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IX International Symposium on
Lightning Protection
26 -30 November 2007 – Foz do Iguaçu, Brazil
LE2I UMR CNRS 5158 université de Bourgogne
[email protected] [email protected]
[email protected]
ADEE electronic
avenue Alain Savary 21048 Dijon, France
Abstract - Surge propagation on cables of electrical or
data lines leads to a major protection problem as the number of
equipments based on solid-state circuits or microprocessors
increases. Sub-microsecond components of real surge
waveform has to be taken into account for a proper protection
even in the case of surges caused by indirect lightning effects.
The response of a model of transient voltage suppressor
diode (TVS) based surge protection device (SPD) to fast front
transient is analytically studied, then compared to simulations,
including the lines connected to the SPD and to the protected
Keywords: SPD, Surge Propagation, Fast transient, low
Voltage Circuit, protection distance.
The lightning surges are highly destructive, especially for
the last generation of electronic systems based on solid-state
circuits and microprocessors. These systems work at very
low voltage and are often ground insulated. These advanced
electronic devices have a weak electrical surge withstand
capability, therefore surge protection devices are widely
used to protect electronic systems.
It has been shown that sub-microsecond rise time is possible
in subsequent return strokes [1], whereas building wiring
does not always have sufficient damping effects for steep
fronts [2] induced by close subsequent strokes, and can even
imply overvoltages due to inductances. Therefore, for
proper protection against steep front transients, surge
protection devices (SPD) with extremely short timeresponse are needed nowadays. In low voltage power
distribution systems, setting up SPD on the distribution
board prevents side effects in the installation, avoiding the
flow of surge currents in the branch circuits. But the way of
setting up the SPD and especially the inductance of long
connections to the protected line or to the grounding system,
degrades SPD performances. Moreover, the length of
connecting cables between the SPD and the load can
produce reflection and resonance. These phenomena can
increase the voltage at the load up to twice the protection
level at SPD connection point notably for fast front
In this paper, an analytical study has been carried out in
order to define the behavior of SPD under fast front
transients. The analytical expressions have been compared
with an electromagnetic transient simulation program
(EMTP) for validation. In section II, we will consider the
SPD between two infinite or matched lines. In section III,
the influence of the load and line length is investigated. In
section IV, a parametric study is conducted focusing on two
2.1 Modelling
Figure 1 describes the electrical system under investigation,
where the load is the electronic system to be protected and
where the generator models an incoming surge by releasing
fast front transients. In between, the SPD is connected to the
lines and to the earth via cables. Earth is modeled by its
inductance and its resistance [3], in the case of grounding by
Figure 1: electrical configuration under study.
About the generator, bi-exponential waves will be
considered. Bi-exponential waves are indeed usually
introduced as the normalized waves induced by a stroke.
In the case of infinite or matched lines, there is no need to
define the load precisely, while section IV will deal with
capacitive ones.
The dynamical characteristics of the SPD are so that :
C is the stray capacity of SPD, ranging from few
hundred pico-Farads to nano-Farads for TVS diodes.
- RV is the nonlinear resistance, allowing deviating
unwanted power.
- L is the inductance of the earthing, which is estimated
to be about 1µH.
- RE is the earth resistance with the standard value of 10
The TVS diode based SPD could be seen as derivating from
the IEEE recommended model for varistors [4] also known
as the Durbak’s model, which is presented in Figure 2. Note
that the second nonlinear resistance A1 is decoupled for the
fast front transients (Figure 2) due to the inductive effect of
L1. Thus, this part of the model is ignored. The resistance R0
has been inserted to avoid numerical instability during
simulations [5] and has therefore no physical meaning. So,
these parameters (grayed on Figure 2) are not taken into
account in the following.
The value of RON is the dynamic resistance of the TVS
diode; it has been chosen from the parameter VBR and
clamping voltage VCL at peak pulse current IPP for a given
surge duration. The dynamic resistance is calculated with
the following formula [6] for pulse with 20µs duration.
I PP20µs
This value lies around few Ω for TVS diodes with VBR
higher than 100 volts, so RON is fixed to 1 Ω.
These values are estimated roughly and correspond to a
wide range of magnitude values, whereas it depends notably
on stand-off voltages and peak pulse power of the
considered TVS diode.
At the SPD location, the lines are considered long enough to
be viewed as infinite or matched ones for the time duration
of the front. Therefore the SPD can be regarded as being
connected to two lossless lines of equal characteristic
impedance RC. For usual installation cables, the
characteristic impedance lies between 50 Ω and 200 Ω. So
in figure 4.a, by a Thevenin and Norton equivalence, the
value of Req (in Figure 4.b) is taken to be equal to half the
characteristic impedance of lines:
Req =
VE1 =
Figure 2 : IEEE recommended model (from [8])
L0 is the inductance of the SPD including its connection to
the line. The value of L0 could be included in L.
TVS diodes present strong nonlinearity so V(I) is
approximated to fit a piecewise linear function : RV(V) is a
piecewise constant function. Therefore, it can only take two
different values : ROFF before spark over and RON after, as
illustrated in Figure 3. The transition between these two
values corresponds to the transition voltage VBR of TVS
ROFF and RON are calculated according to [6] from the TVS
diode parameters given on technical datasheets:
leakage current at the stand-off voltage (IR @ VRM)
breakdown voltage at test current (VBR @ IT)
clamping voltage at maximum peak pulse current for
given waveform (VCL @ IPP for 20µs pulse).
The value of ROFF is assumed to be about 1 MΩ, IR at VRM
and VBR at IT give the values of the static resistance under
nominal voltage and breakdown voltage which lies between
a few tens of kΩ and a few tens of MΩ.
Figure 3 : approximated V(I) function of the RV model.
In the following expressions, the subscript ON denotes that RV
takes the value RON, idem for the subscript OFF. These
subscript are also used to differentiate the voltages and
variables before and after sparkover.
a) Electrical circuit
b) Equivalent electrical circuit
Figure 4 : local modeling of SPD setup.
From the electrical circuit of Figure 4.b, voltages VC and VP
before spark over can be written in the Laplace domain:
V 1 ( s).( s -Z1 )( s -Z 2 )
VP OFF ( s) = E
( s -P1 )( s -P2 )
VC OFF ( s ) =
VE1 ( s )
LC ( s -P1 )( s -P2 )
VEn 1 (t ) = A(e − a (t + tb ) -e − b (t + tb ) )-VBR .
V 1(s).(s-Z1)(s-Z2) +VPON (0)(s-Z3)
(s-P1)(s-P2) En
whereas P1,P2 are the roots of
2.2 Analytical expressions
The usual waveform for lightning surge is the biexponential. If we consider for instance a 0,1/50 µs voltage
wave, it is expressed by the following formula:
VE1 (t ) = A(e-at - e-bt ) ,
where a=1,4.10 s and b=2,3.10 s . The bi-exponential
surge is given in Laplace domain by
VE1 ( s ) = A(
s+a s+b
so the voltage at the nonlinear part of the SPD before spark
over can be written after inverse Laplace transform as:
e-a t
e-b t
 , (9)
VCOFF (t) =
P1 t
P2 t
LC  b − a 
 P1 − P2  (a + P1)(b + P1) (a + P2)(b + P2) 
and the terminal voltage of the SPD is given introducing
(8) in (3) expressed in time domain:
 e-at (a+Z1)(a+Z2) e-bt (b+Z1)(b+Z2)
 (a+P1)(a+P2)
VPOFF (t) = A
P1 t
P2 t
+ b-a  e (P1 -Z1)(P1-Z2) - e (P2 -Z1)(P2 -Z2) 
 P1 - P2  (a+P1)(b+P1)
(a+P2)(b+P2) 
The nonlinear resistance changes from ROFF to RON at tb,
when the voltage at the nonlinear part of SPD (VC in Figure
4) reaches the reference voltage VBR.
The point (tb,VBR) is taken as the new origin for VPON , so the
surge is now expressed in this way with the additional
 dVP
dVEn1 
-VEn (0)(s-Z4) +
 dt
d t 
0 
RV LCs ² + ( L + ( Req + RE ) RV C ) s + RV + Req + RE = 0 . (6)
Additional terms should be inserted here for right inverse
Laplace transform. The values of VEn and VPn and those of
their derivative at the new origin of the new coordinates are
taken into account.
VPON (s) =
In equations (3) and (4), Z1, Z2 are the roots of
RV LCs ² + s( L + RV RE C ) + RV + RE = 0 ,
subscript n used for terms expressed in the new coordinates.
The diagram on the Figure 4.b implies the continuity of VC
and IP. The expression of surge VE and its derivative are
taken as continuous, so the derivative of IP and VP are also
continuous. The continuity of VC implies those of the current
through the nonlinear resistance and then through the
capacity, thus the time derivative of VC is also continuous.
These continuities lead to:
VPON (0) = VPOFF (tb )-VBR , VEn1(0) = VE1(tb )-VBR , (13)
dVEn 1
For equation (12), the roots Z3 and Z4 can be expressed such
 Req
1 
. (15)
Z3 = − 
 and Z 4 = −
ON 
Then the expression of VPON in time domain in the new
coordinates is :
 e- a (t + tb ) (a + Z1 )( a + Z 2 ) e-b (t + tb ) (b + Z1 )(b + Z 2 ) 
(a + P1 )(a + P2 )
(b + P1 )(b + P2 )
 P1 t
-b tb
- a tb
VP ON (t ) = A  +
( P1 - P2 )(a + P1 )(b + P1 )
 P t
-b tb
- a tb
- ae )
 - e ( P2 - Z1 )( P2 - Z 2 )(be
( P1 - P2 )(a + P2 )(b + P2 )
 e P1 t ( P1 - Z1 )( P1 - Z 2 ) e P2 t ( P2 - Z1 )( P2 - Z 2 ) Z1Z 2  (16)
- VBR 
P1 ( P1 - P2 )
P2 ( P1 - P2 )
P1P2 
 e P1 t ( P1 - Z 3 ) - e P2 t ( P2 - Z3 ) 
+VP ON (0) 
( P1 - P2 )
 e P1 t ( P1 - Z 4 ) - e P2 t ( P2 - Z 4 ) 
-VE n 1 (0) 
( P1 - P2 )
 dVP
dVEn 1   e P1 t - e P2 t 
ON n
 dt
d t   P1 - P2 
The expression of VCONn is obtained in the same way.
To connect the two curves correctly, VP corresponds in old
ON n
coordinates to
VPON (t ) = VPON (t − tb ) + VBR .
2.3 Comparison with EMTP
The diagram of the Figure 4 is simulated with EMTP, using
a piecewise linear resistance with the values given in Figure
3 and no flashover.
Expressions of VC and VP are plotted (Figure 5) and show a
very good match with EMTP simulations.
circuit in parallel with the SPD[7]; in the second case, a
wave will travel forward and backward on the line between
SPD and load.
Considering global damping effects for the wave front by
capacitive elements of the circuit and transmission lines
frequency dependent losses, first reflections are expected to
give the highest level for VL(t).
The impedance of protected equipment is considered here to
be capacitive according to [7]. It corresponds indeed to the
case of ground insulated electronic devices viewed in lineto-ground mode. The value of this capacitive load suggested
in [7] is considered to be in the top of the range of values for
The generator being still considered to be matched, the
influence of the left line can be considered only as delayed
constant time before the surge reaches the SPD, that is,
without incidence on voltage evolution at the right part of
Figure 6.a. Therefore, we could consider the left part of
Figure 6.a as a Thevenin generator with Vth (s) expressed as
in equation (4), but with Req=RC and VE instead of VE1. Zth is
expressed such as:
Zth =
Figure 5 : comparison between theoretical curves and EMTP simulation
curves for a 5kV - 0.1/50µs wave with L=1µH, C=1nF and Req =50Ω: plain
line VE , dashed line : VC (squares : EMTP), dash-dotted line : VP (circles :
EMTP), dashed line : IP (triangle : EMTP)
This method could be used with more precisely piecewise
linear V(I) function for the approximation of the nonlinear
resistance RV, if we use more values for the dynamical
resistance of TVS diode. Using a continuous nonlinear V(I)
function to model RV leads to an analysis approach of
dynamical systems.
For usual insulated cables in the low voltage system, the
speed c of the waves in the cables is approximately a third
of the speed of the light (usually more for power cables, less
for data cables). For fast transient, the lines connecting
protected equipment whose length exceeds 10 meters could
be seen as infinite for the first hundred nanoseconds of the
When the line between SPD and protected equipment is
short (less than 10 meters long) or after the first hundred
nanoseconds, the influence of the line and its load has to be
taken into account.
In the first case, the parameters of the line can be considered
as localized ones, so the line behaves like a LC resonant
RC (RV LCs² + s(L + RV REC) + RV + RE )
RV LCs² + s(L + RV (RE + RC )C) + RV + RE + RC
a)system with line
b)system after Thevenin tranform
Figure 6 : Thevenin transform of complete system
In Laplace domain, the reflection ratios at the source and the
load ending the line are respectively:
Γ0 =
Zth − RC
Zth + RC
Γl =
1 − RC CL s
1 + RC CL s
which gives in Laplace domain the voltage at a distance x
from the entrance of the right line whose length is d:
2(d- x)
c ) k
R (e c +Γle
V(x, s) = VTH (s) C
∑e-2nτ ( ΓoΓl )
k is the number of reflexions in the plotted duration.
VP(s) is the voltage at the point x=0. The expression of the
forward traveling wave VP1 before any reflection is given by:
VP1 ( s) = VTH ( s)
Note that introducing (18) in (21) leads to VP (s) then
VP (s), which corresponds to simplification of Figure 4.b.
Considering a suitable earth electrode, parameters L and RE
are respectively fixed to 1µH and 10Ω. Breakdown voltage
of TVS diode is taken at 1kV.
4.1 Stray capacity of SPD
Figure 8 shows that a high value of SPD stray capacity have
benefic impact on protection distance in case of fast front
The time tb when the SPD sparks over depends on the
steepness of the wave and the values of C, RC and L
although the inductance has lesser influence.
Figure 7 : Voltage at the load for a 5kV 0.1/50µs surge at the end of a 6
meters line loaded with CL=100 pF and 1µH, C=10nF, RC=100 Ω and
The expression of VP (s), then VP (s), could be obtained
with the same method used in section II with rather more
tedious calculus. Here VL(t) is obtained using the
convolution product to perform inverse Laplace transform
of (20).
For high values of C and steep front, the first reflection
occurs before spark over (tb) while the stray capacity limit
the voltage variation at the SPD (case of short lines as for
Figure 7). The graph of maximum voltage at the load versus
the distance presents local minima the time tb is a multiple
of the back and forth travel time of the wave on the.
4.2 Load capacity
A high capacity of the load implies resonance on the line
and degrade protection level even with short distances and
small values of C (Figure 9). For both high values of C and
CL global damping effect of capacitive components make
this degradation of protection less effective (Figure 10).
A parametric study is now available. The maximum voltage
at the load ending the line (Vmax on Figure 7) can be found
according to each parameter of the circuit (C, L, CL, RE, d, b,
VBR, ).
Nevertheless, the study which is presented in the following
section only take into account conducted lightning effects on
the line upstream from the SPD.
Figure 9 : Maximum voltage at the load vs. line length for a 5kV 0.1/50µs
wave with load capacity CL as a variable parameter. C=100pF,
CL=[100pF,10nF], L=1µH, RE=10Ω, Rc=100Ω and VBR=1kV.
Figure 8 : Maximum voltage at the load vs. line length for a 5kV 0.1/50µs
wave with stray capacity of SPD C as a variable parameter C=[100pF,
10nF], CL=100pF, L=1µH, RE=10Ω, Rc=100Ω and VBR=1kV.
In the case of long protected line, analytical expression of
maximum voltage at the load end can be obtained including
the line length and load impedance. A very nice agreement
between theoretical calculations and EMTP simulations
validates our analytical approach. Moreover, it becomes
possible to analyze the role of each component of the device
separately. Namely, the influence of the load, line
impedance and SPD characteristics versus line length on the
surge propagation can be studied to evaluate distance of
protection in different cases.
Figure 10 : Maximum voltage at the load vs. line length for a 5kV
0.1/50µs wave with load capacity CL as a variable parameter. C=10nF,
CL=[100pF,10nF], L=1µH, RE=10Ω, Rc=100Ω and VBR=1kV.
We thank the Regional Council of Burgundy (France) for
financial support.
4.3 Wave front steepness.
Figure 11 shows that with a high value of SPD stray
capacity C and a small value of load capacity CL, the
maximum voltage at the end of a 9 meters length line will be
1.7kV for a 5kV 100 ns front duration surge although it is
above 1.8kV for a 5 meter long line.
Figure 11 Maximum voltage at the capacitive load vs. line length for a
5kV wave with wave front duration as variable parameter. plain line :
100ns, dashed line : 300ns, dotted line 600ns. Other parameter : C=10nF,
CL=100pF, L=1µH, RE=10Ω, Rc=100Ω and VBR=1kV.
This work shows an analytical approach of the surge
response of TVS based SPD to fast transient.
It appears that for fast transient, a high value of the stray
capacity of SPD can have a positive effect on protection
level for fast transient.
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