Modelling Of Mathematical Equation for Determining Breakdown Voltage

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1 2013 First International Conference on Artificial Intelligence, Modelling & Simulation Modelling Of Mathematical Equation for Determining Breakdown Voltage Muhammad S. Laili, Noradila Yusof School of Electrical System Engineering University Malaysia Perlis (UniMAP) Perlis, Malaysia Zetty N. Zakaria, Noor A Mohd Razali School of Electrical System Engineering University Malaysia Perlis (UniMAP) Perlis, Malaysia zetty@unimap.edu.my, ashikinrazali@unimap.edu.my Abstract Breakdown voltage in gases is determined in different various electrodes gaps and geometries. There is a mathematical equation had been developed throughout experimental work, but when considering the physical and environmental factor such as humidity, temperature and pressure, the mathematical equation is limited to express the breakdown voltage. Taking into account this problem, Dimensional Analysis (DA) is used to show the relationship between the factors such as physical parameter and environmental condition that contribute to the breakdown voltage. The mathematical equation has been developed from a set of variables into a set of dimensionless product which provide relationship between the parameters. The mathematical equation model is then used to study the characteristics of breakdown voltage. In order to validate and finalize the developed equation, simulations have been conducted using the experimental data from the previous works. Comparative study between previous experimental works with simulation results have been done as the results are almost similar. Keywords- Breakdown voltage; Dimensional analysis; Electrode gaps; physical and environmental conditions I. INTRODUCTION Breakdown voltage is determined as the maximum voltage difference across the material by applying high voltage and then insulator will be collapsed and conducted. By applying maximum voltage to the insulation at the breakdown moment, it also called breakdown voltage. The electrical breakdown in gases occurs because of collisions between photons or electrons and gas molecules. Various phenomena occur in gaseous dielectric when the voltage applied. When the applied voltage is low, small currents flow between the electrodes and the insulation retains its electrical properties. Compare when applying high voltage the current flowing through the insulation increase very sharply and cause electrical breakdown occurs. A short circuit between the electrodes formed during the breakdown. The term of breakdown voltage of insulating material is concerned on four types such as in gases, liquid and solid. Electrode arrangement in such of the affected area had been investigated that caused to the characteristics of dielectric breakdown in vacuum gaps. Various factors contribute to the levels of breakdown voltage are based on experimental observation. Both of environmental conditions and physical factor will affect breakdown phenomenon. Electrode gap length, effective area, and surface roughness give influences from the physical aspect. Meanwhile, the environmental condition such as pressure, humidity, temperature and solar radiation gives significant effects to the phenomena of electrical breakdown [1]. Generally, the breakdown voltage in gases is explained through the experimental observation of the different electrode gap. However, there are some limitations in explaining several parameters when considering environmental conditions and physical factors that affected breakdown voltage [1] [5]. In this paper, Dimensional Analysis (DA) has been applied to develop a mathematical equation that relates all the parameters together. In order to validate and finalize the developed equation, simulations have been conducted using the experimental data from the previous works. This paper outlines as follows; Section I is the overview of research, Section II discussed the mathematical modeling of the problem while Section III explained the results and discussions. Lastly, Section IV concluded the overall paper objectives. II. MATHEMATICAL MODELLING DA technique is applied to develop the relationship between physical and environmental condition contributed to the breakdown voltage. This analytical technique had been applied to lot of applications such as in various experimentally concerns on physical sciences and engineering [6], [7]. In breakdown voltage s (V b ) characteristics, there are some relationships with the several dominant physical and environmental parameters which are identified based on investigating experimental. The parameters are electrode gap length (d)[1], [5], [8], solar radiation (S)[1], effective area (A eff )[1], [5], conductivity (C)[9], pressure (p)[1], [2], humidity (h)[1], [2], [8], number of electrons (N e )[1] and surface roughness (R a )[1], [3], [4]. Each parameter have its own fundamental dimension and represented by M, L, T and Q which is mass, length, time and charge respectively. Table I shows all the parameters associate in this problem /13 $ IEEE DOI /AIMS

2 The relationship between those parameters can be stated as follows; V b = f (d, S, A eff, C, p, h, N e, R a ) (1) where f determines as unknown function. The arrangements of dimensional matrix parameters based on their similar fundamental dimensions are shown in Table II. TABLE I FUNDAMENTAL DIMENSIONS Parameters Dimensions Breakdown voltage (V b ) ML 2 T -2 Q -1 Electrode gap length (d) L Solar radiation (S) MT -3 Effective area (A eff ) L 2 Conductivity (C) ML 2 T -3 Q -1 Pressure (p) ML -1 T -2 Humidity (h) ML -3 Number of electron (N e) Q Surface roughness (R a) L TABLE II DIMENSIONAL MATRIX OF PARAMETER V b d S A eff C p h N e R a M L T Q The rank of the dimensional analysis, r = 4 and the number of parameters, n = 9. Based on Buckingham- theorem [7], the best solution can be expressed in the form of (n - r) = 5 known as the independent dimensionless product, x. As the group is all dimensionless, it has dimension of M 0 L 0 T 0 Q 0. From the dimensional homogeneity s principle, the repeated variable is chosen by taking p, h, N e and R a. For each dimension (M L T Q), the power must be equal on both sides of the equation. Then, the five groups of the dimensionless products of x using dimensional homogeneity can be written as; 1 : 1 D [ML 2 T -3 Q -1 ] a1 [ML -3 ] a2 [Q] a3 [L] a4 [ML 2 T -2 Q -1 ] = [M 0 L 0 T 0 Q 0 ] (2) 2 : 1 D [ML 2 T -3 Q -1 ] a1 [ML -3 ] a2 [Q] a3 [L] a4 [L] = [M 0 L 0 T 0 Q 0 ] (3) 3 : 1 D [ML 2 T -3 Q -1 ] a1 [ML -3 ] a2 [Q] a3 [L] a4 [MT -3 ] = [M 0 L 0 T 0 Q 0 ] (4) 4 : 1 D [ML 2 T -3 Q -1 ] a1 [ML -3 ] a2 [Q] a3 [L] a4 [L 2 ] =[M 0 L 0 T 0 Q 0 ] (5) 5 : 1 D [ML 2 T -3 Q -1 ] a1 [ML -3 ] a2 [Q] a3 [L] a4 [ML 2 T -3 Q -1 ] = [M 0 L 0 T 0 Q 0 ] (6) The equations of the dimensionless products which are from equation (2) (6) have been solved by using the homogeneous linear algebraic equation in Matlab software. The matrix of solution that corresponding to the dimensionless product equations is shown in Table III. TABLE III MATRIX OF SOLUTION FOR DIMENSIONLESS PRODUCTS V b d S A eff C p h N e R a /2 1/ /2 1/2 1-2 Based on the table of dimensionless product above, the power of each dimension has been obtained according to their parameters. Afterward, the equations of the dimensionless product can be written as follows; 1 = (7) 2 = (8) 3 = (9) 4 = (10) 5 = (11) The relationship between the dimensionless product from equation (7) (11) according to Buckingham s theorem can be correlated as; 1 = f ( 2, 3, 4, 5 ) (12) Equation (12) is then rearranged in order to show the relationship of breakdown voltage to the other parameters as depicted in (13). V b = f ( 2 a2, 3 a3, 4 a4, 5 a5 ) (13) by considering the nominal form, that leads to the relationship, it can be expressed as; 1 = D c ( 2 a2, 3 a3, 4 a4, 5 a5 ) (14) where D c is the dimensional constant. Referring to equation (7) (11), it is solved into equation (14) that produced an equation as in (15). V b = D c x x( x x (15)

3 Factor a 2, a 3, a 4, a 5 are all constant and can be assigned by looking through the relationship between the physical, the environmental conditions and the breakdown voltage. Based on the previous experimental work [1] [5], the breakdown voltage is proportional to the gap distance and conductivity while inverse proportional to the solar radiation and effective area of the electrodes. So, those factors that are present as a 2 = 1, a 3 = -1, a 4 = -1, a 5 = 1. Finally, the model of the breakdown voltage can be written as a function of physical factor and environmental conditions given by equation (16); constant D 1 = and S 0 = which have been plotted in Fig. 1. TABLE IV SOLAR RADIATION DATA Solar Radiation, S (hours) Breakdown voltage, Vb (kv) V b = D c (16) III. RESULTS AND DISCUSSIONS The breakdown voltage between electrode gaps is one type of breakdown in gases which is highly influenced by the several essential parameters such as electrode gap length, solar radiation, effective area, conductivity, pressure, humidity, number of electron and surface roughness. By using previous experimental data and equation (16), a curve of a model is developed to fit the graph using MATLAB software. By using the dimensional analysis method, the parameters are having the relationship in one equation which combines of physical factor and environmental condition. There are three parameters that are contributing to the breakdown voltage that have been analyzed in this paper. Those parameters are solar radiation, gap length and pressure. A. Relationship between Breakdown Voltage, V b and Solar Radiation, S The breakdown voltage at various solar radiations with the fixed electrode gap of 2 cm is conducted by experimental work [1]. The other parameters are set to be constant values except the period of the sunlight exposure is varied as to concern the variable solar radiation only. By assuming pressure, p surface roughness, R a conductivity, C electrode gap, d affective area, A eff humidity, h and number of electrons, N e are constant, so then the equation is determined as below; V b = f(s) = (17) where, D 1 and S 0 are the constant to be determined by assuming the solar radiation is the variable based on the general equation (16). S0. Table IV shows the experimental result of breakdown voltage at different solar radiations [1]. The solar radiations varied from 1 hour to 3 hours and the corresponding breakdown voltage is inversely proportional in the range of kv to kv as shown Table IV below. By using equation (17) the Fig. 1. radiation. Graph characteristics of breakdown voltage and solar From the Fig. 1, approximately the minimum value of V b = 25 kv starts at solar radiation of S = 40 hours. Initially the value of the solar radiation is extended until 100 hours and it is exponentially decreasing to infinity. The equation (17) that has been developed is validated with the experimental data and can be concluded that, the result of breakdown voltage shows almost a similar result. B. Relationship between Breakdown Voltage, V b and Electrode Gap (Sphere-sphere and Point-plane) In this project, there are two configurations of electrode gap that have been studied; Sphere-sphere Electrode Gap and Point-plane Electrode Gap. In determining the relationship between the breakdown voltage, V b and the electrode gap, d the parameters of pressure, p surface roughness, R a conductivity, C solar radiation, S effective area, A eff humidity, h and number of electrons, N e are assumed to be constant. The function of sphere gap to the breakdown voltage can be written as equation (18) which depicted below; V b = f(s) = D 2 d (18) The experimental work to investigate the breakdown voltage at the different electrode gaps has been conducted by [9] as shown in Table V. It is used as the defining point to obtain the constant D 2 and by considering the data from Table V, the constant value of D 2 is determined to be TABLE V

4 ELECTRODE GAP LENGTH POINT (SPHERE-SPHERE) Electrode gap length, d (cm) Breakdown voltage, Vb (kv) The same investigation had been done by using different electrode configuration, point-plane electrode gap [9] and the result is shown in Table VI below. TABLE VI ELECTRODE GAP LENGTH POINT (POINT TO PLANE) Electrode gap length, d (cm) Breakdown voltage, Vb (kv) Considering the data from Table VI, the constant value of D 2 is equal to Fig. 2 below is the result of breakdown voltage, V b versus gap length for both types of configurations. The blue line is represented for the pointplane configuration and the green line is for spheresphere gap configuration. [9]. By assuming the parameters of surface roughness, R a electrode gap, d conductivity, C solar radiation, S effective area, A eff h and number of electrons, N e are constant, the function of pressure versus gap length can be written as below; V b = f(s) = D 3 (p) (19) Considering the data from Table VII, the constant value of D 2 is obtained equal to Finally, Fig. 3 shows the characteristic of the breakdown voltage with varying pressure values. TABLE VII BREAKDOWN VOLTAGE VERSUS PRESSURE x GAP Pressure, p (torr) Breakdown Voltage,Vb (kv) Fig. 2. Simulation model for both configurations The graph shows the breakdown voltage is proportional to the increasing of the gap. At point plate of 50 cm, breakdown voltage is roughly at 350 kv but by using a sphere to sphere gap configuration, it nearly approaches 1000 kv. Different level of breakdown is recognized as the factor of uniform field effect across the gap distance. C. Relationship between Breakdown Voltage, V b and Pressure, p In high voltage apparatus, pressure has a significant effect of the breakdown phenomenon. It is identified and proved using the Paschen s law which involves two parallel copper electrodes. Data of the previous experimental work was using pressure, p and electrode gap, d as the varying parameters as shown in Table VII Fig. 3. Simulation model for pressure versus breakdown voltage From experimental observation, solar radiation, S gives influences to the level of breakdown voltage, V b. By varying the UV radiation, the breakdown voltage is continuing to decrease as well. This phenomenon might be because of the development of energized electrons in the electrode. When the high voltage is applied across the electrodes, the electrons are easily released from the electrode material and get excited. Therefore, this condition has decreased the value of the breakdown voltage eventually. Study on the effect of the gap length to the breakdown voltage is found to be proportional by assuming the pressure, p surface roughness, R a conductivity, C solar radiation, S and effective area, A eff are constant. When the gap spacing between the electrodes becomes larger, it needed more electrons in the collision process to form the electron avalanche. For that reason, as the gap length increased, the breakdown voltage will increase as well

5 The influence of pressure gives significant effect to the breakdown voltage. From Fig. 3, it can be observed that the breakdown voltage and pressure are proportional to each other. IV. CONCLUSION Implementation of the dimensional analysis method allows development of the relationship between several parameters in investigating and determining the effect of physical factors and environmental conditions that contributed to the breakdown voltage. The results show that the developed equation (17) (18) have been proven with the previous experimental works. Comparisons between previous experimental works with simulation results have been done as the results are almost similar. As a conclusion, when applying the developed equations in others experimental works, each of the equation is aligned with all experimental results which supported by the theories. Generally, this research is focused on the experimental observation by determining validation of the breakdown characteristic in high voltage apparatus. There are eight parameters including physical factors ad environmental conditions have been studied in this work, however only three parameters have been validated and discussed in the current work. Moreover, the factors of breakdown voltage can be varied depends on influences. Hence, for future work, there are more researches can be done to study the effects of various factors for breakdown voltage to be occurring. REFERENCES [1] M. A. M. Piah, P. A. Ping, and Z. Buntat, Development of mathematical equation for determining breakdown voltage of electrodes gap, 2008 IEEE 2nd Int. Power Energy Conf., [2] L. Ming, F. Sahlen, G. Wu, G. Asplund, and B. Jacobson, Humidity effects on dielectric strength of air-gaps for indoor HV installations, CEIDP Annu. Rep. Conf. Electr. Insul. Dielectr. Phenomena, 2005., [3] A. M. Mahdy, H. I. Anis, and S. A. Ward, Electrode roughness effects on the breakdown of air-insulated apparatus, IEEE Trans. Dielectr. Electr. Insul., vol. 5, [4] K. Kato, Y. Fukuoka, H. Saitoh, M. Sakaki, and H. Okubo, Effect of electrode surface roughness on breakdown conditioning under non-uniform electric field in vacuum, IEEE Trans. Dielectr. Electr. Insul., vol. 14, [5] U. Schumann and M. Kurrat, Break down voltage of electrode arrangements in vacuum in consideration of surface area, 20th Int. Symp. Discharges Electr. Insul. Vac., [6] T. Szirtes, Applied Dimentional Analysis and Modeling. Mc.Graw- Hill Publishing, [7] H.. Langhaar, Dimentional Analysis and Theory of Models. Wiley, [8] K.Feser, Influence of Humidity on The Breakdown Voltage of D.C and A.C Voltage in Air. Haefly Publication, 1972, pp [9] P. B. Sankar, Measurement of Air Breakdown Voltage and Electric Field Using Standard Sphere Gap Method,

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