COMPEL 28,4. Christophe Geuzaine Department of Electrical Engineering and Computer Science, Montefiore Institute, University of Liège, Liège, Belgium - PDF

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The current issue and full text archive of this journal is available at COMPEL 28,4 988 Modelling earthing systems and cables with moment methods Pieter Jacqmaer and Johan Driesen Research Group ELECTA, Department of Electrical Engineering, Katholieke Universiteit Leuven, Heverlee, Belgium, and Christophe Geuzaine Department of Electrical Engineering and Computer Science, Montefiore Institute, University of Liège, Liège, Belgium Abstract Purpose The purpose of this paper is to present a method to model earthing systems subjected to lightning strikes with a one-dimensional moment method. This paper was conducted because an accurate method to model earthing systems subjected to lightning strikes, was deemed necessary. To name a few examples of relevant situations: supply stations of railway systems, from which also critical signalling infrastructure is fed, earthing systems of cellular phone basestations, located in the vicinity of high-antenna towers, prone to lightning strikes, and gas and oil pipelines. There exist already methods to solve this problem, based on circuit theory, but the electromagnetic method of this work is based directly on Maxwell s equations and therefore more accurate. Design/methodology/approach The earthing electrodes and meshes are represented as wire scatterers. First, the method is outlined for scatterers in a single medium. Next, the method is extended to model to presence of the soil-air interface layer. An approximate technique, known as the modified image theory, is used to account for the vicinity of the soil. Finally, a second extension is given so that cables without metal sheets which are in the vicinity of the earthing systems, can be included in the model. Thereafter, it is described how the method can be used to calculate the effects of lightning strikes on earthing structures, and finally a validation of the method is presented. Findings The method is validated by applying it to simple situations which can also analytically be calculated, and by applying it to earthing structures for which the transient voltage was measured or calculated with circuit methods. A good agreement is seen. However, the method is computationally very expensive. Research limitations/implications In order to account for the influence of the air-ground interface, an approximate method was used: the modified image method, and not the exact Sommerfeld theory. This was done because of its simplicity and in order to speed up the calculation process. Furthermore, cables can be included in the model, but they must be of simple structure: a cylindrical core, surrounded by an insulating cladding. Originality/value A few authors have already described this method to simulate lightning strikes on earthing systems. However, in this paper, a new and easy model for underground cables in the vicinity of earthing systems is presented. Keywords Method of moments, Electromagnetism, Lightning, Electric current, Image processing Paper type Research paper COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 28 No. 4, 2009 pp q Emerald Group Publishing Limited DOI / Introduction The accurate modelling of earthing rods and grids is becoming increasingly important because design engineers want to know how adequate their earthing This work was supported by the Belgian Interuniversitary Attraction Poles programme. systems are in safeguarding against lightning, and because present computers are powerful enough to answer this question. In order to predict the effects of a strike, and to determine the minimum requirements of lightning protection equipment (surge arrestors, isolation transformers,...) and because earthing has an important effect on electromagnetic compatibility, the need for good earthing modelling techniques is of great importance. Basically, there exist three families of methods to tackle this problem. First, there are methods based on circuit theory where earthing electrodes or rods are modelled with lumped or distributed passive components (Lorentzou and Hatziargyriou, 2005; Menter and Grcev, 1994; van Waes, 2003; Kraft, 1991). A circuit simulator such as Spice or an EMTP tool is then used to calculate the voltages and currents on the electrodes. These methods provide an easy model for earthing systems. They are fast and work directly in the time-domain. Furthermore, soil ionization and nonlinearities such as in the behaviour of surge arrestors, are easily included in the circuit simulations. However, the accuracy of these methods is rather doubtful, because a uniform filter network or transmission line is used in order to describe an earthing electrode which is inherently non-uniform. Furthermore, this technique cannot accurately model earthing structures other than straight conductors. Second, there are methods based on the direct solution of Maxwell s laws (Otero et al., 1999; Grcev and Dawalibi, 1990; Grcev and Arnautovski-Toseva, 2003; Grcev et al., 2003; Grcev, 1996). For this, they employ numerical techniques such as the finite element method (FEM), the method of moments (MoM) or other numerical methods. They are based on fewer assumptions and simplifications than the circuit methods and thus have the potential to be more accurate. There exist volumetric and one- or two-dimensional techniques. They operate in the time domain or frequency domain. However, they are more time-consuming and more difficult to understand and implement than the circuit theory methods. Third, there exist hybrid methods which try to combine the circuit theory methods and the electromagnetic methods in order to profit from the advantages of both (Liu et al., 2001; Heimbach and Grcev, 1997). In this work, the electromagnetic approach is used to calculate fields and transient voltages or currents, when lightning strikes an earthing system. The MoM is used to solve the electromagnetic equations. 2. Description of the method The problem of modelling earthing systems is tackled by numerically solving Maxwell s equations. This is done with the MoM (Harrington, 1968), which solves the integro-differential equation known as Pocklinton s equation. Because earthing systems consist of rods and meshes, it is fair to model them as wire scatterers. As shown in this paper, a one-dimensional method will already give a good accuracy. In Section 1, the method is reviewed for wires in one space filled with a homogeneous dissipative medium. However, earthing systems are not present in a single space. They are embedded in the soil but are in close vicinity with the air above the ground. Therefore, in Section 2, the method is extended to account for the presence of the air, above the soil. Because cables are often present in the vicinity of an earthing system, a technique was devised the model them. This is described in Section 3. Modelling earthing systems and cables 989 COMPEL 28,4 2.1 MoM in a single homogeneous dissipative medium The MoM is a general technique for solving integral equations. It is handy to express Maxwell s equations in terms of a scalar potential f and a vector potential ~ A: ~B S ¼ ~ 7 ~ A ð1þ 990 ~E S ¼ 2 ~ 7f 2 jv A ~ ð2þ Z 1 fð~rþ ¼ r s ð~r 0 Þ e 2gk~r2~r0k V s 1 eff 4pk~r 2 ~r 0 k d 3 ~r 0 ð3þ ~Að~rÞ ¼ m ZV ~ J s ð~r 0 Þ e 2gk~r2~r0 k s 4pk~r 2 ~r 0 k d 3 ~r 0 : ð4þ The equations give the relationship between charges and currents on the one hand, and the scattered electric field E ~ S and the scattered magnetic induction B ~ S on the other hand. The charges have a volume density r s in volume V s and the currents have a surface density ~ J s in V s. The quantity V s is called the source region and it contains only free charges and currents. The quantity: g ¼ v c pffiffiffiffiffiffiffiffiffiffi ¼ jv m1 eff is the propagation constant. It has a real part, the attenuation constant, and an imaginary part, the phase constant. It is possible to write Maxwell s equations in this concise from equations (1) to (4), only by employing an effective permittivity 1 eff, equal to 1 eff ¼ 1 þðs=jvþ, instead of the permittivity 1. Also, the source region V s has to coincide with the volume of the scatterer. In its easiest form, the MoM is applied to wire scatterers. Many earthing systems consist of rods, loops or grids and can be modelled as wire-like structures. To be able to numerically solve the equations (1)-(4), the wire structure is divided in smaller segments. A thin-wire assumption is made: it is assumed that the radius of the wire is much smaller (more than ten times smaller) than the length of a segment. With this assumption, it is possible to model the current flowing through the wire and the electric charges on the wire surface as being concentrated on the wire axis. This assumption transforms equations (1)-(4) into a one-dimensional form. There also is a second assumption: for reasons of simplicity, a perfect conducting material is assumed. Then, the relationship between the incident electric field ~ E i and the scattered field ~ E S on the outer boundary of the conductor is ð ~ E S þ ~ E i Þ ~n ¼ ~ 0 with ~n a unit-normal vector on the boundary surface. However, it is also possible to take a not perfect conductor and the skin effect into account (see further in this section). The next step is the application of the moment method to solve the equations. The wire scatterers are discretized in N segments. The nth segment is identified by its starting point, n 2, its midpoint, n and its termination point n þ. This segment is denoted by the interval Dl n. It can be shown that the one-dimensional, discretized form of equations (1)-(4) can be written in matrix form as: ½VŠ ¼½ZŠ½I s Š; ð5þ where: 2 3 I S ð1þ I S ð2þ ½I S Š¼ I S ðnþ 2 ~E i! 3 ð1þ Dl 1 ~E i! ð2þ Dl 2 ½VŠ ¼ ~E i! ðnþ Dl N ð6þ Modelling earthing systems and cables 991 are the vectors with the source currents through each segment and the voltages across each segment. A good introduction to this basic formulation of the MoM is given in Harrington (1968). The mutual impedance between elements m and n is defined as the voltage across m divided by the cause of this voltage, the current through n:! E Z mn ; ~ i ðmþ Dl m I s ðnþ ¼ jvmdl Q Q n Dl m cðn; mþ þ 1 jv1 eff ½cðn þ ; m þ Þ 2 cðn 2 ; m þ Þ 2 cðn þ ; m 2 Þþcðn 2 ; m 2 ÞŠ ð7þ It can be seen that the impedance of segment m due to segment n consists of five terms. The first term is the influence of the magnetic vector potential on the electric field and the last four terms represent the influence of the scalar potential on the electric field. The function c(n, m) is given by: cðn; mþ ¼ 1 e Dl n ZDln 2gR 4pR dl ð8þ and should be numerically evaluated (Harrington, 1968). The skin effect and a non-zero wire resistivity can be taken into account by adding the internal impedance of a wire conductor to the self-impedances of the matrix [Z ]. The internal impedance per unit length is given by (Stratton, 1941): sffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi Z int ¼ r jmv I 0 r jmv=r pffiffiffiffiffiffiffiffiffiffiffiffiffi; ð9þ 2pr r I 1 r jmv=r where r is the conductor s radius, r is its resistivity and I 0 and I 1 are modified Bessel function of order 0 and 1, respectively. The diagonal elements of the impedance matrix [Z ] are then changed to: Z k;k ¼ Z k;k þ Z int Dl k ; ð10þ with Dl k the length of the kth segment. 2.2 Vicinity of the air above the soil Modified image theory and modification of the impedance matrix. In previous section, it was assumed that the wire scatterers were radiating in a single homogeneous medium. COMPEL 28,4 992 However, for earthing rods and grids, this is not realistic. They are embedded in the soil but are in close vicinity of the air, a medium with different electromagnetic characteristics. This problem, of the radiation of an antenna suspended over a not perfectly conducting earth, was first solved by Arnold Sommerfeld. The solution involves calculating integrals which are so difficult that a different method is used here, known as the modified image technique. It states that the electric and magnetic field in an observation point due to a radiating charge- and current-carrying element which is suspended in the same half-space as the observation point, are approximately equal to the electric and magnetic fields in the situation of Figure 1(b). There, the scatterer is mirrored with respect to the flat interface between the two media and the charge and current of the image element are equal to the original charge and current, multiplied with a factor F. Furthermore, in the situation of Figure 1(b), there is only one medium, exhibiting the electromagnetic characteristics of medium 1. This is the modified image theory and the factor F is calculated as (cf. next section): F ¼ 1 eff eff 2 : ð11þ 1 eff 1 þ 1 eff 2 This technique is approximate and is valid if the distance R between source element and observation point, satisfies the relationship jgrj,, 1. The mutual impedances of the matrix expression (5) now become, in the presence of the air-ground interface plane: Q Q Z mn ¼ jvm 1Dln Dl m cðn; mþþjvm 1 FDl Q Q ni Dl m cðn I ; mþ þ 1 ½cðn þ ; m þ Þ 2 cðn 2 ; m þ Þ 2 cðn þ ; m 2 Þþcðn 2 ; m 2 ÞŠ jv1 eff 1 ð12þ þ F c n þ I ; m þ 2 c n 2 I ; m þ 2 c n þ I ; m 2 þ c n 2 I ; m 2 : jv1 eff 1 In the previous equation, the index n I denotes the current and charge element, obtained by mirroring element n with respect to the interface plane between the two media. The factor F is also present in this expression Derivation of the factor F of the modified image theory. Consider a space consisting of two half-spaces characterised by the electromagnetic parameters ð1 eff 1 ; m 1 Þ for the upper halve-space and ð1 eff 2 ; m 2 Þ for the lower half-space (Figure 2). An elementary charge and current element exist in only one of the two media (medium 1). Q, I Observation point ε 1, µ 1, σ 1 ε 1, µ 1, σ 1 Q, I Observation point ε 2, µ 2, σ 2 F.Q, F.I (a) (b) Figure 1. Notes: (a) We want to know the electromagnetic situation in the observation point when a scatterer in medium 1 is suspended above a medium 2; (b) the situation of Figure 1(a) is approximately equal to the situation depicted here. This is the modified image method Q d R n E ind Modelling earthing systems and cables ε 1 σ 1 ε 2 σ 2 Ec F Q t Ec 993 QF Q d F l E F l ind Figure 2. Derivation of the factor F of the modified image theory The electric field in any point on the boundary plane is the vectorial sum of the coulombian component E ~ C, due to the charge Q and the rotational component ~E ind, due to the current I dl.! Index 1 represents the complex image coefficients, the image current and charge have to be multiplied with, in order to calculate the electric field in medium 1. The original source elements in medium 1 then remain and the entire space consists of the material medium 1 is made of. Index 2 represents the complex image coefficients the current and charge have to be multiplied with in order to calculate the electric field in medium 2. The original source elements in medium 1 remain and the mirroring does not have to be performed. The entire space consists, in this situation, of the material medium 2 is made of. The continuity condition at the boundary plane for the normal component of the total current density ~ J ¼ 1 E ~ þ D= t ~ ¼ s eff E ~ with seff ¼ s þ jv1, states that: s eff 1 4p ð12f Q1Þ Q! # R ~n ~ dl ~n 1 eff 1 R R 3 þjvm 1ð12F I1 ÞI ¼ s eff 2 4p ! # F Q R ~n ~ Q2 1 eff 2 R 3 þjvm dl ~n 2F I2 I R The continuity condition for the tangential components of the total electric field at the boundary plane, demands that: 1 4p ð1þf Q1Þ Q! # R ~t ~ 1 eff 1 R 3 þjvm dl ~t 1ð1þF I1 ÞI ¼ 1! # R 4p F Q R ~t ~ Q2 1 eff 2 R 3 þjvm dl ~t 2F I2 I R where ~n and ~t are the normal and tangential unit vectors in a point on the interface plane. The quantity ~ R is the distance vector from the source to the observation point on the interface plane. In calculating the image sources, the retardation and attenuation factor e 2gR has been neglected. This is only valid if jgrj, 1. After a few calculations, the boundary conditions yield: F Q1 ¼ F I1 ¼ 1 eff eff 2 1 eff 1 þ 1 eff 2 ¼ F ð13þ F Q2 ¼ 21 eff 2 1 eff 1 þ 1 eff 2 ; F I2 ¼ m 1 m 2 21 eff 1 1 eff 1 þ 1 eff 2 : ð14þ COMPEL 28, Cables: dielectrically insulated wires scatterers A dielectric is a material with a high-specific resistivity. In a dielectric material, there exist electric dipoles which orientate themselves according to an externally applied electric field. When this electric field has a time-varying orientation, the moving charges produce a current, which is called polarization current. The problem of modelling dielectric insulations is thus reduced to finding these sources, i.e. the polarization charges and current. In the derivation of an expression for the polarization charges, the geometrical model of Figure 3 is assumed. The insulating layer has an inner radius r a and an outer radius r b. The notation r þ a denotes the position on the outer boundary of the metal core, just inside the insulating layer. r 2 a denotes the position on the outer boundary of the core, still inside the metal. An analogous notation ðr þ b and r 2 b Þ applies for the boundary between medium 1 and 2. The insulating layer consists of a dielectric material with permittivity 1 1 and conductivity s 1. The scatterer is present in a medium with parameters 1 2 and s 2. It is also assumed that the insulating layer and medium 2 are not magnetically, and thus have the same permeability as the free space. The polarization charges only exist on the boundaries r a and r b. The surface density of the polarization charges at these two boundaries, is given by: S P ðr a Þ¼2^r Pr ~ þ a 2 Pr ~ 2 a S P ðr b Þ¼2^r Pr ~ þ b 2 Pr ~ 2 b ¼ 2^r ð Þ Er ~ þ b ¼ 2ð Þ^r ~E r þ a 2 ð Þ Er ~ 2 b ð15þ : ð16þ A cylindrical reference system is used, where ^r is the unit radial vector. ~ P is the polarization density vector. Equation (16) can be further simplified by substituting ~ Eðr þ b Þ by ~ Dðr þ b Þ=1 2 and making use of the boundary condition ~ Dðr þ b Þ ^r ¼ S f ðr b Þþ^r ~Dðr 2 b Þ. Here, S f ðr b Þ is the surface density of the free charges at boundary r b. This density of free charges is found by demanding the continuity of the radial component of the total current density ~ J tot ¼ðs þ jv1þ ~ E across the boundary r b : ^r ðs 1 þ jv1 1 Þ ~ Eðr 2 b Þ¼^r ðs 2 þ jv1 2 Þ ~ Eðr þ b Þ Medium 0 Medium 2 e 1 s 1 r a Medium 2 e 2 s 2 Figure 3. Geometry of the insulated wire scatterer r b and substituting in this relation Eðr ~ þ b Þ by Dðr ~ þ b Þ=1 2 and again making use of the boundary condition for D, ~ introducing S f ðr b Þin the expressions. We find: 1 1;eff S f ðr b Þ¼ ^r 1 2;eff 1 ~Eðr 2 b Þ ð17þ 2 S P ðr b Þ¼^r ~E r 2 b ð Þþ 1 1;eff ð Þ 1 2;eff : ð18þ Expressions (17) and (18) still contain unknown quantities: the radial component of the electric field inside the dielectric layer. This electric field can be calculated with Ampère s law for an integration path inside the insulation. Let us take as integration path a circle with radius rðr a # r # r b Þ and centre on the conductor s axis. The electric displacement field is radially oriented and therefore vanishes in Ampère s law: H w 2pr ¼ Iðz 0 Þ Ampère s law, written in differential form, is: H w ¼ Iðz0 Þ 2pr : ð19þ ~7 ~ H ¼ ~ J þ jv ~ D ð20þ For a point inside the dielectric, the source and conduction currents are zero: ~ J ¼ ~ 0. Ampère s law therefore becomes: ~7 H ~ ¼ jv D ~ ¼ jv1 2 E ~ ¼ 1 H z r w 2 H w ^r þ H r z z 2 H z ^w þ 1 r r ¼ 2^r H w z þ 1 r r ðrh wþ ^z ¼ 2^r 1 2pr r ðrh wþ 2 H r ^z w Iðz 0 Þ z 0
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