# Problems

# Sections 9.1 and 9.2: Short Dipole and Antenna Radiation Characteristics

1. *9.1 A center-fed Hertzian dipole is excited by a current I 0=20 A. If the dipole is λ/50 in length, determine the maximum radiated power density at a distance of 1 km.

2. 9.2 A 50-cm long center-fed dipole directed along the z direction and located at the origin is excited by a 1 MHz source. If the current amplitude is I 0=10 A, determine:

   1. the power density radiated at 2 km along the broadside of the antenna pattern, and

   2. the fraction of the total power radiated within the sector between θ=85° and θ=95°.

3. 9.3 A 1-m long dipole is excited by a 1 MHz current with an amplitude of 12 A. What is the average power density radiated by the dipole at a distance of 5 km in a direction that is 45° from the dipole axis?

4. *9.4 Determine:

   1. the direction of maximum radiation,

   2. directivity,

   3. beam solid angle, and

   4. half-power beamwidth in the x–z plane,

   for an antenna whose normalized radiation intensity is given by

   F(θ,ϕ)=1 for 0≤θ≤60° and 0≤ϕ≤2 π 0 elsewhere.

   Suggestion: Sketch the pattern prior to calculating the desired quantities.

5. 9.5 Repeat Problem 9.4 for an antenna with

   F(θ,ϕ)=sin 2 θ cos 2 ϕ for 0≤θ≤π and −π/2≤ϕ≤π/2 0 elsewhere.

6. 9.6 A 2-m long center-fed dipole antenna operates in the AM broadcast band at 1 MHz. The dipole is made of copper wire with a radius of 1 mm.

   1. a.Determine the radiation efficiency of the antenna.

   2. *b.What is the antenna gain in decibels?

   3. c.What antenna current is required so that the antenna will radiate 80 W, and how much power will the generator have to supply to the antenna?

7. 9.7 Repeat Problem 9.6 for a 20-cm long antenna operating at 5 MHz.

8. 9.8 Determine the frequency dependence of the radiation efficiency of the short dipole, and plot it over the range from 600 kHz to 60 MHz. The dipole is made of copper, its length is 10 cm, and its circular cross section has a radius of 1 mm.

9. *9.9 An antenna with a pattern solid angle of 1.5 (sr) radiates 60 W of power. At a range of 1 km, what is the maximum power density radiated by the antenna?

10. 9.10 An antenna with a radiation efficiency of 90% has a directivity of 7.0 dB. What is its gain in decibels?

11. *9.11 The radiation pattern of a circular parabolic-reflector antenna consists of a circular major lobe with a half-power beamwidth of 3° and a few minor lobes. Ignoring the minor lobes, obtain an estimate for the antenna directivity in dB.

12. 9.12 The normalized radiation intensity of a certain antenna is given by

    F(θ)=exp(−20 θ 2)for 0≤θ≤π

    where θ is in radians. Determine:

    1. the half-power beamwidth,

    2. the pattern solid angle, and

    3. the antenna directivity.

# Sections 9.3 and 9.4: Dipole Antennas

1. 9.13 Repeat Problem 9.6 for a 1-m long half-wave dipole that operates in the FM/TV broadcast band at 150 MHz.

2. *9.14 Assuming the loss resistance of a half-wave dipole antenna to be negligibly small and ignoring the reactance component of its antenna impedance, calculate the standing-wave ratio on a 50 Ω transmission line connected to the dipole antenna.

3. 9.15 A 50-cm long dipole is excited by a sinusoidally varying current with an amplitude I 0=5 A. Determine the total radiated power if the oscillating frequency is:

   1. 1 MHz,

   2. 300 MHz.

4. 9.16 For a short dipole with length l where l≪λ, instead of treating the current I˜(z) as constant along the dipole, as was done in Section 9-1, a more realistic approximation that ensures the current goes to zero at the dipole ends is to describe I˜(z) by the triangular function

   I˜(z)=I 0(1−2 z/l)for 0≤z≤l/2 I 0(1+2 z/l)for −l/2≤z≤0

   shown in Fig. P9.16. Use this current distribution to determine the following:

   1. *a.the far-field E˜(R,θ,ϕ),

   2. b.the power density S(R,θ,ϕ),

   3. c.the directivity D, and

   4. d.the radiation resistance R rad.

   # Figure P9.16 Triangular current distribution on a short dipole (Problem 9.16).

5. 9.17 For a dipole antenna of length l=3 λ/2,

   1. *a.determine the directions of maximum radiation,

   2. b.obtain an expression for S max,

   3. c.generate a plot of the normalized radiation pattern F(θ), and

   4. d.compare your pattern with that shown in Fig. 9-17(c).

6. 9.18 For a dipole antenna of length l=λ/4,

   1. determine the directions of maximum radiation,

   2. obtain an expression for S max, and

   3. generate a plot of the normalized radiation pattern F(θ).

7. 9.19 Repeat parts (a)–(c) of Problem 9.17 for a dipole of length l=3 λ/4.

8. *9.20 Repeat parts (a)–(c) of Problem 9.17 for a dipole of length l=λ.

9. 9.21 A car antenna is a vertical monopole over a conducting surface. Repeat Problem 9.6 for a 1-m long car antenna operating at 1 MHz. The antenna wire is made of aluminum with μ c=μ 0 and σ c=3.5×10 7 S/m, and its diameter is 1 cm.

10. 9.22 For a dipole antenna of length l=2.5 λ, use Module 9.2 to determine:

    1. approximate directions of maximum radiation,

    2. antenna directivity, and

    3. antenna radiation resistance.

11. 9.23 For a dipole antenna of length l=3 λ, use Module 9.2 to determine:

    1. approximate directions of maximum radiation,

    2. antenna directivity, and

    3. antenna radiation resistance.

12. 9.24 A dipole antenna has a radiation impedance

    Z rad=R rad+j X rad.

    For a dipole of length l=λ/2, R rad=73.13 Ω and X rad=42.55 Ω. Use Module 9.3 to modify the value of l slightly so as to reduce X rad to less than 1 Ω.

    1. What is the value of l?

    2. What is the corresponding value of R rad?

# Sections 9.5 and 9.6: Effective Area and Friis Formula

1. 9.25 Determine the effective area of a half-wave dipole antenna at 100 MHz, and compare it with its physical cross section if the wire diameter is 2 cm.

2. *9.26 A 3 GHz line-of-sight microwave communication link consists of two lossless parabolic dish antennas, each 1 m in diameter. If the receive antenna requires 10 nW of receive power for good reception and the distance between the antennas is 40 km, how much power should be transmitted?

3. 9.27 A half-wave dipole TV broadcast antenna transmits 1 kW at 50 MHz. What is the power received by a home television antenna with a 3 dB gain if located at a distance of 30 km?

4. *9.28 A 150 MHz communication link consists of two vertical half-wave dipole antennas separated by 2 km. The antennas are lossless, the signal occupies a bandwidth of 3 MHz, the system noise temperature of the receiver is 600 K, and the desired signal-to-noise ratio is 17 dB. What transmitter power is required?

5. 9.29 Consider the communication system shown in Fig. P9.29 with all components properly matched. Determine the following if P t=10 W and f=6 GHz.

   1. What is the power density at the receiving antenna (assuming proper alignment of antennas)?

   2. What is the received power?

   3. If T sys=1,000 K and the receiver bandwidth is 20 MHz, what is the signal-to-noise ratio in decibels?

   # Figure P9.29 Communication system of Problem 9.29.

   Figure P9.29 Full Alternative Text

6. 9.30 The configuration shown in Fig. P9.30 depicts two vertically oriented half-wave dipole antennas pointed towards each other with both positioned on 100-m tall towers separated by a distance of 5 km. If the transit antenna is driven by a 50 MHz current with amplitude I 0=2 A, determine the following:

   1. *a.The power received by the receive antenna in the absence of the surface. (Assume both antennas to be lossless.)

   2. b.The power received by the receive antenna after incorporating reflection by the ground surface, assuming the surface to be flat and to have ε r=9 and conductivity σ=10−3 (S/m).

   # Figure P9.30 Problem 9.30.

   Figure P9.30 Full Alternative Text

7. 9.31 Fig. P9.31 depicts a half-wave dipole connected to a generator through a matched transmission line. The directivity of the dipole can be modified by placing a reflecting rod a distance d behind the dipole. What would its reflectivity in the forward direction be if

   1. d=λ/4

   2. d=λ/2

   # Figure P9.31 Problem 9.31.

8. 9.32 The configuration shown in Fig. P9.32 depicts a satellite repeater with two antennas, one pointed towards the antenna of ground station 1 and the other towards the antenna of ground station 2. All antennas are parabolic dishes; antennas A 1 and A 4 are each 4 m in diameter, antennas A 2 and A 3 are each 2 m in diameter, and the distance between the satellite and each of the ground stations is 40,000 km. Upon receiving the signal by its antenna A 2, the satellite transponder boosts the power gain by 80 dB and then retransmits the signal to A 4. The system operates at 10 GHz with P t=1 kW. Determine the received power P r. Assume all antennas to be lossless.

   # Figure P9.32 Satellite repeater system.

   Figure P9.32 Full Alternative Text

# Sections 9.7 and 9.8: Radiation by Apertures

1. *9.33 A uniformly illuminated aperture is of length l x=20 λ. Determine the beamwidth between first nulls in the x–z plane.

2. 9.34 The 10 dB beamwidth is the beam size between the angles at which F(θ) is 10 dB below its peak value. Determine the 10 dB beamwidth in the x–z plane for a uniformly illuminated aperture with length l x=10 λ.

3. *9.35 A uniformly illuminated rectangular aperture situated in the x–y plane is 2 m high (along x) and 1 m wide (along y). If f=10 GHz, determine:

   1. the beamwidths of the radiation pattern in the elevation plane (x–z plane) and the azimuth plane (y–z plane), and

   2. the antenna directivity D in decibels.

4. 9.36 An antenna with a circular aperture has a circular beam with a beamwidth of 3° at 20 GHz.

   1. What is the antenna directivity in dB?

   2. If the antenna area is doubled, what will be the new directivity and new beamwidth?

   3. If the aperture is kept the same as in (a), but the frequency is doubled to 40 GHz, what will the directivity and beamwidth become then?

5. 9.37 Compare directivity D ant of a 1-m diameter antenna aperture operating at 10 GHz with directivity D eye of the eye’s pupil operating in the middle of the visible spectrum at λ=0.5 μ m. Treat the pupil as a circular aperture with a diameter of 4 mm.

6. *9.38 A 94 GHz automobile collision-avoidance radar uses a rectangular-aperture antenna placed above the car’s bumper. If the antenna is 1 m in length and 10 cm in height, determine:

   1. its elevation and azimuth beamwidths, and

   2. the horizontal extent of the beam at a distance of 300 m.

7. 9.39 A microwave telescope consisting of a very sensitive receiver connected to a 100 m parabolic-dish antenna is used to measure the energy radiated by astronomical objects at 20 GHz. If the antenna beam is directed toward the moon and the moon extends over a planar angle of 0.5° from Earth, what fraction of the moon’s cross-section will be occupied by the beam?

8. 9.40 A parabolic dish antenna has a diameter d=5 λ. Use Module 9.4 to evaluate three antenna parameters, namely the directivity D, the beamwidth β, and the level of the first sidelobe next to the mainbeam, all at the following three values of the taper factor: α=0, α=0.5, and α=1.

9. 9.41 Given a parabolic dish antenna with a diameter d=10 λ, use Module 9.4 to select the highest value of the taper factor α without having the beamwidth exceed 6°.

# Sections 9.9 and 9.11: Antenna Arrays

1. 9.42 A two-element array consisting of two isotropic antennas separated by a distance d along the z axis is placed in a coordinate system whose z axis points eastward and whose x axis points toward the zenith. If a 0 and a 1 are the amplitudes of the excitations of the antennas at z=0 and at z=d, respectively, and if δ is the phase of the excitation of the antenna at z=d relative to that of the other antenna, find the array factor and plot the pattern in the x–z plane for the following.

   1. *a.a 0=a 1=1, δ=π/4, and d=λ/2

   2. b.a 0=1, a 1=2, δ=0, and d=λ

   3. c.a 0=a 1=1, δ=−π/2, and d=λ/2

   4. d.a 0=1, a 1=2, δ=π/4, and d=λ/2

   5. e.a 0=1, a 1=2, δ=π/2, and d=λ/4

2. 9.43 If the antennas in part (a) of Problem 9.42 are parallel, vertical, Hertzian dipoles with axes along the x direction, determine the normalized radiation intensity in the x–z plane and plot it.

3. *9.44 Consider the two-element dipole array of Fig. 9-29(a). If the two dipoles are excited with identical feeding coefficients (a 0=a 1=1 and ψ 0=ψ 1=0), choose (d/λ) so that the array factor has a maximum at θ=45°.

4. 9.45 Choose (d/λ) so that the array pattern of the array of Problem 9.44 has a null, rather than a maximum, at θ=45°.

5. 9.46 Find and plot the normalized array factor and determine the half-power beamwidth for a five-element linear array excited with equal phase and a uniform amplitude distribution. The interelement spacing is 3 λ/4.

6. 9.47 Repeat Problem 9.46, but change the excitation to tapered amplitude distribution so that the amplitude of the central element is 1, the amplitudes of the next adjacent elements are both 0.5, and those of the outer elements are both 0.25.

7. 9.48 Repeat Problem 9.46 for a nine-element array.

8. *9.49 A five-element equally spaced linear array with d=λ/2 is excited with uniform phase and an amplitude distribution given by the binomial distribution

   a i=(N−1)!i!(N−i−1)!,i=0,1,…,(N−1),

   where N is the number of elements. Develop an expression for the array factor.

9. 9.50 A three-element linear array of isotropic sources aligned along the z axis has an interelement spacing of λ/4 (Fig. P9.50). The amplitude excitation of the center element is twice that of the bottom and top elements, and the phases are −π/2 for the bottom element and π/2 for the top element relative to that of the center element. Determine the array factor and plot it in the elevation plane.

   # Figure P9.50 Three-element array of Problem 9.50.

   Figure P9.50 Full Alternative Text

10. *9.51 An eight-element linear array with λ/2 spacing is excited with equal amplitudes. To steer the main beam to a direction 60° below the broadside direction, what should be the incremental phase delay between adjacent elements? Also, give the expression for the array factor and plot the pattern.

11. 9.52 A linear array arranged along the z axis consists of 12 equally spaced elements with d=λ/2. Choose an appropriate incremental phase delay δ so as to steer the main beam to a direction 30° above the broadside direction. Provide an expression for the array factor of the steered antenna and plot the pattern. From the pattern, estimate the beamwidth.

12. 9.53 Use Module 9.5 to obtain the radiation pattern for a two-dipole array, each 0.5 λ in length, separated by λ/2, and fed equally in amplitude but with one of them 90° ahead of the other in phase.

13. 9.54 Repeat Problem 9.53, but change the separation between the two dipoles to d=1 λ.

Test Your Understanding of Chapter 9:
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