## 3 Design Principles of the Helical Phased Array  
The array directivity and front-to-back ratio are defined by the amplitude and phase difference between the two helical antennas. This phase difference is determined by the spacing and orientation of the antennas, as well as by the phase shift introduced by the beamformer, which serves both as a splitter and a combiner [28]. The spacing between  
Figure 14: Spectrum of (**a**) HE016 horizontal component, (**b**) helical horizontally polarized. Figure 13: Spectrum of (**a**) HE016 vertical component, (**b**) helical vertically polarized. the helical antennas determines the necessary phase difference to steer the beam in a specific direction. The phase shift required between the two adjacent helical antennas is proportional to the wavelength and the spacing between them. Given that the two helical antennas are placed side by side \(\lambda/8\) apart, which is equivalent to a distance of 1.6 m at 25 MHz, the helical array is illustrated in Figure 15.  
\[\Psi=\frac{2\pi\mathrm{s}_{\lambda}\mathrm{sin}\,\theta}{\lambda}-\Delta\Phi \tag{5}\]  
where:  
\(\Psi\): the phase of the array in a given direction;  
\(\mathrm{S}_{\lambda}\): element spacing;  
\(\Theta\): the angle of radiation;  
\(\Delta\Phi\): phase difference between the two elements;  
\(\lambda\): wavelength. To determine the array phase difference, Equation (5) was applied [28]. The radiation pattern of the array caused by the superposition of the signals from each helical antenna is expressed by Equation (6) below [29]. \[\mathrm{ARRAY FACTOR}=\frac{\mathrm{Sin}\left(\frac{\Psi\mathrm{T}}{2}\right)}{2 \mathrm{Sin}\left(\frac{\Psi\mathrm{T}}{2}\right)} \tag{6}\]  
where:  
N: the number of elements;  
\(\Psi\): the phase of the array in a given direction. Using software simulation based on Equations (5) and (6) with \(\Delta\Phi=135^{\circ}\) and \(\mathrm{S}_{\lambda}=\lambda/8\), the radiation pattern of the two helical array is illustrated in Figure 16.  
Figure 16: Radiation pattern of two helical antennas placed at \(\lambda/8\) apart and at \(135^{\circ}\) phase difference. Figure 15: The helical array. The spacing between the two helical antennas determines the array factor, which describes the combined radiation pattern of the array. The array factor can be computed using a mathematical expression that describes the complex phasor sum of the individual radiation patterns of each helical antenna in the array. It takes into account the amplitude and phase shift of each element's contribution to the overall radiation pattern. The array gain was then computed using Equation (7) below [30]. \[\text{Gain}=[(\text{cos}45+\text{cos}135)^{2}+(\text{sin}45+\text{sin}135)^{2}]^{ 0.5}=1.41\text{ dB or }1.5\text{ dB} \tag{7}\]  
From Equation (7), it follows that the gain of the array is 1.5 dB at 25 MHz with a front-to-back ratio of 23 dB. Alternatively, by simulating the radiation pattern of the array using EZNEC, the front-to-back ratio of the resulting cardioid radiation pattern exceeds 30 dB, as illustrated in Figure 17.  
Using EZNEC software, the radiation pattern of the helical array, when fed by a 135\({}^{\circ}\) phase shift at a distance of \(\lambda/8\) at 25 MHz, was simulated at 15 MHz, 20 MHz, and 30 MHz, with the respective results shown in Figure 18a-c. When the two-helical antennas operate off frequency, they fail to achieve proper phasing towards the desired direction, a fact supported by parameter \(\Psi\) in Equation (5), which varies proportionally to the wavelength of the spacing (S\({}_{\lambda}\)) between them and inversely proportionally to \(\Delta\Phi\). Consequently, this deviation causes a distortion in the  
Figure 13:0 The helical array radiation pattern at 25 MHz using EZNEC modeling software. Figure 13:1 EZNEC software simulation (**a**) at 15 MHz; (**b**) at 20 MHz; (**c**) at 30 MHz. radiation pattern from its ideal value, as depicted in Figure 18a, ultimately resulting in a significant reduction in the front-to-back ratio to only 4 dB at 15 MHz.