1

2006/Mar/14

22

The Reciprocity Theorem

• 1.

• 2.

• 3.

)135.1(

33

The Reciprocity Theorem

• Equivalent of the Tx and Rx patterns ofan Antenna– a set: Antenna 1: Tx, Antenna 2: Rx– b set: Antenna 1: Rx, Antenna 2: Tx– On cross-section of transmission lines

an

a1 H1K a

n1

0a

1m E1K a

na2 H1K a

n1

0a

2m E1K b

nb1 H1K b

n1

0b

1m E1K b

nb2 H1K b

n1

0b

2m E1K

dS

dS

SS

SS

21

21

bm

aba

am

bab

KBKE

KBKE

44

The Reciprocity Theorem

• Equivalent of the Tx and Rx patterns of an Antenna dSdS

SSSS

2121

bn

abn

aan

ban

b E1HH1EE1HH1E

)143.1(

yxzIzyx

yxzVzyx

, ,,

, ,,:moden Propagatio

tang

tang

hH

gE

1 ,,z dSyxyx

Shg1

b

b

a

a

RRn

an

bn

n

bn

an I

VKIVZZZZIVIV

1

2

2

1212211

2

1

2

1 ,,, difiningby

,, 21

21

aa

bb I

VKVI

linear.) bemust (Material antenna. theof or type shape, size, theofindepent

proof general a is This identical. are patterns fieldRx Tx / Normalized

55

The Reciprocity Theorem

• For the power patterns of Tx and Rx antennas

2

2

2

2

1

2

2

11

2

1 ,21,

21

Ra

a

b

R

RR

b RIV

KVRRRI

66

Directivity and Gain

• Directivity

– If the radiation intensity is defined by

• Gain

)160.1(

0

2

0sin,

, 4,ddP

PD

)161.1(

)162.1(

0

2

0

2 sin , ddrKP Lacc P

)163.1(

)164.1(

,, DG

77

Directivity and Gain

• Partial directivity and Partial gain

LK

DG ,,

LKDG 101010 log,log,log :dBin

)165.1(

)166.1(

,,, DDD

)167.1()168.1(

)169.1(

)170.1(

88

Receiving Cross Section

• Equivalent receiving cross-sectional area

receiver) (matchedpower absorbed :matched)ion (polarizat waveplane incoming :

,,

r

rr

PS

ASP

antenna. matchedon polarizati lossless allfor same The4

2

)172.1(

)173.1(

)171.1(

)174.1(

)175.1( )190.1(

99

Receiving Cross Section

S

situation-a

situation-b

2, 1, 12 rrba AAII

situation-a situation-b

Tx :1Ant Rx :1Ant

)(Tx :2Ant 2 dlI b1

)177.1()176.1(

)178.1(

)179.1(

)181.1()180.1(

abbaban

an

bn

n

bn

an

IVIVIV

IVIV

112211

2

1

2

1

r r

lossless ,2 :situation- 111 RVIa ga

22

2

221 RI a

1010

Receiving Cross Section

dlIrEIVIV baabba21111 ,,

2

2

22

11411

211

211211

11

1111

111111

1111111111

4

2

,

dlIEVVZ

R

dlIEVVZ

RZZZZVV

ZVZVIZVI

baba

bababa

bR

bbaa

11211

2

111

2

1 21

21:situation- R

Z

VRIPa

aa

acc

2

1112

11

2

1

2

4,,,

21

rDR

Z

VrE aa

2

21211

2112

1 , 16

dlIDr

RZV bb

:situation-b

, 4

, 42

1

1,

2

2211

211

22

1

1,

2

22

2

rbb

rb

abs

AdlIr

RZkV

AdlIr

kP

,4

, 1

2

1, DAr

4

2

rA

11211

2

111

2

1 21

21 R

Z

VRIP

bb

abs

1111

Polarization

tjeEE 11E

EjEE

EjEE

tEtEtEtE

eEjEEjEtr tj

sin cos sin cos ,,,

1111E Re

tBtAEE

EE

EEBEEA

tBtA

cos cos

tan tan

)()( )()(

in which cos cos

11

2222

11E

11E

1212

Polarization

• Linear Polarization

0,but 0or 0or

0

BAA

B

1

1

1313

Polarization

• Circular Polarization

)-( kwisecoutercloc

sin cos90 and If o

handrightttA

BA

11E

)-e( clockwise

sin cos90 and If o

handftlttA

BA

11E

1

1

1414

Polarization

• Elliptical Polarization

2/1 2222

o

cos cos

0 and

tBtAt

BA

E

2cos2sin2tan

at Extrema

22

2

BABt

1

1

1515

• The Center-Fed Dipole

Dipole Fed-Center Simple 2.1 Fig.

Radiation Patterns of Dipoles,Loops, and Helices

2.2 Fig.

tjm exlkItxI sin,

aveformcurrent w

dl 2

d

1616

The Center-Fed Dipole

cossin

are source The

L

lkIIdI

m

1717

The Center-Fed Dipole

• The half-wavelength dipole, 2l = /2

sincos2cos

2

sincos2cos60

reIjH

reIjE

rktjm

rktj

m

1818

The Center-Fed Dipole

• The half-wavelength dipole, 2l = /2

2

609.02m

radIP

73609.02

609.021

4/ Since2

2

rad

mradradm

R

IPRI

l

1919

The Center-Fed Dipole

• The short dipole, 2l <<

!3)(sin :currentinput The

3klklIklII mm

2

22

20

12)(

LR

IklP

rad

rad

2020

The Center-Fed Dipole

2121

The Center-Fed Dipole

2222

Imagesin a

GroundPlane

2323

A Monopole Above a Ground Plane

2424

A Dipole in Front of a Ground Plane

Homework #1Plot the field pattern of a dipole in front of a ground plane as shown in Fig. 2.7with 2l = / 2, h = / 8a) = 90o, = 0 ~ 180o

b) = 90o, = 0 ~ 180o

2525

The Small Current Loop

dIaIdaa

cossin0 sin cos

yx 11l

2626

The Small Current Loop

: large with loopFor ka

2727

Traveling Wave Current on a Loop

2828

The End-Fire Helix

2929

The End-Fire Helix

2/ 0

0

zyx cossin sin cos

Lj

sj

eII

eIsI

dbaadbaa

111l