1

EMLAB

Radiation principles

2

EMLAB

Radiation field 계산 식 유도

R

l

t

IlI

R

ej

Rt

jkR

/4)(

4)(rE

)(tI

3

EMLAB

Basic laws of EM theory

0

B

D

DJH

BE

t

t1) Maxwell’s equations

0

t

J2) Continuity equation (the relation between current

density and charge density in a space)

)(0

0

MHB

PED

3) Constitutive relation (explains the properties of materials)

4) Boundary conditions ( should be satisfied at the interface of two materials by E, H, D, B.)

4

EMLAB

0)1( B

Potentials of time-varying EM theory

AB 0)( A

tt

tt

AE

AE

AE

AE

0)(

t

BE)2(

t

D

JH)3(

ttt

AJ

EJA

1

AJA

Att

2

22

D)4(

t

t

)(2 A

A

0

0)(

A

Vector identity

5

EMLAB

AB

To find unique value of vector potential A, the divergence and the curl of A should be known.

Only the curl of A is physically observed, divergence of A can be arbitrarily set.

0

t

A

For the above choice, (3), (4) become

Lorentz condition

JA

AJA

A

2

2

22

2

22 1

tct

2

2

22

2

22 1

tct

0

t

A Lorentz condition

Lorentz condition

6

EMLAB

Solution of wave equations in free space

JA

A

2

2

22 1

tc

2

2

22 1

tc

•boundary condition : free space

1. Because the number of variables are as many as four (x, y, z, t), we apply Fourier transform to the above equations.

dedec

detdet

tjtj

tjtj

~

2

1~~

2

1

),(~2

1),(,),(

~

2

1),(

2

22

rrrr

2. For a non-homogeneous differential equation, it is easier to substitute the source term with a delta function located at origin. (effectively it is an impulse response.)

~~~

2

22

c

7

EMLAB

)(22 r gkg

1. The impulse response with the source replaced with a delta function is called a Green function g.

2. For the differential equation with a delta source, a solution is sought first in the region other than origin. Then an integration constant is generated and its value can be found by the delta function.

022 gkg

3. For free space, the green function of the point source is spherically sym-metric. That is, g is a function of r only.

0)()(

0)(1 2

2

22

2

2

rgkr

rggk

r

rg

r

Scalar Green function of free space

ck

where

kr

kr

e

r

Ag

jkr

cos

sinr

Aeg

jkr

8

EMLAB

4. To find the value of A, a volume integral is performed for the sphere at the origin and of radius .

1)(22 VVVdgdkgd r

)0(01)1(4

4

sin

0

2

2

0 0 0

222

jk

jkr

jkr

V

ekA

drreAk

drddrr

eAkgdk

)0(4)1(4

sin)1(

0

22

0 2

2

AejkA

ddrr

ejkrA

dggd

jk

jk

SVa r

ergA

jkr

4)(,

4

1

5. If the source is located at r’, the solution is

',4

)',( rrrr

RR

eg

jkR

Green function of free space

9

EMLAB

~~~ 22 k ')'()'(~)(~

' d

Vrrrr

)'()',()',( 22 rrrrrr gkg ',4

)',( rrrr

RR

eg

jkR

''

'4

)'(~')',(

)'(~),(

~V

jkR

Vd

R

edg

r

rrr

r

6. The source can be represented by an integral of weighted delta functions, the scalar potential is

'

'

)/(

'

'

'4

)/,'(

'),'(~2

1

4

1

''4

),'(~

2

1

),(~

2

1),(

V

V

cRtj

tj

V

jk

tj

dR

cRt

ddeR

dede

det

r

r

rr

r

rr

rr

10

EMLAB

'

'4

)/,'(),(

Vd

R

cRtt

r

r

Retarded potential

(Retarded potential)

'

'4

)/,'(),(

Vd

R

cRtt

rJ

rA

In the same manner, a vector potential is obtained.

The potentials A and is not independent. They are related by Lorentz condition.

0

t

A

In the electro-dynamic solution, the time variable is retarded by the distance from the source.

11

EMLAB

z4

)/(

R

cRtI

)(tI

Observation point

Source current

R

Due to the finite speed of electromagnetic waves, an observed signal at a distance R is delayed by R/c.

Retarded potential

12

EMLAB

Electric field in a phasor form

'

' 4

2

2

2

''

'4

)]ˆ(ˆ[

')()(33)(1

4

1

)(1

'

V

jkR

V

jkR

VV

dR

ej

dR

kRjkR

R

kRjkR

R

ej

dgj

gdj

JRRJ

JRRJ

rJJE

jj

t

)( AA

AE

''

)(1

)]()([1

VVdg

jdRg

jj

rJrJ

A

R

eRgdRg

jkR

V

4)(,')()(

'

rJA

JJRR jkRjkR e

R

jkRe

R

kRjkRg

35

2

4

1)(

4

)(33

13

EMLAB

Example – wire antenna coscos222 zrzrrrR rr

z

o

r

C

zjkjkr

V

jkR

zdezJr

ej

dR

ej

cos)(4

sin

'4

)ˆ(ˆ)( JRRJrE

2/l

2/l

02/)2/(sin

2/0)2/(sin)(

0

0

zlzlkI

lzzlkIzJ

2coscos

2cos

2sin 0 klkl

r

eIj

jkr

14

EMLAB

Green’s function

dtftf )()()(

2

)()|(2 rrrr G

V

dG )|(

)()( rr

rr

V

Vd

V rrP

rrrrP

),(

,0)()( V)(rP

V

V

V

d

dG

dG

)()(

)|()(

)|()(

2

22

rrr

rrr

rrr

15

EMLAB

)()|(2 rrrr G

2

1. G 는 원래 미분 방정식을 만족 .

2. G 는 경계조건도 만족시켜야 한다 .

)()|()( 22 rrrr Gk )( 22 k

Poisson’s equation

Helmholtz’ equation

JEE jk 2

)(2 rrIGG kVector wave equation

rrrr

4

1)|(G

Free space scalar green function

rrrr

rr

4)|(

jkeG

rrIrrG

rr

41

)|(2

jke

k

Green’s function for physical laws

16

EMLAB

Domain

,J

,J

경계가 infinite space 인 경우

17

EMLAB

V

jkRe

e

V

jkRe

0

0

e

dR

eρ

dR

e

μ

j

4

4

1

JA

AH

AE

A

A

Example - Free space

,J

Boundary condition : radiation condition

Source

JAA 22 k

18

EMLAB

Example – Parallel plate

JA

JAA

22

2

2

2

2

2

22

kzyx

k

x

yz

V ee

V e0

0

e

gdρ

gd

μ

j

JA

AH

AE

A

A

1

J0z

dz

nyx

ykxkjyxx

x

dkdkekkAd

znA

JAkzyx

yx )(

02

2

2

2

2

2

2

),(sin

19

EMLAB

Derivation of dyadic Green’s function for vector wave eq.

''

jji,

'

jji

2

2ji,

2

'''1

1

VVV

djdJGjdJxx

g

kgj

kj

jjj

JG

AAA

AAE

222 )'()'()'(,4

zzyyxxRR

eg

jkR

R

xx

R

ejkR

x

R

R

g

x

g jkr )(

4

)1( jj

2jj

R

xx

x

R )( jj

j

jiijjijiji

2

x

R

R

g

xx

R

x

R

R

g

Rx

R

R

g

xx

g

xxx

g

ji3

jjii

2

jjii

2

2

jiijj2

2

))(())((

x

R

R

g

xR

xxxx

R

g

R

xxxx

R

g

x

R

R

g

xx

R

x

R

RR

g

x

R

R

g

JJRR jkRjkR

jkRjkR

eR

jkRe

R

kRjkR

eR

jkRxxxxe

R

kRjkR

RR

g

R

xxxx

R

g

R

gR

35

2

ji,3jjii5

2

ji,3

jjii2

2

4

1)(

4

)(33

4

1))((

4

)(33

))((

'

)'(

'

2

2

dj

k

jk

V

JGE

rrIGG

JEE

안테나의 radiated E field 를 구하기 위한 Green function.

Free space

rrR

20

EMLAB

)()(33)(1

4

1

)(1)(

)(33

4

1

4

)(1)(

4

)(33

1

4

)(1))((

4

)(33

4

2

2

2

2

2

2

4

2

2

3

2

5

2

2

jji,ji

2

2jji,i

ji,3

2

jjii5

2

ji,2

JRRJ

JJRR

JJRR

JG

R

kRjkR

R

kRjkR

R

e

k

R

kRjkR

R

kRjkR

R

e

k

R

kRjkR

R

kRjkR

k

e

Jgxx

g

kJG

eR

kRjkRxxxxe

R

kRjkRGk

jkR

jkR

jkR

jkRjkR

21

EMLAB

Equivalence theorem for Electric field E

)'(, 22 rrIGGJEE kjk

')'()ˆ()ˆ(')|'()'(

')'()ˆ()'(ˆ')|'()'()(

')ˆ()'()'(ˆ

')''()''(

dajdj

dadj

da

d

SV

SV

S

V

GEnGHnrrGrJ

GEnGEnrrGrJrE

BnABAn

BABA

S

JiE

n

V

')|'()'()(

')'()ˆ()ˆ()(

dj

daj

V

i

S

i

rrGrJrE

GEnGHnErE

)(rE

Enˆ

Hnˆ Scatterer 의 표면에서 E 와 H 의 접선 성분만 알면 공간 내의 모든 점에서 E, H 를 구할 수 있다 .

22

EMLAB

'

)'(

'

2

2

dj

k

jk

V

JGE

rrIGG

JEE

Derivation of Equivalence theorem for E-field

arrIaGaG

JEE

)'(2

2

k

jk

aErraGEaGE

JaGaGEEaG

)'()()(

)()()(2

2

k

jk

arEJaGaGEEaG )'()()()(VV

djd

S

V

d

d

aQPPQ

PQQP

arEJaGSEaGaGE )'()()()(VS

djd

SV

ddj SEaGaGEJaGarE )()()()'(

SV

dSdj )ˆ()()()ˆ()()'( EnaGaGEnJaGarE

SV

dSjdj )()ˆ()()ˆ()()'( aGHnaGEnaGJarE

SV

dSjdj ')'|()'ˆ()'|(')'ˆ(')'|()( rrGHnrrGEnrrGJrE

SJiE

'n

V

)(rE

En'ˆ

Hn'ˆ

'n

(Vector Green’s identity)

gxx

g

kG ji,

ji

2

2ji,

1)|'( rrG

SV

dSjdj )|'()ˆ()|'()ˆ()|'()()'( rrGHnrrGEnrrGrJrE

g

xx

g

kG ji,

ji

2

2ji,

1)',( rrG

(a 는 임의의 상수 벡터 .)

23

EMLAB

Equivalence theorem

)'(, 22 rrIGGJEE kjk

')'()'ˆ()'ˆ(')'|()'(

')'()'ˆ()'('ˆ')'|()'()(

')ˆ()'()'(ˆ

')''()''(

dajdj

dSdj

dS

d

SV

SV

S

V

GEnGHnrrGrJ

GEnGEnrrGrJrE

BnABAn

BABA

S

JiE

'n

V

')'|()'()(

')'()ˆ()ˆ()(

dj

daj

V

i

S

i

rrGrJrE

GEnGHnErE

)(rE

Enˆ

Hnˆ

Scatterer 의 표면에서 E 와 H 의 접선 성분만 알면 공간 내의 모든 점에서 E, H 를 구할 수 있다 .

면적 요소 n 의 방향이 바뀜 .

g

gg

ggkgk

gkgk

')'ˆ()',(')'ˆ(

]')'ˆ[()'()'ˆ()',(')'ˆ(

')](')('''[1

)',('

])(''[1

)',(

22

22

EnrrGEn

aEnaEnarrGEn

aaaarrG

aaarrG

24

EMLAB

SV

dajdj ')'|(')'ˆ()'|()'ˆ(')'|()( rrGEnrrGHnrrGJrE

SV

dajdj ')'ˆ()'|(')'ˆ()'|(')'|()( TTT EnrrGHnrrGJrrGrE

g

xx

g

kG ji,

ji

2

2ji,

1)',( rrG

R

eg

jkR

40

; for free space

)'(')(33)(1

4

14

2

2

2

20 JRRJJGR

kRjkR

R

kRjkR

R

e

k

jkR

Equivalence theorem with free space kernel : G0

]'ˆ)'ˆ[(44

)1(]'ˆ)'ˆ[(')'ˆ()',(')'ˆ(

20 REnREnEnrrGEn

R

ejke

R

jkRg

jkRjkR

R

e jkR

4)'ˆ('ˆ

0

JRRJJG

S

jkR

V

jkR

daR

ejd

R

ej '

4

)'ˆ(ˆ)]'ˆ(ˆ[ˆ)'ˆ('

4)'ˆ('ˆ)(

EnR

HnRRHnJRRJrE

Free space Green function 을 이용하여 만든 Green’s theorem 식 .

S

JiE

'n

V

)(rE

Enˆ

Hnˆ

Infinite free space

25

EMLAB

PEC

'S)(rE

,JiE

n

'V

. )(

같다는변하지만

함께도따라선택에의

rE

EiG

G 를 선택할 때 기준은 이 물체의 경계면에서 경계 조건을 만족하는 G 를 쉽게 구할 수 있느냐 이다 . 계산하기 편하게 G 를 정해 놓아도 실제 解와의 오차는 물체의 표면 전류로 보정할 수 있다 .

sm , JJ

air

'S

,JiE

n

'V

sm ',' JJ

두 경우의 G 와 Jm, Je 는 모두 다르다 . 아래 그림의 경우 매질이 균일하므로 free space 의 G 를 선택한다 .

Equivalence theorem

)(rE

26

EMLAB

Dyadic Green’s function for half space – PEC

AE j

2222

2221 )'()'()'(,)'()'()'( zzyyxxRzzyyxxR

tniV iV

V tnV

V

JJdgdg

dJJgdg

dgg

tnJJJ

tnJ

nnttIJttnnnnttA

ˆˆ,''

')ˆˆ('

ˆˆˆˆ,')]ˆˆˆˆ()ˆˆˆˆ([

' 2' 1

' 2' 1

' 21

])()()[(1

)()(11

ttntntntt

ntnt

AAA

AAAH

0H PEC

0nH

0E

0tE

22

112121 4

,4

,,,)ˆˆˆˆ(21

R

eg

R

egggggggdgg

jkRjkR

ntV nt

JnnttA

; image current

전기장의 접선 성분을 0 으로 만들기 위한 Green function 들 .

t 는 지면과 평행한 성분 , n 은 지면에 수직인 성분 .

27

EMLAB

mJ

iJ iJ

Image source for magnetic current mJ

nEJ ˆm

J

iJ

J

iJ

Image source for electric current

HnJ ˆe

Image current due to PEC plane

0][ tan eJG

0][ tan mJG

28

EMLAB

• 실제 상황에서 안테나는 도체판 위에 설치된다 .

• 대부분의 경우 도체 평판은 완전한 평면도 아니고 , 유한한 크기를 가지며 완전도체도 아니다 . • 그러나 문제를 비교적 단순화함으로써 현상에 대한 이해를 분명하게 할 수

있다 . 무한 (Infinite), 평판 (flat), 완전도체 (Perfect conductor) 를 가정 • Image Theory 적용

Antenna

Ground plane

Antenna

Ground plane

Infinite ground plane

Antenna

P.E.C

무한 평판 완전 도체에서의 선형 소자 (1)

29

EMLAB

Image Theory

무한 균일 매질가정 가상의 Imaginary Source 전반사의 법칙 적용 : θ i = θ r

도체속 또는 아래는 관심없음 경계 조건 (Boundary Condition) 무한 도체상에서의 모노폴은 무한균일 매질에서의 다이폴과 동일

무한도체평면

monopole antenna

P.E.C. Equivalent

dipole

30

EMLAB

Love’s equivalence theorem

HE,

Source #1

Source #2 (surface current)

,J

nEJ

HnJ

ˆ

ˆ

m

e

SS HE ,

공간상에서 E, H 를 구하기 위해 공간을 S 를 기준으로 분할한다 .

1. S 의 외부에서 E, H : 실제 source #1 을 적분한 값 + 등가 면에서 source #2 를 면 적분 한 값 .

2. S 의 내부에서 E, H: source #2 를 면적분한 값 .

31

EMLAB

Equivalence theorem

11 HE ,

22 HE ,

D

11 HE ,

0, 22 HE

D

Free space PEC

nEEJ

HHnJ

ˆ)(

)(ˆ

21m

21S

nEJ

HnJ

ˆ'

ˆ'

1m

1S

(1) 무한 공간에 있는 유전체

(2) 도체로 둘러싸여 있는 유전체

•(1) 과 (2) 의 주변 환경은 다르지만 관심 영역인 유전체 안에 존재하는 E, H 는 같게 만들 수 있다 . 이 경우 Js, Jm 을 적절히 영역의 경계에 포함시켜 주면 된다 .

11,

22 ,

11,

32

EMLAB

Schelkunoff’s field equivalence principles

image current

33

EMLAB

iE

1E

2E

1ˆ EnJ m

도체

Electromagnetic 문제

iE

2ˆ EnJ m

Reflected wave 문제 Transmitted wave 문제

2ˆ2 EnJ m

두 문제는 등가이다 . 오른쪽은 도체를 없애고 전류와 전하의 크기를 2 배로 한 것 . :image current 이용함 .

Image current 이용

34

EMLAB

Fourier transform, Fresnel diffraction

'2

)]'ˆ('ˆ[)( daR

ejk

S

jkR

EnnrE

tan)'ˆ'ˆ()'ˆ('ˆ)'ˆ('ˆ EnnEEnnEnn

'2

)( tandaR

ejk

S

jkR

ErE

)(rE

'')','(2

'')','(2

'')','(2

')',','(2

),,(

'

])'()'('2'2[2

0

2

'

])'()'('2'2[2

0

'

])'()'[(2

0

'

22

22

2222

22

1

dydxeyxueR

jk

dydxeyxuR

jke

dydxeyxuR

jke

dazyxuR

ejkzyxu

S

yxyyxxz

kjz

yxzjk

S

yxyyxxyxz

kjjkz

S

yyxxz

kjjkz

S

jkR

z

yyxxz

z

yyxxz

z

yyxxzzyyxxR

zzzyyxxR

22

2

22

2

22222

222

)'()'(

2

1)'()'(

2

11

)'()'(1)'()'(

0',)'()'()'(

: Fresnel diffraction

35

EMLAB

Far field, near field

2

3

22

2

sincos2

'1sin

2

'1cos'

z

r

z

rzrR

36

EMLAB

Far field, near field

37

EMLAB

z

ky

z

kxUe

R

jk

yxuFeR

jk

dydxeyxueR

jkzyxu

z

yxzjk

z

yxzjk

S

yyxxz

kjz

yxzjk

,2

)}','({2

'')','(2

),,(

0

2

0

2

'

)''(

0

2

22

22

22

Fraunhofer diffraction

: Fourier transform

sin

)sin(2

2

'''2

'')','(2

),,(

120

2

0

2

0

)'')('cos(sin'0

2

'

)''(

0

2

22

22

22

ka

kaJaAe

R

jk

ddeAeR

jk

dydxeyxueR

jkzyxu

z

yxzjk

ayyxxjkz

yxzjk

S

yyxxz

kjz

yxzjk

sinsin,cossin

'sin'','cos''

z

y

z

x

yx

)()(,2

1)( 10

2

0

cos0 xxJxdxxJdexJ jx