Modeling Partial Discharges in a Cavity at Different
C. Forssén and H. Edin
KTH Electrical Engineering
100 44 Stockholm, Sweden
Abstract- A model of partial discharges (PD) in an insulated
disc-shaped cavity is presented. The flat cavity walls are covered The model describes PD in an insulated disc-shaped cavity
with copper foil and each PD is assumed to affect the whole cav- with copper foil covering the flat cavity walls. The geometry
ity. The discharge process in the cavity is simulated dynamically of the corresponding test object is shown in Fig. 1. The model
and the model is charge consistent. The model is used to simulate is used to dynamically simulate the sequence of PDs in the
the sequence of PDs in the cavity at two different applied fre-
quencies: 0.02 Hz and 100 Hz. The simulation results are com-
pared with measurements and good agreement is found. As the
applied frequency is increased from 0.02 Hz to 100 Hz, the mean A. Local field in cavity
apparent charge increases and the number of PDs per cycle of The electric field distribution in the test object is calculated
the applied voltage reduces. This is interpreted as an effect of the at each point of time by use of the finite-element based pro-
statistical time lag of PD. gram Comsol Multiphysics® 3.3a . Since the test object is
axi-symmetric a two-dimensional mesh is used. The cavity
I. INTRODUCTION with copper foil is discretized with quadrilateral elements, 12
in the vertical and 200 in the horizontal direction. Triangular
Partial discharges (PD) are usually measured at the power mesh elements are used for the remaining parts of the geome-
frequency 50 (60) Hz. There are however reasons to measure try. The total mesh has 4702 elements and the whole model
PD also at other frequencies. Measuring at a lower frequency has 12040 degrees of freedom.
(usually 0.1 Hz is used) reduces the size of the power supply
needed. Another option is to measure at variable applied fre- B. Electron generation
quency as in the Variable Frequency Phase Resolved Partial For a PD to occur in the cavity the field must exceed the
Discharge (VF-PRPDA) technique . Here the observed fre- critical field for discharge, and there must be a free electron to
quency dependence of PD [1, 2] is utilized to extract more start the first electron avalanche. In the model it is assumed
information from the PD measurements than is possible from that the emission of free electrons from the cavity walls in-
traditional measurements at a single frequency. creases with increasing field . The electron generation in-
It is necessary to physically understand the PD frequency tensity is modeled as
dependence in order to interpret PD measurements at variable
applied frequency. This paper presents a model of PD in an
insulated disc-shaped cavity. The model is used to interpret (
I e (t ) = I e 0 exp U cav (t ) U crit . ) (1)
the frequency dependence in phase resolved PD measurements
at two different applied frequencies (0.02 Hz and 100 Hz) on a Here Ie0 is a constant, Ucav is the voltage over the cavity cen-
corresponding test object. In both the model and the test object tre and Ucrit is the voltage over the cavity centre corresponding
copper foil is placed on the flat cavity walls. Due to the copper to the critical field for discharge.
foil each PD affects the whole cavity. This simplification is
introduced to verify the current model. The intention is to fur-
ther develop the model in the future. Modeling a similar cavity
without copper foil requires a model where each PD may af-
fect only part of the cavity, as discussed in [3,4].
The measurements show that the number of PDs per cycle
reduces and the mean discharge size increases as the applied
frequency is increased. The simulations confirm that this is a
statistical effect related to the statistical time lag of PD.
Fig. 1. Schematic picture of the test object (rotational symmetry, measures in
millimeter). The copper foil is 0.05 mm thick.
The probability that a free electron is generated in the cavity the voltage over the test object is already compensated for in
during the time interval [t,t+Δt] is assumed to be Ie(t)Δt, pro- the software of the measurement system.
vided that |Ucav|>Ucrit. The corresponding distribution function
for PD is F. Simulations
The sequence of PD in the cavity was simulated for 100 peri-
⎛ t ⎞ ods of an applied voltage with amplitude 10 kV and with two
F (t ) = 1 − exp⎜ − ∫ I e (t )dt ⎟ .
⎜ ⎟ (2) different frequencies: 0.02 Hz and 100 Hz. One simulation on
⎝ 0 ⎠ a 3 GHz Xeon EM64T processor with 2 GB RAM took about
The parameters used in the simulations are shown in Table I.
Each time |Ucav| exceeds Ucrit there is a possibility for PD.
In choosing the parameter values it was assumed that the PD
The PD can appear in time from that |Ucav| reaches Ucrit until
activity is deterministic at 0.02 Hz, that is that the influence of
|Ucav| drops below Ucrit again. The model determines if there is
the statistical time lag on the PD activity is negligible at this
a free electron in this time interval and at which point of time
low frequency. This assumption is supported by the measure-
that electron is generated. To do this, a help parameter called
ment results at 0.02 Hz (see Fig. 3a) where the spread in phase
Uno_PD is calculated. This parameter describes the voltage over
and apparent charge is small. Paschen’s law (for gaseous
the cavity center if there is no PD. It is calculated from that
breakdown between plane metal electrodes) predicts that an
|Ucav| reaches Ucrit until |Uno_PD|< Ucrit. From Uno_PD the distri-
air-gap of 0.25 mm (at atmospheric conditions) should dis-
bution function F(t) is calculated. Then a random number R
charge at voltage 1.67 kV. This value was used as an initial
between 0 and 1 is generated and the time of PD is determined
guess for the critical voltage for discharge (Ucrit). With the
as the point of time where F=R. If F is always less than R then
assumption of a deterministic PD activity at 0.02 Hz, Ucrit and
there is no PD.
the critical voltage for extinction of discharge (Ustop) were then
This event-controlled modeling technique, where the time of
chosen to reach agreement with measurement data with re-
each PD is simulated from the distribution function (2), gives
spect to the phase position of the PDs, the apparent charge and
much shorter simulation times than if the FEM-solver is inter-
the number of PDs per cycle of the applied voltage.
rupted in each time step to check if there is any PD. It is desir-
The constant Ie0, which is related to the electron generation
able to keep the simulation time as short as possible since a
intensity (1), was chosen to reach agreement with measure-
simulation of the PD activity in the cavity must extend over
ment data at 100 Hz with respect to the spread in apparent
several periods of the applied voltage to gain reasonable statis-
charge and the number of PDs per cycle of the applied volt-
tics. Otherwise it is not possible to draw a PD pattern or to
study the statistical properties of the PDs.
The conductivity of the copper foil in the model was set
much lower (10-4 S/m) than the conductivity of copper (which
C. Discharge process
is about 6·107 S/m). A higher conductivity in the model would
A discharge in the cavity is modeled by increasing the con-
cause numerical problems due to the resulting fast changes in
ductivity in the whole cavity, from a low value to a high value,
the electric field. The low conductivity used in the model is
so that current flows through the cavity. The discharge process
nevertheless high enough to make the copper foil act as an
is modeled dynamically and a short time step is used. When
equi-potential surface on time-scales relevant for the simula-
|Ucav| drops below a certain level called Ustop, the cavity con-
tions. Similarly the cavity conductivity during discharge is
ductivity is changed back to its low value and the discharge
also set low (10-4 S/m) but is still high enough to make the
stops. Each discharge affects the whole cavity.
cavity discharge in a very short time (less than 0.2 μs) com-
pared to the shortest period time of the applied voltage used in
this study (10 ms).
The dynamical modeling of the discharge process makes the
model charge consistent and the apparent charge can be calcu-
lated numerically. Hence there is no need for λ-functions and
analytical estimations of the apparent charge , which is
common in other PD models as . The current through the
test object is calculated by integrating the current density over
the lower electrode surface. This current is then integrated in
time to give the apparent charge of the PDs.
E. Model of measurement circuit
The measurement circuit is shown in Fig. 2. The model in-
cludes only the test object and the applied voltage while the
Fig. 2. Schematic picture of the measurement circuit.
other components are omitted. The voltage drops over the fil-
ter impedance and measuring impedance are negligible in
comparison to the voltage over the test object. In addition the
phase shift (about 1 degree) between the applied voltage and
Applied voltage 10 kV
Frequency of applied voltage (f) 0.02 Hz and 100 Hz
Critical voltage for discharge 1.57 kV
Critical voltage for extinction of 1.24 kV
Ie0 1000 s-1
Time step (not during discharge) 1/(2000f) s
Time step during discharge 1 ns
Permittivity polycarbonate 3
Conductivity polycarbonate 10-15 S/m
Permittivity epoxy 5.2
Conductivity epoxy 10-15 S/m
Permittivity cavity 1
Conductivity cavity (not during 0
Conductivity cavity during dis- 10-4 S/m
Conductivity copper foil 10-4 S/m
Tolerance for the FEM-solver stop 0.01
conditions |Ucav|=Ucrit, |Ucav|=Ustop
and |Uno_PD|= Ucrit
The PD activity in the test object was measured at an applied
voltage with amplitude 10 kV and with two different frequen-
cies: 0.02 Hz and 100 Hz. The measurement system is de-
scribed in detail in [1, 8]. The measurement results are shown
as PD-patterns in Fig. 3. Fig. 3. a) Measurement at 0.02 Hz with 606 PDs over 100 periods. b) Meas-
At 0.02 Hz (Fig. 3 a) the PD activity is regular with 6 PDs urement at 100 Hz with 2277 PDs over 1000 periods.
per cycle of the applied voltage and the favored phase posi-
tions are visible as six fuzzy spots in the PD-pattern. The IV. SIMULATION RESULTS
spread in apparent charge is small which indicates that the
PDs start at about the same voltage over the cavity. Hence The simulation results are shown in Fig. 4. As described ear-
there is not much influence of the statistical time lag. Due to lier, Ucrit and Ustop were adjusted to match the simulations to
the copper foil each PD affects the whole cavity and this also the measurements at 0.02 Hz. A fairly good agreement was
contributes to the small spread in the apparent charge. reached with respect to phase and apparent charge (see Fig. 4
At 100 Hz (Fig. 3 b) there are only about 2.3 PDs per cycle a), and the simulated number of PDs per cycle was the same as
and the PD activity is no longer regular. There is also a wider the measured. The value used for Ucrit in the simulations (1.57
spread in the apparent charge than for 0.02 Hz, although the kV) is not far from the breakdown voltage level as predicted
lowest apparent charge level is about the same for both fre- by Paschen’s law (1.67 kV).
quencies. Both the spread in the apparent charge and the re- The constant Ie0 was chosen to match the simulations to the
duction in the number of PDs per cycle are probably caused by measurements at 100 Hz and the matching was done mainly
the statistical time lag. Due to a lack of free electrons PDs are with respect to the spread in apparent charge. As seen in Fig. 4
shifted forward in time and phase. This effect has a larger im- b the simulated PD-pattern has about the same lowest and
pact on the PD activity at higher frequency since the rate of highest apparent charge level as the measurement (Fig. 3 b).
change of the applied voltage is faster. In fact, during the However the size distribution of apparent charge is not the
measurement at 100 Hz several consecutive voltage cycles same. In the simulation the majority of the PDs are gathered at
without any PD at all were observed occasionally by only the lowest apparent charge level while in the measurement the
watching the voltage over the measuring impedance on an most common apparent charge is considerably higher. This
oscilloscope. This phenomenon can be explained by that once indicates that the exponential expression used in the model for
a whole voltage cycle has passed without any PD, the supply the electron generation intensity (1) might not be the best
of free electrons reduces. This comes since one important suited.
source of free electrons is charge generated by earlier PDs in The simulated number of PDs per cycle at 100 Hz was 4.3
the cavity, and this charge decay with time. and is lower than that at 0.02 Hz (which was 6 PDs per cycle).
Hence the simulation shows the same trend as the measure-
ment with fewer PDs per cycle at the higher frequency, proba- V. CONCLUSIONS
bly caused by the statistical time lag. The simulated number of
4.3 PDs per cycle at 100 Hz is however far higher than the A model of PD in an insulated disc-shaped cavity is pre-
measured 2.3 PDs per cycle at the same frequency. This dis- sented and used to dynamically simulate the sequence of PDs
crepancy between the simulation and measurement is most in the cavity. The flat cavity walls are covered with copper foil
likely due to that the model does not account for the effect of so that each PD affects the whole cavity. Since the discharge
interruptions in the PD activity in the cavity, with several con- process is modeled dynamically, the model is charge consis-
secutive periods without any PD at all, as was observed in the tent and the apparent charge is calculated numerically.
measurement at 100 Hz. Simulation results for two different applied frequencies (0.02
Hz and 100 Hz) are compared with measurements on a corre-
sponding test object. At 0.02 Hz a regular PD activity was
measured in the test object without much influence from the
statistical properties of PD. The corresponding simulation
showed good agreement with the measurement for reasonable
parameter values. The measurement at 100 Hz showed a wider
spread in phase and apparent charge than at 0.02 Hz, and also
a smaller number of PDs per cycle of the applied voltage. This
is probably an effect of the statistical time lag of PDs. The
simulations showed the same tendency with fewer PDs per
cycle at the higher frequency. At 100 Hz the simulated distri-
bution of apparent charge did not fully reflect the measure-
ment, which indicates that the expression for the electron gen-
eration intensity in the model can be improved.
 H. Edin, Partial Discharges Studied with Variable Frequency of the
Applied Voltage. Ph. D. Thesis, KTH, Stockholm, Sweden, 2001.
 C. Forssén, Partial Discharges in Cylindrical Cavities at Variable Fre-
quency of the Applied Voltage. Licentiate Thesis, KTH Electrical Engi-
neering, Stockholm, Sweden, 2005.
 C. Forssén and H. Edin, “Influence of Cavity Size and Cavity Location
on Partial Discharge Frequency Dependence,” Conf. on Electrical Insu-
lation and Dielectric Phenomena, Boulder, Colorado, USA, 2004, pp.
 K. Wu, Y. Suzuoki, and L. A. Dissado, “The Contribution of Discharge
Area Variation to Partial Discharge Patterns in Disc-Voids,” J. Phys. D:
Applied Physics, vol. 37, pp.1815-1823, 2004.
 COMSOL AB, Tegnérgatan 23, SE-111 40 Stockholm, Sweden.
 L. Niemeyer, “A Generalized Approach to Partial Discharge Modeling,”
IEEE Trans. on Dielectrics and Electrical Insulation, vol. 2, pp.510-528,
 A. Pedersen, G. C. Crichton, and I. W. McAllister, “The Functional
Relation between Partial Discharges and Induced Charge,” IEEE Trans
on Dielectrics and Electrical Insulation, vol. 2, pp.535-543, 1995.
Fig. 4. a) Simulation at 0.02 Hz with 599 PDs over 100 periods. b) Simulation
 J. Giddens, H. Edin, and U. Gäfvert, “Measuring System for Phase-
at 100 Hz with 429 PDs over 100 periods.
Resolved Partial Discharge Detection at Low Frequencies,” 11th Int.
Symp. on High-Voltage Engineering, London, UK, 1999, pp. 5.228-