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iii

**Preface
**
This Instructors’ Manual provides solutions to most of the problems in ANTENNAS:
FOR ALL APPLICATIONS, THIRD EDITION. All problems are solved for which
answers appear in Appendix F of the text, and in addition, solutions are given for a large
fraction of the other problems. Including multiple parts, there are 600 problems in the
text and solutions are presented here for the majority of them.
Many of the problem titles are supplemented by key words or phrases alluding to the
solution procedure. Answers are indicated. Many tips on solutions are included which
can be passed on to students.
Although an objective of problem solving is to obtain an answer, we have endeavored
to also provide insights as to how many of the problems are related to engineering
situations in the real world.
The Manual includes an index to assist in finding problems by topic or principle and
to facilitate finding closely-related problems.
This Manual was prepared with the assistance of Dr. Erich Pacht.
Professor John D. Kraus
Dept. of Electrical Engineering
Ohio State University
2015 Neil Ave
Columbus, Ohio 43210
Dr. Ronald J. Marhefka
Senior Research Scientist/Adjunct Professor
The Ohio State University
Electroscience Laboratory
1320 Kinnear Road
Columbus, Ohio 43212

iv

**Table of Contents
**
Preface iii
Problem Solutions:
Chapter 2. Antenna Basics ............................................................................................1
Chapter 3. The Antenna Family ..................................................................................17
Chapter 4. Point Sources .............................................................................................19
Chapter 5. Arrays of Point Sources, Part I ..................................................................23
Chapter 5. Arrays of Point Sources, Part II .................................................................29
Chapter 6. The Electric Dipole and Thin Linear Antennas .........................................35
Chapter 7. The Loop Antenna .....................................................................................47
Chapter 8. End-Fire Antennas: The Helical Beam Antenna and the Yagi-Uda
Array, Part I ...............................................................................................53
Chapter 8. The Helical Antenna: Axial and Other Modes, Part II .............................55
Chapter 9. Slot, Patch and Horn Antennas ..................................................................57
Chapter 10. Flat Sheet, Corner and Parabolic Reflector Antennas ...............................65
Chapter 11. Broadband and Frequency-Independent Antennas ....................................75
Chapter 12. Antenna Temperature, Remote Sensing and Radar Cross Section............81
Chapter 13. Self and Mutual Impedances....................................................................103
Chapter 14. The Cylindrical Antenna and the Moment Method (MM) ......................105
Chapter 15. The Fourier Transform Relation Between Aperture Distribution
and Far-Field Pattern ...............................................................................107
Chapter 16. Arrays of Dipoles and of Aperture ..........................................................109
Chapter 17. Lens Antennas..........................................................................................121
Chapter 18. Frequency-Selective Surfaces and Periodic Structures
By Ben A. Munk ......................................................................................125
Chapter 19. Practical Design Considerations of Large Aperture Antennas ................127
Chapter 21. Antennas for Special Applications ..........................................................135
Chapter 23. Baluns, etc. By Ben A. Munk ..................................................................143
Chapter 24. Antenna Measurements. By Arto Lehto and
Pertti Vainikainen ....................................................................................147
Index 153

1

**Chapter 2. Antenna Basics
**

**2-7-1. Directivity.
**Show that the directivity *D* of an antenna may be written

( ) ( )

( ) ( ) ∫∫ Ω

= ∗

∗

π

φθφθ π

φθφθ

4 2

2maxmax

,, 4 1

,,

*dr
Z
EE
*

*r
Z
EE
*

*D *

*Solution*:

*avU
UD *max),( φθ=

, 2maxmax ),(),( *rSU *φθφθ = , ∫∫ Ω= π φθπ 4 ),(4
1 *dUU av *

2),(),( *rSU *φθφθ = , ( ) ( )

*Z
EES *φθφθφθ ,,),(

∗

=

Therefore ( ) ( )

( ) ( ) ∫∫ Ω

= ∗

∗

π

φθφθ π

φθφθ

4

2

2maxmax

,, 4 1

,,

*dr
Z
EE
*

*r
Z
EE
*

*D * q.e.d.

Note that =2*r *area/steradian, so 2*SrU *= or (watts/steradian) = (watts/meter2) × meter2

**2-7-2. Approximate directivities.
**Calculate the approximate directivity from the half-power beam widths of a
unidirectional antenna if the normalized power pattern is given by: (a) *Pn* = cos θ, (b) *Pn*
= cos2 θ, (c) *Pn* = cos3 θ, and (d) *Pn* = cosn θ. In all cases these patterns are unidirectional
(+*z* direction) with *Pn* having a value only for zenith angles 0° ≤ θ ≤ 90° and *Pn* = 0 for
90° ≤ θ ≤ 180°. The patterns are independent of the azimuth angle φ.
*
Solution*:

(a) oo1HP 12060 2)5.0(cos2 =×== −θ , 278

)120( 000,40

2 ==*D * (*ans.*)

(b) oo1HP 9045 2)5.0(cos2 =×== −θ , 94.4

)90( 000,40

2 ==*D * (*ans.*)

2

(c) oo31HP 93.7437.47 2)5.0(cos2 =×== −θ , 3.7

)75( 000,40

2 ==*D * (*ans.*)

**2-7-2. continued
**

(d) )5.0(cos2 1HP *n
*−=θ , 21 ))5.0((cos

000,10
*nD *−= (*ans.*)

***2-7-3. Approximate directivities.
**Calculate the approximate directivities from the half-power beam widths of the three
unidirectional antennas having power patterns as follows:

*P*(θ*,*φ) = *Pm* sin θ sin2 φ

*P*(θ*,*φ) = *Pm* sin θ sin3 φ

*P*(θ*,*φ) = *Pm* sin2 θ sin3 φ
*P*(θ*,*φ) has a value only for 0 ≤ θ ≤ π and 0 ≤ φ ≤ π and is zero elsewhere.
*Solution*:
To find *D* using approximate relations,
we first must find the half-power beamwidths.

θ−= 90 2

HPBW or 2

HPBW90 −=θ

For sin θ pattern, 2 1

2 HPBW90sinsin =⎟

⎠ ⎞

⎜ ⎝ ⎛ −=θ ,

⎟ ⎠ ⎞

⎜ ⎝ ⎛− −

2 1sin

2 HPBW90 1 , 90

2 1sin

2 HPBW 1 −⎟

⎠ ⎞

⎜ ⎝ ⎛− − , ∴ oHPBW 120=

For sin2 θ pattern, 2 1

2 HPBW90sinsin 22 =⎟

⎠ ⎞

⎜ ⎝ ⎛ −=θ ,

2 1

2 HPBW90sin =⎟

⎠ ⎞

⎜ ⎝ ⎛ − , ∴ oHPBW 90=

For sin3 θ pattern, 2 1

2 HPBW90sinsin 33 =⎟

⎠ ⎞

⎜ ⎝ ⎛ −=θ ,

3

3 2 1

2 HPBW90sin =⎟

⎠ ⎞

⎜ ⎝ ⎛ − , ∴ oHPBW 74.9=

***2-7-3. continued
**Thus,

70.3 )90)(120(

000,4082.3 )90)(120(

253,41 deg. sq. 253,41

HPHP =≅===

φθ
*D * (*ans.*)

45.4 )9.74)(120(

000,4059.4 )9.74)(120(

253,41
=≅== (*ans.*)

93.5 )9.74)(90(

000,4012.6 )9.74)(90(

253,41
=≅== (*ans.*)

***2-7-4. Directivity and gain.
**(a) Estimate the directivity of an antenna with θHP = 2°, φHP = 1°, and (b) find the gain of
this antenna if efficiency *k* = 0.5.
*Solution*:

(a) 4 HPHP

100.2 )1)(2(

000,40000,40 ×===

φθ
*D * or 43.0 dB (*ans.*)

(b) 44 100.1)100.2(5.0 ×=×== *kDG * or 40.0 dB (*ans*.)

**2-9-1. Directivity and apertures.
**Show that the directivity of an antenna may be expressed as

( ) ( ) ( ) ( )∫∫

∫∫∫∫ ∗

∗

=
*Ap
*

*ApAp
*

*dxdyyxEyxE
*

*dxdyyxEdxdyyxE
D
*

,,

,,4 2λ π

where *E*(*x, y*)is the aperture field distribution.
*Solution*: If the field over the aperture is uniform, the directivity is a maximum (= *D*m)
and the power radiated is *P*′ . For an actual aperture distribution, the directivity is *D* and
the power radiated is *P*. Equating effective powers

for *P*(θ,φ) = sin θsin2φ

for *P*(θ,φ) = sin θsin3φ

for *P*(θ,φ) = sin2 θsin3φ

4

*PDPD *=′m , ( ) ( )
∫∫

∗= ′

=

*Ap
*

*p
*

*p
*

*dxdy
Z
*

*yxEyxE
*

*A
Z
EE
*

*A
P
PDD
*

,, 4

* avav

2m λ π

**2-9-1. continued
**

where ∫∫= *pAp
dxdyyxE
*

*A
E *),(1av

therefore ( ) ( )

( ) ( )2 , ,4

, ,
*Ap Ap
*

*Ap
*

*E x y dxdy E x y dxdy
D
*

*E x y E x y dxdy
*π

λ

∗

∗ =

∫∫ ∫∫ ∫∫

q.e.d.

where ( ) ( ) ( ) ( )

av av av av av 2

av 1 ( ), , , ,

*p e
ap
*

*p
Ap
*

*p
*

*E E A E E E A
E AE x y E x y dxdy E x y E x y dxdy
*

*A
*

ε ∗ ∗ ∗

∗ ∗

= = = = ∫∫ ∫∫

**2-9-2. Effective aperture and beam area.
**What is the maximum effective aperture (approximately) for a beam antenna having half-
power widths of 30° and 35° in perpendicular planes intersecting in the beam axis?
Minor lobes are small and may be neglected.
*Solution*:

o oHP HP 30 35 ,*A *θ φΩ ≅ = ×
22

oo

22 1.3

3530 3.57 λλλ =

× ≅

Ω =

*A
emA * (*ans.*)

***2-9-3. Effective aperture and directivity.
**What is the maximum effective aperture of a microwave antenna with a directivity of
900?

*Solution*: 24 / ,*emD A*π λ=
2

2

6.71 4 900

4 2

λλ ππ

λ ===

*DAem * (*ans.*)

**2-11-1. Received power and the Friis formula.
**What is the maximum power received at a distance of 0.5 km over a free-space 1 GHz
circuit consisting of a transmitting antenna with a 25 dB gain and a receiving antenna
with a 20 dB gain? The gain is with respect to a lossless isotropic source. The
transmitting antenna input is 150 W.

5

*
Solution*:

2 2 8 9/ 3 10 /10 0.3 m, ,

4 4
*t r
*

*et er
D Dc f A A*λ λλ

π π = = × = = =

**2-11-1. continued
**

mW 10.8 W0108.0 500)4(

1003.0316150 )4( 22

2

222

22

22 == ××

=== πλπ

λλ
λ *r
*

*DD
P
*

*r
AA
*

*PP rtt
eret
*

*tr *(*ans.*)

***2-11-2. Spacecraft link over 100 Mm.
**Two spacecraft are separated by 100 Mm. Each has an antenna with *D**= *1000 operating
at 2.5 GHz. If craft A's receiver requires 20 dB over 1 pW, what transmitter power is
required on craft B to achieve this signal level?
*
Solution*:

2 8 9/ 3 10 / 2.5 10 0.12 m,

4*et er
Dc f A A *λλ

π = = × × = = =

12 10

2 2 2 2 2 2 2 16 2 10

2 2 4 2 2 6 2

(required) 100 10 10 W

(4 ) (4 ) 10 (4 )10 10966 W 11 kW ( ) 10 0.12

*r
*

*t r r r
et
*

*P
*

*r r rP P P P ans.
A D D
*λ π λ π π

λ λ

− −

−

= × =

= = = = = ≅

**2-11-3. Spacecraft link over 3 Mm.
**Two spacecraft are separated by 3 Mm. Each has an antenna with *D* = 200 operating at 2
GHz. If craft A's receiver requires 20 dB over 1 pW, what transmitter power is required
on craft B to achieve this signal level?
*Solution*:

2 8 9/ 3 10 / 2 10 0.15 m

4*et er
Dc f A A *λλ

π = = × × = = =

12 10

2 2 2 2 2 2 12 10

2 2 2 4 2

100 10 10 W

(4 ) (4 ) 9 1010 158 W ( ) 4 10 0.15

*r
*

*t r r
et er
*

*P
*

*r rP P P ans.
A A D
*

λ π λ π λ λ

− −

−

= × =

× = = = =

× ×

**2-11-4. Mars and Jupiter links.
**(a) Design a two-way radio link to operate over earth-Mars distances for data and picture
transmission with a Mars probe at 2.5 GHz with a 5 MHz bandwidth. A power of 10-19

6

W Hz-1 is to be delivered to the earth receiver and 10-17 W Hz-1 to the Mars receiver. The Mars antenna must be no larger than 3 m in diameter. Specify effective aperture of Mars and earth antennas and transmitter power (total over entire bandwidth) at each end. Take earth-Mars distance as 6 light-minutes. (b) Repeat (a) for an earth-Jupiter link. Take the earth-Jupiter distance as 40 light-minutes.

**2-11-4. continued***Solution*:
(a) 8 9/ 3 10 / 2.5 10 0.12 m*c f*λ = = × × =

19 6 13

17 6 11

(earth) 10 5 10 5 10 W

(Mars) 10 5 10 5 10 W
*r
*

*r
*

*P
*

*P
*

− −

− −

= × × = ×

= × × = ×

Take )5.0( m 5.31.5(1/2)Mars)( ap
22 === επ*eA *

Take kW 1Mars)( =*tP *

Take )5.0( m 35051(1/2)earth)( ap
22 === επ*eA *

MW 9.6 3505.3

12.0)103360(105)earth(

Mars)(earth)( Mars)()earth(

228 11

22

= ×

×× ×=

=

−
*t
*

*etet
rt
*

*P
*

*AA
rPP *λ

To reduce the required earth station power, take the earth station antenna

22 m 392750 )2/1( == π*eA *(*ans.*)
so

6 2(earth) 6.9 10 (15 / 50) 620 kW ( )*tP ans.*= × =

W108 12.0)103360(

39305.310earth)(Mars)(Mars)()earth( 14228 3

22 −×=

×× ×

==
λ*r
AAPP erettr *

which is about 16% of the required 5 x 10−13 W. The required 5 x 10−13 W could be
obtained by increasing the Mars transmitter power by a factor of 6.3. Other alternatives
would be (1) to reduce the bandwidth (and data rate) reducing the required value of *Pr* or
(2) to employ a more sensitive receiver.

7

As discussed in Sec. 12-1, the *noise power* of a receiving system is a function of its
system temperature *T* and bandwidth *B* as given by *P* = *kTB, *where *k* = Boltzmann’s
constant = 1.38 x 10−23 JK−1.
For *B *= 5 x 106 Hz (as given in this problem) and *T* = 50 K (an attainable value),

W105.310550101.38noise)( 15623 −− ×=××××=*P *

**2-11-4. continued**The received power (8 x 10−14 W) is about 20 times this noise power, which is probably
sufficient for satisfactory communication. Accordingly, with a 50 K receiving system
temperature at the earth station, a Mars transmitter power of 1 kW is adequate.
(b) The given Jupiter distance is 40/6 = 6.7 times that to Mars, which makes the
required transmitter powers 6.72 = 45 times as much *or* the required receiver powers 1/45
as much.
Neither appears feasible. But a practical solution would be to reduce the bandwidth for
the Jupiter link by a factor of about 50, making *B* = (5/50) x 106 = 100 kHz.

***2-11-5. Moon link.
**A radio link from the moon to the earth has a moon-based 5λ long right-handed mono-
filar axial-mode helical antenna (see Eq. (8-3-7)) and a 2 W transmitter operating at 1.5
GHz. What should the polarization state and effective aperture be for the earth-based
antenna in order to deliver 10-14 W to the receiver? Take the earth-moon distance as 1.27
light-seconds.
*
Solution*:

8 9/ 3 10 /1.5 10 0.2 m, *c f*λ = = × × =
From (8-3-7) the directivity of the moon helix is given by

60512 =×=*D * and
π
λ

4 )moon(

2*D
Aet *=

From Friis formula

RCPm 152 602

4)27.110(310)4( 22814 2

2222 =

× ××

=== − π

λ λπλ

*DP
rP
*

*AP
rPA
*

*t
*

*r
*

*ett
*

*r
er * or

about 14 m diameter(*ans.*)

8

**2-16-1. Spaceship near moon.
**A spaceship at lunar distance from the earth transmits 2 GHz waves. If a power of 10 W
is radiated isotropically, find (a) the average Poynting vector at the earth, (b) the rms
electric field **E **at the earth and (c) the time it takes for the radio waves to travel from the
spaceship to the earth. (Take the earth-moon distance as 380 Mm.) (d) How many
photons per unit area per second fall on the earth from the spaceship transmitter?

**2-16-1. continued
***Solution*:

(a) 2218262 aWm 5.5 Wm105.5)10(3804 10

4 earth)(at PV −−− =×=

× ==

ππ *r
Pt *(*ans.*)

(b) *ZES */PV 2== or 2/1)(*SZE *=
or 192/118 nVm 451045)377105.5( −−− =×=××=*E *(*ans.*)
(c) s 27.1103/10380/ 86 =××== *crt *(*ans.*)
(d) Photon = *hf *J 103.11021063.6 24934 −− ×=×××= , where Js 1063.6 34−×=*h *
This is the energy of a 2.5 MHz photon. From (a), 2118 mJs 105.5 PV −−−×=

Therefore, number of photons = 12624 18

sm 102.4 103.1 105.5 −−

−

−

×=
×
× (*ans.*)

**2-16-2. More power with CP.
**Showthat the average Poynting vector of a circularly polarized wave is twice that of a
linearly polarized wave if the maximum electric field **E** is the same for both waves. This
means that a medium can handle twice as much power before breakdown with circular
polarization (CP) than with linear polarization (LP).
*
Solution*:

From (2-16-3) we have for rms fields that o

2 2

2 1PV

*Z
EESav
*

+ ==

For LP, 2

1 2 1

o

(or ) 0, so *av
EE E S
Z
*

= =

For CP, 2

1 1 2

o

2 , so *av
EE E S
Z
*

= =

Therefore LPCP 2*SS *= (*ans.*)

9

**2-16-3. PV constant for CP.
**Show that the instantaneous Poynting vector (PV) of a plane circularly polarized
traveling wave is a constant.
*Solution*:

*tEtEE yx *ωω sincosCP += where o*EEE yx *==

**2-16-3. continued
**2 2 2 2 1/ 2 2 2 1/ 2

CP o o o o( cos sin ) (cos sin )*E E t E t E t t E*ω ω ω ω= + = + = (a constant)

Therefore
*Z
ES
*

2
oous)instantane(or PV = (a constant) (*ans.*)

***2-16-4. EP wave power
**An elliptically polarized wave in a medium with constants σ *= *0, μ*r* = 2, ε*r**= *5 has *H*-
field components (normal to the direction of propagation and normal to each other) of
amplitudes 3 and 4A m-1. Find the average power conveyed through an area of 5 m2
normal to the direction of propagation.
*
Solution*:

2222/12 2

2 1

2/12 2

2 1 Wm2980)43()5/2(3772

1)()/(377 2 1)(

2
1 −=+=+=+= *HHHHZS rrav *εμ

kW 14.9 W1490229805 ==×== *avASP *(*ans.*)

**2-17-1. Crossed dipoles for CP and other states.
**Two λ/2 dipoles are crossed at 90°. If the two dipoles are fed with equal currents, what is
the polarization of the radiation perpendicular to the plane of the dipoles if the currents
are (a) in phase, (b) phase quadrature (90° difference in phase) and (c) phase octature
(45° difference in phase)?
*Solution*:
(a) LP(*ans.*)
(b) CP(*ans.*)
(c) From (2-17-3) δγε sin2sin2sin =

10

1 2 1

1 2

where tan ( / ) 45

45 22

AR cot 1/ tan 2.41 (EP)...( )

*E E
*

*ans.
*

γ

δ

ε ε ε

−= =

=

= = = =

***2-17-2. Polarization of two LP waves.
**A wave traveling normally out of the page (toward the reader) has two linearly polarized
components

*tEx *ωcos2=

( )90cos3 += *tEy *ω

(a) What is the axial ratio of the resultant wave?
(b) What is the tilt angle τ of the major axis of the polarization ellipse?
(c) Does **E **rotate clockwise or counterclockwise?
*
Solution*:
(a) From (2-15-8) , 5.12/3AR == (*ans.*)
(b) τ = 90o (*ans.*)
(c) At 0, ;*xt E E*= = at / 4, *yt T E E*= = − , therefore rotation is CW(*ans.*)

**2-17-3. Superposition of two EP waves.
**A wave traveling normally outward from the page (toward the reader) is the resultant of
two elliptically polarized waves, one with components of **E** given by

*tE y *ωcos2=′ and ( )2cos6 πω +=′ *tEx *

and the other with components given by *tEy *ωcos1=′′ and ( )2cos3 πω −=′′ *tEx *
(a) What is the axial ratio of the resultant wave?
(b) Does **E** rotate clockwise or counterclockwise?
*Solution*:

2cos cos 3cos

6cos( / 2) 3cos( / 2) 6sin 3sin 3sin
*y y y
*

*x x x
*

*E E E t t t
*

*E E E t t t t t
*

ω ω ω

ω π ω π ω ω ω

′ ′′= + = + =

′ ′′= + = + + − = − + = −

(a) *Ex* and *Ey* are in phase quadrature and AR 3/ 3 1 (CP)= = (*ans.*)

11

(b) At ˆ0, 3*t *= =**E y **, at ˆ/ 4, 3*t T*= = −**E x **, therefore rotation is CCW(*ans.*)

***2-17-4. Two LP components.
**An elliptically polarized plane wave traveling normally out of the page (toward the
reader) has linearly polarized components *Ex *and *Ey*. Given that *Ex = Ey *= 1 V m-1 and
that *Ey* leads *Ex* by 72°,
(a) Calculate and sketch the polarization ellipse.
(b) What is the axial ratio?
(c) What is the angle τ between the major axis and the x-axis?
*Solution*:
(b) oo12

1 72 ,45)/(tan === − δγ *EE *
From (2-17-3), o36=ε , therefore 38.1tan/1AR == ε (*ans.*)

(c) From (2-17-3), δε tan/2tanτ2sin = or o45τ = (*ans.*)

**2-17-5. Two LP components and Poincaré sphere.
**Answer the same questions as in Prob. 2-17-4 for the case where *Ey* leads *Ex* by 72° as
before but *Ex *= 2 V m-1 and *Ey**= *1 V m-1.
*Solution*:

1 o

o

(b) tan 2 63.4 72

γ

δ

−= =

=

17.2AR and 8.24 o ==ε (*ans.*)
(c) o2.11τ = (*ans.*)

***2-17-6. Two CP waves.
**Two circularly polarized waves intersect at the origin. One (*y*-wave) is traveling in the
positive *y* direction with **E** rotating clockwise as observed from a point on the positive *y*-
axis. The other (*x*-wave) is traveling in the positive *x* direction with **E** rotating clockwise
as observed from a point on the positive *x*-axis. At the origin, **E **for the *y*-wave is in the
positive *z* direction at the same instant that **E** for the *x*-wave is in the negative *z* direction.
What is the locus of the resultant **E** vector at the origin?
*
Solution*:

12

Resolve 2 waves into components or make sketch as shown. It is assumed that the waves have equal magnitude.

***2-17-6. continued
**

Locus of **E** is a straight line in *xy* plane at an angle of 45o with respect to *x* (or *y*) axis.

***2-17-7. CP waves.
**A wave traveling normally out of the page is the resultant of two circularly polarized
components *tjright eE
*

ω5= and ( )902 += *tjleft eE
*ω (V m-1). Find (a) the axial ratio AR, (b)

the tilt angle τ and (c) the hand of rotation (left or right).
*Solution*:

(a) AR 33.23/7

52 52

−=−= − +

= (*ans.*)

(b) From diagram, τ o45−= (*ans.*)

[Note minus sign for RH (right-handed polarization)]

13

(c) Since **E** rotates counterclockwise as a function of time,RH. (*ans.*)

**2-17-8. EP wave.
**A wave traveling normally out of the page (toward the reader) is the resultant of two
linearly polarized components *tEx *ωcos3= and ( )90cos2 += *tE y *ω . For the resultant
wave find (a) the axial ratio AR, (b) the tilt angle τ and (c) the hand of rotation (left or
right).
*
Solution*:
(a) AR = 3/2 = 1.5(*ans.*)
(b) τ = 0o(*ans.*)
(c) CW, LEP(*ans.*)

***2-17-9. CP waves.
**Two circularly polarized waves traveling normally out of the page have fields given by

*tj
left eE
*

ω−= 2 and *tjright eE
*ω3= (V m-1) (rms). For the resultant wave find (a) AR, (b) the

hand of rotation and (c) the Poynting vector.
*Solution*:

(a) 5
3-2
32AR −=+= (*ans.*)

(b) REP(*ans.*)

(c) 22 22

mWm 34 Wm034.0 377

94PV −− ==+=+=
*Z
*

*EE RL * (*ans.*)

**2-17-10. EP waves.
**A wave traveling normally out of the page is the resultant of two elliptically polarized
(EP) waves, one with components *tEx *ωcos5= and *tEy *ωsin3= and another with

components *tjr eE
*ω3= and *tjl eE
*

ω−= 4 . For the resultant wave, find (a)AR, (b) τ and
(c) the hand of rotation.
*Solution*:
(a)

14

*ttttE
*

*ttttE
*

*y
*

*x
*

ωωωω ωωωω

sin2sin4sin3sin3 cos12cos4cos3cos5

=−+= =++=

**2-17-10. continued
**
AR 62/12 == (*ans.*)
(b) Since *Ex* and *Ey* are in time-phase quadrature with *Ex*(max) > *Ey*(max), τ = 0o.

Or from (2-17-3), δε tan/2tanτ2sin = , o1 46.9)AR/1(tan == −ε
but o90=δ so ∞=δtan
Therefore o0τ = (*ans.*)

(c) At 0 ,12 ,0 === *yx EEt *

At 2 ,0 ),90t( 4/ o ==== *yx EETt *ω
Therefore rotation is CCW, so polarization is right elliptical, REP(*ans.*)

***2-17-11. CP waves.
**A wave traveling normally out of the page is the resultant of two circularly polarized
components *tjr eE
*

ω2= and ( )454 +−= *tjl eE
*ω . For the resultant wave, find (a) AR, (b) τ

and (c) the hand of rotation.
*Solution*:

(a) 3 2 6

24 24AR

1

1 == − +

= − +

=
*r
*

*r
*

*EE
EE *(*ans.*)

(b) When o o10 450, 2 and 4*rt E E*ω ____ −= = = ∠∠
When o o o1 1 12 2 2122 , 2 22 and 4 22*rt E E*ω __ __= − = − = −∠ ∠

so that *rEE *+1 = 6max =*E
*o1

222__−∠ or o

2
122τ −= (*ans.*)

Note that the rotation directions are opposite for *Er* and *E*1
so that for ,*t*ω− 12 but *rE t E t*ω ω__ __= − = +∠ ∠
Also, τ can be determined analytically by combining the waves into an *Ex* and *Ey
*component with values of

o o5.60 30.4 and 2.95 16.3*x yE E*__ __= − =∠ ∠

from which o7.46−=δ

E at *t* = *T*/4

CCW
E at *t* = 0

15

***2-17-11. continued
**
Since from (a) AR = 3, ε can be determined and from (2-17-3), the tilt angle

oτ 22.5 ( .)*ans*= −
(c) E1 > *Er* so rotation is CW (LEP) (*ans.*)

**2-17-12. Circular-depolarization ratio.
**If the axial ratio of a wave is AR, show that the circular-depolarization ratio of the wave
is given by.

AR 1 AR 1

*R *−=
+

Thus, for pure circular polarization AR* = *1 and *R = *0 (no depolarization) but for linear
polarization AR* =* ∞ and *R**= *1.
*Solution*:
Any wave may be resolved into 2 circularly-polarized components of opposite hand, *Er
*and *E*1 for an axial ratio

1

1

min

maxAR
*EE
EE
*

*E
E
*

*r
*

*r
*

− +

==

from which the *circular depolarization ratio*
1AR
1AR1

+ −

==
*rE
*

*ER *

Thus for pure circular polarization, AR = 1 and there is zero depolarization (*R* = 0), while
for pure linear polarization AR = ∞ and the depolarization ratio is unity (*R* =1). When
AR = 3, *R* = ½.

16

17

**Chapter 3. The Antenna Family
**

**3-4-1. Alpine-horn antenna.
**Referring to Fig. 3-4a, the low frequency limit occurs when the open-end spacing > λ/2
and the high frequency limit when the transmission line spacing *d* ≈ λ/4. If *d* = 2 mm and
the open-end spacing = 1000 *d*, what is the bandwidth?
*Solution*:
*
D* = opened end spacing, * d* = transmission line spacing

Bandwidth = 1000

2

2 min

max

==
*d
D
*

λ

λ

(*ans.*)

***3-4-2. Alpine-horn antenna.
**If *d* = transmission line spacing, what open-end spacing is required for a 200-to-1
bandwidth?
*Solution*:
If *d* = transmission line spacing min / 2λ= and *D* = open-end spacing = 2/maxλ ,

for 200-to-1 bandwidth, we must have max

min

2 200, or 200

2

*D D d
d
*

λ

λ= = = (*ans.*)

***3-5-2. Rectangular horn antenna.
**What is the required aperture area for an optimum rectangular horn antenna operating at
2 GHz with 16 dBi gain?
*Solution*:
From Fig. 3-5 for m) 0.15( GHz 2 == λ*f *,

2 2

2

7.5 63.118 dBi 63.1, 0.19 m 7.5

*whD wh *λ
λ

= = = ∴ = = (*ans.*)

18

***3-5-3. Conical horn antenna.
**What is the required diameter of a conical horn antenna operating at 3 GHz with 14 dBi
gain?
*Solution*:
From Fig. 3-5 for m) 0.1( GHz 3 == λ*f *,

2 2 2

2

6.5 15.812 dBi 15.8, 0.09 m , 2 0.18 m 6.5

*rD r d r*π λ
λ π

= = = ∴ = = = = (*ans.*)

**3-7-2. Beamwidth and directivity
**
For most antennas, the half-power beamwidth (HPBW) may be estimated as HPBW =
κλ/*D*, where λ is the operating wavelength, *D* is the antenna dimension in the plane of
interest, andκis a factor which varies from 0.9 to 1.4, depending on the filed amplitude
taper across the antenna. Using this approximation, find the directivity and gain for the
following antennas: (a) circular parabolic dish with 2 m radius operating at 6 GHz, (b)
elliptical parabolic dish with dimensions of 1 m × 10 m operated at 1 GHz. Assume κ =
1 and 50 percent efficiency in each case.
*Solution*:
From Fig. 3-9 for 1600 MHz ( 0.1875 m), *f *λ= =

17 dBi 50 (for 100% efficiency)*G D*= = =

(a) 15 5050, so 3.33 15

*LD L *λ λ
λ

= = = =

If spacing = 105.10 /

turnsofnumber ,/ ≈=== πλ

πλ *Ln * (*ans.*)

(b) Turn diameter = / 0.0596 6 cmλ π = ≈ (*ans.*)

(c) Axial ratio AR 05.1 20 21

2 12

== +

=
*n
*

*n * (*ans.*)

19

**Chapter 4. Point Sources
**

***4-3-1. Solar power
**The earth receives from the sun 2.2 g cal min-1 cm-2.
(a) What is the corresponding Poynting vector in watts per square meter?
(b) What is the power output of the sun, assuming that it is an isotropic source?
(c) What is the rms field intensity at the earth due to the sun’s radiation, assuming all the
sun’s energy is at a single frequency?
Note: 1 watt = 14.3 g cal min-1, distance earth to sun = 149 Gm.
*
Solution*:

(a) 1 2

2 2 1

2.2g cal min cm 0.1539 W cm 1539 W m 14.3 g cal min

*S
*− −

− −
−= = = (*ans.*)

(b) 2 2 22 26P(sun) 4 1539 4 1.49 10 W 4.29 10 W*S r*π π= × = × × × = × (*ans.*)
(c) 2 1 2 1 2 1o o/ , ( ) (1539 377) 762 V m*S E Z E SZ
*

−= = = × = (*ans.*)

**4-5-1. Approximate directivities.
**(a) Show that the directivity for a source with at unidirectional power pattern given by
*U* = *Um* cosn θcan be expressed as *D = 2(n+1)*. *U* has a value only for 0° ≤ θ ≤ 90°. The
patterns are independent of the azimuth angle φ. (b) Compare the exact values calculate
from (a) with the approximate values for the directivities of the antennas found in Prob.
2-7-2 and find the dB difference from the exact values.
*
Solution*:

(a) n 2 n n+1 0

0

2 4 2If cos , 2(n+1)

2 sin cos cos n+1

*mU U D
d
*

π π πθ

π θ θ θ θ = = = =

−∫
(*ans.*)

(b)

.

.

For n=1, 2.78 4.4 dBi

4 6.0 dBi

1.6 dB

*approx
*

*exact
*

*exact approx
*

*D
*

*D
*

*D D
*

≈ ⇒

= ⇒

− =

.

.

For n=2, 4.94 6.9 dBi

6 7.8 dBi

0.9 dB

*approx
*

*exact
*

*exact approx
*

*D
*

*D
*

*D D
*

≈ ⇒

= ⇒

− =

.

.

For n=3, 7.3 8.6 dBi

8 9.0 dBi

0.4 dB

*approx
*

*exact
*

*exact approx
*

*D
*

*D
*

*D D
*

≈ ⇒

= ⇒

− =

20

***4-5-2. Exact versus approximate directivities.
**(a) Calculate the exact directivities of the three unidirectional antennas having power
patterns as follows:

*P*(θ*,*φ) = *Pm* sin θ sin2 φ

*P*(θ*,*φ) = *Pm* sin θ sin3 φ

*P*(θ*,*φ) = *Pm* sin2 θ sin3 φ
*P*(θ*,*φ) has a value only for 0 ≤ θ ≤ π and 0 ≤ φ ≤ π and is zero elsewhere.
(b) Compare the exact values in (a) with the approximate values found in Prob. 2-7-3.
*Solution*:

(a)

4

4 4 , sin
( , )*A n
*

*D d d d
P d
*

π

π π θ θ φ θ φ

= = Ω = Ω Ω∫∫

For *P*(θ*,*φ) = *Pm* sin θ sin2 φ, 2 2 2
0 00 0

4 4
sin sin sin sinsin*m
*

*m
*

*D
P d dd d
*

*P
*

π ππ π

π π θ φ θ φ θ φθ θ φ

= = ∫ ∫∫ ∫

2

0 0

1 4 16sin sin 2 , 5.09 2 4 2

2 2

*d D
*π πθ π πθ θ θ

π π π ⎛ ⎞= − = ∴ = = =⎜ ⎟ ⎛ ⎞⎛ ⎞⎝ ⎠

⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

∫ (*ans.*)

Using the same approach, we find,

for *P*(θ*,*φ) = *Pm* sin θ sin3 φ,
2 3

0 0

4 4 6.0 4sin sin

2 3

*D
d d
*

π π

π π πθ φ θ φ

= = = ⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

∫ ∫
(*ans.*)

for *P*(θ*,*φ) = *Pm* sin2 θ sin3 φ,
3 3

0 0

4 4 7.1 4 4sin sin 3 3

*D
d d
*

π π

π π

θ φ θ φ = = =

⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

∫ ∫
(*ans.*)

(b) Tabulating, we have 5.1 vs. 3.8, 6.0 vs. 4.6, and 7.1 vs. 6.1 (*ans.*)

**4-5-3. Directivity and minor lobes.
**Prove the following theorem: if the minor lobes of a radiation pattern remain constant as
the beam width of the main lobe approaches zero, then the directivity of the antenna
approaches a constant value as the beam width of the main lobes approaches zero.

21

**4-5-3. continued***Solution*:

4 4

*A M m
*

*D *π π= =
Ω Ω + Ω

where total beam area

main lobe beam area minor lobe beam area

*A
*

*M
*

*m
*

Ω = Ω = Ω =

as 0, , so 4 (a constant)*M A m mD *πΩ → Ω → Ω = Ω (*ans.*)

**4-5-4. Directivity by integration.
**(a) Calculate by graphical integration or numerical methods the directivity of a source
with a unidirectional power pattern given by *U* = cos θ. Compare this directivity value
with the exact value from Prob. 4-5-1. *U* has a value only for 0° ≤ θ ≤ 90° and 0° ≤ φ ≤
360° and is zero else where.
(b) Repeat for a unidirectional power pattern given by *U* = cos2 θ.
(c) Repeat for a unidirectional power pattern given by *U* = cos3 θ.
*Solution*:
Exact values for (a), (b), and (c) are: 4, 6, and 8. (*ans.*)

**4-5-5. Directivity.
**Calculate the directivity for a source with relative field pattern *E* = cos 2θ cos θ.
*Solution*:

Assuming a unidirectional pattern, (0 ), 24 2

*D*πθ≤ ≤ = (*ans.*)

22

23

**Chapter 5. Arrays of Point Sources, Part I
**

**5-2-4. Two-source end-fire array.
**(a) Calculate the directivity of an end-fire array of two identical isotropic point sources in
phase opposition, spaced λ/2 apart along the polar axis, the relative field pattern being
given by

⎟ ⎠ ⎞

⎜ ⎝ ⎛= θπ cos

2
sin*E *

where θ is the polar angle.
(b) Show that the directivity for an ordinary end-fire array of two identical isotropic point
sources spaced a distance *d* is given by

( ) ( )λππλ *ddD *4sin41
2

+ = .

*Solution*:
(a) 2*D *= (*ans.*)

**5-2-8. Four sources in square array.
**(a) Derive an expression for *E(*φ*)* for an array of 4 identical isotropic point sources
arranged as in Fig. P5-2-8. The spacing *d* between each source and the center point of
the array is *3*λ*/8.* Sources 1 and 2 are in-phase, and sources 3 and 4 in opposite phase
with respect to 1 and 2.
(b) Plot, approximately, the normalized pattern.

Figure P5-2-8. Four sources in square array.

*Solution*:
(a) ( ) cos ( cos ) cos ( sin )*nE d d*φ β φ β φ= − (*ans.*)

24

**5-5-1. Field and phase patterns.
**Calculate and plot the field and phase patterns of an array of 2 nonisotropic dissimilar
sources for which the total field is given by

ψφφ ∠+= sincos*E *

where ( )1cos 2

cos +=+= φπδφψ *d *

Take source 1 as the reference for phase. See Fig. P5-5-1.

Figure P5-5-1. Field and phase patterns.

*Solution*:
See Figures 5-16 and 5-17.

**5-6-5. Twelve-source end-fire array.
**(a) Calculate and plot the field pattern of a linear end-fire array of 12 isotropic point
sources of equal amplitude spaced λ/4 apart for the ordinary end-fire condition.
(b) Calculate the directivity by graphical or numerical integration of the entire pattern.
Note that it is the power pattern (square of field pattern) which is to be integrated. It is
most convenient to make the array axis coincide with the polar or *z*-axis of Fig. 2-5 so
that the pattern is a function of θ.
(c) Calculate the directivity by the approximate half-power beamwidth method and
compare with that obtained in (b).
*
Solution*:
(b) 17 ( .)
(c) 10 ( .)

*D ans
D ans
*

= =

25

**5-6-7. Twelve-source end-fire with increased directivity.
**(a) Calculate and plot the pattern of a linear end-fire array of 12 isotropic point sources of
equal amplitude spaced λ/4 apart and phased to fulfill the Hansen and Woodyard
increased-directivity condition.
(b) Calculate the directivity by graphical or numerical integration of the entire pattern
and compare with the directivity obtained in Prob. 5-6-5 and 5-6-6.
(c) Calculate the directivity by the approximate half-power beamwidth method and
compare with that obtained in (b).
*Solution*:
(b) 26 ( .)
(c) 35 ( .)

*D ans
D ans
*

= =

**5-6-9. Directivity of ordinary end-fire array.
**Show that the directivity of an ordinary end-fire array may be expressed as

( ) ( )[ ] ( )λππλ *kdkknnd
nD n
*

*k
*

4sin21 1

1 ∑

−

=

−+ =

Note that ( )

( ) ( )∑ −

=

−+=⎥ ⎦

⎤ ⎢ ⎣

⎡ 1

1

2

2 2cos2

2
2sin *n
*

*k
*

*kknnn *ψ
ψ

ψ

*Solution*: Change of variable.
It is assumed that the array has a uniform spacing *d* between the isotropic sources. The
beam area

( ) ( )

2 2

2 0 0

sin 21 sin
sin 2*A
*

*n
d d
*

*n
*π π ψ

θ θ φ ψ

⎡ ⎤ Ω = ⎢ ⎥

⎣ ⎦ ∫ ∫

(1) where angle from array axisθ = . The pattern is not a function of φ so (1) reduces to

( ) ( )

2

2 0

sin 22 sin
sin 2*A
*

*n
d
*

*n
*π ψπ θ θ

ψ ⎡ ⎤

Ω = ⎢ ⎥ ⎣ ⎦

∫ (2)

where / 2 (cos 1)*d*λψ π θ= − (2.1)

26

**5-6-9. continued
**

Differentiating sin 2

*d d d*λ
ψ π θ θ= (3)

or 1sin 2

*d
d*λ

ψθ θ π

=

(4) and introducing (4) in (2)

( ) ( )

2 2

2 0

sin 22 sin 2 2

*d
*

*A
*

*n
d
*

*n d
*λπ

λ

ψ ψ ψ

⎡ ⎤ Ω = ⎢ ⎥

⎣ ⎦ ∫ (5)

Note new limits with change of variable from to / 2.θ ψ
When 0, / 2 0 and when , / 2 2 .*d*λθ ψ θ π ψ π= = = =

Since ( ) ( )

2 1

1

sin 2 2( )cos(2 / 2)

2

*n
*

*k
*

*n
n n k k
*

ψ ψ

ψ

−

=

⎡ ⎤ = + −⎢ ⎥

⎣ ⎦ ∑ (6)

(5) can be expressed 12

2 0 1

2 [ 2( ) cos(2 / 2)] 2

*nd
*

*A
k
*

*n n k k d
n d
*

λπ

λ

ψψ −

=

Ω = + −∑∫ (7)

Integrating (7) 21

2 1 0

2 2( ) sin(2 / 2) 2 2

*dn
*

*A
k
*

*n kn k
n d k
*

λπ

λ

ψ ψ −

=

−⎡ ⎤Ω = +⎢ ⎥⎣ ⎦ ∑ (8)

or 1

2 1

2 2 sin(4 )
*n
*

*A
k
*

*n knd kd
n d k*λ λλ

π π −

=

−⎡ ⎤Ω = +⎢ ⎥⎣ ⎦ ∑ (9)

and 2

1

1

24

2 sin(4 )
*n
*

*A
*

*k
*

*n dD
n knd kd
*

*k
*

λ

λ λ

ππ

π π −

=

= = −Ω + ∑

(10)

Therefore 1

1

1 sin(4 / ) 2

*n
*

*k
*

*nD
n k kd
*

*nd k
*λ π λ

π

−

=

= −

+ + ∑ q.e.d. (11)

We note that when / 4, or a multiple thereof, the summation term is zero and *d D n*λ= =
exactly.
This problem and the next one are excellent examples of integration with change of
variable and change of limits.

27

The final form for *D* in (11) above is well adapted for a computer program.

28

**5-6-10. Directivity of broadside array.
**Show that the directivity of a broadside array may be expressed as

( ) ( )[ ] ( )λππλ *kdkknnd
nD n
*

*k
*

2sin1 1

1 ∑

−

=

−+ =

*Solution*:

The solution is similar to that for Prob. 5-6-9 with cos 2

*d*λ
ψ π θ=

where 0, / 2 and when = , /2=*d d*λ λθ ψ π θ π ψ π= = − so that (8) of Prob. 5-6-9 becomes

1

2 1

2 ( ) sin(2 / 2) 2

*dn
*

*A
k d
*

*n kn k
n d k
*

λ

λ

π

λ π

ψ ψ −−

= +

−⎡ ⎤ Ω = +⎢ ⎥⎣ ⎦

∑

1

2 1

2 2 2 sin(2 )
*n
*

*A
k
*

*n knd kd
n d k*λ λλ

π π −

=

− −⎡ ⎤Ω = +⎢ ⎥⎣ ⎦ ∑

1

2 1

4 sin(2 )
*n
*

*A
k
*

*n knd kd
n d k*λ λλ

π π −

=

−⎡ ⎤Ω = +⎢ ⎥⎣ ⎦ ∑

2

1 1

1 1

4

sin(2 ) 1 sin(2 / )
*n n
*

*A
*

*k k
*

*n d nD
n k n knd kd kd
*

*k nd k
*

λ

λ λ

ππ λπ π π λ

π

− −

= =

= = = − −Ω ⎛ ⎞+ + ⎜ ⎟

⎝ ⎠ ∑ ∑

q.e.d.

Note that when / 2,*d *λ= or a multiple thereof, the summation term is zero and *D n*=
exactly.
See application of the above relations to the evaluation of *D* and of the main beam area

*A*Ω of an array of 16 point sources in Prob. 16-6-7 (c) and (d).

29

30

**Chapter 5. Arrays of Point Sources, Part II
**

**5-8-1. Three unequal sources.
**Three isotropic in-line sources have λ/4 spacing. The middle source has 3 times the
current of the end sources. If the phase of the middle source is 0°, the phase of one end
source +90° and phase of the other end source -90°, make a graph of the normalized field
pattern.
*
Solution*:
Phasor addition

**5-8-7. Stray factor and directive gain.
**The ratio of the main beam solid angle Ω*M* to (total) beam solid angle Ω*A* is called the
*main beam efficiency*. The ratio of the minor-lobe solid angle Ω*m* to the (total) beam
solid angle Ω*A* is called the *stray factor*. It follows that Ω*M*/Ω*A* + Ω*m*/Ω*A* = 1. Show that
the *average directivity gain* over the minor lobes of a highly directive antenna is nearly
equal to the stray factor. The directive gain is equal to the directivity multiplied by the
normalized power pattern [= *D Pn*(θ,φ)], making it a function of angle with the maximum
value equal to *D*.
*Solution*:

Stray factor = *m
A
*

Ω Ω

En 0.6 North

0.6 South

0.2 East

1.0 West

0.24 North-East

0.96 North-West

31

**5-8-7. continued
**where total beam area

main lobe beam area minor lobe beam area

*A
*

*M
*

*m
*

Ω = Ω = Ω =

4

4

( , )

( , )
*M
*

*n
*

*m
*

*A n
*

*P d
*

*P d
*π

π

θ φ

θ φ −Ω

Ω Ω

= Ω Ω

∫∫

∫∫

4

1Average directive gain over minor lobes = (minor) = ( , ) 4

*M
*

*av n
M
*

*DG DP d
*π

θ φ π −Ω

Ω − Ω ∫∫

where 4 / *AD *π= Ω

Therefore 4 4 ( , )

1 4 (minor) = 4 4

*M
*

*n
*

*m
av
*

*M A M A
*

*P d
DG *π

π θ φ π

π π −Ω

Ω Ω

= − Ω Ω − Ω Ω

∫∫

If 4*M *πΩ << (antenna highly directive),

(minor) *mav
A
*

*DG *Ω≅
Ω

(stray factor) q.e.d.

***5-9-2. Three-source array.
**The center source of a 3-source array has a (current) amplitude of unity. For a sidelobe
level 0.1 of the main lobe maximum field, find the Dolph-Tchebyscheff value of the
amplitude of the end sources. The source spacing *d* = λ/2.
*Solution*: Let the amplitudes (currents) of the 3 sources be as in the sketch

/ 2, 10*d R*λ= =
Let amplitude of center source o1 2*A*= =

2 2

2 2 o o

2 o o

1 2, ( ) 2 1

2 1 10 2 11

5.5 2.345

*n T x x R
*

*x x
*

*x x
*

− = = − =

− = =

= = ±

1*A *O2*A *1*A
*

/ 2λ/ 2λ

32

**5-9-2. continued
**

2 2 3 o 1 o 1 o 12 2 cos 2 2 2 (2cos 1) 2 2 (2 1)2 2

*E A A A A A A w*ψ ψ= + = + − = + −

Let o/*w x x*= so
2

2 21 3 o 1 o 12 2

o o

2 2 (2 1) 2 4 2 2 1*AxE A A A x A x
x x
*

= + − = + − = −

2 2
3 1 o 10.728 ( 2 ) 2 1*E A x A A x*= + − = −

1 1

o 1 o

Therefore, 0.728 2 and 2.75 2 2 1 and 2 5.5 1 4.5

*A A
A A A
*

= = − = − = − =

Thus, normalizing o 12 1 and 2.75 4.5 0.61*A A*= = = (*ans.*)
Amplitude distribution is 0.61 1.00 0.61
Pattern has 4 minor lobes. For center source, amplitude 1=
The side source amplitudes for different *R* values are:

*R *8 10 12 15
*A*1 0.64 0.61 0.59 0.57

**5-9-4. Eight source D-T distribution.
**(a) Find the Dolph-Tchebyscheff current distribution for the minimum beam width of a
linear in-phase broadside array of eight isotropic sources. The spacing between the
elements is λ/4 and the sidelobe level is to be 40 dB down. Take φ = 0 in the broadside
direction.
(b) Locate the nulls and the maxima of the minor lobes.
(c) Plot, approximately, the normalized field pattern (0° ≤ φ ≤ 360°).
(d) What is the half-power beam width?
*
Solution*:
(a) 0.14, 0.42, 0.75, 1.00, 1.00, 0.75, 0.42, 0.14
(b) Max. at:
±21o, ±27o, ±36o, ±48o, ±61o, ±84o, ±96o, ±119o, ±132o, ±144o, ±153o, ±159o
Nulls at:
±18o, ±23o, ±32o, ±42o, ±54o, ±71o, ±109o, ±126o, ±138o, ±148o, ±157o, ±162o
(d) HPBW o12= (*ans.*)

33

***5-18-1. Two sources in phase.
**Two isotropic point sources of equal amplitude and same phase are spaced 2λ apart. (a)
Plot a graph of the field pattern. (b) Tabulate the angles for maxima and nulls.
*
Solution*:
(a) Power pattern 2*n nP E*=
*
In
Instructional comment to pass on to students:
*
The lobes with narrowest beam widths are broadside (±90o), while the widest beam width
lobes are end-fire (0o and 180o). The four lobes between broadside and end-fire are
intermediate in beam width. In three dimensions the pattern is a figure-of-revolution
around the array axis (0o and 180o axis) so that the broadside beam is a flat disk, the end-
fire lobes are thick cigars, while the intermediate lobes are cones. The accompanying
figure is simply a cross section of the three-dimensional space figure.

34

**5-18-2. Two sources in opposite phase.
**Two isotropic sources of equal amplitude and opposite phase have 1.5λ spacing. Find
the angles for all maxima and nulls.
*Solution*:
Maximum at: 0o, 180o, ±70.5o, ±109.5o, Nulls at: ±48.2o, ±90o, ±131.8o

35

36

**Chapter 6. The Electric Dipole and Thin Linear Antennas
**

***6-2-1. Electric dipole.
**(a) Two equal static electric charges of opposite sign separated by a distance *L* constitute
a static electric dipole. Show that the electric potential at a distance *r* from such a dipole
is given by

24 cos

*r
QLV
*

πε θ

=

where *Q* is the magnitude of each charge and θ is the angle between the radius *r* and the
line joining the charges (axis of dipole). It is assumed that *r* is very large compared to *L.*
(b) Find the vector value of the electric field **E** at a large distance from a static electric
dipole by taking the gradient of the potential expression in part (a).
*Solution*:

1 2

1 2

(a) (at ) , 4 4

( / 2)cos , ( / 2)cos

*Q QV r
r r
*

*r r L r r L
*πε πε

θ θ

= −

= − = +

2 2 2

2

1 1 4 ( / 2)cos ( / 2)cos

( / 2)cos ( / 2)cos 4 ( / 2) cos

cosFor , q.e.d. 4

*QV
r L r L
*

*Q r L r L
r L
*

*QLr L V
r
*

πε θ θ

θ θ πε θ

θ πε

⎛ ⎞ = −⎜ ⎟− +⎝ ⎠

⎛ ⎞+ − + = ⎜ ⎟+⎝ ⎠

>> =

3 2

3 3

1 1 2cos 1 sinˆ ˆˆ ˆ ˆ ˆ(b) 0 sin 4

cos sinˆˆ 2 4

*V V V QLV
r r r r r r
*

*QL QL
r r
*

θ θ θ θ φ πε

θ θ πε πε

∂ ∂ ∂ − ⎛ ⎞= −∇ = + + = − − +⎜ ⎟∂ ∂ ∂ ⎝ ⎠

= +

**E r θ φ r θ φ
**

**r θ
**

or 3 3 cos sin, , 0

2 4*r
QL QLE E E
*

*r r*θ φ
θ θ

πε πε
= = = (*ans.*)

***6-2-2. Short dipole fields.
**A dipole antenna of length 5 cm is operated at a frequency of 100 MHz with terminal
current *Io *= 120 mA. At time *t* = 1 s, angle θ = 45°, and distance *r* = 3 m, find (a) *Er*, (b)
*E*θ, and (c) *H*φ.

θ

θ
/ 2*L
*

/ 2*L
*

cos
2
*L *θ

cos
2
*L *θ

*L
*

*Q*+

*Q*−

1*r
*

*r
*

2*r*

37

***6-2-2. continued***Solution*:
(a) From (6-2-12)

( ) 6

6 8

( ) o

2 3 o

(2 )100 102 100 10 (1) (3) 3 10

3 o 12 8 2 6 3

2 3 2 o

cos 1 1 2

1 1(120 10 )(0.05) cos 45 2 (8.85 10 ) 3 10 (3 ) (2 )100 10 (3 )

2.83 10 (4.5 10 ) 2.86 10 9 V

*j t r
*

*r
*

*j
*

*I leE
cr j r
*

*e
j
*

*j
*

ω β

ππ

θ πε ω

π π

−

⎡ ⎤⎛ ⎞× × −⎢ ⎥⎜ ⎟⎜ ⎟×⎢ ⎥⎝ ⎠⎣ ⎦

− −

− − −

⎛ ⎞ = +⎜ ⎟

⎝ ⎠

⎛ ⎞ = × +⎜ ⎟× × ×⎝ ⎠

__= × − × = × −∠ /m ( .)*ans
*
(b) From (6-2-13)

( ) 2 2o

2 2 3 o

2 o

sin 1 1 1.41 10 (8.65 10 ) 4

8.77 10 81 V/m ( .)

*j t rI le jE j
c r cr j r
*

*ans
*

ω β

θ θ ω

πε ω

− − −

−

⎛ ⎞ = + + = × + ×⎜ ⎟

⎝ ⎠ __= × ∠

(c) From (6-2-15)

( ) 5 4o

2

4 o

sin 1 3.75 10 (2.36 10 ) 4

2.39 10 81 A/m ( .)

*j t rI le jH j
cr r
*

*ans
*

ω β

φ θ ω

π

− − −

−

⎛ ⎞= + = × + ×⎜ ⎟ ⎝ ⎠

__= × ∠

***6-2-4. Short dipole quasi-stationary fields.
**For the dipole antenna of Prob. 6-2-2, at a distance *r* = 1 m, use the general expressions
of Table 6-1 to find (a) *Er*, (b) *E*θ, and (c) *H*φ. Compare these results to those obtained
using the quasi-stationary expressions of Table 6-1.
*Solution*:
Using the same approach for , , and *rE E H*θ φ as in solution to Prob. 6-2-2, we find for

1 m,*r *=
282 mV/m
242 mV/m
784 mA/m

*rE
E
H
*

θ

φ

= = =

38

***6-2-4. continued
**Using quasi-stationary equations,

3 o 3o o

3 3 2 6 12 3 o o

3o o 3 3

o o

o 2

cos cos 120 10 (0.05cos 45 ) 0 (121 10 ) 2 (2 ) (2 ) (100 10 )(8.85 10 )1

121 mVm ( .)

sin sin 0 (61 10 ) 61 mV/m ( .) 4 (4 )

sin 3.38 1 4

*r
q l I lE j
*

*r j r j
ans
*

*q l I lE j ans
r j r
*

*I lH
r
*

θ

φ

θ θ πε ω πε π

θ θ πε ω πε

θ π

− −

−

−

× = = = = − ×

× × =

= = = − × =

= = × 40 338 μA/m ( .)*ans*− =

***6-3-1. Isotropic antenna. Radiation resistance.
**An omnidirectional (isotropic) antenna has a field pattern given by *E* = 10*I/r* (V m-1),
where *I* = terminal current (A) and *r* = distance (m). Find the radiation resistance.
*Solution*:

2 2

2

10 100 so *I E IE S
r Z r Z
*

= = =

Let 2 power over sphere 4 ,*P r S*π= = which must equal power 2*I R * to the antenna
terminals. Therefore 2 24 and*I R r S*π=

2 2

2 2

1 100 4004 3.33 120 120

*IR r
I r
*

π π

= = = Ω (*ans.*)

***6-3-2. Short dipole power.
**(a) Find the power radiated by a 10 cm dipole antenna operated at 50 MHz with an
average current of 5 mA. (b) How much (average) current would be needed to radiate
power of 1 W?
*Solution*:
(a)

26 3

82 6o

o

(2 )50 10 (5 10 )0.1 3 10( ) 377 2.74 10 W 2.74 μW

12 12
*avI lP
*

π μ β ε π π

−

−

⎛ ⎞×
×⎜ ⎟×⎝ ⎠= = = × = (*ans.*)

39

***6-3-2. continued
**

(b) 1 2

3 6

1For 1 W, 5 10 3.0 A
2.7 10*av
*

*P I *− −
⎛ ⎞= = × =⎜ ⎟×⎝ ⎠

(*ans.*)

**6-3-4. Short dipole.
**For a thin center-fed dipole λ/15 long find (a) directivity *D*, (b) gain *G*, (c) effective
aperture *Ae*, (d) beam solid ΩA and (e) radiation resistance *Rr*. The antenna current tapers
linearly from its value at the terminals to zero at its ends. The loss resistance is 1 Ω.
*Solution*:
(a) ( ) sin*nE *θ θ=

22 2 3

0 0 04

4 4 4 4 sin sin sin 2 sin

4 3 1.5 or 1.76 dBi ( .)4 22 3

*A
*

*D
d d d d
*

*ans
*

π π π

π

π π π π θ θ θ θ π θ θ

π

π

= = = = Ω Ω Ω

= = =

∫∫ ∫ ∫ ∫

(d) From (a), 8 / 3 8.38 sr*A *πΩ = = (*ans.*)

(e) 2 2 2

2

o

1 1From (6-3-14), 790 790 0.878 2 15

*av
r
*

*IR L
I *λ

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = = Ω⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎝ ⎠

(*ans.*)

(b) 0.878 1.5 0.70 or 1.54 dBi 0.878 1

*G kD*= = × = −
+

(*ans.*)

(c) 2

23 where
8*e em em A
*

*A kA A *λ λ
π

= = = Ω

20.878 3Therefore 0.058
0.878 1 8*e
*

*A *λ
π

= × = +

(*ans.*)

***6-3-5. Conical pattern.
**An antenna has a conical field pattern with uniform field for zenith angles (θ) from 0 to
60° and zero field from 60 to 180°. Find exactly (a) the beam solid angle and (b)
directivity. The pattern is independent of the azimuth angle (φ).*
Solution*:

(a) oo o o 60360 60 60

0 0 0 0
2 sin 2 cos sr*A d d*π θ θ π θ πΩ = Ω = = − =∫ ∫ ∫ (*ans.*)

40

***6-3-5. continued
**

(b) 4 4 4
*A
*

*D *π π
π

= = = Ω

(*ans.*)

**6-3-6. Conical pattern.
**An antenna has a conical field pattern with uniform filed for zenith angles (θ) from 0 to
45° and zero field from 45 to 180°. Find exactly (a) the beam solid angle, (b) directivity
and (c) effective aperture. (d) Find the radiation resistance if the *E* = 5 V m-1 at a
distance of 50 m for a terminal current *I* = 2 A (rms). The pattern is independent of the
azimuth angle (φ)*.**Solution*:

(a) o45

0
2 sin 1.84 sr*A d*π θ θΩ = =∫ (*ans.*)

(b) 4 4 6.83
1.84*A
*

*D *π π= = =
Ω

(*ans.*)

(c) 2 2

20.543
1.84*e em A
*

*A A *λ λ λ= = = =
Ω

(*ans.*)

(d) 2 2

2 2 2 2

1 5, 1.84 50 76.3
2 377*r A r
*

*EI R r R
Z
*

= Ω = × = Ω (*ans.*)

***6-3-7. Directional pattern in **θ **and** φ.
An antenna has a uniform field pattern for zenith angles (θ) between 45 and 90° and for
azimuth (φ) angles between 0 and 120°. If *E* = 3 V m-1 at a distance of 500 m from the
antenna and the terminal current is 5 A, find the radiation resistance of the antenna. *E* =
0 except within the angles given above.
*
Solution*:

(a) o o o

o o

120 90 90

0 45 45

2sin cos 1.48 sr
3*A
*

*d d *πθ θ φ θΩ = = − =∫ ∫ (*ans.*)

(c) 2 2

2 2 2 2

1 1 31.48 500 354
5 377*r A
*

*ER r
I Z
*

= Ω = × = Ω (*ans.*)

41

***6-3-8. Directional pattern in **θ **and** φ.
An antenna has a uniform field *E* = 2 V m-1 (rms) at a distance of 100 m for zenith angles
between 30 and 60° and azimuth angles φ between 0 and 90° with *E* = 0 elsewhere. The
antenna terminal current is 3 A (rms). Find (a) directivity, (b) effective aperture and (c)
radiation resistance.
*Solution*:

o o o

o o

90 60 60

0 30 30 (a) sin cos 0.575 sr ( .)

2 4 21.9 ( .)

0.575

*A d d ans
*

*D ans
*

πθ θ φ θ

π

Ω = = − =

= =

∫ ∫

(b) 2 2

21.74
0.575*e em A
*

*A A *λ λ λ= = = =
Ω

(*ans.*)

(c) 2 2

2 2 2 2

1 1 20.575 100 6.78
3 377*r A
*

*ER r
I Z
*

= Ω = × = Ω (*ans.*)

***6-3-9. Directional pattern with back lobe.
**The field pattern of an antenna varies with zenith angle (θ) as follows: *En* (=*Enormalized*) =
1 between θ = 0° and θ = 30° (main lobe), *En* = 0 between θ = 30° and θ = 90° and *En* =
1/3 between θ = 90° and θ = 180° (back lobe). The pattern is independent of azimuth
angle (φ). (a) Find the exact directivity. (b) If the field equals 8 V m-1 (rms) for θ = 0° at
a distance of 200 m with a terminal current *I* = 4 A (rms), find the radiation resistance.
*Solution*:

(a) o o

o

30 180

20 90

22 sin sin 2 (0.134 0.111) 2 (0.245)
3*A
*

*d d*ππ θ θ θ θ π πΩ = + = + =∫ ∫

4 8.16 2 (0.245)

*D *π
π

= = (*ans.*)

(b) 2 2

2 2 2 2

1 1 82 (0.245)200 653
4 120*r A
*

*ER r
I Z
*

π π

= Ω = = Ω (*ans.*)

**6-3-10. Short dipole.
**The radiated field of a short-dipole antenna with uniform current is given by

( )30 sin*E l I r*β θ= , where *l* = length, *I *= current, *r* = distance and θ = pattern angle.
Find the radiation resistance.

42

**6-3-10. continued
***Solution*:

The current *I* given in the problem is a peak value, so we put 2 2
Power 4 Power
input radiated

1
2 *r
*

*I R Sr d
*π

= Ω∫∫

where 2

and is as given*ES E
Z
*

=

so 2 2 2 2

2 3 2 2 2 2 2 0

2 302 sin 80 ( / ) 790( / )
120*r
*

*l IR r d l l
I r
*

πβπ θ θ π λ λ π

= = = Ω∫ (*ans.*)

**6-3-11. Relation of radiation resistance to beam area.
**Show that the radiation resistance of an antenna is a function of its beam area Ω*A* as given
by

*Ar I
SrR *Ω= 2

2

where *S* = Poynting vector at distance *r* in direction of pattern maximum
*I* = terminal current.
*Solution*:

Taking *I* as the rms value we set 2 2
Power Power
input radiated

*r AI R Sr*= Ω , therefore
2

2*r A
SrR
I
*

= Ω q.e.d.

***6-3-12. Radiation resistance.
**An antenna measured at a distance of 500 m is found to have a far-field pattern of *|E| =
Eo*(sinθ)1.5 with no φ dependence. If *Eo* = 1 V/m and *Io* = 650 mA, find the radiation
resistance of this antenna.
*Solution*:
From (6-3-5)

2 2 2

2 2 2 2 0 0 o o

3 4

0

0

120 120 sin sin (0.65) (377)

(6.28 10 )(500)2 sin

3 sin(2 ) sin(4 ) 319.7 19.7 23.2 ( .) 8 4 32 8

*r
s
*

*E
R ds r d d
*

*I Z
*

*d
*

*x x x ans
*

π π

π

π

π π θ θ θ φ

π θ θ

π

−

= =

= ×

⎡ ⎤ ⎛ ⎞= − + = = Ω⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠

∫ ∫ ∫

∫

43

***6-5-1. **λ**/2 antenna.
**Assume that the current is of uniform magnitude and in-phase along the entire length of a
λ*/2* thin linear element.
(a) Calculate and plot the pattern of the far field.
(b) What is the radiation resistance?
(c) Tabulate for comparison:

1. Radiation resistance of part (b) above
2. Radiation resistance at the current loop of a λ*/2* thin linear element with

sinusoidal in-phase current distribution
3. Radiation resistance of a λ*/2* dipole calculated by means of the short dipole

formula
(d) Discuss the three results tabulated in part (c) and give reasons for the differences.
*Solution*:
(a) ( ) tan sin[( / 2)cos ]*nE *θ θ π θ= (*ans.*)
(b) 168 *R *= Ω (*ans.*)
(c) [from (b)] = 168 ( .) *R ans*Ω

(sinusoidal ) 73 ( .) (short dipole) = 197 ( .)

*R I ans
R ans
*

= Ω Ω

(d) 168 Ω is appropriate for uniform current. 73 Ω is appropriate for sinusoidal current.

197 Ω assumes uniform current, but the short dipole formula does not take into
account the difference in distance to different parts of the dipole (assumes >>*L*λ )
which is not appropriate and leads to a larger resistance (197 Ω) as compared to the
correct value of 168 Ω.

**6-6-1. 2**λ** antenna.
**The instantaneous current distribution of a thin linear center-fed antenna *2*λlong is
sinusoidal as shown in Fig. P6-6-1.
(a) Calculate and plot the pattern of the far field.
(b) What is the radiation resistance referred to a current loop?
(c) What is the radiation resistance at the transmission-line terminals as shown?
(d) What is the radiation resistance λ*/8* from a current loop?

44

**6-6-1. continued
**

Figure P6-6-1**.***2*λantenna.
*Solution*:

(a) cos(2 cos ) 1( )
sin*n
*

*E *π θθ
θ

−
= (*ans.*)

(b) max(at ) 259 *R I *= Ω (*ans.*)
(c) (at terminals) *R *= ∞ Ω (*ans.*)
(d) max(at /8 from ) 518 *R I*λ = Ω (*ans.*)

**6-7-1. **λ**/2 antennas in echelon.
**Calculate and plot the radiation-field pattern in the plane of two thin linear λ/2 antennas
with equal in-phase currents and the spacing relationship shown in Fig. P6-7-1. Assume
sinusoidal current distributions.

Figure P6-7-1. λ/2 antennas in echelon.

*Solution*:
cos[( / 2)cos ] 2( ) cos cos[( / 4) ]

sin 4*n
E *π θ πθ π θ

θ ⎛ ⎞

= +⎜ ⎟⎜ ⎟ ⎝ ⎠

45

***6-8-1. 1**λ** and 10**λ** antennas with traveling waves.
**(a) Calculate and plot the far-field pattern in the plane of a thin linear element 1λ long,
carrying a single uniform traveling wave for 2 cases of the relative phase velocity *p* = 1
and 0.5. (b) Repeat for the single case of an element 10λ long and *p* = 1.
*Solution*:

(a) From (6-8-5), sin 1( ) [sin ( cos )]
1 cos*n
*

*E
p p
*

φφ π φ φ

= − −

, patterns have 4 lobes.

(b) Pattern has 40 lobes.

**6-8-2. Equivalence of pattern factors.
**Show that the field pattern of an ordinary end-fire array of a large number of collinear
short dipoles as given by Eq. (5-6-8), multiplied by the dipole pattern sin φ, is equivalent
to Eq. (6-8-5) for a long linear conductor with traveling wave for *p* = 1.
*Solution*:

(1)

nsin 2Field pattern=

sin 2

ψ

ψ (5-6-8)

where cos*d*ψ β φ δ= +

(2) Field pattern = sin[ (1 cos )]

2sin 1 cos

*b p
pc
*

*p
*

ω φ φ

φ

−

− (6-8-5)

For ordinary end-fire, (cos 1)*d*ψ β φ= −

Also if *d* is small (1) becomes
sin (1 cos )

2

(1 cos ) 2

*nd
*

*d
*

β φ

β φ

⎛ ⎞−⎜ ⎟ ⎝ ⎠

−

For larger , .*n nd b*≅ Also multiplying by the source factor sinφ and taking the con-
stant / 2 1*d*β = in the denominator, (1) becomes

sin (1 cos )

2sin 1 cos

*d*β φ
φ

φ

⎛ ⎞−⎜ ⎟ ⎝ ⎠

−

46

**6-8-2. continued
**which is the same as (2) for 1*p *=

since 2 2 2 2

*b fb b
pc f
*

ω π β λ

= = q.e.d.

Note that for a given length *b*, the number *n* is assumed to be sufficiently large that *d* can
be small enough to allow sin / 2ψ in (1) to be replaced by / 2ψ .

47

48

**Chapter 7. The Loop Antenna
**

**7-2-1. Loop and dipole for circular polarization.
**If a short electric dipole antenna is mounted inside a small loop antenna (on polar axis,
Fig. 7-3) and both dipole and loop are fed in phase with equal power, show that the
radiation is everywhere circularly polarized with a pattern as in Fig. 7-7 for the 0.1λ
diameter loop.
*Solution*:
Uniform currents are assumed.

2

2

120 sin( )(loop)= *IAE
r*φ

π θθ λ

(1)

j60 sin( )(dipole)= *ILE
r*θ

π θθ λ

(2)

2

4
2(loop)=320*r
*

*AR *π
λ

⎛ ⎞ Ω⎜ ⎟ ⎝ ⎠

(3)

2 2(dipole)=80*rR L*λπ (4)
For equal power inputs,

2 2
loop dipole(loop) (dipole)*r rI R I R*=

2 2 2 2 loop 2 4 2 2 2 2 2 dipole

80(dipole) (loop) 320 ( / ) 4 ( / )

*r
*

*r
*

*I L LR
I R A A
*

λ λπ π λ π λ

= = = (5)

loop 2
dipole 2 ( / )
*I L
I A
*

λ

π λ = (6)

Therefore

2

dipole dipole 2 2

120 sin 60 sin ( )(loop)=

2 ( / )
*L I A I L
*

*E
r A r
*

λ φ

π θ π θ θ

λ π λ λ = (7)

which is equal in magnitude to ( )*E*θ θ (dipole) but in time-phase quadrature (no *j*).
Since the 2 linearly polarized fields ( *E*φ of the loop and *E*θ of the dipole) are at right
angles, are equal in magnitude and are in time-phase quadrature, the total field of the
loop-dipole combination is everywhere circularly polarized with a sinθ pattern. q.e.d.

49

**7-2-1. continued
**
Equating the magnitude of (1) and (2) (fields equal and currents equal) we obtain

22
*L A*π
λ λ

= (8)

which satisfies (6) for equal loop and dipole currents. Thus (8) is a condition for circular
polarization.
Substituting 2( / 4)*A d*π= , where *d *= loop diameter in (8) and putting *C d*π=

2 2

2 2

12 4 2

*L d C*ππ
λ λ λ

= = (9)

we obtain 1 2(2 )*C L*λ λ= (10)
as another expression of the condition for circular polarization.
Thus, for a short dipole /10λ long, the loop circumference must be
1 2(2 0.1) 0.45*C*λ = × =
(11)

and the loop diameter 0.45 0.14*d *λ λ
π

= =

or 1.4 times the dipole length. If the dipole current tapers to zero at the ends of the dipole, the condition for CP is

24
*L A*π
λ λ

= (12)

and
1 2( )*C L*λ λ=
(13)
For a /10λ dipole the circumference must now be 1 2(0.1) 0.316*C*λ = = and the loop

diameter 0.316 0.1*d *λ λ
π

= ≅ or approximately the same as the dipole length.

The condition of (10) is applied in the Wheeler-type helical antenna. See Section 8-22, equation (8-22-4) and Prob. 8-11-1.

50

**7-4-1. The 3**λ**/4 diameter loop.
**Calculate and plot the far-field pattern normal to the plane of a circular loop 3λ/4 in
diameter with a uniform in-phase current distribution.
*Solution*:

3 2.36 4

*C*λ π= =

From (7-3-8) or Table 7-2, the *E*φ pattern is given by

1( sin )*J C*λ θ
See Figure 7-6.

***7-6-1. Radiation resistance of loop.
**What is the radiation resistance of the loop of Prob. 7-4-1?
*Solution*:
From (7-6-13) for loop of any size

22 20

60 ( )
*C
*

*rR C J y dy
*λ

λπ= ∫
where 3 4 2.36, 2 4.71*C C*λ λπ= = =

From (7-6-16), 2 2

2 o 10 0 ( ) ( ) 2 (2 )

*C C
J y dy J y dy J C*λ λ λ= −∫ ∫

By integration of the o ( )*J y * curve from 0 to 2 ( 4.71)*C*λ = ,
2

o0 ( ) 0.792

*C
J y dy*λ =∫

From tables (Jahnke and Emde), 1 1(2 ) (4.71) 0.2816*J C J*λ = = −

and 2

20 ( ) 0.7920 2 0.2816 1.355

*C
J y dy*λ == + × =∫

Therefore 260 2.36 1.355 1894 (Round off to 1890 )*rR *π= × = Ω Ω (*ans.*)

51

**7-6-2. Small-loop resistance.
**(a) Using a Poynting vector integration, show that the radiation resistance of a small loop
is equal to ( ) Ω320 224 λπ *A * where *A* = area of loop (m2). (b) Show that the effective
aperture of an isotropic antenna equals λ2/4π.
*Solution*:

(a) 2 22 max

2 2
*AA
*

*r
E rSrR
*

*I ZI
*ΩΩ

= =

From (7-5-2) and Table 7-2,

2

max2

120 sin sin*IAE E
r*φ
π θ θ
λ

= =

2

0

4 82 sin sin 2
3 3*A
*

*d
*π

π θ θ θ π πΩ = = =∫

Therefore, 22 4 2 2 2

4 4 2 4 2 2

120 8 320 197
120 3*r
*

*I A r AR C
r I *λ
π π π

λ π λ ⎛ ⎞= = Ω = Ω⎜ ⎟ ⎝ ⎠

q.e.d.

(b) 2 2

4 ,
4*A e A
*

*A *λ λπ
π

Ω = = = Ω

q.e.d.

**7-7-1. The **λ**/10 diameter loop.
**What is the maximum effective aperture of a thin loop antenna 0.1λ in diameter with a
uniform in-phase current distribution?
*Solution*:

*A*Ω is the same as for a short dipole ( 8 / 3 sr).π= See Prob. 6-3-4a.

Therefore, 2

2 23 0.119
8*em A
*

*A *λ λ λ
π

⎛ ⎞= = =⎜ ⎟Ω ⎝ ⎠
(*ans.*)

**7-8-1. Pattern, radiation resistance and directivity of loops.
**A circular loop antenna with uniform in-phase current has a diameter *d*. What is (a) the
far-field pattern (calculate and plot), (b) the radiation resistance and (c) the directivity for
each of three cases where (1) *d* = λ/4, (2) *d* = 1.5λ and (3) *d* = 8λ?

52

**7-8-1. continued***Solution*:
Since all the loops have 1/ 3,*C*λ > the general expression for *E*φ in Table 7-2 must be
used. From Table 7-2 and Figures 7-10 and 7-11, the radiation resistance and directivity
values are:

Diameter *C*λ *rR * Directivity

λ/4 0.785 76 Ω 1.5

1.5λ 4.71 2340 Ω 3.82

8λ 25.1 14800 Ω 17.1

***7-8-2. Circular loop.
**A circular loop antenna with uniform in-phase current has a diameter *d*. Find (a) the far-
field pattern (calculate and plot), (b) the radiation resistance and (c) the directivity for the
following three cases: (1) *d* = λ/3, (2) *d* = 0.75λ and (3) *d* = 2λ.
*Solution*:
See Probs. 7-4-1 and 7-8-1. Radiation resistance and directivity values are:

Diameter *C*λ *rR * Directivity

λ/3 1.05 180 Ω 1.5

0.75λ 2.36 1550 Ω 1.2

2λ 6.28 4100 Ω 3.6

***7-9-1. The 1**λ** square loop.
**Calculate and plot the far-field pattern in a plane normal to the plane of a square loop and
parallel to one side. The loop is 1λ on a side. Assume uniform in-phase currents.

53

***7-9-1. continued***Solution*:
Pattern is that of 2 point sources in opposite phase. Referring to Case 2 of Section 5-2,
we have for / 2 2 ( / 2) ,*rd *π λ π= =

( ) sin( cos )*nE *φ π φ=
resulting in a 4-lobed pattern with maxima at o o60 and 120φ = ± ± and nulls at

o o o0 , 90 and 180 .±

**7-9-2. Small square loop.
**Resolving the small square loop with uniform current into four short dipoles, show that
the far-field pattern in the plane of the loop is a circle.
*Solution*:

The field pattern (1,2)*E * of sides 1 and 2 of the small square loop is the product of the
pattern of 2 point sources in opposite phase separated by *d* as given by

sin[( / 2)cos ]*rd *φ
and the pattern of short dipole as given by cosφ
or (1, 2) cos sin[( / 2)cos ]*rE d*φ φ=
For small *d* this reduces to 2(1, 2) cos*nE *φ=
The pattern of sides 3 and 4 is the same rotated through 90o or in terms of φ is given by

2(3,4) sin*nE *φ=
The total pattern in the plane of the square loop is then

2 2( ) (1,2) (3,4) cos sin 1*n n nE E E*φ φ φ= + = + =
Therefore ( )*E *φ is a constant as a function of φ and the pattern is a circle. q.e.d.

54

**Chapter 8. End-Fire Antennas: The Helical Beam Antenna and
the Yagi-Uda Array, Part I
**

**8-3-1. A 10-turn helix.
**A right-handed monofilar helical antenna has 10 turns, 100 mm diameter and 70 mm turn
spacing. The frequency is 1 GHz. (a) What is the HPBW? (b) What is the gain? (c)
What is the polarization state? (d) Repeat the problem for a frequency of 300 MHz.
*Solution*:

8

9

3 10(a) 0.3 m (0.1) 0.314 10 0.314 0.07 1.047 0.233

0.3 0.3

*C
*

*C S*λ λ

λ π×= = = =

= = = =

From (8-3-4)

O O o

1 2 1 2

52 52HPBW 32.5
( ) 1.047(10 0.233)*C nS*λ λ

= = = ×

(*ans.*)

(b) From (8-3-7), 212 30.7 or 14.9 dBi*D C nS*λ λ≅ = (*ans.*)
If losses are negligible the gain = *D*.
(c) Polarization is RCP. (*ans.*)
(d) At 300 MHz, 8 63 10 / 300 10 1 mλ = × × = , 0.314 /1 0.314.*C*λ = =
This is too small for the axial mode which requires that 0.7 1.4.*C*λ< <

From Table A-1, 2 41000 38.8 or 15.9 dBi 32.5

*D *= = or 1 dB higher.

The lower value is more realistic

**8-3-2. A 30-turn helix.
**A right-handed monofilar axial-mode helical antenna has 30 turns, λ/3 diameter and λ/5
turn spacing. Find (a) HPBW, (b) gain and (c) polarization state.
*Solution*:

(a) From (8-3-4), o o

o 1 2

1 2

52 52HPBW 20.3 ( ) (30 0.2)

3
*C nS*λ λ π

≅ = = ×

(*ans.*)

(b) For zero losses, *G D*=

55

**8-3-2. continued
**
From (8-3-7), 2 212 12( / 3) 30 0.2 79 or 19 dBi*D C nS*λ λ π≅ = × = (*ans.*)
(c) RCP (*ans.*)

**8-3-3. Helices, left and right.
**Two monofilar axial-mode helical antennas are mounted side-by-side with axes parallel
(in the *x* direction). The antennas are identical except that one is wound left-handed and
the other right-handed. What is the polarization state in the *x* direction if the two
antennas are fed (a) in phase and (b) in opposite phase?
*Solution*:
Assuming that *x *is horizontal, (a) LHP (*ans.*) (b) LVP (*ans.*)

56

**Chapter 8. The Helical Antenna: Axial and Other Modes, Part II
**

***8-8-1. An 8-turn helix.
**A monofilar helical antenna has α = 12°, *n* = 8, *D* = 225 mm. (a) What is *p* at 400 MHz
for (1) in-phase fields and (2) increased directivity? (b) Calculate and plot the field
patterns for *p* = 1.0, 0.9, and 0.5 and also for *p* equal to the value for in-phase fields and
increased directivity. Assume each turn is an isotropic point source. (c) Repeat (b)
assuming each turn has a cosine pattern.
*Solution*:
(a) The relative phase velocity for in-phase fields is given by (8-8-9) as

1 cossin

*p
*

*C*λ
αα

= +

The relative phase velocity for increased directivity is given by (8-8-12)

2 1 2

*Lp nS
n
*

λ

λ

= +

+

From the given value of frequency and diameter , *D C*λ can be determined. Introducing
it and the given values of and *n*α

0.802 for in-phase fields 0.763 for increased directivity

*p
p
*

= =

***8-11-1. Normal-mode helix.
**(a) What is the approximate relation required between the diameter *D* and height *H* of an
antenna having the configuration shown in Fig. P8-11-1, in order to obtain a circularly
polarized far-field at all points at which the field is not zero. The loop is circular and is
horizontal, and the linear conductor of length *H* is vertical. Assume *D* and *H* are small
compared to the wavelength, and assume the current is of uniform magnitude and in
phase over the system.
(b) What is the pattern of the far circularly polarized field?

57

***8-11-1. continued
**

Figure P8-5-3. Normal mode helix.
*Solution*:
See solution to Prob. 7-2-1.
(a) 1 2(2 ) /*D H*λ λ π= (*ans.*)
(b) sin*E *θ= (*ans.*)

**8-15-1. Design of quad-helix earth station antenna.
**An array of four right-handed axial-mode helical antennas, shown in Fig. 8-54, can be
used for communications with satellites. Determine (a) the best spacing based on the
effective apertures of the helixes, (b) the directivity of the array. Assume the number of
turns is 20 and the spacing between turns is 0.25 λ.
*Solution*:
(a) From (8-3-7) the directivity of each helix is

212 (1.05) 20 0.25 66.15*D *≅ × × × =
2

266 5.26
4*e
*

*A *λ λ
π

= =

The spacing is then 5.26 2.29λ λ= (b) At 2.29λ spacing the effective aperture for the array is 25.26 4 21.04λ× = so for the array

2

4 21.04 264 (24.2 dBi)*D *π
λ

×
= = (*ans.*)

58

**Chapter 9. Slot, Patch and Horn Antennas
**

**9-2-1. Two **λ**/2 slots.
**Two λ/2-slot antennas are arranged end-to-end in a large conducting sheet with a spacing
of 1λ between centers. If the slots are fed with equal in-phase voltages, calculate and
plot the far-field pattern in the 2 principal planes. Note that the *H* plane coincides with
the line of the slots.
*Solution*:
Thin slots are assumed.
The pattern in the *E plane* is a circle (*E* not a function of angle) or ( ) 1*E *φ = (*ans.*)
In the *H-plane* we have by pattern multiplication that the pattern is the product of 2 in-
phase isotropic sources spaced 1λ and the pattern of a / 2λ slot. The pattern of the / 2λ
slot is the same as for a / 2λ dipole but with and *E H *interchanged.
The pattern of the 2 isotropic sources is given by
( / 2)cos ( / 2)cos or 2cos[( / 2) cos ] 2cos( cos )*j d j dE H e e d*β θ β θ β θ π θ− += + = =
The total normalized pattern in the *H*-plane is then

cos[( / 2)cos ]( ) cos( cos )
sin*n
*

*E *π θθ π θ
θ

= (*ans.*)

***9-5-1. Boxed-slot impedance.
**What is the terminal impedance of a slot antenna boxed to radiate only in one half-space
whose complementary dipole antenna has a driving-point impedance of *Z *= 150 +*j*0 Ω?
The box adds no shunt susceptance across the terminals.
*Solution*:

From (9-5-12) the impedance of an unboxed slot is 35476*s
d d
*

*Z
R jX
*

= +

where *dR * is the resistance and *dX * is the reactance of the complementary dipole.

φ measured in plane perpendicular to page

*d *λ= θ

*H H*

59

***9-5-1. continued
**

Thus, 35476 236.5
150 0*s
*

*Z
j
*

= = +

Ω

Boxing the slot doubles the impedance so 2 236.5 473.0 473 *sZ *= × = ≅ Ω (*ans.*)

***9-5-2. Boxed slot.
**The complementary dipole of a slot antenna has a terminal impedance *Z* = 90 + *j*10 Ω. If
the slot antenna is boxed so that it radiates only in one half-space, what is the terminal
impedance of the slot antenna? The box adds no shunt susceptance at the terminals.
*Solution*:

From (8-5-12) we have a boxed slot 354762 779 87
90 10*s
*

*Z j
j
*

= × = − Ω +

(*ans.*)

**9-5-3. Open-slot impedance.
**What dimensions are required of a slot antenna in order that its terminal impedance be 75
+ *j*0 Ω? The slot is open on both sides.
*Solution*:

From (8-5-11), 35476 35476 473
75*d s
*

*Z
Z
*

= = = Ω

From Fig. 14-8 a center-fed cylindrical dipole with length-to-*diameter* ratio of 37∼ has a
resistance at 4th resonance of 473 Ω∼ (or twice that of a cylindrical stub antenna of a
length-to-*radius* ratio of 37). The width of the complementary slot should be twice the
dipole diameter, so it should have a length-to-width ratio of 181∼ . At 4th resonance the
dipole is 2λ∼ long and the slot should be the same length. The pattern will be midway
between those in Fig. 14-9 (right-hand column, bottom two patterns) but with *E* and *H*
interchanged.
Nothing is mentioned in the problem statement about pattern so the question is left open
as to whether this pattern would be satisfactory.
The above dimensions do not constitute a unique answer, as other shapes meeting the
impedance requirement are possible.

60

**9-7-1. 50 and 100 **Ω** patches.
**What value of the patch length *W* results in (a) a 50 Ω and (b) a 100 Ω input resistance
for a rectangular patch as in Fig. 9-22a?
*Solution*:

From (9-7-7.1), 22

90 1

*r
r
*

*r
*

*LR
W
*

ε ε

⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠

Solving for *W
*

290

( 1)
*r
*

*r r
*

*W L
R
*

ε ε

= −

Since o r

0.49*L *λ
ε

=

o0.49 9.49 ( 1)
*r
*

*r r
*

*W
R
*

ε λ ε

= × −

With 2.27*r*ε =

o o
2.27 1 14.65 6.22
1.27 *r r
*

*W
R R
*

λ λ= =

(a) oFor 50 , 0.88*rR W *λ= Ω =
(b) oFor 100 , 0.62*rR W *λ= Ω =

**9-7-3. Microstrip line.
**For a polystyrene substrate (εr = 2.7) what width-substrate thickness ratio results in a 50-
Ω microstrip transmission line?
*Solution*:
From (9-7-4) (see Fig. 9-21),

o o 377 or 2 2 2.6
[( / ) 2] 50 2.7*c r c r
*

*Z ZWZ
tW t Z*ε ε

= = − = − = +

(*ans.*)

61

**9-7-3. continued
**
2.6 field cells under strip plus 2 fringing cells = 4.6 cells giving

377 50
2.7 4.6*c
*

*Z *== = Ω
×

***9-9-1. Optimum horn gain.
**What is the approximate maximum power gain of an optimum horn antenna with a square
aperture 9λ on a side?
*Solution*:

Assuming a uniform *E * in the *E *-direction and cosine
distribution in the *H *-direction, as in the sketches, and with
phase everywhere the same, the aperture efficiency from
(19-1-50) is

2 2 2 2 2

o o2 (2 / ) /( / 2) 8 / 0.81( )
*av
*

*ap
av
*

*E E E
E
*

ε π π= = = =

A more detailed evaluation of *ap*ε for a similar distribution is
given in the solution to Prob. 19-1-7.

Assuming no losses,

2

4Power gain = *eAD *π
λ

=

where 2 2 20.81 10 81*e ap em ap pA A A*ε ε λ λ= = = × =
and 4 81 1018 or 30 dBi*D *π= =

**E
E
**

o*E
*

o*E
*

**E
**

*W
*

*t
*

Strip line

Ground plane

62

The same gain is obtained by extrapolating the *Ea *λ line in Fig. 9-29a to 10λ . However,
this makes *H Ea a*λ λ> and not equal as in this problem.

***9-9-1. continued
**In an optimum horn, the length (which is not specified in this problem) is reduced by
relaxing the allowable phase variation at the edge of the mouth by arbitrary amounts
( o o90 2 0.25 rad in the -plane and 144 2 0.4 rad in the -plane).*E H*π π= × = × This results
in less gain than calculated above, where uniform phase is assumed over the aperture.
From (9-9-2), which assumes 60% aperture efficiency, the directivity of the 10λ square
horn is

2 27.5 / 7.5 10 750 or 29 dBi*pD A *λ= × = × =

To summarize: when uniform phase is assumed ( 0.81)*ap*ε = as in the initial solution
above, 1018 or 30 dBi*D *= but for an optimum (shorter) horn ( 0.6)*ap*ε = , 750 *D *= or
29 dBi.

**9-9-2. Horn pattern.
**(a) Calculate and plot the *E*-plane pattern of the horn of Prob. 9-9-1, assuming uniform

illumination over the aperture.
(b) What is the half-power beamwidth and the angle between first nulls?
*Solution*:
(a) From (5-12-18) the pattern of a uniform aperture of length *a* is

sin sin( sin )2

sin 2

*n
aE
*

*a
*λ

λ

ψ π θ

ψ π θ

′

= =′ (1)

where aperture length = 10

angle from broadside
*a *λ
θ

= =

(b) From Table 5-8, oHPBW 50.8 /10 5.08= = (*ans.*)
Introducing o5.08 / 2 2.54= into (1) yields 0.707*nE *= which confirms that

o5.08 is the
true HPBW since 2 20.707 0.5*n nP E*= = =

63

Using (5-7-7) and setting *nd a*λ λ= for a continuous aperture,

1 1 oBWFN 2sin (1/ ) 2sin (1/10) 11.48*a*λ
− −= = = (*ans.*)

**9-9-2. continued
**Setting *nd a*λ λ= assumes *n* very large and *d*λ very small, but we have not assumed that
their product *nd*λ is necessarily very large. If we had, we could write

BWFN 2 / *a*λ= rad
and obtain

oBWFN 2 /10 rad = 11.46= for a difference of o0.02 .

**9-9-3. Rectangular horn antenna.
**What is the required aperture area for an optimum rectangular horn antenna operating at
2 GHz with 12 dBi gain?
*Solution*:

From (9-9-2) or Fig. 3-5b, 22 7.5

, 7.5

*p
p
*

*A DD A *λ
λ

≅ =

1.210 15.85, =0.15 m*D *λ= =

2 215.85 (0.15) 0.0475 m

7.5*p
A *= × =

**9-9-4. Conical horn antenna.
**What is the required diameter of a conical horn antenna operating at 2 GHz with a 12 dBi
gain?
*Solution*:

From Fig. 3-5b, 2

2

6.5 *rD *π
λ

≅

The diameter 2*d r*= is 2
6.5

*Dd *λ
π

= , 1.210 15.85*D *= = , 0.15 mλ =

64

15.852 0.15 26.4 cm 6.5

*d
*π

= × =

**9-9-5. Pyramidal horn.
**(a) Determine the length *L*, aperture *aH* and half-angles in *E* and *H* planes for a pyramidal
electromagnetic horn for which the aperture *aE* = 8λ. The horn is fed with a rectangular
waveguide with TE10 mode. Take δ = λ/10 in the *E* plane and δ= λ/4 in the *H* plane.
(b) What are the HPBWs in both *E* and *H* planes?
(c) What is the directivity?
(d) What is the aperture efficiency?
*Solution*:
(a) For a 0.1λ tolerance in the *E*-plane, the relation with dimensions in wavelengths is
shown in the sketch.

From which 2 2 2/ 4 0.2 0.01*EL a L L*λ λ λ λ+ = + +
with 8 (given),*Ea *λ =

2 / .8 80*EL a*λ λ= = (*ans.*)
In the H-plane we have from the sketch that

1 o

1 o

/ 2 6.33 and 12.7

/ 2 tan 4 / 80 2.9 ( .)

/ 2 tan 6.33/ 80 4.5 ( .)

*H H
*

*E
*

*H
*

*a a
*

*ans
*

*ans
*

λ λ

θ

θ

−

−

= =

= =

= =

(c) If the phase over the aperture is uniform 0.81*ap*ε = (see solution to Probs. 19-1-7
and 9-9-1),

4 8 12.7 0.81 1034 or 30.1 dBi*D *π= × × × =

However, the phase has been relaxed to o36 2 0.1π= × rad in the *E*-plane and to
o90 2 0.25π= × rad in the *H*-plane, resulting in reduced aperture efficiency, so *ap*ε must

be less than 0.8. If the *E*-plane phase is relaxed to o90 and the *H*-plane phase to o144 ,
0.6*ap*ε ∼ , which is appropriate for an optimum horn. Thus, for the conditions of this

problem which are between an optimum horn and uniform phase, 0.6 0.8.*ap*ε< < Taking
0.7*ap*ε ≅ ,

4 8 12.7 0.7 894 or 29.5 dBi*D *π= × × × = (*ans.*)

(b) Assuming uniform phase in the *E*-plane,

0.1*L*λ +
/ 2*E*θ

/ 2*Ea *λ

*L*λ

80.25
/ 2*H*θ

/ 2*Ha *λ

80

65

o o o o50.8 50.8(HPBW) 6.35 6.4

8*E Ea *λ
≅ = = ≅ (*ans.*)

and from the approximation

**9-9-5. continued
**41000 41000 894

(HPBW) (HPBW) 6.4(HPBW)*E H H
D *= = =

so o(HPBW) 7.2*H *≅
From Table 9-1 for an optimum horn,

o o

o o

56(HPBW) 7 8 67(HPBW) 5.3 12.7

*E
*

*H
*

≅ =

= =
(*ans.*)

The true (HPBW)*E * for this problem is probably close to

o6.4 . While the true
(HPBW)*H * is probably close to

o5.3 .
(d) 0.7*ap*ε = from part (c). (*ans.*)

66

**Chapter 10. Flat Sheet, Corner and Parabolic Reflector
Antennas
**

**10-2-1. Flat sheet reflector.
**Calculate and plot the radiation pattern of a λ/2 dipole antenna spaced 0.15λ from an
infinite flat sheet for assumed antenna loss resistance *RL* = 0 and 5 Ω. Express the
patterns in gain over a λ/2 dipole antenna in free space with the same power input (and
zero loss resistance).
*
Solution*:
From (10-2-1) the gain over a / 2λ reference dipole is given by

1 2

11

11 12

( ) 2 sin( cos )*f r
L
*

*RG S
R R R
*

φ φ ⎛ ⎞

= ⎜ ⎟+ −⎝ ⎠ (1)

where, spacing of dipole from reflector angle from perpendicular to reflector

*S
*φ

= =

(See Fig. 10-2.)
Note that (1) differs from (10-2-1) in that 0*LR *= in the numerator under the square root
sign since the problem requests the gain to be expressed with respect to a *lossless*
reference antenna.
Maximum radiation is at 0,φ = so (1) becomes,

1 2 73.1( ) 2 sin(2 0.15)

73.1 29.4*f L
G
*

*R
*φ π

⎛ ⎞ = ×⎜ ⎟+ −⎝ ⎠

and for 0*LR *=
( ) 2.09 or 6.41 dB (= 8.56 dBi)*fG *φ = (*ans.*)

Note that 12*R * is for a spacing of 0.3 ( 2 0.15 )λ λ= × . See Table 13-1.
Note that 10 , ( ) 1.89 or 5.52 dB (= 7.67 dBi)*L fR G *φ= Ω = (*ans.*)
Note that ( )*fG *φ is the gain with respect to a reference / 2λ dipole and more explicitly
can be written ( )[ / ].*fG A HW*φ
The loss resistance 10 *LR *= Ω results in about 0.9 dB reduction in gain with respect to a
lossless reference dipole. If the reference dipole also has 10 Ω loss resistance, the gain
reduction is about 0.3 dB.

67

**10-2-1. continued
**The above gains agree with those shown for 0*LR *= and extrapolated for 10 *LR *= Ω
at 0.15*S *λ= in Fig. 10-4. Note that in Fig. 10-4 an equal loss resistance is assumed in
the reference antenna.
The pattern for 0*LR *= should be intermediate to those in Fig. 10-3 for spacings of
0.125 (= /8) and 0.25 (= /4).λ λ λ λ The pattern for 10*LR *= Ω is smaller than the one for

0*LR *= but of the same shape (radius vector differing by a constant factor).

**10-3-1. Square-corner reflector.
**A square-corner reflector has a driven λ/2 dipole antenna space λ/2 from the corner.
Assume perfectly conducting sheet reflectors of infinite extent (ideal reflector).
Calculate and plot the radiation pattern in a plane at right angles to the driven element.
*Solution*:
From (10-3-6) the gain of a lossless corner reflector over a reference / 2λ dipole is given
by

1 2

11

11 14 12

( ) 2 [cos( cos ) cos( sin )]
2*f r r
*

*RG S S
R R R
*

φ φ φ ⎛ ⎞

= −⎜ ⎟+ −⎝ ⎠

For / 2*S *λ= and maximum radiation direction o( 0 )φ = this becomes

1 273.1( ) 4 3.06 or 9.7 dB (=11.9 dBi)
73.1 3.8 2 24*f
*

*G *φ ⎛ ⎞= =⎜ ⎟+ + ×⎝ ⎠

See Table 13-1 and Fig. 13-13 for the mutual resistance values for 14*R * at 1λ separation
and 12 at 0.707*R *λ separation. The above calculated gain agrees with the value shown by
the curve in Fig. 10-11. The pattern should be identical to the one in Fig. 10-12a.

**10-3-2. Square-corner reflector.
**(a) Show that the relative field pattern in the plane of the driven λ/2 element of a square-
corner reflector is given by

( )[ ] ( ) θ

θθ sin

cos90cossincos1 *rSE *−=

where θ is the angle with respect to the element axis. Assume that the corner-reflector sheets are perfectly conducting and of infinite extent.

68

**10-3-2. continued
**
(b) Calculate and plot the field pattern in the plane of the driven element for a spacing of
λ/2 to the corner. Compare with the pattern at right angles (Prob. 10-3-1).
*Solution*:
(a) The pattern in the plane of the dipole (*E* plane) is that of an array of three / 2λ
elements arranged as in the sketch with amplitudes 1:2:1 and phasing as indicated.
By pattern multiplication the pattern is the product of the pattern of an array of 3
isotropic sources with amplitudes and phasing 1: 2 : 1− + − and the pattern of / 2λ dipole
(6-4-4). Thus,

ocos(90 cos )(2 1 sin 1 sin )
sin*r r
*

*E S S *θθ θ
θ

__ __= − − −∠ ∠

or, see phasor sketch, ocos(90 cos )2[1 cos( sin )]

sin*r
E S *θθ

θ = −

Dropping the scale factor 2 yields the results sought, q.e.d.

***10-3-4. Square-corner reflector.
**(a) Calculate and plot the pattern of a 90° corner reflector with a thin center-fed λ/2

driven antenna spaced 0.35λ from the corner. Assume that the corner reflector is of infinite extent.

(b) Calculate the radiation resistance of the driven antenna. (c) Calculate the gain of the antenna and corner reflector over the antenna alone.

Assume that losses are negligible.

1*I *= −

1*I *= −

2*I *= +

*S
*

*S
*

θ

2cos( sin )*rS *θ−

sin1 *rS *θ__− ∠

1 sin*rS *θ__− −∠

o2 0__∠

Phasor sketch

69

***10-3-4. continued
***Solution*:
(a) From (10-3-6) the normalized field pattern for 0.35*S *λ= is

o o[cos(126 cos ) cos(126 sin )] ( )

1.588*n
E
*

φ φ φ

− =

(b) 11 14 122 73.1 24.8 25 73.3 *rR R R R*= + − = − + = Ω (*ans.*)
(c) From (10-3-6) for 0 and 0.35*S*φ λ= =

1 2( ) 2(73.1/ 73.3) 1.588 3.17 or 10.0 dB (=12.1 dBi)*fG *φ = × = (*ans.*)

**10-3-5. Square-corner reflector versus array of its image elements.
**Assume that the corner reflector of Prob. 10-3-4 is removed and that in its place the three
images used in the analysis are present physically, resulting in 4-element driven array.
(a) Calculate and plot the pattern of this array.
(b) Calculate the radiation resistance at the center of one of the antennas.
(c) Calculate the gain of the array over one of the antennas alone.
*Solution*:
(a) 4-lobed pattern as in Fig. 10-9 with shape of pattern of Prob. 10-3-4a.
(b) 73.3 *rR *= Ω (*ans.*)
(c) ( ) 1.59 or 4.0 dB (= 6.1 dBi)*fG *φ = (*ans.*)
since power is fed to all 4 elements instead of to only one (Power gain down by a factor
of 4 or by 6 dB).

***10-3-6. Square-corner reflector array.
**Four 90° corner-reflector antennas are arranged in line as a broadside array. The corner
edges are parallel and side-by-side as in Fig. P10-3-6. The spacing between corners is
1λ. The driven antenna in each corner is a λ/2 element spaced 0.4λ from the corner. All
antennas are energized in phase and have equal current amplitude. Assuming that the
properties of each corner are the same as if its sides were of infinite extent, what is (a) the
gain of the array over a single λ/2 antenna and (b) the half-power beam width in the *H*
plane?

70

***10-3-6. continued
**

Figure P10-3-6. Square-corner reflector array.

*Solution*:
(a) From (10-3-6) the gain of one corner reflector with 0.4*S *λ= is given by

1 2 o o73.1( ) 2 (cos144 cos0 )

73.1 18.6 42 2 0.870 1.81 3.15 or 10 dB (= 12.1 dBi)

*fG *φ
⎛ ⎞= −⎜ ⎟− +⎝ ⎠

= × × = ≅

Under lossless conditions,

( )2( ) 1.64 16.3*fD G *φ= × =
Thus, the maximum effective aperture of one corner is

2 2 216.3 1.3

4 4*em
DA *λ λ λ

π π = ≅ =

The effective aperture of a single corner may then be represented by a rectangle 1 1.3λ λ× as in the sketch below.

1.3λ

1λ terminals

/ 2 dipoleλ

0.4λ

*emA *

90o corner reflector

λ

71

***10-3-6. continued
**In an array of 4 reflectors as in Fig. P10-3-6 the edges of the apertures overlap 0.3λ so
that the reflectors are too close. However, at the 1λ spacing the total aperture is

24 1 4λ λ λ× = and the total gain of the array under lossless conditions is

2

4 4 4 50 or 17 dBi*emAG D *π π
λ

= = = × ≅ (*ans.*)

No interaction between corner reflectors has been assumed. With wider spacing ( 1.3 )λ= the expected gain 16.3 4 65 or 18 dBi.= × = (b) Assuming a uniform aperture distribution, the HPBW is given approximately from Table 5-8 by

o o oHPBW = 50.8 / 50.8 / 4 12.7*L*λ = =
To determine the HPBW more accurately, let us use the total antenna pattern. By pattern
multiplication it is equal to the product of an array of 4 in-phase isotropic point sources
with 1λ spacing and the pattern of a single corner reflector as given by

1 sin(4 sin ) 1( ) [cos(0.8 cos ) cos(0.8 sin )]
4 sin( sin ) 1.809*n
*

*E *π φφ π φ π φ
π φ

= −

The ¼ is the normalizing factor for the array and 1/1.809 for the corner reflector. Thus,
when o0 ,φ = ( ) 1.*nE *φ = Note that φ must approach zero in the limit in the array factor
to avoid an indeterminate result.
Half of the above approximate HPBW is o o12.7 / 2 6.35 .= Introducing it into the above
equation yields o( ) 0.703. For 6.30 , ( ) 0.707*n nE E*φ φ φ= = = as tabulated below.

φ ( )*nE *φ
6.35o 0.703
6.30o 0.707

Thus, oHPBW 2 6.30 12.6= × = (*ans.*)
The 4-source array factor is much sharper than the corner reflector pattern and largely
determines the HPBW.
Returning to part (a) for the directivity, let us calculate its value with the approximate
relation of (2-7-9) using the HPBW of part (b) for the *H*-plane and the HPBW of 78o for
the *E*-plane from Example 6-4.1.

72

***10-3-6. continued
**Thus,

40000 40.7 (= 16 dBi) 12.6 7.8

*D *≅ =
×

as compared to 50 (= 17 dBi) *D *≅ as calculated in part (a).
Although the directivity of 16.3 for a single corner reflector should be accurate, since it is
determined from the pattern via the impedances*, the directivity of 50 for the array of 4
corner reflectors involves some uncertainty (apertures overlapping). Nevertheless, the
two methods agree within 1 dB.
____________________________
*Assuming infinite sides

**10-3-7. Corner reflector. **λ**/4 to the driven element.
**A square-corner reflector has a spacing of λ/4 between the driven λ/2 element and the
corner. Show that the directivity *D* = 12.8 dBi.
*Solution*:
For the case of no losses,

( )2( ) 1.64, and for / 4 and 0,*fD G S*φ λ φ= × = =
(10-3-6) becomes

1 273.1( ) 2 3.39
73.1 12.7 35*f
*

*G *φ ⎛ ⎞= =⎜ ⎟− −⎝ ⎠

Therefore, ( )2( ) 1.64 18.9 or 12.8 dBi*fD G *φ= × = (*ans.*)

**10-3-8. Corner reflector. **λ**/2 to the driven element.
**A square-corner reflector has a driven λ/2 element λ/2 from the corner.
(a) Calculate and plot the far-field pattern in both principal planes.
(b) What are the HPBWs in the two principal planes?
(c) What is the terminal impedance of the driven element?
(d) Calculate the directivity in two ways: (1) from impedances of driven and image

dipoles and (2) from HPBWs, and compare. Assume perfectly conducting sheet reflectors of infinite extent.

73

**10-3-8. continued***Solution*:
(a) From Prob. 10-3-2 the pattern in the *E*-plane is given by

o1 cos(90 cos )( ) [1 cos( sin )]

2 sin*n
E *θθ π θ

θ = − (1)

From (10-3-6) the pattern in the *H*-plane is given by

1( ) [cos( cos ) cos( sin )]
2*n
*

*E *θ π φ π φ= − (2)

Note that ( )*nE *θ = maximum for
o90θ = while ( )*nE *φ = maximum for

o0 .φ = (b) Assuming initially that HPBW( )θ HPBW( )φ≅ and noting from Fig. 10-11 that for

/ 2*S *λ= the directivity is about 12 dBi, we have from (2-7-9) that

o 2

40000 16 or HPBW( ) 50 HPBW( )

*D *θ
θ

≅ ≅ ≅ and o

oHPBW( ) 50 25 2 2

θ = =

Introducing o o o90 25 65θ = − = in (1) yields ( )*nE *θ which is too high. By trial and error,
we obtain o( ) 0.707 when =34.5 .*nE *θ θ≅
Therefore, o oHPBW( ) 2 34.5 69θ ≅ × = (*ans.*)
Introducing o25φ = in (2) yields ( ) 0.60*nE *φ = which is too low. By trial and error, we
obtain ( ) 0.707*nE *φ ≅ when

o21 .φ =
Therefore, o oHPBW( ) 2 21 42φ ≅ × = (*ans.*)
(c) The terminal impedance of the driven element is (see Prob. 10-3-1 solution),

73.1 3.8 2 24 125*TR *= + + × = Ω (*ans.*)
From Prob. 10-3-1 solution,

( )2( ) 1.64 15.4 (= 11.9 dBi) by impedance*fD G *φ= × = (*ans.*)

o o

40000 13.8 (= 11.4 dBi) by beam widths 69 42

*D *≅ =
×

(*ans.*)

74

**10-3-8. continued
**
The 15.4*D *= value is, of course, more accurate since it is based on the pattern via the
impedances. The two methods differ, however, by only 0.4 dB.

***10-7-2. Parabolic reflector with missing sector.
**A circular parabolic dish antenna has an effective aperture of 100 m2. If one 30° sector
of the parabola is removed, find the new effective aperture. The rest of the antenna,
including the feed, is unchanged.
*Solution*:
The full dish has an effective aperture 2100 m .*eA *= Assuming that the dish character-
istics are independent of angle ( ),φ removing one o45 sector reduces the effective aper-
ture to 7/8 of its original value provided the feed is modified and so as not to illuminate
the area of the missing sector. However, the feed is not modified and, therefore, its
efficiency is down to 7/8. Therefore, the net aperture efficiency is (7/8)2 and the net
effective aperture is

2 2(7 / 8) 100 76.6 m× = (*ans.*)

φ 45o

Sector removed

φ

75

76

**Chapter 11. Broadband and Frequency-Independent Antennas
**

***11-2-2. The 2**°** cone.
**Calculate the terminal impedance of a conical antenna of 2° total angle operating against
a very large ground plane. The length *l* of the cone is 3λ/8.
*Solution*:
From (11-2-2) for ,*kZ * and noting that when θ is small

o( 20 )θ <

Then 4cot 4 θ

θ ⎛ ⎞ ≅⎜ ⎟ ⎝ ⎠

or 4120ln*kZ *θ
=

and with (11-2-4) and (11-2-5) for ,and (11-2-3) for *m iZ Z *
so 270 350*iZ j*= + Ω (*ans.*)

**11-5-1. Log spiral.
**Design a planar log-spiral antenna of the type shown in Fig. 11-11 to operate at
frequencies from 1 to 10 GHz. Make a drawing with dimensions in millimeters.
*Solution*:
High frequency limit = 10 GHz, 8 93 10 /10 10 30 mmλ = × × =
Low frequency limit = 1 GHz, 300 mmλ =
Take o77.6 (see Fig. 11-10)β =
From (11-5-5), antiln ( /tan ) = antiln ( /4.55)*r *θ β θ=

77

**11-5-1. continued
**

θ *r R
*

0 rad 1 1.5 mm

/ 2π 1.41 2.12

π 2.00 3.00

3 / 2π 2.82 4.23

2π 4.00 6.00

5 / 2π 5.66 8.50

3π 8.00 12.0

7 / 2π 11.3 17.0

4π 16.0 24.0

9 / 2π 22.6 34.0

5π 32.0 48.0

11 / 2π 45.2 68.0

6π 64.0 96.0

Spiral is like one in Fig. 11-10. If gap *d* at center is equal to /10λ at high frequency
limit, then gap should be 30/10 = 3 mm and radius *R* of actual spiral = 3/2 = 1.5 mm.
If *diameter* of spiral is / 2λ at low frequency limit, then the actual spiral radius should be
300 /(2 2) 75 mm.× =
This requires that 4.55ln(75 /1.5) 17.8 5.7θ π= = =
For good measure we make 6θ π= . Thus, the spiral has 3 turns ( 6θ π= ).
The table gives data for the actual spiral radius R in mm versus the angle θ in rad. The
overall diameter of the spiral is 96 2 192 mm× = which at 1 GHz is 192/300 = 0.64λ .
Calling the above spiral number 1, draw an identical spiral rotated through / 2π rad, a
third rotated through π rad and a fourth rotated through 3 / 2π rad. Metalize the areas
between spirals 1 and 4 and between spirals 2 and 3, leaving the remaining areas open.
Connect the feed across the gap at the innermost ends of the spirals as in Fig. 11-11.

78

**11-7-1. Log periodic.
**Design an “optimum” log-periodic antenna of the type shown in Fig. 11-17 to operate at
frequencies from 100 to 500 MHz with 11 elements. Give (a) length of longest element,
(b) length of shortest element, and (c) gain.
*Solution*:
From Fig. 11-19 let us select the point where o15α = intersects the optimum design line
which should result in an antenna with slightly more than 7 dBi gain. From the figure,
*k*=1.195. The desired frequency ratio is 5 = *F *= 250/50. Thus, from (7) the required
number of elements musts be at least equal to

log log5 9.0 log log1.195

*Fn
k
*

= = =

at 250 MHz, 1.2 m, / 2 0.6 mλ λ= = (*ans.*)
at 50 MHz, 6 m, / 2 3 mλ λ= = (*ans.*)
If element 1 is 0.6 in long, then element 10 (= *n*+1) is 0.6 91.195× = 2.98 m or
approximately 3 m as required.
Adding a director in front of element 1 and a reflector in back of element 10 brings the
total number of elements to 12. (*ans.*)
The length 2 of any element with respect to the length , of the next shorter element is
given by

2 1/ 1.195*k*= = (*ans.*)
From (11-7-5) (note geometry of Fig. 11-18), the spacing *s* between any two elements is
related to the length of the adjacent shorter element by

o

( 1) 0.195 0.364 2 tan 2 tan15

*ks
*α

−
= = = (*ans.*)

[Note that (11-7-6) gives *s* with respect to adjacent *longer* elements.]
Finally, connect the elements as in Fig. 11-17.

79

**11-7-2. Stacked LPs.
**Two LP arrays like in the worked example of Sec. 11-7 are stacked as in Fig. 11-21a.
(a) Calculate and plot the vertical plane field pattern. Note that pattern multiplication
cannot be applied.
(b) What is the gain?
*Solution*:
From the worked example of Sec. 11-7,

o15 , 1.2, 4, 7.6 (8) and 1 9*k F n n*α = = = = + =

Consider that min max1 m and 4 m.λ λ= = Therefore, 81 1 1 90.5 m and 0.5 1.2 2 m

*n
n k*+= = = × = =

From (11-7-6) the distance between elements 1 and 9 is

2

1 1 0

( 1) where 0.5 m 2 tan

*nn
*

*n
*

*k kS
*α

−

=

− = =∑

and

7

o 0

0.2 1.2 =0.187+0.224+0.268+0.322+0.387+0.464+0.557+0.669 = 3.08 m 4 tan15

*n
*

*n
S
*

=

× = ∑

The stacked LPs are shown in side view in the sketch below. Elements 1 through 9 are included in the calculation. Element 1 is / 2λ resonant at 1 m wavelength and element 9 is / 2λ resonant at 4 m wavelength. A director element is added ahead of element 1 and a reflector element is added after element 9 making a total of 11 elements. In the 60 angle stacking arrangement the stacking distance is 0.5 m or / 2λ at 1 m wavelength and 3.65 m or 0.91λ at 4 m wavelength. At the geometric mean wavelength (2 m) the stacking distance is 1.25 m or 0.625 λ .

80

**11-7-2. continued
**
Let us calculate the vertical plane pattern at 2 m wavelength where elements 3, 4 and 5 of
the upper and lower LPs are active. As an approximation, let us consider that the 3 active
elements are a uniform ordinary end-fire array with spacing equal to the average of the
spacing between elements 3 and 4 and between 4 and 5.
For element 4 we take 4 / 2λ= . Then from (11-7-6)

45 o

1 1.2 1 0.187
4 tan 4 tan15
*kS *λ

α − −

= = =

81

**11-7-2. continued
**and

34 0.187 /1.2 0.155*S *λ= =

0.155 .187 0.171
2*av
*

*S *λ+= =

The end-fire array field pattern is given by 1 sin / 2 3 sin / 2

*nE *ψ
ψ

=

where o2 0.171/ 2 (cos 1) 30.8 (cos 1), 3

2
*n*πψ φ φ×= − = − =

Each LP (end-fire array) has a broad cardiod-shaped pattern like the ones shown in Fig. 11-21a with one pattern directed up 30o and the other down 30o. The total field pattern in the resultant of these patterns and a broadside array of 2 in- phase isotropic sources stacked vertically and spaced 0.625λ with pattern given by

ocos[(2 0.625 / 2)sin ] cos(112.5 sin )*E *π φ φ= × =
This pattern is shown in Fig. 16-11. Numerical addition of the LP patterns and multi-
plication of the resultant by the broadside pattern yields the total field pattern for

2 mλ = shown in the sketch. At 1 mλ = the up-and-down minor lobes disappear but the
main beam is about the same. At 4 mλ = the main beam is narrower but the up-and-
down minor lobes are larger.
(b) Each LP has a gain 7≅ dBi. From the equation of Prob. 5-2-3, the directivity of 2
in-phase isotropic sources with 0.625λ spacing is 2.44 or 3.9 dBi. So for 2 mλ = , the
gain may be as much as 7 3.9 10.9 dBi+ ≅ (*ans.*)
At both 1 m and 4 mλ λ= = the gain may be less than this.
The HPBW in the vertical plane is about o47 and in the horizontal plane about o76 .
From the approximate directivity relation, we have, neglecting minor lobes,

41000 11.5 or 10.6 dBi 47 76

*D *= =
×

(*ans.*)

Actual directivity is probably 10 dBi.∼

82

**Chapter 12. Antenna Temperature, Remote Sensing and Radar
Cross Section
**

***12-2-1. Antenna temperature.
**An end-fire array is directed at the zenith. The array is located over flat nonreflecting
ground. If 0.9 Ω*A* is within 45° of the zenith and 0.08 Ω*A* between 45° and the horizon
calculate the antenna temperature. The sky brightness temperature is 5 K between the
zenith and 45° from the zenith, 50 K between 45° from the zenith and the horizon and
300 K for the ground (below the horizon). The antenna is 99 percent efficient and is at a
physical temperature of 300K.
*Solution*:

0.9 is at 5 K 0.08 is at 50 K

*A
*

*A
*

Ω Ω

Therefore 0.02 is at 300 K*A*Ω
From (12-1-8),

1 [5 0.9 50 0.08 300 0.02 ] 4.5 4 6 14.5 K*A A A A
A
*

*T *= × Ω + × Ω + × Ω = + + =
Ω

(*ans.*)

***12-2-2. Earth-station antenna temperature.
**An earth-station dish of 100 m2 effective aperture is directed at the zenith. Calculate the
antenna temperature assuming that the sky temperature is uniform and equal to 6 K.
Take the ground temperature equal to 300 K and assume that 1/3 of the minor-lobe beam
area is in the back direction. The wavelength is 75 mm and the beam efficiency is 0.8.
*Solution*:
From (12-1-8),

1 2 1[6 0.8 6 0.2 300 0.2 ] 4.8 0.8 20 25.6 K
3 3*A A A AA
*

*T *= × Ω + × × Ω + × × Ω = + + =
Ω

(*ans.*)

83

***12-3-4. Satellite TV downlink.
**A transmitter (transponder) on a Clarke orbit satellite produces an effective radiated
power (ERP) at an earth station of 35 dB over 1 W isotropic.
(a) Determine the *S*/*N* ratio (dB) if the earth station antenna diameter is 3m, the antenna
temperature 25 K, the receiver temperature 75 K and the bandwidth 30 MHz. Take the
satellite distance as 36,000 km. Assume the antenna s a parabolic reflector (dish-type) of
50 percent efficiency. (See Example 12-3.1).
(b) If a 10-dB S/N ratio is acceptable, what is the required diameter of the earth station
antenna?
*Solution*:
(a) Satellite ERP = 35 dB (over 1 W isotropic)

ERP = 2 24 / , ERP / 4*t t t et t etPD P A P A*π λ λ π= =
From (12-3-3),

2

2 2 2 2

ERP 4

*t et er er
*

*sys sys
*

*P A A AS
N kT r B kT r B
*

λ λ π λ

= = , ERP = 35 dB or 3162

2 2 21 1 1.5 3.53 m

2 2*er
A r*π π= = = , 25 75 100 K*sysT *= + =

23 2 14 7

3162 3.53 16.6 or 12.2 dB 4 1.38 10 100 3.6 10 3 10

*S
N *π −

× = =

× × × × × × ×
(*ans.*)

(b) If only 10 dB *S*/*N* ratio is acceptable, the dish aperture could be 2.2 dB
( 12.2 10 dB)= − less or 60% of the π 1.52 = 7.07 m2 area specified in part (a).
Therefore acceptable area 2 27.07 0.6 4.24 m *r*π= × = =
For a diameter 1 22 2(4.24 / ) 2.3 m*r *π= = = (*ans.*)

***12-3-5. System temperature.
**The digital output of a 1.4 GHz radio telescope gives the following values (arbitrary
units) as a function of the sidereal time while scanning a uniform brightness region. The
integration time is 14 s, with 1 s idle time for printout. The output units are proportional
to power.

84

***12-3-5. continued
**

Time Output Time Output 31m30s 234 32m45s 229 31 45 235 33 00 236 32 00 224 33 15 233 32 15 226 33 30 230 32 30 239 33 45 226

If the temperature calibration gives 170 units for 2.9 K applied, find (a) the rms noise at
the receiver, (b) the minimum detectable temperature, (c) the system temperature and (d)
the minimum detectable flux density. The calibration signal is introduced at the receiver.
The transmission line from the antenna to the receiver has 0.5 dB attenuation. The
antenna effective aperture is 500 m2. The receiver bandwidth is 7 MHz. The receiver
constant *k *′ =2.
*Solution*:
(a) Noise output readings 234, 235, etc. with respect to average value (231), are
squared, averaged and then square rooted for root-mean-square (rms) noise value 4.71.
The rms noise at receiver is then

4.71 2.9 0.08 K 170

× = (*ans.*)

(b) Transmission line attenuation = 0.5 dB for efficiency of 0.89.
Therefore, min 0.08 / 0.89 0.09 K*T*Δ = = (*ans.*)
(c) From (12-2-3),

1 2 6 1 2 min ( ) 0.09(7 10 14) 445 K

2*sys
T ftT
*

*k
*Δ Δ × ×

= = = ′

(*ans.*)

(d) From (12-1-7),

23 min

min 2 2 1.38 10 0.09 497 mJy 500 mJy (rounded off)

500*e
*

*k TS
A
*

−Δ × × ×
Δ = = = ≅ (*ans.*)

85

**12-3-6. System temperature.
**Find the system temperature of a receiving system with 15 K antenna temperature, 0.95
transmission-line efficiency, 300 K transmission-line temperature, 75 K receiver first-
stage temperature, 100 K receiver second-stage temperature and 200 K receiver third-
stage temperature. Each receiver stage has 16 dB gain.
*Solution*:
From (12-2-1) and (12-2-2),

1 1 100 20015 300 1 75 .95 .95 40 40

15 15.8 78.9 2.6 5.3 117.6 K ( .)

*sysT
*

*ans
*

⎛ ⎞ ⎛ ⎞= + − + + +⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

= + + + + =

***12-3-7. Solar interference to earth station.
**Twice a year the sun passes through the apparent declination of the geostationary Clarke-
orbit satellites, causing solar-noise interference to earth stations. A typical forecast
notice appearing on U.S. satellite TV screens reads:

ATTENTION CHANNEL USERS: WE WILL BE EXPERIENCING

SOLAR OUTAGES FROM OCTOBER 15 TO 26

FROM 12:00 TO 15:00 HOURS
(a) If the equivalent temperature of the sun at 4 GHz is 50,000 K, find the sun’s signal-to-
noise ration (in decibels) for an earth station with a 3-m parabolic dish antenna at 4 GHz.
Take the suns diameter as 0.5° and the earth-station system temperature as 100 K.
(b) Compare this result with that for the carrier-to-nose ratio calculated in Example 12-
3.1 for a typical Clarke-orbit TV transponder.
(c) How long does the interference last? Note that the relation *eA A
*

2λ=Ω gives the solid beam angle in steradians and not in square degrees. (d) Why do the outages occur between October 15 and 26 and not at the autumnal equinox around September 20 when the sun is crossing the equator? (e) How can satellite services work around a solar outage?

86

***12-3-7. continued
***Solution*:

(a) Earth station min
2 *sys
*

*e
*

*kT
S
*

*A
*=

sun 2

22 2 *s sA A A
e e A
*

*kTk T k TS
A A *λ

ΩΔ Δ Ω = = =

Ω

Assuming 50% aperture efficiency as in Prob. 12-3-4,

2 2

3

2

0.075 1.59 10 sr 5.22 sq. deg.1 1.5 2

*A
eA
*

λ

π

−Ω = = = × =

Therefore, 4 o 2

sun

min

5 10 (.25 ) 18.8 or 12.7 dB 100 5.22

*s s
*

*sys A
*

*S TS
N S T
*

πΩ × × = = = =

Ω ×
(*ans.*)

(b) From Prob. 12-3-4,
12.7 12.2 0.5− = dB more than satellite carrier, resulting in degradation of TV picture
quality. The phenomenon may be described as *noise jamming* by the sun.
(c) At half-power, solar noise will be reduced to only 3 0.5 2.5− = dB below carrier.
Assuming low side-lobes, the solar interference should not last more than the time it
takes the sun to drift between first nulls. Using this criterion we have

Solar drift rate

Time 4(min/ deg) BWFN (deg)= ×

From Table 15-1,
HPBW o o o66 / 66 /(3 0.075) 1.65 and Time 4 2 1.65 13.2 min*D*λ= = = ≅ × × = (*ans.*)
Allowing for the angular extent of the sun (approx. o1 2 ) increases the time by about 2
minutes.

sq. deg.

87

***12-3-9. Critical frequency. MUF.
**Layers may be said to exist in the earth’s ionosphere where the ionization gradient is
sufficient to refract radio waves back to earth. [Although the wave actually may be bent
gradually along a curved path in an ionized region of considerable thickness, a useful
simplification for some situations is to assume that the wave is reflected as though from a
horizontal perfectly conducting surface situated at a (*virtual*) *height h.*] The highest
frequency at which this layer reflects a vertically incident wave back to the earth is called
the *critical frequency fo*. Higher frequencies at the vertical incidence pass through. For
waves at oblique incidence (φ>0 in Fig. P12-3-9) the *maximum usable frequency* (MUF)
for point-to-point communication on the earth is given by MUF = *fo*/cos φ, where φ =
angle of incidence. The critical frequency *Nfo *9= , where *N* = electron density
(number m-3). *N* is a function of solar irradiation and other factors. Both *fo* and *h* vary
with time of day, season, latitude and phase of the 11-year sunspot cycle. Find the MUF
for (a) a distance *d* = 1.3 Mm by *F2*-layer (*h* = 3 25 km) reflection with *F2*-layer electron
density *N* = 6 x 1011 m-3; (b) a distance *d* = 1.5 Mm by *F2*-layer (*h* = 275 km) reflection
with *N* = 1012 m-3; and (c) a distance *d* = 1 Mm by sporadic *E*-layer (*h* = 100 km)
reflection *N* = 8 x 1011 m-3. Neglect earth curvature.

Figure P12-3-9. Communication path via reflection from ionospheric layer.

*Solution*:
MUF = Maximum Usable Frequency for communication via ionospheric reflection.
(a)

5 11 1 2 1 1 o

o 5

6.5 109 9(6 10 ) 6.97 MHz, tan [( / 2) / ] tan 63.43 3.25 10

*f N d h*φ − −
⎛ ⎞×

= = × = = = =⎜ ⎟×⎝ ⎠

ocos 0.477, MUF / cos 6.97 / 0.447 15.6 MHz*f*φ φ= = = = (*ans.*)

88

***12-3-9. continued
**5

12 1 2 -1 o o 5

7.5 10(b) 9(10 ) 9 MHz, tan 69.86 , cos 0.344 2.75 10

*f *φ φ
⎛ ⎞×

= = = = =⎜ ⎟×⎝ ⎠

MUF 9 / 0.344 26.1 MHz= = (*ans.*)

5 11 1 2 -1 o

o 5

5 10(c) 9(8 10 ) 8.05 MHz, tan 78.69 , cos 0.196 10

*f *φ φ
⎛ ⎞×

= × = = = =⎜ ⎟ ⎝ ⎠

6MUF 8.05 10 / 0.196 41.0 MHz= × =

**12-3-10. mUF for Clarke-orbit satellites.
**Stationary communication (relay) satellites are placed in the Clarke orbit at heights of
about 36 Mm. This is far above the ionosphere, so that the transmission path passes
completely through the ionosphere twice, as in Fig. 12-3-10. Since frequencies of 2 GHz
and above are usually used the ionosphere has little effect. The high frequency also
permits wide bandwidths. If the ionosphere consists of a layer 200 m thick between
heights of 200 and 400 km with a uniform electron density *N* = 1012 m-1, find the lowest
frequency (or *minimum usable frequency,* mUF) which can be used with a
communication satellite (a) for vertical incidence and (b) for paths 30° from the zenith.
(c) For an earth station on the equator, what is the mUF for a satellite 15° above the
eastern or western horizon?

Figure P12-3-10. Communication path via geostationary Clarke-orbit relay satellite.
*Solution*:

mUF = minimum usable frequency for transmission through ionosphere
(a) 1 2 12 1 2mUF 9( ) 9(10 ) 9 MHz*N*= = = (*ans.*)

89

**12-3-10. continued
**Although MUF = mUF, whether it is one or the other, depends on the point of view. It is
MUF for reflection and mUF for transmission. Below critical frequency, wave is
reflected; above critical frequency, wave is transmitted.
(b) omUF 9 MHz/ cos30 10.4 MHz= = (*ans.*)
(c) omUF 9 MHz/ cos75 34.8 MHz= = (*ans.*)
A typical Clarke orbit satellite frequency is 4 GHz, which is 115 9 6( 4 10 / 34.8 10 )= × ×
times higher in frequency than the mUF, so that at 4 GHz, transmission through the
ionosphere can occur at much lower elevation angles than 15o.

***12-3-11. Minimum detectable temperature.
**A radio telescope has the following characteristics: antenna noise temperature 50 K,
receiver noise temperature 50 K, transmission-line between antenna and receiver 1 dB
loss and 270 K physical temperature, receiver bandwidth 5 MHz, receiver integration
time 5 s, receiver (system) constant 2/π=′*k * and antenna effective aperture 500 m2. If
two records are averaged, find (a) the minimum detectable temperature and (b) the
minimum detectable flux density.
*
Solution*:
(a) For 1 dB loss, 2 1/1.26 0.79ε = =
From (12-2-1),

1 5050 270 1 50 71.8 63.3 185 K
.79 .79*sys
*

*T *⎛ ⎞= + − + = + + =⎜ ⎟
⎝ ⎠

From (12-2-3),

min 1 2 6 1 2

185 0.058 K 0.06 K (rounded off) ( ) (5 10 5 2)2

*sysk TT
f t n
*

π′ Δ = = = ≅

Δ × × ×
(*ans.*)

where *n* = number of records averaged

(b) From (12-1-7), 23

min min

2 2 1.38 10 0.058 320 mJy
500*e
*

*kTS
A
*

−× × ×
Δ = = = (*ans.*)

90

**12-3-12. Minimum detectable temperature.
**A radio telescope operates at 2650 MHz with the following parameters: system
temperature 150 K, predetection bandwidth 100 MHz, postdetection time constant 5 s,
system constant *k’* = 2.2 and effective aperture of antenna 800 m2. Find (a) the minimum
detectable temperature and (b) the minimum detectable flux density. (c) If four records
are averaged, what change results in (a) and (b)?
*Solution*:

(a) min 8 1 2 2.2 50 0.015 K

(10 5)
*T *×Δ = =

×
(*ans.*)

(b) 23

min 2 1.38 10 0.015 52 mJy

800
*S
*

−× × ×
Δ = = (*ans.*)

min

min

0.015(c) 0.008 K 4

52 26 mJy 4

*T
*

*S
*

Δ = =

Δ = =
(*ans.*)

***12-3-13. Interstellar wireless link.
**If an extraterrestrial civilization (ETC) transmits 106 W, 10 s pulses of right-hand circ-
ularly polarized 5 GHz radiation with a 100 m diameter dish, what is the maximum
distance at which the ETC can be received with an SNR = 3. Assume the receiving
antenna on the earth also has a 100 m diameter antenna responsive to right circular
polarization, that both antennas (theirs and ours) have 50 percent aperture efficiency, and
that the earth station has a system temperature of 10 K and bandwidth of 0.1 Hz.
*Solution*: (Note: Problem statement should specify *S*/*N* =3.)

From (12-3-3), 8

2 2 9

3 10, 0.06 m 5 10

*t er et
*

*sys
*

*P A AS c
N r BkT f
*

λ λ

× = = = =

×

6 2 2 2 38 2

2 2 23

10 [ (50) 0.5][ (50) 0.5] 1.03 10 m ( / ) (0.6) (0.1)(1.38 10 )(10)(3)

*t er et
*

*sys
*

*P A Ar
BkT S N
*

π π λ −

= = = × ×

Therefore, 191.02 10 m 1000 light years*r *= × ≅ (*ans.*)

91

***12-3-14. Backpacking penguin.
**This penguin (Fig. P12-3-14) participated in a study of Antarctic penguin migration
habits. Its backpack radio with λ/4 antenna transmitted data on its body temperature and
its heart and respiration rates. It also provided information on its location as it moved
with its flock across the ice cap. The backpack operated at 100 MHz with a peak power
of 1 W and a bandwidth of 1- kHz of tone-modulated data signals. If *Tsys* = 1000 K and
SNR = 30 dB, what is the maximum range? The transmitting and receiving antennas are
λ/4 stubs.

Figure P12-3-14. Antarctic backpacking penguin.

*Solution*:

From (12-3-3), 8

2 2 6

3 10, 3 m 100 10

*t er et
*

*sys
*

*P A AS c
N r BkT f
*

λ λ

× = = = =

×

2 2

2 2 2

2

11 2 2 3 23 3

1.5 1.5 4 4

( / ) ( / )

1.5(3)1 4

9.3 10 m (3) (10 10 )(1.38 10 )(1000)(10 )

*t
t er et
*

*sys sys
*

*P
P A Ar
*

*BkT S N BkT S N
*

λ λ π π

λ λ

π −

⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠= =

⎛ ⎞ ⎜ ⎟ ⎝ ⎠= = ×

× ×

Therefore, 59.6 10 m 960 km*r *= × ≅ (*ans.*)

***12-3-17. Low earth orbit communications satellite.
**A communications satellite in low earth orbit (LEO) has *r*up = 1500 km and *r*down = 1000
km, with an uplink frequency of 14.25 GHz and a down link frequency of 12 GHz. Find
the full-circuit *C/N* if the transmitting earth station ERP is 60dBW and the satellite ERP
is 25 dBW. Assume the satellite receiver *G/T* is 5 dB/K and the earth-station *G/T* is 30
dB/K.

92

***12-3-17. continued
***Solution*:

From Prob. 12-3-15,
2 2
*t et er
*

*sys
*

*P A AC
N r kT*λ

= and using 2
4 *eAG *π

λ = , we have

2

2 2 (4 )
*t t r
*

*sys
*

*PG GC
N r kT
*

λ π

=

We now define effective radiated power = ERP = *t tPG *, Power loss

2 2 2(4 ) /*L r*π λ= =

This gives 1 1(ERP) L

*r
*

*sys
*

*GC
N k T
*

⎛ ⎞⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠

For the uplink,

8

up up9 upup up

(sat)3 10 1 10.021 m, and (ERP) 14.25 10 (sat)

*r
*

*sys
*

*Gc C
f N L k T
*

λ ⎛ ⎞ ⎛ ⎞× ⎛ ⎞ ⎛ ⎞= = = = ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟× ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

Working in dB, ERPup = 60 dBW, 26

up 4 (1.5 10 )10log 179.1 dB

0.021
*L *π

⎛ ⎞× = =⎜ ⎟

⎝ ⎠

2310 log(1.38 10 ) 228.6 dB,*k *−= × = − -1(sat.) 5 dB K

(sat.)
*r
*

*sys
*

*G
T
*

=

So up

60 179.1 228.6 5 114.5 dB*C
N
*

⎛ ⎞ = − + + =⎜ ⎟ ⎝ ⎠

Similarly, down

25 174 228.6 30 109.6 *C
N
*

⎛ ⎞ = − + + =⎜ ⎟ ⎝ ⎠

From Prob. 12-3-1, 1 1 1

11 11.45 10.96

up down

1 1 1.45 10 10 10

*C C C
N N N
*

− − − −⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + = + = ×⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

10

11

1 6.89 10 108.4 dB 1.45 10

*C
N *−

= = × = ×

(*ans.*)

***12-3-18. Direct broadcast satellite (DBS).
**Direct broadcast satellite services provide CD quality audio to consumers via satellites in
geosynchronous orbit. The World Administrative Radio Conference (WARC) has
established these requirements for such services.

93

***12-3-18. continued
**

Frequency band 11.7 to 12.2 GHz (Ku band)
Channel bandwidth 27 MHz
Minimum power flux density -103 dBW/m2
Receiver figure of merit (*G/T*) 6 dB/K
Minimum carrier-to-noise ratio 14 dB

(a) Find the effective radiated power (ERP) over 1 W isotropic needed to produce the
specified flux density at the Earth’s surface from a DBS satellite in a 36,000 km orbit.
(b) If the satellite has a 100 W transmitter and is operated at 12 GHz, what size circular
parabolic dish antenna must be used to achieve the required ERP? Assume 50 percent
efficiency. (c) Does a consumer receiver with circular 1 m dish antenna with 50 percent
efficiency and system noise temperature of 1000 K meet the specified *G/T*? (d) By how
much does the system specified in parts (a) through (c) exceed the required carrier-to-
noise ratio?
*Solution*:

(a) Since ERP *t tPG*= and the power flux density 2( ) ,4
*t tPGr
r
*

ψ π

=

we have 2 7 2 ERP103 dB Wm

4 (3.6 10 )π −− =

×

7 2 10.3 5ERP 4 (3.6 10 ) 10 8.16 10 Wπ −∴ = × = × (*ans.*)

(b) 2

2 2

4 ERP 4 ( )( / 2)
*t
*

*A dG k k
P
*

π π π λ λ

= = =

2 5 2

2 2

ERP( )4 8.16 10 (0.025) 4 1.0 m
4 (100)(4)(3.14) (0.5)*t
*

*d
P k
*

λ π

×
= = = (*ans.*)

(c) For a 1-m receive antenna and 1000 K noise temperature,

2 1 1

2 2

4 4 ( )(1/ 2)0.5 5.48 K 7.4 dB K (0.03) (1000)

*eAG k
T T
*

π π π λ

− −= = = = (yes) (*ans.*)

(d) The *C*/*N* for a channel bandwidth *B* is

2 5

7 23 7

1 1 1 0.03 1 1ERP 8.16 10 (5.5) 4 (3.6 10 ) 1.38 10 2.7 10

52.9 17.2 dB

*C G
N L k T B *π −

⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = × ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟× × ×⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ = =

94

***12-3-18. continued
**
Therefore *C*/*N * exceeds requirement (14 dB) by 17.2 14 3.2 dB− = (*ans.*)

**12-3-19. Simplified expression for C/N.**The expression for

*C/N*provided in P12-3-15 may be simplified by making the following substitutions: Effective isotropic radiated power = ERP =

*PtGt*(W) Link path loss =

*L*link = 4π

*r*2/λ2 The carrier-to noise ratio may then be written as

sys

r

link

11ERP
*T
G
*

*kLN
C
*

=

where *G*r/*T*sys is the receive antenna gain divided by the system noise temperature. This
ratio, referred to as “*G over T*,” is commonly used as a figure of merit for satellite and
earth station receivers. Find the *C/N* ratio for the uplink to a satellite at the Clarke orbit
(*r* = 36,000 km) equipped with a 1 m parabolic dish antenna with efficiency of 50 percent
and a receiver with noise temperature of 1500 K. Assume that the transmitting earth
station utilizes a 1 kW transmitter and a 50 percent efficient 10 m dish antenna and
operates at a frequency of 6 GHz.
*Solution*:

22 7 191 1 4 4 (3.6 10 )ERP , 0.05 m, 8.18 10

0.05
*r
*

*sys
*

*GC rL
N L K T
*

π πλ λ

⎛ ⎞ ⎛ ⎞×⎛ ⎞⎛ ⎞ ⎛ ⎞= = = = = ×⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

2

2 2 8 1

2

4 ( )(5)0.5 4 ( )(5) (0.05)ERP 1000 0.5 1.97 10 W, 1.3 K

(0.05) 1500
*r
*

*t t
sys
*

*GPG
T
*

π π π π −⎛ ⎞= = = × = =⎜ ⎟

⎝ ⎠

8 19 23

11

1 1Therefore, (1.97 10 ) (1.3) 8.18 10 1.38 10

2.3 10 113.5 dB ( .)

*C
N
*

*ans
*

−

⎛ ⎞⎛ ⎞= × ⎜ ⎟⎜ ⎟× ×⎝ ⎠⎝ ⎠ = × =

95

**12-3-22. Galileo’s uncooperative antenna.
**When the Galileo spacecraft arrived at Jupiter in 1995, ground controllers had been
struggling for 3 years to open the spacecraft’s 5 m high-gain communications dish, which
was to operate at 10 GHz (X band). Unable to deploy this antenna because of prelaunch
loss of lubricant, a low-directivity (*G* = 10 dB) S-band antenna operating at 2 GHz had to
be used to relay all pictures and data from the spacecraft to the Earth. For a spacecraft
transmit power of 20 W, distance to Earth of 7.6 x 1011 m, and 70 m dish with 50 percent
efficiency at the receiving station, find (a) the maximum achievable data rate if the 5 m
X-band antenna had deployed and (b) the maximum data rate using the 1 m S-band
antenna.
*Solution*:

1 1ERP *r
sys
*

*GC
N L K T
*

⎛ ⎞⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠

(a) For the 5-m X-band antenna, 2

6 2

4 ( )(2.5)ERP 20 0.5 2.74 10 W
(0.03)*t t
*

*PG *π π
⎛ ⎞

= = = ×⎜ ⎟ ⎝ ⎠

22 11 294 4 (7.6 10 ) 1.01 10 ,

0.03
*rL *π π

λ ⎛ ⎞×⎛ ⎞= = = ×⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠

2

2 4 1

4 ( )(35)0.5 (0.03) 5.37 10 K 500

*r
*

*sys
*

*G
T
*

π π −= = ×

6 4 5

29 23

1 1Therefore, (2.74 10 ) (5.37 10 ) 1.05 10 1.01 10 1.38 10

*C
N *−

⎛ ⎞⎛ ⎞= × × = ×⎜ ⎟⎜ ⎟× ×⎝ ⎠⎝ ⎠

The maximum data rate 51.44 1.44(1.05 10 ) 152 kBits/s*CM
N
*

= = × ≅ (*ans.*)

***12-4-1. Antenna temperature with absorbing cloud.
**A radio source is occulted by an intervening emitting and absorbing cloud of unity
optical depth and brightness temperature 100 K. The source has a uniform brightness
dis-tribution of 200 K and a solid angle of 1 square degree. The radio telescope has an
effective aperture of 50 m2. If the wavelength is 50 cm, find the antenna temperature
when the radio telescope is directed at the source. The cloud is of uniform thickness and
has an angular extent of 5 square degrees. Assume that the antenna has uniform response
over the source and cloud.

96

***12-4-1. continued***Solution*:

From (12-4-1) and (12-1-6), (1 )*c cc sA c s
A A
*

*T T e T e*τ τ− −Ω ΩΔ = − +
Ω Ω

2

2

2 2

5 / 57.3 0.00152 sr

1/ 57.3 0.00030 sr

/ 0.5 / 50 0.005 sr

*c
*

*s
*

*A emA*λ

Ω = =

Ω = =

Ω = = =

Therefore, 1 1.00152 .0003100(1 ) 200 23.6 K
.005 .005*A
*

*T e e*− −Δ = − + = (*ans.*)

**12-4-3. Forest absorption.
**An earth-resource satellite passive remote-sensing antenna directed at the Amazon River
Basin measures a night-time temperature *TA* = 21°C. If the earth temperature *Te* = 27°C
and the Amazon forest temperature *Tf* = 15°C, find the forest absorption coefficient τ*f*.
*Solution*:

From (12-4-2), 294 288 0.5 and 0.693 300 288

*f A f
f
*

*f e
*

*T T
e
*

*T T
*τ τ−

Δ − − = = = =

− −

(*ans.*)

***12-4-4. Jupiter signals.
**Flux densities of 10-20 W m-2 Hz-1 are commonly received from Jupiter at 20 MHz. What
is the power per unit bandwidth radiated at the source? Take the earth-Jupiter distance as
40 light-minutes and assume that the source radiates isotropically.
*Solution*:

1 8 1 1140 light min 60 s min 3 10 m s 7.20 10 m*r *− −= × × × = ×

2

1 4

*tr PP
f f r*π

= × Δ Δ

or 2 20 2 22 14 10 4 7.2 10 65.1 kW Hz*t rP P r
f f
*

π π− −= × = × × × = Δ Δ

(*ans.*)

97

**12-5-1. Radar detection
**A radar receiver has a sensitivity of 10-12 W. If the radar antenna effective aperture is 1
m2 and the wavelength is 10 cm, find the transmitter power required to detect an object
with 5 m2 radar cross section at a distance of 1 km.
*Solution*:

4 2 12 3 4 2

2 2

4 10 4 (10 ) (0.1) 25 mW (1) 5

*r
t
*

*P rP
A
*π λ π

σ

− −× × = = =

×
(*ans.*)

***12-5-3. RCS of electron.
**The alternating electric field of a passing electromagnetic wave causes an electron
(initially at rest) to oscillate (Fig. P12-5-3). This oscillation of the electron makes it
equivalent to a short dipole antenna with *D* = 1.5. Show that the ratio of the power
scattered per steradian to the incident Poynting vector is given by 2 2( sin / 4 )*oe m*μ θ π ,
where *e* and *m* are the charge and mass of the electron and θ is the angle of the scattered
radiation with respect to the direction of the electric field **E** of the incident wave. This
ratio times 4π is the radar cross section of the electron. Such reradiation is called
*Thompson scatter*.

Figure P12-5-3.

*Solution*:

From (6-2-17), the magnitude of the electric dipole far-field is o 2 o

sin
4
*I lE
*

*c r*θ
ω θ

πε =

The force on the electron due to the incident field o*E * is

2
o*F eE ma m l*ω= = =

So, o2
*eEl
m*ω

=

Since

o 2

o 2 o

( ) sin ,

4

*eEe
mI e E
*

*c r*θ
ω ω θ

ωω πε

⎛ ⎞ ⎜ ⎟ ⎝ ⎠= =

Thus, the scattered power density 222 2

o 2

o

4 sin 14
4*scat
*

*r E e ES
Z c m Z
*

θπ θπ πε

⎛ ⎞ = = ⎜ ⎟

⎝ ⎠

98

***12-5-3. continued
**

and since the incident power density
2
o ,*inc
*

*ES
Z
*

= the ratio of scattered to incident power is

( )

2 o

22 22 22 2 o o 2 2 o o o o o

sin 14 4 sin1 sin 1 sin 1

4 4 1/ 4
*scat
*

*inc
*

*e E
c m ZS ee e
*

*ES c m m m
Z
*

θπ πε μ θθ θσ

π ε π ε μ ε π

⎛ ⎞ ⎜ ⎟ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎝ ⎠= = = = =⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠

For 28 2sin 1, 1 10 mθ σ −= ≅ × (*ans.*)

***12-5-5. Detecting one electron at 10 km.
**If the Arecibo ionospheric 300 m diameter antenna operates at 100 MHz, how much
power is required to detect a single electron at a height (straight up) of 10 km with an
SNR – 0 dB? See Fig. P12-5-5. The bandwidth is 1 Hz, *T*sys = 100 K and the aperture
efficiency = 50 percent.

Figure P12-5-5.

*Solution*:

2 4(4 )
*t t erPG AS
*

*N r kTB
*σ

π = ,

8

6

3 10 3 m, 100 10

λ ×= = ×

2 4 20.5 (150) 3.53 10 m*erA *π= = ×

4 4

2 2

4 4 (3.53 10 ) 4.93 10 3

*e
t
*

*AG *π π
λ

× = = = ×

Thus, 2 4 2 4 4 23

16 4 4 28

(4 ) ( / ) (4 ) (10 ) (1.38 10 )(100)(1)(1) 1.25 10 W
(4.93 10 )(3.53 10 )(1 10 )*t t er
*

*r kTB S NP
G A
*

π π σ

−

−

× = = = ×

× × ×
(*ans.*)

99

**12-5-6. Effect of resonance on radar cross section of short dipoles.
**(a) Calculate the radar cross section of a lossless resonant dipole (*ZL* = -*jXA*) with length =
λ/10 and diameter = λ/100. (See Secs. 2-9, 14-12).

(b) Calculate the radar cross section of the same dipole from ( )[ ]24

6

12ln 34

−*bL
L
*

λ
, where *L
*

is the dipole length and *b* is its radius.
(c) Compare both values with the maximum radar cross section shown in Fig. 12-9.
Comment on results.
*Solution*:
(a) 2 20.714 4 0.119 1.5λ λ= × × (*ans.*)

(b) 5 22 10 λ−× (*ans.*)
(c) 20.83λ (*ans.*)
Summary and comparison:

In (a) resonance is obtained by making .*L AZ jX*= − In (c) resonance is obtained
( 0)*AX *= by increasing the length to 0.47λ (with 0,*LZ *= terminals short-circuited).
The cross section in (c) is larger than in (a) because the dipole is physically longer.
In (b) the dipole is non-resonant because 0*LZ *= (terminals short-circuited) and the
length (0.1 )λ is much less than the resonant length (0.47 ),λ resulting in a very small
radar cross section.
It appears that if a short dipole (length 0.1 )λ≤ is resonated by making ,*L AZ jX*= − its
radar cross section 2(0.714 )λ approaches the cross section of a resonant / 2λ dipole
(length =0.47 λ ), regardless of how short it is, provided it is lossless.
A short dipole may be resonated ( )*L AZ jX*= − by connecting a stub of appropriate length
across its terminals,

Case σ Condition
(a) 20.714λ Resonant short dipole, length 0.1 ( )*L AZ jX*λ = −
(b) 5 22 10 λ−× Non-resonant 0.1λ dipole ( 0)*LZ *=
(c) 20.83λ Resonant 0.47 dipole ( 0)*LZ*λ =

(resonant dipole, )*s t L AA Z jX*σ= = −

*(matched dipole, )*em L AA Z Z*=
Directivity

100

**12-5-6. continued
**
Thus,
or by connecting lumped inductance, thus

***12-5-12 Fastball velocity.
**A 20 GHz radar measures a Doppler shift of 6 kHz on a baseball pitcher’s fastball. What
is the fastball’s velocity?
*Solution*:

8 3

9

2 (3 10 )(6 10 ), 45 m/s 162 km/h 101 mi/h 2 2(20 10 )

*f v c fv
f c f
*

Δ Δ × × = = = = = =

×
(*ans.*)

***12-5-13. Radar power for fastball measurement.
**To measure the velocity of the fastball of Prob. 12-5-12 with the 20 GHz radar at a
distance of 100 m, what power is required for an SNR = 30 dB? The radar uses a conical
horn with diameter = 8 cm and aperture efficiency εap = 0.5. The ball diameter = 7 cm
and it has a radar cross section (RCS) half that of a perfectly conducting sphere of the
same diameter.
*Solution*:

3 4 ,(4 )
*t t rPG GS
*

*N r kTB
*σ

π =

8

9

3 10 0.015 m, 20 10

λ ×= = ×

2 2

2 2

6.5 6.5 (0.04) 0.5 72.6
(0.015)*t r
*

*rG G kD k *π π
λ

= = = = = , 2 3 20.5 (0.035) 1.9 10 mσ π −= = ×

Dipole

Stub

101

***12-5-13. continued
**

3 4 3 4 23 6 3

t 2 3

(4 ) ( / ) (4 ) (100) (1.38 10 )(600)(10 )(10 ) 160 mW
(72.6) (1.9 10 )*t r
*

*r kTB S NP
G G
*

π π σ

−

−

× = = ≈

×
(*ans.*)

***12-5-14. Anticollision radar.
**To provide anticollision warnings, forward-looking radars on automobiles, trucks and
other vehicles (see Fig. P12-5-14) can alert the driver of vehicles ahead that are
decelerating too fast or have stopped. The brake light on the vehicle ahead may not be
working or it may be obscured by poor visibility. To warn of clear-distance decrease
rates of 9 m/s or more, what doppler shift must a 20 GHz radar be able to detect?

Figure P12-5-14.

*Solution*:

8

9

2 3 10, 0.015 m, 9 m/s, 20 10

2 2 2(9) 1.2 kHz ( .) 0.015

*f v c v
f c f
*

*vf vf ans
c
*

λ

λ

Δ × = = = = =

×

Δ = = = =

***12-5-18. Police radar.
**A pulsed speed measuring radar must be able to resolve the returns form two cars
separated by 30 m. Find the maximum pulse width that can be used to prevent
overlapping of the returns from the two vehicles.
Note that it takes *t = 2R/c* seconds for a signal to travel from the transmitter to a target
and return, where *t* is the time, *R* is the range and *c *is the velocity in the media. If τ is the
pulse width, then it can be shown that the *range resolution* (the minimum range
difference between objects for which the returns do not overlap in time) is given by

102

***12-5-18. continued
**

212
τ*cRRR *=−=Δ

where the subscripts denote the different objects.
*Solution*:

2or
2
*c RR
*

*c
*τ τ ΔΔ = =

The pulse width must be less than 88 2(10) 6.6 10 s 66 ns 3 10

τ −= = × = ×

(*ans.*)

***12-5-20. Sea clutter.
**Search-and-rescue aircraft using radar to locate lost vessels must contend with
backscatter from the surface of the ocean. The amplitude of these returns (know as *sea
clutter*) depends on the frequency and polarization of the radar waveform, the size of the
illuminated patch on the surface, the angle of incidence, and the sea state. The scattering
geometry is shown in Fig. P12-5-20. To characterize sea cluster independently on the
radar footprint on the surface, the scattering cross section of the ocean may be specified
per unit area. This parameter, designated σo, has dimensions of square meters of dB
above a square meter (dBsm). The total RCS of a patch of ocean surface is found by
multiplying σo by the area of the patch. For area scattering, the radar equation is written

2 2

o patch44
*e
*

*r t
AP P A
*

*r
*λ σ

π = .

For a pulsed radar with pulse width τ and 3 dB antenna beamwidth of θ rad, the
illuminated area for low grazing angle is approximately (*c*τ/2)(*r*θ). The radar equation
may therefore be written as

( ) 2 2 2 2

o o4 34 2 8

*e e
r t r
*

*A A ccP P r P
r r
*λ λ σ τθτσ θ

π π ⎛ ⎞= =⎜ ⎟ ⎝ ⎠

(a) Determine the received power from sea clutter at a range of 10 km for a monostatic pulsed radar transmitting a 1 μs pulse with 1 kW peak power at a frequency of 6 GHz. Assume a 1.5 m circular dish antenna with 50 percent efficiency and sea state 4 (σo = -30 dBsm/m2 at C band). (b) For a receiver bandwidth of 10 kHz and noise figure of 3.5 dB, find the receiver noise power. (c) If this radar used to search for a ship with RCS = 33

103

***12-5-20. continued
**dBsm under these conditions, what signal-to-noise and signal-to-clutter ratios can be
expected?

Figure P12-5-20. Sea search-and-rescue geometry.

*Solution*:

2 2 8 o 3 9

3 10, 0.05 m 8 6 10

*e
t r
*

*A cP P
r
*

λ σ τθ λ π

× = = =

×

(a) 3 dB
0.05 0.033 radians
1.5*D
*

λθ ≅ = =

2 2 2 30 10 8 6

15 4 3

[ (0.75) ] (0.05) (10 )(3 10 )(1 10 )(0.033)(1000) 3.1 10 W
8 (10 )*r
*

*P *π
π

− −
−× ×= = × (*ans.*)

(b) The receiver noise power is

3.5 10 23 4 17(10 )(1.38 10 )(290)10 8.96 10 W*n fP N kTB
*

− −= = × = × (*ans.*)
(c) For a ship with 233 dB m at 10 km.*r*σ = =

2 2 2 2 2 33 10 3

4 4 4

[ (0.75) ] (0.05) (10 )1000 1.24 10 W
4 4 (10 )
*e
*

*r t
AP P
*

*r
*λ σ π
π π

−= = = ×

Thus, the signal-to-noise (*S*/*N*) and signal-to-clutter (*S*/*C*) ratios are:

3 3

17 15

1.24 10 1.2 101384 31.4 dB ( .), 40 16 dB ( .) 8.96 10 3.1 10

*S Sans ans
N C
*

− −

− −

× × = = = = = =

× ×

104

**Chapter 13. Self and Mutual Impedances
**

***13-4-1. A 5**λ**/2 antenna.
**Calculate the self-resistance and self-reactance of a thin, symmetrical center-fed linear
antenna 5λ/2 long.
*Solution*:
From (13-5-2),

11 30[0.577 ln(2 ) Ci(2 ) Si(2 )], where 5*Z n n j n n*π π π= + − + =

Since 2 2 5 10 1,*n*π π π= = >> we have from (13-3-18), sin(10 )Ci(10 ) 0
10

ππ π

= =

And from (13-3-22), cos(10 ) 1Si(10 ) 1.539

2 10 2 10 π π ππ

π π = − = − =

and 11 30[0.577 ln(10 ) 0] 120.7 *R *π= + − = Ω (*ans.*)

11 30 1.539 46.2*X *= × = Ω (*ans.*)

**13-6-1. Parallel side-by-side **λ**/2 antennas.
**Calculate the mutual resistance and mutual reactance for two parallel side-by-side thin
linear λ/2 antennas with a separation of 0.15λ.
*Solution*:
From (13-7-6),

2 2 2 2
21 30{2Ci( ) Ci[ ( )] Ci[ ( )]}*R d d L L d L L*β β β= − + + − + −

where 0.15 , 0.5*d L*λ λ= =
and 21 30[2Ci(0.942) Ci(6.42) Ci(0.138)]*R *= − −
From (13-3-16) or Ci table or from Fig. 13-5 and from (13-3-18), we have

21 30(0.60 0 0.577 ln 0.138) 60.1*R *= − − − = Ω (*ans.*) (compare with Fig. 13-13)

105

**13-6-1. continued
**
From (13-7-7),

2 2 2 2
21 30{2Si( ) Si[ ( )] Si[ ( )]}*X d d L L d L L*β β β= − − + + − + −

and
21 30[2Si(0.942) Si(6.42) Si(0.138)]*X *= − − −

From (13-3-20) or Si table or from Fig. 13-5 and from (13-3-21), we have

11 30(1.8 1.42 0.138) 7.3 *X *= − − − = − Ω (*ans.*) (compare with Fig. 13-13)

***13-6-3. Three side-by-side antennas.
**Three antennas are arranged as shown in Fig. P13-6-3. The currents are of the same
magnitude in all antennas. The currents are in-phase in (a) and (c), but the current in (b)
is in anti-phase. The self-resistance of each antenna is 100 Ω, while the mutual
resistances are: *Rab* = *Rbc* = 40 Ω and *Rac* = -10 Ω. What is the radiation resistance of
each of the antennas? The resistances are referred to the terminals, which are in the same
location in all antennas.

Figure P13-6-3. Three side-by-side antennas.

*Solution*:

100 40 10 50 2 100 80 20 50

*a s ab ac
*

*b s ab
*

*c a
*

*R R R R
R R R
R R
*

= − + = − − = Ω = − = − = Ω = = Ω

(*ans.*)

**13-8-1. Two **λ**/2 antennas in echelon.
**Calculate the mutual resistance and reactance of two parallel thin linear λ/2 antennas in
echelon for the case where *d* = 0.25λ and *h* = 1.25λ (see Fig. 13-16).
*Solution*:
Use (13-9-1), (13-9-2), (13-3-16) and (13-3-20) where 1.25 and 0.25*h d*λ λ= =

106

**Chapter 14. The Cylindrical Antenna and the Moment Method
(MM)
**

**14-10-1. Charge distribution.
**Determine the electrostatic charge distribution on a cylindrical conducting rod with a
length-diameter ratio of 6.
*Solution*:

Divide rod into 6 equal segments. By symmetry charges are as shown. Neglecting end faces, the potential at point P12 is

3 31 2 2 1 12

1(P ) 4 2 2 10 26 50 82

*Q QQ Q Q QV
a a a a a a*πε

⎛ ⎞= + + + + +⎜ ⎟ ⎝ ⎠

Writing similar expression for 23 34(P ) and (P ),*V V * equating them (since the potential is
constant along the rod) and solving for the charges yields:

1 2 3: : 1.582 :1.062 :1.000*Q Q Q *= (*ans.*)

**14-12-2. **λ**/10 dipole impedance.
**Show that the convergence or true value of the self impedance *Zs* of the dipole of Table
14-4 is 1.852-*j*1895 Ω.
*Solution*:
Using *sZ * values of Table 14-4, plot vs. *sR N * to suitable large scale and suppressed zero
and note that *sR * approaches a constant (convergence) value as *N* becomes large which
should agree with Richmond’s value given following (14-12-34). Calculate *sR * for larger
values of *N* than 7, if desired. Do same for *sX *.

1*Q *2*Q * 3*Q *3*Q * 2*Q * 1*Q *

107

108

**Chapter 15. The Fourier Transform Relation Between Aperture
Distribution and Far-Field Pattern
**

***15-3-1. Pattern smoothing.
**An idealized antenna pattern-brightness distribution is illustrated by the 1-dimensional
diagram in Fig. P15-3-1. The brightness distribution consists of a point source of flux
density *S* and a uniform source 2° wide, also of flux density *S*. The point source is 2°
from the center of the 2° source. The antenna pattern is triangular (symmetrical) with a
2° beam width between zero points and with zero response beyond.
(a) Draw an accurate graph of the observed flux density as a function of angle from the
center of the 2° source.
(b) What is the maximum ratio of the observed to the actual total flux density (2*S*)?

Figure P15-3-1. Pattern smoothing.

*Solution*:

Ratio = 1/2 (*ans.*)

**15-6-1. Number of elements.
**In Fig. 15-15 how many elements *n* have been assumed?
*Solution*:

From Fig. 15-15, 2 1 1
4*nd d*λ λ

= ×

or* n* = 8

109

110

**Chapter 16. Arrays of Dipoles and of Apertures
**

***16-2-1. Two **λ**/2-element broadside array.
**(a) Calculate and plot the gain of a broadside array of 2 side-by-side λ/2 elements in free
space as a function of the spacing *d* for values of *d* from 0 to 2λ. Express the gain with
respect to a single λ/2 element. Assume all elements are 100 percent efficient.
(b) What spacing results in the largest gain?
(c) Calculate and plot the radiation field patterns for λ/2 spacing. Show also the patterns
of the λ/2 reference antenna to the proper relative scale.
*Solution*:
(a) From (16-2-27),

1 2
00 11 12( )( / ) [2 /( )] cos[( cos ) / 2]*f rG A HW R R R d*φ φ= +

In broadside direction / 2φ π=
so 1 200 11 12( )( / ) [2 /( )]*fG A HW R R R*φ = +
where 00 11 73.1 *R R*= = Ω
and 12*R * is a function of the spacing as given in Table 13-1 (p. 453). A few values of the
gain for spacings from 0 to 1 λ are listed below:

Spacing λ Gain over / 2λ reference 0.0 1.00 0.1 1.02 0.2 1.08 0.3 1.19 0.4 1.36 0.5 1.56 0.6 1.72 0.7 1.74 0.8 1.64 0.9 1.49 1.0 1.38 etc.

Note that
2( / 2) 4 ( / 2) / 4 (30 / 73.1 ) 1.64 or 2.15 dBi*emD A*λ π λ λ π π= = =

111

***16-2-1. continued
**
so *D* of 2 in-phase / 2λ elements at 0.67 λ spacing is equal to 4.9 2.15 7.1+ = dBi as
above.
(b) By interpolation, the highest gain occurs for a spacing of about 0.67 λ for which the
gain is about 1.76 or 4.9 dB (= 7.1 dBi). At spacings over 1λ no gains exceed this.

**16-3-1. Two **λ**/2-element end-fire array.
**A 2-element end-fire array in free space consists of 2 vertical side-by-side λ/2 elements
with equal out-of-phase currents. At what angles in the horizontal plane is the gain equal
to unity:
(a) When the spacing is λ/2?
(b) When the spacing is λ/4?
*Solution*:
(a) From (16-3-18),

1 2
00 11 12( )( / ) [2 /( )] sin[( cos ) / 2]*f rG A HW R R R d*φ φ= −

When / 2,*d *λ=
1 2

00 11 12( )( / ) [2 /( )] sin[( / 2)cos )]*fG A HW R R R*φ π φ= −
where 00 11 73.1 *R R*= = Ω
and from Table 13-1, 12 12.7 *R *= − Ω
so ( )( / ) 1.31sin[( / 2)cos )]*fG A HW*φ π φ=
For unit gain, sin[( / 2)cos )] 1/1.31 0.763π φ = =
or o ocos 49.8 / 90 0.553φ = = and o o56 , 124φ = ± ± (*ans.*)
(b) When / 4,λ

1 2
00 11 12( )( / ) [2 /( )] sin[( / 4)cos ]*fG A HW R R R*φ π φ= −

where 00 11 73.1 *R R*= = Ω
and from Table 13-1, 12 40.9 *R *= Ω

112

**16-3-1. continued
**
so ( )( / ) 2.13sin[( / 4)cos ]*fG A HW*φ π φ=
For unit gain, sin[( / 4)cos ] 1/ 2.13 0.47π φ = =

o o o ocos 28 / 45 0.62 and 52 , 128φ φ= = = ± ± (*ans.*)

***16-3-2. Impedance and gain of 2-element array.
**Two thin center-fed λ/2 antennas are driven in phase opposition. Assume that the current
distributions are sinusoidal. If the antennas are parallel and spaced 0.2λ,
(a) Calculate the mutual impedance of the antennas.
(b) Calculate the gain of the array in free space over one of the antennas alone.
*Solution*:
(a) This is a single-electron W8JK array.
From Sec. 13-7, 12 52 21 *Z j*= − Ω (*ans.*)
(b) From (16-5-8) and assuming losses,

1 2 o2 73( )(max)( / ) sin 36

73 52 2.64 0.588 1.55 or 3.8 dB (= 6.0 dBi) ( .)

*fG A HW
*

*ans
*

φ ×⎛ ⎞= ⎜ ⎟−⎝ ⎠ = × =

**16-4-3. Two-element array with unequal currents.
**(a) Consider two λ/2 side-by-side vertical elements spaced a distance *d* with currents
related by *I*2 = *aI*1/δ. Develop the gain expression in a plane parallel to the elements and
the gain normal to the elements, taking a vertical λ/2 element with the same power input
as reference (0 ≤ *a* ≤ 1). Check that these reduce to Eq. (16-4-15) and Eq. (16-4-13)
when *a* = 1.
(b) Plot the field patterns in both planes and also show the field pattern of the reference
antenna in proper relative proportion for the case where *d* = λ/4, *a* = ½ and δ = 120°.
*Solution*:
(a) 2 1 2 2 1 211 11 12( ) { /[ (1 ) 2 cos ]} (1 2 cos )*fG R R a aR a a*θ δ ψ= + + × + +

113

**16-4-3. continued
**
where sin*rd*ψ θ δ= +

( ) ( ) but with cos*f f rG G d*φ θ ψ φ δ= = +

***16-6-1. Impedance of D-T array.
**(a) Calculate the driving-point impedance at the center of each element of an in-phase
broadside array of 6 side-by-side λ/2 elements spaced λ/2 apart. The currents have a
Dolph-Tchebyscheff distribution such that the minor lobes have 1/5 the field intensity of
the major lobe.
(b) Design a feed system for the array.
*Solution*:
(a) From Prob. 5-9-5 and the 6 sources have the distribution:

1 2 3 4 5 6 0.93 0.84 1.00 1.00 0.84 0.93

Normalizing the current for element 1, the distribution is

1.00 0.90 1.08 1.08 0.90 1.00

Using impedance data from Chap. 13 and assuming thin elements, the driving point impedance of element 1 is

1

6

73 43 0.9( 12 29) 1.08(3 18) 1.08( 2 12) 0.9(1 10) 1 3 73 10.8 3.2 2 0.9 1 (43 26 19.4 13 9 3) 63 29 = ( .)

*R j j j j j j
j j R ans
*

= + + − − + + + − − + + − − = − + − + − + − + − + − = − Ω

In like manner,
2 5 3 446 2 , ( .) 53 10 , ( .)*R R j ans R R j ans*= = − Ω = = − Ω

**16-6-3. Square array.
**Four isotropic point sources of equal amplitude are arranged at the corners of a square, as
in Fig. P16-6-3. If the phases are as indicated by the arrows, determine and plot the far-
field patterns.

114

**16-6-3. continued
**

Figure 16-6-3. Square array.

*Solution*:
Pattern is a rounded square.

o o o

o o

1.00 at 0 , 90 , 180

0.895 at 45 , 135
*n
*

*n
*

*E
*

*E
*

φ

φ

= = ±

= = ± ±

***16-6-4. Seven short dipoles. 4-dB angle.
**A linear broadside (in-phase) array of 7 short dipoles has a separation of 0.35λ between
dipoles. Find the angle from the maximum field for which the field is 4 dB (to nearest
0.1°).
*Solution*:
The dipoles are assumed to be aligned collinearly so that the pattern of a single dipole is
proportional to sinφ where φ is the angle from the array. Thus
Since the dipoles are in-phase, the maximum field is at o omax90 or ( ) 90 .*E*φ φ= =
The normalized pattern is given by

1 sin / 2 sin
sin / 2*n
*

*nE
n
*

ψ φ ψ

= (1)

where *n* = 7
(2 / ) 0.35 cos , 4 20log , 1.585*x x*ψ π λ λ φ= × = =

0.35λ

φ

115

***16-6-4. continued
**
Therefore, ( 4 dB) 1/1.585 0.631*nE *− = =
Setting (1) equal to 0.631, *n* = 7 and solving (see note below) yields o( 4 dB) 78.3φ − =
Angle from max( )*E*φ is

o o o90 78.3 11.7− = (*ans.*)
Note: Use trial and error to solve (1) for ( 4 dB)φ − or calculate pattern with small
increments in φ .

**16-6-5. Square array.
**Four identical short dipoles (perpendicular to page) are arranged at the corners of a
square λ/2 on a side. The upper left and lower right dipoles are in the same phase while
the 2 dipoles at the other corners are in the opposite phase. If the direction to the right (*x*
direction) corresponds to φ = 0°, find the angles φ for all maxima and minima of the field
pattern in the plane of the page.
*Solution*:
Pattern maxima at o o45 , 135φ = ± ± Pattern minima at o o o0 , 90 , 180φ = ±

***16-6-7. Sixteen-source broadside array.
**A uniform linear array has 16 isotropic in-phase point sources with at spacing λ/2.
Calculate exactly (a) the half-power beam width, (b) the level of the first sidelobe, (c) the
beam solid angle, (d) the beam efficiency, (e) the directivity and (f) the effective
aperture.
*Solution*:
(a) From (5-6-9),

o

o

1 sin(1440 cos )(HP) 0.707 16 sin(90 cos )

*E *φ
φ

= =

By trial and error, o o o o o86.82 and HPBW 2(90 86.82 ) 6.36 6 22 ( .)*ans*φ ′= = − = =
(b) From (5-18-10),
*K* = 1 (first minor lobe)

1 0.215 or -13.3 dB
16sin[(2 1) / 32]*ML
*

*E
*π

≅ = +

116

***16-6-7. continued
**
This is only approximate (becomes exact only for very large *n*).
To determine the level more accurately, we find the approximate angle for the maximum
of the first minor lobe from (5-18-5).

1 1 o(2 1) 3cos cos 79.212 2 16 2

*m
K
nd*λ

φ − −± + ±≅ = = × ×

Then from (5-6-9) we calculate *E* at angles close to 79.2o and find that *E* peaks at 79.7o
with 0.22012 or -13.15 dB ( .)*E ans*=
Although (5-18-5) locates the angle where the numerator of (5-18-5) is a maximum (= 1),
the denominator is not constant. See discussion of Sec. 5-18 (p. 159) and also Fig. 5-47
(p. 100).
(c) From equation for *D* in Prob. 5-6-10, the summation term is zero for / 2*d *λ= so
that *D* = 16 exactly.
Since 4 / , 4 / 4 /16 / 4 sr*A AD D*π π π π= Ω Ω = = = (*ans.*)
(d) HPBW 1/ 1/(16 0.5) 1/ 8 rad in direction*nd*λ φ≅ = × = ,
BW in θ direction 2 radπ=
Therefore, 2 (1/ 8) / 4 sr and / 1 or 100%*M M M A*π π εΩ = × = = Ω Ω =
This result is too large since with any minor lobes *M*ε must be less than unity (or

*M A*Ω < Ω ).
For an exact evaluation, we have from Prob. 5-6-10 that

2

1

1

2 1

2 cos sin(2 cos )
*n
*

*M
k
*

*n kn d kd
n d k
*

θ

λ λ λ θ

π θ π θ −

=

−⎡ ⎤ Ω = +⎢ ⎥⎣ ⎦

∑ o

1 o o

2 o

o

where 90

90 angle to first null

θ γ

θ γ γ

= −

= + =

From (5-7-7), 1 1 1 oo sin (1/ ) sin [1/(16 0.5)] sin (1/ 8) 7.18*nd*λγ

− − −= = × = = Therefore, o o1 282.82 and 97.18θ θ= =

117

***16-6-7. continued
**Thus

o

o

97.181

2 1 90

4 cos sin(2 cos )
*n
*

*M
k
*

*n kn d kd
n d k*λ λλ

π θ π θ −

=

−⎡ ⎤ Ω = +⎢ ⎥⎣ ⎦

∑ 1

2 2 1

2 0.5 4 150.125 sin(0.25 ) sin(0.125 ) 16 16 0.5 1

*n
*

*M
k
*

*n kn d kd
n d k*λ λλ

ππ π π −

=

−⎡ ⎤ ⎡Ω = + = +⎢ ⎥ ⎢× ⎣⎣ ⎦ ∑

1014 13 12 11 25 ) sin(0.75 )sin(0.25 ) sin(0.375 ) sin(0.5 ) sin(0.6 62 3 4 5

π ππ π π ++ + + +

9 8 7 6 5sin(0.875 ) sin(1.00 ) sin(1.25 ) sin(1.375 ) sin(1.5 ) 7 8 9 10 11

π π π π π+ + + + +

4 3 2 1sin(1.625 ) sin(1.75 ) sin(1.875 ) sin(2.00 ) 12 13 14 15

π π π π ⎤+ + + + ⎥⎦

*M*Ω = 0.0982 + 0.03125 [5.740 + 4.950 + 4.003 + 3.000 + 2.033 + 1.179 + 0.492

+ 0 – 0.550 – 0.554 – 0.455 – 0.308 – 0.163 – 0.055 +0]
= 0.0982 +0.03125 19.312 0.702 sr = *M*× = Ω

/ 0.702 /( / 4) 0.894*M M A*ε π= Ω Ω = = (*ans.*)

By graphical integration (see Example 4-5.6 and Fig. 4-8b) *M*ε was found to be
approximately 0.90, in good agreement with the above result. The graphical integration
took a fraction of the time of the above analytical integration and although less accurate,
provided confidence in the result because it is much less susceptible to gross errors.
(e) As noted in (c), 16*D *= (*ans.*)
(f) From 24 /*emD A*π λ= ,

2 2 2/ 4 16 / 4 1.27*emA D*λ π λ π λ= = = (*ans.*)

***16-8-6. Four-tower broadcast array.
**A broadcast array has 4 identical vertical towers arranged in an east-west line with a
spacing *d* and progressive phase shift δ. Find (a) *d* and (b) δ so that there is a maximum
field at φ =45° (northeast) and a null at φ =90° (north). There can be other nulls and
maxima, but no maximum can exceed the one at 45°. The distance *d* must be less than
λ/2.
*Solution*:

φ EW

*d*

118

***16-8-6. continued
**(a)
Null at o90φ = requires that o o90 or 180δ = ± ±
For maximum field (fields of all towers in phase) set

o

maxcos 0 and 90 / 2 rad*d*ψ β φ δ δ π= + = = − = −

so o max

/ 2 0.354 cos (2 / )cos 45

*d *δ π λ
β φ π λ

= − = = (*ans.*)

If o180 rad,δ π= − = − o 0.707 ,(2 / ) cos 45
*d *π λ

π λ = = but this exceeds 0.5λ

(b) Therefore, o90δ = − (*ans.*)

**16-10-1. Eight-source scanning array.
**A linear broadside array has 8 sources of equal amplitude and λ/2 spacing. Find the
progressive phase shift required to swing the beam (a) 5°, (b) 10° and (c) 15° from the
broadside direction. (d) Find BWFN when all sources are in phase.
*Solution*:
Broadside is set at o90 .φ = Set maxcos 0*d*ψ β φ δ= + =

Therefore, o omax 2 2cos cos95 15.7

2
*d*π π λδ φ

λ λ
= − = − = + (*ans.*)

o o o180 cos85 15.7δ = − = − Thus, depending on whether o ois +15.7 or 15.7 ,δ − the beam is o5 left or right of broadside. In the same way, we have

*d d
*

o

o

90 0

φ

γ

=

= γ φ

119

**16-10-1. continued
**

(b) o o31.3 ( .) for beam 10*ans*δ = ± left or right of broadside
(c) o o46.6 ( .) for beam 15*ans*δ = ± left or right of broadside
(d) From (5-7-7) the angle of the first null from broadside, when the sources are in-
phase ( 0),δ = is given by the complementary angle

1 1 o
o o90 sin ( / ) sin (1/ 4) 14.48*nd*γ φ λ

− −= − = = =
Therefore, BWFN o2 14.48 28.96 29= × = ≅ (*ans.*)
From the long broadside array equation (5-7-10),

BWFN o o2 / 1/ 2 rad 180 / 2 28.65*nd*λ π≅ = = =
The HPBW is a bit less than BWFN/2. For long broadside arrays, we have from Table
5-8 (p.155) that
o oHPBW 50.8 / 50.8 / 3.5 14.5*L*λ= = =

***16-16-1. Terminated V. Traveling wave.
**(a) Calculate and plot the far-field pattern of a terminated-V antenna with 5λ legs and 45°
included angle.
(b) What is the HPBW?
*Solution*:
(a) The *field pattern* for each leg of the V is shown at the left and the *combined
field pattern* at the right. Minor lobes are neglected except for the principal side lobe of
the V.
(b) HPBW ≅ 17o (*ans.*)

120

***16-16-2. E-type rhombic.
**Design a maximum *E*-type rhombic antenna for an elevation angle α = 17.5°.
*Solution*:
From Table 16-1 (p. 590) for a maximum *E* rhombic,

o

o o

2

1/(4sin ) 1/(4sin17.5 ) 0.83 ( .)

90 72.5 ( .) 0.5 / sin 5.5 ( .)

*H ans
*

*ans
L ans
*

λ

λ

α

φ α

α

= = =

= − =

= =

**16-16-3. Alignment rhombic.
**Design an alignment-type rhombic antenna for an elevation angle α = 17.5°.
*Solution*:
From Table 16-1 (p. 590) for an alignment rhombic,

o

o o

2

1/(4sin17.5 ) 0.83 ( .)

90 72.5 ( .) 0.371/ sin 4.1 ( .)

*H ans
*

*ans
L ans
*

λ

λ

φ α

α

= =

= − =

= =

***16-16-4. Compromise rhombic.
**Design a compromise-type rhombic antenna for an elevation angle α = 17.5° at a height
above ground of λ/2.
*Solution*:
From Table 16-1 (p. 590) for a compromise rhombic,

o o o

2 o

o o o

0.5 ( .)

90 17.5 72.5 ( .)

tan[( )sin 17.5 ] 1 0.5 sin17.5 2 sin17.5 tan( sin17.5 )

*H ans
*

*ans
*

*LL
*

λ

λ λ

φ

π π π

=

= − =

⎛ ⎞ = −⎜ ⎟

⎝ ⎠

121

***16-16-4. continued
**or

o 0.56tan(16.3 )
*L
*

*L
*λ

λ

=

By trial and error, 5.14*L*λ = (*ans.*)

**16-16-5. Compromise rhombic.
**Design a compromise-type rhombic antenna for an elevation angle α = 17.5° with leg
length of 3λ.
*Solution*:
From Table 16-1 (p. 590),

o

1 o o

1/(4sin17.5 ) 0.83 ( .) 3 0.371sin 67 ( .) 3cos17.5

*H ans
*

*ans
*

λ

φ −

= =

−⎛ ⎞= =⎜ ⎟ ⎝ ⎠

***16-16-6. Compromise rhombic.
**Design a compromise-type rhombic antenna for an elevation angle α = 17.5° at a height
above ground of λ/2 and a leg length of 3λ.
*Solution*:
From Table 16-1 (p. 590, bottom entry),

1 sin tan tan(2 sin ) 4 tan( 2 )

*H L
H L
*

λ λ

λ λφ α π α πψ ψ π = −

where (1 sin cos ) / 2ψ φ α= −
By trial and error, o60φ ≅ (*ans.*)

122

**Chapter 17. Lens Antennas
**

**17-2-1. Dielectric lens.
**(a) Design a plano-convex dielectric lens for 5 GHz with a diameter of 10λ. The lens
material is to be paraffin and the *F* number is to be unity. Draw the lens cross section.
(b) What type of primary antenna pattern is required to produce a uniform aperture
distribution?
*Solution*:

8 9( ) 3 10 /(5 10 ) 0.06 m 60 mm 1 so 10 600 mm ( diameter) 1.4 (see Table 17-1)

*a
F L d d
n
*

λ λ

= × × = = = = = = = =

Therefore from (17-2-7),

(1.4 1)600 1.4cos 1

*R
*θ

− =

−

θ *R * sin*R *θ

o0 600 mm 0 mm
o10 634 110
o20 761 260
o22 805 300 / 2*d*=∼

(b) From (17-2-14), power density at edge of lens is

o 3

2 o o

(1.4cos 22 1) 0.35 or 4.6 dB down (1.4 1) (1.4 cos 22 )

*S
S
*

θ −= = − −

Focus 600 mm*L *= Lens
(lower-half mirror image)

/ 2 300 mm*d *=
o22θ =

*R *

123

**17-2-1. continued
**
To reduce side lobes, this much or even more taper may be desirable. To obtain a
uniform aperture distribution, as requested in the problem, requires a feed antenna at the
focus with more radiation (up about 4.6 dB) at 22o off axis than on axis. This is difficult
to achieve without unacceptable spillover unless the lens is enclosed in a conical horn,
except that at edge locations where *E* is parallel to the edge, *E* must be zero. To reduce
this effect a corrugated horn could be used.

**17-3-1. Artificial dielectric.
**Design an artificial dielectric with relative permittivity of 1.4 for use at 3 GHz when the
artificial dielectric consists of (a) copper spheres, (b) copper disks, (c) copper strips.
*Solution*:
From Table 17-2,
(a) 3(sphere) 1 4 1.4*r Na*ε π= + =
At 3 GHz, 8 1 93 10 ms / 3 10 Hz 0.1 m 100 mmλ −= × × = =
For , take a (radius) 5 mm from which*a *λ<< =

3 3 3 3

1.4 1 0.4 255,000 m 4 4 (5 10 )

*N
a*π π

− −

− = = =

×
(*ans.*)

The dielectric volume per sphere = 1/255,000 6 34 10 m−= ×
While the volume of each sphere is given by 3 3 3 7 3(4 / 3) (4 / 3) (5 10 ) 5.2 10 m*a*π π − −= × = ×

Therefore, 6

7

volume of dielectric 4 10 7.7 volume of sphere 5.2 10

−

−

× = =

×

6 1 3 2(4 10 ) 1.59 10 15.9 mm = side of cube versus sphere diameter = 2 5 10 mm− −× = × = × =

Thus, there is 15.9 10 5.9 mm− = between adjacent spheres in a cubical lattice so there is room for the spheres without touching, provided the lattice uniform.

124

**17-3-1. continued
**
(b) 3(discs)=1+5.33*r Na*ε , and taking (radius) 5 mm (diameter = 10 mm),*a *=

3

3 3

1.4 1 600,000 m 5.33(5 10 )

*N *−−
−

= = ×

(*ans.*)

The dielectric volume per disc 6 31/ 600,000 1.7 10 m−= = × for a cube side length of

6 1 3(1.7 10 ) 12 mm−× ≅ , so that there is 12 10 2 mm− = minimum spacing between
adjactent discs in a uniform lattice.
(c) 2(strips)=1+7.85*r Nw*ε
Taking *w* (width) = 10 mm,

2 2 3

1.4 1 51,000 m 7.85(10 )

*N *−−
−

= = (*ans.*)

as viewed in cross section (see Fig. 17-8a). The square area per strip is then 1/ 51,000

5 22 10 m−= × for a cross-sectional area side length 5 1 2(2 10 ) 4.5 mm.−× =

This is less than the strip width. However, if the square is changed to a rectangle of the same area with side length ratio of 9 as in the sketch, the edges of the strips are separated by 3.5 mm and the flat sides by 1.5 mm. The above answers are not unique and are not necessarily the best solutions.

***17-4-1. Unzoned metal-plate lens.
**Design an unzoned plano-concave *E*-plane type of metal plate lens of the unconstrained
type with an aperture 10λ square for use with a 3 GHz line source 10λ long. The source
is to be 20λ from the lens (*F* = 2). Make the index of refraction 0.6.
(a) What should the spacing between the plates be?
(b) Draw the shape of the lens and give dimensions.
(c) What is the bandwidth of the lens if the maximum tolerable path difference is λ/4?

10 mm

Strip

Rectangular area

Cross section

125

***17-4-1. continued***
Solution*:

8 93 10 /(3 10 ) 0.1 m 100 mm 0.6, 2 so / 2 (Fig.17-13)

*n F A L
*λ = × × = =

= = =

(b) Expressing dimensions in λ , we have from (17-4-4)

(1 ) (1 0.6)20 8 1 cos 1 0.6cos 1 0.6cos

*n LR
n
*

λ λ θ θ θ

− − = = =

− − −

θ *R*λ sin*R*λ θ

o0 20 0 o10 19.6 3.4

o15.25 19.0 5.0
(a)From (17-4-2), 2 1 2 2 1 2o o[1 ( / 2 ) ] or / 2(1 )*n b b n*λ λ= − = −
For (*ans.*)
(c) From (17-4-12),

Bandwidth 2 2

(1 )
*n
n t
*

λ

λ

δ −

ocos 20 19cos15.25 1.67*t L R*λ λ λ θ= − = − =
(a) From (17-4-2), 2 1 2 2 1 2o o[1 ( / 2 ) ] or / 2(1 )*n b b n*λ λ= − = −
For o0.6, 0.625 62.5 mm*n b *λ= = = (*ans.*)
(b) From (17-4-12), Bandwidth 22 /(1 )*n n t*λ λδ= −

ocos 20 19cos15.25 1.67*t L R*λ λ λ θ= − = − =

Therefore, Bandwidth 2 2 0.6 0.25 0.28 or 28% (1 0.6 )1.67

× × = =

−
(*ans.*)

5λ Lens
(lower-half
mirror image) 20*L *λ= o15.25θ =

*t
*

*R *ine source with
to line⊥

**E**

126

**
Chapter 18. Frequency-Selective Surfaces and Periodic
**

**Structures. By Ben A. Munk
**

**18-9-1. Unloaded tripole.
**Determine the approximate length of the legs of an unloaded trislot operating at *f* = 15
GHz with
(a) No dielectric substrate.
(b) Dielectric substrate εr = 2.2 and thickness 0.50 mm located on both sides of the FSS
(use arithmetic average of εr and εo for εeff).
(c) Determine *Dx* just short enough that no grating lobes are present when scanning in the
*xy* – plane for any angle of incidence.
*Solution*:
(a) For one leg of the tripole, i.e., the monopole length

2.0 0.5 cm

4 4 λ

= = (*ans.*)

(b) eff 2.2ε = since it is the same on both sides

eff 2 2 1.35 cm

1.482.2 λ = = =

eff 1.35 0.337 cm 4 4

λ
= = (*ans.*)

(c) For the scattering case, (15-6-1) can be written as

sin sin
/*i s x
*

*m
D
*

η η λ

+ =

Since grating lobes start in the plane of the array, o90 and 1*i s m*η η= = =
So 2 / , / 2 2 / 2 1 cm*x xD D*λ λ= = = = (*ans.*)

**18-9-2. Four-Legged loaded element.
**Determine the approximate dimensions for a four legged loaded element operating at *f* =
15 GHz with
(a) No dielectric substrate.
(b) Dielectric substrate εr = 2.2 and thickness 0.50 mm located on only one side the FSS
(estimate εeff).

127

**18-9-2. continued
**
(c) Leave a separation of 1 mm between adjacent elements (rectangular grid); determine
the lowest onset frequency for grating lobes for any angle of incidence.
*Solution*:
(a) For a loop type element, the size should be eff / 4λ across.
So / 4 2.0 / 4 0.50 cmλ = = (*ans.*)

(b) ff 2.2 1.0 3.2 1.6

2 2*e
*ε += = = , eff 2 / 1.6 2 /1.265 1.58 cmλ = = =

eff / 4 0.4 cmλ = (*ans.*)

(c) As in Prob. 18-9-1, the condition we want to meet is

2 / or 2*x xD D*λ λ= =

With no dielectric, 0.5 0.1 0.6 cm*xD *= + =

so 8

2

3 101.2 cm, 25 GHz 1.2 10

*f*λ −
×

= = = ×

(*ans.*)

With dielectric, 0.4 0.1 0.5 cm*xD *= + =

so 8

2

3 101 cm, 30 GHz 1 10

*f*λ −
×

= = = ×

(*ans.*)

128

**Chapter 19. Practical Design Considerations of Large Aperture
Antennas
**

***19-1-3. Efficiency of rectangular aperture with partial taper.
**Calculate the aperture efficiency and directivity of an antenna with rectangular aperture
*x*1*y*1 with a uniform field distribution in the *y* direction and a cosine field distribution in
the *x* direction (zero at edges, maximum at center) if *x*1 = 20λ and *y*1 = 10λ.
*
Solution*:
From Prob. 19-1-6 solution,
(a) 0.81 or 81%*ap*ε = (*ans.*)
(b) 4 10 20 0.81 2036 or 33 dBi*D *π= × × × = (*ans.*)

***19-1-4. Efficiency of rectangular aperture with full taper.
**Repeat Prob. 19-1-3 for the case where the aperture field has a cosine distribution in both
the *x* and *y* directions.
*
Solution*:
From Prob. 19-1-7 solution,
(a) 0.657 66%*ap*ε = ≅ (*ans.*)
(b) 4 10 20 0.657 1651 or 32 dBi*D *π= × × × = (*ans.*)

**19-1-5. Efficiency of aperture with phase ripple.
**A square unidirectional aperture (*x*1*y*1) is 10λ on a side and has a design distribution for
the electric field which is uniform in the *x* direction but triangular in the *y* direction with
maximum at the center and zero at the edges. Design phase is constant across the
aperture. However, in the actual aperture distribution there is a plus-and-minus-30°
sinusoidal phase variation in the *x* direction with a phase cycle per wavelength. Calculate
(a) the design directivity, (b) the utilization factor, (c) the actual directivity, (d) the
achievement factor, (e) the effective aperture and (f) the aperture efficiency.
*
Solution*:
Referring to Sec. 19-1,

129

**19-1-5. continued
**
Let max max

Design field Actual field
( , ) ( , ) 1*E x y E x y*′ = =

Design: 1 ( , )*av
p
*

*E E x y dxdy
A
*

′ ′= ∫∫ (1)

10 5

0 0

2 1
5 2*p
*

*y dxdy
A
*

λ λ

λ = =∫ ∫

Note: This result can be deduced directly from
the figure by noting that average height of triangle
is ½ max.
(b) Utilization factor, *uk *:

* 1

1 ( , ) ( , )
*u
*

*p av av
*

*k
E x y E x y dxdy
*

*A E E
*

= ⎛ ⎞⎛ ⎞′ ′ ⎜ ⎟⎜ ⎟′ ′⎝ ⎠⎝ ⎠

∫∫ (2)

2 10 5

2 0 0

1 3 ( .) 41 2

(1/ 2) 5*p
*

*ans
y dxdy
*

*A
*λ λ

λ

= = ⎛ ⎞ ⎜ ⎟ ⎝ ⎠∫ ∫

(3)

Note that for in-phase fields (19-1-50) is a simplified form of (2) giving

( )

2 2

2

(1/ 2) 3 / 4 as in (3) 1/ 3

*av
u
*

*av
*

*E k
E
*

= = = (4)

(a) Design directivity, (design):*D *

22 2
4 4(design) (100 )(3 / 4) 940 ( .)*p uD A k ans
*π π λ

λ λ = = = (5)

Turning attention now to the effect of the phase variation:

10 5

0 0

2 2 1cos sin (0.933)
6 5 2*av p
*

*x yE dx dy
A
*

λ λπ π λ λ

⎛ ⎞= =⎜ ⎟ ⎝ ⎠∫ ∫ (6)

*E
*

10λ

10λ

*y
*

*x*

130

**19-1-5. continued
**

Note that from figures above, 1 0.8662 0.933
2*av
*

*E *+≅ =

(d) Achievement factor, :*ak *

1 ( , ) ( , )

1 ( , ) ( , )
*p av av
*

*a
*

*p av av
*

*E x y E x y dxdy
A E E
*

*k
E x y E x y dxdy
*

*A E E
*

∗

∗

⎛ ⎞⎛ ⎞′ ′ ⎜ ⎟⎜ ⎟′ ′⎝ ⎠⎝ ⎠= ⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

∫∫

∫∫

2

2

4 / 3 0.87 ( .)1 1 2 ( / 5 ) [(1/ 2)0.933]

*a
*

*p
*

*k ans
y dxdy
*

*A
*λ

= =

∫∫ (7)

where *( ) ( ) 1*E x E x *=
Note that gain loss due to total phase variation across aperture (*not* surface deviation) is
from (19-2-3)

2 ocos 360*gk
*δ
λ
′⎛ ⎞= ⎜ ⎟

⎝ ⎠ , where

o o

o o

30 0.707 21.2 360 360

δ λ ′ ×

= =

or 2 ocos 21.2 0.87 as in (7)*g ak k*= = =

( , )*E x y **avE *

*x **E *

131

**19-1-5. continued
**
(c) Directivity:

2

4 34 100 0.87 818 ( .) 4

*p
u a
*

*A
D k k ans
*

π π

λ = = × × × =

(e) Effective aperture, :*eA *

2 265.2 ( .)

4*e p u a
A D A k k ans*λ λ

π = = =

(f) Aperture efficiency, :*ap*ε
0.65*ap a uk k*ε = = (*ans.*)

Note: Although phase errors with small correlation distance ( )λ≅ as in Prob. 19-1-5
reduce the directivity and, hence, increase *A*Ω , the HPBW is not affected appreciable.
However, for larger correlation distances ( )λ>> the scattered radiation becomes more
directive, causing the near side lobes to increase and ultimately the main beam and the
HPBW may be affected.

***19-1-6. Rectangular aperture. Cosine taper.
**An antenna with rectangular aperture *x*1*y*1 has a uniform field in the *y* direction and a
cosine field distribution in the *x* direction (zero at edges, maximum at center). If *x*1 = 16λ
and *y*1 = 8λ, calculate (a) the aperture efficiency and (b) the directivity.
*
Solution*:

1

1

8 16

*y
x
*

λ λ

= =

1*x *

o 1

( ) sin *xE x E
x
*

π =

*x
*

*y *

1*y *

o*E **E
*

*E *

o*E *

132

***19-1-6. continued
**
Although the taper in the *x*-direction is described as a cosine taper, let us represent it by a
sine function as follows:

(a) From (19-1-50), ( ) ( )

2

2

( ) ( )

*av
ap
*

*av
*

*E x
E x
*

ε =

where 1

1 1o o o1 0 0

1 1 1 1 1 0

21( ) ( ) sin cos
*x
*

*x x
*

*av
E E Exx xE x E x dx dx
*

*x x x x x
*π π

π π −⎛ ⎞= = = =⎜ ⎟

⎝ ⎠∫ ∫

1 1

2 2 2 2 2o o

0 0 1 1 1

1[ ( )] ( ) sin 2

*x x
*

*av
E ExE x E x dx dx
*

*x x x
*π

= = =∫ ∫

Therefore,

2

o

2 2 o

2 8 0.811 or 81%1

2

*ap
*

*E
*

*E
*

πε π

⎛ ⎞ ⎜ ⎟ ⎝ ⎠= = = ≅

(b) 20.81 8 16 103.7*e ap emA A*ε λ λ λ= = × × =

2 2 2

4 (4 103.7 ) / 1304 or 31.2 dBi*eAD *π π λ λ
λ

= = × =

**19-1-7. Rectangular aperture. Cosine tapers.
**Repeat Prob. 19-1-6 for the case where the aperture field has a cosine distribution in both
the *x* and *y* directions.
*
Solution*:
Let the distribution be represented by

o 1 1

( , ) sin sin*x xE x y E
x y
*

π π =

133

**19-1-7. continued
**

1 1 1 1

1 1

o 0 0 0 0

1 1 1 1 1 1

o o1 1 2

1 1 1 10 0

1(a) ( , ) ( , ) sin sin

4cos cos

*x y x y
*

*av
*

*x y
*

*E x yE x y E x y dxdy dx dy
x y x y x y
*

*E Ex yx y
x y x y
*

π π

π π π π π

= =

⎡ ⎤ ⎡ ⎤ = − − =⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

∫ ∫ ∫ ∫

1 1 1 1

2 2 2 2 2 2o o

0 0 0 0 1 1 1 1 1 1

1[ ( , )] ( , ) sin sin 4

*x y x y
*

*av
E Ex yE x y E x y dxdy dx dy
*

*x y x y x y
*π π

= = =∫ ∫ ∫ ∫ Therefore,

2

o2

4 2 o

4 16 4 0.657 or 66%1

4

*ap
*

*E
*

*E
*

πε π

⎛ ⎞ ⎜ ⎟ ×⎝ ⎠= = = ≅

(b) 20.657 8 16 84.1 *e ap emA A*ε λ λ λ= = × × =

2

2 2

4 4 84.1 1057 or 30.2 dBi*eAD *π π λ
λ λ

× = = =

***19-1-8. A 20**λ** line source. Cosine-squared taper.
**(a) Calculate and plot the far-field pattern of a continuous in-phase line source 20λ long

with cosine-squared field distribution.
(b) What is the HPBW?
*
Solution*:

The field along the line may be represented by

2 1

( ) cos 2

*xE x
x
*

π =

(a) The field pattern ( )*E *θ is the Fourier transform of the distribution ( )*E x * along the
line. Thus,

1

1

10(2 / )cos 2 (2 / )cos

10 ( ) ( ) cos [( / 2)( /10 )]

*x j x j x
*

*x
E E x e dx x e dx*λ

λ

π λ θ π λ θθ π λ + +

− − = =∫ ∫

*E*(*x*)

20λ

*+x*1 −*x*1

134

***19-1-8. continued
**
Let /*s x *λ= from which *dx ds*λ=
Then

10 102 2 cos 2 cos

10 10

10 102 cos 2 cos

10 10

1 cos( /10)( ) cos ( / 20) 2

cos( /10) 2 2

*j s j s
*

*j s j s
*

*sE s e ds e ds
*

*e ds s e ds
*

π θ π θ

π θ π θ

πθ λ π λ

λ λ π

+ +

− −

+ +

− −

+ = =

= +

∫ ∫

∫ ∫

and sin(20 cos ) 1 sin(20cos 1) sin(20cos 1)( )

2 cos 2 [2cos (1/10)] [2cos (1/10)]*n
E *π θ θ π θ πθ

π θ θ π θ π ⎡ ⎤+ −

= + +⎢ ⎥+ −⎣ ⎦

2

2

sin(20 cos ) 4cos1 2 cos 4cos 0.01

π θ θ π θ θ

⎛ ⎞ = −⎜ ⎟−⎝ ⎠

(*ans.*) (1)

(b) From graph or by trial and error from (1),

o o oHPBW 2(90 87.9 ) = 4.2= − (*ans.*)
From Table 4-3 for a 20λ *uniform* aperture,

oHPBW = 50.8/ 50.8 / 20 2.5*L*λ = =
Thus, the cosine-squared aperture distribution has nearly twice the HPBW of the uniform
aperture, but its side lobes are much lower with first side lobe down 31 dB as compared
to only 13 dB down for a 20λ uniform aperture distribution.

135

136

**Chapter 21. Antennas for Special Applications
**

**21-4-2. Horizontal dipole above imperfect ground.
**Calculate the vertical plane field pattern broadside to a horizontal λ/2 dipole antenna λ/4
above actual homogeneous ground with constants ε*r’* = 12 and σ = 2 x 10-3 Ω-1 m-1 at (a)
100 kHz and (b) 100 MHz.
*
Solution*:

3 1 1
o , 12, 2 10 m , / 4*r h*μ μ ε σ λ

− − −′= = = × Ω =

2 1 2

2 1 2

sin ( cos ) sin ( cos )

*r
*

*r
*

α ε αρ α ε α⊥

− − =

+ − (1)

1 cos(2 sin ) sin(2 sin )*E h j h*ρ β α β α⊥ ⊥= + +__∠ (2)

(b) 3

8 12 o

2 10 0.36 at 100 MHz
2 10 8.85 10*r
*

σε ωε π

−

−

×′′ = = = ×

12 0.4 12*r r rj j*ε ε ε′ ′′= − = − ≅

Introducing *r*ε into (1), (1) into (2) and evaluating (2) as a function of α results in the
pattern shown. The pattern for perfectly conducting ground ( )σ = ∞ is also shown for
comparison (same as pattern of 2 isotropic sources in phase opposition and spaced / 2λ ).
For perfectly conducting ground the field doubles ( 2*E *= ) at the zenith ( o90α = ), but
with the actual ground of the problem, it is reduced to about 1.55 (down 2.2 dB) because
of partial absorption of the wave reflected from the ground.

(a) At 100 kHz, 360 and 1*r*ε ρ⊥′′ = ≅ − , so the pattern is approximately the same as for

σ = ∞ in the sketch.

137

**21-9-1. Square loop.
**Calculate and plot the far-field pattern in the plane of a loop antenna consisting of four
λ/2 center-fed dipoles with sinusoidal current distribution arranged to form a square λ/2
on a side. The dipoles are all in phase around the square.
*
Solution*:
Squarish pattern with rounded edges.
Maximum-to-minimum field ratio = 1.14

***21-9-3. DF and monopulse.
**Many direction-finding (DF) antennas consist of small (in terms of λ) loops giving a
figure-of-eight pattern as in Fig. P21-9-3a. Although the null is sharp the bearing
(direction of transmitter signal) may have considerable uncertainty unless the *S*/*N* ratio is
large. To resolve the 180° ambiguity of the loop pattern, an auxiliary antenna may be
used with the loop to give a cardiod pattern with broad maximum in the signal direction
and null in the opposite direction.

The maximum of a beam antenna pattern, as in Fig. P21-9-3b, can be employed to
obtain a bearing with the advantage of a higher *S*/*N* ratio but with reduced pattern change
per unit angle. However, if 2 receivers and 2 displace beams are used, as in Fig.
P21-9-3c, a large power-pattern change can be combined with a high *S*/*N* ratio. An
arrangement of this kind for receiving radar echo signals can give bearing information on
a single pulse (*monopulse radar*). If the power received on beam 1 is *P*1 and on beam 2
is *P*2, then if *P*2 > *P*1 the bearing is to the right. If *P*1 > *P*2 the bearing is to the left and if
*P*1 = *P*2 the bearing is on axis (boresight). (With 4 antennas, bearing information left-
right *and* up-down can be obtained.)
(a) If the power pattern is proportional to cos4 θ, as in Fig. P21-9-3c, determine *P*2/*P*1 if
the interbeam (squint) angle α = 40° for Δθ = 5 and 10°.
(b) Repeat for α = 50°.
(c) Determine the *P*0/*P*1 of the single power pattern of Fig. P21-9-3b for Δθ = 5 and 10° if
the power pattern is also proportional to cos4 θ.
(d) Tabulate the results for comparison and indicate any improvement of the double over
the single beam.

138

***21-9-3. continued
**

Figure P21-9-3. Direction finding: (a) with loop mull, (b) with beam maximum and (c)
with double beam (monopulse).
*
Solution*:
(a) o40α =

4 o o o 2

4 o o 1

4 o o o 2

4 o o 1

cos (20 5 )5 , 1.290 or 1.1 dB cos (20 5 )

cos (20 10 )10 , 1.672 or 2.2 dB cos (20 10 )

*P
P
*

*P
P
*

θ

θ

− Δ = = =

+

− Δ = = =

+

(b) o50α = 4 o o

o 2 4 o o

1 4 o o

o 2 4 o o

1

cos (25 5 )5 , 1.386 or 1.4 dB cos (25 5 )

cos (25 10 )10 , 1.933 or 2.9 dB cos (25 10 )

*P
P
*

*P
P
*

θ

θ

− Δ = = =

+

− Δ = = =

+

4 o

o 0 4 o

1 4 o

o 0 4 o

1

cos 0(c) 5 , 1.015 or 0.06 dB cos 5

cos 010 , 1.063 or 0.26 dB cos 10

*P
P
*

*P
P
*

α

α

= = =

= = =

(d) Over 1 dB more at 5o and about 2 dB more at 10o.

139

***21-10-1. Overland TV for HP, VP and CP.
**(a) A typical overland microwave communications circuit for AM, FM or TV between a
transmitter on a tall building and a distant receiver involves 2 paths of transmission, one
direct path (length *ro*) and one an indirect path with ground reflection (length *r*1 + *r*2), as
suggested in Fig. P21-10-1. Let *h*1 = 300 m and *d* = 5 km. For a frequency of 100 MHz
calculate the ratio of the power received per unit area to the transmitted power as a
function of the height *h*2 of the receiving antenna. Plot these results in decibels as
abscissa versus *h*2 as ordinate for 3 cases with transmitting and receiving antennas both
(1) vertically polarized, (2) horizontally polarized and (3) right-circularly polarized for *h*2
values from 0 to 100 m. Assume that the transmitting antenna is isotropic and that the
receiving antennas are also isotropic (all have the same effective aperture). Consider that
the ground is flat and perfectly conducting.
(b) Compare the results for the 3 types of polarization, and show that circular polarization
is best from the standpoint of both the noncriticalness of the height *h*2 and the absence of
echo or ghost signals. Thus, for horizontal or vertical polarization the direct and ground-
reflected waves may cancel at certain heights while at other heights, where they
reinforce, the images on the TV screen may be objectionable because the time difference
via the 2 paths produces a double image (a direct image and its ghost).
(c) Extend the comparison of (b) to consider the effect of other buildings or structures
that may produce additional paths of transmission.
Note that direct satellite-to-earth TV downlinks are substantially free of these
reflection and ghost image effects.

Figure P21-10-1. Overland microwave communication circuit.

*
Solution*:
(a) and (b) answers in Appendix F, pg. 919-920.
(c) The effect of reflection from other buildings or structures (or from aircraft) can be
minimized by the use of CP transmit and receive antennas of the same hand, particularly
when these structures are many wavelengths in size and reflection is specular. Trouble-
some reflections can be reduced by placing non-reflecting absorbers on the structure.

140

***21-12-1. Signaling to submerged submarines.
**Calculate the depths at which a 1 μV m-1 field will be obtained with *E* at the surface
equal to 1 V m-1 at frequencies of 1, 10, 100 and 1000 kHz. What combination of
frequency and antennas is most suitable?
*
Solution*:
From Table A-6, take 80 and 4*r*ε σ′ = = for sea water. At the highest frequency (1000

kHz), ,σ ωε>> so that / 2α ωμσ= can be used at all four frequencies. At 1 kHz,

3 7 12 10 4 10 4 0.13 Npm

2 π πα

− −× × × ×= =

Since 6

o

6 13.810 , log*yE e y e
E
*

α

α α − −= = = =

and

at 1 kHZ, depth 106 m at 10 kHz, 35 m at 100 kHz, 11 m at 1000 kHz 3.5 m

*y
y
y
y
*

= = = =

(*ans.*)

From the standpoint of frequency, 1 kHz gives greatest depth. However, from (21-2-3)
the radiation resistance of a monopole antenna as a function of its height ( )*ph *is

2

400 *pr
h
*

*R
*λ

⎛ ⎞ = Ω⎜ ⎟

⎝ ⎠

For 300 m at 1 kHz*ph *=

2 4

5

300400 4 10 (or 400 μ )
3 10*r
*

*R *−⎛ ⎞= = × Ω Ω⎜ ⎟×⎝ ⎠

With such a small radiation resistance, radiation efficiency will be poor. At 10 kHz the radiation resistance is a hundred times greater. A practical choice involves a compromise of sea water loss, land (transmitting) antenna effective height, and submarine antenna efficiency as a function of the frequency.

141

***21-13-1. Surface-wave powers.
**A 100-MHz wave is traveling parallel to a copper sheet (|*Zc*| = 3.7 x 10-3 Ω) with **E** ( =
100 V m-1 rms) perpendicular to the sheet. Find (a) the Poynting vector (watts per square
meter) parallel to sheet and (b) the Poynting vector into the sheet.
*
Solution*:

(a) 2 2

2 to sheet

o

100 26.5 Wm 377

*yES
Z
*

−= = = (*ans.*)

(b) 22 3

2 2 int o sheet 2

o

100 3.7 10 182 μWm
377 2*e c e c
*

*ES H R Z R Z
Z
*

− −×⎛ ⎞= = = =⎜ ⎟

⎝ ⎠
(*ans.*)

**21-13-2. Surface-wave powers.
**A 100-MHz wave is traveling parallel to a conducting sheet for which |*Zc*| = 0.02 Ω. If **E**
is perpendicular to the sheet and equal to 150 V m-1 (rms), find (a) watts per square meter
traveling parallel to the sheet and (b) watts per square meter into the sheet.
*
Solution*:

(a) 2 2

2 to sheet

o

150 59.7 Wm 377

*yES
Z
*

−= = = (*ans.*)

(b) 22 2

2 2 int o sheet 2

o

150 2 10 2.24 mWm
377 2*e c e c
*

*ES H R Z R Z
Z
*

− −×⎛ ⎞= = = =⎜ ⎟

⎝ ⎠
(*ans.*)

***21-13-3. Surface-wave power.
**A plane 3-GHz wave in air is traveling parallel to the boundary of a conducting medium
with **H** parallel to the boundary. The constants for the conducting medium are σ= 107
Ω-1 m-1 and μ*r* = ε*r* =1. If the traveling-wave rms electric field *E* = 75 mV m-1, find the
average power per unit area lost in the conducting medium.
*
Solution*:

2 2

int o sheet 2 o

*e c e c
ES H R Z R Z
Z
*

= =

7 9 o

7

4 10 2 3 10 0.034
2 2 10*e c
*

*R Z *μ ω π π
σ

−× × × = = = Ω

×

142

***21-13-3. continued
**

Therefore, 2

2 int o sheet

0.75 0.034 1.35 nWm 377

*S *−⎛ ⎞= =⎜ ⎟
⎝ ⎠

(*ans.*)

**21-13-4. Surface-wave current sheet.
**A TEM wave is traveling in air parallel to the plane boundary of a conducting medium.
Show that if *K* = ρ*sv*, where *K* is the sheet-current density in amperes per meter, ρ*s* is the
surface charge density in coulombs per square meter and *v *the velocity of the wave in
meters per second, it follows that *K* = *H*, where *H* is the magnitude of the **H** field of the
wave.
*
Solution*:

1 1 2

Q m Q 1 A m I m
m s s m*s
*

*K v*ρ − −= = = = =

By Amperes’s law, integral of *H* around strip of width *w* equals current enclosed or

ds = *I wK*=∫ **H**i

and (note that )*wH wK H K H K*= = ⊥ (*ans.*)

***21-13-6. Coated-surface wave cutoff.
**A perfectly conducting flat sheet of large extent has a dielectric coating (ε*r* = 3) of
thickness *d* = 5 mm. Find the cutoff frequency for the TMo (dominant) mode and its
attenuation per unit distance.
*Solution*:

1

o o

2 8.893 1 Np mπα λ λ

−= − = (*ans.*) 0*cf *= (*ans.*)

143

144

**Chapter 23. Baluns, etc. By Ben A. Munk
**

**23-3-1. Balun 200 **Ω**, antenna 70 **Ω**.
**A Type III balun has the characteristic impedance equal to *Zcp* = 200 Ω and the electrical
length is equal to *lp* = 7.5 cm. It is connected to an antenna with impedance *ZA* = 70 Ω.
(a) Find the balun impedance *jXp* at *f* = 500, 1000 and 1500 MHz.
(b) Calculate the parallel impedances *ZA* || *jXp* at 500 1000 and 1500 MHz and plot them
in a Smith Chart normalized to *Zo* = 50 Ω. Check that all these impedances lie on a circle
with a diameter spanning over (0,0) and *ZA* = 70 Ω. Alternatively, you may determine *ZA*
|| *jXp* graphically in a Smith Chart.
(c) Explain what effect it would have on the bandwidth if we changed *Zcp* to 150 Ω or
250 Ω.
*Solution*:

(a) 8

8

3 10 7.560 cm, 0.125 5 10 60

*p
L
*

*L
*

λ λ

× = = = =

×

8

9

3 10 7.530 cm, 0.250 1 10 30

*p
M
*

*M
*

λ λ

× = = = =

×

8

9

3 10 7.520 cm, 0.375 1.5 10 20

*p
H
*

*H
*

λ λ

× = = = =

×

From the Smith Chart, by moving the number of wavelengths around from the short
(zero) position, it is found that
for 500 MHz, 200 *pf jX j*= = Ω (*ans.*)
for 1000 MHz, *pf jX j*= = ∞ Ω (*ans.*)
for 1500 MHz, 200 *pf jX j*= = − Ω (*ans.*)
Alternatively, the transmission line equation can be used.

(b) 2 2

2 2
*A p A p A p
*

*A p
A p A p
*

*Z jX Z X jZ X
Z jX
*

*Z jX Z X
*+

= = + +

For mid frequency, ,*p A p AX Z X Z*= ∞ =

145

**23-3-1. continued
**

For low frequency, 2 2

2 2

70 200 70 200200, 62.36 21.83
70 200*p A p
*

*jX Z jX j*× + ×= = = +
+

Normalized to o 50,*Z *=
o

1.25 0.44*A p
Z jX
*

*j
Z
*

= +

Similarly, for high frequency, 200, 62.36 21.83*p A pX Z jX j*= − = −

o

1.25 0.44*A p
Z jX
*

*j
Z
*

= −

See accompanying figure of Smith Chart

146

**23-3-1. continued
**
To find these values by the Smith Chart, it is a matter of adding the values as admit-
tances. This is accomplished by finding their position as impedances, projecting the
values through the origin an equal distance, adding them, then projecting the added
values an equal distance to the other side of the origin.
(c) For 150 , it is found that 57.48 26.82*cp A pZ Z jX j*= Ω ± = ±
or normalized as 1.15 0.54*j*= ±
For 250 , it is found that 64.91 18.18*cp A pZ Z jX j*= Ω ± = ±
or normalized as 1.30 0.36*j*= ±
It is easily seen that the 150 Ω value decreases the bandwidth and the 250 Ω value
increases the bandwidth. Note: The closer the values are to the origin, the better the
VSWR.

**23-3-5 Stub impedance.
**(a) What is the terminal impedance of a ground-plane mounted stub antenna fed with a
50-Ω air-filled coaxial line if the VSWR on the line is 2.5 and the first voltage minimum
is 0.17λ from the terminals?
(b) Design a transformer so that the VSWR = 1.
*Solution*:

o o

o

tan tan

*T
m
*

*T
*

*Z jZ xZ Z
Z jZ x
*

β β

+ =

+ (1)

min

o

where impedance on line at 0 line impedance 50 0 stub antenna terminal impedance =

*m m
*

*T T T
*

*Z V R j
Z j
Z R jX
*

= = + = = + Ω = +

Rearranging (1) in terms of real and imaginary parts:

*V*min

*ZT
*VSWR = 2.5

*Z*o = 50 Ω

.17λ

147

**23-3-5 continued
**

o

tan*T mm T
X RR R x
*

*R
*β

⎛ ⎞ − = ⎜ ⎟

⎝ ⎠ by equating reals, (2)

and

o o

tan tan*T m T
R R x X R x
*

*R
*β β= + by equating imaginaries (3)

o

o50 / 2.5 20, 50, tan tan(360 .17) 1.82*mR R x*β= = = = × =

From (2), 2020 1.82 0.728
50*T T T
*

*R X X*− = × =

From (3), 20 1.82 50 1.82, 0.728 91
50*T T T T
*

*R X R X*× = + × = +

From which, 56 50 *T T TZ R jX j*= + = − Ω (*ans.*)

148

**Chapter 24. Antenna Measurements.
By Arto Lehto and Pertti Vainikainen
**

**24-3-1. Uncertainty of pattern measurement due to reflected wave.
**The level of a wave reflected from the ground is 45 dB below the level of the direct
wave. How large of errors (in dB) are possible in the measurement of:
(a) main lobe peak;
(b) -13 dB sidelobe;
(c) -35 dB sidelobe?
*Solution*:
From Sec. 24-3b and since the reflected wave is (45/ 20)45 dB or 10 0.0056−− = ,
(a) 1 0.0056 0.9944 or 0.049 dB− = − (*ans.*)
1 0.0056 1.0056 or +0.049 dB+ = (*ans.*)
(b) (13/ 20)13 dB side lobes provides 10 0.2238−− =

so 0.2238 0.0056 0.9749 or 0.22 dB 0.2238

−
= − (*ans.*)

0.2238 0.0056 1.0251 or +0.22 dB 0.2238

+
= (*ans.*)

(c) (35/ 20)35 dB side lobes provides 10 0.0178−− =

so 0.0178 0.0056 0.6838 or 3.30 dB 0.0178

−
= − (*ans.*)

0.0178 0.0056 1.3162 or +2.38 dB 0.0178

+
= (*ans.*)

**24-3-2. Range length requirement due to allowed phase curvature.
**The maximum allowed phase curvature in the measurement of a very low-sidelobe
antenna is 5°. The width of the antenna is 8 m and it operates at 5.3 GHz. Find the
required separation between the source and AUT.

149

**24-3-2. continued
***Solution*:
Similar to Fig. 24-5, let *d* be the distance causing the phase error.

Then 2

2 2( )
2
*DR d R *⎛ ⎞+ = + ⎜ ⎟

⎝ ⎠

2 2 2 2 22 ,

4 8
*D DR dR d R R
*

*d
*+ + = + ≅

For a o5 phase error,

2 5 (rad) 180

*kd d*π π
λ

= =

so, 5 1 360 72

*d
*λ

= =

Therefore, 2 21 972

8
*D DR
*λ λ

= =

Since, 8

9

3 10 9 640.0566 m, 10,176 m 5.3 10 0.0566

*R*λ × ×= = ≥ =
×

**24-4-1. Design of elevated range.
**Design an elevated range (range length, antenna heights, source antenna diameter) for the
measurement of a 1.2 m reflector antenna operating at 23 GHz.
*Solution*:

8

10

3 10 0.013 m 2.3 10

λ ×= = ×

so, 2 22 2 (1.2) 221 m

0.013
*DR
*λ

×
≥ = = (*ans.*)

/ 2*D **R
*

*R
*

*d *

150

**24-4-1. continued
**From combining requirements in (24-4-1) and (24-4-2)

5 5 1.2 6 m*RH D*≅ × = × = (*ans.*)

and similarly for *T RH H*=

From (24-4-1), 1.5 1.5 0.013 221 0.72 m
6*T R
*

*RD
H
*

λ × ×
≥ = = (*ans.*)

**24-4-2. Time required for near-field scanning.
**Estimate the time needed for a planar near-field measurement of a 2 m antenna at 300
GHz. The sampling speed is 10 samples per second.
*Solution*:

8

11

3 10 2 m0.001 m, 2000 3 10 0.001 m

*D*λ λ×= = = =
×

Sample at 2 per wavelength, so samples = 4000 per line per side Total samples = 3 2 62 (4 10 ) 32 10× × = ×

6 632 10 3.2 10 sec 888 hrs 54min 37 days

10 samples/sec
*t *×= = × = ≅ (*ans.*)

**24-5-1. Power requirement for certain dynamic range.
**The AUT has a gain of 40 dBi at 10 GHz. The gain of the source antenna is 20 dBi. The
separation between the antennas is 200 m. The receiver sensitivity (signal level that is
sufficient for measurement) is –105 dBm. Find the minimum transmitted power that is
needed for a dynamic range of 60 dB.
*Solution*:

From (24-5-2) and since 2 2

100.03 1.42 10 98 dB
4 4 200*R
*

λ π π

−⎛ ⎞ ⎛ ⎞= = × = −⎜ ⎟ ⎜ ⎟×⎝ ⎠ ⎝ ⎠

151

**24-5-1. continued
**

dB

40 dBi + 20 dBi 98 dB 38 dB*R
T
*

*P
P
*

⎛ ⎞ = − = −⎜ ⎟

⎝ ⎠

With 105 dBm− needed at a minimum for the reception and a 60 dB dynamic range, then

105 dBm 38 dB 60 dB 7 dBm 0.2 mW ( .)

*t
*

*t
*

*P
P ans
*

= − + + = − =

**24-5-2. Gain measurement using three unknown antennas.
**Three horn antennas, A, B, and C are measured in pairs at 12 GHz. The separation of
antennas is 8 m. The transmitted power is +3 dBm. The received powers are -31 dBm,
36 dBm, and -28 dBm for antennas pairs AB, AC, and BC, respectively. Find the gains
of the antennas.
*Solution*:

From (24-5-2), 2 8

10

3 10, 0.025 m 4 1.2 10

*T
T R
*

*R
*

*PG G
P R
*

λ λ π

− ×⎛ ⎞= = =⎜ ⎟ ×⎝ ⎠

2 2

80.025 6.18 10 or 72 dB
4 4 8*R
*

λ π π

−⎛ ⎞ ⎛ ⎞= = × −⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

then 31 dBm 3 dBm 72 dB 38 dB 36 dBm 3 dBm 72 dB 33 dB 28 dBm 3 dBm 72 dB 41 dB

*A B AB
*

*A C AC
*

*B C BC
*

*G G C
G G C
G G C
*

= = − − + = = = − − + = = = − − + =

2, ,*B AB AB ABB C C BC
*

*C AC AC AC
*

*G C C CG G G C
G C C C
*

⎛ ⎞ = = =⎜ ⎟

⎝ ⎠

So 1 (41 dB + 33 dB 38 dB) 18 dBi 2

*BC AC
C
*

*AB
*

*C CG
C
*

= = − = (*ans.*)

33 dB 18 dB 15 dBi*ACA
C
*

*CG
G
*

= = − = (*ans.*)

152

**24-5-2. continued
**

41 dB 18 dB 23 dBi*BCB
C
*

*CG
G
*

= = − = (*ans.*)

**24-5-3. Gain measurement using celestial radio source.
**At 2.7 GHz the antenna temperature increases 50 K as a 20 m reflector is pointed to
Cygnus A. Find the antenna gain and aperture efficiency.
*Solution*:

From (24-5-7), 23

5 2 26 2

8 8 1.38 10 50 1.79 10 52.5 dBi 785 10 (0.111)

*Ak TG
S
*

π π λ

−

−

Δ × × × = = = × =

× ×

2 5 2

21.79 10 (0.111) 175.5 m
4 4*e
*

*GA *λ
π π

× × = = =

For a 20 m circular reflector,

2

175.5 0.56 or 56% (10)

*e
ap
*

*p
*

*A
A
*

ε π

= = =

**24-5-4. Impedance in laboratory.
**You try to measure the impedance of a horn antenna with 15 dBi gain at 10 GHz in a
normal laboratory room by pointing the main lobe of the antenna perpendicularly towards
a wall 2 m away. The power reflection coefficient of the wall is 0.3 and it can be
assumed to cover practically the whole beam of the AUT. Estimate the uncertainty of the
measurement of the reflection coefficient of the AUT due to the reflection of the wall.
*Solution*:
The normalized received power from the horn to the wall and back into the horn

2

4
*R
*

*T R
T
*

*P G G
P R
*

λρ π

⎛ ⎞= ⎜ ⎟ ⎝ ⎠

2 2

70.030.3 5 dB, 0.03 m, 3.6 10 64 dB 4 R 4 2 2

λρ λ π π

−⎛ ⎞ ⎛ ⎞= = − = = = × = −⎜ ⎟ ⎜ ⎟× ×⎝ ⎠ ⎝ ⎠

153

**24-5-4. continued
**

5 dB 15 dBi 15 dBi 64 dB 39 dB

0.000126 in power =0.01122 in voltage

*R
*

*T
*

*P
P
*

= − + + − = −

=

So the uncertainty is about 1%.

154

**INDEX
**

Index reads as follows: Entry (Problem number) page

**A**

Aperture with phase ripple (19-1-5) 127 with tapered distribution (19-1-3) 127,

(19-1-4) 127, (19-1-6) 130, (19-1-7) 131, (19-1-8) 132

Array broadside (5-6-10) 27, (16-2-1) 109

seven short dipoles (16-6-4) 113 sixteen source (16-6-7) 114

end-fire (16-3-1) 110 four sources in square (5-2-8) 23 ordinary end-fire (5-6-9) 25 square-corner reflector (10-3-6) 68 three-source (5-9-2) 30 three unequal sources (5-8-1) 29 twelve-source end-fire (5-6-5) 24, (5-6-7) 25 two-element, unequal currents (16-4-3) 111 two-source end-fire (5-2-4) 23 two-sources in opposite phase (5-18-2) 33

Artificial dielectric (17-3-1) 122

**B**

Backpacking penguin (12-3-4) 90 Balun (23-3-1) 143 Beamwidth (3-7-2) 18 Broadcast array, four-tower (16-8-6) 116 Broadside array

two element (16-2-1) 109 seven short dipoles (16-6-4) 113 sixteen source (16-6-7) 114

**C**

Carrier-to-noise ratio (*C*/*N*) (12-3-19) 93
Circularly polarized wave (2-16-2) 8, (2-16-3) 8,

(2-17-6) 11, (2-17-7) 12, (2-17-9) 13, (2-17-11) 14,

Coated-surface wave cutoff (21-13-6) 141 Conical antenna (11-2-2) 75 Conical pattern (6-3-5) 38, (6-3-6) 39 Critical frequency (12-3-9) 86

**D**

Depolarization ratio (2-17-2) 15 Detecting one electron (12-5-5) 97 Dielectric, artificial (17-3-1) 122 Dipole

electric (6-2-1) 35 short (6-2-2) 35, (6-2-4) 36, (6-3-2) 37,

(6-3-4) 38, (6-3-10) 40 Direction finding (21-9-3) 136 Directional pattern, with back lobe (6-3-9) 40 Directional pattern in θ and φ (6−3−7) 39,

(6−3−8), 40 Directive gain (5-8-7) 29 Directivity (2-7-1) 1, (2-7-2) 1, (2-7-3) 2, (2-7-4)

3, (2-9-1) 3, (2-9-3) 4, (3-7-2) 18, (4-5-1) 19, (4-5-2) 20, (4-5-3) 20, (4-5-4) 21, (4-5-5) 21, (5-2-4) 23, (5-6-5) 24, (5-6-9) 25, (5-6-10) 27

Dolph-Tchebyscheff array (5-9-4) 31, (16-6-1) 112

Dynamic range, power requirement (24-5-1) 149

**E**

Echelon array (6-7-1) 43 Eelevated range design (24-4-1) 148 Effective aperture (2-9-2) 4, (2-9-3) 4 Elliptically polarized wave (2-16-4) 9, (2-17-3)

10, (2-17-4) 11, (2-17-8) 13, (2-17-10) 13 End-fire array, two element (16-3-1) 110

**F**

Field pattern (5-5-1) 24, (5-6-5) 24 two-sources in phase (5-18-1) 32

Forest absorption (12-4-3) 95 Friis formula (2-11-1) 4

**G**

Gain (2-7-4) 3, (9-9-1) 60 Gain measurement

using celestial radio source (24-5-3) 151 using three unknown antennas (24-5-2) 150

Galileo spacecraft (12-3-22) 94

155

**H**

Helical antennas axial mode (8-3-1) 53, (8-3-2) 53, (8-3-3) 54,

(8-8-1) 55, (8-15-1) 56 normal mode (8-11-1) 55

Horn antenna (3-4-1) 17, (3-4-2) 17, (3-5-2) 17, (3-5-3) 18

Horns (9-9-1) 60, (9-9-2) 61, (9-9-3) 62, (9-9-4) 62, (9-9-5) 63

**I**

Impedance 2-element array (16-3-2) 111 5λ/2 antenna (13-4-1) 103 antennas in echelon (13-8-1) 104 dipole (14-12-2) 105

Impedance Dolph-Tchebyscheff array (16-6-1) 112 in laboratory (24-5-4) 151 open-slot (9-5-3) 58 side-by-side antennas (13-6-1) 103, (13-6-3)

104 slot (9-5-1) 57 stub antenna (23-3-5) 145

Isotropic antenna (6-3-1) 37

**J**

Jupiter signals (12-4-4) 95

**L**

Lenses (17-2-1) 121, (17-4-1) 123 Linearly polarized waves (2-17-2) 10 Link

interstellar wireless (12-3-13) 89 Mars and Jupiter (2-11-4) 5 Moon (2-11-5) 7 satellite (12-3-4) 82 spacecraft (2-11-2) 5, (2-11-3) 5 spaceship near moon (2-16-1) 7

Loaded element (18-9-2) 125 Log-periodic antenna (11-7-1) 77, (11-7-2) 78 Log-spiral antenna (11-5-1) 75 Loop

λ/10 (7-7-1) 50 3λ/4 (7-4-1) 49 circular (7-8-2) 51 directivity (7-8-1) 50 radiation resistance (7-6-1) 49, (7-8-1) 50 square (7-9-1) 51, (7-9-2) 52

Loop-dipole for CP (7-2-1) 47 Low earth orbit satellite (12-3-17) 90

**M**

Maximum useable frequency (MUF) (12-3-9) 86 Microstrip line (9-7-3) 59 Minimum usable frequency (mUF) (12-3-10) 87 Moment method, charged rod (14-10-1) 105 Monopulse (21-9-3) 136

**N**

Near-field scanning, time required (24-4-2) 149

**O**

Overland TV (21-10-1) 138

**P**

Patches, 50 and 100 Ω (9-7-1) 59 Pattern

directional with back lobe (6-3-9) 40 directional in θ and φ. (6-3-7) 39, (6-3-8) 40 elements (15-6-1) 107 factors (6-8-2) 44

horn (9-9-2) 61 measurement, uncertainty (24-3-1) 147 smoothing (15-3-1) 107 Patterns over imperfect ground (21-4-2) 135 Phase pattterns (5-5-1) 24 Poincaré sphere (2-17-5) 11 Polarization (2-16-4) 9, (2-17-1) 9, (2-17-2) 10,

(2-17-3) 10, (2-17-4) 11, (2-17-5) 11, (2-17- 6) 11, (2-17-7) 12, (2-17-8) 13, (2-17-9) 13, (2-17-10) 13, (2-17-11) 14,

Poynting vector (2-16-2) 8, (2-16-3) 8

**Q**

Quad-helix antenna (8-15-1) 56

**R**

Radar cross section (12-5-6) 98 Radar detection (12-5-1) 96, (12-5-12) 99,

(12-5-13) 99, (12-5-14) 100, (12-5-18) 100, (12-5-20) 101

Radiation resistance (6-3-1) 37, (6-3-11) 41, (6-3-12) 41, (6-5-1) 42, (6-6-1) 42, (7-6-1) 49

Range length (24-3-2) 147 RCS of electron (12-5-3) 96 Reflector

flat sheet (10-2-1) 65 parabolic with missing sector (10-7-2) 73 square-corner (10-3-1) 66, (10-3-2) 66,

(10-3-4) 67, (10-3-5) 68, (10-3-6) 68, (10-3-7) 71, (10-3-8) 71

Rhombic

156

alignment (16-16-3) 119 compromise (16-16-4) 119, (16-16-5) 120,

(16-16-6) 120 E-type (16-16-2) 119

**S**

Satellite
downlink (12-3-4) 82, *See* also Link
direct-broadcast (DBS) (12-3-18) 91
low earth orbit (LEO) (12-3-17) 90

Scanning array eight-source (16-10-1) 117

Slots (9-2-1) 57, (9-5-1) 57, (9-5-2) 58, (9-5-3) 58

Solar interference (12-3-7) 84 Solar power (4-3-1) 19 Square array (16-6-3) 112, (16-6-5) 114 Square loop (21-9-1) 136 Stray factor (5-8-7) 29 Stub impedance (23-3-5) 145 Submarines, communication with (21-12-1) 139

Surface-wave current sheet (21-13-4) 141 cutoff (21-13-6) 141 powers (21-13-1) 140, (21-13-2) 140,

(21-13-3) 140

**T**

Temperature with absorbing cloud (12-4-1) 94 antenna (12-2-1) 81, (12-2-2) 81 minimum detectable (12-3-11) 88, (12-3-12)

89 system (12-3-5) 82, (12-3-6) 84

Traveling wave antennas (6-8-1) 44

**U**

Unloaded tripole (18-9-1) 125

**V**

V antenna (16-16-1) 118