1- A quarter-wave monopole (as shown in the figure below) situated above a perfectly conducting ground plane is excited by a sinusoidal source at its base. Find its radiation pattern, radiation resistance, and maximum directivity.

36.57 21.25 36.57 , 3.282in rZ j R D= + Ω → = Ω =

3- Consider two half-wave linear wire antennas aligned, so that their axes are offset by 60° . If the transmit antenna radiates 10 W of power at an operating frequency of 1 GHz, find: (a) the max power that will be available at the receiver antenna when they are 1 km apart, (b) the magnetic and electric field strength at the position of the receive antenna.

6- A linear infinitesimal dipole of length L and constant current is placed vertically a distance h above an infinite ground plane. Find the first five smallest heights (in ascending order) so that a null is formed (for each height) in the far-field pattern at an angle of 60° from the vertical.

6- Yarıçapı a=0.001λ olan bakır (σ = 5.7 x 107 S/m) telden yapılan ve 175 MHz’te çalışan yarım dalga boyu bir dipol anten 300 ohm’luk bir iletim hattına bağlanmıştır. Antenin kazancını (G0) hesaplayın. Antenin radyasyon yoğunluğu 3( ) sinU Aθ θ= olsun. 6- A resonant half-wave dipole with radiation resistance Rr=73 ohm is connected to a transmission line with characteristic impedance is 300 ohm. The dipole is made of copper with σ=5.7x107 S/m and f=175 MHz. Calculate the dipole gain if its field pattern can be approximated by 3( ) sinU Aθ θ= . The dipole radius is a=0.001λ.

8

8

33 10

1.75 10

03

73 30073 300 0

223 4

rad0 0 0

1.714m, 0.86m, 0.001 1.714 10 m

0.860.278

2 2 1.714 1073

0.6 1 0.64, 1 0.6473

3sin sin 2 sin

s L s s

r cd r cd

L

L a

f LR R R R

a

e e e e eR

P A d d A d Aπ π π

λ λ

π µ

σ π π

πθ θ θ φ π θ θ

−××

−

−+

= = = = = ×

= = → = = = Ω× ×

Γ = = → = − Γ = = ≈ → = =+

= = =∫ ∫ ∫max

0 0 0 0

rad

44 16

0.64 1.7 1.083

UD G e D

P

π

π= = → = = × =

1- Yarıçapı 6.35 mm alüminyum (σ = 3.5 x 107 mhos/m) telden yapılan ve 100 MHz’te çalışan yarım dalga boyu bir dipol anten 50 ohm’luk bir iletim hattına bağlanmıştır. Antenin radyasyon, yansıma ve toplam verimini hesaplayın. 1- A half-wave dipole made of aluminum wire (σ = 3.5 x 107 mhos/m) with a diameter of 6.35 mm (0.25 in) is operating in free space at 100 MHz. The antenna is connected to a transmission line with a characteristic impedance of 50 ohm. Determine the radiation, reflection and the overall efficiency and the gain of the antenna.

3

20

0

0 0

100MHz =3m, 3.358 10 0.126252 2 4

730.99827 99.827% 1

73 0.12625

73 42 500.37134 1 0.864 86.4%

73 42 50

0.8625 86.25%

s L s s

radcd

rad ohmic

Ar

A

cd r cd

Lf R R R R

a a

Re

R R

Z Z je

Z Z j

e e e G e D

ωµ λλ

σ π π

−= → = = × Ω → = = = Ω

= = = = ≈+ +

− + −Γ = = → Γ = → = − Γ = =

+ + += = = → = (0.99827)(1.643) 1.64 2.148 dB= = =

5- A z-directed electric dipole of current moment II resides on the y axis at y y′= . In the far zone, this source causes an electric field given below. Derive an expression for the total power radiated by this dipole. Set up the proper expression for power, simplify it, but it is not necessary to perform the last integration unless you wish to do so.

sin sinsin .

4

jkrjkye

E jk Il er

θ φ

θ η θπ

−′

=

4- The magnetic radiation field of a short dipole oriented in the z direction, with dipole moment m is given below. Find the radiation resistance for length 50 10lλ λ≤ ≤ .

4- Dipol momenti m olan ve manyetik alanı aşağıda verilen z yönündeki kısa dipol antenin

50 10lλ λ≤ ≤ iken radyasyon direncini hesaplayın.

sinˆ ˆ( ) ( ) .2

jkrm eH j

rφ

θφ φ

λ

−

= =H r r

221 1

2 20 02

2 222

2 2 3 20 12 0

0 0 0

1( ) ,

2

sin sin 2 sin2 2 4 2 6

l

t rl S

t r r

m I z dz I l P dA I R

m m I l m lP d r d d I R R

r

π π π

η

πηφ θ θ θ π θ θ

λ λ λ

−

−= = = =

= = = → =

∫ ∫∫

∫ ∫ ∫

H

1- A linear dipole antenna of length 10 cm is oriented with its axis along z-direction and driven with a peak current of 10 A at a frequency of 1 MHz. a) Determine whether this is a Hertzian or short dipole antenna at this frequency.

cm m Hertzian Dipole!410 , 300 3.3 10 .c l

lf

λλ

−= = = ⇒ = ×

b) By what factor does the radiated electromagnetic field propagating through free space change (if at all) between two locations situated at R1= 50 km and R2= 150 km from the antenna?

but 1 22

2 1

1 1 1503

50

E RS E

R R E R∝ ∝ ∴ = = =

c) What is the value of power density at R1=50 km. i) along the z-direction ii) perpendicular to z-direction (in the z=0 plane)

2220

2

15( , , ) sin

I lS r

r

πθ φ θ

λ

=

i) For 0 0Sθ = ° → = , (along the z-direction) ii) For 290 0.21 pW mSθ = ° → =

d) Is the state of polarization the same for waves emitted at arbitrary angles θ? If so, state what it is. If not, indicate why not. The radiation is linearly polarized in all directions. The axis of polarization lies perpendicular to k in the kz-plane. e) What is the total (average) power radiated by the antenna?

2 2222 2 2 2 30

TOT 020 0 0

15( , , ) sin sin 30 sin 13.15W

S

I l lP S r d r d d I d

r

π π ππθ φ θ θ θ φ π θ θ

λ λ

= Ω = = = ∫ ∫ ∫ ∫

1- Answer the following questions: a) Why is a half-wave dipole more useful than a short-length dipole, even though the directivities of both antennas are nearly the same? Greater efficiency because: impedance of half-wave dipole easier to match to common system impedance and Rrad>>Rdiss for half-wave dipole. b) Roughly sketch the current distribution along each arm of a dipole whose total length is 5 8λ .

c) A 5 mm long z-oriented dipole is driven by a 3 GHz current of 1.5 A. What is the amplitude of the E –field at a distance of 2 m along the + x axis? What is the corresponding power density and direction of power flow at that location?

22

0

0

40.005m, 0.1m short dipole, 126 1 Farfield

20

2120 7.07 V m 66.3mW m along the -axis

8 2

l l kr

ElE I P x

r

λ πλ

λ

ππλ π η

= = → = → = >> →

= = → = = +

≃

2- Calculate the radiation efficiency for a λ/80 Hertzian dipole operating at 3 MHz in free space. Assume that the dipole is made of copper wire (σ = 5.8 × 107 S/m) with a = 1.024mm.

2 22 2

4

4

3

180 80 0.123 ,

80

1.25m (4.519 10 )4.519 10 0.0878

2 2 (1.024 10 m)

0.1230.583 58.3%

0.123 0.0878

r

s L s

rcd

r L

dlR

dlR f R R

a

Re

R R

π πλ

π µ σπ π

−

−−

= = = Ω

× Ω= = × Ω → = = = Ω

×

→ = = = =+ +

2- A Hertzian dipole of length L=2m operates at 1MHz. Find the radiation efficiency if for copper σ=57MS/m and radius a=1mm.

3

7 6 6

1 1 10

154 10 10 57 10fδ

πµ σ π π

−

−= = =

⋅ ⋅ ⋅ ⋅ ⋅,

6 3 3

2 20.084

2 57 10 2 10 (10 15)L

loss

LR

A aσ σ π δ π − −= = = = Ω⋅ ⋅ ⋅ ⋅

.

2 2 222 22 2 2( ) ( ) 0.0067

20 80 80 ( ) 0.0356 6 1

o av av avr

o o o o

l I l I l IR

I I I

µ β βπ π

ε π π λ

= ⋅ = ⋅ = = = Ω ,

0.0350.29, 29%

0.12rad rad rad

rad

in rad loss rad loss

P P Re or

P P P R R= = = = =

+ +. seemingly, this has

nothing to do with the condition of max power in the antenna circuit

3- Serbest uzayda yarim dalga boyu dipol anten 100 V’luk (peak) ve 50 ohm’luk bir kaynağa bağlanmıştır. Dipol antenin giriş empedansı 73+j42 ohm’dur.Antenden yayılan toplam gücü ve 100m, 45r θ= = ° için elektrik alanı hesaplayın.

3- A half-wave dipole in free space is connected to a 100 V (peak) source having a 50 ohm impedance. The input impedance of the dipole is 73+j42 ohm. Find the total radiated power and the magnitude of the E- field at 100m, 45 .r θ= = °

Magnitude of the current into the halfwave dipole at the feed point is

2

00

731000.796A (peak), 21.58 W

50 73 42 2rad

II P

j= = = =

+ +

10002

0

cos( cos 4)( 100m, 45 ) 60 0.290V m

1000 sin 4

jkeE r j I

π

θ

πθ

π

−

= = ° = =

1- 38 cm uzunluğunda olup z-yönünde ve merkezi orijinde olan bir dipol antendeki fazor akım 1.3 68∠ ° A ve 7.6mλ = ’dir. 2m, 70 , 135r θ φ= = ° = ° noktasında elektrik ve

manyetik alan değerlerini ve güç yoğunluğunu hesaplayın. 1- A 38 cm long dipole oriented and centered on the z-axis is driven by a phasor input current of 1.3 68∠ ° A at 7.6mλ = . Find the E & H fields at 2m, 70 , 135r θ φ= = ° = ° . Also

determine the time average Poynting vector at this location.

0.38 10.05

7.6 20

l

λ= = = use triangular current distribution

2 22 2(0.38)

0.038m 2m7.6

D

λ= = ⇒≪ 2m, 70 , 135r θ φ= = ° = ° is in the far field

2(2)

7.6

0

2(1.38 68 )

7.6sin 376.734 sin 70 (0.38) 2.87637 63.263 V m8 8 (2)

jkr ekI le

E j jr

π

θ

π

η θπ π

−− ∠ °

= = ° = ∠ °

20 1ˆsin 0.007635 63.263 A m Re( ) 0.0109 W m

8 2

jkr

r

kI le EH j

r

θφ θ

π η

−∗= = = ∠ ° → = × =S E H a

1- The expression for the far electric field of two Hertzian dipoles at right angles to each other (below figure), fed by equal-amplitude currents with 90° phase difference, is given below. Find the far zone magnetic field, the radiation intensity, the power radiated, the directive gain, and then directivity.

( ) ˆ ˆ[(sin cos cos ) ( sin ) ]4

j rj Idle j j

r

ββηθ θ φ θ φ φ

π

−= − +E

6- A half-wave dipole placed symmetrically along the z-axis. a) Determine the vector effective length (in terms of λ)

b) Determine the max value of the vector effective length and angle θ where this occurs.

c) Determine the ratio (in %) of the maximum effective length to its total physical length.

4- Find the directivity pattern and maximum directivity of a monopole antenna that has electric field given by sin jkrE e rθ θ −= .

4- Consider two identical z-directed electric dipoles of current moment II which reside on the x-axis, one at x = d and the other at x = -d. Determine an expression for Eθ(r,θ) under the assumption that r is very large. Begin with the expression for Eθ in the far zone due to a dipole at the origin and show your steps.

2- Find the shortest length of a center-driven dipole antenna such that there would be no radiation in the direction that makes a) 90o with the antenna axis; Express the dipole length in terms of the operating wavelength.

b) 75.52o with the antenna axis. Express dipole length in terms of the operating wavelength.

2- Consider a dipole antenna of length 5 2l λ= .

a) Analytically determine the directions of the nulls in the radiation pattern.

( ) ( )( )2 2

2

cos cos cos 2 15 2 5 , 0 cos cos 0 cos

sin 5

kl kl

kln

nl kl

θλ π θ θ

θ

− += → = = = ⇒ = → =E

0 78.5

1 53.1

2 0

n

n

n

θθθ

= → == → == → =

1 101.5

2 180

3 0

n

n

n

θθθ

= − → == − → == − → =

b) Generate a plot of the normalized radiation pattern.

3- Assume that the antenna of a 900 MHz handheld mobile phone can be modeled as a z-directed monopole of length 4 cm on the on the infinite ground plane. The input power to the antenna is 1 W and the radiation efficiency is 80%. Find the amplitude of the radiated electric field at the base station located 1 km away from the antenna over a hill of height 300 m.

4 cm 8 cmm dl l ′= = = P 0.8 Wrad = . From Stutzman

P ( , )rad U dθ φ= Ω∫ ∫ , 2

( , ) ( , )mU U Fθ φ θ φ= , 2

( , )F dθ φΩ = Ω∫ ∫ ⇒ Prad mU= Ω

2 22

r2 2 2

0

( , ) ( , )( , )

2m ad

U F FE U P

r r r

θ φ θ φθ φ

η= = =

Ω ⇒ 0 r

( , ) 2 adF P

Er

θ φ η=

Ω

For a center-fed dipole antenna ( )sinzE j Aθ ω θ= r

02

cos cos cos2 2

2 sin

jkr

z

kl kl

I eA

k r

θµ

π θ

−

− ≈

, cos cos cos

( , )sin

l l

F

π πθ

λ λθ φ

θ

− =

900 MHzf = ⇒ 33.33 cmλ = , 8 cm=0.24 l λ= ,

2 2(1000) (300) 1044.03mr = + = , 1 1000

tan 73.3300

θ −= = ° ⇒ ( 73.3 ) 0.26F θ = ° =

monopole

4

D

πΩ = , monopole dipole( 4 cm) 2 ( 8 cm)D l D l= = =

Directivity increases as the length of a dipole increases from l λ<< to l λ= . For

2l λ= we have 1.64D = . Therefore 1.55D ≈ ⇒ monopole 3.1D = . Therefore

0.26 2 120 0.8 3.1

3mV m1044 4

Eπ

π

× × ×= =

5- A very nice property of linear-reciprocal systems is superposition, and we can use it to calculate quite accurately the pattern of any dipole. For example, consider a 2λ-long center-fed dipole, oriented along the z-axis, with a peak current of Io. a) Draw the current distribution on the dipole.

b) This antenna can be modeled by four distinct short dipoles located at 4z λ= ± and

3 4z λ= ± .

1. Determine the current distribution of each of these 4-dipoles.

2. Calculate and draw (polar/linear) the E-plane pattern of the 2l-long dipole using the approximation of 4-distinct short dipoles.

3. What is its peak directivity at what angle?

c) Compare the pattern calculated in (b) with the exact pattern of the dipole. They are identical

4- Compute and plot the normalized far-field pattern F(θ) of a 3λ/2-length dipole antenna occupying the segment (−3λ /4, 3λ/4) of the z-axis and has a current distribution given by

( )( ) sin 3 4mI z I zβ λ = − .

8- A linear infinitesimal electric dipole of length l and constant current is placed vertically a distance h above an infinite, perfectly conducting electric ground plane. Find the first five smallest lengths (in wavelengths, in ascending order) so that a null is formed (for each height) in the far-field pattern at an angle of 40o from the vertical

2- A dipole has length of L=1 m and is operated at 200 kHz. a) Find the radiation resistance assuming a linearly-tapered current distribution, i.e.

0 2 , 2 2I I L z L z L′ ′= − − ≤ ≤ . (make short dipole assumption).

2 2

8 3 513 10 200 10 1500 m, 120 8.8 10

6 6 1500r

Lc f R

π πλ η π

λ

− = = × × = = = = × Ω

b) Calculate the beam solid angle AΩ . (make infinitesimal dipole assumption).

9- A vertical infinitesimal electric dipole is placed at a distance h = 1.8λ above an infinite, perfectly conducting electric ground plane. a) Determine the angles (in degrees from the vertical) where the array factor of the system will achieve its maximum value

b) Determine the angle θ where the max of the total field will occur

c) The relative (compared to its maximum) field strength (in dB) of the total field at the angles determined in part (a)

1- An infinitesimal x-directed electric dipole is placed horizontally a distance h above an infinite, perfectly conducting electric plane (x-y plane). Determine the E and H far-zone fields.

3- The vector potential due to an x directed electric dipole at the origin is ˆ4

jkreIl

rµ

π

−

=A x

where ˆ ˆˆˆ sin cos cos sin sinθ φ θ φ φ= + −x r θ φ . If we define ( ) (4 )jkrg r e rπ−= then the

components of the vector potential can be written as ( )sin cosrA Ilg rµ θ φ= ,

( )cos sinA Ilg rθ µ θ φ= , and ( )sinA Ilg rφ µ φ= − . From the above, determine the far-zone

magnetic field. (There are two ways to answer this question. One involves computing the far-zone magnetic field from 1

µ= ×H Α∇ by performing all the partial derivatives and,

subsequently, taking the limit of the result as r approaches infinity. The other involves writing 1

µ= ×H Α∇ in terms of the components of the vector potential and eliminating

various components on the basis of their limiting (r → ∞ ) values. The latter requires a little more thought than the former but results in far less wear and tear on the pencil and fingers.)

8- For a very thin center-fed half-wave dipole lying centered at the origin along the z-axis: a) Find the charge distribution on the dipole if current distribution is 0( ) cos( )I z I zβ= .

b) Repeat part (a) for 0( ) (1 4 )I z I z λ= − , 4z λ≤ .

3- A 15 MHz uniform plane wave having a peak electric field intensity E0 = 0.05 V/m is incident on a 2.5 λ long dipole at an angle θ. Assume lossless dipole. a) Plot the current on the dipole.

b) Find the angles of incidence θ that will give a zero open circuit voltage Voc at the terminals of the dipole.

c) Plot the far-field E plane pattern of the dipole (Polar plot).

d) Find the maximum effective length of the dipole.

e) If the dipole is connected to a matched load, what is the maximum power PL that can be delivered to the load?

Or

3- Two half wave dipoles are arranged as shown in Fig. The first antenna is transmitting 300W at 300MHz. Find the open circuit voltage induced at the terminals of the receiving second antenna and its effective area. Find available power at antenna two.

The open circuit voltage at antenna2 will be 2 2 1o eV h E= ⋅ . The antenna2 effective height is

2cos

2

2

sin 2( ) sin ( ) ( )

2

L

j z mm

o oL

L Ih I z e dz F

I Iβ θθθ β θ

β−

−

= ⋅ = ⋅∫ ∓

cos( cos ) cos( )2 2( )

sin

L L

Fβ θ β

θθ

−= ,

or after substitution, for a half wave dipole L=λ/2 and where 2 90oθ θ= = ,

2

22

cos( cos )2 22sin

m m

o o

I I ch

I I f

π θ

β θ β π= ⋅ = = .

The E1 in place of antenna 2 will be

2cos

1

2

sin 60( ) sin ( ) ( )

4 2

L

j zo om

o L

L j IE j I z e dz F

r rβ θµ θθ β θ

ε π−

−

= ⋅ = ⋅∫ ∓ .

For a half wave dipole,

1

cos( cos60)2( )

sin60F

π

θ = . So the electric field will be

1

cos( cos60)2( )

2 sin60

o

oEr

µ πεθπ

= ⋅ , or in the terms of effective antenna 1 height,

1 1, 1

cos( cos60)2 2( )4 sin60

o

oe eE h where h

r

µ πβεθ

π β= = ⋅ .

We can calculate this to be 11 1 1

1cos( )60 22 2( ) 0.6 0.49

100 332

IE I I

π

θ = ⋅ = = .

The current I1 can be found from the given power P1,

2 11 1 1

2

1 2 2 3002.87

2 73 73

PP R I I Aλ

⋅= → = = =Ω

Finally, 8

2 2 1 8

3 100.49 2.87 0.448

3 10o eV h E V

π⋅= ⋅ = ⋅ =⋅

.

The available power comes from the assumption that the antenna has a matched resistance for the rest of the circuit,

2 22 22 2 12

1 1 ( ) 0.448( ) 344

2 2 2 8 8 73e

a ant antant ant

V h EP R I R W

R Rµ= = = = =

⋅.

3- A half-wave dipole at 2 GHz is connected to a generator with Zg=50 Ω. The generator can deliver a maximum of 1W of power to a load under impedance matched conditions: a) Calculate the power delivered to the2λ dipole.

20

0

73 500.186, 1W (1 | | ) 965mW

73 50L

incident delivered incident

L

Z ZP P P

Z Z

− −Γ = = = = → = − Γ =

+ +

b) Calculate the current at the antenna terminal and the far-field electric field at a distance of 200m from the antenna (observation point) in the direction of maximum radiation.

20 0

8

9

0

(41.89)(200)3 8378

1 2 2 0.965162.6mA

2 73

3 10 22GHz , = 0.15 m , 41.89 rad/m

2 10

cos( 2 cos )60 max radiation at 90

sin

60(0.1626) 48.8 10200

radrad rad

rad

jkRo

jj

PP I R I

R

f k

eE j I

R

eE j j e

θ

θ

πλ

λ

π θθ

θ

−

−− −

×= ⇒ = = =

×= = = =

×

= =

= = ×

c) If a small solenoid with N = 10 turns is placed at the observation point: i) How would you orient the solenoid to pick up the antenna radiated field?

ii) If the radius of the solenoid loops is 1 cm, calculate the induced rms voltage across the terminals of the solenoid.

5- A half-wave dipole along the z-axis is described by the constant current density

0ˆ( ) cos( )z I kz=J r for 4 4zλ λ− ≤ ≤ .

a) Find the magnetic vector potential in the far field.

4 4cos cos0 0 0 0 1

24 4

ˆ ˆcos( ) ( )4 4

jkr jkrjkz jkz jkz jkzI e I e

z kz e dz z e e e dzr r

λ λθ θ

λ λ

µ µ

π π

− −− −′ ′ ′ ′−

− −′ ′ ′≈ = +∫ ∫A

20 02

cos( cos )ˆ

2 sin

jkrI ez

k r

πµ θ

π θ

−

≈A

(b) Find the electric far field.

(c) Find the radiated power. Use the fact that 2

0

cos [( 2)cos ]1.22

sind

π π θθ

θ∫ ≃ .

(d) Find the antenna gain pattern.

2- An antenna operating at 1200 MHz is composed of a system of four elementary dipoles all located on and parallel to the z axis, 50 cm apart. Thus the current density J is described as

Recall from lecture that, in the far field, the effect of moving an antenna is calculated quite simply, namely if ( ) ( )→J r E r then 1( )

1( ) ( )je β ⋅− → r rJ r r E r . Using this fact, address the

following questions: a) compute the vector potential A and electric field E.

b) compute the Poynting vector S, and intensity U.

c) compute the normalized power pattern, plot it.

d) compute the directivity. In what direction(s) does the peak occur?

3- In the previous problem, move all the antenna elements in the positive direction by an amount 75 cm, and repeat the calculations. Which answers change? Which don’t? Why?

2- What antenna current is required so that the antenna will radiate 20 W if a) the antenna is a 2 m long dipole antenna operating in the AM broadcast band at 1 MHz.

1 MHzf = , 300 mc fλ = = , point dipole2 m 50 6 m short dipole: 4l P Pλ= < = ⇒ =

2

2 20 0

140 67.56 A

4

zP I Iπ

λ∆ = ⇒ =

b) the antenna is a 1 m long dipole antenna operating in the FM/TV broadcast band at 150 MHz.

150 MHzf = , 2 mc fλ = = , 20 01 m 2 half-wave dipole: 36.6 0.74 Al P I Iλ= = ⇒ = ⇒ =

c) For the above two antennas, find the maximum distance from the antenna in the horizontal plane at which the FCC standard of 25 mV/m field strength is maintained. Assume antennas are vertical.

Short dipole 0 0( 2)sin 25 mV/m

4

k I z

r

η θπ

∆= ≥E , 0 0 sin8 0.025

kI zr

η θπ

∆≤×

Horizontal plane 2θ π= , sin 1θ = , 0 0 1.7 Km8 0.025

kI zr

ηπ

∆≤ =×

Half-wave dipole ( )20 0

cos cos25 mV/m

2 sin

I

r

π θηπ θ

= ≥E ,

2θ π= , sin 1θ = , cos 0θ = , ( )2cos cos 1π θ =

0 0 1.8 Km2 0.025

Ir

ηπ

≤ =×

2- Consider an elementary, z-directed electric dipole of current moment II on the x-axis at x=d. Assume that d is very large. a) What is the direction and magnitude of the electric field at the origin due to this dipole?

b) Let the dipole continue to be z-directed but move it to the point (d,d,0) in the xy plane. What is the direction of the dipole’s electric field now?

Alternate solution method by reciprocity

3- Assume a cross dipole antenna as shown below figure. It is made of two half-wavelength dipoles along z and y directions. Find the radiated fields assuming that the antennas are excited by current sources with equal amplitude and phase. What is the polarization of electric field in the far-zone region in the xz plane?

3- The transmitting antenna of a radio navigation system is a vertical metal mast of height h=40 m insulated from the earth. A source operating at f=180 KHz sends a current with amplitude Im=100 mA into the base of the mast. Assuming the current amplitude in the antenna to decrease linearly toward zero at the top of the mast and the earth to be a perfectly conducting plane, determine

(a) the effective length of the antenna (The effective length of a linear antenna of length L is defined as the length le of an equivalent linear antenna with uniform current distribution I(0)

such that it radiates the same far-zone fields in the H-plane, i.e. 2

1(0) 2

( )L

e I Ll I z dz

−

−′ ′= ∫ .)

(b) the maximum field intensity at a distance of R=160 km from the antenna,

(c) the time-average radiated power,

(d) the radiation resistance.

4- A thin linear dipole of length L is placed symmetrically about the z-axis. Find the far-zone electric and magnetic fields assuming the current distribution is given by

7- A dipole antenna, with a triangular current distribution, is used for communicating with a submarine at f =110 kHz. The overall length of the dipole is L = 230 m and the wire radius is a = 1.5 m. Assume a loss resistance of 3 Ω. (a) Evaluate the input impedance of the antenna, including the loss resistance. The input reactance can be approximated by

(b) (3 points) Evaluate the radiation efficiency of the antenna

Hint: The far-field pattern of the above dipole has a term sin2(0.25βLcosθ). Use a one term Maclaurin (Taylor) series expansion of this function before you use it in part (a).

9- Consider the double-dipole antenna. The dipoles are oriented along the x-axis and are spaced a distance d from each other on the y-axis. The dipoles are fed in phase and with the same amplitude (for cases 1-4). The dipole length is 0.625 lambda. a) Calculate the antenna pattern (normalized). Define the E-plane and the H-plane, and write their patterns mathematically.

b) For d=0.5 lambda, plot the E-plane and H-plane co-pol and x-pol patterns (linear in power, radial in angle). Plot also the 45 degree co-pol and x-pol patterns. Clearly label the polarization in the E-plane and H-plane patterns and on the z-axis.

c) The double-dipole is now placed a height h (h=0.25 lambda) above the ground plane (x-y plane). Calculate the new pattern, determine the 3-dB beamwidth, and estimate the directivity of the antenna (do not use the equation in Balanis). What do you think happens to the cross-polarization level in the presence of the ground plane (assume the ground plane is infinite in extent)?

d) Calculate the driving point impedance for one dipole antenna (in the double-dipole antenna) in the presence of the ground plane. Use the graphs in the notes.

e) Which f/d reflector would you choose to match the antenna of case (3) above (with a finite but small ground plane). Clearly state your reasons. Also, draw a realistic feeding mechanism for the double-dipole antenna.