# A Compact Wideband Active Two-Dipole HF Phased Array
## 2 Theoretical Model of the Antenna Array  
Without minimizing the mutual coupling between the two driven elements of a traditional quarter wavelength phased array with 90\({}^{\circ}\) phase difference feed may cause a serious deviation from the expected specifications. For the aforementioned, this fact is manifested in the different simulated and real radiation patterns as discussed below. It can be demonstrated by modelling the array first in ideal conditions by assuming that both driven elements maintain a 50 \(\Omega\) constant impedance across the entire band, as illustrated in the equivalent circuit in Figure 1. The two driven elements of the array are denoted by the two equal unity voltage sources V1 and V2, both of which have a 50 \(\Omega\) purely resistive impedance. Furthermore, both antennas are connected to a 90\({}^{\circ}\) hybrid that employs three ports of a 50 \(\Omega\) purely resistive impedance and acts as a beamformer. Figure 1: The equivalent circuit of a quarter-wavelength phased array with 90\({}^{\circ}\) difference feed under ideal conditions. At the input ports of the hybrid illustrated in Figure 1, V1 and V2 can then be expressed by Equation (1), where \(A\) is the sum of their voltages at the output of the 90\({}^{\circ}\) hybrid:  
\[A=\sqrt{\left(1+cos90\right)^{2}+\left(sin0+sin90\right)^{2}}=\sqrt{2} \tag{1}\]  
The current magnitudes I\({}_{1}\) and I\({}_{2}\) can be computed by the use of Kirchhoff's law, where R1 and R2 are the 50 \(\Omega\) output impedance of each driven element that is terminated at the 50 \(\Omega\) input ports of the combiner, as given by Equations (2) and (3) below:  
\[\text{V1}=\text{I}_{1}\left(\text{R1}+50\right) \tag{2}\]  
\[\text{V2}=\text{I}_{2}\left(\text{R2}+50\right) \tag{3}\]  
Since currents I\({}_{1}\) and I\({}_{2}\) are of identical magnitude, the radiation pattern of the array (assuming that the two driven elements are isotropic radiators) can be plotted by adding vectors using Equations (4) and (5), which have been derived from Figure 2 as follows:  
\[\Psi=\frac{\pi}{2}sin\theta+\Delta\Phi \tag{4}\]  
\[AF=\sqrt{\left(1+cos\psi\right)^{2}+\left(sin\psi\right)^{2}} \tag{5}\]  
where:  
\(AF=\) Array factor  
\(\theta=\) The radiation angle  
\(\text{d}=\frac{\pi}{2}\)  
\(\Delta\Phi=\) The phase difference feed  
According to [PERSON] [10], the normalized value of the total field of two isotropic sources referred to as the 'array factor' can also be expressed by Equation (6) as follows [6]:  
\[\text{E}=\frac{sin(2\psi/2)}{2 sin(\psi/2)} \tag{6}\]  
Under the condition that both driven elements have a 50 \(\Omega\) purely resistive impedance and a constant 90\({}^{\circ}\) phase difference feed across the desired band, the only parameter that affects the array's radiation pattern, based on the above given equations, is the element's spacing because varies according to the wavelength which the antenna is tuned. It is illustrated in Figure 3, where the radiation patterns of two isotropic sources spaced from 0.05 \(\lambda\) to 0.5 \(\lambda\) with a 90\({}^{\circ}\) phase difference feed have been simulated. The radiation patterns below have been simulated at different wavelengths using Equations (4) and (5) in order to determine the range at which the array has a usable F/B. In this regard, the mutual coupling between the two elements of the array has not been considered as it affects the active impedance of the array's driven elements and consequently the array factor, as discussed in the next section. Figure 2: The geometry of A and B phased-array isotropic radiators. It is important to note that the value of the self-impedance of an isolated driven element changes under the following conditions. Particularly, according to [PERSON], it is defined as the active impedance when two driven elements which are excited installed side by side as a phased array [11; 12; 13; 14]. The mutual resistance and reactance for the case of two parallel, side-by-side antennas, as a function of spacing, was also demonstrated by [PERSON].