A Physical Interpretation of the Equal Area Rule - IEEE Xplore

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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 48, NO. 2, MAY 2006

A Physical Interpretation of the Equal Area Rule Abraham Rubinstein, Member, IEEE, Marcos Rubinstein, and Farhad Rachidi, Senior Member, IEEE

Abstract—This paper presents a derivation and a physical interpretation of the equal area rule (EAR) for wire-grid simulation of surfaces. We propose a new procedure that leads to a formulation for obtaining the radii for an arbitrarily meshed surface. A simple comparison of the classical EAR and the new equal area rule (NEAR) is presented in which the electric field inside a closed metallic surface, which is known to be identical to zero, is calculated using the radii predicted by the two methods. The results show that the proposed new equal area rule predicts a smaller field for the frequency range considered, suggesting an improvement over the classical EAR. Index Terms—Electromagnetic modeling, electromagnetic pulse (EMP) simulator, equal area rule (EAR), method of moments (MoM), numerical electromagnetics code (NEC).

I. INTRODUCTION HE Numerical Electromagnetics Code (NEC) is a useroriented computer program based on the Method of Moments (MoM) and written in FORTRAN for the analysis of the electromagnetic response of antennas and other metal structures [1]–[3]. It has been widely used with great success for radio communications testing, antenna design, and the simulation of large structures such as cars and ships. With its ability to represent models by means of wires, the code allows the simulation of very complex three-dimensional (3-D) structures [1]. In spite of the fact that NEC was written over 20 years ago and in a language such as FORTRAN, it continues to be used today since it is open source, free, and has been widely tested and verified over the years. Examples of the current fields of application of NEC can be found in the numerous publications in recent years (e.g., [4]–[6]). Modeling of a surface using the MoM and, in particular, using NEC, can be achieved by two different approaches. Though the NEC manual [1] states in the Structure Modeling Guidelines that straight segments can be used for the modeling of wires and flat patches for the modeling of surfaces, it also mentions the use of wire-grid modeling with “varying success.” Since patches can only be used to model closed surfaces, they are not applicable to a wide range of practical applications, hence the importance of the wire-grid representation of surfaces. Since computer power by the time NEC was developed did not allow the computation of large problems, the application of

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Manuscript received April 21, 2005; revised December 20, 2005. This work was carried out as part of the GEMCAR project, a collaborative research project supported by the European Commission under the competitive and Sustainable Growth Programme of Framework V under EC Contract G3RD-CT-1999- 00024 and by the Swiss Federal Office for Education and Science under Grant 99.0377. A. Rubinstein and F. Rachidi are with the Swiss Federal Institute of Technology, Lausanne CH 1015, Switzerland. M. Rubinstein is with the University of Applied Sciences of Western Switzerland, Yverdon-les-bains 1401, Switzerland. Digital Object Identifier 10.1109/TEMC.2006.873861

the code was devoted mainly to antenna simulations. Attempts to model very complex structures using a wire-grid approach exceeded available memory and calculating power. The geometrical considerations for the construction of a successful model are clearly stated in the manual for the modeling of simple wire structures, such as antennas. Some of these considerations, however, do not apply to the use of wires for the simulation of surfaces and, consequently, a new set of guidelines were introduced in 1991, by Trueman and Kubina [7], based on the so-called Equal Area Rule (EAR) [8]. The original formulation of the EAR comes from empirical observation and, as of today, no physical interpretation has ever been given to the rule. Comparison of experimental measurements with numerical simulations suggest that the EAR, when applied to arbitrarily shaped meshes, does not yield optimum results [9]. An understanding of the EAR is therefore essential for the correct simulation of surfaces. The aim of this paper is to obtain a physical interpretation of the EAR. Our results show that, contrary to the classical EAR, the radius of the segments needed for a correct representation of a surface is dependent on the polarization (except for the case of a square mesh). Although our derivation implies a greater preconditioning effort for the case of an arbitrarily shaped mesh due to the dependence on the polarization, this is an important result since it affects greatly the accuracy of the simulations. In Section II of the paper, a derivation is presented that leads to the rectangular form of the EAR proposed by Miller (see, for example, [8]). Although this derivation uses a number of rough approximations, it represents, to the best of our knowledge, the first attempt at a physical interpretation of the EAR. The same methodology, when applied to a nonrectangular mesh, yields a new expression that differs from the results given by the generalization of the EAR formula proposed in [7]. II. THEORETICAL BASIS FOR THE CLASSICAL EAR AND PROPOSED NEW EQUAL AREA RULE (NEAR) As already mentioned in Section I, the derivation presented here represents a first attempt at providing a physical interpretation for the empirical EAR. Although crude approximations are made throughout the derivation, the results are encouraging and they are presented here as a basis for future work. When an incident field Ei impinges on a perfectly conducting surface, currents are induced on it. These induced currents produce, in turn, a scattered electric field that cancels out the tangential electric field on the conductor’s surface. This is so, since the total (incident plus scattered) tangential field on the surface of a perfect conductor must always equal zero. The situation is illustrated in Fig. 1 for a metallic slab. We now assume that the slab is made of a material with a finite but high conductivity. In that case, most of the induced

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RUBINSTEIN et al.: PHYSICAL INTERPRETATION OF THE EQUAL AREA RULE

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Fig. 1. Scattered field generated by a perfect conductor in response to an incident electric field.

current flows close to the surface of the conductor due to the skin effect. The current can be written as Is = σ A E s w = σ A E i w

Fig. 2.

Wire-grid representation of the slab using rectangles.

Fig. 3.

Geometry of the problem for one of the segments.

(1)

where w is the width of the slab, σA is the surface conductivity of the slab (defined below), and Es and Ei are the magnitudes of the scattered and the incident field, respectively. The surface conductivity can be defined as σA = σδ

(2)

where σ is the conductivity of the material, and δ is the skin depth of the conductor. Note that σA is measured in Ω−1 or Siemens. Radiation from a small electric dipole is proportional to the length of the dipole and to the current in the dipole (e.g., [10]). We call here the product of these two quantities the radiation dipole moment. The radiation dipole moment, as defined here, equals the time derivative of the electrostatic dipole moment, which itself equals the product of the charge and the length. For the scattered current, the radiation dipole moment equals the product of the current σA Es w [see (1)] times the length l of the slab as long as the length and the width are much smaller than the wavelength. As we will see, the condition that the dimensions be smaller than the wavelength will be violated in our derivation. We will however discuss this crude approximation later. The slab’s total dipole moment pSlab tot can be calculated as follows: pSlab

tot

= σA Es wl.

(3)

Note that the magnitude of pSlab tot equals the product of the surface conductivity times the incident electric field (which is identical to the scattered field) times the surface area of the slab, wl. Let us now repeat the exercise of finding the total moment, this time for a surface represented by a rectangular mesh of cylindrical wire segments as shown in Fig. 2. We will assume that the total dipole moment is the vector sum of the dipole moments from all the segments, where the direction of the vectors is given by the direction of the current, which is identical to the direction of the thin segments. We will first concentrate on one of the segments of length ∆1 that

compose the wire grid. The geometry is illustrated in Fig. 3, to which the following derivation refers. Let us call the incident electric field Ei as in the case of the slab. In this case, we will assume that the incident field and the cylindrical conductor’s axis are not necessarily parallel. Let us call the angle between the conductor and the incident electric field θ. The component of the incident electric field tangential to the surface of the conductor is given by Et = Ei cos(θ).

(4)

The scattered electric field has the same magnitude but opposite direction. Its magnitude is therefore Es = Ei cos(θ).

(5)

Assuming that the scattered field is uniform around the circumference of the conductor, and using the fact that the induced current flows near the surface of the conductor, we can calculate the induced current as follows: Ii = 2πaσA Ei cos(θ)

(6)

where a is the radius of the conductor, and σA is the surface conductivity of the wire, whose definition was already given

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above, and which is assumed to be equal to that of the slab. The induced current in (6) is the current that is induced on the segment we are considering as a result of the incident electric field. The radiation dipole moment of the wire segment is given by the product of the scattered current and the length of the segment PSegment = 2πaσA ∆1 Ei cos(θ).

(7)

The component of the dipole moment in the direction of the incident field is PSegment cos(θ) = 2πaσA ∆1 Ei cos2 (θ).

(8)

We can easily obtain the dipole moment in the direction of the incident field for the segments of length ∆2 (see Fig. 2) as follows: PSegment sin(θ) = 2πaσA ∆2 Ei sin2 (θ)

Fig. 4. Surface chosen for the derivation of the NEAR formula for a triangular mesh.

(9)

where we have used the fact that the angle between the present segment’s axis and the incident electric field is π/2 − θ. We calculate the total moment in the direction of the incident field for a grid that has a width of w = m∆2 and a height of l = n∆1 by multiplying the magnitude of one of them by the number of dipoles in the mesh. The result is PGrid tot = 2m(n + 1)πaσA ∆1 Ei cos2 (θ) + 2n(m + 1)πaσA ∆2 Ei sin2 (θ).

(10)

The calculation of the total radiation dipole moment in (10) neglects any phase or amplitude differences between the dipole moments. We will now assume that, to approximate the scattering behavior of the slab by way of the wire-grid model, the total moments must be equal. Equating (3) and (10) and canceling out terms that appear on both sides of the equation, we obtain

III. APPLICATION TO OTHER PATTERNS

mn∆1 ∆2 = 2m(n + 1)πa∆1 cos2 (θ) + 2n(m + 1)πa∆2 sin2 (θ).

(11)

Let us assume that the grid is square, ∆1 = ∆2 = ∆, and that m = n. Under such conditions, we can rewrite (11) as m2 ∆2 = 2m(m + 1)πa∆(cos2 (θ) + sin2 (θ))

(12)

which, after solving for the radius a, can be written as a=

m∆ . (m + 1)2π

(13)

If m 1 so that m + 1 ∼ = m, a reduces to the well-known EAR formula a=

∆ . 2π

Fig. 5. (a) Triangular mesh corresponding to the surface under consideration and (b) the geometry of one triangle.

The derivation presented in Section II can be applied to wire grids made of polygons other than squares. Here, we illustrate the derivation of the NEAR for a triangular grid in response to a vertically polarized incident electric field. Consider the surface shown in Fig. 4. As in the case of the slab discussed in Section II, the total moment is the product of the surface conductivity times the incident electric field times the surface area. To calculate the surface area, we divide the surface into triangles as shown in Fig. 5. The total surface area is given by the product of the number of triangles times the area of each individual triangle. The number of triangles is readily 2mn. The area of a triangle, on the other hand, is given by

(14)

Note that this result represents a special case where the mesh is square. It is only in this case that the angle dependence drops out in (12), where the sum cos2 (θ) + sin2 (θ) equals 1. As we will see in Section III, application of the current derivation to other mesh shapes leads to polarization-dependent results.

s=

∆2 tan(α) 4

(15)

and the total moment of the surface is PSurface

total

= 2mnσA Ei

∆2 tan(α) . 4

(16)


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Let us now calculate the total moment for the wire-grid version of the surface. The wire-grid consists of horizontal and nonhorizontal segments. The moment from the horizontal segments is zero as can be seen from (7), where, in this case, θ = π/2. For the nonhorizontal segments, on the other hand, the length of the segments is (∆/2 cos(α)) and the angle between the incident electric field and the axis of the segments is θ = π/2 − α. The moment in the direction of the incident field is therefore given by PSegment = 2πaσA

∆ Ei sin2 (α) 2 cos(α)

= πa∆σA Ei tan(α) sin(α).

(17)

The total moment for the wire grid is obtained by multiplying the moment given in (17) by the total number of nonhorizontal segments which, from Fig. 5, can be seen to be 2mn. The result is PGrid

total

= 2mnπa∆σA Ei tan(α) sin(α).

Fig. 6.

Model of an octahedron meshed with a triangular grid.

Fig. 7.

Equal Area Rule for an arbitrarily shaped mesh. Adapted from [9].

(18)

Equating now the moments for the surface and for its wiregrid representation, we obtain 2mnπa∆σA Ei tan(α) sin(α) = 2mnEi σA

∆2 tan(α) (19) 4

from which the radius a can be found to be ∆ . a= 4π sin(α)

(20)

For equilateral triangles, α = π/3, and ∆ . (21) 2π Interestingly, the radius is the same as that obtained for the square grid and, as we will see in Section IV, it is different from that predicted by the classical EAR. a=

IV. COMPARISON BETWEEN THE CLASSICAL AND THE NEW EARS The objective here is to test the merits of the NEAR relative to the classical rule. A closed surface was chosen for the simulations since the field inside is known to be zero. Consider the octahedron model shown in Fig. 6. The field point to be used for the comparison was chosen inside the octahedron, whose sides were modeled using equilateral triangles formed by individual segments of 1 cm length each. The range of frequencies used for the analysis is such that the segments satisfy the NEC’s length requirements. The following straightforward test was performed: The field inside the octahedron was calculated, using the radius predicted by both the classical equal area rule and the new rule described in Section III, and the results were compared. A. Radius Predicted by the Classical EAR Trueman and Kubina [7] proposed (22) as the general formula for the calculation of the wire radius for an arbitrarily shaped

mesh A1 + A2 (22) 4·π·∆ where A1 and A2 are the surface areas of the two shapes adjacent to the segment for which the radius is required (Fig. 7), and ∆ is the segment length. Since the triangles are equilateral of side ∆, their surface areas, A1 and A2 , can be obtained as follows: 1√ 2 A1 = A2 = 3∆ . (23) 4 Substituting (22) into (23) and using the fact that ∆ = 0.01 m, we obtain the segment radius a predicted by the EAR a=

a = 6.8916 · 10−4 m.

(24)

B. Radius Predicted by the NEAR To find the radius of the segments, we followed the procedure presented in Section III. Refer to the triangle in Fig. 8, which represents one of the faces of the octahedron. The total moment for a solid triangular surface can be found by multiplying the incident electric field Ei times the surface conductivity σA times the surface area of the triangle pTotalTriangle = Ei σA

l2 sin(60) 2

(25)

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Fig. 8.



where l is the length of one side of the triangle. We will now find the total moment from the segments. Each segment has a length of ∆ = l/8. The contribution of the horizontal segments is zero since the field is vertically polarized [see (7)]. The moment in the direction of the incident field for each one of the other segments, of which there are 72, is l pSegment = Ei cos(30)2 σA 2πa . 8 The total moment for the 72 segments is

(26)

l p72 Segments = 72 · Ei cos(30)2 σA 2πa . (27) 8 Equating the total moments from the solid surface and from the grid, we have l2 sin(60) l = 72 · Ei cos(30)2 σA 2πa . 2 8 Solving for the radius a, we get Ei σA

a=

l √ . 18 3

(28)

(29)

Since the side of a face of the octahedron is composed of eight segments, l = 0.08 m and the radius predicted by the NEAR is a = 8.1678 · 10−4 m.

(30)

We ran NEC using the model from Fig. 6 and the values of the radii obtained from (22) and (29). The calculated total E field at the center of the octahedron is displayed in Fig. 9. As we can see from Fig. 9, the NEAR formula seems to be more accurate than the original EAR, that is, the radius calculated using the NEAR yields field values that are closer to zero. V. SUMMARY AND CONCLUSIONS We have presented a theoretical development that leads, for the case of a square-grid representation of a surface, to the same formula proposed by the EAR. Although this development involves a number of crude approximations, it is, to the best of our knowledge, the first mathematical derivation and physical interpretation of the EAR as of today. A new formulation has been derived for the calculation of the radii for meshes other than square. Our development gives a different value for the radius of the segments if the representation of the surface uses a triangular mesh. To compare the accuracy

Fig. 9.


of the two methods (the traditional versus the new EAR), we carried out a simple numerical test using an octahedron. The NEAR yields better results for the case studied in this paper, that is, the field inside the model for that case is closer to zero, which seems to indicate that the new formulation is better than the EAR for the representation of triangular meshes. The original EAR, for square grids does not depend on the polarization of the source. This makes this formula relatively easy to apply as it depends only on segment lengths. On the other hand, geometries from CAD data are difficult to adapt and very sophisticated software must be used to modify the mesh. The square representation also fails in reproducing the very fine details of the geometry, which has a negative impact on near field results. The developed NEAR is more difficult to apply than the classical formula as it depends on the angle of polarization. This means that a change in the position or even the polarization of the source may require a new calculation of the parameters. A body-fitted geometry from CAD data could be used but it requires preconditioning. Furthermore, since the radius must be individually calculated for segment groups as a function of several parameters, including the angle of polarization, the results may lead to conflicts with NEC’s general geometry construction guidelines. Although the application of the NEAR requires more effort than the application of the classical EAR, the improved accuracy and the applicability to arbitrarily shaped models is of great interest, especially if one considers the fact that much of the preprocessing can eventually be made automatic. Further work is in progress. ACKNOWLEDGMENT The authors would like to thank all the partners of the project for their valuable contribution. The assistance of Volvo Car Corporation (Sweden) is acknowledged for permission to use vehicle CAD data in the project.


REFERENCES [1] G. Burke and A. Poggio, “Numerical Electromagnetics Code—Method of Moments,” Lawrence Livermore Nat. Lab., Livermore, CA, Rep. No. UCID-18834, 1981. [2] A. J. Poggio and E. K. Miller, Computer Techniques for Electromagnetics. New York: Summa, 1987, ch. 4. [3] T. R. Ferguson, T. H. Lehman, and R. J. Balestri, “Efficient solution of large moments problems: Theory and small problem results,” IEEE Trans. Antenna Propag., vol. AP-24, no. 2, pp. 230–235, Mar. 1976. [4] R. Pokharel, M. Ishii, and Y. Baba, “Numerical electromagnetic analysis of lightning-induced voltage over ground of finite conductivity,” IEEE Trans. Electromagn. Compat., vol. 45, no. 4, pp. 651–656, Nov. 2003. [5] A. Sarolic, B. Modlic, and D. Poljak, “Measurement validation of ship wiregrid models of different complexity,” in Proc. 2001 IEEE Int. Symp. EMC, vol. 1, 2001, pp. 651–656. [6] M. McKaughan, “Coast guard applications of NEC,” in Proc. IEEE Antennas Propag. Soc. Symp., vol. 3, 2004, pp. 2879–2882. [7] C. W. Trueman and S. J. Kubina, “Fields of complex surfaces using wire grid modeling,” IEEE Trans. Magnetics, vol. 27, no. 5, pp. 4262–4267, Sep. 1991. [8] A. C. Ludwig, “Wire grid modeling of surfaces,” IEEE Trans. Antennas Propag., vol. AP-35, no. 9, pp. 1045–1048, Sep. 1987. [9] A. Rubinstein, F. Rachidi, and M. Rubinstein, “On wire-grid representation of solid metallic surfaces,” IEEE Trans. Electromagn. Compat., vol. 47, no. 1, pp. 192–195, Feb. 2005. [10] J. William and H. Hayt, Engineering Electromagnetics. New York: McGraw-Hill, 1991, ch. 13.

Abraham Rubinstein (S’02–M’03) was born in Bogot´a, Colombia, in 1971. He received the B.S. degree in electrical engineering from the University of Zulia, Maracaibo, Venezuela, in 2000 and the Ph.D. degree from the Swiss Federal Institute of Technology, Lausanne, Switzerland, in 2004. From 1998 to 2000, he worked at Coplan (a contractor for PDVSA, the Venezuelan National Oil Company) Maracaibo, where he was the Head of the Design and Development team for Human Machine Interfacing. He was also in charge of the Protocol and Communication Integration Division for natural gas compressor plants. He is currently with the EMC group at the Power Systems Laboratory of the Swiss Federal Institute of Technology. He is also a Lecturer in Teleinformatics at the University of Applied Sciences of Western Switzerland, Yverdon-les-bains, Switzerland. He is author or coauthor of more than ten scientific papers published in reviewed journals and presented at international conferences. His research interests concern fast algorithms and parallel methods for computational electromagnetics, electromagnetic compatibility, and electromagnetic field interactions with large structures. Dr. Rubinstein is a Member of the IEEE EMC society. He appeared on the List of Honor of the College of Engineering. He also received the Maraven Prize of Academic Excellence and obtained the award of Best Thesis in Venezuela for the degree of electrical engineering granted by INELECTRA.

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Marcos Rubinstein was born in San Cristobal, Venezuela, in 1960. He received the B.S. degree in electronics from the Universidad Simon Bolivar, Caracas, Venezuela, in 1982 and the M.S. and Ph.D. degrees in electrical engineering from the University of Florida, Gainesville, in 1986 and 1991, respectively. He has worked in Hybrid Circuit manufacturing technology at the Venezuelan Engineering Research Institute, Caracas. In 1992, he joined the Swiss Federal Institute of Technology, Lausanne, Switzerland, where he worked in the fields of lightning physics, lightning electromagnetics, and lightning location systems in close cooperation with the former Swiss PTT. In 1995, he joined Swisscom (formerly Swiss PTT), where he was involved in research in the fields of electromagnetic compatibility, numerical electromagnetics, and biological effects of electromagnetic radiation. In 2001, he joined the University of Applied Sciences of Western Switzerland, Yverdon-les-bains, Switzerland, where he is currently a Professor in the field of communications and a Member of the Institute for Information and Communication Technologies team. He is the author or co-author of over 70 scientific publications in reviewed journals and international conferences. His current research interests include numerical electromagnetics, advanced wireless technologies, power line communications, EMC/coexistence in communication systems, and lightning physics. Prof. Rubinstein is the recipient of the best Master’s Thesis award from the University of Florida and co-recipient of NASA’s recognition for innovative technological work.

Farhad Rachidi (M’93–SM’02) was born in Geneva, Switzerland, in 1962. He received the M.S. degree in electrical engineering and the Ph.D. degree from the Swiss Federal Institute of Technology, Lausanne, Switzerland, in 1986 and 1991, respectively. He has worked at the Power Systems Laboratory, Swiss Federal Institute of Technology, until 1996. In 1997, he joined the Lightning Research Laboratory, University of Toronto, Toronto, ON, Canada, and from April 1998 until September 1999, he was with Montena EMC, Rossens, Switzerland. He is currently the Head of the EMC Group at the Swiss Federal Institute of Technology. He is the author or coauthor of over 200 scientific papers published in reviewed journals and presented at international conferences, and the convener of the joint CIGRE-CIRED Working Group “Protection of MV and LV networks against Lightning.” His research interests concern electromagnetic compatibility, lightning electromagnetics, and electromagnetic field interactions with transmission lines. Dr. Rachidi is a Member of various IEEE, CIGRE, and CIRED working groups dealing with lightning, the scientific committees of various international symposia (International Conference on Lightning Protection, International Zurich Symposium on EMC, etc.), and the editorial board of the Journal of Lightning Research, as well as the Vice-Chair of the European COST Action on the Physics of Lightning Flash and its Effects. In 2005, he was the recipient of the IEEE Technical Achievement Award and the CIGRE Technical Committee Award.