Transient Voltage Distribution in Bushing

DEGREE PROJECT IN ELECTRICAL ENGINEERING, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2020

MD NAZMUS SHAKIB KHAN

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE

KTH Royal Institute of Technology Stockholm, Sweden

Abstract An electrical bushing is one of the most important elements in a power transformer. Steep front surges such as transient impulse voltage from lightning strikes is an inevitable electromagnetic transient mostly happening in power transmission and distribution system. The bushing might lead to be degraded due to such kind of surge. This project deals with overvoltage stress distribution on the transformer bushing under the effect of electromagnetic transient response such as lightning impulse. To understand the behavior of transient response on the bushing, a proper model of power transformer bushing is built-in Comsol multiphysics to authenticate the stress distribution. The electromagnetic wave of impulse propagates onto the overhead line that connects with the transformer. Some understanding of the transient behavior of a conductor bushing has been achieved through studying the influence of inductance property and the skin effect characteristics of a multi-layer coaxial cable on the wave propagation, which has been structured in this project to simplify the model. On the other hand, the skin effect analysis on the conductor of the bushing has been taken also into account in this project using real conductor simulation in the Comsol model. Thus, it will be interesting to compare the real conductor model with the perfect conductor of the bushing through analyzing the current density effect on it. In this project, multi-layer of coaxial cable and transformer bushing are simulated. The simulation is carried out for time domain and frequency domain in Comsol based on the model characteristics. Keywords: Coaxial cable, Overvoltage transient, Power transformer bushing, Skin effect

Sammanfattning En elektrisk genomföring är ett av de viktigaste elementen i en transformator. Spänningsvågor med branta fronter som impulsspänningar från blixtnedslag är ett oundvikligt elektromagnetiskt övergående fenomen som oftast sker i kraftöverförings- och distributionssystem. Genomföringen kan leda till att degraderas på grund av en sådan våg. Detta projekt handlar om fördelning av överspännings på transformatorgenomföringen under påverkan av elektromagnetisk transient respons, såsom blixtimpuls. För att förstå beteendet hos övergående respons på genomföringen är en korrekt modell av transformatorgenomföring inbyggd Comsol-flerfysik för att autentisera spänningsfördelningen. Den elektromagnetiska impulsvågen fortplantas från luftledningen som ansluter till transformatorn. Viss förståelse för det övergående beteendet hos en ledargenomföring har uppnåtts genom att studera påverkan av induktansegenskaper och hudeffektegenskaperna hos en flerskikts koaxialkabel på vågutbredningen, vilket har strukturerats i detta projekt för att förenkla modellen. Å andra sidan har hudeffektanalysen på genomföringens ledare beaktats i detta projekt med användning av verklig ledarsimulering i Comsol-modellen. Således blir det intressant att jämföra den riktiga ledarmodellen med den perfekta ledaren för genomföringen genom att analysera strömtäthetseffekten på den. I detta projekt simuleras flerskikt av koaxialkabel och transformatorgenomföring. Simuleringen utförs för tidsdomän och frekvensdomän i Comsol baserat på modellegenskaperna. Nyckelord: Koaxialkabel, överspänningsövergående, transformatorgenomföring, hudeffekt

Acknowledgements Thanks to Almighty Allah for giving me enough strength to complete this thesis with patient and peaceful mind. My dream become reality to achieve the degree of Electric Power Engineering in KTH Royal Institute of Technology, Stockholm, Sweden through finishing this footstep of my life. I submit my heartfelt gratitude first to my respected Examiner Mr. Hans Edin, Professor in KTH Royal Institute of Technology for his tremendous support and guideline, without him my dream would not have perceived the glare. With my deepest thankfulness, I must wholeheartedly acknowledge the funder of my masters’ studies in KTH by the Swedish Institute (SI). I would like to extend my appreciation to Mrunal Parekh, PhD student in Electromagnetic Engineering Division for his untiring enthusiasm and inspiration kept me engaged in Electromagnetic Theory to complete this endeavor, and in particular the Comsol supervision that I have got from him. My joy knows no bounds in articulating my gratitude to my friends in KTH whose encouragement was truly helpful throughout this journey especially of Mohammad Alharbi and Aravind S Kumar. I am truly grateful to my colleague Mr. Rashedul Amin Tuhin, Senior Lecturer of East West University in Bangladesh for his unceasing support from the beginning of my masters’ studies. Finally, I would like to thank KTH Royal Institute of Technology for making me appreciate studying In Electric Power Engineering. Last but not the least, I am decidedly indebted to my father Mr. Md. Amirul Islam Khan (Saiful) and my mother Mrs. Rowshon Ara (Jesmine) whose value grow to me with age, their continuous motivation and support made me fruitful in this work. I owe them my life for this attainment.

Md Nazmus Shakib Khan Stockholm, 2020

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Table of Contents Abstract ...... i Sammanfattning ...... ii Acknowledgements ...... iii List of Table ...... v LIST OF FIGURES ...... vi Acronyms ...... viii Chapter 1 ...... 1 Introduction ...... 1 1.1 Project motivation: ...... 1 1.2 Project objective and Methodology: ...... 2 1.2.1 Objective ...... 2 1.2.2 Methodology: ...... 2 1.3 Project organization: ...... 2 Chapter 2 ...... 3 Theory ...... 3 2.1 Theoretical Background ...... 3 2.1.1 Transformer in power system ...... 3 2.1.2 Transformer Bushing: ...... 6 2.1.3 Multi-conductor transmission lines: ...... 14 2.1.4 Overvoltage transient ...... 16 2.1.5 Lumped model of overhead transmission line and bushing: ...... 17 2.1.6 Self and mutual inductance of coaxial cable: ...... 20 2.1.7 Skin effect of a circular conductor: ...... 23 Chapter 3 ...... 25 Result part with simulation ...... 25 3.1 Skin effect of coaxial cable ...... 25 3.2 Inductance phenomenon and skin effect of the coaxial cable: ...... 27 3.3 Transformer bushing: ...... 34 3.3.1 The Skin effect in transformer bushing: ...... 38 Conclusion ...... 45 Future work ...... 46 Bibliography ...... 47

List of Table Table 1 Cutline 2d 1 direction for sheath 2 ...... 31 Table 2: Mutual inductance for center conductor-sheath 1...... 31 Table 3: Cutline 2d 1 direction for sheath 3 ...... 32 Table 4: Mutual inductance for center conductor-sheath 2...... 32 Table 5: Mutual inductance for center conductor-sheath 2 after changing the current ...... 32 Table 6: Cutline 2d 1 direction for the center conductor ...... 33 Table 7: Self-inductance values ...... 33 Table 8: Geometrical values of the bushing……………………………………………………………………...35 Table 9: Transformer oil parameters ...... 35 Table 10: Parameters for bushing...... 36 Table 11: Cutline 2d 1 direction in the bushing conductor ...... 40

List of Figures Figure 1: Power Transformer ...... 3 Figure 2: Design Of Power Transformer ...... 4 Figure 3: Transformer Conservator Tank ...... 5 Figure 4: Transformer Bushing ...... 6 Figure 5 (a): Bushing Construction ...... 7 Figure 5 (b): Oil filled Condenser Bushing (OIP) ...... 7 Figure 6: Solid Type Bushing [11]...... 8 Figure 7: Oil Filed Capacitance Graded Bushing ...... 9 Figure 8: SF6 In Oil Bushing ...... 10 Figure 9: SF6 To Air Bushing ...... 11 Figure 10: Oil Impregnated Paper Bushing (a) Total Structure (b) Condenser Body Of Oip ...... 12 Figure 11: Resin Impregnated Paper Bushing ...... 13 Figure 12: Coaxial Cable Structure ...... 14 Figure 13: Transient Overvoltage ...... 16 Figure 14: Equivalent Circuit of Transmission Line ...... 17 Figure 15: Coaxial Cable Construction ...... 18 Figure 16: Lightning Impulse Propagation On OHL ...... 20 Figure 17: Single-Core Coaxial Cable ...... 20 Figure 18: Cross-Section Of Coaxial Cable ...... 21 Figure 19 Two Concentric Coplanar Loop ...... 22 Figure 20: Cross-Section Of Conductor That Carried Current For Dc ...... 23 Figure 21: Current At The Surface Of The Conductor For Ac ...... 23 Figure 22: Skin Depth Of The Conductor ...... 24 Figure 23 (a): Skin Effect For 50 Hz ...... 25 Figure 23 (b): Skin Effect For 500 Hz ...... 26 Figure 23 (c): Skin Effect For 5000 Hz...... 26 Figure 23(d): Skin Effect Plot Of Current Density Vs Length...... 26 Figure 24: 5 Layers Coaxial Cable...... 27 Figure 25: Material Put In The Coaxial Cable ...... 28 Figure 26 (a): Coil 1...... 28 Figure 26 (b): Coil 2...... 28 Figure 26 (c): Coil 3 ...... 29 Figure 27: Finer Mesh Of 5 Layers Coaxial ...... 29 Figure 28 (a): Skin Effect Of Coaxial Cable For 50 Hz ...... 30 Figure 28 (b): Skin Effect Of Coaxial Cable For 500 Hz ...... 30 Figure 28 (c): Skin Effect Of Coaxial Cable For 5000 Hz ...... 31 Figure 29 (a): Cutline 2D In Coil 1...... 32 Figure 29 (b): Cutline 2D In Coil 3...... 32 Figure 29 (c): Cutline 2d In Coil 1for Self-Inductance ...... 33 Figure 30: Schematic Diagram Of Section Of A Coaxial Cable For A Transmission Line Connected To Source Voltage And Loads ...... 34 Figure 31: Input Voltage Pulse ...... 35 Figure 32: Bushing Rectangle Built-In Comsol ...... 35 Figure 33: Bushing Material As Transformer Oil Built-In Comsol ...... 36 Figure 34 (a): Transformer Bushing With Foils...... 36 Figure 34 (b): Finer Mesh For Bushing...... 36 Figure 35 (a): Simulation Of Transformer Bushing With Perfect Electric Conductor Condition In The Surface ... 37 Figure 35 (b): Input-Output Pulse With Damping Effect On Perfect Conductor ...... 37 Figure 36 (a): Electric Filed In Bushing With Real Conductor...... 38 Figure 36 (b): Input-Output Pulse With Real Conductor...... 38

Figure 37 (a) Electric Field...... 38 Figure 37 (b): Input-Output Pulse ...... 38 Figure 38 (a): Electric Field Of Normal Component...... 38 Figure 38 (b): Input-Output Pulse...... 38 Figure 39 (a): Electric Field Of Normal Component ...... 39 Figure 39 (b): Input-Output Pulse...... 39 Figure 40 (a) Electric Field Of Normal Component ...... 39 Figure 40 (b): Input-Output Pulse...... 39 Figure 41 (a): Electric Field Of Normal Component...... 40 Figure 41 (b): Input-Output Pulse...... 40 Figure 41 (c): Plot For Current Density Vs Length ...... 40 Figure 42 (a): Electric Field Of Normal Component ...... 41 Figure 42 (b): Input-Output Pulse ...... 41 Figure 42 (c): Plot For Current Density Vs Length ...... 41 Figure 43 (a): Electric Field Of Normal Component ...... 42 Figure 43 (b): Input-Output Pulse ...... 42 Figure 43 (c): Plot For Current Density Vs Length ...... 42 Figure 44 (a): Electric Field Of Normal Component ...... 43 Figure 44 (b): Input-Output Pulse ...... 43 Figure 44 (c): Plot For Current Density Vs Length ...... 43 Figure 45 (a): Electric Field Of Normal Component ...... 44 Figure 45 (b): Input-Output Pulse ...... 44 Figure 45 (c): Plot For Current Density Vs Length ...... 44

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Acronyms

TEM Transverse Electromagnetic Wave

RF Radio Frequency

AC Alternating Current

DC Direct Current

ESD Electrostatic Discharge

OHL Overhead Line

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Chapter 1 Introduction

The bushing is an insulated device of a transformer that allows an electrical conductor passing through the oil containing grounded tank. There are low and high voltage windings in a power transformer. Bushing is provided to make a connection between the windings of the transformer and the power system such as transmission, and distribution lines. An incoming electrical transient such as a lightning impulse is a type of step surge that may cause high stress on transformer insulation. Studying of transient propagation in electrical component are of importance from different aspects of the power system. In the case of the bushing might fail due to the overvoltage transient that enters through back flashover [1]. It is important to find the effect of stress distribution on transformer bushing during transient response which provides valuable information on transformer bushing’s condition [2]. Therefore a model of transformer bushing has been studied in this project to investigate the voltage distribution in bushing when it comes to electromagnetic wave propagation of lightning impulse during transient overvoltage. To the stress distribution, a model of the bushing is built in Comsol multiphysics where all the parameters for the bushing are entered numerically according to the geometrical perspective of the Comsol model. The behaviors of the overvoltage stress distribution proposed convenient information about the state of transient impulse in the bushing. Some of literature have been explored by various functions such as admittance or skin effect analysis to have an accurate model [2].

1.1 Project motivation: The idea behind this work is to investigate overvoltage transient effect on the transformer bushing at the high voltage side of the power transformer. Lightning impulse as step surge on the bushing might damage such an insulating part of the transformer resulting in failure of the system. Coaxial cable is a medium of conductor where electromagnetic wave propagates from one end to the other. The main purpose of this work is about investigating the effect of transient behaviors on transformer bushing during the overvoltage situation. This thesis work also study the inductance phenomenon of the coaxial cable structure and the current density on its outer surface to study the characteristics of the cable when it connects to the transformer bushing. Also, the skin effect of the bushing comes to attention during working on the report obviously a good attempt for this thesis.

1.2 Project objective and Methodology: 1.2.1 Objective The purpose of this project is to investigate the stress distribution along with the transformer bushing when transient lightning impulse propagates during an overvoltage state. 1.2.2 Methodology: To obtain the project goal, the total task was performed in two steps. 1. To find the numerical and analytical value of self-inductance of the core of multi- number of coaxial layers and the mutual inductances between core to the sheath of the layers. In the Comsol model, investigation of the inductances was designed for 2-axial cables with single core and two sheaths.

2. The final task was to design a transformer bushing in Comsol in the electromagnetic wave transient domain. The model was limited to three floating foils added in the oil- impregnated paper bushing where the last foil has been connected to the transformer tank with short- circuited ground to observe the desired voltage impulse distribution along the bushing conductor. As mentioned above, the real conductor simulation has been performed at a later step to compare the result with the perfect conductor to study the skin effect observation.

3. Before performing the tasks mentioned above, electric field distribution and current density function of a single layer coaxial cable of the Comsol model were built in the frequency domain as a starting point. 1.3 Project organization: This project report is arranged into three chapters. Where: • In 1st chapter, the introduction, motivation, project methodology, and purpose are overviewed.

• In the 2nd chapter, the theoretical background behind this project are described. At the starting, some basic parts of transformer bushing are mentioned. Different parts of the bushings are given with an explanation. After that, the effect of the stress distribution concept is added with the bushing and discussed also how it can greatly influence on the power system. The skin effect and the analytical equations of mutual inductances between the core and sheath of the coaxial cable are also explained in this chapter.

• The results of the simulation taken from the Comsol model are explained in chapter 3. The plotting for input and output impulse, current distribution along the conductor are figure out graphically. Here at the beginning, the analytical equations that suited to build a Comsol model for the skin effect in coaxial cable and the mutual inductances of a single core with multiple layers of sheaths are discussed. Then the design of transformer busing in the transient state due to overvoltage is briefed finally. The implementation process for the model analysis is also given in this chapter. Finally, the skin effect phenomenon of a real conductor bushing has been simulated.

• At last, the future implementation has been given for further work.

Chapter 2 Theory 2.1 Theoretical Background The theory deals with three different parts. First, a section on the importance of the transformer in the power system is discussed in the first part. The bushing and its application are discussed in the second part. Finally, the effect of transient stress on transformer bushing are discussed in the third part. 2.1.1 Transformer in power system A transformer is an equipment in the power system that transforms electric power by electromagnetic induction from one circuit to another. It converts ac voltage without changing its frequency. The transformer transfers energy through the circuit in between generator and distributed electrical or electronic primary circuit. It acts to step up and step down the voltages in a distribution system. A liquid immersed type of transformer is a very common design. Other insulations systems may be dry type transformers in the form of resin cast windings or even SF6 gas-insulated transformers. A critical aspect of a transformer is its technical lifetime, which is typically 30 years around or more [3].

Bushing

Figure 1: Power Transformer [3]

There are three types of power transformers according to their ranges: 1. Small power transformer, 2. Medium power transformer, 3. Large power transformer.

The range of small power transformer is from 500-7500 kVA. The range of medium power transformer is from 7500 kVA-100 MVA and the range of large power transformer is above 100 MVA or beyond. Based on Faraday’s electromagnetic induction law, a power transformer holds a high current and a lower voltage at one end of the transformer, and at the other end, it holds high voltage with low current. A power transformer magnetic core is designed by metallic laminated sheets of either core or shell type [3]. The windings consist of copper wound by the conductor that makes available as three-phase or single-phase units. The transformer has also a small footprint.

Figure 2: Design of power transformer [3] Transformer core: The core of the transformer acts as a structural support for windings. The core provides a path of low reluctance to flow the electromagnetic flux. The laminated core reduces the loss of eddy current and also the hysteresis loss. The diameter of the core is proportional to the copper losses of the power transformer and inversely proportional to the iron losses [4]. The steel weight of the transformer core is minimized if the core diameter is decreased. But vice versa when the diameter of the core is increased. Why winding made of copper: In a power transformer, copper reduces the losses due to high conductivity. Also, it minimizes the need for metal in the transformer windings. It is very easy to bend the conductors into windings due to the ductility of the copper. So it minimizes the necessity of copper and the total winding volume [4].

Transformer insulating material: Cardboard and paper as an insulating material are used to isolate the winding of primary and secondary winding of a power transformer from each other, also from the core of the transformer. Another insulating material of the power transformer is transformer oil. The oil performs two function: it keeps core cool and enhance the dielectric withstand strength. The core of the transformer must be immersed into the oil. As the transformer oil, mineral oil of hydrocarbon is used [4]. The contamination of oil reduces the dielectric properties, also renders it for useless as insulating medium. Transformer Conservator tank: The conservator of power transformer conserves oil. It may be a cylindrical, airtight, or metallic drum fitted on the transformer. At the top, the tank conservator vented to the atmosphere. The oil level of the transformer is in the middle of the tank which allows expanding of the oil and contract with the variation of temperature [4]. The conservator tank is connected with the main tank that is filled with the oil of the transformer by a pipeline.

Figure 3: Transformer Conservator Tank [4]

Cooling Tubes To cool the transformer oil, the cooling tube is used where the oil of the transformer is circulated by this tube. The oil circulation may be natural or it may be forced. The hot oil rises naturally to the top due to the rises of temperature of the oil in natural circulation. The cold oil on the other hand sinks downward [4]. An external pump is required to use in forced circulation.

2.1.2 Transformer Bushing:

When the phase conductor is fed through the grounded transformer tank, it creates an electric field around it of high field strengths. The bushing of a transformer controls the electric field strength and the curve shape of the field lines, reduces overvoltage stresses along the insulating part of the transformer [5]. A bushing is used in high voltage apparatus in order to bring high voltage conductor over a grounding structure. To avoid surface discharges it is important that the high electric field near the area of grounded part of bushing needs to be graded [6].

Figure 4: Transformer Bushing [7] As a vital component of a power transformer, bushing is used to isolate the center conductor from the transformer grounded tank to make a connection between the supply line and transformer winding. But it is one of the feeblest components in power transformer from the view of reliability. One of the major causes of outages of the transformer is the failure of bushing [8]. When the windings of the transformer are connected with the high voltage lines, the end connection of a transformer must be taken care in order to avoid lightning from the high voltage connection that comes in contact with the transformer. For low voltage distribution, cable connections are taken from the cable boxes which is in the secondary side of the transformer. The bushing is consisting of a current carrying conducting part (rod) keeps in the center of the bushing, cable , porcelain cylinder that is installed in the hole of a transformer cover uses to isolate the current carrying part [9]. The high quality of glazed porcelain insulated bushing is used for medium to high voltage or up to 33 kV with having finned or smooth surface. The upper part (outside) of this bushing which is used for the transformer to work outdoors makes with the petticoats for the protection of the fins (lower one) against water during rainy weather. The bushings of the transformers usually used for the voltage levels above 36 kV are mostly oil filled condenser type bushing such as OIP (Oil Impregnated Paper condenser bushing). There are a hollow porcelain cylinder with two parts in the oil filled bushing with a conductor that passes through its center axis. The oil is contained in the space between the inner surface of the porcelain and the conductor, and remains separate from the transformer tank [9].

External connecting terminal

Top arcing horn

Poercelain Insulator Center Conductor

Bottom arcing horn

Filled with oil or other insulating materials

Figure 5 (a): Bushing construction [10] Figure 5 (b): Oil filled condenser bushing (OIP) [11]

The upper part of the bushing of the transformer is mainly connected to an expansion chamber that requires to accommodate the distinctions in the oil temperature volume. In a current transformer has a provision such that, it is possible to remove the bushing without any disturbance of the current transformer [9]. Purpose of Electrical bushing: The main purpose of the bushing is that it transmits the power that comes from voltage and current out or in the busing enclosure. It must be capable for the conductor to carry the rated current and the bushing insulation must be also capable of voltage withstanding where it applies without producing overheating in the nearby insulation part [12]. Practically, bushing is not relevant to transmit electric power through it, rather it is rated by maximum current and voltage based on the design. Types of Electrical Bushings There are several ways for the classification of bushings which are based upon some practical or real reasons, and that are discussed in below [12]: 1. Based on construction: There are two types of bushings based on the construction • Bulk type or solid type of bushing • Condenser type or capacitance-graded bushing

2. Based on the media of insulation on ends: in this classification, bushing depends upon its application primarily. It is classified accordingly:

• Air to oil bushing • Air to SF6 bushing • Air to air bushing

• Oil to oil bushing • SF6 to oil bushing

3. Based on the insulation around bushing of the transformer: they are classified as:

• Air insulated bushing • Gas insulated bushing • Oil filled or oil insulated bushing • Oil impregnated paper insulated bushing • Cast insulation bushing

1. Bushing based on the construction:

• Solid type (Bulk type) bushing Usually, this type of bushing is used for all voltage ratings below 25 kV. The porcelain insulators are used at either end of the bushing.

Figure 6: Solid Type Bushing [13] This type of bushing is used for applications which range from a small circuit switch and distributed transformer to the very large equipment of the electrical circuit such as hydro-power generator or on the primary (low voltage) side of a step-up power transformer. The limitation of such type of bushing is the capability to withstand the voltages above 90 kV of 60-Hz [12].

So the applications range are limited to the voltages of 25 kV apparatus ratings with the test voltages of 70 kV. During the test of the transformer in recent applications, it needs low partial discharge which limits on the voltages of 25 kV terminals and have been caused restriction on using of such type of bushing. In that case, either high expensive capacitance graded bushing, or a special designed of solid type bushing with grading shielded (unique) that can enable small partial discharge levels should be used [12].

• Capacitance graded bushing This type of bushing is currently used for all voltage ratings of nearly 25 kV or above of the system voltage.

Figure 7: Oil filed capacitance graded bushing [12] This type of transformer bushing has conducting sheets within some insulating material such as oil-impregnated paper at prearranged radial intervals which is in between insulator and central conductor. The capacitance graded bushing is designed by using different types of materials depends on the manufacturers. The old method was inserting metallic surfaces of concentric cylindrical porcelain or inserted conductive layers of laminated tubes. After that, conductive foils of copper or aluminum used in later methods in the oil-impregnated kraft paper [12]. Printing semiconductive toner on oil-impregnated paper kraft paper wraps with different conductivities is an alternative method.

The main elements of capacitance graded bushing are the circular central conductor where the core of the capacitance graded wounds, lower and top insulators, the bottom and top terminals, mounting of flange, oil, and enlargement oil cap. The manufacturing and technical details include such as insulation or conducting layer, winding equipment for the capacitance core, impregnating the paper insulation by oil. The dimension of the radial is much smaller than of solid type construction in capacitance graded bushing which saves on bushing materials as well as the apparatus where the bushing uses [12].

2. Types of electrical bushing based on end insulation Based on the insulating medium of the bushings at the ends, this type of bushings are categorized into several types and are explained some of them in the following section [12]:

• Air to oil bushing This type of bushing has insulation of air at one end and insulation of oil at another end. Due to strong dielectric behavior, oil is twice strong at the atmospheric pressure as air and the air end is twice as long or less as the oil end. Usually, this type of bushing is used between the atmospheric air and the oil-filled device or apparatus.

• Air to air bushing Insulation of air is used at both ends of the bushing. Generally, this type of bushing is used for building application purpose where one of the ends exposes to the outdoor atmospheric situation and another is to indoor. Exceptional application of bushing has the limitation to use and these are included as follows:  SF6 to oil bushing: this type of bushing is used in between oil-filled apparatus and bus ducts of SF6 as transitions.

Figure 8: SF6 in oil bushing [14]

 Air to SF6 bushing: used usually in insulated SF6 circuit breakers.

Figure 9: SF6 to air bushing [15]

 Oil to oil bushing: used in between of the oil-filled apparatus and oil bus ducts.

3. Based on the insulation around bushing of the transformer This type of bushing is applicable to the insulating material inside of the bushing. The material can either be used in capacitance graded bushing or solid type construction. More than one insulating materials are used in conjunction.

The following text is a brief explanation of these types [12]:

• Air insulated bushing

This type of bushing is generally used with the apparatus that are air-insulated and of solid construction which employs air between conductor bushing and insulators at atmospheric pressure.

• Oil Insulated bushing In a solid type of bushing, electrical graded mineral oil is used between bushing conductor and insulators called oil-filled bushing. The oil can also be shared with other apparatus where the bushing is used. In capacitance graded bushing, mineral oil uses which contained within the transformer bushing and holds between the insulators and insulating materials to impregnate the paper kraft and heat transferring of the conducting lead.

• Oil-Impregnated Paper-Insulated Bushing To produce a composite material that has a greater dielectric characteristics, this type of bushing is used. So oil-impregnated paper insulated bushing uses electric graded kraft paper and mineral oil. These materials are using around 50 years in the core of capacitance graded bushing as an insulating material. This type of bushing must have voltage gradient from the center of the conductor to the ground. Different layers of paper insulation and foils of aluminum or copper make filled with the oil of the transformer as insulating fluid. The layers form capacitors to grade-down the voltage [16].

Figure 10: Oil impregnated paper bushing (a) total structure (b) condenser body of OIP [17]

• Impregnated Paper Insulated Bushing or Resin Bonded bushing To fabricate the core of the capacitance graded bushing, resin bonded paper insulated bushing is used with coated resin kraft paper. On the other hand, paper impregnated with the resin is used in resin impregnated paper insulated bushing and then uses for the fabrication of capacitance graded core.

Figure 11: Resin Impregnated paper bushing [18]

• Cast insulation bushing This type of bushing is built in a solid cast type of material without or with inorganic (mineral) filler. This bushing either can be capacitance graded type or solid type where former type is held as more representative in the recent technology.

• Gas insulated bushing This type of bushing uses pressurized gas (such as SF6) for the insulation between the flange and the central conductor and is used usually in SF6 Circuit Breakers. There is no capacitance grading used in this type of bushing. The electric fields are controlled using dimensions and ground placement.

2.1.3 Multi-conductor transmission lines:

Electromagnetic waves propagate along the transmission line such as coaxial cable used in the Television cable system. The propagation works according to the dielectric properties of the material between the center of the cable and the surrounding metallic cylinder. During the breaking of the dielectric material uniformity, the wave partially reflected as a disruption. Connectors between electronic apparatus and transmission lines disrupt always Transverse Electromagnetic waves (TEM). This effect can be minimized by well-designed connectors (possibly expensive) [19]. In other application wave reflection as physical effects due to dielectric changes can be considered as advantages. For dc transmission over a medium has two conductors including go and return wire to complete the circuit. To transmit RF (Radio Frequency), the center wire of the coaxial cable is considered as go wire and the outer conductor which is consisted of shielding surrounded the center conductor is called the return wire. The wave nature of high-frequency alternating current namely as RF is taken into account when the coaxial cable is designed. Unshielded cable for RF transmission creates power loss during radio wave generation and induces spurious transmissions on the wire [20]. The concept of the return wire in the coaxial cable as a shielding mechanism causes an electromagnetic field between innermost and outermost conductors. Construction of the coaxial cable The construction of coaxial cable shows the mechanical structure of the components that has an impact on electrical performances when impulse propagates through the cable.

Figure 12: Coaxial cable structure [21] • Center conductor The center conductor is used for AC power transmission consisted of multi strands twisted wire made by copper or solid copper wire. But multi strands wire is flexible than solid copper wire counterparts, experience attenuation significantly per meter of cable because of the proximity effect. The surface area or the diameter of the inner conductor is critical to reduce ohmic losses

through the skin effect. Due to the large surface area in multi strands wire than the solid cable, the proximity effect at high frequencies causes much more losses than the skin effect [20]. For better electrical performance, solid copper often called bare copper is offered typically. Taking into account of the advantages of skin effect, manufacturers provide a center hollow conductor or a core of aluminum with a copper jacket. This process in electrical performance makes a tradeoff for a significant cost reduction [20]. Key points: 1. Bigger Diameter: Reduces losses, Greater handling of power, Reduces Flexibility 2. Solid Copper in the center: Higher electrical performance, Substantial Mass, high cost 3. Strand conductor: flexibility is increased, losses increase.

• Dielectric Insulator

The dielectric insulator in a coaxial cable separates the center conductor and the outer conductor by minimizing the ohmic losses at the same time when arises from the contact with conductors. The dielectric is made of materials such as PTFE, or polyethylene. An insulator of a perfect dielectric would consist of vacuum or inert gas. Dielectric is also used for achieving constant impedance. It isolates the conductors at a fixed distance [20]. Usually, the dielectric of a coaxial cable is an extremely high resistive material apply to the conductor for resisting electric current flowing to the shield [22].

Key points:

1. Dielectric has to choose with lower possible density 2. Halogen: Better performance in electrical efficiency 3. Non-Halogen: low acidity, low smoke, but minimizes electrical performance

• Outer Conductor & Shielding

The outer conductor is kept connected with ground potential which provides shielding electromagnetically- isolating the innermost electromagnetic impulse from the exterior interference, restraining power to the dielectric confines. The outermost conductor as a form of braiding metal wire while affords higher effect, the wires gaps cause radio frequency interference and leakage. Avoiding this effect, higher graded cables would be often double- shielded as aluminum tape or APA with metal foils. The shielding efficiency provides by cables specification would give percentage braid coverage through comparative metric [20].

Key points:

1. Effective shield 2. Mechanical strength 3. Higher Flexibility 4. Ease of termination and stripping 5. Erosion of resistance

Outer Jacket

The outer jacket of a coaxial cable does not have electrical function rather the purpose to use the outer jacket is to provide mechanical and environmental protection. The material commonly included as TPFE, PVC, and FEP. Additional chemicals are added for ultra-violate sustainability, minimize toxicity due to smoke (lower smoke causes zero halogen), protection against water access and oil to have direct burial [20].

The following characteristics is chosen for the application purpose [20].

• Elongation: cable stretching quantity or how much it is stretched before it can be breaking. • Thermoset vs Thermoplastic. • Tensile strength of outer jacket: force is required to break physically or splitting the jacket. • Weather ability: the capability to withstand weather, water chemicals, UV. • Flammability: resistance of ignition. • Temperature rating: cable range without degradation can be performed. • Flexibility: bending ability of cable or minimum radius to bend. • Specific gravity: weight and density.

2.1.4 Overvoltage transient

Transient overvoltage is an electrical short-duration impulse of high energy induces onto electrical or electronic devices from external sources. It is also called spike voltage. These are sometimes random or repetitive. Random transient is like striking of lightning impulse or ESD (Electrostatic Discharge) induces spike on vulnerable apparatus. On the other hand, repetitive transient includes switch off or on of heavy machinery or other electrical devices. The risk for overvoltage transient has been increasing our concern on the electrical system such as power transformer which is one of the most essential elements in power system [23]. Transient voltage can be protected through designing a circuit which limits current, voltage, or transient time.

Overvoltage has tend to stress the insulating part of electrical equipment might cause damage when occurs frequently. Overvoltage transient due to surges can causes flashover and spark between line and ground in the weakest point of the network, also the breakdown of insulation result in failure of the power transformer and other rotating machines [24].

Overvoltage transient are surges and reach for a short duration of order of millisecond or microsecond. Despite having short duration of time results in some serious problem on the equipment as [25]:

1. Service interruption 2. Serious disruption or damage 3. Financial losses.

Figure 13: Transient overvoltage [26]

External lines and longer cables are exposed mostly to overvoltage transient receive high induction level. It can be also due to non-weather occurrences like transformer center switching or inductive loads cause voltage. Overvoltage transient does not arise merely at power distributed lines but also at metal conductor like telephony [25]. Overvoltage transient is one of the major causes of outage and faults in the power transmission line. Magnitude and overvoltage rise rate due to any surge such as lightning impulse strikes on power lines are taken as consideration for insulating substation and this strategy is adopted to limit the overvoltage. The fault in the transmission line due to lightning surge is classified as back flashover caused by the failure of shielding. Both the events result in overvoltage travel from the striking point to the substation. Very high frequency comes from the nature of overvoltage lightning causes attenuation by skin effect and corona losses. So lightning strokes nearly to the power substation are considered during evaluating the protection requirements of overvoltage and the related risk of substation equipment failure. Insulation faults over power transmission lines can incite a high magnitude of short circuit current. A very steep front surge is formed for insulator flashover. This flashover enters the substation and creates insulation stress on the power transformer [27]. In this thesis, overvoltage stress distribution along the bushing of the transformer will be performed at a later step. So the lightning overvoltage is a fast front transient mainly occurred due to the effect of overvoltage lightning return strokes to the overhead lines. However, it is possible to improve the performance of lightning of an overhead transmission line by shielding wires which should be installed at the top of the poles or tower, where the shielding wires prevent the effect of return strokes to the active phase conductor. Most of the transmission lines are shielded by shielding wires. Thus, the lightning overvoltages are due to return strokes to the phase conductor in these lines, to a shield wire, or to a tower. On the other hand, most of the distribution lines are not protected through shielding. Therefore the lightning flashovers might be occurred due to direct strokes to the conducting lines or might be induced by lightning strokes to earth in the vicinity of transmission lines [28]. 2.1.5 Lumped model of overhead transmission line and bushing:

The equivalent circuit of a coaxial cable transmission line is shown as a lumped element model where the LC sections have an effective substitute of a transmission line carrying out the experiments on the characteristics of electromagnetic wave propagation along the line.

Figure 14: Equivalent circuit of Transmission line [29]

To propagate electromagnetic wave through a coaxial cable where any pairs of wires separated by an insulating medium as air create capacitance between those wires. On the other hand, each loop of wires by definition acts as an inductor. There are mutual inductances between two conductors of a coaxial cable which has been figure out in section 2.1.6 and in the result section of chapter 3 below using Comsol multiphysics. Here in the figure above, it is assumed that all the values of inductances are equal (L1, L2, L3, L4) in Henry as well as the capacitances (C1, C2, C3, C4) in Farad. Now the behaviors of a lumped model element in a transmission line are discussed in below when electromagnetic wave propagates onto the cable. When a voltage is applied between two separate conductors, an electric field is created between those conductors result in stored electric energy in this field. Therefore this energy storage causes an opposition to changing voltage. The following equation [30] describes the response of capacitance on this change of voltage which shows the current flows in the conductors is proportional to the rate of change of voltage over time. i = C (dv/dt) (1) During closes of the switch in figure (14), the capacitance reacts to the sudden increase of voltage by charging up, draws electric current from the main source. An instant increase of applied voltage according to the equation (1) will increase infinite charging current. But the current drawn by two conductors will not be infinite due to inductance exists in the conductor. The flowing current in the conductors creates a magnetic field inside it where magnetic energy is stored, causes an opposition of changing the current. Therefore each of the conductors develops a magnetic field due to the charging current of the capacitance between the conductors. This results in voltage drops according to the following equation. This dropping voltage limits the rate of change of voltage across the distributed capacitances which prevent the current reaching to infinite magnitude [30]. v = L (di/dt) (2) Applying a voltage to one end of a power transmission line causes the propagation of voltage and current wave along the conductor at nearly the speed of light. But for a DC input impulse voltage at one of the end of a conductor for an infinitely long transmission line, the electric current will draw from the DC source due to constant resistance [30]. For a coaxial construction of a transmission line, the following equation shows the characteristics impedance.

Figure 15: Coaxial cable construction [30]

= log (3) 138 𝑑𝑑1 0 = line characteristics impedance𝑍𝑍 √𝜀𝜀 𝑑𝑑2

0= inner diameter of the outermost conductor 𝑍𝑍 = outer diameter of the innermost conductor 1 𝑑𝑑ε is the relative permittivity of insulation of the conductors 2 𝑑𝑑 The transmission line characteristics impedance is the resistance exhibits for infinite length. But this impedance is different entirely from the dielectric leakage resistance that separates the 0 conductors and the metallic resistance of the wires.𝑍𝑍 Characteristic impedance is the function of distributed inductance and capacitance along the length of line that exist even though the dielectric is perfect and the conductor has zero series resistance as superconducting. When the spacing between the conductors is increased, the characteristics impedance is also increased. The distributed capacitance will decrease when the conductors move away as a long gap from each other, which means the greater space between the plates of the capacitor [30]. On the other hand, inductance will increase due to the small cancellation of opposing the two magnetic fields. Therefore, more series inductance and small parallel capacitance in a transmission line draw very less amount of current for a given applied voltage. But when the conductors bring closes, the value of the capacitance increases, and the inductance value decreases result in more current drawn from the given applied voltage. Neglecting the dissipative effect of conductor resistance and dielectric leakage, the following equation has shown as the characteristic impedance of a transmission line [30].

= (4) 𝐿𝐿 0 = line characteristics impedance𝑍𝑍 �𝐶𝐶 L= inductance per unit length, in H 0 C𝑍𝑍 = capacitance per unit length, in F

The wave propagation velocity and the characteristics impedance of a coaxial cable is effected when the insulating material is vacuum or other than air. On the other hand, if the angular frequency along the transmission line increases, the propagation velocity decreases. The propagation velocity is zero when the angular frequency becomes greater than its cutoff frequency 𝜔𝜔 [2].

ω𝑐𝑐 = (5) 1 𝑐𝑐 So when ≥ , the waveω cannot√𝐿𝐿𝐿𝐿 propagate inside the bushing of the transformer.

𝜔𝜔 ω𝑐𝑐

The following figure shows how the electric wave as lightning impulse propagates from a coaxial cable to a power transformer bushing.

Figure 16: Lightning impulse propagation on OHL

According to the given figure 16, the impulse response has been shown in the result section for a lightning waveform as a step surge. It can be seen in the figure that, lightning impulse stroked on a three-phase overhead line. The impulse then is propagating to the bushing of the transformer for a three-phase system which is connected with a grounded ending. Also, a surge arrester is connected with each of the transformers. Therefore the main goal of this thesis is to find out the stress distribution on the bushing due to lightning surge has been shown in chapter 3 using Comsol multiphysics. The inductance phenomenon of a coaxial cable is investigated first and then connected to the bushing.

The whole idea of simulating the coaxial structure to find out the mutual inductance has taken from the lumped element model explained above. The wave cannot propagate to the transformer bushing without distortion. Due to the existence of parallel capacitance and mutual inductance in a coaxial cable, there may have changes in the form of complex wave when penetrating inside the bushing [2].

2.1.6 Self and mutual inductance of coaxial cable:

Inductance phenomenon of coaxial cable is an important property that influences the wave propagation of the electric signal through the cable.

Figure 17: Single-core coaxial cable [31]

A voltage source is connected to one end of the cable and at the other end, a load resistor flows the output of the electric signal. Here the electric charges flow out from the source and through the innermost conductor of the cable and then return to the outermost conductor of the coaxial to the e.m.f (electromotive forced) source [31]. So the innermost and outermost conductors of the cable are formed as a conducting ring or loop. A changing of electric current in this loop induces e.m.f which opposes the change.

To find the numerical value of self and mutual-inductance using Comsol software, a single core coaxial cable for multi-layer of sheaths are studied in the result section. For the comparison with analytical value, the equations of self and mutual inductances are studied for two concentric coils arrangement of single turn are explained in this report, using DC analysis for steady-state and AC analysis for frequency domain. The coils are then excited to have the induction matrix [32].

Figure 18: Cross-section of coaxial cable [32] In the figure 18 above the coils have different radius of R1 and R2. The excitation is given to the center conductor in turns with a 1A current. The induced current exists if there is a variation of driving current in the coils with respect to time. The self-inductance is defined as the total magnetic flux that passes through a surface. On the other hand, the edges of the surface are defined as primary and secondary coil respectively. Taking R1>>R2, the analytical expression for inductance is given as below [32]:

. = (6) ∬𝑆𝑆1 𝐵𝐵 𝑛𝑛 𝑑𝑑𝑑𝑑 Here n is the normal vector to the surface and is the current of the primary or center coil. The 𝐿𝐿11 𝐼𝐼1 integral over the surface is taken which is well-defined by the coil. Here the magnetic field B 1 is computed from the following term of vector𝐼𝐼 potential [32].

B= ×A (7)

According to the Stroke theorem, ∇the surface integral of the curl of the vector field is equal to the line integral of that field over the edge of the surface [32].

( × ). = ∬𝑆𝑆1 𝛁𝛁 𝐀𝐀 𝑛𝑛 𝑑𝑑𝑑𝑑 𝐿𝐿11 . 𝐼𝐼1 = (8) ∮𝜞𝜞𝟏𝟏 𝑨𝑨 𝒕𝒕𝒕𝒕𝒕𝒕

𝑰𝑰𝟏𝟏

t is the unit tangent vector over the surface. An alternative method to solve the self-inductance of coaxial cable is the energy method by stationary situation based on the total magnetic energy of the system. The equation can be defined as [32]:

2 = 2 dΩ (9) 1 11 𝐿𝐿 ∫Ω 𝑚𝑚 𝐼𝐼 𝑊𝑊 Here, = magnetic energy density. = current feeding in the system 𝑚𝑚 Is equal𝑊𝑊 to zero 1 𝐼𝐼 2 Similarly𝐼𝐼 , the mutual inductance between two coils of a single core coaxial cable of multi number of sheaths can be calculated as the line integral equation no. (8) of self-inductance which is mentioned above. The equation of mutual inductance between coil 1 and coil 2 is given in below [32]:

. = (10) ∮𝜞𝜞𝟐𝟐 𝑨𝑨 𝒕𝒕𝒕𝒕𝒕𝒕 12 𝐿𝐿 𝟏𝟏 Similarly, the mutual inductance𝑰𝑰 between coil and 1 and coil 3 is defined as:

. = (11) ∮𝜞𝜞𝟑𝟑 𝑨𝑨 𝒕𝒕𝒕𝒕𝒕𝒕 13 𝐿𝐿 𝟏𝟏 It can be mentioned that, changing𝑰𝑰 current to compute mutual inductance is independent of current and the analytic value is included in the result section. The following equation of mutual inductance for two concentric coplanar loops show that mutual inductance does not depend upon the change of current.

Figure 19: Two concentric coplanar loops [33] Considering two concentric single turn coplanar coils with radius R1 and R2 where R1>>R2. The mutual inductance between two coils is computed from the following equation [33].

= (12) 𝜇𝜇0𝐼𝐼1 The magnetic field is at the center of the conductor ring due to the current of the outer 1 1 coil. As R1>>R2, the magnetic𝐵𝐵 2𝑅𝑅 field can be approximated through the whole inner loop by the 1 1 magnetic field . So𝐵𝐵 the flux linkage through the inner coil (with R2 radius) is 𝐼𝐼[33]:

𝐵𝐵1 22

= = ( ) π (13) 𝜇𝜇0𝐼𝐼1 We know that mutual inductance is the rate of change2 of flux of second coil proportional to the 21 1 2 1 2 rate of change of currentΩ of center𝐵𝐵 𝐴𝐴 coil with2𝑅𝑅 the current𝑅𝑅 .

1 𝐼𝐼 M= = 2 (14) Ω21 𝜇𝜇0π𝑅𝑅2 The equation (14) above shows that mutual inductance between two coils depends only upon 1 1 its geometrical factors, radius𝐼𝐼 R1 and2 𝑅𝑅R2, also is independent of coil current [33].

1 2.1.7 Skin effect of a circular conductor: 𝐼𝐼

The skin effect is an important phenomenon of the coaxial cable of any electric power network. This effect is on a metallic conductor of the cable. For a coaxial cable, skin effect inside it and other apparatus ensure the signals of any frequency (radio frequency) through assuming that the shielding is intact. For DC application the entire cross-section of the wire carries an electric current [34].

Figure 20: Cross-section of conductor that carried current for DC [34] As the frequency increases when applying alternating current, the tendency of the electric current becomes distributed to the surface of the conductor is called the skin effect of the coaxial cable. The region is much shallower when the frequency is higher as proportional where the electric current is conducted (nearer to the surface). Thus the density of the current is larger at the surface and exponentially decreases with the higher depths of the conductor [32].

Figure 21: Current at the surface of the conductor for AC [34]

The current mainly flows at the skin or surface of the conductor that is between a level and the outer surface. This level is called the skin depth and depends upon the frequency of the AC (Alternating Current). As the frequency increases, the flow of electric current change the direction and moves towards the surface of the conductor causes small skin depth. On other hand, Skin effect increases the effective resistance of the conductor through decreasing its effective cross-section. The skin effect of the conductor is due to opposing the eddy currents that induce by change of magnetic field causes due to the AC current [34].

Figure 22: Skin depth of the conductor [34]

The formula to find the skin depth of a circular cross-section of a coaxial cable is [34]:

δ= ( )( ) (15) 2𝜌𝜌 � 2𝜋𝜋𝜋𝜋 𝜇𝜇0𝜇𝜇𝜇𝜇 Here,

δ= skin depth in meters

f =frequency of the electric current in Hertz

ρ =conductor resistivity in Ω·m

µr =relative permittivity (0.999991, for copper conductor)

µ0= permittivity of the free space (4π10-7 H/m)

Skin effect impact attenuation:

Attenuation is a function of frequency for the coaxial cable related to four factors as given in below [34]:

1. Radioactivity of outermost of the cable for imperfect shielding 2. Absorption of electric signal or pulse in the dielectric 3. Resistive losses inside the conductors. 4. Misalliances between the conductor cable and termination results in pulse reflection.

Chapter 3 Result part with simulation

In this project, the perfect electrical conductor is preferred 1st and then real conductor in the bushing model, and compared both to study the behavior of skin effect. The mutual inductances are calculated both analytically and Comsol to understand its phenomenon in coaxial cable and transformer. A future work has been proposed also in this project for further work. 3.1 Skin effect of coaxial cable

Usually, the cause of skin effect is due to electromagnetic induction. A magnetic field of time- varying is escorted by an induced time-varying electric field. This creates a secondary time- varying induced current and also a secondary time-varying electric field. According to the Lenz law, the magnetic flux produced by the induced current opposes the external flux that produced the induced current, result in the total flux reduction. The induced current is larger when the conductivity is larger. Also, the flux reduction is pronounced more when the permeability is larger [35].

The inner conductor of a coaxial cable is surrounded by an insulating layer, followed by a conducting shield. The advantage of coaxial cable is that the electric field exits in the space of the inner and outermost conductor. So the cable can be set up next to metal entities without any power losses. An application of coaxial cable is power transmission lines for frequency or any signal transmission. This thesis investigates the surface current density of a single core coaxial cable for three different frequency (50, 500, and 5000) in Hz connect in the transformer bushing terminal [35].

Figure 23 (a): Skin effect for 50 Hz

Figure 23(b): Skin effect for 500 Hz

Figure 23 (c): Skin effect for 5000 Hz

Figure 23 (d): Skin effect plot of current density vs length.

Figures 23 above show the current density inside a single core coaxial cable. The skin effect is visible clearly and it shows that tendency of electric current is distributed near the surface area of the conductor (figure 23 (d)). The current density depends on the frequency. The skin effect becomes more and more visible as the frequency increases from 50 Hz-5000 Hz.

3.2 Inductance phenomenon and skin effect of the coaxial cable:

The self and mutual inductance has been calculated both numerically and analytically using Comsol simulation. To find the self and mutual inductance of five layers coaxial cable, it is important to define 1st the magnetic flux which is needed to pass through a surface, whose edge should be a coil. Here coils are used to have inducing voltage from one coil to another, or from primary to a secondary coil. The turn of the coils is made up of conducting material such as aluminum or copper. The coil feature can be used to excite it using electric current for any kind of open or closed circuit.

To figure out the geometrical part for a five layers coaxial cable, the length is put in meter unit. Using this geometry, five circles has been designed in Comsol with the radius of 10, 40, 50, 80, and 90 in mm respectively. The rotation of the vector field is kept 0. It is possible to change the rotation angle to 90 degree also. Next, the most important part is the magnetic field in the model. The ampere circuital law is used to assign electric current in the center conductor of the geometry using conductivity and permeability of the material. The equations are given in below. ×H=J (16) B= ×A (17) J=∇ σ× E (18) H= Magnetic field intensity ∇ B= Magnetic induction J= Current density E= Electric field strength A= Magnetic vector potential σ =Specific conductivity

σ is a scalar function that is almost constant in most materials.

Figure 24: 5 layers coaxial cable The material can be assigned manually from the add physics part of the Comsol mode. Here the center conductor and the sheaths are given aluminum with different conductivity, the permittivity is equal to 1. The insulation is given as air with zero conductivity. The blue colors in the figure below indicate the materials put in the center conductor and the sheaths as aluminum. The other parts indicate the insulating material as air.

Sheath 2 Air insulation Air Insulation Center conductor Sheath 1

Figure 25: Material put in the coaxial cable

The magnetic insulation is needed to define the boundary of the geometry. The normal component of the magnetic vector potential is zero.

n ×A=0 (19)

As mentioned above that, the total magnetic flux B must be passed through the surface whose edge must be defined as a boundary. So the magnetic flux is within this boundary. The vector normal is used to define a surface integral of a vector field. In equation (19), A is magnetic vector potential as an infinitesimal rotation of vector field. So A=0 means no flux is coming out and confined inside this boundary.

Next, three coils of the same feature are assigned in the circles. One coil is for the center conductor and another two coils are for sheaths, therefore voltage will be induced from one coil to another so that mutual inductance can find out easily. Generally it’s a trick which means there is no coil as such in this model. A straight conductor is also known as coil which is a one turn or single-turn in design and this turn is made of conductive material like copper or aluminum. In coil 1, the current is given to 1 A. In coil 2 and coil 3, no current is given externally as an induced current will be in the sheaths due to the skin effect.

Coil in sheath 1 Coil in center conductor

Figure 26 (a): Coil 1 Figure 26 (b): Coil 2

Coil in sheath 2

Figure 26 (c): Coil 3 The mesh is basically defined as the size of the geometry. It depends on how fast the result or simulation can capture. If it is put a smaller size, the accuracy will be much better as compared to other size. As eddy current is concentrated to a small part or small tiny surface (in mm), it is important to make sure that the surface mesh is really fine. Otherwise, it will not capture the eddy current. From the skin depth equation 15, the element size should be smaller than the skin depth for better mesh size.

Figure 27: Finer Mesh of 5 layers coaxial The result parts in the Comsol are divided by two functional domains. One is in the frequency domain and another part is in the time domain or stationary domain. The frequency domain shows the current density or skin effect of these coaxial layers. Skin effect as known that it reduces the effective radius of the conductor thus increase the effective resistance also. This skin effect is caused by opposing the eddy current induced by changing the magnetic field which is caused by the alternating current. The eddy current is in the center of the conductor shows in figure 28 (a) below. So with the increase of frequency, the current will choose the minimum resistance path, shows in figure 28 (b) and figure 28 (c). So it will move towards the surface area.

Figure 28 (a): Skin effect of coaxial cable for 50 Hz

Figure 28 (b): Skin effect of coaxial cable for 500 Hz

Figure 28 (c): Skin effect of coaxial cable for 5000 Hz

From the figure above it can be seen that the current concentration is mostly on the surface of the conductor. For 50 Hz the current is uniformly distributed, which means the entire current is utilized. When changes the frequency from 50 Hz -5000 Hz, the blue color indicates that there is no current or small amount of current inside the conductor. On the other hand, the red color in the color code of the given figure 28 (c) is the highest current density. In the inner part of sheath 1, it can be seen that a small part of the current has been induced, also in the outermost sheath of the coaxial cable (sheath 2) due to current in the primary coil 1.

• Mutual inductance of coaxial cable:

The mutual flux between coil 1 and coil 2 can be evaluated by the surface integral of the magnetic flux through that coil, or the line integral of magnetic vector potential (figure 29 (a)). Using cutline 2D 1 in Comsol model, it has been assumed the values of point 1 and point 2 in the Y direction as given in the following table below:

Point X Y Point 1 0 40 mm Point 2 0 50 mm

Table 1 Cutline 2D 1 direction for sheath 2 Now using line integral in Comsol, the mutual inductance value between coil 1 and coil 2 is found 1.3822 e-9 H-m. For the verification of this value, the path integral of the magnetic vector potential is used according to equation no. (10)

Current in Current in Mutual inductance coil 1 [A] coil 2 [A] between coil 1-2 [H-m] [wb] [ wb] [wb] 𝐴𝐴𝑥𝑥 𝐴𝐴𝑦𝑦 𝐴𝐴𝑧𝑧 1 0 1.3822*10 0 0 1.3904*10

−9 −9 Table 2: Mutual inductance for center conductor-sheath 1

Where , and are the x, y, and z components of magnetic vector potential respectively.

𝐴𝐴𝑥𝑥𝐴𝐴𝑦𝑦 𝐴𝐴𝑧𝑧

Figure 29 (a): Cutline 2D in coil 1 Figure 29 (b): Cutline 2D in coil 3 Similarly to find out the mutual inductance between coil 3 to coil 1, the cut line 2D 1 method is used in Comsol for line integration as given in figure 29 (b) above.

Point X Y Point 1 0 80 mm Point 2 0 90 mm

Table 3: Cutline 2D 1 direction for sheath 3

It is seen that the mutual inductance between coil 1 and coil 3 is 1.1316*10 H-m. For the verification of this value, equation no. (11) is used as given in the following −table.10

Current in coil 1 Current in coil 3 Mutual inductance [A] [ A] between coil 1-3 [wb] [wb] [wb] [H-m] 𝐴𝐴𝑥𝑥 𝐴𝐴𝑦𝑦 𝐴𝐴𝑧𝑧 1 0 1.1316*10 0 0 1.1547*10 −10 −10 Table 4: Mutual inductance for center conductor-sheath 2

Where , are the x, y and z components of the magnetic vector potential respectively.

It has been𝐴𝐴𝑥𝑥𝐴𝐴 𝑦𝑦men𝐴𝐴𝑧𝑧tioned in the equation (14) that, the mutual inductance is independent of the current. Therefore changing the value of current in coil 1, the value of mutual inductance between coil 3 and 1 remains the same. For the verification, equation no (11) is used for the magnetic vector potential of z component.

Current in coil 1 Current in coil 3 Mutual inductance [A] [A] between coil 1-3 [wb] [wb] [wb] [H-m] 𝐴𝐴𝑥𝑥 𝐴𝐴𝑦𝑦 𝐴𝐴𝑧𝑧 2 0 1.1316*10 0 0 2.3094*10 −10 −10

Table 5: Mutual inductance for center conductor-sheath 2 after changing the current

• Self-inductance of the center conductor in coaxial cable:

The self-inductance value for coil 1 can be found using cutline 2D1.

Figure 29 (c): Cutline 2D in coil 1for self-inductance

Point X Y Point 1 0 0 mm Point 2 0 10 mm

Table 6: Cutline 2D 1 direction for the center conductor So the value of self-inductance is found through line integration in the Comsol model. For the verification of this value, the equation (9) has been used as given in the following table.

Self-inductance of the center = dΩ conductor using line integral 2 in Comsol [H-m] [H2 -m] 𝐿𝐿11 𝐼𝐼1 Ω 𝑊𝑊𝑚𝑚 4.8944*10 4.8944*∫10 −9 −9 Table 7: Self-inductance values An alternating formula can also be used to find the self-inductance of the coaxial cable, where the energy equation is:

= = . (20) 1 2 1 2 1 11 1 1 𝑆𝑆1 So the self-inductance𝑊𝑊 value2 𝐿𝐿 is the𝐼𝐼 vector2 𝐼𝐼 ∬ dot 𝐵𝐵product𝐻𝐻 𝑑𝑑𝑑𝑑 of the normal component of magnetic field density. Using Comsol the value of magnetic field intensity= 5 e-8 2 The self and mutual inductance are needed to calculate several analytical 𝐻𝐻values𝐻𝐻 of transformer bushing or the effectiveness of the sheath of the coaxial cable. On the other hand, to change the values of alternating current and voltage, the phenomenon of mutual inductance is used in the transformer device. In alternating transmission and distribution system, the alternating voltage with ease can be increased or decreased by the transformer.

3.3 Transformer bushing:

The electric field distribution is an imperative parameter to evaluate the performance of high- voltage transformer bushing. A study [36] has shown that 25%-30% of transformer failures are related to the failure of the bushing. However, to see the voltage distribution and output plots on different simulations of the transformer bushing, 1st the circuit diagram can be shown below with an explanation which will have a generic idea of the bushing in Comsol.

Figure 30: Schematic diagram of a section of coaxial cable for a transmission line connected to source voltage and loads

When an electrical wave travels through a coaxial cable and meets an open circuit in the remote end, the electrical wave reflects back to the main source. As mentioned that, a cable is generally a medium where an electric signal passes through one point to another and comprises a center conductor enclosed by metallic shielded dielectric material. When a wave propagates through a coax, it undisturbed till it encounters the change of impedance. The load impedance and the characteristics impedance of the cable determines the behavior of the wave propagation. When the load impedance is higher than the characteristics impedance of the coaxial cable, a portion of the wave reflects back to the source. On the other hand, there is no reflection of the wave back to the source when the load impedance is smaller than or equal to the characteristics impedance of the cable. Attenuation may occur due to the presence of resistances. For optimal output, the input load impedance and the characteristics impedance of the cable should be equal to the load impedance while delivering maximum power to the output load and prevents signal reflections. So to maximize power transfer to the load, impedance matching is important while propagating a single wave through the coax.

In the figure below, the voltage pulse is used as step pulse voltage of 1 volt applied to the input.

Figure 31: Input voltage pulse.

The bushing geometry is designed by a rectangle in 2D axial-symmetric model in Comsol.

Inner radius Outer radius Bushing width Length Corner base r_coax R_coax R_coax-r_coax In meter (m) position of center [mm] [mm] In mm conductor [mm] 25 500 475 3 25

Table 8: Geometrical values of the bushing

Figure 32: Bushing rectangle built in Comsol In the figure above, initially, the material is put as air with zero electric conductivity as shown in the table below. The relative permittivity is 1 for the air and the relative permeability is 1.

Material name Conductivity Permittivity Permeability Air 0 1 1

Table 9: Transformer oil parameters

Figure 33: Bushing material as transformer oil built in Comsol

Now three stress grading foils as parallel rectangle are added in the above bushing. The last foil is connected to the ground with the transformer tank. The transformer tank is also connected to the ground. The figure below is the simplified model of transformer bushing. The mesh size is taken as finer mesh.

Figure 34 (a): Transformer bushing with foils Figure 34 (b): finer mesh for the bushing.

The electromagnetic equation used to simulate in Comsol is given below:

×µ ( ×A) +µ + µ ( ) =0 (21) −1 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕𝜕𝜕 This equation (21∇ ) is𝑟𝑟 a second∇ -or0der𝜎𝜎 𝜕𝜕𝜕𝜕 partial0 𝜕𝜕𝜕𝜕 differential𝜀𝜀𝑟𝑟𝜀𝜀0 𝜕𝜕𝜕𝜕 equation used in Comsol by means of magnetic vector potential. The equation relates to space and time derivative of magnetic vector potential. The magnetic field B is the curl of the magnetic vector potential as mentioned in equation (17). A temporal gauge of electric field E has assumed for . It can be added here that, the equation (21) is used for a time-independent material. The permittivity𝜕𝜕𝜕𝜕 cannot take outside of time derivative for a time dependent-material [37]. 𝜕𝜕𝜕𝜕

The simulation of this model has taken 7 hours 22 minutes. The following parameter has been taken to figure out the desired output.

relative tolerance time step Input resistance [ohm] Output resistance [ohm] 0.0001 1e-11 0 10000

Table 10: parameters for bushing

It means that there is no input resistance and input lightning impulse voltage is directly applied to the conductor. The normal component of the electric field model after the simulation and the output line graph has been shown as below:

Figure 35 (a): Simulation of transformer bushing with perfect electric conductor condition in the surface

Due to very high impedance at the output side, the output is twice the input pulse and acts as a capacitive circuit. Here the boundary condition for the foils is a real electric conductor. The output shows damping effect in below.

Noted that: In all the results of input-output pulse given in below, the green color indicates the input pulse and the blue color indicates the output pulse.

Figure 35 (b): Input-output pulse with damping effect on a perfect conductor

Making the boundary of the foils as perfect electric conductor and the main conductor of the bushing is a real conductor, the following outputs are obtained.

Figure 36 (a): Electric filed in Bushing with real conductor Figure 36 (b): Input-output pulse with real conductor Now putting the material as transformer oil inside the bushing instead of air with permittivity 4, the following outputs are obtained from the figure as shown below:

Figure 37 (a): Electric field Figure 37 (b): Input-output pulse 3.3.1 The Skin effect in transformer bushing:

To see the skin effect in the bushing, it can be seen using the current density of z component vs length of the radius (keeping the permittivity of oil to 4). The behavior of output pulse (without foils) is explained with the change of conductivity of the conductor. The skin effect shows for matched load condition when the characteristics impedance of the bushing and output impedance are equal. The input-output pulse has been shown for the permittivity of oil

1. Oil conductivity=0.001 S/m, Aluminum conductivity= 3.774e7 S/m

Figure 38 (a): Electric field of normal component Figure 38 (b): Input-Output Pulse

Figure 38 (b) shows the effect of an extremely high and in reality unrealistic high conductivity in oil. The conductivity used is in the range of tap water as the conductivity of the transformer oil is 0.001 S/m, but not saltwater when the conductivity is 1 S/m. Keeping the conductivity of transformer oil to a constant value 0 (given in below), there will have some reflection in the output voltage with the change of the conductivity of aluminum started from a lower value to a higher one. The output pulse will bounce back and forth due to attenuation or skin effect. On the other hand, to study the skin effect the plots of current density vs length of radius of the conductor have been given according to the Cutline 2D method in Comsol. It can see there that the current is in the center of the conductor initially and uniformly distributed. It means that at the beginning, the value of the current is very low and it is increasing exponentially and reaches to the surface of the conductor. As soon as the applied transient input DC voltage as step pulse reaches to its DC value, the value of the current will also reach to the DC value and the bouncing of the output pulse will disappear gradually with the increase of conductivity which is accurate according to the characteristics of the skin effect in the conductor.

2. Transformer oil conductivity =0 S/m, using perfect conductor :

Figure 39 (a): Electric field of normal component Figure 39 (b): Input-Output Pulse

3. Transformer oil conductivity=1 S/m, using perfect conductor

Figure 40 (a): Electric Field of normal component Figure 40 (b): Input-Output Pulse

4. Transformer oil conductivity=0 S/m, Aluminum conductivity= 1 S/m

Making the conductivity very low of the aluminum, which is unrealistic in a real situation, it behaves like a distributed RC circuit (figure 41 (b)). With this low conductivity in the conductor, it charges up to cable capacitance with very high impedance in the output side of 10000 ohm. The current density curve has been figure out for different time which shows that, current is moving towards the surface with the increase of time due to skin effect. In Comsol, the following values are used in Cutline 2D of the geometry of the bushing conductor.

Point X Y Point 1 0 0.1 m Point 2 0.025 m 0.1 m

Table 11: Cutline 2D 1 direction in the bushing conductor

Figure 41 (a): Electric field of normal component Figure 41 (b): Input-output pulse

Figure 41 (c): Plot for current density vs length

5. Transformer oil conductivity=0 S/m, Aluminum conductivity= 10 S/m

Figure 42 (a): Electric field of normal component

Figure 42 (b): Input-output pulse

Figure 42 (c): Plot for current density vs length

6. Transformer oil conductivity=0 S/m, Aluminum conductivity= 100 S/m

Figure 43 (a): Electric field of normal component

Figure 43 (b): Input-output pulse

Figure 43 (c): Plot for current density vs length

7. Transformer oil conductivity=0 S/m, Aluminum conductivity= 1000 S/m

Figure 44 (a): Electric field of normal component

Figure 44 (b): Input-output pulse

Figure 44 (c): Plot for current density vs length

1. Transformer oil conductivity=0 S/m, Aluminum conductivity= 3.774e7 S/m

Figure 45 (a): Electric field of normal component

Figure 45 (b): Input-output pulse

Figure 45 (c): Plot for current density vs length

Conclusion In this paper 2D (two-dimensional) axial-symmetric model of geometry describes overvoltage stress and the effect of Electromagnetic transient on transformer bushing which has been developed successfully using Comsol software. The model is able to simulate the stress distribution on bushing. It can be seen that overvoltage stress distribution and surface current density as skin effect are influenced by some factors including the conductivity of the conductor, permittivity of the bushing, length and width of the model, coaxial cable input and output impedance, and so on. Applying input voltage as a step pulse, the output shows the accurate value and is twice the input voltage when the output impedance is very high and acts as an open circuit. Changing the conductivity of the conductor from a very lower value to a higher one makes the voltage curve bouncing back and forth due to getting attenuation or skin effect. But it will settle as soon as the conductivity of the conductor will take place to a higher value and the bouncing will be disappeared with loss-free curve. On the other hand, the skin effect for changing the conductivity of real conductor approaches the dc situation. It means that when the applied step voltage reaches to dc value, the current density becomes uniform (like at DC) which is accurate according to the applied input pulse and the characteristics of the skin effect on a real conductor.

Future work

The explanation or the more detailed understanding of the behaviors of mutual inductance phenomenon on lightning pulse propagation, skin effect of multi-layers coaxial cable, and the damping effect of output pulse of the transformer bushing simulations should be done in future for further work.

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