1. science
2. mathematics
3. geometry

Evaluation of a Novel Tubular Sample Holder for
Dielectric Measurements with Abdulnour’s Method
André Rotavaa, Luiz E. P. Borgesb, Maurício H. C. Diasa, José C. A. Santosa
a
b
Electrical Engineering Department
Chemical Engineering Department
Military Institute of Engineering – IME
Rio de Janeiro, Brazil
{rotava, luiz, mhcdias, araujo}@ime.eb.br
Abstract—This paper assesses the accuracy of a rectangular
waveguide pierced by a tubular load sample holder for
measuring liquid materials at high temperatures with a
formulation based on Abdulnour’s method. The measurement
technique is derived from the transmission line method. The
sample holder is adapted from a section of a rectangular
waveguide pierced by a tubular load for high temperature
measurements at 2.45 GHz. Simulations of the loaded sample
holder are carried out with CST Microwave Studio® for several
values of sample permittivity. The accuracy of the technique is
evaluated by comparing these values to those obtained from
Abdulnour’s extraction model. It is shown that by properly
choosing a set of calibration standards the technique can be quite
accurate.
Keywords— permittivity measurements; high temperature
measurements; transmission line method; Abdulnour’s method
I.
INTRODUCTION
during measurements. However, it was found that it is not fully
compliant with measurements at the frequency range of
interest, i.e., at low microwave spectrum, due dimensional
problems. Related to that, there come all the difficulties of
measuring liquid materials with this technique, particularly, but
not exclusively, at very high temperatures.
The traditional transmission line method is not appropriate
for the purposes of this work, both for measuring liquid
materials, and for measuring materials at high temperatures.
In order to overcome these problems, different types of
sample holders and techniques were investigated.
In this context, a variant of the transmission line method
developed by Abdulnour et al. [2] which uses a double layered
cylindrical sample holder confined in a rectangular waveguide
was of interest. Although their apparatus is not adequate for
high temperature measurements, their formulation is quite
simple and has been shown to be very precise.
Measuring dielectric properties of liquid materials at high
temperatures poses a series of challenges. Although the
fundamentals of any permittivity measurement method can be
applied, many technical issues arise from their effective use at
very high temperatures, i.e., temperatures well above room
temperature. Thermal isolation between material under test
(MUT) and measuring setup, heat dissipation, thermal
expansion, and temperature monitoring are just some of them.
The sample holder introduced by Nishikata [3] brought
more flexibility to the measurement system. His apparatus
propitiates easy handling of the sample, especially for liquid
materials, and has a considerable potential for high temperature
applications. In fact, with some slight modifications,
considering thermal isolation between sample and measuring
setup, and proper heat dissipation, this sample holder can be
used for the purposes of the current work.
At the Military Institute of Engineer (IME), a recent work
lead by the Microwave Group has dealt with this kind of
measurements. The main interest was on high temperature
permittivity measurements of liquid materials at a frequency
range around 2.45 GHz. As part of this effort, an evaluation of
the coaxial probe method has been performed [1]. In this
context, some national oil samples were measured at
temperatures up to 180°C.
One of the greatest difficulties in dielectric measurements
when a discontinuity in a transmission line is present is to solve
the inverse problem, i.e., to extract the dielectric permittivity of
the MUT from the measured S-parameters of the sample
holder. So far, a number of approaches have been proposed
[2][3][4][5]. The problems with these techniques lie in their
intrinsic complexity, the relatively long CPU time for their
computation, and various convergence difficulties for some of
them.
The coaxial probe method is known to be very appropriate
for measuring liquid materials. However, since the probe must
be in direct contact with the sample, measurements are limited
to temperatures dictated by the probe, which usually are not too
high.
The free space method has also been investigated for the
purposes of this work. The great advantage of this method is
that the MUT does not make contact with the measuring setup
In this paper we investigate if the approximate analytical
formulation derived in [2] would still be valid for a sample
holder similar to that presented in [3].
Section II presents the fundamental concepts of
Abdulnour’s method. Section III describes the tubular sample
holder in use. Section IV explains the methodology to assess
the accuracy of the modified sample holder with Abdulnour’s
formulation. Section V presents some significant results of this
evaluation. And Section VI concludes the work.
II.
ABDULNOUR’S METHOD
The measurement technique described in [2] is based on the
scattering analysis of a cylindrical sample inserted in a
rectangular waveguide. The sample can be confined in a
dielectric cylindrical tube, as shown in Fig. 1.
For the sample holder in question, it is not possible to
obtain a complete analytical solution for extracting the
permittivity of the MUT (εMUT). On the other hand, the direct
problem, which consists of determining the S parameters of the
tubular structure, given the permittivities of the tube and of the
MUT, can be analytically formulated and numerically solved.
In [2], a combination of the boundary integral equation
technique with a modal expansion approach is used to
specifically equate the direct problem.
Fig. 2. Typical diagram of S21 as a function of the relative permittivity of the
MUT.
It must be emphasized that the extraction model proposed
in [2] is not a proper analytical solution for the problem, but an
approximation. Even though, in general, it allows reproducing
the actual response with great precision. Abdulnour et al. [2]
state that for waveguides with a useful frequency range
between 8 and 26 GHz, if 1.2 < f ⁄ fc < 1.8, where fc is the
waveguide cutoff frequency, and D1 < a ⁄ 5, the method’s
approximation error is always better than 1%.
III.
Fig. 1. Top (a) and front (b) view of a rectancular waveguide with a
dielectric tube containing a sample of material under test.
Fig. 2 shows a typical diagram of the forward scattering
parameter S21 obtained from the solution of the direct problem
for various values of the relative permittivity of the MUT. It is
found that the geometrical locations of the real and imaginary
parts of εMUT (ε′MUT and ε″MUT, respectively) resemble circular
arches in the complex plane of S21. In this example, the sample
holder is composed of a WR90 waveguide section
(a = 22.86 mm, b = 11.43 mm), and the tube has relative
permittivity εt = 4.5, inner diameter D1 = 1.08 mm, and outer
diameter D2 = 1.732 mm. The analysis frequency is f = 8 GHz.
A detailed geometrical analysis of the diagram of Fig. 2
allows one to deduce simple analytical equations that express
ε′MUT and ε″MUT directly as a function of the S21 parameter of
the cylindrical obstacle [2]. These equations constitute an
approximate analytical solution for the inverse problem, i.e.,
for directly extracting the relative permittivity of the MUT
from the sample’s scattering parameters.
In Abdulnour’s technique, at least three solutions to the
direct problem must be known for computing the unknown
parameters of its extraction model. In a traditional calibration
procedure, one’s would need three standards, or materials,
whose permittivities are known beforehand.
THE TUBULAR SAMPLE HOLDER DEFINITION
In Abdulnour’s method, the sample is completely confined
within the waveguide. This leads to a series of practical
difficulties for adapting this technique to high temperature
measurements. Besides the problem of heating the sample
without affecting the measurement setup, it would be necessary
to disconnect the sample holder from the setup to load, unload,
or replace the sample. This would substantially increase
measurement errors.
A similar solution, but with a few improvements to the
previous sample holder, was presented in [3]. The cylindrical
sample is made to pierce through the center of the wider walls
of the rectangular waveguide, without making contact with
them. To avoid escaping microwave energy, the holes at the
waveguide walls are followed by metallic sleeves, or chimneys,
which in practice act as circular waveguides with cutoff
frequency well above the effective bandwidth of the
rectangular waveguide. Fig. 3 shows a sketch of the described
sample holder. The tubular load is composed by a cylindrical
tube of linear, isotropic, and homogeneous dielectric material,
entirely filled with the material under test.
Fig. 3. Rectangular waveguide pierced by tubular load.
The main advantage of this sample holder is that one can
load, unload, or replace the MUT through the holes without
disconnecting the waveguide joints.
In the analysis performed in [3], the S parameters of the
sample holder are rigorously formulated in terms of the modal
scattering coefficients of the cylindrical object, a complex and
costly solution.
The dimensions that specify the sample holder of Fig. 3 are
shown in Fig. 4: a, b and L are the width, height and length of
the rectangular waveguide; HC and DC are the internal height
and internal diameter of the chimneys; D1 and D2 are the
internal and external diameters of the dielectric tube,
respectively. The structure is symmetric, i.e., the tubular load is
centered at the waveguide. The thickness of the waveguide
walls and the extensions of the tubular load outside the
chimneys do not interfere with the electromagnetic response of
the device.
remaining values of S21 to extract the corresponding relative
permittivity of the MUT (εo). This process is illustrated in Fig.
5. Comparing the input and output permittivities, we can check
if Abdulnour’s formulation can reproduce the response of the
sample holder’s model.
The errors are expressed in three forms:
ET = ε o − ε i ε i
• Real part error: E R = (ε o′ − ε i′ ) ε i′
• Imaginary part error: E I = (ε o′′ − ε i′′) ε i′′
• Total error:
Fig. 5. Methodology of evaluation.
Fig. 4. Dimensions for specifing the sample holder.
Since the sample holder is meant to be used in a frequency
band around 2.45 GHz, it was built upon a WR340 waveguide
standard, with dimensions a = 86.36 mm and b = 43.18 mm.
The effective bandwidth of this waveguide goes from 2.2 to
3.3 GHz. Additional modifications were made to allow precise
positioning of the dielectric tube and thermal isolation from the
measurement setup that do not affect the electromagnetic
behavior of the measurement system.
IV.
METHODOLOGY
The primary objective of this work is to use Abdulnour’s
method with the modified sample holder. For this to be
possible, the accuracy of the method must be assessed. The
idea behind this is to evaluate how Abdulnour’s extraction
model reproduces the actual relation between the MUT’s
permittivity and the tubular load S parameters. So, to reach that
objective, the following methodology is adopted.
Firstly, we obtain, with CST Microwave Studio® [6], the
sample’s S21 parameters relative to a large set of relative
permittivity values of the MUT (εi), in a given frequency.
Then, with Matlab®, we arbitrarily pick three of these
solutions to determine the coefficients of the model equation
proposed in [2]. Also in Matlab®, that equation is applied to the
The results are organized in graphical format, according to
some error intervals, on a complex Cartesian plane. In the
following figures, the crosses indicate de tubular load’s S21
parameter obtained from CST, and the small black circles
indicate the three points used for determining Abdulnour’s
extraction model. The blue arches represent the calculated
model, by mapping constant ε′MUT and constant ε″MUT values
into the complex plane of S21. The precision for each
simulation point is indicated by the color of the cross,
according to the following legend:
• Lilac → ER < 1% and EI < 1%;
• Red → ER > 1% or EI > 1% and ER < 5% and EI < 5%;
• Green → ER > 5% or EI > 5%.
V.
RESULTS
A. Single Frequency Evaluation
Fig. 6 shows the evaluation of the sample holder described
in Section III. The first observation about this figure is that the
extraction model does not agree equally with the sample holder
response for all values of MUT’s relative permittivity.
In this example, the simulation was at f = 2.5 GHz; with
(typical value for laboratorial glass);
L = 100 mm; D1 = 10 mm; D2 = 12 mm; HC = 25 mm;
DC = 14 mm; ε′i = 1, 3, 5...21; and ε″i = 0, 2, 4...20. The three
MUT relative permittivity values used for computing the
extraction model coefficient are ε1 = 1, ε2 = 1 – j20 and ε3 = 21.
εt = 4.6 – j0.017
B. Swept Frequency Evaluation
Fig. 7(d) shows the evaluation of the same sample holder at
2.5 GHz, with ε1 = 21 – j20, ε2 = 9 – j20 and ε3 = 15 – j10.
Fig. 8 repeats the analysis for the frequencies of 2 and 3 GHz.
It is noticed that the region of error below 1% changes when
the frequency sweeps from 2 to 3 GHz.
Fig. 6. Precision evaluation of the sample holder at 2.5 GHz..
At first, depending on the expected permittivity value for
the MUT, Fig. 6 may indicate that Abdulnour’s modeling is not
adequate for representing the sample holder in question.
However, by changing the three points that determine the
extraction model, the precision of the method can change
substantially. This is shown in Fig. 7.
It can be noticed that it is possible to generate an adequate
extraction model for different ranges of MUT permittivities by
properly choosing the calibration standards for the extraction
model. Although this may restrict the use of the technique for
measuring a broad range of arbitrary materials, this limitation
may be overcome in the case of high temperature
measurements. In fact, the extraction model can be tuned for a
specific material by using its low temperature permittivity as
reference.
As this case shows, the precision of the extraction model is
not maintained when the frequency of analysis is changed, if
the three calibration permittivity values are kept constant. In
other words, if at a certain frequency of analysis one specific
trio of permittivities generates a model with good precision for
a given interval of εMUT, this will not be guaranteed for other
frequencies. If a certain precision level is required for a broad
band of frequencies, the choice of the calibration standards for
the sample holder must be subject to further analysis.
C. A Case Study for Methanol
One way to ascertain if the choice of ε1, ε2 and ε3 is
adequate for the interval of εMUT of interest is through
simulation. Fig. 9 shows the simulation of a measurement with
methanol, in which were used ε1 = 21 – j20, ε2 = 9 – j20 e
ε3 = 15 – j10. In the figure, the red curves represent the
methanol’s theoretical relative permittivity and the blue ones
represent the approximate response from the extraction model.
(a)
(b)
(c)
(d)
Fig. 7. Precision evaluation of the sample holder at 2.5 GHz by switching the points for calculating the extraction equation coeficients (a) ε1 = 1, ε2 = 3 – j4 and
ε3 = 5 – j6, (b) ε1 = 1 – j10, ε2 = 1 – j14 and ε3 = 5 – j20, (c) ε1 = 21, ε2 = 15 and ε3 = 21 – j6, (d) ε1 = 21 – j20, ε2 = 9 – j20 and ε3 = 15 – j10.
In this example, it is noticed that the approximation was
good along the entire frequency band of interest, which
validates the choice of ε1, ε2 e ε3 for calibrating the sample
holder in this case.
VI.
CONCLUSION
In this work, an alternative approach for permittivity
measurements of liquid materials with a tubular sample
holder was evaluated.
(a)
Abdulnour’s method [2] was used. Although it was
developed for the analysis of a sample holder that is not
suitable for high temperature measurements, it comprises a
very simple solution for the reverse problem, i.e., for
computing MUT permittivity from measured S parameters.
We investigated if Abdulnour’s extraction model could
be applied to a sample holder made of a cylindrical sample
piercing through the walls of a rectangular waveguide, with a
WR340 standard.
The sample holder was thoroughly simulated for various
values of sample permittivity. The approximation errors were
computed for each point of simulation. The results were
represented in a graphical format.
(b)
Fig. 8. Precision evaluation of the sample holder changing the simulation
frequency from (a) 2 GHz to (b) 3 GHz.
It was found that by properly choosing calibration
standards it’s possible to apply Abdulnour’s extraction model
to the proposed tubular sample holder with good accuracy.
ACKNOWLEDGMENT
The authors would like to thank Petrobras for the
financial funding to this work.
REFERENCES
[1]
[2]
(a)
[3]
[4]
[5]
(b)
[6]
Fig. 9. Simulation of measurement of (a) real and (b) imaginary parts of
the relative permittivity of methanol.
Santos, J. C. A., Dias, M. H. C., Aguiar, A. P., Borges, I., e Borges,
L. E. P. Using the coaxial probe method for permittivity
measurements of liquids at high temperatures. Journal of Microwaves
and Optoelectronics, v. 8, p. 78S-91S, 2009.
Abdulnour, J., Akyel, C., e Wu, K. A generic approach for
permittivity measurement of dielectric materials using discontinuity
in a rectangular waveguide or a microstrip line. IEEE Transactions on
Microwave Theory and Techniques, n. 43 (5), p. 1060-1066, 1995.
Nishikata, A. Scattering analysis for layered cylindrical object
perpendicularly piercing the wider walls of a rectangular waveguide
and its application to εr and µ r measurement. IEEE Transactions on
Microwave Theory and Techniques, vol. 57, n. 6, p. 1602-1611, June
2009.
Akhtar, M. J., Feher, L., e Thumm, M. Noninvasive procedure for
measuring the complex permittivity of resins, catalysts, and other
liquids using a partially filled rectangular waveguide structure. IEEE
Transactions on Microwave Theory and Techniques, vol. 57, n. 2, p.
458-470, February 2009.
Chen, L. F., Ong, C. K., Neo, C. P., Varadan, V. V., e Varadan, V. K.
Microwave electronics: measurement and materials characterization.
John Wiley & Sons, 2004.
Computer Simulation Technology, Inc., CST Microwave Studio®, in
http://www.cst.com, accessed in may 2012..