Download V. Optimization of welding transformer

Welding Transformer Analysis and Optimization
by Finite Element Method
István Király and Nándor Burány
Subotica Tech – College of Applied Sciences
[email protected]
Abstract - The development of computers made possible the
magnetic circuit analysis with finite element method (FEM).
This methodology is suitable for transformer inductance
evaluation and lumped model developement for further
analysis with simulation software. In this paper, the
modeling, simulation and optimization of welding
transformers based on FEM is discussed. This kind of
transformer differs from the classical power transformer
where good coupling of windings and low magnetization
current is mandatory. By welding transformers, the aims
are acceptable open circuit voltage and short circuit current
at the secondary to obtain good welding arc stability. To
fulfill these requirements, large flux leakage have to be
tolerated. Load current regulation is achieved by magnetic
shunt which changes the leakage, or by thyristor phase
control. The transformer currents can be calculated by use
of step by step numerical integration methods (for example
Runge - Kutta) based on lumped parameters obtained from
FEM modeling and circuit modeling of the welding arc.
Determined values of
transformer currents can be
substituted back in the finite element model to recalculate
the magnetic field distribution. The solution of FEM model
gives a visual view of the magnetic field, allows the
determination of flux via appropriate areas. Different core
forms and winding techniques are analyzed to optimize
transformer parameters.
Figure 1. Change of current by constant voltage welding with the
length of arc
Keywords: welding transformer, finite element method,
numerical simulation, leakage inductances.
I.
INTRODUCTION
In our highly industrialized world, welding is the most
used binding method since the first patent issued in the
late nineteenth century. Many types of welding are
known. The common in every method is that heat is
generated at the point of binding, which melt the base
material. Heat can be generated by electric arc. Two types
of the arc welding apparatus can be distinguished, they
behaves approximately as a constant current (CC) source,
or as a constant voltage (CV) source. In both cases
electrical arc exists, but the arc behavior is different. The
voltage drop on arc is proportional to its length, and
slightly changes with arc current Fig 1 [1].
By constant voltage welding sources, the arc current
changes dramatically with arc voltage change, Fig. 2 [1],
so current changes abruptly with arc length. This
fluctuation of current has a great influence on the quality
of welding. The behavior is very different by machines,
which provide constant current during the welding. This
Figure 2. Change of current by constant voltage welding
Figure 3. Change of current by constant current welding
way the influence of the voltage change, which occurs by
change of the arc length is damped Fig. 3 [1].
Constant current welding apparatus have to include a
self regulation mechanism to maintain constant current.
The CV welding machines need a quick external
regulation, because their current changes abruptly with
distance between the electrode and the base metal. The
CC welding machines with appropriately designed
welding transformers have a current limiting feature. This
behavior is achieved by abrupt change of the leakage
inductance of the transformer by current change. This fact
made difficult their mathematical modeling and
simulation, which differs significantly from the approach
used by ordinary power transformers. By
power
transformers the lines of leakage flux close in air, while at
the welding transformers the leakage flux lines go mainly
trough the magnetic shunt).
In the forthcoming part of this work the welding
transformer will be discussed trough calculation of its
leakage inductance and by simulation of its behavior.
Taking into account the advantages of the magnetic
shunt, the magnetic circuit of the welding transformer can
be modeled in two-dimensions to calculate the leakage
inductance by Finite Element Method (FEM), as the
leakage flux lines mainly close through the magnetic
shunt. The results obtained this way, can be used to
determine the characteristics influenced by leakage
inductance by use of other mathematical software
packages.
II.
THE TWO-DIMENSION MODEL OF MAGNETIC
CIRCUIT
One of the main steps of the electrical machine desing
is the determination of the characteristics of theirs
magnetic circuit. Earlier analytical models were used,
which did not consider the exact distribution of magnetic
field and the core saturation. Nowadays, instead of
analytical methods, the inductivities are determined by
FEM software. In this paper FEM calculations were made
by FEMM 4.2 [2] software. This software includes
graphical pre- and post-processors, which can be
manipulated by the LUA script language [3]. The
program is designed so that axis-symmetrical and twodimensional magnetic problems can be solved. The scope
of problems which can be treated by this program is
restricted to the cases, where the lines of flux density are
parallel to a plane. This condition is realized at welding
transformers, which FEM model is shown in Fig. 4., and
Figure 5. Distribution of the magnetic field in the welding
transformer
a solution of magnetic filed distribution for this case is
shown in Fig. 5.
The primary winding (denoted in the Fig 4. by
Primary + and -) generates a Magneto Motive Force
(MMF) which develops the flux shown in Fig. 5. The
greater part of flux (resultant mutual flux) links the
secondary winding, while the other part of flux closes
through the magnetic shunt and the air gap and links only
primary winding (it is the primary leakage flux). The
resultant mutual flux depends on the difference of the
MMF of primary and secondary windings. The leakage
flux depends on the sum of the two MMF, it leads to the
saturation of the shunt, and the abrupt fall of leakage
inductance of primary and secondary windings as it is
shown in Fig 6. for a given magnetic shunts and air gap.
III. CHARACTERISTICS OF LEAKAGE INDUCTANCE
By solving of FEM model of magnetic circuit, the flux
linkages of the primary and secondary windings can be
determined. They are relevant quantities to the
determination of leakage inductance, which varies with
the magnetic saturation of the shunt, and can be
determined by following expression:
L 
(1)
Where the:
- L - is the leakage inductance,
-  - the difference between two calculated
flux linkages,
- I - the difference of two currents which were
used to excite the windings by calculation of the two flux
linkages.
The inductance values between two calculated values
can be determined by a linear function which is defined by
the known end points of the interval.
L ( I ) 
Figure 4. The configuration of FEM model

I
L ( I1 )  L ( I 2 )
( I  I1 )  L ( I1 )
I1  I 2
(2)
-
N1 / N 2 - winding turns ratio.
After the determination of the transformer parameters,
the reduced differential equation can be written. During
the simulation the primary and the transformed secondary
currents to be equal. This assumption simplifies the
whole process of calculation.
In the equation (5) the leakage inductance LK is a
function of the current, it decreases by increase of the
current.
di u p (t )  RK i p  u s (t )

dt
LK (i p )
Figure 6. The characteristic of leakage inductance for various
magnetic shunt
These continuous curves make possible further
numerical simulations of the welding process.
Here:
L ( I 1 ) and L ( I 2 ) are the calculated values
of leakage inductivities, while the current I is in the
interval, determined by currents I 1 and I 2 . The leakage
-
inductances can be adjusted by change of air gap at the
magnetic shunt. By increase of the air gap the leakage
inductance decreases, at the same time the saturation
current increases.
IV. THE VOLTAGE DIFFERENTIAL EQUATION OF THE
WELDING TRANSFORMER
By simulation of ordinary power transformers theirs
differential voltage equations are used [3]. In this case it is
not so relevant. The leakage inductance changes abruptly,
which dictates the decrease of the actual time step during
the numerical simulation. This increases the duration of
modeling but the result will not differ significantly from
the result calculated by the simplified model. The
simplified model consists of a reactance and an
inductance, their values can be determined by a short
circuit test.
If the parameters of the transformer are known, then by
the transforming the resistance and leakage inductance of
the secondary to the primary, the components of the
equivalent circuit will be obtained as follows:
RK  R1  ( N1 / N 2 ) 2 R2
(3)
LK  L1  ( N1 / N 2 ) 2 L2
(4)
The low value of leakage inductance is very useful to
decrease the DC current, which appears by switching
transients [5].
V. OPTIMIZATION OF WELDING TRANSFORMER
After the definition of the magnetic problem, the
conditions, which define the purpose of design, have to be
defined. These are the maximum and minimum values of
the secondary current.
The maximum appears when the shunt is pulled out,
while the minimum value can be measured, when the
shunt is entirely pulled in on its place. The first operation
point helps to determine the number of turns of the
secondary winding, while the second point is used to
evaluate the air-gap length.
A. Determination of the number of turns of secondary
This process starts with the definition of excitations. At
this stage, the best solution is to presume the same number
of turns at the primary and the secondary windings (for
example: 150), and the current increases (for instance
from 0 A to 100 A). The directions of currents are
assumed so, that their MMF-s point in the opposite
direction in the magnetic shunt. This way, as equal MMF
forces were assumed on the primary and the secondary
side, the magnetizing MMF is neglected, because it makes
only 1-2% of the whole excitation. When this analysis is
finished, from the calculated flux leakages, the leakage
inductances of primary and secondary winding can be
evaluated. This is almost constant, as the flux density lines
Where:
- R K - resistance obtained by short circuit test,
- LK - inductance obtained by short circuit test,
- R1 - resistance of primary winding,
-
R2 - resistance of secondary winding,
(5)
Figure 7. The voltage versus current characteristics
close in the air, and it results a constant leakage
inductance. The voltage to current ratio can be calculated
at the given number of turns by use of leakage inductance
and reactance. The maximum value of current can be
determined from the allowed current density and the
whole area reserved for conductors. The product of he
winding window cross section and the filling factor
divided by the number of turns gives the maximum
conductor cross section and the maximum working
current. The necessary welding current, and the presumed
number of turns make possible to determine the number of
turns of secondary winding. From this result, the number
of turns of the primary windings can be evaluated by use
of the ratio of the secondary and primary voltage and the
number of turns of secondary.
B. Determination of air gap length
The determination of air gap length consists of two
steps which are repeated until they do not give an
adequate result. The magnetic shunt is now filling the
whole space between the primary and secondary
windings. The used excitation is same as in the previous
case, only the current changes from 0 A to the maximum
value of previously determined current. When the
magnetizing curve is determined, the next step follows. It
is the circuit simulation by use of the determined leakage
inductance. If the current at zero secondary voltage is less
than the prescribed value, then the air gap has to be
increased, otherwise it has to be decreased. These steps
have to be repeated, with new air gap length, until the
result does not correspond to minimum value of secondary
current.
VI. THE SIMULATION OF THE TRANSFORMER
The FEM analysis allows to use functions determined
by previous circuit simulation. This way that processes,
which are described by non traditional mathematical
functions can be investigated.
The results of simulation help to understand the
behavior and working of welding transformers, and allow
their optimization. By the substitution of numerical values
of the current obtained by circuit simulation back to the
FEM transformer model, the changes of magnetic field
can be observed visually. The results of simulation,
besides the optimization of magnetic circuit, are useful for
developing the regulators, which are used in welding
apparatus.
The current-voltage characteristic of the transformer
Figure 9. Secondary voltage or arc voltage waveforms for different
arc lengths, a simplified model.
shown in Fig 7, is created by use of RMS values of
voltage and current. It describes the dependence of voltage
on current.
As can be seen, the voltage versus current static
characteristics show the same features as usual constant
current welding transformers reported in the literature [1]
The validity of these static characteristics can be proven
also by analysis of arc voltage and arc current waveforms.
To this purpose the instantaneous values of current,
leakage inductance and the secondary windings voltage
are shown in Fig. 8, 9 and 10 respectively. By this
analysis the arc is simply modeled by a bidirectional
constant voltage source. The voltage polarity is the same
as the current polarity in each moment.
The evaluated characteristics clearly describe the
behavior of welding transformers. As it is shown in Fig 8.
the transformer works as an alternating current generator.
The current waveform is almost sinusoidal. The current
amplitude or the current RMS value changes by arc
voltage but the relative changes of current in Fig 8 are
much damped compared to the voltage changes in Fig 9.
This way the arc current will be almost independent of arc
length.
Another critical factor is the secondary voltage, which
has a great influence on the welding quality. Its polarity
changes rapidly which helps to minimize the low voltage
period, and to provide enough voltage to maintain the arc.
Figure 10. The leakage inductance variations during time.
Figure 8. Current waveforms for different arc voltages and arc
lengths
This special performance of the leakage inductance is
ensured by its rapid increasing and decreasing. This helps
the commutation and forces the current to an adequate
value.
This great leakage inductivity at small secondary and
primary currents, helps the open circuit current to restrict
together with magnetizing inductivity to an adequate
value.
The result of simulation compared to the parameters of
real transformer proves the closeness of model to the real
welding transformer. These parameters are:
- the open circuit secondary voltage varies from 43
V to 49 V.
- the welding current varies form 40 A to 120 A
The arc voltage during the welding varies from 25 V to
about 30 V. These parameters can be read from Fig 7. The
neglectiof of magnetizing current and iron losses cause
some difference between the simulation results and the
real transformer behavior.
CONCLUSION
In this paper the FEM analysis approach to a
traditional, line frequency, welding transformer is shown.
The transformer core form allows 2D modeling. The
FEM analysis results are incorporated in circuit analysis
to show the transformer behavior under different load
conditions.
Further work in this field will include the use of FEM
for three phase welding transformer analysis and design,
the analysis of rectifier influence on transformer voltage
and current waveforms and also 2D and 3D modeling of
high frequency transformers incorporated in modern
inverter based welding apparatus.
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