Faculty of Engineering, Blekinge Institute of Technology, 371 79 Karlskrona, Sweden Master of Science in Mechanical Engineering
October 2020
Finite element modelling of LV
transformer winding to simulate dynamic events occurring under short circuit
M.V.S Prudhvi Bikkina
ii
This thesis is submitted to the Faculty of Engineering at Blekinge Institute of Technology in partial fulfilment of the requirements for the degree of Master of Science in Mechanical Engineering. The thesis is equivalent to 20 weeks of full-time studies.
The authors declare that they are the sole authors of this thesis and that they have not used any sources other than those listed in the bibliography and identified as references. They further declare that they have not submitted this thesis at any other institution to obtain a degree.
Contact Information:
Author(s): M.V. S Prudhvi Bikkina E-mail: mady18@student.bth.se
University advisor:
Shafiqul Islam, PhD Mechanical Engineering
Advisor at Hitachi ABB Power Grids Julia Forslin
Faculty of Engineering Blekinge Institute of Technology
SE-371 79 Karlskrona, Sweden SE-371 79 Karlskrona, Sweden
Internet : www.bth.se Phone : +46 455 38 50 00 Fax : +46 455 38 50 57
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Abstract
The ability to withstand a short circuit is the most essential feature of a power transformer. The most important reason to design short-circuits proof transformers is to ensure the reliability of the power grid (avoiding black outs etc.) and safety (fire and explosion in case of failure). During short circuit, the most effected winding is the LV winding due to the flow high currents even during the normal working condition. So during a short circuit large forces are generated which act on the winding and these forces can reach hundreds of tons in fraction of a second, so the transformer must be properly designed in order to withstand these forces or the transformer can fail in different ways. One of the possible failure modes called “Spiraling” is discussed and analyzed in this thesis. Spiraling Occurs when the LV winding twists tangentially in the opposite direction at the ends due to radial short circuit forces. From literature study the transient forces acting on the winding during a 3-phase short circuit was determined and these transient forces were used to perform simulations on the model. The axial and radial forces applied on the model were such that it has a uniform magnitude per each turn. Various analysis was performed on the model which includes the Static, Modal and Transient Structural analysis in Ansys Workbench and each analysis involved parametric analysis where the deformations and the torsional mode shapes were determined
Keywords: Ansys Mechanical, LV winding, Power Transformer, Short-circuit, Spiraling, Torsional Mode Shapes
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Acknowledgements
This work was carried out at Hitachi-ABB power grids Ludvika, Sweden and Blekinge Institute of Technology (BTH), Karlskrona Sweden from March 2020 to September 2020.
I would like to thank my thesis supervisor Julia Forslin of Research and Development at Hitachi-ABB power grids. The door to Julia was always open whenever I ran into trouble spot or had a question about my research or writing. She allowed this paper to be my own work but steered me in the right direction whenever she thought I needed it.
I would also like to thank Nima-Sadr Momtazi for his positive support and guidance during my stay at Ludvika.
I would also like to thank Shafiqul Islam of Mechanical Engineering department at BTH for his continuous support, guidance, and encouragement during my work.
Finally, I must express my very profound gratitude to my father Bikkina Sree Rama Prasad and my mother Bikkina Surya Bhavani for providing me with unfailing support and continuous encouragement throughout my years of study. This accomplishment would not have been made without them.
Thank You
Madhu Venkata Sri Prudhvi Bikkina
5 Table of Contents
NOTATIONS 10
1. INTRODUCTION 11
1.1. BACKGROUND 11
1.1.1. NEED FOR A FINITE ELEMENT MODEL 11
1.2. AIM AND OBJECTIVE 12 1.3. METHOD 12 1.4. RESEARCH QUESTION 13 1.5. LIMITATIONS 13 2. LITERATURE STUDY 14
2.1. SHORT CIRCUIT CURRENTS 14 2.2. EFFECT OF TEMPERATURE 15 2.3. SHORT CIRCUIT FORCES 15
2.3.1. GENERAL 15
2.3.2. NATURE OF SHORT CIRCUIT FORCES 16
2.3.3. AXIAL SHORT CIRCUIT FORCES 18
2.3.4. RADIAL SHORT CIRCUIT FORCE 18
2.3.4.1. General 18
2.3.4.2. Tangential stress in conductor 19
2.3.4.3. Spiraling force 19
2.3.5. TANGENTIAL FORCE DUE TO AXIAL CURRENT COMPONENT 20
2.4. FAILURE 20
2.4.1. SPIRALING 21
2.5. AXIAL PRE-STRESS 21 3. FINITE ELEMENT MODEL 23
3.1. GEOMETRY OF THE FE MODEL 23 3.2. GEOMETRIC PROPERTIES: 24 3.3. MATERIAL PROPERTIES 24
3.3.1. CONDUCTOR, INNER CYLINDER: 24
3.3.2. INSULATION MATERIAL (AXIAL STICKS, SPACER RING,UPPER AND LOWER RING) 26
3.4. CONTACT: 26 3.5. COORDINATE SYSTEM: 27 3.6. BOUNDARY CONDITIONS AND MESHING: 28 3.7. GRAVITY 29 4. RESULTS AND ANALYSIS 30
4.1. STATIC STRUCTURAL ANALYSIS 30
4.1.1. AVERAGE STRESS 31
4.1.2. PARAMETRIC ANALYSIS 32
4.2. MODAL ANALYSIS 34
4.2.1. NATURAL FREQUENCIES 34
4.2.2. EFFECT OF AXIAL PRE-STRESS IN THE NATURAL FREQUENCY 36
4.3. IMPLICIT DYNAMICS 37
4.3.2. STRESS INDUCED 41
4.3.3. SPIRALING PHENOMENON 45
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4.3.4. PARAMETRIC ANALYSIS 46
5. DISCUSSION AND CONCLUSIONS 48 6. FUTURE WORK 49
REFERENCES 50
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List of Figures
Figure 1.1: CTC Cable...13
Figure 2.1: Equivalent circuit for short circuit analysis ...15
Figure 2.2: Current split up into two components: i) A steady state sinusoidal component and ii) unidirectional decay component ...15
Figure 2.3: Current carrying conductor placed in magnetic field ...16
Figure 2.4: Magnetic field around core viewed in 2D plane[8] ...16
Figure 2.5: Time varying short circuit current with time constant τ =0,0949 at 50hz frequency ...17
Figure 2.6: Time varying Short circuit force with time constant τ =0.0949 and 50Hz frequency ...18
Figure 2.7: Steady state and transient state axial force in LV winding[9] ...18
Figure 2.8: Steady state radial short circuit forces along the winding height[9]...19
Figure 2.9: Transient state radial short circuit forces along winding height[9] ...19
Figure 2.10: Buckling failure ...20
Figure 2.11: Spiraling failure ...21
Figure 2.12: The transformer as a spring mass system ...22
Figure 3.1: The FE model ...23
Figure 3.2: Yield strength vs strain of the material...25
Figure 3.3: Bilinear isotropic hardening curve ...25
Figure 3.4: Configuration for frictional and bonded contacts ...27
Figure 3.5: Figure showing cylindrical coordinate system ...28
Figure 3.6: Meshed with Solid 187 tetrahedral element ...28
Figure 3.7: Solid 187 tetrahedral element ...29
Figure 4.1: Tangential deformation at the end of steady state radial short circuit load ...31
Figure 4.2: : Axial Pre-stress vs tangential deformations upper exit due to static short circuit radial Pressure ...33
Figure 4.3: Axial Pre-stress vs tangential deformations of lower exit due to static short circuit radial Pressure ...33
Figure 4.4: Undeformed winding the exits are aligned along vertical axis ...34
Figure 4.5: 1.6Mpa Pre-stressed 20turns winding first torsional mode shape at 18Hz ...35
Figure 4.6: 1.6Mpa Pre-stressed 20 turns winding second torsional mode shape at 41Hz the node is formed at the center of the winding ...35
Figure 4.7: 1.6Mpa Pre-stressed 20 turns winding third torsional mode shape at 74Hz ...35
Figure 4.8: 1.6Mpa pre-stressed winding for 4 turns winding first mode shape at 78Hz ...36
Figure 4.9: Axial Pre-stress vs natural frequency of 20 turns winding ...36
Figure 4.10: Axial Pre-stress vs natural frequency of 4 turns winding ...37
Figure 4.11: Short circuit pressure Vs time ...38
Figure 4.12:Face on which radial short circuit pressure was applied ...38
Figure 4.13:Axial short circuit force acting direction in the model such that it compresses the winding ...39 Figure 4.14: Deformed winding due to transient short circuit radial and axial loads .39
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Figure 4.15: Deformation of exits for a 20 turn winding due to transient axial and
radial pressure ...40
Figure 4.16: The deformation of exits with transient axial and radial short circuit pressure for 4 turn model ...41
Figure 4.17: Average Von-mises stress in 4 turn winding over time ...41
Figure 4.18: Average tangential stress in 4 turn winding over time ...42
Figure 4.19: Average Radial stress in 4 turn winding over time ...42
Figure 4.20: Average Axial stress in 4 turn winding over time ...43
Figure 4.21: Average Von-mises for a 20 turn winding over time ...43
Figure 4.22: Average tangential stress for 20 turn winding over time ...44
Figure 4.23: Average Radial Stress for 20 turn winding over time ...44
Figure 4.24: Average Axial Stress for 20 turn winding over time ...45
Figure 4.25: Force reaction at exits ...46
Figure 4.26: Spiraling and plastic deformation of upper exit due radial short circuit pressure ...47
Figure 4.27: Spiraling and plastic deformation of lower exit due radial short circuit pressure ...47
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List of Tables
Table 3.1:Geometric Properties ...24
Table 3.2: Physical Properties of Copper ...26
Table 3.3: Physical Properties of Pressboard ...26
Table 4.1: The tangential deformation of exits for 20 turn winding ...30
Table 4.2: Average von-mises stress in 1.6Mpa pre-stressed winding ...31
Table 4.3: Average stress due to radial short circuit pressure ...31
Table 4.4: Average stress after removing the radial short circuit pressure ...32
Table 4.5: Natural frequencies for the 1.6Mpa prestressed winding ...34
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Notations
LV- Low Voltage HV-High Voltage ms-millisecond
= | |=
= + =
= =
2 =
=
=
=
= 2 =
E= Modulus of Elasticity R= Resistance
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1. Introduction
1.1. Background
The ability to withstand a short-circuit is the most essential feature of a power transformer in order to ensure the reliability and safety. Power Transformers are designed for 100 percent efficiency but due to losses they work at 97.78 efficiency.
During a survey of short-circuit tests, 48% of the transformers with power rating of 25 MVA and above fail to pass short-circuit tests[1] . A transformer works on the principle of mutual inductance. Due to the varying magnetic field and current flowing in the winding there will be a force generated, the nature of force generated is proportional to square of current which is why the forces acting are very large during short circuits.
Standard power transformers have three basic components which are its Primary winding (LV winding), Secondary winding (HV winding winding) and the core[2].
Considering a Step-Up type power transformer, the LV winding is helically wound around the core these windings can also have multiple conductors connected in parallel.
The secondary winding is connected to HV side, their structure is unique as they can be of disc type or cross-over type. During short-circuit the most affected component of the transformer is the Low Voltage winding due to its helical structure and flow of large currents whose magnitude is usually about 10 times the magnitude of current that usually flows in it during operation. These large currents combined with changing magnetic flux generate large mechanical forces. These forces act in both axial and radial direction whose magnitudes can reach hundreds of tons in fraction of a second[3]. For a reliable transformer, it is very important that none of these deformations persist after the short circuit is broken.
During short circuit, forces are directed such that the primary windings are compressed radially inwards, and secondary windings are pulled outwards, this is due to the opposite direction of flow of current in the windings. One of the possible failure modes can be spiraling. This deformation can be seen in the LV helical winding. Spiraling failure occurs when the winding is compressed radially which causes the winding to twist tangentially.
1.1.1. Need for a Finite element model
A power transformer should go through the short circuit test before putting them into service. These testing laboratories are very few in the world one of them is in Netherlands[4]. So, in case these transformers fail due to their poor design or any other factors during the test it would cause a huge economical damage to the company as they need to replace the whole winding.
Even though design criteria exist which suggests keeping the maximum tangential stress below a certain limit to avoid spiraling failure, but it cannot be the perfect choice all the time. This raises a need for developing a good model that can be studied to understand the physical mechanisms behind the failure. A 3D finite element model can help in understanding the causes for certain failures.
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1.2. Aim and Objective
The aim of the present work is to develop a mechanical 3D finite element model of helical type transformer winding so that it can simulate dynamic events occurring during short circuit. The main objective of the problem is to study the effects of various parameters that effect the failure of the winding. The objective is determined using the following steps
x Literature study of Short Circuit Forces on transformer windings
x Create a finite element model in Ansys Mechanical by defining properties of material and perform static, implicit simulation with the short circuit forces with prestress conditions.
x Studying the effect of axial pre-stress on resonance frequencies of the structure x Analyzing the Axial, Radial and tangential and von mises stresses
x Parametric study of winding deformations using implicit dynamics and in static analysis
1.3. Method
In this thesis a static analysis and a dynamic analysis was performed in Ansys, the dynamic analysis involves implicit time integration. In Ansys there are two methods to solve dynamic analysis one is using the modal super position method and the other is using the transient structural analysis[5]. In modal-superposition method initially a modal analysis is performed, and later dynamic loading is applied, this method is very fast, but this removes non linearities, so this method was not used. The latter method involves a very high computational time but would include all the non-linear effects so this would be the best method proceed with the dynamic analysis. The below flow chart would give a better insight to the method
Ansys Workbench
Design modeller CAD Geomet ry
St at ic St ruct ural Transient St ruct ural
Assign mat erial
Def ine Cont act t ype
Choose mesh element t ype
Solve t he problem in t hree st eps Assign Boundary Condit ions and ramp t he loads
Import file
Post Processing
Paramet ric Analysis
Def ine mat erial Propert ies
Assign mat erial
Def ine Cont act t ype
Choose mesh element t ype
Assign Boundary Condit ions
Applying t ime varying pressure load wit h implicit t ime int egrat ion
Post Processing
Paramet ric Analysis
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1.4. Research Question
What is the effect of Axial –Prestress on the Torsional Natural frequency of a Layer Type transformer winding?
How do the exits of a winding deform with the applied axial-prestress due to short-circuit loads?
1.5. Limitations
x The conductor used in power transformers are CTC cables[6], these cables have many conductors connected in parallel and the strands are often epoxy bonded in order to improve the mechanical strength of the conductor.
Figure 1.1: CTC Cable
x Efforts made in this thesis is to simplify the model, so the cooling ducts are left bonded to the inner shell and the inner structure which includes the core, LV insulation sheets were not modelled.
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2. Literature study
2.1. Short Circuit Currents
Short circuits can occur due to various faults such as single line to ground faults, line to line faults with or without a simultaneous ground fault, and three-phase faults with or without a simultaneous ground fault. The most severe short circuit failure is due to three phase short circuit except for some special cases. So, the transformer is usually designed to withstand the three phase short circuit faults. The zero-voltage interval is when there is flow of high magnitude short circuit currents. The nature of short circuit current is asymmetric and in most of the cases the short circuit currents are calculated using an equivalent circuit analysis as shown in Figure 2.1
The steady state short circuit current, is given by
= Where =
= ℎ
By operating the make switch in Figure 2.1 we simulate a short circuit at the instant t=0, The resulting current ( ) [7] is given by the equation 2.1
( ) = ( + − ) − ( − ) 2.1
= | |
= +
= =
2 =
For an asymmetric short circuit, the current consists of two components an alternating steady state component at fundamental frequency and a unidirectional component decreasing exponentially with time See Equation 2.3 also see Figure 2.2. The exponential decay rate is described using the X/R ratio and for power transformer according to IEC 60076-5 (third edition: 2006-02) for X/R > 14, its value is stated as 1.8 for transformers below 100 MVA rating, whereas it is 1.9 for transformers above 100 MVA rating.
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Figure 2.1: Equivalent circuit for short circuit analysis[7]
Figure 2.2: Current split up into two components: i) A steady state sinusoidal component and ii) unidirectional decay component[7]
2.2. Effect of temperature
During short circuit there will be currents having high magnitudes flowing in the winding for a very short interval of time as small as 30-40 ms duration as there will be circuit breakers that break the circuit during short circuit. So, the effect of temperature does not cause any observable damage to the winding
2.3. Short circuit forces 2.3.1. General
The governing equation for the force acting on a current carrying conductor placed in a magnetic field is given by
= ( ) 2.2
Here theta ( ) is the angle made by the current carrying conductor with the magnetic field . Considering 2D scenario where the magnetic field density is acting in the Z direction the leakage flux at any point is resolved into two components directing in the
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radial and axial directions. The axial leakage flux causes the radial forces and radial leakage flux causes axial forces.
The force direction can be determined using Flemings left hand rule where index finger points the current direction, middle finger shows direction of magnetic field and thumb finger indicates the force direction.
Figure 2.3: Current carrying conductor placed in magnetic field
2.3.2. Nature of short circuit forces
The transformer Low voltage and high voltage winding are wound on the same core concentrically such a configuration allows the current to flow in opposite directions in the respective windings. The below Figure 2.4 shows the magnetic field lines in a transformer, considering a 2-dimesional scenario as explained in 2.1 there will axial and radial forces these forces are act such that it compresses the LV winding and expands the HV winding radially.
Figure 2.4: Magnetic field around core viewed in 2D plane[8]
The uni-directional component attains a highest value when
= ±
The main interest is the peak value of the current that occurs by addition of the two 2 components which in turn generate the force peaks. The current attains its highest magnitude at zero voltage instant independent with the value of i.e. when = 0 We can assume the value of to be approximately because in power systems the circuit reactance is greater than resistance so the short circuit current can be written as
( ) = − + 2.3
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Where = ℎ
Since the nature of short circuit force is the square of short circuit currents
( ) = − + 2.4
( ) = + − ( ) + ( ) 2.5
The short force has two alternating component one at fundamental frequency decreasing with time and other at double the fundamental frequency and two uni-directional components one with a constant value and other decreasing with time.
-1,5 -1 -0,5 0 0,5 1 1,5 2 2,5
0 0,05 0,1 0,15 0,2 0,25 0,3 0,35
Ampere
Time in Seconds
Figure 2.5: Time varying short circuit current with time constant =0,0949 at 50hz frequency
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2.3.3. Axial Short Circuit forces
For Low voltage winding the radial leakage flux is such that it is directed opposite at the ends of the winding this means that the axial force will act in the opposite direction on the top and bottom ends. The Figure 2.7 shows the variation of axial force along the axis of the axis of the winding. The top end has an axial force pointed downwards i.e.
in the positive direction and the bottom end has the force directed upwards i.e. in the negative direction this compresses the winding at the center. The Figure 2.7 shows the variation of the steady state and transient forces acting along the winding height.
Figure 2.7: Steady state and transient state axial force in LV winding[9]
2.3.4. Radial Short Circuit force 2.3.4.1.1. General
The current flow in the LV and HV windings are opposite in direction this causes the LV winding to compress and the HV to expand. The below Figure 2.8 and Figure 2.9 shows the direction of radial force along the axis of the winding, as we can see that the
-0,5 0 0,5 1 1,5 2 2,5 3 3,5 4
0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45
Force In N
Time in Seconds
Figure 2.6: Time varying Short circuit force with time constant =0.0949 and 50Hz frequency
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radial force is constant along the middle section if the winding and reduces at the end of the winding. This is because of the bending magnetic flux lines at the ends of the winding. The force is considered positive if they are directed inwards towards the core.
Figure 2.8: Steady state radial short circuit forces along the winding height[9]
Figure 2.9: Transient state radial short circuit forces along winding height[9]
2.3.4.2. TANGENTIAL STRESS IN CONDUCTOR
If the LV winding structure is assumed as a thin cylinder, the inwards acting radial short circuit forces converts to a tangential compressive stress in the conductor. This tangential stress in the conductor is usually called hoop stress and it is calculated according to equation 2.6.
=( )( )
∗ 2.6
2.3.4.3. SPIRALING FORCE
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In a helical-shaped winding, the tangential stress in the conductor (as described in section 2.3.4.2) may cause the winding to deform in a screwing pattern. This is called Spiraling. The magnitude of tangential force generated due to radial short circuit force is determined by the equation 2.7. This spiraling force is responsible of the spiraling failure as described in section 2.4.1
= ∗ 2.7
2.3.5. Tangential force due to axial current component
Due to the helical pitch there is a small current component in the axial direction too.
This axial current combined with the radial flux at the winding ends produces a small tangential force on the conductor. This component is usually neglected in design since the magnitude is significantly smaller than the axial and radial forces acting on the conductor.
2.4. Failure
The helical windings when subjected to short circuit forces can fail. For example, bending between supports, buckling or spiraling. When LV winding is firmly supported with spacers and the support structure as a whole has higher stiffness than conductors in such a situation the conductor bends along the circumference as shown in Figure 2.10.
However, this failure mode may only occur in windings that are designed with very weak conductor and a stiff support. If the winding conductors are thin or have a very high stiffness compared to the inner cylinder and if the inner cylinder is not firmly fixed the conductors bulge outwards and inwards at one or more locations. Some of the factors that contribute to buckling are the looseness in winding and a material having low modulus of elasticity is used or if radial width of conductors is less or if the strands and epoxy bonding are not properly glued the conductor would buckle and fail. This buckling failure is sequential as if it starts at one particular region it bulges along the height of the winding
Figure 2.10: Buckling failure
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2.4.1. Spiraling
This failure is seen in LV winding and the failure is due to the radial forces originating from the axial leakage flux in the winding. If we consider the transformer winding to have three sections namely the top, middle and bottom sections the axial leakage flux is lower at the top and bottom section and the at the middle section the axial leakage flux is relatively high due to which the radial forces act such that it has a maximum magnitude in the middle section and less magnitude at the top and bottom section. This radial force results in the twisting of the winding such that it tightens up like a screw and at some rare cases the winding tends to untwist if the radial forces are tensile in nature. The compressive load acting on the winding can be calculated by assuming the structure as a thin cylinder where the load can be calculated using the compressive hoop stress developed see section 2.3.4.3
Figure 2.11: Spiraling failure
2.5. Axial Pre-stress
The pressure applied on the winding after assembling the core winding is termed as Axial pre-stress. Axial pre-stress plays a very important role in maintaining the mechanical integrity of the structure and plays an important role in response of winding during short circuits.[10]
A proper pre-stressing during short circuit can remove the possibility of Spiraling failure in the windings due to radial short-circuit forces. When the winding is properly pre- stresses there will be transfer of force through the spacer so this would provide a mechanical integrity to the structure which could eliminate the possibility of spiraling.
The pre-stress plays a significant role in deciding the natural frequency of the winding but the relation between them is highly nonlinear. A high pre-stress reduces the oscillations as it increases the stiffness thereby also increasing the resonance
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frequencies. If a non-prestressed winding has natural frequency greater than the excitation frequency, the vibrations are reduced at higher pre-stress and vice versa.
Some pressure is lost mainly in the first hours after clamping due to relaxation of insulation material since the insulation material is made up of cellulose sulphate. After initial pressure loss, the pressure will remain relatively stable over time. The transformer winding can be represented as a system of spring and mass vibrating between supports see Figure 2.12.
Figure 2.12: The transformer as a spring mass system
Each mass represents a winding, and the soft spacer material is treated as spring the governing equation for the above system is
+ + = ( , ) +
Where M, C, K are the mass, damping and stiffness matrices, F is the short circuit electromagnetic forces and x is the displacement matrix along axial directions. The frequency depends on the spacer material properties[11]
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3. Finite element model
3.1. Geometry of the FE model
The main aim of the FE model is to investigate the spiraling failure mode of a LV transformer winding during short circuit events. For the analysis a model of a layer type transformer winding having 20 turns, wound around a shell with cooling ducts between them is created. Cooling ducts are used for providing a better oil flow along the winding.
In order to reduce computational time and enable dynamic analysis of the problem several simplifications of the geometry were made:
x Winding conductor was modelled as solid copper with rectangular cross-section shaped as a helix
x Insulation between turns (paper and/ or pressboard) was modelled as solid material with rectangular cross-section shaped as a helix between turns (this mostly resembles insulation in layer-type winding)
x Number of axial sticks on the circumference were significantly reduced compared to a real design
x Thickness of inner shell was increased in order to compensate for the reduced number of axial sticks on the circumference
x Vertical winding exit leads and fastening of those were not modelled x Top press ring was not included in the model
The geometry of the Low Voltage helical type transformer winding was created in Ansys Design Modeler See Figure 3.1
Figure 3.1: The FE model
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Note: The spacer thickness does not exceed beyond 3-5mm but in our model the spacer dimension is chosen such that it fits exactly between turns, such a dimension is chosen in the model is to reduce the non-linearity.
3.2. Geometric properties:
The model is built in design modeler using the below geometrical properties, the loading rings dimension are adjusted so that the model converges, any contact gap is closed in Ansys Mechanical using adjust to touch option.
Table 3.1: Geometric Properties
Parameter Value Unit
Inner Diameter 1627 mm
Nominal winding Height 732 mm
Axial Number of turns in Helix
20
Pitch 36.6 mm
Conductor Height 28 mm
Radial Copper width 45 mm
Spacer Thickness 8 mm
Thickness of Axial Sticks 16,5 mm
Thickness of inner cylinder 160 mm
Area of upper loading ring 233,860 mm2
Note: The inner cylinder is modelling the inner supporting structure which includes inner winding shell,
Cooling ducts, cylinders, and the core support.
Note: Cross-sectional area per turn is calculated Area per turn= x(Mean Diameter)x(Radial copper width)
3.3. Material properties
3.3.1. Conductor, inner cylinder:
The conductor is usually made of copper, the main properties that are to be considered are the young's modulus and the proof stress. The proof stress is defined as the stress which produces, while the load is still applied. This is also called the yield point. The mechanical properties of the conductor can be significantly improved by means of cold working process some of these include rolling stretching and bending. The greater the yield point the higher is the linear elastic region in the stress/strain curve see Figure 3.2
25
and Figure 3.3. In my model the elastic-plastic behavior of the conductor was described by a bilinear isotropic hardening curve.
Figure 3.2: Yield strength vs strain of the material
Figure 3.3: Bilinear isotropic hardening curve
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Table 3.2: Physical Properties of Copper Physical Properties of Copper
Property Value Unit
Density 8900
Modulus of Elasticity 110
Poisson Ratio 0.35
Bulk Modulus 122.22
Shear modulus 40.74
Yield Strength 230
Tangent Modulus 8,4 Tensile Ultimate
Strength
320
3.3.2. Insulation Material (Axial sticks, spacer ring, Upper and lower ring):
The insulation material is made of sulfate cellulose. The mechanical stability is obtained by hot pressing process which is effective in achieving high compressive, tensile, and bending strength and, with good stabilization processes clamping pressure relaxation can be minimized. They could serve as axial sticks as shown in the Figure 3.1. They could be cut into small strips or kept in a continuous pitch along the winding for providing frictional support and stability for the structure. The insulation material used in the model is linearly elastic and its material properties are given in the Table 3.3
Table 3.3: Physical Properties of insulation material Physical Properties of Insulation material
Property Value Unit
Density 1300
Modulus of Elasticity 540
Poisson Ratio 0.35
Bulk Modulus 600
Shear modulus 200
Compressive Ultimate Strength
200
3.4. Contact:
Except for the cooling ducts being bonded to the inner shell all interactions are frictional with a frictional coefficient of 0.2. For a frictional contact, the two geometries can carry shear stresses up to a certain magnitude across their interface before they start sliding relative to each other. The model defines an equivalent shear stress at which sliding on the geometry begins as a fraction of the contact pressure. When the shear stress is
27
surpassed, the two components will slide comparative with one another for any non- negative coefficient of friction value. All geometric gaps can be closed using the adjust to touch command see Figure 3.4.
A bonded contact restricts any movement between the surfaces, and it is modelled as a linear contact as there is no movement at all. If any bodies come in contact during deformation and if their interaction contact is not defined the two bodies would penetrate so such interactions were kept frictionless.[12]
Figure 3.4: Configuration for frictional and bonded contacts
3.5. Coordinate system:
A cylindrical coordinate system was defined in Ansys such that the principle axis assigned x-axis directed in the global x direction and orientation about principle axis is assigned as Z axis defined along the global Y axis direction. Such a coordinate system was necessary for to determine the tangential deformation of the exit in Y direction. The exits deformation is such that the clockwise movement would give a negative value and an anti-clockwise movement would give a positive value.
Such a coordinate system would help is determining the hoop stress, tangential stress and axial stress in the model. A global coordinate system was also defined to measure the equivalent von-misses stress.
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Figure 3.5: Figure showing cylindrical coordinate system
3.6. Boundary conditions and meshing:
The whole winding structure with exits is placed on a fixed base such a boundary condition was chosen to check the movement of the whole structure based on their respective contact status.
Figure 3.6: Meshed with Solid 187 tetrahedral element
For meshing the model Solid 187 tetrahedral elements were used. A mesh size of 30mm was used, if a lower mesh size was used the computational time would take a very long time than usual so an optimal value was used[13].
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Figure 3.7: Solid 187 tetrahedral element
3.7. Gravity
A standard gravitation force having a magnitude of 9.81 was applied on all bodies during the simulation acting in the negative Y direction.
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4. Results and analysis
In this section various analysis were presented which include parametric studies, stress results and implicit and static dynamics.
4.1. Static structural Analysis
A static analysis was performed on the model having 20 turns in Ansys static structural.
The load includes only the radial short circuit pressure but not the axial short circuit pressure because the axial pressure acts such that it compresses the winding which provides an extra compression to the windings making it more stable and rigid. In static analysis such a conclusion would overestimate the strength of the windings as short circuit forces are highly dynamic in nature. The problem was solved in three steps each having 1second duration
1. In the first step i.e. 0-1second interval the axial pressure(pre-stress) with a magnitude of 1.6Mpa corresponding to 375KN of force was applied on the upper loading ring with a step size of 0.5. For higher values of pre-stress if Ansys shows an un-converged solution the step size can be reduce to 0.1 or 0.05 based.
This ramped load at the end of 1 second was kept constant in the second and third load step to ensure a constant compression.
2. A radial short circuit pressure was applied on the outer face of the winding directed inwards towards the shell. The pressure was ramped from 0 to 6.5Mpa in 1-2 second duration. The radial pressure here was kept constant throughout the winding height neglecting the variation due to fringing effect
3. In the third step the radial pressure was removed to see the retained deformation of the exits. This was performed to check for any permanent deformation magnitude
Table 4.1: The tangential deformation of exits for 20 turn winding
Exit Tangential Deformation at the end of ramped short circuit pressure (End of 1- 2nd load step)
Tangential Deformation after removing the short circuit pressure load (End of 2- 3rd load step)
Upper
Exit 17 mm 9 mm
Lower
Exit -11mm -5 mm
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4.1.1. Average stress
The above 20 turn helical winding was stressed by ramping an axial pressure of 1.6Mpa and the average value of the equivalent von mises stress in the winding, spacer and the axial cooling ducts were determined.
Table 4.2: Average von-mises stress in 1.6Mpa pre-stressed winding
Element Von-mises
stress Tangential
stress Radial stress Axial stress
Winding 1.95 MPa 0.4 MPa 0.0014 MPa -1.6 MPa
Spacer 1.4 MPa -0.55 MPa -0.049 MPa -1.53 MPa
Cooling ducts 0.024 MPa -0.001Mpa -0.02 MPa -0.015MPa
Table 4.3: Average stress due to radial short circuit pressure
Element Von-mises stress Tangential stress Radial stress Axial stress
Winding 95.4 MPa -93 MPa -4.1 MPa -1.4 MPa
Spacer 1.25 MPa -0.9 MPa -0.13 MPa -1.4 MPa
Cooling ducts 2.8 MPa -1.48 MPa -3.9 MPa -1.9 MPa
Figure 4.1: Tangential deformation at the end of steady state radial short circuit load
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Table 4.4: Average stress after removing the radial short circuit pressure
Element Von-mises stress Tangential stress
Radial stress Axial stress
Winding 4.6 MPa 2.46 MPa -0.04 MPa -1.5 MPa
Spacer 1.38 MPa -0.5 MPa -0.05 MPa -1.5 MPa
Cooling ducts 0.4 MPa -0.18 MPa -0.28 MPa -0.17 MPa
4.1.2. Parametric Analysis
A parametric analysis was performed on a winding model having 4 turns. Number of turns were reduced from 20 to 4 in parametric study in order to reduce computational time.
The goal of the analysis is to investigate how calculated magnitude of spiraling deformation of the exits varies with the applied magnitude of axial pre-stress. The cylindrical coordinate system is defined such that an anti-clockwise movement of an exit has a +vee sign and vice versa. The below graph shows the deformation at exits with axial pre-stress. The below graphs involve two curves showing the spiraling deformation due to short circuit pressure and retained plastic deformation after removing the short circuit pressure
From the graphs we can see the magnitudes of the tangential deformation of the exits at the end of ramped short circuit pressure load are reduced by increasing applied axial- pre-stress, At lower pre-stress values we see a permanent damage of the winding since the retained deformation was close to 1 cm. Spiraling deformation may cause change in reactance. If deformation is large it may damage for example paper insulation (worst case scenario electrical fault and arc), if spiraling deformation is very large it may cause the spacers to slide of axial support spacers, the winding can freely move in axial direction and this will cause a severe failure. However, there is no single value defined, what spiraling deformation is acceptable or not. Only resulting change in impedance is limited, which is easy to measure. There should also be a visual inspection after short circuit test which would say winding failed test if exits have moved significantly.
However, it is important to note that this is a static calculation model that is very simplified and does not include for example the fastening of the winding exits that may have prevented the movement, or the dynamic effects.
Though the spiraling force is equal at both the exits the lower exit deformation’s maximum value is less than the maximum value of the upper exit this is due to the effects of gravity i.e. the weight of the winding so the lower exit experiences not only the magnitude of prestress but also the weight of the winding which provides extra frictional force
It can be seen from Figure 4.3 that at higher prestress value there is almost no retained plastic deformation at all, but we see a constant deformation curve due to short circuit pressure but with the same axial pre-stress in Figure 4.2 we see some plastic deformation still left in the winding. At very low prestress values of magnitude 0.4Mpa Ansys couldn’t converge the problem due to increase in the non-linearity of the model.
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Figure 4.2: : Axial Pre-stress vs tangential deformations upper exit due to static short circuit radial Pressure
Figure 4.3: Axial Pre-stress vs tangential deformations of lower exit due to static short circuit radial Pressure
0 5 10 15 20 25
0 2 4 6 8 10 12 14 16 18
Deformation mm
Axial Pre-stress Mpa
Tangential Displacement of due to short circuit pressure
Retained plastic deformation after removing short circuit pressure load
-16 -14 -12 -10 -8 -6 -4 -2 0 2
0 2 4 6 8 10 12 14 16 18
Deformation mm
Axial Pre-stress Mpa
Tangential Deformation Due to static short circuit pressure
Retained plasic deformation after rmoving the short circuit pressure
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4.2. Modal Analysis
A modal analysis was performed on the winding having 20 turns and 4 turns with a 1.6 MPa axial pre-stress and the resulting six modes were determined.
4.2.1. Natural Frequencies
From the Table 4.5 it can be seen that the natural frequencies are closer to the excitation frequency for the winding having 20 turns in the second mode this results in a vibration mode as shown in the below Figure 4.6 the node is formed at the center part of the winding and the exits move in opposite directions and Figure 4.7 shows the third mode shape where it has 2 nodes in the winding i.e. the blue region and the exits move in the same direction not like in the second mode shape where exits move in opposite directions.
Table 4.5: Natural frequencies for the 1.6Mpa prestressed winding Mode Natural frequency for 20 turn
winding in Hz Natural frequency for 4 turn winding in Hz
1 18 78
2 41 110
3 74 114
4 97 130
5 98 135
6 100 150
Figure 4.4: Undeformed winding the exits are aligned along vertical axis
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Figure 4.5: 1.6Mpa Pre-stressed 20turns winding first torsional mode shape at 18Hz
Figure 4.6: 1.6Mpa Pre-stressed 20 turns winding second torsional mode shape at 41Hz the node is formed at the center of the winding
Figure 4.7: 1.6Mpa Pre-stressed 20 turns winding third torsional mode shape at 74Hz
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4.2.2. Effect of Axial pre-stress in the natural frequency
A parametric study was performed on the winding having 20 turns, and the mode shapes at different axial pre-stress values was plotted. The Figure 4.9 and Figure 4.10 shows the variation of the natural frequency with the axial prestress at lower pre-stress values the second natural frequency is close to the excitation frequency at higher axial pre- stress the first mode is having a value closer to the natural frequency
Figure 4.9: Axial Pre-stress vs natural frequency of 20 turns winding
From the Figure 4.9 Figure 4.10 it can be seen that the relation between axial pre- stress and the mode shapes is highly nonlinear because for a winding having less number of turns would have higher resonance frequency than the excitation force and for a taller winding the resonance frequency would be close to excitation frequency.
0 20 40 60 80 100 120
0 2 4 6 8 10 12
Frequency Hz
Axial Pre-stress
Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6
Figure 4.8: 1.6Mpa pre-stressed winding for 4 turns winding first mode shape at 78Hz
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Figure 4.10: Axial Pre-stress vs natural frequency of 4 turns winding
4.3. Implicit Dynamics
For implicit dynamics, the analysis was performed in Ansys transient structural this analysis was performed in three steps. For implicit dynamics unlike the static structural analysis requires a damping so the damping value was chosen to be 0.08. The three steps in which it was solved is explained below
1. In the first step the axial-prestress pressure of 0.3Mpa corresponding to 233KN was ramped from 0-1 second duration by switching off the time integration this pressure is then kept constant during the next steps.
2. In the second load step for the time varying short circuit forces initially the short-circuit force having the peak magnitude close to value chosen in static analysis was chosen this was obtained by dividing the maximum short circuit force magnitude in static analysis by the maximum peak amplitude of time function with its respective frequency and the X/R ratio.
The magnitude of short circuit forces at time interval 0-18ms was calculated in excel and the magnitudes were then applied in second load step from 1second to 15ms duration as 0-1 second interval was for ramping the axial pre-stress.
( ) = 1
2+ − 2 cos( ) +1
2cos(2 ) Here = 0,0949,frequency=50Hz
The pressure used in static analysis was 6.5Mpa so the peak value of the function was 3.61 so the in order to have peak value of 6.5Mpa for the first cycle the pressure used in static analysis should be divided by 3.61 in this case so the resulting equation would be
0 20 40 60 80 100 120 140 160 180 200
0 1 2 3 4 5 6 7 8
Frequency in Hz
Axial Pre-stress in Mpa
Mode 6 Mode 5 Mode 4 Mode 3 Mode 2 Mode 1
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( ) = 1.8 10 1
2+ , − 2 cos(2 50 ) +1
2cos(4 50 )
Figure 4.11: Short circuit pressure Vs time
The radial and axial pressure was applied on the faces as shown in the Figure 4.12 and Figure 4.13. For the axial short-circuit forces to be applied on the winding due its nature of compressing the winding to its center the winding was split in the mid-section of the conductor so that axial forces can act in one direction in one face and another direction in other face as shown in Figure 4.13. A uniform radial pressure was assumed even though the radial pressure is higher near the exits.
Figure 4.12:Face on which radial short circuit pressure was applied
-1 0 1 2 3 4 5 6 7
0 0,005 0,01 0,015 0,02 0,025 0,03
Pressure in Mpa
Time in Seconds
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3 In the third step the retained plastic deformation in the conductor was determined this was done by reducing the load to 0pa this was kept constant for both axial and radial short circuit pressure throughout each sub-step from 18ms to 25ms duration and the retained plastic deformation was determined.
Figure 4.14: Deformed winding due to transient short circuit radial and axial loads Figure 4.13:Axial short circuit force acting direction in the model such that it compresses the
winding
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Figure 4.15: Deformation of exits for a 20 turn winding due to transient axial and radial pressure
From Figure 4.15 we can see that the deformation of exits moves forward and backwards tangentially due to the transient short circuit loads. After removing the loads i.e. from 15ms -25ms duration we can see that the deformations still exit this is due to the resonance of the structure being close to the excitation load so even after removing the load the structure would still vibrate.
The whole system of the winding which includes the conductor, insulation, spacers and clamping structure can be considered as a mechanical system having mass and elasticity.
Since the short circuit electromagnetic forces that are acting on the system are oscillatory in nature, the transmitted dynamic forces in the structure can be very different from the applied forces. In radial direction the elasticity of copper high and this will result into natural frequencies that are much higher. The dynamic effects on radial vibrations are small and static calculation may be used to estimate elastic radial vibrations. However, in the axial direction the winding acts like a spring due to the large amount of axially stacked insulation material. Therefore, the actual forces occurring in the winding during a short-circuit do not equal the electromagnetic short-circuit forces due to dynamic effects. Dynamic effects may occur also for tangential vibrations. This will depend on the eigenfrequencies of the torsional resonance modes of the winding. If eigenfrequencies are close to the excitation frequency of the applied loads, dynamic amplifications of vibrations and stresses will occur.
Figure 4.16 shows the deformation of exits due transient short circuit axial and radial pressure loads and the deformations after removing the loads for 4 turns model. A 1.6Mpa axial pre-stress was applied to the model. The transient loads were applied till 100ms duration and later the short circuit load was removed, and the retained deformation was determined. Since the natural frequency for a 4 turn winding see section 4.2.1 is high the deformations are instantaneous with no lag this is because the natural frequency is higher than the excitation frequency. There is a permanent
-8 -6 -4 -2 0 2 4 6 8 10
0 0,005 0,01 0,015 0,02 0,025 0,03
Magnitude
Time in Seconds
Transient Axial and Radial Pressure in Mpa Deformation of upper exit mm Deformation of Lower Exit mm
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deformation of 4mm at the upper exit, but the lower exits retain its deformation at the end of 20ms.
Figure 4.16: The deformation of exits with transient axial and radial short circuit pressure for 4 turn model
4.3.2. Stress Induced
The average tangential, radial, Axial and Von-mises stress in the conductor, spacer ring and cooling duct were calculated and their instantaneous values over the transient Axial and radial short circuit load duration and after removing the transient loads were plotted below.
Figure 4.17: Average Von-mises stress in 4 turn winding over time
-6 -4 -2 0 2 4 6 8
0 0,05 0,1 0,15 0,2 0,25
Magnitude Mpa, mm
Time in seconds
Transient axial and radial Short-circuit pressure Deformation of upper exit Deformation of lowerexit
0,00 10,00 20,00 30,00 40,00 50,00 60,00 70,00 80,00 90,00 100,00
0 0,05 0,1 0,15 0,2 0,25
Von Mises stress in Mpa
Time in seconds
Conductor Cooling Ducts Spacer ring
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Figure 4.18: Average tangential stress in 4 turn winding over time
Figure 4.19: Average Radial stress in 4 turn winding over time
-100 -80 -60 -40 -20 0 20
0 0,05 0,1 0,15 0,2 0,25
Tangential Stress in Mpa
Time in seconds
Conductor Cooling duct Spacer ring
-4,5 -4 -3,5 -3 -2,5 -2 -1,5 -1 -0,5 0 0,5
0 0,05 0,1 0,15 0,2 0,25
Radial Stress in Mpa
Time in seconds
Conductor Cooling Duct Spacer Ring
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Figure 4.20: Average Axial stress in 4 turn winding over time
Figure 4.21: Average Von-mises for a 20 turn winding over time
-12 -10 -8 -6 -4 -2 0
0 0,05 0,1 0,15 0,2 0,25
Axial Stress In Mpa
Time in Seconds
Conductor Cooling Duct Spacer Ring
0 10 20 30 40 50 60 70 80 90 100
0 0,005 0,01 0,015 0,02 0,025 0,03
Von mises Stress in Mpa
Time in seconds
Conductor Spacer ring Cooling Ducts
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Figure 4.22: Average tangential stress for 20 turn winding over time
Figure 4.23: Average Radial Stress for 20 turn winding over time
-120 -100 -80 -60 -40 -20 0 20
0 0,005 0,01 0,015 0,02 0,025 0,03
Tangential stress in Mpa
Time in seconds
Conductor Spacer Ring Cooling Ducts
-7 -6 -5 -4 -3 -2 -1 0 1
0 0,005 0,01 0,015 0,02 0,025 0,03
Radial Stress Mpa
Time in Seconds
Conductor Spacer Ring Cooling Ducts
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Figure 4.24: Average Axial Stress for 20 turn winding over time
4.3.3. Spiraling Phenomenon
In order to determine the forces acting on the exits as a result of the applied radial pressure, the axial, tangential and radial force reactions were analyzed for winding model with 4 turns by fixing the exits. Calculated tangential force reaction corresponds to the so-called spiraling force or end thrust force acting on the exit lead. The axial, tangential and radial force reactions at the exits were determined for a winding having 4 turns. The winding was initially pre-stressed by applying a axial pressure of 0.8Mpa in the first step and in the second step the transient short circuit pressure load was applied radially along the winding for 57ms duration the short circuit pressure load has a magnitude as that explained in section 4.3. The radial, axial and tangential force reactions were determined at each transient radial pressure load i.e. at every 0.001 second from 0 to 0.057 duration.
From Figure 4.25. We can see the axial force reaction is completely negligible and only the magnitude of tangential force reaction is the highest so the exit would spiral tangentially as there since the axial force dropped to zero the exit would spiral tangentially as there since the axial force dropped to zero
-40 -35 -30 -25 -20 -15 -10 -5 0
0 0,005 0,01 0,015 0,02 0,025 0,03
Axial Stress in Mpa
Conductor Spacer Ring Cooling Ducts
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Figure 4.25: Force reaction at exits
As can be seen in Figure 4.18 and Figure 4.19, the applied radial pressure of with peak amplitude 6.5 MPa converts to tangential end thrust forces (spiraling force) with peak amplitude of about 120 kN. The thrust force acting on the exits is traditionally assumed to be equal to the product of mean hoop compressive stress on the winding and cross- sectional area of the lead exit. According to Figure 4.18 in section 4.3.1, mean tangential (hoop) stress in conductor is about 95 MPa. Cross-sectional area of the FEM exits equals (28 mm x 45 mm) 1260 mm2 (see Table 3.1). Thus, calculated tangential force reaction at exits agree well with the expected spiraling force at exits, which equals about 120 kN (95 MPa x 1260 mm2).
4.3.4. Parametric analysis
A parametric analysis was performed on a winding containing four turns the tangential deformations of the exits was determined and plotted in Figure 4.27 and Figure 4.26.
The graphs contain the deformation curves for both the exits. The radial transient load as explained in 4.3 was applied for 15ms duration and the resulting deformation was plotted against axial pre-stress. Calculated tangential displacements of the exits are significantly reduced with increasing axial pre-stress. This shows the importance of applied axial clamping pressure on the winding to avoid spiraling deformation.
The lowest axial pre-stress was 0.6Mpa and the highest was 14 Mpa the deformations of the exits at the end of radial short pressure i.e. end of 15ms duration and the retained plastic deformation of exits at 25ms duration i.e. end of third load step after removing the transient radial load was determined and was plotted against the axial-prestress. The deformations of both the exits reduce while increasing the prestress at 14Mpa axial pre- stress the deformation of lower exits approaches zero but there is still deformation left in the upper exits
-20 0 20 40 60 80 100 120 140
0 0,01 0,02 0,03 0,04 0,05 0,06
Force Reaction KN
Time in Seconds
Axial Force Radial force Tangential force
