ECE-GY 9423: Design and Analysis of Communication Circuits and Components (Fall 2024)
Homework # 2 (Due on Tuesday, October 1st, 2024 before 11:59 pm EST.)
HW: All your calculations should have parametric expressions (wherever possible) and numerical answers (wherever required).
Problem 1 (Antenna Arrays):
In this problem, we will understand how an N-element antenna array works and the trade-offs associated with it. Since the transmitters and receiver antennas are reciprocal (we have seen with examples in class), we will just consider the Tx case. The results will be valid for receiver beam control as well.
a. Consider an array of N identical isotropic Tx antennas driven identically in phase. Prove that the array pattern is given by (10pt)
b. Plot the array factor AF (φ) for N=10 for d varying between d = λ/4, d = λ/2, d = λ and d = 2λ . Comment on your choice for d if you would like to direct a beam with maximum efficiency from the Tx to Rx. (10pt)
c. If this array was used as a car radar for autonomous driving, comment on what might happen with d = 2λ? (5pt)
d. For d = λ/2, plot AF (θ) against N varying from N=2 to 10 in one plot and comment on what you observe. Plot the beam bandwidth (where the flux drops by 3 dB) against N. (5pt)
e. Consider now an array with a linear phase gradient as shown below. Write down the expression of AF (θ). (10pt)
f. For d = λ/2 and N = 10 plot AF (θ) for β=0, β =π/4, β =π/2 and β =π in both linear and polar plots. Comment on what you observe for β =π . (10pt)
g. Now consider each antenna in the array to be an infinitesimal (L <<λ) dipole antenna. Write the expression for the total pattern for the array. You can use your results from HW1 for the dipole antenna. (10pt)
h. For d = λ/2 and N = 10 plot the total pattern for the array for β =0, β =π/4, β =π/2 and β =π in both linear and polar plot. Comment on what difference you notice from part f. and whether individual antenna patterns should be directive or not. (7pt)
i. Repeat Part “h” for a full-wavelength dipole antenna (7pt)
j. Also, comment on what you observe for β =π and compare it to part f. (7pt)
k. Calculate the phase difference β 1 and β 2 to turn the beam by -60° and +60° . (6pt)
l. Since the system is linear with respect to excitation and electric field, superposition principle should hold. Therefore, if each element is simultaneously excited by the linear phase gradients of β 1 and β 2, plot the total beam pattern and explain what you observe. (7pt)
m. In this case, plot the amplitude and phase for each element for N=10. (6pt)