Computed parameters of conformal cylindrical array for various radii of curvatures.

## Abstract

Phase compensation techniques based on projection method and convex optimization (phase correction only) for comparing the maximum gain of a phase-compensated conformal antenna array have been discussed. In particular, these techniques are validated with conformal phased array antenna attached to a cylindrical-shaped surface with various radii of curvatures. These phase compensation techniques are used to correct the broadside radiation pattern. It is shown that the maximum broadside gain compensated is still less than the gain of a linear flat array for any surface deformation. This fundamental maximum compensated gain limitations of the phase compensation techniques can be used by a designer to predict the maximum broadside obtainable theoretical gain on a conformal antenna array for a particular deformed surface.

### Keywords

- conformal antennas
- phased arrays
- antenna radiation patterns
- convex optimization
- microstrip arrays

## 1. Introduction

Conformal antenna arrays are beneficial for applications that need an antenna to be placed on a non-flat surface, for example, on the fuselage of a UAV/airplane in the aerospace industry [1, 2, 3], implantable sensors in wearable networks [4, 5, 6, 7, 8], and satellite communications [9, 10, 11]. One of the main advantages of using conformal antenna is its structural integration ability on singly curved (e.g., a wedge/cylinder) [12, 13, 14, 15] and doubly curved (like a sphere) surfaces [16]. This can be very useful in applications where using definite flat surface may not be a practical design choice. Another exciting application of conformal antenna array is at the base station in a cellular mobile communication system. Today, mobile service providers are utilizing three separate antenna panels (dipole or monopole array) in a cell for a 120° sector coverage. What about, if one cylindrical array is used instead of three dipole arrays [17]? This can result in a much smaller transmitter requirement with 360° azimuthal coverage plus reduced base station size at a lower cost (specifically beneficial in crowded residential areas where cellular companies have to rent the space for base station installation).

On the other hand, these curved surfaces may be subjected to intentional (e.g., flexing wings of a UAV/aircraft) and/or unintentional (bending of aircraft wings due to severe weather conditions/vibrations) forces that change the shape of the surface [12]. As a result, the radiation pattern of the conformal antenna array is changed as shown in Figure 1. The results in [18] indicate that directivity of conformal antenna array can be reduced by 5–15 dB. In the literature, various methods have been proposed to compensate the reduction in directivity and to improve/correct the radiation pattern of a conformal antenna array. In [1, 2, 3, 11, 19, 20, 21], mechanical calibration techniques have been used to steer the main beam on a conformal surface in the desired direction. In [12, 13, 15, 16, 22, 23, 24, 25], projection method of [26] is used to correct the main beam direction of a deformed/flex surface. In [27, 28, 29, 30], various optimization algorithms have been used to control the radiation pattern of conformal antenna arrays. In summary, it has been shown that the radiation pattern of a conformal antenna array can be improved with different calibration techniques, signal processing algorithms, sensor circuitry, and phase and amplitude adjustments.

This chapter will focus on phase compensation of four-element conformal cylindrical antenna array using (1) projection method and (2) convex optimization method. First, a brief introduction and working principle of phase compensation is presented using projection method. Then, array factor expression will be derived to compensate the radiation pattern of conformal cylindrical array. Then, the convex optimization algorithm will be discussed to compute the array weights for pattern recovery of conformal cylindrical array. Then, compensated gain using both the methods will be compared to linear flat array to explore the gain limitations of these compensated techniques for conformal antenna arrays. Finally, conclusion and future work are presented.

## 2. Projection method for pattern recovery of conformal antenna array

The projection method in [26] and its further exploration in [12, 15, 22] are adopted here to describe the behavior of the conformal antenna array shown in Figure 2. For discussion, consider the problem where the flat antenna array is placed on the singly curved surface shaped as a cylinder with radius r as shown in Figure 2. The position of each antenna element on the cylinder is represented as

To compute this compensated phase, the antenna elements are projected on the reference plane and then the distances from antenna elements on cylindrical surface (shown as black dots) to the projected elements on the reference plane (shown as dashed circles) are calculated. Suppose

### 2.1 Computing the distance to the projected elements

The distance from the antenna elements on the cylinder to the projected elements on the reference plane can be computed using:

where

### 2.2 Computing the compensated phase

The required compensated phase to correct the broadside radiation pattern of a conformal cylindrical antenna array in Figure 2 is then given by:

where

### 2.3 Array factor expression

To analytically compute the corrected (compensated) radiation pattern and validate with simulation results, the following compensated array factor

where

where

For this work, the phase difference

## 3. Convex optimization for pattern recovery of conformal antenna array

The broadside gain maximization problem of a conformal antenna array in Figure 2 can be formulated as a linear constrained quadratic programming problem [32], i.e.,

where

Iterative optimization algorithms are normally adopted to find this minimum. Although second-order Newton method is able to find the solution in fewer steps, it requires evaluation of complex quadratic norm defined as the Hessian matrix ^{th} iteration for Gradient descent method is given as

where

## 4. Analytical and simulation results

To analytically compute the corrected (compensated) radiation pattern and validate with simulation results, the compensated array factor

### 4.1 Cylindrical surface with radius of curvature r = 8 cm

The four-element microstrip patch antenna array on a cylindrical surface with radius **Table 1** that the broadside compensated gain

Parameter | Projection method | Convex optimization | |
---|---|---|---|

8 | [121.67,0,0,121.67] | [−89.49,146.95,146.95,-89.49] | |

6.17 | 6.17 | ||

0.8 | 0.8 | ||

10 | [101.84,0,0,101.84] | [−170.44,85.95,85.95,−170.44] | |

4 | |||

0.53 | |||

12 | [86.95,0,0,86.95] | [114.61,26.05,26.05,114.61] | |

2.74 | |||

0.4 | |||

15 | [70.95,0,0,70.95] | [9.66,−62.66,−62.66,9.66] | |

1.765 | |||

0.24 | |||

30 | [36.42,0,0,36.42] | [−102.2,−139.3,−139.3,−102.2] | |

0.5 | |||

0 |

### 4.2 Cylindrical surface with radii of curvatures r = 10 , 12 , 15 cm

When radii of curvatures of cylindrical surface array increase (less deformation), both the projection and convex optimization (phase correction only) methods recover the broadside radiation pattern with decreasing gap between corrected and uncorrected gains as shown in Figures 5**–**7 and are also tabulated in Table 1. The gain

### 4.3 Cylindrical surface with radii of curvatures r = 30 cm

In the limiting case, when the radius of curvature of conformal cylindrical array increases up to 30 cm and above (approaching flat array), the compensated gains achieved from both projection and convex optimization methods nearly reaches the linear flat array gain, which is demonstrated in Figure 8 and is shown in Table 1.

## 5. Conclusion

In this chapter, phase compensation techniques based on projection method and convex optimization (phase correction only) have been discussed for recovery of broadside radiation pattern on a conformal cylindrical-shaped antenna array. The compensated gains of both the methods have been compared with linear flat antenna array. It is shown that the maximum broadside gain recovered with both the methods is less than the linear antenna array for severe deformation cases and approaches the gain of linear antenna array for less conformal deformation surfaces. The analytical expressions and convex optimization algorithm used can be used by a designer to predict the maximum possible compensated gain of conformal antenna array.

## 6. Future work

The proposed techniques can be extended for broadside pattern correction of conformal antenna arrays on other deformed surfaces (spherical nose of plane, flexing wings of UAV, etc.). Another interesting research can be to extend these techniques for beamforming on conformal deformed surfaces (e.g., on base station/tower of cellular companies) to improve the signal-to-noise ratio (SNR) and its capacity. In this work, convex optimization is used to compute the compensated phases (with uniform amplitudes constraint) for recovery of broadside radiation pattern only. In practice, the technique is robust and can be used to calculate complex weights (amplitude tapering as well as phase correction), which can be further explored for beamforming applications and side lobe level control of conformal antenna arrays.

## Acknowledgments

This work is supported by Ignite (NTF), Ministry of IT & Telecom, Government of Pakistan via project no. ICTRDF/TR&D/2015/04.