RING RESONATOR METHOD FOR DIELECTRIC PERMETTIVITY MEASUREMENT

OF FOAMS

Introduction
In this essay I will re-explain the contents of a thesis which was written by Isaac Waldron in 2006
regarding to the measurement of dielectric permittivity from foam using the ring resonator method.
Dielectric properties are useful data for imaging and radar experimentation. In particular, the dielectric
properties of expanded polystyrene (EPS) foams are interesting as they drive the use of EPS foam as
physical supports for radar cross-section measurements and other applications. These foams are aslo
utilized in modern antenna research as supporting materials for antenna construction and for radomes.
Electromagnetic imaging applications require the dielectric properties of materials to predict the
interaction between fields used for imaging and the materials. Few examples are available for research
directed at the measurement of foam permittivity, or dielectric constant and loss tangent.
The author decribe a new microstrip-based resonant device that is specifically intended for permittivity
measurements of low dielectric constant materials at frequencies 1 to 10 GHz. The device is based on
a microstrip ring resonator with the ring and the ground plane/feed network physically located in
separate planes. This enables the placement of an arbitrary sample of material to be measured between
the ring resonator and ground plane. A transmission line method (TLM) is used to predict the frequency
response including feed and radiation effects. Experimental data in L- and S-bands are presented and
compared with theory estimates for the foams.

The material properties dielectric constant and loss tangent play important roles in the propagation of
electromagnetic energy in dielectric media. In a low-loss case, the dielectric constant er slows the
propagation of energy through a media by altering the phase velocity. The loss tangent tand specifies
how much energy is absorbed as a wave travels through a medium. The propagation constant, which
takes er and tand as parameters, defines the intrinsic impedance of the medium and the behavior of
waves at the interfaces between different media.

Measurement Methods

M. Plonus established a theory for predicting the RCS of EPS foam supports used for radar
measurements [1]. In this paper, Plonus models EPS foam as a collection of “randomly arranged, closely
packed spherical shells.” He finds the volume ratio of air to polystyrene in the foam as a function of
foam density and uses this to determine the dielectric constant of the foam.

E. F. Knott in August 1993 published his model of plastic foams as a regular cubic lattice [2]. Knott
derives the dielectric constant of a foam from the capacitance of a unit cell of the cubic lattice model.
Ultimately, he provides a formula based on the dielectric constants of the base polymer and inclusion
gas and the volumetric fraction of polymer in the foam. For an air-blown foam of density 26 kg/m3
Knott’s formula predicts a dielectric constant of 1.03. He compares his result to a logarithmic prediction
of dielectric constant by Cuming as well as dielectric constants derived from backscatter measurements
of extruded polystyrene forms.

There are several methods that can be used to measure the dielectric constant of a material, including
methods based on free-space, waveguide, and resonator measurements. These techniques have been
described in both the scientific literature and in United States utility patents.

U.S. Patent 3,965,416, issued June 22, 1976, describes a method of dielectric constant measurement
using a pulse delay oscillator [7]. The material under test is used to alter the phase velocity of signals
traveling along a shorted transmission line. This transmission line is the frequency determining
component of the pulse delay oscillator, and so the frequency of oscillation is used to calculate the phase
velocity.

U.S. Patent 5,132,623, issued July 21, 1992, describes a broadband dielectric property measurement
technique specifically applied to the problem of oil-bearing strata determination [8]. The apparatus
described consists of broadband transmitting and receiving antennas which are used to measure
dielectric constant from frequency- or time-domain methods. The inventors claim a measurement
frequency range of 2 kHz-1 GHz.

U.S. Patent 5,157,337, issued October 20, 1992, describes a probe for dielectric constant measurements
of thin materials [9]. The device is essentially a resonator constructed from an open coaxial transmission
line. Fringing fields from the open end of the resonator penetrate into the material under test, and the
resonant frequency is affected by the dielectric constant of that material. The inventors claim accuracy
of better than one percent.

U.S. Patent 6,496,018, issued December 17, 2002, describes a method for determining a calibration
curve for an open, or radiating, resonator that relates the resonant frequency to the dielectric constant
and thickness of a sheet of sample material of known dielectric constant [10]. This calibration curve is
then applied to find the dielectric constant of a material with unknown properties.

Free-space methods for measuring dielectric constant rely primarily on reflection and transmission of
electromagnetic waves through a sample of the material under test. Reference [11] describes a method
using dielectric lenses to focus a signal on a small piece of material under test. The dielectric constant
and loss tangent are then found from the scattering parameters of the complete system. The authors of
[12] use a similar experimental setup to that of [11], but establish a relation between the reflection
coefficient and the complex permittivity of the material under test to determine dielectric constant and
loss tangent. Reference [13] describes a method to measure the properties of large slabs of material
using two standard horn antennas. The authors also use some algebraic manipulation to reduce the
problem from a 2-D search of a complex space to a search on a real-valued function.

Waveguide methods for dielectric property measurement involve the comparison of empty waveguide
with a waveguide including the sample material or the comparison of measured cattering parameters
and numerical electromagnetic solutions. The waveguide may be either fully or partially filled with the
material under test. In [14], the material under test partially fills a shorted waveguide and the measured
reflection coefficient is compared to a FEM solution of the system.

The authors of [15] fit an expression for the effective complex dielectric constant to measurements of a
system composed of two rectangular waveguides separated by a relatively thin piece of sample material.
In [16] the complex permittivity of a cylindrical rod is calculated from the resonant frequency and
bandwidth of the transmission spectrum. The authors of [17] measure the reflection coefficient of a thin
sample in a matched waveguide and compare to a model of the system using an infinitesimally thin
resistive sheet as the sample.

The methods for measuring dielectric properties using resonators are well developed and most use the
scattering parameters of a one- or two-port resonator system as a basic measurement. These
measurements are then compared with numerical or analytical solutions of the system to find the
dielectric constant and loss tangent values. Of interest is a paper that uses a large open resonator with a
multilayered dielectric load to determine the properties of an unknown layer in the load.

The authors develop an analytical formula for the loss tangent of an unknown sample material. The
technique is aimed at the measurement of high-permittivity layers in multilayer systems. In general, the
full-wave methods will be very computationally intensive, while the methods based on perturbation of
a resonant cavity will require standards of known dielectric constant and loss tangent to calibrate
against. Therefore, there exists a need for a reasonably accurate method that neither is computationally
intensive nor requires samples of known permittivity for calibration.
Theoretical Basis

This measurement use resonator-based method for the measurement of complex dielectric permittivity
of materials. A ring resonator structure on a printed circuit board (PCB) can be used to determine the
complex permittivity of the substrate material. The Author use a measurement of the S21 parameter of
a two-port ring resonator to determine the permittivity of the board substrate. The ABCD parameter
formulation of the network parameter is used, where in this formulation, the response of a system can
be simply calculated as the product of the ABCD parameter matrices of cascaded elements.

The structure of a ring resonator device consists entirely of printed microstrips on a rigid substrate. A
twolayer board with one dielectric material is used. The ground plane occupies the entire lower surface
of the board. The feed lines and ring resonator are printed transmission lines with width chosen for 50
W characteristic impedance. A small gap D is included between the ring and each feed line; this gap is
included to separate the resonant behavior of the ring from the feed network and ranges from 0.1 to 1.0
times the width of the feed microstrip. SMA connectors are used to connect the device to a network
analyzer for measurement.

In the Planar Ring circuit figure it can be seen that there is a small gap of width D between each feed
line and the ring resonator. This gap slightly affects the resonant frequencies of the ring resonator but
greatly affects the peak amplitudes of the S21 parameter of the device. The author’s analysis of the ring
resonator follows Yu and Chang’s work. In [28], they cite a previous work by Owens that concludes
that curvature effects can be ignored for microstrip ring resonators of sufficiently large diameter as
compared to their width [39]. In this case, Yu and Chang point out that a microstrip ring resonator of
large diameter can be modeled as two straight sections of microstrip that are connected in parallel.

The device in Planar Ring cannot be used to measure the dielectric permittivities of polymer foams due
to the difficulty of printing circuits on foams sehingga the author propose a suspended ring resonator
structure that places material to be measured between the microstrip ring and the ground plane. This
ensures that the resonator is strongly affected by the properties of the sample material.

In the suspended ring resonator, the material to be measured is placed between two supports of known
dielectric constant and loss tangent. The lower surface of the lower support is fully metallized to provide
a ground plane for the feed lines and microstrip ring resonator. A small air gap is included between the
top surface of the sample and the microstrip ring to allow easy insertion and removal of the sample. The
sample is larger than the ring to account for fringing fields around the microstrip ring; a square
dimension of 1.5R2 is sufficient. The feed structure is similar to that of the planar resonator; two 50 W
microstrip transmission lines are printed on the upper surface of the lower support and direct energy
into and out of the ring resonator. The primary difference between the analysis of the device Planar ring
resonator and Suspended ring resonator is in the calculation of the propagation parameters for the
microstrip ring.

In the suspended ring design, the microstrip ring is in a different plane than that of the feed lines. So
the author has chosen to consider the feed gap D as if the ring were projected onto the upper surface of
the lower support. Though not an ideal model, this approximation captures the essence of the capacitive
coupling in the sense that the fringing fields at the end of the feed line interact with the fringing fields
of the microstrip ring. This interaction allows one feed line to excite fields in the ring and the other to
sense fields in the ring.

If we observed, the ring resonator microstrip structure on Suspended ring resonator is largery similar to
Planar ring resonator. What distinguishes between these two devices is that the suspended ring resonator
has a multilayer dielectric substrate as well as a dielectric superstrate with a non-unity dielectric
constant. The primary difference the analysis of these two structures is the calculation of the complex
propagation constant g and characteristic impedance Z0.
In my opinion, the measurement of permittivity and dielectric constant using the resonator-based
method is correct. However, what needs to be considered at the time of measurement is the uncertainty
analysis so that the maximum error value can still be accepted. After I read this thesis in its entirety, I
can give the advantages and disadvantages of the measured materials.

In ring resonator planar materials, due to this simple circuit it is possible to apply simple micristrip
theory to determine the propagation constant and character impedance. However, due to the structure
of the circuit, which is the area of the microstrip ring that is concentrated under and around the ring, it
is very difficult to print the circuit on the foam. This makes it impossible to use the planar ring resonator
to measure the permittivity of the foam dielectric.

Whereas in the hanging ring resonator, because it uses a multilayer it is a little more complicated. we
cannot apply simple microstrip theory to determine the propagation constant and its characteristic
impedance, instead we use multilayer dielectric microstrip transmission line theory using conformal
mapping method. The advantage of this device is that we can measure the dielectric permetivity of the
polymer foam.

Due to the simple circuit shape of the planar ring resonator, the author was able to apply simple
microstrip theory to determine the propagation constant and characteristic impedance, while in the the
multilayer dielectric case is somewhat more complicated. In [42], Svačina develops a theory of
multilayer dielectric microstrip transmission lines using a conformal mapping method. I have applied
this theory to the problem of determining the propagation characteristics of the microstrip ring resonator
in the suspended ring resonator device. The dielectric environment for the microstrip ring consists of
five layers. Layer 1 is lower support (e1, tan d1), layer 2 is the sample (e2, tan d2), layer 3 is the air gap
(eair, tan dair), layer 4 is upper support (e1, tan d1) and layer 5 is atmosphere (eair, tan dair).

Layers 1-3 are below the ring and layers 4 and 5 are above the ring. Each layer is assigned a filling
factor based on the thickness of the layer as compared to the height of the microstrip above the ground
plane and the width of the microstrip compared to its height over the ground plane. The filling factor of
the ith layer is designated qi, the width of the microstrip ring is designated W, the height of the
microstrip ring above the ground plane is designated h, and the ratio of the height above the ground
plane of the upper surface of the ith dielectric layer to h is designated Hi. The effective dielectric
constant of the multilayer microstrip ring is calculated from the dielectric constant of each layer material
and the filling factors, where ei is the complex dielectric constant of the ith layer and ej is the complex
dielectric constant of the jth layer.

It is important to note the relationship in (46) between the dielectric constant of the microstrip mode ee
and the individual layer dielectric constants ei; the terms of ee vary as the reciprocal of the sum of the
reciprocals of the ei. This means that ee is dominated by low dielectric constant layers, such as the air
gap above the sample. A close analogy is given by a parallel resistor circuit where individual resistances
play similar roles to the ei. A consequence of this is that high dielectric constant samples will be
measured less accurately by this device. Low dielectric constant samples, such as EPS foam, will be
accurately measured.

For thick samples, the height of the microstrip ring resonator above the ground plane may be relatively
large compared to the width of the microstrip and the guided wavelength of waves at the resonant
frequencies. As a result, the ring may exhibit relatively significant radiation losses that will affect the
frequency and bandwidth of the ring resonances. In order to take this radiation into account, The author
use semi-analytical techniques presented by Hill, Camell, Cavcey, and Koepke [44]. They develop
methods for calculating the electric far-field of a straight length of microstrip. To apply this model, the
microstrip ring can again be modeled as two parallel microstrips due to its low curvature [39]. The
authors of [44] first calculate the so-called array.
The author included the additional loss due to radiation as an additional real term in the expression for
the propagation constant g. The fraction of applied power that reaches the end of a microstrip is reduced
by the radiation efficiency of the microstrip.

As for the planar ring resonator device, the overall response of the suspended ring resonator is calculated
as the product of the ABCD parameter formulations of the left feed gap, the microstrip ring, and the
right feed gap. Once again, the S21 scattering parameter is calculated from the ABCD parameter
formulation of the suspended ring resonator. where Z0 is the characteristic impedance of the
measurement system and feed lines, Zring is the characteristic impedance of the microstrip ring
resonator, and L is pRm.

The S21 parameter of the ring resonator device shown in suspended ring resonator circuit can be applied
to find unknown dielectric constants and loss tangents. In particular, the location and bandwidth of
peaks in a plot of the S21 parameter versus frequency will vary with the dielectric constant and loss
tangent of the material in the sample layer.

Experimental Verification

The author present descriptions of test setups for the planar and suspended ring resonator concepts as
well as dielectric measurement results obtained with both types of resonator. Unless otherwise noted,
all measurements were taken at room temperature using a HP/Agilent 8722ET network analyzer.

In the Planar Ring Resonator Test Setup, the author designed and constructed a planar ring resonator
similar to the draft circuit of the planar ring resonator in order to verify the network parameter theory
could be used to determine the dielectric constant and loss tangent of a known substrate. The ring and
feed lines are 2.2 mm wide, and the ring resonator has a mean radius of 25.9 mm. The gap between
each feed line and the ring is 0.25 mm. The substrate is 62 mil thick FR-4, an Isola datasheet lists the
dielectric constant 35 and loss tangent of this material as 4.25 and 0.016 at 1 GHz. The dimensions of
the substrate are 136x90 mm, and the bottom surface of the substrate is fully metallized. SMA
connectors enable attachment to measuring equipment.

The device of constructed planar ring resonator was connected to a network analyzer and subjected to
a frequency sweep from 800 MHz to 2.4 GHz at intervals of 1 MHz. The magnitude of the S21
parameter was recorded at each frequency point. The author simulated the S21 response of the planar
ring resonator at the same frequency points. The microstrip and gap capacitance parameters were
calculated according to the formulas. Performance values for the first two resonances are listed in Table
5, and simulated and measured S21 parameters for this device. As the authors of previous works [21]-
[23], the simulated S21 parameter matches well with the measured data, validating the use of network
parameter formulations for the simulation of planar ring resonators.

In order to verify the suspended ring resonator concept, the author designed and constructed the device
similar to the draft circuit of the suspended ring resonator. This device was used to determine the
dielectric constant and loss tangent of expanded polystyrene (EPS) foam samples. The ring and feed
lines are 2.2 mm wide, and the ring resonator has a mean radius of 25.9 mm. The gap between each
feed line and the ring is 0.25 mm. The upper and lower supports are 62 mil thick FR-4. The lower
support attaches to a laboratory jack, part number NT54-687; the top board mounts independently in a
fixed position using two two-part posts, part numbers NT54- 939/956. The jack and posts attach directly
to a bench plate, part number NT54-638. As with the planar ring resonator, the feed lines and microstrip
ring are 2.2 mm wide, and the lower surface of the lower support is fully metallized.

In figure of constructed suspended ring resonator, a sample of EPS foam is shown underneath the ring;
this sample is smaller than the actual samples measured. The laboratory jack is used to assign the
vertical position of the sample with respect to the ring on the lower surface of the upper support. The
four long screws that mount the upper support also provide support for a copper shield to partially
protect the resonator from the room’s EM environment.
Using MATLAB, the author simulated the suspended ring resonator to determine the relationship
between sample er and the shift in the first and second resonances. For this exercise, the author assumed
6.22 mm thick sample with 0.75 mm air gap. Both resonances exhibit nearly linear dependence on the
sample dielectric constant, fit in a least-squares sense, a 3rd-order polynomial to the data for 38 each
resonance. This results in a maximum error, over the simulated dielectric constants, of 3.9 x 10-7 in er
for the first resonance and 3.1 x 10-7 in er for the second resonance.

Measurements of S21 were made over a band of 40 MHz with 1601 frequency points, vertical resolution
of 5dB/div, and reference -30 dB. The sweep time was manually set to 5 sec and the averaging and
smoothing options of the network analyzer were disabled. The peak magnitude of S21 corresponding
to a resonance was recorded at intervals of 5 sec using automatic marker tracking.

Fig. 15 shows excellent agreement at the first resonance of my predictions of dielectric constant with
Knott’s formula. The second resonance results shown in Fig. 16 are slightly lower than those for the
first resonance but generally still fall between the predictions by Cuming and Knott.

Using the measured dielectric constant for the foam sample, the author simulated the suspended ring
resonator to determine the relationship between a shift in loaded Q factor and the loss tangent for the
first two resonances. Again, the author calculated 3rd-order polynomials to approximate the simulation
results. As for the dielectric constant, the simulations exhibit a mostly linear dependence of tand on the
shift in Q factor. The maximum errors over the simulated values of loss tangent are 5.2 x 10-7 for the
first resonance and 1.7 x 10-7 for the second resonance.

Given the shift in Q factor between the air-filled resonator and the sample-filled resonator, the
polynomial coefficients in Table 7 can be used to predict the loss tangent of a 6.22 mm (245 mil) thick
sample material at the resonances. Measurements of S21 were made over a band of 20 MHz for the first
resonance and 40 MHz for the second resonance with 1601 frequency points, vertical resolution of
1dB/div, and reference -30 dB. The sweep time was manually set to 5 sec and the averaging and
smoothing options of the network analyzer were disabled. The Q factor, which is automatically
calculated by the network analyzer, corresponding to a resonance was recorded at intervals of 5 sec
using automatic marker tracking.

For these measurements, a shield consisting of solid copper is placed above the ring resonator. This
shield serves to reduce radiation losses dramatically and allows comparison of measurements with
simulations that do not include radiation losses. The results for the first resonance match very well with
the loss tangent of the bulk material. However, the second resonance loss tangent measurements are
much higher. This is likely due to radiation at the second resonance that is not taken into account; it is
unlikely that the loss tangent undergoes such a dramatic increase between the first two resonances.

Conclusion

The dielectric constant and loss tangent of materials are important inputs to RF engineering tasks. Many
methods for the measurement of these properties are available, and these methods are based on a diverse
set of tools including direct scattering parameter measurements, transmission line and waveguide
methods, and resonant structure analysis. I have developed a method for determining the dielectric
constant and loss tangent of arbitrary low dielectric constant materials based on a suspended ring
resonator device.

The suspension of the ring above the sample material under test maintains strong interactions between
the fields of the ring resonator and the sample material and produces accurate results. Using basic
network circuit analysis techniques I have analyzed the behavior of the suspended ring resonator with
respect to the S21 parameter. The S21 model includes the effects of feed gaps and radiation from the
ring. The magnitude of the calculated S21 parameter is compared to measurements of a real device to
determine the dielectric constant and loss tangent of EPS foam. The measured values of dielectric
constant match closely with other sources from the literature.
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