Given a 10 isotropic element mounted one above another so that the axis of the array is along the z-axis with a separation of 0.6 \lambda\ between the elements. All elements are fed with the same amplitudes and phases.(a)  Write the expressions for the normalized array factor.(b)  Calculate the angle(s) of maximum radiation for the array. What type of radiation is produced by this array?(c)  Calculate the angle(s) of null radiation for the antenna array.(d)  Calculate and sketch the first-null beamwidth and half-power beamwidth for the principal lobe of the array.(e)  Neglecting mutual impedance effects, calculate the directivity of the array.

(a) The normalized array factor for an isotropic antenna array can be given by the following expression:

AF = [sin(Nψ/2)] / [Nsin(ψ/2)]

where N is the number of elements in the array (in this case, 10), and ψ is the phase difference between adjacent elements, given by:

ψ = βd cos(θ) + α

where β is the phase constant, d is the distance between the elements (0.6λ in this case), θ is the angle from the axis of the array, and α is the progressive phase shift (which is 0 in this case as all elements are fed with the same phase).

(b) The angle of maximum radiation (θ) for the array can be calculated by setting the derivative of the array factor with respect to θ equal to zero and solving for θ. Since all elements are fed with the same phase and amplitude, this array produces a broadside radiation pattern, with the maximum radiation perpendicular to the line of array (θ = 90°).

(c) The angles of null radiation can be found by setting the array factor equal to zero and solving for θ. This will give the angles at which the radiation from the array is zero.

(d) The first-null beamwidth and half-power beamwidth can be calculated using the standard formulas for these quantities for an isotropic antenna array. The first-null beamwidth is the angle between the first nulls on either side of the main lobe, and the half-power beamwidth is the angle between the points on the main lobe where the power drops to half its maximum value. These can be calculated and then sketched on a radiation pattern diagram.

(e) The directivity of the array, neglecting mutual impedance effects, can be calculated using the formula:

D = 4π / Ω

where Ω is the solid angle subtended by the main lobe of the radiation pattern. This can be calculated using the beamwidths calculated in part (d).