Conference Record of the 1998 IEEE International Symposium on Electrical Insulation, 7-10 June 1998, Washington, DC, USA. Vol. 2., pp. 607--610.

Lavrentyev Institute of Hydrodynamics SB RAS

Novosibirsk, 630090, RUSSIA

All growth criteria considered below are based on two main
assumptions. Firstly, the growth is stochastic in time. Secondly,
the probability of streamer growth is proportional to some
function of local electric field ** r(E)** depending on dielectric
properties of a substance. Thus, the generation of new bonds is
governed by some stochastic growth criteria in each time step.

The electric field potential was obtained by solving the Poisson equation. Charge transport along the streamer branches was computed according to Ohm's law for each conductive element.

In single-element models, we actually have to take both the new bond and the "physical" time interval in a stochastic way.

** p(E_{i} ) = r(E_{i} ) / $\sum _{i=1}^n$
r(E_{i} )**.

For the first time this model allowed one to simulate the formation of branching streamer structures having fractal dimension.

*E _{i} > E_{*} - *d

a new region of a conductive phase arises. A random value
**d**
is assumed to take into account inhomogeneities in dielectrics,
thermal and other fluctuations, including local microfields acting
on the molecules.

The following probability distribution for fluctuations
**d**
is used

**j(d) =
exp( -d / g) / g ,
(2)**

that is **d = - g ln(x)**.
Hereinafter

*r(E) = A* e^{E/g} ,
(3)

where

*A *= $\frac ${1}{t}
exp(*-E _{*} / g*) ,

where ** A** and

To each random value
**d _{i}**
corresponds the time of appearance
of the

**t _{i} =
- $\frac ${ln(1-exp(-d_{i} / g))}
{r(E_{i})}.
(4)**

The distribution of times
**t _{i}**
is

**j (t _{i} ) =
r(E_{i} ) exp(-r(E_{i} )
t_{i} ).
(5) **

Thus, condition (1) is exactly equivalent to the inequality
**t _{i}** <

** p(E_{i} ,t) = $\int $\limits_{t}^{$\infty $}
r(E_{i} ) exp(-r(E_{i} ) t) dt =
exp(-r(E_{i} ) t)**.

Let there be ** n** dielectric sites near the structure. Then the
probability that no bonds arise in time

** P(t) = $\prod $\limits_{i=1}^{n}
p(E_{i} ,t) = exp(-S_{n}t)**,

where ** S_{n} =
S_{i=1}^{n} r(E_{i} )** . The
corresponding
distribution function of the time of appearance of the first new
bond is

At the same time, a new bond in the single-element model can be selected in several ways:

- for instance, by adding, as in [5],
the bond that has minimum delay time

**t**to the conductive structure (see in [6]);_{i}= - ln(x_{i}) / r(E_{i}) - using the NPW criterion;
- using the FFC. A bond that has the shortest of the generation times
**t**calculated from Eq. (4) is chosen._{i}

- The times
**t**is computed for all candidate bonds as in Biller's model. All the bonds having_{i}= - ln(x_{i}) / r(E_{i})**t**arise._{i}< t - Several time steps
**t**are sequentially performed as in SSTM as long as_{Sj}= -ln(x_{j}) / S_{i=1}^{n}*r(E*)_{i}**S**._{j=1}^{m}t_{Sj}< t

The remaining growth criteria with the "physical" time that were
considered above (FFC, SSTM, Biller's, and modifications of them)
use the same distribution (5) of the stochastic time and,
therefore, describe correctly, in particular, the statistical time
lag of breakdown. Thus, these models are essentially equivalent,
provided that the function ** r(E) = A e^{-E/g}**
is the same.

The basic difference of single-element models from multi-element models is that they consider the competition between candidate bonds in different ways. The competition is neglected in multi-element models, and the first bond that appears is assumed to suppress the growth of other bonds in single-element models. All multi-element models allow one to introduce a constant time interval for all steps of the growth. This fact is important for gas-dynamics flow simulation.

The time step is decreased because of the competition among the great number of branches during simulation of the streamer propagation by single-element models. It is reasonable to continue the computations using one of the multi-element models in this case.

The FFC criterion of conductive phase generation satisfies a
specific form (3) of the function ** r(E)**, which is considered to
describe dissociation and ionization in a strong electric field.
All other models allow us to use any other dependences

**L(j) = - 4 p Q/h^{3}** ,

**$\frac{ dQ}{dt}$ = - $\Div $(I_{m})** ,

** I_{m} =
- s_{m} s
D j_{m} / l_{m} ** ,

where **L** is the finite-difference Laplace operator,
** h** is the
lattice bond length,

We used time-implicit finite-difference method for the equation of charge transport to obtain the equation for computation of electric potential

**L(j ^{n+1})h^{3} +
4 p t s
\Div(s_{m}
Dj_{m}^{n+1} / l_{m}) =
- 4 p Q^{n}_{i,j,k} **,

where **t** is a step in time and
** n** is its number. Charge
transport is computed using new values of the electric-field
potential

*Q ^{n+1}_{i,j,k} - Q^{n}_{i,j,k} =
s *t
S

to ensure charge conservation.

The calculation method developed is stable over a wide range of
parameter
**4 p s t s / h^{2}**
up to 10

The simulation was carried out on lattices with sizes up to
90x90x90. The conductivity **s** was assumed to be
constant and equal for all branches of the streamer structure. Streamer
propagation velocity was ensured equal for all bonds including
both types of diagonals, provided that the electric potential
difference between their ends was the same. This statement was
tested specially.

For example, typical streamer structures obtained with
multi-element FFC model (Fig. 1) and single-element Biller's model
(Fig. 2) are shown. A denser streamer structure is formed when the
applied voltage ** V** increases, provided the conductivity is the
same (Figs. 1,a--c and 2,a--c). Increase in

Figures 4 and 5 show the graphs of the logarithm of the conductive
bond density averaged over 15 streamer patterns versus the
logarithm of the distance * r* from the centre. There is no fixed
value of the fractal dimension for the streamer structure at
finite

**Figure 4.**
Differential fractal dimensions of streamer structures
for FFC model. ** V = **10(a) and 30(b)
(

Some discrepancies between single-element and multi-element models
exist in the high-field region in the vicinity of the central
electrode. There multi-element models give a higher density of
streamer channels (Fig. 5b). These differences practically vanish
in the case where the time step **t**
in multi-element models is small, and, hence, the competition of
neighboring bonds becomes insignificant.

**Figure 5.**
Differential fractal dimensions of streamer structures.
(a) Biller`s model, (b) FFC (·) and Biller's (+) models,
** V** = 5,

The same simulations were performed with SST models and new multi-element modification of Biller's model. These results agree with those described above.

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