Calculation and Measurement of Transient Fields from Voluminous Olbjects R. Zaridze, D. Karkashadze: R.G. Djobava: D.Pommerenke, M. Aidam:
Tbilisi State University, Republik of Georgia Suchuini University, Republik of Georgia Technical University Berlin
Revaz Zaridze, telex: 212 130 beno su, email:
[email protected]
Abstract The paper reports on an ongoing research about numerical calculations of transient fields from ESD. Results fiom a thin wire code which includes an arc model and preliminary results for scattering on voluminous objects are presented.
I
Introduction 2
In the recent years a coupIe of numerical methods for the calculation of transient fields from ESD have been published [ 1,2,4,7]. The methods simplify the problem in one or the other way: The arc is neglected [2,4], or the method can only calculate simple geometries [7].
A complete solution would require the calculation of a lossy 3-dimensional non linear problem with numerically hard to handle dimension: arc lengths < 1 mm and body sizes up to 2 m. The calculation should combine the breakdown physics with electrodynamical processes. Due to the non linearity of the arc, time domain methods are preferable. The ongoing research is aiming at the development of such a method. Although presently only simplified structures can be calculated, some important characteristics of the discharge process can be understood from the results. The paper is structured the following way: Section 2 treats the influence of the approach on the static charge distribution and voltage. Section 3.1 presents results of a thin wire Method of Moments calculation and Section 3.2 outlines the numerical method for the calculation of voluminous objects presently investigated.
!Static Problem
For the transient calculation the static charge distribution is needed as the initial charge value. Objects are charged by triboelectric processes. These processes introduce a charge Ion the body, not a voltage. Once the triboelectric process has come to an end, the voltage is determined by the capacitance, by corona losses or by charge drainage through high impedance ground paths. The capacitance to ground will change, as the object approaches other objects. As the arc length is determined by the voltage and the statistical time lag, the voltage change due to the approach must be considered. Besides the voltage change the charge wilI redistribute during the approach. A considerable fraction of the charge concentrates at the parts of the body which are facing the ground plane especially if the gap is very small compared to the body. During the breakdown the charge distribution on the body determines the behaviour of the current in the gap. The more charge is concentrated close to the gap, the bigger the portion of charge that flows and leaves the body during the first moments. The electrostatic problem can be formulated by a simplified (static) version of the frequency domain electromagnetic Method of Auxiliary Sources (MAS) [5].
2.6.1
EOS/ESD SYMPOSIUM 95-95
In case of bodies of revolution the unknown potential fimction f(x,z) equals some given potential V on the surface of the body. The unknown potential can be approximated by a sum of potentials of elementary auxiliary sources which are located on an auxiliary surface inside the body:
Using this method the dependence of voltage on distance between the body and the ground plane and dependence of the capacitance were calculated different objects. The accuracy was tested against analytical solution of a sphere above ground [13]:
the the for the
with:
LQ
= 2(h+R) R: radius of the sphere h distance between sphere and ground -
Function (2) satisfies Laplace's equation outside the body and provides zero potential on the ground plane. From boundary conditions we find the coefficients ai (i=l, ...N); f(x,z). They represent an approximate solution to the problem in the area z>O outside the body.
22
To calculate the coefficients ai some collocation points (xj,yj)('j=17...7M;M2N) on the surface of the body are
20
chosen leading to a system of linear equations
18
N ai Gi ( x j ,z j ) = 471E V ,j = 1, ...,M ( 3 ) i=l
16
Using (1) it is possible to calculate the potential in every point (z20 and outside the body). The electric field can be calculated by N
Ex>' = Ca, Gf.z(x,z) (4) i=l
with
f x
- xi
x + xi
\
14
10'
1oo
IO1
1oL
10'
distance [mm] Figure 1: Change of the capacitance due to approach t i ground; spheroid of 5 cm and 31 cm semiruces. The free space capacitance equals 13.549pF Assuming an initial voltage of 10 kV at a distance of 1 m to ground, the voltage will drop to 7.3 kV at a distance of 1.8 mm. For any distance smaller than 1.8 mm a discharge can be initiated. The change has three consequences for the ESD:
The location of the auxiliary surface which holds the auxiliary sources is important. Some particular areas inside the body must be included by the auxiliary surface otherwise the fimctions given by (2) can not provide the solution of (3). These particular areas depend on the geometry of the problem [6]. For a spheroid this area degenerates to a line between the two foci of the spheroid. The accuracy of the solution can be calculated by the fulfilment of the boundary conditions and by the erroneous change of the total charge calculated at different distances to ground.
EOS/ESD SYMPOSIUM 95-96
The reduced voltage will cause a faster rising discharge current.
- The charge accumulation close to ground will cause a higher peak current value.
- A voltage value measured in a position which is not the discharge position may not represent the voltage at the beginning of the sparking. 3 Method ofMoments in the Time Domain 3.1 Thin wire code
The moment a discharge current starts to flow, an electrostatic description can not be used anymore. As
2.6.2
long as the discharging structure supports a TEM-wave (e.g. a coax cable, a conical line), the discharge current can be calculated by transmission lines and reflection coefficients. But most real structures do not support TEM-mode, i.e. the object must be seen as an antenna which is excited by the discharge current. As the excitation is nonlinear, the numerical analysis of the travelling wave should be done in the time domain. This allows the combination of the antenna algorithm with a physical description of the ionization process in the arc [7,21. A general picture of the process can be gained by analysing a simple structure, such as the ESD of a vertical rod. This was done by a thin wire code [9, 71 combined with an arc model [7]. The algorithm provides the time depending current and charge distribution.
20
,
2.53311s 1
0.5
1
3.811s
1
charge
0.5
0
1
0.5
1
1.5
2
5.067ns
charge
I
time [ns]
6.33311s
Figure 2: Discharge current of a vertical rod (76cm long, 6 m m diameter) charged to 5 kV prior to the discharge, assuming 0.5" arc length. The arc was modelled by the law of Rompe and Weizel [I 2,7]
1
0.5
The first maximum of the current is caused by the charge which has been redistributed during the approach prior to the discharge. The current drop at 6 ns is caused by the wave which had been reflected at the upper tip. The current fall time is slower than the rise time in the initial moment. Such a change of the wave form would not be expected if TEM waves could be assumed (e.g. discharge of a coax cable). The change is caused by radiation. On the antenna a rapid charge wave is travelling along the surface at nearly the speed of light. Its wave form will be changed due to radiation and reflection. The charge wave causes an electric and a magnetic field. The resulting electric field is shown in the next pictures.
0.5
1 rm1
1.5
0 2
7.611s 1
0.5
0.5
1
1.5
0
2
static field
2.6.3
EOSIESD SYMPOSIUM 95-97
simulator. The rise time of a contact mode simulator is fixed to 0.7..lns. In real ESD the rise time is determined by the arc length. From arc simulation [I 11 it is known that at 5 kV an arc length of 0.8 mm will give a 1 ns current rise time, Using these parameters, the current was calculated and compared to the simulator.
8.867ns
.5
10.1311s 1
charge
0.5
0.5
1
1.5
time [ns]
0 2
Figure 3: Electric field lines and charge distribution during the discharge of a rod (76cm long, 6 m m diameter). The thick lines show the approximate position of the wavefionts. Note that the le? boundary ofthe$eld lines is at a distance of 10 cmfiom the rod. The initial charge on the rod was positive. A wave with a negative charge originates at the arc and travels upwards (The discharge is not initiated at 0 ns). It is superposed to the static charge distribution (0...3.8 ns). At approximately 4 ns the wave reaches the upper tip and is reflected causing a downward travelling wave of negative charge. The reflection creates a second wave front at the top of the rod. At approximately 7 n s the downward travelling wave meets an upward travelling wave which originated from the lower tip of the dipole. Both charge waves nearly cancel out at 10.13 ns. The oscillating process will go on. It is attenuated by radiation and by the remaining arc resistance. The electromagnetic fields are caused by the complete structure not only by the arc as assumed in [ 2 ] . Thus, the complete ESD-simulator must be seen as an antenna, not only the spark region.
time [ns]
Figure 4: Calculated (dashed line) and measured (solid line) initial peak current of a simulator at 5 kV (le?) and its magnetic Jield in 10 cm distance (right). For the measurement the ground strap was substituted by a high impedance plastic wire.
-3 0 2 +
5 ,
22
I
I
-5
-10
W
-15
time [ns]
time [ns]
Figure 5: Calculated (dashed line) and measured (solid line) electric field of a simulator at 5 kVat I O cm distance (le3) and 90 cm distance (right). For the measurement the ground strap was s u b s t z e d by a high impedance plasTc wire. The numerical results can reproduce the major characteristics of the real transient fields. From measurement it is known that different simulators yield different fields. Thus, a simple model provides a sufficient estimation of real fields. A precise calculation can not be done this way.
3.1.1 Simulation of the initial peak by thin wire approximation
3.2
The initial peak of an ESD is often the main cause for disturbances. Its transient fields can be calculated approximately by the thin wire code. A simple simulator model was used. It was approximated by a rod of the same length with a diameter of 20 mm. This diameter was chosen, because it stores the same amount of charge as the initial peak drains of the
EOS/ESD SYMPOSIUM 95-98
Magnetic field equation code for voluminous objects
As the thin wire code concentrates the surface current into a wire, it is not useful for voluminous objects.
For any transient process on voluminous, perfectly conducting objects the boundary value problem can be transformed into a magnetic field integral equation (MFIE) [7]:
2.6.4
where J is the current density on the surface of the body, the exciting field, r' the point of observation on the surface of thebody, t the time of obrvation, r" the point of integration, 3 = 7 - 7 , R = 17 - 7'1 the distance between the points of observation and integration, R t' = t - - the retarded time, n' the normal of the surface C
and c is the speed of light.
3.2.4 Scattering problem Only the MFIE on the surface of the body is needed. may be any incident field.
3.2.1 Numerical treatment For the solution of integral equations a technique proposed by Bennett, Miller and others [7] was used. In the case of a body of revolution the MFIE is transformed into the following equation
Now the following physical problems can be formulated:
3.2.1 Simulation of an electric field source charging the system through a thin perfectly conducting wire. For this problem the boundary conditions are given by:
(:Eexc( t )+ gsey( t) + J!?body (t)),g MFIE Idere
OSVSX
,
7c OSWS-
2 bsim(v')cos(w') , y'= bsin(v')sin(w') , z'= acos(v') x=bsin(v) , y=O , z=acos(v)
XI=
R = d(x - x')2 + y' 2 +(z - 2 )2
= 0 on the wire and
on the surface of the spheroid.
gexC is a given exciting field and ,!7,~
>witha = -rg
where orthogonal surface coordinates v, w are introduced
In
h + , / m
- h + d m
a2 at
=-a-
1
, where 2h is the length
G!(v;v',w') =
(x - x')G - (z - Z')T-
R2
and ro the radius of the wire, is the self influence of the ,wire. Ebody is the electric field caused by the body on the wire and Z?inc in the MFIE is the magnetic field radiated by the wire. ___ 3.2.2 Simulation of a magnetic field source charging the system through a thin perfectly conducting wire. This can be formulated the following way:
3 = 2Z x BeXC + PJ+ 2E x #...
Now the surface is divided into small patches. Furthermore, it is assumed that the current is constant on every patch during each time-step. For simplification we approximate the integration over a patch by multiplication of the influence of the current in the middle of the patch by its area. For the self-patches Bennett's approximation is used:
// ...
on the wire and
body
MFIE
p5
on the surface of the spheroid.
is the self influence of current on the wire.
= PJ
self-patch 1
#... body
yields the magnetic field radiated by the spheroid.
3.2.3 Simulation of the discharge
s
- k,) P = -J-k 2 47c where kv and k , are the principle curvatures of the surface and AS is the area of the self-patch. For calculation of the retarded time it is necessary to know the time delay between the event which caused the field and the reaction at the point of observation. This is given by:
R
In this case we suppose the existence of a channel with time-dependent conductivity c(t) in the gap. Therefore
o ( t ) J ( t >= Eseq-(t)+ Ebody(t) in the channel and on the surface of the spheroid. MFIE
- = ( N + y)dt C
where N is an integer and 0 Iy 51. A two-point smoothing approximation was used for the current and its derivative:
2.6.5
EOS/ESD SYMPOSIUM 95-99
5
J(v',t')= ( I - y ) J ( v ' , ( j - N)dt)+yJ(v',(j- N - 1)dt) Due to problems in the link between the arc model and the electrodynamic model only preliminary results can be shown presently. An example for the reaction of a simple structure to an indirect ESD is given in Figure 6. It shows lines of constant magnetic field strength for a spheroid which is excited by the far field of the initial peak of an ESD simulator (the impulse was approximated by the derivative of a Gauss-impulse). The original electromagnetic wave has already passed the spheroid. It excited a secondary (creeping) wave which is just leaving the spheroid.
time-step 100 1
I
0.5
zo U
-0.5 sour
