RAMAHI: ARTIFICIAL MAGNETIC MATERIALS USING FRACTAL HILBERT CURVES 2619

Fig. 6. Imaginary part of the resultant permeability at frequencies below the

resonance. The frequency is normalized to the resonance frequency.

As shown in Fig. 4, and Fig. 5, for

exhibit lower resonance frequency compared to the split spiral

with the same size. Therefore, as predicted by the circuit model,

for a given resonance frequency, using Hilbert inclusions with

orderof3orhigherresultsinasmallersizefortheunitcellcom-

pared to the split spiral. Furthermore, as shown in Fig. 4, when

the order of the fractal Hilbert curve increases, the response of

the permeability becomes smoother. Therefore, as predicted by

the circuit model, using fractal structures decreases the rate of

the permeability variation with respect to frequency leading to

lower dispersion in the artificial substrate.

From Fig. 4, the maximum value achievable for the resultant

permeabilityissmallerthanthatofthespiral,andthismaximum

valuedecreasesastheorderofHilbertinclusionsincreases.This

fact was also predicted by the circuit model as explained in

Section II-A. In terms of loss comparison, as shown in Fig. 5,

at the resonance frequency Hilbert inclusions yield lower value

for theimaginary part,so theyprovidelowerloss attheresonant

frequency. Notice that it is difficult to compare the imaginary

parts at the other frequencies using the scale shown in Fig. 5.

For this, we have presented the imaginary part of the perme-

ability at the frequencies below the resonance in Fig. 6. In this

figure, the numerical results are used, and for the sake of com-

parison, the frequency is normalized to the resonance frequency

for all inclusions. As shown in Fig. 6, for frequencies below

resonance, the imaginary part of the permeability is higher for

Hilbert inclusions than that of the spiral inclusion, and as the

order of Hilbert increases the imaginary part increases. There-

fore, as predicted by the circuit model, at the frequencies below

the resonance Hilbert inclusions provide higher loss when com-

pared to spirals.

, fractal structures

III. EXPERIMENTAL RESULTS

To verify our analytical and numerical results, an artificial

magnetic material composed of 2nd order Hilbert inclusions

was fabricated and characterized. A unit cell of this structure,

and its dimension are shown in Fig. 7.

To measure the constitutive parameters of the artificial sub-

strate, we use a novel microstrip-line based characterization

Fig. 7. One unitcell ofthe fabricatedartificialsubstrate, ? ? ? ? ???????,

?

? ?

? ?

? ?

? ?

? ?

? ???? ??, ?

????? ??, ?? ? ?? ? ?? ??.

? ???? ??, ?? ?

Fig. 8. A single strip containing 6 unit cells of inclusions fabricated using

printed circuit board technology.

method reported in [40]. In this method, the permeability of

the artificial substrate is extracted by measuring the input

impedance of a shorted microstrip line which is implemented

over the artificial substrate [40]. The advantage of this method

over previously developed techniques is that the characteriza-

tion can be performed by using a simple inexpensive fixture.

Furthermore, no sample preparation is needed for characteriza-

tion, and the same substrate designed for any microstrip device

can be used for characterization as well.

Using printed circuit technology, a strip of 6 unit cells of

Fractal Hilbert2 inclusions was fabricated on an FR4 substrate

with

and

strips were then stacked in the y direction to provide a three-di-

mensional substrate. Due to the thickness of the metal inclu-

sions, an average air gap of 50

in the stacking process. The air space, while unavoidable in

the fabrication process, is nevertheless measurable so it can be

easily included in the design.

(See Fig. 8). Forty of these

develops between the strips

A. Permittivity Measurement

Using the engineered substrate and two conducting plates,

a parallel-plate metamaterial capacitor was fabricated, and by

measuring its capacitance, the permittivity of the artificial sub-

strate was calculated [18].

According to the classical image theory, using only one

period of the artificial unit cells, in the area between the two

metallic parallel plates, can mimic the behavior of an infinite

array of unit cells, which is the default assumption in the

analysis and design of artificial structures. This property, there-

fore, makes the method presented in [18] highly robust and

well-suited for metamaterial characterization. It is interesting

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2620IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 8, AUGUST 2010

Fig. 9. Simulation results for extracted permittivity (real part).

to note, however, that the authors of [18] observed a rather large

difference between the analytically estimated and measured

permittivity values (a difference of more than 30%). The large

discrepancy is in fact due to the approximations used in the

derivation of the analytical formula. For the same artificial

structure measured in [18], we have performed a full wave

numerical simulation using the simulation setup discussed in

Section II-B and obtained a difference between simulation and

measurements of less than 7%.

Usingtheaforementionedmethod,thex-directedpermittivity

for this artificial magnetic substrate was measured, at the low

frequency of 10 MHz, as

ulation setup explained in Section II and Ansoft HFSS10, the

permittivity of the artificial magnetic substrate was numerically

calculated as

, thus showing good agreement with

measurement.Inthesimulation,the50

However; the simulation results show that the resultant permit-

tivity changes with frequency in such a way that at the reso-

nance frequency of 630 MHz, the permittivity decreases to 7.6

(see Fig. 9). The method used in [40] is not suitable to mea-

sure the permittivity as a function of frequency; however, since

the measurement results at low frequency are close to the sim-

ulation result, the subsequent calculation of the permeability is

expected to yield a comparable level of accuracy.

[40]. By using the sim-

airgapwasincluded.

B. Permeability Measurement

The same substrate that is used for permittivity measurement

is used as a substrate of the shorted microstrip line to extract

the permeability. The fabricated fixture used for permeability

measurement is shown in Fig. 10. The fabricated substrate has

dimensions of 12, 8.2, and 1.1 cm in the y, z, and x directions,

respectively. For the quasi-TEM dominant mode, the

fields in the substrate will be in the y and x directions, re-

spectively.Thereforethis configurationcan be used for retrieval

of

.

Using a vector network analyzer, the complex input

impedance of the shorted microstrip line shown in Fig. 10

is measured over the frequency range of 500–680 MHz. Then

using the data shown in Fig. 9 for

is extracted by the method reported in [40]. The real and

and

and measured impedance,

Fig. 10. The fabricated fixture used for permeability measurement.

Fig. 11. The measured and numerically simulated real part of the permeability

for the artificial magnetic material shown in Fig. 10.

Fig. 12. The measured and numerically simulated imaginary part of the per-

meability for the artificial magnetic material shown in Fig. 10.

imaginary parts of the measured y-directed permeability are

shown in Fig. 11 and Fig. 12. The measurement data shown in

these figures are from [40]. In these figures, the measurement

results are compared with the numerical simulation results.

The 50

air gap was also included in the simulation. Unfor-

tunately this air gap cannot be modeled within the analytical

formulas presented in Section II. The analytical model, which

dose not consider the air gap, results in a resonant frequency

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YOUSEFI AND RAMAHI: ARTIFICIAL MAGNETIC MATERIALS USING FRACTAL HILBERT CURVES2621

of 520 MHz, corresponding to a 16% shift when compared to

the simulation results that included the air gap. A similar shift

in the resonance frequency due to the air gap has also been

reported in pervious works [18].

The shaded area in Fig. 11 and Fig. 12 determines the fre-

quenciesoverwhichtherealpartofthepermeabilityisnegative.

As explained in [40], over this frequency range corresponding

to frequencies higher than 638 MHz, the measurement results

are not valid due to the restriction of the formulas used for mi-

crostrip effective permeability in [40]. As explained in [40],

the formulas exist in literature for microstrip effective perme-

ability are derived using conformal mapping technique where

the permeability was assumed positive. Therefore, these equa-

tions cannot be used at over the frequency range where the per-

meability is negative. However, since artificial magnetic mate-

rials are designed to operate at frequencies over which the per-

meability is positive, characterization of the permeability be-

havior at those frequencies would be sufficient for application

purposes. Over the frequency range where the real part of the

permeability is positive, good agreement is observed between

the simulation and measurement results.

As briefly discussed above and more extensively in [40], in

the numerical analysis, periodic boundary conditions are used

to mimic an infinite number of unit cells. However, in practice

we can only realize a finite number of unit cells. For example in

the setup used in thiswork (see Fig. 10),the fabricatedsubstrate

contains 6 unit cells of inclusions in the z direction and only

one unit cell in the x direction. Therefore; we do not expect a

very strong agreement between simulation and measurements.

By analyzing the induced magnetic field distribution within the

inclusion, we observed that at the resonance frequency, a strong

magnetic field is present mainly at the center of the inclusion.

However, at frequencies below resonance, we observe a rela-

tively weak field at the sides of the inclusion (the sides normal

to the x-direction in Fig. 9). Accordingly, at the resonance fre-

quency the adjacent unit cells in the x direction have a minor ef-

fectontheresultantpermeability,whileatthefrequenciesbelow

resonance, this effect is not negligible. This observation could

potentially explain the fact that we have good agreement be-

tween simulation and measurement at the resonance frequency,

but the agreement becomes weaker for frequencies below reso-

nance (see Fig. 11).

It is most likely that increasing the number of unit cells in the

x-direction provides higher homogeneity in the fabricated sub-

strate which will resultin a better agreement between numerical

and measurement results. On the other hand, in a wide class of

applications such as antenna miniaturization, only one unit cell

is used in the x direction [21]–[24], and therefore, the measure-

ment results would be of strong relevance.

IV. CONCLUSIONS

The use of Fractal curves to miniaturize artificial magnetic

materials was investigated. Novelconfigurationswere proposed

to realize artificial magnetic materials with smaller unit cells

by combining the square split loop configuration with Fractal

Hilbert curves. The new designs are desirable when designing

low profile miniaturized antennas in which the engineered mag-

neticmaterialsareusedasthesubstrate.Analyticalmodelswere

introduced to design and analyze the new structures. The ana-

lytical model was validated through both full wave simulation

andexperimentalcharacterization.Itwasshownthatusingforth

order of Fractal Hilbert curve, it is possible to realize inclusions

as small as 0.014 of the wavelength in the dielectric. This size

is 63% of the size of spiral inclusion and 32% of the size of

SRR. Using higher order Hilbert curves results in even further

miniaturization of the unit cell. In terms of the electromagnetic

properties, the new structures provide lower frequency disper-

sionandlowermagneticlossattheresonancefrequencyincom-

parison to the simple square spiral inclusions. This advantage

comes at the expense of lower permeability and higher mag-

netic loss at frequencies below resonance.

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Leila Yousefi (M’09) was born in Isfahan, Iran, in

1978. She received the B.Sc. and M.Sc. degrees

in electrical engineering from Sharif University of

Technology, Tehran, Iran, in 2000 and 2003, respec-

tively, and the Ph.D. degree in electrical engineering

from the University of Waterloo, Waterloo, ON,

Canada, in 2009.

Currently she is working as a Postdoctoral Fellow

at the University of Waterloo. Her research interests

include metamaterials, miniaturized antennas, elec-

tromagneticbandgap structures,and MIMO systems.

Omar M. Ramahi (F’09) received the B.S. de-

grees in mathematics and electrical and computer

engineering (summa cum laude) from Oregon State

University,Corvallis,andthe M.S.andPh.D.degrees

in electrical and computer engineering from the

University of Illinois at Urbana-Champaign.

From 1990–to 993, he held a visiting fel-

lowship position at the University of Illinois at

Urbana-Champaign. From 1993 to 2000, he worked

at Digital Equipment Corporation (presently, HP),

where he was a member of the alpha server product

development group. In 2000, he joined the faculty of the James Clark School of

Engineering, University of Maryland at College Park, first as an Assistant Pro-

fessor, later as a tenured Associate Professor, and where he was also a faculty

member of the CALCE Electronic Products and Systems Center. Presently,

he is a Professor in the Electrical and Computer Engineering Department and

holds the NSERC/RIM Industrial Research Associate Chair, University of

Waterloo, ON, Canada. He holds cross appointments with the Department of

Mechanical and Mechatronics Engineering and the Department of Physics and

Astronomy. Previously, he served as a consultant to several companies and was

a co-founder of EMS-PLUS, LLC and Applied Electromagnetic Technology,

LLC. He has authored and coauthored over 240 journal and conference papers.

He is a coauthor of the book EMI/EMC Computational Modeling Handbook

(Springer-Verlag, 2001).

Prof. Ramahi serves as an Associate Editor for the IEEE TRANSACTIONS ON

ADVANCED PACKAGING and as the IEEE EMC Society Distinguished Lecturer.

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