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Partial Discharges in Cavities and their Connection 
with Dipoles, Space Charges, and Some Phenomena 

Below Inception Voltage 
 

George E. Vardakis 
Department of Electrical and Computer 

Engineering 
Democritus University of Thrace 

Thrace, Greece 

Michael G. Danikas  
Department of Electrical and Computer 

Engineering 
Democritus University of Thrace 

Thrace, Greece 

Anastasios Nterekas 
Department of Electrical and Computer 

Engineering 
Democritus University of Thrace 

Thrace, Greece 
 

 

Abstract—This paper tries to relate Pedersen’s model on partial 

discharges and work carried out by Bruning and co-workers on 

the possibility of the existence of charging phenomena below 

inception voltage, which may eventually cause deterioration of 

polymeric insulation. Moreover, with the aid of the 
Electromagnetic theory, some aspects of the Pedersen’s model are 

tried to be clarified, especially those which are correlated with 

space charges, electric dipoles, charge distribution, charge 
dynamics, and partial discharge activity.  

Keywords-Pedersen’s model; partial discharges; dipoles; 

inception voltage; space charges 

I. INTRODUCTION  

Pedersen’s model [1] was proposed as an alternative to the 
traditional capacitive model [2, 3] for the interpretation and/or 
prediction of partial discharges in enclosed cavities in solid 
dielectrics. This model is based on electromagnetic theory and 
gives the magnitude q induced on the measuring electrode by 
the partial discharge in a cavity, in terms of a variety of 
parameters, as is shown in (1):  

q = kΩεrε0 (Ei – El) ∇�����    (1) 

where k is the geometrical cavity factor, Ω the cavity volume, 
Ei the inception electric field for streamer inception, El the 
limiting electric field for ionization, εr and ε0 are the relative 
permittivity of the surrounding dielectric material and the 
permittivity of the free space respectively, and λ0 is the function 
giving the ratio of the electric field at the position of the cavity 
(in the absence of the cavity) to the voltage between the 
electrodes. According to [1], the charge deposited on the cavity 
surface S can be considered as an electric dipole, the moment 
of which µ, is given as: 

µ = ∫ rσdS    (2) 

where r is a radius vector which locates the position of the 
surface element dS. The induced charge which will eventually 
arise from the dipole is given as: 

� = −	� ∙ ∇����    (3) 

with λ being a dimensionless scalar function which depends on 

the position of dS only. Function λ is given by Laplace’s 
equation:  

∇�����∇����
 = 0    (4) 

where ε is the permittivity of the insulating material and with 
the following boundary conditions: 

• λ= 1 at the electrodes where q is distributed,  

• λ=0 at all other electrodes.  

Moreover, authors in [4] utilized the principle of 
superposition with the calculation of D-field (Maxwellian 
approach) and the calculation of P-field (quasi-molecular 
description). In addition to this, the induced charge, according 
to [4], can be expressed as the difference between the charge on 
an electrode following discharge activity and the charge on the 
same electrode prior to the activity. At the Maxwellian 
approach (D-field) and the corresponding establishment of λ-
function, all electrodes are supposed to hold at zero potential 
and the resulting electric field owned only to the space charges 
(and surface charges) in the interelectrode volume. Also in [5], 
the PD event is separated into two distinct time intervals: 

• The 1st time interval is determined as the duration of the 
void discharge development 0<t<T1 

• The 2nd time interval is the time following the cessation of 
discharge development T1<t<T2 

where t denotes time, T1 denotes the duration of the void 
discharge development, and T2 is the time of occurrence of the 
next discharge. For T1<t<T2, the induced charge q is given by 
either (12) for D-field analysis or (13) for P-field analysis     

� = −∬����    (12) 

� = −∭�� ∙ ∇���− ∬ �����     (13)  

It is evident from the above that the notion of dipole and/or 
dipole moment plays a decisive role in Pedersen’s model which 
tries to relate the partial discharge taking place in a cavity with 
the dipole moment of the charge distribution on the surface (or 
in the interior) of the cavity. 

Corresponding author: Michael G. Danikas



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II. PEDERSEN’S MODEL RE-CONSIDERED 

A criticism leveled against Pedersen’s model was that it is 
confined to a certain type of partial discharges, namely that of 
streamer-type discharges and it cannot be applied to all types of 
discharges. It is a model rather for initial experimental 
conditions and not for an estimate of the long-term developed 
discharges [6]. Moreover, as was noted in [6], λ does not seem 
to be a function without physical meaning but it is the relative 
potential and can be expressed in terms of a percentage. 
However, even though there is a disagreement as to whether 
the net charge within the cavity remains zero [1, 6], it is evident 
from [1] that Pedersen’s model tries to relate dipole moment 
with charge dynamics inside the cavity and thus with even 
minute discharge currents. The resulted electric field at the void 
exerts forces causing charge generation, which consequently 
has as a result, electric dipoles appearance within the cavity. 
The partial discharge activity inside the void leads to charge 
distribution and redistribution inside the void, and thus, 
induced charge appears at the electrodes [1]. Shortly, dipole 
moments within the cavity are supposed to induce charges and 
eventually cause the recorded partial discharge currents. 
Consequently, it is reasonable to assume that, agreeing with 
[6], the net charge within the cavity is not zero. If the opposite 
is valid (i.e. net charge is zero), this cannot explain the findings 
of [7] as well as some ideas put forward by other people [8]. 
Charge dynamics means that even elementary charge motion 
can cause somehow minute currents that may circulate inside 
the insulating material. Such minute currents, related to 
charging phenomena below inception voltage, were noticed in 
[7], where it was remarked that chemical by-products below 

and above the inception voltage were quite similar. Further 
work on polymers and nanocomposite polymers reinforced this 
viewpoint, namely that there were some below-inception 
charging phenomena [9-12]. To be sure, previous published 
work does not prove that such phenomena exist in all insulating 
materials but, at least, there are some indications that this may 
be the case. Further evidence that that may be the case comes 
from other sources as well [13].  

The topic of very small partial discharges (barely detectable 
from partial discharge detectors) was dealt with in numerous 
publications [14-19], where it was discussed whether such 
small events are of pulsive or of non-pulsive nature. In [10, 20], 
it was remarked that, having an enclosed cavity in 
polyethylene, a conducting path caused the magnitude of quite 
low current pulses (1-10mA) which is substantially different 
and quite lower than current pulses of about 1A measured with 
other more conventional arrangements [21]. It was mentioned 
in [6] that there are doubts whether the net charge in a cavity is 
zero or can supposed to be equal to zero. However the fact that 
inside the cavity we have a dipole (or a number of dipoles or a 
distribution of dipoles) does not necessarily mean that the total 
electric charge is zero. Various approaches point out the 
possibility of having small currents (or small charge 
displacements) inside the dielectric causing phenomena that 
may not be detected, but which may still provoke damage. 
Thus the discussion of the pulsive or non-pulsive nature of 
partial discharge phenomena at and/or above inception voltage 
is shifted to a more fundamental question whether charge 
movement inside the dielectric causes deterioration, even 
below the inception voltage. 

TABLE I.  SHORT PRESENTATION OF THE D-FIELD AND P-FIELD ANALYSIS [4] 

D-field (Maxwellian analysis) P-field (Quasi-Molecular analysis) 

The induced charge depends in a unique way on the location and magnitude 
of the space charge. An infinitesimal dQ located anywhere in the interelectrode 
region induces charges dq at all over electrodes: 

dqi=-λdQi    (5) 

The polarization P in the dielectrics is an important property, especially in 
insulating materials containing polarizable regions. The polarization effect is 
included in the λ–function in the D-field analysis. The whole interelectrode 
region is considered as a vacuum supposing that the solid dielectric itself is a 
distribution of electric dipoles, with polarization (density) P. 

dµ = Qdr    (6) 
Applying Green’s reciprocal theorem (7), two states are discriminated. In the 

first state, all electrodes are held at ground potential and the only charge left on 
the electrode will be the induced charge associated with the space charge, 
deposited in the space subtended by these electrodes. In the second state, ρ=0, 
σ=0 everywhere, so all electrodes are at zero potential (8), (9). 

∭� ∙ ���� = ∭�′ ∙ ���                  (7) 
�� ∙  !� + ∭�!���� + ∬�!���� = 0    (8) 
�� = −∭����� − ∬�����             (9) 

The polarization P in a point of a solid dielectric depends on : 
Applied voltages (Pa) and space charge formed by partial discharge (∆P). 

P=Pa+∆P       (10) 
The resulting induced charge on the i-th electrode is given by (11): 

�� = −∭#��� + $� · &��'�� − ∬�����		    (11) 

 

III. REMARKS ON CAVITIES IN RELATION TO PEDERSEN’S 
MODEL: DISCUSSION AND COMMENTS 

There is a large amount of experimental data on cavities 
enclosed in insulating materials. Very often, however, the 
experimentalists do not go below 1mm or 0.1mm in radius [1, 
22]. There is though some evidence that cavities may become 
dangerous even below such dimensions [23]. Certainly this is a 
topic needing further investigation. Further confirmation of the 
above is offered in [24], where computer calculations indicated 
that, in the case of electrical machine insulation, cavities as 
small as 0.01mm (or even smaller) in radius near a conductor, 
there may be electric fields as high as 25kV/mm. Other 

evidence regarding the existence and effect of small cavities (in 
the range of up to 120µm) was recorded in [25], where work 
was carried out with photovoltaic modules and components. 
Moreover, a lot of emphasis was given in the past decades on 
the role of partial discharge detection by electrical means. In 
[25] offers another perspective, since it stresses the importance 
of both optical and electrical measurements. In [26] stresses 
also the limitations of the electrical measurements of PD. 
According to [26], simultaneous imaging and PD 
measurements may reveal the successive stages of some treeing 
growth phenomena. Studying a cavity, one should not forget 
that its surface may not necessarily be smooth. In other words, 
its surface may have protrusions, not necessarily metallic but 



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consisted of dielectric material. Recent research reveals that 
dielectric protrusions may be the sites of high field 
intensifications and thus contribute to the breakdown 
mechanism [27]. Such data agree with preliminary efforts in [9, 
10], where it was indicated that even small irregularities on the 
surface of the cavity can contribute to the rise of minute 
currents below inception. A further remark concerning the PD-
extinction voltage: this can be determined with limited 
accuracy because of the influence from temperature and 
humidity [28]. Even without such parameters, PD-extinction 
voltage may somehow vary from measurement to measurement 
due to the highly statistical nature of PD phenomena. This may 
be a further indication as to the possibility of charging events 
below inception. 

The whole issue of damage below inception voltage (which 
we try to relate with charge packets) is not unrelated to the 
uncertainties of defining the level of inception [29]. Moreover, 
material composition plays a crucial role in determining the 
inception voltage [30]. Cavity position affects also the 
inception voltage [31]. Yet on another question related to 
considerations developed in [7], namely that charging 
phenomena are possible below inception voltage (because of 
some minute irregularities on the cavity surface), there is some 
indirect confirmation in [32], where the morphology and 
surface topography of polyethylene may affect space charge 
packet characteristics. Although the authors of [32] do not 
mention partial discharges and related phenomena, it is evident 
that surface treatment plays a role in determining charge 
packets. In agreement to the above, in [33] it is noted that the 
formation of packet-like charge is a result of a high 
conductivity region that is caused when traps are filled by 
electronic carriers (electrons or holes). In [34], various partial 
discharge models are presented. In all models, the role played 
by surface charge accumulation is evident. Furthermore, gas 
conductivity inside a cavity is also important. Such comments 
– on gas conductivity – were made earlier [7,9]. However, gas 
conductivity may appear even in the absence of partial 
discharges, conductivity which will not necessarily be 
“translated” (or transformed) into something detectable. Such 
ideas conform to [35], where it was reported that in minute 
cavities partial discharges may have very long statistical time 
lag and the number of initial electrons may indeed be very 
small. In relatively low voltages, ionization processes of low 
energy may occur, which means that charges appearing in the 
cavity may result to clusters of space charges on the cavity 
walls. Such a space charge results in an electric field which – in 
the case of AC fields - is added to the applied electric field on 
the insulation [36]. Although what was described is the normal 
process of a PD, no one can exclude the possibility of having 
such events even below the inception voltage. 

A careful reader can observe that the whole approach of 
space charge phenomena is related to the model proposed many 
years ago by some distinguished researchers [37], namely that 
the electrodes, under AC conditions, inject and extract charges. 
Some electrons are emitted or injected into the dielectric during 
the negative half cycle for a short distance, limited by the 
declining stress away from the points. They will be drawn back 
into the point on the positive half cycle and re-injected in the 
following cycle. On each cycle some of the electrons will gain 

sufficient energy to cause some polymer decomposition and 
thus create space charges in the bulk of the polymer. The 
estimated distance within which injected electrons can interact 
with the material to produce electrical trees near the tip of a 
needle is thought to be less than 20µm. Tree initiation time (t) 
is related to the electric field (E), the effective work function φ 
(i.e. the difference between the work function of a metal and 
the electron affinity of a dielectric) with the following equation 

ln(t) = Bφ3/2 /E + ln (C/A)    (14) 

where A, B, and C are constants. 

Moreover, in [1], it is stated that λ function is given by:  

∇�����∇����
 = 0    (4) 
where ε is the permittivity of the dielectric. Solving the above 
equation in the appropriate area of interest, λ can be evaluated. 
Initially, none can ignore that the space where the above 
equation is solved, includes, conducting (electrode, ionized air 
void), non-conductive (dielectric) and of course interfaces 
between the aforementioned materials. The analysis in [1, 5], 
needs more clarification, because (4) is Laplacian, while in all 
interfaces, in the interior of the solid dielectrics, at the 
electrodes and inside the air void, space charges are present, 
determining the electric potential distribution. Shortly, in [1, 5], 
the authors prove with a Maxwellian approach for λ function : 

 
Fig. 1.  After PD activity, the presence of space charge in the void is 
considered to be Poissonian (left rectangle). In [5], after a maxwellian 
analysis, (4) with the conditions (17) which are laplacian, are finally utilized 
(right rectangle). 

So, in [5], it is stated clearly that the defining equation of λi 
is: 

�� � �∭����� � ∬�����    (9) 
The first integral corresponds to the space charge in all 

space Ω between the electrodes (consequently including air 
void) and the second integral calculates the surface charge 
between dielectrics (at the interfaces between them). The 
discrimination of electric field into basic Laplace and Poisson 
(general and basic) field also needs more clarification since, 
among others, uses in [5]:   

� � &�������    (18) 
while the divergence of electric displacement D depends only 
on the free charge density [38]: 

�)*)+, � � � �- " �. � �- � &�� ∙ ���    (19) 

∮���� ∙ ������� � 0- � 1 �-2 ��    (20) 

∮���� ∙ ������� � 1 &��2 ⋅ ������    (21) 
Comparing (20) and (21), results to: 

�- � &�� ∙ ����    (22) 



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In addition to this, in [5], the electric field is assumed to be 
Laplacian, before the PD activity and after PD activity, the 
presence of space charges in the void dictates us to solve 
Poisson’s equation. In a mixed system, constituting from at 
least 2 electrodes, one solid dielectric and one air void enclosed 
in the aforementioned solid insulator, the voltage application U 
at the electrodes, creates space charges. The existence of space 
charges does not depend on the occurrence of a PD activity or 
not. Generally speaking, space charges are localized states of 
charge, deriving from various sources. 

A first source of space charges is the polarization that took 
place at the dielectric. Every molecule of the solid dielectric in 
the interelectrode area, before the electric field application, 
may exist in two states. Either the molecule has symmetrical 
distribution of the electric charge, with a coincidence of the 
negative and positive charge (zero dipole moment), or it is 
polarized due to the asymmetry in the negative charge 
distribution (non-zero dipole moment or permanent dipole 
moment). After the electric field application, the molecules of 
the first category gain an induced dipole moment due to the 
separation of the centers of positive and negative charge 
distributions. Molecules of the second category rotate into the 
direction of the field. As it is known from Fundamental 
Electromagnetism, dielectric hysteresis loop appear in solid 
dielectrics, but in the present paper relaxation time for solid 
dielectrics is not taken into account, similarly with [1, 5]. 
Moreover, the interelectrode system includes the air void. It is 
known that the electric field inside an air slit enclosed in a solid 
dielectric is k times bigger than the mean electric field.  

E0=ε·Emean     (23) 

where Eo is the value of the electric field inside the air void, ε is 
the relative permittivity of the dielectric (dielectric constant) 
and Emean is the mean value of the electric field inside the 
dielectric. The bigger value of E0 is due to the polarization 
charges in the interior of air void. Of course, (23) is valid 
whether partial discharge sparks inside the air void or not. 
Thus, a dielectric after a voltage application shows polarization 
and/or polar charges at various positions, which in turn is itself 
one of the major factors for the space charge existence, 
independently or not from the PD activity. 

�. � �&�� ∙ ���    (24) 
A second source, contributing to space charges, 

independently again from the PD occurrence or not, are the free 
charges. In our system (electrodes, solid dielectric, air void, 
interfaces), free charges exist at: 

• electrodes (before PD as charge q and after PD as q+qinduced) 
and  

• air void (before PD and after PD). 

at the region in front of the injecting electrode. This procedure 
is most common and usual from point electrode. The space 
charge injection distorts the electric field and reduces or 
enhances the local electric strength, in the region around the tip 
of the needle electrode. In case of homocharges, reduces the 
electric field E and in case of heterocharges enhances the local 
electric field. This category is free charges which have lost 

their kinetic energy to break bonds and they have converted to 
thermal ones.  

TABLE II.  TWO SITUATIONS FOR POTENTIAL AND CHARGE BEFORE 
AND AFTER INDUCED CHARGE APPEARANCE ON THE ELECTRODE.  

1
st
 state (before PD) 2

nd
 state (after PD) 

Volume and surface density 
are equal to zero ρ=0, σ=0. 

All electrodes are held in ground potential. 

One electrode (the i-th 
electrode) has potential Uci. 

Almost all charges (linked with partial 
capacitances) are zero. 

All other electrodes are at zero 
potential. 

The only charges left on the electrode will 
be the induced charges associated with 
space charges. 

 
So, it is evident, that space charges (free charges and 

polarization charges), exist at the interelectrode area, after the 
voltage application and their existence is not dependent from 
the occurrence or not of the partial discharge activity in the 
interior of the enclosed air void. Thus, the electric field, after 
voltage application is not Laplacian but it is clearly Poissonian. 
The electric field, of course, is Poissonian before and after PD 
activity. It must be pointed out that in [4], a fundamental 
theorem of Electromagnetism, Green’s Reciprocal Theorem, 
[39, 40] is utilized according the following:   

According to [39], the Green reciprocal theorem states that 
if separate charge densities ρ and ρ’ then give rise to electric 
potentials V and V’ respectively having as a result : 

∭� ∙ ���� = ∭�′∙ ���    (25) 
In case, the charge resides solely to the surfaces of n fixed 

conductors: 

∑ �� ∙ ��
� =5�67 ∑ ��′ ∙��

5
�67     (26) 

where charges qi on the conductors correspond to respective 
potential Vi  and charges qi’ correspond to Vi’. 

Applying Green’s reciprocal Theorem in a system 
containing electrodes, void, dielectric and interfaces before and 
after partial discharge activity (Table I), the following equation 
is used in [39]: 

�� ∙  !� + ∭�!���� + ∬�!���� = 0    (27) 
where  !� is the applied Voltage and �!� is the scalar potential 
at dΩ and dS. 

A point of further investigation is the application of (26), 
which refers to charges residing solely to conductors (e.g. 
electrodes). At conductors, normally there are free charges ρf 
and not polarized charges. In (27), volume charge density ρ 
includes, among others, the polarized charges (bound charges 
per unit area) in the interior of the solid insulating material. 
These charges are not free charges. Furthermore in (27), 
charges before PD activity (either in the form of volume 
density ρ or in the form of surface density σ are equal to zero 
ρ=0, σ=0). This is the reason which the second term in (27) (or 
the second term in (25), (26)), is equal to zero. This hypothesis 
neglects the various charges, mostly bound polarized charges 
which form and appear inside solid dielectric in the time 
period, after the voltage application and before the partial 
discharge activity. Finally, in [1] it is stated that the dipole 



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moment of a charge distribution left on the surface of an 
ellipsoid is given by the following equation : 

	� � 89:;< 	[>������ � ?1 "
+.!A#:BC:'

D:
E>,����]    (28) 

where a, b, c are the semiaxes of the ellipsoid, E0 is the ambient 
field when the internal field is equal to the inception field Ei 
and El is the limiting field when the discharge is quenched. 
Equation (28), is one of the basic equations leading finally to 
(1). However, in [41], there is a clear discrimination for the 
potential calculation φ as can be seen in Table III. 

TABLE III.  POTENTIAL EXPRESSIONS Φ FOR TWO KINDS OF 
ELLIPSOID (CONDUCTING AND DIELECTRIC) INSIDE A PARALLEL 

ELECTRIC FIELD [41] 

Conducting ellipsoid in parallel field Dielectric ellipsoid in parallel field 

The potential at the ellipsoid is a 
constant φs 

�G � ��[1"
HI
JK
1 LM#ξNαI'OP
Q
� 	]						(29) 

The potential at any interior point of 
the ellipsoid: 

�C � � RBS∙T
7N

UVW
IXI

#YKCYI'ZK
   (30) 

And the field intensity in the interior 
of the dielectric ellipsoid is: 

[\
C =

]B^

7N
_`a
IbI

#:KC:I'<K
   (31) 

 
Equation (31), belonging to the case of a dielectric ellipsoid 

inside a parallel electrical field, according to [41], is apparently 
a part of (28), or has close relation with the analysis about a 
dielectric sphere inside a parallel field. Equation (28), in [1], 
which calculates the dipole moment inside an ellipsoid needs 
further clarification because it seems to correspond to the case 
of dielectric ellipsoid and how it affects the electric field inside 
and outside the ellipsoid. The ellipsoid after PD activity, is 
conducting, due to the electrons, positive and negative ions 
produced during the ionization activity. The ellipsoid, before 
the PD activity can considered to be insulating (dielectric e.g. 
air) but (28) and the final important equation (1), are utilized 
for induced charge calculation, which is a phenomenon which 
clearly happened after PD activity. It seems that (28), as 
referred to [1], is not compatible with the conducting behavior 
of the void after pd activity.    

Recapitulating we may say that the present paper tries to 
clarify some aspects of Pedersen’s model about the induced 
charge after PD activity. There is no proposal of a new model 
but a criticism is presented of a very important model from two 
different aspects. The first aspect is concerned with 
experimental data below the so-called inception voltage 
whereas the second aspect is related to the electromagnetic 
theory and mostly to the conducting behavior of dielectric 
materials. An effort was made – in the context of the present 
work – to enlighten some aspects of Pedersen’s model. 
Furthermore, the present paper – as is evident from the title- 
tries to correlate PD activity with space charges and electric 
dipoles under the prism of electromagnetic theory.  

It is true that a criticism may be levelled in the fact that 
Pedersen’s model is rather old. However, it remains one of the 
most prominent and used models regarding PD activity and the 
authors consider that it has significant explanatory power. It is 
also true that the authors use many references published long 
time ago. It is also true that more recent criticism of Pedersen’s 

model has been published [42, 43], their comments, however, 
are beyond the scope of the present paper since they are 
concerned with the notion of capacitance. 

IV. CONCLUSION 

The present paper tries to enlighten some aspects of the 
Pedersen model relating partial discharge activity inside air 
voids and the induced charge at the electrodes after PD activity 
and charge redistribution. Electromagnetic analysis is 
fundamental in Pedersen’s model and in the present paper some 
points which need further investigation are demonstrated. The 
discrimination between conducting and insulating air void, the 
presence or not of space charges in the interelectrode area, the 
precise description of the charge kinetics after the voltage 
application and before partial discharge activity, the role of 
dipoles and dipole moments, the partial activity below 
inception voltage, the cavity dimension (especially those with 
dimensions less than 0.1mm), the PD extinction voltage are 
some points which are analyzed in the present paper and an 
effort of a correlation with Pedersen’s model is being made.  

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