While indepth derivations are not provided, this section will consider the SuperNEC formu- lation - an application of the integral equation theory covered in chapter 3. The inverse (integral) operator used in the approximation of a thin, cylindrical wire has been the subject of much analysis for over 60 years, with its origins traced back to the works of Pockling- ton, more than a century ago. The methods of Pocklington [41] and H´allen [43] form the first rigorous analysis of the linear cylindrical antenna problem; the texts “The Theory of Linear Antennas” by King [164] and “Advanced Antenna Theory” by Schelkunoff [165] are considered catalysts to the development of antenna theory in the later half of the twentieth century.
Consider a thin, conducting cylindrical wire segment in free space. Using Maxwell’s equations, and the theory developed in chapter 3, we can state without further proof the z-directed form of the EFIE, known as Pocklington’s integral equation. For radiation from a wire of length
L [166], with surface S: Z L I(z0) · ∂2 ∂z2G(z, z0) + k2G(z, z0) ¸
dz0= −i ω²Ezinc(z) z ∈ L (6.1.1)
where k is the wavenumber, G(z, z0), the free-space Green’s function and Einc
z the z-directed
field produced by the generator. For the exact formulation,
G(u) = 1
2π Z 2π
0
exp£−ik¡u2+ 4a2sin2¡1 2θ
¢¢¤
u2+ 4a2sin2¡1 2θ
¢ dθ u ∈ R (6.1.2)
where a denotes the radius of the wire, and θ the cross-sectional angle. The simplification presented here, as derived by Schelkunoff [165] is the z-directed formulation of the Electric Field Integral Equation (EFIE), eqn (5.1.2). While a full derivation is not given here, some analysis of the inverse operator problem with respect to Green’s functions is given in sec- tion B.1. Mei [166] provides a pedestrian prescription for a general curved wire in terms of a distance parametric independent variable.
For the thin-wire approximation, we generate the reduced-kernel form. This approximation assumes that the current can be represented by a filament on the z-axis. Transverse currents and circumferential variations are ignored and the required boundary conditions are only enforced in the axial region. The free space Green’s function, G(z, z0) that is consistent with
these assumptions, is given by
G(z, z0) = e−jkR 4πR where R = ((z − z 0)2+ a2)1/2 (6.1.3) Let l(z, z0) = h∂2 ∂z2G(z, z0) + k2G(z, z0) i
, denote the kernel. This approximation is valid for wire radii significantly less than wire length and wavelength.
For these derivations, the reader is referred to the work of Poggio et al [6]. For brevity, discussions relating to the Numerical Electromagnetic Code (NEC2) in this document will be limited to the SuperNEC implementation, which formed the basis of this work. The electromagnetic framework presented in the Fortran code, NEC2, is still present, in a more advanced object-oriented form in SuperNEC [5].
The formulation of (6.1.1) as well as the equations to follow are valid at a single frequency. Viewing (6.1.1) as a general linear operator equation
Lh = e (6.1.4)
with h the unknown response and e the excitation vector. Consistent with the definitions of section 4.1, h, e ∈ L2(S)1, where S is the surface of the cylindrical wire segment; the z-directed
formulation integral is evaluated over the length of the wire segment. The unknown function
h, in (6.1.4), the z-directed current density, is a point in an infinite dimensional Lebesgue
square-integrable function space. A projection of the function from an infinite-dimensional space to a finite P -dimensional “approximation”, can be represented as a sum of P basis functions, such that
h =
P
X
p=1
αphp (6.1.5)
where αpis a scalar coefficient. The kernel of (6.1.1), l(z, z0) ∈ L2(S×S)1×1has corresponding
integral operator L : L2(S)1 → L2(S)1. Authors often drop the superscript “1”, the spatial
dimension of the geometry.
wm, we have P X p=1 αphwm, Lhpi = hwm, ei for m = 1, 2, . . . , N (6.1.6) where hx, yi = Z S
x(r)y(r) dS for SuperNEC
hx, yi defines the inner product of vectors x and y. For the Fredholm integral equation used
in the EFIE formulation, a wire of length L, divided into N segments of length ∆n, with
dirac-delta weighting functions applied at rm is given by
Z L/2 −L/2 P X p=1 αnphnp(rm, r0; sβ)Γ(rm, r0; sβ)dr0 = −Etan(rm; sβ) for m = 1, 2, . . . , N (6.1.7)
where Γ(·) is the kernel of the Fredholm equation and sβ the frequency at which the integral is formulated. Etan is the tangential electric field at the surface of the wire segment1. Sub-
sectional bases [3] have been used, implying that P basis functions are defined over each of the N segments with different coefficients αpn for each segment. Equation (6.1.5) becomes
hn= P
X
p=1
αpnhpn (6.1.8)
on segment n. Applying the weighting functions at the midpoints of each of the segments comprising the wire gives N equations in N unknowns. The SuperNEC basis functions are represented by 3 terms: a sine, cosine and constant term, viz. on segment n [4],
hn(d) = An+ Bnsin k(d − dn) + Cncos k(d − dn) and |d − dn| < ∆n/2 (6.1.9)
where d is a distance parameter along the wire axis at r, dn the distance parameter at
the centre of the n-th segment and k the wavenumber. The amplitudes of the 3 terms are related such that their sum satisfies physical conditions on the local behaviour of current and charge on segment ends; it allows the 3 coefficients in (6.1.9) to be reduced to 1 unknown [4]. Expressing (6.1.9) in terms of the single unknown as
hn(d) = Inh0(d, dn) = Inh0(r, r0) (6.1.10)
the integral equation of (6.1.7) is then
N X n=1 In Z ∆n h0(rm, r0)Γ(rm, r0)dr0 = −Etan(rm) for m = 1, 2, . . . , N (6.1.11) 1The explicit dependence of frequency, s
where the integrand is now completely known and for each ∆n, the integration of r0 is per-
formed over the length of the nth segment. This representation is also given as
N
X
n=1
ZmnIn= Vm for m = 1, 2, . . . , N (6.1.12)
Zmnis the system impedance matrix and Vm, the excitation vector. Clearly, Vm= −Etan(rm)
and Zmn =
R
∆nf 0(r
m, r0)Γ(rm, r0)dr0.
The derivations of equations (6.1.1) through (6.1.7) are based on a time-harmonic representa- tion of the electric field, current and Green’s function, implying that the equations are defined at a single frequency only.
The general derivations in Appendix B.7 use the complete representation. The frequency dependence of a segmented structure in SuperNEC is discussed in section 7.1.1, in the interest of completeness.
6.1.2 Lebesgue-Integrability and Sobolev Spaces for the Scattering Prob-