Cross Antenna: An Experimental and Numerical Analysis
| Scientific Paper | |
|---|---|
| Title | Cross Antenna: An Experimental and Numerical Analysis |
| Read in full | Link to paper |
| Author(s) | Jose Luis Lopez-Bonilla |
| Keywords | Pocklington equation, Cross antenna, Method of analysis |
| Published | 2006 |
| Journal | Apeiron |
| Volume | 13 |
| Number | 2 |
| No. of pages | 14 |
| Pages | 274-287 |
Read the full paper here
Abstract
The cross antenna is a medium gain and circular polarization structure made of a conductor or strip line over a ground plane, following a cross contour of four or more branches. One end is feed by a generator and the other one is charged with a load impedance. This paper presents a theoretical and experimental analysis of an eight arms cross antenna, loaded with four different impedances. The theoretical study is made via the computational solution of Pocklington's equation applied to the structure; experimental results are obtained over an antenna of a 12 AWG wire over a ground plane, working in 3.2 GHz. We present radiation efficiency, gain, field pattern, and axial rate results.
Overview
This is an ordinary applied-electromagnetics engineering paper rather than a foundational one, and it should be read as such: it appeared in Apeiron but takes no position on any contested question in physics. The authors — J. Sosa-Pedroza, A. Lucas-Bravo and J. López-Bonilla of SEPI-ESIME-Zacatenco at the Instituto Politécnico Nacional in Mexico City — model and then build an eight-arm cross antenna operating at 3.2 GHz, and compare the computed radiation pattern against measurements taken in an anechoic chamber.
The motivation is that the cross structure is nearly unstudied. It belongs to the family of travelling-wave current-distribution antennas and was introduced by Antoine Roederer in an IEEE paper of 1990; the authors remark that "it seems that no other article has been written about the subject". They consider this a gap worth filling because the geometry is small, light, easy to reproduce, and delivers around 14 dB of gain with circular polarization — attractive for mobile communications or as a primary radiator feeding a parabolic reflector. The contribution is therefore twofold: an independent numerical treatment of the structure via the generalized Pocklington equation, and a set of measurements on physically constructed antennas made with four different terminating load impedances.
The analysis
The generalized Pocklington equation
Pocklington's equation relates the current distribution on a thin straight wire to the electric field impressed on its surface. The model assumes a perfect conductor of electrically small radius and replaces the actual current density by an equivalent filament parallel to the antenna axis, treating the remainder of the conductor as free space. For a bent geometry the authors use the generalized form, obtained from Maxwell's equations together with the magnetic and electric potentials in the Lorenz gauge, in which the local coordinate system attached to each small wire segment is rotated relative to the reference frame. That rotation enters through the dot product s · s′ of the tangential unit vectors of the two curves.
The geometric specification is the distinctive part of the method. The wire axis is given by a vector equation r(s) = x(s)i + y(s)j + z(s)k, and the equivalent current filament by r′(s′) = r(s′) + an(s′), where n is the unit normal to the axis and a the wire radius. Because the filament is a parallel curve offset by exactly a from the axis, source and observation points can never coincide, and the authors note the useful consequence: "there is no possibility of any singularities". This is what allows the double derivative ∂2/∂s∂s′ acting on the Green function to be expanded, converting the integro-differential equation into a pure integral equation with a kernel in R−5 containing terms in R2k2, jkR and the projections R·s, R·s′.
Method of Moments solution
The integral equation is solved by the Method of Moments using simple point matching: pulse functions as basis functions and Dirac deltas as weighting functions. This yields the standard matrix system [Zmn](In) = (Vm), with Zmn the impedance matrix formed by integrating the expanded kernel over segment n, Vm the voltage matrix formed from the impressed tangential field, and the currents recovered as (In) = [Zmn]−1(Vm).
The cross itself is broken into 23 straight segments, each described parametrically by r′j(s) = rj + s t̂, where t̂ is the unit vector of the current propagation direction and rn points to the start of each segment. The reference vector is (1.2A + B)i + 0.5Aj, with A = 0.136λ the arm width and B = 0.543λ the arm length, and s0 = 0 taken as the feed point.
Dimensions
The geometry follows Roederer's prescription, scaled to the effective wavelength λe:
| Parameter | Value |
|---|---|
| Arm length | 0.543 λe |
| Arm width | 0.136 λe |
| Cross diameter | 1.42 λe |
| Wire diameter | 0.02 λe |
| Height over ground plane | 0.0625 to 0.1 λe |
Numerical results
The computed current distribution falls off exponentially with position along the segments, which the authors take as confirmation that the structure carries a travelling-wave rather than a standing-wave current. Superimposed undulations are attributed to reflections at the antenna corners. The current distribution is then used to compute radiation efficiency and gain as functions of the terminating load. The best numerical results come from the open-circuit and short-circuit terminations, both giving 15 dB of gain.
Experimental results
Several antennas were built, in both wire and strip-line form. The wire versions use Teflon standoffs over the ground plane so that the height can be varied; measurements are reported at the tabulated height for direct comparison with the simulation. Impedance and VSWR were measured with an HP8510 network analyzer, and gain and field pattern with an Agilent E8254A signal generator and E4407B spectrum analyzer, in an anechoic chamber.
For the short-circuit termination the input impedance near 3.2 GHz is Z = 48 − j40 Ω, with VSWR reported across 2.9 to 3.5 GHz. Horizontal (φ = 0°) and vertical (φ = 90°) polarization patterns together give an axial ratio of about 1.5 dB at maximum gain around 3.2 GHz. Maximum measured gain by load impedance:
| Polarization | Short | Open | 50 Ω | Z0 |
|---|---|---|---|---|
| Horizontal | 13.5 | 14 | 13 | 13 |
| Vertical | 15 | 15 | 13.5 | 13.5 |
The measured and computed field patterns are reported as nearly coincident, the main divergence being near 60°; following Roederer, the authors attribute the peak at that angle to the feed end. Their conclusion is simply that there is "a very good coincidence between the digital simulation and the experiment".
Assessment
The strength of this paper is that it does the unglamorous thing properly. The generalized Pocklington formulation for arbitrarily bent thin wires, developed by the same group in earlier work, is a genuinely clean piece of technique: offsetting the current filament by exactly the wire radius makes the kernel non-singular by construction rather than by numerical regularization, which removes the most common source of error in Method-of-Moments wire codes. Applying it to a structure that has attracted essentially no literature since 1990, and then validating against measurements taken with named commercial instruments in an anechoic chamber, is exactly the right order of operations. The reported agreement between the simulated and measured patterns, with the one discrepancy identified and physically attributed to the feed end, is the kind of result that can be checked by anyone with the same equipment.
The limitations are those of a short conference-style report rather than errors. No error bars, tolerances or repeatability spread are given, even though the authors state that several antennas were constructed specifically "to compare its repeatability" — the comparison itself is never reported. The claim of "very good coincidence" between simulation and experiment is asserted from a single overlaid figure rather than quantified by any residual or correlation measure. Nor is the number of segments justified: 23 straight segments at 3.2 GHz is a coarse discretization for a structure 1.42 λ across, and no convergence study is offered to show that the computed gain has stabilized with respect to segment count.
There is also a mild inconsistency in the numbers that the paper does not address. The numerical section reports 15 dB gain for both open and short terminations, while the measured table gives 15 dB only for vertical polarization and 13.5–14 dB for horizontal; the introduction meanwhile advertises "14 dB gain in a very small antenna". Since the antenna is sold on its circular polarization, a 1–1.5 dB gap between the two polarization gains is precisely the quantity of interest — it is what the 1.5 dB axial ratio expresses — and it would have been worth stating whether the simulation reproduces that asymmetry or only the higher of the two figures. The impedance result Z = 48 − j40 Ω likewise indicates a substantially reactive input that is not commented on, though it bears directly on the practicality of feeding the structure from a 50 Ω line.
None of this bears on any dispute in fundamental physics, and the paper makes no attempt to make it do so. Within the mainstream framework it assumes, the work appears sound on its own terms; it is a competent piece of computational and experimental antenna engineering whose chief value is that it independently reproduces and extends a result that had stood unexamined for sixteen years.