CHAPTER 2 OVERVIEW OF THE LITERATURE

Transcription

1 13 CHAPTER 2 OVERVIEW OF THE LITERATURE 2.1 INTRODUCTION In this chapter, a review of earlier research investigations carried out by several researchers on Computation of electric fields by a) Charge Simulation method. [CSM] b) Optimized charge simulation method c) Finite Difference Method [FDM] d) Finite Element Method [FEM] are presented in the beginning. Subsequently, earlier work on a) Ionization & breakdown in uniform electric fields, and b) Corona investigations under influence of electric fields are reported. 2.2 COMPUTATION OF ELECTRIC FIELD INTENSITY BY CHARGE SIMULATION METHOD Accurate computations of electric field intensities between nonuniform field configuration electrodes are useful in design of HV equipment insulation. In view of Innumerable possibilities of electrode geometry configurations in equipments, and the electric fields being complex in these regions, analytical solutions for Electric Field Intensity are extremely difficult. However, mathematical expressions for electric field intensities between selected few common type of electrode

2 14 configurations are given in reference[43 ]. For more accurate calculations of electric field intensities, numerical methods are suitable. L.B.Loeb, J.H.Parker, E.E.Dodd and W.N.English [ 1,2 ], and Abou - Seada and Nasser [ 3,4 ] have attempted numerical field solutions employing the CSM, for rod - plane [ 1,3 ] and cylindrical conductor - plane Configurations [ 4 ]. The fundamentals of CSM and calculations of electric field intensities for models having rotational symmetry have been presented by Singer, Steinbigler and Weiss [ 5 ]. The basic idea of charge simulation method is the introduction of fictitious charges in the region other than where the field solutions are desired so as to simulate the field in the region of interest with the boundary conditions met as appropriate to the problem. The position and magnitude of charges are selected to compute the potential and field distribution in the region using classical theory of electric fields. According to superposition theorem, n Vi= j 1 PijQj 2.1 Where Vi is the potential at any point, [ known conductor potential and point at which potential has to be estimated ], which is the summation of potentials due to n number of individual simulated charges Qj located outside the region where potential and electric field intensities are calculated. Pij are the potential co-efficients obtained by the analytical solution of Laplace Equation for many types of charges.

3 15 The application of equation (2.1) for m points on the boundary leads to the following m linear equations. 2.2 Equation [2.2] is solved for simulated charges Q1,Q2,..Qn. It is necessary to ascertain whether the simulated charges evaluated using Potential at n points produce actual potential at other boundary Points other than the values used in equation [2.2]. The above accuracy criterion is ensured by altering the number, type and location of simulation charges. Even the location of Contour points used in equation

4 16 [2.2] might be altered. Once accuracy criterion is met to a desired degree, the potential and field at any point outside the contour points can be estimated. Electrode surfaces or Interfaces of different insulators which are not highly irregular like curved and rounded surfaces have been simulated by this method. Sharps bends invariably found in high voltage equipments have been found difficult to be simulated by this analysis. In this method, the experience of the investigator plays an important role in determining the accuracy. Requirement of a minimum gap distance between the location of a charge and electrode Contours is one of the Constraints in this method [5]. 2.3 OPTIMIZED CHARGE SIMULATION METHOD FOR THE CALCULATION OF HIGH VOLTAGE FIELDS. Modeling of practical electrostatic apparatus with improved accuracy in a routine fashion, with minimum reliance on personal experience and accuracy optimally related to computing resources available has been reported by A.Yializis and his colleagues [6]. The objective function employed is the accumulated squared error m 2 [ j (, )] 2.3 j 1 U V r z where V is the value of potential at Contour points, фj (r,z) the value of the simulated potential, which is a function of cylindrical coordinates, r,z and m are the number of points [ rj, Zj ] on the electrode contour. The

5 17 variables of optimization are the position of the charges and their magnitudes subject to the equality and inequality constraints relating to the physical system. The authors have used Rosenbrock s optimization technique for the optimization of the objective function modified accordingly to account for equality and inequality constraints.

6 18 Several Investigators have suggested Criteria for determining simulation accuracy. The potential error can be computed at a number of points for each conductor boundary. From the computed values of potential error, the average value of, the maximum Δфm or the mean a or the mean squared [ ] 2 a can be determined. If capacitance has to be determined, Δфa is useful since per unit capacitance is proportional to Δфa [ 7 ]. Δфm is taken into account when maximum error in the electric field is to be kept low[11]. It has been suggested that Em= k L m where L is the distance between two consecutive contour Points and the constant K has a value between 1 and 2 [ 12 ]. Some researchers have suggested a very small ratio Et / En, Et and En referring to tangential and normal component of electric field at the interface of conductor di-electric boundary in view of error in the electric field being higher than error in the potential [ 13 ]. Some authors have suggested that deviation angle as a more sensitive indicator of simulation accuracy [ 14,15 ]. The derivative of potential gradient perpendicular to the conductor surface divided by the gradient itself must be equal to the Curvature at the test point is a very severe accuracy criterion as reported by H.Singer and his colleagues and A.Mahsen.[5,7] In multidielectric systems potential discrepancy defined as the difference in solutions for potential at the dielectric boundary as

7 19 evaluated in the medium on each side of the interface [ 16 ] should be small. Similarly, discrepancy in the tangential electric field or the normal flux density is proposed by some Authors [13]. In an ideal solution, such discrepancies should be zero. Other methods to check the accuracy of the CSM include Comparisons of the CSM solution with solutions obtained from other numerical and analytical methods or experimental measurements where all possible. 2.4 FINITE DIFFERENCE METHOD FDM The finite difference method [FDM] [17] is a simple numerical technique. In this method, the field space between the electrodes is covered with uniform spaced mesh which may be square, rectangular, a combination of the two or even triangular or hexagonal grid (figure 2.2). This gives rise to a large number of nodes. The original Laplace differential equation is then transformed into difference equations for potential Φ at these nodes. To cover an irregular 3 dimensional field so that these nodes are laid upon the boundaries become extremely difficult. Further, to cover such fields by a proposed mesh, an unlimited number of Φ(x,y) values of potential distribution are necessary requiring a large memory and time for computation. An electrostatic problem is uniquely defined by 3 steps. a) A partial different equation such as LAPLACE S or POISSON S equation. b) A solution region.

8 20 c) Boundary and / or initial conditions. A finite difference solution to Poisson s or Laplace s equation proceeds In three steps as detailed below: 1. Dividing the solution region into a grid of nodes. 2. Approximating the differential equation and boundary conditions by a set of linear algebraic equations ( called difference equations ) on grid points within the solution region and, 3. Solving this set of algebraic equations. The above method requires large memory and time for computation. The FDM is therefore found suitable only for two dimensional symmetrical fields. This method is rarely applied for electrostatic fields anymore.

9 FINITE ELEMENT METHOD The field in the region of interest is divided into triangular elements for two dimensional representation and tetrahedron for three dimensional representation. Fig. 2.3 A Tetrahedron and a Triangular Finite Element

10 22 Regions with higher electric field intensity are covered with triangles of Smaller dimensions. Quadrilateral elements are useful for problems whose boundaries are sufficiently regular.whereas in FDM, direct solution of Laplace s equation is attempted to, in FEM, the method is based on the inherent characteristics of an electrostatic field that the total energy associated with the entire field region acquires a minimum value. This implies that the potential Ф Should make the enclosed energy function to be a minimum for a given dielectric volume V. Therefore, 2.4 should be minimum. We is the electrostatic energy in the volume of the dielectric material. Ф would be determined only by the metal electrodes in case of Space charge free region. 1 We 2 x y z [ ] dxdydz In Cartesian coordinate system. In a two dimensional case where potential does not vary in the Z direction, We 1 2 z x y dxdy 2 2 [ ]

11 23 where z is a constant. We/z is the energy density per elementary area da. The most common approximation for Ф within an element is polynomial approximation ( x, y) a a x a y for a triangular element. For a more exact estimation of the field, higher order polynomials can be used. For example, a square or a cubic equation[18] ( x, y) a a x a y a x a xy a y a x a x y a xy a y Considering equation (2.7) a a x a y i 1 2 i 3 i a a x a y 2.9 j 1 2 j 3 j a a x a y k 1 2 k 3 k 1 a1 ( i i j j k k ) 2 e 1 a2 ( i i j j k k ) 2 e a3 ( i i j j k k ) 2 e where i x j yk xk y j i yj yk

12 24 y y j xk yi xi yk j k i k xi y j x j yi k i j y y 2.11 x x and i k j x x 2 e i j k j i k i j j i The symbol e represents area of the triangular element I, j, k under Consideration. From equation 2.7 a2 f( i, j, k ) x and a3 f( i, j, k ) and 2.12 y W W / z = Energy enclosed in the triangular element e e [ ] x y W e e 2.13 e dxdy = Area of the element considered. A W e is minimum when W e i = 0 1 a a E (2 a2 ) i i

13 25 1 ( a2 i a3 i ) [( i i ) i ( i j i j ) j ( i k i k ) k ] e 2.14 The set of all three equations with respect to фi, фj and фk can be expressed in matrix form W e e [ h] ( ) e e The matrix [h]e is known as the Stiffness factor for the individual element (e), geometrically locating the element and containing functional sensitivity with respect to the potentials. It also contains the permittivity of the dielectric material. Therefore, for every unknown potential, a corresponding equation has to be set up.

14 IONIZATION AND BREAKDOWN IN UNIFORM ELECTRIC FIELDS In gaseous medium, electrons and ions are the electric charge carriers. Ions are produced from neutral molecules or atoms by ejection or attachment of an electron. Ejection of an electron from a neutral molecule leaves behind a positive ion, where as absorption of an electron by a molecule produces negative ion.[19,20] During an electrical breakdown, the insulating gas between the electrodes is bridged by a conducting discharge (channel). The Charge carriers required in order to build this discharge channel are not only produced within the gaseous dielectric across the gap (primary or α - process ) [Fig. 2.4 ] but are also released from the electrode surfaces (secondary or γ processes ). [ Fig. 2.5 ] [ 22,23,27,28,29] Fig 2.4 Ionization by electron impact of neutral gas molecule or atom ( -process) ` Fig 2.5 Ionization by +ve ion impact ( -process) (At very high electric fields )

15 27 The electric field directly influences the movement of the charge carriers during the electrical breakdown of a gas. For a discharge process to begin with, some charge carriers must be present in the gas before applying the electric field. If not present, they must be generated externally. A considerably strong multiplication of charge carriers must take place in order to reach the required conductivity or electrical breakdown conditions in the gas. The production of charge carriers from the neutral gas molecules by electron collisions is known as ionization process. The amount of energy required for releasing an electron from neutral gas atom or molecule is known as Ionization Energy. For the process of breakdown to develop, the electrons should attain energy higher than the ionization potential of gas molecules between successive collisions as it travels in the direction of electric field. The average number of ionizing collisions made by one electron per centimeter drift across the gap in the direction of the field in a uniform electric field is defined as α or Townsend s primary ionization coefficient which represents basically a probability process. For gas discharges, α is a very important coefficient strongly dependent upon the electric field intensity E and gas pressure P at constant temperature. The governing criterion for Ionization is 1 m V 2 2 e e I eu 2.17

16 28 ee e where e is the charge of electron, Ui is the ionization potential of the atom or molecule of the gas and e is the mean free path of an electron for ionization collisions in the gas in the direction of field intensity E. The mean free path being inversely proportional to gas pressure, the ratio of electric field intensity (E) to gas pressure (P) signifies an important concept termed the mean energy. Under uniform electric field conditions, the energy attained by electrons between successive collisions and averaged over innumerable number of such collisions is termed mean energy. For certain range of values of E/p, α/p can be expressed as a function of E/p given by the relation.[20] BP E Ae 2.18 P where A and B are constants for the range of E/p values for the gas under consideration GROWTH OF IONIZATION CURRENT IN UNIFORM ELECTRIC FIELDS :- In case of a uniform field gap, the electric stress is the same everywhere In the gap and hence the ionization and deionization parameters are constant. The current in an uniform field without secondary effects is given by the expression

17 29 I d Ioe 2.19 where IO is the initial current and I is the current reaching the anode placed at a distance d from the cathode. The above constitutes a single avalanche process. During the amplification of electrons in the field by α process, additional electrons are being liberated in the gap by other (secondary) Processes as well when electric field intensities are relatively higher.. The secondary electrons thus produced create their own avalanches. The secondary processes are 1. Positive ion effect γion :- While the positive ions produced in the primary avalanche cannot gain enough kinetic energy in the electric field to ionize molecules, they may have sufficient potential energy to cause ejection of electrons upon striking the cathode. 2. Photon effect γp :- Excited molecules in the avalanche may emit photons on returning to their ground state. This radiation falling on cathode may produce photo emission of electrons. 3. Metastable effect γm :- Metastable molecules may diffuse back to the cathode and cause electron emission on striking it. The three processes of cathode effect are described quantitatively by a coefficient γ as given by γ = γion + γp + γm 2.20

18 30 γ is known as Townsend s secondary ionization coefficient. It is defined as the number of secondary electrons on an average produced at the cathode per electron generated by the primary process, that is, per ionizing collision in the gap. γ strongly depends upon the cathode material and is a function of field intensity and pressure of the gas. E f P 2.21 Like α, γ also represents a probability process. If the mean number of secondary electrons per avalanche produced are µ, then ( e d 1) 2.22 The influence of secondary process on current growth can be given by the following equation I I exp( d) 1 {exp( d) 1} BREAKDOWN UNDER UNIFORM FIELD CONDITIONS When distance d between uniform field electrodes is increased keeping E/p constant ( i.e. keeping mean energy attained by electrons between Successive collisions in direction of field constant), at a certain value of d, the increased value of γ gives rise to condition ( e d 1) Under these conditions the current in the gap tends to become enormously high ( near infinity, ideally) and the current in the external circuit is limited by external resistance in the circuit. Also, the current is

19 31 maintained in the gap even in the absence of initiatory current IO. The situation can be explained by the concept that as an electron leaves the cathode, it makes e αd number of ionizing collisions due to which average number secondary electrons released from cathode because of secondary effects is atleast one.

20 IONIZATION PHENOMENA IN NONUNIFORM ELECTRIC FIELDS In a nonuniform field gap, α and γ are no longer constants. These vary with the field between the two electrodes and hence equations for current growth have to account for such a position related dependence of α and γ. The equation for growth of ionization current can be written as :- I I e o d x dx in the absence of γ effects and I I 0 exp d dx 0 d 1 {exp dx 1} 0 x x 2.26 considering secondary effects under the condition that the electrons attain mean energy corresponding to electric field intensity between successive collisions. With increase in nonuniformity of electric field, the electric field values which exist between hemispherically capped cylinder and plane ( radius of hemispherical cap being in the range of few millimeters ), the variation of electric field adjacent to the hemispherical electrode becomes considerably high so that ionization by electron can exist over a short

21 33 distance from the hemispherical tip electrode ( also called as point electrode ). As voltage applied between such electrodes placed in gaseous medium is gradually increased, at a certain magnitude of voltage, a small value of current in range of fraction of microamp is observed. The phenomena does not cause conduction of current between the gap as an actual breakdown can cause. It is only just beginning of partial conductivity in the gas. This is termed Corona. Loeb (1965) identified the positive streamers in point-to-plane gap in 1936 and almost at the same time Raether (1964) [25,34] found them in his cloud Chamber studies. Following this was the discovery of burst pulses in Positive corona and of the negative pulses in negative point corona by Trichel (1939). Later investigations (Loeb,1965) showed that positive corona in a point-to-plane gap can take three different forms, namely, burst Pulses, streamer pulses, or a pulseless glow. Bandel undertook a very careful study of both negative and positive Point coronas in clean, dry, filtered air over a range of pressures and point diameters[ 32,35]. The point diameters used were 0.25, 0.5 and 1.0mm and the pressures ranging from 10 to 750mm. The burst pulse Threshold current jump starts from currents of the order of ampere and the steady negative ion conditioned glow corona sets in at about 10-8 ampere to 10-7 ampere as pressures increase to 750mm.

22 34 Hermstein (1960) explained a new type of steady or glow corona in dry air as being caused by the formation of negative ions in the region between the positive space charge and the anode. Loeb (1969) has clarified that such a type of corona does not exist because the assumed negative ion sheath close to the anode gives a short high field region very near the anode which prevents streamer formation. Miller and Loeb(1951) have defined the terms generally used to describe the corona discharge phenomena. By and large the same terminology has been adopted in the present study. For the purpose of clarity a few important terms are described below. Just preceding the initial appearance of the corona discharge, generally a sudden large current jump to order of fraction of a microamp from low value (range of A ) takes place and the associated value of potential is termed corona threshold or onset potential. At onset the current is fluctuating or oscillatory and with further increase of voltage the corona current changes to a steady current. This is termed as steady corona. Usually in the region of steady corona the discharge region is covered by glow and the corona mode is also termed glow corona. Above positive corona threshold voltage, current pulses of low amplitude with duration of several tens of microsecond appear and are called burst pulses. Below onset of steady corona and near positive corona threshold, in some cases, single large amplitude (much larger than burst pulse

23 35 amplitude) pulses appear. The duration of these pulses is generally in the range of several microseconds. They are designated as pre-onset streamers whose rate of rise are required to be measured with present day equipment. They often reappear just before spark breakdown of the gap. They are then longer and more intense. These are called breakdown streamers. 2.8 REFERENCES 1. L.B.Loeb, ELECTRICAL CORIONAS university of California Press, Berkley, California, L.B.Loeb, J.H.Parker, E.E.Dodd and W.N.English, The choice of suitable gap forms for the study of corona breakdown and the field along the axis of a hemispherically Capped cylindrical point-to-point gap, Rev. Sci.instr; Vol 21, 1950, PP M.S.Abou-Seada and E.Nasser, Digital computer Calculation of the electric potential and field of a rod gap, Proc. IEEE, Vol.56, No.5, 1968, PP M.S.Abou - Seada And E.Nassear, Digital Computer Calculation of the potential and its gradient of a twin cylindrical Conductor, IEEE Trans. PAS. VOL. 88, 1969, PP H.Singer, H.Steinbigler and P.Weiss, A charge simulation method for the calculation of high Voltage fields, IEEE Trans PAS, Vol. 93, 1974, PP

24 36 6. An optimized charge simulation method for the calculation of high Voltage fields by A.Yializis, E.Kuffel, P.H.Alexander IEEE trans. PAS Vol. 97 No.97, no A.Mohsen, Justification of the charge Simulation technique and its Applications, Proc. IEEE Canadian Communication and power Conference, 1980, PP Y.L.Chow and C.Chara Lambus Static field Computation by method of optimized simulated images, Proc. IEEE, Vol. 126, No.1, 1979, PP H.Anis, A.Zeitoun, M.El-Ragheb and M.El-Desouki, Field Calculations around non-standard electrodes using regression and their spherical [equivalence, IEEE Trans. Pas., VOL.96, No.6, 1977, PP M.Khalifia, M.Abdel Salam, F.Aly and M.Abou Seada, Electric fields around conductor bundles of EHV transmission lines, IEEE PES paper AT , S.Sato and S.Menju, Digital calculation of electric field by Charge Simulation method, Electrical Engine ring in Japan, Vol.100, No.2, 1980, PP S.Kato, An Estimation method for the electric field error of a Charge simulation method, 3 rd ISH symp, Milan, 1979, paper

25 A.A Elmoursi and N.H.Malik, Field uniformity of a high Voltage test electrode system, IEEE Trans. Electrical Insulation, Vol.18, No.1, 1983, PP N.H.Malik and A.Al - Arainy, Charge simulation modeling of three core belted cables, IEEE Trans. Electrical Insulation, Vol.20, No.3, 1985, PP M.R.Iravani and M.R.Raghuveer, Accurate field solution in the entire interelectrode space of a rod - plane gap using optimized charged simulation, IEEE Trans.Electrical Insulation, Vol.17, no.4, 1982, PP M.J.Khan and P.H.Alexander, Charge Simulation modeling of practical insulator geometries, IEEE Trans.Electrical Insulation, Vol. 17 no PP G.D.Smith, Numerical solution of partial differential equations: finite difference methods, 2 nd edition : oxford : Clarendon, Speck J Computation of electrostatic fields with FEM using cubic equations Technical journal of TU Dresden, 28 (1979) 3, PP ( in German ). 19. High Voltage Insulation Engineering by Ravindra Arora, Wolfgang Mosch, New Age International Publishers 2007, ISBN: X. 20. E.Kuffel, W.S.Zaengl and J.Kuffel. High Voltage Engineering fundamentals Newness ( Butterworth - Heinemann ) 2000.

26 Qiu. Simple Expressions of field non uniformity factor for hemispherically Capped rod plane gaps, IEEE Trans on E.I Vol 21, PP , Townsend, J.S. Nature, 62 (1900), P Townsend, J.S. Philosophical Magazine, 1 (1900), P Engelmann, Eberhard A contribution to the discharge behavior of large electrodes having field distortions in air with positive switching surge Dissertation, TU Dresden ( 1981 ) ( in German ). 25. Raether, H. Die Eutwicklung der Elektronen in den Funkenkanal Zeitschrift fur Angewande Physik, 112 ( 1939 ) PP (in German) 26. Wagner, K.H. Investigation of the development of electron avalanche in plasma channel by highgain image Converter streak shutter technique, Zeitschrift fur physik, 189 ( 1966 ) PP ( in German ). 27. Meek, J.M. and Craggs, J.D. Electrical Breakdown in Gases John Wiley & Sons ltd ( 1978 ). 28. Alston, J.S Electricity in Gases oxford university press ( 1968 ). 29. Townsend, J.S. Electricity in gases oxford ( 1914 ). 30. Rees, D.B; PhD. Thesis, university of Wales, U.K. (1963). 31. Kluckow, R. on the variation of current with time for a gas Discharge in hydrogen Zeitschrift fur physic, 189 (1966 ) PP ( in German ).

27 Bandel, H.W. Measurement of the current during the formative time lag of sparks in uniform fields in air, Physical Review 95 [1954 ] 5, PP Hoger, H. Dielectrics, 1 ( 1963 ) P Raether H. Electron Avalanches and Breakdown Voltage in gases Butterworths, London [ 1964 ]. 35. H.W.Bandel, Phys. Rev. 84, 92 ( 1951 ). 36. Primary Ionization Coefficients for dry and moist Air; Secondary Ionization Coefficients for a water surface, By R.R.Stout and G.A.Dawson, Pageoph. Vol. 116 PP ( 1978 ), Birkhauser. Verlag, Basel. 37. S.Rajapandyan and G.R.Govindaraju Ionization Current multiplication in a non uniform electric field in dry air, J.Phys. D:APPL Phys ; Vol.5, L6 L8.Printed in Great Britain. 38. H.M.Ryan, Electric Field of a rod-plane spark gap, IEE proceedings, vol.117,pp, , A.A.Azer, R.P.Comsa, Influence of field nonuniformity on the breakdown Characteristics of sulfur hexafluoride, IEEE Transactions on Electrical Insulation, vol. EI-8,pp , R.Brambilla, A.pigini, Electric field strength in typical high voltage Insulation,International symposium on High voltage Engineering,Zurich, September Y.Safar, N.H.Malik, A.H.Qureshi, P.H.Alexander, Effect of grounded enclosure On the field distribution of Rod-Plane gaps, Gaseous

28 40 Dielectrics 111 (Edited by L.G.Christophorou), pergamon press, pp , E.S.Kolechitsky, Calculation of Electric Field of high voltage installations, Energoatom press, Moscow,pp.46-49,1983,(in Russian) 43. L.L.Alston H.V.Technology Book oxford university press (1968 ).