Chung-Ang University Field & Wave Electromagnetics 8-6 Normal Incidence at a Plane Conducting...

Chung-Ang University Field & Wave Electromagnetics

8-6 Normal Incidence at a Plane Conducting Boundary

• The incident wave travels in a lossless medium • The boundary is an interface with a perfect conductor.

Medium 11( 0) 2( )

Medium 2

Perfect conductor

Hi

Ei

ani

Hr

Er

anr

Incident

wave

Reflected

wave

y .

x

z

z=0

10ˆ( ) j z

i x iE z a E e

10

1

ˆ( ) j zii y

EH z a e

0 1

1

: the magnitude of : phase constant

: intrinsic impedance of medium 1i iE E

Incident wave ( inside medium 1 )



Reflected wave ( inside medium 1 )

10ˆ( ) j z

r x rE z a E e

1

1ˆ( ) ( )r nr rH z a E z

1

1ˆ( ) ( )z ra E z

10

1

1ˆ j z

y ra E e

Inside medium 2 , both electric and magnetic fields vanish 2 20, 0

E H

No wave is transmitted across the boundary into the z > 0 .



01 1

1

ˆ( ) ( ) ( ) 2 cos

ii r y

EH z H z H z a z

Total wave in medium 1

1 11 0 0ˆ( ) ( ) ( ) ( )j z j z

i r x i rE z E z E z a E e E e

- continuity of tangential component of the E-field at the boundary z = 0

1 0 0 2ˆ(0) ( ) (0) 0x i rE a E E E

0 0r iE E

1 11 0ˆ( ) ( )

j z j zx iE z a E e e 0 1

ˆ 2 sin x ia j E z



The space-time behavior of the total field in medium 1

1 1 0 1ˆ( , ) Re[ ( ) ] 2 sin sinj tx iE z t E z e a E z t

01 1 1

1

ˆ( , ) Re[ ( ) ] 2 cos cosj t iy

EH z t H z e a z t

11

1

Zeros of ( , ) occur at , or , 0,1,2,...

2Maxima of ( , )

E z tz n z n n

H z t

11

1

Maxima of ( , ) occur at (2 1) , or (2 1) , 0,1,2,...

2 4 Zeros of ( , )

E z tz n z n n

H z t

01 1

1

ˆ( ) ( ) ( ) 2 cos

ii r y

EH z H z H z a z

1 11 0ˆ( ) ( )

j z j zx iE z a E e e 0 1

ˆ 2 sin x ia j E z



The total wave in medium 1 is not a traveling wave.

Note following three points

Standing wave.

ⅰ) vanishes on the conducting boundary. 1

E

ⅱ) a maximum on the conducting boundary.

1

H

ⅲ) The standing waves of and are in time quadrature ( 90˚ phase difference ).

1

H1

E

z

x

2

0t 5 / 4t

3 / 2t

/ 4,3 / 4t

/ 2t

z

0t / 4t

/ 2t 3 / 4t

t

4

3

4

E1 versus z

H1 versus z

1 1 0 1ˆ( , ) Re[ ( ) ] 2 sin sinj tx iE z t E z e a E z t

01 1 1

1

ˆ( , ) Re[ ( ) ] 2 cos cosj t iy

EH z t H z e a z t


8-7 Oblique Incidence at a Plane Conducting Boundary

8-7.1 Perpendicular Polarization

Medium 11( 0) 2( )

Medium 2

Perfect conductor

Hi

Ei

ani

Hr

Er

anr

Incident

wave

Reflected

wave

y .

x

z

z=0

Direction of propagation of incidence wave

i

r


ˆ ˆ ˆsin cosni x i z ia a a

1

1ˆ( , ) ( , )i ni iH x z a E x z

1 ˆ0ˆ( , ) nij a R

i y iE x z a E e

1 ( sin cos )0

1

ˆ ˆcos sin i ij x zix i z i

Ea a e

1 ( sin cos )0ˆ i ij x z

y ia E e




Direction of propagation of reflected wave

ˆ ˆ ˆsin cosnr x r z ra a a

0 0 &r i i rE E

1 ( sin cos )0ˆ( , ) r rj x z

r y rE x z a E e

Boundary condition, z = 0

1( ,0) ( ,0) ( ,0)i rE x E x E x

1 1sin sin0 0ˆ ( ) 0i rj x j x

y i ra E e E e , for all x

Snell’s law of reflection

» the angle of reflction equals the angle of incidence.



Magnetic field of the Reflected wave

1 ( sin cos )0ˆ( , ) r rj x z

r y rE x z a E e

1 ( sin cos )0 ( snell's law of reflection )ˆ i ij x z

y ia E e

1

1ˆ( , ) ( , )r nr rH x z a E x z

1 ( sin cos )0

1

ˆ ˆ cos sin i ij x zix i z i

Ea a e



Total field in medium 1

1( , ) ( , ) ( , )i rE x z E x z E x z

1 sin0 1ˆ 2 sin( cos ) ij x

y i ia j E z e

1

1

sin01 1

1

sin1

ˆ( , ) 2 cos cos( cos )

ˆ sin sin( cos )

i

i

j xix i i

j xz i i

EH x y a z e

a j z e

1 1 1cos cos sin0ˆ ( )i i ij z j z j x

y ia E e e e



In z-direction ( x=constant )

90 out of phase

1 1sin( cos ) j ty iE jA z e

no is propagate.avP

In x-direction ( z=constant )

1 1cos( cos ) j tx iH B z e

traveling wave

1( sin )1

ij t xyE Ce

11

1 sin sinxi i

uu

1

1 sinxi

C = f(z) , D = g(z)1( sin )

1ij t x

zH De



In x-direction

( ) ( )C f z D g z

1

1

( sin )1

( sin )1

i

i

j t xy

j t xz

E Ce

H De

nonuniform plane wave

when 1sin( cos ) 0iz

or when

11

2cos cos , 1, 2,3,...i iz z m m

1 , 1, 2,3,...2cos i

mz m

1 0E

1( , ) ( , ) ( , )i rE x z E x z E x z

1 sin0 1ˆ 2 sin( cos ) ij x

y i ia j E z e



In x-direction , wave propagate

1 0xH But, ( not transverse )

∴ TE wave

- At A, A’’, O (intersection of the long and short dashed lines) → E=0

- At B ( intersections of two long dashed line) → E ( out of page ) is maximum .

11sing

i

- At B’ ( intersections of two short dashed line) → E ( into the page ) is maximum .

- Thes points travel in the +x direction

Guided wavelength


8-7.2 Parallel Polarization

Medium 11( 0) 2( )

Medium 2

Perfect conductor

Hi

Ei

ani

Hr

Eranr

Incident

wave

Reflected

wave

y .

x

z

z=0

Direction of propagation of incidence wave

i

rIncident wave ( inside medium 1 )

ˆ ˆ ˆsin cosni x i z ia a a

1 ˆ0 ˆ ˆ( , ) ( cos sin ) nij a R

i i x i z iE x z E a a e

1 ( sin cos )0

1

ˆ( , ) i ij x zii y

EH x z a e

1 ( sin cos )0 ˆ ˆ( cos sin ) i ij x z

i x i zE a a e



Direction of propagation of reflected wave


ˆ ˆ ˆsin cosnr x i z ia a a

1 ( sin cos )0 ˆ ˆ( , ) ( cos sin ) r rj x z

r r x i z rE x z E a +a e

1 ( sin cos )0

1

ˆ( , ) r rj x zrr y

EH x z a e


? ?r roE Boundary Condition 1 0 ( 0)tE z

1 1sin sin ( cos ) ( cos ) 0i rj x j xio i ro rE e E e ( for all x)

ro io r iE E


0 1ˆ2 [ cos sin( cos )i x i iE a j z

1 sin01

1

ˆ 2 cos( cos ) ij xiy i

Ea z e


Total field in medium 1

1 sin1ˆ sin cos( cos )] ij x

z i i+a z e

1( , ) ( , ) ( , )i rE x z E x z E x z

1( , ) ( , ) ( , )i rH x z H x z H x z

In z-direction : standing-wave1 ,xE 1yH

In x-direction : traveling wave. ( phase velocity : )

1 ,zE 1yH1 1 / sinx iu u

nonuniform plane wave

where1 1 / 2cos ( 1,2,3,...)iz m m 1 ( for all x)0xE ∴ TM wave


8-8 Normal Incidence at a Plane Dielectric Boundary

Medium 11 1( , )

Hi

Ei

ani

Incident

wave

y .

x

z

z=0

10ˆ( ) j z

i x iE z a E e

10

1

ˆ( ) j zii y

EH z a e


Hr

Er

anr

Reflected

wave

Medium 22 2( , )

Ht

Et

ant

Transmitted

wave


10ˆ( ) j z

r x rE z a E e

10

1 1

1ˆ ˆ( ) ( ) ( ) j zr

r z r y

EH z a E z a e

Transmitted wave ( inside medium 2 )

20ˆ( ) j z

t x tE z a E e

20

2 2

1ˆ ˆ( ) ( ) j zt

t z t y

EH z a E z a e

1 2 1 20 ,



The tangential components (the x-components)of the electric and magnetic field

intensities must be continuous. ( at interface z=0 )

1tan 2 tanE E 1tan 2 tanH H

(0) (0) (0)i r tE E E

io ro toE E E

(0) (0) (0)i r tH H H

1 2

1( ) to

io ro

EE E

2 1

2 1ro ioE E

2

2 1

2to ioE E

2 1

2 1

ro

io

E

E

Reflection coefficient ( + or - ) ≤ 1

1 2 : 0

2 0( ) 1short E/H, E=0 perfect conductor !!

2 ( ) 1open H(I)=0 No current !!



2

2 1

2to ioE E

2

2 1

2to

io

E

E

Transmission coefficient ( + always )

1

1 , 0 If medium 2 Perfect conductor2 0

0 0 0 , 0r i tE E E

Totally reflected . Standing wave produced in medium 1 .



If medium 2 is not a perfect conductor ,

partial reflection will result .

1( ) ( ) ( )i rE z E z E z

1 1ˆ ( )j z j zx ioa E e e

1 1 1ˆ [(1 ) ( )]j z j z j zx ioa E e e e ( 0)z

11ˆ [(1 ) ( 2sin )]j z

x ioa E e j z traveling standing

1 121 ˆ ( ) (1 )j z j z

x ioE z a E e e

( 0)z

12 2

1(1 2 cos 2 )ioE z

1( )E z 1 1

12 2 * 2(1 )(1 )j z j z

ioE e e



For dissipationless media are real .

However, can be positive or negative.

1 2, , ,

ⅰ) 2 12 1

2 1

0 ( )

- Maximum value of is ,

1( )E z

0 (1 )iE

which occures when 1 max2 2 ( 0,1,2,...)z n n

1max

1

, 0,1, 2,...2

nnz n

- Minimum value of is ,

1( )E z

0 (1 )iE

which occures when 1 min2 (2 1) ( 0,1,2,...)z n n

1min

1

(2 1)(2 1), 0,1, 2,...

2 4

nnz n



ⅱ) 2 10 ( )

- Minimum value of is ,

0 (1 )iE 1( )E z

1max

1

, 0,1,2,...2

nnz n

at

- Maximum value of is ,

1( )E z

0 (1 )iE

1min

1

(2 1)(2 1), 0,1,2,...

2 4

nnz n

at

max

min

1

1

ES

E

Standing wave Ratio (SWR)

if = 0, S=1 : No reflection, full power transmission

if = 1, S= : Total reflection, no power transmission

1

1

S

S

( 1 1 , 1 )S



Transmitted wave

1 11 ˆ( ) ( )j z j z

x ioE z a E e e

1 11

1

ˆ( ) ( )j z j zioy

EH z a e e

1 12

1

ˆ (1 )j z j zioy

Ea e e

1 12ˆ (1 )j z j zx ioa E e e

1 1max minE H

2ˆ( ) j zt x ioE z a E e

2

2

ˆ( ) j zt y ioH z a E e

Chung-Ang University Field & Wave Electromagnetics 8-6 Normal Incidence at a Plane Conducting...

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