IEEE Transactions on Electrical Insulation Vol. EI-22 No.5, October 1987

A CONCEPT FOR THEINTRINSIC DIELECTRIC STRENGTH OF

ELECTRICAL INSULATION MATERIALS

Edward F. Cuddihy

Jet Propulsion LaboratoryCalifornia Institute of Technology

Pasadena, California

ABSTRACT

There are analytical solutions of Laplace's Field equationwhich can be used to calculate the maximum potential gradientEMAX on an electrode surface as a function of voltage differ-ence V, separation t, and radius R. Such solutions can beseries-expanded and, after combining the first two terms ofthe series algebraically, they yield the general expressionV/t=EMAXa7 (t+a)J~, where (V/t)=VA is the average voltageacross the separation t, and a is a term involving radius ofcurvature only. If K is defined as EMAXa , then the expres-sion becomes VA=K(t+a)n which is strikingly similar to theexpression VA=K(t)-n, an historically observed data corre-lation between VA and t, if a is ignored. The experimentalobservation that VA decreases with increasing t has been in-terpreted as a material property.

However, the similarity with the series expression suggeststhat, at constant EMAX, this is a manifestation of thethickness-dependence of the spatial distribution of the elec-tric field. If true, then voltage breakdown of an insulationmaterial is occurring whenever a critical but constant valueof the potential gradient EMAX is reached or exceeded on anelectrode surface. It is suggested that this critical valueof EMAX may be the intrinsic dielectric strength of an in-sulation material, herein designated as S. From experimentalvoltage breakdown data measured as a function of thickness t,the constants K, a, and n in the series expression can be de-rived by least-squares techniques and, therefore, for t=O,S=K(a)-n which is identically equal to EMAX, since K=EMAXac.

Further, from experimental data, the term a yields a nearconstant value of R=30.5 pm, independent of actual electrodegeometries. Although unexplained, it can be speculated thatthis may be associated with electrode surface features suchas nicks, scratches, and/or needle-like asperities which wouldtend to have small radii of curvature.

As a practical application, this new concept for the intrinsicdielectric strength, and the use of R=30.5 pm, was used tooffer a possible explanation of the cause of failures in buriedHV cables with polyethylene insulation, and as a possible ex-planation of the origin of electrical trees.

In addition, efforts to use this concept for correlating dcvoltage breakdown data were unsuccessful. For dc testing, itwas observed that the surfaces of test specimens acquired an

electrostatic charge, similar to a capacitor. It is specu-lated herein that the charged surfaces mimic a parallel-plateelectrode configuration, thus offering an explanation for thelinear relationship between V and t for de testing.

001 8-9367/8_7/I 5 I.--0,e0 co. 1987 IEEE

573

IEEE Transactions on Electrical Insulation Vol. EI-22 No.5. October 1987

The Jet Propulsion Laboratory (JPL) has been assign-ed responsibility by the U.S. Department of Energy(DOE) for managing the Flat-Plate Solar Array Project(FSA). The responsibilities include planning and sup-porting national research programs to reduce the cost,and increase the outdoor service life, of terrestrialphotovoltaic (PV) modules, with a goal of 30 years ofservice. One phase of this project is concerned withthe development of materials that are to function aslow-cost, long-life encapsulants for solar cells [1].Fig. 1 illustrates the essential material componentsof state-of-the-art commercial photovoltaic module en-capsulation systems.

MODULE SUNSIDE LAYER DESIGNATION

J SURFACE11 MATERIAL2) MODIFICATION

I_ Zl FRONT COVER

POTTANT

FUNCTION

* LOW SOILING* EASY CLEANABILITY* ABRASION RESISTANT* ANTIREFLECTIVE

* UV SCREENING* STRUCTURAL SUPERSTRATE

* SOLAR CELL ENCAPSULATIONAND ELECTRICAL ISOLATION

However, a mathematical analysis [4] of the computerdata led serendipitously to an unexpected finding re-lated to the dielectric strength of electrical insula-tion materials. In essence, it appeared that thefundamental definition of an intrinsic dielectricstrength of insulation materials had been identified,which could be stated as a basic material property,independent of any test technique or service environ-ment. This is similar to other pure material proper-ties such as Young's modulus, index of refraction, andcoefficient of thermal expansion. The concept is de-scribed herein.

COMPUTER ANALYSIS

Fig. 2 is an illustration of a solar cell with arounded edge of radius R, encapsulated in a pottantwith thickness t isolating the solar cell from an elec-trically conducting ground plane. The voltage differ-ence between the cell and ground is V. The computer

SOLAR CELL

I/ /Z / / ZX POROUS SPACER * AIR RELEASE* MECHANICAL SEPARATION

SUBSTRATE

BACK COVER

* STRUCTURAL SUPPORT

* MECHANICAL PROTECTION* WEATHERING BARRIER* INFRARED EMITTER

Fig. 1: Construction elements of photovoltaicencapsulation systems.

The central core of an encapsulation system is thepottant, a transparent, elastomeric material that isthe actual encapsulation medium in a module. This mate-rial totally encloses and embeds all of the solar cellsan.d their associated electrical circuitry, and func-tions as electrical insulation.

With engineering advances leading to reductions inthe cost of photovoltaic modules, these devices are be-coming increasingly attractive to electric utilities asalternative sources of commercial power generation.Therefore, FSA is becoming increasingly concerned withthe long-term (30 years) electrical insulation quali-ties of pottant materials in outdoor weathering en-vironments.

A review of published literature and journal articlesreveals that researchers and workers in the field ofelectrical insulation have been seeking an understand-ing of electrical aging mechanisms as well as the de-velopment of life-prediction methodologies [2]. But,despite considerable progress, there are no immediatelyavailable methods or techniques to assess the electri-cal insulation life-potential of encapsulation mate-rials.

As part of FSA research activities, a computer pro-gram was developed to model the level of electricalfield intensities and stresses associated with typicalgeometries of encapsulated solar cells [3]. This waspart of the FSA technical program related to acceler-ated aging of encapsulation pottant materials, andknowledge was sought on typical electrical stress levelsto which pottants would be subjected in service.

GROUND PLANE

Fig. 2: Encapsulated solar-ceZZ geometry.

program calculates the maximum electrical voltagegradient EMAX that will occur on the rounded edge, asa function of V, t, and R. It was found [3] that thecomputer results could be expressed as two dimension-less reduced variables:

(t2R) and (VA/EMAX) (1)

where VA=V/t is the average voltage across the sepa-ration t.

For the example illustrated in Fig. 2, the computer-generated numerical values of the two dimensionlessreduced variables are given in Table 1. With thesedata, an effort was made to generate a graphical datacorrelation, intended for convenient interpolation [4].The result of the effort led to the log-log plot shownin Fig. 3, of VA/E4AX versus the term 2R/(t+2R). Thislatter term was generated using the dimensionless var-iable t/2R and the following algebraic conversion:

1 2R(2 + 1) (t + 2R)2R

(2)

It was observed that the initial portion of the log-log trace for small values of t, or large values of R,which is on the right, is virtually linear, and thetrace for large values of t, or small values of R,gradually develops curvature. A solid line and a dash-ed line are used in Fig. 3 to define the linear andcurved portions, respectively.

US

C,)

0z

(n

C)00i

v

574

L-1

Cuddihy: The intrinsic dielectric strength of electrical insulation materials5

Table 1

Computer-Generated Values of VA/EMAX and (t/2R)for the Solar-Cell Geometry Illustrated in Fig. 2

breakdown voltage of insulation materials, here beingVA, and insulation thickness t:

VA = K(t)-12 (7)

VA/EMAX (t/2R)

0.9117 0.25

0.8200 0.50

0.7004 1.00

0.5648 2.00

0.3916 5.00

0.2968 10.00

0.2305 20.00

0.1898 50.00

o.0.

0.

0.

0.1x 0.

0.

0.

0.01

2 3 4 5 6 7 8 91

0.1

2R/(t + 2R)

2 3 4 5 6 789

Fig. 3: log-log plot of Table 1 as VA/EMAX versusthe variable 2R/(t+2R).

The slope of the linear portion is 0.54, thereforeresulting for the linear portion in the following con-nective relationship

2R 0.54

VA/EMAX = |t + 2R) (3)

Eq. (3) can be expressed as

VA = EMXi (2R) 0'54 (t + 2R)-0.54 (4)

and if K is defined as

K = EMAX (2R) 054 (5)

then Eq. (4) becomes, finally,

VA = K(t + 2R)- 0.54 (6)

This empirical relationship is commented on, or uti-lized for data correlation in a diverse cross sectionof published papers and articles on electrical insula-tion behavior, and is also described in the appendix-of the standard ASTM-D-149-64 [5] test procedure formeasuring the dielectric strength of insulation mate-rials. In general, the average breakdown voltage VA,is assigned to be the electric strength, and it is re-quired that, as a minimum, the experimental test con-ditions, the environment, material thickness, andelectrode geometries must be specified.

Using Eq. (7), Fig. 4 is a data correlation betweenthe average breakdown voltage VA and sample thickness

E lOool-

CDz

LA

U? 100

C-)

0 1.4 i

F-_DATA POINTS FROM 3MTECHNICAL LITERATURE .

FOR X-22416 AND X-22417IN-SCREENING ACRYLICFILMS

0.01 0.1

GENERAL RELATIONSHIPK -0.50VA K(t)

THESE DATA:

VA K(t)-55DATA POINTS FROM -

DU PONT TECHNICALLITERATURE FOR

" LUCITE

1.0 10.0THICKNESS t, mm

VA - (VOLTAGE AT BREAKDOWNW/(THICKNESS V/It

Fig. 4: The ac dielectric strength of PEM acrylic.

0O t for polymethyl methacrylate (PMMA), using data valuesextracted from technical bulletins as remarked upon inthe Figure. A least-squares fit of these data withEq. (7) conforms to the empirical expectation.

One explanation for the empirically observed rela-tionship between VA and t appears to be rooted in a

material flaw theory, remarks on which are found inpublished articles on insulation behavior [2]. Theconcept is that material flaws of whatever kind act tocause or contribute to voltage breakdown, and as theseflaws are expected to be randomly distributed through-out the bulk of the insulation material, therefore,statistically more flaws become available in thickermaterials, thus reducing the voltage at breakdown (di-electric strength). Conceptually, a flaw-free stateoccurs for thickness t=O; therefore, VA in the limitof t=0 would constitute a flaw-free measurement of theintrinsic dielectric strength of insulation materials.

However, Eq. (6), and therefore Eq. (7), if the term2R is ignored, was generated on the basis of the elec-trical field distribution throughout a space of thick-ness t, originating from a curved surface of radius R(which could also be an edge or corner), and withoutconsideration of any insulation materials at all.

If the term 2R in Eq. (6) is ignored, the form of (6)is strikingly similar to an historically observed em-

pirical square-root relationship between the average

9

.7-6 -

_5

4

.2-

II lI I I

LI - -1- I

575i

IEEE Transactions on Electrical Insulation Vol. EI-22 No. 5, October 1987

THE CONCEPT

It is to be noted, for Eqs. (3), (4), or (6), whichare merely different algebraic forms, that when t=O,then

VA = EMAX (8)

It is here that the new concept states that EMAX, a

potential gradient, is the intrinsic dielectric strengthof insulation materials. The principle is that, as Vincreases, the material in test will fail when a poten-

tial gradient on the electrode surface reaches and ex-

ceeds a critical value of EMAX. This critical value isdesignated as S, the intrinsic dielectric strength,which is a material property and which is associatedwith a specific insulation material.

Before proceeding further, it is important to deter-mine if the form of Eq. (6) is specific to the geometry

depicted in Fig. 2, or is a general consequence of any

electrode geometry and associated field distributions.Table 2 details two analytical solutions of Laplace'sfield equations for electrode geometries consisting of

Table 2

being dictated by electrode geometries and pairings.Pairings are the use of two equivalent electrodes asin Eq. (9), or two non-equivalent electrodes as in Eqs.(4) and (10). It is convenient to refer to theseelectrode pairings as symmetric or asymmetric.

In general, for small values of t or large values ofR, Eqs. (4), (9), and (10) are of the general form

VA = K(t+a-n (11)

where a and n are dictated by electrode geometries andpairings, and when t=O,

S = Ka-n = EMAX (12)

Recently, Pillai and Hackam [8] reported on computer-generated values of the electric field and potentialdistributions for unequal spheres separated by variousgap lengths, and presented their results in Tables.Analysis of their tabulated data revealed that, forsmall gaps, their data also converged to fit the formof Eq. (11). This same general convergence at smallvalues of t of solutions to Laplace's field equationsis found in very early literature, dating back to 1924[9], 1928 [10], and 1941 [11].

Analytical Solutions of Laplace's FieldEquations For Needle Electrodes

Tip-to-Ground Plane (Reference 6)

El,, = 2V t P/Ln (Q)

P =(1 + R/t) /R

Q = t + R + 2t 1/2 (t + R) 1] /R

Tip-to-Tip (Reference 7)

VA t(l + 2R/t) /E A

2R tanh [/t+ 2R)]

a needle-tip-to-ground-plane configuration and for aneedle-tip-to-needle-tip configuration [6,7]. Both ofthese analytical solutions can be series-expanded (Ap-pendix A) and, when the first two terms of each oftheir respective series expansions are algebraicallycombined, the result is:

Tip- to-7ip

VA = EMAX (3R) (t+3R)-1 (9)

Tip- to-Ground

VA = EMAX (R)2"3 (t+RJ-2/3 (10)

Eqs. (9) and (10), derived from analytical solutions,are identical in form to Eq. (4), which, in turn, wasderived from a computer solution of Laplace's fieldequation for the geometry in Fig. 2. What is differentfor Eqs. (4), (9), and (10) are the integer multipliersof R, and the value of the exponent, both of which are

Given that Eq. (11) is indeed a general convergentsolution, at small values of t, for electrical fielddistributions, this leads to speculation on what mustbe the natural, or intrinsic, insulation property ofa dielectric for Eq. (11) behavior to be reflected inthe experimental testing, as in Eq. (7) for solid di-electrics. The working concept is that electricalbreakdown of an insulator involves two distinct stages,an initiation stage and a propagation stage. It isfurther assumed that the stress required for initia-tion is greated than the stress required for prop-agation. This parallels, for example, the mechanicaltear behavior of polymers, which have a high initia-tion stress and a low propagation stress. Now theelectrode geometries establish the potential gradientrelationship throughout the insulator, which for thegeometry of Fig. 2, has the maximum potential gradientat the surface of the rounded edge and the minimum atthe surface of the opposite, flat ground plate. Sincethe maximum potential gradient is always at the round-ed edge for any voltage level, initiation of electricalbreakdown in the insulator material will occur when in-creasing voltage increases the potential gradient levelat the curved surface to be equal to the required ini-tiation value of the insulation material. The initia-tion value of the potential gradient EMAX., for the in-sulation failure is the intrinsic dielectric strengthS.

In summary, this concept states that an electricalfield imposes a mechanical stress on the insulationmaterial, and that mechanical failure is initiated atthe site of the highest potential gradient, which is,therefore, also the site of highest mechanical stress.In this concept, initiation is considered to occur es-sentially on the surface of an insulation material or,if the material encloses a conductor, at the inter-facial surface between the conductor and the insula-tion. Therefore, it might be expected that the en-vironmental and material properties at the surface ad-jacent to the site of the high-potential gradient aremore critical than average bulk material properties,or environments some distance removed.

It is also well known that voltage breakdown of in-sulation materials is associated with measurable

576

Cuddihy: The intrinsic dielectric strength of electrical insulation materials

detection of electrical current. Recently, Dickensonet al. [12], have reported that mechanical fracture ofpolymeric materials results in a release of chargedparticles and electrons at the fractured interface.Electrodes with a potential difference are positionedon opposite sides of the test material, and as the elec-tric-field-induced mechanical stresses initiate mechan-ical fracture, the fracture-released charge particlesand electrons could be attracted to these electrodesand constitute 4 detectable current flow. In otherwords, current flow and detection would be a consequenceof material fracture and breakdown, as opposed to beinga cause of the material failure by electrode injectionof electrons.

EXPERIMENTAL DATA

An ethylene vinyl acetate (EVA) elastomer, based onDuPont's ELVAX 150® [13], is a commonly employed photo-voltaic pottant material, for which interest centerson its electrical insulation qualities. Data on acvoltage breakdown have been measured for this EVA pot-tant, and for this test, it was experimentally conven-ient to use a symmetric pairing of electrodes, notdictated by any of the concepts or theories being de-scribed in this paper. It turned out to be a fortunatechoice. The test results measured on three thicknessesof EVA film are given in Table 3, along with the cal-culated average dielectric strength VA.

Table 3

Average ac Breakdown Voltage of EVAFor Three Film Thicknesses

Average AverageThickness ac Breakdown Dielectric Strength

t,mm Voltage, kV VA = V/t, kV/mm

0.119 11.7 98.32

0.152 13.0 85.52

0.398 17.6 44.22

Using the VA and t data given in Table 3, Eq. (11)was solved for K, a, and n by a least-squares techniqueto yield the following:

VA = 22.2 (t + 0.095)Y0.96 (13)

and therefore for t=0,

S = K(a)-n = 212. 7 kV/Irrn (14)

In light of the concept being described here, it istempting to assign the value of S=212. 7 kV/mm as theintrinsic dielectric strength of EVA, and to state thatwhenever this potential gradient is reached on an elec-trode surface in contact with EVA, the EVA material willexperience voltage breakdown. Fig. 5 is a log-log plotof VA versus the thickness term (t+0. 095). This issimilar to the historically empirical data-correlationtechnique of plotting VA versus thickness t on log-logpaper, except that here the term a is included alongwith t in the abscissa. Again, VA decreases with in-creasing values of t, not because of any materialcharacteristic but because of the behavior of the elec-trical field distribution associated with increasingthe gap thickness between electrodes (which happenshere to be filled with EVA).

EE

C-1

z

-A

XLZ-

100

lo0.04 0.1 1.0

THICKNESS TERM (t + 0.095), mmFig. 5: The ac dielectric strength of EVA.

It is to be recalled that this EVA test, fortunately,was carried out with symmetric electrodes and that, inEq. (13), the value of the exponent n is 0.96, or verynearly 1. This may be compared with Eq. (9), which isthe convergence solution for small values of t, forsymmetric tip-to-tip electrodes, which happens to havean exponent n=1. If similarities continue, then theeffective radius of curvature R associated with thisvoltage breakdown is found in the a value, by dividingby 3. Hence, R is equal to 0.095/3=0.0316 mm.

In a 1955 paper [6], Mason reported experimental re-sults of the measurement of the average dielectricstrength VA of low-density polyethylene (PE) as afunction of sample thickness. For his test, Mason usedan asymmetric electrode pairing, with the ground elec-trode being a flat plane. Using his published VA andt data for PE, Eq. (4) was solved by a least-squarestechnique for K, a, and n, yielding the following re-sult:

VA = 28. 02 (t + 0. 0305)- 0.6 7

and for t=0

S = K(aJ -n = 290.4 kV/mm

(15)

(16)

His data, plotted as VA versus the term (t+0.0305 mm),are shown in Fig. 6.

Note the striking similarity of Eq. (5) for Mason'sPE data, measured with asymmetric electrodes, and Eq.(10), the convergence solution for the asymmetric tip-to-ground electrode configuration. Not only are thevalues of the exponent n essentially the same but,also, for Mason's data, the value of a, which is equalto R in Eq. (10), is almost the same value of R de-rived from the EVA data using symmetric electrodes.It is tempting to assign the value of S=290.4 kV/mm asthe intrinsic dielectric strength of PE.

Last, the data for PMMA shown in Fig . 4, which wasextracted from separate technical data bulletins, wasalso fit to Eq. (11) by a least-squares technique,yielding

VA = 31.16 (t + 0. 0221)- 0(63

VA -22.2 (t + 0. 95) 096Satt-0 -212.7 KVJmm

imn.

5;77

(17)

IEEE Transactions on Electrical Insulation Vol. EI-22 No.5, October 1987

10.0

Fig. 6: The ac dielectric strength of PE.

and for t=0

S = K(a)-n = 344. 06 kV/mm (18)

The separate technical bulletins reported that the volt-age breakdown testing was carried out with asymmetricelectrodes. With the recognized possibility of inac-curacy that may result from merging separate experi-mental data, Eq. (17) reflects the behavior now ex-pected for asymmetric electrodes.

The three materials (EVA, PE, and PMMA) are a softelastomer, a semi-hard thermoplastic, and a rigid plas-tic, respectively. It is noted for each that their re-spective value of S also increases in the same order.In itself this is not a new observation, as the recog-nition of a relationship between material hardness andaverage dielectric strength can be found in early lit-erature on electrical insulation studies [14]. Thus,if S is the intrinsic dielectric strength, the observ-ations reported here agree with historical observations.

It is interesting to note that, for the three mate-rials, the range of S from 212.7 to 344.06 kV/mm issurprisingly narrow, considering that the materialsrange from a soft elastomer above its glass transitiontemperature Tg to a rigid plastic below its Tg. In1976, Swanson et al. [15], reported dielectric strengthmeasurements made on a wide variety of polymeric mate-rials, ranging from soft, to semi-hard, to rigid. Theyconcluded that Tg had only a slight effect on dielectricstrength values.

.iHE EFFECTIVE RADIUS OF CURVATURE

The voltage breakdown data reported above were ob-tained using symmetric and asymmetric electrodes con-tacting the opposite surfaces of a test material, asshown in Fig. 7. Using those data and Eq. (11), an ef-fective radius of curvature R=30.5 pm nominal value wascalculated for the EVA and PE data, and a nominalR=22.1 pm was calculated for the PMMA data. This lat-ter value may be inaccurate due to the use of mergeddata from different technical data sources.

Nevertheless, there seems to be no immediate correla-tion between these values of R and those quoted for ex-ample in ASTM-D-149-64, for the radii of standard test

,ELECTRODE ELECTRODE

GROUND PLANE

Fig. 7: Symmetric and asyrmnetric electrodepairings.

electrodes. It is interesting to comment that perhapsthere should be no correlation at all, if it can bespeculated that this radius of curvature may be asso-ciated with conductor surface features such as nicks,scratches, and/or needle-like asperities, all of whichwould tend to have small radii of curvature.

Along these lines, Mason reported data [6] on meas-urement of the average voltage-at-breakdown VA of PEby needle electrodes, inserted into the test material.The needle electrodes had a range of tip radii fromabout 1.5 to 40 pm. His experimental results are shownin Fig. 8, from his Fig. 7(G) in [6]. The averagevoltage-at-breakdown decreased as the radii of theneedle tips decreased from 40 to 10 pm, and, there-after, essentially no further reduction in voltage-at-breakdown occurred with continuing decreases in theradii of the needle tips. This follows from inspectionof his data line in Fig. 8, but he also has two datapoints closer to 20 pm, which appear to be in theasymptotic minimum region for the average breakdownvoltage.

What his data imply is that the potential gradientpeaked to a limit value, when tip radii became lessthan 10 to 20 pm, no matter how much smaller the tipradii. It is interesting to observe that these tipradii are in the order of magnitude of those generatedby the least-squares fit of Eq. (11) for the voltagebreakdown data described above.

From the foregoing, it is suspected that some nat-ural, but not yet clearly understood, law of physicsis regulating this effective radius of curvature tobe somewhere in the nominal range of perhaps 10 to 30pm. If this is accepted, even without current under-standing, having knowledge of that value of R that canbe associated with the edges, corners, nicks, and sur-face features of electrically conductive devices, aswell as a value of S that can be associated with theinsulation material, and a value of n determined bysymmetric or asymmetric device configurations, newvistas are opened on electrical stress failure anal-ysis and on safe electrical stress design.

Assuming that R=30. 5 pm can be regarded as a funda-mental constant for ac voltage breakdown testing, thenan impressive illustration of its utility can be shown.The 1976 article by Swanson, et al. [15], reported avalue of 29.2 kV/mm as the average dielectric strength(VA) for low-density PE. The measurement was made ona 1. 39 mm thick specimen, using two 12. 7 mm diameterstainless steel balls as electrodes. This electrodepairing is symmetrical. Using a value of R=30.5 pm,and the symmetrical tip-to-tip equation given in Table2, a value of S=301. 3 kV/mm can be calculated as the

1000O III

100-

EE5:

0

in

0

tn

Xw

SYMMETRIC

VA * 28.02 (t + 0.0305) 0.67

\ ~~~~Sat t - 0 - 290.4 KVl/mm

0N

EXPERIMENTAL DATA REPRODUCEDFROM REFERENCE 6

ASYMMETRIC

10

0.01 0.1 1.0THICKNESS TERM (t + 0.0305), mm

1-11

I

Cuddihy: The intrinsic dielectric strength of electrical insulation materials

1.0 ELECTRODE SEPARATION, 500 pm 14010-XXX POSITIVEPOINTS/.E 00 NEGATIVE POINTSu

> 0.9E

> 0.8 120

0.7 -XX

0.6 X I I 0O /1 2 4 6 8 10 20 40 100

TIP RADIUS, pm

Fig. 8: Dependence of average breakdown voltage VAof embedded-needle electrodes at 600C (Fig. 7b in 80[6]).

8

< YIELD POINTintrinsic dielectric strength of low-density polyethy-lene, in virtual agreement with the value of LUS=290.4 kV/mm calculated using Mason's 1955 data. 60

APPLICATION TO BURIED CABLE

It has been reported [16] that buried HV cables arefailing in service from voltage breakdown of the PE in- 40sulation. Despite considerable research, an explana- (LINEAR PLOT)tion for these failures has not been satisfactorilyachieved. Therefore, corrective actions based on afundamental understanding of the failure cause has notbeen implemented. It is the intent of this Section tooffer a possible explanation for the cause of these 20failures, based on the intrinsic dielectric concept S,and the acceptance of the radius of curvature R=30.b5 pmas a fundamental constant.

THE PROPORTIONALLJ MI 0 I

The intrinsic dielectric strength concept, as ad- 0 10 20vanced, states that an electric field imposes a mech- STRAIN, %anical stress on the insulation material, and thatmechanical failure is initiated at the site of thehighest electric stress, which is, therefore, also the (B)site of highest mechanical stress. In this concept,initiation is considered to occur essentially on thesurface of an insulation material, or, if the materialencloses a conductor, at the interface between the con-ductor and the insulation. Therefore, it might be ex-pected that the environmental and material properties,at the surface adjacent to the site of the high field, X 100are more critical than average bulk material propertiesor environments some distance removed.

However, the value of S at voltage breakdown is iden- YIELD POINTtified with ultimate failure, identically as the ulti-mate tensile stress in mechanical testing. Mechanical PROPORTIONAL LIMITengineers do not design load-carrying structures to thelimit of the ultimate tensile stress of the construc-tion materials, but to a much lower value that is typ- 10 _ically less than the material's yield stress. If, in- (LOG-LOG PLOT)deed, and electric field imposes a mechanical stresson electrical insulation materials, then mechanical ___property considerations may provide a clue to the safe o 1 1.0 10.0engineering design limits for insulation materials.

STRAIN, %Fig. 9(a) is a conventional linear plot of a uni-

axially-measured stress-strain curve, here illustrated Fig. 9: Comparative stress/strain pZots.

579

IEEE Transactions on Electri cal Insulation Vol. EI-2- No.5, October 1987

for Mylar® polyester film (polyethylene terephthalate,E.I. DuPont). The stress is calculated as the forcecausing extension, divided by the initial cross-sec-

tional area AD of the test specimen. This convention-al plotting format reveals an initial near-linear re-

lationship between stress and strain, followed by de-parture from linear behavior at a yield point, andthen ductile behavior to the ultimate stress at fail-ure. As the cross-sectional area decreases with in-creasing extension, Fig. 9(a) does not reflect the true

stress, which is the load divided by the actual cross-

sectional area at any extension of the test specimen.

Fig. 9(b) is a replot of the same load-extension dataon log-log coordinates, as true stress versus strain.Although there are similarities with Fig. 9(a), it issignificant that the stress at departure from initiallinearity is not at the conventionally accepted yieldpoint. The actual stress at departure in Fig. 9(b) iscalled the proportional limit, and, typically, is a

value about 60% of yield. For Mylar, the proportionallimit is about 58.6 MPa [17], as compared to its yieldstress of near 93.1 MPa, which can be extracted fromFig. 9(a).

The significance of the proportional limit, again il-lustrated with Mylar, is shown in Fig. 10. Fig. 10(a)

(a)

L-i

-)(-)

es

L.X

-i

MAXIMUM STRESS AMPLITUDE, MPa

(b)120 If_

1101

100

90

80

70

60

50

YIELD STRESS

MATERIAL: MYLAR "A"

ENVIRONMENTAL STRESS LIMIT

400 10 20 30 40 50 60 70 80

TIME TO FAILURE, s

ENVIRONMENTAL STRESS CRACKING

Fig. 10: Significance of the proportional Limit(proportional Zimit = fatigue-endurance Zimit =

environmentaL-stress limit).

is a plot of the flexural fatigue properties of Mylar,plotted as number of cycles-to-failure versus the peak-stress amplitude. The fatigue-endurance limit is thatstress level above which an abrupt decrease in cyclelifetime occurs. For Mylar, this occurs at 58.6 MPa,which is also its proportional limit.

Mylar, as are many engineering plastic materials[18], is susceptible to environmental-stress cracking,a phenomenon wherein life-under-stress is reduced dra-matically when the material is exposed to certain gasesor liquids which are specific to the stressed material.Mylar is highly susceptible to environmental-stresscracking when stressed in the presence of methyl ethylketone (MEK) solvent [17]. Fig. 10(b) is a plot ofMylar lifetime, when simultaneously stressed and ex-posed to MEK. Note that the susceptibility to en-vironmental-stress cracking vanishes for stress levelsbelow the proportional limit; that is, the solid dataline merges asymptotically with the dotted line.

The proportional limit is the upper design limit formechanical service. It is the stress level at whichmaterial behavior departs from its elastic, Hookian,behavior to ductile characteristics. It can be viewedas the beginning point for mechanical service problems,however they may manifest themselves. Service-stresslevels in the region between the proportional limitand ultimate stress can lead to voids, cracks, pre-mature failures, and time-dependent losses in per-formance.

If indeed electric fields impose a mechanical stresson electrical insulation materials, then the upperlimit of a field for electrical insulation design couldbe regarded as that value scaled down from knowledgeof S at voltage breakdown, and the mechanical propertyvalues of ultimate tensile stress and proportionallimit. With this, a possible explanation for polyethy-lene insulation failures in buried V cables can be of-fered, and a possible explanation of the origin ofboth electrical trees and water trees.

POLYETHYLENE-INSULATED CABLES

Orientation Properties

Using Mason's data for PE [6], a value of 5=290.4kV/mm was calculated above for the material's intrinsicdielectric strength at voltage breakdown. This valueis in sharp contrast with values at or in excess of787.4 kV/mm, reported as the intrinsic dielectricstrength of PE in papers dating back to the 1940's[19-22]. One paper by Austen [22] reported on measure-ments of the dielectric strength of oriented paraffinwaxes, in directions parallel with and perpendicular tothe orientation axis. The dielectric strengths were

significantly higher when measured in the directionperpendicular to the orientation axis, as compared withthose measured parallel with the orientation axis.

This clue led to the strong suspicion that the poly-ethylene samples, on which the early dielectric test-ing was performed, had become highly oriented as a con-

sequence of the method of sample preparation then inuse. Further, the measurements were then made in thedirection perpendicular to the orientation axis, whichwould tend to yield the highest values for dielectricstrength. Fig. 11 is adapted from Figs. 1 and 2 of[i19), and illustrates recessing devices used to preparetest specimens.

cL

:-5-

Cuddihy: The intrinsic: dielectric strength of electrical insulation materials5

140

120

100

0

0

C0a:>

80 F

60F

40hTEST THICKNESS < 0.05 mm

20F

TEST THICKNESS> 0.05 mm

Fig. 11: Devices used by researchers before 1950to prepare recessed test specimens for electrical-property testing (Figs. 1 and 2 in [19].

The approach was to position a test specimen betweena flat base plate and a ball or mandrel having a largeradius of curvature. The base plate was then heatedabove the softening point of the test specimen and theball or mandrel was pressed into the sample to gen-erate a large, indented recess. When cooled, the re-cess was filled with an electrode material which, bygeometry, acquired a large radius of curvature. Next,voltage was applied until breakdown, which occurred atthe thinnest point in the sample, typically at thebottom of the recess.

Two situations then may develop. First, the samplecould become oriented by this method of preparation,and the test then measures the voltage-at-breakdown inthe direction perpendicular to the orientation. Second,the electrode in the recess has a large radius of cur-vature and, under that condition, the device approxi-mates a parallel-plate electrode configuration, thusgenerating a uniform field through the sample thickness.

It was practice with the recessed-specimen techniqueto measure the voltage-at-breakdown V as a function ofsample thickness t, which, because of the large R, re-sulted in a-linear relationship. Fig. 12, adapted fromFig. 7 of [19], is typical of the measured relationshipbetween V and t for PE, having *a slope that is approxi-mately 787.4 kV/mm.

By the very nature of the recessed technique, thetest results are an intrinsic dielectric strength inaccordance with the concept described here. What ap-parently could be different between data measured onrecessed specimens in the 1940's and those of Mason

0

0 0000

0

- 0f00 0

0

0

GRADE 20 POLYETHYLENE AT 200 CI~~~~~~~~~~~~I

0.05 0.10THICKNESS, mm

0.15 0.20

Fig. 12: Representative illustration of the Zinearre lationship between breakdown voltage and sarnplethickness, using recessed specimens (after Fig. 7,[19]).

reported in 1955 is orientation effects. Mason madehis measurements using contact electrodes and withoutuse of any recessing; therefore, it can be presumedthat he tested unoriented, isotropic PE, resulting ina value of S=290.4 kV/mm. A similar value of S=306kV/mm was calculated, using experimental data publish-ed by Swanson, et al., in 1976 [15]. By taking theratio (787.4/290.4), it would appear that the recessedPE specimens were being oriented about three times inthe thinnest portion of the recess, assuming a linearrelationship between orientation factor and dielectricstrength increases. This level of orientation iseasily accommodated by ductile polymers. For example,commercial Mylar "A" film is oriented three times inboth its length and width directions [23], and mostpolymer fibers are easily oriented to higher orders[24]. Since the PE insulation used on a HV cable isextruded uniaxially, similarly as a polymer fiber, itslevel of orientation aligned with the core might easilyexceed three times. Therefore as manufactured, itwould be expected to have an artificially high, orien-tation-induced, dielectric strength in the directionperpendicular to the cable axis.

Proportional Limit

Table 4 is a tabulation of properties of PE resinsand is reproduced from [25]. For the three PE mate-rials having the highest molecular weight, D-100,D-130, and D-145, the average tensile strength (ulti-mate strength at failure) is 21.1 MPa, and the averageyield strength is 12.1 MPa. Since the proportionallimit is approximately 60% of yield, a value of 7.26MPa is therefore estimated as the proportional limitfor polyethylene; this corresponds to 35% of the ulti-mate strength (100x7.26/21.1). Invoking a linear re-lationship between intrinsic dielectric strength prop-erties and mechanical properties, and using the iso-tropic dielectric strength at failure S=290.4 kV/mm,it is estimated for PE that its intrinsic dielectricstrength associated with the proportional limit is0.35x290.4=101.64 kV/mm. Stated another way, thethreshold voltage of PE for long-life service would beabout 35% of the initially measured, short-time volt-age at breakdown if, experimentally, the same electrodesand sample thicKness were consistently employed.

u III

58 1

Al

IEEE Transactions on Electrical Insulation Vol. EI-22 No.5, October 1987

Table 4

Properties of PE resins (adapted from Table LXXXI, [25].

(Grade Designation of Polyethylene Resins)D-55

Properties D-40 (DYNH) D-70 D-85 D-100 D-130 D-145 Test Method

Molecular weight, average 14- 18- 20- 24- 26- 28- 30- --18,000 20,000 22,000 26,000 28,000 30,000 32,000

Specific gravity 0.92 0.92 0.92 0.92 0.92 0.92 0.92Stiffness in flexure, p.s.i.

250C 18,000 18,000 18,000 18,000 18,000 18,000 18,000 A.S.T.M. D747-43T00C 30,000 30,000 30,000 30,000 30,000 30,000 30,000 A.S.T.M. D747-43T

-250C 66,000 66,000 66,000 66,000 66,000 66,000 66,000 A.S.T.M. D747 (Tentative)-500C 160,000 160,000 160,000 160,000 160,000 160,000 160,000 A.S.T.M. D747 (Tentative)

Yield strength at 250C, p.s.i. 1,430 1,480 1,490 1,600 1,700 1,830 1,720 A.S.T.M. D412-41Tensile strength, p.s.i. 1,430 1,825 1,965 2,435 2,965 3,160 3,060 A.S.T.M. D412-41Compressive strength, p.s.i. -- 3,000 -- -- -- -- -- --Ultimate elongation at 250C, % 305 560 550 560 580 605 625 A.S.T.M. D412-41Brittle temperature, OC -55 Below Below Below Below Below Below A.S.T.M. D746-43T

-70 -70 -70 -70 -70 -70Impact strength, ft.-lb./in.

of notch -- >3 -- -- -- -- -- A.S.T.M. D256-43T (A)Tear strength, p.s.i. 440 500 540 560 580 605 690 A.S.T.M. D256-41TAbrasion volume loss

(standard butyl rubber = 100) 85 55 50 45 40 35 30 142aliardness Durometer D at 250C 52-54 52-54 52-54 52-54 52-54 52-54 52-54 --

Effect of Water

Fig. 13 illustrates the effect of water on reducingthe dielectric strength of Mylar. This Figure is re-produced from DuPont Technical Bulletin M-4D, whichprovides technical data on the electrical properties ofMylar. Analysis of the data in Fig. 13 indicates that

400

300 2001o R H

>200 35% RH

2100

50 I I0.025 0.04 0.06 0.08 0.1 0.2 0.25

THICKNESS, mm

Fig. 13: Dielectric strength of Mylar fiZm materiaZat various humidities (after DuPont Co. TechnicaZBulletin M-4D, "EZectricaZ Properties of MyZarPoZyester FiZm").

about a 15% reduction in dielectric strength occursupon going from 20 to 80% RH. If linearity is assumed,this same reduction would be expected upon going from40% RH to 100% RH, where 40% RH would be a typical roomhumidity in which dielectric testing is carried out,and 100% RH would be the environment of a buried cable

in moist soil. It will be assumed that the PE insula-tion in a buried cable experiences a 15% reduction indielectric strength properties, compared with thosefound in laboratory testing.

Buried CabZe

Most commercial cables intended for buried appli-cation are rated at, and operate at, 15 kV and 60 Hz[26]. The PE insulation surrounding the conductingcore is about 4. 45 mm thick. When new, the averagefield at breakdown VA of the insulation is about 31.5kV/mm, and in service, the cables operate with anaverage field (V/t) across the insulation of about3.34 kV/mm, for a safety margin of about 10 to 1.

With this information, an estimate of the initial in-trinsic dielectric strength of the PE insulation per-pendicular to the cable axis can be made, using thetip-to-ground equation given in Table 2. For thiscalculation, VA=31.5 kV/mm, t=4.45 mm, and R=0.0305mm. Thus, S11446.5 kV/mm is estimated, which, whencompared with the isotropic, unoriented value of 290.4kV/mm, indicates about a five-fold orientation of thepolyethylene material which may be associated with thecable manufacturing process.

Next, the average field that is associated with theproportional limit can be calculated. This would bethe maximum service level to keep the electric-field-induced mechanical stresses at or below the proportion-al limit, which also keeps the mechanical stress frombeing in the ductile region of PE. For this calcula-tion, the proportional limit is 101.6 kV/mm, t=4.45mm, and R=0.0305 mm. Carrying out the calculation, theaverage field associated with the proportional limit is2.2 kV/mm. Despite the conventional l0-to-l safetymargin, this calculation strongly suggests that theservice environment may be imposing electrical-field-induced mechanical loads above the proportional limitof PE, and in the ductile region.

5:2

Cuddihy: The intrinsic diielectric strength of electrical insulation materials

Under these mechanical loads, it can be expected thattime-dependent ductile response will occur, causing areorientation of the PE insulation from its initialdirection parallel with the cable axis to a directionperpendicular to the cable axis. It can further besuspected that the rate of reorientation will be high-est at the interfacial surface of PE immediately ad-jacent to the site of the high field, that is, thehighest mechanical stress, and that it will fall offprogressively through the bulk in relation with out-ward spatial tapering of the field intensity.

This reorientation would constitute a physical agingprocess, with at least two associated consequences.First, reorientation to a direction perpendicular tothe cable axis would cause a gradual reduction in thedielectric strength in that direction. Given a near-five-times manufacturing orientation, and if bulk re-orientation were to approach the isotropic state, thenit can be predicted that the average voltage-at-break-down measured across the insulation could graduallydecay from an initial value of 31.5 kV/mm to a lowervalue in the order of 6.3 kV/mm. However, local sur-face reorientation at the site of high field intensitywould be expected to be greater than the average bulkreorientation.

Second, the reorientation would be expected to pro-duce a change in the optical birefrigence of the poly-ethylene material. It can be strongly speculated thatelectrical trees, as first reported by Kitchin [27]and later by others [28,29] and which are revealed onstained specimens in optical microscopy, are a mani-festation of reorientation-induced optical birefrigence.As such, they would reflect morphological changes atthe surface and in the bulk material, and would not becavities, voids, or filamentary tunnels. Indeed, theappearance of an electrical tree defined by this mor-

phological consideration could bridge across the en-tire thickness of the insulation, but would not be initself any manifestation of actual electrical break-down. It precedes electrical breakdown by revealingstress reorientation, which would also result in alower value of dielectric strength as compared withthat of a freshly manufactured cable. Dissado et al.[30] have reported that electrical trees (they calledthem water trees) can cross the entire thickness ofpolyethylene specimens without an associated breakdown.

With respect to actual failure, it can be expectedthat the PE immediately adjacent to the site of thehigh field would experience the greatest degree oflocal reorientation, compared with locations away fromthe site, where the field becomes progressively lower.With continuing local reorientation, the associated in-trinsic dielectric strength for voltage breakdown con-tinues to decrease until a critical value is reached.This initiates breakdown, which then propagates catas-trophically. As observed in Fig. 13, absorbed water inMylar reduces its voltage-at-breakdown by about 15%.Thus, absorbed water in buried cables can act to ex-acerbate failure behavior by lowering the criticalvalue of the field required for failure initiation.This becomes especially serious if ground water findsits way to the interface between the conducting coreand the PE, for it is also at this interface that themaximum field is expected to be found.

It should not be overlooked that the proportionallimit of PE, as herein estimated, assumed similaritieswith Mylar, also a ductile polymer, whose proportionallimit is 35% of its ultimate strength. If the percent-age for PE is different, say higher, then the critical

value of the intrinsic dielectric strength for voltagebreakdown of the locally reoriented PE may be at amarginal level when addressing underground serviceoperations as compared with those above ground. Inthe ground, absorbed water reduces the critical valuebelow the service stresses, thus failures occur; where-as above ground in drier conditions, the critical valuemay remain marginally above the service stresses. Whatis intended is to have shown that the in-ground servicecondition of 3.34 kV/mm is marginally near the esti-mated proportional limit of 2.2 kV/mm which, in turn,may have a 15% tolerance range associated with locally-moist environments. If the physical aging mechanismas postulated is valid, then it is suggested that thecorrective approach is to increase the thickness ofthe PE insulation such that the service stress (cur-rently 3.34 kV/mm) is dropped below the proportionallimit electrical stress of polyethylene at 100%relative humidity.

Lastly, there are three other matters to be consider-ed: temperature, crosslinking, and anti-tree agents.First, increases in cable operating temperatures wouldincrease the rate of ductile response leading to fast-er stress reorientation. Thus, times-to-failure wouldbe expected to decrease with increasing cable operatingtemperature, as reported in [26].

Second, cross linking of PE occurs throughout thebulk, but the local reorientation leading to failureinitiation would occur at the surface of the PE imme-diately adjacent to the site of the high field. Bulkcrosslinking may not be effective in retarding orstopping surface behavior. Along these lines, itshould be noted that the physical state or morphologyof crystalline polymers such as PE, polypropylene,etc., can be quite different at surfaces as comparedwith those in the bulk. It often happens that ifmelted polymers are adjacent to metallic surfaces dur-ing cooling from the melt, the crystalline characterof the surface is different from the bulk cyrstallinestate. This surface behavior has been termed trans-crystallinity, and Shaner and Corneliussen [31] ob-served such behavior for polypropylene insulationaround a copper core. The surface properties and bulkproperties were different. Dissado, et al. [30], havereported similar observations for PE. The point isthat the dielectric strength of the surface materialadjacent to the conducting core, where failure wouldinitiate, may be not only different from the bulk, butsurface transcrystallinity effects may resist cross-linking efforts. Thus, the failure potential is es-sentially unchanged as compared with that of uncross-linked PE. Bahder, et al. [26], report that cross-linked PE cables fail in service.

Third, anti-tree agents might be compounding addi-tives that act to raise the dielectric strength of PE,thus countering any reductions that would be causedby water. Fig. 14 is adapted from DuPont TechnicalBulletin M-4D, illustrating for Mylar that transformeroil decreases its dielectric-strength, but Freon C-318increases its dielectric-strength.

AC VERSUS DC TEST DATA

The data correlation

VA = K(t + a) n (19)

has been found to work with ac voltage breakdown datapublished for PE and PMMA, and for experimental datameasured for EVA, a commonly employed lamination

58----

IEEE Tra.nsactions on Electrical Tnsulation Vol. EI-22 No.5, October 1987

0.025 0.04 0.06 0.08 0.1

THICKNESS, mm

0.2 0.25

Fig. 14: Effect of eZectrode size on dieZectricstrength of MyZar (after DuPont Co. TechnicalBuZZetin M-4D, "ElectricaZ Properties of MyZarPoZyester FiZm").

EVA film due to high electric resistance. There de-veloped a concern that time-effects associated withthe spreading of charge on the top surface, presumablyproceeding outward to match the area-of-charge asso-ciated with the ground electrode, would or could re-sult in erroneous or false dc test data. Reversingdc polarity with the asymmetric electrodes resulted inthe same electrostatic behavior.

Therefore the asymmetric electrode approach was re-placed with a symmetric electrode pairing which con-sisted of two 6.35 mm diameter brass rods, each with ahemispherical round tip. The concept was to reducedrastically the surface-charging time by reducing elec-trode size on both sides of the test film.

Thereafter because of the small-area symmetric elec-trode pairing, no visual evidence of broad-area sur-face charging could be observed, although it may havebeen occurring, and dc breakdown voltage values werethen determined on EVA film specimens ranging between0.048 to 0.224 mm thick. The dc test machine is lim-ited to 32 kV full-scale, and it was noted for thesesymmetric electrodes that film thicknesses greaterthan 0.228 mm would reach full-scale without dc break-down.

THE DC TEST DATA

material for commercial photovoltaic modules. For thislatter application however, voltage is dc and not ac,and thus dc voltage breakdown data of EVA as a functionof sample thickness was measured. The intent was tofit the data to the correlation expression (17) andderive therefrom a dc intrinsic dielectric strength.It did not work. This Section describes the dc experi-mental effort and test results, derives a dc intrinsicdielectric strength for comparison with the ac value,and offers an explanation as to why the dc data couldnot be fitted to the above expression.

EXPER-IMENTAL OBSERVATION

The first experimental effort with EVA employed anasymmetric electrode pair consisting of a large-area,flat-plate electrode as ground and a 6.35 mm diameterbrass rod with a hemispherically rounded tip as thepositive electrode. The EVA test specimen was a 15 mmsquare film about 0.15 mm thick, which was positionedon the ground electrode and which could be seen throughthe transparent EVA film.

Upon applying dc voltage in a constant rate-of-risetest (z500 V/s), it became immediately apparent thatthe transparent EVA film was being charged like a ca-pacitor. Without voltage, the view of the ground platethrough the EVA film had a slight hazy quality asso-ciated with the typical non-smooth, air-filled, inter-facial contact between two surfaces. With voltage,electrostatic attraction caused the EVA film to betightly pressed against the ground plate, resulting ina clear view. This action began as a small circularspot on the ground electrode just below the positiverod electrode, and then radially spread outward on theground electrode as a growing circle. Eventually dcbreakdown occurred.

It was then speculated, rightly or wrongly, that theground electrode being metallic, developed its capac-itor-like charge almost instantaneously over its wholearea with rising dc voltage, but that the oppositecharge, causing the visual electrostatic clinging, didnot spread instantaneously on the top surface of the

Fifty values of the dc breakdown voltage V were meas-

ured for EVA samples over the above-mentioned thicknessrange t, and it was intended to fit these data to thecorrelation equation [19].

Interestingly this did not match; instead the dcdata yielded a near-straight-line relationship betweenV and t as shown in Fig. 15. The slope of the straightline is 143.7 kV/mm, which could be considered as theintrinsic dc dielectric strength of EVA.

It is to be remarked however, that a straight-linerelationship between V and t is expected when break-down occurs between parallel-plate electrodes, whereradius R is infinity, and the electrode field distri-bution between the plates is linear. But for thistest, the rod electrodes can hardly be considered as

parallel-plate electrodes.

It is speculated for this dc test that the effectiveelectrodes are the opposite surfaces of the EVA film,which have acquired a capacitor-like charge from con-

tact with the rod electrodes. Stated another way, thesurfaces of the uniformly thick film mimic a parallel-plate electrode configuration when oppositely charged,and therefore result in a straight-line relationshipbetween V and t. In ac testing, this does not occur

because polarity reversal prevents this surface-charg-ing behavior.

Eq. (17) is the general form of expressions whichcan be derived from algebraically combining the firsttwo terms of the series expansion of analytical solu-tions to Laplace's field equation. When R=c, as forparallel-plate electrodes, then the series expansionssimply reduce to

VA = V/t (20)

because the field throughout the thickness directionis everywhere constant. Thus Eq. (20) is a form of

Eq. (19) for a radius equal to infinity.

EE

C-,)

C-,

L4,j

*FROM DU PONT TECHNICAL BULLETIN M-40,"ELECTRICAL PROPERTIES OF MYLAR POLYESTER FILM'

_Cuddihy: The intrinsic dielectric strength of electrical Insulation materials

0

,

0LJgm

u

t, EVA FILM THICKNESS, mm

Fig. 15: The dc dielectric strength of EVA.

For comparison, ac test data for the same EVA mate-rial could be fit to Eq. (17), yielding

V/t = 22.2 (t+0.095 0)-96 (21)

and for t=O, S=Kacn = 212.7 kV/mm

Thus it is observed that the ac and dc intrinsic di-electric strengths, 212.7 and 143.7 kV/mm respectively,are numerically of the same magnitude, with the acvalue being the higher of the two.

MYLAR AC/DC DATA

It is interesting to analyze published ac and dcbreakdown voltage data, interpreted in terms of the in-trinsic dielectric concept for ac testing, and surfacecharging for dc. DuPont technical bulletins for Mylarpolyester film report a value of 550 kV/mm for thematerial's dc dielectric strength at 250C. It will beassumed that dc testing of the Mylar film also resultedin surface charging as similarly observed with EVA, andthat this value of 550 kV/mm is indeed the intrinsic dcdielectric strength.

Since Mylar is industrially oriented about 3 x [23],and if linearity can be assumed between property in-creases (or decreases) and the orientation factor, aswas assumed for the PE insulation, then the isotropicdc dielectric strength of lMylar can be estimated at550/3=183 kV/mm. This can be compared with the valueof 144 kV/mm, herein determined for EVA which is un-oriented. As Mylar is a plastic, it is expected tohave a higher dielectric strength than an elastomericmaterial such as EVA [15].

Previously, the intrinsic ac dielectric strengths ofEVA and PMMA, a plastic, were determined to be 213kV/mm and 344 kV/mm, respectively. It is interestingto observe from these limited data that the intrinsicdc dielectric strength values tend to be lower than theintrinsic ac dielectric strength values when comparingelastomer to elastomer and plastic to plastic. ForEVA, its dc-to-ac ratio is 144/213=0.67, or about two-thirds.

The same DuPont technical bulletins for Mylar reporta value of near 750 kV/mm as the average ac dielectricstrength VA of 6 pm Mylar. This ac test used asym-metric electrodes, therefore

V/t = K(t + 0.0305)-213 (22)

is used to derive S, the intrinsic ac dielectricstrength of Mylar, where V=748 and t=6 pm.

From these calculations, the intrinsic ac dielectricstrength S is found to be about 850 kV/mm, as comparedto the reported dc value of near 550 kV/mm for the sameorientation state. Note that not only is the dc valuelower than the ac value, but it is lower by almost two-thirds, similar to the elastomeric EVA. Further, di-viding the ac intrinsic dielectric strength by 3,Mylar's orientation factor, a value of 282 kV/mm isestimated for its isotropic dielectric strength. Thisvalue is in the same order of magnitude as previouslyobserved for EVA, PE, and PMMA.

SUMMARY

The relationship VA=Kt-n, where VA=V/t, has been ob-served historically as an empirical data correlationfor measurement of the ac breakdown voltage V as afunction of insulation thickness t. It is indicatedin this paper that this relationship may be a limitedform of Eq. (17), which can be derived mathematicallyfrom the first two terms of the series expansion ofanalytical solutions to Laplace's field equation.Thus, the observed correlation would reflect thethickness-dependence of the spatial distribution ofan electric field through an insulation material and,therefore, may not be associated with any propertiesor flaw-states of an insulation material.

If the observed correlation between VA and t is amanifestation of the thickness-dependence of the elec-tric field, then voltage breakdown would occur when-ever a critical value of the field on a conductor sur-face is reached or exceeded. It is suggested that thiscritical value of the field is the intrinsic dielectricstrength of an insulation material.

From the series expansion, the term a is an integermultiple of R, the radius of curvature, and from ex-perimental data, R appears to behave as a fundamentalconstant having a numerical value equal to 30.5 vm.This is observed, but unexplained. Nevertheless, itcan be speculated that if this value of R is associat-ed with surface features on conductors such as nicks,scratches, and/or needle-like asperities, then thisvalue can be used in analytical solutions of Laplace'sfield equation to predict relationships between volt-age and thickness for a diversity of conductor/insula-tor systems such as, for example, insulated HV cables.

Along this line, the intrinsic dielectric strengthconcept has been used to offer a possible explanationfor the voltage breakdown failures of the PE insula-tion in buried HV cables. The dielectric strength of

=-IR=,j E- j

iEEE T.ranE= tion on EectricaI Insul-at on Vo. EII-2 No.5, October 1987

insulation materials appears to be orientation sensi-tive, and this feature, along with the use of thefield concept as the intrinsic dielectric strength,the effective radius of curvature, and a mechanicalproperty known as the proportional limit were all uti-lized to generate a physical aging failure mechanismas the cause of failures in buried PE HV cables andfor water trees produced in the material during service.This concept indicates that the intrinsic dielectricstrengths of electrical insulation materials have nu-merically the value of a field, which can be generatedfrom experimental measurements of average breakdownvoltage as a function of sample thickness t. The actionof an electric field is to impose a local mechanicalstress on the material surface, which becomes the in-itiating mechanism of failure when the generated stressequals or exceeds the intrinsic dielectric strength.

The measurement of dc voltage breakdown of an EVAelastomer as a function of thickness yielded a straight-line relationship between V and t, rather than conform-ing to Eq. (17), which was found to accommodate acvoltage breakdown data. For ac data, the intrinsic di-electric field strength S is equal to the limit valueof (V/t) at t=O, that is, Kan.

Observation of the dc testing suggested that the sur-faces of the EVA test specimens were becoming electro-statically charged, like a capacitor, and it is postu-lated that this generates in effect a parallel-plateelectrode configuration. This would explain the linearrelationship between breakdown voltage V and samplethickness t. In addition, for an infinite radius ofcurvature (parallel-plates), the above expression be-comes simply V/t=dV/dt and thus the slope of V versust is a field which can be designated as the dc intrinsicdielectric strength.

This paper presents the results of one phase of re-search conducted at the Jet Propulsion Laboratory,California Institute of Technology, for the U.S. De-partment of Energy, through an agreement with theNational Aeronautics and Space Administration.

APPENDIX A

Series Expansions Of The Tip-To-Tip andTip-To-Ground AnalyticalExpressions In Table 2

Tip-to-Tip

The tip-to-tip equation is

VA t(1 + 2R/t) 0'5EMAX = 2R- arctanh [t/(t + 2R) ] %5 (A-1)

which can be rearranged algebraically to

EMAX (t/2R) (1 + 2R/t) 05

VA arctanh[- -12 5

It is convenient to define the variable

x = (1 + 2R/t) ° 5

(A-2)

(A- 3)

which, when substitubed into Eq. (A-2) yields

EMAX _ (t/2R) (l/x)VA arctanh (x) (A-4)

It is interesting to summarize the ac and dc datacorrelations as shown in Fig. 16, along with the math-ematical expressions for data correlation and for cal-culation of their respective intrinsic dielectricstrengths.

From reference handbooks, the series expression for theinverse hyperbolic tangent is

arctanh (x) = x+ + 5 + x7 + .....3 ' 5 7 (A-5)

AC

* DATA CORRELATION

V *(V/t) - K (t + a)-nA

* ac INTRINSIC DIElECTRIC STRENGTH

S - K (a) n

_- S @ t 0

LOG (VA))

LOG (t + a)

which can be divided into the numerator term l/x, ofEq. (A-4) to yield

(l/x) _ 1 1 4 2 (higher powers (A-6)arctanh (x) -2 3 4 of x)

From Eq. (A-3), x =(1+2R/t)-1 or 1/x2=(1+2Rt), which,when substituted into Eqs. (A-6), and (A-4), will yield

DC

* DATA CORRELATION

V - Kt

* dc INTRINSIC DIELECTRIC STRENGTH

(A- 7)EMAX t 4 22Rf/24__ 2R) t- 3 45(1 + 2R/t)

/V- SLOPE - SVX

S - (Vlt)- K

Fig. 16: Electrical insuZation: data correlationfor ac and dc intrinsic dielectric strengths.

ACKNOWLEDGMENTS

The author thanks John Winslow, Ron Ross, Gordon Mon,and Carl Solloway of the Jet Propulsion Laboratory fortheir positive contributions to and commentaries on thetechnical scope and mathematical content of this paper.

When t/2R is multiplied into each term of the seriesexpression in Eq. (A-7), the third term as well as allhigher terms become numerically negligible for small tor large R. Therefore, Eq. (A-7) reduces to

EMAX = 1 + t/3RVA

(A- 8)

Taking the reciprocal of Eq. (A-8), yields the finalexpression shown as Eq. (12) in the text:

VA = EMAX 3R (t + 3R)-1 (A-9)

Cud_ihy: The intrinsic dielectric strength of electrical insulation materials

Tip-to-Ground

The tip-to-ground equation is

EMAX = 2VA tP/ln(Q)

where

P = (1 + 1/oW)05/R, a = t/R

Q = [t + 2t°'5 (t + R) + (t + R)]/R,

(A- 10)

(A-li)

(A- 12)

which can be recognized as the binomial expression

Q = [La'5 + (a, - 1J5 ]2 (A- 13)

Exclusive of VA, the numerator term of Eq. (A-12), is2tP, which corresponds to

2tP = 2a00X5 (a + 1) 05 (A- 14)

Upon substituting Eqs. (A-13) and (A-14) into Eq. (A-10),the following can be derived

EM4XVA

a05 (Ol + 1) 0.5

O. 51n (a + 1) + ln[1 + a -05(A- 15)

The term (cX+1)0'5 and the denominator terms can be a

series expanded which results in

EMAX 1 2 a4 a2+VA 3 45 (A- 16)

Now ignoring all terms ol2 and higher, which becomenumerically negligible for small values of t or largevalues of R, and substituting for a, Eq. (A-16) becomes

EMAX 2t1 +

VA 3R

or from a series expansion

EM4X = (1 + t/R)23

VA

(A- 1 7)

(A- 18)

Taking the reciprocal of Eq. (A-18), yields the finalEq. (10) in the text:

VA = EMAX (R) 23 (t + RJ-2P (A- 19)

SYMMETRY

It should be noted that the tip-to-tip and tip-to-ground equations used are symmetrically related, withthe midpoint (see Fig. A-1) between the two needle tipsbeing the ground plane for the tip-to-ground equation.

This is demonstrated by substituting V/2 and the gapvariable T=t/2 in the tip-to-tip Eq. (A-1), yielding

d(V/2)/d(T/2) = dV/dT T(1 + ?I0((V/2)/(T/2) v/t arctanh [L/(T +- R)]0

The right-hand term of Eq. (A-20) can be rearrangedwhich results in

(dV/dT)MAX 2 T(1+R/lT) °5/R(V/-l)-lnr(1+R/) 0'5 + 1

(1+Rl 0.5 - 1

(A-21)

UPPER TIP AT POTENTIAL V

LOWER TIP AT POTENTIAL 0

Fig. A-1: Geometry of symmetry between the tip-to-tip and tip-to-ground equations.

The numerator is now recognized as 2tP observed withthe tip-to-ground equation, and the remaining step isto transform the denominator algebraically to theln(Q) form also observed with the tip-to-groundequation.

This then becomes

ln(Q) = ln [2T + R + 2T15 (T + R)0 (A-22)

Therefore, the convergence expressions for smallvalues of t or large values of R in Eqs. (A-8) and(A-17) are interchangeable using the gap variablet=2T. The tip-to-tip convergence expression was foundto be

(dV/dy)MAX 1 2TVA 3R

(A- 23)

as derived above, whe' e EMAX=(dV/dy)MAX.

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®Registered Trademark

Manuscript was received on 24 March 1986, in finalform 11 August 1987.

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