1- Consider an array of six elements with element spacing d = 3λ/8. a) Assuming all elements are driven uniformly (same phase and amplitude), calculate the null beamwidth. b) If the direction of maximum radiation is desired to be at 30o from the array broadside direction, specify the phase distribution. c) Specify the phase distribution for achieving an end-fire radiation and calculate the null beamwidth in this case.

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 2d λ= .

a)

b)

1- Give the array factor for the following identical isotropic antennas with N and d.

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 25θ= ° ,

10φ= ° and provide the widest possible beamwidth.

nulls at7 22 7

27

sin cos 0 sin25 cos10 0.4162

, 1,2, , 1,2, 3,4,5,6

( 1) 0.634 0.1 0.264

x

null

kd kd kd

n n n n

kd n kd dπ

θ φ α α α

π π

α λ α

Ψ

Ψ = + ⇒ = ° ° + ⇒ = −

= ± = ⋅⋅ ⋅ ⇒ Ψ = ± =

− = − = ⇒ = ⇒ = ⇒ =−

2- Two-element uniform array of isotropic sources, positioned along the z-axis 4λ apart is

seen in the figure below. Give the array factor for this array. Find the interelement phase shift, α , so that the maximum of the array factor occurs along 0θ = ° (end-fire array).

[ ]

[ ] [ ]

Let

max at since

sin (cos 1)sin (cos 1)1 12( ) , 2 (cos 1)

2sin (cos 1) sin (cos 1)2 2

1 sin 2 1 2 sin cos( ) cos cos (cos 1) cos cos )

2 sin 2 sin

0 ( )2 2

n

n

Nkd

kdAF x kd

kd kdN

x x xAF x kd kd

x x

kd

θθ

θ

θ θ

θ θ α

π πθ α

− − = = = −

− −

= = = = − = +

= ⇒ = − =

1- Aralarındaki faz farkı 2π olan iki tane izotropik anten y ekseninde 4y λ=±

noktalarına dizilmiştir. Hangi φ açılarında bu anten dizsinin elektrik alanı sıfır olur?

1- Two isotropic antennas are placed on the y-axis, at 4y λ=± with an intrinsic phase shift of 2π . 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?

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

12

2

12 2 1 1

212 4 2

sin ( cos )2, ( ) ,

2 sin ( cos )

sin ( sin ) sin ( sin )( ) , sin sin 30 ,150

sin ( sin ) 2 sin ( sin ) 2

n

n

N kdkd AF

N kd

N kdAF

N kd

π

π

π π

α θπ λπ θ θ φ

λ α θ

α φ π φ πφ π φ φ

α φ φ

−

+= = = = − →

+

+ += = = → = = ° °

+ +

1- or 2k β π λ= is called the Phase Constant.

The formula sin( 2)

sin( 2)

nψ

ψ was first introduced by Schelkunoff.

The phase change along a distance of a quarter wavelength is 2π .

As the sign of the formula sin( 2)

sin( 2)

nψ

ψ changes from positive to negative, the phase of

the pattern changes by π .

The maximum value of sin( 2)

sin( 2)

nψ

ψ) is n .

The maximum value of sin( 2)

sin( 2)

nψ

ψ occurs in any direction for which ψ =0.

For a Broadside Array: the phase difference, α between elements of a “Broadside Array” is 0α = . 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 kdα = ± . as d changes α must change to keep the main beam of the array in the same direction. when the element spacing is 0.5d λ≥ grating lobes are introduced. to increase the directivity one can increase the phase difference, α between elements.

2- 5 tane izotropik antenden oluşan anten dizisi aşağıda gösterilmektedir. a) antenler arasındaki faz farkı α ’yı bulun.

b) Radyasyonun 110θ= ° ’de maksimum yapması için gereken d’yi λ cinsinden bulun.

c) N=5 için ( )n

AF 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 110θ= ° .

c) Sketch ( )n

AF Ψ for N=5.

d) Find the visible range in Ψ . e) Sketch the polar plot of the radiation.

24 4 4 4

4 4

52 4

12 4

, cos 0 cos cos110 0.365

0.73 0.73 0.48 0.98

sin(5 2) sin ( cos )5 ( ) ( )

5 sin( 2) 5 sin ( cosn n

kd kd d d

kd kd

kdN AF AF

kd

π π π π π

λ

π π

π

π

α θ θ λ

α α π π π π

θθ

θ

= Ψ = + = ⇒ = − ⇒ ° = − ⇒ =

− + ≤ Ψ ≤ + → − + ≤ Ψ ≤ + → − ≤ Ψ ≤

Ψ + = ⇒ Ψ = → = Ψ +

−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 broadside (θο=90o).

sin(7 2)1 12 67 , 0,

7 sin( 2) 7 7

7 7nulls at sin 0 & sin 0 , 1,2, 3,,, & 1,2,3,

2 2 2 7N

N AF kd d

mm m

πα λ

π

Ψ= → = = = → =

Ψ

Ψ Ψ Ψ = ≠ → = = ≠

−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).

sin(7 2)17 , cos 80 0 cos 80 0.1736

7 sin( 2)

120.1736 0.73 0.1736 0.797

7visible range 0 5.38 3.79

N AF kd kd kd

kd kd d kd

kd kd

α α

πλ α

θ π α α

Ψ= → = Ψ = ° + = → =− ° = −

Ψ

− − ≥− → ≤ → = − = −

≤ ≤ → − + ≤ Ψ ≤ + → − ≤ Ψ ≤

−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.

cos 0 cos75 cos 75 0.31 .

0.31 0.725 0.3623 0.31(0.725 ) 0.225 40.432

kd kd kd kd

kd kd kd d

θ α α α

ππ λ α π π

Ψ = + → = ° + → = − ° = −

− = ⇒ = ⇒ = ⇒ =− = − = °

−6 −5 −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 45θ = �and a null in the back lobe ( 180θ = � ).

cos 45 0 cos 45 .2

8 8 0.940.937 0.47 , 0.66

5 2 5 2 2

kdkd kd

kd kdkd kd kd d

α α

π π πα π λ α π

Ψ = + = → =− ° = −

− = − ⇒− − =− → = → = = − = − = −

�

−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

2d λ= along the z-axis so that the main beam occurs at 0 4θ π= . Sketch the polar plot.

3- z-ekseninde yarım dalga boyu aralıkla dizili 5-elemanlı bir anten dizisinin ana huzmesinin

0 4θ π= ’da gerçekleşmesi için faz farkı α ’yı bulun. Dizi faktörünü kutupsal çizin.

4 4

2, cos 0 cos 0.87

2 2 2 2 2

kdd kd kdπ πλ πλ π

α α α πλ

= Ψ = + = ⇒ = − → = − = − = − = −

−6 −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 5θ = ° and to provide the widest possible beamwidth.

since we want a wide beamwidth, the dipoles should be as close together as possible.

(any closer thay would touch) so

cos 0 cos cos 5

2 cos5

kd kd kd

d kd kd

θ α α θ

λ π α π

Ψ= + = ⇒ = − = − °

⇒ = = → =− ° −≃

−6 −5 −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 shift α. The spacing between elements is d. For 2d λ= and 2α π=− . Sketch the array factor ( )AF Ψ

vs.Ψ and sketch also the far field polar plot. 12

12 2

sin( ) sin( (cos( ) ))( ) cos( 2) ( )

2 sin( 2) 2 sin( (cos( ) ))AF a

π

π θθ

θ

Ψ −Ψ = = Ψ → =

Ψ −

−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 4θ π= 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. 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.

a) max when

b)

sin( 2) 2 cos 42 cos1 2cos

sin( 2) 2

2cos 2 2 0.5858

1.712 2 14 5.7 180

4 126180 5.7

m

n m

m

N dd dAF

N d

kdkd kd kd d

NN

π ππ θλα πθ α

π λ λ λ

λ λθ π π λ

π ππ

π

Ψ= = ⇒ = = =

Ψ

+ ≤ → + ≤ ⇒ ≤ = =+

= → = =

3- Consider a 4-element uniformly excited equally spaced broadside array along the z-axis with 2d λ= . Where does the main beam point? Plot the radiation pattern (polar plot).

max max max4, 2 , cos , Broadside 90 0N d d dλ β π α β θ θ α= = ⇒ = Ψ = + → = ° ⇒ =

−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 2, 2d λ α π= = . Where does the main beam point? Sketch the polar plot for a) N=4 b) N=5.

1

(radius), 2 (center), arccos arccos 1202 2

mkdd

λαπ α π θ

π

= = = − = − = °

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) , 0d λ α= = b) 4, 0d λ α= = c) 4, 2d λ α π= =− d) 4, 2d λ α π= =−

sin( )( ) cos( 2)

2 sin( 2)AF

ΨΨ = = Ψ

Ψ

sin(2 cos( ))

( )2 sin( cos( ))

aπ θ

θπ θ

=

2

4

sin( (cos( ) 1))( )

2 sin( (cos( ) 1))a

π

π

θθ

θ

−=

− 2

4

sin( (cos( ) 1))( )

2 sin( (cos( ) 1))a

π

π

θθ

θ

+=

+

1- A 4-element array antenna consists of z-oriented half-wave dipole radiators separated by a distance 4d λ= along the x-axis. We are interested in radiation in the x-y plane only.

a) Assuming initially equal excitation (i.e. 0α = ), 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 70φ = ° ?

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.

max max

3 1 1, PSLL , PSLL( 2) 0.707,

sin( 2) sin(3 2 )

PSLL( 3) 0.333, PSLL( 4) 0.271,PSLL( 2) 0.259 5

end fire cos 0

For No Grating Lobes 2 2 2 5 0.8 2 0.8 0.4

SLL

SLL

NN N N N

N N N N

d d

d d d d

π

π

α β θ α β

β π π β π π λ π λ

Ψ = = = = =Ψ

= = = = = = ⇒ =

→ Ψ = + = ⇒ =−

⇒ ≤ − → ≤ → ≤ → ≤

−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 20θ = ° . Can this array produce a beam directed at 0θ = ° (i.e., along the z-axis)? Why or why not?

cos 0 cos cos20 0.94kd kd kd kdθ α α θ αΨ = + = ⇒ =− ⇒ =− ° = −

No. Individual loops do not radiate towards 0θ = °

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

2 3α π= , 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. 2 10 10 4

0.53 6 6 6

2cos 0 cos arccos arccos 131.8

2 3

5(visible range)

3 3

m m m

kd kd d

kdkd d

kd kd

π π π ππ λ

α λαθ α θ θ

π

π πα α

+ ≤ → ≤ − = → =

Ψ = + = → = − → = − = − = °

→ − ≤ Ψ ≤ + → − ≤ Ψ ≤

−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 0, 3 2kdα π= = 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.

a) b) c)2 2

3 3 30, 3 0.75, 1 0.25

2 4 2 4

1,2, 3,....1 sin(2 )( ) , nulls sin(2 ) 0 cos ,

0, 4,8,12,.....4 sin( 2) 2 2n n

kd d kd dkd kd

nn nAF kd

n

π π

π πα

λ λ

π πθ

= = → = → = = = → = → = =

=Ψ = ⇒ Ψ = ⇒ Ψ = ± ⇒ = ± ≠Ψ

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 2α π= 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 & 2kd π α π= = .

a)

b)

c) For main lobe angle

3 2 2 3 2 3 4

2 2 2 4

0 cos cos 2 120

kd d d

kd d d

kd

π π λ π λ

π π λ π λ

θ α π θ π θ

= ⇒ = ⇒ =

= ⇒ = ⇒ =

Ψ = = + = + ⇒ = �

2

π π 3

2

π 2π2

π−π−3

2

π−2π− Ψ

grating lobe

side lobes

main lobe

θ

2α π= kd π=

1- Three half-wave dipoles are aligned parallel to the z-axis, but have their centers located at 2,0, 2x λ λ=− 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.

sin(3 2)( ) 1, , 2, 0

3 sin( 2)nf AF dθ λ α

Ψ= = = =

Ψ, array and the total pattern is the same.

2- Answer true of false: a) The array factor 2 2sin( ) sin( )N ψ ψ 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 4x λ= ± noktalarına yerleştirilmi ştir. 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

4x λ= ± 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) (c)

2cos( sin )nAFπ φ=

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

( 1)

2sin( 2)

sin( 2)

njn

AF e− ΨΨ=

Ψ for n=4

3

2sin(2 )

sin( 2)

jAF e

ΨΨ=Ψ

b) What is the expression for the total field pattern, E(φ) of the four dipole array?

ETotal = Edipole x Earray factor = 3 2sin(2 )cos( )

sin( 2)jeφ ΨΨ

Ψ

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.

3- Three half-wave dipoles are aligned parallel to the x-axis, but have their centers located at 2z λ= ± and 0z = 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.

y

z

x

y

z

32

2

sin( cos )2, 0, cos cos , ( ) 1, 3 ( )

3 sin( cos )nd kd f N AF

π

π

θλ α θ π θ θ θ

θ= = Ψ = = = = → =

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 is90� . 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.

y

z

x

2jI e πI

2

πθ = : ( ) ( )2 1 cos 2 cos sin 2 2n

kl kdϕ= − + ΨE , 2 2kl π= , 2kd π= ,

2πΨ =

42 cos( sin ) 0n

ππ ϕ= + =E when sin 1 4, 0, 1, 2n nϕ = + = ± ±

0 14.5 , 165.5

1 311.4 , 228.6

n

n

ϕ ϕ

ϕ ϕ

= → = =

= → = =

� �

� �

4- Consider 3 short dipoles, pointed in the x-direction, located on the z-axis at 8z λ= ± , and 0z = as shown in the figure. Assume that they are driven in phase, with equal amplitudes.

y

z

x

a) Sketch the far-field pattern for yz-plane and for xz-plane.

2 2

3 38 8

8 8

sin( 2)1( , ) 1 sin cos , , 3, 8, cos cos

sin( 2) 4

plane ( , 90 ) 1 plane ( , 0 ) cos

sin( cos ) sin( cos )1 1( , ) ( , ) cos

3 sin( cos ) 3 sin( cos )

Nf AF N d kd

N

yz f xz f

F F

π π

π π

πθ φ θ φ λ θ θ

θ φ θ φ θ

θ θθ φ θ φ θ

θ θ

Ψ= − = = = Ψ = =

Ψ

− = ° = − = ° =

= =

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.

[ ]2 21

22 cos ( cos ) , 2 2 cos( cos 2)

cos 0 cos 2 cos 80 1.09 rad( 62.5 )

j jAF e kd kd AF e

kd kd

α αθ α π π θ α

α θ α θ π

= + = → = +

Ψ = + = → = − =− ° = − − °

b) Determine the array factor defined in part (a).

0.5452 cos( cos 0.545)jAF e π θ−= − c) Determine the nulls of the array factor of part (a).

{ } { }1

3 1 3cos 0.545 , cos 0.173, 0.173,

2 2 2 2

cos 0.327, 0.673 1.904,0.833 rad 109.1 , 47.7

π ππ θ θ

θ −

− = ± ± ⋅⋅⋅ ⇒ = ± + ± + ⋅⋅ ⋅

⇒ = − = ° °

d) Determine the far field electric field of the array.

( 0.545) 20 0 cos( cos )ˆcos( cos 0.545)

sinj krj I

er

π

θ

µ θπ θ

π θ

− += −E a

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; plane: 0 or 180 sin 0x z ϕ ϕ ϕ− = = → =� � ,

( ) ( ) ( )2 22

cos cos cos2 cos

sin

kl kl

n

θθ

Ψ−

=E , 2 2kl π= , ( ) ( )2

4

cos cos2 cos

sinn

ππθ

θ=E

0

n=E when cos 2 1nθ = + 1 180n θ= → = � , 1 180n θ= → = �

y

z

x

2jI e πI

x

z

b) the y-z plane; plane: 90 or 270y z ϕ ϕ− = =� �

( ) ( )2 2cos cos cos2 cos sin sin

sin 2 2

kl kl

n

kdθθ ϕ

θ

− Ψ= + E , 2 2kl π= , 2kd π= ,

2πΨ =

0n

=E when ( )2cos cos 0π θ = or ( )cos sin sin 4 0π θ ϕ π+ =

( )2cos cos 0π θ = for 0 , 180θ = � �

for 90ϕ = � , sin 1 4nθ = + , 0 14.5 , 165.5n θ θ= → = =� � , 1n θ π= − → >

for 270ϕ = � , sin 1 4nθ = − − , 0n θ π= → > , 1 48.6 , 131.4n θ θ= − → = =� �

9- At points A and B, the signal strength, when a single dipole antenna operates, is 1AE and

1BE , respectively. How will the electric field magnitude change if a second identical antenna is added a distance 2d λ= 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 2π . 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 1AE and

1BE , respectively.

{ 2d λ=

I 2jIe π

x

y

B

A

30�

One antenna : 1E

Two antenna : 1=E F E

2 cos sin sin2 2

kd θ ϕ Ψ = +

F 2 2 2

kdd

λ π= ⇒ = 2 2 4

π πΨΨ = ⇒ =

horizontal plane (x-y plane) 2θ π= , sin 1θ =

2 cos sin sin2 4

π πθ ϕ = +

F

Point A : 30Aϕ = →� 2 cos sin 30 02 4A

π π = + =

F � , 1 0A A A= =E F E

Point B : 210Bϕ = →� 2 cos sin 210 22 4B

π π = + =

F � , 1 12B B B B= =E F E E

1- Sadece tek anten kullanıldığında A ve B noktalarındaki elektrik alan şiddeti 1AE ve

1BE ’dir. Şekilde görüldüğü gibi 2d λ= uzaklıkta ikinci bir anten ilave edildiğinde A ve B

noktalarındaki alan şiddetini 1AE ve 1BE cinsinden hesaplayın. İki antenin akımlarının

mutlak değeri eşit fakat ikinci antenin faz farkı 2π ’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 1AE and

1BE , respectively. How will the electric field magnitude change if a second identical antenna

is added a distance 2d λ= 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 2π . 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 1AE and

1BE , respectively. (Hint: use unnormalized antenna factor expression).

{2d λ=I

2jIe π

y

z

B

A

30�

One antenna: Point A : 60θ = →� 1A AE=E Point B : 120θ = →� 1B BE=E

Two antenna: Point A : 60θ = →� 1( 60 )A AAF Eθ= =E � Point B : 120θ = →�

1( 120 )B BAF Eθ= =E �

( )( )

( )( )

sin 2 sin 2 cos, 2 cos

sin 2 sin 4 2 cos

NAF AF

π π θπ π θ

π π θ

Ψ += Ψ = + → =

Ψ +

Point A : ( )

( )sin 2 cos60 0

( 60 ) 01sin 4 2 cos60

AFπ π

θπ π

+= = = =

+

�

�

�

, 0A =E

Point B : ( )

( )

sin 2 cos120 0( 120 )

0sin 4 2 cos120AF

π πθ

π π

+= = =

+

�

�

�

, take the derivative

( )( )

sin120 cos 2 cos120 3 2( 120 ) 2

3 42 sin120 cos 4 2 cos120AF

π π π πθ

ππ π π

+= = = =

+

� �

�

� �

, 12B BE=E

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 1 2 3, 2,I j I I j= = = − .

( )

( ) ( ) ( )

( )

( )

2 2cos cos2

21 12 2 2 2 2 4 4max

2 2 2

2 2 2

2, 2 2 2 sin cos

4 ( ) 1 sin cos 1 cos cos sin cos

Nulls: sin cos 1 cos 0

Max: sin cos 1 cos

j j

n

kd AF je je

AF AF

π πθ θ π

π π π π π

π π π

π π π

π θ

θ θ θ

θ θ θ

θ θ θ π

−= = + − = −

= ⇒ = − = − − = −

= ⇒ = ⇒ =

= − ⇒ = − ⇒ =

,

2- Consider the three element array depicted below where the spacing between the elements is d = λ/4. Assume that 1 2 3, 2,I j I I j=− = = .

a) Find the normalized array factor. b) Find all the nulls for 0 ≤ θ ≤ π.

( ) ( )

In the far - zone

For the phas

a

e terms

)1 2 3

1 2 1

1 2 3 1 2 3

1 2 3

2 2cos cos cos cos

1 2 3

1 1 1 1.

cos , , cos .

2

2 2 cos(

j r j r j r

j r j rj jj d j d j d j d

j r

e e eAF I I I

r r r r r r r

r r d r r r r d

e eAF I e I I e e e e e

r re

AFr

β β β

β βπ πβ θ β θ β θ β θ

β

θ θ

β

− − −

− −−− −

−

= + + ≈ ≈ ≈

= − = = +

= + + = + +

= +( )

b)

2

max

2 2

2

coscos 2) 4 cos 4

2 4

coscos , 2 ( ) cos cos

2 4 4 4

( ) cos cos 0 cos 0, 1, 2, 3,4 4 4 4 2

3cos co

4 4

j r

n

e dd AF

r

dAF d f

f n n

n

β β θ πθ π

β θ π π πβ π θ θ

π π π π πθ θ θ π

π πθ π

− − = − ⇒ = = − = → = − = − = → − = + = ± ± ± ⋅⋅ ⋅

= + ⇒ s 3 4 , 1 cos 1 180n nθ θ θ= + = − ⇒ = − ⇒ = °

.

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? b) Plot the magnitude (in dB) of the normalized array factor for 0 ≤ θ ≤ π. Determine the half-power beamwidth (in degrees) and the maximum level (dB) of the first minor lobe.

( )First minor lobe side lob

a)

b)

HPBW e

0 0

92

2

4

218, 50 , 4, cos cos50 cos50 1.01rad

4 2sin( 2) sin( (cos cos50 ))

cos (cos cos50 ), ( )sin( 2) 18 sin( (cos cos50 ))

5 7 42 15 .

N d d

Nd f

N

π

π

π

π λ πθ λ α β θ

λ

θβ θ α θ θ

θ

= = ° = = − = ° = − ° = −

Ψ − °Ψ = + = − ° = =

Ψ − °

≈ °− ° = ° ≈ − dB and13.2 @ =16 =71θ θ° °

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?

( )2cos cosEF

sin

π θ

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 x-z 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?

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