research article transformation algorithm of dielectric...

Research ArticleTransformation Algorithm of Dielectric Response inTime-Frequency Domain

Ji Liu1 Daning Zhang1 Xinlao Wei1 and Hamid Reza Karimi2

1 State Key Laboratory Breeding Base of Dielectrics Engineering Harbin University of Science and TechnologyHarbin Heilongjiang 150080 China

2Department of Engineering Faculty of Engineering and Science University of Agder 4898 Grimstad Norway

Correspondence should be addressed to Ji Liu liujihrbusteducn

Received 21 March 2014 Revised 23 April 2014 Accepted 25 April 2014 Published 1 June 2014

Academic Editor Kwok-WoWong

Copyright copy 2014 Ji Liu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A transformation algorithm of dielectric response from time domain to frequency domain is presented In order to shortenmeasuring time of lowor ultralow frequency dielectric response characteristics the transformation algorithm is used in this paper totransform the time domain relaxation current to frequency domain current for calculating the low frequency dielectric dissipationfactor In addition it is shown from comparing the calculation results with actual test data that there is a coincidence for bothresults over a wide range of low frequencies Meanwhile the time domain test data of depolarization currents in dry and moistpressboards are converted into frequency domain results on the basis of the transformation The frequency domain curves ofcomplex capacitance and dielectric dissipation factor at the low frequency range are obtained Test results of polarization anddepolarization current (PDC) in pressboards are also given at the different voltage and polarization time It is demonstrated fromthe experimental results that polarization and depolarization current are affected significantly by moisture contents of the testpressboards and the transformation algorithm is effective in ultralow frequency of 10minus3Hz Data analysis and interpretation ofthe test results conclude that analysis of time-frequency domain dielectric response can be used for assessing insulation system inpower transformer

1 Introduction

Most of dangerous breakdowns in high voltage (HV) appa-ratus are caused by the aging effects of HV insulationsystem Reliable diagnostics for insulation system are cor-respondingly based on changes of the dielectric proper-ties Generally insulation diagnosis methods for large oil-impregnated power transformer include frequency domainmethod (frequency domain spectroscopy FDS) [1] timedomain voltagemethod (return voltagemeter RVM) [2] andcurrent method (polarization and depolarization currentPDC) [3] RVMcan only be used for analyzing depolarizationprocess of dielectric Nevertheless the change of dielectricconductivity cannot be shown in RVM Many researchersfocus their attention on the analysis of time domain andfrequency domain dielectric responses which has been usedfor diagnosing insulation aging of high voltage apparatus[4 5] As for FDS dielectric dissipation factor tan 120575(120596) and

dielectric capacitance 119862(120596) are investigated at the range oflow and ultralow frequency which can represent relaxationcharacteristics such as interfacial polarization and spacecharge polarization widely found in most of the dielectricsWith regard to PDC comparing the PDC curves betweennew factory oil and aged transformer oil indicates theoperating conditions of on-site power transformers [6]

The presentation of dielectric response in frequencydomain has advantages The real and imaginary part of thecomplex capacitance can be separated and the dissipationfactor is defined in frequency domain Transformation oftime domain to frequency domain is helpful to interpret databetter [7 8] Compared with FDS fast on-site measurementto dielectric characteristics can be accomplished by the useof PDC test Unlike the limited accuracy to PDC in higherfrequency range FDS analysis can be utilized over a widerange of frequency In traditional FDS nevertheless weneed a long time at very low frequency measurement In

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 547105 7 pageshttpdxdoiorg1011552014547105

2 Mathematical Problems in Engineering

this paper the principles of dielectric response are analyzedtheoretically on the basis of the analysis of polarization anddepolarization current and frequency domain properties ofdielectric dissipation factor will be obtained according totransform algorithm of depolarization current data whichprovide a convenient and fast measurement for the lowfrequency characteristics of dielectric dissipation factor

2 Transform Algorithm

21 Dielectric Response in Time Domain Assuming a homo-geneous electric field 119864(119905) is applied over a vacuum-insulatedelectrode arrangement the electric displacement 119863(119905) isproportional to the electric field 119864(119905)

119863 (119905) = 1205760

119864 (119905) (1)

where 1205760

is vacuum permittivityIf any kind of isotropic dielectric materials replaces the

vacuum the electric displacement 119863(119905) will grow as a resultof additional electrical (macroscopic) polarization 119875(119905) and(1) can be rewritten as

119863 (119905) = 1205760

119864 (119905) + 119875 (119905) (2)

The electrical polarization 119875(119905) can be divided into twoparts one represents ldquorapidrdquo polarization processes and theother represents ldquoslowrdquo polarization processes The electricalpolarization can be written as

119875 (119905) = 1205760

(120576infin

minus 1) 119864 (119905) + 1205760

int

119905

minusinfin

119891 (119905 minus 120591) 119864 (120591) 119889120591 (3)

where 120576infin

is optical frequency permittivity The ldquorapidrdquopolarization follows the applied electric field whereas ldquoslowrdquopolarization is built up from a convolution integral betweenthe applied electric field and a function which is called thedielectric response function 119891(119905) The dielectric responsefunction represents the ldquomemoryrdquo effects in a dielectricmaterial and has the following characteristics

119891 (119905) equiv 0 forall119905 lt 0 (4)

In accordance with Amperersquos law [9 10] the electric field119864(119905) generates a total current density 119869(119905) which can bewritten as a sum of conduction vacuum and polarizationdisplacement current

119869 (119905) = 1205900

119864 (119905) + 1205760

120576infin

119889119864 (119905)

119889119905

+ 1205760

119889

119889119905int

119905

0

119891 (119905 minus 120591) 119864 (120591) 119889120591

(5)

According to the theory of dielectric response when fixedelectric field 119864(119905) generated by an external voltage 119880(119905) isapplied to isotropic dielectric material the total current 119894(119905)in dielectric can be written as

119894 (119905) = 1198620

[1205900

1205760

119880 (119905) + 120576infin

119889119880 (119905)

119889119905+

119889

119889119905int

119905

0

119891 (119905 minus 120591)119880 (120591) 119889120591]

(6)

where 1205900

is dielectric volume conductivity applied voltage119880(119905) = 119864(119905) sdot 119889 119862

0

is geometry capacitance betweenelectrodes and 119889 is the spacing between the two electrodes

For characterizing dielectric response in time domain astep dc ldquocharging voltagerdquo with magnitude 119880

119862

which mustbe constant and free of ripple is suddenly applied to thetest sample which has been discharged previously Then thepolarization current 119894pol(119905) flowing through the dielectric canbe recorded as

119894pol (119905) = 1198620

119880119862

[1205900

1205760

+ 120576infin

120575 (119905) + 119891 (119905)] (7)

where 120575(119905) is the delta function arising from the suddenlyapplied step voltage at 119905 = 0 The charging current 119894pol(119905)contains three parts the first one is related to the inherentconductivity of the test object and is independent of anypolarization process the middle part with the delta functioncannot be recorded in practice and is always ignored inthe calculation due to the large dynamic range of currentamplitudes arising from the very fast polarization processesat first and the last one represents all the ldquoslowrdquo polarizationprocesses during the applied voltage Therefore (7) can berewritten as

119894pol (119905) = 1198620

119880119862

[1205900

1205760

+ 119891 (119905)] (8)

If the step voltage119880119862

is removed and the testing dielectricis short-circuited when 119905 = 119905

119888

the depolarization current119894depol can be measured As shown in Figure 1 119879

119888

is thetime duration when the step voltage was applied Figure 1shows the changing curve of polarization and depolarizationcurrent According to the superposition principle the suddendecrease of119880

119862

can be regarded as a negative step voltage minus119880119862

applied at time 119905 = 119905119888

we get the depolarization current 119894depol

119894depol (119905) = minus1198620

119880119862

[119891 (119905) minus 119891 (119905 + 119905119888

)] (9)

The second part in (9) can be neglected if the chargingtime 119905

119888

is long enough to complete all the polarizationprocesses Then the depolarization current becomes propor-tional to the dielectric response function 119891(119905)

119891 (119905) asymp119894depol

1198620

119880119862

(10)

If the test period is long enough the conductivity 1205900

caneasily be calculated from the polarization and depolarizationcurrents

1205900

asymp1205760

1198620

119880119862

[119894pol (119905) + 119894depol (119905)] (11)

22 Dielectric Response in Frequency Domain Assuming thatvoltage 119880(120596) is given frequency domain current will beobtained by applying the Fourier transform to total current119894(119905) Consider

119868 (120596) = 1198620

[1205900

1205760

119880 (120596) + 119895120596120576infin

119880 (120596) + 119895120596119865 (120596)119880 (120596)]

= [1205900

1205760

+ 119895120596 (120576infin

+ 119865 (120596))]1198620

119880 (120596)

(12)

Mathematical Problems in Engineering 3

Charging DischargingUc

Tc

t

ipol

idepol

0tc

Figure 1 Polarization and depolarization current

Obviously 119865(120596) is Fourier transform value of dielectricresponse function 119891(119905) namely complex dielectric suscep-tibility The Fourier transform is the link between timeand frequency domain The principle of superposition iseffective if the dielectric material is linear homogenous andisotropic Formula (13) shows the relationship between timeand frequency domain Consider

120594 (120596) = 119865 (120596) = 1205941015840

(120596) minus 11989512059410158401015840

(120596) = int

infin

0

119891 (119905) 119890minus119895120596119905

119889119905 (13)

According to (12) we have

119868 (120596) = [1205900

1205760

+ 119895120596 (120576infin

+ 1205941015840

(120596) minus 11989512059410158401015840

(120596))]1198620

119880 (120596)

= [1205900

1205760

+ 12059612059410158401015840

(120596) + 119895120596 (120576infin

+ 1205941015840

(120596))]1198620

119880 (120596)

(14)

Therefore frequency domain expression of total currentin dielectric material under 119880(120596) is

119868 (120596) = 1198951205961198620

[120576infin

+ 1205941015840

(120596) minus 119895 (1205900

1205760

120596+ 12059410158401015840

(120596))]119880 (120596)

= 1198951205961198620

120576 (120596)119880 (120596)

(15)

where complex permittivity is

120576 (120596) = 1205761015840

(120596) minus 11989512057610158401015840

(120596) (16)

In many cases it is more convenient to use the complexpermittivity instead of the complex dielectric susceptibilityThe dielectric dissipation factor in frequency domain cantherefore be defined as follows

tan 120575 (120596) =12057610158401015840

(120596)

1205761015840 (120596)=

(1205900

1205760

120596) + 12059410158401015840

(120596)

120576infin

+ 1205941015840 (120596) (17)

Uc

Cinfin

R1

C2 CnC1

R2 Rn

R

idepol

middot middot middot


Figure 2 Equivalent circuit of a linear dielectric

23 Transformation of Dielectric Response As mentionedbefore the Fourier transform is the link between time andfrequency domain [11 12] If the dielectric response function119891(119905) follows the ldquoCurie-von Schweidlerrdquo model its Fouriertransform can be calculated as follows

120594 (120596) = 119865 (120596) = int

infin

0

119860119905minus119899

119890minus119895120596119905

119889119905 =119860 sdot Γ (1 minus 119899)

(119895120596)1minus119899

= 119860 sdot Γ (1 minus 119899) sdot 120596119899minus1

(sin(119899120587

2) minus 119895 cos(119899120587

2))

(18)

where Γ is the Gamma function Many results confirm thatthis method is effective with a narrow frequency range [13]

If the measured depolarization current can be approxi-mated by a piecewise ldquoCurie-von Schweidlerrdquo model Hamonapproximation [14] is an alternative for fast calculation ofthe time domain data into frequency domain But onlythe imaginary part of the complex permittivity 120576

10158401015840 can becalculated by the method If n is in the range of 03 lt 119899 lt 12the imaginary part of the complex permittivity can be writtenas

12057610158401015840

(120596) asympminus119894depol (01119891)

21205871198911198620

119880119888

(19)

It is shown that the imaginary part of the complexpermittivity at a frequency of 119891 cycles per second can beobtained directly from the current at a time 119905 equal to01f seconds after applying a direct step voltage Since thedepolarization current is easily measured if 119905 is large enoughthe method is adapted to the rapid evaluation of loss factorat frequency below 001 cycles per second where directmeasurement would be tedious or time consuming

According to dielectric Debye model [15] as shownin Figure 2 the depolarization current of linear insulationsystem can be expressed by the superposition of differentrelaxation current components When charging time is longenough dielectric response function 119891(119905) is proportional topolarization current Consider

119891 (119905) =1

1198620

119880119888

119899

sum

119894=1

119860119894

119890minus119905120591119894 (20)

The complex susceptibility can be obtained as

120594 (120596) = int

infin

0

119891 (119905) 119890minus119895120596119905

119889119905 =1

1198620

119880119888

119899

sum

119894=1

119860 i120591i minus 119895119860119894

1205961205912

119894

1 + (120596120591119894

)2

(21)


Thus real part and imaginary part of susceptibility can beobtained in (21) by

1205941015840

(120596) =1

1198620

119880119888

119899

sum

119894=1

119860 i120591i

1 + (120596120591119894

)2

12059410158401015840

(120596) =1

1198620

119880119888

119899

sum

119894=1

119860119894

1205961205912

119894

1 + (120596120591119894

)2

(22)

Therefore the dielectric dissipation factor in (17) can berewritten as

tan 120575 (120596) =12057610158401015840

(120596)

1205761015840 (120596)=

(1205900

1205760

120596) + 12059410158401015840

(120596)

120576infin

+ 1205941015840 (120596)

=(1205900

1205760

120596) + (11198620

119880119888

)sum119899

119894=1

(119860119894

1205961205912

119894

(1 + (120596120591119894

)2

))

120576infin

+ (11198620

119880119888

)sum119899

119894=1

(119860119894

120591119894

(1 + (120596120591119894

)2

))

(23)

3 Test System

The test samples are pressboards with the thickness of 1mmAfter polarization and depolarization current measurementthe frequency domain dielectric response is obtained throughthe above transformation thus dielectric dissipation factorcurves in low frequency can be illustrated The experimentalsystem mainly consists of high-voltage dc power supplyKeithley 6517B electrostatic meter testing electrode boxcontrol switch and PC The schematic diagram of experi-mental circuit is shown in Figure 3 A dc high voltage can bechanged continuously with maximum output voltage of 5 kVThe three-electrode system is used with 50mm diameter ofmeasure electrode and 2mm protection gap

4 Results and Discussions

For verifying the transform of dielectric response fromtime to frequency domain the pressboard samples withthickness of 1mm were used Applied polarization voltagesrespectively were 250V 500V and 1000V with polarizationtime of 100 s For comparing conveniently the polarizationcurrent was given in the absolute value Experimental resultsare shown in Figure 4 which prove that the amplitude ofpolarization and depolarization current will be affected bydifferent polarization voltages Moreover there are the samedeclined trends to the curves of polarization and depolariza-tion current individually The higher the polarization voltageis the bigger the current amplitude is It is shown fromtest results that there are better linear characteristics for thepressboards

Because of the long polarization time for insulationmate-rial the process of dielectric polarization is more sufficientand there is more information related to polarized patternwhich is more suitable for insulation condition assessment ofthe insulation system The phenomena can be called herebyldquomemory effectrdquo In order to explain it the experimentswere carried out with the pressboard of 1mm thicknesspolarization voltage of 500V and polarization time of 100 s

200 s and 400 s individually as shown in Figure 5 Thedependence of depolarization current is roughly coincidentwithin initial 30 s but the curves of depolarization currenthave upward tendency with the increase of polarization timeThe phenomena illustrate adequately that more boundedcharge will appear with full polarization process in dielectricand more internal information can be obtained

As a strong polar molecule for ageing by-products ofpressboard polarization and depolarization currents are verysensitive to moisture Two similar pressboards of 1mm wererespectively marked as A and B Samples A and B were putinto drying oven for heating and drying under the sameconditions Then sample B was put in damp environmentmaintaining up to 24 hours with absorption of moistureThe polarization and depolarization currents of A and Bwere tested with applied voltage of 500V polarization anddepolarization time of 2000 s As shown in Figure 6 currentamplitude of sample B becomes higher than that of sampleA due to the absorption of moisture In addition when thevoltage is applied because of strong polarity of moisturethe more moisture is produced the more bounded chargesappear therefore the depolarization process will slow downAccording to the above theoretical analysis discrete signalsof depolarization current in dry and moisture pressboardwere probed at first Test data were fitted according topolynomial curve to generate the time domain formula andwere calculated by the use of the transform algorithm toget the frequency domain properties of pressboards Thegeometric capacitance of model is 174 pF The dielectricdissipation factor real part and imaginary part of complexcapacitance in frequency domain are shown in Figures 7ndash9

As shown in Figure 7 dielectric dissipation factor ofsample B is higher than that of sample A and curves ofdielectric dissipation factor decline slowly with increasingfrequency As shown in Figures 8 and 9 the calculated realpart of complex capacitance is identical with both discreteand continuous transformations The complex capacitanceof dry pressboard A is not changed obviously in the wholefrequency range

Nevertheless the real part of complex capacitanceincreases clearly for the moist pressboard B and the differ-ence between A and B becomes very little with the increaseof frequency Calculated curves of discrete transform arecoincident with value of continuous transform in the formerpart of the curves The former is less than the latter partand there is no excessive error With regard to the real partof capacitance the change dependence between pressboardsA and B was basically identical and only the amplitude ofpressboard B is higher than that of A Therefore imaginarypart of the complex capacitance will be increased due toexisting moisture

According to the theoretical and experimental resultshere presented transformation algorithm and derived for-mula can not only carry out measurements more rapid incontrast to the PDC test but also show more changingpatterns of dielectric parameters from the response curvesAs shown in Figure 8 different effects of moisture ondependence of real part complex capacitance are obviousThe distinctions are more evident at the frequency range


Keithley6517

Measurementelectrode

Sample

PC

Protectionelectrode

Charging

Discharging

GroundedK1

K2

Test electrodebox

RS-232

DC

Test

HVelectrode

Figure 3 Schematic diagram of test system

1 10 100

Curr

ent (

A)

Time (s)

10minus11

10minus10

10minus9

10minus8

10minus7

250V polarization current250V depolarization current500V polarization current

500V depolarization current1000V polarization current1000V depolarization current

Figure 4 Effect of different polarization voltage on PDC

1 10 100 1000

Curr

ent (

A)

Time (s)

Polarization current

Depolarization current

10minus12

10minus11

10minus10

10minus9

10minus8

10minus7

100 s200 s400 s

Figure 5 Effect of different polarization time on PDC

1 10 100 1000

Curr

ent (

A)

Time (s)

AB



10minus12

10minus11

10minus10

10minus9

10minus8

Figure 6 Effect of different moisture on PDC

of lower 10minus2Hz However for the measurement results

of depolarization current in PDC method [6 13 16] therelaxation current cannot be used for distinguishing differentdielectric relaxation behavior unlike the FDS curves at certainfrequency ranges

5 Conclusions

Based on theoretical analysis in this paper the test resultsof polarization and depolarization current in pressboards aregiven at different voltage and polarization times It is shownfrom the experimental results that the polarization anddepolarization currents are affected significantly by moisturecontents of the pressboard samples In addition linear char-acteristics of pressboard are explained and it is shown thatmore sufficient polarization process can help obtain furtherinternal information of dielectric The dependence of curvesof dielectric dissipation factor in frequency domain whichis obtained by calculation shows that dielectric dissipation


Frequency (Hz)

A multi expA FFT

B multi expB FFT

100

10minus2

10minus1

10minus2

10minus3

10minus1

tan120575

Figure 7 Dependence of dielectric dissipation factor in frequencydomain

40

60

80

100

120

Frequency (Hz)10

minus210

minus310

minus1

A multi expA FFT

B multi expB FFT

C998400(p

F)

Figure 8 Dependence of real part of complex capacitance

factor will shift upwards with the increase of moisture and beflat with increased frequency

Comparing the calculating results with actual test datathere is a coincidence for both results in low frequency rangewhich demonstrates the feasibility and practicability It is con-cluded that dielectric dissipation factor in the low frequencyrange can be calculated by the use of the transform algorithmto the depolarization current in time domain which canreplace test value in low frequency range consequently it is arapid assessment method for dielectric diagnosis

Frequency (Hz)10

minus110

minus210

minus3

C998400998400

(pF)

103

102

101

100

10minus1

A multi expA FFT

B multi expB FFT

Figure 9 Dependence of imaginary part of complex capacitance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project was supported by the National Basic ResearchProgram of China (2012CB723308) The authors wish tothank the Key Laboratory of Engineering Dielectric and ItsApplication of Ministry of Education State Key LaboratoryBreeding Base of Dielectrics Engineering for their support

References

[1] J Blennow C Ekanayake K Walczak B Garcıa and S MGubanski ldquoField experiences with measurements of dielectricresponse in frequency domain for power transformer diagnos-ticsrdquo IEEETransactions on PowerDelivery vol 21 no 2 pp 681ndash688 2006

[2] R G Zhang M Dong G J Zhang and Z Yan ldquoInvestigationof return voltage measurement for the assessment of powertransformersrdquo in Proceedings of the International Conference onCondition Monitoring and Diagnosis (CMD rsquo08) pp 902ndash905Beijing China April 2008

[3] T K Saha ldquoReview of time-domain polarizationmeasurementsfor assessing insulation condition in aged transformersrdquo IEEETransactions on Power Delivery vol 18 no 4 pp 1293ndash13012003

[4] A A Shayegani H Borsi E Gockenbach and H MohsenildquoTransformation of time domain spectroscopy data to fre-quency domain data for impregnated pressboardrdquo in Proceed-ings of the Annual Report Conference on Electrical Insulation andDielectric Phenomena (CEIDP rsquo04) pp 162ndash165 Boulder ColoUSA October 2004

[5] M Farahani H Borsi and E Gockenbach ldquoDielectric responsestudies on insulating system of high voltage rotating machinesrdquo


IEEE Transactions on Dielectrics and Electrical Insulation vol13 no 2 pp 383ndash393 2006

[6] S A Bhumiwat and P Phillips ldquoVerification of on-site oilreclamation process by means of polarisation depolarisationcurrent analysisrdquo inProceedings of theConference Record of IEEEInternational Symposium on Electrical Insulation pp 105ndash108San Juan Puerto Rico September 2004

[7] N Issarachai ldquoDynamic events analysis of Thailand andMalaysia power systems by discrete wavelet decompositionand short term fourier transform based on GPS synchronizedphasor datardquo International Journal of Innovative ComputingInformation and Control vol 9 no 5 pp 2203ndash2228 2013

[8] M Fallahpour and D Megias ldquoHigh capacity robust audiowatermarking scheme based on FFT and linear regressionrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 4 pp 2477ndash2489 2012

[9] J C Maxwell A Treatise on Electricity and Magnetism Claren-don Press Oxford UK 1981

[10] W S Zaengl ldquoDielectric spectroscopy in time and frequencydomain for HV power equipment part I theoretical consider-ationsrdquo IEEE Electrical Insulation Magazine vol 19 no 5 pp5ndash19 2003

[11] N Zakaria A J Pal and S NM Shah ldquoStagewise optimizationof distributed clustered finite difference time domain (FDTD)using genetic algorithmrdquo International Journal of InnovativeComputing Information and Control vol 9 no 6 pp 2303ndash2326 2013

[12] M Nachidi F Tadeo and A Benzouia ldquoController designfor Takagi-Sugeno systems in continuous-timerdquo InternationalJournal of Innovative Computing Information and Control vol8 no 9 pp 6389ndash6400 2012

[13] V Der Houhanessian Measurement and analysis of dielectricresponse in oil-paper insulation system [PhD thesis] SwissFederal Institute of Technology ETH Zurich 1998

[14] B V Hamon ldquoAn approximate method for deducing dielectricloss factor fromdirect-currentmeasurementrdquoProceedings of theIEE II Power Engineering vol 99 no 27 pp 151ndash155 1952

[15] T K Saha R Middleton and A Thomas ldquoUnderstanding fre-quency amp time domain polarisation methods for the insulationcondition assessment of power transformersrdquo in Proceedings ofthe IEEE Power and Energy Society General Meeting (PES rsquo09)Calgary Canada July 2009

[16] VDerHouhanessian andW S Zaengl ldquoTime domainmeasure-ments of dielectric response in oil-paper insulation systemsrdquo inProceedings of the IEEE International Symposium on ElectricalInsulation pp 47ndash52 Montreal Canada 1996

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Decision SciencesAdvances in

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Stochastic AnalysisInternational Journal of


this paper the principles of dielectric response are analyzedtheoretically on the basis of the analysis of polarization anddepolarization current and frequency domain properties ofdielectric dissipation factor will be obtained according totransform algorithm of depolarization current data whichprovide a convenient and fast measurement for the lowfrequency characteristics of dielectric dissipation factor

2 Transform Algorithm

21 Dielectric Response in Time Domain Assuming a homo-geneous electric field 119864(119905) is applied over a vacuum-insulatedelectrode arrangement the electric displacement 119863(119905) isproportional to the electric field 119864(119905)

119863 (119905) = 1205760

119864 (119905) (1)

where 1205760

is vacuum permittivityIf any kind of isotropic dielectric materials replaces the

vacuum the electric displacement 119863(119905) will grow as a resultof additional electrical (macroscopic) polarization 119875(119905) and(1) can be rewritten as

119863 (119905) = 1205760

119864 (119905) + 119875 (119905) (2)

The electrical polarization 119875(119905) can be divided into twoparts one represents ldquorapidrdquo polarization processes and theother represents ldquoslowrdquo polarization processes The electricalpolarization can be written as

119875 (119905) = 1205760

(120576infin

minus 1) 119864 (119905) + 1205760

int

119905

minusinfin

119891 (119905 minus 120591) 119864 (120591) 119889120591 (3)

where 120576infin

is optical frequency permittivity The ldquorapidrdquopolarization follows the applied electric field whereas ldquoslowrdquopolarization is built up from a convolution integral betweenthe applied electric field and a function which is called thedielectric response function 119891(119905) The dielectric responsefunction represents the ldquomemoryrdquo effects in a dielectricmaterial and has the following characteristics

119891 (119905) equiv 0 forall119905 lt 0 (4)

In accordance with Amperersquos law [9 10] the electric field119864(119905) generates a total current density 119869(119905) which can bewritten as a sum of conduction vacuum and polarizationdisplacement current

119869 (119905) = 1205900

119864 (119905) + 1205760

120576infin

119889119864 (119905)

119889119905

+ 1205760

119889

119889119905int

119905

0

119891 (119905 minus 120591) 119864 (120591) 119889120591

(5)

According to the theory of dielectric response when fixedelectric field 119864(119905) generated by an external voltage 119880(119905) isapplied to isotropic dielectric material the total current 119894(119905)in dielectric can be written as

119894 (119905) = 1198620

[1205900

1205760

119880 (119905) + 120576infin

119889119880 (119905)

119889119905+

119889

119889119905int

119905

0

119891 (119905 minus 120591)119880 (120591) 119889120591]

(6)

where 1205900

is dielectric volume conductivity applied voltage119880(119905) = 119864(119905) sdot 119889 119862

0

is geometry capacitance betweenelectrodes and 119889 is the spacing between the two electrodes

For characterizing dielectric response in time domain astep dc ldquocharging voltagerdquo with magnitude 119880

119862

which mustbe constant and free of ripple is suddenly applied to thetest sample which has been discharged previously Then thepolarization current 119894pol(119905) flowing through the dielectric canbe recorded as

119894pol (119905) = 1198620

119880119862

[1205900

1205760

+ 120576infin

120575 (119905) + 119891 (119905)] (7)

where 120575(119905) is the delta function arising from the suddenlyapplied step voltage at 119905 = 0 The charging current 119894pol(119905)contains three parts the first one is related to the inherentconductivity of the test object and is independent of anypolarization process the middle part with the delta functioncannot be recorded in practice and is always ignored inthe calculation due to the large dynamic range of currentamplitudes arising from the very fast polarization processesat first and the last one represents all the ldquoslowrdquo polarizationprocesses during the applied voltage Therefore (7) can berewritten as

119894pol (119905) = 1198620

119880119862

[1205900

1205760

+ 119891 (119905)] (8)

If the step voltage119880119862

is removed and the testing dielectricis short-circuited when 119905 = 119905

119888

the depolarization current119894depol can be measured As shown in Figure 1 119879

119888

is thetime duration when the step voltage was applied Figure 1shows the changing curve of polarization and depolarizationcurrent According to the superposition principle the suddendecrease of119880

119862

can be regarded as a negative step voltage minus119880119862

applied at time 119905 = 119905119888

we get the depolarization current 119894depol

119894depol (119905) = minus1198620

119880119862

[119891 (119905) minus 119891 (119905 + 119905119888

)] (9)

The second part in (9) can be neglected if the chargingtime 119905

119888

is long enough to complete all the polarizationprocesses Then the depolarization current becomes propor-tional to the dielectric response function 119891(119905)

119891 (119905) asymp119894depol

1198620

119880119862

(10)

If the test period is long enough the conductivity 1205900

caneasily be calculated from the polarization and depolarizationcurrents

1205900

asymp1205760

1198620

119880119862

[119894pol (119905) + 119894depol (119905)] (11)

22 Dielectric Response in Frequency Domain Assuming thatvoltage 119880(120596) is given frequency domain current will beobtained by applying the Fourier transform to total current119894(119905) Consider

119868 (120596) = 1198620

[1205900

1205760

119880 (120596) + 119895120596120576infin

119880 (120596) + 119895120596119865 (120596)119880 (120596)]

= [1205900

1205760

+ 119895120596 (120576infin

+ 119865 (120596))]1198620

119880 (120596)

(12)



Tc

t

ipol

idepol

0tc



120594 (120596) = 119865 (120596) = 1205941015840

(120596) minus 11989512059410158401015840

(120596) = int

infin

0

119891 (119905) 119890minus119895120596119905

119889119905 (13)


119868 (120596) = [1205900

1205760

+ 119895120596 (120576infin

+ 1205941015840

(120596) minus 11989512059410158401015840

(120596))]1198620

119880 (120596)

= [1205900

1205760

+ 12059612059410158401015840

(120596) + 119895120596 (120576infin

+ 1205941015840

(120596))]1198620

119880 (120596)

(14)


119868 (120596) = 1198951205961198620

[120576infin

+ 1205941015840

(120596) minus 119895 (1205900

1205760

120596+ 12059410158401015840

(120596))]119880 (120596)

= 1198951205961198620

120576 (120596)119880 (120596)

(15)


120576 (120596) = 1205761015840

(120596) minus 11989512057610158401015840

(120596) (16)


tan 120575 (120596) =12057610158401015840

(120596)

1205761015840 (120596)=

(1205900

1205760

120596) + 12059410158401015840

(120596)

120576infin

+ 1205941015840 (120596) (17)

Uc

Cinfin

R1

C2 CnC1

R2 Rn

R

idepol





120594 (120596) = 119865 (120596) = int

infin

0

119860119905minus119899

119890minus119895120596119905

119889119905 =119860 sdot Γ (1 minus 119899)

(119895120596)1minus119899


(sin(119899120587

2) minus 119895 cos(119899120587

2))

(18)




12057610158401015840


21205871198911198620

119880119888

(19)



119891 (119905) =1

1198620

119880119888

119899

sum

119894=1

119860119894

119890minus119905120591119894 (20)


120594 (120596) = int

infin

0

119891 (119905) 119890minus119895120596119905

119889119905 =1

1198620

119880119888

119899

sum

119894=1

119860 i120591i minus 119895119860119894

1205961205912

119894

1 + (120596120591119894

)2

(21)



1205941015840

(120596) =1

1198620

119880119888

119899

sum

119894=1

119860 i120591i

1 + (120596120591119894

)2

12059410158401015840

(120596) =1

1198620

119880119888

119899

sum

119894=1

119860119894

1205961205912

119894

1 + (120596120591119894

)2

(22)


tan 120575 (120596) =12057610158401015840

(120596)

1205761015840 (120596)=

(1205900

1205760

120596) + 12059410158401015840

(120596)

120576infin

+ 1205941015840 (120596)

=(1205900

1205760

120596) + (11198620

119880119888

)sum119899

119894=1

(119860119894

1205961205912

119894

(1 + (120596120591119894

)2

))

120576infin

+ (11198620

119880119888

)sum119899

119894=1

(119860119894

120591119894

(1 + (120596120591119894

)2

))

(23)

3 Test System











Keithley6517


Sample

PC

Protectionelectrode

Charging

Discharging

GroundedK1

K2

Test electrodebox

RS-232

DC

Test

HVelectrode


1 10 100

Curr

ent (

A)

Time (s)

10minus11

10minus10

10minus9

10minus8

10minus7




1 10 100 1000

Curr

ent (

A)

Time (s)



10minus12

10minus11

10minus10

10minus9

10minus8

10minus7

100 s200 s400 s


1 10 100 1000

Curr

ent (

A)

Time (s)

AB



10minus12

10minus11

10minus10

10minus9

10minus8




5 Conclusions



Frequency (Hz)

A multi expA FFT

B multi expB FFT

100

10minus2

10minus1

10minus2

10minus3

10minus1

tan120575


40

60

80

100

120

Frequency (Hz)10

minus210

minus310

minus1

A multi expA FFT

B multi expB FFT

C998400(p

F)




Frequency (Hz)10

minus110

minus210

minus3

C998400998400

(pF)

103

102

101

100

10minus1

A multi expA FFT

B multi expB FFT




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Tc

t

ipol

idepol

0tc



120594 (120596) = 119865 (120596) = 1205941015840

(120596) minus 11989512059410158401015840

(120596) = int

infin

0

119891 (119905) 119890minus119895120596119905

119889119905 (13)


119868 (120596) = [1205900

1205760

+ 119895120596 (120576infin

+ 1205941015840

(120596) minus 11989512059410158401015840

(120596))]1198620

119880 (120596)

= [1205900

1205760

+ 12059612059410158401015840

(120596) + 119895120596 (120576infin

+ 1205941015840

(120596))]1198620

119880 (120596)

(14)


119868 (120596) = 1198951205961198620

[120576infin

+ 1205941015840

(120596) minus 119895 (1205900

1205760

120596+ 12059410158401015840

(120596))]119880 (120596)

= 1198951205961198620

120576 (120596)119880 (120596)

(15)


120576 (120596) = 1205761015840

(120596) minus 11989512057610158401015840

(120596) (16)


tan 120575 (120596) =12057610158401015840

(120596)

1205761015840 (120596)=

(1205900

1205760

120596) + 12059410158401015840

(120596)

120576infin

+ 1205941015840 (120596) (17)

Uc

Cinfin

R1

C2 CnC1

R2 Rn

R

idepol





120594 (120596) = 119865 (120596) = int

infin

0

119860119905minus119899

119890minus119895120596119905

119889119905 =119860 sdot Γ (1 minus 119899)

(119895120596)1minus119899


(sin(119899120587

2) minus 119895 cos(119899120587

2))

(18)




12057610158401015840


21205871198911198620

119880119888

(19)



119891 (119905) =1

1198620

119880119888

119899

sum

119894=1

119860119894

119890minus119905120591119894 (20)


120594 (120596) = int

infin

0

119891 (119905) 119890minus119895120596119905

119889119905 =1

1198620

119880119888

119899

sum

119894=1

119860 i120591i minus 119895119860119894

1205961205912

119894

1 + (120596120591119894

)2

(21)



1205941015840

(120596) =1

1198620

119880119888

119899

sum

119894=1

119860 i120591i

1 + (120596120591119894

)2

12059410158401015840

(120596) =1

1198620

119880119888

119899

sum

119894=1

119860119894

1205961205912

119894

1 + (120596120591119894

)2

(22)


tan 120575 (120596) =12057610158401015840

(120596)

1205761015840 (120596)=

(1205900

1205760

120596) + 12059410158401015840

(120596)

120576infin

+ 1205941015840 (120596)

=(1205900

1205760

120596) + (11198620

119880119888

)sum119899

119894=1

(119860119894

1205961205912

119894

(1 + (120596120591119894

)2

))

120576infin

+ (11198620

119880119888

)sum119899

119894=1

(119860119894

120591119894

(1 + (120596120591119894

)2

))

(23)

3 Test System











Keithley6517


Sample

PC

Protectionelectrode

Charging

Discharging

GroundedK1

K2

Test electrodebox

RS-232

DC

Test

HVelectrode


1 10 100

Curr

ent (

A)

Time (s)

10minus11

10minus10

10minus9

10minus8

10minus7




1 10 100 1000

Curr

ent (

A)

Time (s)



10minus12

10minus11

10minus10

10minus9

10minus8

10minus7

100 s200 s400 s


1 10 100 1000

Curr

ent (

A)

Time (s)

AB



10minus12

10minus11

10minus10

10minus9

10minus8




5 Conclusions



Frequency (Hz)

A multi expA FFT

B multi expB FFT

100

10minus2

10minus1

10minus2

10minus3

10minus1

tan120575


40

60

80

100

120

Frequency (Hz)10

minus210

minus310

minus1

A multi expA FFT

B multi expB FFT

C998400(p

F)




Frequency (Hz)10

minus110

minus210

minus3

C998400998400

(pF)

103

102

101

100

10minus1

A multi expA FFT

B multi expB FFT




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1205941015840

(120596) =1

1198620

119880119888

119899

sum

119894=1

119860 i120591i

1 + (120596120591119894

)2

12059410158401015840

(120596) =1

1198620

119880119888

119899

sum

119894=1

119860119894

1205961205912

119894

1 + (120596120591119894

)2

(22)


tan 120575 (120596) =12057610158401015840

(120596)

1205761015840 (120596)=

(1205900

1205760

120596) + 12059410158401015840

(120596)

120576infin

+ 1205941015840 (120596)

=(1205900

1205760

120596) + (11198620

119880119888

)sum119899

119894=1

(119860119894

1205961205912

119894

(1 + (120596120591119894

)2

))

120576infin

+ (11198620

119880119888

)sum119899

119894=1

(119860119894

120591119894

(1 + (120596120591119894

)2

))

(23)

3 Test System











Keithley6517


Sample

PC

Protectionelectrode

Charging

Discharging

GroundedK1

K2

Test electrodebox

RS-232

DC

Test

HVelectrode


1 10 100

Curr

ent (

A)

Time (s)

10minus11

10minus10

10minus9

10minus8

10minus7




1 10 100 1000

Curr

ent (

A)

Time (s)



10minus12

10minus11

10minus10

10minus9

10minus8

10minus7

100 s200 s400 s


1 10 100 1000

Curr

ent (

A)

Time (s)

AB



10minus12

10minus11

10minus10

10minus9

10minus8




5 Conclusions



Frequency (Hz)

A multi expA FFT

B multi expB FFT

100

10minus2

10minus1

10minus2

10minus3

10minus1

tan120575


40

60

80

100

120

Frequency (Hz)10

minus210

minus310

minus1

A multi expA FFT

B multi expB FFT

C998400(p

F)




Frequency (Hz)10

minus110

minus210

minus3

C998400998400

(pF)

103

102

101

100

10minus1

A multi expA FFT

B multi expB FFT




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Keithley6517


Sample

PC

Protectionelectrode

Charging

Discharging

GroundedK1

K2

Test electrodebox

RS-232

DC

Test

HVelectrode


1 10 100

Curr

ent (

A)

Time (s)

10minus11

10minus10

10minus9

10minus8

10minus7




1 10 100 1000

Curr

ent (

A)

Time (s)



10minus12

10minus11

10minus10

10minus9

10minus8

10minus7

100 s200 s400 s


1 10 100 1000

Curr

ent (

A)

Time (s)

AB



10minus12

10minus11

10minus10

10minus9

10minus8




5 Conclusions



Frequency (Hz)

A multi expA FFT

B multi expB FFT

100

10minus2

10minus1

10minus2

10minus3

10minus1

tan120575


40

60

80

100

120

Frequency (Hz)10

minus210

minus310

minus1

A multi expA FFT

B multi expB FFT

C998400(p

F)




Frequency (Hz)10

minus110

minus210

minus3

C998400998400

(pF)

103

102

101

100

10minus1

A multi expA FFT

B multi expB FFT




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Frequency (Hz)

A multi expA FFT

B multi expB FFT

100

10minus2

10minus1

10minus2

10minus3

10minus1

tan120575


40

60

80

100

120

Frequency (Hz)10

minus210

minus310

minus1

A multi expA FFT

B multi expB FFT

C998400(p

F)




Frequency (Hz)10

minus110

minus210

minus3

C998400998400

(pF)

103

102

101

100

10minus1

A multi expA FFT

B multi expB FFT




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article transformation algorithm of dielectric...

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