Design a two-element dipole array that will radiate equal intensities in the \(\phi=0, \pi / 2, \pi\), and \(3 \pi / 2\) directions in the \(H\) plane. Specify the smallest relative current phasing, \(\xi\), and the smallest element spacing, \(d\).

A two-element dipole array's array factor in the H-plane is given by the following equation: Array Factor (AF) = \(1 + \cos(\beta d \sin \phi + \xi)\) Where \(\beta = 2\pi / \lambda\) is the phase constant, \(d\) is the element spacing, and \(\xi\) is the phase difference between the currents of the two elements.

We need to find the values of \(d\) and \(\xi\) for which the array factor would have equal intensities in the directions \(\phi = 0, \pi / 2, \pi\), and \(3 \pi / 2\). For \(\phi = 0\): AF(\(0\)) = \(1 + \cos(\beta d \sin 0 + \xi)\) For \(\phi = \pi / 2\): AF(\(\pi / 2\)) = \(1 + \cos(\beta d \sin(\pi / 2) + \xi)\) For \(\phi = \pi\): AF(\(\pi\)) = \(1 + \cos(\beta d \sin \pi + \xi)\) For \(\phi = 3\pi / 2\): AF(\(3 \pi / 2\)) = \(1 + \cos(\beta d \sin(3 \pi / 2) + \xi)\)

To find ξ and d that satisfy the above conditions, we will equate the array factors and then solve the equations. Equating array factors for \(\phi = 0\) and \(\phi = \pi\): \(1 + \cos(\beta d \sin 0 + \xi) = 1 + \cos(\beta d \sin \pi + \xi)\) \(\cos(\xi) = \cos(\beta d - \xi)\) This gives us the first equation: \(\xi = \beta d - \xi\) Now, equating array factors for \(\phi = \pi / 2\) and \(\phi = 3\pi / 2\): \(1 + \cos(\beta d \sin(\pi / 2) + \xi) = 1 + \cos(\beta d \sin(3 \pi / 2) + \xi)\) \(\cos(\beta d + \xi) = \cos(-\beta d + \xi)\) This gives us the second equation: \(\beta d + \xi = -\beta d + \xi\) Let's solve the two equations: 1. \(\xi = \beta d - \xi\) 2. \(\beta d + \xi = -\beta d + \xi\) Adding equations 1 and 2: \(2\xi = 0\) Solving for ξ, we obtain \(\xi = 0\). Substituting ξ back into either Equation 1 or 2, we get: \(\beta d = 0\) As the phase constant \(\beta\) is non-zero, it means that \(d = 0\). However, in a two-element dipole array, the element spacing (d) cannot be zero. Therefore, we should look for the "smallest non-zero" value of d. This occurs when the phase difference between the adjacent elements is an integral multiple of \(2\pi\). Hence, \(\beta d = n (2\pi)\), where \(n\) is an integer and \(n \neq 0\). For the "smallest" value of \(d\), let's take \(n = 1\). Therefore, \(d = \frac{2\pi}{\beta} = \frac{\lambda}{2}\) So, the smallest relative current phasing, ξ, is 0, and the smallest element spacing, d, is \(\lambda / 2\).