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《天线理论》(朱兰成[国际权威电磁波与雷达专家]) Word版讲义

Chapter 8 Antenna Theory

8-1 Calculation of EM Fields of Antennas

L. J. Chu (朱蘭成): 朱蘭成院士生於民國2年，民國62年逝世。民國47年4月當選第二屆中研院院士，是電磁波及雷達研究方面的三大國際權威之一。 Y. T. Lo (羅遠梓): Yuen Tze Lo (MSEE’49, PhD’52) died May 10, 2002. He was 82. An expert in antennas, Lo invented the broadband television receiving antenna, and he

developed the cavity model theory for microstrip patch antennas now used in global positioning systems (GPS). In 1986, Lo was

elected to the National Academy of Engineering for inventions and innovative ideas that significantly advanced the theory and design of antennas and arrays.

Case 1 Given ),,,(t z y x J

, ???-=

'4v jkR

dv R e

J A π

μ, A H ??=μ1, H j E ??=ωε1 Case 2 Given ),,,(t z y x ρ,???-=''41V jkR

dv R e

V ρπε, t J ??-=??ρ , H j t A V E

??=??--?=ωε1 Eg. (a) Assume the spatial distribution of the current on a very thin center-fed half-wave dipole lying along the z -axis to be I 0cos(βz ), where β=2π/λ. Find the charge distribution on the dipole.

(Sol.) z I j dz z dI j j J βωβ

ωρωρsin )(0-==

?-=??

Stratton-Chu formulas for calculating EM Fields of antennas: (by L. J. Chu)

']')?(')?()?([']''[)('

dS G E a G E a H a G j dV G J G J G j r E n S n n m V ???+??+?-+??-?+-=?????

ωμε

ρωμ

????????+??+?+??+?+-='

']')?(')?()?([']''[)(S n n n V m m dS G H a G H a E a

G j dV G J G J G j r H ωεμ

ωε where r

e G jkr

π4-= is Green ’s function in the free space.

Elemental electrical dipole (Hertzian dipole):

Qd z

p ?= ω

ωj I

Q Q j dt dQ I ±

=±=±

=, R e Id a a R e Id z A R

j R R j βθβπμθθπμ--?-=?=?4)sin ?cos ?(4?00 φφθθA a A a A a

R R ???++= ?????

????=-=-===?--0sin )(4sin cos )(4cos 00φβθβθπμθθπμθA R e Id A A R e Id A A R

j z R

j z R ? R

j R e R j R j Id a A RA R R a A H βφφφβββπθμμ-+-=??-??=??=])

(11sin[4?])([1?12200

)](1?)sin (sin 1?[1100φθφθθθωεωεRH R

R a H R a j H j E R ??

-??=??=

????

???

???Ω?==++-=+-=--)

(120/,0])(1)(11[sin 4])(1)(1[cos 240

003

2203220πεμηβββθβηπββθβηπφβθβwhere E e R j R j R j Id E e R j R j Id E R

j R j R

Far field of a Hertzian dipole: if βR =2πR /λ>>1

θβπβφsin )(4R e Id j H R j -= , θβηπβθsin )(40R

e Id j E R

j -=

Elemental magnetic dipoles: m z IS z

m ??==

m z b I z m ??2

==π '41

01?-=? d R e I A R j βπμ

)](1[1)(111R R j e e e e R j R R j R j R j --?=-----βββββ

]''

)1[(41

0??-+=-

d j R d R j

e I A R j ββπμβθβπμβφsin )1(4?20R j e R j R m a

A -+=? A H ??=0

μ and H j E ??=01ωε

????

???????++-=+-=+=?---R j R j R R j e R j R j R j m j H e R j R j m j H e R j R j m j E βθββφ

βββθβπηωμββθβπηωμββθβπωμ])(1

)(11[sin 4])(1)(1[cos 24])(11[sin 43

22003

22002

20 Far field of an elemental magnetic dipoles: θβπωμβφsin )(40R e m E R j -=, θβπηωμβθsin )(400R e m H R

j --=

Eg. A small filamentary rectangular loop of dimensions L x and L y lies in the

xy -plane with its center at the origin and sides parallel to the x - and y -axes. The loop carries a current i (t )=I 0cos(ωt ). Assuming L x and L y to be much less than the wavelength, find the expressions for the following quantities at a point in the far zone: (a) vector magnetic potential, (b) electric field intensity, (c) magnetic field intensity.

(Sol.) (a) y x L tL I z

m ωcos ?0=

, θβπωμβφsin )1(4cos ?00?+=-R j y x e R j R

L tL I a A (b) θβπωωμβφsin )(4cos 00R e L tL I E R

j y x -=

j y x --=

Eg. A composite antenna consists of an elemental Hertzian electric dipole of length L along the z -axis and an elemental magnetic dipole of area S lying in the xy -plane. Equal time-harmonic currents of amplitude I 0 and angular frequency ω flow in the dipoles. (a) Verify that the far field of the composite antenna is elliptically polarized. (b) Determine the condition for circular polarization.

(Sol.) (a) φφθθβφβθθβπηωμθβηπE a E a R e S I a R e L jI a E R

j R j ??sin 4?sin 4?00000+=+=-- (b) 0

00ηωμηφθS

L E E =?=

Duality between elemental electric and magnetic dipoles: (E e , H e ) due to electric dipole and (E m , H m ) due to magnetic dipole

m e H E 0η?, 0

ηm e E

H -?, S d IS j m j Id βββ?=? ,

8-2 Radiation Patterns of Antennas

Half-power beam width: Angular width of main beam between the half-power (-3dB) points

Sidelobe level: (|E max | in one

sidelobe)/( |E max | in main beam)

Null positions: Directions which have no radiations in the far-field zone.

Directivity: D =?

πφ

θθφθππ0

max

sin ),(44d d E E P U r

, where U =R 2

P av 2

E R ∝

and P r =??Ω=Ud dS P av φθθπ

d d E R sin 2

∝ is the time-average radiated power

Directivity gain: G D (θ,φ)=

?Ω

Ud U P U r

)

,(4),(4φθπφθπ, ∴ D =(G D )max Power gain: G P =

P U max

4π, where P i = P r +P l , P i : total input power, P l : loss Radiation efficiency: ηr = G P /D =P r /P i

Eg. Find the directive gain and the directivity of a Hertzian dipole.

(Sol.) φθH E H E P av 21*Re 21=?=, θβηπ

202

2sin 32)( Id U =. θφ

θθθθ

πφθπ

π2

220

2sin 2

3sin )(sin sin 4),(=

?d d G D , ),2(φπD G D ==1.5=1.76 (dB ).

Eg. Find the radiation resistance of a Hertzian dipole.

(Sol.) ?

=π

φθπ

φθθ0

sin 21d d R H E P r

r R I d I d I d d d I 2

])(80[212)(sin 32)(22

2220223

20202

22===?

?λπβηπφθθβηππ

∴ 22)(

80λ

d R r =

Eg. Find the radiation efficiency of an isolated Hertzian dipole made of a metal wire of radius a , length d , and conductivity σ. (Sol.) The ohmic power loss is R I P 221=

. The radiated power is r r R I P 22

= )

/(11

r r r r R R P P P +=

η, )2(a d R R s π =, where σμπ0

f R s =

?))((16011

d a R s r λ

λπη+

= Assume that a =1.8mm , m d 2= , MHz f 5.1=, and σ =)/(1080.57m S ?

)(200m f c ==λ, )(1020.31080.5)104()1050.1(47

76Ω?=????=--ππs R , )(057.0)10

8.122(

1020.33

4Ω=???=--π R , )(079.0)2002(8022

Ω==πr R and %58057

.0079.0079

.0=+=

r η

Eg. A 1MHz uniform current flows in a vertical antenna of the length 15m . The antenna is a center-fed copper rod having a radius of 2cm . Find (a) the radiation resistance, (b) the radiation efficiency, (c) the maximum electric field intensity at a distance of 20km , the radiated power of the antenna is 1.6kW .

(Sol.) d m m =>>=?=153********

8λ, a =0.02m , σcopper =5.8×107

, c

s f R σμπ=

=2.6×10-4 (a) Ω==97.1)300/15(8022πr R , (b) %98))

/)(/(1601/(13

= d a R s

r λλπη (c) 202

212)(βηπ

d I P r =

=1600?m V R

Id E /109.1)4(22

220

2max

-?≈=βηπθ

Eg. A time-harmonic uniform current I 0cos(ωt ) flows in a small circular loop of radius b (<<λ) lying in the xy -plane. (a) Find the radiation resistance R r of the magnetic dipole. (b) Obtain an expression for its radiation efficiency ηr if the loop is made of radius a .

(Sol.) (a) Duality 46222

)(320)(80λ

πλβππβπb

b R b d r ==??? (b) ))((16011

2b a R s r βπλ

λπη+=

Eg. The amplitude of the time-harmonic current distribution on a center-fed short dipole antenna of length 2h (h <<λ) can be approximated by a triangular function )1()(0h

I z I -

=. Find (a) the far-zone electric and magnetic field

intensities, (b) the radiation resistance, and (c) the directivity.

(Sol.) (a) R

e dz z jI dE R

j βθθ

βηπ-=sin 4)(0, (R =R 0-z cos θ in e -jβR ), θπββφsin 40R j e I R R j H -≈ θβθ

θβθβθβηπβββθsin 30cos )cos cos(1sin 60sin )1(402000)(0000R j R j h h

z R j e I R h

j h e h R jI dz R e h z jI E -----≈-?≈??-≈?

(b)2222200

)(80])(80[2sin 21λ

πλπφθθπ

φθπ

R h I d d R H E P r r =?==?

max

?π

θπ

θθπd d E E D

8-3 Linear Dipole Antennas and Effective Lengths

Assume I (z )=I m sin β(h -|z |)=???<+>-0),(sin 0

),(sin z z h I z z h I m

m ββ

E θ=η0H φ=?--h

h R j R e dz z I θβπηβsin '

4)('

0 (θθcos )cos 2(',2/122z R Rz z R R h R -≈-+=>>)

?E θ)(60)(sin 4sin cos 0θβπθβηβθ

ββF e R

I j dz e z h e R I j R j m h h

z j R j m ??=-≈---? where F (θ)=

βθβsin cos )cos cos(h

h -

Half-wave dipole: 2h =λ/2, βh =π/2

???

??????????=??????=--θθππθθπβφ

βθ

sin cos )2/cos[(2sin cos )2/cos[(60R j m R j m e R jI H e R I j E ??

??????==?2

22*sin ]cos )2/cos[(1521θθππφθR I H E P m

Half-power beam width of a half-wave dipole: ?≈-=?7821θθθ, where θ1 and θ2 are two roots of

sin )cos )2/cos((=θθπ.

P r =r m m m

av R I w I d I d d R P 2

)(54.36sin ]cos )2/[(cos 30sin 22

022

20===???θθθπφθθπ

)(1.73Ω=?r R R r =73.1Ω and U max =R 2P av (π/2)=2

15m I πr

P U D max 4π=

?=1.64>1.5

Radiation patterns of linear dipoles:

E-plane radiation patterns for center-fed dipole antennas

Effective length of a transmitting linear dipole antenna , l e (θ):

{}

??----=-==h h z j R j h h z j R j m dz

e z I e R

j dz e z h e R I j H E θββθ

ββφθθββπθβηηcos cos 00)(sin 30)(sin 4sin

)()0(30θββ R j e R I j -=, where l e (θ)=dz e z I I z j h h

θβθcos )()0(sin ?- is the effective length . Maximum of l e (θ) occurs when θ=π/2?l e (θ=π/2)=

dz z I I h

?-)()0(1

Note: l e =-V oc /E i is the effective length of a receiving linear dipole antenna = that of transmitting one.

Eg. Assume a sinusoidal current distribution on a center-fed, thin, straight half-wave dipole. Find its effective length. What is its maximum value? (Sol.) I (0)=I m , h =λ/4,

?--=

/cos )4

(sin )

0(sin )(λλ

βλ

βθ

θdz e

z I I z j m e ]sin )cos 2cos([

θπ

, πλβπ==

2)2(e

Eg. A 1.5MHz uniform plane wave having a peak electric field intensity E 0 is incident on a half-wave dipole at an angle θ. (a) Find the expression for the open-circuit voltage oc V at the terminals of the dipole. (b) If the dipole is connected to a matched load, what is the maximum power L P delivered to the load?

(Sol.) (a) ]sin )

cos 2cos([00θ

θπ

πλE E V e oc -=-= , m 200=λ

(b) r

oc L L r c

L R V R R R V P 82122

0=+=

Eg. The transmitting antenna of a radio navigation system is a vertical metal mast 40m in height insulated from the earth. A 180kHz source sends a current having an amplitude of 100A into the base of the mast and the earth to be a perfectly conducting plane, determine: (a) the effective length of the antenna (b) the maximum field intensity at a distance 160km from the antenna (c) the total radiated power.

(Sol.) (a) m h

dz h z I I h e 202

)1(1000==-=? , (b) R h I R I E ββηπθ000max 3024=?=

20020

14.1)4(32sin 221π

βηπ

θθπηkW I d R E P r

Monopole antenna: P

r =18.27I m 2W , and R r =2P r /I m 2=36.54Ω is exactly one-half of the radiation resistance of a half-wave antenna in the free space. D =2πU max /P r =1.64 is the same as the directivity of a half-wave antenna.

8-4 Traveling –wave Antenna

j e

I z I β-=0)(, θβηπβsin )(400R

e Idz j dE R

j -= for a small dipole Idz ?-=L z j R j dz e z I e r j E 0

cos 0)(4sin θ

ββθπθβη)(60)]cos 1)(2/([0θθβF e R I j L R j -+-=, where

θβθθcos 1]

2/)cos 1(sin[sin )(--=

L F

Some examples of coplanar antennas (by H. –C. Chen and Dr. I-Fong Chen):

Test Result

Unit ：dBi

2.4G~2.5GHz 的量測結果表

The Impedance of the Tab Monopole in the Smith Chart

The S11 parameter of the Tab Monopole

The VSWR of Tab Monopole

Test Result

2.4G~2.5GHz的量測結果表

The Impedance of the Semi-Circular Tab Monopole in the Smith Chart

The S11 parameter of the Semi-Circular Tab Monopole

The VSWR of the Semi-Circular Tab Monopole

Normal mode (s , 2b <<λ): Its behavior is like an electric dipole Axial mode (s , 2b ≈λ): Its mainbeam placed in the endfire direction. φφθθE a E a

E ??+= θβππωμφθβsin )??)((420b a js a R e I N R

j +=-: Elliptically-polarized. If s =βπb 2 or b =

s , it becomes circularly-polarized.

Eg. A helical antenna operating in the normal mode has N turns with diameter 2b and interturn spacing s . Both 2b and s are very small in comparison to N /λ and are adjusted to radiate circularly polarized waves. Find (a) its directive gain and directivity, (b) its radiation resistance. (Sol.) (a)

θβππωμ?θβsin ]??)[(40?+=-b a js a R e I N E R j , θβππβηθφβsin ]??)[(4?10?-=?=-b a js a

e I N E a H R

j R Circularly polarized: s =βπb 2,

θπ

ηβ2

0222sin )(16]Re[21??NIs H E a R P a R U R av R =?=?= 202020)(sin NIs b d d U P r πηβφθθππ==??θπ2

sin 2

34==?r D P U G , 5.1)2(==πD G D

(b) 2222

02)(403)(2b N NIs I

P R r r πβπη==

Note: Receiving antenna ’s pattern is identical with transmitting one ’s.

Two-element antenna array: (In case of no coupling between antennas) φθcos sin 01d R R -?

])[,(1

0101

0R e e R e F E E E E R j j R j m βξβφθ--+=+=

]1[),(cos sin 0

ξφθββφθj d j R j m e e e R F E +=-

cos 2(),(2/00ψ

φθψβj R j m

e e R F E -=, where Ψ=βd sin θcos φ+ξ ?2

cos ),(20.ψ

φθ?=

F R E E m =Element Factor×

Array Factor

Eg. Plot the H -plane radiation patterns of two parallel dipoles for the following two cases: (a) 0,2/==ξλd , (b) 2/,4/πξλ-==d . (Sol.) Let the dipole is z -directed In the H -plane )2/(πθ=: )cos (2

cos 2cos

)(ξφβψ

φ+==d A (a) )cos 2cos()(φπφ=A , (b) )1(cos 4

cos )(-=φπ

φA

General Uniform Linear Arrays:

Normalized array factor in the xy-plane (θ=π/2):

ψ-ψψ++++=

ψ)1(2 (11)

)(n j j j e e e N

A ψψ--=

j jN e e N 111=)2/sin()

2/sin(1ψψN N , where Ψ=βd sin(π/2)cos φ+ξ=βd cos φ+ξ Mainbeam direction, φ0: ∵ Max at Ψ=0, ∴ βd cos φ0+ξ=0?d

βξ

φ-=0cos Null locations:

πψ

k N ±=2

, k =1,2,3,… Sidelobe locations:

)12(2π

+±=ψm N , m =1, 2, 3, …

The first sidelobe level:

232π±=ψN , )(212.0)

3/2sin(1

1)(∞→==ψN as N N A π Broadside array )0,2

(0=±

=ξπ

φ: |E max | occurs at a direction ⊥ the line of arrays. Endfire array ),0(0d βξφ-==: |E max | occurs at a direction // the line of arrays.

Beamwidth between two first nulls:

N N π

ππ42,22121=ψ-ψ?-=ψ=ψ (2φ?N

d d d π

φφβξφβξφβ4)cos (cos )cos ()cos (2121=-=+-+? Let φφφφφφ?-=?+=0201,

)(sin )2(10Nd λ

φπφ-=??= for a broadside array.

φφ2)0(0≈

??= for an endfire array.

Eg. For a uniform linear array of 12 elements spaced λ/2 apart. Sketch the normalized array pattern )(ψA .

(Sol.) 2λ

d , πβ=d , )

2/sin()

6sin(121)2/sin()2/sin(1)(ψψ=

ψψ=

ψN N A Endfire )1(cos cos cos ,-=-=+=ψ-=?φππφπξφβπξd Broadside φπξφβξcos cos ,0=+=ψ=?d Half-power point: ???=??=ψψd d /78.46)/(55.9221

)2/sin(12)6sin(λλφ array

broadside for ree array endfire for ree deg deg

Eg. Consider a five-element broadside binomial array. (a) Determine the relative

excitation amplitudes in the array elements. (b) Plot the array factor for d =λ/2. (c) Determine the half-power beamwidth and compare it with that of a five-element uniform array having the same element spacings. (Sol.) 1:4:6:4:1, broadside 0=?ξ (a) ψψψψ++++=

ψ4324641161)(j j j j e e e e A ψ+ψ+=2cos 2cos 8616

1, where ξφβ+=ψcos d

(b) 2

=d , πβ=d , and 0=ξ 2)]cos cos(1[4

)(φπ+=

ψ?A (c)

1)]cos cos(1[412=+φπ, ?=86.74φ, ∴ ?=?-?=?28.30)86.7490(22φ

Phased Array: ∵d

βξ

φ-=

0cos , ∴Vary ξ electrically ?Vary φ0 (the direction of the

main beam). It can be utilized as a military radar system to scan and track a target.

Eg. Draw the far-field pattern of a phased array with N =5, d =λ/2.

(Sol.) The effective scan range is about from 600=φ to 1200=φ

as follows.

2π

ξ=

?30πφ=

0=ξ?20πφ= 2πξ-=?3

20πφ=

Eg. Obtain the pattern function of a uniformly excited rectangular array of N 1×N 2 parallel half-wave dipoles. Assume that the dipoles are parallel to the z -axis and their centers are spaced d 1 and d 2 apart in the x - and y -directions, respectively.

(Sol.) )()(sin )cos 2cos(),(y y x x A A F ψ?ψ?=θθπφθ, where )2

sin()

2sin(1)(11x x x x N N A ψψ=

ψ, )2sin(

)

2sin(

)(22

y y y y N N A ψψ=

ψ, φθβcos sin 2

d x =

ψ, and φθβcos sin 2

d y =