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Polymer is one of the world's most used electrical insulators. They are used as power cables insulator, dielectric in capacitors and many more. However, despite their widespread use and long history, not all is understood about their mechanisms of electrical breakdown or the process in which an applied voltage may cause eventual degradation a certain period of exposure. In certain insulating systems where the material is completely crystalline or amorphous, specific values of breakdown voltage can be obtained given the identical initial condition and testing procedures. This is not the case for polymeric insulators. Seemingly identical polymeric insulators may produce different breakdown strengths and at a different time.
In this project a program is used to simulate the effect of spherulites in a polymer and to model the growth of an electrical tree. It will be written using Matlab and simulated in 2-D format. The basic functions of the program are to calculate the estimated path of the tree growth and calculate the electrical breakdown strength. The user will be prompt to input various parameters in the program. Parameters include the thickness of the sample, density of spherulites, sizes of spherulites, frequency and ramp rate of an AC electrical input source.
The program will enable a better understanding towards the electrical breakdown process in polymeric insulators at the convenience of repeating an experiment virtually.
Polymeric insulators such as polyethylene (PE) and polypropylene (PP) provide excellent electrical insulating properties (typically> 1016 Ω m), high breakdown strengths (typically up to ~109 V m-1) and low dielectric losses (typically tan ∂ < 10-3). They are further low in cost, resistance to corrosion, hydrophobic, have good mechanical strengths and relatively easy to be produced and shaped. For this reasons, they are widely used as electrical insulators and cables. However, the electrical strength properties of polymers deteriorate for various reasons, for example aging in the electrical field. These issues have been the subject of interest as they have not been fully understood.
Polymers are made up of long-chain macromolecules with repeating monomer units. During formation, a semi-crystalline polymer is cooled from its liquid state through the melting temperature into crystalline structures. Fast cooling rates result in more amorphous regions (less crystallization). Crystallization continues until the available crystallizable material is used up or spherulites impinge leaving the amorphous regions in rest of the volume. Spherulites are sphericalsemi-crystallineregions occurring in certain polymers such as polyethylene. They first form as a lamellae which then acts as a site for further growth.
Spherulites process high electrical strengths (3-4 times) to that of the surrounding material and therefore an unlikely path for tree growth. The electrical tree growth is not strictly guided by the direction of the maximum electric field; it tends to prefer a weaker path of less resistance, running between adjacent lamellas and avoiding regions of high crystallinity. The electric field varies considerably through the thickness of the dielectric because of the electrical and physical inhomogeneities. Therefore, in imperfect materials, there is always a distribution of breakdown strength, not a single breakdown strength. The breakdown field also decreases with increasing time of application of the electric field and with increasing sample thickness.
Using this trend to our advantage, lamellae would serve to deflect the growing tree away from the direction of maximum electric field and, as a consequence, the breakdown strength is also affected by the arrangement of lamellas in the volume. An increase of spherulites in the polymer is welcomed; however, there exist a saturation value for each polymer. Therefore, in terms of morphology, lamellas can inhibit tree growth rate by slowing it down but not stopping it entirely.
Other factors that affect the breakdown strength of polymer include contamination, mechanical stress, temperature, aging, etc. These factors will not be of interest in this report.
2.2 Technical Background
2.2.1 Numerical Analysis
As tree grows, potentials in the lattice need to be recalculated. Finite Difference Method (FDM) is used to calculate the lattice points in the sample. This is done by iteration until a certain pre-set tolerance (0.1%) is met. Gauss-Seidel equation:
The discharge starts at an electrode needle (105 KV/mm) on a small void at the surface with a depth of 1 micron and grows on a square lattice by one lattice bond per growth step. A bond connects two adjacent lattice points. Once a given point is connected to the discharge structure by a bond, it becomes part of the structure. Each point of the structure is on electrode potential i.e., the discharge structure has zero internal resistance.
2.2.2Tree branch propagation
The model of the tree generation is stochastic, based on Wiesmann and Zeller (W-Z) model. The tree grows in a stepwise manner. The probability of growth to any particular point adjacent to the existing tree structure is random with the requirement of a minimum local field Ec at that point for further propagation. Tree segments are allowed to grow diagonally, horizontally or vertically. The probability of being chosen is governed by the equation:
Where V is the potentials on each lattice points
2.2.3 Tree Initiation Time
For a tree growth to occur, a minimum field Ec must be present. Tree initiation time is calculated by:
Where R is the ramp rate Vrms/s, V is potentials on each lattice points and 'a' a constant.
Spherulites in the program are allowed to impinge but not completely over lapping one another. The region that defines a spherulite has higher electrical strengths than the rest of the surrounding. If the local field within a spherulite is high, the electrical tree may penetrate it.
The first objective was to convert a BASIC program written by Dr. Ian Hoiser into a Matlab program that simulate the actual process of electrical breakdown of polymeric insulators when high voltage is applied between the material. The Matlab program will maintain all of its original functions and subsequent add-on functions will be implemented.
3.1 Running the Original Program
DosBox emulator is used to run Microsoft QuickBasic (QB) 7.1 in Windows, which in turns run the original program.
BASIC (Beginner's All-purpose Symbolic Instruction Code) was very popular in the 70's to the early 90's. It is getting obsolete; few books, materials and internet websites remain to allow for a good overall study of the language. The language was not too difficult to understand.
The other programming language to get familiarised is Matlab. Matlab is a powerful mathematical software and is the ideal choice to simulate a numerical model. It is very convenient for matrix manipulation and often offers one-keyword command solution. With a very huge library of functions, the correct use of a keyword is necessary as there are many alternative keywords with almost the same function.
3.4 Numerical Method
Understanding on Finite Element Method (FDM) and Successive-Over- Relaxation (SOR) is essential.
3.5 BASIC to Matlab: Goto Statements
A problem encountered while converting the program was the usage of numerous "goto" statements in the BASIC program. They cause confusion in the flow of the program, as the "goto" statements allow programmers to specify a transfer of control to one of many possible destinations in a program. The way to replace them is to use "while" statements for a command which returns to a pervious function and "if" statements to advance to a later function. The structure of the program altered greatly; however, the overall flow of the program remains the same.
3.6 Cost Factor and Mixed Spherulites Function
New functions that are being implemented, more will be elaborated in later sections.
A General Overview of Program Algorithm
Outputs of Original Program in BASIC
Input parameters are Thickness of sample (d): 75 µmeter, Ramp Rate (r):70 Vrms/s
Frequency (f): 50 Hz, spherulites size: 12 µmeter; Spherulites Number Density: 1
Output: Tree initiation Time (Ti) = 120.0 seconds
Electrical strength of material: 118.5 KV/mm
Similar inputs with Spherulites Number Density changed to 3/1000 µmeter2
Output: Tree initiation Time (Ti) = 120.0 seconds
Electrical strength of material: 119.4 KV/mm
Similar inputs with Spherulites Number Density changed to 6/1000 µmeter2
Output: Tree initiation Time (Ti) = 120.0 seconds
Electrical strength of material: 119.0 KV/mm
Output of Program in Matlab
Input parameters are identical to 1. The applied voltage is from the top electrode.
Electrical strength of material: 118.4 KV/mm (very close to 1)
Input parameters are identical to 2.
Electrical strength of material: 119.9 KV/mm (very close to 2)
Input parameters are identical to 3.
Electrical strength of material: 120.1KV/mm (very close to 3)
Additional functions will be included into the program, programming imperfections will be eliminated, and there might also be a change of electrical treeing model in the future.
A Mix of Spherulites of Different Shapes and Sizes
This function will allow the simulation to be closer to real-life polymer material composition where it consists of a range of different shapes and sizes. Shapes could be oval, rectangular, or irregular instead of the usual circular.
An insulation failure in power cables causes financial losses. This function will calculate the cost of consumption of energy needed for a electrical tree to penetrate a specific path in the polymeric material.
This is important as it can demonstrate to us how their arrangement actually determines the breakdown characteristic and the ease of penetration. The cost factor will rate the difficult of penetration and thereby if a certain arrangement is cost viable.
As a result that this program is converted from one language to another, the spherulites displayed in the simulation were drawn slightly over the boundary due to the fact that a workable technique in BASIC is meaningless in Matlab. The problem will be solved. The problem does not affect the accuracy of the results as the lattice of the spherulites remains within the boundaries.
An improvement to consider is changing the model of the electrical tree breakdown characteristic. Using the simulation model mentioned in , we will replace the current Wiesmann and Zeller (W-Z) model. The W-Z model is based on probability, a stochastic process in which the tree branches propagate in randomness based on the most preferable local electric field at a the end of every branch of the electrical tree. Deterministic model will eliminate the need of a random variable.
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