Investigation of the Asymptotic Properties of Active Impedance in Large Finite Array Antennas

Abstract: This paper presents an improved full-wave solver for finite array antennas, and uses this solver to examine asymptotic properties of active impedance for large regular arrays. The improved solver is based on a preconditioning scheme that has been adapted for use on more general finite geometries, and an improved data structure. These two improvements result in a fast and stable solver with a lower memory footprint. By investigating the active impedance of finite arrays it is found that, even for arrays with 1000 elements, asymptotic behavior may differ from infinite arrays. Tied to these properties, different predictors on how the active impedance of the center element in the array behaves are presented. The two best predictors work very well for the wideband arrays investigated, and may be used to decide when array approximations, such as unit cell methods, are appropriate, rather than general statements on array size.

• The paper develops a preconditioned conjugate-gradient MBT/MLFFT solver that is at least 8× faster than its predecessor and scales to 1,024-element arrays while preserving accuracy.
• The paper finds finite-array coupling decays approximately as r^-2.4 to r^-2.6 rather than the infinite-array r^-2 law, showing that unit-cell approximations may remain unreliable even for very large arrays.
• The paper introduces impedance- and scattering-based predictors that estimate center-element active-impedance error more practically than generic array-size rules across element types, spacings, and scan conditions.

Introduction and scope

The paper investigates a question of persistent practical importance in array antenna design: how large must a regular finite array be before the active impedance (and hence the Active Reflection Coefficient, ARC) of its center element behaves as in an infinite array, justifying unit-cell-based approximations (2608.17844). Rather than offering blanket rules such as "a $10\times10$ array suffices," the authors couple an improved full-wave Method of Moments (MoM) solver with quantitative error predictors based on passive inter-element impedance. Their central finding is that asymptotic behavior derived for infinite arrays applies only weakly to finite arrays: measured coupling decays as $r^{-2.4}$ to $r^{-2.6}$ rather than the theoretical $r^{-2}$, even for arrays approaching 1000 elements.

Improved MBT/MLFFT solver

The solver builds on the multilevel block-Toeplitz (MBT) structure that arises when a planar, fully populated array on a uniform rectangular grid is decomposed into interior elements ($Z_A$), margin-interior coupling ($Z_B$), and margin self-coupling ($Z_C$). Because $Z_A$ contains only $O(N_1 N_2)$ unique blocks instead of $O((N_1N_2)^2)$ elements, storage scales linearly with element count. Matrix-vector products are accelerated by embedding the MBT matrix in a multilevel block-circulant matrix evaluated via a multilevel FFT (MLFFT).

Two contributions improve on the authors' earlier GMRES-based solver. First, a new preconditioner for fully three-dimensional RWG-discretized geometries is constructed from the block-diagonal part of the normal-equation matrix $r^{-2.4}$0, extended to include the border blocks $r^{-2.4}$1 and $r^{-2.4}$2. Since $r^{-2.4}$3 is Hermitian positive definite, this enables solution by Preconditioned Conjugate Gradient (PCG) rather than GMRES; PCG's memory grows linearly in problem size only, independent of iteration count, unlike GMRES's Krylov subspace storage. Second, a commutativity result is proven: for symmetric MBT matrices, the MLFFT commutes with Hermitian conjugation,

$r^{-2.4}$4

so only one transformed first-column needs to be stored, roughly halving preconditioner memory. Additionally, a four-dimensional data arrangement aligns the MLFFT with optimized FFT implementations, avoiding costly row-wise data gathering.

Numerical performance. On bowtie arrays at 150 MHz, the PCG and "new M1" solvers are at least 8× faster than the previous M1 solver for arrays above 64 elements; the PCG solver reaches a $r^{-2.4}$5 (1024-element) array in about 38,000 s while maintaining stable scaling. An important accuracy caveat is documented: solving the normal equations directly with GMRES inflates the relative rms error in the passive impedance matrix by roughly a factor of 10 ($r^{-2.4}$6 versus $r^{-2.4}$7) at the same convergence tolerance, because squaring worsens conditioning. Applying the same spectrum via the split-preconditioned PCG form preserves accuracy at the $r^{-2.4}$8 level, comparable to the old scheme. The center-element relative error is $r^{-2.4}$9, well within the range relevant to convergence studies.

Asymptotic behavior of finite-array coupling

Infinite-array theory (Wu, Galindo, Amitay et al.) establishes that column-to-column coupling coefficients decay as $r^{-2.6}$0 for planar arrays and $r^{-2.6}$1 for parallel-plate waveguide openings, and asserts this decay is "universal for all phased arrays." The theory, however, is proven only for infinite structures; the paper tests whether it transfers to finite arrays using full-wave solutions of up to $r^{-2.6}$2 bowtie, $r^{-2.6}$3 body-of-revolution (BoR), and Vivaldi arrays.

The empirical results contradict the universality claim for finite arrays:

Array	Observed slope of $r^{-2.6}$4	Infinite-array prediction
Bowtie ($r^{-2.6}$5, 100 MHz)	$r^{-2.6}$6	$r^{-2.6}$7
BoR ($r^{-2.6}$8, 18 GHz)	$r^{-2.6}$9	$r^{-2}$0
Vivaldi	no clear convergence	—

Moreover, coupling coefficients predicted purely from the center element's ARC via the Fourier relation do not reproduce either slope or amplitude reliably: for the bowtie the ARC-derived prediction overestimates decay, while for the BoR the slope is closer but amplitude deviates substantially. The authors attribute part of the discrepancy to the mechanism underlying the infinite-array result—a discontinuity in the directional derivative of ARC at grazing lobes—which is absent or softened in finite arrays where energy radiates and scatters at the array edge rather than returning entirely through ports. The implication is direct: the asymptotic regime assumed by Floquet- and unit-cell-based methods may not be reached even at 1000 elements for wideband elements, so array-size thumb rules alone are unreliable indicators of approximation validity.

Error predictors for center-element active impedance

Two practical predictors of the relative deviation $r^{-2}$1 between the true center-element active impedance and its unit-cell approximation are developed. Both exploit that changes in active impedance upon adding border elements scale with the weakly coupled entries of the scattering matrix, via a Neumann-series splitting $r^{-2}$2 applied to the $r^{-2}$3-to-$r^{-2}$4 conversion.

Impedance-based predictor $r^{-2}$5: the ratio of the largest passive mutual impedance between the center element and nearest border element to the self-impedance. It requires only two inexpensive simulations per array size and tracks the simulated deviation $r^{-2}$6 closely for both bowtie and Vivaldi arrays across boresight and steered excitations, including spacings other than $r^{-2}$7.

Scattering-based predictor $r^{-2}$8: built from an idealized scattering matrix assuming monotonic $r^{-2}$9 off-center decay, requiring no simulations beyond a single reference. It performs comparably to $Z_A$0 for the wideband cases studied, though it shows larger isolated deviations for the Vivaldi array.

A naive triangle-inequality bound on the impedance change is shown to be uselessly conservative—predicting errors on the order of 100% where actual deviations lie between $Z_A$1 and $Z_A$2 for arrays exceeding a hundred elements—motivating the sharper estimators. The practical consequence is that designers can quantitatively assess when unit-cell approximations are acceptable for their specific element, frequency, and scan condition, replacing generic size heuristics such as the generalized $Z_A$3 rule, which the Vivaldi results at 12 GHz show to be inconsistent.

Limitations and open questions

Several limitations are acknowledged. The asymptotic $Z_A$4 law is not established for finite arrays and, per the authors' argument, likely cannot be proven in general; the observed steeper slopes ($Z_A$5 to $Z_A$6) mean the theoretical predictor $Z_A$7 rests on an assumption contradicted by the simulations it aims to summarize, yet it still predicts active-impedance error well—an unexplained asymmetry between the decay of $Z_A$8 and that of $Z_A$9. The Vivaldi array did not reach convergent behavior or a clear coupling slope within the sizes simulated, leaving its asymptotic characterization open. The predictors were validated on three element types over finite ground planes; behavior under other element classes, polarizations, scan volumes, or without ground planes remains untested. Finally, the anomaly where $Z_B$0 dips near 3–4λ from the edge is reported only anecdotally, without a physical explanation.

Conclusion

The paper delivers both a computational tool and a physics finding. Computationally, the block-diagonal normal-equation preconditioner, the MLFFT–Hermitian commutativity property, and a four-dimensional data layout yield a stable PCG-based full-wave solver that is at least 8× faster than its predecessor, scales to ~1000 elements, and retains numerical accuracy where a naive application would degrade it by an order of magnitude. Physically, full-wave evidence shows that infinite-array asymptotics for inter-element coupling do not hold in large finite wideband arrays, undermining size-based justifications of unit-cell methods. The proposed predictors $Z_B$1 and $Z_B$2 offer a constructive alternative: per-design, quantitative estimates of approximation error, applicable also to non-half-wavelength spacings.