Pocklington Equation Method Versus Curved Segments Technique for the Numerical Study of Circular Antennas

Scientific Paper
Title	Pocklington Equation Method Versus Curved Segments Technique for the Numerical Study of Circular Antennas
Read in full	Link to paper
Author(s)	Jose Luis Lopez-Bonilla
Keywords	Curved segments, vector potential, Pocklington equation, method of moments.
Published	2006
Journal	Apeiron
Volume	13
Number	2
No. of pages	14

Read the full paper here

Abstract

We present a mathematical model applying the general Pocklington equation to arbitrary shaped thin wire antennas. In order to simplify the antenna analysis this approach uses the point matching technique and a simplified kernel form. By means of this technique it is possible to increase the Method of Moments solution convergence and reduce computational time and effort. To exemplity this the procedure is applied to the well-known circular loop antenna. The obtained results are compared with those of Champagne method, which uses quadratic segments, in order to get a numerical solution for the Electric Field Integral Equation.

Overview

This paper by J. Sosa-Pedroza, V. Barrera-Figueroa and J. López-Bonilla of the Instituto Politécnico Nacional in Mexico City is a piece of straightforward computational electromagnetics rather than a challenge to established physics. It appeared in Apeiron 13(2) in April 2006. The question it addresses is a practical one: when a thin-wire antenna is bent — the circular loop being the canonical case — what is the cheapest numerical route to its current distribution?

The authors' answer is that the general Pocklington integral equation, written for an arbitrarily bent wire and solved by the method of moments with point matching, gives results comparable to the more elaborate curved-segment technique of Champagne, Williams and Wilton, while requiring far less machinery. Their method uses pulse basis functions, Dirac delta weight functions and Simpson's rule; Champagne's uses piecewise-linear expansion functions, the Galerkin procedure and Gaussian quadrature. The claim is that the simpler apparatus reproduces the same current distribution to within a uniform 5 per cent, with fewer integrations.

The argument

The general Pocklington equation

Pocklington's classical equation relates the current distribution on a straight perfectly conducting wire to the tangential impressed electric field on its surface, on the assumption that the wire radius is small compared with the wavelength and that the surface current density can be replaced by a filament parallel to the antenna axis. The generalisation to bent wires, obtained formally from Maxwell's equations via the magnetic and electric potentials and the Lorenz gauge, introduces the dot product s·s′ between tangential unit vectors at the observation and source points, which encodes the rotation of the local coordinate frame along the wire.

The geometry is fixed by two parallel space curves: r(s) = x(s)i + y(s)j + z(s)k for the wire axis, and r′(s′) = r(s′) + an(s′) for the equivalent current filament, where a is the wire radius and n the unit normal. The tangential unit vectors follow by differentiation with respect to arc length.

Kernel simplification and the removal of singularities

The paper's technical contribution is the expansion of the mixed operator ∂²/∂s∂s′ acting on the Green's function e^−jkR/4πR. Carrying the differentiation through analytically converts the integro-differential equation into a pure integral equation whose kernel involves the combination (k²R² − 1 − jkR)s·s′ + (3 + 3jkR − k²R²)(s·R)(s′·R) divided by R⁵, with R the distance between observation and source points.

The authors emphasise a consequence that matters computationally: because the filament curve lies a distance a from the axis curve, observation and source points never coincide, "thus, there are no singularities". This is the pivot of the whole comparison. Champagne's formulation, which works with the full surface integral and admits any wire radius, has singular kernels whenever the observation point falls on the source segment, and must treat those singularities separately by approximation and then add the non-singular remainder — a step the authors describe as complicating the solution. Specialising the general equation to a straight wire recovers the familiar textbook form.

Solution by the method of moments

Equation (5) is discretised with pulse functions i_n(s′), equal to unity on the nth segment and zero elsewhere, and delta functions w_m(s) = δ(s − s_m) as testing functions — that is, point matching. This yields the standard matrix system [Z_mn](I_n) = (V_m) with impedance matrix, voltage matrix and current matrix (I_n) = [Z_mn]⁻¹(V_m).

The circular loop test case

The loop is parameterised by r(s) = Acos(s/A)i + Asin(s/A)j, fed by a unit delta-gap source at φ = 0°, with f = 3 GHz, N = 17 segments, loop radius A = 0.0637λ and wire radius a = 0.0027λ. Champagne's published comparison uses 8 quadratic segments of three points each — also 17 points — with four-point Gaussian quadrature. Real and imaginary parts of the current distribution are plotted against each other; the authors report "an uniform error of 5% in the whole current distribution", which "for practical uses, it will be neglected".

Computational comparison

Both codes were written in Visual C++ 6.0, with complex numbers stored as two doubles, sixteen bytes per matrix element. Total storage is 16(N² + 2N) bytes: 1280 for N = 8, and 5168 for N = 17. Champagne's method, because of the Galerkin testing and the finite-radius surface integrals, requires of order 10N² + 2N integrations — 504 in the case reported — while the Pocklington point-matching scheme needs only N² + N = 306. Run at equal segment counts (N = 17), Champagne's would require 2924. From this the authors conclude that their method "is more rapid".

Assessment

The methodological point is a fair one and is made cleanly. Moving the double arc-length differentiation inside the kernel analytically, and exploiting the offset between axis curve and filament curve so that R never vanishes, genuinely removes the singularity handling that dominates the bookkeeping in surface-integral formulations. The observation that most practical antennas are thin compared with the operating wavelength — so that Champagne's generality with respect to wire radius buys little in practice — is reasonable engineering judgement. And the vector parameterisation of the wire axis is written so that straight and curved antennas fall out of the same expression, which is a real convenience.

The weaknesses are those of a short conference-style comparison. The 5 per cent agreement is asserted from a figure rather than quantified: no norm, no per-point table, and no statement of which method is taken as the reference. Since neither is exact, "uniform error of 5%" describes a discrepancy, not an accuracy, and the paper offers no independent benchmark — no measurement, no analytic loop solution, no highly converged reference computation — against which either could be judged. Nor is any convergence study given: the abstract promises to "increase the Method of Moments solution convergence", but only a single value N = 17 is ever run, so the claim is untested. Point matching with pulse bases is well known to converge more slowly than Galerkin with linear bases, which is precisely the trade the paper is making; whether the 5 per cent shrinks or persists as N grows is the question a reader would most want answered.

The cost accounting is also loose. The byte totals are correct as arithmetic — 16(64 + 16) = 1280 and 16(289 + 34) = 5168 — but are then reported as "1.25 MBytes" and "5.04 MBytes" when they are kilobytes, an error of three orders of magnitude that would have been caught on any reading. The integration counts are inconsistent with the memory figures: the memory estimate for Champagne is computed with N = 8 while the stated 504 integrations follow from 10N² + 2N only at N = 7. More fundamentally, counting integrations is a poor proxy for run time when the two schemes evaluate integrands of very different cost, and no actual timings are reported despite both codes having been written and run. The conclusion that the Pocklington route "is more rapid" is therefore plausible but not demonstrated by the evidence given.

None of this touches foundational physics; the paper is an applied-electromagnetics note that stands or falls on its numerics, and within those limits it is honest about what it did.

See also

• Jose Luis Lopez-Bonilla
• Apeiron
• Maxwell's Equations
• Electrodynamics
• Electromagnetism
• Electric Current