Chapter 9. Electric Field Calculations

# Chapter 9. Electric Field Calculations - 9 ELECTRIC FIELD...

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329 9. ELECTRIC FIELD CALCULATIONS Summary The electric field is calculated for some idealized insulation structures which can often be useful approximations to actual insulation systems. This can provide insight into the parameter dependence of such fields and can suggest ways of reducing such fields if necessary. The electric fields in the oil gaps between pairs of disks in a transformer winding are strongest at the corners. These corner fields therefore determine the gap spacing and paper insulation required to avoid breakdown. We determine these fields by means of a conformal mapping technique for a 2-D geometry consisting of 2 conductors at different potentials separated by a gap (disk— disk spacing). Both conductors are separated by another gap from a ground plane. This latter gap could be a winding—winding gap or a winding— core or tank gap. The analytic solution does not include the effect of the paper insulation. This leads to an enhancement of the field in the oil over the situation without insulation. A method is proposed to account for the insulation based on a comparison with a finite element solution. Finite element solution methods are discussed for complex geometries. 9.1 SIMPLE GEOMETRIES It is often possible to obtain a good estimation of the electric field in a certain region of a transformer by idealizing the geometry to such an extent that the field can be calculated analytically. This has the advantage of exhibiting the field as a function of several parameters so that the effect of changing these and how this affects the field can be appreciated. Such insight is often worth the price of the slight inaccuracy which may exist in the numerical value of the field. As a first example, we consider a layered insulation structure having a planar geometry as shown in Fig. 9.1 . This could represent the major insulation structure between two cylindrical windings having large radii. We are further approximating the disk structure as a smooth surface so that the resulting field calculation would be representative of the field away from the corner of the disks. We treat this corner field in the next section. © 2002 by CRC Press

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ELECTRIC FIELD CALCULATIONS 330 We use one of Maxwell’s equations in integral form to solve this, (9.1) where D is the displacement vector, dA a vectorial surface area with an outward normal, and q the charge enclosed by the closed surface S. Because of the assumed ideal planar geometry, the surface charge density on the electrode at potential V is uniform and is designated σ in the figure. An opposite surface charge of - σ exists on the ground electrode. Note that both electrode potentials could be raised by an equal amount without changing the results. Only the potential difference matters. We will also assume that the materials have linear electrical characteristics so that D = ε E (9.2) holds within each material where ε , the permittivity, can differ within the various layers as shown in the figure. Because of the planar geometry, the D and E fields are directed
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## This note was uploaded on 10/19/2010 for the course ENGINEERIN ELEC121 taught by Professor Tang during the Spring '10 term at University of Liverpool.

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Chapter 9. Electric Field Calculations - 9 ELECTRIC FIELD...

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