Question

Problem 7 An eigth-element linear array with half-wavelength spacing is excited with equal amplitude, as shown. To steer the main beam to a direction 60° below the broadside and toward the +x direction, what should be the incremental phase delay $\delta$ between adjacent elements? Also, give the expression for the normalized array factor $Array(\theta,\phi)$ in spherical coordinates $\theta$ and $\phi$, and plot the normalized power pattern for $0 \le \theta \le \pi$ and $\phi = 0$. $\lambda$ $d = \frac{\lambda}{2}$ $z$ $x$ $\rightarrow$ $Ae^{-j7\delta/2} \quad Ae^{-j5\delta/2} \quad Ae^{-j3\delta/2} \quad Ae^{-j\delta/2} \quad Ae^{j\delta/2} \quad Ae^{j3\delta/2} \quad Ae^{j5\delta/2} \quad Ae^{j7\delta/2}$ Answers: (a) $\delta = -\frac{\sqrt{3}\pi}{2} \approx -156^\circ$, (b) $Array(\theta,\phi) = \frac{\sin(\frac{7}{2}(\theta - \gamma))}{8\sin(\frac{1}{2}(\theta - \gamma))}$, ...

          Problem 7
An eigth-element linear array with half-wavelength spacing is excited with equal amplitude,
as shown. To steer the main beam to a direction 60° below the broadside and toward the +x
direction, what should be the incremental phase delay $\delta$ between adjacent elements? Also,
give the expression for the normalized array factor $Array(\theta,\phi)$ in spherical coordinates $\theta$
and $\phi$, and plot the normalized power pattern for $0 \le \theta \le \pi$ and $\phi = 0$.
$\lambda$
$d = \frac{\lambda}{2}$
$z$
$x$
$\rightarrow$
$Ae^{-j7\delta/2} \quad Ae^{-j5\delta/2} \quad Ae^{-j3\delta/2} \quad Ae^{-j\delta/2} \quad Ae^{j\delta/2} \quad Ae^{j3\delta/2} \quad Ae^{j5\delta/2} \quad Ae^{j7\delta/2}$
Answers: (a) $\delta = -\frac{\sqrt{3}\pi}{2} \approx -156^\circ$, (b) $Array(\theta,\phi) = \frac{\sin(\frac{7}{2}(\theta - \gamma))}{8\sin(\frac{1}{2}(\theta - \gamma))}$, ...

Problem 7
An eigth-element linear array with half-wavelength spacing is excited with equal amplitude,
as shown. To steer the main beam to a direction 60° below the broadside and toward the +x
direction, what should be the incremental phase delay δ between adjacent elements? Also,
give the expression for the normalized array factor Array(θ,ϕ) in spherical coordinates θ
and ϕ, and plot the normalized power pattern for 0 ≤θ≤π and ϕ = 0.
λ
d = (λ)/(2)
z
x
→
Ae^-j7δ/2 Ae^-j5δ/2 Ae^-j3δ/2 Ae^-jδ/2 Ae^jδ/2 Ae^j3δ/2 Ae^j5δ/2 Ae^j7δ/2
Answers: (a) δ = -(√(3)π)/(2)≈ -156^∘, (b) Array(θ,ϕ) = (sin((7)/(2)(θ - γ)))/(8sin((1)/(2)(θ - γ))), ...

Added by Teresa B.

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