**2.3 Array factor expression**

To analytically compute the corrected (compensated) radiation pattern and validate with simulation results, the following compensated array factor AF*c* is used [18, 31]:

$$\mathbf{AF}\_c = \mathbf{AF}e^{j\mathbf{\hat{5}}\_{\ast n}},\tag{3}$$

where δ±n*<sup>c</sup>* can be computed using Eq. (2) and the array factor (AF) expression for antennas on conformal surfaces is [18]:

$$\mathbf{AF} = \sum\_{n=1}^{N} F\_n(\Theta, \Phi) \,\mathrm{w}\_n e^{jk\left[\mathbf{x}\_n \sin \theta \star \mathbf{z}\_n \cos \theta\right]},\tag{4}$$

where *k* is the free space wave number, *N* is number of antenna elements, (*xn*, *zn*) is the location of nth antenna element on the conformal surface, and *Fn*(θ,ϕ) is the individual element pattern in **Figure 2**. *wn* = *In e <sup>j</sup>*∆ϕ is the complex weighting function required to drive the nth antenna element.

For this work, the phase difference ∆ϕ between adjacent antenna elements was made zero, and the amplitude tapering coefficient *In* was kept equal to 1 (uniform excitation). Using Eqs. (1)–(4), the expression for corrected array factor is given by:

$$\begin{array}{l} \text{AF}\_{\varepsilon} = F\_{1}(\Theta, \phi) \, e^{jk\,\Lambda\_{-1}} e^{jk\,[x\_{1}\sin\Omega + x\_{1}\cos 0]} + F\_{2}(\Theta, \phi) \, e^{jk\,\Lambda\_{-2}} e^{jk\,[x\_{2}\sin\Omega + x\_{2}\cos 0]}\\ \text{+ } F\_{3}(\Theta, \phi) \, e^{jk\,\Lambda\_{+2}} e^{jk\,[x\_{j}\sin\Omega + x\_{j}\cos 0]} + F\_{4}(\Theta, \phi) \, e^{jk\,\Lambda\_{+1}} e^{jk\,[x\_{4}\sin\Omega + x\_{4}\cos 0]} \end{array} \tag{5}$$

*F*1(θ,ϕ) = cos (θ + ϕ−1) and *F*2(θ,ϕ) = cos (θ + ϕ−2) are the element patterns for the two left antenna elements and *F*3(θ,ϕ) = cos (θ − ϕ+1) and *F*4(θ,ϕ) = cos (θ − ϕ+2) are the element patterns for the two right antenna elements on the cylindrical surface in **Figure 2**. It should be noted that for linear array, the element pattern is cos (θ) as all the elements are pointing towards zenith (that is towards ϕ*s*=0° ). However on conformal surfaces, the individual element patterns become geometry dependent. For example, as can be seen in **Figure 2**, the look directions of antenna elements *A*−1 and *A*−2 are towards the angular directions ϕ−1 and ϕ−2 respectively, while the look directions of antenna elements *A*1 and *A*2 are towards the angular directions ϕ1 and ϕ2 respectively. Therefore, the element patterns are cos (θ + ϕ−1), cos (θ + ϕ−2), cos (θ − ϕ1), and cos (θ − ϕ2) respectively.
