higher-order mode electromagnetic analysis of a material...

Research ArticleHigher-Order Mode Electromagnetic Analysis of a MaterialSample between Two Flanged Coaxial Probes for BroadbandModelling of Dielectric Measurement Setups

Raúl V. Haro-Báez,1,2 Juan Córcoles ,2 Jorge A. Ruiz-Cruz ,2

José R. Montejo-Garai,3 and Jesús M. Rebollar3

1Universidad de las Fuerzas Armadas ESPE, Departamento de Eléctrica y Electrónica, Quito 171103, Ecuador2Universidad Autónoma de Madrid, Escuela Politécnica Superior, C/Francisco Tomás y Valiente 11, Madrid 28049, Spain3Universidad Politécnica de Madrid, Departamento de Señales, Sistemas y Radiocomunicaciones, Ciudad Universitaria s/n,Madrid 20840, Spain

Correspondence should be addressed to Juan Córcoles; [email protected]

Received 5 November 2018; Accepted 25 December 2018; Published 10 January 2019

Academic Editor: John D. Clayton

Copyright © 2019 Raúl V. Haro-Báez et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The characterization of the dielectric properties of amaterial requires a measurement technique and its associated analysis method.In this work, the configuration involving two coaxial probes with a material for dielectric measurement between them is analyzedwith a mode-matching approach. To that effect, two models with different complexity and particularities are proposed. It will beshown how convergence is sped up for accurate results by using a proper choice of higher-order modes along with a combinationof perfect electric wall and perfect magnetic wall boundary conditions. It will also be shown how the frequency response is affectedby the flange mounting size, which can be, rigorously and efficiently, taken into account with the same type of approach. Thisnumerical study is validated through a wide range of simulations with reference values from another method, showing how theproposed approaches can be used for the broadband characterization of this well-known, but with a recent renewed interest fromthe research community, dielectric measurement setup.

1. Introduction

The behavior of a material under the exposure to electro-magnetic fields is determined by its dielectric properties. Theaccurate knowledge of these properties is essential in scienceand engineering, since they are used in a variety of industries[1, 2]. The properties of a material can be tracked during anindustrial process, and, thus, they can be used to monitora product, incorporating that information into an improvedquality control. They are also important in diverse fields suchas remote sensing, heating, molecular analysis, and creationof liquid standards [3, 4].

More specifically, in microwave engineering [5, 6], thedielectric properties of materials are key data for evaluatingthe performance of circuit substrates, printed antennas,radomes, dielectric resonators, etc. [7]. In modern computer

aided design, these properties are incorporated in the so-called full-wave methods (techniques to solve the Maxwellequations starting from the full electromagnetic model) inorder to provide accurate predictions for each application.In the telecommunications industry, with a massive use ofthe microwave spectrum, the precise characterization of thematerials involved in the microwave components is essentialto optimize and make designs with stringent electrical speci-fications, which are becoming progressively more demandingto address new systems with higher data capacity.

Regardless of the specific area where the dielectric prop-erties are required, the process to characterize them is basedon two techniques: (i) a measurement technique extractinga parameter related directly or (more commonly) indirectlywith the dielectric properties and (ii) an electromagneticanalysis technique (circuit-based or full-wave) modelling the

HindawiAdvances in Mathematical PhysicsVolume 2019, Article ID 6404812, 17 pageshttps://doi.org/10.1155/2019/6404812

http://orcid.org/0000-0002-9595-3992http://orcid.org/0000-0003-3909-8263https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/6404812

2 Advances in Mathematical Physics

1

x

y

z

(a)

204,1

2,12b 2a

d

1

MUT

z

(, )

(b)

Figure 1: Schematic of the one-port open coaxial used for dielectric measurement, showing the material under test (MUT) with its dielectricproperties: (a) 3D schematic view; (b) longitudinal cut (revolution symmetry with respect to z).

measurement, i.e., providing estimations of the results of thatmeasurement for a given material.

With respect to the measurement techniques, there is aplurality of options, such as those based on coaxial probes,transmission lines, free space methods, use of resonators (asin microwave filter characterization), etc. [1]. This varietyof measurement techniques provides different alternativesfor dealing with constrains in the frequency range, therequirements of preparation for the sample, type of expected(low/high) losses in the material, etc. In this work, the coaxialprobe will be used, since it provides a good trade-off betweenmany of these features, can be used for biological tissues, andcan be considered a broadband method, which is one of thegoals of this paper. In addition, its use is very simple andwell-known, especially in microwave engineering, where ithas been used for a long time [8–11].

With respect to the analysis technique used in thedielectric characterization process, when the electromagneticanalysis is based on a full-wave approach, the results willbe more precise, at the expense of a higher computationalburden than simpler circuit analyses. This paper will use anaccurate full-wave electromagnetic method based on mode-matching methods [12, 13], which provide more accurateestimations than circuitalmethods. In comparisonwith otherfull-wave methods [14], it also shows a great efficiency, whichwill be further improved by exploiting the specific featuresof the geometry associated with the two-port coaxial probemeasurement problem.

In fact, this kind of problem has received very recentattention for different types of materials and with differenttypes of probes for the input/output [15–17]. In this paper,two different models based on mode-matching along thelongitudinal direction are proposed for characterizing thetwo-port problem associated with the coaxial probe fordielectric measurement of homogeneous samples. The firstmodel will benefit from different types of boundary condi-tions that can be used to model the simplest structure, madeup of two coaxial lines with the material to be measuredinserted in a virtual circular waveguide that contains the

sample.This circular waveguide will bemodelled with perfectelectric or magnetic wall boundary conditions and with acombination of both cases, which will be shown to providefaster convergent results. The second model will also takethe flanges of the setup into account. The convergence ofthese models with respect to different parameters will bestudied, resulting in a very efficient tool that can be used forthe broadband analysis of the two-port scattering problemof a material sample placed between two coaxial probes fordielectric measurement.

2. The Coaxial Probe forDielectric Measurement

2.1. One-Port Open Coaxial Probe. The coaxial probe open atone end, whose basic schematic version is shown in Figure 1,is one of the preferred techniques for measuring materials.It is a very simple setup, where the sample material can beinserted in the test unit with almost no preparation, fillingthe space just after the coaxial probe. This can be easily donefor a large variety of malleable materials such as liquids orsemisolids whose shape can be adjusted. However, for solids,contact surfaces must be flat enough and tighten to the testunit in order to avoid air gaps between the sample and theprobe that will degrade the measurement.

From the electromagnetic point of view, the problem inFigure 1 is a coaxial probe radiating in a semi-infinite space(the orange block in the figure) whose dielectric propertiesare those of the material under test (MUT), which may havelosses: complex dielectric permittivity 𝜖=𝜖-j𝜖 and complexmagnetic permeability 𝜇=𝜇-j𝜇. The coaxial transmissionline has two conductors of internal and external radiusa and b, respectively, with a supporting dielectric (bluecolor in Figure 1). These parameters are typically adjustedfor providing 50 Ω characteristic impedance, as in classicmicrowave equipment. In the analysis technique, a semi-infinite space is assumed for the region related to the MUT,and, thus, the thickness (d) and the transversal size of thesample (Rsample) must be electrically large enough in order

Advances in Mathematical Physics 3

MUT

x

y

z

1 2

(, )

(a)

MUT1

2

(, )

(b)

Figure 2: Schematic of the two-port coaxial holder for dielectric measurement, showing the material under test (MUT): (a) 3D schematicview; (b) longitudinal cut (revolution symmetry with respect to z).

to confine the incoming electromagnetic field excited by theprobe within the MUT limits. Within the same modelling,the metallic flange is assumed to be large enough for avoidingbackward radiation.

The measurement technique for this case extracts thereflected electromagnetic wave at the one-port problem. Thesignal excited by the generator connected to the coaxialpropagates through the transmission line and reaches thematerial, where it is partially reflected back. This reflectioncoefficient depends on the electromagnetic properties of thedielectric material. Thus, from this measurement, the valuesof the dielectric properties of the material in contact withthe probe can be estimated. However, there is not a directclosed formula relating the 𝜖, 𝜇 of the MUT (this is the goalof the dielectric characterization process) to the measuredreflection coefficient, and this is where the different analysistechniques, from equivalent circuits to full-wave approaches,play a key role. It is important to note that this is also aclassic problem in microwave engineering when designingantenna feed systems, and, thus, there has been continuousresearch in this subject from different perspectives during thelast decades [8–11, 18–21].Thesemodels have evolved betweenthe complexity of the lumped and distributed elements in theinitial equivalent circuits, to the way of approximating theelectromagnetic field functions in the more recent full-waveapproaches.

2.2. Proposed Models for the Characterization of the Two-PortProblem. The two-port measurement is a method directlyrelated to the coaxial probe [22–24], where a thin sample ofthe MUT is inserted between two coaxial lines as in Figure 2,allowing a quick measurement of the return and insertionloss as a function of frequency. The analysis methods arevery closely related to the coaxial probe discussed in theprevious subsection. Nevertheless, different formulationshave been the subject of research in very recent years [15–17].

In this work, the two models shown in Figure 3 will beproposed for MUTs with homogeneous dielectric properties.The simplest model A is shown in Figures 3(a) and 3(b). Itwill allow a straight analysis through mode-matching, sinceit involves a problem with two waveguide discontinuities

between coaxial and circular waveguides [25] surrounded byperfect electric wall (PEW) at its boundaries. A variation willbe introduced to account for the particularities of the circularwaveguide that contains the sample. The computations willbe also done with a waveguide having perfect magneticwall (PMW) at its boundaries and with a combination ofboth PEW/PMW.Their convergence properties will be tested,allowing us to decide the faster method for the two-portproblem characterization.

An alternative model B is shown in Figures 3(c) and3(d), which takes the finite flanges associated with theinput/output coaxial lines into account. This model can alsobe simulated with mode-matching, as long as the problemof waveguide bifurcations is included in the formulation.This type of approach has been used in the past for probesand, more recently, to improve the mode-matching accuracyin radiation problems [11, 26, 27]. In this work, the modelwhich incorporates the finite flange information will beproposed for the two-port coaxial holder. It will exploit, asmodel A, the advantage of the circular revolution, since onlymodes with no circular variation will be needed to solve thisproblem.

3. Electromagnetic Analysis of the ProposedModels through Mode-Matching

The proposed two models are analyzed with the mode-matching method, comprised of two main stages. First, themodes of all the waveguides involved in the problem arecalculated.Then, the modal series to expand the fields in eachwaveguide are matched at each side of the step discontinuity,providing the Generalized Scattering Matrix (GSM) of eachstep. Finally, the GSMs of each step are cascaded [14, 28].Themain steps are briefly described now.

3.1. Characterization of the Waveguides. The electromagneticfield inside homogeneous waveguides can be described interms of transversal electromagnetic (TEM), transversalelectric (TE), and transversal magnetic (TM) modes [29].The electromagnetic field of a TE or TM mode is derivedfrom a scalar function Φ which is the solution to the


26 2,1

MODEL B d

MUT

2b

2b 2b

2a

2a 2a

d

MUT

MODEL A z

z

(b)(a)

(d)(c)

(, )

(, )

Figure 3: Schematic proposedmodels for the two-port characterization (additional details will be given in Figures 6 and 14).Model A: circularwaveguide with material sample within two coaxial lines: (a) 3D view; (b) longitudinal cut with dimensions defining the problem. Model Btaking the flange into account: (c) 3D view; (d) longitudinal cut with main dimensions.

bS

C

PEW

S

PEW(PMW)

CC

, ,

aa

Figure 4: Cross-section of the type of homogenous waveguides used in this work: circular (with PEW and PMW boundary conditions) andcircular coaxial.

Helmholtz equation with Neumann or Dirichlet boundaryconditions:

∇2𝑡 Φ + 𝑘𝑐2Φ = 0, 𝑖𝑛 𝑆, 𝑤𝑖𝑡ℎ ∇𝑡Φ ∙ 𝑛⌋𝐶 = 0 (TE) ,∇2𝑡 Φ + 𝑘𝑐2Φ = 0, 𝑖𝑛 𝑆, 𝑤𝑖𝑡ℎ Φ⌋𝐶 = 0 (TM) ,

(1)

where ∇𝑡 is the transversal to z (two dimensional) nabladifferential operator, S is the cross-section with normal 𝑛,and C is its contour. In our case for the models in Figure 3,only coaxial and circular waveguides are needed, as shown inFigure 4. In (1), function Φ is the longitudinal z-component,up to a constant factor (not depending on the spatial point),of themagnetic or electricmodal field, of theTE orTMmode,

respectively. The cutoff frequency of the mode is directlyrelated to the cutoff wavenumber of the mode 𝑘𝑐 [7, 29].

For the coaxial waveguide, the TEM mode is also neces-sary to build the complete mode spectrum of the waveguide,and it can be easily computed analytically by solving theLaplace equation, where Φ represents the potential functionthat provides the electric field:

∇2𝑡 Φ = 0, 𝑖𝑛 𝑆, 𝑤𝑖𝑡ℎ Φ⌋𝐶𝑖 = V𝑖, 𝑖 = 1, 2 (TEM) . (2)OnceΦ is solved for TE, TM, andTEMmodes, the transversalcomponents are obtained using basic gradient operations[7, 29]. When the cross-section of the waveguide belongsto a canonical case (such as rectangular or circular), thefunction Φ is readily obtained by applying the technique


z

z = 0

z

z = 0

PEWPE

WPE

W

(PMW

)

(s) (s)

()

() am(w)

bm(w)

am(w)

bm(w)

bn(s)bn

(s)

an(s)an

(s)

(, )

(, )

(s, s)

(s, s)

= +

(a) (b)

Figure 5: Basic waveguide step between circular and coaxial waveguides for the models in Figure 3.

of the separation of variables. In our case, it is well-knownthat the longitudinal field of the TE and TM modes for thewaveguides in Figure 4 is written in terms of the Besselfunction of the first (𝐽𝑛) and second kind (𝑌𝑛):

Φ = 𝐾 𝐶𝑛 (𝑘𝑐𝜌)(cos (𝑛𝜙)sin (𝑛𝜙)) , (3)where 𝐾 is related to the power carried by the mode andis thus used as a normalization factor. For a hollow circularwaveguide, 𝐶𝑛(𝑢) = 𝐽𝑛(𝑢) in (3). The cutoff wavenumbers ofthe modes for a PEW circular waveguide are 𝑘𝑇𝐸𝑐 = 𝑝𝑚𝑛/𝑎,𝑘𝑇𝑀𝑐 = 𝑝𝑚𝑛/𝑎, and 𝑝𝑚𝑛, 𝑝𝑚𝑛 are the zeros of 𝐽𝑛(𝑢) and𝐽𝑛(𝑢), respectively. For a PMW boundary at the circularwaveguide, the dual solution with 𝑘𝑇𝐸𝑐 = 𝑝𝑚𝑛/𝑎, 𝑘𝑇𝑀𝑐 =𝑝𝑚𝑛/𝑎 is obtained [29]. This case for PMW is also foundin [27], where the virtual waveguide with PMW is used forimproving the field approximation in horn problems. For acoaxial waveguide in (3), 𝐶𝑛(𝑢) = 𝐴𝐽𝑛(𝑢) + 𝐵𝑌𝑛(𝑢) is acombination of Bessel functions whose amplitudes and thecutoff wavenumber are obtained by imposing the boundarycondition at the contour [7, 29].

It is important to note that the cutoffwavenumbers for thecoaxial waveguide must be computed for each specific aspectratio b/a. Nevertheless, in our problem, since the structurewill be excited with a TEM mode, which does not haveangular variation and has PMWsymmetry at both xz- and yz-planes, all the higher-order modes generated in the structureare TM0n modes in both the circular and coaxial waveguides.

3.2. Mode-Matching Formulation. As every waveguide in thetwo models of Figure 3 is surrounded by PEW (or PMW),the transverse electromagnetic field in these structures canbe expanded in a set of orthogonal TE, TM, and TEMmodes

formulated in the previous section (although TE modes willnot be needed as discussed in previous section, they will bekept in this formulation for the sake of the generality). Withthem, the electromagnetic field in each point of the structureis represented as a modal series.

In the reference plane 𝑧 = 0 (see Figure 5), the fields canbe written as a sum of orthonormal modes with the incidentand reflected amplitudes at the discontinuity plane. In eachwaveguide (index g=w, for the larger, g=s for the smaller), thefollowing can be written:

→E (𝑔)𝑡 ⌋𝐴𝑔,𝑧=0

=𝑁𝑔∑𝑚=1

(𝑎(𝑔)𝑚 + 𝑏(𝑔)𝑚 ) →e (𝑔)𝑚 ,

→H(𝑔)𝑡 ⌋𝐴𝑔,𝑧=0

=𝑁𝑔∑𝑚=1

(𝑎(𝑔)𝑚 − 𝑏(𝑔)𝑚 ) →h (𝑔)𝑚(4)

where →e (𝑔)𝑚 , →h (𝑔)𝑚 are the transverse modal vectors corre-sponding to them-th TE, TM, or TEMmode of the guide (g),which can be propagating or evanescent. These fields take thelateral boundary conditions into account while the electricand magnetic modal functions are related by the modeimpedance as usual, which are also typically normalized tounity as follows.

∬𝐴𝑔

→e (𝑔)𝑛 × →h (𝑔)𝑚 ∙ ẑ𝑑𝑆 = {{{1, 𝑚 = 𝑛0, 𝑚 ̸= 𝑛 , 𝑔 ≡ 𝑤, 𝑠 (5)

The electric and magnetic field boundary conditions (EFBCandMFBC, respectively) for this problem relate the transver-sal fields in (4) at both sides of the discontinuity.


𝐸𝐹𝐵𝐶 𝑖𝑛 𝐴𝑤: ẑ × →E (𝑤) = {{{0 𝑖𝑛 𝐴𝑐, 𝑧 = 0ẑ × →E (𝑠) 𝑖𝑛 𝐴𝑠, 𝑧 = 0

𝑀𝐹𝐵𝐶 𝑖𝑛 𝐴𝑠 𝑖𝑛: ẑ × →H(𝑤) = ẑ × →H(𝑠) 𝑖𝑛 𝐴𝑠, 𝑧 = 0(6)

These equations are imposed by means of a classic Galerkinmethod where the EFBC is tested by 𝑁w magnetic modalfields

→h(𝑤)

𝑚 from the larger waveguide, and theMFBC is testedwith 𝑁s electric modal fields →e (𝑠)𝑚 in the smaller waveguide.This field-matching procedure also requires truncating theseries (4) to 𝑁g terms and collecting the modal amplitudesin column vectors. The final equations expressed in matrixform become

EFBC: (a𝑤 + b𝑤) = X𝑡 (a𝑠 + b𝑠) , (𝑁𝑤 𝑒𝑞𝑠.)MFBC: X (a𝑤 − b𝑤) = (b𝑠 − a𝑠) , (𝑁𝑠 𝑒𝑞𝑠.) ,

(7)

where matrix X is filled with the interaction between theelectric andmagnetic modal functions fromboth waveguidesin the step:

[𝑋𝑚𝑛] = ∬𝐴𝑠

→e (𝑠)𝑚 × →h (𝑤)𝑛 ∙ ẑ𝑑𝑆. (8)The equation system presented in (7) provides, after itsresolution, the Generalized Scattering Matrix (GSM) S of theproblem in the form:

b = Sa,bw = S𝑤𝑤a𝑤 + S𝑤𝑠a𝑠bs = S𝑠𝑤a𝑤 + S𝑠𝑠a𝑠

(9)

with:

S𝑤𝑤 = X𝑡FX − I𝑤S𝑤𝑠 = S𝑡𝑠𝑤=X𝑡FS𝑠𝑠 = F − I𝑠

F = 2 (I𝑠 + XX𝑡)−1 .

(10)

This formulation has been used in the past many times.Examples of the inner-cross product computation for differ-ent waveguide steps can be found in [25, 30, 31]. After thiscomputation, the cascading of the GSM of each step in theproblem is carried out systematically [14, 28], as it will beexplained in the next section for the two proposed models.

4. Results for the Proposed Models

The mode-matching technique described in the previoussection has been applied for the two proposed models inFigure 3, further detailed in this section. The first model usestwo coaxial waveguides attached to both sides of a virtualcircular waveguide filled by the sample material, which is

Table 1: Basic data for the test cases.

Item ValueRouter IO coaxial (b) 1.75 mmRinner IO coaxial (a) 0.46 mm𝜖r,coax 2.54𝜖r,sample (MUT) 10-j0.1 (tan 𝛿e=0.01)𝜇r,sample (MUT) 5-j0.05 (tan 𝛿m=0.01)Sample Thickness (d) 0.2 mm/1.0 mmFrequency range 100 MHz – 20 GHz

denoted asmodel A.The second model takes into account thefinite flanges used in the measurement, and it is called modelB.

The test cases will be evaluated under the two models Aand B, for the broadband frequency range from 0.1 GHz to20 GHz, varying their constituent parameters. The coaxialwaveguide used for the measurement has an inner and anouter radius of a=0.46mm and b=1.75mm, respectively, filledby Teflon (𝜖r,coax = 2.5) for providing 50 Ω. The samplematerial for the tests has 𝜖r,sample = 10, 𝜖r,sample = 0.1(tan 𝛿e = 0.01) and 𝜇𝑟,sample = 5, 𝜇𝑟,sample = 0.05 (tan 𝛿m =0.01). This material is prepared with a thickness of d=0.2mmand d=1mm. These main data are shown in Table 1.

The reference values are provided by themethod from theDielectric Assessment Kit (DAK) product line manufacturedby the Swiss company SPEAG [32], used in high-precisionsystems for measurements of dielectric properties. Theseinstruments allow dielectric parameter measurements over avery broad frequency range.

4.1. Model A with the Step between the Coaxial and theLarge Circular Waveguide Containing the Sample. Model Ais detailed in Figure 6. This model will be also dividedinto three cases (A1, A2, A3), depending on the boundaryconditions used for the outer boundary of the virtual circularwaveguide containing the sample. In all these cases, fromthe analysis point of view, the model only requires thecharacterization of two GSMs, called SA and SB accordingto Figure 6(b). However, SA and SB are only different inthe port numbering, but both represent a step between thesame types of waveguides with the same dimensions; thecomputational cost reduces to computing a single GSM, sinceonce SA is obtained, SB is also available.Then, they have just tobe cascaded through well-known operations to get the GSMrepresenting the full structure.

4.1.1. Case Model A1: Large Virtual Circular Waveguide withPEW. Since the problem has no angular variation, under theTEM excitation by the coaxial line, the series (4) in the mode-matching will only require the TEMmode at the coaxial lines,and TM0n modes at the coaxial and the circular waveguides.The field in the circular waveguide will be always representedin both models A and B by TM0n modes. However, inorder to see the importance of the higher-order TM0n modesat the input/output, we will first compute in Figure 7 the


2b2b

d

MUT

Model A1) PEW

Model A2) PMW

Model A3) PEW/PMW

PEW

2a2a

1 2

z

(, )

coax

(a)

COAXIAL CIRCULAR COAXIAL

WAVEGUIDE

1 2

[SA]

[SB]

(b)

Figure 6: (a) Dimensions defining the model A (with revolution symmetry): case A1, PEW; case A2, PMW; case A3, combination of PEWand PMW; (b) Block diagram of the analysis by mode-matching, showing the symmetry plane.

0 5 10 15 20Frequency (GHz)

−35

−30

−25

−20

−15

−10

−5

0

S11 Ref DAKRSample =20 bRSample =30 bRSample =40 bRSample =100 bRSample =200 b


−35

−30

−25

−20

−15

−10

−5

0

S21 Ref DAKRSample =20 bRSample = 30 bRSample = 40 bRSample =100 bRSample =200 b

３ 11

(dB)

||

３ 21

(dB)

||

Figure 7: Magnitude (dB) of S11 and S21 for model A1 using only the TEM mode at the input/output coaxial lines, for different values of thevirtual circular waveguide containing the sample (thickness d=0.2 mm).



−25

−20

−15

−10

−5

0

S11 Ref DAKRSample =10 bRSample =20 bRSample =30 bRSample =40 bRSample =100 bRSample =200 b


−35

−30

−25

−20

−15

−10

−5

0

S21 Ref DAKRSample =10 bRSample =20 bRSample =30 bRSample =40 bRSample =100 bRSample =200 b


−140

−120

−100

−80

−60

−40

−20

0

S11 Ref DAKRsample = 20 bRsample = 30 bRsample = 40 bRsample = 100 bRsample = 200 b


−150

−100

−50

0

50

100

Phas

e (S 2

1) (d

egre

es)

Phas

e (S 2

1) (d

egre

es)


３ 11

(dB)

||

３ 21

(dB)

||

Figure 8: Magnitude (dB) and phase (degrees) of S11 and S21 for model A1 with TEM and TM0n modes at the input/output coaxial lines, fordifferent values of the virtual circular waveguide containing the sample (thickness d=0.2 mm).

results by only using the TEMmode to approximate the fieldwithin the coaxial lines, as initial pioneer works with probesdid.

In the simulation in Figure 7, the magnitude of the S11and S21 parameters is shown for different values of the radiusRsample of the virtual circular waveguide between the inputand output coaxial lines. The analysis starts with Rsample equalto 20 times the outer radius of the input/output coaxial,which shows a ripple that starts to decrease when this circularwaveguide radius is enlarged. In this case, when Rsampleincreases, the results in both reflection and transmissionare flatter and converge to the larger case of Rsample=200b.

However, the results do not reproduce the results of the DAKreference values, showing a relevant difference in the curves.

In order to overcome this issue, Figure 8 shows the sametype of analysis as Figure 7, but using also TM0n modesin the coaxial transmission lines in addition to the TEMmode. Although the ripple also appears as in the previouscase and decreases when Rsample increases as well, now theresults converge to those of the reference case obtainedby DAK. Thus, the response improves the results obtainedfrom the previous case, having a very good agreement. Thephase is also shown in this graph, showing also a very goodagreement. The phase reference is taken at the interface



−35

−30

−25

−20

−15

−10

−5

0



−35

−30

−25

−20

−15

−10

−5

0



−150

−100

−50

0



−150

−100

−50

0

50

100

S 21 Ref DAKS sample = 20 bS sample = 30 bS sample = 40 bS sample = 100 bS sample = 200 b

Phas

e (S 2

1) (d

egre

es)

Phas

e (S 1

1) (d

egre

es)

３ 11

(dB)

||

３ 21

(dB)

||

Figure 9: Magnitude (dB) and phase (degrees) of S11 and S21 for model A1 with TEM and TM0n modes at the input/output coaxial lines, fordifferent values of the virtual circular waveguide containing the sample (thickness d=1mm).

between the coaxial and the circular waveguide in all thecases.

The same type of analysis is now done with the samplehaving a thickness of 1mm. From now on, the analysis will bealways performed with TEM and TM0n modes in the coaxiallines, since its need has been clearly justified in Figures 7and 8. The results for the new thickness d=1mm are shownin Figure 9, where the same type of performance as with thethinner sample can be seen: ripple in the curves related tothe size of the large virtual circular waveguide containing thesample. This ripple reduces when the size of Rsample increases

till achieving convergent results matching those of the DAKreference.

4.1.2. Case Model A2: Large Virtual Circular Waveguide withPMW. InmodelA,we are expecting that the electromagneticfield at the lateral limit of the circular waveguide (i.e., for large𝜌) is small enough to represent a suitable model for the actualmeasurement, where it is assumed that the field is confinedwithin the MUT limits. In order to confirm that, the modelshould not provide very different results when the lateral



−35

−30

−25

−20

−15

−10

−5

0

Ref DAKRsample = 10 bRsample = 20 bRsample = 30 bRsample = 40 bRsample = 100 bRsample = 200 b


−35

−30

−25

−20

−15

−10

−5

0

Ref DAKRsample = 10 bRsample = 20 bRsample = 30 bRsample = 40 bRsample = 100 bRsample = 200 b

３ 11

(dB)

||

３ 21

(dB)

||

Figure 10: Magnitude (dB) of S11 and S21 for model A2 with TEM and TM0n modes at the input/output coaxial lines, for different values ofthe virtual circular waveguide containing the sample (thickness d=1mm).

boundary conditions of the circularwaveguide are changed toPMWand the waveguide steps in Figure 6(b) are recomputedwith these new modes. This is confirmed with the resultsshown in Figure 10 for a sample thickness of 1mm, wherethe magnitude of S11 and S21 has a very similar behavior tothat of the case of model A1 with PEW in Figure 9. There isalso a ripple in the mode-matching results, more importantfor smaller Rsample, which disappears when Rsample is largeenough.

In all the simulations done so far, we have presented theresults with a number of modes large enough to achieveconvergent results with respect to the number of termsin the series (4). In this type of problems, as a generalrule to overcome the relative convergence problem [14], thenumber of modes in the modal series is selected followinga criterion related to the size of the involved waveguides.One possible solution is keeping a relation between themodes of the different waveguides as the square root oftheir areas. This means that for a step between circularwaveguides, the ratio of the modes between two waveg-uides in a step discontinuity would be the ratio of theirradii.

The convergence has been verified with several analyses,varying the number ofmodes while keeping a fixed geometry.One of those analyses is shown in Figure 11. It shows theresults of this last case in Figure 10, although the behaviorwould be very similar for other geometries and for bothmodels A or B. In Figure 11, the black solid line represents thereference and the other lines represent the results for differentnumber of modes in the coaxial lines Ncoax and in the largecircular waveguide Ncirc.

4.1.3. Case Model A3: Large Virtual Circular Waveguide withCombination of PEW and PMW. Figure 12 shows the resultsofmodels A1 and A2 with Rsample=10b plotted within the samegraph. It can be seen that the ripple of the responses for thetwo cases (PEW/PMW at the outer lateral boundary of thecircular waveguide, respectively) are interleaved. This wouldalso happen in the other results of Figures 9 and 10, althoughit is more difficult to see when the ripple starts to decreasefor large Rsample . In addition, since both boundary conditionsin the circular waveguide are approximations of the actualmeasurement problem, the combination of both results couldbe also another suitable analysis model. Moreover, this typeof strategy has also been used in [26].

Following this rationale, Figure 12 also shows the combi-nation of the PEW analysis in model A1 with those using thePMW in model A2, which will be the model A3 proposed inthis subsection:

𝑆11 = (𝑆𝑃𝐸𝑊11 + 𝑆𝑃𝑀𝑊11 )

2 ,

𝑆21 = (𝑆𝑃𝐸𝑊21 + 𝑆𝑃𝑀𝑊21 )

2 .(11)

In order to verify model A3, Figure 13 shows the results fordifferent values of𝑅𝑠𝑎𝑚𝑝𝑙e. It can be seen how the combinationof the two boundary conditions converges to the referenceresults with smaller 𝑅𝑠𝑎𝑚𝑝𝑙e, in both magnitude and phase.This translates into a higher computational efficiency, sincea smaller number of modes will be needed to get the resultsusing this smaller circular waveguide (in comparison withmodels A1, A2) containing the sample.



−35

−30

−25

−20

−15

−10

−5

0

S11 Ref DAKNcoax = 4; Ncirc=830Ncoax = 4; Ncirc=747Ncoax = 3; Ncirc=664Ncoax = 3; Ncirc=581Ncoax = 2; Ncirc =498Ncoax = 2; Ncirc =415Ncoax = 1; Ncirc = 500


−35

−30

−25

−20

−15

−10

−5

0

S21 Ref DAKNcoax = 4; Ncirc=830Ncoax = 4; Ncirc=747Ncoax = 3; Ncirc=664Ncoax = 3; Ncirc=581Ncoax = 2; Ncirc=498Ncoax = 2; Ncirc=415Ncoax = 1; Ncirc=500

３ 11

(dB)

||

３ 21

(dB)

||

Figure 11: Convergence analysis. Magnitude (dB) of S11 and S21 for model A2 with TEM and TM0n modes at the input/output coaxial lines,for different values of the number of modes Ncoax and Ncirc (thickness d=1 mm with Rsample=200b).


−35

−30

−25

−20

−15

−10

−5

0

S11 Ref DAKRsample = 10 b; Case A1Rsample = 10 b; Case A2Rsample = 10 b; Case A3


−35

−30

−25

−20

−15

−10

−5

0

S21 Ref DAKRsample = 10 b; Case A1Rsample = 10 b; Case A2Rsample = 10 b; Case A3

３ 11

(dB)

||

３ 21

(dB)

||

Figure 12: Magnitude (dB) of S11 and S21 comparing models A1, A2, and A3 with TEM and TM0n modes at the input/output coaxial lines,for Rsample=10b and sample thickness d=1 mm.

4.2. Model B with the Steps for Taking the Finite Flangesinto Account. The second model to compute the two-portproblem is shown in Figure 14. This geometry takes thefinite flanges into account, as well as the outer size ofthe coaxial lines involved in the measurement. This maybe relevant for some cases where the assumption that all

the fields are confined within the MUT limits starts tobe not fully compliant with the experimental data. Thestructure has been plotted symmetrically with respect to themiddle plane for the sake of the simplicity, but data fromthe input coaxial could be made different from the outputcoaxial.



−35

−30

−25

−20

−15

−10

−5

0

S11 Ref DAKRsample = 10 bRsample = 20 bRsample = 30 bRsample = 40 b


−35

−30

−25

−20

−15

−10

−5

0

S 21 Ref DAKRsample = 10 bRsample = 20 bRsample = 30 bRsample = 40 b


−150

−100

−50

0

S11 Ref DAKRsample = 10 bRsample = 20 bRsample = 30 bRsample = 40 b


−100

−50

0

50

100

Ref DAKRsample = 10 b

Rsample = 20 b

Rsample = 30 b

Rsample = 40 b

Phas

e (S 2

1) (d

egre

es)

Phas

e (S 1

1) (d

egre

es)

３ 11

(dB)

||

３ 21

(dB)

||

Figure 13: Magnitude (dB) and phase (degrees) of S11 and S21 formodel A3 with TEM and TM0n modes at the input/output coaxial lines, fordifferent values of the virtual circular waveguide containing the sample (thickness of d=1mm).

From the mode-matching point of view, the new geom-etry involves two new virtual coaxial transmission lines ateach side of the problem, associated with the incorporationof the finite flange with radius Rflange and length 𝐿𝑓𝑙𝑎𝑛𝑔eto the model. They have an inner radius of Ro and Rflange,respectively, both with outer radius of Rsample. They are filledby air.

This geometry involves a new step discontinuity betweenthe two virtual air-filled coaxial lines associated with theouter part of the structure, as is shown in Figure 14(b)with the block [Scoax]. Model B also involves a bifurcationbetween, at one side, the Teflon coaxial (radii a and b) and

one air-filled coaxial (radii Rflange and Rsample) and, at theother side, the larger circular waveguide of radius Rsample. Inthis problem, two additional virtual ports appear, carryingthe signal not confined within the MUT limits and nottaken into account by model A. Once the GSM of thesteps and bifurcations are computed, they are also easilycascaded.

For the following tests, in addition to the data inTable 1, the flange length will be Lflang𝑒=1mm, with acover for the input/output coaxial of 0.5mm thickness,i.e., Ro=b+0.5mm. As was done for model A, the firstsimulation in Figure 15 is done using a fixed geometry


2a 2b

d

MUT

z

1 2

3 4

PEW

coax

00

(, )

(a)

CIRCULARCOAXIAL COAXIAL

WAVEGUIDE1 2

43

[SA]

[SB]

VIRTUALPORT

VIRTUALPORT

[S]

coax

[S]

coax(b)

Figure 14: (a) Dimensions defining the model B (with revolution symmetry) taking the finite flanges into account. (b) Block diagram of theanalysis by mode-matching, showing the symmetry plane.


−35

−30

−25

−20

−15

−10

−5

0



−35

−30

−25

−20

−15

−10

−5

0


３ 11

(dB)

||

３ 21

(dB)

||

Figure 15: Magnitude (dB) of S11 and S21 formodel B with TEM and TM0n modes at the input/output coaxial lines, for different values of theRsample (thickness d=0.2 mm, Rflange=5.4b).



−35

−30

−25

−20

−15

−10

−5

0|S

11| (

dB)

S11 Ref DAKRflange = 5.4 b; Ncoax = 4Rflange = 28.2 b; Ncoax = 4Rflange = 28.2 b; Ncoax = 10


−35

−30

−25

−20

−15

−10

−5

0

|S21

| (dB

)

S21 Ref DAKRflange = 5.4 b; Ncoax = 4Rflange = 28.2 b; Ncoax = 4Rflange = 28.2 b; Ncoax = 10

Figure 16: Magnitude (dB) of S11 and S21 formodel B with TEM and TM0n modes at the input/output coaxial lines, for different values of theflange size, and number of modes (thickness d=0.2 mm, Rsample=50b).

with finite Rflange =5.4b, varying the radius of the Rsampleassociated with the circular waveguide between 10b and100b. In this case, although the results are very similarfor the different Rsample , the finite size of the flange isleading to results different from those of the DAK refer-ence, regardless of the size of the virtual circular waveg-uide required in the mode-matching simulations. The DAKvalues corresponded to the typical geometry with a largeenough flange and electromagnetic field fully confinedwithinthe MUT limits, which is coherent with results of modelA.

This effect of the size of the flange is further analyzedin Figure 16, where the simulations are run with differentnumber of modes and different values for Rflange. The rippleis now associated with Rflange since these simulations havea fixed value of Rsampl𝑒. It is also important to note thatadditional terms in series (4) are needed for having con-vergent results in terms of number of modes, since nowthe structure is more complex involving more waveguidesteps.

In this case, since the new air-filled coaxial lines havea large size, many modes will be needed to approximatethe field within them, with a high aspect ratio between theouter and inner conductor. Thus, their modes will have tobe computed carefully to avoid instabilities related to thebehavior of the𝑌𝑛(𝑢) function in (3)when it is evaluated closeto zero.

Figure 17 shows the results of model B for differentvalues of Rflange in Figure 14(a), with a fixed value of 𝑅𝑠𝑎𝑚𝑝𝑙e.Results show that for a large flange, the DAK reference isachieved. Figure 17 also shows the backward and forwardradiation for different values of the flange, characterized

by the S-parameter between the input (or output) Tefloncoaxial port and the virtual air-filled coaxial ports 3 and 4in Figure 14. As expected, for a smaller flange, large peaksof signal at specific frequencies are not confined within theMUT limits. Figure 18 shows the same type of analysis ofFigure 17 for a thickness of 1mm, where it can be seenthat these peaks of forward and backward radiation are lesssignificant.

5. Conclusions

The mathematical formulation of the mode-matchingmethod applied to the two-port scattering problemarising in a material placed between two coaxial probeshas been presented. This well-known configuration hasreceived renewed attention in very recent years. In thisline, this paper has developed two different approachesto characterize this dielectric measurement setup forbroadband applications. The analysis of the problemwith the MUT considering both PMW and PEW hasbeen studied, while their combination has been shown toimprove the efficiency of the method. In all cases, the use ofhigher-order modes in the coaxial waveguide at the MUTinterface is essential to obtain fast and accurate results.The effect of the finite flanges has been also efficientlyand rigorously assessed, proving that a flatter frequencyresponse is achieved when the field is confined withinthe MUT, i.e., when large flanges are considered. Twosetups have been analyzed and their results have beencompared with the reference data provided by the DAKcharacterization method, showing excellent agreementbetween them.



−35

−30

−25

−20

−15

−10

−5

0

S11 Ref DAKRflange = 5.4 bRflange = 11.1 bRflange = 16.8 bRflange = 22.5 bRflange = 28.2 b


−35

−30

−25

−20

−15

−10

−5

0

S21 Ref DAKRflange =5.4 bRflange = 11.1 bRflange = 16.8 bRflange = 22.5 bRflange = 28.2 b


−90

−80

−70

−60

−50

−40

−30

−20

Rflange = 5.4 bRflange = 11.1 bRflange = 16.8 bRflange = 22.5 bRflange = 28.2 b


−90

−80

−70

−60

−50

−40

−30

−20


３ 31

(dB)

||

３ 41

(dB)

||

３ 11

(dB)

||

３ 21

(dB)

||

Figure 17: Magnitude (dB) of S31 and S41 formodel B with TEM and TM0n modes at the input/output coaxial lines, for different values of theflange size (thickness d=0.2 mm, Rsample=50b).

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper.

Acknowledgments

This work was supported in part by the Spanish Govern-ment (Agencia Estatal de Investigación, Fondo Europeo deDesarrollo Regional) under Grant TEC2016-76070-C3-1/2-R (AEI/FEDER, UE) and in part by the program of Comu-nidad de Madrid under Grant S2013/ICE-3000. The authorswant to acknowledge SPEAG company, Zurich, Switzerland,especially Pedro Crespo-Valero and FerencMuranyi, for theirassistance with the DAK results.



−35

−30

−25

−20

−15

−10

−5

0

S11 Ref DAKRflange = 5.4 bRflange = 11.1 bRflange = 16.8 bRflange = 22.5 bRflange = 28.2 b


−35

−30

−25

−20

−15

−10

−5

0

S 21 Ref DAKRflange = 5.4 bRflange = 11.1 bRflange = 16.8 bRflange = 22.5 bRflange = 28.2 b


−100

−80

−60

−40

−20

0



−100

−80

−60

−40

−20

0


３ 31

(dB)

||

３ 41

(dB)

||

３ 11

(dB)

||

３ 21

(dB)

||

Figure 18: Magnitude (dB) of S31 and S41 formodel B with TEM and TM0n modes at the input/output coaxial lines, for different values of theflange size (thickness d=1 mm, Rsample=50b).

References

[1] Keysight Technologies, “Basics of measuring the dielectricproperties of materials,” Application Note 5989-2589EN, USA,March 7, 2017.

[2] V. Komarov, S. Wang, and J. Tang, “Permittivity and measure-ment,” in Encyclopedia of RF and Microwave Engineering, pp. 1–20, John Wiley & Sons, 2005.

[3] U. Kaatze and Y. Feldman, “Broadband dielectric spectrometryof liquids and biosystems,” Measurement Science and Technol-ogy, vol. 17, no. 2, pp. R17–R35, 2006.

[4] A. P. Gregory and R. N. Clarke, “A review of RF and microwavetechniques for dielectric measurements on polar liquids,” IEEE

Transactions on Dielectrics and Electrical Insulation, vol. 13, no.4, pp. 727–743, 2006.

[5] R. W. Bruce, “New Frontiers in the Use of Microwave Energy:Power and Metrology,” in Proceedings of the Materials ResearchSociety Symposium, vol. 124, pp. 3–15, 1990.

[6] L. F. Chen, C. K. Ong, C. P. Neo, V. V. Varadan, and V. K.Varadan, Microwave Electronics: Measurement and MaterialsCharacterization, John Wiley & Sons, Ltd, 2004.

[7] D. M. Pozar,Microwave Engineering, John Wiley, Hoboken, NJ,USA, 2011.

[8] R. W. P. King and L. D. Scott, “The Cylindrical Antenna as aProbe for Studying the Electrical Properties of Media,” IEEE


Transactions on Antennas and Propagation, vol. 19, no. 3, pp.406–416, 1971.

[9] E. C. Burdette, F. L. Cain, and J. Seals, “In vivo probe mea-surement technique for determining dielectric properties atVHF through microwave frequencies,” IEEE Transactions onMicrowave Theory and Techniques, vol. 28, no. 4, pp. 414–427,1980.

[10] J. R. Mosig, J.-C. E. Besson, M. Gex-Fabry, and F. E. Gardiol,“Reflection of an open-ended coaxial line and application tonondestructive measurement of materials,” IEEE Transactionson Instrumentation andMeasurement, vol. IM-30, no. 1, pp. 46–51, 1981.

[11] J. D. Baños-Polglase and J. D. Rebollar, “Coaxial probes withfinite ground plane,” IEEE Electronics Letters, vol. 24, no. 5, pp.291-292, 1988.

[12] P. J. B. Clarricoats and K. R. Slinn, “Numerical method for thesolution of waveguide-discontinuity problems,” IEEE Electron-ics Letters, vol. 2, no. 6, pp. 226–228, 1966.

[13] A. Wexler, “Solution of waveguide discontinuities by modalanalysis,” IEEE Transactions on Microwave Theory and Tech-niques, vol. 15, no. 9, pp. 508–517, 1967.

[14] T. Itoh, Numerical Techniques for Microwave and Millimeter-Wave Passive Structures, John Wiley, New York, NY, USA, 1989.

[15] M. W. Hyde and M. J. Havrilla, “A broadband, nondestructivemicrowave sensor for characterizingmagnetic sheet materials,”IEEE Sensors Journal, vol. 16, no. 12, pp. 4740–4748, 2016.

[16] M. W. Hyde, M. J. Havrilla, and A. E. Bogle, “Nondestructivedetermination of the permittivity tensor of a uniaxial materialusing a two-port clamped coaxial probe,” IEEE Transactions onMicrowave Theory and Techniques, vol. 64, no. 1, pp. 239–246,2016.

[17] M. H. Hosseini, H. Heidar, and M. H. Shams, “Widebandnondestructive measurement of complex permittivity and per-meability using coupled coaxial probes,” IEEE Transactions onInstrumentation and Measurement, vol. 66, no. 1, pp. 148–157,2017.

[18] D. K. Misra and M. Chabbra, “A quasi-static analysis of open-ended coaxial lines,” IEEE Transactions on Microwave Theoryand Techniques, vol. 35, no. 10, pp. 925–928, 1987.

[19] J. Baker-Jarvis, M. Janezic, P. Domich, and R. Geyer, “Analysisof an open-ended coaxial probe with lift-off for nondestructivetesting,” IEEE Transactions on Instrumentation and Measure-ment, vol. 43, no. 5, pp. 711–718, 1994.

[20] D. V. Blackham and R. D. Pollard, “An improved techniquefor permittivity measurements using a coaxial probe,” IEEETransactions on Instrumentation and Measurement, vol. 46, no.5, pp. 1093–1099, 1997.

[21] B. Garcı́a-Baños, J. M. Catalá-Civera, A. J. Canós, and F.Peñaranda-Foix, “Design rules for the optimization of thesensitivity of open-ended coaxial microwave sensors for mon-itoring changes in dielectric materials,” Measurement Scienceand Technology, vol. 16, no. 5, pp. 1186–1192, 2005.

[22] R. A.Weck, “Thin-Film Shielding forMicrocircuitApplicationsand a Useful Laboratory Tool for Plane-Wave Shielding Evalua-tions,” IEEE Transactions on Electromagnetic Compatibility, vol.10, no. 1, pp. 105–112, 1968.

[23] A. R. Ondrejka and J. W. Adams, “Shielding effectiveness (SE)measurement techniques,” in Proceedings of the IEEE NationalSymposium on Electromagnetic Compatibility, pp. 1–6, 1984.

[24] J. Baker-Jarvis andM.D. Janezic, “Analysis of a two-port flangedcoaxial holder for shielding effectiveness and dielectric mea-surements of thin films and thin materials,” IEEE Transactionson Electromagnetic Compatibility, vol. 38, no. 1, pp. 67–70, 1996.

[25] A. P. Orfanidis, G. A. Kyriacou, and J. N. Sahalos, “A mode-matching technique for the study of circular and coaxial waveg-uide discontinuities based on closed-form coupling integrals,”IEEETransactions onMicrowaveTheory andTechniques, vol. 48,no. 5, pp. 880–883, 2000.

[26] Z. Shen and R. H. MacPhie, “Input admittance of a multilayerinsulated monopole Antenna,” IEEE Transactions on Antennasand Propagation, vol. 46, no. 11, pp. 1679–1686, 1998.

[27] L. Polo-Lopez, J. A. Ruiz-Cruz, J. R. Montejo-Garai, and J.M. Rebollar, “Analysis of waveguide discontinuities with lateraland transverse perfect magnetic wall boundary conditions,” inProceedings of the 2016 URSI International Symposium on Elec-tromagnetic Theory, EMTS 2016, pp. 811–814, Finland, August2016.

[28] J. M. Rebollar and J. A. Encinar, “Field theory analysis ofmultiport-multidiscontinuity structures: application to short-circuited E-plane septum,” IEE Proceedings Part H Microwaves,Antennas and Propagation, vol. 135, no. 1, pp. 1–7, 1988.

[29] R. E. Collin, Field Theory of Guided Waves, IEEE Press, NewYork, NY, USA, 1991.

[30] Z. Shen andR. H.MacPhie, “Scattering by aThickOff-CenteredCircular Iris in Circular Waveguide,” IEEE Transactions onMicrowaveTheory andTechniques, vol. 43, no. 11, pp. 2639–2642,1995.

[31] A. Morán-López, J. Córcoles, J. A. Ruiz-Cruz, J. R. Montejo-Garai, and J. M. Rebollar, “Electromagnetic Scattering at theWaveguide Step between Equilateral Triangular Waveguides,”Advances inMathematical Physics, vol. 2016, Article ID 2974675,pp. 1–16, 2016.

[32] DAK, “DAK - Dielectric Assessment Kit Product Line,” https://speag.swiss/products/dak/dielectric-measurements.

https://speag.swiss/products/dak/dielectric-measurementshttps://speag.swiss/products/dak/dielectric-measurements

Hindawiwww.hindawi.com Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwww.hindawi.com Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwww.hindawi.com Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation http://www.hindawi.com Volume 2013Hindawiwww.hindawi.com

The Scientific World Journal

Volume 2018

Hindawiwww.hindawi.com Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com

Di�erential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwww.hindawi.com
https://www.hindawi.com/journals/jmath/https://www.hindawi.com/journals/mpe/https://www.hindawi.com/journals/jam/https://www.hindawi.com/journals/jps/https://www.hindawi.com/journals/amp/https://www.hindawi.com/journals/jca/https://www.hindawi.com/journals/jopti/https://www.hindawi.com/journals/ijem/https://www.hindawi.com/journals/aor/https://www.hindawi.com/journals/jfs/https://www.hindawi.com/journals/aaa/https://www.hindawi.com/journals/ijmms/https://www.hindawi.com/journals/tswj/https://www.hindawi.com/journals/ana/https://www.hindawi.com/journals/ddns/https://www.hindawi.com/journals/ijde/https://www.hindawi.com/journals/ads/https://www.hindawi.com/journals/ijanal/https://www.hindawi.com/journals/ijsa/https://www.hindawi.com/https://www.hindawi.com/

higher-order mode electromagnetic analysis of a material...

Documents