Antenna Theory and Random Signals, Exercises for Electrical Engineering

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Antenna Fundamentals

( )

( )

( )

[ ]

[ ]

[ ] [ ] ˆ ˆ D D R 4

- 1 - 1 e e P P 16.

ˆ ˆ 4

D - 1 e A 15.

4

D A 14.

1 - SWR 1 SWR ˆ and

ˆ 1

ˆ 1 SWR 13

Z Z Z- Z

ˆ 12.

- 1 e e 11.

D e P U 4

e G 10.

P e P 9.

P

, U 4 G 8.

A 4

D and P U 4

D 7.

P

, U 4 D 6.

ˆ ˆ PLF 5.

d d sin ) ,( U P 4.

E E 2 1 Wr , U3.

E E 2 1 â W 2.

E

â E

â - ) , (r, Ĥ

E â E â ) , (r, E 1.

2 rt0r0t

2 2

r 2

tcdrcdt t

r

2 rt

2

0 2

cdem

2

0em

0L

0L

2 cdt

max t0

in g

2 em

max 0

2 rt

0

2

2 2 av

2

2 2 rav

ρ•ρ 

  π λ

ΓΓ=

ρ•ρ 

  

π λ

Γ=

 

  

π λ

=

+ =Γ

Γ−

Γ+ =

+ =Γ

Γ=

= 

  

 π =

=

φθπ =

λ

π =

π =

φθπ =

ρ•ρ=

∫ φθθφθ∫=

 

  +

η ==φθ

 

  +

η =

η +

η =φθ

+=φθ

ππ

φθ

φθ

θ φ

φ θ

φφθθ

Array Theory

( )[ ]

( )[ ]

sin sin dk and cos sin dk

2 sin

2 N

sin

N 1

2

sin

2 Msin

M 1 AF :ArrayPlanar 6.

cos dk 2 1 u

u 1 -n 2 cos a AF

u 1 -n 2 cos a AF

:ArrayLinear uniformNon 5.

array) endfire odyard(Hansen Wo dk N 2 1.789 D

array) endfire(ordinary dk N 2 D

array) (broadside dk N D 4.

N

kd - :array endfire dyardHansen Woo 3.

cos â sin sin â cos sin â â cos â sin â - â

sin â cos â â 2.

â â cos and cos kd where

2 sin

2 Nsin

N 1 AF 1.

yyyxxx

y

y

x

x

1M

1n n12M

M

1n n2M

0

0

0

zyxr

yx

yx

rarrayN

β+φθ=Ψβ+φθ=Ψ     

    

 

  

 Ψ

 

  

 Ψ

   

   

  

   Ψ

  

   Ψ

=

θ=

=

=

  

   π

=

π =

π =

  

   π+=β

θ+φθ+φθ=

φ+φ=

φ+φ=

•=γβ+γ=Ψ   

   Ψ

  

   Ψ

=

∑ +

= +

=

φ

ρ

Pascal’s Triangle

Tschebyscheff Polynomials

Antenna Synthesis 1. Schelkunoff’s Polynomial

( ) ( ) ( )

locations null are ........ ,z ,z ,z

β θ cos dk Ψ and e z re whe

z - z .............. z - z z - z a e a AF

321

Ψ j

N

1n 1-N21n

Ψ j n

+==

== ∑ =

2. Fourier Transform Method

( )

( ) zd e ) z(I SF

d e SF 2 1 ) z(I

Source Line (i)

z j a

2-

2 -

z j - da

2

1

′′=θ

ξξ π

=′

′ξ

ξ′ ζ

ζ

∫ 

( )

( ) e a AF

β θ cos dk Ψ where

d e AF 2 1 a

ArrayLinear ii)

M

-Mm

m j m

m j dm

2

1

=

ψ

ψ ζ

ζ

+=

ψΨ π

=

3. Woodward Lawson Method

( )

( ) ( )

( )

( )

( ) ( )

( )

( )

( )

( ) ( )

( )

cos - cos 2

dNk

cos - cos 2

dNksin b f AFfactor array Total

cos - cos

2 dNk

cos - cos 2

dNksin b f sampleeach For

ArrayLinear (ii)

cos - cos

2 k

cos - cos 2 ksin

b s SFfactor space Total

cos - cos

2 k

cos - cos 2 ksin

b s sampleeach For

:Source Line (i)

arraylinear for AF b source linefor SF b

:sampleeach for efficient -co excitation Sample 1 M 2 N samples ofNumber

............... 2, 1, 0, m m cos location Sample

separation Sample

m

m M

-Mm m

M

-Mm m

m

m

mm

m

m M

-Mm m

M

-Mm m

m

m

mm

mdm

mdm

1 - m

   

   

θθ

  

   θθ

∑=∑=θ

   

   

θθ

  

   θθ

=

   

   

θθ

  

   θθ

∑=∑=θ

   

   

θθ

  

   θθ

=

θ=θ= θ=θ=

+= ±±=∆=θ

λ =∆

==

== 

Apertures

[ ]

[ ]

[ ]

[ ]

[ ]

[ ] L N r 4

ek j - H L N r 4

ek j - E

L N r 4

ek j H L N r 4

ek j - E

0 H 0 E 7.

â â â

0 cossin -

sin -sin cos cos cos cossin sin cos sin

â â â

6.

â r cos r 5.

Sd e cos M sin M- L

Sd e sin M - sin cos M cos cos M L

Sd e cos J sin J- N

Sd e sin J - sin cos J cos cos J N 4.

Sd e M L e wherL r 4

e Sd R e M

4 F 3.

Sd e J N e wherN r 4

e Sd R e J

4 A 2.

E n̂ - M and H n̂ J 1.

rk j -rk j -

rk j -rk j -

rr

z

y

xr

r

S

cos rk j yx

S

cos rk j zyx

S

cos rk j yx

S

cos rk j zyx

S

cos rk j S

rk j -

S

Rk j - S

S

cos rk j S

rk j -

S

Rk j - S

SS

 

  

 η

+ π

=−η π

=

 

  

 η

− π

=+η π

=

==

  

  

  

  

φφ θφθφθ θφθφθ

=   

  

•′=ψ′

′φ+φ=

′θφθ+φθ=

′φ+φ=

′θφθ+φθ=

′= π

ε =′

π ε

=

′= π

µ =′

π µ

=

×=×=

φ θφθφφ

θ φθφθθ

φ

θ

ψ′ φ

ψ′ θ

ψ′ φ

ψ′ θ

ψ′

ψ′

∫∫

∫∫

∫∫

∫∫

∫∫∫

∫∫∫

Horns

( )

( ) ( ) ( )

( )

( ) ( ) ( )

2 b and 3 ay directivit optimumFor .4

4 1 -

a )aa ( p and

4 1 -

b )bb ( p 3.

cos and cos .2

- x 0 where 2 x x 4.

x k variationPhase :horn sectoral-H 3.

- y 0 where 2 y y 2.

y k variationPhase :horn sectoral-E 1.

1 12 1

2

1

h 1H

2

1

e 1E

hh2ee1

2h 2

2

1e 1

2

ρλ=ρλ=

 

  

 ρ −=

  

 ρ −=

ψρ=ρψρ=ρ

ρρ≤′δ≤⇒ ρ ′

=′δ

′δ=

ρρ≤′δ≤⇒ ρ ′

=′δ

′δ=

Some Standard Integrals

8 c 3 dx x

c cos 4.

c

2

c 2

sin

c 2

-

2 c dx e x

c cos .3

c

2

c 2

sin c dx e 2.

3 4 d sin 1.

2 / c

2 / c -

4

2 2

2 x j

2 / c

2 / c -

2

x j 2 / c

2 / c -

0

3

=  

   π

   

   

  

   α

  

   α

    

    

  

   απ

π =

 

   π

   

   

  

   α

  

   α

=∫

=θθ∫

α

α

π

Corner Reflector

sin â cos â â 1. yx φ+φ=ρ