Shape and excitation optimization for conformal array antennas

Authors


Abstract

[1] We present a procedure for the optimization of array antennas that conform to circular cylinders in two space dimensions. For a phase-mode excitation of the antenna, a large number of antenna element designs are compared in terms of two objectives: (1) the active reflection coefficient and (2) the far-field distortion with respect to the excited phase-mode. This analysis yields Pareto fronts with respect to the two objectives and appropriate designs are selected based on the information collected for the useful phase modes given a specific frequency band. This antenna element design is used for the pattern synthesis, where we optimize a linear combination of (1) requirements on the directive gain and (2) similar requirements on the power reflected from the array antenna aperture.

1. Introduction

[2] Conformal antennas are desirable when integration of the antenna with a platform is required, e.g. antennas integrated in the skin of an aircraft reduce the aerodynamic drag and lower the fuel consumption. Array antennas can conform to a given platform and they offer the possibility of electronic beam steering and pattern synthesis. Important examples can be found in mobile communications where the base station antenna is required to have 360° coverage, which can be achieved by an array antenna that conforms to a circular cylinder. (Frequency-invariant pattern synthesis is also feasible for circular array antennas [Steyskal, 1989].)

[3] The design and usage of conformal array antennas involve several aspects and, here, we mention (1) the design of the array antenna's geometry and (2) the selection of excitation for the individual antenna elements. Given a supporting structure for the array antenna, the number of antenna elements must be chosen together with their positions and orientations, see Josefsson and Persson [2006] for a number of studies and further references to the literature. In practice, this process is subject to a number of additional constraints such as the feeding of the antenna elements and the mechanical strength of the supporting structure. Another aspect of the geometrical design relates to the actual shape (and materials) used for the individual antenna elements. For planar arrays, it is common to design the antenna element when used in an infinite periodic array [Chio and Schaubert, 2000], where the design must fulfill requirements on the active reflection coefficient and beam steering capabilities for a given frequency interval. Planar array antennas for operation over broad frequency-bands can be constructed by Vivaldi antenna elements [Chio and Schaubert, 2000] and similar solutions [Holter, 2007].

[4] In this article, we conduct a study of array antennas conformal to circular cylinders. The antenna elements are horn shaped and excited by the fundamental mode of a feeding parallel plate waveguide. We solve the electromagnetic field problem to high accuracy with a stable hybrid [Rylander and Bondeson, 2002] between the finite-difference time-domain (FDTD) scheme [Taflove and Hagness, 2005] and the finite element method (FEM) [Jin, 2002]. Given one time-domain computation, we compute the embedded element pattern and the full scattering matrix for a large frequency-interval. We perform an extensive parameter study with respect to the shape of the horn antenna elements and, for phase-mode excitations, we characterize the results in terms of (1) the active reflection coefficient and (2) the distortion of the far-field with respect to the excited phase-mode. This allows us to choose a design of the antenna element that is optimal in the Pareto sense [Tamaki et al., 1996] (provided the sampled parameter space that we use) for a given frequency band and a given range of phase modes. Given a good design of the antenna element, we perform pattern synthesis by gradient based optimization of the phase-mode amplitudes such that a linear combination of two goals is minimized: (1) the directive gain outside an upper and lower mask; and (2) the power reflected at the array antenna aperture. This approach allows us to only use the phase-mode amplitudes that radiate well (given the frequency at hand) as design variables in the pattern synthesis. Alternatively, all phase-mode amplitudes can be included together with an additional term in the goal function that penalizes the reflected power, which typically stems from the upper range of phase modes.

[5] This article contains three substantial contributions: (1) an attempt to co-optimize the geometry and excitation to achieve a desired radiation pattern and a good antenna efficiency; (2) gradient based pattern synthesis that incorporates requirements on the efficiency of the array antenna; and (3) a design process for conformal array antennas that is motivated by means of phase-mode analysis for uniform array antennas conformal to circular cylinders. In addition, we consider a range of questions for the specific case of two-dimensional array antennas that conform to a circular cylinder: (1) extensive parameter studies by means of accurate and unbiased computational models for the mutual coupling and the embedded element pattern; (2) Pareto optimal geometries and excitations; and (3) comparisons with conventional pattern synthesis techniques such as the Dolph-Chebyshev synthesis [Dolph, 1946; Lau and Leung, 2000; Vescovo, 1999].

2. Circular Array

[6] As a relatively representative model problem, we study a two-dimensional circular array antenna with horn-shaped antenna elements. This particular situation does not feature all design challenges that occur for general conformal array antennas but we consider it to be sufficiently good for testing purposes of various optimization techniques. An example of such an array antenna is shown in Figure 1, where integers are used to index the individual antenna elements. (This type of antenna can be considered a two-dimensional version of the dual-polarized broadband array antenna with body-of-revolution elements [Holter, 2007].)

Figure 1.

The geometry of the array antenna that conforms to a cylinder of circular cross section and radius a = 0.6 m. The horn shaped antenna elements are fed by parallel plate waveguides excited by the fundamental TEM mode.

[7] Given the cross section of the supporting circular cylinder perfect electric conductor (PEC) core of radius a = 0.6 m, we remove the parts of this cylinder that correspond to the horn antenna elements and part of their feeding waveguides. Figure 2 shows the geometry of the reference horn. It is described by the five points P1 = (0, wh/2), P2 = (x2, y2), P3 = (x3, ww/2), P4 = (lh, ww/2) and P5 = (lh + lw, ww/2). Here, we denote the horn's aperture width by wh and its length by lh. Further, the points P2 = (x2, y2) and P3 = (x3, ww/2) are used to control the shape of the horn described by the rational Bézier curve [Farin, 1992] of degree three that ends at the point P4 = (lh, ww/2), which connects the horn to its feeding waveguide of width ww and length lw. We use ww = 0.02 m throughout this article. Given the boundary curve for the reference horn in the region y > 0, we simply use its mirror image with respect to the x-axis to define the corresponding lower part in the region y < 0, which yields a symmetrical reference horn with respect to the x-axis. The reference horn is rotated and translated to fit the supporting cylindrical structure considered. In the following, we consider the transverse electric (TE) case and let the parallel plate waveguides be excited by the fundamental transverse electromagnetic (TEM) mode.

Figure 2.

The geometry of the reference horn antenna element described by the five points P1 = (0, wh/2), P2 = (x2, y2), P3 = (x3, ww/2), P4 = (lh, ww/2) and P5 = (lh + lw, ww/2). Here, the horn's aperture width is wh and its length is lh. The feeding waveguide has the width ww and length lw.

2.1. Phase Mode Analysis

[8] The embedded element pattern equation image(equation image, ω) and the elements Spq = Vp/Vq+ of the scattering matrix yield a rather complete electromagnetic characterization of a radiating array antenna. Here, Vq+ = wwEq+ is the voltage of the incident wave on port q and Vp = wwEp is the corresponding reflected voltage on port p. However, it is also instructive to use phase-mode excitations for arrays that are conformal to circular cylinders. For a single phase-mode excitation with the mode number m, the incident voltages Vq+ are given by

equation image

where equation imagem is the phase-mode amplitude and the azimuthal angle of the q-th antenna element (with respect to the x-axis) is denoted ϕq.

2.1.1. Active Reflection Coefficient

[9] A useful quantity for the characterization of an array antenna is the active reflection coefficient [Josefsson and Persson, 2006]. In order to facilitate comparisons of different antenna element designs, we average the active reflection coefficient over a frequency interval [ω0, ω1] and the 2M + 1 lowest phase modes −MmM, since this yields a single number that characterizes the array's ability to radiate. Thus, the root-mean-square (RMS) value of the reflection coefficient is given by

equation image

The average (1) can be modified to include a weight function that emphasizes modes and frequency intervals of particular importance or interest.

2.1.2. Far-Field Distortion

[10] For the phase mode m, the far-field amplitude equation imagem(equation image, ω) varies approximately as ejmϕ with respect to the azimuthal angle ϕ. (A deviation from the ideal ejmϕ-variation is an effect of the finite number of antenna elements and this is further discussed by Josefsson and Persson [2006].) In the following, we use an excitation of unit amplitude and relate the far-field amplitude to the electric field by equation image(equation image, ω) = r−1/2 exp(−jkr) equation image(equation image, ω). We use the normalized far-field amplitude equation imagem(equation image, ω) = equation image · equation imagem(equation image, ω)ejmϕ, in combination with its azimuthal average equation image[equation imagem](ω) = (2π)−102πequation imagem(equation image, ω)dϕ, to evaluate the RMS value of the far-field distortion

equation image

Consequently, equation imagem > 0 implies a deviation of the electric field in the far-field zone from the ideal azimuthal variation ejmϕ. The RMS value of equation imagem with respect to phase-modes and frequency is

equation image

A large amount of distortion may be undesirable in the context of pattern synthesis: (1) an undistorted azimuthal far-field variation ejmϕ could be exploited by the pattern-synthesis algorithm; and (2) a severely distorted far-field pattern may reduce the pattern-synthesis abilities for the array antenna. In the work of Josefsson and Persson [2006], the ripple in the magnitude of the far-field is used as a measure for the quality of the far-field pattern, when a circular array is excited by a phase mode. The average (2) can be modified to include a weight function that emphasizes modes and frequency intervals of particular importance or interest.

3. Shape Optimization of the Antenna Elements

[11] A well-designed array antenna yields, simultaneously, a low reflection equation image and a low far-field distortion equation image for a given frequency range and an appropriate set of phase modes. In this section, we evaluate equation image and equation image for a large number of different antenna element geometries by means of an extensive parameter study. Finally, three Pareto-optimal [Tamaki et al., 1996] candidates are chosen based on this parameter study. (In the next section, we investigate their abilities for pattern synthesis by means of optimizing the excitation of the array antenna.)

3.1. Parameter Space for the Element Shape

[12] First, we consider the cylinder with circular cross section shown in Figure 1. It is equipped with N = 20 horn antenna elements distributed uniformly along its circumference. (Josefsson and Persson [2006] discuss the choice of N for different cylinder radii, element spacings and wavelengths.) We use the cylindrical array antenna for an extensive parameter study where the shape of the antenna element is varied. The aperture width is parametrized as wh = 2 + 16(n − 1)/3 cm, where n = 1, 2, …, 5. The position of the control point P2 is parametrized in a similar manner and we use x2 = 0.5 + 19(n − 1)/3 cm, where n = 1, 2, …, 5. The value for y2 is chosen from the set of discrete parameter values used for wh and the value for x3 is chosen from the parametrization set used for x2. This gives a four dimensional parameter space with 45 = 625 geometries. We explore all combinations of these parameters that satisfy the constraint y2wh/2 (to avoid horns with shapes that resemble leaking cavities) and we exclude geometries where neighboring horns intersect. Despite the coarse grid used for the geometry parameters, the goal functions of interest are rather well-resolved.

3.2. Reflection and Radiation Properties

[13] For characterization purposes, we evaluate ∣equation imagem(f)∣ and equation imagem(f) for each individual design. In particular, we focus on the frequencies that correspond to d/λ = 0.1, 0.2, …, 0.6, where d = 2πa/N is the element separation distance. Figure 3 shows an overview chart of the results, where each sub-figure has ∣equation imagem∣ on the horizontal axis (with a range from 0 to 1) and equation imagem on the vertical axis (with a range from 0 to 0.2). The dots indicate the computed ∣equation imagem∣ and equation imagem for each design in the parameter space. As indicated to the left and on the top of Figure 3, the rows of subplots correspond to different frequencies and the columns to different mode numbers m. This information is mainly used to assess the overall characteristics associated with the designs in the parameter study, and it will be used later to select appropriate antenna element designs.

Figure 3.

Parameter study of shape variations for the antenna element when a = 0.6 m and N = 20: the reflection ∣equation image∣ on the horizontal axis (range from 0 to 1) and the far-field distortion equation image on the vertical axis (range from 0 to 0.2) for each subplot. The rows (from top to bottom) correspond to d/λ = 0.1, 0.2, …, 0.6 and the columns (from left to right) correspond to m = 0, 1, 2, …, 10.

[14] Phase modes that radiate well satisfy ∣m∣ ≤ ka = Nd/λ = 2, 4, 6, 8, 10 and 12 for d/λ = 0.1, 0.2, …, 0.6. This is confirmed by the results in Figure 3, where a mode number ∣m∣ that is considerably larger than ka yields a large active reflection coefficient for all antenna element designs. Independent of the phase mode (i.e. also for the region ∣m∣ < ka), all the antenna element designs in our parameter space yield rather large reflections for the case d/λ = 0.1, which is a result of the small size of the antenna element aperture in relation to the wavelength.

[15] In Figure 3, the far-field distortion is significant for ka + ∣m∣ greater than approximately 17 due to the excitation of the first harmonic ±(∣m∣ − N). The higher-order harmonics (with mode numbers found in the sequence m ± N, m ± 2N, etc) are also present but they yield relatively small amplitudes and their contribution to the far-field distortion is negligible. The phase-mode number ∣m∣ for the excitation of the array antenna can not exceed N/2, which makes it difficult to control and mitigate far-field distortions that are associated with harmonics that feature a relatively large amplitude. It is desirable to choose a geometry of the antenna elements that yields a small far-field distortion and, given the results from the parameter study, designs with the largest possible aperture width wh provide the best results.

3.3. Average With Respect to Phase Modes

[16] From a more practical viewpoint, a pattern synthesis algorithm applied to a specific frequency exploits the modes ∣m∣ ≤ ka (and possibly a couple of modes above the limit ka). Consequently, it is useful to average the active reflection and the far-field distortion with respect to the phase modes −MmM for Mka in order to identify designs that are good on average for a rather arbitrary pattern synthesis case. For the frequency that corresponds to ka = M = 8, Figure 4 shows equation image and equation image for all designs and the corresponding Pareto front when the averages are evaluated for ∣m∣ ≤ M. The corresponding results for M = 9 and 10 yield Pareto fronts that are very similar in shape. However, the far-field distortion increases significantly (in both magnitude and range) with the value of ka = M while the reflection is relatively unchanged. With reference to Figure 4, we notice that the active reflection coefficient varies substantially for the designs in the parameter study as compared to the far-field distortion.

Figure 4.

Values of equation image and equation image for ka = M = 8 (with a = 0.6 m and N = 20): dots, geometries in the parameter study; filled circle, representative design with low reflection; filled square, compromise between reflection and far-field distortion; and filled diamond, representative design with relatively low far-field distortion.

3.4. Selection of Representative Designs

[17] It is possible to select designs that are appropriate for operation over a frequency band in a number of different ways. Here, we use the following procedure. Given the results for M = 8, shown in Figure 4, we assign the designs on the Pareto front the rank 1. Among the remaining designs (excluding the designs with rank 1), we determine the new Pareto front and assign these designs the rank 2. This procedure is repeated until all designs are assigned a rank that, in a sense, corresponds to their individual distances to the actual Pareto front in Figure 4. The next step is to perform the same procedure for M = 9 and 10 individually. Given these results, we identify the set of designs that have a (total) rank less than or equal to 4 for M = 8, 9 and 10.

[18] Given these results, we select three antenna element designs that we find representative and these designs are shown in Figure 5: (a) one design that yields particularly low reflection equation image; (b) one design that provides a compromise between equation image and the far-field distortion equation image; and (c) one design that is characterized by a relatively low far-field distortion equation image. It should be mentioned that the above selection procedure is robust in the sense that the selected designs are still among the top candidates if we also include smaller values for M = ka. If we choose each design's average Pareto rank as a final fitness value, the designs shown in Figure 5 are again among the top candidates, where the selection based on the average rank is also rather insensitive with respect to value of M = ka. Figure 4 also shows the three designs selected based on the parameter study: the dot indicates design (a); the square indicates design (b); and the triangle indicates design (c). We anticipate that the low reflection coefficient associated with design (a) makes this antenna element very attractive, since the far-field distortion is rather low for all the three designs.

Figure 5.

Antenna element designs selected from the parameter study: (a) particularly low reflection equation image; (b) compromise between low reflection equation image and low distortion in the far-field equation image; (c) relatively low far-field distortion equation image.

4. Optimization of the Array Excitation

[19] There is a large number of techniques for and publications on pattern synthesis for array antennas. Chiba et al. [1989] used a projection method that provides a low sidelobe pattern for conformal array antennas. Prasad [1980] developed a method of alternating orthogonal projections for synthesis by means of feasible and desired patterns. Shi et al. [1997] used techniques from signal processing to synthesize antenna patterns for arbitrary arrays. Guy [1988] implemented pattern adjustments by means of a least-mean-squares (LMS) method that iteratively update an initial set of excitations, which is also computed by the LMS method. Yan and Lu [1997] used genetic algorithms [Haupt, 1995] for pattern synthesis. These algorithms only make explicit use of the goal function itself. Here, we use a different technique that exploits the goal function in combination with its gradient with respect to the design variables, which in general yields considerably better convergence properties. Lebret and Boyd [1997] used similar techniques for pattern synthesis problems that are convex. However, realistic synthesis problems are in general not convex and, therefore, they require more unbiased optimization techniques.

[20] Here, we first compare some different formulations of the pattern synthesis problem for the antenna element design labeled (b) in Figure 5, which features a compromise between the far-field distortion and the reflection objectives according to the Pareto front shown in Figure 4. The comparison between the different designs (a), (b) and (c) in Figure 5 is postponed to section 4.4.

4.1. Pattern Synthesis

[21] For pattern synthesis, we formulate the goal function in terms of the directive gain D(ϕ). (In the following, the directive gain is expressed in terms of decibels.) Given a fixed geometry, the objective is to find an excitation that yields a directive gain such that it is less than an upper mask Dup(ϕ) and greater than a lower mask Dlo(ϕ), i.e. Dlo(ϕ) ≤ D(ϕ) ≤ Dup(ϕ) for all azimuthal angles ϕ. We find this approach useful since the optimization problem is then formulated in terms of the shape of the radiation pattern.

4.1.1. Optimization Problem

[22] For pattern synthesis, we use the goal function

equation image

where W(ϕ) is a weight function. We exploit the ramp function

equation image

to quantify the parts of the directive gain that is outside the masks. Instead of the ramp function R, it could be desirable to choose a function with a continuous derivative. However, we find that equation (3) works well despite its simplicity and the discontinuity in its derivative does not cause any noticeable problems in practice.

[23] The pattern synthesis is formulated as an optimization problem by

equation image

where v is the vector [equation imageM, equation imageM+1, …, equation image−1, equation image1, equation image2, … equation imageM]. The zeroth phase-mode amplitude equation image0 is removed from the design space. Instead, equation image0 is forced to be real valued and expressed in terms of the higher phase-mode amplitudes by

equation image

where Ptot+ = 1 W is the predefined input power per unit length. It deserves to be mentioned that, in general, equation image0 > 0 since it requires a rather special situation for the zeroth phase-mode amplitude to be zero. We exploit a gradient based method to solve the optimization problem (4) and it converges to a (possibly local) minimum in only a few iterations.

4.1.2. Specification on the Directive Gain

[24] For testing purposes, we consider the pattern synthesis based on

equation image
equation image
equation image

We have deliberately chosen a mask that can not be satisfied by the directive gain. (Given the mask Dlo(ϕ) in equation (5), a directive gain that satisfies D(ϕ) ≥ Dlo(ϕ) has a total radiated power that exceeds the input power.) As a consequence, the goal function is positive for all possible excitations. The masks are symmetric with respect to ϕ = 180° and we seek a directive gain that is symmetric. Therefore, we choose the phase-mode amplitudes Vm = V*+m, where z* is the complex conjugate of z.

4.1.3. Test 1: All Phase Modes

[25] Most synthesis algorithms exploit the port excitations directly for the optimization of the directive gain. This is equivalent to include all the phase modes in the formulation of the optimization problem in this article. Thus, we deliberately use all the phase modes in order to study the effects of directly optimizing the port excitations, despite the fact that the higher order phase modes may not radiate very well.

[26] Figure 6 shows the optimized directive gain for antenna element design (b) with d/λ = 0.3, where we use M = 10. Given the phase-mode amplitudes, we evaluate the incident and reflected power shown in Figure 7: light gray bars, incident power; and dark gray bars, reflected power. We note that the reflected power is rather large and the active reflection coefficient exceeds unity for two ports. The phase-mode amplitudes are listed in Table 1 and we notice that in particular the phase mode m = 9 is assigned a rather large amplitude.

Figure 6.

Optimized directive gain as a function of azimuthal angle at d/λ = 0.3 with M = 10 for antenna element design (b): solid curve, synthesized directive gain; and dashed curves, upper and lower mask.

Figure 7.

Incident and reflected power associated with the synthesized pattern shown in Figure 6: light gray bars, incident power; and dark gray bars, reflected power.

Table 1. Optimized Excitation Amplitudes equation imagem for Three Different Test Casesa
ModeTest 1Test 2Test 3
  • a

    Test 1: pattern synthesis based on all phase modes; Test 2: pattern synthesis based on phase modes that radiate well; and Test 3: pattern synthesis for all phase modes with a penalty on the reflected power. The absolute values of the excitation amplitudes are shown in parenthesis.

07.36(7.36)12.64(12.64)9.15(9.15)
1−7.09 + j0.55(7.11)−10.53 + j0.74(10.56)−9.02 + j0.60(9.04)
26.58 − j2.23(6.95)9.35 − j3.08(9.84)8.18 − j2.60(8.58)
3−4.72 + j4.40(6.45)−6.35 + j5.85(8.63)−5.96 + j4.97(7.76)
41.24 − j5.57(5.71)1.58 − j7.11(7.29)1.78 − j6.66(6.89)
52.79 + j4.16(5.01)3.23 + j4.81(5.80)3.10 + j4.91(5.81)
6−4.44 + j0.35(4.45)−6.53 + j0.55(6.55)−4.46 + j0.37(4.48)
70.39 − j3.98(4.00)  −0.05 − j4.88(4.88)
84.95 − j0.28(4.96)  6.18 + j1.11(6.28)
93.97 + j12.0(12.7)  −4.55 + j3.08(5.49)
10−6.42 + j2.80(7.01)  5.20 − j3.19(6.10)

4.1.4. Test 2: Phase Modes That Radiate Well

[27] The relatively large active reflection coefficients in Figure 7 result from that the optimization algorithm attempts to improve the far-field pattern with phase modes that do not radiate very well, i.e. ∣m∣ > ka = 6 shown in Figure 3. We try to reduce the reflected power by excluding the phase modes with ∣m∣ > ka, which also reduces the number of degrees of freedom for the optimization problem and prevents too rapid azimuthal variations in the excitation and aperture fields of the array antenna. (Rapid azimuthal variations could also be prevented by means of another parametrization of the port excitation, e.g. a spline curve that is sampled at the ports.) For the case M = 6, the directive gain and reflected power are shown in Figures 8 and 9, respectively. The degradation in the directive gain can be considered small in comparison to the improvements for the reflected power. The excitation amplitudes associated with the phase modes are listed in Table 1.

Figure 8.

Optimized directive gain as a function of azimuthal angle at d/λ = 0.3 with M = 6 for antenna element design (b): solid curve, synthesized directive gain; and dashed curves, upper and lower mask.

Figure 9.

Incident and reflected power associated with the synthesized pattern shown in Figure 8: light gray bars, incident power; and dark gray bars, reflected power.

4.2. Pattern Synthesis With Reflection Control

[28] For broad frequency-band pattern synthesis it may be difficult to select the appropriate phase modes for the optimization of the directive gain, and simultaneously achieve a low reflected power. Again, we refer to Figure 3 and notice that, for 0.2 ≤ d/λ ≤ 0.6, the useful phase modes form a non-trivial region if both the active reflection coefficient and the far-field distortion are taken into account simultaneously. (As described in section 3.2, phase modes with ∣m∣ ≤ ka and d/λ ≥ 0.2 radiate well but will have significant far-field distortion for ka + ∣m∣ greater than approximately 17.) In the case of a cylinder of arbitrary shape that is only partially covered by an array antenna, the information that corresponds to Figure 3 may be difficult to derive.

4.2.1. Optimization Problem

[29] Next, we introduce a linear combination of the pattern synthesis goal function equation imageP and a corresponding goal function for the reflected power, given by

equation image

where Pi is the power associated with the wave that propagates from the antenna aperture towards port i. Similar to the construction of equation imageP, we have a weighting function Wi and an upper mask Pup. Thus, we solve the optimization problem

equation image

where the zeroth phase-mode amplitude is forced to be real valued and expressed in terms of the higher phase modes amplitudes as above.

[30] Given the optimization problem (9), we include again all phase modes in the pattern synthesis problem and exploit equation imageR to penalize phase modes that yield a large contribution to the reflected power. (The term equation imageR can be used for regularization if the requirement on the radiation pattern can be fulfilled for more than one excitation.) An advantage with this solution is that it would be feasible to optimize the port excitations directly, rather than parameterizing the excitation of the array antenna in terms of phase modes. The direct optimization of the port excitations can also be used for cases where a representation in terms of phase modes or similar constructions is not available.

4.2.2. Test 3: All Phase Modes and Penalty Term

[31] Figure 10 shows the directive gain when M = 10 and α = 0.8, which mainly differs from Test 1 (that is equivalent to α = 1) by a slightly wider main lobe. (The choice α = 0.8 gives a reasonable balance between the requirements on the reflected power and directive gain.) Here, we use Pup = −5 dBm and Figure 11 shows the reflected power, which is significantly lower as compared to Test 1 that does not include any penalty for the reflected power. Mainly the optimization procedure yields strong excitation amplitudes for a few of the elements located on the front part of the circular cylinder, with respect to the lobe direction. The antenna elements on the back part of the circular cylinder are only weakly excited.

Figure 10.

Optimized directive gain as a function of azimuthal angle for d/λ = 0.3 for α = 0.8 for antenna element design (b): solid curve, directive gain; and dashed curves, upper and lower mask.

Figure 11.

Incident and reflected power associated with the synthesized pattern shown in Figure 10: light gray bars, incident power; and dark gray bars, reflected power.

4.3. Comparisons of Synthesis Techniques

[32] Dolph-Chebyshev pattern synthesis [Dolph, 1946] is a conventional technique for designing a directive gain with a narrow main beam and a prescribed side-lobe level. It is developed for linear arrays, but here we employ the technique for uniform circular arrays [Lau and Leung, 2000; Vescovo, 1999]. Figure 12 shows the Dolph-Chebyshev synthesized directive gain for d/λ = 0.3, where we use M = 9. The peak-to-side-lobe ratio is 27 dB and this yields a good fit to the prescribed mask. The reflected power levels shown in Figure 13 are rather large and, all together, these results are very similar to Test 1.

Figure 12.

Dolph-Chebyshev synthesized directive gain as a function of azimuthal angle for d/λ = 0.3 for antenna element design (b): solid curve, directive gain; and dashed curves, upper and lower mask.

Figure 13.

Incident and reflected power associated with the synthesized pattern shown in Figure 12: light gray bars, incident power; and dark gray bars, reflected power.

[33] Table 2 shows the two terms equation imageP and equation imageR that feature in the goal function αequation imageP + (1 − α)equation imageR for the different cases considered. (Here, equation imageR is again evaluated with Pup = −5 dBm.) For the synthesis problem under consideration, we notice that it is feasible to reduce the reflected power Ptot significantly and this is achieved in both Test 2 and Test 3, where phase modes that do not radiate well are either excluded or penalized. The Dolph-Chebyshev pattern synthesis does not take the reflected power into account at all and it yields about twice the reflected power. The differences in the pattern synthesis objective equation imageP are relatively small when the different techniques are compared.

Table 2. Goal Functions equation imageP and equation imageR and the Relative Reflected Power
 Test 1Test 2Test 3Dolph-Chebyshev
equation imageP2.623.512.852.67
equation imageR18.611.610.617.7
Ptot/Ptot+0.550.240.360.53

4.4. Comparison of Antenna Element Designs

[34] We use designs (a), (b) and (c) shown in Figure 5 for pattern synthesis by means of the optimization problem (9) when d/λ = 0.4 and M = 10. (Larger values of d/λ could possibly make problems with far-field distortions somewhat easier to identify.) Again, we used the specification (5)–(7) for equation imageP in combination with Pup = −5 dBm in equation (8). We explore a range of linear combinations with α = n/20 and n = 1, 2, …, 20 to construct a Pareto front for the pattern synthesis algorithm, where we start the gradient based optimization algorithm with a number of initial guesses in order to mitigate problems with local minima. The results are shown in Figure 14: black dots, antenna element design (a); dark gray dots, antenna element design (b); and light gray dots, antenna element design (c).

Figure 14.

Pattern synthesis requirement linearly combined with requirement on the reflected power for d/λ = 0.4 and 0 < α ≤ 1: black dots, antenna element design (a); dark gray, antenna element design (b); and light gray, antenna element design (c).

[35] Antenna element design (a) clearly outperforms the other two designs labeled (b) and (c). In particular, we notice that switching from design (c) to design (a) yields a lowered reflection of about 6 dB, which improves the efficiency of the array antenna significantly. Thus, we conclude that optimization of the shape of the antenna elements is mainly useful for improved efficiency. The quality of the radiation pattern is not much influenced by the specific geometry of the antenna element. A corresponding test for the case of d/λ = 0.6 shows that the first harmonic introduces far-field distortions that reduce the array antenna's ability to achieve the desired radiation pattern. However, the deterioration of the pattern synthesis abilities is similar for all the three antenna elements (a), (b) and (c). We conclude that the geometry of the antenna element does not influence the array's pattern synthesis ability to a great extent.

5. Conclusion

[36] We have presented a procedure for the optimization of array antennas that consist of identical antenna elements uniformly distributed around the circumference of a circular cylinder. In particular, we focus on the optimization of the shape of the antenna elements in combination with pattern synthesis. As a model problem in two space dimensions, we used horn shaped antenna elements fed by parallel plate waveguides excited with the fundamental mode.

[37] We perform a large parameter study with respect to the shape of the horn antenna elements. Given a phase-mode analysis, we characterize the different shapes in terms of their (1) active reflection coefficient and (2) deviation in the far-field from the azimuthal variation of the excited phase-mode. A few Pareto-optimal antenna elements are selected for pattern synthesis, where we use the phase-mode amplitudes as design degrees of freedom.

[38] For the pattern synthesis procedure, we exploit a goal function that combines two different objectives: (1) find a directive gain that is inside an upper and a lower mask; and (2) minimize the reflected power from the array antenna aperture. This allows us to use all the available phase modes, where the optimization algorithm then assigns relatively small amplitudes to phase modes that contribute to a large reflection from the antenna aperture. Equivalently, the port amplitudes can be used directly as degrees of freedom by the optimizer and this is a clear advantage for the general case, where a parameterization that corresponds to phase modes may be difficult to find. For the case where the requirements on the directive gain are less demanding, a penalty term in the goal function that is associated with the reflected power provides regularization for the optimization problem and it also warrants an optimum with better efficiency. Otherwise, the trade-off in-between radiation qualities and efficiency concerns can be established in terms of a Pareto front. We demonstrate that pattern synthesis algorithms that do not involve requirements on the reflected power can yield relatively poor efficiency. The pattern synthesis technique that we describe in this article can be used for arbitrary conformal array antennas in 3D and it allows for a number of generalizations such as polarimetric pattern synthesis.

Acknowledgments

[39] This work was supported by the Strategic Research Center CHARMANT, financed by the Swedish Foundation for Strategic Research. The authors thank Lars Josefsson and David Degerfeldt for rewarding and fruitful discussions.

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