d
d
d
Physics 214 Problem 5 Week 2
Phased Array
We can use interference effects to make a very directional antenna (either for transmitting or
receiving). As an example, consider the following situation: 4 radio antennas are equally spaced
in a line, separated by intervals of length d. The antennas all emit radiation coherently, with the
same intensity I1 and the same wavelength λ. Initially the antennas all emit in phase.
1. For concreteness, assume d = λ/4. Draw phasor diagrams corresponding to the net amplitude
far away from the antennas in the directions indicated. Calculate the intensity in each of these
directions. Hint: In considering radiation to the east, the signal from antenna #1 takes longer to
arrive, so its phasor will lag behind that of antenna #2, which will lag behind that of antenna #3,
etc.
N,S: φ = 2πδ/λ If we are far away, then in the up (N) or down (S) directions, all the path lengths
are essentially equal. Therefore, we have δ = 0 = φ:
Atot = 4A1 Itot = 16I1
E: φ = 2πδ/λ In this case, δ = r1 – r2 = d = λ/4. Therefore, φ = 2π(λ/4)/λ = π/2:
Atot = 0 Itot = 0.
W: φ = 2πδ/λ In this case, δ = r1 – r2 =−d = −λ/4. Therefore, φ = 2π(−λ/4)/λ = −π/2:
Atot = 0 Itot = 0.
To the west, the antenna #2 signal lags the antenna #1 signal, so A2 will lag behind A1, and so
on.
Solution
A1
A2
A3
A4
A1
A2
A3
A4
