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PROGRAM FOR COMPUTATION OF OPERATION
PARAMETERS OF LARGE TRANSFORMERS
Ivo DOLEŽEL
CZECH TECHNICAL UNIVERSITY, PRAHA, CZECH REPUBLIC
Pavel KARBAN
UNIVERSITY OF WEST BOHEMIA, PLZEŇ, CZECH REPUBLIC
ABSTRACT
About fifteen years ago Prof. Doležel started developing a relatively large program
package for solution of a wide spectrum of tasks associated with operation of large power
transformers. The program ordered by one of the principal Czech manufacturers – ŠKODA
Plzeň – was written in FORTRAN 77 for environment MS-DOS supplemented with memory
extender DBOS and contained the following modules:
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Solution of the 2D magnetic field distribution in the window. It could be calculated either
analytically (provided that the window contains no ferromagnetic subdomains such as
shunts) or numerically, using the finite element method.
Determination of radial and axial forces in particular windings, their self- and mutual
inductances and short-circuits voltage.
Determination of additional losses in the windings.
Building of the equivalent circuit of the transformer in the form of a complete polygon (that
respects both resistances and inductances of particular windings).
Investigation of steady-state operation regimes using the equivalent circuit.
Solution of the 2D electric field distribution in the window. FEM solution of the timedependent 2D magnetic field distribution in the magnetic cores (eventual transversal
magnetic flux is neglected) and corresponding magnetization losses.
Computation of the time-dependent magnetization currents in particular phases.
ABSTRACT
But even when the results mostly very well corresponded with measured values and
the package was successfully used for more than 10 years, it was not able to carry out
some fine computations due to the maximum permissible number of nodes in discretization
mesh (under DOS this number was only several thousand). That is why were asked to
completely rewrite the code for the WINDOWS environment, improve (as far as it is
possible) the algorithms and build in it some more modules containing:
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Computation of all transversal capacitances between particular windings and between the
individual windings and tank.
Computation of the longitudinal capacitances of the windings.
The time dependence of distribution of voltages and currents in the windings for a step
voltage or voltage wave brought to one of them (for specified initial conditions).
Export of results to special html sheets suitable for handling via Internet.
The paper presents and describes the basic structure of the program written in
Borland Delphi, algorithms used for the computations, way of visualisation of the results
and possibilities of their storing and next processing. Particular steps of the solution are
illustrated on a typical example.
STRUCTURE OF THE PROGRAM - PREPROCESSOR
The program written in Borland Delphi consists of three basic modules.
Preprocessor containing
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input of the complete
set of geometrical
data (dimensions of
the cores, window,
coils and other
elements such as
shunts,
information about
particular coils
(arrangements,
structures and
numbers of partial
turns in the coils,
voltages of currents),
relevant material data
(for example
permittivities or
permeabilities).
Main window of the preprocessor
STRUCTURE OF THE PROGRAM - PREPROCESSOR
The sheet for input the corresponding coil data
Properties of the transformer
STRUCTURE OF THE PROGRAM
PROCESSOR AND POSTPROCESSOR
Processor that allows (March 2005)
Postprocessor that permits
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assembling data for generation of the
mesh,
computing magnetic field and electric
field,
determining distribution of forces and
total losses in particular coils (the total
losses consist of the basic and
additional losses),
determining parameters of the
equivalent circuit with respecting
resistances of coils,
evaluating (from the viewpoints of
delivered power and voltage drops) any
operation regime defined by connection
of terminals of particular coils and
values of loads,
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visualising distribution of various quantities
(coloured maps of distributions of magnetic
vector potential and magnetic flux density
in magnetic fields, electric potential and
electric field strength in electric fields,
distribution of losses and forces in
particular coils and writes into files other
integral quantities such as total energy of
magnetic and electric fields, inductances,
capacitances or short-circuit voltage),
exporting all calculated data in the form of
tables into Excel and MatLab for the
purpose of their eventual further
processing,
exporting the input and output data to
special sheets for their further handling via
Internet.
STRUCTURE OF THE PROGRAM
Processor
list of computations
Main window of the postprocessor
STRUCTURE OF THE PROGRAM
Distribution of magnetic field
along axis x
Distribution of the additional losses in a winding
DESCRIPTION OF THE ALGORITHMS
USED FOR COMPUTATIONS
Magnetic field
Magnetic field in the window is
supposed to be described by equation (in
2D cylindrical coordinate system) for
magnetic vector potential. This equation
reads (the vector potential as well as current
density in the windings has only one
nonzero component in the circumferential
equation that are denoted as A and J,
respectively)
 2 A
r 2
1 A A  2 A
 
 2 
  0 J
r r
r
z 2
The boundary condition along the
window are of the Neumann type, so that it
is necessary to prescribe its value at an
arbitrary point.
While equation on the rectangular area of
the window can be solved analytically (the
results are given by an infinite sum of the
Bessel, Struve and other higher functions),
equation must be solved numerically, using the
finite element techniques.
Electric field
Distribution of electric field is described by
the Laplace equation for scalar potential in the
form
1      2
 r
 2  0
r r  r  z
with the Dirichlet boundary condition along the
window that sounds (the magnetic circuit is
supposed to be grounded). It is, moreover,
necessary to describe distribution of voltage
along the present windings that depends on the
operation regime solved and input the relative
permittivity of all elements. Equation must be
solved numerically.
DESCRIPTION OF THE ALGORITHMS
USED FOR COMPUTATIONS
Forces and losses in the windings
The force acting on a turn of given
dimensions that carries current in magnetic
field is given by integral
F =  J  B  dV
V
which in components gives
Fr =  J Bz dV , Fz =   J Br dV
V
V
Here denotes the volume of the turn. As
the skin-effect at industrial frequencies is
rather low, computations of forces are
performed on the assumption that is
constant over its cross-section (this,
however, is not the case of the additional
losses!!). For a ring of rectangular crosssection we obtain
Fr = 2 J  Bz r  dS , Fz = 2 J  Br r  dS
S
S
On the other hand, the losses the same
turn is calculated from relation
J2
1
2
2
P=
 dV   J2  dV 
 J r  dS
V


V

V
where denotes the electrical conductivity. Now,
of course, the nonuniform distribution of the
current density must be respected, which is
realized using (relative complicated, but very
accurate) analytical expressions. In fact,
computations are carried out not for the turns,
but for partial conductors in them (which is
more accurate).
EXPORT OF RESULTS
A typical sheet with basic information about the transformer
CONCLUSION
The results are automatically stored in corresponding files. But, at request, they may
be exported to Excel or Matlab for further numerical processing. Moreover, they can be
printed (in common with various input data) into special sheets (for the purpose of
Internet handling, presentations etc.).
The program is in the stage of permanent development. The results of each module
were carefully tested by comparison with experiments or values calculated by some
other method. Nowadays, a new module is being completed containing computations of
the time-dependent distribution of voltages and currents along the windings when one of
them carries a wave of given shape. This necessitates additional calculation of the
transversal and longitudinal capacitances of all windings, evaluation of their
conductances and solution of a system of partial differential equations providing the
required distribution.