**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

**93**

the cylindrical surface.

*Broadside Pattern Correction Techniques for Conformal Antenna Arrays*

as *A*±n, *n* = 1, 2 (for four antenna elements on the cylindrical surface). If each antenna element on the cylindrical surface is excited with uniform amplitudes and

radiated from the antenna elements will arrive at the reference plane with different phases. This phase difference is due to the different path lengths experienced by the radiated E-fields in the z direction while reaching to the reference plane. Because phases of E-fields radiated from *A*±1 are not the same as those radiated by *A*±2, the radiation may not necessarily be constructively added broadside to

). If the E-fields *E*<sup>±</sup>*<sup>n</sup> e<sup>j</sup>*ϕ<sup>±</sup>*<sup>n</sup>*

ensured to arrive at the reference plane with the same phase, then constructive interference will result in a broadside radiation pattern and therefore pattern recovery toward the broadside is possible. The delayed phase of E-fields due to free-space propagation (−*k* ∆±n) from antenna elements on the conformal surface to the reference plane can be compensated (corrected) by exciting the elements with phase advance (+*k* ∆±n). This will cause the E-fields from antenna elements to arrive at the reference plane with same phase and will cause constructive broadside

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 ∆−1, ∆−2, ∆2, and ∆1 denote the distances from the elementsA−1, A−2, A2, and A1, respectively, to the reference plane (or the

The distance from the antenna elements on the cylinder to the projected ele-

where **r** is the radius of the cylinder and *±n* is the angular position of antenna elements on the cylinder. Using the relation of **L =** r, the angular positions of antenna elements can be calculated, where **L =** /**2** is the inter-element distance on

∆±n = *r* − *rsin* ϕ±n (1)

°

put on all elements, then the E-fields

from antenna elements are

*DOI: http://dx.doi.org/10.5772/intechopen.90957*

phases, that is, if excitation weights *wn* = 1 *e<sup>j</sup>*<sup>0</sup>

*Phase compensation of a conformal cylindrical antenna array.*

the array (i.e., with ϕ*s* = 90°

**Figure 2.**

radiation pattern in the +z direction.

projected elements on the reference plane).

**2.1 Computing the distance to the projected elements**

ments on the reference plane can be computed using:

*Broadside Pattern Correction Techniques for Conformal Antenna Arrays DOI: http://dx.doi.org/10.5772/intechopen.90957*

**Figure 2.**

*Advances in Array Optimization*

**Figure 1.**

*Array on a conformal surface.*

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 [1–3, 11, 19–21], mechanical calibration techniques have been used to steer the main beam on a conformal surface in the desired direction. In [12, 13, 15, 16, 22–25], projection method of [26] is used to correct the main beam direction of a deformed/flex surface. In [27–30], 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

**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

**92**

future work are presented.

*Phase compensation of a conformal cylindrical antenna array.*

as *A*±n, *n* = 1, 2 (for four antenna elements on the cylindrical surface). If each antenna element on the cylindrical surface is excited with uniform amplitudes and phases, that is, if excitation weights *wn* = 1 *e<sup>j</sup>*<sup>0</sup> ° put on all elements, then the E-fields radiated from the antenna elements will arrive at the reference plane with different phases. This phase difference is due to the different path lengths experienced by the radiated E-fields in the z direction while reaching to the reference plane. Because phases of E-fields radiated from *A*±1 are not the same as those radiated by *A*±2, the radiation may not necessarily be constructively added broadside to the array (i.e., with ϕ*s* = 90° ). If the E-fields *E*<sup>±</sup>*<sup>n</sup> e<sup>j</sup>*ϕ<sup>±</sup>*<sup>n</sup>* from antenna elements are ensured to arrive at the reference plane with the same phase, then constructive interference will result in a broadside radiation pattern and therefore pattern recovery toward the broadside is possible. The delayed phase of E-fields due to free-space propagation (−*k* ∆±n) from antenna elements on the conformal surface to the reference plane can be compensated (corrected) by exciting the elements with phase advance (+*k* ∆±n). This will cause the E-fields from antenna elements to arrive at the reference plane with same phase and will cause constructive broadside radiation pattern in the +z direction.

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 ∆−1, ∆−2, ∆2, and ∆1 denote the distances from the elementsA−1, A−2, A2, and A1, respectively, to the reference plane (or the projected elements on the reference plane).

### **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:

$$
\Delta\_{\pm \mathbf{n}} = r - r \sin \Phi\_{\pm \mathbf{n}} \tag{1}
$$

where **r** is the radius of the cylinder and *±n* is the angular position of antenna elements on the cylinder. Using the relation of **L =** r, the angular positions of antenna elements can be calculated, where **L =** /**2** is the inter-element distance on the cylindrical surface.

## **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:

$$
\delta \square\_{\pm \mathbf{n}}{}^{c} = \star \not{k} \Delta\_{\pm \mathbf{n}} \tag{2}
$$

where *k* is the free-space wave number.
