Power transformer is one of the major and key apparatus in electric power system. Monitoring and diagnosis of transformer fault is necessary for improving the life period of transformer. The failures caused by short circuits are one of the causes of transformer outages. The short circuit currents induce excessive forces in the transformer windings which result in winding deformation affecting the mechanical and electrical characteristics of the winding. In the present work, a transformer producing only the radial flux under short circuit is considered. The corresponding axial displacement profile of the windings is computed using Finite Element Method based transient structural analysis and thus obtained displacements are compared with the experimental result. The change in inter disc capacitance and mutual inductance of the deformed windings due to different short circuit currents are computed using Finite Element Method based field analyses and the corresponding Sweep Frequency Responses are computed using the modified electrical equivalent circuit. From the change in the first resonant frequency, the winding movement can be quantified which will be useful for estimating the mechanical withstand capability of the winding for different short circuit currents in the design stage itself.
1. Introduction
High magnitudes of currents in the transformer windings due to short circuit events induce excessive forces in transformer and the magnitude of the fault current depends upon the short circuit reactance of the transformer
[1]
. In general, the winding currents produce leakage flux which can be resolved into radial and axial components. These radial and axial fluxes interact with winding current to produce axial and radial forces respectively. Depending on the distribution of forces and mechanical strength of the windings, the winding undergoes deformation. If these forces are not properly restrained, a major failure is likely to occur. For example, 1% difference in the heights of HV and LV windings leads to an axial force of nearly 50 kN/m in windings of a 5 MVA three phase transformer during short circuit
[2]
.
As the details about the deformation in terms of displacements is necessary, the literature on computation/measurement of the distribution of forces, displacements and analysis using SFRA are surveyed. G.B. Kumbhar and S.V. Kulkarni
[3]
computed the force distribution due to shortcircuit current of the splitwinding transformers using Finite Element Analysis (FEA). J. Faiz, B. M. Ebrahimi, and T. Noori
[4]
analysed the radial and axial electromechanical forces developed by the inrush current and short circuit current using 2D and 3D Finite Element Models (FEM). HyunMo Ahn and, YeonHoOh
[5]
compared the simulated and measured forces in transformer windings. Gopalakrishna. S
[2]
presented the force developed in top disc of the winding and the distribution of forces in each disc using Finite Element Analysis. A.P. Purnomadi and D. Fransisco
[6]
analysed axial and radial winding deformation using sweep frequency response analysis for the manually displaced windings. In
[7]
, the mechanical damage to a transformer winding is analysed with the help of sweep frequency response analysis using cross correlation method. K. Ludwikowski, K. Siodla and W. Ziomek
[8]
determined the frequency ranges for detection of the winding deformation for buckling phenomenon in a high voltage power transformer.
In the present work, force distribution and the winding displacements are computed using FEA and compared with the measured values. The change in winding capacitance and inductance due to deformation are calculated using FEA and the corresponding shift in resonance frequencies are obtained from sweep frequency responses. From the change in resonant frequency due to short circuit current, it is easy to quantify the winding displacement.
2. Transformer under Analysis
A 722VA, 10V aluminium wound transformer used by Gopalakrishna
[2]
is considered for the analysis. The transformer has two identical windings (W1 and W2) with 2 discs per winding as shown in
Fig. 1(a)
and
1(b)
. Each disc has 10 turns. The windings are connected in such a way to simulate a transformer with the winding currents in the opposite directions resulting in only axial displacement due to repulsive force.
(a) Winding model (b) Winding connection
3. Transient Structural Analysis using Finite Element Analysis (FEA)
As the shortcircuit events result in high mechanical forces in the transformer windings, the short circuit withstand capability of a transformer is of a mechanical nature. As the short circuit current varies with time, the induced force also varies with time resulting in continuous movement of discs. Hence, it is essential to estimate the displacements using transient structural analysis using winding equivalent structural model
[9]
. To do the transient structural analysis, each disc is represented as lumped mass (M), spacer as spring (K) and the insulation between the discs as dashpot (C). The winding equivalent structural model is shown in
Fig. 2
.
Winding and equivalent structural model
The governing equation for transient structural analysis is given as,
where:

[M] = structural mass matrix

[C] = structural damping matrix

[K] = structural stiffness matrix

{u} = nodal displacement vector

{Fa(t)} = applied load vector
To carry out the analysis, the following parameters are required, Short circuit force vector
F^{a} (t)
Mechanical parameters of the winding like mass (
M
), stiffness (
K
) and damping coefficient (
C
) of the winding
Parameters from (3.1) and (3.2) are given as input to Transient Structural Analysis to simulate the transient displacement of the winding
 3.1 Short circuit forcesfa(t)using fem (simulation)
The electromagnetic force distribution in the windings is computed using FEM based Magnetostructural analysis. Both the windings are energised with same current density in the opposite directions. The magnetic vector potential is calculated from magnetic field analysis (2).
where
μ
is the magnetic permeability,
A
represents the magnetic vector potential and
J
(A/m
^{2}
) is the current density. Electromagnetic forces are produced due to winding currents and the leakage flux in the winding regions and the force on the windings is given by the Lorentz force as,
where
F
represents the force (N).
The magnetic field analysis on the windings is carried out for short circuit current density of 12 A/mm
^{2}
.
Fig. 3
shows the magnetic flux plot of the winding.
Magnetic Flux plot of the winding
From the magnetic flux plot, it is observed that the axial flux is negligible (as the axial flux produced by W1 and W2 gets cancelled) compared to the radial flux. As a result, the forces on the windings are only in axial directions and repulsive.
Fig. 4(a)
and
4(b)
show the axial force distribution of the transformer.
Distribution of axial force; (b) Axial force along line CD
In general, for the energising current
the corresponding force is given by,
where
F_{max}
is obtained from Magnetostructural analysis and
F^{a} (t)
is given as an input to the transient structural analysis.
 3.2 Measurement of structural parameters (experimental)
Modal testing is a form of vibration test on an object by which natural frequencies, stiffness, masses and damping ratios can be determined. This test is carried out in the Council of Scientific & Industrial Research (CSIR), Government of India, Chennai and the response is recorded. The typical instrumentation setup is shown in
Fig. 5
.
Block diagram for modal analysis using Hammer Method (measured)
The natural frequency is the rate at which an object vibrates when it is not disturbed by an outside force. Each degree of freedom of an object has its own natural frequency, expressed as
ω_{n}
(radians per second). The mass (
M
) of the winding is 15 kg. The measured natural frequency of the test object is 21.52 Hz (
Fig. 6
).
Natural frequency of the test object (measured)
By using (6), the stiffness (
K
) of the total system is calculated as 282.47×10
^{3}
N/m.
A convenient way to measure the amount of damping present in a system is to measure the rate of decay of free oscillations. The larger the damping, the greater is the rate of decay. The rate of decay of free oscillations of the test object is recorded (
Fig. 7
).
Output of vibration analysis (measured)
The damping ratio is calculated as 811.5 using (7).
where:

ζ= damping ratio

C= damping

Cc= damping coefficient
Thus measured structural parameters are used in structural model (
Fig. 2
) to estimate the winding displacement.
 3.3 Displacement due to short circuit force
 3.3.1 Displacement using transient structural analysis (simulation):
In the present work, displacement in windings are computed using load transfer (from electromagnetic to structural analysis) method using FEM based transient structural analysis. Transient analysis is a technique used to determine the dynamic response of a structure (here windings) under the action of any general timedependent loads (force due to short circuit current). This type of analysis is used to determine the timevarying displacements and is solved by transient structural equation is given in (1). Here, the force on each disc (
F^{a}
(
t
)) calculated from Magnetostructural analysis (section 3.1) and measured structural parameters (section 3.2) act as a load for each and every node in transient structural analysis as given in
Fig. 2
. The displacements of all the discs are computed for an excitation period of 0.3s and the
Fig. 8
shows the dynamic displacement of the top disc of W1.
Displacement of top disc of W1 (simulation)
 3.3.2 Measurement of winding displacement (measured):
To validate the simulated results, short circuit test as per IEC Standard 600765
[10]
is conducted on the test object at On Load Gears, Ambattur, Chennai (
Fig. (9)
) and the outputs are recorded. The linear displacement transducer with 1.5 mm linear stroke length placed on top of the winding (on Disc 1 of W1) is used to measure the displacement.
Block diagram of experimental setup
The test is conducted for a period of 0.3s and the movement of the top disc is recorded and shown in
Fig. 10(a)
and
10(b)
. The maximum displacement is found to be 3.02mm at 0.11s.
(a) Excitation current (measured); (b) Displacement of top disc of W1 (measured)
The dynamic movement of the winding
Fig. 10(b)
is compared with the simulated response
Fig. 8
and observed that the disc movement pattern is similar and the percentage error in the maximum displacement is 4% for the same current density. As the simulation methodology is validated using experimental results, further analyses are carried out for different currents using simulation methodology.
Fig. 11
shows the displacement profile for different current densities.
Current density Vs displacement
In the next section, the change in the equivalent circuit parameters of the winding due to axial displacement for different current densities is incorporated by modifying the existing equivalent circuit.
4. Change in Winding Parameters and Resonance Frequencies due to Winding Deformation
 4.1 Electrical equivalent circuit and change in resonant frequencies (simulation)
Electrical equivalent circuit of transformer winding is represented by a combination of winding resistance, selfinductance, mutual inductance, interturn capacitance, interdisc capacitance and stray capacitance
[11

12]
. The basic electrical equivalent circuit of a transformer winding is shown in
Fig. 12
.
Electrical equivalent circuit of transformer inding
For axial deformation of transformer winding, both the inter turn capacitance and the self inductance do not vary with different short circuit currents. In the basic equivalent circuit, the interturn and interdisc capacitances are represented as single capacitance (C
_{s}
) as given in
[13]
. Due to axial deformation, there will be a change in inter disc capacitance which necessitates the separation of the series winding capacitances (C
_{s}
) into inter turn (C
_{t}
) and interdisc capacitance (C
_{d}
) and C
_{d }
is placed across the two consecutive discs. The change in the mutual inductance can be incorporated through coupling factor (k).
Fig. 13
shows the proposed ‘modified electrical equivalent circuit’ with separated C
_{t}
and C
_{d}
.
Modified electrical equivalent circuit for 2Disc winding
In the subsequent section, the change in equivalent circuit parameters of the winding due to displacement is calculated using Finite Element Method (FEM).
 4.2 Determination of winding parameters due to different displacement
Axisymmetric 2D modeling of the transformer winding is done using FEM and the transformer winding model is shown in
Fig. 14
. Due to axial displacement, the inter disc capacitance and the mutual inductances change.
Modeling of transformer winding using FEM
As both the inter turn capacitance and the self inductance do not vary with short circuit currents and the same are computed as 10.58pF and 32.27μH using FEM using electrostatic and magnetostatic solver respectively. Changes in interdisc capacitance and magnetic coupling coefficient (for mutual inductance) for different distances between the discs are computed and given in
Table 1
. Stray / Shunt capacitance (C
_{g}
) is also calculated using Electrostatic solver.
Interdisc capacitance and coupling coefficient (k) of W1 and W2 for different current densities
Interdisc capacitance and coupling coefficient (k) of W1 and W2 for different current densities
 4.3 Resonance frequencies due to winding deformation
The change in resonant frequencies due to displacements for different short circuit currents is computed using Circuit Simulation Package (OrCAD PSpice) and shown in
Fig. 15(a)
and
15(b)
. It is observed that the first resonant frequency (f
_{r1}
) increases with the increase in current for W1 and decreases with the increase in current for W2.
(a) Change in resonant frequency for W1 (simulated); (b) Change in resonant frequency for W2 (simulated)
The simulated results are checked with measurements done using sweep frequency response analyser ( FRAX 101 of Megger make) and shown in
Fig. 16(a)
and
16(b)
. The maximum difference in first resonance frequency between the measured and the simulated is less than 4%.
(a) Change in resonant frequency for W1 (measured); (b) Change in resonant frequency for W2 (measured)
The percentage shift in f
_{r1}
(with reference to unexcited case) for both the windings is calculated (
Fig. 17
) from which the displacement profile can be predicted for any short circuit current from
Fig. 11
.
Percentage change in resonant frequency for W1 and W2 (simulation)
From the change in first resonant frequency for different currents, the axial displacement and the corresponding change in equivalent circuit parameters of each and every disc can be predicted. The above analysis can also be used to predict the withstand capability of transformer by calculating the winding impedance of the deformed winding due to short circuit at the design stage itself as per IEC Standard 600765
[10]
.
5. Conclusions
The Finite Element Method is used to compute the electromagnetic force and displacement profiles acting on the transformer winding. In the present work, the transient deformation profile of the winding due to short circuit current is simulated and compared with the measured value. Changes in equivalent circuit parameters (inter disc capacitance and mutual inductance) due to displacements are calculated. The transformer equivalent circuit is modified to incorporate the axial displacement of each disc and the resonant frequencies for different short circuit currents are computed. Thus computed results are checked with measured resonant frequencies using sweep frequency response analyser. From the change in the first resonant frequency due to short circuit current, it is easy to quantify the winding movements. Though the methodology is applicable for transformers with both radial and axial displacements of continuous disc windings, the same need to be checked for other types of winding.
Acknowledgements
The authors are thankful to Mr. B. Babu, ExecutiveTesting, On Load Gears, Circuit Breaker Manufacturer, Ambattur, Chennai and Dr. N. Gopalakrishnan, Senior Principal Scientist, Advanced Seismic Testing and Research Laboratory, CSIR Campus, Taramani, Chennai for their support in carrying out the experiments successfully. The authors are also thankful to late Dr. V. Jayashankar for the transformer winding and his useful suggestions.
BIO
M. Arul Sathya She received the B.E degree in Electrical and Electronics Engineering from Shanmugha College of Engineering, Bharathidasan University in 1994. She received the M.S. By Research from College of Engineering, Guindy, Anna University in 2002. At present she is a research student in the Division of High Voltage Engineering, Anna University, Chennai, India.
S. Usa She received the B.E., M.E., and Ph.D. degrees in Electrical and Electronics Engineering from the College of Engineering, Guindy, Anna University, Chennai, India in 1986, 1989 and 1995, respectively. In 1992, she joined the Department of Electrical and Electronics Engineering at Anna University and at present she is Head for Electrical Department and Professor in the Division of High Voltage Engineering. She is also one of the working group members for CIGRE A2/C4.39 Electrical transient interaction between transformers and power systems.
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