Biuletyn Instytutu Spawalnictwa No. 01/2012

1 dr
r dt
- in the resistance form
1 ⎛ u
⎜1
−
⎝ U
2
=
kol
2
θC
C
⎞
⎟
⎠
(3)
where ϴ C
– time constant of the model, U C
–
model voltage, g = 1/r – conductance and resistance
of the arc column.
Similarly, on the basis of the power balance
equation (1) and after adopting appropriate
simplifying assumptions and transformations
[3], one can obtain the Mayr-Kulakov models
in the conductance form
1
g
dg
dt
1 ⎛ u ⎞
koli
= ⎜ −1
⎟
θM
⎝ PM
⎠
or in the resistance form
1
r
dr
dt
1 ⎛ u
⎜1
−
⎝ P
=
kol
θM
M
i ⎞
⎟
⎠
(4)
(5)
where U stat
(i) – static voltage-current characteristics,
G stat
(i) – static nonlinear conductance,
R(3)
stat
(i) – static nonlinear resistance.
After substituting (8) and (9) to (6) and (7)
one obtains a generalised Mayr-Kulakov equation
in the conductance form
1 dg 1 ⎡Gstat
() i ⎤
= ⎢ −1
g dt θ
⎥
(10)
Ms ⎣ g ⎦
or in the resistance form
1
r
(11)
(4) When conductance does not change in time,
the static characteristics of the arc in this model
are as follows:
U
(5)
dr
dt
1 ⎡ r
() ⎥ ⎤
= ⎢1
−
θ
Ms ⎣ Rstat
i ⎦
() i
stat
=
P
i
M
(12)
(1
(
where ϴ M
– time constant of the model, P M
–
power of Mayr-Kulakov model.
The Mayr-Kulakov arc model can be transformed
into another, general conductance form
1
g
dg
dt
() t
1 ⎡ P
() ⎥ ⎥ ⎤
kol
= ⎢ −1
θ Ms ⎢⎣
Pdys
t ⎦
or the resistance form
(6)
(13)
Therefore, on the basis of these formulas
one can write models (6) and (7) in the conductance
form
(6)
1 dg 1 ⎡ i ⎤
= ⎢ −1⎥ g dt θ Ms ⎣ g ⋅U
stat
() i ⎦
1 dr 1 ⎡ P () ⎤
or in the resistance form
= ⎢ −
kol
t
1 ⎥
(7) (7)
r dt θ
Ms ⎢⎣
Pdys
() t ⎥ ⎦
1 dr 1 ⎡ ri
()
where ϴ Ms
– corresponds to relaxation time of
⎥ ⎤
= ⎢1
−
(15)
r dt θ
Ms ⎣ U
stat
i ⎦
thermal process, and the supplied electric power
amounts to
U stat
(i) = - U stat
(-i).
The application of the appropriate approximation
of static characteristic U
2
i 2
P
(8)
stat
(i) offers more
kol
() t = ukoli
= = i r
(8)
g
precise determination of arc dynamic characteristics
if compared with hyperbolic static characteristic,
pre-set only by one constant Mayr-Ku-
As the processes of heat dissipation slowly
respond to external disturbance, one can assume
that the power of losses is basically deterlakov
power value. Such an approach extends
somewhat the range of the model applicability to
mined by static characteristics [4] i.e. include stronger currents, when a characteristic
2
i 2
P () t U () i i i R () i
is no longer drooping [5] but becomes flat and, in
dys
=
stat
⋅ = =
stat
()
(9)
(9)
Gstat
i
the case of stronger currents, is even rising.
G
stat
() i
i
=
P
2
M
=
U
i
stat
2
=
1
() i ⋅i
R () i
stat
(14)
(
NR 01/2012
BIULETYN INSTYTUTU SPAWALNICTWA
17