A New Method for Calculating of Electric Fields Around or Inside Any Arbitrary Shape Electrode Configuration

Transcription

1 Proceedings of the 5th WSEAS Int. Conf. on Power Systems and Electromagnetic Compatibility, Corf, Greece, Agst 3-5, 005 (pp43-48) A New Method for Calclating of Electric Fields Arond or Inside Any Arbitrary Shape Electrode Configration M. FARSADI Department of Electrical Engineering Urmia Uniersity Urmia IRAN Ö. KALENDERLI Department of Electrical Engineering Istanbl echnical Uniersity Istanbl URKEY Abstract: - In this paper a new method for calclating of electric fields intensity arond or inside any arbitrary shape electrode systems or condctors with the same or different configration hae been presented. As we know that in design and constrction of a high oltage eqipment, analysis of electric field intensity arond and or inside that eqipment has ery important role therefore, according to precision and ability of this new compter program or new compter code, we can se this code in research centers for design new eqipments in electric indstries. According to complexity of some electrodes or condctors configration, soling of Laplace's or Poisson's eqations by different nmerical methods are ery difficlt in actal field. In this new method we transfer actal field (electrode or condctors and their oter bondary) to rectanglar plane. In general this compter code sole two cople Poisson's or Laplace's eqations in general crilinear coordinate. For soling elliptic differential eqations we hae been sed M (HOMPSON, HAMES & MASIN) algorithm. Key-Words: - Electric fields, Eqipotential lines, Nmerical method, Arbitrary shape electrode configration, M algorithm, Conformal mapping, Crilinear coordinate 1 Introdction In this paper, we can obtain electric field intensity and also eqipotential lines arond or inside any arbitrary shape electrodes with M algorithm. With this compter code we can sole two cople Poisson or Laplace eqations in general crilinear coordinate. For finding electric field intensity and also eqipotential lines arond any electrode, we transfer actal field or actal coordinate system (srface of any arbitrary shape electrode) and also its oter bondary to rectanglar calclating field or rectanglar coordinate system. Soling of Poisson and also Laplace eqations are easy in rectanglar field than actal field [1]. As we know that, by sing conformal mapping, we can sole electric fields and also eqipotential lines arond some definite electrodes []. In conformal mapping method we define any complex transfer fnction and then, by sing this fnction, for eery point in actal complex plane z with coordinates x and y we can find one or many points in an other complex plane w with coordinates and. he relationship between points in plane z and plane w hae been determined by complex analytic fnction w = f(z). his complex analytic fnction is named transfer fnction also. Since definition of transfer fnction is not possible for all type electrode system therefore In general we can se conformal mapping techniqe nder some special conditions []. In this compter code, similar to conformal mapping techniqe for calclating electric field and eqipotential lines arond or inside any arbitrary shape electrodes first of all we transfer actal field to rectanglar calclating field and after soling Laplace or Poisson eqations in rectanglar coordinate system we again transfer the reslts to actal field also. he constraints of the conformal mapping method hae been not seen in this code. In mathematical method of this compter program, we will explain transfer fnction briefly. In the past seeral methods sch as finite difference, finite element, bondary element and charge simlation hae been sed for soling electric field distribtion [1-16]. According to some difficlties in the aboe mentioned different methods we hae obtained a paper that is comparing finite difference, finite element and charge simlation methods with each other [4]. In finite difference method if the shape of the electrode is complicated therefore the reslts obtained by this method will hae a large errors. In this new method we can analysis electric field and also eqipotential lines arond or inside any arbitrary shape electrode with minimm errors [17, 18].

2 Proceedings of the 5th WSEAS Int. Conf. on Power Systems and Electromagnetic Compatibility, Corf, Greece, Agst 3-5, 005 (pp43-48) Mathematical Discssion Since calclation of electric field intensity in rectanglar plane is qite simple therefore in this compter program we transfer actal comptational field and its oter bondary to any comptational rectanglar plane. Henceforth, we show actal field with E and calclating field with E. Figre 1 shows actal field (body or electrode and its oter bondary) in complex z-plane with x and y coordinates. Oter bondary 3 In the aboe discssion we assmed only one electrode or body in actal field. In this code we can assme many electrodes in actal field therefore by this new method electric field intensity and eqipotential lines arond three phase transmission line can be inestigated. Figre 3 shows actal field and figre 4 shows comptational field of two electrodes configration Electrode Electrode Electrode y Figre 1. One electrode and its oter bondary (actal field) Figre shows calclating field in complex w-plane with coordinates and respectiely. In general, electrode srface and oter bondary transfer to constant lines ( 1 and ) that is shown in figre. In figre 1, 1 shows body or electrode srface and shows oter bondary in the comptational field, in this figre also we assme two ctting lines or cres 3 and 4 respectiely these hypothesis lines or cres connect body or electrode to oter bondary in actal field. hese hypothesis lines or cres in rectanglar comptational field hae been shown by lines 3 and 4. Notice that 3 and 4 in actal field hae been coincide to each other therefore coordinates x(, ) and y(, ) for 3 and 4 will be eqal. 4 z x 1 4 = 3 y Figre 3. wo electrodes and their bondary (actal field) 4 z = 3 = 1 Figre 4. Comptational field of two electrodes and their oter bondary From the point of iew of mathematics, transfer from actal field to comptational field and ice ersa, can be shown by the following eqations. 7 1 = 1 (x, y) = (x, y) (1) Figre. Comptational field of one electrode and its oter bondary x(, ) = y y(, ) ()

3 Proceedings of the 5th WSEAS Int. Conf. on Power Systems and Electromagnetic Compatibility, Corf, Greece, Agst 3-5, 005 (pp43-48) he matrices of this transformation are as follows: x y x x J 1 =, J = x y (3) y y According to aboe eqation we can say that [J 1 ] = [J ] -1. In general the elements of matrix J 1 are obtained as follows: x = y / J, y = x / J, (4a) x = y / J, x / J y = (4b) In the aboe relationships J, shows the determinant of matrix J, following eqation shows the determinant of matrix J : [ J ] = x y x y J = det (5) he soled eqations in this compter program are Laplace or Poisson. At first step we consider Laplace eqations soltion and at the second step we inestigate Poissons eqations soltion. In general, in crilinear coordinate system Laplace eqations are shown as follows: xx + yy = 0 (6) xx + yy = 0 (7) With respect to Dirichlet bondary conditions can be obtained following eqations: 1(x, y) =, [x, y] 1 = (x, y), [x, y] 1 (8) (9) In the aboe eqations 1 and are known fnctions that are obtained from cres 1 and (electrode srface and oter bondary) separately. In comptational field (rectanglar coordinate), we se following eqations: α x βx + γx 0 (10) = α y βy + γy 0 (11) = In eqations (10) and (11), constants α, β and γ can be obtained from following eqations: α = x + y (1) β = x x + y y (13) γ = x + y (14) Under transferred bondary conditions, we obtain following eqations: f = y f 1 (, 1) (, ), [, ] 1 g1(, ), y = g (, ) [, ] 1 (15) (16) Fnctions f 1, f, g 1, g in eqations (15) and (16) are well defined, since these fnctions are the coordinates of region ( 1 and ) that we define in the actal field. Notice that the systems of the aboe eqations are qasi-linear and elliptic, therefore the soltion of these fnctions are ery complicated than Laplace eqations in actal field coordinates, bt in this code we sole these eqations in rectanglar coordinate system that make easy or calclations. If we hae two electrodes in actal field for second electrode also we can obtain following eqations: h 1 (, 1 ) =, [, y h (, 1 ) 1 ] o 1 (, 1 ) =, [, y o (, 1 ) 1 ] 7 (17) 8 (18) In general, in crilinear coordinate system Poisson eqations are shown as follows: xx + yy = ε 1 (19) xx + yy = ε (0) In the aboe eqations ε 1 and ε can be eqal or different. According to aboe discssion, we obtain following eqations in transformed field or plane: 1 = α x βx + γx + J ( ε x + ε x ) 0 (1) 1 = α y βy + γy + J ( ε y + ε y ) 0 ()

4 Proceedings of the 5th WSEAS Int. Conf. on Power Systems and Electromagnetic Compatibility, Corf, Greece, Agst 3-5, 005 (pp43-48) 3 Flowchart and Preparation of he Code he figre 5 shows the flowchart of this compter code. As we hae mentioned preiosly, by this compter code we can analysis electric field intensity and also eqipotential lines arond or inside any arbitrary shape electrode configration. Start Inpt Data Oter Bondary Calclation N 4 Comparison of he Reslts Figre 6 and figre 7 shows the electric field intensity and also eqipotential lines arond or inside any arbitrary shape electrode configration. In Figre 6 electric field intensities and also eqipotential lines arond a complicated shape electrode, and also in Figre 7 electric field and eqipotential lines arond three condctors with the same shape As like transmission lines, hae been obtained by this new method. Comparison these reslts with analoge electrode systems electric field intensity and eqipotential lines, that obtained with different methods in gien references, shows that we can se this new algorithm for analyzing electric field intensity arond any arbitrary shape electrode or condctors with the same or different configration. Y Calclate Oter Bondary Atomatically Initial Gess Inpt Oter Bond Calclate Initial Vale Use SOR for Elliptic Eqations Soltion Figre 6. Electric field intensities and eqipotential lines arond arbitrary shape electrode Reslts Conerge N Y Print Reslts End Figre 5. Flowchart In this compter code, first of all we define inpt file (GEMDIN) according to gien electrode configration. After exection of this compter code, we obtain two different otpt files (ZOU, GEMEC.DA). ZOU shows specification of inpt file with gien initial bondary conditions, errors in each iteration and also final reslts that are obtained. we se GEMEC.DA file in ECPLO software to plot electric field and eqipotential lines inside or otside any arbitrary shape electrode configrations. Figre 7. Electric field intensities and eqipotential lines arond three condctors Figre 8 shows electric field intensities and also eqipotential lines inside a system as like rod-plane electrode configration and also figre 9 shows electric field intensities and eqipotential lines inside a system as like sphere-plane electrode configration with srface

5 Proceedings of the 5th WSEAS Int. Conf. on Power Systems and Electromagnetic Compatibility, Corf, Greece, Agst 3-5, 005 (pp43-48) roghness on plane electrode. As we know that analysis of srface roghness on electrode has ery important role in design and application of high oltage apparats. In general if there is a symmetry inside any electrode system configration then, in analysis of the electric field and eqipotential lines inside the system we can assme only one half of the system as shown in figre 8 and 9 respectiely. With this new method we can calclate electric field intensity and also eqipotential lines arond or inside any arbitrary shape electrode or condctors with the same or different configrations. As we know that in high oltage engineering, analysis of electric field arond or inside any eqipment has ery important role in the design of that eqipment therefore, we can se this new compter code in the design and constrction of any HV eqipment that are sed in electric indstry. Concerning to precision and ability of this new method we can se this code also in research and edcational actiities. Figre 8. Electric field intensities and eqipotential lines inside a system as like rod-plane electrode configration Figre 9. Electric field intensities and eqipotential lines inside a system as like sphere-plane electrode configration with electrode srface roghness on plane electrode After comparison these reslts with the electric field and eqipotential lines of the same electrode configration that are obtained with different nmerical methods as mentioned in different references, we can say that in general, we can se also this new compter code for analysis of electric field intensities inside any high oltage eqipment. 5. Conclsion We hae described here a new compter program with the ability to sole Poisson's and Laplace's eqations. References: [1] John D. Anderson, Jr. Comptational Flid Dynamics, International Ed., McGraw-Hill, [] H. Mohseni, Adanced High Voltage Engineering, ehran Uniersity Pblications, ehran, Iran, [3] H. Mohseni, Fndamentals of High Voltage Engineering, ehran Uniersity Pblications, ehran, Iran, [4] H. Steinbigler, D. Haller, A. Wolf, "Comparatie Analysis of Methods for Compting -D and 3-D Electric Fields'', IEEE rans. on Electrical Inslation, Vol. 6, No. 3, 1991, pp [5] S. S. Bamji, A.. Blinski, K. Mprasad, "Electric Field Calclation With the Bondary Element Method'', IEEE rans. on Electrical Inslation, Vol. 8, No. 3, 1993, pp [6] P. L. Lein, A. J. Hansen, D. Beatoic, and H. Gan, "A Unified Bondary Element Finite Element Package'', IEEE rans. on Electrical Inslation, Vol. 8, No., 1993, pp [7] M. G. Mohemmedi, High Voltage Engineering, Amirkabir Uniersity of echnology, ehran, Iran, [8] S. Naid, V. Kamaraj, High Voltage Engineering, International Ed., ata McGraw-Hill, New Delhi, [9] E. Kffel, W. S. Zaengl, High Voltage Engineering Fndamentals, Newness, 000. [10]. J. Gallagher, A. J. Pearman, High Voltage Measrement esting and Design, J. Wiley& Sons Chichester, [11] J. N. Hoffman, P. Plino, "New deelopments on he Combined Application of Charge Simlation and Nmerical Methods for he Comptation of Electric Fields'', IEEE rans. on Power Deliery, Vol. 10, No., 1995, pp [1] J. Gartner, E. Gockenbach, H. Borsi, "Improement of Electrical Field Analysis by Use of Compter Aided Software Engineering ools", Proceedings 11 th International Symposim on High Voltage Engineering, Paper No. 467, 1999.

6 Proceedings of the 5th WSEAS Int. Conf. on Power Systems and Electromagnetic Compatibility, Corf, Greece, Agst 3-5, 005 (pp43-48) [13]. Zaho, M. G. Comber, "Calclation of Electric Field and Potential Distribtion Along Nonceramic Inslators Considering the Effect of Condctors and ransmission owers", IEEE rans on Power Deliery, Vol. 15, No. 1, 000. [14] S. Chakraorti, H. Steinbigler, "Bondary Element Stdies on Inslator Shape and Electric Field arond HV Inslators With or Withot Polltion'', IEEE rans. on Dielectrics and Electrical Inslation, Vol. 7, No., 000, pp [15] W. Qe and S. A. Sebo, "Electric Field and Potential Distribtions along Dry and Clean Nonceramic Inslators", IEEE Electrical Inslation Conference and Electrical Manfactring & Coil Winding Conference Proceedings, 001, pp [16] W. Qe and S. A. Sebo, "Electric Field and Potential Distribtions along Non-ceramic Inslators With Water Droplets Inslators", IEEE Electrical Inslation Conference and Electrical Manfactring & Coil Winding Conference Proceedings, 001, pp [17] M. Farsadi, Ö. Kalenderli, "A New Method for Calclating of Electric Fields Arond Any Arbitrary Shape Electrodes", XIII th International Symposim on High Voltage Engineering, Netherlands, 003, pp [18] M. Farsadi, Ö. Kalenderli, "A New Method for Calclating of Electric Fields Arond ransmission Lines", nd International Conference on echnical and Physical Problems in Power Engineering, 6-8 September 004, ebriz, Iran. Athor address: Mrtaza Farsadi Urmia Uniersity, Electrical Engineering Department, Urmia-Iran