IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 12, DECEMBER 2007 2865
Equivalent-Circuit Models for the Design of
Metamaterials Based on Artificial
Magnetic Inclusions
Filiberto Bilotti, Senior Member, IEEE, Alessandro Toscano, Member, IEEE, Lucio Vegni, Member, IEEE,
Koray Aydin, Student Member, IEEE, Kamil Boratay Alici, and Ekmel Ozbay
Abstract—In this paper, we derive quasi-static equivalent-circuit
models for the analysis and design of different types of artificial
magnetic resonators—i.e., the multiple split-ring resonator, spiral
resonator, and labyrinth resonator—which represent popular in-
clusions to synthesize artificial materials and metamaterials with
anomalous values of the permeability in the microwave and mil-
limeter-wave frequency ranges. The proposed models, derived in
terms of equivalent circuits, represent an extension of the
models presented in a recent publication. In particular, the ex-
tended models take into account the presence of a dielectric sub-
strate hosting the metallic inclusions and the losses due to the finite
conductivity of the conductors and the finite resistivity of the di-
electrics. Exploiting these circuit models, it is possible to accurately
predict not only the resonant frequency of the individual inclu-
sions, but also their quality factor and the relative permeability of
metamaterial samples made by given arrangements of such inclu-
sions. Finally, the three models have been tested against full-wave
simulations and measurements, showing a good accuracy. This re-
sult opens the door to a quick and accurate design of the artifi-
cial magnetic inclusions to fabricate real-life metamaterial samples
with anomalous values of the permeability.
Index Terms—Artificial magnetic inclusions, labyrinth res-
onators, metamaterials, miniaturization, multiple split-ring
resonators, split-ring resonators.
I. INTRODUCTION
ARTIFICIAL materials exhibiting anomalous values of thepermeability, (e.g., materials with values of the relative
permeability greater than one [1], with negative values of the
permeability: the so-called mu-negative materials [2], with a
zero value of the permeability, with an absolute value of the
relative permeability less than one: the so-called mu-near-zero
materials [3]) are of interest in many applications at different
frequency ranges.
Such materials are usually obtained at microwaves by
printing suitable metallic resonating inclusions on supporting
dielectric boards and stacking the boards to form a medium.
Manuscript received May 2, 2007; revised July 27, 2007. This work was sup-
ported by the European Union under the Network of Excellence METAMOR-
PHOSE.
F. Bilotti, A. Toscano, and L. Vegni are with the Department of Applied
Electronics, University of “Roma Tre,” Rome 00146, Italy (e-mail: bilotti@
uniroma3.it).
K. Aydin, K. B. Alici, and E. Ozbay are with the Department of Physics and
the Department of Electrical and Electronics Engineering, Nanotechnology Re-
search Center, Bilkent University, Bilkent 06800, Turkey.
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.2007.909611
The inclusions react to the impinging magnetic field, and
provided that the dimensions of and the separation between the
inclusions are small compared to the wavelength, the magnetic
response of the artificial material can be expressed in terms of
a macroscopic permeability function [4]–[6].
In most cases, the main reason to employ artificial magnetic
metamaterials in the design of transmissive and radiating mi-
crowave components is related to the possibility of squeezing
the component dimensions [6]–[10] by using electrically small
samples of artificial magnetic materials (planar slabs, cylin-
drical shells, rods, etc.). Since in some cases [6]–[9] there is
no theoretical limitation for the achievable miniaturization
(i.e., the dimensions of the materials can be, in principle,
even infinitesimal), the only intrinsic limit is represented by
the dimensions of the inclusions used to fabricate the needed
metamaterial samples. However, the most common type of
resonator used to achieve anomalous values of the permeability
at microwaves, the split-ring resonator [11], exhibits a physical
dimension of the order of , which may represent, thereby,
a significant limitation.
In order to overcome this drawback and reduce the dimen-
sions of the resonant inclusions, multiple split-ring and spiral
resonators may be used. In [12], we have presented suitable
circuit models for a quick design of both of these artifi-
cial magnetic inclusions. Such models are limited to the ideal
cases of: 1) absence of losses and 2) resonators immersed in air.
Moreover, the models presented in [12] are able to accurately
predict only the resonant frequency of the individual inclusions
without giving any information about their quality factor, which
may give a good indication of the bandwidth of operation of the
metamaterial constituted by those inclusions. Finally, the
circuit models proposed in [12] do not give the designer any in-
formation on how to get a specific value of the real part of the
permeability needed to fabricate a given component and on the
losses related to the imaginary part of the permeability in real-
istic layouts.
The aim of this paper is to present a significant extension
of the work done in [12], proposing complete circuit models
that also take into account the presence of a dielectric substrate
where the inclusions are printed on and the losses both in the
dielectrics and metallic conductors. Moreover, in this paper,
we compare the analytical results obtained from the presented
models and both the full-wave simulations and measurements.
On the other hand, when moving towards higher microwave
frequencies, miniaturization of the inclusions might not be an
0018-9480/$25.00 © 2007 IEEE
2866 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 12, DECEMBER 2007
issue any more. Sometimes, indeed, miniaturization has to be
avoided, due to the technological limitations to print sufficiently
small inclusions. In this case, new concepts have to be employed
in the design of the inclusions and a possible solution consists in
the employment of the so-called labyrinth resonators [13], i.e.,
inclusions characterized by multiple splits along the ring strips.
The labyrinth resonator has been experimentally investigated in
[13], while in [14], for the first time, we have derived an accurate
analytical model of this type of inclusion based again on a suit-
able resonant circuit. Also in this case, the model is limited
to a labyrinth resonator immersed in air and made by an ideal
conductor. In this paper, we extend the formulation presented
in [14] and propose a complete model, which takes into account
the presence of the dielectric substrate and the losses in the con-
ductors and dielectrics. Also in this case, the analytical model is
tested through a proper comparisons with full-wave numerical
results and the measurements.
This paper is organized as follows. In Section II, we present
the extended analytical models of the multiple split-ring, spiral,
and labyrinth resonators in terms of suitable equiva-
lent circuits, validated through the comparison with proper
full-wave numerical simulations and experimental measure-
ments. In Section III, we analytically derive the quality factor
of the isolated inclusions proposed in Section II. A discussion
of the quality factor is also conducted with the perspective of
the possible applications in real-life artificial magnetic metama-
terials. Finally, an analytical representation of the macroscopic
permeability function of a given arrangement of resonators
is derived and compared with the permeability numerically
retrieved through the scattering parameters [15], [16].
II. MULTIPLE SPLIT-RING, SPIRAL, AND LABYRINTH
RESONATORS: ANALYTICAL MODEL, DESIGN, AND
EXPERIMENTAL VERIFICATION
A. Multiple Split-Ring Resonators
In [12], it has been shown that the electromagnetic behavior
in the quasi-static regime (i.e., when the inclusions are much
smaller than the operating wavelength, which is, indeed, the
case when artificial magnetic inclusions are used to fabricate
metamaterial samples) of the lossless multiple split-ring res-
onator immersed in air is described by the equivalent series
circuit depicted in [12, Fig. 1(b)].
The presence of a dielectric substrate, upon which the mul-
tiple split-ring resonator is printed [see Fig. 1(a)], does not affect
the inductance. Therefore, its expression is the same as the one
reported in [12]
(1)
where is the vacuum permeability, is the side length of the
external ring, is the width of the strips, is the separation
between two adjacent strips,
is the average strip length calculated over all the rings, and
is the so-called
filling ratio [12].
Fig. 1. (a) Sketch and geometrical dimensions of a multiple split-ring res-
onator. (b) Quasi-static equivalent circuit model of the multiple split-ring
resonator depicted in (a).
The expression of the capacitance is also given by the formula
derived in [12]
(2)
but this time, the per-unit-length capacitance between two par-
allel strips having width and separation in the presence of a
dielectric substrate of height and relative permittivity dif-
fers from the one given in [12] and reads
(3)
where is the vacuum permittivity, is the complete elliptic
integral of the first kind, , and the effective rela-
tive permittivity related to the dielectric filling the substrate
is given by
(4)
BILOTTI et al.: EQUIVALENT-CIRCUIT MODELS FOR DESIGN OF METAMATERIALS BASED ON ARTIFICIAL MAGNETIC INCLUSIONS 2867
This formula has been heuristically determined through a
proper fit of different sets of numerical data under the assump-
tion that the substrate thickness is not very large compared
to the inclusion details (i.e., .1 In
particular, four constraints have been imposed to derive (4),
which are: 1) when , the effective
permittivity is expected to be a linear function of ; 2) a
further increase of the substrate thickness should correspond
to a smoother growing of the effective permittivity; 3)
should go to 1 when the thickness of the dielectric substrate
vanishes; and 4) should go to 1 when the substrate is filled
by air. Formula (4) satisfies all four of these constraints in the
range of validity.
Although mutual interactions between nonadjacent segments
and rings are neglected, (1) and (2) describe in an accurate
manner the behavior of the multiple split-ring resonator in the
quasi-static limit when perfect conductors and dielectrics are
considered [12].
In the case of lossy conductors and dielectrics, the correct
equivalent circuit of the multiple split-ring resonator is now the
one reported in Fig. 1(b). We added a series resistance
to take into account the losses in the conductor and a shunt resis-
tance to describe the losses in the dielectric substrate.
The equivalent resistance is in series with the induc-
tance and, thus, should be calculated along the same
domain (i.e., the equivalent loop described in [12]). Inductance
in (1) can be cast in the form ,
where is a per-unit-length inductance and
is the average length of the loop with being a
correction function depending on the filling ratio. Therefore, the
total series resistance can be cast in the same form as
, where is the per-unit-length resis-
tance, with being the electrical resistivity of the metal and
being the thickness of the metallic strips. Thereby, the final ex-
pression of the total series resistance reads
(5)
The shunt resistance is in parallel with the total ca-
pacitance and can be determined as follows. Let us con-
sider at first the most external pair of rings. Following the dis-
cussion about the current and voltage distribution on the rings
referred in [12], the conductances associated to the two symmet-
rical halves of the ring pair are connected in series to each other.
Therefore, the total conductance of the most external pair of
rings is given by , where is the
total length of the gap between the two rings and is
the per-unit-length conductance, with being the conductivity
of the dielectric substrate. The shunt resistance associated to the
most outer pair of rings is thus given by .
When adding the other inner rings, the shunt resistances associ-
ated to each pair of adjacent rings are all connected in parallel.
However, since these resistances are not equal to each other,
due to the different lengths of the rings, it is not straightfor-
ward to derive a formula for the total equivalent shunt resis-
1When talking about metamaterials, in fact, we should keep in mind that the
separation between the inclusions is assumed to be much smaller than the wave-
length, as it happens also for the inclusion dimensions.
Fig. 2. Resonant frequency of a multiple split-ring resonator as a function of
the number of rings N . In the inset, we show the experimental setup used to
measure the transmission features and, thus, the resonant frequency of the iso-
lated inclusion. Data: ` = 8 mm, w = 0:1 mm, s = 0:1 mm, t = 30 m,
h = 0:2 mm, " = 3:85; tan  = 0:01;  = 0:017 
 10 
m.
tance . We just know that this resistance should have the
form , with being a decreasing func-
tion of . Under the assumption that is not too close to the
maximum number of the rings allowable for a given inclusion,2
a suitable expression for has been determined through a
proper fitting of numerical data as . There-
fore, the final expression for is given by
(6)
Using (1)–(6) and the circuit representation depicted in
Fig. 1(b), it is now possible to analytically calculate the res-
onant frequency of the individual inclusion. The comparison
with the values obtained through the full-wave numerical
results performed through CST Microwave Studio and the
measurements is reported in Fig. 2.
The inset of Fig 2 shows how the measurements have been
performed. As already done in [13] for a different type of res-
onator, the isolated multiple split-ring resonator has been placed
between two electrically small monopole antennas connected
to the HP-8510C network analyzer to measure the transmission
coefficient. The antennas and resonator are arranged in such a
way that the magnetic field produced in the near zone by the an-
tennas excites the multiple split-ring resonator (see the inset of
Fig. 2). In order to calibrate the network analyzer, we measured
at first the transmission spectra in free space (i.e., without the
multiple split-ring resonator unit cell). We then have inserted
the multiple split-ring resonator unit cell between the monopole
antennas, and we performed the transmission measurements by
maintaining the distance between the transmitter and receiver
monopole antennas fixed.
2The reader may find in [12] a quantitative definition of the maximum number
of rings that a given inclusion can host. It should be noticed that the assump-
tion adopted here is in line with the practical employment of the inclusions in
metamaterial samples. As already pointed out in [12] and later in this paper, the
number N of the rings to be used depends on the filling ratio and it is far from
the maximum.
2868 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 12, DECEMBER 2007
The numerical simulation of the behavior of the inclusion has
been performed in two ways, which are: 1) we have considered
a plane wave impinging on the isolated inclusion depicted in
Fig. 1(a) with the impinging magnetic field aligned along the
axis of the magnetic inclusion (in this case, in order to detect
the magnetic resonant frequency of the inclusion, a magnetic
field probe revealing the normal component of the -field with
respect to the multiple split-ring resonator has been placed in
the center of the inclusion) and 2) we have simulated the exper-
imental setup previously described with the two antennas and
the multiple split-ring resonator in the middle). The simulations
performed through these two methods return very close values
of the resonant frequency (the discrepancy is always within a
couple of percentage points).
The good agreement between measured, simulated, and an-
alytical data is well evident from Fig. 2. In addition, in Fig. 2,
the expected saturation of the resonant frequency (see [12]) is
also evident. From the practical point of view, we learn that a
few rings are enough to obtain a good reduction of the resonant
frequency, giving a typical miniaturization rate of the order of
in the linear dimensions of the inclusion.
B. Spiral Resonators
Further miniaturization may be obtained using the spiral res-
onator [17], [18] depicted in Fig. 3(a). Such an inclusion, which
is the planar version of the “Swiss roll” (i.e., a metallic sheet
wound as a coil to form a cylindrical resonator) proposed in
[11] is currently used in several experimental layouts (including
low-profile antennas [19] and ultrathin microwave absorbers
[20]) that foresee the employment of magneto-dielectrics and
mu-negative metamaterials.
In parallel to what has been done for the multiple split-ring
resonator, we have already derived in [12] an accurate
equivalent model for the lossless isolated spiral resonator
immersed in air. The presence of a dielectric substrate affects
only the distributed capacitance between the turns of the spiral
and, thus, the final expressions of the total inductance and
capacitance read as follows:
(7)
(8)
where this time is the number of the turns of the spiral, is
the side length of the external turn, is the width of the strips,
is the separation between two adjacent turns, is defined as
for the multiple split-ring resonator, and
is the average length of the spiral turn.
It is worth noticing that, as in the case of the multiple split-
ring resonator, we have also considered in this case the squared
version of the spiral resonator. Anyway, the same formulation
also applies to the circular counterparts, simply by changing the
length of the straight segments with the length of the circular
ones.
Fig. 3. (a) Sketch and geometrical dimensions of a spiral resonator. (b) Quasi-
static equivalent circuit model of the spiral resonator depicted in Fig. 3(a).
Now assuming the presence of the losses in the metallic con-
ductor and in the dielectric, the equivalent-circuit model of the
spiral resonator is the one depicted in Fig. 3(b). The series re-
sistance taking into account the losses in the conductor has been
determined analogously to the multiple split-ring resonator case
and it is given by
(9)
In the same way, the shunt resistance, which takes into account
the dissipation in the lossy dielectric, can be written as
(10)
Also in this case, from the equivalent circuit depicted in Fig. 3
and (7)–(10), it is possible to analytically derive the resonant
frequency of the spiral resonator. A comparison between the
results obtained using the proposed analytical model, the full-
BILOTTI et al.: EQUIVALENT-CIRCUIT MODELS FOR DESIGN OF METAMATERIALS BASED ON ARTIFICIAL MAGNETIC INCLUSIONS 2869
Fig. 4. Resonant frequency of a spiral resonator as a function of the number
of the turns N . Data: ` = 8 mm, w = 0:1 mm, s = 0:1 mm, t = 30 m,
h = 0:2 mm, " = 3:85; tan  = 0:01;  = 0:017 
 10 
m.
wave simulations carried out through CST, and the measure-
ments is reported in Fig. 4. A good agreement between the dif-
ferent sets of data is observed.
Please also note in this case the saturation behavior obtained
when increasing the number of the turns. Again, there is no need
to fill in the entire free area in the center of the resonator. The
achievable miniaturization here is of the order of , which
is a useful result for a lot of interesting applications.
C. Labyrinth Resonators
When metamaterials with anomalous values of the perme-
ability are to be designed for the higher microwave frequen-
cies, miniaturization is not always desired. To this end, a suit-
able inclusion, called the labyrinth resonator, has been proposed
in [13]. The details on how to derive from the circuit model
the total inductance and capacitance when the inclusion is in
free space and losses are neglected have already been reported
in [14].
The inductance is not affected by the presence of the dielec-
tric substrate [see Fig. 5(a)] and should be the same as in [14].
However, we found out that, in parallel to what happens for the
multiple split-ring and spiral resonators [see (1) and (7)], it is
better to replace the total external length of the inclusion with
the average length of the ring as
(11)
The total capacitance of the labyrinth resonator is given by
two contributions, which are: 1) the distributed capacitance as
for the previous inclusions and 2) the capacitances associated
to the cuts. In the case of the labyrinth resonator, in fact, it is
not advised to neglect the cut capacitances anymore. Since the
effect of the distributed capacitance is very much reduced in
the labyrinth resonator, the length of the gap starts playing
a significant role. The gap capacitance, in fact, may be of the
same order of magnitude of the total distributed capacitance.
Considering the series and parallel connections of the capac-
itances in Fig. 5(b), the first contribution is given by the sum of
Fig. 5. (a) Sketch and geometrical dimensions of a labyrinth resonator.
(b) Quasi-static equivalent-circuit model of the labyrinth resonator depicted
in (a).
the distributed capacitances between any pair of adjacent rings
as
(12)
where this time is the number of the concentric rings, is the
side length of the external ring, is the width of the strips,
is the separation between two adjacent rings, is the length of
the cuts, and is defined as for the multiple split-ring and the
spiral resonator.
The second contribution, instead, is given by the sum of the
capacitances of the gaps as
(13)
where the usual formula of the capacitance between two printed
strips with separation and width has been amended here by
using (4).
2870 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 12, DECEMBER 2007
Fig. 6. Resonant frequency of a labyrinth resonator as a function of the gap
length g. Data: ` = 5:65 mm, w = 0:9 mm, s = 0:2 mm, t = 30 m,
h = 1:6 mm, " = 3:85; tan  = 0:01;  = 0:017 
 10 
m.
Finally, since it can be shown that the gap capacitance is con-
nected in parallel to the distributed capacitance [11], the total
capacitance is given by
(14)
The series resistance of the labyrinth resonator is obtained in
the same way as for the previous inclusions as
(15)
The shunt resistance is instead given by the two following
contributions:
(16)
(17)
where the average length of the ring reads
(18)
, which is the resistance associated to the dielectric losses
between the strips, has been calculated as for the multiple split-
ring and the spiral resonator, while , which represents the
dielectric losses in the cuts, is derived straightforwardly from
the definition of resistance and considering that all the con-
tributions are equal and all connected in parallel. The final ex-
pression of the equivalent shunt resistance thus reads
(19)
Again, using the equivalent circuit depicted in Fig. 5(b) and
(11)–(14), it is possible to analytically derive the resonant fre-
quency of the individual labyrinth resonator. In Fig. 6, we show
the comparison between the results obtained through the model
proposed here, CST full-wave simulations, and measurements.
Fig. 7. Quality factor of individual multiple split-ring and spiral resonators as
a function of the number of the rings/turns N . The quality factor is normalized
to the one of the split-ring resonator with the same space occupancy (i.e., the
multiple split-ring resonator with N = 2). Simulated results are from [12].
Data: ` = 5 mm, w = 0:1 mm, s = 0:1 mm, t = 30 m, h = 0:2 mm,
" = 1:01; tan  = 0:001;  = 0:017 
 10 
m.
III. QUALITY FACTOR AND PERMEABILITY FUNCTION
A. Quality Factor of the Isolated Inclusion
One of the main issues related to real-life metamaterials is
their inherent narrow bandwidth of operation. The bandwidth
of the fabricated metamaterial is strongly related to the reso-
nance behavior of the inclusions used to implement the mate-
rial. Since the resonant behavior of the inclusions can be easily
derived from the accurate resonant circuits we have presented in
Section II, it is also possible to analytically express the expected
operative bandwidth of a metamaterial made up by a certain set
of inclusions.
A good estimation of the bandwidth may be derived as the
inverse of the quality factor related to the resonance of the in-
dividual inclusion. Considering generic equivalent circuit
representations as the ones proposed in Section II, the quality
factor in the presence of the losses both in the dielectric sub-
strate and metallic conductor can be written as [21]
(20)
where is the quality factor related to the losses in the metallic
conductor and is given by
with (21)
is the quality factor related to the losses in the dielectric
substrate and is given by
with (22)
with being the angular resonant frequency
of the circuit.
In Fig. 7, we show the behavior of the quality factor of in-
dividual multiple split-ring and spiral resonators as a function
BILOTTI et al.: EQUIVALENT-CIRCUIT MODELS FOR DESIGN OF METAMATERIALS BASED ON ARTIFICIAL MAGNETIC INCLUSIONS 2871
of the number of rings/turns . The quality factor is normal-
ized to the one of the split-ring resonator with the same space
occupancy.
The comparison between the analytical evaluation of the
quality factor using the proposed circuit models and the numer-
ical results presented in [12] clearly shows the accuracy of the
present formulation.
It is worth noticing that, as expected, in the cases of multiple
split-ring and spiral resonators, the electrically smaller the in-
clusions (i.e., the higher the miniaturization rate is), the higher
the and the narrower the operation bandwidth of the meta-
material. This is, of course, an unavoidable optimization issue,
whose solution is left to the designer and depends on the spe-
cific application for which the metamaterial has to be tailored.
B. Equivalent-Circuit Model and Permeability Function
Here we present a compact and accurate formulation for the
effective permeability of a medium densely filled with mul-
tiple split-ring, spiral, and labyrinth resonators. As a first-order
approximation, the permeability function is derived using the
Clausius–Mosotti [10] equation as
(23)
where is the number of inclusions per unit volume and mm is
the magnetic polarizability of the individual magnetic inclusion
defined as [22]
(24)
The effective impedance associated to the individual mag-
netic inclusion can be determined through the proposed
circuits shown in Figs. 1(b), 3(b), and 5(b) for the multiple
split-ring, spiral, and labyrinth resonator, respectively,
with (25)
The analytical expression of the permeability derived through
(23) is compared here with the permeability function derived
through a direct numerical retrieval from the calculated scat-
tering parameters [15], [16]. The corresponding results for the
case of a multiple split-ring and a spiral resonator are reported
in Figs. 8 and 9, respectively.
The results obtained through the employment of analytical
formula (23), which is based on the models derived in Section II,
are in a rather good agreement with the numerical data retrieved
from the calculated scattering parameters.
Fig. 8. Relative permeability of a multiple split-ring resonator medium
as a function of frequency. Comparison between the analytical model pro-
posed in this paper and the direct extraction from S-parameters. Data:
N = 12; ` = 8 mm, w = 0:1 mm, s = 0:1 mm, t = 30 m,
h = 0:2 mm, " = 3:85; tan  = 0:01;  = 0:017 
 10 
m,
n = 1=8:8 mm 8.8 mm 4.2 mm.
Fig. 9. Relative permeability of a spiral resonator medium as a function of
frequency. Comparison between the analytical model proposed in this paper
and the direct extraction from S-parameters. Data: N = 12; ` = 8 mm, w =
0:1 mm, s = 0:1 mm, t = 30 m, h = 0:2 mm, " = 3:85; tan  =
0:01;  = 0:017 
 10 
m, n = 1=8:8mm 8.8 mm 4.2 mm.
IV. CONCLUSION
In this paper, we have presented accurate analytical circuit
models for the design of different magnetic inclusions, such as
the multiple split-ring, spiral, and labyrinth resonators, to im-
plement real-life metamaterials with anomalous values of the
permeability. Starting from existing models considering only
lossless metallic conductors and inclusions immersed in air, we
have derived a complete formulation, which is able to take into
account the presence of a dielectric substrate, where the inclu-
sions are printed on and the presence of the losses in both the
dielectric substrate and in the metallic conductor. The extended
models presented here are able to predict with a very good ac-
curacy the resonant frequency of individual multiple split-ring,
spiral, and labyrinth resonators.
Moreover, the complete circuit model allows to derive an an-
alytical expression of the quality factor of the individual inclu-
sions, thus giving an estimation of the bandwidth of operation
2872 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 12, DECEMBER 2007
of the corresponding metamaterial sample. The circuit models
presented here also allow to analytically determine an expres-
sion of the permeability function, which has been shown to be
in good agreement with the commonly used numerical retrieval
techniques. The results obtained through the employment of the
proposed models have been compared to both full-wave numer-
ical results and to the measurements, showing good agreement.
REFERENCES
[1] H. Mosallaei and K. Sarabandi, “Magneto-dielectrics in electromag-
netics: Concept and applications,” IEEE Trans. Antennas Propag., vol.
52, no. 6, pp. 1558–1567, Jun. 2004.
[2] A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a
mu-negative slab: Anomalous tunneling and transparency,” IEEE
Trans. Antennas Propag., vol. 51, no. 10, pp. 2558–2570, Oct. 2003.
[3] A. Alù, F. Bilotti, N. Engheta, and L. Vegni, “Compact leaky-wave
components using metamaterials,” in IEEE MTT-S Int. Microw. Symp.
Dig., Long Beach, CA, Jun. 12–17, 2005, [CD ROM].
[4] A. Ishimaru, L. Seung-Woo, Y. Kuga, and V. Jandhyala, “General-
ized constitutive relations for metamaterials based on the quasi-static
Lorentz theory,” IEEE Trans. Antennas Propag. (Special Issue), vol.
51, no. 10, pp. 2550–2557, Oct. 2003.
[5] L. Landau and E. M. Lifschitz, Electrodynamics of Continuous
Media. Oxford, U.K.: Pergamon, 1984.
[6] F. Bilotti, “Application of metamaterials for miniaturized components,”
presented at the Metamater. Ind., Jouy-en-Josas, France, Nov. 28–30,
2005, Short Course for Ind. and SMEs.
[7] A. Alù, F. Bilotti, N. Engheta, and L. Vegni, “Sub-wavelength, com-
pact, resonant patch antennas loaded with metamaterials,” IEEE Trans.
Antennas Propag., vol. 55, no. 1, pp. 13–25, Jan. 2007.
[8] R. W. Ziolkowski and A. D. Kipple, “Application of double negative
materials to increase the power radiated by electrically small antennas,”
IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2626–2640, Oct.
2003.
[9] N. Engheta, “An idea for thin subwavelength cavity resonators using
metamaterials with negative permittivity and permeability,” IEEE An-
tennas Wireless Propag. Lett., vol. 1, no. 1, pp. 10–13, 2002.
[10] F. Martín, F. Falcone, J. Bonache, T. Lopetegi, R. Marqués, and M.
Sorolla, “Miniaturized coplanar waveguide stopband filters based on
multiple tuned split ring resonators,” IEEE Microw. Wireless Compon.
Lett., vol. 13, no. 12, pp. 511–513, Dec. 2003.
[11] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Mag-
netism from conductors and enhanced nonlinear phenomena,” IEEE
Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2075–2081, Nov.
1999.
[12] F. Bilotti, A. Toscano, and L. Vegni, “Design of spiral and multiple
split-ring resonators for the realization of miniaturized metamaterial
samples,” IEEE Trans. Antennas Propag., to be published.
[13] K. Aydin, I. Bulu, K. Guven, M. Kafesaki, C. M. Soukoulis, and E.
Ozbay, “Investigation of magnetic resonances for different split-ring
resonator parameters and designs,” New J. Phys., vol. 7, no. 168, pp.
1–15, 2005.
[14] F. Bilotti, A. Toscano, L. Vegni, K. Aydin, K. M. Alici, and E. Ozbay,
“Theoretical and experimental analysis of magnetic inclusions for the
realization of metamaterials at different frequencies,” in IEEE MTT-S
Int. Microw. Symp. Dig., Honolulu, HI, Jun. 3–8, 2007, pp. 1835–1838.
[15] D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Elec-
tromagnetic parameter retrieval from inhomogeneous metamaterials,”
Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol.
71, pp. 1–11, 2005, 036617.
[16] X. Chen, T. Gregorczyk, B.-I. Wu, J. Pacheo, Jr., and J. A. Kong, “Ro-
bust method to retrieve the constitutive effective parameters of meta-
materials,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip.
Top., vol. 70, pp. 1–7, 2004, 016608.
[17] K. Buell, H. Mosallaei, and K. Sarabandi, “A substrate for small patch
antennas providing tunable miniaturization factors,” IEEE Trans. Mi-
crow. Theory Tech., vol. 54, no. 1, pp. 135–146, Jan. 2006.
[18] J. D. Baena, R. Marques, F. Medina, and J. Martel, “Artificial magnetic
metamaterial design by using spiral resonators,” Phys. Rev. B, Condens.
Matter, vol. 69, pp. 1–5, 2004, 014402.
[19] F. Bilotti, A. Alù, N. Engheta, and L. Vegni, “Miniaturized circular
patch antenna with metamaterial loading,” in Proc. 1st Eur. Antennas
Propag. Conf., Nice, France, Nov. 6–10, 2006, [CD ROM].
[20] F. Bilotti, A. Alù, N. Engheta, and L. Vegni, “Compact microwave ab-
sorbers utilizing single negative metamaterial layers,” in Proc. IEEE
AP-S Int. Symp./USNC/URSI Nat. Radio Sci. Meeting, Albuquerque,
NM, 2006, [CD ROM].
[21] S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Com-
munication Electronics. New York: Wiley, 1994.
[22] P. Ikonen, “Artificial electromagnetic composite structures in selected
microwave applications,” Ph.D. dissertation, Radio Lab., Helsinki
Univ., , Helsinki, Finland, 2007.
Filiberto Bilotti (S’97–M’03–SM’06) was born
in Rome, Italy, on April 25, 1974. He received the
Laurea (summa cum laude) and Ph.D. degrees in
electronic engineering from the University of “Roma
Tre,” Rome, Italy, in 1998 and 2002, respectively.
Since 2002, he has been with the Department
of Applied Electronics, University of “Roma Tre,”
where he is currently an Assistant Professor of
electromagnetic field theory. He has authored or
coauthored approximately 200 papers in interna-
tional journals and conference proceedings. Since
2003, he has been a Technical Reviewer of the European Community for
scientific projects in the fields of metamaterials and antennas. He is a member
of the Editorial Board of Metamaterials and a Technical Reviewer for major
international journals related to electromagnetic field theory. His main research
interests are microwave applications of complex media, metamaterials and
metasurfaces, analysis and synthesis of planar and conformal integrated com-
ponents and phased antenna arrays, and development of improved numerical
algorithms for an efficient analysis of printed antennas and circuits.
Dr. Bilotti has been a national expert of the European actions COST260 and
COST284 on antenna technology and design (1999–2006). Since 2004, he has
been a member of the governing bodies of METAMORPHOSE, the European
Network of Excellence on Metamaterials, where he acts as the coordinator of
spreading activities, with a particular emphasis on training events. He is also
a member of the Steering Committee, European Doctoral School on Metama-
terials. He served as a member of the Technical Program, Steering Committee,
and Organizing Committee of several national and international conferences and
training events related to metamaterials, and as organizer and chairman of spe-
cial sessions focused on the applications of metamaterials at microwave frequen-
cies. He was the recipient of the 2007 Raj Mittra Travel Grant Senior Researcher
Award.
Alessandro Toscano (M’90) was born in Capua,
Italy, on June 26, 1964. He received the Electronic
Engineering degree and Ph.D. degree from the
University of Rome “La Sapienza,” Rome, Italy, in
1988 and 1993, respectively.
Since 1993, he has been with the Department of
Applied Electronics, University “Roma Tre,” Rome,
Italy, where he is currently an Associate Professor.
His research interests are microwave and millimeter-
wave circuits and antennas, electromagnetic compat-
ibility, and general techniques in electromagnetics of
complex material media such as metamaterials.
Lucio Vegni (M’73) was born in Castiglion
Fiorentino, Italy, on June 20, 1943. He received the
Electronic Engineering degree from the University
of Rome, Rome, Italy.
Following a period as an Antenna Designer with
Standard Elektrik Lorenz, Stuttgart, Germany, he
joined the Istituto di Elettronica, University of Rome,
where he was a Researcher in applied electronics.
From 1976 to 1980, he was Researcher Professor of
applied electronics with the University of L’Aquila.
From 1980 to 1985, he was a Researcher Professor
of applied electronics, and from 1985 to 1992, he was an Associate Professor
of electromagnetic compatibility with the University of Rome “La Sapienza.”
Since 1992, he has been with the University of “Roma Tre,” Rome, Italy,
where he is currently a Full Professor of electromagnetic field theory. He has
authored or coauthored over 500 international papers appearing in journals,
BILOTTI et al.: EQUIVALENT-CIRCUIT MODELS FOR DESIGN OF METAMATERIALS BASED ON ARTIFICIAL MAGNETIC INCLUSIONS 2873
transactions, and conferences. His research interests are in the areas of mi-
crowave and millimeter-wave circuits and antennas with a particular emphasis
on electromagnetic compatibility problems. He was specifically active in
studies on partial coherence radio links with particular attention on multipath
electromagnetic propagation effects up until 1977. He then moved to the area of
integrated microwave circuits, where he studied the electromagnetic modeling
of microstrip planar circuits and antennas. In cooperation with industry, he
was engaged in the development of integrated microstrip antennas for satellite
applications and in study of radiating system electromagnetic compatibility
problems from 1985 to 1990. Since 1990, he has been actively involved with
theoretical and numerical aspects of new planar antennas modeling involving
unconventional materials. In these recent studies, he has offered new contri-
butions to equivalent-circuit representations of planar microwave components
and new variational formulations for their numerical simulations. Finally, in
the area of unconventional materials, he has given noteworthy contributions to
the study of chiral and omega grounded dielectric slab antennas. His recent
research activities are in the field of metamaterials.
Prof. Vegni has been the organizer and chairman of the second and third edi-
tion of the International Workshop on Metamaterials and Special Materials for
Electromagnetic Applications and TLC, Rome, Italy (2004 and 2006). He is
the local chairman of Metamaterials 2007—The First International Congress on
Advanced Electromagnetic Materials in Microwaves and Optics, Rome, Italy
(2007). He is a member of the European Chiral Group, the Network of Ex-
cellence METAMORPHOSE, and the Italian Electrical and Electronic Society
(AEI).
Koray Aydin (S’03) was born in Ankara, Turkey,
on April 10, 1980. He received the B.S. and M.S.
degrees in physics from Bilkent University, Ankara,
Turkey, in 2002 and 2004, respectively, and is cur-
rently working toward the Ph.D.. degree in physics
at Bilkent University. His M.S. thesis focused on the
characterization of left-handed metamaterials and
negative refractive media.
He has authored or coauthored 28 refereed journal
papers and 26 conference proceedings and abstracts.
His research activities are conducted at the Nanotech-
nology Research Center, Bilkent University, Bilkent, Turkey. His research in-
terests are left-handed metamaterials, sub-wavelength imaging and resolution,
resonances in split-ring resonator structures, tunable metamaterials and nega-
tive refractive index in photonic crystals. He is listed in Who’s Who in Science
and Engineering.
Mr. Aydin is a member of the Optical Society of America (OSA), the IEEE
Antennas and Propagation Society (IEEE AP-S), and The International Society
for Optical Engineers (SPIE). He was the recipient of a 2007 SPIE Educational
Scholarship.
Kamil Boratay Alici was born in Sivas, Turkey, on
January 12, 1981. He received THE B.S. degree in
physics from Bilkent University, Ankara, Turkey in
2004, and is currently working toward the Ph.D. de-
gree at Bilkent University.
He has authored or coauthored eight papers in sci-
entific journals. His scientific interests are electro-
magnetic metamaterials: antenna and superlens ap-
plications, acoustic metamaterials, and negative re-
fraction in photonic crystals.
Mr. Alici was the recipient of the Undergraduate
Scholarship (2001–2004), and Graduate Scholarship (2004–2009) of The Sci-
entific and Technological Research Council of Turkey (TUBITAK).
Ekmel Ozbay was born in Ankara, Turkey, on March
25, 1966. He received the B.S. degree in electrical en-
gineering from the Middle East Technical University,
Ankara, Turkey, in 1983, and the M.S. and Ph.D. de-
grees in electrical engineering from Stanford Univer-
sity, Stanford, CA, in 1989 and 1992, respectively.
His doctoral dissertation concerned high-speed reso-
nant tunneling and opto-electronic devices.
From 1992 to 1994, he was a Scientist with the De-
partment of the Environment (DOE), Ames National
Laboratory, Iowa State University, where he was in-
volved in the area of photonic-bandgap materials. In 1995, he joined Bilkent
University, Ankara, Turkey, where he is currently a Full Professor with the
Department of Electrical and Electronics Engineering and the Department of
Physics. He is the Founding Director of the Nanotechnology Research Center,
Bilkent University. He has authored or coauthored over 320 papers in scientific
journals, conference proceedings, and books. His research involves nanopho-
tonics, metamaterials, MOCVD growth and fabrication of GaN-based electronic
and photonic devices, photonic crystals, and high-speed opto-electronics. He has
been a Topical Editor of Optics Letters since 2002.
Dr. Ozbay was the recipient of the 1997 Adolph Lomb Medal of the Optical
Society of America and the 2005 European Union Descartes Science Award.