# Antenna Array Questions and Answers – Array of N-Isotropic Sources

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This set of Antenna Array Questions and Answers for Aptitude test focuses on “Array of N-Isotropic Sources”.

1. The direction of nulls for broadside array of N –Isotropic sources is given by _____
a) $$cos^{-1}⁡([±\frac{nλ}{Nd}])$$
b) $$cos^{-1}⁡([±\frac{2nλ}{Nd}])$$
c) $$cos^{-1}⁡([±\frac{2πnλ}{2Nd}])$$
d) $$cos^{-1}⁡([±\frac{nλ}{Nd}])$$

Explanation: The nulls of the N- element array is given by $$θ_n=cos^{-1}⁡(\frac{λ}{2πd} [-β±\frac{2πn}{N}])$$
Given it’s a broadside array so β=0
$$θ_n=cos^{-1}⁡(\frac{λ}{2πd} [±\frac{2πn}{N}])= cos^{-1}⁡([±\frac{nλ}{Nd}]).$$

2. The direction of first null of the broadside array of N-Isotropic sources is _____
a) $$cos^{-1}([±\frac{λ}{Nd}])$$
b) $$cos^{-1}⁡([±\frac{πλ}{Nd}])$$
c) $$cos^{-1}⁡([±\frac{2πλ}{Nd}])$$
d) $$cos^{-1}⁡([±\frac{λ}{2Nd}])$$

Explanation: The nulls of the N- element array is given by $$θ_n=cos^{-1}⁡(\frac{λ}{2πd} [-β±\frac{2πn}{N}])$$
Given it’s a broadside array so β=0 and n=1 for first null
$$θ_n=cos^{-1}⁡(\frac{λ}{2πd} [±\frac{2πn}{N}])= cos^{-1}⁡([±\frac{nλ}{Nd}])=cos^{-1}⁡([±\frac{λ}{Nd}])$$

3. The direction of nulls for end-fire array of N –Isotropic sources separated by λ/4 is given by ____
a) $$θ_n=cos^{-1}⁡([∓1±\frac{4n}{N}])$$
b) $$θ_n=sin^{-1}⁡([∓1±\frac{4n}{N}])$$
c) $$θ_n=cos^{-1}⁡([∓1±\frac{2n}{N}])$$
d) $$θ_n=cos^{-1}⁡([∓1±\frac{n}{N}])$$

Explanation: The nulls of the N- element array is given by $$θ_n=cos^{-1}⁡(\frac{λ}{2πd} [-β±\frac{2πn}{N}])$$
Since its given broad side array $$β=±kd=±\frac{2πd}{λ}=±\frac{π}{2},$$
$$θ_n=cos^{-1}⁡(\frac{2}{π} [∓\frac{π}{2}±\frac{2πn}{N}])$$
$$θ_n=cos^{-1}⁡([∓1±\frac{4n}{N}])$$
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4. The necessary condition for the direction of maximum side lobe level of the N-element isotropic array is _______
a) $$ᴪ=±\frac{2s+1}{N} π$$
b) $$ᴪ=±\frac{2s+2}{N} π$$
c) $$ᴪ=±\frac{2s}{N} π$$
d) $$ᴪ=±\frac{2(s+1)}{N} π$$

Explanation: The secondary maxima occur when the numerator of the array factor equals to 1.
⇨ $$sin(\frac{Nᴪ}{2})=±1$$
⇨ $$\frac{Nᴪ}{2}=±\frac{2s+1}{2} π$$
⇨ $$ᴪ=±\frac{2s+1}{N} π .$$

5. The necessary condition for the direction of maximum first side lobe level of the 8-element isotropic array is _______
a) $$\frac{3}{8} π$$
b) $$\frac{3}{4} π$$
c) $$\frac{1}{8} π$$
d) $$\frac{5}{8} π$$

Explanation: The secondary maxima occur when the numerator of the array factor equals to 1.
⇨ $$sin(\frac{Nᴪ}{2})=±1$$
⇨ $$\frac{Nᴪ}{2}=±\frac{2s+1}{2} π$$
⇨ $$ᴪ=±\frac{2s+1}{N}π=\frac{2+1}{8} π=\frac{3}{8} π.$$

6. The necessary condition for the direction of maximum second side lobe level of the 4-element isotropic array is _______
a) $$\frac{5}{4} π$$
b) $$\frac{3}{4} π$$
c) $$\frac{1}{8} π$$
d) $$\frac{5}{8} π$$

Explanation: The secondary maxima occur when the numerator of the array factor equals to 1.
⇨ $$sin(\frac{Nᴪ}{2})=±1$$
⇨ $$\frac{Nᴪ}{2}=±\frac{2s+1}{2} π$$
⇨ $$ᴪ=±\frac{2s+1}{N}π=\frac{2(2)+1}{4} π=\frac{5}{4} π.$$

7. The Half-power beam width of the N-element isotropic source array can be known when _____
a) $$ᴪ=\frac{2.782}{N}$$
b) $$ᴪ=\frac{1.391}{N}$$
c) $$ᴪ=\frac{1.414}{N}$$
d) $$ᴪ=\frac{3}{N}$$

Explanation: Normalized array factor is given by $$AF=\frac{sin(Nᴪ/2)}{N \frac{ᴪ}{2}}=\frac{1}{√2}$$
⇨ $$\frac{Nᴪ}{2}=1.391$$
⇨ $$ᴪ=\frac{2.782}{N}$$

8. Which of the following is the necessary condition to avoid grating lobes in N-Isotropic element array?
a) $$\frac{d}{λ}≤\frac{1}{1+|cosθ_m |}$$
b) $$\frac{d}{λ}≥\frac{1}{1+|cosθ_m |}$$
c) $$\frac{λ}{d}≤\frac{1}{1+|cosθ_m |}$$
d) $$\frac{λ}{d}=\frac{1}{1+|cosθ_m |}$$

Explanation: Grating lobes are the minor and unnecessary lobes other than the major lobe.
To avoid grating lobes, kd(cosθ-cosθm) ≤ 2π
θm – Direction of maximum radiation
⇨ $$\frac{2πd}{λ} (cosθ-cosθ_m)≤2π$$
$$\frac{d}{λ}≤\frac{1}{cosθ-cosθ_m}$$
$$\frac{d}{λ}≤\frac{1}{1+|cosθ_m |}$$

9. Which of the following is the necessary condition to avoid grating lobes in N-Isotropic element broadside array?
a) d < λ
b) d > λ
c) d=λ
d) d < 2λ

Explanation: Grating lobes are the minor and unnecessary lobes other than the major lobe.
To avoid grating lobes, kd(cosθ-cosθm)≤2π
$$\frac{d}{λ}≤\frac{1}{1+|cosθ_m |}$$
For broadside to avoid grating lobes (θm=90)
⇨ $$\frac{d}{λ}$$ < 1
⇨ d < λ

10. Which of the following is the necessary condition to avoid grating lobes in N-Isotropic element end-fire array?
a) d < λ/2
b) d < λ
c) d > λ/2
d) d=λ

Explanation: Grating lobes are the minor and unnecessary lobes other than the major lobe.
To avoid grating lobes, kd(cosθ-cosθm)≤2π
$$\frac{d}{λ}≤\frac{1}{1+|cosθ_m|}$$
For broadside to avoid grating lobes (θm=0)
⇨ $$\frac{d}{λ}$$ < 1/2
⇨ d < λ/2

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