Advanced Calculations of Magnetic Leakage Fields in Transformers
O.W. Andersen
Norwegian Inst. of Technology, N-7034 Trondheim, Norway
andersen@elkraft.ntnu.no
http://www.elkraft.ntnu.no/~andersen/
Abstract
Analysis of magnetic leakage fields in transformers is
necessary in order to calculate important characteristics,
such as reactance, eddy current losses and short circuit
forces and stresses. Analytical methods are unable to
cope with the complexities of such fields.
1 Introduction
The author has developed three finite element
computer programs specifically for this purpose, two for
round cores and one for rectangular cores. Vector
potentials must be calculated as complex numbers when
there are phase shift connections (zig-zag, polygon or
extended delta), and when the leakage field is affected
by induced currents in sheet windings.
2 Sample Calculations
2.1 Simple, two winding transformer
The flux plot in Fig.1 is for a simple, two winding
transformer with wire windings and symmetry about the
radial centerline, so that only the upper half needs to be
calculated.
The left boundary is the surface of a cylindrical core
leg, and the bottom boundary is the radial centerline.
The top boundary could represent the upper yoke, but is
usually further out, so that the position represents a
weighted average of conditions around the periphery.
Calculations assume that the geometry is axi-symmetric.
The right boundary could represent the tank wall, but
again the position is usually further out, as a weighted
average. The position is not very critical.
Analytical methods usually assume that the flux lines
are purely axial, and that they stop at the winding ends.
Magnetic energy is calculated too high in and between
the windings and ignored outside. The two errors tend to
cancel, so that reactance is often calculated reasonably
accurately.
Radial forces and eddy current losses due to axial
flux will be less accurate, but still within reason.
However, since radial flux is ignored, the analytical
methods are unable to evaluate axial forces and eddy
current losses due to radial flux. Also, if the transformer
is more complicated, perhaps with parts of a winding
disconnected due to tap connections, with unequal
winding lengths and so on, there are even more reasons
to apply more advanced calculation methods.
Fig.1. Simple, two winding transformer
2.2 Sheet winding transformer
Fig.2. Sheet winding transformer
When radial flux tries to penetrate a sheet winding,
induced eddy currents will be set up to prevent it,
according to Lenz's law. As a result, flux lines will be
straightened out through the winding and will be almost
purely axial, except at the very ends. This is shown in
Fig.2 in a sheet winding with six turns.
Induced currents are phase shifted with respect to the
main current in the winding, so that vector potentials
must be calculated as complex numbers.
Current densities will usually be very high at the
winding ends, often several times the average. However,
this is concentrated in a small volume, and losses and
temperature rise are usually tolerable.
Current density distribution can be improved by
making the low voltage sheet winding slightly longer
than the high voltage wire winding in a transformer such
as the one in Fig.2.
Current density distribution is calculated in each turn.
The distribution is shown in Fig.3 for the inner turn of
the sheet winding in Fig.2 from the radial centerline to
the end of the winding. Each vertical bar represents the
current density calculated in a finite element mesh. The
horizontal dashed line represents the current density for
a uniform distribution.
this, and now often require documentation from manufacturers that their transformers are designed to withstand short circuits, based on computer programs that
they both are familiar with and have confidence in.
Radial forces are dominant and due to axial flux.
They usually produce tensile stress in the outer winding
and compressive stress in the inner winding. Compressive stress can cause buckling, and the winding must be
properly supported by axial spacer bars.
Axial forces due to radial flux are usually compressive. Accumulated forces acting on the spacer blocks
between disks are usually maximum near the radial
centerline, whereas forces per unit volume are usually
highest at the winding ends. In disk and helical windings
they tend to bend disks between spacers.
5 Eddy Current Losses
In wire windings, eddy currents are usually confined
within such small conductors that their influence on the
magnetic leakage field is negligible. Losses are proportional to (fdB)2 times volume, where f is frequency and d
is the conductor dimension perpendicular to either axial
or radial flux density B. If parallel strands are not properly transposed, circulating current losses will also be
present. Eddy current losses in wire windings are
inversely proportional to resistivity and will decrease as
the winding temperature increases.
In sheet windings, induced currents counteract radial
flux, as mentioned earlier. Current density distribution is
strongly influenced by the length and thickness of the
winding and varies from turn to turn. Eddy current losses
are nearly constant as a function of resistivity, and
therefore also of temperature. The losses often become
excessively high when the winding is thicker than 50-60
mm, which makes it impractical to use sheet windings in
large transformers. The calculation of the losses is
impossible by analytical methods.
Fig.3. Current density distribution, inner turn
6 Crossover Points
3 Reactance
Short circuit reactance is calculated from magnetic
energy, which can be found by integrating either ½HB or
½JA times volume.
In the calculation based on current density J times
vector potential A, the integration must only be performed in winding regions. The integration of field
strength H times flux density B times volume must be
performed in the whole region of the field, but the added
time this takes is insignificant.
Both methods give the same magnetic energy. For the
transformer in Fig.1, in seven digit numbers the first six
digits were identical.
4 Forces and Stresses
Forces at short circuit are calculated as flux density
times current times length. The windings must be
designed to withstand such forces, but this is usually not
validated by tests, and accurate calculations are essential. Utilities are becoming increasingly concerned with
Disk and helical windings often have a large number n
parallel conductors. For them to share the current
equally, they must change positions at n-1 transposition
or crossover points. These points have often been
equally spaced in the past, but their spacing should really
be inversely proportional to average axial flux density
times turns/mm axially. Placing them in proper positions
as given by the computer output involves no extra cost
and can save significant losses due to circulating
currents.
7 Parallel Circuits
Windings often have parallel connected circuits, where
current distribution and circulating currents are not known
in advance, when magnetic leakage fields are to be
calculated.
A three winding rectifier transformer is used as an
example, as shown in Fig.4. It has a primary high voltage
winding consisting of two parallel connected parts H1 and
H2. There are two secondary windings
above each other, one wye connected and one delta connected. Both are designed for the same rated MVA and
kV.
LY
H1
8 Phase Shift Connections
In windings with phase shift connections, currents are
displaced 60 degrees between the two winding parts on
one core leg if the connection is zig-zag or polygon, 30
degrees if it is extended delta (Ref.3). An accurate
calculation of the magnetic leakage field is only possible
if the phase shift is properly accounted for, with vector
potentials calculated as complex numbers.
9 Rectangular Cores
LD
H2
Fig.4. Three winding rectifier transformer
One problem here is to calculate short circuit reactance and forces for a short circuit in one of the secondary windings, such as LY, not knowing ahead of time
what the current distribution is between H1 and H2.
The MVA is zero for LD and 100% for LY, but it is
uncertain initially what the MVAs should be for H1 and
H2, except that the sum should be the same as the MVA
for LY. Two methods for finding the current distribution
between H1 and H2 will now be explained.
Sufficient accuracy can be obtained with calculations
in two dimensions. If the core is round, the field is axisymmetric. With rectangular cores, the field is assumed
to be flat along the straight parts of the coils, axisymmetric around the corners. Reactance and eddy current losses are calculated from weighted averages.
Large shell type transformers can be analyzed with
the program made for rectangular cores.
10 Program Input
Fig.5 shows a transformer without symmetry about
the radial centerline, so that the full height must be
calculated. It has three layers (or windings) belonging to
two terminals (high and low voltage). The layers are
divided up into segments, since they are not uniform
axially in this case.
7.1 Minimizing magnetic energy
Current distributions and circulating currents always
adjust themselves to give minimum magnetic energy.
This is often the easiest way of finding the correct
currents.
100% MVA can be specified initially for H1, zero for
H2. Then gradually MVA is decreased for H1 and
increased for H2 until minimum calculated magnetic
energy is reached within a certain tolerance. The current
distribution will then be correct.
7.2 Equalizing flux linkages
Since H1 and H2 are in parallel, for the correct
current distribution they should have the same flux
linkages. Flux linkages are linear functions of currents.
Again, 100% MVA can be specified initially for H1,
zero for H2. The difference in flux linkages is recorded.
Then current distribution can be changed by say 1%, to
99% in H1, 1% in H2. That will probably make the
difference in flux linkages closer to, but not quite zero.
Linear extrapolation down to zero establishes the correct
current distribution for a third calculation, where flux
linkages will be equal.
If desired, the result can be checked by observing how
the calculated magnetic energy is changed with small
deviations from the calculated currents. The changes
should always be positive.
Fig.5. Transformer without axial symmetry
The programs can handle up to 30 layers, 6 terminals
and 200 segments, which is intended to cover all practical cases.
The finite element grid is generated automatically,
and only the most essential information needs to be
specified about the geometry, connections and current
distribution.
An input file must be made up containing this information, and there are different ways of doing this.
Numbers can be changed and inserted directly in an
input file, using a text editor, based on information in the
user's manual of what the numbers mean.
Headings can be brought in first, using a simple
command. Then the numbers can be changed and
inserted, using a text editor. Finally, the headings must
be removed, again using a simple command.
An input subprogram can be run, which creates new
input, based on earlier input. Detailed explanations are
given when the program asks for the input items, one at
a time. When an item is the same as in the earlier input,
the user simply strikes ENTER. The new input file is
created in the right format automatically.
A design program can create the input file for the
finite element leakage field calculation completely automatically. Such a program has also been made by the
author, and is described in Ref. 4.
11 Demo Program
A demo program can be downloaded from the
author's web page (address in heading). It contains
information about input and output and allows a user to
analyze a simple two winding transformer of his own,
like the one in Fig.1. Most of the information in this
paper is also contained in the demo.
Like the main programs, the demo runs under
MS-DOS, which is an option under all the versions of
Windows, currently used.
12 Conclusions
The calculation of magnetic leakage fields in transformers by finite elements is rapidly becoming not only
a desirable option, but a necessity for transformer manufacturers who want to stay in business. In the absence of
tests, customers more and more demand that the ability
of transformers to withstand short circuits is documented
this way. Also, reactance and eddy current losses can be
predicted much more accurately than previously possible, using analytical methods.
References
[1] O.W. Andersen, "Transformer Leakage Flux
Program Based on the Finite Element Method", IEEE
Transactions on Power Apparatus and Systems, Vol.
PAS-92, 1973, pp. 682-689.
[2] O.W. Andersen, "Magnetic Leakage Fields in
Rectifier Transformers", Conference Proceedings ICEM,
Paris September 1994, pp. 738-740.
[3] O.W. Andersen, "Large Transformers for Power
Electronic Loads", IEEE Transactions on Power
Delivery 1997, October 1997, pp. 1532-1537.
[4] O.W. Andersen, "Optimized Design of Electric
Power Equipment", IEEE Computer Applications in
Power, April 1992, pp. 34-38.