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 2 possible effective area (m ) at 8 GHz.
0. 1 cos4 θ W sr, 0 ≤ θ ≤ π 2 and 0 ≤ φ ≤ 2π U (θ, φ) = 0, elsewhe re. elsewhere. 0.1 cos4 θ HPBW = → 0.5 = ⇒ θ = cos−1 (0.50 .25 ) = 32.7651° U max 0.1 U max = 0 @ θ = 0, HPBW = 2(32.7651°) = 65.5302° U (θ, φ)
Pra d
= ∫∫
ΩUdΩ =
D(θ, φ) =
π2
∫ ∫ 0. 1 cos
4πU (θ , φ) P rad
2π
0
0
4
θ si n θd θd φ = 0. 04π
4π (0. 1 cos4 θ) = = 10 c os4 θ → D 0 = 10 = 10 dBi @ θ = 0 0.04 π
G (θ, φ) = ecd D(θ, φ) = 0.95(10 co cos4 θ ) = 9.5 co cos4 θ
= ecd
Aem
λ2
4π
D 0
= 0.125664 W
→ G0 = 9.5 = 9.777 dBi
= 1.06169 × 10− 3 m2
2- An antenna ant enna in free space has has an input inp ut p ower of 20 W and radiates radiates 19.0608 W. Determine 2 t he radiated radiated power p ower density density (in W/m ), the radiation intensity (in W/sr), the directivity and maximum maximum directivity (in dBi), and and the t he radiation radiation efficiency efficiency if its p hasor far-zone electric electric and magnetic fields are given by e − j 8 r e − j 8 r 3 E = aˆφ 50 si n θ (V m), H = −ˆa θ 0. 13272 si n3 θ (A m) . r r
Pav e
6 6 = 1 Re {E × H ∗} = ˆar 3.318 2si n θ W m2 , Pave = E = 3.318 s2 in θ W m2
2
U (θ, φ) = r 2 Pave D0
r
= 3.318 sin6 θ W sr,
Bi , e = = 2.1875 = 3.39945 ddB
P rad P i n
D (θ,φ ) =
= 95. 3%
2η 4πU (θ , φ) P rad
r
= 2.1875 sin6 θ
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 directivi directivity ty D(θ,φ). What is the maximum directivity?
e − j kr cos θ (V m) θ ≤ 90° aˆθ 4πr 80 co E(θ , φ) = e − jkr 20 cos θ (V m) 90° ≤ θ ≤ 180° aˆθ 4πr 400 cos2 θ θ ≤ 90° 0. 05379 co 2ηπ2 cos 2 θ θ ≤ 90° 2 r 2 2 U (θ, φ)= W sr sr = E θ + E φ = 25 2 2η 0 . 0 0 3 3 6 2 c co o s θ 9 0 θ 1 8 0 2 ° ≤ ≤ ° 90° ≤ θ ≤ 180° 2ηπ2 cos θ 90 2π π2 π Prad = ∫ ∫ 0. 05379 c os 2 θ sin θ d θ = ∫ 0. 003362 co cos 2 θ sin θ d θ dφ = 0.1197 W, 0 0 π2 cos 2 θ θ ≤ 90° 5.647 co 4πU D (θ, φ) = = 104.98U (θ, φ) = → D max = 5. 647 = 7.518 dBi 0.3529 co P ra d cos 2 θ 90 9 0° ≤ θ ≤ 180° Balanis Balanis 2.10. The radiat radiation ion intensit intensity y of an antenna is represent represent ed by by
1 0° ≤ θ ≤ 30° 0.5 30° ≤ θ ≤ 60° U = 0.1 60° ≤ θ ≤ 90° 0 90° ≤ θ ≤ 180° (a) What What is the t he directivity directivity (above isotropic) of the t he antenna (in (in dB)? π 3 π 2 π 6 Pra d = ∫ ∫ U sin θ d θ dφ = 2π ∫ sin θ d θ + ∫ 0.5 si n θ d θ + ∫ 0. 1 s i nθ dθ = 0. 734π = 2.3059 0 0 0 π6 π 3 5.4496 = 7. 7 . 3636 dB 0° ≤ θ ≤ 30° 2.7250 = 4. 3530 dB 30° ≤ θ ≤ 60° 4πU ma x U ma x D0 = = = D (θ ) = 0. 54496 = −2. 636 dB 60° ≤ θ ≤ 90° Pra d U 0 0.0000 = −∞ dB 90° ≤ θ ≤ 180 °
2π
π
(b) What is t he directivity directivity (above infinitesim infinitesimal al dip ole) ole) of t he antenna antenna (in dB)?
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 directivi directivity ty D(θ,φ). What is the maximum directivity?
e − j kr cos θ (V m) θ ≤ 90° aˆθ 4πr 80 co E(θ , φ) = e − jkr 20 cos θ (V m) 90° ≤ θ ≤ 180° aˆθ 4πr 400 cos2 θ θ ≤ 90° 0. 05379 co 2ηπ2 cos 2 θ θ ≤ 90° 2 r 2 2 U (θ, φ)= W sr sr = E θ + E φ = 25 2 2η 0 . 0 0 3 3 6 2 c co o s θ 9 0 θ 1 8 0 2 ° ≤ ≤ ° 90° ≤ θ ≤ 180° 2ηπ2 cos θ 90 2π π2 π Prad = ∫ ∫ 0. 05379 c os 2 θ sin θ d θ = ∫ 0. 003362 co cos 2 θ sin θ d θ dφ = 0.1197 W, 0 0 π2 cos 2 θ θ ≤ 90° 5.647 co 4πU D (θ, φ) = = 104.98U (θ, φ) = → D max = 5. 647 = 7.518 dBi 0.3529 co P ra d cos 2 θ 90 9 0° ≤ θ ≤ 180° Balanis Balanis 2.10. The radiat radiation ion intensit intensity y of an antenna is represent represent ed by by
1 0° ≤ θ ≤ 30° 0.5 30° ≤ θ ≤ 60° U = 0.1 60° ≤ θ ≤ 90° 0 90° ≤ θ ≤ 180° (a) What What is the t he directivity directivity (above isotropic) of the t he antenna (in (in dB)? π 3 π 2 π 6 Pra d = ∫ ∫ U sin θ d θ dφ = 2π ∫ sin θ d θ + ∫ 0.5 si n θ d θ + ∫ 0. 1 s i nθ dθ = 0. 734π = 2.3059 0 0 0 π6 π 3 5.4496 = 7. 7 . 3636 dB 0° ≤ θ ≤ 30° 2.7250 = 4. 3530 dB 30° ≤ θ ≤ 60° 4πU ma x U ma x D0 = = = D (θ ) = 0. 54496 = −2. 636 dB 60° ≤ θ ≤ 90° Pra d U 0 0.0000 = −∞ dB 90° ≤ θ ≤ 180 °
2π
π
(b) What is t he directivity directivity (above infinitesim infinitesimal al dip ole) ole) of t he antenna antenna (in dB)?
2- Dekanlık Dekanlık ile i le Bolü Bolü m arasınd arasında a 70 m’lik kablosuz bir link kurulacaktır. Vericinin maksimum çıkış çıkış gücü 100 W, frekansı 5 GHz ve alıcının alıcının alma hassasiy eti de -80 dBm’dir. dBm’dir. a) 1.5 dB/m dB/m zayıfla zay ıflatması tması olan olan bir b ir koaksiy koaksiyel el kablo kablo lu link link kulla ku llan n arak arak bu iş i ş gerçekleş gerçekleşir mi? b) 3 dB kazancı olan iki antenden olu ş an kablosuz bir link ile bu iş i ş gerçekleş gerçekleşir mi? c) (a) ve (b)’dek (b)’dek i senaryolarda maksim maksimum um mesafeleri mesafeleri hesap layın. 2- You wish t o implement implement a 5 GHz GH z communi communica cation tion link b etween the t he Elec Electt ric-Eng ric-En g. Buildin g and the Library Library (70 m apart). apart ). A 100 mW transmitt transmitt er po p ower is available available and the minimum detectable signal at the receiver is -80 dBm. (a) Would Would a “wired “w ired conn connec ection” tion” consistin g of a coa xial line wit h att enuation enuation of 1.5 dB/m be capable of doing the job? Explain.
7 0 m = 105 dB dB, Pi = 1 00 mW = 10 l oogg(100 1) = 20 d Bm Bm = 1.5 dB m ×70 dBm< − 80 dBm dBm wo won't n't wor work! ⇒ Pr = Pi − P loss = 20 − 105 = −85 dBm
Ploss
(b) Would a “wirel “w ireless ess connection” connect ion” consist consisting ing of identical identical t ransmit ransmit and receive receive antennas with 3 dB of gain be capable of doing the job? Explain.
Pr
λ 2 0.06 2 −9 1 . 8 6 1 1 0 W = −57.3 dB dB m > −80 dB d Bm O K! = PtGtGr = (0.1)(2)(2) = × 4πr 4π × 70
(c) Also, compare t he maximum maximum transmi t ransmission ssion rang ran ges of sce s cenarios narios (a) and (b).
100dB
(0.1)(4)(0.06)2 rm ax ( wired ) = = 67 m, r max(w ireles s) = = 955 m 1. 5 dB m (10−1 1 )( 4π )2 2-Uz ak alan alan ortalam ort alamaa gü gü ç y o ğunluğ unlu ğu Pavg =
2sin θ cos φ
π
rˆ ( W /m 2 ) , 0 <θ < π, 0 < φ < olan r 2 bir antenin direktif kazancını ( D ( θ, φ ) ve D 0 ) hesaplayın. 2
2- Calculat Calculatee the direct direct ive gain and dir dir ectivit ectivity y of an antenna that produces at far field the t ime 2sin θ cos φ π average average vector p ower density density given iven by Pavg = rˆ ( W /m 2 ) , 0 <θ < π, 0 < φ < . 2 r 2 D(θ, φ) =
U ( θ, φ) Ua vg
=
r 2Pa vg U a vg
=
r 2Pa vg
P rad 4π
=
8π s in θ cos φ π
π2
∫ ∫ 0
0
2sin 2sin θ cos φ sin sin θdθd φ
= 8 s in θ cos φ → D 0 = 8
− jkr
e 1- The far electric field of a certain antenna is given by E(θ, φ) = θˆ Ε0
cos(2φ) . r Determine the power radiated by the antenna and the directivity . Would this antenna be potentially useful as a broadcast antenna? Why or why not? U (θ, φ) =
2 2 1 2 1 2 E E r E 0 cos 2 (2φ) + = θ φ 2η 2η
2π π
Pr
=∫
π
∫ U (θ,φ )sin θ d θ dφ = 2η ∫ cos (2φ)d φ ∫
0
D0
2π
E 02
2
0
0
= U max = Uavg
U max Pr 4 π
sin θ d θ =
0
E 02 η
2π
∫ cos (2φ )dφ = 2
0
πE 20 η
= 4πU max = 2, Since D is small, the antenna could be useful as a P r
broadcast antenna except that the pattern has nulls. e − jkr j φ ˆ e . Determine the power 2- The far E- field of an antenna is given by E(θ, φ) = θ E 0 r
radiated by the antenna and the directivity.
1 Pr = 2η D0
=
2π π
∫∫E 0
U max Pr 4π
2 θ
2
r sin θ d θ dφ
=
E0
2η
0
=
2 2π π
− jkr 2
e
∫ ∫ r 0
e
j φ
r sin θ d θ d φ = 2
2π E 0
2
η
0
1 2 1 1 max E 0 2 r 2 = 1 2η Pr 4π r − jkr
e θ ηI 0 3- Akim kaynağı I 0 olan bir antenin uzak elektrik alanı E(r, θ ) = ˆ
(1 − cos θ ) V/m
r ise bu antenin yaydığı toplam gücü ve bu antenin radyasy on direncini hesaplayın. . 2 2π π 2 ηI 8πηI 02 2P 16πη 0 2 Prad = (1 − cos θ) sin θ d θ dφ = W, Rrad = rad = 0 0 I 02 2 3 3
∫ ∫
Ω,
4- An antenna, with inp ut impedance of 73 ohm, is t o be connected to a transmission line whose characteristic impedance is 50 ohm. Assume that t he pattern of the antenna is given by
U (θ, φ) = B 0 sin3 θ . Find t he directivity and the overall maximum gain of t he antenna. 2π π
Pr
=∫ 0
Γ=
2π π
U 3π 2 16 B0 sin θ sin θd θ d φ = B 0 → D 0 = 4π max = = 1.698 P rad 4 3π
∫ U (θ, φ) sin θ d θ dφ = ∫ ∫ 0
0
0
3
73 − 50 = 0.186 → er = 1 − Γ 2 = 0.965 → G0 = er D = 0.965 × 1.698 = 1.639 = 2.14 dB 73 + 50
5- The far E-field of a short vertical current element Idl located at the origin of a spherical coordinate sy stem in free space is given below. Find the total average power radiated by this 60 πI dl e − jβ r sin θ θ current element. E(r, θ ) = j θˆ (V/m). r λ
H(r , θ) =
1 η0
rˆ × E(r, θ ) =
E θ (r , θ ) ˆ φ η0
=
j
Idl
2λr
ˆ A m sin θ e − j β r φ
1 Idl 2 2 ˆ ∗ sin θ r Pav (r , θ ) = Re {E× H } = 15π λr 2 Pr
=
2π
π
∫ P (r, θ) ⋅d s = 15π ∫ ∫ 0
S
0
2 Idl 2 sin2 θ r 2 sin θ d θ dφ = 40 2 Idl π W λr λ
6- Compute the radiation resist ance of an antenna whose far-zone e lectric field of in free ηI 0 e − jkr ˆ (1 − cos θ) V/m . space with input current I 0 is given by E = θ θ 10 r − jkr
1
I e H = ˆr × E = φˆ 0 (1 − cos θ), S = 12 Re {E × H ∗ } η 10 r 2π π
Prad
= lim S ⋅ ˆr r sin θ d θ d φ = r →∞ ∫ ∫ 2
0
0
2 2π π
ηI0
200
η 4π 2πηI 02 → Rrad = 2P rad = (1 − cos θ) sin θ d θ d φ = 2 75 75 I 0
∫∫ 0
0
2
ˆθ 7- The far E- field of an antenna is given by E(r) = a
jωµ 0I 0L e − jkr sin [kL cos θ ]
sin θ . r kL cos θ Determine the power radiated, the maximum directivity and the radiation resist ance of the π sin 2 ( π cos θ) sin3 θd θ ≃ 0.8 . antenna if kL = π . Hint: 2 0 (π cos θ ) 4π
∫
Prad D
1 = 2η
=
2π π
∫∫E 0
U max Prad 4π
0
2 θ
2π ωµ0I 0 L2 r sin θ d θ dφ = 2η 4π 2
= 2.5 at
θ
= 90 , Rrad = 0
2Prad I 02
π
∫ 0
sin2 [ π cos θ ] 3 0.8π ωµ0 I 0L 2 sin θ = 2 (π cos θ) η 4π
1.6π ωµ0 L 2 = η 4π
1- Bir antenin uzak elektrik alanı E (r, θ ) = 100I 0
e − jkr
cos4 θ V/m, 0° ≤ θ ≤ 90° olsun.
r 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 E (r, θ )= 100I 0
e − jkr
cos4 θ V/m, 0°≤ θ ≤ 90° .
r a) Find the radiation intensity and the total power radiated. r 2
U (θ, φ) =
Prad
=∫
E
2η0
2π
∫
0
π
0
2
= 13.263 I 0
2
cos4 θ, 0° ≤ θ ≤ 90° 0, elsewhere
2π(100)2 I 0 U (θ, φ)sin θ dθ dφ = 2η0
2
π2
∫ 0
cos8 θ sin θd θ = 9.259 I 0
2
W
b) What is t he radiation resistance and the directivity of this antenna?
Rrad
=
2Prad I 0
2
= 18.5 Ω, D 0 =
4 πU max P rad
(100)2 I 0 2 = 4π 2η0
π(100)2 9η0
I 0
2
= 18 = 12.5 dB
b) What is the maximum effective area of this antenna in square wavelengths?
Aem
=
λ2
4π
D 0 = 1.432λ2
2- The radiation intensity of an antenna is given below for. Determine the maximum effective 2 aperture (in m ) of t he 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 Z A = 55 + j8 Ω and that the antenna terminals are connected to a load of Z L=50 Ω.
cos3 θ, 0° ≤ θ ≤ 90° U (θ, φ) = 0, 90° ≤ θ ≤ 180° D 0
= 4π
2π
∫ ∫ 0
λ
π2
0
U ma x
cos3 θ sin θd θd φ
1
= 4π 2π
∫ 0
π2
cos3 θ sin θd θ
= 4π1 = 8 2π 4
−1.4 dB = 10− 0.14 = 0.7244 = 0.03 m, ecd = 1, PLF =
4RARL λ2 −4 er = 1 − Γ = m2 = 0.992, Aem = erecdPLFD 0 = 4.1175 × 10 2 2 (RA + RL ) + X A 4π 2
4- An antenna over ground is driven by a terminal current of I 0 resulting in a radiation intensity function defined by
U (θ, φ) = 4I 20 sin2 θ cos2 φ (W rad2 ), 0 ≤θ ≤ π 2, 0 ≤ φ ≤ 2π . (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 I 0 =2A. (d) Determine the magnitude of the magnetic field at the point(s) defined in part (d). (a) Prad
=∫
(b) Rrad
=
2π
∫
0
π 2
0
2Prad I 0
= 16.8 Ω, D0 =
2
U (θ, φ)
(c) Pavg
=
(d) Pavg
=
U (θ, φ)sin θd θd φ = 4I 20
r2
→ (P avg )ma x =
4πU max P rad
2π
∫ ∫ 0
π2
0
8 sin3 θ cos2 φ d θdφ = π I 02 = 8.38I 02 W 3
= 4π ( 4 I 0 2 )
[U (θ ,φ )]max r 2
(
8 3
π I0
2
)= 6
2
4 I 0 = = 16mW m2 2 (1000)
2P avg 1 η0H 2 → H = = 291 mA m 2 η0
2- Radyasyon yoğunluğu U (θ, φ) = 1.5cos θ, 0 < θ < π 2 , 0 < φ
< 2π , olan bir anten için
toplam radyasy on gücünü ve directivity’yi hesaplayın.
Prad
=∫
2π 0
π2
∫ 0
U (θ, φ) sin θ d θ d φ
=
U 3π 1.5 W, D0 = max = = 4 (θ = 0) Prad 4π 2 1.5π 4π
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 p ower density?
D
=
4π
Ω
= 16 →
Gt
= 0.7 × 16 = 11.2, S max =
GtP t 2
4πr
= 8.9 × 10− 7 W m2 = −60.5 dBm
1- A satellite S transmits an electromagnetic wave at 10 GHz via its transmitting antenna. The characteristics of the sat ellite-based transmitter are: (a) the power radiated away by the satellite antenna is 10 W, (b) t he distance between the satellite antenna and a point A on the earth’s surface is 37,000 km, and (c) the satellite transmitt ing 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 p oint A.
D0
=
4πU max Prad
= 10 ,U max = r 5
2
E max
2η
2
→
E max
=
2ηPrad D 0 = 2.09 × 10− 4 V m 2 4πr
2- 500 MHz’te çalışan bir 1 kW’lık bir UHF vericisin anten kazancı 10 dB’dir. 100 km öt ede 3 dB kazancı olan alıcı bir antende elde edilen maksimum güç ne kadardır? 2- A UHF transmitter op erating at 500 M Hz delivers 1 kW into an antenna with 10 dB of power gain. A receiver antenna is 100 km away from the t ransmitt er (direct line of sight). What is the maximum received power to be expected if the receiver antenna gain is 3 dB?
Pr (dB) = Pt (dB) + Gt (dB) + Gr (dB) − 20 log R(k m) − 20 log f (MHz ) − 32.44 Pr (dB) = 10 log(1000) + 10 P r
+ 3 − 20 log R(100) − 20 log f (500) − 32.44
= 30 + 10 + 3 − 40 − 53.98 − 32.44 = − 83.42 dB=10
-(83.42 10)
= 4.55 ×10− 9 W
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 p ower must be at least -100 dBm. Assume that transmit and receive antennas are directed towards each other.
λ 4πr
Gr (dB) = Pr (dB) − Pt (dB) −G t(dB) − 20 log
= −130 − 10 − 31 − 20 log
0.029 = 3.32 dB = 2.148 4π × 1200 × 103
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 d egrees with resp ect t o vertical, what is the new maximum range? If the scouts were 9 km apart at the time, can they still communicate?
λ Pr
=
c f
= 0.648 m, R = 12.872 × 103 m, Pt = 1W, Γ = 0, PLF = 1,G = 1.6406 = ecdDt = ecdDr
λ 2 D (θ , φ )D (θ , φ ) PLF = 43.19573 pW = Pt ecdtecdr (1 − Γt )(1 − Γr ) 4πR t t t r r r
PLF = ˆρt i ρˆr
2
2
2
= cos Ψ 2 = cos 35° = 0.6701
2 0.648 100.215 100.215(0.671) → Rmax,sit = 10, 544.1m 43.19573 × 10− 12 = (1W)(1)(1) 4πRmax,sit
1- Two vertical half-wavelength dip oles are used in a communication link at f = 400 MHz. The distance between the transmitter and receiver is 5 Km. The transmitt er 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.
73 − 50 2 er = 1 − Γ = 1 − 73 + 50 = 0.965, λ = c f = 0.75 m, G 1 = G2 = 1.64 λ 2 0.75 2 Prec = Pt G1G 2er 1er 2 = 2 (1.64)2 (0.965)2 = 7.14 × 10− 10 W 4π R 4π × 5000 2
b) Calculate the received p ower if the receiver antenna is a short dipole of length l = 7.5 cm. 2 2 l 7. 9 50 − = 0.471, G 2 = 1.5 Rr = 80π = 7.9 Ω → er 2 = 1 − λ 7.9 + 50 0.75 2 P rec = 2 (1.64)(1.5)(0.965)(0.471) = 3.19 ×10 − 10 W 4π × 5000 2
o
c) What will be the reduction in the received power if the receive antenna is rotated by 30 in a plane perpendicular to the direction of incidence.
Prec
∼
2 Erec
∼
(Ein cos θ )2 → Prec (30 °) =
1 P 2 max
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 90I j ωt −jkr e maximum effective aperture of t he antenna. E = V/m r
G
=
U max Pin 4π
=
1 2 1 2
2
E r 2 2
I R 4π
= 5.4, Ae =
λ2G
4π
= 0.43λ2 ,
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 transmitt er 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?
λ 4πd
2
3 × 10 8 915 × 106 2 = 6.807 × 10−10 = −91.67 dB = 3 4 π × 10
b) What are antenna gains in dB?
Gdipole
= 10 log10 (1.64) = 2.14dB, G horn = 10 log10(12) = 10.79 dB
c) How much power is received in dB and in watts?
P R
= 0 dBW + 2.14 + 10.79 − 91.67 − 5 − 4 = − 81.74 dBW=10
− 81.74 10
= 1.68 ×10−9 W
9- For a lossy antenna it is found that the G = 0.8 D. Find the ratio of the antenna’s loss resistance to it s radiation resistance.
D(θ, φ) = G
=
D
S (r, θ, φ ) Pr 4πr
Rr Rr
+ RL
2
,G =
S (r , θ,φ ) Pin 4 πr
= 0.8 → 1 +
RL Rr
2
=
,Pin = Pr + PL ⇒
G D
=
Pr Pin
, Pr =
1 2 1 2 I Rr ,PL = I R L 2 2
1 = 1.25 → RL = 0.25Rr 0.8
1- A low orbit (LEO) satellite syst em transmits at 1.62 GHz using a 29 dB gain antenna with sp ot beams directed toward users on t he earth that are a maximum of 1500 km away. Find the required satellite transmit p ower in order for the power received by a user at t he maximum distance be at least -100 dBm if the user has a 1 dB gain antenna directed toward the sat ellite.
f
= 1.62GHz → λ = 0.185 m, Gt = 29dB, r = 1500 km, Pr ≥ − 100 dBm, G r = 1dB
λ 2 Pt (dBm) = Pr (dBm) − Gt (dB) −G r (dB) − 10log = 30 dBm = 1 W 4π r 2
2- A radio link has 20 W att transmitter connected to an antenna of 0.5 m effective aperture at 2 9 GHz. The receiving antenna has an effective aperture 2.0 m and is located 20 km line of sight from the t ransmitt ing antenna. Assuming the antennas have efficiencies equal to unity , find the power delivered to the receiver.
P r
=
PA t et Aer λ 2r 2
= 45 µ W
2- For t he following communication link the aperture efficiency is 60%. Calculate the received power by the load. And t he power reradiated from the horn.
Pt
= 1 W, Gt = 10 dBci, R = 10m, Z a = 80 Ω, Z L = 50 Ω,Aa = 300 cm2 y y
x
Gt
Pt
Aa
Aperture efficiency
Z a
R RHCP Tx antenna
Rx horn y -polarized
Z L
Pr
2 λ 2 e (1 )Dr (θr , φr ) ˆρt i ρˆr = Pt ecdt (1 − Γt ) Dt (θt , φt ) − Γ cdr r 4πR
Pr
2 ˆ ˆ ± x y (A × 0.6)(1 − Γ r ) ˆρt i ρˆr = (A × 0.6)(1 − Γ r ) = ⋅ yˆ 2 4π R 2 a 4 πR 2 a
Pr
=
2
PG t t
2
2
PG t t
2
2
10 80 2 −5 P P r = 10. 9 × 10−5 W (0.03 0.6) (1 0.23 ) 0.5 6.78 10 W, × × − × = × = scat 2 4π × (10) 50
4- A receiving antenna with a loss resistance of 7 Ω , er = 0 .9 and reactance X A =35 Ω has its terminals connected to a load of Z L = 50 Ω . If the open circuit voltage is V oc = j0.5 V : a) Draw the equivalent circuit. Determine the values of all the circuit.
er
=
Rr Rr
+ Rloss
= 0.9 → Rr = 63 Ω → Z A = Rr + Rloss +
jX A
= 70 + j 35 Ω
b) Determine the average power delivered to the load (in W) 2
j 0.5 j 0.5 j 0.5 1 2 1 PL = I L 50 → I L = 50 = 0.4 mW = → PL = 2 50 + Z A 120 + j 35 2 120 + j 35 2
c) Ass uming the avera ge p ower density of the incident field at t he antenna is 0.06 W/m , 2 determine the effective area Aeff , (in m ).
0.4 × 10− 3 Ae ff = = = 6.67 × 10− 3 m2 S i 0.06 P L
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 p ower for t he last measurement. 1- Birbirinden farklı A ve B antenleriyle 10 GHz’te aşağıdaki 3 ayrı deney yapılıyor. Bütün deneylerde antenler b irbirinden 15 m uzaklıkta olup konumları maksimum alış gücü için ayarlanmıştır. Buna göre aşağıdaki 3. deneyde alış gücü kaç dBm’dir? (P re c =? dBm)
Pin
= 0 dBm → (A ← − − − − 15 m − − − − → A) → P rec = −22 dBm = 0 dBm → (B ← − − − −15 m − − − − → B) → P rec = − 34dBm
Pin
= 0 dBm → (A ← − − − − 15 m − − − − → B) → P re c = − ?? dBm
Pin
λ 2 m = Gd → G d = 4πr Pr , λ = c = 0.03 4πr Pt f λ 4π × 15 − 22 10 1 2 4π × 15 −34 10 1 2 Gd 1 = G 10 499, = = ( ) (10 ) = 125.4 d 2 Pr Pt
2
0.03
P r Pt
λ 2 = Gd 1 G d 2 4πr
0.03 0.03 2 = 499 × 125.4 = 1.59 × 10− 3 = −28 dB 4π × 15
4- f = 1 GHz ’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 olu şan gücü hesaplayın.
Gt
= 20 dB = 100, G r = 15 dB = 31.623,
Pr
2 λ 2 0.3 PG G = (100)(31.6)(150) = 270.344 µW = −5.68dB = 4πR t t r 4π × 103
5- Pt = 1 00W,Gt
f
= 1GHz → λ = 0.3m
= 10 dB,G r = 2 dB, f = 150 MHz, e 0 = 1, R = 30 km : ise alıcı antenin
efektif açıklık alanını ve a lıcı antendeki gücü hesaplayın. 5- Given Pt = 1 00W,Gt = 10 dB,G r = 2 dB, f = 150 MHz, e 0 = 1, R = 30 km : What is the effective aperture area of the receivin g antenna? What is the received power? 2 2 3 × 10 8 2 10 2 λ f = 150 MHz → λ = = 2m, Ae = D0 = 10 = 0.5044 m 2 (D 0 = 1.5849 =2 dB) 6 150 × 10 4π 4π −6 PG 10 10 10 Pr = t t 2 Ae = (100)(10 ) (0.5044) = 0.04460 µW (2.112 mV to 50 Ω) 4πR 4 π × 302 Pr (dB) = −73. 51 dBW = −43. 51 dBm = 10 log10(Pr 1W ) for dBW
6- 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 Z g=55+ j10 Ω via a λ /2 long lossless transmission line with characteristic impedance of 50 Ω . Determine the average power supplied by the generator, the power radiated and p ower dissipated by the antenna. Z g
λ 2 Z L
Z 0 , β
V oc Z in
= 20V ,
Voc
Zg
= 55 + j 10 Ω,
Rr
= 64 Ω,
Voc V oc∗
RL
= 4 Ω,
XA
= − 65 Ω,
βl
= π, Z in (λ /2) = Z A
2
V oc
1 1 1 1 (20)2 × (55 + 68) ∗ Ps = RL (VocI g ) = Re( ∗ Re(Z g + Z in ) = = 1.35 W ∗ ) = Z g + Z in 2 2 2 Z g + Z in 2 2 55 + 68 − j 55 2 Prad
1 2 1 (20)2 × 64 1 2 1 (20)2 × 4 = I g Rr = = 0.705 W, PL = I g RL = = 0.044 W 2 2 (123)2 + (55)2 2 2 (123)2 + (55)2
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 Z g=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 p ower dissipated by the antenna.
Z g
λ 4 Z L
Z 0 , β
V oc Z in
Voc = 20V , Z g
= 55 + j 10Ω,
Rr = 64Ω, R L = 4Ω, X A = − 65Ω,
βl = π 2, Z in (λ / 4)= Z 20 Z A = 19.2 + j 18.6 Ω ∗
Voc V oc 1 1 1 ∗ Ps = RL (VocI g ) = Re( ∗ V ) = ∗ 2 2 Z g + Z in 2 oc er PL
2
Re(Z g + Z in )
Zg
+ Z in
2
=
200 × 74.211 = 2.35 W (74.211)2 + (28.264)2
2 = 64 , Pi n = 1 I g Re {Z in } = 1 Voc 2 Re(Z in ) 2 = 0.6087 W → P in 68 2 2 Z g + Z in = Pi n − P rad = 0.0358 W
=
P r
Pr
W = Pin er = 0.5729
1- Radyasyon direnci Rrad = 48 Ω kayıp direnci Rloss = 2 Ω ve reaktif empedansı + j X = j 50 Ω olan bir anten varsayalım. Voltajı V g = 10 V ve empedansı Z g = 50 Ω olan bir de voltaj kayna ğı olsun. Bu anten ile voltaj kaynağı, karakteristik empedansı Z 0 = 100 Ω ve uzunluğu λ 4 olan kayıpsız bir iletim hatt ı ile birbirine bağlanmıştır. Buna göre: a) Eşdeğer devrey i çizin (radyasy on, kayıp direnci, reaktif e mpedansı seri ba ğlı varsayın). 50 Ω
λ / 4 Z L = 2 + 48 + j 50 Ω
Z 0 = 100 Ω
10 V Z in
b) Determine the p ower supplied by the generator (p ower input to the line).
Z in
=
2
Z0
Z ant
(100)2 50 1 − 2 j = = 200 = 100(1 − j ), Γ in = Z in − = Z in + 50 50 + j 50 1 + j 2 − 2j
102 Pav = = 1 W, Pin = P av (1 − Γin 2 ) = 0.25(1 − 5 13) = 0.154 W 8 × 50 4 c) Antenin radyasyon gücünü bulun.
Prad
= P in
Rrad Rrad
+ Rloss
= 0.148 W
3- Consider an earth station. a) What is the required EIRP of the earth station if necessary S/N is 12 dB (in dBW)? -23
7
Prec = k T sy s B = 1.38 x 10 x 132 x 10 = 1.82 x 10 Ps req = -137 + 12 = -12.5 dBW EIRP = P s req + LS + Gr = -125 +214 -34 = 55 dBW
-14
W = -137 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) + M argin (dB)= 55 + 6 = 61 dBW
Balanis 2.3. T he 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 t he directivity and gain (in dB) when the radiated p ower is 125.66 mW
Balanis 2.4. The p ower radiated by a lossless antenna is 10 W. The antenna are represented by the following radiation intensity: U
= B 0 cos3 θ W sr, 0 ≤ θ ≤ π 2, 0 ≤ φ ≤ 2π 2
a) Find the maximum p ower density (in W/m ) at a distance of 1000 m (assume far field distance). Specify the angle where this occurs.
b) Find t he directivity (max) of t he antenna (in dB)
c) Find the gain of the antenna (in dB)
Balanis 2.11. The radiation intensity of an antenna is given by U (θ ,φ ) = cos4 θ sin 2 φ for 0 ≤ θ ≤ π 2 and 0 ≤ φ ≤ 2π (i.e., in the upper half-sp ace). It is zero in the lower half-space. a) Find the directivity. 2π π
Pra d
=∫ 0
U max
∫
π 2
2π
U (θ) sin θ d θ d φ = 2π
0
∫
sin2 φ d φ
0
∫ 0
= U (θ = 0o , φ = π 2) = 1, Dmax = 4πU max
π
Prad
= 20 = 13 dB
cos 4 θ sin θ d θ = 5
b) Find t he elevation p lane half-power beamwidth (in degrees). elevation plane φ = constant , choose φ = π 2
U = cos 4 θ , 0 ≤ θ
≤ π 2 → cos4 (HPBW 2)= 1 2 ⇒ HPBW = 2cos− 1 ( 0.5) = 65.5o
Balanis 2.12. T he normalized radiation intensity of an antenna is symmetric, can be approximated by 0° ≤ θ ≤ 30° 1
cos(θ ) U (θ ) = 0.866 0
30° ≤ θ ≤ 90° 90° ≤ θ ≤ 180°
(a) Find the exact directivity by integrating the function π2 π 6 cos θ = 3.63 W → D = 4πU max = 3.53 = 5.47 d U d d d d ( θ )sin θ θ φ 2 π sin θ θ sin θ θ = + max ∫0 ∫ ∫π 6 0.866 3.563 0
2π π
Prad =
∫ 0
(b) Find t he approximate directivity using Kraus’ formula
Balanis 2.41. An antenna with radiation a r esistance of 48 ohms, loss resist ance 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 λ 4 long transmission line with a ch aracteristic impedance of 100 ohms. a) Draw t he equivalent circuit
b) Determine the p ower supplied by the generator
c) Determine the p ower radiated by the antenna
Balanis 2.58. T wo lossless, p olarization matched antennas are aligned for maximum radiation between them, and are separated by a distance of 50 λ. The antennas are matched to t heir transmission lines and have directivities of 20 dB. Assuming that t he power at the input terminals of the transmitt ing 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 ≤ θ ≤ π 4 0 π 4≤θ ≤ π 2 E = 1 2 π 2 ≤ θ ≤ π a) What is the directivity of this antenna?
b) What is t he radiation resistance of the antenna at 200 m from it if t he field is equal to 10 V/m (rms) for θ = 0° at that distance and the terminal current is 5 A (rms)? When the field is equal to 10 V/m (rms) for θ = 0°
2
Balanis 2.48. An antenna has a maximum effective aperture of 2.147 m at 100 M Hz. It has no conduction, dielectric or p olarization 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 includin g the reflection due to mismatch between the antenna and transmission line.
75 − 50 3 × 108 ec = 1 , ed = 1 , PLF = 1, Γ = = 0.2, λ = = 3 m, Aem = 2.147 75 + 50 100 × 106 λ 2 2.147 2 Ae m = eced PLF D D (1 | | ) − Γ → = = 3.125 2 max max 2 3 4π 1 (0.2) − 4π
Balanis 2.49. An incoming wave with a uniform power density of 10−3 W/m 2 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.
3 × 108 D ma x = 20 dB = 100, λ = = = 0.03 m f 10 × 109 , 2 2 λ (0.03) Ae m = Dmax = 100 = 7.16 × 10−3 → Prec = 10−3 Aem = 7.16 × 10−6 W 4π 4π c
,
Balanis 2.68. T ransmitt ing and receiving antennas op erating at 1 GHz with gains (over isotropic) of 20 and 15 dB, resp ectively, are separated by a distance of 1 km. The input p ower is 150 miliwatt. Find the maximum power delivered to the load when the a) antennas are polarization matched.
Gt Pr
2
= 20 dB = 100, G r = 15 dB = 31.623, λ = 0.3 m, R = 1000 m, PLF = ˆρt i ρˆr = 1 = Pt
2 0.3 2 λ −3 ˆt i ρ ˆr G G ρ (100)(31.623)(150 10 ) = 270.3 µW = −5.68 dBm = × 0 t 0 r 3 4π × 10 4πR 2
b) transmitt ing antenna is circularly p olarized and the receivin g antenna is l inearly polarized.
ˆt i ρˆr PLF = ρ
2
xˆ ± yˆ ˆ 2 1 = = ⇒ i x 2 2
P r
= 135.2 µW = − 8.68 dBm
1- Birbirine uzaklığı 25 mil (1 mil =1.61 km) ve özellikleri birbirinin ay nı olan iki tane anten ile bir radyo-link hattı oluşturulmuştur. Pt G t = 8 Watt olup alış gücü P r = -70 dBm’dir. 2 a) Antenlerin açıklık verimi (aperture efficiency) 0.8 ise antenin fiziksel alanı kaç m dir? b) Antenler için Ω Α = 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 = P t G t ) is equal to 8, and t he received power is -70 dBm. a) Assuming matched horn antennas with an ap erture efficiency of 0.8, what is the p hysical 2 aperture size (m ) of t he antennas?
−70 dBm=10−10 W, Pr =
PG t t
4π R
2
Aer
=
EIRP A 4 πR2 e r
⇒ Aer = 0.2545 m2 = 0.8Aphy s ⇒ Aphys = 0.32m 2
b) Find t he frequency of op eration if the beam solid angles of the antennas are 0.04 sr.
ΩA =
4π D
=
λ2 Ae
→ λ 2 = ΩAAe → f =
c
ΩAAe
= 3GHz
c) Assume that the frequency is doubled, while keeping the same transmitted p ower, what would be received power in dBm?
Pr
Aer 2 f 2 = P t doubling f ⇒ R c
P r
Note that if EIRP was kept constant
= 4 × 10− 10 W = −64 dBm .
⇒ 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 app roximate e xpression for the half p ower beam width (in degrees): θr = 65λ d . (Assume symmetric narrow major lobe with negligible minor lobes)
1- The Voyager satellite transmitt ed some spectacular pictures back to Earth from a dist ance 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 an gle and gain for this antenna.
π 2 Ω = 1° = 3.05 × 10− 4 sr → G = 4π Ω = 41253 ≅ 46 dB 180° b) Calculate the power density at Earth.
S
=
P rad
(12)(52600)
G = 4πr 2 4π(4.32 × 1012 )2
= 2.11 × 10−21 W m 2
c) The Earth receiving station used p arabolic dish antennas with diameters of 60m. Assuming the effective area is about half t he physical area, how much p ower is received?
Aphys
1 60 2 = π = 2827 m 2 , Aeff = Aphys = 1414 m2 → Prec = PAeff = 2.98 × 10−18 W 2 2
d) The effective noise temperature of the receiving system was about 30 K. What bandwidth of data transmission would t his allow in order to maintain a SNR of at least 10?
Pnoi se S N
≅ kT ∆f = (1. 38 × 10−23 )(30K )( ∆f ) = 4. 14 ×10− 22 ∆f −19
≥ 10 → N ≤ 2.98 × 10
2.98 × 10− 19 → ∆f = ≅ 720 Hz 4.14 × 10−22
e) A typical low-loss op tical fiber transmission sy stem can have an attenuation rate as low as 0.1 dB/km. Compare this loss with the transmission loss of the satellite link.
FiberLoss = (0.1dB km) × 4.32 × 109 km = 4.32 × 108 dB
12W = 186 dB or 4.2 × 10− 8 dB km Link Loss = 10log − 18 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 -7 2 greater than 10 watts/m and the corresp onding amplitude of t he electric field? 2 E 100 × 5 −7 S = Sisotrpoic G = = ≥ 10 → r ≤ 20 km → S = → E = 2ηS = 8.7 × 10− 3 V m 2 2 4πr 4πr 2η
PG t
b) Estimate t he vertical beamwidth (in degrees).
∆φ = 2π, ∆θ = ? → ∆Ω ≃ ∆φ ∆θ sr, G = Formally d Ω = (r sin θd φrd θ)
1 2
r
4π
∆Ω 2π
→ ∆Ω = ∫ 0
= 5 → ∆Ω =
∫
π
2
π
2
θ −∆
2
+ ∆θ
4π → ∆θ 5
≃
sin θ d θ d φ = 4π sin
2
4π 10π
2 5
= rad = 23°
∆ θ = 4π ⇒ ∆ θ 2
5
≃
0.403
2- If the Earth appears as a uniform 300 K disk in the lunar sky, what is the antenna temperature, T A, of the lunar antenna (in Kelvin)? Earth radius is 6400 km and the moon distance is 400,000 km.
ψ
=
12800 4 × 105
Ω E ΩE = 3.2 × 10 rad = 1.8°, small wrt β 1 2 use TA = T earth + 1 − ΩP Ω P −2
2 ψ 4π ΩE = π = 8.04 × 10− 4 sr, ΩP = 4π = = 5.32 × 10− 3 sr → T A = 45 + 3 = 48 K 3 2 2.36 × 10 D 0
d) What is T sy s of the lunar receiving system (in Kelvin)? TL NA TE
= (1.26 − 1)290 = 75K ,TEtl = (Ltl − 1)290 = (1.58 − 1)290 = 170K ,TErec = (2.51− 1)290 = 438K
= TLNA +
TEtl GLNA
+
T Erec = 84K, Tsys GLNAG t l
= TA′ +TE ;TA′ = TA = 48K → Tsys = 48 + 84 = 132K
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 r eceived 30 channels, while he bought an excellent Ya gi-Uda/log-periodic antenna (with a gain of 13) and received only 12 channels! T hey bot h live on adjacent flats in t he 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 p attern solid angle of π /4 (sr) and a radiation efficiency of 70%. The input power to the antenna is 100 W. What is t he max power density at a range of 10 k m? 2
-7
2
2
D = 4π / Ω p = 16, G = eD = 11.2, Smax = Pt G / 4 π R = 8.9 x 10 W/m (60.5 dBW/m )
3- a) A uniformly illuminated apert ure 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 p lanes?
b) What is the antenna directivity if the physical efficiency (the ratio between max effective area, Aem, to t he physical are. Ap hys) is 80%?
1- The far-zone magnetic field of an antenna (located at the origin) is given by e − jβ r 2 ˆ ˆ H = I in φ cos θ (1 + sin 2φ ) + j θ 3 sin θ (2 3 + cos 2φ) (A m) 4 πr
{
}
a) Determine the effective height.
e − j βr e − jβ r 2 ˆ ˆ ˆ = jωµI in h E = ηH × r = I i nη θ cos θ (1 + sin 2φ) + jφ 3 sin θ (2 3 + cos 2φ ) 4πr 4πr 1 ˆ 2 →h= θ cos θ (1 + sin 2φ) + j φˆ 3 sin θ (2 3 + cos 2φ )
{
j β
}
{
}
b) Determine the radiation intensity.
Sav =
rˆ E 2η
2
2
= ˆr
I in η
{cos 32π r 2 2
2
θ (1 + sin 2φ)2
+ 3 sin2 θ (2 3 + cos2 2φ)2 }
2
U (θ, φ) =
I in η
32π
2
{cos θ (1 + sin2φ) 2
2
+ 3sin2 θ (2 3 + cos2 2φ)2 }
c) Determine the polarization vector of the antenna in the direction θ linear, RCP, LCP, REP or LEP?
= π 3 , φ = π 4 . Is it
9- A lossless antenna is completely specified by its vector effective length as hφ = 0 , hθ = h0 cos2 θ for 0 ≤ θ ≤ π 2 , hθ = 0 for π 2 ≤ θ ≤ π .
a) Find the radiated electric field vector as distance r , when it is excited with current I 0. Integrate the associated Poy nting flux (over the hemisp here), to find the t otal radiated p ower.
∫ cos
n
θ sin θ d θ
= − cosn +1 θ (n + 1) .
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 p olar coordinates.
sin θ π 9 < φ < 5π 9 F (θ, φ) = 0 elsewhere.
D 0
=
4π π
5π 9
∫ ∫ 0
π9
2
sin θ sin θ dθ d φ
= 6.75 = 8.293 dBi
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 resist ance, find its gain in the same direction. 5- Bir anten için D = 40 olsun. Bu antenin kayıp direnci radyasyon direncin beste biri ise ( Rloss Rrad = 1 5 ) antenin kazancı nedir( G = ? )
G (θ, φ) = G =
S Pi n 4πr
, D (θ, φ ) = 2
S Pr 4πr 2
→
G D
=
Prad Pin
=
1 2 1 2
I 2Rrad
I 2 (Rrad
+ Rloss )
=
1 1 + Rloss Rrad
5 × 40 = 33.3 = 15.2 dB 6
1- Consider the antenna pattern for a lossless antenna.
a) What is the value of the peak sidelobe level? 25 dB b) M ark the half p ower 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?
=
5 6
1- A 1 km radio link operates in free-space at 10 GHz. The transmitting and receiving antennas are identical, lossless, p olarization and impedance matched. The t ransmitt er 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 2 the capture area (in m ) of the receiving antenna when it is misaligned by half the 3 dB beamwidth.
Pr
=
Pt
4πr
Gt Ar 2
=
Pt
4 πr
Gt 2
λ2
4π
Gr
=
2 PG t 0
(4π r λ)2
G
⇒ G0 =
Pr Pt
4.19 × 103 = 36.2 dBi 4πr λ =
= G0 − 3 = 33.2 dBi → Aeff =
λ2
4π
G
2 = 0.15m
2- You are designing a 3 GHZ communications station for a lunar base. The distance from 5 the Moon to Earth is 4x10 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? 8
2
21
r = 4 x 10 m, λ = 0.1 m, L = (4πr / λ) = 2.53 x 10 = 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 radiatin g a power of 100 W. a) Calculate t he power density of the incoming signal from t he satellite.
S max
=
G t P t
4πr 2
= 8500
100 = 52.19 × 10− 12 W m 2 7 2 4π(3.6 × 10 )
b) Calculate the p ower received by the antenna of the handheld wireless telephone.
Pr
− 20 log(36000) − 20 log(2400) − 32.44 = −99.9 dBm = 0.103 pW = Pt (dBm) + Gt (dB) + Gr (dB)
c) Calculate the open-circuit voltage developed at the t erminals of t he receiving antenna given t hat t he impedance of the antenna of t he handheld telephone is Z A = 75 + j100 ohms. Voc = 8 RaPr = 8 × 75 × 0.103×10-12 = 7.86 µ W
d) Calculate the effective areas Ae for both the transmitting and the receiving antennas.
=
Ae t
λ2
4π
G t
= 10.57 m
2
, Ae r
=
λ2
4π
Gr
30 2 1.585 = = 19.7 c m2 240 4π
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 p hysical explanation for t his power reduction. Ae
4π0.7π t = = At = 0.7Aphys = 0.7π m2, λ = c = 0.02m, Gt = 4πA = 6.91 × 10 4 2 2
Prec
f
= Sinc Ae =
PG t t
4π R 2
Ae
=
1W (6.91× 10 4) 0.7π 4π(50 × 10 3 )2
λ
λ
−6
= 4.84 × 10 W
lower the frequency, the wavelength increases resulting a decrease in the antenna gain.
If
we
1- The antenna for the base-station of a cell phone system have a horizontal ( φ-direction) o o beamwidth of 120 and a vertical ( θ-direction) beamwidth of 5 at 1 GHz. a) Estimate its maximum gain.
∆φ = 120° =
2π π 4π 36 × 3 × 4π 216 rad, ∆θ = 5° = rad → G max = = = = 68.75 3 36 2π2 π ∆θ∆φ
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?
E
= 0.01 V m → S =
E2 2η
= 1.33 × 10−7 W m2 =
PG t t
4πr 2
→ P t ≥ 21.7 W
c) The typical mobile user has a handset with a dipole antenna which we can model as having a gain G=1 in a ty pical orientation. What is the maximum power available from his receiving antenna when the conditions of (a) & (b) app ly for a user at 30 km range? f
= 1GHz → λ = 0.3 m, G =
Prec
−7
= SAe = 1.33 × 10 ×
λ2
4π
4 πAe λ2
= 1 → Ae =
λ2 4π
= 9.5 × 10 − 10 W = − 60dBm
d) What other factors might reduce the actual power received below t he value from part (c)? Polarization mismatch due to misalignment of polarization in the incident wave with the polarization of the receiving antenna. 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 2 30dB. The radar is tracking a target with a radar cross section of 2m at a ran ge of 25 k m. a) Determine the p ower density of the incident field at t he target.
b) Determine the magnitude of t he incident electric field intensity at the target.
c) Determine the t otal power intercepted by t he target.
d) Determine the p ower density of the scatt ered field at the radar.
e) Determine the magnitude of the scatt ered electric field intensity at the radar.
f) Determine the t otal p ower received by the radar.
1- The impedance of a short antenna is modeled by a resistance Rra d in series with a capacitance C ant; 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 t he 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 ori ented along the φ-axis (user lying on couch watching TV). Given that the portable antenna is linearly p olarized of effective length ho p arallel to its antenna and the e − j kr sin θ h0 (θˆ + φˆj 0.2) . vector E-field radiated from the base unit is Erad = j ωµI 4πr
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 t hese cases cover the best and worst conditions, what dynamic range in dB is needed for t he receiver amplifier? (The power level would be stabilized by an automatic gain control c ircuit t hat 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 ρˆa
= (3xˆ + jy ˆ) 10 .
a) Find the p olarization mismatch factor (p) (in dB) when t he incident wave is LCP.
3xˆ + jyˆ xˆ + jyˆ 2 p = eˆa ⋅ e ˆw = ⋅ = 0.2 = −6.99 dB 10 2 b) Find t he polarization mismatch factor (p ) (in dB) when the incident wave is RCP.
3xˆ + jyˆ xˆ − jyˆ 2 p = eˆa ⋅ e ˆw = ⋅ = 0.8 = −0.969 dB 10 2 2- A linearly p olarized wave traveling in the negative z-direction has a tilt angle of +55 with respect to the x-axis. It is incident up on an antenna whose polarization is given by
ρˆa
o
= (xˆ − j 4y ˆ) 17 . Find the polarization mistmatch factor (p) in dB.
3- Verici bir antenin elektrik alanı Ei = xˆ E0 ( x , y) e − jkz olsun. A şağıda polarizasyonları verilen alıcı antenler kullanıldığında antenlerin polarizasyonlarının kaynaklanan güç kaybı ne kadar olacaktır? a) Ea ≅ xˆ E ( r ,θ ,φ ) gü ç kay bı olmaz b) Ea ≅ yˆ E (r ,θ ,φ ) güç tam amen k aybolur c) Ea ≅ (xˆ + yˆ ) E (r , θ , φ ) 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 capt ain really mad, and he sent his helicopters to check out t he 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 r adar sy stem is the backbone of the ship air defense system. Its modulation and frequency hopping procedure are classified information. However, let us use some typ ical numbers for t he 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 -23 Pn =kTB, where k is Boltzman Constant k =1.38x10 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 2 radar cross section of 2 m (approximately.) a) Calculate t he range of the US radar without the AUB microwave source. Find t he 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 t he radar range in (a). Assume PLF=0.5.
c) Calculate the power received by the US radar where the AUB source is t urned 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 t he US radar. Calculate the new range and warning time of the US fleet.
e) Think of some ECM (Electronic Counter M easure) 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 t he transmitting antenna is 2.8 relative to a halfwave dipole. Determine the p ower sent by the TV stations.
b) Determine the signal strength that t hese 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 t he table of 29-2 for t he correction factor of channel 2.
c) Determine the rms volta ge across a matched 75 Ω load when a Yagi-Uda antenna with a gain of 12 dB over isotrop ic is used. Derive equation 29-2 in handouts.
d) Determine the antenna noise voltage for Ch annels 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 T V receiver is 3 dB (T=290K) at both channels.
f) Calculate the maximum ran ge for a "passable" p icture quality for Stations 2 and 12 (S/N = 30 dB). M ax passable range found fro m Fig 29-1: we have drop ped 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 t he half p ower beamwidth. b) Calculate the space loss factor between the satellite and the earth receiver. c) Calculate the received S/N ratio if the t ransmitting antenna has a gain of 33 dB. What is its size (εap=85%)? What is ap proximately 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 t he 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 typ ical units of the vertical axis? Pattern plot, angle, dB c) Name the three different regions t hat describe the type of fields around an antenna as a function of dist ance from the antenna. reactive near f ield, radiating near field, far field d) At what distance r , does the far-field of antenna start? Remember to write y our answer in 2 terms of an antenna dimension and the wavelength of the frequency of op eration. 2D / λ f) What is t he type of polarization of most antenna systems t hat must communicate between space and earth? Circular g) What are the units of Radiation Intensity Watts/steradian 2 h) What are the units of Radiation density Watt/m k) Gain = kD, what is k? antenna efficiency e) What field component determines t he polarization of t he antenna? E-field b) Which antenna has higher directivity , an omni directional antenna or a d ipole? dipole n) For circularly polarized waves the phase difference between E x and Ey is +π /2 or -π /2. o) What is t he polarization of a wave is its Axial Ratio is infinit e? Linear c) What is the total solid angle of a sphere? 4π x) The pattern solid angle of an isotrop ic radiator is Ω A = 4π. True False a) The directivity of an isotropic source. D = 4 π / Ω A, Ω A = 4π, so D = 1 c) What does it mean to e xpress 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 discontinuit ies like a sp iral or log periodic antenna is typically wide p) A vertical linearly polarized antenna transmits to a linearly p olarized receive antenna tilted 45 degrees from vertical. The p olarization 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 2
t) 10 W/m is an accept able 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 t o 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 I in is t he terminal current) e − j β r 2 ˆ ˆ Ea = I in η θ10cos φ(1 − cos θ ) + φ j 15sin φ(1 + cos θ) Vm r
{
}
a) Determine the vector effective height h(θ,φ)
b) Find t he open circuit voltage (ma gnitude and phase) for t his antenna when it is in the receiving mode. Assume that t he 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 p art (b). Assume the antenna is lossless, has an input impedance of Z A=58+ j10 Ω and a load of Z L=50 Ω is connected directly to its terminals.
