ANSYS HFSS: Split Ring
Resonator
The advent of metamaterials has represented a paradigm shift in how engineers design
new electromagnetic devices and even media with unusual properties. One canonical
example of a metamaterial is the split ring resonator, the first to experimentally verify the
possibility of a negative refractive index. This App Note and accompanying example
project show how to simulate a split ring resonator with negative refractive index near 11
GHz using ANSYS HFSS.
ANSYS HFSS: Split Ring Resonator
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ANSYS HFSS: Split Ring Resonator
Introduction
The split ring resonator is the classic example of a metamaterial that can achieve negative
refractive index. First verified experimentally in 2001 [1], much of the motivation for
development of such materials can be attributed to the prospect of a “perfect lens” [2],
however many other interesting applications have been identified in the last decade.
Since metamaterials are usually complex composite structures, analytical solutions to their
scattering properties quickly become unfeasible. Therefore, numerical simulation is a
crucial step in the design process. In this note and accompanying example project, we show
how to model and analyze a split ring resonator [3] in ANSYS HFSS. The example shows
some of the most important points of metamaterial simulation and characterization,
including the boundary and port definitions, solution setup, and even effective parameter
retrieval.
Introduction  1
ANSYS HFSS: Split Ring Resonator
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Split Ring Resonator in HFSS
Using the geometric parameters in Ref. [3], the geometry of the split ring resonator was
modeled in HFSS. Given the relative simplicity of the structure, there are several valid
ways to draw it. In this case, we first started with a box to represent the FR4 substrate.
Next, polylines were used to trace out the metallic pieces on both sides of the substrate,
then thickened to the desired trace dimensions by setting the cross section to “Rectangle.”
After drawing the substrate, resonator, and wire structures, a region was added to define
the boundaries of the computational domain. This region is important so that the correct
unit cell size along x and y is defined, and so that the ports can be placed sufficiently away
from the near-fields induced on the structure. This ensures that the scattering parameters
are calculated properly. A picture of the resulting geometry is shown below.
Split Ring Resonator in HFSS  2
ANSYS HFSS: Split Ring Resonator
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Master/Slave Boundaries and Floquet Ports
After drawing the geometry and defining the materials, the next step was to assign
boundaries and excitations. Since we modeled a single unit cell which is to be situated in a
periodic lattice, Master/Slave boundaries are the correct choice for the boundaries on the
x-y faces of the model.
Floquet ports were then added as excitations to the ±z faces of the model domain.
Master/Slave Boundaries and Floquet Ports  3
ANSYS HFSS: Split Ring Resonator
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Detailed instructions for the correct definition of the Master/Slave boundaries and Floquet
ports can be found in the HFSS help documentation. A crucial setting worth noting for the
Floquet ports in this case is the deembedding. Since accurate calculation of the S-parameter
phase is important for the effective parameter retrieval, the ports must be deembedded to
the surface of the unit cell, as indicated by the blue arrows in the above figures.
Effective Parameter Retrieval in HFSS
Perhaps the most important and difficult step in metamaterial analysis is retrieval of the
effective material parameters from the frequency-dependent S-parameters obtained by the
simulation or scattering experiment. Thanks to the Output Variables functionality, it is
possible to perform this step in HFSS without having to export any data to an external
script, though one may do so if desired.
Ref. [3] gives the relevant equations needed for performing the effective parameter
retrieval. Specifically Eqs. 9 and 10,
𝑛=
1
𝑘𝑑
cos −1 [
1
2𝑆21
(1 − 𝑆11 2 − 𝑆21 2 )]
Effective Parameter Retrieval in HFSS  4
ANSYS HFSS: Split Ring Resonator
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and
𝑧=√
(1+𝑆11 )2 −𝑆21 2
(1−𝑆11 )2 −𝑆21 2
.
In the Output Variables window of the example project, the equations in HFSS for the
effective parameters can be found. They are also shown below.
Note that this is a complicated step and some care must be taken when defining these
equations. There are some conditional statements which are required to avoid
discontinuities and make the resulting effective parameters physically meaningful and
passive. As example, view the below results for the retrieved parameters that were
calculated before applying the required conditionals, i.e. naively applying Eqs. 9 and 10
from Ref. [3].
Effective Parameter Retrieval in HFSS  5
ANSYS HFSS: Split Ring Resonator
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Effective Parameter Retrieval in HFSS  6
ANSYS HFSS: Split Ring Resonator
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Note the unphysical discontinuities in the retrieved index, permittivity, and permeability.
Clearly there is an issue with the signs of the retrieved index and impedance. These clues
say that we have not calculated physical and passive material parameters.
The final results for the correct retrieved parameters are shown below, confirming the
negative sign of the real part of the refractive index near the resonance. Also, the magnetic
resonance of the structure is clearly visible from the retrieved permeability. It may appear
strange that there is an antiresonance with negative imaginary part in the retrieved
permittivity, however this can be attributed to the fact that the homogenization limit has
not quite been reached. More details are given in the text of Ref. [3].
Effective Parameter Retrieval in HFSS  7
ANSYS HFSS: Split Ring Resonator
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Effective Parameter Retrieval in HFSS  8
ANSYS HFSS: Split Ring Resonator
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References
[1] R. A. Shelby, D. R. Smith, S. Schultz, Science 292 (5514), 77-79 (2001).
[2] J. B. Pendry, Phys. Rev. Lett. 85, 3966-3969 (2000).
[3] D. R. Smith, D. C. Vier, Th. Koschny, C. M. Soukoulis, Phys. Rev. E 71, 036617
(2005).
References  9