Neural networks forFDTD-backed permittivity

reconstructionVadim V Yakovlev

Department of Mathematical Sciences Worcester Polytechnic InstituteWorcester Massachusetts USA

Ethan K MurphyDepartment of Mathematics Colorado State University Colorado USA

E Eugene EvesThe Ferrite Company Inc Hudson New Hampshire USA

Abstract

Purpose ndash To outline different versions of a novel method for accurate and efficient determining thedielectric properties of arbitrarily shaped materials

Designmethodologyapproach ndash Complex permittivity is found using an artificial neural networkprocedure designed to control a 3D FDTD computation of S-parameters and to process theirmeasurements Network architectures are based on multilayer perceptron and radial basis functionnets The one-port solution deals with the simulated and measured frequency responses of thereflection coefficient while the two-port approach exploits the real and imaginary parts of the reflectionand transmission coefficients at the frequency of interest

Findings ndash High accuracy of permittivity reconstruction is demonstrated by numerical andexperimental testing for dielectric samples of different configuration

Research limitationsimplications ndash Dielectric constant and the loss factor of the studiedmaterial should be within the ranges of corresponding parameters associated with the databaseused for the network training The computer model must be highly adequate to the employedexperimental fixture

Practical implications ndash The method is cavity-independent and applicable to the samplefixtureof arbitrary configuration provided that the geometry is adequately represented in the modelThe two-port version is capable of handling frequency-dependent media parameters For materialswhich can take some predefined form computational cost of the method is very insignificant

Originalityvalue ndash A full-wave 3D FDTD modeling tool and the controlling neural networkprocedure involved in the proposed approach allow for much flexibility in practical implementation ofthe method

Keywords Numerical analysis Neural nets Dielectric properties

Paper type Technical Paper

1 IntroductionRecently microwave power engineers have taken a particular interest in complexpermittivity 1 frac14 10 2 i100 While modern electromagnetic simulators allow theengineers to extensively characterize a constructed device prior to making a physicalprototype in order to perform a trustworthy simulation it is necessary to have reliableknowledge of the dielectric properties of the materials being modeled

The Emerald Research Register for this journal is available at The current issue and full text archive of this journal is available at

wwwemeraldinsightcomresearchregister wwwemeraldinsightcom0332-1649htm

FDTD-backedpermittivity

reconstruction

291

Received August 2004

COMPEL The International Journalfor Computation and Mathematics inElectrical and Electronic Engineering

Vol 24 No 1 2005pp 291-304

q Emerald Group Publishing Limited0332-1649

DOI 10110803321640510571318

Determination of dielectric constant 10 and loss factor 100 of practical materials is adifficult problem Perturbation and transmissionreflection techniques and otherknown methods may give satisfactory results under conditions which are eitherdifficult to follow or simply not acceptable samples typically require the laboriouspreparation to comply with strict dimensional tolerance requirements

With more progress in numerical methods it has become feasible to developtechniques in which the more difficult tasks are assigned to a simulator while theexperimental part is reduced to an elementary measurement This approach has beentaken in the methods using finite element method (FEM) (Deshpande and Reddy 1995Coccioli et al 1999 Thakur and Holmes 2001) and finite difference time domain(FDTD) (Wappling-Raaholt and Risman 2003) and modeling the entire experimentalfixtures To further explore this trend the present paper outlines the principal aspectsof a novel efficient technology for permittivity reconstruction

In our previous work (Eves et al 2004) we have proposed an approach involving anexperimental setup (a closed cavity with an embedded measured sample) whoseS-parameters are computed by the FDTD method and measured by a networkanalyzer Dielectric constant and the loss factor are determined in the course ofprocessing of simulated and measured data by an optimization procedure based onartificial neural networks (ANN)

In the present contribution we generalize the capabilities of this method by consideringother network architectures built on the multilayer perceptron (MLP) and the radial basisfunction (RBF) ANN checking different options in network training and expanding theclass of suitable materials to the ones with frequency-dependent media parametersSince the underlying modeling technique easily handles arbitrary samplefixturegeometry and ANN technology is capable of generalizing the processed data andadjusting to the physical characteristics of the cavity our method is presented as a flexibleand efficient technique of permittivity reconstruction well suited to practical applications

2 Method of permittivity reconstruction21 Network architecturesWe consider two basic one-hidden-layer architectures associated with two types of theexperimental setups ndash a one-port structure intended for measurement of the reflectioncoefficient S11 and a two-port system which also quantifies the transmissioncoefficient S21 The configurations of the closed systems considered in our analysis areshown in Figure 1 while the corresponding networks are shown in Figures 2 and 3

For network training and testing we use information generated in the modelingphase of the method the latter is powered by the 3D FDTD method In the firstapproach the network input receives the simulated values of jS11j at n points of theinterval around the frequency of interest f0 while the network output is associated with10 and 100 In the second architecture the input layers get (and the output layersgenerate) either the simulated values of Re(S11) Im(S11) Re(S21) Im(S21) or the valuesof 10 and 100 for which S-parameters are computed at the modeling stage When thenetwork is well trained it is supplied with the measured values of S-parameters anddetermines 10 and 100 of the sample in question

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22 One-port solutionIn order to describe the computation of complex permittivity with the presentednetworks we introduce the vectors S frac14 frac12 S1 Sn

T frac14 frac12jS11jeth f 1THORN jS11jeth f nTHORNT

and 1 frac14 frac1211 12T frac14 frac1210 100T Then the one-port networks generate the following

output

1l frac14XNjfrac140

w3ljs

Xnifrac140

w2jiSi

l frac14 1 2 eth1THORN

where sethmiddotTHORN frac14 tanhethmiddotTHORN is the activation function used for the hidden neurons and w2j3pq

represents the network weights of the links between the qth neuron in the first orsecond layer and the pth neuron in the second or third layer the activation function forthe output neurons is a linear function In the 2pound n-input network N frac14 NA andN frac14 NB for Net A and Net B respectively

The training data are pairs of eth SkEkTHORN k frac14 1 P where Ek is the desiredoutputs of the network with input Sk (ie the values of dielectric constant and the lossfactor for which Sk have been simulated) and P is the number of training vectorsThe aim is to adjust the vector of network weights w in order to reduce the errorsdefined as

e1l frac141

2

XPkfrac141

j1leth SkwTHORN2 Ekj2

eth2THORN

Figure 1One- (a) and two-port

closed systems (b) withdielectric samples of

arbitrary configuration

FDTD-backedpermittivity

reconstruction

293

Figure 2MLP networks for theone-port system (a) n- and(b) 2 pound n-inputarchitectures

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where 1leth SkwTHORN is the ANN output for input Sk The errors depend on the way thenetwork is trained as well as on its configuration ie on the number of hidden neuronsTo minimize the errors (and improve the quality of learning) we determine thisnumber by a standard trial-and-error process applied to the same training data set

Two training algorithms namely back propagation technique and the second-ordergradient-based technique are implemented with the use of the gradient method (iterationsfrom 1 to 200) and the Levenberg-Marquardt method (iterations beyond 200) respectively

Since the one-port approach deals with the frequencies different from f0 we have afundamental restriction on the accuracy of this version of the method applied to thematerials with frequency-dependent media parameters FDTD computation of afrequency response is performed for 10 and 100 at f0 and measurement of the reflectioncoefficient is conducted everywhere in ( f1 fn) hence the measured values may

Figure 3MLP and RBF networks

for the two-port structure(a) 4- and (b) 2-input

architectures

FDTD-backedpermittivity

reconstruction

295

correspond only at f0 This provides motivation for considering alternative networkarchitectures processing the related information only at f0 and dealing with moreparameters representing the system behavior ie with the complex reflectioncoefficient (S11) and the transmission coefficient (S21)

23 Two-port solutionWith introduction of the vector S frac14 frac12S1 S4

T frac14 frac12ReethS11THORN ImethS21THORNT the output of

the two-port MLP and RBF networks is represented by the formulas

1l frac14XNjfrac140

w3ljs

X4

ifrac140

w2jiSi

l frac14 1 2 eth3THORN

and

Sl frac14XNjfrac140

w3ljs

X2

ifrac140

w2ji1i

l frac14 1 4 eth4THORN

associated with the 4- and 2-input networks the hidden neuron activation functions arethe hyperbolic tangent and Gaussian function sethgTHORN frac14 e2g 2

for MLP and RBF ANNrespectively A linear activation function is used for the output layer in the networks ofboth types

The training data for the 4-input MLP and RBF architectures are pairs of ethSkEkTHORNand the training error is defined as

e1l frac141

2

XPkfrac141

j1lethSkwTHORN2 Ekj2

eth5THORN

where 1lethSkwTHORN is the ANN output for input Sk In the 2-input networks the trainingdata are pairs of eth1kSkTHORN where Sk is the desired outputs of the network for inputs1k (ie the values of S-parameters simulated for given 1k) Computation of error in thiscase is preceded by minimization of the function

Gk frac14 jS leth1kwTHORN2 Skj2 k frac14 1 P and l frac14 1 4 eth6THORN

where Sleth1kwTHORN is the ANN output for input 1k The solution to this minimizationproblem is a set of approximated complex permittivity values Therefore the networkerror is determined from

eSl

frac141

2

XPkfrac141

frac12MinethGkTHORN2 Ek2 eth7THORN

For the training the backpropagation technique and the second-order gradient-basedtechnique are used in the two-port networks just as in the one-port ones

3 Numerical testing31 One-port structureAll the above ANN algorithms have been implemented in a MATLAB 6 environmentFor modeling we use the full-wave 3D conformal FDTD simulator QuickWave-3D

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(QW-3D) (QuickWave-3D 1997-2004) Data required for network training are collectedby a special procedure that repeatedly runs QW-3D to compute S-parameters forvarious values of 10 and 100 of the sample

The one-port scheme has been tested numerically for a section of 72 pound 34 mmwaveguide with a rectangular eth20 pound 20 pound 30 mmTHORN dielectric block in the corner near theshorting wall The FDTD model representing this scenario was built with anonuniform mesh with 75 and 3 mm cubic cells in air and in a dielectric samplerespectively (8463 cells total)

The networks were trained using vectors of jS11j frequency responses with n frac14 3f 1 frac14 24 GHz f 2 frac14 f 0 frac14 245 GHz f 3 frac14 25 GHz and with 27 values of complexpermittivity from the intervals 5 10 9 and 02 100 10 The graphs in Figure 4show the typical sum-squared error produced by the n- and 2pound n-input networks fordifferent number of neurons in the second layer It is seen that for more than ten hiddenneurons the networks are characterized by errors not larger than 1025

When tested with the training sets of 51 vectors the networks demonstratedsufficiently accurate permittivity reconstruction The desired and actual responses fromthe 2 pound n-input MLP are shown in Figure 5 mean square error (MSE) is of order 1023

32 Two-port structureThe two-port scheme dealing with the S-parameters at f0 has been numerically testedwith vectors of Re(S11) Im(S11) Re(S21) and Im(S21) at f 0 frac14 915 MHz for the 497 mmsection of a 248 pound 124 mm waveguide containing a rectangular dielectric sample (Table I)

We built the training sets for the values of relative complex permittivity in the ranges54 10 74 and 6 100 30The 4- and 2-input MLP and the 4-input RBF ANNs weretrained with the sets obtained for 48 equally spaced points in the complex (10 100)-plane andadditional points on the border (68 samples total) For the 2-input RBF where the numberof vectors in the training set is equal to the number of hidden neurons the decision as tohow many vectors (ie points from the (10 100)-plane) in the database to use was madedynamically The network was given a small database and the error was computedThe three test points with the greatest error were chosen and for each point an averagewas taken between the supposed and the ANN-generated values This average was thentaken for the computation of the next sample for the database For example for sample B inposition B the optimal number of training vectors (and hidden neurons) turned out to be 57(Figure 6) In the 4- and 2-input MLP N was taken 13 and 14 respectively

Although all MLPRBF 4-2-input networks have demonstrated good performancesome of them were found to be more accurate In Figure 7 the desired and actualresponses are shown for the 2-input RBF network with a corresponding MSE 0013while for the 4- and 2-input MLP ANNs MSEs are 0029 and 0073 respectively

Training sets for the ranges of 36 10 56 and 4 100 26 have also beencreated The MLP and RBF networks were trained as described above The 2-inputnets have again shown somewhat lower errors In Figure 8 typical examples of thedesired and actual responses from the MLP networks are presented in both casesthe MSE values are of order 1023

The detailed error analysis has been carried out to evaluate the accuracy of thetwo-port systems with a ^2 mm divergence in the samplersquos geometry in eachdimension Numerical experimentation has been performed for 1 frac14 57 2 i8 (apple 88percent moisture contents) 1 frac14 68 2 i14 (cantaloupe 92 percent) 1 frac14 62 2 i22

FDTD-backedpermittivity

reconstruction

297

Figure 4Training and testing errorof n-input MLP (a) andNet A (b) and Net B (c) of2 pound n-input MLP

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(potato 79 percent) and 1 frac14 55 2 i16 (sweet potato 80 percent) (Nelson and Datta2001) A typical example of this computation is shown in Figure 9

Generalizing the results of the analysis conducted for these materials as samplesA-D at Positions A-D we conclude that the 2-input networks can give an error in 100

less than 5 percent if the samplersquos geometrical deviation in the longitudinal andtransverse directions does not exceed 05-10 mm A 10 percent error results from a12-15 mm deviation For 10 the error is less than 5 percent when the deviation is less

Figure 5Complex permittivity

reconstructed with the2 pound n-input MLP withNA frac14 NB frac14 10 circles

and crossed circles markthe test data and the actual

responses respectively

Sample x- y- z-dimensions (mm) Position Distance (mm) from

A 50 pound 50 pound 20 A the second port 120 central line 0B 42 pound 30 pound 50 B the second port 120 central line 30C 20 pound 25 pound 62 C the second port 120 central line 60D 20 pound 25 pound 20 D the second port 150 central line 30

Table IDielectric samples used in

numerical testing of thenetworks for the two-port

scheme

Figure 6MSE of the 2-input RBF

with the number oftraining samples from 48

to 69 with step 3

FDTD-backedpermittivity

reconstruction

299

Figure 8Complex permittivity ofsample C in position Breconstructed by the4-input MLP with N frac14 13(a) and the 2-input MLPwith N frac14 14 the test dataand the actual responses

Figure 7Complex permittivity ofsample B in position Breconstructed with the2-input RBF withN frac14 57 the test data andthe actual responses

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than 12-15 mm and always less than 10 percent in the considered 2 mm deviationSimultaneously a notable variation of accuracy is observed when the samplersquos heightchanges ndash even if variation in the vertical dimension is quite small So for highaccuracy the experimental setup should be constructed to minimize accidentaldeviations of the sample size in the z-direction

4 Experimental testingTo show the method in full operation we have designed the experimental fixtureimplementing the concept of the one-port solution and thus measuring reflections froma cavity with a dielectric sample (Figure 10) Using a rectangular eth70 pound 70 pound 50 mmTHORNTeflon block with a cylindrical cutout (radius 25 mm height 40 mm) suitable for

Figure 9Percent error in getting

right 10 (a) and 100 (b) as afunction of deviation of

training data for thesample dimension in the

x-direction potato assample B in position B

FDTD-backedpermittivity

reconstruction

301

holding liquids we have determined complex permittivity of tap and saline waterThe container filled with water was placed on the center line of the waveguide sectionat 40 mm from the waveguidersquos shorting wall in the opposite end with respect to thecoaxial-waveguide transition

We used a QW-3D model consisting of 71442 cells with a non-uniform mesh(cell sizes in air Teflon and water are 15 5 and 2 mm respectively) for theentire cavity and dielectric inclusions The permittivity of Teflon was taken as206 2 i0

The database of the training and testing sets was created with n frac14 3 f 1 frac14091 GHz f 2 frac14 f 0 frac14 0915 GHz and f 3 frac14 092 GHz for 60 10 90 and 1 100 20and included 108 and 224 vectors respectively For the 2pound n-input network theoptimal structure was found as having NA frac14 15 and NB frac14 19 The normalized sum ofsquared differences between the desired and actual network responses at the trainingstage was less than 1024 for both Net A and Net B

The values of jS11j measured at f 1 frac14 091 GHz f 2 frac14 0915 GHz f 3 frac14 092 GHz forthe Teflon container filled with water were given to the trained network and it

Figure 10Diagram (a) and photo (b)of the experimental setupfor the one-port solution

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generated waterrsquos dielectric constant and the loss factor For the sample of knowntemperature and salinity 10 and 100 have been also determined from the model whoseaverage error is 03 percent for 10 and 18 percent for 100 (Eves and Yakovlev 2002)As one can see from Table II the results are in very close agreement This confirms thecapability of the proposed ANN-based method for accurate reconstructing of complexpermittivity of materials

5 Computer resourcesThe computational cost of the method is primarily determined by the time required tocreate a database for network training and testing The time spent on the training itselfis nearly negligible For example when working with the two-port scheme and usingthe 15 and 33 mm cells in air and dielectric respectively we dealt with the modelcontaining 26796 cells and the simulation of one point on a PC with Pentium IV25 GHz processor took 25 s Hence the database with 149 samples outlined in Section 3was created within 62 min Clearly the accuracy of permittivity reconstruction couldbe generally improved by increasing the number of samples in the database (and thusagreeing on a higher computational cost)

Also the precision of our method may depend on the accuracy of modeling which inits turn is conditioned by the FDTD mesh used In order to virtually exclude aninfluence of discretization and to make sure that the applied cell sizes for all mediainvolved are adequate we performed a sensitivity analysis prior to building thedatabases subsequently simulating the scenario with slightly smaller cells as long asno substantial change in the results was noticed All cellsrsquo sizes mentioned above areresults of this type of an analysis

6 ConclusionOur novel technology of permittivity reconstruction which employs FDTD modelingan ANN-based optimization technique and elementary measurement of S-parametersplaces minimal physical requirements on fixture and sample geometry and issufficiently accurate for practical use Further developments of the method mayinclude its adjustment to non-homogeneous dielectrics and a refinement to allowsample preparation to less strict dimensional tolerances

The practical advantages of the method are obvious It does not depend on theassociated closed system and thus can be used with any available cavity and anysuitable FDTD simulator not necessarily QW-3D While a relatively largecomputational effort may be required for creation of a database the subsequentprocesses of training and determination of complex permittivity require nearlynegligible time Whenever we work at a fixed frequency with materials that can take

Proposed method Model (Eves and Yakovlev 2002) Divergence (percent)

10

806 805 012100

425 430 12

Note Complex permittivity of fresh water with salinity 0033 percent at temperature 1868Cdetermined by the one-port method and the 2 pound n-input MLP ANN Table II

FDTD-backedpermittivity

reconstruction

303

some pre-defined form the database is created only once One can do that prior toactual experimental testing and each new material can be processed thereafterpractically in real time ndash provided that 10 and 100 of this material are within the rangesspecified in the database and that the computer model is based upon the measuredexperimental fixture

References

Coccioli R Pelosi G and Selleri S (1999) ldquoCharacterization of dielectric materials with thefinite-element methodrdquo IEEE Transactions on Microwave Theory and Techniques Vol 47No 10 pp 1106-12

Deshpande MD and Reddy CJ (1995) ldquoApplication of FEM to estimate complex permittivity ofdielectric material at microwave frequency using waveguide measurementsrdquo NASAContractor Report CR-198203 p 23

Eves EE and Yakovlev VV (2002) ldquoAnalysis of operational regimes of a high power waterloadrdquo Journal of Microwave Power amp Electromagnetic Energy Vol 37 No 3 pp 127-44

Eves EE Kopyt P and Yakovlev VV (2004) ldquoDetermination of complex permittivity withneural networks and FDTD modelingrdquo Microwave Optical Technology Letters Vol 40No 3 pp 183-8

Nelson S and Datta AK (2001) ldquoDielectric properties of food materialsrdquo in Datta AK andAnantheswaran RC (Eds) Handbook of Microwave Technology for Food ApplicationsMarcel Dekker Inc New York NY pp 69-114

QuickWave-3D (1997-2004) QWED ul Zwyciezcow 342 03-938 Warsaw Poland wwwqwedcompl

Thakur KP and Holmes WS (2001) ldquoAn inverse technique to evaluate permittivity of materialin a cavityrdquo IEEE Transactions on Microwave Theory and Techniques Vol 49 No 10pp 1129-32

Wappling-Raaholt B and Risman PO (2003) ldquoPermittivity determination of inhomogeneousfoods by measurement and automated retro-modeling with a degenerate mode cavityrdquoProceedings of the 9th Conference on Microwave and HF Heating Loughborough UKpp 181-4

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22 One-port solutionIn order to describe the computation of complex permittivity with the presentednetworks we introduce the vectors S frac14 frac12 S1 Sn

T frac14 frac12jS11jeth f 1THORN jS11jeth f nTHORNT

and 1 frac14 frac1211 12T frac14 frac1210 100T Then the one-port networks generate the following

output

1l frac14XNjfrac140

w3ljs

Xnifrac140

w2jiSi

l frac14 1 2 eth1THORN

where sethmiddotTHORN frac14 tanhethmiddotTHORN is the activation function used for the hidden neurons and w2j3pq

represents the network weights of the links between the qth neuron in the first orsecond layer and the pth neuron in the second or third layer the activation function forthe output neurons is a linear function In the 2pound n-input network N frac14 NA andN frac14 NB for Net A and Net B respectively

The training data are pairs of eth SkEkTHORN k frac14 1 P where Ek is the desiredoutputs of the network with input Sk (ie the values of dielectric constant and the lossfactor for which Sk have been simulated) and P is the number of training vectorsThe aim is to adjust the vector of network weights w in order to reduce the errorsdefined as

e1l frac141

2

XPkfrac141

j1leth SkwTHORN2 Ekj2

eth2THORN

Figure 1One- (a) and two-port

closed systems (b) withdielectric samples of

arbitrary configuration

FDTD-backedpermittivity

reconstruction

293

Figure 2MLP networks for theone-port system (a) n- and(b) 2 pound n-inputarchitectures

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294

where 1leth SkwTHORN is the ANN output for input Sk The errors depend on the way thenetwork is trained as well as on its configuration ie on the number of hidden neuronsTo minimize the errors (and improve the quality of learning) we determine thisnumber by a standard trial-and-error process applied to the same training data set

Two training algorithms namely back propagation technique and the second-ordergradient-based technique are implemented with the use of the gradient method (iterationsfrom 1 to 200) and the Levenberg-Marquardt method (iterations beyond 200) respectively

Since the one-port approach deals with the frequencies different from f0 we have afundamental restriction on the accuracy of this version of the method applied to thematerials with frequency-dependent media parameters FDTD computation of afrequency response is performed for 10 and 100 at f0 and measurement of the reflectioncoefficient is conducted everywhere in ( f1 fn) hence the measured values may

Figure 3MLP and RBF networks

for the two-port structure(a) 4- and (b) 2-input

architectures

FDTD-backedpermittivity

reconstruction

295

correspond only at f0 This provides motivation for considering alternative networkarchitectures processing the related information only at f0 and dealing with moreparameters representing the system behavior ie with the complex reflectioncoefficient (S11) and the transmission coefficient (S21)

23 Two-port solutionWith introduction of the vector S frac14 frac12S1 S4

T frac14 frac12ReethS11THORN ImethS21THORNT the output of

the two-port MLP and RBF networks is represented by the formulas

1l frac14XNjfrac140

w3ljs

X4

ifrac140

w2jiSi

l frac14 1 2 eth3THORN

and

Sl frac14XNjfrac140

w3ljs

X2

ifrac140

w2ji1i

l frac14 1 4 eth4THORN

associated with the 4- and 2-input networks the hidden neuron activation functions arethe hyperbolic tangent and Gaussian function sethgTHORN frac14 e2g 2

for MLP and RBF ANNrespectively A linear activation function is used for the output layer in the networks ofboth types

The training data for the 4-input MLP and RBF architectures are pairs of ethSkEkTHORNand the training error is defined as

e1l frac141

2

XPkfrac141

j1lethSkwTHORN2 Ekj2

eth5THORN

where 1lethSkwTHORN is the ANN output for input Sk In the 2-input networks the trainingdata are pairs of eth1kSkTHORN where Sk is the desired outputs of the network for inputs1k (ie the values of S-parameters simulated for given 1k) Computation of error in thiscase is preceded by minimization of the function

Gk frac14 jS leth1kwTHORN2 Skj2 k frac14 1 P and l frac14 1 4 eth6THORN

where Sleth1kwTHORN is the ANN output for input 1k The solution to this minimizationproblem is a set of approximated complex permittivity values Therefore the networkerror is determined from

eSl

frac141

2

XPkfrac141

frac12MinethGkTHORN2 Ek2 eth7THORN

For the training the backpropagation technique and the second-order gradient-basedtechnique are used in the two-port networks just as in the one-port ones

3 Numerical testing31 One-port structureAll the above ANN algorithms have been implemented in a MATLAB 6 environmentFor modeling we use the full-wave 3D conformal FDTD simulator QuickWave-3D

COMPEL241

296

(QW-3D) (QuickWave-3D 1997-2004) Data required for network training are collectedby a special procedure that repeatedly runs QW-3D to compute S-parameters forvarious values of 10 and 100 of the sample

The one-port scheme has been tested numerically for a section of 72 pound 34 mmwaveguide with a rectangular eth20 pound 20 pound 30 mmTHORN dielectric block in the corner near theshorting wall The FDTD model representing this scenario was built with anonuniform mesh with 75 and 3 mm cubic cells in air and in a dielectric samplerespectively (8463 cells total)

The networks were trained using vectors of jS11j frequency responses with n frac14 3f 1 frac14 24 GHz f 2 frac14 f 0 frac14 245 GHz f 3 frac14 25 GHz and with 27 values of complexpermittivity from the intervals 5 10 9 and 02 100 10 The graphs in Figure 4show the typical sum-squared error produced by the n- and 2pound n-input networks fordifferent number of neurons in the second layer It is seen that for more than ten hiddenneurons the networks are characterized by errors not larger than 1025

When tested with the training sets of 51 vectors the networks demonstratedsufficiently accurate permittivity reconstruction The desired and actual responses fromthe 2 pound n-input MLP are shown in Figure 5 mean square error (MSE) is of order 1023

32 Two-port structureThe two-port scheme dealing with the S-parameters at f0 has been numerically testedwith vectors of Re(S11) Im(S11) Re(S21) and Im(S21) at f 0 frac14 915 MHz for the 497 mmsection of a 248 pound 124 mm waveguide containing a rectangular dielectric sample (Table I)

We built the training sets for the values of relative complex permittivity in the ranges54 10 74 and 6 100 30The 4- and 2-input MLP and the 4-input RBF ANNs weretrained with the sets obtained for 48 equally spaced points in the complex (10 100)-plane andadditional points on the border (68 samples total) For the 2-input RBF where the numberof vectors in the training set is equal to the number of hidden neurons the decision as tohow many vectors (ie points from the (10 100)-plane) in the database to use was madedynamically The network was given a small database and the error was computedThe three test points with the greatest error were chosen and for each point an averagewas taken between the supposed and the ANN-generated values This average was thentaken for the computation of the next sample for the database For example for sample B inposition B the optimal number of training vectors (and hidden neurons) turned out to be 57(Figure 6) In the 4- and 2-input MLP N was taken 13 and 14 respectively

Although all MLPRBF 4-2-input networks have demonstrated good performancesome of them were found to be more accurate In Figure 7 the desired and actualresponses are shown for the 2-input RBF network with a corresponding MSE 0013while for the 4- and 2-input MLP ANNs MSEs are 0029 and 0073 respectively

Training sets for the ranges of 36 10 56 and 4 100 26 have also beencreated The MLP and RBF networks were trained as described above The 2-inputnets have again shown somewhat lower errors In Figure 8 typical examples of thedesired and actual responses from the MLP networks are presented in both casesthe MSE values are of order 1023

The detailed error analysis has been carried out to evaluate the accuracy of thetwo-port systems with a ^2 mm divergence in the samplersquos geometry in eachdimension Numerical experimentation has been performed for 1 frac14 57 2 i8 (apple 88percent moisture contents) 1 frac14 68 2 i14 (cantaloupe 92 percent) 1 frac14 62 2 i22

FDTD-backedpermittivity

reconstruction

297

Figure 4Training and testing errorof n-input MLP (a) andNet A (b) and Net B (c) of2 pound n-input MLP

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298

(potato 79 percent) and 1 frac14 55 2 i16 (sweet potato 80 percent) (Nelson and Datta2001) A typical example of this computation is shown in Figure 9

Generalizing the results of the analysis conducted for these materials as samplesA-D at Positions A-D we conclude that the 2-input networks can give an error in 100

less than 5 percent if the samplersquos geometrical deviation in the longitudinal andtransverse directions does not exceed 05-10 mm A 10 percent error results from a12-15 mm deviation For 10 the error is less than 5 percent when the deviation is less

Figure 5Complex permittivity

reconstructed with the2 pound n-input MLP withNA frac14 NB frac14 10 circles

and crossed circles markthe test data and the actual

responses respectively

Sample x- y- z-dimensions (mm) Position Distance (mm) from

A 50 pound 50 pound 20 A the second port 120 central line 0B 42 pound 30 pound 50 B the second port 120 central line 30C 20 pound 25 pound 62 C the second port 120 central line 60D 20 pound 25 pound 20 D the second port 150 central line 30

Table IDielectric samples used in

numerical testing of thenetworks for the two-port

scheme

Figure 6MSE of the 2-input RBF

with the number oftraining samples from 48

to 69 with step 3

FDTD-backedpermittivity

reconstruction

299

Figure 8Complex permittivity ofsample C in position Breconstructed by the4-input MLP with N frac14 13(a) and the 2-input MLPwith N frac14 14 the test dataand the actual responses

Figure 7Complex permittivity ofsample B in position Breconstructed with the2-input RBF withN frac14 57 the test data andthe actual responses

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than 12-15 mm and always less than 10 percent in the considered 2 mm deviationSimultaneously a notable variation of accuracy is observed when the samplersquos heightchanges ndash even if variation in the vertical dimension is quite small So for highaccuracy the experimental setup should be constructed to minimize accidentaldeviations of the sample size in the z-direction

4 Experimental testingTo show the method in full operation we have designed the experimental fixtureimplementing the concept of the one-port solution and thus measuring reflections froma cavity with a dielectric sample (Figure 10) Using a rectangular eth70 pound 70 pound 50 mmTHORNTeflon block with a cylindrical cutout (radius 25 mm height 40 mm) suitable for

Figure 9Percent error in getting

right 10 (a) and 100 (b) as afunction of deviation of

training data for thesample dimension in the

x-direction potato assample B in position B

FDTD-backedpermittivity

reconstruction

301

holding liquids we have determined complex permittivity of tap and saline waterThe container filled with water was placed on the center line of the waveguide sectionat 40 mm from the waveguidersquos shorting wall in the opposite end with respect to thecoaxial-waveguide transition

We used a QW-3D model consisting of 71442 cells with a non-uniform mesh(cell sizes in air Teflon and water are 15 5 and 2 mm respectively) for theentire cavity and dielectric inclusions The permittivity of Teflon was taken as206 2 i0

The database of the training and testing sets was created with n frac14 3 f 1 frac14091 GHz f 2 frac14 f 0 frac14 0915 GHz and f 3 frac14 092 GHz for 60 10 90 and 1 100 20and included 108 and 224 vectors respectively For the 2pound n-input network theoptimal structure was found as having NA frac14 15 and NB frac14 19 The normalized sum ofsquared differences between the desired and actual network responses at the trainingstage was less than 1024 for both Net A and Net B

The values of jS11j measured at f 1 frac14 091 GHz f 2 frac14 0915 GHz f 3 frac14 092 GHz forthe Teflon container filled with water were given to the trained network and it

Figure 10Diagram (a) and photo (b)of the experimental setupfor the one-port solution

COMPEL241

302

generated waterrsquos dielectric constant and the loss factor For the sample of knowntemperature and salinity 10 and 100 have been also determined from the model whoseaverage error is 03 percent for 10 and 18 percent for 100 (Eves and Yakovlev 2002)As one can see from Table II the results are in very close agreement This confirms thecapability of the proposed ANN-based method for accurate reconstructing of complexpermittivity of materials

5 Computer resourcesThe computational cost of the method is primarily determined by the time required tocreate a database for network training and testing The time spent on the training itselfis nearly negligible For example when working with the two-port scheme and usingthe 15 and 33 mm cells in air and dielectric respectively we dealt with the modelcontaining 26796 cells and the simulation of one point on a PC with Pentium IV25 GHz processor took 25 s Hence the database with 149 samples outlined in Section 3was created within 62 min Clearly the accuracy of permittivity reconstruction couldbe generally improved by increasing the number of samples in the database (and thusagreeing on a higher computational cost)

Also the precision of our method may depend on the accuracy of modeling which inits turn is conditioned by the FDTD mesh used In order to virtually exclude aninfluence of discretization and to make sure that the applied cell sizes for all mediainvolved are adequate we performed a sensitivity analysis prior to building thedatabases subsequently simulating the scenario with slightly smaller cells as long asno substantial change in the results was noticed All cellsrsquo sizes mentioned above areresults of this type of an analysis

6 ConclusionOur novel technology of permittivity reconstruction which employs FDTD modelingan ANN-based optimization technique and elementary measurement of S-parametersplaces minimal physical requirements on fixture and sample geometry and issufficiently accurate for practical use Further developments of the method mayinclude its adjustment to non-homogeneous dielectrics and a refinement to allowsample preparation to less strict dimensional tolerances

The practical advantages of the method are obvious It does not depend on theassociated closed system and thus can be used with any available cavity and anysuitable FDTD simulator not necessarily QW-3D While a relatively largecomputational effort may be required for creation of a database the subsequentprocesses of training and determination of complex permittivity require nearlynegligible time Whenever we work at a fixed frequency with materials that can take

Proposed method Model (Eves and Yakovlev 2002) Divergence (percent)

10

806 805 012100

425 430 12

Note Complex permittivity of fresh water with salinity 0033 percent at temperature 1868Cdetermined by the one-port method and the 2 pound n-input MLP ANN Table II

FDTD-backedpermittivity

reconstruction

303

some pre-defined form the database is created only once One can do that prior toactual experimental testing and each new material can be processed thereafterpractically in real time ndash provided that 10 and 100 of this material are within the rangesspecified in the database and that the computer model is based upon the measuredexperimental fixture

References

Coccioli R Pelosi G and Selleri S (1999) ldquoCharacterization of dielectric materials with thefinite-element methodrdquo IEEE Transactions on Microwave Theory and Techniques Vol 47No 10 pp 1106-12

Deshpande MD and Reddy CJ (1995) ldquoApplication of FEM to estimate complex permittivity ofdielectric material at microwave frequency using waveguide measurementsrdquo NASAContractor Report CR-198203 p 23

Eves EE and Yakovlev VV (2002) ldquoAnalysis of operational regimes of a high power waterloadrdquo Journal of Microwave Power amp Electromagnetic Energy Vol 37 No 3 pp 127-44

Eves EE Kopyt P and Yakovlev VV (2004) ldquoDetermination of complex permittivity withneural networks and FDTD modelingrdquo Microwave Optical Technology Letters Vol 40No 3 pp 183-8

Nelson S and Datta AK (2001) ldquoDielectric properties of food materialsrdquo in Datta AK andAnantheswaran RC (Eds) Handbook of Microwave Technology for Food ApplicationsMarcel Dekker Inc New York NY pp 69-114

QuickWave-3D (1997-2004) QWED ul Zwyciezcow 342 03-938 Warsaw Poland wwwqwedcompl

Thakur KP and Holmes WS (2001) ldquoAn inverse technique to evaluate permittivity of materialin a cavityrdquo IEEE Transactions on Microwave Theory and Techniques Vol 49 No 10pp 1129-32

Wappling-Raaholt B and Risman PO (2003) ldquoPermittivity determination of inhomogeneousfoods by measurement and automated retro-modeling with a degenerate mode cavityrdquoProceedings of the 9th Conference on Microwave and HF Heating Loughborough UKpp 181-4

COMPEL241

304

Figure 2MLP networks for theone-port system (a) n- and(b) 2 pound n-inputarchitectures

COMPEL241

294

where 1leth SkwTHORN is the ANN output for input Sk The errors depend on the way thenetwork is trained as well as on its configuration ie on the number of hidden neuronsTo minimize the errors (and improve the quality of learning) we determine thisnumber by a standard trial-and-error process applied to the same training data set

Two training algorithms namely back propagation technique and the second-ordergradient-based technique are implemented with the use of the gradient method (iterationsfrom 1 to 200) and the Levenberg-Marquardt method (iterations beyond 200) respectively

Since the one-port approach deals with the frequencies different from f0 we have afundamental restriction on the accuracy of this version of the method applied to thematerials with frequency-dependent media parameters FDTD computation of afrequency response is performed for 10 and 100 at f0 and measurement of the reflectioncoefficient is conducted everywhere in ( f1 fn) hence the measured values may

Figure 3MLP and RBF networks

for the two-port structure(a) 4- and (b) 2-input

architectures

FDTD-backedpermittivity

reconstruction

295

correspond only at f0 This provides motivation for considering alternative networkarchitectures processing the related information only at f0 and dealing with moreparameters representing the system behavior ie with the complex reflectioncoefficient (S11) and the transmission coefficient (S21)

23 Two-port solutionWith introduction of the vector S frac14 frac12S1 S4

T frac14 frac12ReethS11THORN ImethS21THORNT the output of

the two-port MLP and RBF networks is represented by the formulas

1l frac14XNjfrac140

w3ljs

X4

ifrac140

w2jiSi

l frac14 1 2 eth3THORN

and

Sl frac14XNjfrac140

w3ljs

X2

ifrac140

w2ji1i

l frac14 1 4 eth4THORN

associated with the 4- and 2-input networks the hidden neuron activation functions arethe hyperbolic tangent and Gaussian function sethgTHORN frac14 e2g 2

for MLP and RBF ANNrespectively A linear activation function is used for the output layer in the networks ofboth types

The training data for the 4-input MLP and RBF architectures are pairs of ethSkEkTHORNand the training error is defined as

e1l frac141

2

XPkfrac141

j1lethSkwTHORN2 Ekj2

eth5THORN

where 1lethSkwTHORN is the ANN output for input Sk In the 2-input networks the trainingdata are pairs of eth1kSkTHORN where Sk is the desired outputs of the network for inputs1k (ie the values of S-parameters simulated for given 1k) Computation of error in thiscase is preceded by minimization of the function

Gk frac14 jS leth1kwTHORN2 Skj2 k frac14 1 P and l frac14 1 4 eth6THORN

where Sleth1kwTHORN is the ANN output for input 1k The solution to this minimizationproblem is a set of approximated complex permittivity values Therefore the networkerror is determined from

eSl

frac141

2

XPkfrac141

frac12MinethGkTHORN2 Ek2 eth7THORN

For the training the backpropagation technique and the second-order gradient-basedtechnique are used in the two-port networks just as in the one-port ones

3 Numerical testing31 One-port structureAll the above ANN algorithms have been implemented in a MATLAB 6 environmentFor modeling we use the full-wave 3D conformal FDTD simulator QuickWave-3D

COMPEL241

296

(QW-3D) (QuickWave-3D 1997-2004) Data required for network training are collectedby a special procedure that repeatedly runs QW-3D to compute S-parameters forvarious values of 10 and 100 of the sample

The one-port scheme has been tested numerically for a section of 72 pound 34 mmwaveguide with a rectangular eth20 pound 20 pound 30 mmTHORN dielectric block in the corner near theshorting wall The FDTD model representing this scenario was built with anonuniform mesh with 75 and 3 mm cubic cells in air and in a dielectric samplerespectively (8463 cells total)

The networks were trained using vectors of jS11j frequency responses with n frac14 3f 1 frac14 24 GHz f 2 frac14 f 0 frac14 245 GHz f 3 frac14 25 GHz and with 27 values of complexpermittivity from the intervals 5 10 9 and 02 100 10 The graphs in Figure 4show the typical sum-squared error produced by the n- and 2pound n-input networks fordifferent number of neurons in the second layer It is seen that for more than ten hiddenneurons the networks are characterized by errors not larger than 1025

When tested with the training sets of 51 vectors the networks demonstratedsufficiently accurate permittivity reconstruction The desired and actual responses fromthe 2 pound n-input MLP are shown in Figure 5 mean square error (MSE) is of order 1023

32 Two-port structureThe two-port scheme dealing with the S-parameters at f0 has been numerically testedwith vectors of Re(S11) Im(S11) Re(S21) and Im(S21) at f 0 frac14 915 MHz for the 497 mmsection of a 248 pound 124 mm waveguide containing a rectangular dielectric sample (Table I)

We built the training sets for the values of relative complex permittivity in the ranges54 10 74 and 6 100 30The 4- and 2-input MLP and the 4-input RBF ANNs weretrained with the sets obtained for 48 equally spaced points in the complex (10 100)-plane andadditional points on the border (68 samples total) For the 2-input RBF where the numberof vectors in the training set is equal to the number of hidden neurons the decision as tohow many vectors (ie points from the (10 100)-plane) in the database to use was madedynamically The network was given a small database and the error was computedThe three test points with the greatest error were chosen and for each point an averagewas taken between the supposed and the ANN-generated values This average was thentaken for the computation of the next sample for the database For example for sample B inposition B the optimal number of training vectors (and hidden neurons) turned out to be 57(Figure 6) In the 4- and 2-input MLP N was taken 13 and 14 respectively

Although all MLPRBF 4-2-input networks have demonstrated good performancesome of them were found to be more accurate In Figure 7 the desired and actualresponses are shown for the 2-input RBF network with a corresponding MSE 0013while for the 4- and 2-input MLP ANNs MSEs are 0029 and 0073 respectively

Training sets for the ranges of 36 10 56 and 4 100 26 have also beencreated The MLP and RBF networks were trained as described above The 2-inputnets have again shown somewhat lower errors In Figure 8 typical examples of thedesired and actual responses from the MLP networks are presented in both casesthe MSE values are of order 1023

The detailed error analysis has been carried out to evaluate the accuracy of thetwo-port systems with a ^2 mm divergence in the samplersquos geometry in eachdimension Numerical experimentation has been performed for 1 frac14 57 2 i8 (apple 88percent moisture contents) 1 frac14 68 2 i14 (cantaloupe 92 percent) 1 frac14 62 2 i22

FDTD-backedpermittivity

reconstruction

297

Figure 4Training and testing errorof n-input MLP (a) andNet A (b) and Net B (c) of2 pound n-input MLP

COMPEL241

298

(potato 79 percent) and 1 frac14 55 2 i16 (sweet potato 80 percent) (Nelson and Datta2001) A typical example of this computation is shown in Figure 9

Generalizing the results of the analysis conducted for these materials as samplesA-D at Positions A-D we conclude that the 2-input networks can give an error in 100

less than 5 percent if the samplersquos geometrical deviation in the longitudinal andtransverse directions does not exceed 05-10 mm A 10 percent error results from a12-15 mm deviation For 10 the error is less than 5 percent when the deviation is less

Figure 5Complex permittivity

reconstructed with the2 pound n-input MLP withNA frac14 NB frac14 10 circles

and crossed circles markthe test data and the actual

responses respectively

Sample x- y- z-dimensions (mm) Position Distance (mm) from

A 50 pound 50 pound 20 A the second port 120 central line 0B 42 pound 30 pound 50 B the second port 120 central line 30C 20 pound 25 pound 62 C the second port 120 central line 60D 20 pound 25 pound 20 D the second port 150 central line 30

Table IDielectric samples used in

numerical testing of thenetworks for the two-port

scheme

Figure 6MSE of the 2-input RBF

with the number oftraining samples from 48

to 69 with step 3

FDTD-backedpermittivity

reconstruction

299

Figure 8Complex permittivity ofsample C in position Breconstructed by the4-input MLP with N frac14 13(a) and the 2-input MLPwith N frac14 14 the test dataand the actual responses

Figure 7Complex permittivity ofsample B in position Breconstructed with the2-input RBF withN frac14 57 the test data andthe actual responses

COMPEL241

300

than 12-15 mm and always less than 10 percent in the considered 2 mm deviationSimultaneously a notable variation of accuracy is observed when the samplersquos heightchanges ndash even if variation in the vertical dimension is quite small So for highaccuracy the experimental setup should be constructed to minimize accidentaldeviations of the sample size in the z-direction

4 Experimental testingTo show the method in full operation we have designed the experimental fixtureimplementing the concept of the one-port solution and thus measuring reflections froma cavity with a dielectric sample (Figure 10) Using a rectangular eth70 pound 70 pound 50 mmTHORNTeflon block with a cylindrical cutout (radius 25 mm height 40 mm) suitable for

Figure 9Percent error in getting

right 10 (a) and 100 (b) as afunction of deviation of

training data for thesample dimension in the

x-direction potato assample B in position B

FDTD-backedpermittivity

reconstruction

301

holding liquids we have determined complex permittivity of tap and saline waterThe container filled with water was placed on the center line of the waveguide sectionat 40 mm from the waveguidersquos shorting wall in the opposite end with respect to thecoaxial-waveguide transition

We used a QW-3D model consisting of 71442 cells with a non-uniform mesh(cell sizes in air Teflon and water are 15 5 and 2 mm respectively) for theentire cavity and dielectric inclusions The permittivity of Teflon was taken as206 2 i0

The database of the training and testing sets was created with n frac14 3 f 1 frac14091 GHz f 2 frac14 f 0 frac14 0915 GHz and f 3 frac14 092 GHz for 60 10 90 and 1 100 20and included 108 and 224 vectors respectively For the 2pound n-input network theoptimal structure was found as having NA frac14 15 and NB frac14 19 The normalized sum ofsquared differences between the desired and actual network responses at the trainingstage was less than 1024 for both Net A and Net B

The values of jS11j measured at f 1 frac14 091 GHz f 2 frac14 0915 GHz f 3 frac14 092 GHz forthe Teflon container filled with water were given to the trained network and it

Figure 10Diagram (a) and photo (b)of the experimental setupfor the one-port solution

COMPEL241

302

generated waterrsquos dielectric constant and the loss factor For the sample of knowntemperature and salinity 10 and 100 have been also determined from the model whoseaverage error is 03 percent for 10 and 18 percent for 100 (Eves and Yakovlev 2002)As one can see from Table II the results are in very close agreement This confirms thecapability of the proposed ANN-based method for accurate reconstructing of complexpermittivity of materials

5 Computer resourcesThe computational cost of the method is primarily determined by the time required tocreate a database for network training and testing The time spent on the training itselfis nearly negligible For example when working with the two-port scheme and usingthe 15 and 33 mm cells in air and dielectric respectively we dealt with the modelcontaining 26796 cells and the simulation of one point on a PC with Pentium IV25 GHz processor took 25 s Hence the database with 149 samples outlined in Section 3was created within 62 min Clearly the accuracy of permittivity reconstruction couldbe generally improved by increasing the number of samples in the database (and thusagreeing on a higher computational cost)

Also the precision of our method may depend on the accuracy of modeling which inits turn is conditioned by the FDTD mesh used In order to virtually exclude aninfluence of discretization and to make sure that the applied cell sizes for all mediainvolved are adequate we performed a sensitivity analysis prior to building thedatabases subsequently simulating the scenario with slightly smaller cells as long asno substantial change in the results was noticed All cellsrsquo sizes mentioned above areresults of this type of an analysis

6 ConclusionOur novel technology of permittivity reconstruction which employs FDTD modelingan ANN-based optimization technique and elementary measurement of S-parametersplaces minimal physical requirements on fixture and sample geometry and issufficiently accurate for practical use Further developments of the method mayinclude its adjustment to non-homogeneous dielectrics and a refinement to allowsample preparation to less strict dimensional tolerances

The practical advantages of the method are obvious It does not depend on theassociated closed system and thus can be used with any available cavity and anysuitable FDTD simulator not necessarily QW-3D While a relatively largecomputational effort may be required for creation of a database the subsequentprocesses of training and determination of complex permittivity require nearlynegligible time Whenever we work at a fixed frequency with materials that can take

Proposed method Model (Eves and Yakovlev 2002) Divergence (percent)

10

806 805 012100

425 430 12

Note Complex permittivity of fresh water with salinity 0033 percent at temperature 1868Cdetermined by the one-port method and the 2 pound n-input MLP ANN Table II

FDTD-backedpermittivity

reconstruction

303

some pre-defined form the database is created only once One can do that prior toactual experimental testing and each new material can be processed thereafterpractically in real time ndash provided that 10 and 100 of this material are within the rangesspecified in the database and that the computer model is based upon the measuredexperimental fixture

References

Coccioli R Pelosi G and Selleri S (1999) ldquoCharacterization of dielectric materials with thefinite-element methodrdquo IEEE Transactions on Microwave Theory and Techniques Vol 47No 10 pp 1106-12

Deshpande MD and Reddy CJ (1995) ldquoApplication of FEM to estimate complex permittivity ofdielectric material at microwave frequency using waveguide measurementsrdquo NASAContractor Report CR-198203 p 23

Eves EE and Yakovlev VV (2002) ldquoAnalysis of operational regimes of a high power waterloadrdquo Journal of Microwave Power amp Electromagnetic Energy Vol 37 No 3 pp 127-44

Eves EE Kopyt P and Yakovlev VV (2004) ldquoDetermination of complex permittivity withneural networks and FDTD modelingrdquo Microwave Optical Technology Letters Vol 40No 3 pp 183-8

Nelson S and Datta AK (2001) ldquoDielectric properties of food materialsrdquo in Datta AK andAnantheswaran RC (Eds) Handbook of Microwave Technology for Food ApplicationsMarcel Dekker Inc New York NY pp 69-114

QuickWave-3D (1997-2004) QWED ul Zwyciezcow 342 03-938 Warsaw Poland wwwqwedcompl

Thakur KP and Holmes WS (2001) ldquoAn inverse technique to evaluate permittivity of materialin a cavityrdquo IEEE Transactions on Microwave Theory and Techniques Vol 49 No 10pp 1129-32

Wappling-Raaholt B and Risman PO (2003) ldquoPermittivity determination of inhomogeneousfoods by measurement and automated retro-modeling with a degenerate mode cavityrdquoProceedings of the 9th Conference on Microwave and HF Heating Loughborough UKpp 181-4

COMPEL241

304

where 1leth SkwTHORN is the ANN output for input Sk The errors depend on the way thenetwork is trained as well as on its configuration ie on the number of hidden neuronsTo minimize the errors (and improve the quality of learning) we determine thisnumber by a standard trial-and-error process applied to the same training data set

Two training algorithms namely back propagation technique and the second-ordergradient-based technique are implemented with the use of the gradient method (iterationsfrom 1 to 200) and the Levenberg-Marquardt method (iterations beyond 200) respectively

Since the one-port approach deals with the frequencies different from f0 we have afundamental restriction on the accuracy of this version of the method applied to thematerials with frequency-dependent media parameters FDTD computation of afrequency response is performed for 10 and 100 at f0 and measurement of the reflectioncoefficient is conducted everywhere in ( f1 fn) hence the measured values may

Figure 3MLP and RBF networks

for the two-port structure(a) 4- and (b) 2-input

architectures

FDTD-backedpermittivity

reconstruction

295

correspond only at f0 This provides motivation for considering alternative networkarchitectures processing the related information only at f0 and dealing with moreparameters representing the system behavior ie with the complex reflectioncoefficient (S11) and the transmission coefficient (S21)

23 Two-port solutionWith introduction of the vector S frac14 frac12S1 S4

T frac14 frac12ReethS11THORN ImethS21THORNT the output of

the two-port MLP and RBF networks is represented by the formulas

1l frac14XNjfrac140

w3ljs

X4

ifrac140

w2jiSi

l frac14 1 2 eth3THORN

and

Sl frac14XNjfrac140

w3ljs

X2

ifrac140

w2ji1i

l frac14 1 4 eth4THORN

associated with the 4- and 2-input networks the hidden neuron activation functions arethe hyperbolic tangent and Gaussian function sethgTHORN frac14 e2g 2

for MLP and RBF ANNrespectively A linear activation function is used for the output layer in the networks ofboth types

The training data for the 4-input MLP and RBF architectures are pairs of ethSkEkTHORNand the training error is defined as

e1l frac141

2

XPkfrac141

j1lethSkwTHORN2 Ekj2

eth5THORN

where 1lethSkwTHORN is the ANN output for input Sk In the 2-input networks the trainingdata are pairs of eth1kSkTHORN where Sk is the desired outputs of the network for inputs1k (ie the values of S-parameters simulated for given 1k) Computation of error in thiscase is preceded by minimization of the function

Gk frac14 jS leth1kwTHORN2 Skj2 k frac14 1 P and l frac14 1 4 eth6THORN

where Sleth1kwTHORN is the ANN output for input 1k The solution to this minimizationproblem is a set of approximated complex permittivity values Therefore the networkerror is determined from

eSl

frac141

2

XPkfrac141

frac12MinethGkTHORN2 Ek2 eth7THORN

For the training the backpropagation technique and the second-order gradient-basedtechnique are used in the two-port networks just as in the one-port ones

3 Numerical testing31 One-port structureAll the above ANN algorithms have been implemented in a MATLAB 6 environmentFor modeling we use the full-wave 3D conformal FDTD simulator QuickWave-3D

COMPEL241

296

(QW-3D) (QuickWave-3D 1997-2004) Data required for network training are collectedby a special procedure that repeatedly runs QW-3D to compute S-parameters forvarious values of 10 and 100 of the sample

The one-port scheme has been tested numerically for a section of 72 pound 34 mmwaveguide with a rectangular eth20 pound 20 pound 30 mmTHORN dielectric block in the corner near theshorting wall The FDTD model representing this scenario was built with anonuniform mesh with 75 and 3 mm cubic cells in air and in a dielectric samplerespectively (8463 cells total)

The networks were trained using vectors of jS11j frequency responses with n frac14 3f 1 frac14 24 GHz f 2 frac14 f 0 frac14 245 GHz f 3 frac14 25 GHz and with 27 values of complexpermittivity from the intervals 5 10 9 and 02 100 10 The graphs in Figure 4show the typical sum-squared error produced by the n- and 2pound n-input networks fordifferent number of neurons in the second layer It is seen that for more than ten hiddenneurons the networks are characterized by errors not larger than 1025

When tested with the training sets of 51 vectors the networks demonstratedsufficiently accurate permittivity reconstruction The desired and actual responses fromthe 2 pound n-input MLP are shown in Figure 5 mean square error (MSE) is of order 1023

32 Two-port structureThe two-port scheme dealing with the S-parameters at f0 has been numerically testedwith vectors of Re(S11) Im(S11) Re(S21) and Im(S21) at f 0 frac14 915 MHz for the 497 mmsection of a 248 pound 124 mm waveguide containing a rectangular dielectric sample (Table I)

We built the training sets for the values of relative complex permittivity in the ranges54 10 74 and 6 100 30The 4- and 2-input MLP and the 4-input RBF ANNs weretrained with the sets obtained for 48 equally spaced points in the complex (10 100)-plane andadditional points on the border (68 samples total) For the 2-input RBF where the numberof vectors in the training set is equal to the number of hidden neurons the decision as tohow many vectors (ie points from the (10 100)-plane) in the database to use was madedynamically The network was given a small database and the error was computedThe three test points with the greatest error were chosen and for each point an averagewas taken between the supposed and the ANN-generated values This average was thentaken for the computation of the next sample for the database For example for sample B inposition B the optimal number of training vectors (and hidden neurons) turned out to be 57(Figure 6) In the 4- and 2-input MLP N was taken 13 and 14 respectively

Although all MLPRBF 4-2-input networks have demonstrated good performancesome of them were found to be more accurate In Figure 7 the desired and actualresponses are shown for the 2-input RBF network with a corresponding MSE 0013while for the 4- and 2-input MLP ANNs MSEs are 0029 and 0073 respectively

Training sets for the ranges of 36 10 56 and 4 100 26 have also beencreated The MLP and RBF networks were trained as described above The 2-inputnets have again shown somewhat lower errors In Figure 8 typical examples of thedesired and actual responses from the MLP networks are presented in both casesthe MSE values are of order 1023

The detailed error analysis has been carried out to evaluate the accuracy of thetwo-port systems with a ^2 mm divergence in the samplersquos geometry in eachdimension Numerical experimentation has been performed for 1 frac14 57 2 i8 (apple 88percent moisture contents) 1 frac14 68 2 i14 (cantaloupe 92 percent) 1 frac14 62 2 i22

FDTD-backedpermittivity

reconstruction

297

Figure 4Training and testing errorof n-input MLP (a) andNet A (b) and Net B (c) of2 pound n-input MLP

COMPEL241

298

(potato 79 percent) and 1 frac14 55 2 i16 (sweet potato 80 percent) (Nelson and Datta2001) A typical example of this computation is shown in Figure 9

Generalizing the results of the analysis conducted for these materials as samplesA-D at Positions A-D we conclude that the 2-input networks can give an error in 100

less than 5 percent if the samplersquos geometrical deviation in the longitudinal andtransverse directions does not exceed 05-10 mm A 10 percent error results from a12-15 mm deviation For 10 the error is less than 5 percent when the deviation is less

Figure 5Complex permittivity

reconstructed with the2 pound n-input MLP withNA frac14 NB frac14 10 circles

and crossed circles markthe test data and the actual

responses respectively

Sample x- y- z-dimensions (mm) Position Distance (mm) from

A 50 pound 50 pound 20 A the second port 120 central line 0B 42 pound 30 pound 50 B the second port 120 central line 30C 20 pound 25 pound 62 C the second port 120 central line 60D 20 pound 25 pound 20 D the second port 150 central line 30

Table IDielectric samples used in

numerical testing of thenetworks for the two-port

scheme

Figure 6MSE of the 2-input RBF

with the number oftraining samples from 48

to 69 with step 3

FDTD-backedpermittivity

reconstruction

299

Figure 8Complex permittivity ofsample C in position Breconstructed by the4-input MLP with N frac14 13(a) and the 2-input MLPwith N frac14 14 the test dataand the actual responses

Figure 7Complex permittivity ofsample B in position Breconstructed with the2-input RBF withN frac14 57 the test data andthe actual responses

COMPEL241

300

than 12-15 mm and always less than 10 percent in the considered 2 mm deviationSimultaneously a notable variation of accuracy is observed when the samplersquos heightchanges ndash even if variation in the vertical dimension is quite small So for highaccuracy the experimental setup should be constructed to minimize accidentaldeviations of the sample size in the z-direction

4 Experimental testingTo show the method in full operation we have designed the experimental fixtureimplementing the concept of the one-port solution and thus measuring reflections froma cavity with a dielectric sample (Figure 10) Using a rectangular eth70 pound 70 pound 50 mmTHORNTeflon block with a cylindrical cutout (radius 25 mm height 40 mm) suitable for

Figure 9Percent error in getting

right 10 (a) and 100 (b) as afunction of deviation of

training data for thesample dimension in the

x-direction potato assample B in position B

FDTD-backedpermittivity

reconstruction

301

holding liquids we have determined complex permittivity of tap and saline waterThe container filled with water was placed on the center line of the waveguide sectionat 40 mm from the waveguidersquos shorting wall in the opposite end with respect to thecoaxial-waveguide transition

We used a QW-3D model consisting of 71442 cells with a non-uniform mesh(cell sizes in air Teflon and water are 15 5 and 2 mm respectively) for theentire cavity and dielectric inclusions The permittivity of Teflon was taken as206 2 i0

The database of the training and testing sets was created with n frac14 3 f 1 frac14091 GHz f 2 frac14 f 0 frac14 0915 GHz and f 3 frac14 092 GHz for 60 10 90 and 1 100 20and included 108 and 224 vectors respectively For the 2pound n-input network theoptimal structure was found as having NA frac14 15 and NB frac14 19 The normalized sum ofsquared differences between the desired and actual network responses at the trainingstage was less than 1024 for both Net A and Net B

The values of jS11j measured at f 1 frac14 091 GHz f 2 frac14 0915 GHz f 3 frac14 092 GHz forthe Teflon container filled with water were given to the trained network and it

Figure 10Diagram (a) and photo (b)of the experimental setupfor the one-port solution

COMPEL241

302

generated waterrsquos dielectric constant and the loss factor For the sample of knowntemperature and salinity 10 and 100 have been also determined from the model whoseaverage error is 03 percent for 10 and 18 percent for 100 (Eves and Yakovlev 2002)As one can see from Table II the results are in very close agreement This confirms thecapability of the proposed ANN-based method for accurate reconstructing of complexpermittivity of materials

5 Computer resourcesThe computational cost of the method is primarily determined by the time required tocreate a database for network training and testing The time spent on the training itselfis nearly negligible For example when working with the two-port scheme and usingthe 15 and 33 mm cells in air and dielectric respectively we dealt with the modelcontaining 26796 cells and the simulation of one point on a PC with Pentium IV25 GHz processor took 25 s Hence the database with 149 samples outlined in Section 3was created within 62 min Clearly the accuracy of permittivity reconstruction couldbe generally improved by increasing the number of samples in the database (and thusagreeing on a higher computational cost)

Also the precision of our method may depend on the accuracy of modeling which inits turn is conditioned by the FDTD mesh used In order to virtually exclude aninfluence of discretization and to make sure that the applied cell sizes for all mediainvolved are adequate we performed a sensitivity analysis prior to building thedatabases subsequently simulating the scenario with slightly smaller cells as long asno substantial change in the results was noticed All cellsrsquo sizes mentioned above areresults of this type of an analysis

6 ConclusionOur novel technology of permittivity reconstruction which employs FDTD modelingan ANN-based optimization technique and elementary measurement of S-parametersplaces minimal physical requirements on fixture and sample geometry and issufficiently accurate for practical use Further developments of the method mayinclude its adjustment to non-homogeneous dielectrics and a refinement to allowsample preparation to less strict dimensional tolerances

The practical advantages of the method are obvious It does not depend on theassociated closed system and thus can be used with any available cavity and anysuitable FDTD simulator not necessarily QW-3D While a relatively largecomputational effort may be required for creation of a database the subsequentprocesses of training and determination of complex permittivity require nearlynegligible time Whenever we work at a fixed frequency with materials that can take

Proposed method Model (Eves and Yakovlev 2002) Divergence (percent)

10

806 805 012100

425 430 12

Note Complex permittivity of fresh water with salinity 0033 percent at temperature 1868Cdetermined by the one-port method and the 2 pound n-input MLP ANN Table II

FDTD-backedpermittivity

reconstruction

303

some pre-defined form the database is created only once One can do that prior toactual experimental testing and each new material can be processed thereafterpractically in real time ndash provided that 10 and 100 of this material are within the rangesspecified in the database and that the computer model is based upon the measuredexperimental fixture

References

Coccioli R Pelosi G and Selleri S (1999) ldquoCharacterization of dielectric materials with thefinite-element methodrdquo IEEE Transactions on Microwave Theory and Techniques Vol 47No 10 pp 1106-12

Deshpande MD and Reddy CJ (1995) ldquoApplication of FEM to estimate complex permittivity ofdielectric material at microwave frequency using waveguide measurementsrdquo NASAContractor Report CR-198203 p 23

Eves EE and Yakovlev VV (2002) ldquoAnalysis of operational regimes of a high power waterloadrdquo Journal of Microwave Power amp Electromagnetic Energy Vol 37 No 3 pp 127-44

Eves EE Kopyt P and Yakovlev VV (2004) ldquoDetermination of complex permittivity withneural networks and FDTD modelingrdquo Microwave Optical Technology Letters Vol 40No 3 pp 183-8

Nelson S and Datta AK (2001) ldquoDielectric properties of food materialsrdquo in Datta AK andAnantheswaran RC (Eds) Handbook of Microwave Technology for Food ApplicationsMarcel Dekker Inc New York NY pp 69-114

QuickWave-3D (1997-2004) QWED ul Zwyciezcow 342 03-938 Warsaw Poland wwwqwedcompl

Thakur KP and Holmes WS (2001) ldquoAn inverse technique to evaluate permittivity of materialin a cavityrdquo IEEE Transactions on Microwave Theory and Techniques Vol 49 No 10pp 1129-32

Wappling-Raaholt B and Risman PO (2003) ldquoPermittivity determination of inhomogeneousfoods by measurement and automated retro-modeling with a degenerate mode cavityrdquoProceedings of the 9th Conference on Microwave and HF Heating Loughborough UKpp 181-4

COMPEL241

304

correspond only at f0 This provides motivation for considering alternative networkarchitectures processing the related information only at f0 and dealing with moreparameters representing the system behavior ie with the complex reflectioncoefficient (S11) and the transmission coefficient (S21)

23 Two-port solutionWith introduction of the vector S frac14 frac12S1 S4

T frac14 frac12ReethS11THORN ImethS21THORNT the output of

the two-port MLP and RBF networks is represented by the formulas

1l frac14XNjfrac140

w3ljs

X4

ifrac140

w2jiSi

l frac14 1 2 eth3THORN

and

Sl frac14XNjfrac140

w3ljs

X2

ifrac140

w2ji1i

l frac14 1 4 eth4THORN

associated with the 4- and 2-input networks the hidden neuron activation functions arethe hyperbolic tangent and Gaussian function sethgTHORN frac14 e2g 2

for MLP and RBF ANNrespectively A linear activation function is used for the output layer in the networks ofboth types

The training data for the 4-input MLP and RBF architectures are pairs of ethSkEkTHORNand the training error is defined as

e1l frac141

2

XPkfrac141

j1lethSkwTHORN2 Ekj2

eth5THORN

where 1lethSkwTHORN is the ANN output for input Sk In the 2-input networks the trainingdata are pairs of eth1kSkTHORN where Sk is the desired outputs of the network for inputs1k (ie the values of S-parameters simulated for given 1k) Computation of error in thiscase is preceded by minimization of the function

Gk frac14 jS leth1kwTHORN2 Skj2 k frac14 1 P and l frac14 1 4 eth6THORN

where Sleth1kwTHORN is the ANN output for input 1k The solution to this minimizationproblem is a set of approximated complex permittivity values Therefore the networkerror is determined from

eSl

frac141

2

XPkfrac141

frac12MinethGkTHORN2 Ek2 eth7THORN

For the training the backpropagation technique and the second-order gradient-basedtechnique are used in the two-port networks just as in the one-port ones

3 Numerical testing31 One-port structureAll the above ANN algorithms have been implemented in a MATLAB 6 environmentFor modeling we use the full-wave 3D conformal FDTD simulator QuickWave-3D

COMPEL241

296

(QW-3D) (QuickWave-3D 1997-2004) Data required for network training are collectedby a special procedure that repeatedly runs QW-3D to compute S-parameters forvarious values of 10 and 100 of the sample

The one-port scheme has been tested numerically for a section of 72 pound 34 mmwaveguide with a rectangular eth20 pound 20 pound 30 mmTHORN dielectric block in the corner near theshorting wall The FDTD model representing this scenario was built with anonuniform mesh with 75 and 3 mm cubic cells in air and in a dielectric samplerespectively (8463 cells total)

The networks were trained using vectors of jS11j frequency responses with n frac14 3f 1 frac14 24 GHz f 2 frac14 f 0 frac14 245 GHz f 3 frac14 25 GHz and with 27 values of complexpermittivity from the intervals 5 10 9 and 02 100 10 The graphs in Figure 4show the typical sum-squared error produced by the n- and 2pound n-input networks fordifferent number of neurons in the second layer It is seen that for more than ten hiddenneurons the networks are characterized by errors not larger than 1025

When tested with the training sets of 51 vectors the networks demonstratedsufficiently accurate permittivity reconstruction The desired and actual responses fromthe 2 pound n-input MLP are shown in Figure 5 mean square error (MSE) is of order 1023

32 Two-port structureThe two-port scheme dealing with the S-parameters at f0 has been numerically testedwith vectors of Re(S11) Im(S11) Re(S21) and Im(S21) at f 0 frac14 915 MHz for the 497 mmsection of a 248 pound 124 mm waveguide containing a rectangular dielectric sample (Table I)

We built the training sets for the values of relative complex permittivity in the ranges54 10 74 and 6 100 30The 4- and 2-input MLP and the 4-input RBF ANNs weretrained with the sets obtained for 48 equally spaced points in the complex (10 100)-plane andadditional points on the border (68 samples total) For the 2-input RBF where the numberof vectors in the training set is equal to the number of hidden neurons the decision as tohow many vectors (ie points from the (10 100)-plane) in the database to use was madedynamically The network was given a small database and the error was computedThe three test points with the greatest error were chosen and for each point an averagewas taken between the supposed and the ANN-generated values This average was thentaken for the computation of the next sample for the database For example for sample B inposition B the optimal number of training vectors (and hidden neurons) turned out to be 57(Figure 6) In the 4- and 2-input MLP N was taken 13 and 14 respectively

Although all MLPRBF 4-2-input networks have demonstrated good performancesome of them were found to be more accurate In Figure 7 the desired and actualresponses are shown for the 2-input RBF network with a corresponding MSE 0013while for the 4- and 2-input MLP ANNs MSEs are 0029 and 0073 respectively

Training sets for the ranges of 36 10 56 and 4 100 26 have also beencreated The MLP and RBF networks were trained as described above The 2-inputnets have again shown somewhat lower errors In Figure 8 typical examples of thedesired and actual responses from the MLP networks are presented in both casesthe MSE values are of order 1023

The detailed error analysis has been carried out to evaluate the accuracy of thetwo-port systems with a ^2 mm divergence in the samplersquos geometry in eachdimension Numerical experimentation has been performed for 1 frac14 57 2 i8 (apple 88percent moisture contents) 1 frac14 68 2 i14 (cantaloupe 92 percent) 1 frac14 62 2 i22

FDTD-backedpermittivity

reconstruction

297

Figure 4Training and testing errorof n-input MLP (a) andNet A (b) and Net B (c) of2 pound n-input MLP

COMPEL241

298

(potato 79 percent) and 1 frac14 55 2 i16 (sweet potato 80 percent) (Nelson and Datta2001) A typical example of this computation is shown in Figure 9

Generalizing the results of the analysis conducted for these materials as samplesA-D at Positions A-D we conclude that the 2-input networks can give an error in 100

less than 5 percent if the samplersquos geometrical deviation in the longitudinal andtransverse directions does not exceed 05-10 mm A 10 percent error results from a12-15 mm deviation For 10 the error is less than 5 percent when the deviation is less

Figure 5Complex permittivity

reconstructed with the2 pound n-input MLP withNA frac14 NB frac14 10 circles

and crossed circles markthe test data and the actual

responses respectively

Sample x- y- z-dimensions (mm) Position Distance (mm) from

A 50 pound 50 pound 20 A the second port 120 central line 0B 42 pound 30 pound 50 B the second port 120 central line 30C 20 pound 25 pound 62 C the second port 120 central line 60D 20 pound 25 pound 20 D the second port 150 central line 30

Table IDielectric samples used in

numerical testing of thenetworks for the two-port

scheme

Figure 6MSE of the 2-input RBF

with the number oftraining samples from 48

to 69 with step 3

FDTD-backedpermittivity

reconstruction

299

Figure 8Complex permittivity ofsample C in position Breconstructed by the4-input MLP with N frac14 13(a) and the 2-input MLPwith N frac14 14 the test dataand the actual responses

Figure 7Complex permittivity ofsample B in position Breconstructed with the2-input RBF withN frac14 57 the test data andthe actual responses

COMPEL241

300

than 12-15 mm and always less than 10 percent in the considered 2 mm deviationSimultaneously a notable variation of accuracy is observed when the samplersquos heightchanges ndash even if variation in the vertical dimension is quite small So for highaccuracy the experimental setup should be constructed to minimize accidentaldeviations of the sample size in the z-direction

4 Experimental testingTo show the method in full operation we have designed the experimental fixtureimplementing the concept of the one-port solution and thus measuring reflections froma cavity with a dielectric sample (Figure 10) Using a rectangular eth70 pound 70 pound 50 mmTHORNTeflon block with a cylindrical cutout (radius 25 mm height 40 mm) suitable for

Figure 9Percent error in getting

right 10 (a) and 100 (b) as afunction of deviation of

training data for thesample dimension in the

x-direction potato assample B in position B

FDTD-backedpermittivity

reconstruction

301

holding liquids we have determined complex permittivity of tap and saline waterThe container filled with water was placed on the center line of the waveguide sectionat 40 mm from the waveguidersquos shorting wall in the opposite end with respect to thecoaxial-waveguide transition

We used a QW-3D model consisting of 71442 cells with a non-uniform mesh(cell sizes in air Teflon and water are 15 5 and 2 mm respectively) for theentire cavity and dielectric inclusions The permittivity of Teflon was taken as206 2 i0

The database of the training and testing sets was created with n frac14 3 f 1 frac14091 GHz f 2 frac14 f 0 frac14 0915 GHz and f 3 frac14 092 GHz for 60 10 90 and 1 100 20and included 108 and 224 vectors respectively For the 2pound n-input network theoptimal structure was found as having NA frac14 15 and NB frac14 19 The normalized sum ofsquared differences between the desired and actual network responses at the trainingstage was less than 1024 for both Net A and Net B

The values of jS11j measured at f 1 frac14 091 GHz f 2 frac14 0915 GHz f 3 frac14 092 GHz forthe Teflon container filled with water were given to the trained network and it

Figure 10Diagram (a) and photo (b)of the experimental setupfor the one-port solution

COMPEL241

302

generated waterrsquos dielectric constant and the loss factor For the sample of knowntemperature and salinity 10 and 100 have been also determined from the model whoseaverage error is 03 percent for 10 and 18 percent for 100 (Eves and Yakovlev 2002)As one can see from Table II the results are in very close agreement This confirms thecapability of the proposed ANN-based method for accurate reconstructing of complexpermittivity of materials

5 Computer resourcesThe computational cost of the method is primarily determined by the time required tocreate a database for network training and testing The time spent on the training itselfis nearly negligible For example when working with the two-port scheme and usingthe 15 and 33 mm cells in air and dielectric respectively we dealt with the modelcontaining 26796 cells and the simulation of one point on a PC with Pentium IV25 GHz processor took 25 s Hence the database with 149 samples outlined in Section 3was created within 62 min Clearly the accuracy of permittivity reconstruction couldbe generally improved by increasing the number of samples in the database (and thusagreeing on a higher computational cost)

Also the precision of our method may depend on the accuracy of modeling which inits turn is conditioned by the FDTD mesh used In order to virtually exclude aninfluence of discretization and to make sure that the applied cell sizes for all mediainvolved are adequate we performed a sensitivity analysis prior to building thedatabases subsequently simulating the scenario with slightly smaller cells as long asno substantial change in the results was noticed All cellsrsquo sizes mentioned above areresults of this type of an analysis

6 ConclusionOur novel technology of permittivity reconstruction which employs FDTD modelingan ANN-based optimization technique and elementary measurement of S-parametersplaces minimal physical requirements on fixture and sample geometry and issufficiently accurate for practical use Further developments of the method mayinclude its adjustment to non-homogeneous dielectrics and a refinement to allowsample preparation to less strict dimensional tolerances

The practical advantages of the method are obvious It does not depend on theassociated closed system and thus can be used with any available cavity and anysuitable FDTD simulator not necessarily QW-3D While a relatively largecomputational effort may be required for creation of a database the subsequentprocesses of training and determination of complex permittivity require nearlynegligible time Whenever we work at a fixed frequency with materials that can take

Proposed method Model (Eves and Yakovlev 2002) Divergence (percent)

10

806 805 012100

425 430 12

Note Complex permittivity of fresh water with salinity 0033 percent at temperature 1868Cdetermined by the one-port method and the 2 pound n-input MLP ANN Table II

FDTD-backedpermittivity

reconstruction

303

some pre-defined form the database is created only once One can do that prior toactual experimental testing and each new material can be processed thereafterpractically in real time ndash provided that 10 and 100 of this material are within the rangesspecified in the database and that the computer model is based upon the measuredexperimental fixture

References

Coccioli R Pelosi G and Selleri S (1999) ldquoCharacterization of dielectric materials with thefinite-element methodrdquo IEEE Transactions on Microwave Theory and Techniques Vol 47No 10 pp 1106-12

Deshpande MD and Reddy CJ (1995) ldquoApplication of FEM to estimate complex permittivity ofdielectric material at microwave frequency using waveguide measurementsrdquo NASAContractor Report CR-198203 p 23

Eves EE and Yakovlev VV (2002) ldquoAnalysis of operational regimes of a high power waterloadrdquo Journal of Microwave Power amp Electromagnetic Energy Vol 37 No 3 pp 127-44

Eves EE Kopyt P and Yakovlev VV (2004) ldquoDetermination of complex permittivity withneural networks and FDTD modelingrdquo Microwave Optical Technology Letters Vol 40No 3 pp 183-8

Nelson S and Datta AK (2001) ldquoDielectric properties of food materialsrdquo in Datta AK andAnantheswaran RC (Eds) Handbook of Microwave Technology for Food ApplicationsMarcel Dekker Inc New York NY pp 69-114

QuickWave-3D (1997-2004) QWED ul Zwyciezcow 342 03-938 Warsaw Poland wwwqwedcompl

Thakur KP and Holmes WS (2001) ldquoAn inverse technique to evaluate permittivity of materialin a cavityrdquo IEEE Transactions on Microwave Theory and Techniques Vol 49 No 10pp 1129-32

Wappling-Raaholt B and Risman PO (2003) ldquoPermittivity determination of inhomogeneousfoods by measurement and automated retro-modeling with a degenerate mode cavityrdquoProceedings of the 9th Conference on Microwave and HF Heating Loughborough UKpp 181-4

COMPEL241

304

(QW-3D) (QuickWave-3D 1997-2004) Data required for network training are collectedby a special procedure that repeatedly runs QW-3D to compute S-parameters forvarious values of 10 and 100 of the sample

The one-port scheme has been tested numerically for a section of 72 pound 34 mmwaveguide with a rectangular eth20 pound 20 pound 30 mmTHORN dielectric block in the corner near theshorting wall The FDTD model representing this scenario was built with anonuniform mesh with 75 and 3 mm cubic cells in air and in a dielectric samplerespectively (8463 cells total)

The networks were trained using vectors of jS11j frequency responses with n frac14 3f 1 frac14 24 GHz f 2 frac14 f 0 frac14 245 GHz f 3 frac14 25 GHz and with 27 values of complexpermittivity from the intervals 5 10 9 and 02 100 10 The graphs in Figure 4show the typical sum-squared error produced by the n- and 2pound n-input networks fordifferent number of neurons in the second layer It is seen that for more than ten hiddenneurons the networks are characterized by errors not larger than 1025

When tested with the training sets of 51 vectors the networks demonstratedsufficiently accurate permittivity reconstruction The desired and actual responses fromthe 2 pound n-input MLP are shown in Figure 5 mean square error (MSE) is of order 1023

32 Two-port structureThe two-port scheme dealing with the S-parameters at f0 has been numerically testedwith vectors of Re(S11) Im(S11) Re(S21) and Im(S21) at f 0 frac14 915 MHz for the 497 mmsection of a 248 pound 124 mm waveguide containing a rectangular dielectric sample (Table I)

We built the training sets for the values of relative complex permittivity in the ranges54 10 74 and 6 100 30The 4- and 2-input MLP and the 4-input RBF ANNs weretrained with the sets obtained for 48 equally spaced points in the complex (10 100)-plane andadditional points on the border (68 samples total) For the 2-input RBF where the numberof vectors in the training set is equal to the number of hidden neurons the decision as tohow many vectors (ie points from the (10 100)-plane) in the database to use was madedynamically The network was given a small database and the error was computedThe three test points with the greatest error were chosen and for each point an averagewas taken between the supposed and the ANN-generated values This average was thentaken for the computation of the next sample for the database For example for sample B inposition B the optimal number of training vectors (and hidden neurons) turned out to be 57(Figure 6) In the 4- and 2-input MLP N was taken 13 and 14 respectively

Although all MLPRBF 4-2-input networks have demonstrated good performancesome of them were found to be more accurate In Figure 7 the desired and actualresponses are shown for the 2-input RBF network with a corresponding MSE 0013while for the 4- and 2-input MLP ANNs MSEs are 0029 and 0073 respectively

Training sets for the ranges of 36 10 56 and 4 100 26 have also beencreated The MLP and RBF networks were trained as described above The 2-inputnets have again shown somewhat lower errors In Figure 8 typical examples of thedesired and actual responses from the MLP networks are presented in both casesthe MSE values are of order 1023

The detailed error analysis has been carried out to evaluate the accuracy of thetwo-port systems with a ^2 mm divergence in the samplersquos geometry in eachdimension Numerical experimentation has been performed for 1 frac14 57 2 i8 (apple 88percent moisture contents) 1 frac14 68 2 i14 (cantaloupe 92 percent) 1 frac14 62 2 i22

FDTD-backedpermittivity

reconstruction

297

Figure 4Training and testing errorof n-input MLP (a) andNet A (b) and Net B (c) of2 pound n-input MLP

COMPEL241

298

(potato 79 percent) and 1 frac14 55 2 i16 (sweet potato 80 percent) (Nelson and Datta2001) A typical example of this computation is shown in Figure 9

Generalizing the results of the analysis conducted for these materials as samplesA-D at Positions A-D we conclude that the 2-input networks can give an error in 100

less than 5 percent if the samplersquos geometrical deviation in the longitudinal andtransverse directions does not exceed 05-10 mm A 10 percent error results from a12-15 mm deviation For 10 the error is less than 5 percent when the deviation is less

Figure 5Complex permittivity

reconstructed with the2 pound n-input MLP withNA frac14 NB frac14 10 circles

and crossed circles markthe test data and the actual

responses respectively

Sample x- y- z-dimensions (mm) Position Distance (mm) from

A 50 pound 50 pound 20 A the second port 120 central line 0B 42 pound 30 pound 50 B the second port 120 central line 30C 20 pound 25 pound 62 C the second port 120 central line 60D 20 pound 25 pound 20 D the second port 150 central line 30

Table IDielectric samples used in

numerical testing of thenetworks for the two-port

scheme

Figure 6MSE of the 2-input RBF

with the number oftraining samples from 48

to 69 with step 3

FDTD-backedpermittivity

reconstruction

299

Figure 8Complex permittivity ofsample C in position Breconstructed by the4-input MLP with N frac14 13(a) and the 2-input MLPwith N frac14 14 the test dataand the actual responses

Figure 7Complex permittivity ofsample B in position Breconstructed with the2-input RBF withN frac14 57 the test data andthe actual responses

COMPEL241

300

than 12-15 mm and always less than 10 percent in the considered 2 mm deviationSimultaneously a notable variation of accuracy is observed when the samplersquos heightchanges ndash even if variation in the vertical dimension is quite small So for highaccuracy the experimental setup should be constructed to minimize accidentaldeviations of the sample size in the z-direction

4 Experimental testingTo show the method in full operation we have designed the experimental fixtureimplementing the concept of the one-port solution and thus measuring reflections froma cavity with a dielectric sample (Figure 10) Using a rectangular eth70 pound 70 pound 50 mmTHORNTeflon block with a cylindrical cutout (radius 25 mm height 40 mm) suitable for

Figure 9Percent error in getting

right 10 (a) and 100 (b) as afunction of deviation of

training data for thesample dimension in the

x-direction potato assample B in position B

FDTD-backedpermittivity

reconstruction

301

holding liquids we have determined complex permittivity of tap and saline waterThe container filled with water was placed on the center line of the waveguide sectionat 40 mm from the waveguidersquos shorting wall in the opposite end with respect to thecoaxial-waveguide transition

We used a QW-3D model consisting of 71442 cells with a non-uniform mesh(cell sizes in air Teflon and water are 15 5 and 2 mm respectively) for theentire cavity and dielectric inclusions The permittivity of Teflon was taken as206 2 i0

The database of the training and testing sets was created with n frac14 3 f 1 frac14091 GHz f 2 frac14 f 0 frac14 0915 GHz and f 3 frac14 092 GHz for 60 10 90 and 1 100 20and included 108 and 224 vectors respectively For the 2pound n-input network theoptimal structure was found as having NA frac14 15 and NB frac14 19 The normalized sum ofsquared differences between the desired and actual network responses at the trainingstage was less than 1024 for both Net A and Net B

The values of jS11j measured at f 1 frac14 091 GHz f 2 frac14 0915 GHz f 3 frac14 092 GHz forthe Teflon container filled with water were given to the trained network and it

Figure 10Diagram (a) and photo (b)of the experimental setupfor the one-port solution

COMPEL241

302

generated waterrsquos dielectric constant and the loss factor For the sample of knowntemperature and salinity 10 and 100 have been also determined from the model whoseaverage error is 03 percent for 10 and 18 percent for 100 (Eves and Yakovlev 2002)As one can see from Table II the results are in very close agreement This confirms thecapability of the proposed ANN-based method for accurate reconstructing of complexpermittivity of materials

5 Computer resourcesThe computational cost of the method is primarily determined by the time required tocreate a database for network training and testing The time spent on the training itselfis nearly negligible For example when working with the two-port scheme and usingthe 15 and 33 mm cells in air and dielectric respectively we dealt with the modelcontaining 26796 cells and the simulation of one point on a PC with Pentium IV25 GHz processor took 25 s Hence the database with 149 samples outlined in Section 3was created within 62 min Clearly the accuracy of permittivity reconstruction couldbe generally improved by increasing the number of samples in the database (and thusagreeing on a higher computational cost)

Also the precision of our method may depend on the accuracy of modeling which inits turn is conditioned by the FDTD mesh used In order to virtually exclude aninfluence of discretization and to make sure that the applied cell sizes for all mediainvolved are adequate we performed a sensitivity analysis prior to building thedatabases subsequently simulating the scenario with slightly smaller cells as long asno substantial change in the results was noticed All cellsrsquo sizes mentioned above areresults of this type of an analysis

6 ConclusionOur novel technology of permittivity reconstruction which employs FDTD modelingan ANN-based optimization technique and elementary measurement of S-parametersplaces minimal physical requirements on fixture and sample geometry and issufficiently accurate for practical use Further developments of the method mayinclude its adjustment to non-homogeneous dielectrics and a refinement to allowsample preparation to less strict dimensional tolerances

The practical advantages of the method are obvious It does not depend on theassociated closed system and thus can be used with any available cavity and anysuitable FDTD simulator not necessarily QW-3D While a relatively largecomputational effort may be required for creation of a database the subsequentprocesses of training and determination of complex permittivity require nearlynegligible time Whenever we work at a fixed frequency with materials that can take

Proposed method Model (Eves and Yakovlev 2002) Divergence (percent)

10

806 805 012100

425 430 12

Note Complex permittivity of fresh water with salinity 0033 percent at temperature 1868Cdetermined by the one-port method and the 2 pound n-input MLP ANN Table II

FDTD-backedpermittivity

reconstruction

303

some pre-defined form the database is created only once One can do that prior toactual experimental testing and each new material can be processed thereafterpractically in real time ndash provided that 10 and 100 of this material are within the rangesspecified in the database and that the computer model is based upon the measuredexperimental fixture

References

Coccioli R Pelosi G and Selleri S (1999) ldquoCharacterization of dielectric materials with thefinite-element methodrdquo IEEE Transactions on Microwave Theory and Techniques Vol 47No 10 pp 1106-12

Deshpande MD and Reddy CJ (1995) ldquoApplication of FEM to estimate complex permittivity ofdielectric material at microwave frequency using waveguide measurementsrdquo NASAContractor Report CR-198203 p 23

Eves EE and Yakovlev VV (2002) ldquoAnalysis of operational regimes of a high power waterloadrdquo Journal of Microwave Power amp Electromagnetic Energy Vol 37 No 3 pp 127-44

Eves EE Kopyt P and Yakovlev VV (2004) ldquoDetermination of complex permittivity withneural networks and FDTD modelingrdquo Microwave Optical Technology Letters Vol 40No 3 pp 183-8

Nelson S and Datta AK (2001) ldquoDielectric properties of food materialsrdquo in Datta AK andAnantheswaran RC (Eds) Handbook of Microwave Technology for Food ApplicationsMarcel Dekker Inc New York NY pp 69-114

QuickWave-3D (1997-2004) QWED ul Zwyciezcow 342 03-938 Warsaw Poland wwwqwedcompl

Thakur KP and Holmes WS (2001) ldquoAn inverse technique to evaluate permittivity of materialin a cavityrdquo IEEE Transactions on Microwave Theory and Techniques Vol 49 No 10pp 1129-32

Wappling-Raaholt B and Risman PO (2003) ldquoPermittivity determination of inhomogeneousfoods by measurement and automated retro-modeling with a degenerate mode cavityrdquoProceedings of the 9th Conference on Microwave and HF Heating Loughborough UKpp 181-4

COMPEL241

304

Figure 4Training and testing errorof n-input MLP (a) andNet A (b) and Net B (c) of2 pound n-input MLP

COMPEL241

298

(potato 79 percent) and 1 frac14 55 2 i16 (sweet potato 80 percent) (Nelson and Datta2001) A typical example of this computation is shown in Figure 9

Generalizing the results of the analysis conducted for these materials as samplesA-D at Positions A-D we conclude that the 2-input networks can give an error in 100

less than 5 percent if the samplersquos geometrical deviation in the longitudinal andtransverse directions does not exceed 05-10 mm A 10 percent error results from a12-15 mm deviation For 10 the error is less than 5 percent when the deviation is less

Figure 5Complex permittivity

reconstructed with the2 pound n-input MLP withNA frac14 NB frac14 10 circles

and crossed circles markthe test data and the actual

responses respectively

Sample x- y- z-dimensions (mm) Position Distance (mm) from

A 50 pound 50 pound 20 A the second port 120 central line 0B 42 pound 30 pound 50 B the second port 120 central line 30C 20 pound 25 pound 62 C the second port 120 central line 60D 20 pound 25 pound 20 D the second port 150 central line 30

Table IDielectric samples used in

numerical testing of thenetworks for the two-port

scheme

Figure 6MSE of the 2-input RBF

with the number oftraining samples from 48

to 69 with step 3

FDTD-backedpermittivity

reconstruction

299

Figure 8Complex permittivity ofsample C in position Breconstructed by the4-input MLP with N frac14 13(a) and the 2-input MLPwith N frac14 14 the test dataand the actual responses

Figure 7Complex permittivity ofsample B in position Breconstructed with the2-input RBF withN frac14 57 the test data andthe actual responses

COMPEL241

300

than 12-15 mm and always less than 10 percent in the considered 2 mm deviationSimultaneously a notable variation of accuracy is observed when the samplersquos heightchanges ndash even if variation in the vertical dimension is quite small So for highaccuracy the experimental setup should be constructed to minimize accidentaldeviations of the sample size in the z-direction

4 Experimental testingTo show the method in full operation we have designed the experimental fixtureimplementing the concept of the one-port solution and thus measuring reflections froma cavity with a dielectric sample (Figure 10) Using a rectangular eth70 pound 70 pound 50 mmTHORNTeflon block with a cylindrical cutout (radius 25 mm height 40 mm) suitable for

Figure 9Percent error in getting

right 10 (a) and 100 (b) as afunction of deviation of

training data for thesample dimension in the

x-direction potato assample B in position B

FDTD-backedpermittivity

reconstruction

301

holding liquids we have determined complex permittivity of tap and saline waterThe container filled with water was placed on the center line of the waveguide sectionat 40 mm from the waveguidersquos shorting wall in the opposite end with respect to thecoaxial-waveguide transition

We used a QW-3D model consisting of 71442 cells with a non-uniform mesh(cell sizes in air Teflon and water are 15 5 and 2 mm respectively) for theentire cavity and dielectric inclusions The permittivity of Teflon was taken as206 2 i0

The database of the training and testing sets was created with n frac14 3 f 1 frac14091 GHz f 2 frac14 f 0 frac14 0915 GHz and f 3 frac14 092 GHz for 60 10 90 and 1 100 20and included 108 and 224 vectors respectively For the 2pound n-input network theoptimal structure was found as having NA frac14 15 and NB frac14 19 The normalized sum ofsquared differences between the desired and actual network responses at the trainingstage was less than 1024 for both Net A and Net B

The values of jS11j measured at f 1 frac14 091 GHz f 2 frac14 0915 GHz f 3 frac14 092 GHz forthe Teflon container filled with water were given to the trained network and it

Figure 10Diagram (a) and photo (b)of the experimental setupfor the one-port solution

COMPEL241

302

generated waterrsquos dielectric constant and the loss factor For the sample of knowntemperature and salinity 10 and 100 have been also determined from the model whoseaverage error is 03 percent for 10 and 18 percent for 100 (Eves and Yakovlev 2002)As one can see from Table II the results are in very close agreement This confirms thecapability of the proposed ANN-based method for accurate reconstructing of complexpermittivity of materials

5 Computer resourcesThe computational cost of the method is primarily determined by the time required tocreate a database for network training and testing The time spent on the training itselfis nearly negligible For example when working with the two-port scheme and usingthe 15 and 33 mm cells in air and dielectric respectively we dealt with the modelcontaining 26796 cells and the simulation of one point on a PC with Pentium IV25 GHz processor took 25 s Hence the database with 149 samples outlined in Section 3was created within 62 min Clearly the accuracy of permittivity reconstruction couldbe generally improved by increasing the number of samples in the database (and thusagreeing on a higher computational cost)

Also the precision of our method may depend on the accuracy of modeling which inits turn is conditioned by the FDTD mesh used In order to virtually exclude aninfluence of discretization and to make sure that the applied cell sizes for all mediainvolved are adequate we performed a sensitivity analysis prior to building thedatabases subsequently simulating the scenario with slightly smaller cells as long asno substantial change in the results was noticed All cellsrsquo sizes mentioned above areresults of this type of an analysis

6 ConclusionOur novel technology of permittivity reconstruction which employs FDTD modelingan ANN-based optimization technique and elementary measurement of S-parametersplaces minimal physical requirements on fixture and sample geometry and issufficiently accurate for practical use Further developments of the method mayinclude its adjustment to non-homogeneous dielectrics and a refinement to allowsample preparation to less strict dimensional tolerances

The practical advantages of the method are obvious It does not depend on theassociated closed system and thus can be used with any available cavity and anysuitable FDTD simulator not necessarily QW-3D While a relatively largecomputational effort may be required for creation of a database the subsequentprocesses of training and determination of complex permittivity require nearlynegligible time Whenever we work at a fixed frequency with materials that can take

Proposed method Model (Eves and Yakovlev 2002) Divergence (percent)

10

806 805 012100

425 430 12

Note Complex permittivity of fresh water with salinity 0033 percent at temperature 1868Cdetermined by the one-port method and the 2 pound n-input MLP ANN Table II

FDTD-backedpermittivity

reconstruction

303

some pre-defined form the database is created only once One can do that prior toactual experimental testing and each new material can be processed thereafterpractically in real time ndash provided that 10 and 100 of this material are within the rangesspecified in the database and that the computer model is based upon the measuredexperimental fixture

References

Coccioli R Pelosi G and Selleri S (1999) ldquoCharacterization of dielectric materials with thefinite-element methodrdquo IEEE Transactions on Microwave Theory and Techniques Vol 47No 10 pp 1106-12

Deshpande MD and Reddy CJ (1995) ldquoApplication of FEM to estimate complex permittivity ofdielectric material at microwave frequency using waveguide measurementsrdquo NASAContractor Report CR-198203 p 23

Eves EE and Yakovlev VV (2002) ldquoAnalysis of operational regimes of a high power waterloadrdquo Journal of Microwave Power amp Electromagnetic Energy Vol 37 No 3 pp 127-44

Eves EE Kopyt P and Yakovlev VV (2004) ldquoDetermination of complex permittivity withneural networks and FDTD modelingrdquo Microwave Optical Technology Letters Vol 40No 3 pp 183-8

Nelson S and Datta AK (2001) ldquoDielectric properties of food materialsrdquo in Datta AK andAnantheswaran RC (Eds) Handbook of Microwave Technology for Food ApplicationsMarcel Dekker Inc New York NY pp 69-114

QuickWave-3D (1997-2004) QWED ul Zwyciezcow 342 03-938 Warsaw Poland wwwqwedcompl

Thakur KP and Holmes WS (2001) ldquoAn inverse technique to evaluate permittivity of materialin a cavityrdquo IEEE Transactions on Microwave Theory and Techniques Vol 49 No 10pp 1129-32

Wappling-Raaholt B and Risman PO (2003) ldquoPermittivity determination of inhomogeneousfoods by measurement and automated retro-modeling with a degenerate mode cavityrdquoProceedings of the 9th Conference on Microwave and HF Heating Loughborough UKpp 181-4

COMPEL241

304

(potato 79 percent) and 1 frac14 55 2 i16 (sweet potato 80 percent) (Nelson and Datta2001) A typical example of this computation is shown in Figure 9

Generalizing the results of the analysis conducted for these materials as samplesA-D at Positions A-D we conclude that the 2-input networks can give an error in 100

less than 5 percent if the samplersquos geometrical deviation in the longitudinal andtransverse directions does not exceed 05-10 mm A 10 percent error results from a12-15 mm deviation For 10 the error is less than 5 percent when the deviation is less

Figure 5Complex permittivity

reconstructed with the2 pound n-input MLP withNA frac14 NB frac14 10 circles

and crossed circles markthe test data and the actual

responses respectively

Sample x- y- z-dimensions (mm) Position Distance (mm) from

A 50 pound 50 pound 20 A the second port 120 central line 0B 42 pound 30 pound 50 B the second port 120 central line 30C 20 pound 25 pound 62 C the second port 120 central line 60D 20 pound 25 pound 20 D the second port 150 central line 30

Table IDielectric samples used in

numerical testing of thenetworks for the two-port

scheme

Figure 6MSE of the 2-input RBF

with the number oftraining samples from 48

to 69 with step 3

FDTD-backedpermittivity

reconstruction

299

Figure 8Complex permittivity ofsample C in position Breconstructed by the4-input MLP with N frac14 13(a) and the 2-input MLPwith N frac14 14 the test dataand the actual responses

Figure 7Complex permittivity ofsample B in position Breconstructed with the2-input RBF withN frac14 57 the test data andthe actual responses

COMPEL241

300

than 12-15 mm and always less than 10 percent in the considered 2 mm deviationSimultaneously a notable variation of accuracy is observed when the samplersquos heightchanges ndash even if variation in the vertical dimension is quite small So for highaccuracy the experimental setup should be constructed to minimize accidentaldeviations of the sample size in the z-direction

4 Experimental testingTo show the method in full operation we have designed the experimental fixtureimplementing the concept of the one-port solution and thus measuring reflections froma cavity with a dielectric sample (Figure 10) Using a rectangular eth70 pound 70 pound 50 mmTHORNTeflon block with a cylindrical cutout (radius 25 mm height 40 mm) suitable for

Figure 9Percent error in getting

right 10 (a) and 100 (b) as afunction of deviation of

training data for thesample dimension in the

x-direction potato assample B in position B

FDTD-backedpermittivity

reconstruction

301

holding liquids we have determined complex permittivity of tap and saline waterThe container filled with water was placed on the center line of the waveguide sectionat 40 mm from the waveguidersquos shorting wall in the opposite end with respect to thecoaxial-waveguide transition

We used a QW-3D model consisting of 71442 cells with a non-uniform mesh(cell sizes in air Teflon and water are 15 5 and 2 mm respectively) for theentire cavity and dielectric inclusions The permittivity of Teflon was taken as206 2 i0

The database of the training and testing sets was created with n frac14 3 f 1 frac14091 GHz f 2 frac14 f 0 frac14 0915 GHz and f 3 frac14 092 GHz for 60 10 90 and 1 100 20and included 108 and 224 vectors respectively For the 2pound n-input network theoptimal structure was found as having NA frac14 15 and NB frac14 19 The normalized sum ofsquared differences between the desired and actual network responses at the trainingstage was less than 1024 for both Net A and Net B

The values of jS11j measured at f 1 frac14 091 GHz f 2 frac14 0915 GHz f 3 frac14 092 GHz forthe Teflon container filled with water were given to the trained network and it

Figure 10Diagram (a) and photo (b)of the experimental setupfor the one-port solution

COMPEL241

302

generated waterrsquos dielectric constant and the loss factor For the sample of knowntemperature and salinity 10 and 100 have been also determined from the model whoseaverage error is 03 percent for 10 and 18 percent for 100 (Eves and Yakovlev 2002)As one can see from Table II the results are in very close agreement This confirms thecapability of the proposed ANN-based method for accurate reconstructing of complexpermittivity of materials

5 Computer resourcesThe computational cost of the method is primarily determined by the time required tocreate a database for network training and testing The time spent on the training itselfis nearly negligible For example when working with the two-port scheme and usingthe 15 and 33 mm cells in air and dielectric respectively we dealt with the modelcontaining 26796 cells and the simulation of one point on a PC with Pentium IV25 GHz processor took 25 s Hence the database with 149 samples outlined in Section 3was created within 62 min Clearly the accuracy of permittivity reconstruction couldbe generally improved by increasing the number of samples in the database (and thusagreeing on a higher computational cost)

Also the precision of our method may depend on the accuracy of modeling which inits turn is conditioned by the FDTD mesh used In order to virtually exclude aninfluence of discretization and to make sure that the applied cell sizes for all mediainvolved are adequate we performed a sensitivity analysis prior to building thedatabases subsequently simulating the scenario with slightly smaller cells as long asno substantial change in the results was noticed All cellsrsquo sizes mentioned above areresults of this type of an analysis

6 ConclusionOur novel technology of permittivity reconstruction which employs FDTD modelingan ANN-based optimization technique and elementary measurement of S-parametersplaces minimal physical requirements on fixture and sample geometry and issufficiently accurate for practical use Further developments of the method mayinclude its adjustment to non-homogeneous dielectrics and a refinement to allowsample preparation to less strict dimensional tolerances

The practical advantages of the method are obvious It does not depend on theassociated closed system and thus can be used with any available cavity and anysuitable FDTD simulator not necessarily QW-3D While a relatively largecomputational effort may be required for creation of a database the subsequentprocesses of training and determination of complex permittivity require nearlynegligible time Whenever we work at a fixed frequency with materials that can take

Proposed method Model (Eves and Yakovlev 2002) Divergence (percent)

10

806 805 012100

425 430 12

Note Complex permittivity of fresh water with salinity 0033 percent at temperature 1868Cdetermined by the one-port method and the 2 pound n-input MLP ANN Table II

FDTD-backedpermittivity

reconstruction

303

some pre-defined form the database is created only once One can do that prior toactual experimental testing and each new material can be processed thereafterpractically in real time ndash provided that 10 and 100 of this material are within the rangesspecified in the database and that the computer model is based upon the measuredexperimental fixture

References

Coccioli R Pelosi G and Selleri S (1999) ldquoCharacterization of dielectric materials with thefinite-element methodrdquo IEEE Transactions on Microwave Theory and Techniques Vol 47No 10 pp 1106-12

Deshpande MD and Reddy CJ (1995) ldquoApplication of FEM to estimate complex permittivity ofdielectric material at microwave frequency using waveguide measurementsrdquo NASAContractor Report CR-198203 p 23

Eves EE and Yakovlev VV (2002) ldquoAnalysis of operational regimes of a high power waterloadrdquo Journal of Microwave Power amp Electromagnetic Energy Vol 37 No 3 pp 127-44

Eves EE Kopyt P and Yakovlev VV (2004) ldquoDetermination of complex permittivity withneural networks and FDTD modelingrdquo Microwave Optical Technology Letters Vol 40No 3 pp 183-8

Nelson S and Datta AK (2001) ldquoDielectric properties of food materialsrdquo in Datta AK andAnantheswaran RC (Eds) Handbook of Microwave Technology for Food ApplicationsMarcel Dekker Inc New York NY pp 69-114

QuickWave-3D (1997-2004) QWED ul Zwyciezcow 342 03-938 Warsaw Poland wwwqwedcompl

Thakur KP and Holmes WS (2001) ldquoAn inverse technique to evaluate permittivity of materialin a cavityrdquo IEEE Transactions on Microwave Theory and Techniques Vol 49 No 10pp 1129-32

Wappling-Raaholt B and Risman PO (2003) ldquoPermittivity determination of inhomogeneousfoods by measurement and automated retro-modeling with a degenerate mode cavityrdquoProceedings of the 9th Conference on Microwave and HF Heating Loughborough UKpp 181-4

COMPEL241

304

Figure 8Complex permittivity ofsample C in position Breconstructed by the4-input MLP with N frac14 13(a) and the 2-input MLPwith N frac14 14 the test dataand the actual responses

Figure 7Complex permittivity ofsample B in position Breconstructed with the2-input RBF withN frac14 57 the test data andthe actual responses

COMPEL241

300

than 12-15 mm and always less than 10 percent in the considered 2 mm deviationSimultaneously a notable variation of accuracy is observed when the samplersquos heightchanges ndash even if variation in the vertical dimension is quite small So for highaccuracy the experimental setup should be constructed to minimize accidentaldeviations of the sample size in the z-direction

4 Experimental testingTo show the method in full operation we have designed the experimental fixtureimplementing the concept of the one-port solution and thus measuring reflections froma cavity with a dielectric sample (Figure 10) Using a rectangular eth70 pound 70 pound 50 mmTHORNTeflon block with a cylindrical cutout (radius 25 mm height 40 mm) suitable for

Figure 9Percent error in getting

right 10 (a) and 100 (b) as afunction of deviation of

training data for thesample dimension in the

x-direction potato assample B in position B

FDTD-backedpermittivity

reconstruction

301

holding liquids we have determined complex permittivity of tap and saline waterThe container filled with water was placed on the center line of the waveguide sectionat 40 mm from the waveguidersquos shorting wall in the opposite end with respect to thecoaxial-waveguide transition

We used a QW-3D model consisting of 71442 cells with a non-uniform mesh(cell sizes in air Teflon and water are 15 5 and 2 mm respectively) for theentire cavity and dielectric inclusions The permittivity of Teflon was taken as206 2 i0

The database of the training and testing sets was created with n frac14 3 f 1 frac14091 GHz f 2 frac14 f 0 frac14 0915 GHz and f 3 frac14 092 GHz for 60 10 90 and 1 100 20and included 108 and 224 vectors respectively For the 2pound n-input network theoptimal structure was found as having NA frac14 15 and NB frac14 19 The normalized sum ofsquared differences between the desired and actual network responses at the trainingstage was less than 1024 for both Net A and Net B

The values of jS11j measured at f 1 frac14 091 GHz f 2 frac14 0915 GHz f 3 frac14 092 GHz forthe Teflon container filled with water were given to the trained network and it

Figure 10Diagram (a) and photo (b)of the experimental setupfor the one-port solution

COMPEL241

302

generated waterrsquos dielectric constant and the loss factor For the sample of knowntemperature and salinity 10 and 100 have been also determined from the model whoseaverage error is 03 percent for 10 and 18 percent for 100 (Eves and Yakovlev 2002)As one can see from Table II the results are in very close agreement This confirms thecapability of the proposed ANN-based method for accurate reconstructing of complexpermittivity of materials

5 Computer resourcesThe computational cost of the method is primarily determined by the time required tocreate a database for network training and testing The time spent on the training itselfis nearly negligible For example when working with the two-port scheme and usingthe 15 and 33 mm cells in air and dielectric respectively we dealt with the modelcontaining 26796 cells and the simulation of one point on a PC with Pentium IV25 GHz processor took 25 s Hence the database with 149 samples outlined in Section 3was created within 62 min Clearly the accuracy of permittivity reconstruction couldbe generally improved by increasing the number of samples in the database (and thusagreeing on a higher computational cost)

Also the precision of our method may depend on the accuracy of modeling which inits turn is conditioned by the FDTD mesh used In order to virtually exclude aninfluence of discretization and to make sure that the applied cell sizes for all mediainvolved are adequate we performed a sensitivity analysis prior to building thedatabases subsequently simulating the scenario with slightly smaller cells as long asno substantial change in the results was noticed All cellsrsquo sizes mentioned above areresults of this type of an analysis

6 ConclusionOur novel technology of permittivity reconstruction which employs FDTD modelingan ANN-based optimization technique and elementary measurement of S-parametersplaces minimal physical requirements on fixture and sample geometry and issufficiently accurate for practical use Further developments of the method mayinclude its adjustment to non-homogeneous dielectrics and a refinement to allowsample preparation to less strict dimensional tolerances

The practical advantages of the method are obvious It does not depend on theassociated closed system and thus can be used with any available cavity and anysuitable FDTD simulator not necessarily QW-3D While a relatively largecomputational effort may be required for creation of a database the subsequentprocesses of training and determination of complex permittivity require nearlynegligible time Whenever we work at a fixed frequency with materials that can take

Proposed method Model (Eves and Yakovlev 2002) Divergence (percent)

10

806 805 012100

425 430 12

Note Complex permittivity of fresh water with salinity 0033 percent at temperature 1868Cdetermined by the one-port method and the 2 pound n-input MLP ANN Table II

FDTD-backedpermittivity

reconstruction

303

some pre-defined form the database is created only once One can do that prior toactual experimental testing and each new material can be processed thereafterpractically in real time ndash provided that 10 and 100 of this material are within the rangesspecified in the database and that the computer model is based upon the measuredexperimental fixture

References

Coccioli R Pelosi G and Selleri S (1999) ldquoCharacterization of dielectric materials with thefinite-element methodrdquo IEEE Transactions on Microwave Theory and Techniques Vol 47No 10 pp 1106-12

Deshpande MD and Reddy CJ (1995) ldquoApplication of FEM to estimate complex permittivity ofdielectric material at microwave frequency using waveguide measurementsrdquo NASAContractor Report CR-198203 p 23

Eves EE and Yakovlev VV (2002) ldquoAnalysis of operational regimes of a high power waterloadrdquo Journal of Microwave Power amp Electromagnetic Energy Vol 37 No 3 pp 127-44

Eves EE Kopyt P and Yakovlev VV (2004) ldquoDetermination of complex permittivity withneural networks and FDTD modelingrdquo Microwave Optical Technology Letters Vol 40No 3 pp 183-8

Nelson S and Datta AK (2001) ldquoDielectric properties of food materialsrdquo in Datta AK andAnantheswaran RC (Eds) Handbook of Microwave Technology for Food ApplicationsMarcel Dekker Inc New York NY pp 69-114

QuickWave-3D (1997-2004) QWED ul Zwyciezcow 342 03-938 Warsaw Poland wwwqwedcompl

Thakur KP and Holmes WS (2001) ldquoAn inverse technique to evaluate permittivity of materialin a cavityrdquo IEEE Transactions on Microwave Theory and Techniques Vol 49 No 10pp 1129-32

Wappling-Raaholt B and Risman PO (2003) ldquoPermittivity determination of inhomogeneousfoods by measurement and automated retro-modeling with a degenerate mode cavityrdquoProceedings of the 9th Conference on Microwave and HF Heating Loughborough UKpp 181-4

COMPEL241

304

than 12-15 mm and always less than 10 percent in the considered 2 mm deviationSimultaneously a notable variation of accuracy is observed when the samplersquos heightchanges ndash even if variation in the vertical dimension is quite small So for highaccuracy the experimental setup should be constructed to minimize accidentaldeviations of the sample size in the z-direction

4 Experimental testingTo show the method in full operation we have designed the experimental fixtureimplementing the concept of the one-port solution and thus measuring reflections froma cavity with a dielectric sample (Figure 10) Using a rectangular eth70 pound 70 pound 50 mmTHORNTeflon block with a cylindrical cutout (radius 25 mm height 40 mm) suitable for

Figure 9Percent error in getting

right 10 (a) and 100 (b) as afunction of deviation of

training data for thesample dimension in the

x-direction potato assample B in position B

FDTD-backedpermittivity

reconstruction

301

holding liquids we have determined complex permittivity of tap and saline waterThe container filled with water was placed on the center line of the waveguide sectionat 40 mm from the waveguidersquos shorting wall in the opposite end with respect to thecoaxial-waveguide transition

We used a QW-3D model consisting of 71442 cells with a non-uniform mesh(cell sizes in air Teflon and water are 15 5 and 2 mm respectively) for theentire cavity and dielectric inclusions The permittivity of Teflon was taken as206 2 i0

The database of the training and testing sets was created with n frac14 3 f 1 frac14091 GHz f 2 frac14 f 0 frac14 0915 GHz and f 3 frac14 092 GHz for 60 10 90 and 1 100 20and included 108 and 224 vectors respectively For the 2pound n-input network theoptimal structure was found as having NA frac14 15 and NB frac14 19 The normalized sum ofsquared differences between the desired and actual network responses at the trainingstage was less than 1024 for both Net A and Net B

The values of jS11j measured at f 1 frac14 091 GHz f 2 frac14 0915 GHz f 3 frac14 092 GHz forthe Teflon container filled with water were given to the trained network and it

Figure 10Diagram (a) and photo (b)of the experimental setupfor the one-port solution

COMPEL241

302

generated waterrsquos dielectric constant and the loss factor For the sample of knowntemperature and salinity 10 and 100 have been also determined from the model whoseaverage error is 03 percent for 10 and 18 percent for 100 (Eves and Yakovlev 2002)As one can see from Table II the results are in very close agreement This confirms thecapability of the proposed ANN-based method for accurate reconstructing of complexpermittivity of materials

5 Computer resourcesThe computational cost of the method is primarily determined by the time required tocreate a database for network training and testing The time spent on the training itselfis nearly negligible For example when working with the two-port scheme and usingthe 15 and 33 mm cells in air and dielectric respectively we dealt with the modelcontaining 26796 cells and the simulation of one point on a PC with Pentium IV25 GHz processor took 25 s Hence the database with 149 samples outlined in Section 3was created within 62 min Clearly the accuracy of permittivity reconstruction couldbe generally improved by increasing the number of samples in the database (and thusagreeing on a higher computational cost)

Also the precision of our method may depend on the accuracy of modeling which inits turn is conditioned by the FDTD mesh used In order to virtually exclude aninfluence of discretization and to make sure that the applied cell sizes for all mediainvolved are adequate we performed a sensitivity analysis prior to building thedatabases subsequently simulating the scenario with slightly smaller cells as long asno substantial change in the results was noticed All cellsrsquo sizes mentioned above areresults of this type of an analysis

6 ConclusionOur novel technology of permittivity reconstruction which employs FDTD modelingan ANN-based optimization technique and elementary measurement of S-parametersplaces minimal physical requirements on fixture and sample geometry and issufficiently accurate for practical use Further developments of the method mayinclude its adjustment to non-homogeneous dielectrics and a refinement to allowsample preparation to less strict dimensional tolerances

The practical advantages of the method are obvious It does not depend on theassociated closed system and thus can be used with any available cavity and anysuitable FDTD simulator not necessarily QW-3D While a relatively largecomputational effort may be required for creation of a database the subsequentprocesses of training and determination of complex permittivity require nearlynegligible time Whenever we work at a fixed frequency with materials that can take

Proposed method Model (Eves and Yakovlev 2002) Divergence (percent)

10

806 805 012100

425 430 12

Note Complex permittivity of fresh water with salinity 0033 percent at temperature 1868Cdetermined by the one-port method and the 2 pound n-input MLP ANN Table II

FDTD-backedpermittivity

reconstruction

303

some pre-defined form the database is created only once One can do that prior toactual experimental testing and each new material can be processed thereafterpractically in real time ndash provided that 10 and 100 of this material are within the rangesspecified in the database and that the computer model is based upon the measuredexperimental fixture

References

Coccioli R Pelosi G and Selleri S (1999) ldquoCharacterization of dielectric materials with thefinite-element methodrdquo IEEE Transactions on Microwave Theory and Techniques Vol 47No 10 pp 1106-12

Deshpande MD and Reddy CJ (1995) ldquoApplication of FEM to estimate complex permittivity ofdielectric material at microwave frequency using waveguide measurementsrdquo NASAContractor Report CR-198203 p 23

Eves EE and Yakovlev VV (2002) ldquoAnalysis of operational regimes of a high power waterloadrdquo Journal of Microwave Power amp Electromagnetic Energy Vol 37 No 3 pp 127-44

Eves EE Kopyt P and Yakovlev VV (2004) ldquoDetermination of complex permittivity withneural networks and FDTD modelingrdquo Microwave Optical Technology Letters Vol 40No 3 pp 183-8

Nelson S and Datta AK (2001) ldquoDielectric properties of food materialsrdquo in Datta AK andAnantheswaran RC (Eds) Handbook of Microwave Technology for Food ApplicationsMarcel Dekker Inc New York NY pp 69-114

QuickWave-3D (1997-2004) QWED ul Zwyciezcow 342 03-938 Warsaw Poland wwwqwedcompl

Thakur KP and Holmes WS (2001) ldquoAn inverse technique to evaluate permittivity of materialin a cavityrdquo IEEE Transactions on Microwave Theory and Techniques Vol 49 No 10pp 1129-32

Wappling-Raaholt B and Risman PO (2003) ldquoPermittivity determination of inhomogeneousfoods by measurement and automated retro-modeling with a degenerate mode cavityrdquoProceedings of the 9th Conference on Microwave and HF Heating Loughborough UKpp 181-4

COMPEL241

304

holding liquids we have determined complex permittivity of tap and saline waterThe container filled with water was placed on the center line of the waveguide sectionat 40 mm from the waveguidersquos shorting wall in the opposite end with respect to thecoaxial-waveguide transition

We used a QW-3D model consisting of 71442 cells with a non-uniform mesh(cell sizes in air Teflon and water are 15 5 and 2 mm respectively) for theentire cavity and dielectric inclusions The permittivity of Teflon was taken as206 2 i0

The database of the training and testing sets was created with n frac14 3 f 1 frac14091 GHz f 2 frac14 f 0 frac14 0915 GHz and f 3 frac14 092 GHz for 60 10 90 and 1 100 20and included 108 and 224 vectors respectively For the 2pound n-input network theoptimal structure was found as having NA frac14 15 and NB frac14 19 The normalized sum ofsquared differences between the desired and actual network responses at the trainingstage was less than 1024 for both Net A and Net B

The values of jS11j measured at f 1 frac14 091 GHz f 2 frac14 0915 GHz f 3 frac14 092 GHz forthe Teflon container filled with water were given to the trained network and it

Figure 10Diagram (a) and photo (b)of the experimental setupfor the one-port solution

COMPEL241

302

generated waterrsquos dielectric constant and the loss factor For the sample of knowntemperature and salinity 10 and 100 have been also determined from the model whoseaverage error is 03 percent for 10 and 18 percent for 100 (Eves and Yakovlev 2002)As one can see from Table II the results are in very close agreement This confirms thecapability of the proposed ANN-based method for accurate reconstructing of complexpermittivity of materials

5 Computer resourcesThe computational cost of the method is primarily determined by the time required tocreate a database for network training and testing The time spent on the training itselfis nearly negligible For example when working with the two-port scheme and usingthe 15 and 33 mm cells in air and dielectric respectively we dealt with the modelcontaining 26796 cells and the simulation of one point on a PC with Pentium IV25 GHz processor took 25 s Hence the database with 149 samples outlined in Section 3was created within 62 min Clearly the accuracy of permittivity reconstruction couldbe generally improved by increasing the number of samples in the database (and thusagreeing on a higher computational cost)

Also the precision of our method may depend on the accuracy of modeling which inits turn is conditioned by the FDTD mesh used In order to virtually exclude aninfluence of discretization and to make sure that the applied cell sizes for all mediainvolved are adequate we performed a sensitivity analysis prior to building thedatabases subsequently simulating the scenario with slightly smaller cells as long asno substantial change in the results was noticed All cellsrsquo sizes mentioned above areresults of this type of an analysis

6 ConclusionOur novel technology of permittivity reconstruction which employs FDTD modelingan ANN-based optimization technique and elementary measurement of S-parametersplaces minimal physical requirements on fixture and sample geometry and issufficiently accurate for practical use Further developments of the method mayinclude its adjustment to non-homogeneous dielectrics and a refinement to allowsample preparation to less strict dimensional tolerances

The practical advantages of the method are obvious It does not depend on theassociated closed system and thus can be used with any available cavity and anysuitable FDTD simulator not necessarily QW-3D While a relatively largecomputational effort may be required for creation of a database the subsequentprocesses of training and determination of complex permittivity require nearlynegligible time Whenever we work at a fixed frequency with materials that can take

Proposed method Model (Eves and Yakovlev 2002) Divergence (percent)

10

806 805 012100

425 430 12

Note Complex permittivity of fresh water with salinity 0033 percent at temperature 1868Cdetermined by the one-port method and the 2 pound n-input MLP ANN Table II

FDTD-backedpermittivity

reconstruction

303

some pre-defined form the database is created only once One can do that prior toactual experimental testing and each new material can be processed thereafterpractically in real time ndash provided that 10 and 100 of this material are within the rangesspecified in the database and that the computer model is based upon the measuredexperimental fixture

References

Coccioli R Pelosi G and Selleri S (1999) ldquoCharacterization of dielectric materials with thefinite-element methodrdquo IEEE Transactions on Microwave Theory and Techniques Vol 47No 10 pp 1106-12

Deshpande MD and Reddy CJ (1995) ldquoApplication of FEM to estimate complex permittivity ofdielectric material at microwave frequency using waveguide measurementsrdquo NASAContractor Report CR-198203 p 23

Eves EE and Yakovlev VV (2002) ldquoAnalysis of operational regimes of a high power waterloadrdquo Journal of Microwave Power amp Electromagnetic Energy Vol 37 No 3 pp 127-44

Eves EE Kopyt P and Yakovlev VV (2004) ldquoDetermination of complex permittivity withneural networks and FDTD modelingrdquo Microwave Optical Technology Letters Vol 40No 3 pp 183-8

Nelson S and Datta AK (2001) ldquoDielectric properties of food materialsrdquo in Datta AK andAnantheswaran RC (Eds) Handbook of Microwave Technology for Food ApplicationsMarcel Dekker Inc New York NY pp 69-114

QuickWave-3D (1997-2004) QWED ul Zwyciezcow 342 03-938 Warsaw Poland wwwqwedcompl

Thakur KP and Holmes WS (2001) ldquoAn inverse technique to evaluate permittivity of materialin a cavityrdquo IEEE Transactions on Microwave Theory and Techniques Vol 49 No 10pp 1129-32

Wappling-Raaholt B and Risman PO (2003) ldquoPermittivity determination of inhomogeneousfoods by measurement and automated retro-modeling with a degenerate mode cavityrdquoProceedings of the 9th Conference on Microwave and HF Heating Loughborough UKpp 181-4

COMPEL241

304

generated waterrsquos dielectric constant and the loss factor For the sample of knowntemperature and salinity 10 and 100 have been also determined from the model whoseaverage error is 03 percent for 10 and 18 percent for 100 (Eves and Yakovlev 2002)As one can see from Table II the results are in very close agreement This confirms thecapability of the proposed ANN-based method for accurate reconstructing of complexpermittivity of materials

5 Computer resourcesThe computational cost of the method is primarily determined by the time required tocreate a database for network training and testing The time spent on the training itselfis nearly negligible For example when working with the two-port scheme and usingthe 15 and 33 mm cells in air and dielectric respectively we dealt with the modelcontaining 26796 cells and the simulation of one point on a PC with Pentium IV25 GHz processor took 25 s Hence the database with 149 samples outlined in Section 3was created within 62 min Clearly the accuracy of permittivity reconstruction couldbe generally improved by increasing the number of samples in the database (and thusagreeing on a higher computational cost)

Also the precision of our method may depend on the accuracy of modeling which inits turn is conditioned by the FDTD mesh used In order to virtually exclude aninfluence of discretization and to make sure that the applied cell sizes for all mediainvolved are adequate we performed a sensitivity analysis prior to building thedatabases subsequently simulating the scenario with slightly smaller cells as long asno substantial change in the results was noticed All cellsrsquo sizes mentioned above areresults of this type of an analysis

6 ConclusionOur novel technology of permittivity reconstruction which employs FDTD modelingan ANN-based optimization technique and elementary measurement of S-parametersplaces minimal physical requirements on fixture and sample geometry and issufficiently accurate for practical use Further developments of the method mayinclude its adjustment to non-homogeneous dielectrics and a refinement to allowsample preparation to less strict dimensional tolerances

The practical advantages of the method are obvious It does not depend on theassociated closed system and thus can be used with any available cavity and anysuitable FDTD simulator not necessarily QW-3D While a relatively largecomputational effort may be required for creation of a database the subsequentprocesses of training and determination of complex permittivity require nearlynegligible time Whenever we work at a fixed frequency with materials that can take

Proposed method Model (Eves and Yakovlev 2002) Divergence (percent)

10

806 805 012100

425 430 12

Note Complex permittivity of fresh water with salinity 0033 percent at temperature 1868Cdetermined by the one-port method and the 2 pound n-input MLP ANN Table II

FDTD-backedpermittivity

reconstruction

303

some pre-defined form the database is created only once One can do that prior toactual experimental testing and each new material can be processed thereafterpractically in real time ndash provided that 10 and 100 of this material are within the rangesspecified in the database and that the computer model is based upon the measuredexperimental fixture

References

Coccioli R Pelosi G and Selleri S (1999) ldquoCharacterization of dielectric materials with thefinite-element methodrdquo IEEE Transactions on Microwave Theory and Techniques Vol 47No 10 pp 1106-12

Deshpande MD and Reddy CJ (1995) ldquoApplication of FEM to estimate complex permittivity ofdielectric material at microwave frequency using waveguide measurementsrdquo NASAContractor Report CR-198203 p 23

Eves EE and Yakovlev VV (2002) ldquoAnalysis of operational regimes of a high power waterloadrdquo Journal of Microwave Power amp Electromagnetic Energy Vol 37 No 3 pp 127-44

Eves EE Kopyt P and Yakovlev VV (2004) ldquoDetermination of complex permittivity withneural networks and FDTD modelingrdquo Microwave Optical Technology Letters Vol 40No 3 pp 183-8

Nelson S and Datta AK (2001) ldquoDielectric properties of food materialsrdquo in Datta AK andAnantheswaran RC (Eds) Handbook of Microwave Technology for Food ApplicationsMarcel Dekker Inc New York NY pp 69-114

QuickWave-3D (1997-2004) QWED ul Zwyciezcow 342 03-938 Warsaw Poland wwwqwedcompl

Thakur KP and Holmes WS (2001) ldquoAn inverse technique to evaluate permittivity of materialin a cavityrdquo IEEE Transactions on Microwave Theory and Techniques Vol 49 No 10pp 1129-32

Wappling-Raaholt B and Risman PO (2003) ldquoPermittivity determination of inhomogeneousfoods by measurement and automated retro-modeling with a degenerate mode cavityrdquoProceedings of the 9th Conference on Microwave and HF Heating Loughborough UKpp 181-4

COMPEL241

304

some pre-defined form the database is created only once One can do that prior toactual experimental testing and each new material can be processed thereafterpractically in real time ndash provided that 10 and 100 of this material are within the rangesspecified in the database and that the computer model is based upon the measuredexperimental fixture

References

Coccioli R Pelosi G and Selleri S (1999) ldquoCharacterization of dielectric materials with thefinite-element methodrdquo IEEE Transactions on Microwave Theory and Techniques Vol 47No 10 pp 1106-12

Deshpande MD and Reddy CJ (1995) ldquoApplication of FEM to estimate complex permittivity ofdielectric material at microwave frequency using waveguide measurementsrdquo NASAContractor Report CR-198203 p 23

Eves EE and Yakovlev VV (2002) ldquoAnalysis of operational regimes of a high power waterloadrdquo Journal of Microwave Power amp Electromagnetic Energy Vol 37 No 3 pp 127-44

Eves EE Kopyt P and Yakovlev VV (2004) ldquoDetermination of complex permittivity withneural networks and FDTD modelingrdquo Microwave Optical Technology Letters Vol 40No 3 pp 183-8

Nelson S and Datta AK (2001) ldquoDielectric properties of food materialsrdquo in Datta AK andAnantheswaran RC (Eds) Handbook of Microwave Technology for Food ApplicationsMarcel Dekker Inc New York NY pp 69-114

QuickWave-3D (1997-2004) QWED ul Zwyciezcow 342 03-938 Warsaw Poland wwwqwedcompl

Thakur KP and Holmes WS (2001) ldquoAn inverse technique to evaluate permittivity of materialin a cavityrdquo IEEE Transactions on Microwave Theory and Techniques Vol 49 No 10pp 1129-32

Wappling-Raaholt B and Risman PO (2003) ldquoPermittivity determination of inhomogeneousfoods by measurement and automated retro-modeling with a degenerate mode cavityrdquoProceedings of the 9th Conference on Microwave and HF Heating Loughborough UKpp 181-4

COMPEL241

304