## 3 Design Principles of the Helical Phased Array
### The Beamformer Design and Implementation  
The beamformer is crucial to the overall array performance as it controls both the phase and amplitude differences in each helical antenna relative to the phase delay of the distance between them when operating collectively as an array. The beamformer functions as a splitter and combiner circuit, consisting of a cascade phase shift network and an active circuit. The phase shift between the two helices, as illustrated earlier in Equation (5), determines \(\Delta\Phi\). Furthermore, the phase delay caused by the spacing between the two helical antennas (S\({}_{\lambda}\)) is a variable dependent on their distance in terms of wavelength. For instance, in scenarios where the array maintains a fixed \(\Delta\Phi\) value (such as with a quarter-wavelength array employing a fixed 90-degree hybrid feed), the phase difference (\(\Psi\)) outlined in Equation (5) only varies at each operating frequency due to changes in S\({}_{\lambda}\). The phase difference between the two helical antennas at each operating frequency is illustrated in Figure 19 below. The blue curve at the top represents the phase difference between the two helical antennas when the array radiates in the backward direction. At 25 MHz, the 180\({}^{\circ}\) phase difference between the two helical antennas results in a theoretically infinite front-to-back ratio. As depicted in Figure 19, the highest phase error occurs at the extremes of the operating frequency range at around +5%. Conversely, the blue graph at the bottom represents the phase difference between the two helical antennas when the array radiates in the forward direction. A maximum gain of 1.89 dB occurs when the phase difference between the two helical antennas is 78.8\({}^{\circ}\) at 20 MHz. To minimize the phase error in the backward direction, \(\Psi\) must be constant at all operating frequencies. Therefore, a compensation circuit was implemented for this linearization. The circuit diagram of the beamformer is depicted in Figure 20. The output impedance of the two LNAs, along with  
Figure 19: The phase relationship between the two helical antennas. the pi attenuator circuits connected to the inputs of the beamformer, is only modeled for simulation purposes. The design philosophy of the beamformer linearization circuit is based on a consistent 180\({}^{\circ}\) phase difference (backward direction) between the helical antennas across all operating frequencies. This is essential for optimizing antenna performance and phase error minimization, particularly at the frequency range extremes. The aim of the compensation circuit is to maintain a stable phase relationship (\(\Psi\)) between the antennas, thus enhancing the overall array performance and ensuring reliable operation across the desired frequency range. As an example, a variable phase 45\({}^{\circ}\) high-pass phase shifter, integrated in series with the fixed 90\({}^{\circ}\) hybrid, was incorporated into one of the two inputs of the beamformer [31]. This setup aims to establish a consistent phase difference of 180\({}^{\circ}\) between the two helical antennas across all operating frequencies. To ensure reliability, Mini Circuits' hybrid PSCQ-2-51W (Brooklyn, NY, USA) was utilized to introduce a fixed 90\({}^{\circ}\) phase shift to the network [32]. Furthermore, the common-base amplifier topology was chosen on the basis of its low input impedance, which matched the 50 \(\Omega\) output impedance of the hybrid. Consequently, the common-base topology provides voltage gain, while the common collector compensates for the current gain of the two-stage amplifier [33] since its low output impedance effectively matches the 50 \(\Omega\) input port (R5) of the receiver. The simulation of the entire system at 25 MHz is depicted in Figure 21 below. The system consists of the two LNAs driven by helical antennas, connected to a 45-degree high-pass phase shifter cascaded to the quadrature hybrid. Moreover, the output of the quadrature coupler is amplified by the final amplifier of the beamformer.