PD

510 IEEE Transactions on Dielectrics and Electrical Insulation Vol. 2 No. 4, August 1995

The ALESSANDRO VOLTA Lecture

A Generalized Approach to Partial Discharge Modeling

Lutz Niemeyer

ABB Corporate Research, Switzerland

ABSTRACT An important tool for improving the reliability of HV insulation systems are partial discharge (PD) measurements. The interpretation of such measurements aims at extracting from the measured data information about insulation defects which then are used for estimating the risk of insulation failure of the equipment. Because the physical understanding of PD has made substantial progress in the last decade, it can now be exploited to support interpretation. In this paper a concept is presented which merges the available physical knowledge about various PD types into a generalized model which can be applied to arbitrary insulation defects. This approach will be restricted to PD of the streamer type in gases and at gas-insulator interfaces which cover a large fraction of the cases encountered in technical insulation systems. The generalized model allows us to derive approximate relations between defect characteristics, insulation design parameters and test conditions on one side, and measurable PU characteristics on the other. The inversion of these relations yields rules for extracting defect information from the PD data. The application of the generalized model is illustrated by two simple examples, namely, spherical voids in an insulator and electrode protrusions in SF6.

1. INTRODUCTION ARTIAL discharges (PD) in insulation systems com- P prise a large variety of physical phenomena, ranging

from low intensity phenomena like charge carrier emission from surfaces over leakage currents along insulator surfaces, glow discharges] sub-critical avalanching, and charge carrier injection into liquids and solids over phenomena of medium intensity such as electric treeing and streamers, to intense discharge types such as leaders, sparks, and partial arcs. Most of these discharge types contribute to insulation degradation and some of them may trigger breakdown. Understanding the physics of PD phenomena therefore has become a major field of research and is being implemented as an important tool for the interpretation of PD measurements, i.e. for the

identification of insulation defects, their quantification, and the assessment of the risk of failure they comport.

The pioneering work on PD physics in the sixties and early seventies (e.g. [l-41) mainly concentrated on understanding void discharges. From the late seventies on- ward, substantial progress was made in also understanding other P D mechanisms [5-131 and in developing meth- ods for their quantitative numerical simulation [14-201. A further group of publications provided the basis for relating the physical processes to measurable PD signals [21]. A basis has thus been prepared on which a generalized approach to PD modeling can be attempted. The physical background knowledge on PD can be subdivid- ed in three fields: PD models, aging and degradation models, and breakdown models.

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IEEE Transactions on Dielectrics and Electrical Insulation Vol. 2 No . 4 , August 1995 till

Table 1 Symbols. Defining equations are given in parenth

sxis of the equivalent ellipsoid parallel to the background field E, Emitting surface area (8) pararnrtrr in approximation for effective ionization coefficient (13) Parameter in volume ionization law (7) rlemcntary charge electric field average field erillancement due t o applied voltage U,

field along channel of propagating streamer (14) background field at (and in the absence of) the defect

peak background field at defect location during ac cy - cle (55) field change inside void (32) dimensionless streamer inception function (19) dimensionless factor for apparent charge (37) Boltnmann cmstant Dimensioriless function (32) streamer propagation length (26)

(3)

( 1 )

parameter in streamer criterion (46) first electron production rate (6) physical (<true') charge (30) minimal charge (55) maximal charge (57) defect scale perpendicular to background field E, characteristic radius of conducting surface (42) function in surface emission law ( 8 )

average PD inception delay (47) ac half period voltage applied to insulation system (1) potential difference driving PD ( 2 ) potential difference due to space or surface charge (4) potential collapse by PD (29) residual potential difference at end of PD (25) effective ionization volume (6) distance at which E = E,, (19) surface ionization coefficient exponent in approximation for effective ionization coefficient (13) relative permittivity of dielectric gas attachment coefficient factor in volume ionization law (7) surface conductivity (42 ) dimensionless factor for streamer propagation (26)

ionic mobility (40) gas density drift decay time constant (39) surface conduction decay time constant (41) effective work function (8) ratio of applied voltage to inception voltage (50)

The present paper will concentrate on the first field and will be restricted to discharges in gases and along gas- insulator interfaces. For the modeling of PD in insulating liquids and solids the reader is referred to recent reviews such as [22] and [23]. The main objective of this paper

axis of the equivalent ellipsoid perpendicular to the background field E , parameter in streamer criterium (46) constant relating ionic mobility pi to gas pressure (40)

parameter in Richardson-Schottky emission law (9) dimensionless field distribution (17) average field enhancement a t defect (5) average field enhancement due to space/surface charge

= ( E / P ) ~ ? ~ critical field (13) average field at streamer inception (53)

background field a t streamer inception (21)

(4)

dimensionless average field enhancement factor (3) dimensionless factor for true charge (30) surface emission current density ( 8 ) logarithm of critical electron number in avalanche (16) defect scale parallel to background field E , scale of conducting surface in the direction of current Bow (42) number of detrappable charge carriers (10) gas pressure reduced ('apparent') charge (36) minimal induced charge (58) stored charge (4) radius of spherical void spatial co-ordinate time after PD event temperature reducedpotential (1) PD inception voltage (24) potential difference due to background field potential difference driving PD at inception minimal potential collapse (54) ion drift velocity (40)

3)

. . spatial co-ordinate along streamer path (16) gas ionization coefficient effective ionization coefficient (13) ratio of streamer channel field to critical field (14)

permittivity of vacuum function controlling volume ionization (6) surface attachment coefficient limit value of surface conductivity (59) dimensionless scalar field describing coupling of defect to PD measurement electrode (35) fundamental phonon frequency (10) surface charge distribution on ellipsoidal void decay time constant for detrappable charges (12) ac phase ionizing quantum flux density (7) proportionality factor (11)

is to provide a unified representation of a large number of seemingly different discharge phenomena by exploiting the following facts. 1. Although defects in insulation systems may exhibit a

large variety of shapes, the PD activity associated with

512 Niemeyer: A Generalized Approach to P D Modeling

them has many features in common, from a physical point of view. This allows a systematic defect classification, a unified definition of defect parameters, and a generalized modeling procedure.

2. While the sensitivity and accuracy of PD measurements in technical equipment is usually limited, the precision of the model needs not be high so that, simplifying approximations can be generously used in the physical models.

3. Frequently, the derivation of physical relations can be simplified considerably by using dimensional reasoning which consists in establishing dimensionally cor- rect proportionality relations between the quantities involved .in the model. The quantitative details of the specific defect geometry and discharge structure are then accounted for by dimensionless proportionality factors which can be estimated or determined from experiments or derived from detailed (mostly numerical) discharge models. The paper is organized in the following way. After

a brief general discussion of PD types in Section 2 , a physically based defect classification scheme is proposed and ‘universal’ defect parameters are defined in Section 3. Section 4 to 6 describe the elements of the generalized PD model, namely, the production of first electrons which control the statistical PD characteristics (Section 4), a model of the streamer process (Section 5), and the quantification of the charges associated with the PD (Sec- tion 6). The application of the model is then illustrated in Section 7 by two examples, namely, spherical voids in solid insulators and conducting electrode protrusions in SF6. Section 8 gives some conclusions.

2. GENERAL CLASSIFICATION OF PARTIAL DISCHARGE

MECHANISMS The PD activity caused by insulation systems defects

may be roughly classified into three categories, namely, discharges in gases and a t gas-insulator surfaces, discharges in liquids and a t liquid-solid interfaces, and discharges in solids (charge injection, electrical trees and water trees).

In this paper, only the first category will be discussed. Table 2 shows a summary of the major gaseous PD types together with a characterization of their main features, their detectability in technical equipment, and their sig- nificance for insulation aging and failure.

The weakest discharges are the emission of charge carriers from surfaces and leakage currents along weakly conducting insulator surfaces. Examples are thermion- ic or field emission from cathodic conductors [24], charge

carrier detrapping from insulator surfaces [25,26], current flow along insulator surfaces [27,28,30], and glow discharges [14]. These discharge types are characterized by very low and quasi-continuous currents which are un- detectable by the usual pulsed PD measuring techniques. Nevertheless they may induce breakdown by the accumu- lation of field distorting charges [29] , thermal runaway [30], and other effects.

A further class of discharges with relatively low intensity comprise such phenomena as Townsend discharges [5,13,31] and swarming partial microdischarges (SPMD) [6]. Although of pulsed nature, their intensity is usually so low that they can only be observed in highly sensitive laboratory experiments and cannot be detected by standard P D measurement techniques in technical equipment. It can be noted from Table 2 that not all discharge types that may be relevant for insulation aging or breakdown are necessarily detectable in technical equipment.

The discharge type to be discussed in this paper is the streamer [5,15,17-191. I t plays a particular role for two reasons: Firstly, it normally causes signals on a level which can be detected in technical equipment. Secondly, it is in many cases the necessary prestadium of more intense discharge types like leaders, sparks, and ultimate insulation breakdown. Transitions of this kind occur if the streamer discharge exceeds critical intensity levels [12,15,16], an issue that will not be further discussed in this paper.

3. DEFECT CLASSIFICATION AND CHARACTERIZATION

3.1. D E FE CT CLASS I F I CAT I 0 N The defect classification scheme proposed in this paper

is based on the nature of the two major boundaries limiting the discharge. The first boundary limits the PD in the direction of the electric field and is represented along the horizontal co-ordinate of the classification matrix in Figure 1. It ranges from both boundaries being conductors (labeled ‘cond-cond’), to both boundaries being insulators (labeled ‘insul-insul’). The second classification criterion is the degree to which solid insulator surfaces parallel to the electric background field E, contribute to the discharge processes. It is represented along the ver- tical co-ordinate in Figure 1 and ranges from negligible (i.e. the discharge develops essentially through the gas without surface contribution) to dominant (i.e. the gas discharge is dominated by surface phenomena). Elec- tric trees can be considered an extreme case of the latter type [8,12]. Any specific insulation system may contain a subset of the defect matrix in Figure 1 which is determined by such factors as design, manufacturing, assem- bling, and service stress history.

IEEE !Z'ransactions on Dielectrics and Electrical Insulation Vol. 2 No. 4 , August 1995 513

Table 2. Survey of gaseous PD types caused by insulation defects with order-of-magnitude characterization, estimates of detectability, an'

PI3 type

surface emission surface conduct.

Townsend, SPMD streamer leader spark partial arc

glow

relevance for aging and failure. order of magnitude

contin.,lO-10 . . . A/m2 con t in . , 10 -~~ . . . IO-* S contin., - 10 A/m2 pulsed, - 100 ns pulse, 1 . . . 100 ns, > 10 pC pulse, 0.01 . . . 1 ms, 1 . . . 10 A 0.001 . . . 1 ms, 0.1 . . . 10 A 0.1 . . . 1 s , 1 . . . 1 0 A

cond. - cond. cond. - gas cond. - insul.

PD detect

relevance for failure

insul. - gas insul. - insul.

Figure 1. Defect classification matrix.

In compressed SFG insulation the defect types a, b, d , f and g may be encountered, a 2 , b, f and g being related to conducting particle contamination and a1 to conductive components a t floating potential. In bulk resin insulation such as spacers, machine insulation, or dry-type transformer insulation, defects of types c, e, and k to o may occur, c, k, 1 and m pertaining to conductor-resin interfaces and e, n and o to voids enclosed in the resin. In mixed insulation systems such as bushings the defect types h and i may occur in addition to those of type c, e, and k to 0 . In outdoor insulation exposed to atmospheric pollution, aggregates of conductive substances such as salt fog or rain droplets may cause PD of type j .

3.2. DEFECT PARAMETERS The parameters characterizing a defect should be cho-

sen such that, only the major scales of the defect are quantified whereas details of the defect geometry are ig- nored; that the local background field E, a t the defect location (as determined by the design of the insulation system) is accounted for quantitatively; and that the materials involved in the PD process, i.e. gas, solids, and the interfaces between them are characterized by a minimal number of adequately chosen integral parameters.

A set of parameters fulfilling these requirements is listed in Table 3. The parameters in the first group are ge- ometrical in nature and characterize the location of the


category geometry

bulk materials

surface

defect within the insulation system and its size. The defect location is characterized by two dimensionless scalar field quantities U , and A,. The first one is the electrostatic potential U a t the defect site normalized to the applied voltage U,, i.e. U , = U/U,. It quantitatively de- termines how the design of the insulation system controls the electric field a t the defect location. The product

E, = U,Vu, (1)

gives the background field E, a t the defect (in the absence of the defect) when a voltage U , is applied to the system. The quantity Au, = E,/U, will be referred to as the reduced background field. The dimensionless scalar field U , is obtained by solving the Laplace equation with the boundary conditions U , = 1 a t the HV electrode and U, = 0 a t the grounded electrode (s).

Table 3 Survey of generalized defect parameters and their controlling factors.

Parameter Symbol Factors background field Vu0 = Eo/Uo design PMC function vx0 design defect scale 1 1 Vu0 1 manufact. defect scale I Vu0 T quality permittivity cl-

pressure P materials gas ionization surf. ionization electron emission 9, Xi, 7 d t materials

( E l P ) c T , B, n, Y

surf. conduct. n,

The third group of parameters characterizes the interfaces involved in the discharge and includes the modi- fication of the gas ionization characteristics by surface contributions, charge emission from surfaces, and charge transport along surfaces. They will be discussed in detail in Section 4.2. These interface da t a are not only determined by the choice of materials but can also be affected by various surface physical and chemical processes. This makes them difficult t o quantify theoretically and normally requires experimental data.

3.3. FIELD ENHANCEMENT AT T H E DEFECT P D is driven by a local field enhancement caused by

the defect. This enhanced field is, in general, composed of two contributions. The first one is due to the applied voltage U , and is the local enhancement of the background field E, by the defect. The second one is associated with local space or surface charges tha t have been left by previous P D activity. Both contributions can be integrally expressed by the potential differences they cause over the P D gap

AU = A U , + A U , (2) where AV is the total potential difference available across the P D gap to drive the discharge. The contribution AUa can be dimensionally related to the background field E, and to the defect scale 1 in the direction of E, by

A U , = fE, l= f l U , V u , = E,l (3)

Here f is a dimensionless factor which integrally accounts for the field enhancement by the defect and E , = AU,/l is the field enhancement averaged over the PD gap. The factor f has to be determined by averaging the distribution of the enhanced field in the P D gap, the latter having to be obtained by solving the Laplace equation with the boundary conditions given by the defect geometry and under consideration of the relative permittivity of the surrounding dielectric. Examples for such solutions for different defect types can be found in [33-351.

The contribution A U , by a local space or surface charge q s can be dimensionally expressed as

(4)

where E, = 8.85 x 10l2 F / m is the permittivity of vacu- u m and g is a dimensionless proportionality factor which accounts for the defect geometry and the spatial distribution of the charge and will be discussed in Section 6.1.

With Equation (3)and (4) a total average field E across the P D gap can be defined as

- - + AUq E = E , + E , = 1 (5)

IEEE Transactions on Dielectrics and Electrical

DT PE 11 PE FE

a b Figure 2.

First electron production mechanisms. ume production, (b) surface production.

(a) vol-

4. INITIAL ELECTRON GENERATION A necessary but not sufficient condition for the occur-

rence of a P D is an initiatory ‘first’ electron to start the first avalanche of the ionization process. The supply of such first electrons controls the statistical characteristics of the P D activity such as inception delay, frequency of occurrence, and distribution with respect to the phase of the applied ac voltage. Two main groups of first electron generation mechanisms can be distinguished, namely, volume and surface processes (Figure 2).

4.1. VOLUME GENERATION

Volume generation (Figure 2(a)) includes radiative gas ionization by energetic photons (PI) and field detachment of electrons from negative ions (FD). In both cases the rate a t which first electrons are produced a t the PD site scales as

N e = Ti(gas7 E , . . )Pv,,, (1 - %) (6)

where p is the gas pressure and V.,, is the effective gas volume exposed to the radiation and t o the field. The function describes the particular ionization mechanism and generally depends on the kind of gas, the electric field E , and further parameters. The last term is the Legler function [36] which gives the probability that a single eIectron develops into an avaIanche. In the case of PD the local field in which the avalanches develop exceeds the critical field of the gas by far (see Section 5.2 and Figure 5). Under this condition a >> r] so that the Legler function is close to unity. Hereafter, it will therefore be assumed to be equal to one. For the case of radiative ionization it was shown in [26] that 77, has the form

%rad Crad@rad(P/p)o (7)

where C T a d characterizes the interaction between radiation and gas, @rad is the quantum flux density of the

Insulation Vol. 2 No. 4 , August 1995 515

radiation, and ( p / ~ ) ~ is the pressure reduced density of the gas.

Radiative ionization plays a role in voids enclosed by insulating materials such as the defect types e, n and o in Figure 1. An example will be given in Section 7.1.

For the case of field detachment from negative ions the function 7, is proportional to the negative ion concentra- tion in the gas and is a strong function of the electric field E [32]. This process controls the P D statistics a t defects of the types a, b, f and g in Figure 1 when the high field electrode is a t positive polarity.

4.2. SURFACE EMISSION First electron production from surfaces (Figure 2(b))

includes field emission from cathodic conductors (FE), detrapping of electrons from traps a t the insulator surface (DT), electron release by ion impact (11), and by the surface photo effect (PE) a t both conducting and insulating surfaces. Experimental data suggest that the surface emission process approximately obeys Richardson- Schottky scaling. The rate i’?, at which successful electrons are emitted from a surface of area A would then scale as

. A 7 7 . Ne = -(1- e -)I CY

Here e is the elementary charge, j the emission current density, @ a n effective work function, E the electric field a t the emitting surface, k the Boltzmann constant, and T the temperature. The lowering of the work function by the electric field E is accounted for by the Schott- ky term d m in the exponent of the e-function. The function S characterizes the specific surface material, structure, and state.

familiar thermoelectric emission law as At a conducting surface the function S is given by the

s CthT2 (9)

where Cth = 1 . 2 ~ 1 0 ~ Am-’K-’ is an universal constant.

Figure 3 shows a Richardson-Schottky plot of measured emission current densities. The full data points were measured a t a technical aluminum surface in SF6 [24] and the open points a t a weakly conducting Pyrex? surface [25]. These data suggest that the effective work function Q, is in the range 1 to 1.3 eV. These low values are consistent with the thermal nature of the emission process. Figure 3 also demonstrates that the emission current density is highly sensitive to the exact value of the effective work function Q, i.e. to the material and


1 2 5 10 20 50 100 E ( kV / mm ) Figure 3.

Richardson-Schottky plot of surface emission current density j in dependence of electric field E . T = 300 K. Full points: Measurements of technical aluminum surface in SFe. Open points: Mea- surements of glass surface in air.

state of the emitting surface. Because surfaces in technical objects are normally not well controlled in a surface physical sense, it can be expected tha t the statistical characteristics of the P D activity when controlled by surface emission processes may exhibit strong scatter and variations.

Surface emission can be expected to play a dominant role when conducting or weakly conducting surfaces de- limit the defect, e.g. for the defect types a, b, c, f, g, h, k, 1 and m in Figure 1. In the case of insulator surfaces, an emission process similar t o the above one may be expected: It is known that a t the surface of insulators traps with depths of the order of eV are present [27,37,43]. If electrons or ions originating from a P D have been trapped in such surface states they may be detrapped by a process

of detrappable surface charges is reset a t each P D event in proportion to the deployed surface charge q and is de- pleted between the P D events by loss mechanisms such as emission, diffusion into deep t r a p , and other processes. The latter are described by an exponential decay time constant ‘j-dt. The detrappable reservoir Ndt is then controlled by the two rate equations

4 Ndt = E -

a t each PD event, and

bctwcen P D cvents, where ( is the fraction of the P D charge q which is stored in detrappable states. It will be shown in Section 7.1 tha t this simple concept describes the observed statistical characteristics of P D in resin- enclosed voids quite well if the three parameters (, P, and Tdt are properly chosen.

Detrapping of charge carriers from insulator surfaces can be expected to affect the P D statistics at defects of type c, d , e, h, i, j , 1, m , n and o in Figure 1. It may also play a role a t defects of types f and g.

Figure 4 Elementary processes controlling streamer propagation along an insulator surface. a, ??: gas ionization and attachment. ad! vs: surface ioniea- tion and attachment.

5. MODEL OF THE STREAMER which, due to the order-of-magnitude of the trap depth, should also obey Richardson-Schottky scaling. The function S would, in this case, have the form [26] PROCESS

Ndt S = v , e A 5.1. BASIC FEATURES AND PARAMETERS OF STREAMERS

(10)

where to s - l is the phonon The streamer process in gases is a self-channeling ion- frequerlcy and Ndt is the surface density Of detrappable ization phenomenon driven by a space charge distorted

electric field and controlled by the interplay of ioniza- charge carriers.

The number Ndt of detrappable charges a t a PD site can be determined approximately by the following phe- nomenological balance consideration: The reservoir Ndt

tion and electron attachment [5,15,17,18]. If a streamer propagates along an insulator surface, the ionization and attachment coefficients a and r ] of the gas are modified

517 IEEE Transactions on Dielectrics and Electrical Insulation Vol. 2 No. 4, August 1995

by contributions CY, and qs from the surface as schemat- ically represented in Figure 4 [19]. Electron avalanches are started by photoionization processes both in the gas (a) and a t the insulator surface (os). In the case of positive (cathode directed) streamers they propagate towards the positive ionic space charge in the streamer head, neutralize i t , and drift along the streamer channel to the electrode. During their drift they may be attached to either gas molecules ( q ) or to the insulator surface (q5). At negative polarity qualitatively similar processes occur with quantitatively different characteristics. A simplified model of the positive streamer propagation was first described in [15]. Detailed numerical simulations in various gases were published later [17,18]. An attempt to simulate the propagation of a streamer along an insulator surface is described in 1191. For the purpose of approximate PD modeling, the streamer process will be quantified in the following way.

Ionization is controlled by the effective ionization coefficient 6 = a-q, i.e. the difference between the ionization coefficient CY and the attachment coefficient q. For a given gas, the dependence of 6 on the gas pressure p and the electric field E is approximated by a power law

with the three parameters C , ( E / P ) , ~ , and p characterizing the gas or gas-surface combination. (Elp), , is the pressure reduced critical field a t which CY = q. The proportionality constant C and the exponent /? characterize the steepness with which 6 increases with E once E,, is exceeded. Numerical values for the parameters C , ( E / P ) , ~ and p are known for many gases but their modi- fication by insulator surfaces has, hitherto, only been determined approximately for a few surface types [19] and remains to be measured for most surfaces of technical interest.

An important feature of the streamer process is the field Ech which is established along the streamer channel during propagation. It is determined by a complex interplay of ionization, attachment, and charge carrier drift [15,18] and is found to be approximately proportional to the critical field E,,

Ech E yEcr (14)

The dimensionless proportionality factor y depends on the gas or gas surface combination and on the streamer polarity

y = y(polarity, gas, interface) (15) Some values for the parameters C, p, (Elp)., and y are listed in Table 4.

They were taken from [l l , 15,381.

5.2. STREAMER INCEPTION

The necessary condition for streamer inception is the well known critical avalanche criterion [15,39]

T & [ E ( z ) ] d z 3 K,, (16) 0

where E ( z ) is the field distribution along the streamer path and Kc, is the logarithm of a critical number of electrons that has to accumulate in the avalanche head to make the avalanche self propagating by its own space charge field. Values for K,, taken from [42] are given in Table 4. The integral is to be extended over the distance x,, within which the effective ionization coefficient 6 exceeds zero, i.e. within which the field E ( z ) exceeds the critical value E,,, Equation (13).

In the absence of space or surface charges E ( z ) is proportional to the applied background field E , and can be expressed in the normalized form

E ( z ) = Eoeo(z / l ) (17)

where e o ( z / I ) is a dimensionless field distribution function which is determined by the defect geometry. The reduced co-ordinate (./a) = 0 denotes the starting point of the avalanche. The integration range z,, of Equa- tion (15) is determined by the condition

the inversion of which yields

( ~ / l ) , ~ = e i l ( E o / E c r ) (19) where e;' is the inverse function of e , ( z / l ) . With Equa- tion (13) , (17) and (19), Equation (16) can be put into the dimensionless form

This is an implicit relation between the ratio Eo/E,,, the product p l , the defect geometry as represented by the dimensionless field distribution e o ( z / I ) , and the gas parameters (Elp), , , C, p, and K,, . Solving this relation for E,/E,, yields the streamer inception criterium in the general form

( E , / E , , ) ~ ~ ~ = E?'/E,, = F[(pI), gas surf,E,,defect geom] (21)

F is a dimensionless function of the product pl which moreover depends on the properties of the gas or gas- surface combination, on the relative permittivity E, of


Table 4. Streamer relevant parameters of some gases and gas-insulator interfaces.

89

the dielectric in the vicinity of the defect, and on the defect geometry.

1 n u 9.

L v

0.1

1 - 1 I t 0.01 L ' I 1 1 I 1

10 i o 2 1 o3 1 o4 ( p e t ) Pasm Figure 5.

Dimensionless streamer inception function F according to Equation (21) in dependence of ( p i ) . Upper curves: Void defects with air and SFe (geometry factor f = l). Lower curves: electrode protrusions in SFe.

Figure 5 shows the function F in dependence of the product pi for conducting electrode protrusions in SF6 (defect type b in Figure 1) and for ellipsoidal voids enclosed in dielectrics (defect types e and o in Figure 1). In the first case, a protrusion in SF6, the defect geometry is represented by the aspect ratio 1/r of the protrusion [34]

In the second case, an ellipsoidal void, F has the form

(23)

as will be shown in Section 7.1. Here, the defect geometry enters via the dimensionless field enhancement factor f defined in Equation (3). In Figure 5 the factor f has been set equal to unity for simplicity.

The P D inception voltage U:"" is obtained from E:tr by making use of Equation (1) and results, after inserting Ecr = ( E / p ) c r p , in

U:"' = ( E / p ) , , p F [ ( p l ) defect geom]/Vu, (24)

It is seen that for known defect geometry, gas, and pressure p the inception voltage U:,' is only determined by the defect scale 1 and the reduced background field Vu, = Eo/Uo a t the defect location. Hence, a measurement of the inception voltage provides 'mixed' quantitative information about the defect scale 1 and the reduced background field Vu, in which the defect is embedded.

5.3. STREAMER PROPAGATION

The field E,h in the channel of the propagating streamer has two influences on the P D process. Firstly, it de- termines the residual value AVr,, to which the voltage across the P D gap breaks down a t the end of the PD. With Equation (14) one obtains

AUT, , E Echl Y(E/P)cr (P l ) (25)

Secondly, Ech controls the maximal length l J t r to which streamers can propagate with an available potential difference A U across the P D gap if no obstacles limit their propagation. It results

l s t r M XAU/Ech (26)

where X is a dimensionless factor characterizing the defect geometry and the discharge structure

X = X(defect geometry, discharge structure) (27)

X is found to be in the range 0.4 < X < 1; the lower limit pertaining t o streamers propagating through the gas [ll] (e .g . defect types a, b, c and e in Figure 1) and the upper limit to sliding discharges along thin insulator layers [38] (e.g. defect type h). With Equation (3), (26) can be written as

lstr ( X / ~ ) f ( E o / E m ) l (28)

which relates the streamer propagation length l s t r to the defect scale 1.

5.4. DISCHARGE INTERFACES T O BO UN DARl ES

The nature of the boundaries limiting the P D perpen- dicularly to the direction of the background field E, was chosen as one of the classification cr i ter ia i n Figure 1. Four basic discharge interface modes to this boundary were distinguished, namely the interface to a conductor

IEEE Transactions on Dielectrics and Electrical Insulation Vol. 2 No. 4, August 1995 519

(‘cond’); ‘open end’ of the discharge in the gas (‘gas’); landing of the discharge on an insulator surface (‘cond- insul’); and starc of the discharge from an insulator surface (‘insul-insul’).

Discharge attachment modes. (a) ‘Landing’ on an insulator surface. (b) Start from an insulator surface.

The first two modes are familiar and will not be discussed further. In the third mode (Figure 6(a)) a streamer having started from a conducting electrode (1) propagates through the gap (2)’ impinges on an insulating ‘electrode’, and continues to propagate along its surface as a surface discharge (3) until it reaches its maximal lateral extension (4). This interface mode occurs a t defects of types c and m in Figure 1. A numerical simulation of this discharge type was described in [ZO].

In the fourth mode (Figure 6(b)) the discharge starts from the cathodic insulator surface with an initial avalanche (I). It then develops into two discharges propagating simultaneously and in opposite directions into the gap and along the insulator surface (2)’ the surface discharge capacitively providing the current feed for the gap discharge. Upon reaching the opposite insulating ‘electrode’ (3)’ the gap discharge forms another surface attachment of the type shown in Figure 6(a). Both surface discharges propagate to their maximal extension (4). This discharge interface mode occurs a t defects which are completely enclosed in insulating material such as the types e, i, n and o in Figure 1. Experimental evidence for this discharge mechanism was presented in [lo]. A detailed numerical simulation has not yet been attempted.

The maximal extension t o which the discharges propagate along the insulator surface can be roughly estimated with the help of the streamer propagation criterion Equation (28). If streamers of both polarity propagate simultaneously like in Figure 6(b) the average of both

polarities has to be used for the factor 7, as a first ap- proximat ion.

(b) I

Figure 7. Dipolar charge distribution deployed by a PD. Real distribution (a ) and equivalent ellipsoidal void (b).

6. PDCHARGE 6.1. PHYSICAL CHARGE

Viewed in an integral manner PD causes the collapse A U ~ D of a potential difference Ai7 across the PD gap to a residual value AU,,,

(29) A u p D = nu - AUT,,

This potential collapse is associated with a charge sepa- ration which results in the deployment of a dipolar space and/or surface charge distribution. The latter can be integrally represented by two equal charges iq of opposite sign separated by an average distance d (Figure 7(a)).

The value of these charges can be dimensionally related to the defect scale I and to the potential collapse A v p ~ by

4 = z k g T E , l A u p D (30) where E , is the vacuum permittivity and g a dimensionless proportionality factor which integrally accounts for the form of the charge distribution, the defect geometry, and the influence of the relative permittivity E, of the dielectric in the vicinity of the defect. The charge q is the physical (‘true’) charge that has flown through the PD. It controls the damage induced by the P D and therefore is responsible for insulation aging. It also is that quantity which controls the transition of a streamer PD to more intense discharge types and to breakdown.

Equation (30) gives a relation between a potential difference A U and a dipolar charge distribution iq. It thus provides, in particular, the potential difference A U , in a P D gap caused by a dipolar space or surface discharge +qs that has remained from a previous PD. This is the relation that was given without proof in Equation (4).


In order t o quantify the dimensionless proportionality factor g for a specific defect and P D structure, the Pois- son equation has to be solved for a prescribed change A U ~ D of the potential difference across the PD gap. The only defect type for which this problem has been treated hitherto are ellipsoidal voids in a dielectric having the axis of rotation parallel t o the electric background field E, [21,33,40]. This special defect type will be used as an equivalent structure to describe arbitrary defects. This concept is illustrated by Figure 7( b) where the space/surface charge distribution of Figure 7(a) is represented by an equivalent surface charge distribution a ( 3 on the inside of an ellipsoidal void. The actual defect is thus assigned the two equivalent ellipsoidal axes a and b.

(4 (b) (4 Figure 8

Examples for the approximate assignment of equivalent ellipsoidal axes a and b for some defect types.

The assignment of these axes to arbitrary defects is illustrated by the three examples in Figure 8: In the case of voids which are prolate in the direction of the background field E,, the assignment of a and b is straight- forward (Figure 8(a)). In the case of very oblate voids (Figure 8(b)) the axis a is assigned half of the defect scale a = 112. For assigning the axis b it has to be considered that the extension of the discharge perpendicular to E, is not limited by the lateral extension T of the defect but by the shorter lateral propagation length l s t r of the discharge. The axis b can therefore be identified approximately with the streamer propagation length l,tr as given by Equation (28), b FZ l s t r . If the defect is adjacent to an electrode as in the case of the electrode protrusion in Figure 8(c), the space charge +q deployed ahead

electric field E inside the void by the value

(31) A U P D A E = -

2a The solution of this problem [33,40] can be expressed in the form

kq = ~e ,b ’ [ l+ er (K(a /b ) - l ) ]AE (32)

Here, K(a /b ) is the inverse of the polarization factor and is a dimensionless function of the axis ratio a/b. A plot of K(a /b ) and a detailed discussion are given in [21]. For the purpose of order-of-magnitude estimates K can be roughly approximated for oblate ellipsoids, sphere, and prolate ellipsoids by

- 1 a / b < 1

K { = 3 a / b = 1 (33) - 4a/b 1 < a / & < 10

Replacing A E in Equation (32) by AUPD from Equa- tion (31) and introducing the ratios a/b and al l , one obtains by comparison to Equation (30) the dimensionless geometry factor g as

6.2. INDUCED CHARGE

The dipolar charge distribution deployed by the PD induces charges on the conductors of the insulation system which will be denoted by q’ and will be referred to as induced charges. They cause signals to travel from the defect location to that part of the conductor to which a PD measurement device is coupled. The response generated in this device produces the measured PD quantity and is usually referred to as the apparent charge. If the signal transmission from the defect location to the coupling device is loss free, and if the signal is adequately processed, the measured apparent charge is equal to the induced charge. In this paper the signal transmission will not be considered and only the induced charge will be discussed.

According to Equation (19) in [21], the induced charge q’ associated with a field change A E in an ellipsoidal void is

q’ = - ~ ~ ~ E , ~ ~ ~ E , K A E V A ~ (35) of the protrusion tip induces a mirror charge -4 in the electrode. In this case the axes of the equivalent ellipsoid become a - I and b - I s t r . Here VA, is the gradient of the dimensionless scalar field

A, which characterizes the coupling of the defect location Once the axes of the equivalent ellipsoid have been as-

signed, the charge transfer kq associated with the PD can be obtained by integrating the surface charge distribution u ( q which is required to cause a voltage breakdown A U ~ D inside the void, i.e. to change the uniform

to the electrode a t which the induced charge is measured. As pointed out in detail in [21] the field A, is obtained by solving the Laplace equation for the defect free insulation system with the boundary conditions A, = 1 a t the measuring electrode and A, = 0 at all other electrodes.


VX, will be referred to as the PMC (Pedersen McAllister Crichton) function throughout this paper.

In technical insulation systems the PD signal is frequently measured a t the HV electrode of a two-electrode system. The boundary conditions for A, are then iden- tical with those for the reduced background potential U, (Equation (1)). In the special case that q' is measured a t the HV electrode of a 2-electrode system

VX, = vu, (36)

Introducing the dimensionless ratio a l l into Equation (35) and comparing the latter to Equation (32) the ratio between induced and true charge becomes

q ' / q = g'lVX, (37) with the dimensionless proportionality factor

(38) 4 €,K(a/b) a

q ' (a /b ,~ , ) = - 3 1 + &,(K(a/b) - 1 ) 7

The ratio q ' / q is thus proportional to the dimensionless product lVXo of the defect scale and the PMC function a t the defect location and the proportionality factor g' depends on the two dimensionless shape parameters a/Z and a/b and on the relative permittivity E, of the dielectric.

Equation (37) is a key relation for the interpretation of PD measurements. It relates the physically significant but immeasurable true PD charge q to the measurable induced charge q' , thus providing a quantitative link between a measurable PD magnitude and a physical PD quantity.

(b) (C) Figure 9

Charge decay mechanisms. (a) Decay of space charge in gas by ambipolar ion drift. (b) and (c) decay of surface charge by ion drift (full arrows) and surface conduction (broken arrows).

6.3, CHARGE DECAY AND MEMORY EFFECTS

The charges deployed by a PD have a finite lifetime during which they constitute a memory of the PD event in two ways. They contribute to the potential difference AV across the PD gap by the associated field (term AU, in Equation (2)) and they are sources for some of the first electron generation processes discussed in Section 4. The

charges deployed by the PD can decay in three different ways depending on the defect type and on the properties of the materials and interfaces a t the defect location (Figure 9).

At defects of types a, b and g in Figure 1 the PD produces space charge clouds q in the gas (Figure 9(a)). These consist of highly mobile positive and negative ions which decay by ambipolar drift in the field. The associated decay time constant 'rdr is of the order of the time the ions need to drift along the characteristic length scale l S t r of the ion distribution, i.e.

Tdr l s t r / V d i (39) where the product of the ion mobility and the electric field E

is the ion drift velocity. The latter is given by

= & E (40)

Ion mobilities in gases scale approximately inversely proportional to the pressure pi = C J p . With fields of the order of the streamer channel field E - Ech - -,E,, (Equation (14)), with streamer propagation lengths l,,, according to Equation (28), and field enhancement factors f of the order of unity one obtains an order-of- magnitude for the drift decay time constant

For a given gas (represented by the parameters in the square brackets) t d T is seen to be essentially proportional to the field ratio (E,,/Ecr) and to the defect scale I.

In another group of defects the P D forms surface charges on insulators along the discharge path (Figure 9(b)) or perpendicular to it (Figure 9(c)). Examples are the defect types f, g, h, i? 1, m and o in Figure 1 and c and e, respectively. In these cases charge decay may occur, in addition to ion drift through the gas, by conduction along the insulator surface (broken arrows in Figure 9(b) and (c)). The corresponding RC decay time constant is of the order

where IC, is the surface conductivity, L the scale of the conducting surface in the direction of the surface current flow, and rc an equivalent radius of the circumference of the conducting surface. The first and second square brackets are the order-of-magnitudes of a surface resistance and a capacitance, respectively. The scales L and rc have to be determined from the geometry of the conducting surface. For the configuration in Figure 9(b), L and T , are of the order of the streamer propagation length Zstr. For the geometry of Figure 9(c), L and T, are of the order of the longitudinal and lateral defect scales 1 and


T , respectively. An exact treatment of the surface conduction in spherical voids can be found in [41].

Surface emission, ion drift through the gas, and conduction through the bulk of the dielectric additionally may contribute to charge memory decay but will not be discussed here.

7 . EXAMPLES The generalized PD model outlined above can be ap-

plied to PD of the streamer type a t arbitrary defects. It yields relations between the insulation design characteristics, the test conditions, and the defect parameters on one side and the measurable PD characteristics on the other. These relations quantitatively involve the test conditions such as the applied voltage U, and the gas pressure p , the design related quantities such as Vu, and VX,, and the material properties such as the ionization characteristics of the gas or gas-solid interface and the relative permittivity E, of solid insulation materials.

The application of the model will be illustrated by dis- cussing two defect types, namely spherical voids embedded in insulating material (defect type e in Figure 1) and oblong electrode protrusions in SF6 (defect type b).

7.1. SPHERICAL VOIDS IN INSULATING M AT E R I A LS

Spherical voids in insulating materials are gas filled bubbles generated by faulty processing. Their characteristic parameters are their diameter 1 = 2R, the contained gas and its pressure p , and the relative permittivity E, of the surrounding dielectric.

In the absence of surface charges the potential difference driving the PD is given by Equation (3) as

A U = fE,l = flU,Vu, (43)

The geometry factor f has the value [33]

3E, f=- 2~ + 1 (44)

In the absence of space or surface charges the field inside the void is uniform so that the dimensionless form of the streamer inception criterion Equation (20) can be evaluated with a uniform reduced distribution .,(./I) = 1. The dimensionless function F in Equation (19) then is

(45)

as given in Equation (23). The parameters B and n are related to the gas properties C, ( E / P ) ~ , , ,O and K,, in

n = - P

With Equation (45) inserted into (21), the background field above which streamer inception can occur becomes

The corresponding P D inception voltage becomes, with Equation (24)

The inception voltage U:"' is thus seen to provide quantitative information on the void diameter 1 and the reduced background field Vu, = E , / U , a t the void location if the gas in the void and its pressure p are known.

The P D inception delay, i.e. the waiting time for a first electron in a virgin void, is controlled by the ionizing effect of cosmic and radioactive radiation if no artificial irradiation (e.g. by X-rays) is provided. In spherical voids with their small surface-to-volume ratio, volume ionization in the gas is the dominating effect. Ionization in the bulk dielectric with subsequent escape of the generated electrons into the void is negligible because the mean free path of electrons in solids is only of the order of 100 nm

The average inception delay Atin, is the inverse of the electron generation rate f i e according to Equation (6) and (7). The effective volume V,,, in this case is that fraction of the void volume $ x ( Z / ~ ) ~ from which avalanches can grow to the critical size. V,,, is approximately given by Equation (11) in [26] and can be expressed] after some algebraic manipulation] in the form

P71.

v,,, E $T(1/2)3(1 - v - p )

where

(49)

is the ratio of the applied voltage to the inception voltage which will be referred to as the overvoltage ratio. The average inception delay time then becomes

Atinc M [ ( . . / ~ ) C , = ~ ~ P , , , ( P / P ) , P ~ ~ ( 1 - v-')]-' (51)

and results inversely proportional to the third power of the void diameter 1. Inspection of the v-dependence shows that the latter becomes negligible for v > 2 when the exponent ,L? is in the range 1/2 < p < 1 (Table 4).

The solid line in Figure 10 shows a plot of Equation (51) for spherical voids filled with air

IEEE Transactions on Dielectrics and Electrical Insulation Vol. 2 No. 4 , August 1995 523

"1 z Q .c 3 10- c

v) S 0

a aJ

5 10- - CI

L v) .- -

0.1 1 10 void diameter C / mm

Figure 10. Calculated (full curve) and measured (points) average PD inception delay in spherical voids (air, p = 1 bar) under natural irradiation in dependence of void diameter. Ordinate scales: average number of first electrons per ac half period (left) and average inception delay (right).

((pip), M l o p 5 kg ~ n - ~ P a - ' , p = 1 / 2 ) a t atmospheric pressure ( p = 100 kPa) and exposed to natural irradiation C r a d Q r a d M 2 x lo6 kg-ls-l [26] , and to an overvoltage ratio w >> 1. The average inception delay Atinc is scaled on the right side of the diagram. The left side is scaled top to bottom in terms of the average number of first electrons per ac half period which is given by ( T / 2 ) / A t Z n c . Experimental data with w % 2 are represented by points with error bars. They were obtained by measuring the time delay between voltage application and PD inception. It can be seen that there is order- of-magnitude agreement between measurement and pre- diction, indicating that volume ionization is in fact the controlling mechanism. This is in contrast with what was experimentally observed with flat voids [5] in which due t o the large surface-to-volume ratio electron production from the surface is probably the dominant mechanism.

Once a first P D has been initiated in a void, an ad- ditional first electron generation mechanism becomes ac- tive, namely, the detrapping of electrons from surfaces where they have been put into detrappable states by a previous PD. The corresponding generation rate N , can be quantified approximately by the phenomenologi- cal Equations (a), ( lo) , (11) and ( 1 2 ) . For void diameters of 1 t o 2 mm, i.e. for emitting areas in the range of 1 to 3 mm2, and for work functions of the order of 1 eV the detrapping rates are found to be much higher than the volume ionization rate due to natural irradiation. As a consequence, void discharges exhibit a 'switch-on' behav-

ior: They are ignited after a relatively long inception delay by natural irradiation and then continue with a much lower statistical time lag controlled by surface charge detrapping.

In order to determine the true P D charge with the help of Equation (30) the geometry factor g and the voltage collapse A U ~ D have to be determined. Because the equivalent ellipsoid of a spherical void is a sphere of radius R, one has a = b = R and hence all = 1 / 2 and a/b = 1. Inserting these values into Equation (33) gives K = 3 with which Equation (34) and (38) yield the geometry factors

2Er + 1 g = - 4

For determining A U p , we consider inception conditions, i.e. the limiting case that the maximum field in the void is just equal to the streamer inception field E s t ' . The latter is, according t o Equation (3), Estr = f E : t r . The associated potential difference AUinC across the void pri- or to the PD then results with Eitr from Equation (47)

Auinc - - f l E i t T ~ IEStT

(53) = l ( E / p ) c r p [ l + B/(~l)" l

The residual voltage after the P D is given by Equa- tion ( 2 5 ) as AU,,, = yEcrZ = y(E/p),,pZ. The potential collapse a t inception then results in

(54) A u r e s AU&y AUi"C -

= ( E / P ) c r P [ l - Y + B/(Pl )" I l

and the minimal true charge results from Equation (30)

The maximal charge qmaz is deployed when a first PD occurs a t one maximum U,,,, of the applied voltage, the charge deployed by it does not decay, and a subsequent P D occurs a t a voltage maximum of opposite polarity. The potential collapse by the P D then becomes

and the ratio of the maximal t o the minimal charge results by combining Equation ( 2 5 ) , (50)' ( 5 4 ) and ( 5 6 ) as

This ratio is seen to increase approximately linearly with the overvoltage ratio U . The induced charge is obtained


from the true charge with the help of the factor g' from Equation (52) as

4kin - $ T E o E r ( E / p ) c r p [ l - - Y + B / ( p l ) " ] 1 3 V X o (58)

It is seen t o be particularly sensitive to the void diameter 1 . It also is proportional to the PMC function VX, and thus depends on the location of the void in the insulation system.

Charge decay and memory effects in void discharges are controlled mainly by the conductivity n, of the void surface (see Figure 9(c)). The surface conductivity of voids in polymeric materials was shown to vary over many orders of magnitude, depending on the aging state of the void [13,28]. It is therefore interesting to derive an order-of-magnitude limit of that surface conductivity 6; above which the charge decay time constant t,, becomes shorter than the ac half period T/2 , i.e. above which a substantial charge decay occurs between two subsequent half periods. For a spherical void the scales L and T ,

in Equation (42) are of the order of the void diameter 1 and radius R = 1 / 2 , respectively, so that T ~ , - E ~ Z / ( ~ I C , ) . Introducing T/2 for r,, and solving for n, one obtains

K: - &,l/T (59)

For void diameters in the m m range n: results of the order

.: - 5 ~ 1 0 - l ~ (60)

The surface conductivity of polymers which have not suf- fered PD activity is typically in the range S , i.e. much smaller than IC:. This means that in non- aged voids the surface charges can survive for many half periods and thus exert memory effects via their field contribution and as source for detrappable charge carriers. Correlation effects over many subsequent ac half periods can thus be expected. The surface conductivities measured in aged voids of oblate shape may reach values as high as 10-l' t o S [13,28] which exceed the limit given by Equation (60).

to

In a spherical void such high conductivities would lead to a complete suppression of the discharge activity because the conducting void surface would shield the void interior from the electric field. In the transition range around 6: charge decay by surface conduction will en- tail various effects on the P D characteristics such as a reduction of the charge magnitude, a reduction of the number of P D events per half period, and a change of the correlation of the pulses with the ac phase. There are also indications that the surface conductivity may cause a transition to another discharge mechanism [13].

The phase dependent statistical features of the P D activity in a void can be determined by integrating the

+4 experiment

--. c Y I - 0 U

- 4 la I I I

simulation 6 = l e v T = l m s

0 7c 27t ac phase cp Figure 11.

(a) Experimental phase distribution histogram of PD activity in a spherical void. Parameters 1 = 3.2 mm, air 100 kPa, EP resin, e , = 3.7, (b) simulated histogram with same parameters as in experiment and surface parameters if = 1, @ = 1 e v , and Tdt = 1 ms and n, = 0.

elements of the discharge model into a numerical Monte- Carlo simulation as described in [ 26 ] . Figure I l ( b ) shows the results of such a simulation for the parameters given in its caption. The da ta are presented in the form of a statistical histogram in which each P D event is represented as a point in a charge - phase (4'-p) plane. The frequency of occurrence of P D pulses with a given q'-p combination is encoded as point density. Figure l l(a) shows, for comparison, an experimental histogram obtained with an epoxy void with the same parameters as assumed in the simulation. The effective work function was assumed to be @ = 1 eV consistent with the expected order-of-magnitude discussed in Section 4.2. The basic phonon frequency is vo - 1014 s- l for epoxy resin. The parameter (. in Equation (11) was assumed to be ( = 1 and the charge decay time constant was chosen r d t = 1 ms.

With these data the simulation reproduces the main features of the experimental data fairly well, both qualitatively and quantitatively. The only lack of agreement is noted in the fine structure of the curved feature on the left side of each half wave pattern. I t is believed to be due to the inadequate description of the detrapping process by the simplified rate Equation (11) and (12). The physical background of this process needs better understanding.


7.2. PROTRUSION ON AN ELECTRODE IN SFG

Electrode protrusions (Figure 8(c)) may occur in SF6 insulation in the form of surface roughness or as parti- cles erected by electrostatic forces. The corresponding defect parameters are the length of the protrusion I , its t ip curvature radius T , the SF6 pressure p a n d the reduced background field Vu, = E,/U, and the PMC function VX, a t the location of the protrusion.

It was shown in [7] and [9] that different discharge types may occur simultaneously or sequentially a t such a defect and that these may interact in a complex way. Here, the discussion will be restricted to the strongest of these, namely an impulsive streamer corona under positive polarity developing into a space charge free gas. This discharge type is a limiting case which can occur because ionic space charges are highly mobile and therefore are swept away rapidly from the PD zone after a discharge. The related charge decay time constant can be estimated by Equation (41). For SF6 the mobility constant is C, E 6 m2Pa/Vs [9], the factor y E 1 (Table 4), and the dimensionless factor in Equation (26) is X - 0.5. In gas insulated equipment the ratio E,/Ecr under service stress is typically in the range 0.1 to 0.2. With these values and for protrusion lengths 1 of the order of 5 mm one obtains drift controlled memory decay times r d T in the ps range. As these are much shorter than the ac half period T/2 , many PD pulses can follow each other during one half period without space charge memory interaction. Their characteristics are therefore only controlled by the instantaneous value of the applied voltage U,. The maximal charge values therefore occur a t the peak of the ac voltage.

The potential difference driving the PD in the absence of space charge is given by Equation (3) as

A U = A U , = fE,1 M E,l (61)

with the geometry factor f M 1.

For the evaluation of the dimensionless inception criterion Equation (20), the normalized distribution e , ( z / l ) of the enhanced field a t the protrusion t ip can be approximated by a power law. The detailed calculations are given in [34] and lead to the dimensionless function F given by Equation (22) and plotted in Figure 5. The PD inception voltage results from Equation (24) as

It is seen that for known SF6 pressure p the inception voltage is quantitatively related t o the length 1 of the protrusion, t o its aspect ratio 1 / T , and to the reduced

background field Vu, a t its location. The measured inception voltage thus provides ‘mixed’ information about these three defect characteristics.

1 DE

0 50

U

0)

m r F

20 ”

10 10 20 50 too

background field E kVcm-1 Figure 12.

Calculated (solid curve) and measured (points) maximal true PD charges from electrode protrusions in SFs, 400 kPa, in dependence of applied background field E,. Protrusion lengths 1 = 5 and 10 mm.

In order to estimate the charges deployed by the PD the dimensionless factors g and g’ have t o be determined from the parameters of the equivalent ellipsoid. The latter can be assigned as shown in Figure 8(c) so that a x I and b x I s t r , hence ( a l l ) x 1 and ( a l b ) x ( l / l s t r ) . The streamer propagation length l s l r is given by Equa- tion (28) with X M 0.5 and y M 1 for SFG (Table 4). The surrounding dielectric is the SF6 gas with relative permittivity E~ = 1. The geometry factors g and g’ then result from Equation (34) and (38)

Y (Eo/Ecr) (631 g‘ RZ 413

With AU from Equation (60) and neglecting AUT,, the true charge q deployed by the PD follows from Equa- tion (30) a

Figure 12 shows a plot of this relation for an SF6 pressure p = 400 kPa and for the protrusion lengths 5 and 10 m m (solid curves). For comparison, experimental data are given as points. For the special case of a conducting electrode protrusion the physical charge is accessible to direct measurement as described in [9]. By covering a protrusion on a grounded electrode with a grounded shield and leaving only its t ip open, the physical PD current entering the t ip can be directly measured. The ex- cellent agreement between estimate and measurement is probably incidental.


The induced charge q’ follows from the true charge q with the factor g‘ from Equation (63) and results, after replacing E, by UoVu,, as

q1 is seen to scale with the third power of the protrusion length I , the square of the applied voltage U,, the square of the reduced field Vu,, linearly with the PMC function VX,, and inversely with the SF6 pressure p . For a given location of the protrusion in the system, i.e. known values of Vu, and VX,, the induced charge q‘ provides quantitative information on the length scale 1 of the protrusion.

8. CONCLUSIONS The generalized P D model described in this paper is

based on the following concepts.

The partial discharge activity caused by different insulation defects is, t o a large degree, controlled by similar physical processes. This permits an efficient unified modeling approach.

The complexity of the model can be reduced signifi- cantly by introducing simplifying approximations without sacrificing the physical content.

As a particularly efficient modeling tool, dimensional reasoning is used which consists in establishing dimensional relations between the major integral parameters of the problem and accounting for the details of defect geometry, discharge structure, and involved materials by dimensionless proportionality factors. The latter can be estimated, measured or calculated from detailed discharge models.

In the context of P D interpretation the generalized modeling approach fulfills the following functions.

It predicts such measurable P D characteristics as inception voltage, inception delay, measurable charges, statistical characteristics, and the distribution of the PD events over the ac phase.

It yields scaling relations between defect parameters, design features, and test conditions on one side and measurable P D characteristics on the other.

It provides the basis for a simulation of random se- quences of P D pulses. This allows to numerically ex- plore various forms of P D da ta representations such as phase resolved histograms and to identify in them those features which are defect specific or which indicate the aging state of the insulation system.

The scaling relations are frequently defect specific and can therefore be used to identify defect types. The model

thus contributes to the first step of P D interpretation, namely, defect identification.

The model furthermore provides relations which allow recovering quantitative defect information from the measured data. This is a n essential element of the second step of PD interpretation, namely, defect quantification.

The quantitative defect information thus obtained is the key prerequisite for the third and final step of PD interpretation, namely, risk of failure assessment.

At present, the major deficiency of the generalized model is the insufficient knowledge of several important material parameters most of which are related to gas- insulator interfaces. These da t a will have to be determined for the materials of interest by laboratory experiments.

The general character of the modeling procedure shows promise to be applicable to other gas discharge types such as Townsend discharges, leaders, or sparks. It may also contain useful elements for modeling P D in solid and liquid insulators in a similar fashion.

ACKNOWLEDGMENT The author gratefully acknowledges many valuable dis-

cussions with P. Biller, B. Fruth, F. Gutfleisch, I. McA1- lister, A. Pedersen and H. J. Wiesmann.

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This paper is based on a presentation given at the 1994 Volta Collo- quium on Partial Discharge Measurements, Como, Italy, 31 August - 2 September 1994.

Manuscript was received on 12 November 1994, in final form 8 May

1995.

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