Application of Combined Integral Equation for Electromagnetic ...

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R E S E A R C H C O M M U N II C A T II O N 

____________________________ 

VSRD-TNTJ, Vol. 3 (2), 2012, 69-78 

Application of Combined Integral Equation for 

Electromagnetic Spreading Problems 

ABSTRACT 

1 Nazneen Khan* and 2 Pankaj Singh 

In this paper we derive a novel CFIE formulation in which the electric and magnetic field integral equations 

(EFIE and MFIE) are combined in the usual manner outside of the object. Inside the object CFIE is composed 

by changing the roles of EFIE and MFIE. Solution of combined field integral equation (CFIE) for 

electromagnetic spreading by an arbitrary shaped dielectric object is analyzed and described in brief after 

completely study. In addition, we consider numerical evaluation of the singular impedance matrix elements with 

the singularity extraction technique. Numerically stable electric field integral equations (EFIE) are presented for 

electromagnetic scattering problems that may include both electrically small geometrically complex and 

electrically large regions. A reduced integrand is achieved by implementing quasi-static assumptions in the 

electrically small regions, full-wave methods in the electrically large regions, and applying appropriate coupling 

relations between the regions. Use of the method provides computational efficiency as well as insight into the 

conditions under which the electromagnetic fields within electrically small regions of the problem can be 

assumed to be primarily capacitive or inductive in nature. The theoretical development of the method is 

highlighted in this communication and then applied to examples of electrically small, inductively-loaded, and 

capacitive-loaded monopole antennas. The accuracy of the results is verified with two independent methods. 

Here we present a new formulation using the time-domain electric field integral equation TD - EFIE to obtain a 

transient scattering response from arbitrarily shaped conducting bodies. 

In the solution of the magnetic field integral equation (MFIE) by the method of moments (MOM) on planar 

triangulations, singularities arise both in the inner integrals on the basis functions and also in the outer integrals 

on the testing functions. A singularity-extraction method is introduced for the efficient and accurate computation 

of the outer integrals, similar to the way Inner-integral singularities are handled. 

In this paper a solution of combined field integral equation (CFIE) for electromagnetic scattering by an arbitrary 

shaped dielectric object is studied. We derive a novel CFIE formulation in which the electric and magnetic field 

integral equations (EFIE and MFIE) are combined in the usual manner outside the object. Inside the object CFIE 

1 Research Scholar, Department of Mathematics, JJTU, Jaipur, Rajasthan, INDIA. 

2 Professor, Department of Mathematics, Kanpur Institute of Technology, Kanpur, Uttar Pradesh, INDIA. 

*Correspondence : knaznnen.khan786@gmail.com

Nazneen Khan et al / VSRD Technical & Non-Technical Journal Vol. 3 (2), 2012 

is composed by changing the roles of EFIE and MFIE. In addition, we consider numerical evaluation of the 

singular impedance matrix elements with the singularity extraction technique. 

Keywords: Singular Impedance Matrix, Electric Field Interpretation, Magnetic Field Integration Equation, 

Combined Field Integration Equation, Time Difference EFIE. 

1. INTRODUCTION 

There are several alternative ways to formulate boundary integral equations for electromagnetic scattering by 

homogeneous dielectric bodies. The most widely applied formulations are the PMCHW formulation and the 

combined field formulation (CFIE) [2], [3]. In [3], the authors observed that if the RWG functions [4] are taken 

as both basis and test functions in the method of moment’s solution of the CFIE, the traditional form of CFIE 

leads to a very unstable solution. The reason is that only the electric surface current is well tested. As a remedy 

they suggest to test by both RWG and n*RWG functions, where n is the outer unit normal of the object. The 

most convenient formulation was obtained when the electric field part of CFIE is tested by RWG + n*RWG 

functions and the magnetic field part is tested by RWG functions. This formulation was named a TENENH 

formulation. 

An application of the method of moments with Galerkin's method to solve electromagnetic integral equations 

requires calculation of double integrals with singular kernels. Singular terms can be considered either by 

numerical methods (e.g. Duffy's method) or by singularity extraction technique [5]. 

In many cases numerical accuracy of the singular integrals is crucial for having an efficient algorithm, especially 

in the near field computing. The aim is of this paper is twofold. Firstly we present a new type of CFIE 

formulation. In this formulation we use traditional CFIE outside the object but inside the object the electric and 

magnetic field integral equations (EFIE and MFIE) are combined by changing their roles. This formulation 

gives TENH formulation outside the object and NETH formulation inside the object. This formulation has 

several nice properties: It is formally simpler that the TENENH formulation, it is free of interior resonances and 

it gives a stable solution. Secondly we present accurate and robust methods to evaluate singular integrals 

appearing in the CFIE formulations, when the integral equations are tested by both RWG and n*RWG functions 

and the basic functions are RWG functions. 

Our method is based on the singularity extracting technique. First we extract enough terms from the singular 

kernel so that the remaining function is at least once continuously differentiable and allows numerical 

integration. In evaluating the gradient of the Green's function the accuracy is easily lost although the singularity 

has been extracted. The reason is that in computing the impedance matrix elements we have to consider double 

integrals and the remaining outer integral may still have a logarithmical singularity. Therefore, in the terms 

including the gradient we modify the integrand and change the order of integration. By these modifications we 

can integrate all singular functions in closed form and the remaining terms are regular enough for numerical 

integration. Thus, Duffy's transformation or any other special integration quadratures are not required. 

Developed formulas and formulations are verified by considering numerical examples. 

Page 70 of 78

2. EFIE INTERPRETATION 


The EFIE describes a radiated field E given a set of sources J, and as such it is the fundamental equation used 

in antenna analysis and design. It is a very general relationship that can be used to compute the radiated field of 

any sort of antenna once the current distribution on it is known. The most important aspect of the EFIE is that it 

allows us to solve the radiation/scattering problem in an unbounded region, or one whose boundary is located 

at infinity. For closed surfaces it is possible to use the Magnetic Field Integral Equation or the Combined Field 

Integral Equation, both of which result in a set of equations with improved condition number compared to the 

EFIE. However, the MFIE and CFIE can still contain resonances. 

In scattering problems, it is desirable to determine an unknown scattered field Es that is due to a known incident 

field Ei Unfortunately; the EFIE relates the scattered field to J, not the incident field, so we do not know 

what J is. This sort of problem can be solved by imposing the boundary conditions on the incident and scattered 

field, allowing one to write the EFIE in terms of Ei and J alone. Once this has been done, the integral equation 

can then be solved by a numerical technique appropriate to integral equations such as the method of moments. 

3. GENERAL TD – EFIE 

Let S denote a perfectly conducting surface, which may be closed or open, illuminated by a transient 

electromagnetic wave. This incident wave induces a surface current J (r, t) on S which then reradiates. We have 

E s (J)= - ∂A / ∂t - φ …(1) 

where A and φ are the magnetic vector potential and the electric scalar potential given by 

A (r, t) = μ/4π ∫s J (r, τ)/R dS` …(2) 

Φ (r, t) = 1/4πε ∫s q (r`, τ)/R dS` …(3) 

τ = t - (R/c) …(4) 

R = | r – r` | …(5) 

and where μ and ε are the permeability and permittivity of space, c is the velocity of wave propagation in that 

space, and r and r` are the arbitrarily located observation point and source point. The surface-charge density q is 

related to the surface divergence of J through the equation of continuity: 

Utilizing (6), we have from (3) 

t 

. J (r, t) = - ∂q (r, t)/ ∂t or q (r, t) = - ∫0 . J (r, t`) dt` …(6) 

τ 

Φ (r, t) = - 1/4πε ∫s ∫0 ` . J (r, t`)/R dt`/dS` …(7) 

Since the total tangential electric field is zero on the conducting surface for all times, we have 

[E i + E s (J)]tan – 0, r € S …(8) 



And [∂A/∂t + Φ]tan = [E i ]tan …(9) 

where E i is the incident electric field on the scattered and the subscript ‘‘tan’’ denotes the tangential component. 

Equation (9) with (2) and (7) constitutes a TD-EFIE from which the unknown current J (r, t) may be determined. 

4. THE MAGNETIC FIELD INTEGRAL EQUATION 

Let an electromagnetic field with angular frequency ω be incident upon an object with a surface S of arbitrary 

shape. For perfectly-conducting material, all induced current is surface current i(r) . The magnetic field B(r) at 

point r can then be written as 

B(r) = B (r)inc – µ0/4π ∫dS` i(r`) * g (r, r`), (r off S) …(1) 

with B(r)inc the given, but further arbitrary, incident field. The second term on the right hand side equals the 

magnetic field generated by the current, with 

g (r – r`) = e ik |r – r`| 

0 / |r – r`|, k0 = ω/c, …(2) 

the free-space Green's function for the Helmholtz equation. The integral in Eq. (1) runs over the surface S, and 

r' indicates a point in S. It is essential that the field point r is not in S, since ∇g(r − r') has a singularity for r'→r. 

Let us now consider the field points r+ and r- , just outside and inside the material, respectively, and near the 

point r in S. with nˆ(r) the unit normal vector on S at r, directed from the medium into the vacuum, we can 

then write r = r ± ε nˆ(r) ± with ε small. As long as ε is finite, Eq. (2) applies with r replaced by r± . For the 

integral over S, we leave out a circle with radius δ around r, and then let ε approach zero. In the limit ε → 0 , 

there is a finite contribution from the singularity at r'= r , and this contribution remains finite for δ →0 . The 

result of this procedure is [20] : 

B (r±) = B(r)inc ±1/2 µ0 I(r) * n(r) – µ0 / 4π .p∫ dS` i(r`) * g (r – r`) (r in S) …(3) 

Expressing the total magnetic field just outside and inside the material as the sum of the incident field at r and 

the contribution from the current density i(r) . The integral over S is now a Cauchy Principal Value integral, and 

the second term on the right-hand side is the finite contribution from the singularity. Taking the difference 

between the plus and minus equations yields 

B(r+) – B(r-) = µ0 i(r) *n(r) …(4) 

Which is the usual boundary condition for an interface carrying a surface current density i(r) . 

When the material is a perfect conductor, the electromagnetic field inside vanishes. In particular at the point r− 

we have B(r−)=0 and Eq. (3) with the lower sign becomes 

I(r) + 1/2π. n(r) * P ∫ dS i(r`) * g(r – r`) =2/µ0. n(r) * B(r)inc …(5) 

after taking the cross product with nˆ(r) . This integral equation for the unknown current density i(r) is the 

Magnetic Field Integral Equation, originally due to Maue [21]. After solving Eq. (5) for i(r), if possible, the 



magnetic field at a field point r outside the material follows by integration from Eq. (1). For any r inside the 

medium, the term with the integral in Eq. (1) should cancel exactly the incident field inc B(r) when i(r) is a 

solution of Eq. (5). Equation (4) becomes 

B(r+) = µ0 i(r) *n(r) …(6) 

for a perfect conductor, and the cross product with nˆ(r) gives 

i(r) = 1/µ0. n(r) * B(r+) …(7) 

since i(r) is in the tangent plane of S at r. 

5. CFIE 

The combined-field integral equation (CFIE) is a linear combination of the H-field and the E-field integral 

equations. Previously, the weighting parameter of the E-field equation in the CFIE had been assumed constant 

along the generating curve of the body of revolution. However, it is shown that the weighting parameter can 

take a variable distribution along the generating curve or on a part of it only. In the latter case, a reduction in the 

computational time of 40-50% is achieved. 

6. FORMULATION 

Consider the problem of electromagnetic scattering by a homogeneous dielectric body D in IR 3 . Let S denote the 

surface of D and let n denote the outer unit normal of D. The traditional form of CFIE reads [2]. 

p EFIE + q n * MFIE …(1) 

Where p = α; q = (1-α) η, 0 < η < 1; η= Sqrt. μ0/ε0 and EFIE and MFIE denote the electric and magnetic field 

integral equations. Another possible CFIE formulation is to combine EFIE and MFIE as 

p n* EFIE + q MFIE: …(2) 

Using RWG functions [4] as both basis and test functions to discretize (1) and (2), gives the following equations 

p∫s fm. EFIE dS - q∫s (n * fm) MFIE dS, 

- p∫s (n * fm). EFIE dS + q∫s fm MFIE dS, 

for all m = 1------N, where fm denotes an RWG function. Equation (1) gives a TENH formulation and (2) gives 

a NETH formulation [3]. As pointed out in [3] none of these two equations lead to a stable solution. As a 

remedy, in [3] the authors suggest to test EFIE by fm +n * fm. The resulting formulations are called TENENH 

and TENETH, respectively. However, the testing procedure can be simplified if EFIE and MFIE are combined 

as follows: 

a EFIE-O + q n * MFIE-O 

p n * EFIE-I + q MFIE-I: …(3) 



Here O stands for outside D and I stand for inside D. This formulation gives TENH formulation outside D and 

NETH formulation inside D, so called TENH/NETH formulation. In TENH/NETH formulation the electric 

current is well tested by the TENH equation outside D and the magnetic current is well tested by the NETH 

equation inside D. Thus, both currents will be well tested. Furthermore, because both TENH and NETH 

formulations are free of interior resonances [3] also their combination shares the same property. 

7. NUMERICAL IMPLEMENTATION 

Let fn denote an RWG function with the support Sn and let gm denote an RWG function or p n * RWG function 

with the support Sm. An application of the method of moments to solve CFIEs (1) and (2) by the RWG 

functions requires calculation of the following double integrals [3]. 

I1 := ∫sm gm(r). ∫sn G(r; r`) `s .fn(r`) dS` dS, 

I2 := ∫sm gm(r). ∫sn G(r; r`) fn(r`) dS` dS, 

I3 := ∫sm gm(r). ∫sn G(r; r`) *fn(r`) dS` dS, 

I4 := ∫sm gm(r). (n (r)*fn(r`)) dS, 

Where G is the free space Green's function. 

If the supports of gm and fn are far away from each other, the above integrals are regular and they can be 

calculated numerically. Let us consider the case when Sm and Sn are close to each other or they have common 

points and let R = | r – r`|. Because the RWG functions are composed on a triangle pair it suffices to consider 

calculation of the above integrals over single triangles. We begin by modifying integral I1 as 

{ I 1 = -∫Tm s f m(r). ∫Tn G(r; r`) `s .f n(r`) dS` dS }, if g m = f m 

{I 1= ∫ ∂Tm m (r) .( n(r)*. f m(r)) ∫Tn G(r; r`) `s .f n(r`) dS` dl }, if g m = n * f m, 

Where m is the unit vector of the boundary ∂Tm of Tm pointing into the exterior of Tm. We may conclude the 

following: 

� Integrals I1 and I2 have a singularity of order 1/R. 

� After evaluating the inner integrals of I1 and I2 with respect to r`, the outer integral with respect to r is 

regular. 

� Integral I3 vanishes if Sm and Sn are on the same plane. 

� If Sm and Sn are not in the same plane, integrand of I3 has a singularity of order 1=R 2 . 

� If Sm and Sn are not in the same plane, after evaluated the inner integral of I3, the outer integral may still 

have a logarithmic singularity. 

� Integral I4 is regular. 



Thereafter integrals I1 and I2 are calculated by extracting two terms from the Green's function 

G(r, r`) = (G(r, r`) – 1/4πR+k 2 R/8π) + 1/4πR - k 2 R/8π. 

Here the term in the brackets has two continuous derivatives and a standard integration routine on triangles such 

as the Gaussian quadrature gives accurate results. The last two terms on the right hand side (RHS) can be 

integrated in closed form [4], [5]. Consider next term I3 with an RWG testing function. Using the same idea as 

above, we write [6] 

G(r; r`) = (G(r, r`) – 1/4πR + k 2 R/8π) + 1/4π 1/R - k / 8π R …(4) 

The first term on the RHS has a continuous derivative and therefore, it can be integrated numerically. The 

extracted terms, i.e. the second and third terms on the RHS of (4), can be integrated analytically over Tn using 

the formulas presented in [7] and [5]. Thereafter, the last term can be integrated numerically over Tm, because 

the integrand is regular. The problem is to integrate the second term over Tm. Let us consider this term more 

carefully. 

Since an RWG function is given as +L / (2A) (r - p) [3], where L and A are the length of an edge and the area of 

a triangle, it suffices to consider the following singular double integral 

Cmn ∫Tm (r - p) . ∫Tn G0(r, r`) *(r`- q) dS` dS, ….(5) 

Where G0 = 1- (4πR), Cmn = +LmLn = (4AmAn) 

and p and q are the free vertices of the test and basis triangles. We consider this integral in two parts by 

separating the normal and surface derivatives. Before applying analytical formulas, we apply the Gauss 

divergence theorem to the surface gradient term and translate the integral over Tn into a line integral over the 

edges of Tn and then change the order of integration. This gives us the following integral [5] 

Cmn ∫∂Tn m (r`) . ∫Tm (r – p)/4πR dS dl` . 

Now the inner integral (with respect to r) is evaluated analytically and the outer integral is regular and allows 

numerical integration. Term I3 with an n*RWG testing function is considered by applying the same idea. More 

details are presented in [5]. 

8. EXAMPLE 

In order to verify the stability of TENH/NETH formulation, we consider a dielectric sphere with εr = 4, r = 1=k0; 

k0 = w sqrt. ε0 µ0 illuminated by an axially incident plane wave. The figures on the LHS of Figure -1 show the 

equivalent electric and magnetic surface current densities on the surface of a sphere. The results show a good 

agreement with the present method and the other methods. Next we study the accuracy of the developed 

integration routines. We consider singular integral (5) with Cmn = 4π in the case where Tm and Tn share an edge. 

We see that the Gaussian quadrature without extracting the singularity leads to a significant error (solid line with 

circles on the RHS of Figure 1). 



Electric Current 

Ө Degrees λ 

Magnetic Current Number of Points 

θ Degrees Number of points 

Figure 1 



Fig. 1: On the LHS is the electric and magnetic current densities on a dielectric sphere. The figure displays CFIE 

(TENH/NETH) solution (solid line with circles), PMCHW solution (dashed line) and analytical solutions (solid 

line). On the horizontal axis is the angle from the axis of the sphere. The figures on the RHS show the geometry 

and the integral (5) in the case where triangles Tm and Tn have a common edge. The value of the integral is 

computed with three methods: the method of this paper (solid line), traditional singularity extraction and 

Gaussian (dashed line), double Gaussian without singularly extraction (solid line with circles). Horizontal axis 

shows the number of integration points. 

9. REFERENCES 

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