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# Analysis of Spiral Resonator Filters

- Created: Sunday, 01 September 2013

### Improved performance coupled with reduced size requires the development of novel filter designs for use in advanced wireless systems.

Increasing demand for more advanced wireless systems necessitates the introduction of novel designs that are capable of simultaneously fulfilling multiple operating and performance criteria. The implementation of high-datarate transmission systems has required the development of innovative designs for microwave filters that must fit within a reduced volume to allow integration of multiple filters in more compact wireless systems. Additionally, the filter’s specific passband frequencies and quality factors must be achieved within the system’s geometrical and topological constraints.

Microstrip Filter based on rectangular type spiral resonators etched on the center line of the microstrip." class="caption" align="right">Spiral resonator filters offer one option for significantly reduced size compared to conventional ring resonators. An array of spiral resonators can be directly fabricated on a printed circuit board and because of their characteristic response, they can be designed to occupy minimal volume.

To characterize the operation of these
devices, a mathematical construct named
the scattering matrix (S-matrix) is used
that describes how the RF signal interacts
with the device. The signal may reflect,
exit other ports, and dissipate via heat or
electromagnetic radiation; the S-matrix
represents each of these signal paths.
The order of this matrix is *n* × *n* with *n*
equaling the number of ports in the system;
thus, S_{ij} represents the scattering for
the *j* input port and the *i* output port
such that S_{11} specifies the ratio of signal
reflected from port 1 for an input on
port 1, and S21 specifies the response at
port 2 due to a signal at port 1.

Bandstop Resonator Filter. Some exterior faces are removed for visualization" class="caption" align="right">A compact microstrip filter (see Figure 1) using spiral resonators was designed to produce a resonant frequency of 7.2 GHz (Lim et al.). A model was set up using COMSOL Multiphysics (see Figure 2) in which the microstrip line is represented as a perfect electric conductor (PEC) surface on a dielectric substrate, with another PEC surface on the bottom of this substrate acting as a ground plane. Two lumped ports are modeled as small rectangular faces that bridge the gap between the PEC faces of the ground plane and the microstrip line at each port. A small air domain bounded by a scattering boundary (SBC) surface is added to avoid back reflection of radiated fields and reduce the size of the modeling domain. The model includes the dielectric substrate defined as a volume with the relative permittivity of the dielectric.

Frequency Response of the bandstop spiral resonator filter comparing experimental measurement (Lim et al.) with COMSOL simulation." class="caption" align="right">The experimental and simulation
results for and over a range of frequencies
of interest are shown in Figure 3,
where S_{11} specifies the ratio of signal
reflected from port 1 for an input on port 1, while S_{21} specifies the response at
port 2 due to a signal at port 1.

The simulation results agree well with
experimental data for the transmitted and
reflected signals, and demonstrate rejection
of frequencies outside the required
frequency cutoff. The resonant frequency
is 7.2 GHz and the bandwidth of the stopband
is 0.5 GHz ( 7.1-7.6 GHz), with the
reference level of |S_{21}| = -10 dB. A deep
rejection band (S_{21} > -50 dB) is obtained
at the resonant frequency with a steep cutoff;
a flat passband (S_{21}< 1.2 dB) is
observed, suggesting
the proposed spiral
filter design has low
insertion losses, thus
limiting its effect on
transmitted signal
when integrated into
a circuit.

Electric Field below (left) and at (right) **Resonant Frequency**." class="caption" align="right">The data can also
be visualized by the
electric field distribution
below and
at the resonant frequency
(see Figure
4). Below the resonant
frequency, a
high level of signal is
transmitted through
the device; at the resonant
frequency of
7.2 GHz, a high level
of signal is attenuated, thus demonstrating
the degree of signal selectivity developed
by the filter design.

Metamaterials Fractal Spiral Resonator: two fractal resonators are connected anti-symmetrically along the feeding line." class="caption" align="right">A fractal spiral resonator developed by
Palandöken & Henke is shown in Figure
5. The filter is composed of two unit cells
of electrically small artificial magnetic
metamaterials formed with the direct connection
of two concentric Hilbert fractal
curves. Operation is based on the excitation
of two electrically coupled fractal spiral
resonators through direct connection
with the feeding line. Simulation results
for the transmission (S_{21}) and reflection
(S_{11}) losses are shown in Figure 6; the
selectivity of the filter is 100 dB/GHz with
a 3 dB reference insertion loss.

The electric field distribution developed by the fractal spiral resonator is shown in Figure 7; below the resonant frequency, signal passes through the filter; at resonance, the signal is highly attenuated with an extremely low level of signal transmitted.

The performance of a spiral resonator filter has been analyzed using COMSOL Multiphysics, and shown to demonstrate agreement with experimental data. A compact microstrip-based spiral resonator filter with a resonant frequency of 7.2 GHz shows low insertion losses with a high level of performance and sharp cutoff over the specified frequency range. Analysis of a fractal spiral resonator consisting of two unit cells of magnetic metamaterials operating at a resonant frequency of ~1.3 GHz also shows a high level of selectivity at 100 dB/GHz. Analyses of this type can be extended to assess the performance of other filter designs prior to fabrication and integration into operating circuits.

*Electric Field at a frequency below the resonant frequency (left) and at the resonant frequency (right)." class="caption" align="right">This article was written by Sergei P.
Yushanov, Jeffrey S. Crompton, and Kyle C.
Koppenhoefer of AltaSim Technologies. For
more information, visit http://info.hotims.com/45607-127.*