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Unformatted text preview: r hams,1 PROBLEMS 679 13.8 A receiving antenna in an airport has a maximum dimension of 3 m and operates at
100 MHZ. An aircraft approaching the airport is 0.5 km from the antenna. The aircraft is
in the far ﬁeld region of the antenna. (a) True (b) False 13.9 A receiving antenna is located 100 m away from the transmitting antenna. If the effective
area of the receiving antenna is 500 cm2 and the power density at the receiving location
is 2 mW/mz, the total power received is: (a) 10 nW (c) 1MW
(b) 100 nW (d) 10 MW (e) 100 ,uW 13.10 Let R be the maximum range of a monostatic radar. If a target with radar cross section of
5 m2 exists at R/2, what should be the target cross section at 3R/2 to result in an equal
signal strength at the radar? (a) 0.0617 m2
(b) 0.555 m2 (c) 15 m2
(d) 45 m,2 (e) 405 m2 Answers: 13.1d, l3.2a, 13.3b, 13.4d, 13.56, 13.6c, 13.7d, 13.8a, 13.96, 13.10e. EEBTIDN 13.2 HERTZIAN DIF‘DLE 13.1 The magnetic vector potential at point P(r, 6, <15) due to a small antenna located at the
origin is given by _ 50 e‘ﬂ’”
— r where r2 = x2 + y2 + Z2. Find E(r, 0, ¢, 1‘) and H(r, 0, ¢, t) at the far ﬁeld. A. ax 13.2 Prove that in a source—free region, the electric ﬁeld intensity E can be expressed in terms of
the magnetic vector potential as V(V  AS)
jwns 13.3 A Hertzian dipole at the origin in free space has d€ = 20 cm and I = 10 cos 27r107t A.
Find [E63] at the distant point (1 00, 0, 0). Es = ~ijS + 13.4 A 2 A source operating at 300 MHZ feeds a Hertzian dipole of length 5 mm situated at the
origin. Find Es and H. at (10°, 30°, 90°). 13.5 (a) Instead of a constant current distribution assumed for the short dipole of Section 13.2, 2
assume a triangular current distribution IS = IO < — shown in Figure 13.25.
Show that
e 2
Rra = 20 2 —
d ” iii which is onefourth of that in eq. (13.13). Thus Rrad depends on the current distribution.
(b) Calculate the length of the dipole that will result in a radiation resistance of 0.5 0. 13.6 Calculate the radiation resistance of a 20 cm rod operating at 85 MHz. 680 CHAPTER 13 ANTENNAS Figure 13.25 Short dipole antenna with triangular current distri—
bution; for Problem 13.5. SECTION 13.3 HALF—WAVE DIPOLE ANTENNA 13.7 A half—wave dipole is fed by a 50 9 transmission line. Calculate the reﬂection coefﬁcient
and the standing wave ratio. 13.8 A car radio antenna 1 m long operates in the AM frequency of 1.5 MHZ. How much cur
rent is required to tranSmit 4 W of power? 13.9 A dipole antenna (6 = )x/8) operating at 400 MHz is used to send a message to a satellite
in space. Find the radiation resistance of the antenna. *13.10 (a) Show that the generated far—ﬁeld expressions for a thin dipole of length (3 carrying si—
nusoidal current Io cos Bz are _Br cos cos 6) — cos _ jloe 2 27rr sin 0 H¢S , E0s = Til—Lbs [Hint: Use Figure 13.4 and start with eq. (1314).]
(b) On a polar coordinate sheet, plot f(6) in part (a) for 6 = )x, 3N2, and 2A. BEETIDN 1:3. 5 SMALL Lump ANTENNA
13.11 A 100turn loop antenna of radius 20 cm operating at 10 MHZ in air is to give a 50 mV/m
ﬁeld strength at a distance 3 m from the 100p. Determine (a) The current that must be fed to the antenna
(b) The average power radiated by the antenna 13.12 A circular loop antenna has a mean radius of 1.2 cm and N turns. If it operates at 80 MHz,
ﬁnd N that will produce a radiation resistance of 8 (2. 5EBTIUN 13~6 ANTENNA BHARABTERISTlBES 13.13 Sketch the normalized Eﬁeld and H—ﬁeld patterns for (a) A half—wave dipole
(b) A quarter—wave monopole 681 PRDBLEMS ; 13.14 Based on the result of Problem 13.10, plot the vertical ﬁeld patterns of monopole
antennas of lengths 6 = 3N2, A, 5N8. Note that a 5N8 monopole is often used in
practice. 13.15 At the far ﬁeld, the electric ﬁeld produced by an antenna is E9 = E [Mr cos 6 cos d) az
r Sketch the vertical pattern of the antenna. Your plot should include as many points as
possible. You may use MATLAB. 13.16 An antenna located at the origin has a farzone electric ﬁeld as cos 26 WW ES: e a9 V/m
r (a) Obtain the corresponding HS ﬁeld.
(b) Determine the power radiated.
(c) What fraction of the total power is radiated in the belt 60° < 6 < 120°? *13.17 An antenna located on the surface of a ﬂat earth transmits an average power of 200 kW.
Assuming that all the power is radiated uniformly over the surface of a hemisphere with
the antenna at the center, calculate (a) the time—average Poynting vector at 50 km and
(b) the maximum electric ﬁeld at that location. 13.18 For an Hertzian dipole, show that the time—average power density is related to the radia—
tion power according to ‘
_ 1.5 sin2 6 P —— P
ave 2 rad
47rr 13.19 At the far ﬁeld, an antenna produces @ave = 2 sin 62cos qb r arW/m2, 0<0< 7r,0<¢< 7r/2 Calculate the directive gain and the directivity of the antenna. 13.20 From Problem 13.10, show that the normalized ﬁeld pattern of a full—wave (6 = )x)
antenna is given by cos (71' cos 6) + 1 J10) = sin 6
Use Matlab to plot the ﬁeld pattern. 13.21 For a thin dipole N16 long, ﬁnd (a) the directive gain, (b) the directivity, (c) the effective
area, ((1) the radiation resistance. 13.22 Repeat Problem 13.21 for a circular thin loop antenna )x/ 12 in diameter. 13.23 A halfwave dipole is made of copper and has a diameter of 2.6 mm. Determine the efﬁ
ciency of the dipole if it operates at 15 MHZ.
Hint: Obtain Re from Re/Rdc = a/26; see Section 10.6. 13.24 Determine the directivity of a small loop antenna. 682 CHAPTER 1 3 ANTENNAS 13.25 Find Uave, Umax, and D if: (a) U(0, ¢) : sin2 20,
(b) U(6, a) = 4 csc2 6,
(c) U(6, (1)) = 2 sin2 6 sin2 ¢, 0<6<7r,0<¢<27r
7r/3<6<7r/2,0<¢<7r
O<0<7r,0<¢<7r 13.26 For the following radiation intensities, ﬁnd the directive gain and directivity:
(a) me, a) = sin2 6, (b) U(6, <75) = 4 sin2 0 c032 49,
(c) U09, (1)) = 10 cos2 0 sin2 ¢/2, 0<0<7r,0<¢<27r
0<6<7r,0<¢<7r
0<0<7r,0<q.’><7r/2 13.27 An antenna has a farﬁeld electric ﬁeld given by I _ ..
Es=7°e Jﬁ’srndag where I0 is the maximum input current. Determine the value of IO to radiate a power of
50 mW. SEETIDN 13.7 ANTENNA ARRAY 13.28 Derive E at far ﬁeld due to the two—element array shown in Figure 13.26. Assume that
the Hertzian dipole elements are fed in phase with uniform current IO cos wt. 13.29 An array comprises two dipoles that are separated by one wavelength. If the dipoles are
fed by currents of the same magnitude and phase, (a) Find the array factor.
(b) Calculate the angles where the nulls of the pattern occur.
(c) Determine the angles where the maxima of the pattern occur. ((1) Sketch the group pattern in the plane containing the elements. 13.30 An array of two elements is fed by currents that are 180° out of phase with each other.
Plot the group pattern if the elements are separated by (a) d = N4, (b) d = )x/2. Figure 13.26 Two—element array of Problem 13.28. N PROBLEMS 683 I Q 1 [2 I E 1 Q Figure 13.27 For Problem 13.34.
e @ ® ®
H— i/2—>l<—i/2——>l<——A/2—+l (a) 1g 1% [a [[317/2
® @ $ $
f<——— A/4—++~—7\/4——><—~ A/4—>{ (b) 13.31 Sketch the group pattern in the xz—plane of the twoelement array of Figure 13.10 with (a) d= Nor: 7r/2
(b) d = N4, 0: = 37r/4
(c) d = 3N4,ct = O 13.32 An antenna array consists of N identical Hertzian dipoles uniformly located along the z—axis and polarized in the zdirection. If the spacing between the dipole is N4, sketch the
group pattern when (a) N = 2, (b) N = 4. 13.33 An array of isotropic elements has the group pattern FOP) =y Sin (27F cos it) 2 003 1,0) cos
_ 7r
sm (2 cos 1,0) Use Matlab to plot F (1%) for 0° < 30 < 180°. 13.34 Sketch the resultant group patterns for the fourelement arrays shown in Figure 13.27. SECTIDN 13.8 EFFEETIVE AREA AND THE FEMS EQUATIBN 13.35 For a lOturn loop antenna of radius 15 cm operating at 100 MHZ, calculate the effective
area atO = 30°, q!) = 90°. 13.36 An antenna receives a power of 2 MW from a radio station. Calculate its effective area if
the antenna is located in the far zone of the station where E = 50 mV/m. 13.37 (a) Show that the Friis transmission equation can be written as Pr _ AerAel PI )\2r2 (b) Two half—wave dipole antennas are operated at 100 MHZ and separated by 1 km. If
80 W is transmitted by one, how much power is received by the other? 684 CHAPTER 13 ANTENNAS 13.38 The electric ﬁeld strength impressed on a halfwave dipole is 3 mV/m at 60 MHz. Ca1
culate the maximum power received by the antenna. Take the directivity of the half—wave dipole as 1.64. 13.39 The power transmitted by a synchronous orbit satellite antenna is 320 W. If the antenna
has a gain of 40 dB at 15 GHz, calculate the power received by another antenna with a
gain of 32 dB at the range of 24,567 km. 13.40 The directive gain of an antenna is 34 dB. If the antenna radiates 7.5 kW at a distance of
40 km, ﬁnd the time—average power density at that distance. 13.41 Two identical antennas in an anechoic chamber are separated by 12 m and are oriented for
maximum directive gain. At a frequency of 5 GHz, the power received by one is 30 dB
down from that transmitted by the other. Calculate the gain of the antennas in dB. 13.42 What is the maximum power that can be received over a distance of 1.5 km in free space
with a 1.5 GHZ circuit consisting of a transmitting antenna with a gain of 25 dB and a re—
ceiving antenna with a gain of 30 dB? The transmitted power is 200 W. SECTION 13.9 THE RADAR EQUATION 13.43 An L—band pulse radar with a common transmitting and receiving antenna having a di
rective gain of 3500 operates at 1500 MHz and transmits 200 kW. If an object is 120 km
from the radar and its scattering cross section is 8 m2, ﬁnd (a) The magnitude of the incident electric ﬁeld intensity of the object
(b) The magnitude of the scattered electric ﬁeld intensity at the radar (c) The amount of power captured by the object
(d) The power absorbed by the antenna from the scattered wave 13.44 A radar system has a common transmitting and receiving antenna with a gain of 40 dB.
The system can transmit a power of 8 kW at 6 GHz and is able to detect a signal of 10 pW. Find the maximum range for detecting a 1 m2 target. 13.45 A monostable radar operating at 6 GHZ tracks a 0.8 m2 target at a range of 250 m. If the gain
is 40 dB, calculate the minimum transmitted power that will give a return power of 2 MW. 13.46 In the bistatic radar system of Figure 13.28, the ground—based antennas are separated by
4 km and the 2.4 m2 target is at a height of 3 km. The system operates at 5 GHZ. For Gd;
of 36 dB and Gdr of 20 dB, determine the minimum necessary radiated power to obtain a
return power of 8 X 10—12 W. EEETIDN 13.10 APPLIEATIDN MUTE: ELECTRDMABNETIC
INTERFERENBE AND EDMPATIBILITY 13.47 An electrostatic discharge (ESD) can be modeled as a capacitance of 125 pF charged to
1500 V and discharging through a 2 km resistor. Obtain the current waveform. PRUBLEMS 685 Target 0 Figure 13.28 For Problem 13.46. Scattered wave
3 km Incident wave
‘ \éransmitting Receiving I ,I antenna
antenna 4km *13.48 The insertion loss of a ﬁlter circuit can be calculated in terms of its A, B, C and D pa~
rameters when terminated by Zg and ZL as shown in Figure 13.29. Show that AZL + B + CZgZL + ng
zg + ZL IL = lOglO 13.49 A silver rod has rectangular cross section with height 0.8 cm and width 1.2 cm. Find (a) The dc resistance per kilometer of the conductor
(b) The ac resistance per kilometer of the conductor at 6 MHz Figure 13.29 For Problem 13.48. ...
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 Fall '08
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 Dipole antenna, Loop antenna

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