5- Four isotropic sources are placed along the z-axis as shown below. Assuming that the
amplitudes of elements #1 and #2 are +1, and the amplitudes of #3 and #4 are -1, find:
a) the array factor in simplified form b) the nulls when d = λ 2 .
a)
b)
1- Give the array factor for the following identical isotropic antennas with N and d.
1- k or β = 2π λ is called the Phase Constant.
The formula
sin(nψ 2)
sin(ψ 2)
was first introduced by Schelkunoff.
The phase change along a distance of a quarter wavelength is π 2 .
As the sign of the formula
sin(nψ 2)
sin(ψ 2)
changes from positive to negative, the phase of
the pattern changes by π .
The maximum value of
The maximum value of
sin(nψ 2)
sin(ψ 2)
sin(nψ 2)
sin(ψ 2)
) is n .
occurs in any direction for which ψ = 0.
For a Broadside Array:
the phase difference, α between elements of a “Broadside Array” is α = 0 .
in general, as the element spacing is increased, the main lobe beamwidth is decreased.
when the element spacing is d ≥ λ grating lobes are introduced
as the number of elements increases, the main lobe beamwidth is decreased.
For an End Fire Array:
the array radiates in the direction along the line of the array.
the phase difference, α between elements of a “Endfire Array” is α = ±kd .
as d changes α must change to keep the main beam of the array in the same direction.
when the element spacing is d ≥ 0.5λ grating lobes are introduced.
to increase the directivity one can increase the phase difference, α between elements.
1- Aralarındaki faz farkı π 2 olan iki tane izotropik anten y ekseninde y = ± λ 4
noktalarına dizilmiştir. Hangi φ açılarında bu anten dizsinin elektrik alanı sıfır olur?
1- Two isotropic antennas are placed on the y-axis, at y = ± λ 4 with an intrinsic phase shift
of π 2 . Given that the azimuthal angle, φ , is defined as the angle from the x-axis, for what
values of φ are there nulls for the electric field pattern in the xy plane?
kd =
sin [ 12 N (α + kd cos θ)]
2π λ
= π, AF (θ) n =
,θ =
λ 2
N sin [(α + kd cos θ)]
AF (φ) n =
π
2
−φ →
sin [ 21 N (α + kd sin φ)]
sin [( π2 + π sin φ)]
π
=
, π sin φ = → φ = sin−1 21 = 30°,150°
π
π
1
2 sin [( 4 + 2 sin φ)]
2
N sin [ 2 (α + kd sin φ)]
2- 5 tane izotropik antenden oluşan anten dizisi aşağıda gösterilmektedir.
a) antenler arasındaki faz farkı α ’yı bulun.
b) Radyasyonun θ = 110° ’de maksimum yapması için gereken d’yi λ cinsinden bulun.
c) N=5 için AF n (Ψ) grafiğini çizin.
d) Radyasyonun olacağı başlangıç ve bitiş Ψ açılarını bulun.
e) Radyasyonu kutupsal koordinatlarda çizin.
2- A 5-element array of isotropic sources is shown below.
a) Find the interelement phase shift α .
b) Find the value of d λ if the main beam is directed in the direction θ = 110° .
c) Sketch AF n (Ψ) for N=5.
d) Find the visible range in Ψ .
e) Sketch the polar plot of the radiation.
π
π
2π
π
= 0 ⇒ kd cos θ = − ⇒
d cos110° = − ⇒ d = 0.365 λ
4
4
λ
4
π
π
−kd + α ≤ Ψ ≤ kd + α → − 0.73π + 4 ≤ Ψ ≤ 0.73π + 4 → − 0.48π ≤ Ψ ≤ 0.98π
α = π 4, Ψ = kd cos θ +
N = 5 ⇒ AFn (Ψ) =
−6
−5
−4
sin(5 Ψ 2)
5 sin(Ψ 2)
−3
−2
→ AFn (θ) =
−1
sin [ 25 (kd cos θ + π4 )]
5 sin [ 21 (kd cos θ + π4 ]
0
1
2
3
4
5
6
3- Design a 7-element, uniformly excited, equally spaced linear array along the z-axis. Select
the element spacing d and linear phasing α such that the beam width is as small as possible
and also so that no part of a grating lobe appears in the visible region. Provide the angles
where pattern nulls occur, and provide computed-generated polar plots of the patterns when
the main beam at broadside (θο=90o).
N = 7 → AF =
1 sin(7 Ψ 2)
12π
6
→d = λ
, α = 0, kd =
7 sin(Ψ 2)
7
7
7Ψ 
Ψ
7ΨN
m
= m π, m = 1,2, 3,,, &
≠ 1, 2, 3,
nulls at sin   = 0 & sin   ≠ 0 →
 2 
2
2
7
−6
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
3- Design a 7-element, uniformly excited, equally spaced linear array along the z-axis. Select
the element spacing d and linear phasing α such that the beam width is as small as possible
and also so that no part of a grating lobe appears in the visible region. Provide the angles
where pattern nulls occur, and provide computed-generated polar plots of the patterns when
the main beam at (θο=80o).
N = 7 → AF =
1 sin(7 Ψ 2)
, Ψ = kd cos 80° + α = 0 → α = −kd cos 80° = −0.1736kd
7 sin(Ψ 2)
12π
→ d ≤ 0.73λ → α = −0.1736kd = −0.797
7
visible range 0 ≤ θ ≤ π → − kd + α ≤ Ψ ≤ kd + α → − 5.38 ≤ Ψ ≤ 3.79
−0.1736kd − kd ≥ −
−6
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
3- Design a 4-element uniformly excited, equally spaced linear array along the z-axis. The
main beam maximum should point to θ = 75o. Find the d and α such that the beam width is as
large as possible and that full main lobe should be visible. Sketch the polar plot of the
radiation pattern.
Ψ = kd cos θ + α → 0 = kd cos 75° + α → α = −kd cos 75° = −0.31 kd .
kd − 0.31 kd =
−6
−5
π
⇒ kd = 0.725π ⇒ d = 0.3623λ ⇒ α = −0.31(0.725π) = −0.225π = 40.43°
2
−4
−3
−2
−1
0
1
2
3
4
5
6
6- Array factor of a 5-element uniformly excited, equally spaced linear array is drawn below.
Find α , the progressive phase shift and d, the inter-element spacing for a pattern with exactly
3 side-lobes and the main beam at θ = 45 and a null in the back lobe ( θ = 180 ).
kd
Ψ = kd cos 45 + α = 0 → α = −kd cos 45° = −
.
2
8π
kd
8π
kd
0.94π
α − kd = −
⇒−
− kd = −
→ kd = 0.937 π → d = 0.47λ, α = −
=−
= −0.66π
5
2
5
2
2
−6
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
3- Determine the progressive phase shift α for a 5-element array equally spaced with
d = λ 2 along the z-axis so that the main beam occurs at θ0 = π 4 . Sketch the polar plot.
3- z-ekseninde yarım dalga boyu aralıkla dizili 5-elemanlı bir anten dizisinin ana huzmesinin
θ0 = π 4 ’da gerçekleşmesi için faz farkı α ’yı bulun. Dizi faktörünü kutupsal çizin.
d=
−6
λ
kd
2πλ
π
, Ψ = kd cos π4 + α = 0 ⇒ α = −kd cos π4 → α = −
=−
=−
= −0.87 π
2
2
λ2 2
2
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
3- Design a 7-element array along the z-axis consisting of λ/2 wave dipoles collinear with the
z-axis. Specifically, determine the interelement phase shift α and the dipole center-to-center
spacing d to point the main beam at θ = 5° and to provide the widest possible beamwidth.
Ψ = kd cos θ + α = 0 ⇒ α = −kd cos θ = −kd cos 5°
since we want a wide beamwidth, the dipoles should be as close together as possible.
⇒ d = λ 2 (any closer thay would touch) so kd = π → α = −kd cos 5° ≃ −π
−6
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
1- Consider the design of a linear uniformly-spaced phased array of N identical antenna
elements with spacing d. The main beam is to be aimed at an angle θ = π 4 radians from the
axis of the array. The beamwidth is to be 5.7 degrees and there are to be no grating lobes in
the visible range of angle 0 ≤ θ ≤ π . In considering the design ignore the element beam-shape
(i.e, assume that the elements radiate isotropically)
a) Specify the progressive phase shift α between the elements in terms of d and λ.
AFn =
2πd cos π 4
1 sin(N Ψ 2)
λα
2πd cos θm
max when cos θm =
⇒α=
=
=
N sin(Ψ 2)
2πd
λ
λ
2πd
λ
b) In order to use the smallest number of elements, you need the greatest spacing d consistent
with the absence of grating lobes. Determine this maximum spacing as a fraction of λ and the
associated minimum number of elements.
kd + kd cos θm ≤ 2π → kd +
kd
λ 2
λ
≤ 2π ⇒ d ≤
=
= 0.5858λ
2
2 + 1 1.71
4π
5.7 π
180
=
→ N = 4π
= 126
N
180
5.7 π
3- Consider a 4-element uniformly excited equally spaced broadside array along the z-axis
with d = λ 2 . Where does the main beam point? Plot the radiation pattern (polar plot).
N = 4, d = λ 2 ⇒ βd = π, Ψ max = α + βd cos θmax , Broadside → θmax = 90° ⇒ α = 0
−6
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
3- Consider an N-element uniformly excited equally spaced array along the z-axis with
d = λ 2, α = π 2 . Where does the main beam point? Sketch the polar plot for a) N=4 b) N=5.
 λα 
 1
kd = π (radius), α = π 2 (center), θm = arccos −
= arccos −  = 120°


 2
 2πd 
3- Consider an array consisting of two isotropic radiators positioned symmetrically about z=0
along the z-axis. The elements have uniform amplitude and progressive phase shit α. The
spacing between elements is d. For d = λ 2 and α = −π 2 . Sketch the array factor AF (Ψ)
vs. Ψ and sketch also the far field polar plot.
AF (Ψ) =
−6
−5
sin(Ψ )
sin(π(cos(θ) − 21 ))
= cos(Ψ 2) → a(θ) =
2 sin(Ψ 2)
2 sin( π2 (cos(θ) − 21 ))
−4
−3
−2
−1
0
1
2
3
4
5
6
3- Consider an array consisting of two isotropic radiators positioned symmetrically about z=0
along the z-axis. The elements have uniform amplitude and progressive phase shit α. The
spacing between elements is d. Sketch AF (Ψ) and sketch also the far field polar plot for:
a) d = λ, α = 0 b) d = λ 4, α = 0 c) d = λ 4, α = − π 2 d) d = λ 4, α = − π 2
AF (Ψ) =
a(θ) =
a(θ) =
sin(Ψ )
= cos(Ψ 2)
2 sin(Ψ 2)
sin(2π cos(θ))
2 sin(π cos(θ))
sin( π2 (cos(θ) − 1))
2 sin( π4 (cos(θ) − 1))
a(θ) =
sin( π2 (cos(θ) + 1))
2 sin( π4 (cos(θ) + 1))
1- A 4-element array antenna consists of z-oriented half-wave dipole radiators separated by a
distance d = λ 4 along the x-axis. We are interested in radiation in the x-y plane only.
a) Assuming initially equal excitation (i.e. α = 0 ), what would be the direction of maximum
radiation and the directivity of this array antenna? (To help solve this part, you may want to
compare radiated power densities of the array and an isotropic radiator of equivalent Prad,
assuming a feed current of 1 [A] and a range of 1 [m]).
b) What excitation phase progression α of the dipoles would be required to obtain maximum
radiation at φ = 70° ?
3- Design a uniform linear array with minimum number of elements and no grating-lobes:
a) Find the number of elements such that the side lobe level peak is less than 0.26. The
definition of the sidelobe level is the level of the peak of the normalized array function of the
side lobe next to main beam.
The definition of a sidelobe level is the relative intensity level of the pattern between the peak
of the main beam and the peak of the sidelobe in question.
b) Plot the polar pattern if the array is endfire.
Ψ SLL =
3π
1
1
, PSLL =
=
, PSLL(N = 2) = 0.707,
N
N sin(Ψ SLL 2) N sin(3π 2N )
PSLL(N = 3) = 0.333, PSLL(N = 4) = 0.271, PSLL(N = 2) = 0.259 ⇒ N = 5
end fire → Ψ max = α + βd cos θmax = 0 ⇒ α = −βd
For No Grating Lobes ⇒ 2βd ≤ 2π − 2π 5 → βd ≤ 0.8π → 2πd λ ≤ 0.8π → d ≤ 0.4λ
−6
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
4- Consider an array of small loop antennas positioned along the z-axis with interelement
spacing d and interelement phase shift α. The loops are oriented such that the plane of each
loop is parallel to the xy-plane. Determine the required interelement phase shift α to point the
main beam at θ = 20° . Can this array produce a beam directed at θ = 0° (i.e., along the zaxis)? Why or why not?
Ψ = kd cos θ + α = 0 ⇒ α = −kd cos θ ⇒ α = −kd cos 20° = −0.94kd
No. Individual loops do not radiate towards θ = 0°
1- Draw the magnitude of the normalized array factor versus Ψ in radians for a 6-element
uniformly excited, equally spaced array along the z-axis. If the interelement phase shift is
α = 2π 3 , determine the interelement spacing d to provide the narrowest possible beam
without inducing grating lobes. Where does the main beam point? Draw the polar plot of the
array factor.
2π
10π
10π 4π
+ kd ≤
→ kd ≤
−
= π → d = 0.5λ
3
6
6
6
 λα 
α
 2
Ψ = kd cos θm + α = 0 → cos θm = −
→ θm = arccos −
 = arccos −  = 131.8°

kd
2πd
3
π
5π
(visible range) → α − kd ≤ Ψ ≤ α + kd → − ≤ Ψ ≤
3
3
−6
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
4- Dört tane ayni olan anten z ekseninde broadside yayın yapmak için dizilmişlerdir.
a) antenler arasındaki faz farkı α ne olmalıdır?
b) grating denen loblar istenmiyorsa d’nin alacağı maksimum değer ne olmalı?
c) side denen loblar istenmiyorsa d’nin alacağı maksimum değer ne olmalı?
d) Dizi faktörünün polar diyagramını α = 0, kd = 3π 2 için çizin.
3- Determine the elements spacing d for a 4-element equally spaced broadside array along the
z-axis. Determine the nulls. Sketch the polar plot.
3π
d
3
kd
π
d
3
→ = = 0.75, c) π = 1 → kd = → = = 0.25
2
λ
4
2
λ
4
2
1 sin(2 Ψ)
nπ
n π n = 1, 2, 3,....
(AF )n =
, nulls ⇒ sin(2 Ψ) = 0 ⇒ Ψ = ±
⇒ kd cos θn = ±
,
4 sin(Ψ 2)
2
2 n ≠ 0, 4, 8,12,.....
a) α = 0, b)
kd
π
2
= 3 → kd =
3- Design a 5-element uniformly excited, equally spaced linear broadside array. Select the
element spacing such that the beam width is as small as possible and also so that no part of
the grating lobe appears in the visible region. Show a polar plot of the radiation pattern.
8
kd ≤ π → d ≤ 0.8λ
5
5- 4 tane birbirinin ayni olan anten z ekseninde dizilmişlerdir. Antenler arasındaki mesafe d
ve faz farkları α olsun. Bu dizinin dizi faktörünün aşağıdaki gibi çizildiğini kabul edelim.
Antenlerin dizildiği eksene dik ışıma yapılması isteniyor.
a) “grating” denen loblar istenmiyorsa d’nin alacağı maksimum değer ne olmalı?
b) “side” denen loblar istenmiyorsa d’nin alacağı maksimum değer ne olmalı?
c) Dizi faktörünün polar diagramını kd = π ve α = π 2 için çizin.
5- Consider a broadside array of 4 identical antennas positioned along the z-axis with
interelement spacing d and interelement phase shift α . The array factor is drawn below.
a) What is the max. value of d if the grating lobes must be avoided?
b) What is the max. value of d if the side lobes must be avoided?
c) Sketch the far-field polar pattern due to the array factor if kd = π & α = π 2 .
a) kd = 3π 2 ⇒ 2πd λ = 3π 2 ⇒ d = 3λ 4
b) kd = π 2 ⇒ 2πd λ = π 2 ⇒ d = λ 4
c) For main lobe angle Ψ = 0 = kd cos θ + α = π cos θ + π 2 ⇒ θ = 120
main lobe
grating lobe
side lobes
− 2π
−
3π
2
−π
−
π
π
2
2
α =π 2
π
3π
2
Ψ
2π
kd = π
θ
1- Three half-wave dipoles are aligned parallel to the z-axis, but have their centers located at
x = − λ 2, 0, λ 2 on the x-axis. The dipoles are driven in phase, with equal amplitudes.
a) Sketch the element pattern in the x-y plane.
b) Determine the AF and sketch the polar plot of the AF and the total pattern in the x-y plane.
f (θ) = 1, AFn =
sin(3 Ψ 2)
3 sin(Ψ 2)
, d = λ 2, α = 0 , array and the total pattern is the same.
2- Answer true of false:
a) The array factor sin(N ψ2 ) sin( ψ2 ) applies to arrays with non-uniform current amplitudes.
FALSE
b) An array with non-uniform (binomial, Chebyshev, etc.) current amplitudes has higher
directivity than the same array with uniform current amplitudes.
FALSE
c) An array of aperture antennas can be analyzed using pattern multiplication.
TRUE
d) An array of log-periodic dipole arrays can be analyzed using pattern multiplication.
TRUE
1- İki tane kısa dipol anten aşağıdaki şekildeki gibi y-eksenine paralel ve x-ekseninde
x = ± λ 4 noktalarına yerleştirilmiştir. Dipoller aynı genlik ve faza sahipler.
a) Çok kısa dipol antenin eleman patternini x-y düzleminde çizin.
b) Dizi faktörünü bulun ve dizi patternini x-y düzleminde çizin.
c) Toplam patterni x-y düzleminde çizin.
1- Two short dipoles are aligned parallel to the y-axis, but have their centers located at
x = ± λ 4 on the x-axis as shown. Dipoles are driven in phase, with equal amplitudes.
a) Sketch the element pattern in the x-y plane.
b) Determine the array factor and sketch the polar plot of the array factor in the x-y plane.
c) Sketch the total pattern in the x-y plane.
(a)
(b)
AFn = cos( π2 sin φ)
(c)
3- Design a 7-element array along the x-axis. Specifically, determine the interelement phase
shift α and the element center-to-center spacing d to point the main beam at
θ = 25°, φ = 10° and provide the widest possible beamwidth.
1- Four short dipoles are arranged in a linear array along the x-axis. The dipoles have a field
pattern Edipole = cos(φ). The dipoles are spaced d=λ/2 and fed in phase.
a) Derive an expression for the array factor of the four element linear array.
since δ= 0
AF =
sin(nΨ 2) j ( n −21) Ψ
e
sin(Ψ 2)
for n=4 AF =
sin(2Ψ ) j 32Ψ
e
sin(Ψ 2)
b) What is the expression for the total field pattern, E(φ) of the four dipole array?
ETotal = Edipole x Earray factor = cos(φ )
sin(2Ψ ) j 3Ψ 2
e
sin(Ψ 2)
c) Explain the principle of pattern multiplication. Show a sketch illustrating this principle.
when you have similar sources, like the dipoles, the total field pattern can be expressed as the
product of element pattern times the array factor as expressed in part (b).
d) Does the pattern multiplication apply if the sources in the array are not the same? If yes,
demonstrate. If not, how is the field pattern determined?
No! Pattern multiplication does not apply if the sources are dissimilar. In this case you must
sum the sources at each point in space to get the total field.
1- Consider a 90° corner reflector with a λ/4 dipole antenna spacing of s from the apex
(origin). The main directions of propagation is along the y-axis and the reflectors are at φ =
45° and φ = 135°.
a) Calculate the array factor. Write the total system pattern. Plot the E & H patterns for s =
0.5 λ (radial-linear) and calculate the directivity (numerical integration).
b) Calculate the field on the y-axis for s = 1.0 λ. Give a physical explanation of the result.
When θ = π/2, φ = π/2, s = λ, EF = 0 then E=0. Out of phase elements propagate 2πn in phase
to reach in phase elements, and thus cancel in that direction
c) For s = 0.5 λ, plot the E-pln pattern at f = 0.66 f0, 0.80 f0, 1.2 f0, 1.5 f0 (all on the same
page and normalized to the peak of s = 0.5λ). If you have to design a corner reflector for
maximum pattern bandwidth, then which spacing would you use at f0?
Fix s = λ0/2, L = λ0/4. D(0.66 f0) = 7.82, D(0.8 f0) = 11.7, D(1.2 f0) = 11.9, D(1.5 f0) = 7.04
3- Three half-wave dipoles are aligned parallel to the x-axis, but have their centers located at
z = ± λ 2 and z = 0 on the z-axis. Assuming that they are driven in phase, with equal
amplitudes, use pattern multiplication to sketch the far-field pattern in the zy plane.
z
z
y
y
x
d = λ 2, α = 0, Ψ = kd cos θ = π cos θ, f (θ) = 1, N = 3 → AF (θ) n =
sin( 32π cos θ)
3 sin( π2 cos θ)
7- Consider a uniform linear array antenna consisting of two vertical half-wave dipoles that
are one wavelength apart. The phase difference between the driving currents on the two
elements is 90 . For the horizontal plane (x-y plane) analytically determine the directions of
the nulls in the radiation pattern, and generate a plot of the normalized radiation pattern.
z
I
I e jπ 2
y
θ=
π
2
x
: E n = 2 1 − cos ( kl 2 ) cos ( sin ϕ kd 2 + Ψ 2 ) , kl 2 = π 2 , kd 2 = π ,
Ψ =π 2
E n = 2 cos(π sin ϕ + π4 ) = 0
when sin ϕ = n + 1 4, n = 0, ± 1, ± 2
n = 0 → ϕ = 14.5 , ϕ = 165.5
n = 1 → ϕ = 311.4 , ϕ = 228.6
4- Consider 3 short dipoles, pointed in the x-direction, located on the z-axis at z = ± λ 8 , and
z = 0 as shown in the figure. Assume that they are driven in phase, with equal amplitudes.
z
y
x
a) Sketch the far-field pattern for yz-plane and for xz-plane.
1 sin(N Ψ 2)
π
f (θ, φ) = 1 − sin2 θ cos2 φ , AF =
, N = 3, d = λ 8, Ψ = kd cos θ = cos θ
4
N sin(Ψ 2)
yz − plane f (θ, φ = 90°) = 1
xz − plane f (θ, φ = 0°) = cos θ
3π
1 sin( 8 cos θ)
F (θ, φ) =
3 sin( π8 cos θ)
3π
1 sin( 8 cos θ)
cos θ
F (θ, φ) =
3 sin( π8 cos θ)
θ
θ
3- A two-element vertical (z-directed) half-wave dipole array are located one wavelength
apart symmetrically along the z-axis.
a) Determine the phase shift between the elements to maximize array factor at θ = 80o.
AF = 2e
jα 2
cos [ 21 (kd cos θ + α)], kd = 2π → AF = 2e
jα 2
cos(π cos θ + α 2)
Ψ = α + kd cos θ = 0 → α = −kd cos θ = −2π cos 80° = −1.09 rad( − 62.5°)
b) Determine the array factor defined in part (a).
AF = 2e − j 0.545 cos(π cos θ − 0.545)
c) Determine the nulls of the array factor of part (a).
π
3π
1
3
π cos θ − 0.545 = ± , ±
⋅ ⋅⋅ ⇒ cos θ = ± + 0.173, ± + 0.173, ⋅ ⋅ ⋅
2
2
2
2
−1
⇒ θ = cos {−0.327, 0.673} = 1.904, 0.833 rad {109.1°, 47.7°}
d) Determine the far field electric field of the array.
E=
j µ0I 0 − j (kr +0.545)
cos( π2 cos θ)
e
cos(π cos θ − 0.545)
aˆθ
πr
sin θ
8- Consider a uniform linear array antenna consisting of two vertical half-wave dipoles that
are one wavelength apart. The phase difference between the driving currents on the two
elements is 90o. Analytically determine the directions of the nulls in the radiation pattern and
sketch the normalized radiation field pattern in
a) the x-z plane; x − z plane: ϕ = 0 or ϕ = 180 → sin ϕ = 0 ,
cos ( kl2 cos θ ) − cos ( kl2 )
En =2
sin θ
E n = 0 when cos θ = 2n + 1
z
I
I e jπ
cos ( Ψ2 ) , kl 2 = π 2 ,
n = 1 → θ = 180
2
E n =2
cos ( π2 cos θ )
sin θ
cos ( π4 )
, n = 1 → θ = 180
z
y
x
x
b) the y-z plane; y − z plane: ϕ = 90 or ϕ = 270
E n =2
cos ( kl2 cos θ ) − cos ( kl2 )
sin θ
 kd
Ψ
cos  sin θ sin ϕ +  , kl 2 = π 2 , kd 2 = π ,
2
2
Ψ =π 2
E n = 0 when cos ( π2 cos θ ) = 0 or cos (π sin θ sin ϕ + π 4 ) = 0
cos ( π2 cos θ ) = 0 for θ = 0 , 180
for ϕ = 90 , sin θ = n + 1 4 , n = 0 → θ = 14.5 , θ = 165.5 , n = −1 → θ > π
for ϕ = 270 , sin θ = − n − 1 4 , n = 0 → θ > π , n = −1 → θ = 48.6 , θ = 131.4
9- At points A and B, the signal strength, when a single dipole antenna operates, is E1A and
E1B , respectively. How will the electric field magnitude change if a second identical antenna
is added a distance d = λ 2 from the first antenna, a shown in the figure? The two antennas
are driven by currents with equal magnitudes. The current on the second antenna is phase
shifted by π 2 . The distance of each points of A and B to the antennas are much larger than
the spacing between the antennas. Express the signal at points A and B in terms of E1A and
E1B , respectively.
B
{
d =λ 2
Ie jπ
I
y
2
30
A
x
One antenna : E1
Two antenna : E = F E1
Ψ
 kd
F = 2 cos  sin θ sin ϕ + 
2
 2
d=
λ
2
⇒
kd π
=
2 2
Ψ=
π
2
⇒
Ψ π
=
2 4
horizontal plane (x-y plane) θ = π 2 , sin θ = 1
π
π
F = 2 cos  sin θ sin ϕ + 
4
2
Point A : ϕ A = 30 →
Point B : ϕ B = 210 →
π
π
FA = 2 cos  sin 30 +  = 0 , E A = FA E1 A = 0
4
2
π
π
FB = 2 cos  sin 210 +  = 2 , E B = FB E1B = 2 E1B
4
2
1- Sadece tek anten kullanıldığında A ve B noktalarındaki elektrik alan şiddeti E1A ve
E1B ’dir. Şekilde görüldüğü gibi d = λ 2 uzaklıkta ikinci bir anten ilave edildiğinde A ve B
noktalarındaki alan şiddetini E1A ve E1B cinsinden hesaplayın. İki antenin akımlarının
mutlak değeri eşit fakat ikinci antenin faz farkı π 2 ’dir. A ve B noktalarının uzak alanda
bulunduğunu varsayın.
1- At points A and B, the signal strength, when a single dipole antenna operates, is E1A and
E1B , respectively. How will the electric field magnitude change if a second identical antenna
is added a distance d = λ 2 from the first antenna, a shown in the figure? The two antennas
are driven by currents with equal magnitudes. The current on the second antenna is phase
shifted by π 2 . The distance of each points of A and B to the antennas are much larger than
the spacing between the antennas. Express the signal at points A and B in terms of E1A and
E1B , respectively. (Hint: use unnormalized antenna factor expression).
z
A
d =λ 2
{
Ie jπ
2
30
y
I
B
One antenna: Point A : θ = 60 →
Two antenna: Point A : θ = 60 →
Point B : θ = 120 →
E A = E1 A
E B = E1B
E A = AF (θ = 60 ) E1 A Point B : θ = 120 →
E B = AF (θ = 120 ) E1B
AF =
(
) , Ψ = π 2 + π cos θ → AF
sin N Ψ 2
( )
sin Ψ 2
Point A : AF (θ = 60 ) =
Point B :
sin (π 2 + π cos 60
AF (θ = 120 ) =
)
sin (π 4 + π 2 cos 60
AF (θ = 120 ) =
(
sin π 2 + π cos θ
=
)
(
sin π 4 + π 2 cos θ
(
(
)
sin π 4 + π 2 cos120
π 2 sin120
)
=
0
, take the derivative
0
(
) =
cos (π 4 + π 2 cos120 )
π sin120 cos π 2 + π cos120
)
0
= 0 , EA = 0
1
=
sin π 2 + π cos120
)
3π 2
3π 4
= 2 , E B = 2 E1B
1- Six vertical dipoles each of length 0.5 λ are mounted on a z-directed vertical mast with a
center-to-center spacing d = 0.8 λ and are excited in phase.
a) Assuming that the antenna array is in free space, write a complete expression for the total
radiated field as a function of angles θ and φ. Define angles θ and φ for this antenna array.
b) Neglecting mutual impedance effects, calculate the directivity for this 6- element antenna
array.
c) Calculate the angles θ for directions of maximum radiation.
d) Calculate the BWFN for this 6-element antenna array.
2- A 20-element equally spaced, uniformly excited antenna array uses an interelement
spacing d=0.25λ. For this antenna array, calculate the following:
a) The progressive phase shift needed for the major lobe to be in the end fire direction.
b) The phase shift needed for the major lobes to be launched at angles of ± 60° relative to the
line of the array.
c) The beam widths between first nulls for cases a and b.
d) The directivities of the antenna for cases a and b assuming that the directivity Do of each of
the elements is 1.95. Neglect mutual impedance effects for calculation of the directivity.
2- Consider the three element array depicted below where the spacing between the elements
is d = λ/4. Find the normalized array factor, find the positions of nulls, and maxima of the
array factor and sketch the array factor as a function of θ in polar form. Assume that the
excitation coefficients are given by I 1 = j, I 2 = 2, I 3 = −j .
π
π
kd = π 2, AF = 2 + je j 2 cos θ − je − j 2 cos θ = 2 − 2 sin ( π2 cos θ )
AFmax = 4 ⇒ (AF )n = 21 1 − sin ( π2 cos θ ) = 12 1 − cos ( π2 cos θ − π2 ) = sin2 ( π4 cos θ − π4 )
,
Nulls:
sin ( π2 cos θ ) = 1 ⇒ π2 cos θ = π2 ⇒ θ = 0
Max: sin ( π2 cos θ ) = −1 ⇒
π
2
cos θ = − π2 ⇒ θ = π
2- Consider the three element array depicted below where the spacing between the elements
is d = λ/4. Assume that I 1 = −j, I 2 = 2, I 3 = j .
a) Find the normalized array factor.
AF = I 1
e − j βr1
e − j βr2
e − j βr3
1
1
1
1
+ I2
+ I1
. In the far-zone ≈ ≈ ≈
r1
r2
r3
r1
r2
r3
r
For the phase terms r1 = r − d cos θ, r2 = r, r3 = r + d cos θ
e − j βr
e − j βr − j π 2 j βd cos θ
jπ 2
e
e
+ 2 + e e − j βd cos θ
(I 1e j βd cos θ + I 2 + I 3e −j βd cos θ ) =
r
r
− j βr
− j βr
 βd cos θ π 
e
e
AF =
2 + 2 cos(βd cos θ − π 2) =
4 cos2 
−  ⇒ AF max = 4
 2
r
r
4
 βd cos θ π 
π
π
AF n = cos2 
− , βd = π 2 → f (θ) = cos2  cos θ − 

 2

4
4
4
(
AF =
(
)
)
b) Find all the nulls for 0 ≤ θ ≤ π.
π
π
π
π
π
f (θ) = cos2  cos θ −  = 0 → cos θ − = + n π n = 0, ±1, ±2, ±3, ⋅ ⋅ ⋅
4
4
4
4
2
π
3π
cos θ =
+ n π ⇒ cos θ = 3 + 4n, n = −1 ⇒ cos θ = −1 ⇒ θ = 180°
4
4
3- Design an 18 element uniform linear array with a spacing of λ/4 between the elements.
a) What is the progressive phase shift (α , radians) between the elements so that the maximum
of the array factor is θ=50o from the line (z-axis) where the elements are placed?
N = 18, θ0 = 50°, d = λ 4, α = −βd cos θ0 =
2π λ
π
cos 50° = − cos 50° = −1.01 rad
λ 4
2
b) Plot the magnitude (in dB) of the normalized array factor for 0 ≤ θ ≤ π. Determine the halfpower beamwidth (in degrees) and the maximum level (dB) of the first minor lobe.
Ψ = βd cos θ + α = π2 (cos θ − cos 50°), f (θ) =
sin(N Ψ 2)
N sin(Ψ 2)
=
sin( 92π (cos θ − cos 50°))
18 sin( π4 (cos θ − cos 50°))
HPBW ≈ 57° − 42° = 15° . First minor lobe (side lobe) ≈ −13.2 dB @ θ=16° and θ=71°
3- Uniform linear array antenna consists of two antenna elements. Determine the array
parameters (element length, l, elements separation, d, and current phase shift, ψ ) so that the
signal at point A is doubled compared to the signal when only one antenna element operates
and there is no signal towards point B. Give the length, l, and the separation, d, in terms of the
operating wavelength.
3- A very common and low-cost HF & VHF antenna is a single λ/2 (folded) dipole backed by
a ground plane.
a) Calculate the directivity of this antenna for a ground-plane spacing of h = λ/4 and 3λ/4
(dipole is z-oriented and the ground plane is the x-z plane). Also, plot the E and H-planes for
both cases (radial, linear) on the same graph. Explain why the h = 3λ/4 case has high
sidelobes. What are the sidelobes in dB?
EF =
cos ( π2 cos θ )
sin θ
Low side lobe in E-plane due to element pattern
b) One way to get more directivity is to use an array of two dipoles spaced 0.5λ from each
other and fed in phase (dipoles are z-oriented at y = λ/4, x = +- λ/4; the ground plane is the xz plane). The ground plane is at h = λ/4 away from the dipole. Calculate the E & H radiation
patterns (radial, dB to -20 dB) and the resulting directivity.
Element pattern same as above
2- A common 860 MHz cellular antenna is a 3-element dipole array (on the z-axis) with each
element being λ/2 long. The array is placed on top of a cellular tower. (The "dipole" is really
a folded-dipole and we will study later that a folded dipole has a much wider impedance
bandwidth.) Calculate and plot the radiation pattern of this array for a spacing of 0.6 λ.
Determine the HPBW. Calculate the directivity of the array. Why do you think that the
designers chose a 3-element array and not a single antenna or a 5-element array? (P.S.: The
plot should be radial and linear in power.)
4- For a planar array in xy-plane, state what the progressive phase shift (αx, αy) should be to
point the main beam at (θo, φo) for a given (dx, dz).
3- A five element linear array of isotropic radiators with spacing of λ/2 are excited with equal
amplitudes and equal phase.
a) What is the functional form of the array factor?
b) What is the beamwidth between the null points around the main lobe?
c) If we had instead adjusted the amplitude such that it followed a linear taper with the
middle element being the largest value (e.g. 0.25, 0.5, 1, 0.5, 0.25) what would have
happened to the beamwidth, i.e., would it have been larger or smaller? Why?
Beamwidth would get larger because transform of anything less that uniform illumination has
larger beamwidth.
d) Imagine that you just turned on the transmitter to the antenna array at an instant, T1 (all
elements start radiating at the same time). An observer is located at some angle off of
broadside or end-fire. Why does the signal amplitude take some time to build up to its final
value?
It takes different time for the signal from each transmitter to arrive at the observation point
since they are at different distances.
3- An array of four identical isotropic radiators has its elements located at the points
(±l/2,±l/2, 0) (the corners of a square) in the plane z = 0. If l = λ /4, what is the maximum
directivity along the z-axis (i.e., θ = 0) that can be achieved by a suitable choice of excitation
currents I1 through I4?