Analysis of common-mode effects in a dual-polarized planar connected-array antenna

Authors


Abstract

[1] An analysis is presented of the efficiency of a dual-polarized planar connected-array antenna, with particular reference to possible use in wideband focal-plane-array applications. The modeling approach is described, and the results of different numerical techniques are compared. The analysis includes the effects of transmission lines used to load the array at a groundplane. Different types of loading are compared including optimum single-ended loading and optimum differential-mode loading with short-circuit and open-circuit common-mode terminations. The results indicate that high efficiency over useful bandwidths should be possible. In the case of differential-mode loading, the efficiency may be reduced at certain frequencies due to resonances of common-mode currents on the transmission lines. The current distributions on the array are analysed, and observations are made which may be useful for modified designs.

1. Introduction

[2] Considerable interest currently exists in the development of antenna technologies for future radiotelescopes such as the Square Kilometre Array (SKA) [Hall, 2004]. The SKA requires large collecting area for high sensitivity. In addition, high survey speed is desired, particularly over the frequency range from around 0.7 to 1.8 GHz. One approach currently being pursed is the use of reflector systems with focal-plane arrays (FPAs), thus forming antenna elements with substantial effective area and substantial field of view. A contiguous, fully-sampled field of view is highly desirable [Fisher and Bradley, 2000], and one approach to the wideband problem is based on FPAs of tapered-slot or Vivaldi elements [Ivashina et al., 2004; Veidt et al., 2007].

[3] There exist other concepts for wideband arrays [Lee et al., 2006a] that may be considered for FPA applications. Of particular interest are planar and low-profile arrays that may have advantages in terms of cost or ease of integration with low-noise receivers. Such arrays may also be easier to model than Vivaldi arrays, and provide another point of view that may assist in understanding the limitations of dense arrays in such applications. One possibility is the capacitor-coupled dipole concept developed by Munk [2003]. Another is the connected-array concept, in either the dipole form proposed by Hansen [2004] or the slot form developed by Lee et al. [2006b]. As discussed by Lee et al. [2006a], connected arrays offer wide bandwidth by approximating Wheeler's continuous current sheet model. However the planar connected-array structures developed so far are single-polarization antennas, whereas dual polarization is required in the FPA applications of current interest.

[4] In this paper, we analyze the efficiency of a planar dual-polarized connected array in relation to wideband FPA applications. The array consists of conducting patches in a checkerboard form, which can be viewed as connected dipoles or connected slots. Alone in free space a large array of such patches is self-complementary and therefore well matched to frequency-independent loads connected between the patch corners [Mushiake, 1992]. Previous work on similar structures includes analysis of TEM horn arrays [McGrath and Baum, 1999], groundplane effects [Dardenne and Craeye, 2003] and bandwidth-enhancement by dielectric loading [Gustafsson, 2006]. Here we extend this work by analyzing a structure that includes transmission lines used to divert the array current to loading circuits conveniently located at the groundplane [Hay et al., 2007]. The power transferred into the loads from focal-region and plane wave fields is analyzed and simple forms of loading are compared, including optimum single-ended loading and optimum differential-mode loading with short- and open-circuit common-mode terminations. With differential-mode loading, common-mode currents on the transmission lines are shown to reduce efficiency at certain frequencies. We analyze the distributions of the corresponding currents on the array and the results give some insight into the relationship between the resonances and the array geometry.

2. Array Loading Analysis

[5] Figure 1 illustrates the prototype planar dual-polarized connected-array antenna. The structure consists in a planar array of conducting patches on one side of a groundplane, with neighboring patch corners connected via two-wire transmission lines, passing through small apertures in the grounplane where load or source circuits can be applied. Such transmission lines exist between all neighboring patch corners except those of neighboring patches at the edge of the array. The characteristic impedance of the transmission lines is 377Ω. This value was selected as a starting point for analysis of the array in the FPA application, and was suggested by the self-complementary property of a large array of such patches and the special case of plane wave incidence from the direction normal to the plane of the array [Mushiake, 1992; Dardenne and Craeye, 2003]. In describing results in the following paragraphs, we refer to the z axis which is orthogonal to the x and y axes shown in Figure 1 and directed from the groundplane to the patches.

Figure 1.

Geometry of the prototype planar dual-polarized connected-array antenna. (a) Array groundplane and coplanar patches and (b) cylindrical feed conductors and grounplane apertures. The element spacing is 90 mm, the patches are 65 mm from the groundplane, g = 10 mm, a = 0.45 mm and b = 1.5 mm.

[6] The receive array would be used in conjunction with a digital beamformer, which produces signals of the form

equation image

where VL and W are column vectors of load voltage phasors and complex weighting coefficients respectively, and the superscript denotes the transpose operation. In principle, the loading circuits can take various forms including optimal matching networks [Warnick and Jensen, 2007]. Here we consider two simple forms that can be represented by the equivalent circuits shown in Figure 2. In the case referred to as single-ended loading, the beamformed signal (1) takes the form

equation image

where i± are the two conductors of transmission-line i and 2N is the total number of beamformed signals. In the case referred to as differential-mode loading, beams are formed from only differential-mode load voltages, ie

equation image

Analysis and comparison of the performance obtainable with these two types of loading is important because for a given array geometry the differential-mode loading involves half the number of beamformer inputs and hence substantially less beamformer cost.

Figure 2.

Equivalent circuits of (a) single-ended and (b) differential-mode loading applied to each pair i± of transmission-line conductors feeding the array.

[7] The performance of the loaded array has been analyzed by numerical techniques. A modular approach has been adopted where the array and the loading circuits may be analyzed separately and then the results combined by enforcing the necessary continuity of voltages and currents at the groundplane. Thus the array analysis produces a mutual admittance matrix YA that relates the vector VA of voltages across the groundplane apertures and the corresponding vector IA of feed-line currents at the groundplane according to the matrix equation IA = YAVA. The load-circuit analysis produces an analogous representation IL = YLVL where YL is the admittance presented by the loads at the groundplane. The results are combined in the solution for the voltages

equation image

where IS is a vector of equivalent currents representing the field incident on either the transmit or receive array and is equal to the feed-line currents at the groundplane when the groundplane apertures are short-circuited. In the case of single-end loading the matrix YL is diagonal and for simplicity we take all such load admittances to be the same. To deal with the differential-loading case, it is convenient to resolve all currents and voltages into differential- and common-mode components; eg,

equation image

where i± denote the two transmission-line conductors forming transmission line i. The load voltages are then readily found to be given by

equation image

where YLD = (ZLD)−1, YLC = (ZLC)−1 are diagonal matrices, the array admittance matrix elements are given by

equation image

and the differential- and common-mode equivalent currents are given by

equation image

The transformation (3) allows the differential-mode loading to be represented in terms of an equivalent circuit of the same general form as that used for the single-ended case.

[8] An important parameter of the array performance in practice is the efficiency of power transfer into the beamformed received signals (1). This can be defined as

equation image

where P2 is the available power of the source of the field incident on the array and

equation image

where GL = ℜ{YL} and the overbar denotes the complex conjugate. The reciprocity theorem implies that the efficiency remains unchanged when the roles of the transmitter and the receiver are reversed and the array is excited by current sources IS = W in parallel with the array loads, in which case P1 becomes the available power. If the field incident on the receive antenna is a plane wave and P2 in (4) is replaced by the power density of this field then (4) becomes Ae the effective area of the receive antenna. The effective area is maximized when the beamformer coefficients satisfy the conjugate-match condition

equation image

giving the maximum effective area

equation image

where PL = 1/2equation imageLtGLVL is the total power absorbed by the loads of the beamformed signals. In practice, other beamforming criteria related to radiation patterns, noise and interference are also important but it will not be necessary to address these in this paper.

[9] Figure 3 compares computed results for the maximum efficiency of the array and a rotationally symmetric paraboloidal reflector. The groundplane of the array is in the focal plane of the reflector with the grounplane's center at the reflector's focal point. The reflector is illuminated by a plane wave propagating parallel to the focal axis and with its electric field vector polarized in the y = 0 plane. The field incident on the array is taken to be the corresponding scattered field of the reflector, as evaluated by physical optics and illustrated in Figure 4. In practice, this field would be perturbed by array and strut blockage but these effects are small and the approximation provides a convenient reference for comparing different arrays and array loading. The efficiency of the array and reflector is defined as Ae,max/Ap, where Ae,max is the maximum effective area computed from (6) and Ap = π(D/2)2 where D is the diameter of the reflector. The array has been analyzed by the time-domain finite-integration method (FIM) (CST Microwave Studio) and also the frequency-domain method of moments (MoM) [Rao et al., 1982; Makarov, 2002]. In the latter case, the cylindrical feed conductors illustrated in Figure 1b have been approximated by planar strips of width w = 4a, with infinitesimal-gap apertures between these strips and the groundplane. Despite these approximations and the different computational approaches of the techniques, good agreement is obtained for the maximum efficiency of the array, giving some confidence in the numerical techniques. Initial experimental work on the array has included measurement of the mutual coupling between various pairs of load ports where good agreement with the computed results has also been obtained. The results in Figure 3 apply to the case of differential-mode loading of the array, with the open-circuit common-mode load impedance ZLC = ∞, and with the differential-mode load impedance ZLD shown in Figure 3b. The impedance ZLD has been obtained by applying computer search techniques to maximize the efficiency, and due to the small differences in efficiency as computed from the FIM and MoM analyses, ZLD has been taken as the average of the maxima from both analyses.

Figure 3.

(a) Calculated maximum efficiency and (b) optimum differential load impedance of array and paraboloidal reflector of 14 m diameter and 0.4 focal-length-to-diameter ratio.

Figure 4.

Focal-plane electric field of paraboloidal reflector of 14 m diameter and 0.4 focal-length-to-diameter ratio at 1.2 GHz.

[10] The computational burden of the array analysis has been greatly reduced by applying the characteristic-basis-function method of moments (CBFMoM) [Prakash and Mittra, 2003]. One very efficient version of the technique uses an image Green's function corresponding to an infinite groundplane approximation and, as shown in Figure 3, produces good approximations for efficiency in the focal-plane-array application. A second version of the technique represents the current on the finite groundplane of the array, and gives radiation patterns in close agreement with the complete MoM results over the entire radiation sphere [Hay et al., 2008]. These techniques have enabled more detailed study of the surface currents and radiation patterns of the prototype array.

[11] Figure 5 compares the computed maximum efficiencies of the array and reflector with different types of array loading. The differential loading with the open-circuit common-mode termination (ZLC = ∞) produces greater efficiency than with the short-circuit common-mode termination (ZLC = 0) over much of the frequency range. Moreover the former results closely approach those obtained with the more complex single-ended loading configuration (ZL = ZL,opt). However, around the frequencies of 0.9 and 1.7 GHz, large reductions in efficiency occur rapidly in the open-common case. In this case, as shown in Figure 6, similar frequency-sensitivity exists around these frequencies in the computed results for the total effective area of the array without the reflector. The desire for frequency ranges in excess of an octave and the potential cost savings of differential loading warrant further investigation of the efficiency reductions.

Figure 5.

Calculated efficiency of array and reflector with different types of array loading and reflector focal-length-to-diameter ratios (f/D) of (a) 0.4 and (b) 0.6.

Figure 6.

Total effective area (dB m2) of the array without the reflector. The effective area is plotted as a function of frequency and angle between the propagation direction of the incident plane wave and the negative-z axis. At positive angles the plane wave's propagation direction and electric field vector are contained in the y = 0 plane whereas at negative angles the propagation direction and magnetic field vector of the plane wave are contained in the x = 0 plane.

3. Effects and Structure of Common-Mode Currents

[12] The literature on unconnected arrays contains references [Bayard et al., 1993; Hansen, 1998; Munk, 2003] that associate common-mode currents on feeding transmission lines with deleterious effects, such as reduction in power transfer in certain beam directions, a phenomenon often referred to as scan blindness. This led to the hypothesis that such currents may also be associated with the computed efficiency reductions in the dual-polarized connected array.

[13] The reciprocity theorem allows us to study the array efficiency conveniently from the point of view of the transmitting array, excited with the conjugated-matched current sources (5). The MoM analysis of the array provides a solution for the current density on all surfaces of the array and in particular along the length of each transmission-line conductor. The differential- and common-mode currents on transmission line i at distance z from the groundplane may be defined analogously to their counterparts at the groundplane, or

equation image

where the components are the currents on the two conductors i± forming the transmission line.

[14] Figure 7 shows the computed ratio

equation image

of the sum-of-squares of common-mode and differential-mode currents on the array, when the array is excited by current sources that are conjugate-matched to the load voltages produced by the paraboloidal reflector, where the corresponding efficiency is given in Figure 3. These results show correlation between the common-mode current and the reduced array efficiencies. Figure 8 shows the same quantity in the cases where the array is conjugated-matched directly to plane waves where the corresponding effective areas are shown in Figure 6. The results in Figures 6 and 8 show that significant common-mode current and efficiency reduction occurs at 0.9 GHz even in the case where the plane wave propagates in the direction normal to the groundplane. In this case, the common-mode currents are excited by the edges of the array and the results highlight the importance of including such effects in the array analysis.

Figure 7.

Ratio of array common-mode and differential-mode currents when the array is excited by current sources conjugate-matched to the receive-mode voltages produced the paraboloid focal-region field (f/D = 0.4).

Figure 8.

Ratio (dB) of array common-mode and differential-mode currents when the array is transmitting and conjugate-matched to incident plane waves. The current ratio is plotted as a function of frequency and the angle between the propagation direction of the plane wave and negative-z axis. At positive angles, the plane wave's propagation direction and electric field vector are contained in the y = 0 plane (E-plane) whereas at negative angles the plane wave's propagation direction and magnetic field vector are contained in x = 0 plane (H-plane).

[15] The common-mode currents on the transmission lines will radiate energy and contribute to the radiation patterns of the array. Figure 9 shows the computed beamformed radiation patterns of the array when the beamformer coefficients are set for conjugate match to the paraboloidal reflector. The resonances of the common-mode current are seen to be associated with rapid variations in the radiation pattern with respect to frequency. The power-transfer characteristics of an array may be due to both impedance and radiation-pattern effects and the results in Figure 9 indicate significant radiation-pattern effects in this case.

Figure 9.

Gain radiation pattern (dBi) of the array when conjugate-matched to the focal-region field of the paraboloidal reflector (f/D = 0.4). The gain is plotted as a function of frequency and angle between the radiation-pattern direction and the z axis. At positive angles, the radiation-pattern direction is in the y = 0 plane (E-plane) and at negative angles the radiation-pattern direction is in the x = 0 plane (H-plane). The gain patterns are symmetric about the z axis in both the x = 0 and y = 0 planes.

[16] In the MoM analysis the common-mode current on the transmission lines can be suppressed by defining additional terminals or ports on the transmission lines and applying high-impedance loading to the common-mode components of the current. As shown in Figure 10, this suppression increases the computed efficiency of the array and reflector in the regions of the notches, confirming the significance of the common-mode current. However, at low frequencies where the array is comparable in size to the wavelength, the existence of the common-mode current appears to have some positive impact on efficiency.

Figure 10.

Array and reflector efficiency with and without suppression of the common-mode current (f/D = 0.4).

[17] Some insight into the resonant frequencies of the common-mode currents is obtained by examining the structure of the current distributions on the patches of the array. Figure 11 shows the magnitude of the patch currents and the direction of current flow on the transmission lines when the transmit array is conjugate-matched to the paraboloid. The currents contain some differential-mode component but also large common-mode components, which are terminated by the open-circuit condition at the groundplane. At 0.9 GHz, the common-mode currents flow in opposite directions on the x- and y-polarized transmission lines of each patch, and have high concentrations along the edges of the patch. At 1.75 GHz, the resonant common-mode currents flow in the same direction on the x- and y-polarized transmission lines associated with each patch, and the current has a standing-wave pattern on the patch, with nodes midway along the patch edges. As noted in the efficiency results with focal-region and plane wave incidence, the resonant frequencies are not highly dependant on the form of the incident field, and this allows the common-mode current distributions to be visualized more readily by exciting the array with uniform common-mode voltages at the groundplane, where similar resonant frequencies of 0.9 and 1.71 GHz are seen. The lower resonant frequency is roughly approximated by the path length along the x- and y-polarized feed conductors of each patch and the joining patch edge, being half the free-space wavelength. Numerical experiments with capacitive common-mode loading at the groundplane indicate that any additional capacitance produced by loading circuits will tend to lower the resonant frequencies.

Figure 11.

Patch currents (dB) when the array is excited by current sources conjugate matched to the receive-mode voltages produced by the paraboloid (f/D = 0.4) at (a) 0.9 GHz and (b) 1.75 GHz. The arrows represent the magnitude and direction of the transmission line currents at the patch corners. Arrows pointing up or left represent current flowing toward the groundplane whereas arrows pointing down or right represent current flowing from the groundplane to the patch at an arbitrarily selected instant in the time-harmonic cycle.

4. Conclusion

[18] The paper has described the results of numerical modeling of a planar dual-polarized connected checkerboard array including the effects of transmission lines used to feed or load the array from a groundplane. The power-transfer characteristics into beamformed signals with optimized single-ended and differential-mode loading at the goundplane have been analyzed and the results indicate that high efficiency can be obtained over useful bandwidths for some FPA applications of current interest. In the case of differential-mode loading, the efficiency may be reduced at certain frequencies due to resonances of common-mode currents on the transmission lines. The structure of these currents has been analyzed and the results provide some insight into the resonances, which may be useful in modifying the design.

Acknowledgments

[19] The authors thank C. J. Granet for generating the FIM data on the array and J. S. Kot for developments with the MoM software.

Ancillary