C
1
=
o
2 -
ln
r
r - d
=
1
2 -
o
c
o
2
ln
r
r - d
. (25)
Finally, the effective permeability of inner ring shorted split ring
resonator is calculated as,
eff
= 1 -
r
2
a
2
1 + i
2r - xl
o
r
2
-
2lc
o
2
2
r
2
2r - x
ln
r
r - d
. (26)
Equation (26) is plotted in Figure 3, which shows that the effective
permeability becomes negative at 2.617 GHz. Comparing Figure
1(b) and Figure 3, it is observed that SRR has been tuned and the
frequency at which effective permeability becomes negative has
shifted from 2.8 to 2.617 GHz.
The resonant frequency of inner ring shorted SRR is given by
o
2
=
2lc
o
2
r
2
2r - x
ln
r
r - d
, (27)
and can be calculated for the values in Figure 3 as,
= 2 2.614 GHz). (28)
The effective permeabilities for different lengths of split parts of
outer ring of inner ring shorted SRR are plotted in Figure 3, which
shows that the resonant frequency decreases with decrease in
length of split part. These results are compared with published
results, shown in Table 1 [8] and a good match is observed for
higher split parts.
4. CONCLUSION
The analytical expression for the effective permeability of inner
ring shorted SRR using Pendry’s technique has been derived,
plotted and compared with conventional SRR. Comparison shows
that the expression derived in this paper is good for higher length
of split parts but does not give good results for small split lengths.
The technique used in this article to calculate the effective perme-
ability and resonant frequency can also be used in calculating the
same for other configurations of SRR, spiral resonators, and S-
shaped resonators, etc.
REFERENCES
1. V. Veselago, The electrodynamics of substances with simultaneously
negative values of å and , Soviet Physics Uspekhi, 10 (1968), 509 –
514.
2. D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Schultz,
Composite medium with simultaneously negative permeability and per-
mittivity, Phys Rev Lett 84 (2000), 4184 – 4187.
3. J.B. Pendry, A.J. Holden, W.J. Stewart, and I. Youngs, Extremely low
frequency plasmons in metallic mesostructure, Phys Rev Lett 76 (1996),
4773– 4776.
4. J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, Magnetism
from conductors and enhanced nonlinear phenomena, IEEE Trans Mi-
crow Theory Tech 47 (1999), 2075–2084.
5. A.K. Iyer and G.V. Eleftheriades, Negative refractive index metamate-
rials supporting 2-D waves, Proc IEEE MTT Int Symp 2 (2002),
412– 415.
6. I. Gil, J. Bonache, J. Garcia-Garcia, F. Martin, Tunable metamaterial
transmission lines based on Varactor-loaded split-ring resonators, IEEE
Trans Microw Theory Tech 54 (2006), 2665–2674.
7. I.V. Shadrivov, S.K. Morrison, and Y.S. Kivshar, Tunable split ring
resonators for nonlinear negative index metamaterials, Opt Exp 14
(2006), 9344 –9349.
8. M. Shamonin, E. Shamonina, V. Kalinin, and L. Solymar, Properties of
a metamaterial element: Analytical solutions and numerical simulations
for a singly split double ring, J Appl Phys 95 (2004), 3778 –3784.
9. J.D. Baena, R. Marques, and F. Medina, Artificial magnetic metamate-
rial design by using spiral resonators, Phys Rev B 69 (2004), 014402.
© 2008 Wiley Periodicals, Inc.
PARTICLE SWARM OPTIMIZATION
WITH TOURNAMENT SELECTION FOR
LINEAR ARRAY SYNTHESIS
J. R. Pe ´ rez and J. Basterrechea
Department of Communications Engineering, ETSIIT, University of
Cantabria, Avda. de Los Castros s/n. 39005 Santander, Spain;
Corresponding author: perezjr@unican.es
Received 23 July 2007
ABSTRACT: The modern heuristic particle swarm optimization tech-
nique (PSO) has received great attention in many applications and re-
search areas over the last few years, and novel PSO-based algorithms
are continuously emerging to improve its overall performance. In this
article, classical PSO schemes are modified by introducing a selection
operator widely used in evolutionary algorithms such as genetic algo-
rithms, and these novel hybrid algorithms are applied to linear array
synthesis using complex weights and considering directive element pat-
terns to weigh up the improvements achieved. Up to eight classical PSO
schemes are addressed by the PSO suite tested in this work, including
PSO with synchronous and asynchronous updates of the swarm along
with global and local topologies for both, real-valued and binary encod-
ing algorithms. Furthermore, representative results comparing both
classical PSO schemes and the new hybrid approaches are reported and
discussed, demonstrating that the selection operator clearly increases
search pressure over the swarm, speeding up convergence for certain
specific schemes. © 2008 Wiley Periodicals, Inc. Microwave Opt
Technol Lett 50: 627– 632, 2008; Published online in Wiley InterScience
(www.interscience.wiley.com). DOI 10.1002/mop.23148
Key words: array synthesis; evolutionary algorithms; particle swarm
optimization; tournament selection
1. INTRODUCTION
As an alternative to traditional gradient-based local-search optimi-
zation methods, modern heuristics such as simulated annealing,
genetic algorithms (GA), memetic algorithms, ant colony optimi-
zation, cultural algorithms, and more recently, particle swarm
optimization (PSO), have aroused great interest among the elec-
tromagnetics community in a wide variety of fields and applica-
TABLE 1 Comparison of Published Results of Resonant
Frequencies of Inner Ring Shorted Split Ring Resonator with
Analytical Results for Different Lengths of the Split Part
Having d 0.5 e–3 m
x (mm) Published Results [8] (GHz) Analytical Results (GHz)
0.1 2.25 2.558
0.5 2.5 2.567
1.0 2.618 2.58
2.5 2.62 2.617
DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 50, No. 3, March 2008 627