FP_B.1_ABB_ DYNAMIC DESIGN TOOLS FOR IMPROVED SHORT
CIRCUIT DESIGN OF POWER TRANSFORMERS
Marco M Milone, ABB Power Grids Division Transformers, +49 2224 14261, marco.m.milone@de.abb.com
Rob Milledge, ABB Power Grids Division Transformers, +61 2 9753 7119, rob.milledge@au.abb.com
Julia Forslin, ABB Power Grids Division Transformers, +46 240 782 288, julia.forslin@se.abb.com
Abstract
While well-known that short circuit forces are dynamically sinusoidal, direct determination of the dynamic
transformer winding response has previously been available only with research tools.
This paper demonstrates that modern dynamic calculation tools defining both force and displacement, integrated
into normal design tools, can be used by designers to directly determine the individual transformer’s short
circuit response behaviour.
The paper describes and examines the knowledge gained from these tools, needed by the designer or reviewer,
to correctly assess the transformer performance.
Illustrated are the consequences of clamping pre-stress, winding resonances where the peak force is not
necessarily the initial peak, the location, duration and magnitude of internal winding displacements or even
separations.
Also demonstrated, is the understanding and dependency of friction dependent winding withstands, such as the
spiraling behaviour, on dynamic localized winding displacements and local friction force reductions. Dynamic
assessment allows the designer to identify and then evaluate changes to eliminate these friction reductions.
This dynamic behaviour knowledge is particularly significant when the transformer is subject to very onerous
short circuit duty or frequent short circuits.
Further testing and measurements are to be performed, using the tools and concepts described. This will deliver
greater accuracy and security of the power transformer withstand to short circuit events by using these dynamic
short circuit response calculations.
Introduction
Traditionally, short circuit forces in power transformers were determined by performing field plots of the
leakage field based on the first offset peak of the short circuit current. This current was treated as a constant
value. The force distribution was then calculated from this field plot result. The developed forces, particularly
the axial forces, are very sensitive to the axial magnetic balance of the windings. It was therefore common to
incorporate into the field plot geometry an axial offset of the windings of opposing ampere turns based on the
constructional tolerances. The more sophisticated of such analyses also incorporated the effects of axial
magnetic imbalance due to the pitch of any winding with an uncompensated helix. These analyses are known as
static, or more correctly electromagnetic, force calculations since the force distribution is assumed to be static,
not varying with time and the windings is assumed to be a rigid structure.
The reality is that the current and the resultant forces are dynamic with sinusoidal and exponential components.
Further, the winding is not a rigid object but rather a network damped spring elements that respond to the
applied forces in a complicated manner. The static, electromagnetic only calculation ignores many aspects of the
force behaviour and even overlooks some forces entirely.
A comparatively obvious example of this is the axial “bounce back” force behaviour of windings subject to an
initial axial compression force regime towards their centre without any initial end thrust to the transformer
structure. The traditional electromagnetic analysis considers forces within the winding but for this case does not
consider any forces applied by the winding to the active part clamping structure. When one considers the
winding as a spring with a time varying applied force, it becomes clear that the winding (spring) response will
be to “bounce back” and apply a response force to the clamping structure as the applied force reduces. This
“bounce back” force can be considerable, often much greater than the electromagnetically calculated end thrust
forces in cases with axial magnetic imbalance. By way of this example it is clear that the calculation of short
circuit forces and winding responses including the dynamic behaviour is very important. Without the correct
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assessment, the strength of constructional parts, such as press rings may be inadequate, leading to failures such
as shown in Figure 1.
Figure 1: Potential Damage due to inadequate consideration of Bounce-Back Forces
Until recently, for those that considered it, these dynamic behaviours were included by multiplying the results of
the electromagnetic calculation by “dynamic factors”. The dynamic factors were based on research and
development calculations and measurements. The dynamic correction factors were typically assumed to be the
same for all windings of a given type or arrangement. These “one size fits all” dynamic factors are typically in
the range of 1.2 to 1.5. In the case of “bounce back” the dynamic factor was often assumed to be ≈0.75 times the
peak axial force within the winding. Using such general dynamic factors can lead to either over-estimation or
under-estimation of the developed forces for individual cases. The individual dynamic characteristic depends on
the winding’s mass, stiffness, resonant frequency and damping response to the time varying forces and hence
will be different for different windings.
Today, a calculation tool has been developed that is accessible to power transformer designers for every design
to determine the individual and particular dynamic response for each winding considering the individual design
and its actual constructional conditions. It also allows the designer to eliminate some of the vulnerabilities that
exist in particular cases that would otherwise remain undetected.
Short Circuit Current and Applied Force Distribution
Before the dynamic behaviour of the forces and winding response can be evaluated, it is necessary to calculate
the peak steady state short circuit current, , and corresponding distribution of the peak steady state applied
force, Fmax on each elemental part of the winding. This is done in the same manner as for the traditional static,
electromagnetic analysis.
That is, is determined using normal circuit solution techniques, including use of symmetrical components for
asymmetrical faults and the Fmax, distribution along the winding by using the two dimensional, axi-symmetric
finite element, field plotting software with winding current, , as the driving current.
The time varying aspects of the short circuit current is calculated in the traditional way by solving an L-R circuit
differential equation as per Figure 2.
Figure 2: Time Varying Current Model for Short Circuit Current
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Hence,
sin
∙ sin
(1)
Where α = switching angle of the voltage at the instant of the short circuit
= phase angle of the circuit impedance
tan
= Peak steady state short circuit current
Ik = RMS steady state short circuit current
k = Asymmetry factor
∙
The short circuit current is greatest when α = ±
sin
Then equation (1) becomes
(2)
The usual application of Maxwell’s equations to a varying current carrying element in a varying magnetic field
gives the time varying force equation for an elemental piece of the winding.
2
That is,
2
(3)
The shape of the current and corresponding force are shown in Figure 3a and 3b
Figure 3: Short Circuit Current and Corresponding Applied Force Characteristic
The initial peak of i(t) occurs at
and has a value ∙
=
∙ √2 ∙
where
1
. In the absence of
other input, a value of k = 1.9 is used, in accordance with IEC 60076.5.
Fmax is the force distribution determined from the field plot and F(t) is the time varying, dynamic force that will
be applied to the dynamic winding model
One Dimensional Winding Model
The transformer winding is mechanically modelled as a lumped spring-mass system with damping dashpots.
The winding model is lumped in order to reduce the degrees of freedom of the system since this reduces
calculation time.
In the mechanical model, the insulation material is considered to be purely elastic elements (springs) and the
conductive material (copper) is represented by lumped incompressible masses. The damping of the dashpots
uses a Rayleigh damping model. The Rayleigh damping matrix consists of a mass-proportional part (external
damping, Ce) and a stiffness-proportional part (internal damping, Ci). In Figure 4, a schematic picture of the
lumped spring-mass model is shown.
The winding is considered to be clamped between the core yokes. The end springs represent the end insulation
to the assumed rigid core clamping structure and the inner springs represent the axial insulation between the
conductors.
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Figure 4: Schematic picture of calculation model of transformer winding
In the model, the number of elements (N) could be made equal to the number of axial sections (turns or discs) of
the winding. However, it was found that computational difference errors led to a loss of calculation precision
and computation time may be excessive. The model is generally found to give sufficient accuracy for most
windings with 20 mass elements. Each winding shell is modelled and calculated separately.
Typically, it has been found that the Young’s Modulus of insulation is in the range 0.4 to 0.8 GPa and that of
copper conductor is approximately 120 MPa. Hence the conductors are regarded as incompressible masses in
the calculation.
It is also found that the insulation characteristic varies with applied compressive force together with some mild
hysteresis as shown in Figure 5.
Figure 5: Measured stress-strain curve of pressboard
The stress-strain characteristic is modelled as a kth order polynominal function σ ε . For any individual winding
element this is converted to a force-compression polynominal F δ as shown in equations (4) and (5)
respectively.
∑
(4)
,
Where σ
ε
Fins
δ(t, i)
aj, αj
∑
,
(5)
= stress in MPa
= strain in percent
= compression force in an insulation element in N
= time varying compression distance of the ith insulation element in m
= known polynomial coefficients (constants)
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One Dimensional Dynamic Force Calculation - Winding Response
The winding model is excited with the typical time-function of the electromagnetic force during short-circuit as
described in equation (3) and Figure 3 above.
The peak electromagnetic force distribution, Fmax, is taken from the maximum forces calculated by the two
dimensional finite element field plot. This field plot also includes the traditional offsets for constructional
tolerances and pitch of the helical windings. The corresponding accumulated axial electromagnetic force
calculated by the field plot at different distances (y) along the winding is modelled with an mth order
polynomial, Faccum, fit(y), using a least squares best fit approach.
The maximum electromagnetic peak force of the ith element of the winding model (Pmax) is then calculated
according to equation (6).
P
i
F
,
y
F
y
,
(6)
The electromagnetic short-circuit force applied to each mass element in the model is then given by equation (7).
P t, i
P
e
i
2e cosωt
cos2ωt , i = 1, 2, …, N
(7)
The initial state of the winding, at the instant of the fault initiation, is determined from the initial compression of
each insulation element by solving the force equations (8) for δ(0, i).
∑
,
= P(0,i) + Mi above·g + Fps , i =1, 2, …, N+1
(8)
Where Mi above = the mass of winding above the ith element
g
= aceleration due to gravity
Fps
= the applied pre-stress (clamping force)
The displacements, x(t, i), of each the mass elements, i=1, 2, …, N, are calculated by solving the following
equations of motion.
m i ∙ x t, i
∑
i ∙ x t, i
m i ∙ g F t, i
α δ t, i
1
C i
1 ∙ δ t, i
1
∑
α δ t, i
C i ∙ δ t, i
(9)
Equation (9) can be re-written into equation (10)
i ∙ x t, i
x t, i
C i
1 ∙ δ t, i
∑
α δ t, i
1
∑
1
α δ t, i
,
g
i
1, 2, … , N
(10)
C i ∙ δ t, i
Equation (10) can be readily rewritten as a system of ordinary differential equations and solved using an explicit
Runge-Kutta method to give x(t, i).
The compression of the insulation elements (δ t, i ) given in equation (11) can then be determined
δ t, i
δ 0, i
δ 0, i
δ 0, i
x t, i
x t, i
x t, i
1
1
x t, i
i
i
i
1
2, 3, … , N
N 1
(11)
This is followed by determination of the relative velocities (δ t, i ) from equation 12.
δ t, i
x t, i
x t, i 1
x t, i
t, i 1
i
i
i
1
2, 3, … , N
N 1
(12)
The internal time varying dynamic forces for each mass element cause incremental displacement (compression)
of the adjacent insulation spring elements. Hence by using equation (8) for initial spring compressions and the
time dependent equation (11) insulation element compressions, these time varying dynamic forces (Fd(t, i)), are
calculated from equation (13)
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F t, i
∑
α δ t, i
C i ∙ δ t, i
∑
0
α δ t, i
C i ∙ δ t, i
δ t, i
δ 0, i
0 δ t, i
δ 0, i
δ t, i
0
i
1, 2, … , N
1
(13)
With this, the dynamic motion behaviour and dynamic forces of the winding are fully determined.
Three Dimensional Dynamic Force Calculation - Winding Response
In addition to the axial forces assessed above, short circuit events generate winding forces of considerably
greater magnitude in the radial direction. The inner windings are subjected to inward forces and the outer
windings are subjected to radial outward hoop stresses.
Generally, the windings have much less compressible insulation within their radial dimension than in the axial
dimension. It is typically only the conductor insulation. Hence it has been considered that the dynamic
assessment of forces in the radial dimension may not be necessary since there are essentially no compressible
springs to consider in this radial dimension. The traditional electromagnetic (static) calculation of radial forces
is then sufficient.
The validity of the above consideration is further confirmed when the radial resonant frequency of the windings
is calculated. They are typically >500 Hz which is far away from the exciting 50/60 Hz frequency. See the
section below on resonant frequencies of windings.
Two cautions are needed here in the use of electromagnetic forces only for the radial forces:
Caution 1: The are some winding types that include significant pressboard within their radial build. These
include windings with one or more radial cooling ducts and also multi-layer windings with interlayer insulation
between layers. For these windings, each layer or radial segment between cooling ducts needs to be considered
as a separate winding shell, otherwise dynamic modelling of the radial direction should be considered.
Caution 2: The authors’ company designs and calculates windings to withstand inward radial forces with a self
supporting principle. That is, the winding radial force withstand does not rely on the insulation materials inside
them to offer radial strength or support. In the alternate technology situation where the winding requires support
from the insulation structures and/or core inside them, then these elastic insulation material’s behaviour needs to
be considered and modelled with a dynamic model in the radial direction also.
However, the modelling of the combined electromagnetic radial forces and the dynamic axial forces was
undertaken and assessed with a three dimensional model, as shown in Figure 6.
Figure 6: The Three Dimensional Model
It is shown in many references, that the radial force is translated into tangential forces (stresses) in the
circumferential direction of the winding conductors. This gives rise to spiralling behaviour that needs to be
modelled and considered. The consequence of incorrect assessment is shown in Figure 7. It is this combined
tangential and axial behaviour that was modelled and dynamically assessed with the three dimensional
calculation.
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Figure 7: Spiralling Behaviour
The developed spiralling forces are restrained by the shearing withstand of the pressboard radial spacers and the
friction between the spacers and ribs against the conductors.
The shear stiffeness (modulus) of the pressboard spacers is a function of the axial compressive force on them
and measurements indicate a characteristic as shown in Figure 8.
Figure 8: Pressboard Shear Modulus vs Axial Pressure (negative stress corresponds to compression)
Note that it is found, as described below, that the results indicate that shear strength is not the limitation in
spiralling but rather the friction withstand characteristic.
For the three dimensional model, both the radial and axial electromagnetic forces along the winding were
calculated using the finite element field plotting software and this was applied the conductor elements in an
equivalent manner to the one dimensional model calculation.
The elastic pressboard connectors (springs) were also modelled with a simple shear stress characteristic in the
tangential direction and axial modelling as per the one dimensional modelling calculation.
In Figure 9, some snapshot results of an (exaggerated) animation of the spiralling effect for a helical wnding
from the three dimensional modelling is illustrated.
Figure 9: Results of the Three Dimensional Model showing Tendency of the Winding to Spiral
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One Dimensional versus Three Dimensional Models
A comparison was made between the results of dynamic axial force response of the one dimensional model and
the three dimensional modelling.
This is shown with time based comparisons of the displacement and force responses from the top, middle and
bottom of the analysed winding. The outcomes are shown in Figure 10 to Figure 15. The trace labelled SC tool
is the one dimensional model response and that labelled 3D FEM is the three dimensional model response.
Figure 10: Calculated displacements of top disc
Figure 11: Calculated displacements of centre disc
Figure 12: Calculated displacements of bottom disc
3000
4000
SC‐tool
3000
3D FEM
SC‐tool
Center [kN]
Top [kN]
5000
2000
1000
0
‐1000 0
0,02
0,04
0,06
0,08
0,1
0,12
0,14
0,16
0,18
0,2
2000
3D FEM
1000
0
0
0,02
0,04
0,06
‐1000
Figure 13: Calculated forces of top insulation
Bottom [kN]
3000
2500
2000
1500
1000
500
0
‐500 0
‐1000
0,08
0,1
0,12
0,14
0,16
0,18
0,2
Time [s]
Time [s]
Figure 14: Calculated forces of center insulation
SC‐tool
3D FEM
0,02
0,04
0,06
0,08
0,1
0,12
0,14
0,16
0,18
0,2
Time [s]
Figure 15: Calculated forces of bottom insulation
As can be seen in figures 10-12, the axial displacements agree quite closely between the one and three
dimensional models. Similarly from Figures 13-15 there is good agreement between the developed forces of the
one and three dimensional models.
It is also found that the shear stress of the pressboard spacers does not significantly affect the dynamic axial
force analysis.
Consequently, it was determined that one dimensional dynamic analysis is sufficient to appropriately analyse the
dynamic behaviour of the winding.
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Use of the Dynamic Short Circuit Analysis as a Design Tool
The dynamic analysis results have been found to be very important as an everyday design tool for each
transformer. The dynamic results are able to directly give a stress based on the dynamic behaviour of the
particular winding in question rather than relying on an “assumed” electromagnetic to dynamic correction
factor. Consequently, better accuracy of the short circuit forces and withstand behaviour is gained.
Additionally, with some additional post processing, the integrity and risk profile of the winding to withstand
short circuit is better understood and controlled.
This includes the analysis of axial peak force distributions, displacement versus time and force versus time
graphs for given locations, determination of resonant frequencies, comparison of electromagnetic and dynamic
performance for given locations, maximum displacement distributions along the winding, analysis of quantity ,
duration and location of gaps in the winding. The impact on these parameters to various pre-stress (clamping)
forces applied to the winding is also analysed.
Some examples of these behaviours and insights is described in the following clauses.
Axial and Radial Resonance
If the resonant frequency is far from the 50/60 Hz exciting frequency then the dynamic increase factor is close to
1.0, while if the resonant frequency is very close to 50/60 Hz then the dynamic increase factor could be as high
as 2.0 (subject to the amount of damping in the system).
The axial resonant frequency of the winding can be calculated by performing a fast Fourier transform (FFT) of
the free response after the applied short-circuit force has been released. The frequency of the maximum peak in
the calculated frequency spectrum can be readily determined. The frequency of the first and second axial
resonance mode is tracked. If applied forces are perfectly symmetric only the second axial resonance mode will
be excited. However, in most cases there will be an end force in the winding and the asymmetric first axial
resonance mode will be strongly excited.
The first axial resonance mode is often in the range of 40-100 Hz for windings in large power transformers. This
is the reason for traditionally accounting for dynamic resonance amplification when designing windings for
short-circuit forces. Some examples are shown in Figure 16.
Figure 16: Examples of Axial Resonant Frequencies with dominant 1st resonance or 2nd resonance respectively
The axial resonant frequency is generally closer to 50/60 Hz in larger, heavier windings. Smaller units tend to
have higher resonant frequencies.
The radial resonant frequency can also be calculated and is typically in the range of 500 Hz to 1000 Hz and
hence the electromagnetic force and dynamic response co-incide.
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Axial Gaps within the Winding
The dynamic calculation demonstrates the existence and location of gaps occurring within the winding between
turns/discs. Depending upon the magnitude, quantity and the time duration of the gaps, these may pose risks to
winding integrity. The design of the winding or the pre-stress applied to the winding (see the “Effects of PreStress” below) can be modified to mitigate this. The gaps can also occur between the winding and the end
insulation to the clamping structure. An example of gap identification within a winding is shown in Figure 17.
Figure 17: Example of a winding with multiple gaps occurring both within the winding and to the end insulation
Given the quantity of gaps, analysis of the stress magnitude and duration of the gaps may be needed in order to
determine if mitigation measures are needed to remove the gaps.
Effects of Pre-Stress (Clamping Force)
It is well known that the pre-stress (clamping force) applied to a winding has a significant effect on its short
circuit behaviour. It is widely acknowledged that unclamped windings are at risk and undesirable. The dynamic
analysis of windings demonstrates this in a quantitative way. One of the dynamic analysis post processing
design tools available is the analysis of consequences of changes in pre-stress.
In Figure 18, the calculated number of gaps due to oscillatory vibration occurring between turns in a particular
winding is plotted versus applied clamping pressure on winding. Note that in some cases a gap may occur on a
number of occasions (different times) at the same location and the gap count in the figure is increased for every
occasion. Above a specific pre-stress, gaps are avoided between the turns for this example design. It is noted
that the gap duration may sometimes be of sufficiently short duration that there is no significant loss of friction.
A duration analysis is shown for a different winding in Figure 19.
Figure 18: Number of gaps versus pre-stress
Figure 19: Total duration of gap separations
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The winding pre-stress also makes a significant change in the peak displacement values of the oscillatory
vibrations that occur. An example of this is shown at the top (blue) and bottom (red) of a winding in Figure 20.
The corresponding peak stresses versus pre-stress of the winding are shown in Figure 21. In Figure 21 the
applied pre-stress is shown with the blue line, the dynamic oscillatory response stress in green and the total
summated stress in black.
Figure 20: Top & Bottom Peak Displacement vs Pre-Stress
Figure 21: Maximum Axial Stress vs Pre-Stress
As can be seen in Figure 21, the compressive stress originating from the oscillatory vibration of the winding
during short-circuit dominates and the applied compressive pre-stress is less important for this particular
winding. The total axial compressive force acting on the winding insulation is in general reduced when the prestress is increased. At a given pre-stress, an increase in calculated compressive stress can be seen due to winding
resonance amplification. However, one should resist the temptation to reduce the pre-stress force in order to
reduce the winding maximum stress. This would lead to a winding that is loosely clamped with large
displacement oscillations and in excess of 100 gaps.
Figure 22 and Figure 23 show another winding from the same transformer design where the minimum stress
area occurs for a pre-stress corresponding to very few or no gaps.
Figure 22: Number of gaps vs Pre-Stress
Figure 23: Maximum Axial Stress vs Pre-Stress
Resonant frequency changes with pre-stress also, this is shown in Figure 24. Note that this graph shows the
dominant resonant frequency and the sudden step change which is due to the change in dominant resonance
from first to second winding resonance.
Figure 24: Resonant Frequency versus Pre-Stress
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Friction Aspects and Spiralling
The mechanism of spiralling due to tangential (radial) forces is illustrated in Figure 7 and the assessment of
three dimensional dynamic modelling above. It is an issue mainly for helical and layer windings rather than disc
windings (unless they have very few turns per disc) due to the much greater friction between the many turns per
disc.
Spiralling behaviour is predominantly determined by the friction withstand provided by the axial pressure on the
spacers. Spiralling can be assessed with the electromagnetic tangential forces together with frictional restraint
due to the dynamic axial forces on the spacers. If the axial pressure and resultant friction is not sufficient, the
conductor can slide on the spacers and spiralling occurs. The occurrence of gaps between winding turns
particularly towards the ends of the winding increases the risk of loss of friction and the occurrence of spiralling.
The time duration of the gaps is also a consideration in this risk.
Note that in some cases, it has been historically requested that the clamping force (pre-stress) should be greater
than either the electromagnetic end force from the winding or the electromagnetic peak axial force within the
winding. The dynamic response calculation of the winding demonstrates that this is not necessary. While a
significant pre-stress is necessary, the value of prestress to minimise or event eliminate the gaps (avoid
spiralling) and minimise the axial stresses (to within the acceptable stresses) can be determined via analysis such
as shown in Figures 17-24.
Case Study Example: 1000 MVA
The Transformer was assessed both with electromagnetic (static) and dynamic analysis. Figure 25 shows the
axial force distribution from the electromagnetic calculations and Figure 26 shows the dynamic peak axial force
distribution.
Figure 25: Peak Axial Electromagnetic Forces (Red)
Figure 26: Peak Axial Dynamic Forces (Blue)
Note that not only is the maximum dynamic axial force greater than the static but also the dynamic axial end
forces are much greater (bounce back) than the static values. However, the bounce back force ratio is ≈30%
(Winding A), significantly less than the tradition 75% factor from empirical dynamic conversion from
electromagetic calculation.
An example of the Winding A dynamic displacement versus time curves of the top, middle and bottom discs is
shown in Figure 27.
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Figure 27: A Dynamic Axial displacement of Discs for Winding A
In Figure 28, the resulting dynamic compressive axial forces in the top, middle and bottom discs are displayed
in blue. The equivalent values from the electromagnetic (static) only calulation are shown in red on the same
graph.
Figure 28: Dynamic and Static Forces vs Time for Winding A
In this example it can be seen that the dynamic forces are larger and the time phasing of the force at the centre
disc is very similar to the electromagnetic calculation. Additionally, the greatest force occurs at the first peak as
per the electromagnetic calculation. This is not always the case. Figure 29 shows an example of another
transformer in which the maximum dynamic force is developed at a later peak and the time phases is different to
the electromagnetic analysis.
Figure 29: Dynamic and Static Forces vs Time with differing Phasing and Maximum Peak
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Summary
The concept of the dynamic short circuit modelling of windings in transformers is very powerful and has given
significant insights into the short circuit behaviour of transformers. It identifies and quantifies characteristics not
available to electromagnetic calculations, such as the appearance of internal gaps, preferred pre-stress
(clamping) forces, friction based performance for the assessment of spiralling.
The calculation when embodied into the normal design calculation routines facilitates the risk reduction that
comes from having the individual and specific dynamic performance on each design and winding rather than
using the “one size fits all” static to dynamic response factors previously used.
The calculation also facilitates understanding and adjustments to “at risk” designs to make them more secure.
Bibliography
1.
2.
3.
4.
5.
6.
Bertagnolli G., “Power Transformers & Short Circuits, Evaluation of the short circuit performance of
power transformers”, ABB 2014
Forslin J., “Dynamic calculation of axial short-circuit forces in transformer windings”, SECRC/DT
Report ABB 2015
Waters M., “The short circuit strength of power transformers” Macdonald & Co 1966
Anger J., Forslin J., Lizunova M., “Short Circuit Calculations, 1000 MVA”, Report No SETFO 2015241 ABB 2015
Anger J., Forslin J, Missing R., “TrACE Short Circuit Design Tool EDS 6.7”, Presentation ABB 2015
“Ability to withstand short circuit”, IEC Standard 60076.5, Edition 3, 2006
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