Proceedings of the XIVth International Symposium on High Voltage Engineering,
Tsinghua University, Beijing, China, August 25-29, 2005
B-49
Numerical modelling of metal oxide varistors
Boris Žitnik1*, Maks Babuder1, Michael Muhr2, Mihael Žitnik3 and Rajeev Thottappillil3
1
Milan Vidmar Electric Power Research Institute, 1000 Ljubljana, Slovenia
2
Institute of High-Voltage Engineering and System Management, Graz, Austria
3
Division for Electricity and Lightning Research, Uppsala University, Sweden
*
E-mail: boris.zitnik@eimv.si
requirement is that the model should be capable of
reproducing the waveshapes of the voltage across and
the current through the varistor faithfully. Usually,
voltage-current characteristics (V-I curves) of varistors
are reproduced in the model by using non-linear
resistance with some accompanying elementary
elements (inductance, capacitance). For improved
accuracy of the models another non-linear resistance [4]
or non-linear inductance [2] are sometimes added. The
way how the varistor is modelled influences also the
usable frequency range. In case of two non-linear
resistances the proposed model [4] gives good results
for the front time of surge current through the surge
arresters in the range from 0,5 µs to 45 µs. In this paper,
standard surge current impulses 8/20 µs were
considered and therefore the models with one and two
non-linear resistances were studied.
From the literature we consider two different varistor
models that seem very promising in their ability to
predict voltage-current characteristics [5, 6]. The
models were implemented in the ATP-EMTP program
using MODELS. Model predictions were then
compared with experimental results.
In the paper, a model of the impulse generator actually
employed in experiments is also used in our simulations
of varistors to take into account the possible effects of
the generator characteristics in the dynamic responses of
the varistors. This is important because the generator is
a combination of physical components with its internal
impedance, which can be different for different
frequencies.
Amplitudes and waveshapes of the current through and
the voltage across the varistors from the simulations and
measurements were then compared and results are
presented and discussed in the following sections. A
new varistor model is suggested to resolve the
discrepancy between the simulated and measured
results.
Abstract: Transient protection is playing an
important role in normal operation of modern
electronic devices. The ZnO varistor is one of the
basic components of transient protection. The paper
describes numerical modelling of metal oxide
varistors for fast transient impulses. Two existing
models of ZnO varistors and surge arresters are
evaluated by comparing the model predictions with
the experimental results. The impulse generator
influences the time domain waveshapes of the
current through and the voltage over the varistor
and therefore the model of the impulse generator
used in experiments was also used in the simulation.
Simulation results showed many deviations from the
measured results and an improved varistor model is
proposed to rectify these problems.
Key Words: varistor, varistor modelling, fast transient
impulses;
INTRODUCTION
ZnO varistors are frequently used for the surge
protection circuits in low-voltage power installations
and electronic devices. They can be used as single
protective devices or may be installed as a component in
a complex protection circuit containing different stages.
When used in a complex protection circuit, the voltage
and current characteristics of the protective devices
have to be co-ordinated [1] to share the surge energy
efficiently according to the energy handling capacity of
protective devices. Improper co-ordination between
protective devices may result in their destruction or
large peak voltages at the vulnerable equipment. The
task of protection co-ordination can be performed
manually by considering the nominal parameters of
protective elements, like current rating and clamping
voltage. However, manual co-ordination is coarse and
difficult when many stages and different surge
waveforms are involved. Knowing the models for surge
protective devices, co-ordination can be performed
using computer simulations [1]. Their advantage is that
the whole varistor element characteristics can be used to
get better accuracy. Besides, all other elements of the
protective device circuit, connecting cables, lumped
components, and circuit parasitic elements, can be
included in the simulation model.
VARISTOR MODELS
The basic varistor model is a general model for metaloxide protective devices. The model can be used for
simulation of varistors, surge arresters or parts of surge
arresters. Fig. 1 shows an equivalent circuit for a
varistor, L is an inductance of conducting leads and C
capacitance of the device package and zinc oxide
material. From the conduction mechanism of the
varistor microstructure the resistive component of the
voltage-current characteristic can be divided into three
A number of different varistor models has been
developed in the last few years [2-4]. The main
1
Proceedings of the XIVth International Symposium on High Voltage Engineering,
Tsinghua University, Beijing, China, August 25-29, 2005
B-49
significant thus directing more current into the first nonlinear section V1.
regions: low, medium and the high-current region. At
low currents, the varistor can be treated as a high value
resistor RL and at very large currents the low value bulk
resistance RB of the zinc oxide grains dominates the
varistor response. In between, RI, has the ideal varistor
property, which is strongly non-linear.
Fig. 2. IEEE frequency-dependant varistor model
Non-linear elements V1 and V2 are defined with V-I
curves given in the IEEE document [4]. We found that
both curves can be fitted with the following equation:
U = kb I I c
Fig. 1. Electrical conduction model of a metal oxide varistor
(basic model)
where:
U
I
k, b, c
The ideal varistor characteristic (V-I curve) in the range
from few µA to tens of kA is approximated by the
interpolation formula [3]:
log(u ) = B1 + B2 log(i ) + B3 exp( − log(i )) + B4 exp(log(i ))
(1)
- is the varistor voltage,
- is the current through the varistor,
- are coefficients obtained from fitting of
the curves given by the IEEE working
group.
Values of the fitting coefficients for Equation (2) are
given in Table 2.
where i is the current through the varistor and u is the
voltage across the varistor. The parameters B1, B2, B3
and B4 are unique for each varistor type.
Table 2. Values of the fitting coefficients
The interpolation parameters B1, B2, B3, B4 and circuit
parameters C and L are given by the manufacturer for
every varistor type and nominal voltage. Parameters
used in the simulation for the disk varistor S20K250 are
listed in Table 1.
V1
V2
k
b
c
1.2968167
0.9713959
1.000005
1.000004
0.0376332
0.0510025
These two fitted curves were implemented with
MODELS in ATP-EMTP. As these two curves present
only relative per unit ratio, they have to be additionally
multiplied with a constant, specific for every varistor
and obtainable from the manufacturer or by
measurements. For the purpose of this investigation the
constant was obtained from the measurements. To
complete the model, one has to calculate the linear
elements (L1, L2, R1, R2, C). They are calculated from
the varistor geometry. As given in the IEEE working
group paper, L1 represents inductance of the varistor, R1
is used for stabilisation of the numerical calculation, C
represents varistor capacitance and L2, R2 comprises the
filter between the two non-linear resistances. The values
calculated according to [4] for the IEEE varistor models
are given in Table 3.
Table 1. Parameters used for the basic varistor model
Coefficient
B1
B2
B3
B4
C [pF]
L [nH]
(2)
Varistor S20K250
2.6830619
0.0261918
-0.0006173
0.0045183
700
13
The second varistor model evaluated was the
recommended model for lightning studies suggested by
the IEEE working group [4]. The model is intended for
station class arresters primarily. The aim here was to
check if the model could be adapted for the varistors.
The model is refereed as the frequency-dependent
model. The non-linear V-I varistor characteristic is
represented with two sections of non-linear resistances
separated by an R-L filter (L2-R2). The circuit diagram
of the model is given in Fig. 2. For fast front surges the
impedance of the second R-L filter becomes more
Table 3. Values of the IEEE model parameters
S20K130
S20K250
2
R1
[Ω]
L1
[nH]
R2
[Ω]
L2
[nH]
C
[pF]
d
[mm]
0.2
0.3
0.4
0.6
0.13
0.195
30
45
0.2
0.3
2
3
Proceedings of the XIVth International Symposium on High Voltage Engineering,
Tsinghua University, Beijing, China, August 25-29, 2005
B-49
to 400 MHz was used for voltage measurements. The
current was measured with current coil (Pearson
Electronic, pulse current transformer model 1049)
terminated with a 50 Ω resistor. The ratio for the current
was 500 A/V.
To reduce effects of the surrounding circuit varistors
were put in a specially designed closed metal container
(5 cm long, 4 cm in diameter) and were connected using
short leads (1 cm). The container was connected to the
source using short conductors (10 cm).
IMPULSE GENERATOR MODEL
The Schaffner impulse generator was used for testing of
the varistors. The internal or fictive impedance of the
generator which is the ratio of the peak value of the
open circuit generator voltage divided by the peak value
of the short-circuit generator current used for the
combination wave tests of surge protective devices,
class III was 2 Ω.
The voltage and current waveshapes during varistor
operation are influenced by the test generator
characteristics. Therefore, generator models also have to
be included in the computer simulation of varistor
operation. For that purpose a generator model was
created.
The generator model was created according to the
manufacturer specifications. Its circuit diagram is
shown in Fig. 3.
Fig. 4. Measuring circuit diagram for the measurement
of the varistor voltage and current
The voltage and current waveshapes of different
varistors were measured. Varistors had different
diameters (5 to 20 mm) and different nominal operating
voltages (17 to 680 V). A series of pulses with different
voltage amplitudes were applied to each varistor.
Fig. 3. Circuit diagram for the Schaffner generator
The generator model was adjusted to a particular
generator used for these measurements and the model
parameters are shown in Table 4.
Measuring results
Measurements described in the previous section were
performed on two sets of varistors to see if the varistors
are performing as specified by the manufacturer. As the
applied magnitudes of the pulses were below the
manufacturer ratings, the degradation of the varistor
material should not affect the results. During the
measurements no substantial difference was observed
between the two sets of measured varistors, which also
confirms that no substantial change had happened to the
varistor material.
Parameter
C1
Parameter
R2
Value
10 µF
Value
C2
10 pF
L1 = L2
12 µH
R
19.8 Ω
L3
20 µH
R1
1.125 Ω
L4
4.5 µH
72 Ω
MEASUREMENTS
1000
Measuring set-up
3,0
Voltage
Voltage [V]
800
The measuring circuit diagram for obtaining the voltage
and current waveshape characteristics is shown in Fig.
4. The Schaffner generator NSG 650 was used as an
impulse source.
2,4
600
1,8
400
1,2
Current
200
0,6
0
The Schaffner generator can give a 1.2/50 µs impulse
voltage in an open circuit and an 8/20 µs impulse
current in a short circuit. The charging voltage and
triggering of the generator were controlled with a
computer via a serial connection.
The voltage and current were recorded with the LeCroy
LC574AL oscilloscope with the sampling rate of 4
Gsa/s. LeCroy PPE 6 kV 1000-times voltage probe with
inner impedance of 50 MΩ and the frequency band up
Current [kA]
Table 4. Parameters for the Schaffner generator model
0,0
0
5
10
15
20
25
30
35
40
Time [µs]
Fig. 5. Measured voltage over and the current through
the varistor. The varistor was SIOV S20K250.
In Fig. 5 measured voltage and current waveshapes for
the varistor S20K250 are shown (the generator was
charged to 6 kV). The voltage peak occurs before the
3
Proceedings of the XIVth International Symposium on High Voltage Engineering,
Tsinghua University, Beijing, China, August 25-29, 2005
current peak. If the varistor during operation is purely
resistive, both current and voltage peaks would have
occurred at the same time.
Waveshapes similar to the ones shown in Fig. 5 were
obtained for other types of the tested varistors. The time
delay between the voltage and current peak was
observed in all measurements, the voltage peaking
before the current. Variations in the voltage and current
waveforms from different samples of varistors of the
same type were negligible.
B-49
900
3000
[V]
[A]
Current
750
2500
600
2000
Voltage
450
1500
300
1000
150
500
0
0
5
10
15
(file nIEEEmodel-arrester.pl4; x-var t) v:VOLT
20
25
30
[us]
0
35
c:CURR -C
VARISTOR MODELLING
Fig. 7. Simulated voltage over and the current through
the varistor, the IEEE surge arrester model, varistor
S20K250
As mentioned earlier, the varistor models were
implemented in the circuit simulation program ATPEMTP using MODELS language.
The above described models were used to replicate the
measured current and voltage waveshapes. First the
basic varistor model with a single non-linear resistance
was tested. The non-linear V-I characteristic was
described with Equation (1) and its coefficients. Fig. 6
shows the current and voltage waveshapes obtained
from simulations of the S20K250 varistor and the
Schaffner generator.
1200
The model suggested by the IEEE working group
reproduces the current and voltage waveshapes quite
well, too. It tends to underestimate the voltage
magnitudes for similar currents.
Another possibility to represent the behaviour of the
varistor and to compare the results from the
measurements as well as from simulations is to plot
instantaneous values of the voltage against the current to
obtain the V-I characteristic. Looking at the V-I
characteristics (Fig. 8), one can see quite a substantial
difference among them. First, the area of the V-I curve
is almost zero in the case of the basic varistor model. It
can be seen also that the basic model curve lies in
parallel to the decaying portion of the measured V-I
curve (the lower part of the measured characteristic)
down from the current peak. Also seen is that the basic
model overestimates the peak voltages over the varistor
by about 9 % for a given current.
3000
[V]
[A]
Current
1000
2500
800
2000
600
1500
Voltage
400
1000
200
500
0
0
5
10
(f ile S20K250Basic1.pl4; x-v ar t) v :VOLT
15
c:CURR -X
20
25
30
[us]
0
35
Fig. 6. Simulated voltage over and the current through
the varistor, the basic varistor model, varistor S20K250
900
800
700
600
Voltage [V]
The voltage and current peak are at the same time. This
is due to how the varistor is modelled. The non-linear
resistance is modelled with a non-linear element,
causing that behaviour. It seems that the impedance
values of the linear elements such as inductance and
capacitance given by the manufacturer are not
significant enough to have much of an effect on the time
delay between the voltage and current peak.
500
400
300
200
100
0
0
500
1000
1500
2000
2500
3000
Current [A]
Basic model
The second model used was the model suggested by the
IEEE working group. The model parameters were
selected according to Tables 2 and 3. The same V-I
characteristics as given in [4], but scaled to fit the
dimensions of the varistors used here, were adopted for
both non-linear elements V1 and V2. The underlying
assumption here is that all varistor material behaves by
the same V-I curves as given in [4], which is not true in
reality. The simulation results are given in the following
diagram, Fig. 7.
IEEE arrester model
Measurement
Fig. 8. Comparison of the V-I characteristics from
different varistor simulation models with the measured
characteristic
Comparing the V-I characteristic of the IEEE model and
the measured V-I curve (Fig. 8), some differences can
be noted. The area of the V-I curve for the IEEE model
is smaller and also the slope is smaller. The IEEE model
underestimates the voltage over the varistor for about
8.5 % for a given current for a S20K250 varistor and for
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Proceedings of the XIVth International Symposium on High Voltage Engineering,
Tsinghua University, Beijing, China, August 25-29, 2005
B-49
18.5 % for the smallest measured varistor (small
diameter and small nominal voltage). This difference
can be explained with the scaling factor for non-linear
resistances V1 and V2 (it was selected according to the
S20K250 varistor).
are very close to the measured values. For the varistors
having the diameter and nominal voltage similar to the
ones of the S20K250 varistor (scaling factor was
optimised to that varistor) that difference is around 1 %
of the measured values for voltages and currents.
NEW VARISTOR MODEL
The proposed varistor model can properly simulate
voltage and current peaks as well as dynamic behaviour
but the V-I characteristic is still different from the
measured V-I characteristic (Fig. 11, compare the black
and the red line). This is due to general V-I
characteristics for non-linear resistances V1 and V2,
from the IEEE surge arrester model.
As shown, none of the two compared varistor models
was able to reproduce all the features of the voltage and
current waveshapes. From the V-I curves in Fig. 8,
some problems in estimating the voltage over the
varistor accurately for a given current can be observed
with both models. Discrepancies between the simulated
and measured results can have some impact on
modelling the protection co-ordination between
protective devices.
For achieving a better accuracy, both non-linear
resistances should be fitted to the V-I characteristics of
a particular varistor (the improved varistor model). Both
non-linear resistances are modelled using the same
equation (1). But since there are two different non-linear
resistances, two sets of parameters are needed for each
varistor.
One set of the parameters can be obtained from the
manufacturer’s catalogue and is the same as for the
basic model. The parameters for the second non-linear
resistance are obtained by fitting the measured voltagecurrent characteristic of the simulated varistor.
Therefore, a new varistor model is proposed for
improving the simulation accuracy. To obtain the time
delay between the voltage and current peak, two nonlinear resistances should be used as in the IEEE model.
Other improved varistor model parameters are selected
almost in the same way as for the new varistor model.
The values of capacitance C and inductance L1 do not
change. The value for the inductance L2 was chosen
according to the measurements by fitting together both
non-linear resistances. The scaling factor for this
presentation of the V-I characteristics for non-linear
resistances is not needed because the real V-I
characteristics are used. Parameters used in the
simulation of varistor S20K250 are listed in Table 5.
Fig. 9. Proposed varistor model
In the first step, the same non-linear resistances were
used as proposed by the IEEE working group.
Parameters of the proposed model (Fig. 9) are
calculated as follows:
Table 5. Improved varistor model parameters
The inductance L1 [nH] = inductance value
from the catalogue / 2
Coefficient
B1
B2
B3
B4
C [pF]
L [nH]
The inductance L2 [nH] = round up (the
operating voltage / 100) * 18 nH
Inductance L1 can also represent the inductance of
connecting leads and in this case the influence of L1 on
the new varistor model behaviour can be much bigger.
Capacitance C can be read directly from the
manufacturer’s catalogue. The correct value of the
scaling factor for non-linear resistances V1 and V2 can be
obtained with some iterations from varistor data which
is not time consuming since the whole procedure can be
easily done with the computer.
First part of the
model
2.6830619
0.0261918
-0.0006173
0.0045183
700
6.5
Second part of the
model
2.6229461
0.01295638
-0.00019911
0.00856252
700
260
In the following figure (Fig. 10) results of the varistor
voltage and current simulated with the improved
varistor model are presented. Behaviour of the time
delays between the voltage and current peaks for the
improved varistor model is very similar to that of the
time delays obtained by measurements (Fig. 5). Also the
values of the delays are the same as those obtained by
measurements.
The accuracy of the voltage and current peaks
simulations with the improved model is in the range of
1 % of the measured values.
The simulations with the new varistor model were made
for all varistors for which the measurements were made.
The results from simulations were compared with
measurements and simulated voltage and current peaks
5
Proceedings of the XIVth International Symposium on High Voltage Engineering,
Tsinghua University, Beijing, China, August 25-29, 2005
900
CONCLUSIONS
3000
[V]
[A]
750
2500
600
2000
1500
300
1000
150
500
0
0
5
10
Co-ordination of protective devices is very important
for designing proper transient protection of the
vulnerable systems. By using computer simulations, coordination efficiency and accuracy can be greatly
improved. This requires accurate models of protective
devices as well as models of accompanying units
(impulse sources, conductors).
Current
Voltage
450
15
20
(file nVARmodel-S20K250-1.pl4; x-var t) v:VOLT
25
30
[us]
B-49
0
35
The investigation shows that the existing varistor and
surge arrester models do not reproduce the varistor
behaviour accurately. The basic varistor model can not
adequately describe the dynamic behaviour of varistors.
The IEEE surge arrester model can be used for varistor
simulations but the voltage and current waveshapes do
not agree completely with measurements.
c:CURR -C
Fig. 10. Simulated voltage over and the current through
the varistor S20K250, the improved varistor model
Finally, by comparing the V-I characteristics obtained
with the new varistor model, improved varistor model
and measured V-I characteristics (Fig. 11), the
improvement of both previously tested models can be
seen (compare Fig. 8 and 11). The V-I curve simulated
with the improved varistor model is very close to the
measured V-I curve. The difference between both
curves is in the range of a few percent.
The varistor model proposed here combines the
advantages of the two investigated models and can quite
accurately predict the varistor behaviour. The results are
especially good when the correct V-I characteristics for
the varistor are used.
By assuring better model accuracy, protection coordination can be performed more precisely thus better
predicting dangerous system overvoltages.
800
700
Voltage [V]
600
REFERENCES
500
400
[1]
300
200
100
0
0
500
1000
1500
2000
2500
3000
[2]
Current [A]
New var. model
Improved var. model
Measurement
Fig. 11. Comparison of V-I characteristics from
different varistor simulation models with the measured
characteristic
[3]
The simulation results, peak voltages and peak currents
from the models used in this research are collected in
Table 6. To allow for a comparison, there are also
measurement results. All the results are for the
S20K250 varistor.
[4]
Table 6. Comparison of simulated and measured
voltages and currents for different models
[6]
Model or measurement
Peak
voltage
[V]
Peak
current
[A]
Measurement
Basic varistor model
IEEE surge arrester model
New varistor model
New - improved varistor model
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798
666
728
729
2525
2500
2571
2536
2541
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