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10- A two way communication link has a receiver with a minimum detectable signal of -55 dBm. The receiver antenna gain is 35 dB and the transmitter antenna gain is 40 dB. The carrier frequency is 24 GHz, and the distance between the transmitter and the receiver is 60 km. Suppose that it starts raining and the attenuation through the atmosphere is 0.2 dB/km. What should the minimum transmitter power be so that the signal can be detected by the receiver?
( )
dBm nW dB dB
m km, Loss km 0.2dB km dB
W dBm dB dBm12
4
55 3.16 , 35 3162, 40 10000,
0.0125 , 60 (60 )( ) 12
0.364 25.6 12 37.6
r r t
dt r r t t t
P G G
d
P P GG P Pλπ
λ
−
= − = = = = =
= = = =
′ ′= = = → = + =
1- A hypothetical isotropic antenna is radiating in free-space. At a distance of 100 m from the antenna, the total electric field (Eθ ) is measured to be 5 V/m. Find the power density and power radiated.
( ) 2W m
W
2
12
02 2
2
rad 0 0 0 0
ˆ ˆ0.033152
sin 0.03315 sin 4166.7
av r r
av
ES
P S r d d d dπ π π π
η
θ θ φ θ θ φ
∗= × = =
= = =∫ ∫ ∫ ∫
E H a a
3- An antenna has its far field (electric field) in the free space given below where r is the distance of the observation point from the antenna. a) Find the total radiated power of the antenna. b) Find the radiation resistance if the feeding current for the antenna inI is 0.01 A.
ˆ( ) cos (V m)jkre
rθθ θ
−
=E a
W
2 2
20 0
2 22 2 2
rad0 0 0 0 0
0 0
2 radrad 2
cos1
2 2
1 2sin cos sin cos sin 0.0056
2 2
1 2112
2
av
av
in r r
in
ES
r
P S r d d d d d
PP I R R
I
θ
π π π π π
θ
η η
πθ θ φ θ θ θ φ θ θ θ
η η
= =
= = = =
= → = = Ω
∫ ∫ ∫ ∫ ∫
2- An RFID reader device transmits at 30 dBm. This signal is received by the tag antenna and reradiated back to RFID antenna with 5 dB loss (The tag has a reflected power loss of -5 dB.) The RFID device needs a power level of at least -90 dBm to properly detect the signal reflected from a tag. The frequency of operation is 928 MHz. Both the reader and the tag have gains of 1.64 dB. Determine the maximum range that the reader can support. 2- Bir RFID cihazının anteni 30 dBm ile yayın yapmaktadır. Bu sinyal TAG alıcısıyla alındıktan sonra gücü 5 dB azalarak RFID cihazına geri yansıtılmaktadır. RFID ve TAG antenlerinin kazancı 1.64 dB’dir. RFID cihazı geri yansıyan sinyali algılamak için enaz -90 dBm sinyal seviyesine ihtiyaç duymaktadır. Bu durumda aradaki mesafe en fazla ne kadar olmalıdır? Sinyalin frekansı 928 MHz.
Power received by tag : dBm dBm
Loss dB dB dB
Power reflected by tag : dBm
Power received by receiver : dBm
1
2 1
2
( ) ( )
( ) ( ) ( ) 147.6 20 log 20 log( ) 28.5 20 log( )
( ) 5 5 3.5 20 log( )
( )= 5
t F
F t r
t F
r F t F
P P
G G f R R
P P P R
P P P
α
α
α
α α
= −
= − − − + + = +
= − = − − =− −
− = − −
dBm m
32 40 log( )
90 32( )= 32 40 log( ) 90 log( ) 28.2
40
F
r
R
P R R R
α− = − −
−− − >− ⇒ < ⇒ <
2- An RFID reader device transmits at 30 dBm. This signal is received by the tag antenna and reradiated back to RFID antenna with 5 dB loss (The tag has a reflected power loss of -5 dB.) The RFID device needs a power level of at least -90 dBm to properly detect the signal reflected from a tag. The frequency of operation is 928 MHz. Both the reader and the tag have gains of 1.64 dB. R= 10 meter. a) Determine the power received by the tag antenna. b) Determine the power reradiated back to RFID antenna. c) Determine the power received by the RFID antenna.
Loss dB dB dB dB
a) Power received by tag : dBm dBm dBm
b) Power reflected by tag : dBm dBm
c) Power received by receiver : d
1
2 1
( ) ( ) ( ) 147.6 20 log 20 log( ) 28.5 20 log( ) 48.5
( ) ( ) 30 48.5 18.5
( ) 5 23.5
(
F t r
t F
r
G G f R R
P P
P P
P
α
α
= − − − + + = + =
= − = − = −
= − = −
Bm dBm2)= 72FP α− =−
2- Serbest uzaydaki bir antenden yayılan uzak elektrik alan ifadesi aşağıdaki gibidir. Radyasyon yoğunluğu ( , )U θ φ ’yi W/sr olarak hesaplayın. 1 km ötede ve ve90 60φ θ= ° = °
iken radyasyon yoğunluğunu (W/m2) olarak hesaplayın 2- In the far-field, the time harmonic electric field of an antenna system operating in free space is given below. Determine the radiation intensity (in W/sr). Find the radiation intensity (in W/m2) at a distance of 1 km when and90 60φ θ= ° = ° .
100 sin 200 sin
ˆ ˆ (V m)4 4
jkr jkre er r
θ φ
θ θ
π π
− −= +E a a
2 2 2 2 2 2
2 2
2 2 2 2
27 2
2 2
100 sin 200 sin0.420228117 sin
2 2 (4 ) (4 )
0.420228117 sin 603.1517 10
(1000)rad
r rU W sr
r r
UW W m
r
θ θθ
η η π π
−
= = + =
°= = = ×
E
4- 100MHz ve 1000 W monostatik bir radar anteninin kazancı 75 olup alınan güç 0.01 W’tır. Buna göre 700 m ötedeki hedefin radar kesitini bulun.. 4- Consider that we want to determine the radar cross section of an unknown target that is 700m away from our radar. Our radar system transmits a 100 MHz signal. The total transmitted power is 1000 W and the total received power 0.01 W. Consider that the radar antenna has a gain of 75. What is the RCS of the target? (assume that the signals are polarization matched and the PLF=1)
228r 1 2
80 0
3 10 4 4 0.01(4 ) 4 (700)(700)3 , 94114.5
1 10 ( )( ) 1000(75)(75) 3t t r
P RRm
P G G
π π π πλ σ
λ
×= = = = =
× 2- Terminal akımı I0, input direnci 50 ohm olan kayıpsız bir antenin belirli bir yöndeki maksimum uzak elektrik alan ifadesi aşağıdadır. Bu antenin maksimum efektif alanını (Ae) dalga boyu cinsinden hesaplayın. 2- Suppose a transmitting antenna produces a maximum far-zone electric field in a certain direction given below where I0 is the peak value of the terminal current. The input resistance of the lossless antenna is 50 ohms. Find the maximum effective aperture of the antenna, Aem. Hint: your answer will be a number times wavelength squared.
max 090 (V m)jkre
E Ir
−
=
2
2
22 (90)222 2 2110max 2 2 2 2max
0 2 21 1r 2 20 0
4 2 (90)0.86
4 4r
em
ad
r Ir EUA D
P RI R I R
ηηλ λ πλ λ λ λ
π π η
×= = = = = =
1- The radiated field from an antenna is given below. Find the total radiated power.
ˆ sin (V m)A
rθ θ=E a
2 2 2 22
2 2 2rad2
0 0
2 223 2
0 0
1ˆ ˆsin ( ) sin sin sin
2 2 2 2
4sin W ( )=1.5sin
2 3
rad r r
E A A AW U P d d
r
A Ad d D
π πθ
π π
θ θ θ θ θ θ φη η η η
πθ θ φ θ θ
η η
= = → = → =
= = →
∫ ∫
∫ ∫
a a
2- Aşağıda uzak elektrik alanı verilen bir dipol antenin radyasyon yoğunluğunu, yaydığı
toplam gücü ve directivity’sini (D0) hesaplayın. ˆ sin (V m)jkrAe
rθ θ −=E a
2- Calculate the directivity D0 for small dipole antenna given that the radiated electric field is
given by ˆ sin (V m)jkrAe
rθ θ −=E a .
2 2
2 2
2
23 2
rad0 0
rad
sinˆ ˆ ( ) sin
2
4 4 ( )sin 2 W ( ) 1.5 sin
3
r r
EW B U r W B
r
UP B d d B D
P
θ
π π
θθ θ
η
π θθ θ φ π θ θ
= = → = =
= = → = =∫ ∫
a a
3- f=10 GHz’te çalışan Pt=200W’lık bir antenin kazancı G0t=30dB’dir. 36000 km ötede G0r=40dB kazancı olan alıcı bir antende elde edilen maksimum güç dB olarak ne kadardır? 3- Calculate the power received by an antenna in dB with Pt=200W, G0t=30dB, R=36000km, G0r=40dB, f=10GHz for the earth station.
2 22 2
0 0 0 0
2 2
0 0 3
(1 )(1 ) PLF4 4
3 10(dB) (dB) (dB) 10 log 10 log200 30 40 20 log
4 4 36000 10
23dB 30dB 40dB 203dB 110dB
r t cdt cdr t r t r t t r
r t t r
P P e e D D P G GR R
P P G GR
λ λ
π π
λ
π π
−
= − Γ − Γ =
× = + + + = + + + × × + + − = −
5- A lossless resonant half-wavelength dipole antenna with input impedance of 73 ohm is to be connected to a transmission line whose characteristic impedance is 50 ohm. Find overall gain of this antenna assuming that 3
0 sinU B θ= .
max 0 00 2 43
rad0 00 0
2
0 0 0 0
4 4 4 21.69
P 3 82 sinsin sin
231 0.965, 1 0.965 1.64 2.14 dB
123r cd r cd
U BD
dB d d
e e e e e G e D
ππ π
π π π
ππ θ θθ θ θ φ= = = = =
Γ = → = − Γ = = → = = → = = =
∫∫ ∫
3- The radiation intensity of an antenna (95% efficient at 8 GHz) is given below. Find the HPBW (degrees), total radiated power (W), directivity (dBi), gain (dBi), and maximum possible effective area (m2) at 8 GHz.
40.1cos W sr, 0 2 and 0 2( , )
0, elsewhere.U
θ θ π φ πθ φ
≤ ≤ ≤ ≤=
4
1 0.25
max
max
2 24
rad0 0
44
0
rad
( , ) 0.1cosHPBW 0.5 cos (0.5 ) 32.7651
0.1
0@ 0, HPBW 2(32.7651 ) 65.5302
0.1cos sin 0.04 0.125664 W
4 ( , ) 4 (0.1cos )( , ) 10 cos 10 10
0.04
U
U
U
P Ud d d
UD D
P
π π
θ φ θθ
θ
θ θ θ φ π
π θ φ π θθ φ θ
π
−
Ω
= → = ⇒ = = °
= = = ° = °
= Ω = = =
= = = → = =
∫∫ ∫ ∫
4 40
23 2
em 0
dBi@ 0
( , ) ( , ) 0.95(10 cos ) 9.5 cos 9.5 9.777 dBi
1.06169 10 m4
cd
cd
G e D G
A e D
θ
θ φ θ φ θ θ
λ
π
−
=
= = = → = =
= = ×
2- An antenna in free space has an input power of 20 W and radiates 19.0608 W. Determine the radiated power density (in W/m2), the radiation intensity (in W/sr), the directivity and maximum directivity (in dBi), and the radiation efficiency if its phasor far-zone electric and magnetic fields are given by
8 83 3ˆ ˆ50 sin (V m), 0.13272 sin (A m)
j r j re e
r rφ θθ θ
− −
= = −E a H a .
6 6
2 2
2 2
2 6 6
0
1 3.318 sin 3.318 sinˆRe W m , W m
2 2
4 ( , )( , ) 3.318 sin W sr, ( , ) 2.1875 sin
2.1875 3.39945dBi, 95.3%
ave r ave
ave
rad
rad
in
r r
UU r D
P
PD e
P
θ θ
η
π θ φθ φ θ θ φ θ
∗= × = = =
= = = =
= = = =
EP E H a P
P
2-Uzak alan ortalama güç yoğunluğu aşağıda verilen bir antenin direktif kazancını hesaplayın. 2- Calculate the directive gain and directivity of an antenna that produces at far field the time average vector power density given below.
2
2
2sin cosˆ (W/m ), 0 , 0
2avg
r
θ φ πθ π φ= < < < <P r
2 2
02
0 0
( , ) 8 sin cos( , ) 8 sin cos 8
4 2 sin cos sin
avg avg
avg avg rad
r P r PUD D
U U P d dπ π
θ φ π θ φθ φ θ φ
π θ φ θ θ φ= = = = = → =
∫ ∫
4- The far-zone, time harmonic, electric field of an antenna operating in free space is given below. Determine the radiation intensity U(θ,φ) (in W/sr). What is the maximum radiation intensity (in W/sr)? Find the time average power radiated by the antenna. Then find the directivity D(θ,φ). What is the maximum directivity?
ˆ 80cos (V m) 904
( , )
ˆ 20 cos (V m) 90 1804
jkr
jkr
e
r
e
r
θ
θ
θ θπ
θ φ
θ θπ
−
−
≤ °= ° ≤ ≤ °
a
E
a
222
( , )2
rU E Eθ φθ φ
η= +
22
2
22
2
2 22 2
0 0 2
400cos 90 0.05379cos 902
W sr25 0.003362cos 90 180cos 90 180
2
0.05379cos sin 0.003362cos sin 0.1197 W,
4( , )
rad
ra
P d d d
UD
P
π π π
π
θ θθ θηπ
θ θθ θ
ηπ
θ θ θ θ θ θ φ
πθ φ
≤ ° ≤ ° = = ° ≤ ≤ ° ° ≤ ≤ ° = = =
=
∫ ∫ ∫2
max2
5.647 cos 90104.98 ( , ) 5.647 7.518 dBi
0.3529 cos 90 180d
U Dθ θ
θ φθ θ
≤ °= = → = = ° ≤ ≤ °
Balanis 2.10. The radiation intensity of an antenna is represented by
1 0 30
0.5 30 60
0.1 60 90
0 90 180
U
θ
θ
θ
θ
° ≤ ≤ ° ° ≤ ≤ °= ° ≤ ≤ ° ° ≤ ≤ °
(a) What is the directivity (above isotropic) of the antenna (in dB)?
2 6 3 2
rad0 0 0 6 3
max max0
rad 0
sin 2 sin 0.5 sin 0.1 sin 0.734
5.4496 7.3636 dB 0 30
2.7250 4.3530 dB 30 604( )
0.54496 2.636 dB 60 90
0.0000 dB 90
P U d d d d d
U UD D
P U
π π π π π
π πθ θ φ π θ θ θ θ θ θ π
θ
θπθ
θ
θ
= = + + = = ° ≤ ≤ °
= ° ≤ ≤ °= = = =
= − ° ≤ ≤ °
= −∞ ° ≤ ≤
∫ ∫ ∫ ∫ ∫
180
° (b) What is the directivity (above infinitesimal dipole) of the antenna (in dB)?
1- The far electric field of a certain antenna is given by 0ˆ( , ) cos(2 )
jkre
rθ φ Ε φ
−
=E θ .
Determine the power radiated by the antenna and the directivity. Would this antenna be potentially useful as a broadcast antenna? Why or why not?
Since D is small, the antenna could be useful as
222 2 2
0
2 2 22 2 2
2 20 0 0
0 0 0 0 0
max max max0
1 1, cos (2 )
2 2
, sin cos (2 ) sin cos (2 )2
42,
4
r
avg r r
U E E r E
E E EP U d d d d d
U U UD
U P P
θ φ
π π π π π
θ φ φη η
πθ φ θ θ φ φ φ θ θ φ φ
η η η
π
π
= + =
= = = =
= = = =
∫ ∫ ∫ ∫ ∫
( )
( )
a
broadcast antenna except that the pattern has nulls.
2- The far E- field of an antenna is given by 0ˆ,
jkrje
E er
φθ φ−
=E( ) θ . Determine the power
radiated by the antenna and the directivity.
22 22 22 0 02 2
0 0 0 0
2 2max0 0 2
21sin sin
2 2
1 1 1max 1
4 4 2
jkrj
r
r r
E EeP E r d d e r d d
r
UD E r
P P r
π π π πφ
θ
πθ θ φ θ θ φ
η η η
π π η
−
= = =
= = =
∫ ∫ ∫ ∫
3- Akim kaynağı 0I olan bir antenin uzak elektrik alanı 0ˆ( , ) (1 cos )
jkrer I
rθ η θ
−
= −E θ V/m
ise bu antenin yaydığı toplam gücü ve bu antenin radyasyon direncini hesaplayın. .
2 22 220 0
20 0
0
8 2 16(1 cos ) sin W,
2 3 3rad
rad rad
I I PP d d R
I
π π η πη πηθ θ θ φ= − = = = Ω∫ ∫
4- An antenna, with input impedance of 73 ohm, is to be connected to a transmission line whose characteristic impedance is 50 ohm. Assume that the pattern of the antenna is given by
30( , ) sinU Bθ φ θ= . Find the directivity and the overall maximum gain of the antenna.
22 2
3 max0 0 0
0 0 0 0r
2
0
3 16, sin sin sin 4 1.698
4 3
73 500.186 1 0.965 0.965 1.698 1.639 2.14 dB
73 50
r
ad
r r
UP U d d B d d B D
P
e G e D
π π π π πθ φ θ θ φ θ θ θ φ π
π= = = → = = =
−Γ = = → = − Γ = → = = × = =
+
∫ ∫ ∫ ∫( )
5- The far E-field of a short vertical current element Idl located at the origin of a spherical coordinate system in free space is given below. Find the total average power radiated by this
current element. 60
ˆ( , ) sinj rI dl e
r jr
βπθ θ
λ
−
=E θ (V/m).
0 0
22
2 22
2 2 2
0 0
1 ( , ) ˆ ˆˆ( , ) ( , ) sin A m2
1ˆ( , ) Re 15 sin
2
( , ) 15 sin sin 40 W
j r
av
r
E r Idlr r j e
r
Idlr
r
Idl Idlr d r d d
r
βθ
π π
θθ θ θ
η η λ
θ π θλ
θ π θ θ θ φ πλ λ
−
∗
= × = =
= =
= ⋅ = = ∫ ∫ ∫
H E
E×H
r
P r
P s
φ φ
SP
6- Compute the radiation resistance of an antenna whose far-zone electric field of in free
space with input current 0I is given by 0ˆ (1 cos ) V/m10
jkrI e
r
ηθ
−
= −E θ .
0 12
2 22 22 20 0
0 0 0 0
20
1 ˆˆ (1 cos ), Re10
2ˆlim sin (1 cos ) sin
200 752 4
75
jkr
radr
radrad
I e
r
I IP r d d d d
PR
I
π π π π
θη
η πηθ θ φ θ θ θ φ
η π
−∗
→∞
= × = − = ×
= ⋅ = − =
= =
∫ ∫ ∫ ∫
H r E S E H
S r
φ
7- The far E- field of an antenna is given by [ ]0 0 sin cos
ˆ( ) sin4 cos
jkr kLj I L e
r kLθ
θωµθ
π θ
−
=E r a .
Determine the power radiated, the maximum directivity and the radiation resistance of the
antenna if kL π= . Hint: ( )
23
20
sin ( cos )sin 0.8
cosd
π π θθ θ
π θ∫ ≃ .
[ ]
( )
2 222 2 2 30 0 0 02
0 0 0
20max 0
20
sin cos1 2 0.8sin sin
2 2 4 4cos
2 1.62.5 90 ,
4 4
rad
radrad
rad
I L I LP E r d d d
U P LD at R
P I
π π π
θ
π θπ ωµ π ωµθ θ φ θ θ
η η π η ππ θ
π ωµθ
π η π
= = =
= = = = =
∫ ∫ ∫
1- Bir antenin uzak elektrik alanı 40( , ) 100 cos V/m, 0 90
jkreE r I
rθ θ θ
−
= ° ≤ ≤ ° olsun.
a) Bu antenin radyasyon yoğunluğunu ve yaydığı toplam gücü hesaplayın. b) Bu antenin radyasyon direncini ve directivity’sini hesaplayın. c) Bu antenin maksimum efektif alanını hesaplayın.
1- The far E-field of an antenna is 40( , ) 100 cos V/m,0 90
jkreE r I
rθ θ θ
−
= °≤ ≤ ° .
a) Find the radiation intensity and the total power radiated. b) What is the radiation resistance and the directivity of this antenna? c) What is the maximum effective area of this antenna in square wavelengths?
422 2
0
0
22
2 2 20 8
00 0 00
22
0max02
00
cos , 0 90) ( , ) 13.263
0, elsewhere2
2 (100)( , )sin cos sin 9.259 W
2
(100) (12 4b) 18.5 , 4
2
rad
rad
rad
rad
ra U E I
IP U d d d I
IP UR D
PI
π π π
θ θθ φ
η
πθ φ θ θ φ θ θ θ
η
πππ
η
° ≤ ≤ °= =
= = =
= = Ω = =
∫ ∫ ∫2
2
0
0
22
0
00)18 12.5dB
9
) 1.4324em
I
c A D
η
λλ
π
= =
= =
2- The radiation intensity of an antenna is given below for. Determine the maximum effective aperture (in m2) of the antenna if its frequency is 10GHz. Assume that the antenna is lossless, the polarization loss PLF = −1.4 dB, the input impedance of the antenna is ZA = 55 + j8 Ω and that the antenna terminals are connected to a load of ZL=50 Ω.
3cos , 0 90
( , )0, 90 180
Uθ θ
θ φθ
° ≤ ≤ °= ° ≤ ≤ °
max
0 2 2 2 13 34
0 0 0
0.14
22 4 2
02 2
1 44 4 8
2cos sin 2 cos sin
0.03 m, 1, 1.4 dB 10 0.7244
41 0.992, 4.1175 10 m
( ) 4
cd
A Lr em r cd
A L A
UD
d d d
e PLF
R Re A e e PLFD
R R X
π π π
ππ π
πθ θ θ φ π θ θ θ
λ
λ
π
−
−
= = = =
= = = − = =
= − Γ = = = = ×+ +
∫ ∫ ∫
4- An antenna over ground is driven by a terminal current of I0 resulting in a radiation intensity function defined by
2 2 2 20( , ) 4 sin cos (W rad ), 0 2, 0 2 .U Iθ φ θ φ θ π φ π= ≤ ≤ ≤ ≤
(a) Determine the total radiated power. (b) Determine the antenna radiation resistance and the directivity. (c) Determine the power density in the direction of maximum radiation at a distance of 1 km from the antenna when I0 =2A. (d) Determine the magnitude of the magnetic field at the point(s) defined in part (d).
(a)2 2 2 2
2 3 2 2 20 0 0
0 0 0 0
8( , )sin 4 sin cos 8.38 W
3radP U d d I d d I I
π π π π
θ φ θ θ φ θ φ θ φ π= = = =∫ ∫ ∫ ∫
(b) ( ) ( )2 2max 830 0 02
0
2 416.8 , 4 4 6rad
rad
rad
P UR D I I
PI
ππ π= = Ω = = =
(c)2
0 2maxmax2 2 2
4( , ) [ ( , )]( ) 16mW m
(1000)avg avg
IU UP P
r r
θ φ θ φ= → = = =
(d) 20
0
21291mA m
2
avg
avg
PP H Hη
η= → = =
2- Radyasyon yoğunluğu ( , ) 1.5 cos , 0 2 , 0 2U θ φ θ θ π φ π= < < < < , olan bir anten için
toplam radyasyon gücünü ve directivity’yi hesaplayın.
2 2
00 0
3 1.5( , )sin W, 4 ( 0)
2 4 1.5 4rad
UP U d d D
π π πθ φ θ θ φ θ
π π π= = = = = =∫ ∫ max
radP
3- An antenna has a pattern solid angle of π/4 and a radiation efficiency of 70%. The input power to the antenna is 100 W. At a range of 10 km, what is the maximum power density?
7 2max 2
416 0.7 16 11.2, 8.9 10 W m 60.5 dBm
4t t
t
G PD G S
r
π
π
−= = → = × = = = × =−Ω
1- A satellite S transmits an electromagnetic wave at 10 GHz via its transmitting antenna. The characteristics of the satellite-based transmitter are: (a) the power radiated away by the satellite antenna is 10 W, (b) the distance between the satellite antenna and a point A on the earth’s surface is 37,000 km, and (c) the satellite transmitting antenna’s maximum directivity in the direction SA is 50 dB. Ignoring ground effects and assuming free space propagation determine the magnitude of the E-field at point A.
2
max5 2 4max 00 max max 2
4 210 , 2.09 10 V m
2 4rad
rad
EU P DD U r E
P r
π η
η π
−= = = → = = ×
4- A low earth orbit satellite system at a radius of 1200 km transmits at 10.2 GHz using a 31 dB gain antenna. Determine the required receive antenna gain if the transmit power is 10 W, and the received power must be at least -100 dBm. Assume that transmit and receive antennas are directed towards each other.
3
(dB) (dB) (dB) (dB) 20 log4
0.029130 10 31 20 log 3.32dB 2.148
4 1200 10
r r t tG P P Gr
λ
π
π
= − − −
= − − − − = = × ×
2- Two scouts communicate using identical FRS walkie-talkies operating at 462.6375 MHz with linearly polarized antennas with gains of 2.15 dBi. The walkie-talkies are specified to have a maximum range of 12.872 km (8 miles) under ideal conditions with a transmit power of 1 W. Assuming the antennas are matched to their transceiver electronics and are both vertically-oriented, what is the minimum received power necessary for operation? When one scout sits down, the walkie-talkie is at an angle of 35 degrees with respect to vertical, what is the new maximum range? If the scouts were 9 km apart at the time, can they still communicate?
λ
λθ φ θ φ
π
π
−
= = = × = Γ = = = = =
= − Γ − Γ =
= = Ψ = ° =
× =
i
3
22 2
22
12
0.648 , 12.872 10 m, 1W, 0,PLF 1, 1.6406
(1 )(1 ) ( , ) ( , ) PLF 43.19573pW4
ˆ ˆPLF cos cos 35 0.6701
0.64843.19573 10 (1W)(1)(1)
4
t cd t cd r
r t cdt cdr t r t t t r r r
t r
cm R P G e D e D
f
P P e e D DR
ρ ρ
→ =
2
0.215 0.215max,
max,
10 10 (0.671) 10,544.1msit
sit
RR
1- Two vertical half-wavelength dipoles are used in a communication link at f = 400 MHz. The distance between the transmitter and receiver is 5 Km. The transmitter antenna is connected to a source with 2 W of maximum available power via a transmission line with characteristic impedance 50 Ω. The receiver has an input impedance of 50 Ω and is connected to the receiver antenna. The antennas are assumed lossless.
a) Find the power received by the receiver.
22
1 2
2 22 2 10
1 2 1 2
73 501 1 0.965, 0.75 m, 1.64
73 50
0.752 (1.64) (0.965) 7.14 10 W
4 4 5000
r
rec t r r
e c f G G
P P GG e eR
λ
λ
π π
−
− = − Γ = − = = = = = +
= = = × ×
b) Calculate the received power if the receiver antenna is a short dipole of length l = 7.5 cm.
222
2 2
210
7.9 5080 7.9 1 0.471, 1.5
7.9 50
0.752 (1.64)(1.5)(0.965)(0.471) 3.19 10 W
4 5000
r r
rec
lR e G
P
πλ
π
−
− = = Ω→ = − = = +
= = × ×
c) What will be the reduction in the received power if the receive antenna is rotated by 30o in a plane perpendicular to the direction of incidence.
2 2max
1( cos ) (30 )
2rec rec in recP E E P Pθ → ° =∼ ∼
2- The maximum far-field electric field intensity of a transmitting antenna in a certain direction is given below where I is the peak value of antenna current. The radiation resistance (or input resistance) of the lossless antenna is 50 ohm. Find the maximum gain and the
maximum effective aperture of the antenna. 90 j t jkrI
E er
ω −= V/m
2 2 21
22max21
2
5.4, 0.434 44
e
in
E rU GG A
P I R
λλ
π ππ= = = = = ,
1- Two antennas are located 1 km apart. One is a dipole antenna with gain of 1.64. The other is a horn with gain of 12. the losses in the transmitter circuitry are 5 dB, and the losses in the receiver circuitry are 4 dB. The transmitter sends 1 W of power at 915 MHz. a) What is the free space path loss in dB? b) What are antenna gains in dB? c) How much power is received in dB and in watts?
22 8 6
10
3
10 10
81.74910
3 10 915 10) 6.807 10 91.67 dB
4 4 10
b) 10 log (1.64) 2.14 dB, 10 log (12) 10.79dB
) 0dBW 2.14 10.79 91.67 5 4= 81.74dBW=10 1.68 10 W
dipole horn
R
ad
G G
c P
λ
π π
−
− −
× × = = × = − × = = = =
= + + − − − − = ×
9- For a lossy antenna it is found that the G = 0.8 D. Find the ratio of the antenna’s loss resistance to its radiation resistance.
2 2
2 2
( , , ) ( , , ) 1 1( , ) , , , ,
4 4 2 2
10.8 1 1.25 0.25
0.8
rin r L r r L L
r in in
r LL r
r L r
S r S r G PD G P P P P I R P I R
P r P r D P
G R RR R
D R R R
θ φ θ φθ φ
π π= = = + ⇒ = = =
= = → + = = → =+
1- A low orbit (LEO) satellite system transmits at 1.62 GHz using a 29 dB gain antenna with spot beams directed toward users on the earth that are a maximum of 1500 km away. Find the required satellite transmit power in order for the power received by a user at the maximum distance be at least -100 dBm if the user has a 1 dB gain antenna directed toward the satellite.
2
1.62 0.185m, 29dB, 1500km, 100dBm, 1dB
(dBm) (dBm) (dB) (dB) 10 log 30dBm 1W4 r
t r r
t r t r
f GHz G r P G
P P G G
λ
λ
π
= → = = = ≥− =
= − − − = =
2- A radio link has 20 Watt transmitter connected to an antenna of 0.5 m2 effective aperture at 9 GHz. The receiving antenna has an effective aperture 2.0 m2 and is located 20 km line of sight from the transmitting antenna. Assuming the antennas have efficiencies equal to unity, find the power delivered to the receiver.
2 245 Wt et er
r
PA AP
rµ
λ= =
2- For the following communication link the aperture efficiency is 60%. Calculate the received power by the load. And the power reradiated from the horn.
21W, 10dBci, 10m, 80 , 50 , 300 cmt t a L aP G R Z Z A= = = = Ω = Ω =
R
y
yx
aZ
LZRHCP
Tx antenna Rx horn
y-polarized
Aperture efficiencyaA
tG
tP
222 2
222 2
2 2
2 5
2
ˆ ˆ(1 ) ( , ) (1 ) ( , )4
ˆ ˆˆ ˆ ˆ( 0.6)(1 ) ( 0.6)(1 )
4 4 2
10(0.03 0.6) (1 0.23 ) 0.5 6.78 10 W,
4 (10)
r t cdt t t t t cdr r r r r t r
t t t tr a r t r a r
r scat
P P e D e DR
PG PGP A A
R R
P P
λθ φ θ φ
π
π π
π
−
= − Γ − Γ
± = × − Γ = × − Γ ⋅
= × × − × = ××
i
i
ρ ρ
ρ ρ x yy
58010.9 10 W
50rP
−= = ×
4- A receiving antenna with a loss resistance of 7 Ω, er = 0.9 and reactance XA =35 Ω has its terminals connected to a load of ZL = 50 Ω. If the open circuit voltage is Voc = j0.5 V : a) Draw the equivalent circuit. Determine the values of all the circuit. b) Determine the average power delivered to the load (in W) c) Assuming the average power density of the incident field at the antenna is 0.06 W/m2, determine the effective area Aeff, (in m2 ).
22
33 2
) 0.9 63 70 35
1 0.5 0.5 1 0.5) 50 50 0.4 mW
2 50 120 35 2 120 35
0.4 10) 6.67 10 m
0.06
r
r r A r loss A
r loss
L L L L
A
L
eff
i
Ra e R Z R R jX j
R R
j j jb P I I P
Z j j
Pc A
S
−−
= = → = Ω → = + + = + Ω+
= → = = → = =+ + +
×= = = ×
1- Three measurements are made at 10 GHz in free space with antennas A and B oriented for maximum response and located 15m apart. Calculate the measured received power for the last measurement. 1- Birbirinden farklı A ve B antenleriyle 10 GHz’te aşağıdaki 3 ayrı deney yapılıyor. Bütün deneylerde antenler birbirinden 15 m uzaklıkta olup konumları maksimum alış gücü için ayarlanmıştır. Buna göre aşağıdaki 3. deneyde alış gücü kaç dBm’dir? (Prec=? dBm)
( )
( )
( )
0dBm A 15m A 22dBm
0dBm B 15m B 34dBm
0dBm A 15m B ??dBm
in rec
in rec
in rec
P P
P P
P P
= → ←−−−− −−−−→ → =−
= → ←−−−− −−−−→ → =−
= → ←−−−− −−−−→ → =−
( ) ( )
22
1 2 1 222 10 34 10
1 2
2 23
1 2
4, 0.03 m
4
4 15 4 1510 499, 10 125.4
0.03 0.03
0.03499 125.4 1.59 10 28 dB
4 4 15
r rd d
t t
d d
rd d
t
P r P cG G
P r P f
G G
PG G
P r
λ πλ
π λ
π π
λ
π π
− −
−
= → = = =
× ×= = = =
= = × = × = − ×
4- 1 GHzf = ’te kazançları 20 dB ve 15 dB olan alıcı ve verici antenlerin arasındaki mesafe 1 Km’dir. Verici antenin gücü 150 W iken alıcı antenin yükünde oluşan gücü hesaplayın.
2 2
3
20dB 100, 15dB 31.623, 1GHz 0.3m
0.3(100)(31.6)(150) 270.344 W 5.68 dB
4 4 10
t r
r t t r
G G f
P PGGR
λ
λµ
π π
= = = = = → =
= = = = − ×
5- 0100W, 10dB, 2dB, 150MHz, 1, 30km :t t rP G G f e R= = = = = = ise alıcı antenin
efektif açıklık alanını ve alıcı antendeki gücü hesaplayın. 5- Given 0100W, 10dB, 2dB, 150MHz, 1, 30km :t t rP G G f e R= = = = = = What is the
effective aperture area of the receiving antenna? What is the received power?
8 2 22 10 2
0 06
610 10
2 2
10
3 10 2150MHz 2m, 10 0.5044 ( 1.5849 2dB)
150 10 4 4
10(100)(10 ) (0.5044) 0.04460 W (2.112 mV to 50 )
4 4 30
(dB) 73.51dBW 43.51dBm 10 log ( 1 )for dBW
e
t tr e
r r
f A D m D
PGP A
R
P P W
λλ
π π
µπ π
−
×= → = = = = = = =
×
= = = Ω×
= − = − =
7- An antenna with a radiation resistance of 64 Ω, a loss resistance of 4 Ω and a reactance of -65 Ω is connected to a generator with open-circuit voltage of 20 V (peak) and internal impedance of Zg=55+j10 Ω via a λ/4 long lossless transmission line with characteristic impedance of 50 Ω . Determine the average power supplied by the generator, the power radiated and power dissipated by the antenna.
ocV
4λ
LZ0,Z β
inZ
gZ
20
2
2 2 2
2
20 , 55 10 , 64 , 4 , 65 ,
2, ( / 4) 19.2 18.6
Re( )1 1 1 200 74.211( ) Re( ) 2.35 W
2 2 2 (74.211) (28.264)
64 1, Re
68 2
oc g r L A
in A
oc oc g in
s L oc g oc
g in g in
rr in g
in
V V Z j R R X
l Z Z Z j
V V Z ZP R V I V
Z Z Z Z
Pe P I Z
P
β π λ
∗
∗∗ ∗
= = + Ω = Ω = Ω =− Ω
= = = + Ω
+ ×= = = = =
+ ++
= = = 2
2
1 Re( )0.6087 W 0.5729 W
2
0.0358 W
inin oc r in r
g in
L in rad
ZV P P e
Z Z
P P P
= = → = =+
= − =
1- Radyasyon direnci 48radR = Ω kayıp direnci 2lossR = Ω ve reaktif empedansı
+j 50X j= Ω olan bir anten varsayalım. Voltajı 10 VgV = ve empedansı 50gZ = Ω olan
bir de voltaj kaynağı olsun. Bu anten ile voltaj kaynağı, karakteristik empedansı 0 100Z = Ω
ve uzunluğu 4λ olan kayıpsız bir iletim hattı ile birbirine bağlanmıştır. Buna göre: a) Eşdeğer devreyi çizin (radyasyon, kayıp direnci, reaktif empedansı seri bağlı varsayın).
/ 4λ
2 48 50LZ j= + + Ω0 100Z = Ω
inZ
50 Ω
10V
b) Determine the power supplied by the generator (power input to the line).
2 20
22
av av
(100) 200 50 1 2100(1 ),
50 50 1 50 2 2
10 1W, (1 ) 0.25(1 5 13) 0.154 W
8 50 4
inin in
ant in
in in
Z Z jZ j
Z j j Z j
P P P
− −= = = = − Γ = =
+ + + −
= = = − Γ = − =×
c) Antenin radyasyon gücünü bulun.
radrad in
rad loss
0.148 WR
P PR R
= =+
3- Consider an earth station. a) What is the required EIRP of the earth station if necessary S/N is 12 dB (in dBW)? Prec = k Tsys B = 1.38 x 10-23 x 132 x 107 = 1.82 x 10-14 W = -137 dBW Ps req = -137 + 12 = -12.5 dBW EIRP = Ps req + LS + Gr = -125 +214 -34 = 55 dBW b) What is the desired EIRP of the earth station if you want to allow 6 dB circuit margin (in dBW)? EIRP (desired) = EIRP (required) + Margin (dB)= 55 + 6 = 61 dBW
Balanis 2.3. The maximum radiation intensity of a 90% efficient antenna is 200 mW/unit solid angle. a) Find the directivity and gain (in dB) when the input power is 125.66 mW
b) Find the directivity and gain (in dB) when the radiated power is 125.66 mW
Balanis 2.4. The power radiated by a lossless antenna is 10 W. The antenna are represented by the following radiation intensity: 3
0 cos W sr, 0 2, 0 2U B θ θ π φ π= ≤ ≤ ≤ ≤
a) Find the maximum power density (in W/m2) at a distance of 1000 m (assume far field distance). Specify the angle where this occurs.
b) Find the directivity (max) of the antenna (in dB)
c) Find the gain of the antenna (in dB)
Balanis 2.11. The radiation intensity of an antenna is given by 4 2( , ) cos sinU θ φ θ φ= for
0 2θ π≤ ≤ and 0 2φ π≤ ≤ (i.e., in the upper half-space). It is zero in the lower half-space.
a) Find the directivity.
22 2
2 4rad
0 0 0 0
max max max
( )sin 2 sin cos sin5
( 0 , 2) 1, 4 20 13dBo
rad
P U d d d d
U U D U P
ππ π ππ
θ θ θ φ π φ φ θ θ θ
θ φ π π
= = =
= = = = = = =
∫ ∫ ∫ ∫
b) Find the elevation plane half-power beamwidth (in degrees). elevation plane constantφ = , choose 2φ π=
4 4 1cos , 0 2 cos (HPBW 2) 1 2 HPBW 2cos ( 0.5) 65.5oU θ θ π −= ≤ ≤ → = ⇒ = =
Balanis 2.12. The normalized radiation intensity of an antenna is symmetric, can be approximated by
1 0 30
cos( )( ) 30 90
0.8660 90 180
U
θθθ θ
θ
° ≤ ≤ °= ° ≤ ≤ °
° ≤ ≤ °
(a) Find the exact directivity by integrating the function
2 6 2
rad0 0 0 6
maxmax
cos( )sin 2 sin sin 3.63W
0.866
43.53 5.47 dB
3.563
P U d d d d
UD
π π π π
π
θθ θ θ φ π θ θ θ θ
π
= = + =
= = =
∫ ∫ ∫ ∫
(b) Find the approximate directivity using Kraus’ formula
Balanis 2.41. An antenna with radiation a resistance of 48 ohms, loss resistance 2 ohms, and reactance of 50 ohms is connected to a generator with an open circuit voltage of 10 V and internal impedance of 50 ohms via 4λ long transmission line with a characteristic impedance of 100 ohms. a) Draw the equivalent circuit
b) Determine the power supplied by the generator
W1 12 2Re 10 0.05546 cos(33.7 ) 0.231S g gP V I ∗= = × × × ° =
c) Determine the power radiated by the antenna
W
W
2 2(100) 21 150 50 2 2100 100 Re (0.05546) 100 0.1538
0.96 0.1538 0.148
jin A g in
rad cd A
Z j P I Z
P e P
+= = − Ω→ = = × × =
= = × =
Balanis 2.58. Two lossless, polarization matched antennas are aligned for maximum radiation between them, and are separated by a distance of 50λ. The antennas are matched to their transmission lines and have directivities of 20 dB. Assuming that the power at the input terminals of the transmitting antenna is 10 W, find the power at the terminals of the receiving antenna.
Balanis 2.45. The E-field pattern of an antenna, independent of φ, varies as follows;
1 0 4
0 4 2
1 2 2
E
θ π
π θ π
π θ π
≤ ≤= ≤ ≤ ≤ ≤
a) What is the directivity of this antenna?
b) What is the radiation resistance of the antenna at 200 m from it if the field is equal to 10 V/m (rms) for 0θ = ° at that distance and the terminal current is 5 A (rms)? When the field is equal to 10 V/m (rms) for 0θ = °
Balanis 2.48. An antenna has a maximum effective aperture of 2.147 m2 at 100 MHz. It has no conduction, dielectric or polarization losses. The input impedance of the antenna is 75 ohm and is connected to a 50 ohm transmission line. Find the maximum directivity of the antenna including the reflection due to mismatch between the antenna and transmission line.
2
8
6
22
max max 234
75 50 3 101 , 1 , 1, 0.2, 3m, 2.147
75 50 100 10
2.147(1 | | ) 3.125
4 1 (0.2)
c d em
em c d
e e PLF A
A e e PLF D Dπ
λ
λ
π
− ×= = = Γ = = = = =
+ × = − Γ → = = −
Balanis 2.49. An incoming wave with a uniform power density of 3 210 W/m− is incident upon an antenna whose directivity is 20 dB. Determine the maximum possible power that can be delivered to a load connected to this antenna at 10 GHz. Assume that there are no losses between the antenna and the receiver or load.
8
max 9
2 23 3 6
max
3 1020dB 100, 0.03 m
10 10
(0.03)100 7.16 10 10 7.16 10 W
4 4em rec em
cD
f
A D P A
λ
λ
π π
− − −
×= = = = =
×
= = = × → = = ×
, ,
Balanis 2.68. Transmitting and receiving antennas operating at 1 GHz with gains (over isotropic) of 20 and 15 dB, respectively, are separated by a distance of 1 km. The input power is 150 miliwatt. Find the maximum power delivered to the load when the a) antennas are polarization matched.
μ
2
2 223
0 0 3
ˆ ˆ20dB 100, 15dB 31.623, 0.3 m, 1000 m, PLF 1
0.3ˆ ˆ (100)(31.623)(150 10 ) 270.3 W 5.68dBm
4 4 10
t r t r
r t t r t r
G G R
P P G GR
λ
λ
π π
−
= = = = = = = =
= = × = = − ×
i
i
ρ ρ
ρ ρ
b) transmitting antenna is circularly polarized and the receiving antenna is linearly polarized.
μ
22 ˆ ˆ 1
ˆ ˆ ˆPLF 135.2 W 8.68 dBm2 2
t r rP ± = = = ⇒ = = −
i iρ ρ x yx
1- Birbirine uzaklığı 25 mil (1 mil =1.61 km) ve özellikleri birbirinin aynı olan iki tane anten ile bir radyo-link hattı oluşturulmuştur. Pt Gt = 8 Watt olup alış gücü Pr = -70 dBm’dir. a) Antenlerin açıklık verimi (aperture efficiency) 0.8 ise antenin fiziksel alanı kaç m2 dir? b) Antenler için ΩΑ = 0.04 sr ise çalışma frekansı ne olmalıdır? c) Verici gücü sabit olup frekans iki katına çıktığında alış gücü kaç dBm olur? 1- A microwave link is formed by two similar horn antennas separated by 25 miles (1 mile =1.61 km). The effective isotropic radiated power (EIRP = Pt Gt ) is equal to 8, and the received power is -70 dBm. a) Assuming matched horn antennas with an aperture efficiency of 0.8, what is the physical aperture size (m2) of the antennas?
10 2 2
2 270dBm=10 W, 0.2545m 0.8 0.32m
4 4t t
r er er er phys phys
PG EIRPP A A A A A
R Rπ π
−− = = ⇒ = = ⇒ =
b) Find the frequency of operation if the beam solid angles of the antennas are 0.04 sr.
224
3GHzA A e
e A e
cA f
D A A
π λλΩ = = → = Ω → = =
Ω
c) Assume that the frequency is doubled, while keeping the same transmitted power, what would be received power in dBm?
2 2
err t
A fP P
R c
= doubling f ⇒ 104 10 W 64dBmrP
−= × = − .
Note that if EIRP was kept constant ⇒ no change in received power as frequency changes. d) Assuming that the horns has circular aperture with diameter d and unity aperture efficiency, derive the following approximate expression for the half power beam width (in degrees): 65r dθ λ= . (Assume symmetric narrow major lobe with negligible minor lobes)
1- The Voyager satellite transmitted some spectacular pictures back to Earth from a distance of about 4 light-hours. The data link involved a 12 Watt transmitter at 8.4 GHz and an antenna with HPBW=1° .
a) Calculate the beam solid angle and gain for this antenna.
241 3.05 10 sr 4 41253 46dB
180G
ππ−
Ω = ° = × → = Ω = ≅ °
b) Calculate the power density at Earth.
21 2
2 12 2
(12)(52600)2.11 10 W m
4 4 (4.32 10 )radP
S Grπ π
−= = = ××
c) The Earth receiving station used parabolic dish antennas with diameters of 60m. Assuming the effective area is about half the physical area, how much power is received?
22 2 1860 1
2827 m , 1414m 2.98 10 W2 2
phys eff phys rec effA A A P PAπ − = = = = → = = ×
d) The effective noise temperature of the receiving system was about 30 K. What bandwidth of data transmission would this allow in order to maintain a SNR of at least 10?
23 22
1919
22
(1.38 10 )(30 )( ) 4.14 10
2.98 1010 2.98 10 720Hz
4.14 10
noiseP kT f K f f
S N N f
− −
−−
−
≅ ∆ = × ∆ = × ∆
×≥ → ≤ × → ∆ = ≅
×
e) A typical low-loss optical fiber transmission system can have an attenuation rate as low as 0.1 dB/km. Compare this loss with the transmission loss of the satellite link. Fiber Loss dB km km dB
WLink Loss dB dB km
9 8
8
18
(0.1 ) 4.32 10 4.32 10
1210 log 186 or 4.2 10
2.98 10−
−
= × × = ×
= = × ×
2) In a "cellular" phone system, a transmitter transmits 100 watts into a beam pattern that has a maximum antenna gain G=5 horizontally in any direction. a) What is the maximum horizontal distance at which the average radiated power density is greater than 10-7 watts/m2 and the corresponding amplitude of the electric field?
27 3
2 2
100 510 20km 2 8.7 10 V m
4 4 2t
isotrpoic
EPGS S G r S E S
r rη
π π η
− −×= = = ≥ → ≤ → = → = = ×
b) Estimate the vertical beamwidth (in degrees).
22 2
20
2 2
4 4 4 22 , ? sr, 5 rad 23
5 10 5
1 4Formally ( sin ) sin 4 sin 0.403
2 5
G
d r d rd d dr
π θπ
π θ
π π πφ π θ φ θ θ
π
θ πθ φ θ θ θ φ π θ
∆−
∆+
∆ = ∆ = → ∆Ω ∆ ∆ = = → ∆Ω = → ∆ = = °∆Ω
∆Ω = → ∆Ω = = = ⇒ ∆∫ ∫
≃ ≃
≃
2- If the Earth appears as a uniform 300 K disk in the lunar sky, what is the antenna temperature, TA, of the lunar antenna (in Kelvin)? Earth radius is 6400 km and the moon distance is 400,000 km.
21 25
24 3
30
128003.2 10 rad 1.8 , small wrt use 1
4 10
4 48.04 10 sr, 5.32 10 sr 45 3 48K
2 2.36 10
E EA earth
P P
E P A
T T
TD
ψ β
ψ π ππ
−
− −
Ω Ω = = × = ° = + − × Ω Ω
Ω = = × Ω = = = × → = + = × d) What is Tsys of the lunar receiving system (in Kelvin)?
(1.26 1)290 75 , ( 1)290 (1.58 1)290 170 , (2.51 1)290 438
84 , ; 48 48 84 132
LNA Etl tl Erec
Etl Erec
E LNA sys A E A A sys
LNA LNA tl
T K T L K T K
T TT T K T T T T T K T K
G G G
= − = = − = − = = − =
′ ′= + + = = + = = → = + =
2- A school friend of mine called one day and complained that he and his father bought the same TV. His father used the rabbit-ear antenna and received 30 channels, while he bought an excellent Yagi-Uda/log-periodic antenna (with a gain of 13) and received only 12 channels! They both live on adjacent flats in the same building. Why? Just think. The issue is directivities. Lower directivity allows more spatial coverage. 1- An antenna has a radiation efficiency of 80% and a directivity of 7.3 dB. What is the gain of the antenna in dB? D = 7.3 dB (5.37 numeric), G = eD = 0.8*5.37 = 4.3 (6.3 dB) 2- An antenna has pattern solid angle of π/4 (sr) and a radiation efficiency of 70%. The input power to the antenna is 100 W. What is the max power density at a range of 10 km? D = 4π/Ωp = 16, G = eD = 11.2, Smax = PtG / 4πR2 = 8.9 x 10-7 W/m2 (60.5 dBW/m2)
3- a) A uniformly illuminated aperture has a length of lx=50 λ and ly =100 λ. What is the beamwidth between first nulls in the x-z and y-z planes?
b) What is the antenna directivity if the physical efficiency (the ratio between max effective area, Aem, to the physical are. Aphys) is 80%?
1- The far-zone magnetic field of an antenna (located at the origin) is given by
2ˆ ˆcos (1 sin2 ) 3 sin (2 3 cos 2 ) (A m)4
j r
in
eI j
r
β
φ θ φ θ θ φπ
−
= + + +H
a) Determine the effective height.
2
2
ˆ ˆˆ cos (1 sin2 ) 3 sin (2 3 cos 2 )4 4
1 ˆ ˆcos (1 sin2 ) 3 sin (2 3 cos 2 )
j r j r
in in
e eI j j I
r r
jj
β β
η η θ θ φ φ θ φ ωµπ π
θ θ φ φ θ φβ
− −
= × = + + + =
→ = + + +
E H r h
h
b) Determine the radiation intensity.
222 2 2 2 2
2 2
2
2 2 2 2 2
2
ˆˆ cos (1 sin2 ) 3 sin (2 3 cos 2 )
2 32
( , ) cos (1 sin2 ) 3 sin (2 3 cos 2 )32
in
av
in
I
r
IU
ηθ φ θ φ
η π
ηθ φ θ φ θ φ
π
= = + + +
= + + +
r ES r
c) Determine the polarization vector of the antenna in the direction 3θ π= , 4φ π= . Is it
linear, RCP, LCP, REP or LEP?
9- A lossless antenna is completely specified by its vector effective length as 0hφ = , 2
0 cosh hθ θ= for 0 2θ π≤ ≤ , 0hθ = for 2π θ π≤ ≤ .
a) Find the radiated electric field vector as distance r, when it is excited with current I0. Integrate the associated Poynting flux (over the hemisphere), to find the total radiated power.
1cos sin cos ( 1)n nd nθ θ θ θ+= − +∫ .
c) Give an expression for its radiation resistance in terms of η, h0, and λ.
b) Write an expression for its gain G(θ). Describe and/or sketch its beam shape and calculate its beam width (at half maximum gain).
7- Determine the directivity of the antenna with the radiation pattern given below. Specify your result on a linear scale, in dB relative to an isotropic radiator (i.e. dBi). Sketch the horizontal and vertical radiation patterns on a linear scale in polar coordinates.
sin 9 5 9( , )
0 elsewhere.F
θ π φ πθ φ
< <= 0 5 9
2
0 9
46.75 8.293dBi
sin sinD
d dπ π
π
π
θ θ θ φ= = =∫ ∫
5- Given that an antenna has a directivity of 40 in a certain direction and that its loss resistance is one fifth of its radiation resistance, find its gain in the same direction. 5- Bir anten için 40D = olsun. Bu antenin kayıp direnci radyasyon direncin beste biri ise ( 1 5loss radR R = ) antenin kazancı nedir( ?G = )
21
2
2 2 212
1 5( , ) , ( , )
4 4 ( ) 1 6
540 33.3 15.2 dB
6
rad rad
in r in rad loss loss rad
S S G P I RG D
P r P r D P I R R R R
G
θ φ θ φπ π
= = → = = = =+ +
= × = =
1- Consider the antenna pattern for a lossless antenna.
a) What is the value of the peak sidelobe level? 25 dB b) Mark the half power points on the plot. c) How many sidelobes are shown in the plot? 28 d) How many grating lobes are shown in the plot? 0 f) What is difference in gain between the pattern maximum and first sidelobe in dB? 25 dB g) What is the difference in gain between the pattern maximum and first sidelobe, as a ratio of field intensities?
1- A 1 km radio link operates in free-space at 10 GHz. The transmitting and receiving antennas are identical, lossless, polarization and impedance matched. The transmitter delivers 10 mW to its antenna and the receiving antenna must deliver 1 µW to the receiver. Find the minimum gain in (dBi) required for the antennas when they are perfectly aligned. Find also the capture area (in m2) of the receiving antenna when it is misaligned by half the 3 dB beamwidth.
2 230
02 2 24 4.19 10 36.2dBi
4 4 4 (4 )t t t r
r t r t r
t
P P PG PP G A G G G r
r r r P
λπ λ
π π π π λ= = = ⇒ = = × =
22
0 3 33.2 dBi 0.15m4
effG G A Gλ
π= − = → = =
2- You are designing a 3 GHZ communications station for a lunar base. The distance from the Moon to Earth is 4x105 km and the Earth’s diameter is 12800 km. Your lunar antenna is a lossless, 2 m diameter dish with an aperture efficiency of η=0.6. Immediately behind the antenna, you mount an LNA whose noise figure and gain are 1 dB and 20 dB, respectively. The LNA is connected to a receiver in the lunar base by a coaxial cable whose total loss is 2 dB. The receiver has noise figure of 4 dB. The physical temperatures of the receiver, coax, and LNA are all 290 K. Assume that video and data from the Earth to the moon requires a bandwidth of 10 MHz and 12 dB S/N ratio. a) What is the free space link loss? r = 4 x 108 m, λ = 0.1 m, L = (4πr/λ)2 = 2.53 x 1021 = 214 dB b) You wish to point the receiving antenna within its 3 dB points. What is the pointing accuracy requirement (in degrees)?
5- A handheld wireless telephone using an antenna with a gain of 2 dBi is to be used for communication with a satellite at a distance of 36000 km. It is given that the gain of the transmitting antenna in the satellite at 2.4 GHz is 8500 and the transmitting antenna is radiating a power of 100 W. a) Calculate the power density of the incoming signal from the satellite. b) Calculate the power received by the antenna of the handheld wireless telephone. c) Calculate the open-circuit voltage developed at the terminals of the receiving antenna given that the impedance of the antenna of the handheld telephone is ZA = 75 + j100 ohms. d) Calculate the effective areas Ae for both the transmitting and the receiving antennas.
a)
b)
c)
d)
12 2max 2 7 2
-12
2 22
1008500 52.19 10 W m
4 4 (3.6 10 )
(dBm) (dB) (dB) 20 log(36000) 20 log(2400) 32.44 99.9 dBm 0.103 pW
8 8 75 0.103 10 7.86 W
3010.57 m ,
4 4 240
t t
r t t r
oc a r
et t er r
G PS
r
P P G G
V R P
A G A G
π π
µ
λ λ
π π
−= = = ××
= + + − − − =− =
= = × × × =
= = = =
221.585
19.7 m4
cπ
= 1- Consider a pair of identical (lossless) parabolic dish antennas of 2 meter diameter, configured for a horizontal microwave link at 15 GHz. They are pointed toward each other at a separation of 50 km. The first antenna transmits a power of 1 watt. Calculate the power received in the second antenna given the following. The aperture efficiency of each antenna is 70%. Antennas are polarization matched to each other and impedance matched to receiver. If the same calculation were done at a lower frequency, you should predict a decrease in the power received; give a physical explanation for this power reduction.
2 4
2 2
46
2 3 2
4 4 0.70.7 0.7 m , 0.02m, 6.91 10
1 (6.91 10 )0.7 4.84 10 W
4 4 (50 10 )
te t phys t
t trec inc e e
c AA A A G
f
PG WP S A A
R
π π ππ λ
λ λ
ππ π
−
= = = = = = = = ×
×= = = = ×
×
If we
lower the frequency, the wavelength increases resulting a decrease in the antenna gain.
1- The antenna for the base-station of a cell phone system have a horizontal (φ-direction) beamwidth of 120o and a vertical (θ-direction) beamwidth of 5o at 1 GHz. a) Estimate its maximum gain. b) The radiated electric field strength is specified to be a minimum of 10 millivolts/m at ranges of up to 30 km from the base-station - in directions covered by the maximum gain. What is the minimum power that should be transmitted? c) The typical mobile user has a handset with a dipole antenna which we can model as having a gain G=1 in a typical orientation. What is the maximum power available from his receiving antenna when the conditions of (a) & (b) apply for a user at 30 km range? d) What other factors might reduce the actual power received below the value from part (c)?
a)
b)
c)
d) Polarization mism
max 2
27 2
2
2 27 10
2
2 4 36 3 4 216120 rad, 5 rad 68.75
3 36 2
0.01V m 1.33 10 W m 21.7 W2 4
4 0.3m, 1 , 1.33 10 9.5 10 W 60dBm
4 4
t tt
ee rec e
G
PGS P
r
AG A P SA
π π π πφ θ
θ φ π π
η π
π λ λλ
λ π π
−
− −
× ×∆ = ° = ∆ = ° = → = = = =
∆ ∆
= → = = × = → ≥
= = = → = = = × × = × = −
EE
atch, impedance mismatch, and possible ohmic losses in antenna.
4- A 150 kW monostatic x-band radar (f=10 GHz) uses an antenna with a directive gain of 30dB. The radar is tracking a target with a radar cross section of 2m2 at a range of 25 km. a) Determine the power density of the incident field at the target.
b) Determine the magnitude of the incident electric field intensity at the target.
c) Determine the total power intercepted by the target.
d) Determine the power density of the scattered field at the radar.
e) Determine the magnitude of the scattered electric field intensity at the radar.
f) Determine the total power received by the radar.
1- The impedance of a short antenna is modeled by a resistance Rrad in series with a capacitance Cant; its vector effective length h is known. Draw the equivalent circuit for the antenna receiving an incident electric field given by a complex vector Einc (arriving from the direction in which h is given). Label the circuit components.
2- The sketch below shows a portable phone in three different positions (A,B&C) relative to its base unit, which is at the origin of the coordinate system. All three positions are in the horizontal plane (z=0); B & C are 10 times further than A; A & B are oriented parallel to the z-axis; C is oriented along the φ-axis (user lying on couch watching TV). Given that the portable antenna is linearly polarized of effective length ho parallel to its antenna and the
vector E-field radiated from the base unit is 0ˆ ˆsin ( 0.2)
4
jkr
rad
ej I h j
rωµ θ θ φ
π
−
= +E .
a) Draw an equivalent circuit for the portable phone receiver and antenna. Give an expression for the power (PA) received in case A assuming the receiver is impedance matched to its antenna impedance Ra.
b) Find the ratio of the powers received by the portable phone PB/PA and PC/PA. If these cases cover the best and worst conditions, what dynamic range in dB is needed for the receiver amplifier? (The power level would be stabilized by an automatic gain control circuit that must function over this dynamic range).
1- A circularly polarized wave, traveling in the +z direction, is received by an elliptically polarized antenna whose polarization is given by ˆ ˆ ˆ(3 ) 10a x jyρ = + .
a) Find the polarization mismatch factor (p) (in dB) when the incident wave is LCP. 2
ˆ ˆ ˆ ˆ3ˆ ˆ 0.2 6.99dB
10 2a w
x jy x jyp e e
+ += ⋅ = ⋅ = = −
b) Find the polarization mismatch factor (p) (in dB) when the incident wave is RCP.
2ˆ ˆ ˆ ˆ3
ˆ ˆ 0.8 0.969dB10 2
a w
x jy x jyp e e
+ −= ⋅ = ⋅ = = −
2- A linearly polarized wave traveling in the negative z-direction has a tilt angle of +55o with respect to the x-axis. It is incident upon an antenna whose polarization is given by ˆ ˆ ˆ( 4 ) 17a x j yρ = − . Find the polarization mistmatch factor (p) in dB.
3- Verici bir antenin elektrik alanı 0ˆ ( , ) jkz
i E x y e−=E x olsun. Aşağıda polarizasyonları
verilen alıcı antenler kullanıldığında antenlerin polarizasyonlarının kaynaklanan güç kaybı ne kadar olacaktır? a) ˆ ( , , )a E r θ φ≅E x güç kaybı olmaz
b) ˆ ( , , )a E r θ φ≅E y güç tamamen kaybolur
c) ˆ ˆ( ) ( , , )a E r θ φ≅ +E x y gücün %50’si kaybolur
1- When the US fleet came to Beirut in 1983, they anchored 20 kms off-shore. The American University of Beirut had some really dumb students. They took 1 W X-band klystron (10 GHz) from the microwave lab, and shined it with a 20 dB gain horn on the US fleet. This made the captain really mad, and he sent his helicopters to check out the source. Finally, after several hide and seek games, the US command met with the AUB-EE department and took the X-band source. Nowadays, a big ship is vulnerable to a torpedo/mine or an air attack. (To study the torpedo/mine problem, take an acoustic course.) The radar system is the backbone of the ship air defense system. Its modulation and frequency hopping procedure are classified information. However, let us use some typical numbers for the US fleet radar. f=10GHz, Pt=100kW, BW=100MHz (10 ns Pulse), Pulse Repetition Rate=10KHz, Averaging=10 (1 msec integration time-update), Antenna Gain= 45 dB (εap=70%), System Noise temp=300K (3 dB Noise Figure). The system temperature is a measure of the noise of the radar system. The noise is given by Pn =kTB, where k is Boltzman Constant k=1.38x10-23 J/K, T is the system temperature in Kelvins, and B is the bandwidth of the radar receiver. In order to have a high probability of detectivity, the received signal should be 4 times higher than the noise level of the radar (S/N=0.6 dB). The main threat in this area is an attacking Syrian airplane for example, a MIG23. This plane can fly at Mach-2 (2000 ft/sec). When loaded with firepower, it has a radar cross section of 2 m2 (approximately.) a) Calculate the range of the US radar without the AUB microwave source. Find the warning time the US fleet has against an attacking MIG23.
b) The attacking plane has an antenna with a gain of 32 dB at 10 GHz and a noise temperature of 1000K. Calculate the distance the plane will hear the radar. Compare with the radar range in (a). Assume PLF=0.5.
c) Calculate the power received by the US radar where the AUB source is turned on. Assume antennas are aligned for maximum gain and the PLF is equal to 0.5.
d) The amount in (2) becomes the new noise power of the US radar. Calculate the new range and warning time of the US fleet.
e) Think of some ECM (Electronic Counter Measure) that you could use if you were the Captain. Use frequency hopping techniques.
1- Consider two TV transmitting stations, Channel 2 (54-60 MHz) and Channel 12 (204-210 MHz). Both stations are located in Detroit, 40 miles away from Ann Arbor. a) Determine the maximum ERP that these stations can use knowing that their transmitters are located 500 ft. above ground. The gain of the transmitting antenna is 2.8 relative to a half-wave dipole. Determine the power sent by the TV stations.
b) Determine the signal strength that these stations produce in Ann Arbor at a receiving antenna 30 ft. above ground (typical roof-mounted antenna). It is assumed that the ground roughness is 50 m between Detroit and Ann Arbor. Use the table of 29-2 for the correction factor of channel 2.
c) Determine the rms voltage across a matched 75 Ω load when a Yagi-Uda antenna with a gain of 12 dB over isotropic is used. Derive equation 29-2 in handouts.
d) Determine the antenna noise voltage for Channels 2 and 12 (which is delivered to a matched impedance). Use table 29-3 for the galactic noise component.
e) Calculate S/N ratio (in dB) at the input antenna terminals when the antenna is connected with a 40 ft. cable of RG-59/U to the TV. The noise figure of the TV receiver is 3 dB (T=290K) at both channels.
f) Calculate the maximum range for a "passable" picture quality for Stations 2 and 12 (S/N = 30 dB). Max passable range found from Fig 29-1: we have dropped 20 dB after moving 25 miles more. Total range to get 30 dB S/N=65 mil
1- The Direct Broadcast Satellite (DBS) is a digital version of the analog-TV located above the U.S. It is placed at 36,000 km above the equator, at the geostationary orbit. It can transmit up to 75 channels, each 6 MHz wide in the allocated bandwidth (there are actually two satellites covering around 150 channels). The receiver is an 12” (30 cm) dish with an aperture efficiency of 80% and a noise figure of 1 dB (noise temperature of 75 K). The frequency of operation is 11.7 GHz-12.2 GHz (500 MHz) and the total radiated power is 100 W from the satellite. a) Calculate the gain of the receiver antenna in dB. Estimate the half power beamwidth. b) Calculate the space loss factor between the satellite and the earth receiver. c) Calculate the received S/N ratio if the transmitting antenna has a gain of 33 dB. What is its size (εap=85%)? What is approximately the footprint of this antenna on earth? d) Can we use an antenna on the satellite with a gain of 43 dB?
1- Quick Answers, Write one or two words for each item. a) What is the graphical depiction of the variation of the field strength of an antenna as a function of angle more normally called? What are the typical units for the horizontal axis? What are the typical units of the vertical axis? Pattern plot, angle, dB c) Name the three different regions that describe the type of fields around an antenna as a function of distance from the antenna. reactive near field, radiating near field, far field d) At what distance r, does the far-field of antenna start? Remember to write your answer in terms of an antenna dimension and the wavelength of the frequency of operation. 2D2/λ f) What is the type of polarization of most antenna systems that must communicate between space and earth? Circular g) What are the units of Radiation Intensity Watts/steradian h) What are the units of Radiation density Watt/m2 k) Gain = kD, what is k? antenna efficiency e) What field component determines the polarization of the antenna? E-field b) Which antenna has higher directivity, an omni directional antenna or a dipole? dipole n) For circularly polarized waves the phase difference between Ex and Ey is +π/2 or -π/2. o) What is the polarization of a wave is its Axial Ratio is infinite? Linear c) What is the total solid angle of a sphere? 4π x) The pattern solid angle of an isotropic radiator is ΩA = 4π. True False a) The directivity of an isotropic source. D = 4π/ΩA, ΩA = 4π, so D = 1 c) What does it mean to express Directivity in dBi? Decibels over isotropic j) The HPBW of an antenna radiation pattern is the angle where the normalized E-field En(θ) = 0.707 l) The relative BW of an antenna with large abrupt discontinuities like a dipole or monopole is typically narrow m) The relative BW of an antenna with small and gradual discontinuities like a spiral or log periodic antenna is typically wide p) A vertical linearly polarized antenna transmits to a linearly polarized receive antenna tilted 45 degrees from vertical. The polarization loss equals ½ or -3 dB r) EIRP can be kept constant while reducing transmitted power by a factor of 4 if antenna gain is increased by 6 dB. True False s) Free space loss for a fixed wireless link in vacuum is less at 3 GHz than at 1 GHz. True False t) 10 W/m2 is an acceptable power density for human exposure at all frequencies between the AM broadcast band and millimeter waves. True False u) The electric polarization of a medium is a measure of the medium’s adjustment to an imposed electric field. True False v) If the electric field at time t is zero, then the electric polarization of a medium at time t must also be zero. True False y) For any integer n, the main lobe of a dipole antenna of length n*(λ/2) is normal to the axis of the antenna. True False
2- An antenna located at the origin of a coordinate system has a far-zone radiation pattern given by (f=25 MHz and Iin is the terminal current)
2ˆ ˆ10 cos (1 cos ) 15 sin (1 cos ) V mj r
a in
eI j
r
β
η θ φ θ φ φ θ−
= − + +E
a) Determine the vector effective height h(θ,φ)
b) Find the open circuit voltage (magnitude and phase) for this antenna when it is in the receiving mode. Assume that the field incident (f= 25 MHz) from the direction θ = π /4, φ = π/3 is a plane wave whose magnetic field at the antenna location is given by
c) Find the effective area (in dB) for part (b). Assume the antenna is lossless, has an input impedance of ZA=58+j10 Ω and a load of ZL=50 Ω is connected directly to its terminals.
