# Simple array of two antennas Consider two identical ... · PDF fileSimple array of two antennas Consider two identical parallel Hertzian dipoles separated by a distance d I2 I1 z x

• View
219

0

Embed Size (px)

### Text of Simple array of two antennas Consider two identical ... · PDF fileSimple array of two...

• Antennas

Amanogawa, 2006 Digital Maestro Series 64

Simple array of two antennas

Consider two identical parallel Hertzian dipoles separated by a distance d

I1I2

z

x

12

r1r 2 r

d 2 d 2

i

i1

i2

• Antennas

Amanogawa, 2006 Digital Maestro Series 65

The dipole currents have the same amplitude and total phase difference

The electric farfield components at the observation point are

( )

( )

21 1

22 2

cos( 2) I

cos( 2) I

jo o

phasorj

o ophasor

I t I t I e

I t I t I e

= + =

= =

j r jo

j r jo

j I z eir

j I z eir

1

1

2

2

2

1 11

2

2 22

E sin4

E sin4

+

• Antennas

Amanogawa, 2006 Digital Maestro Series 66

At long distance, we have

and the field components can be written as

1 2

1 2

1 2cos cos2 2

i i i

d dr r

d

r

r

r

+

>>

j r d joj I z ei

r d

( ( 2)cos ) 21E 4 ( 2)cos

+

( )j r d j

oj I z eir d

( ( 2)cos ) 22

sin

E4 ( 2)cos

+

+( ) sin

• Antennas

Amanogawa, 2006 Digital Maestro Series 67

After applying the approximations, the two components can be combined to give the total electric field

The final result is

( ) ( )( )j ro

j d j d

j I zi er

e e

1 2

( 2)cos 2 ( 2)cos 2

sinE E E4

+ +

= +

+

j roj I z di er

field of a Hertzian dipole locatedat the center of the arra

array y

factor

sin cosE 2 cos4 2

+

• Antennas

Amanogawa, 2006 Digital Maestro Series 68

The resultant radiation pattern of the electric field is proportional to

The unit pattern is proportional to the radiation pattern of the individual antennas, assumed to be identical.

The group pattern is proportional to the radiation pattern the array would have with isotropic antennas.

Note: on the xy plane, coincides with the azimuthal angle .

Following are examples of twoantenna arrays with specific values of dipole distance and current phase difference.

d

group patterunit

patt rn ne

cossin cos2

+

• Antennas

Amanogawa, 2006 Digital Maestro Series 69

Broad-side / p2 e0 att rnd = =

z

x x

z

x

y

x

yUnit Pattern Group Pattern

x

yResultant Pattern

x

z

• Antennas

Amanogawa, 2006 Digital Maestro Series 70

d Broad-side p/ rn0 t2 at e = =Radiation Pattern for E and H Power Radiation Pattern

• Antennas

Amanogawa, 2006 Digital Maestro Series 71

d End-fir/ 2 1 e pattern80 = =Radiation Pattern for E and H Power Radiation Pattern

• Antennas

Amanogawa, 2006 Digital Maestro Series 72

d Cardioid/ 4 90 pattern = =Radiation Pattern for E and H Power Radiation Pattern

• Antennas

Amanogawa, 2006 Digital Maestro Series 73

d Cardioid patter4 9 n/ 0 = = Radiation Pattern for E and H Power Radiation Pattern

• Antennas

Amanogawa, 2006 Digital Maestro Series 74

d / 2 90 = =Radiation Pattern for E and H Power Radiation Pattern

• Antennas

Amanogawa, 2006 Digital Maestro Series 75

d 0 = =Radiation Pattern for E and H Power Radiation Pattern

• Antennas

Amanogawa, 2006 Digital Maestro Series 76

d 180 = =Radiation Pattern for E and H Power Radiation Pattern

• Antennas

Amanogawa, 2006 Digital Maestro Series 77

CASE STUDY - 1) A radio broadcast transmitter is located 15 km West of the city it needs to serve. The FCC standard is to have 25 mV/m electric field strength in the city. How much radiation power must be provided to a quarter wavelength monopole? We consider =90 for transmission in the plane perpendicular to the antenna

In a monopole, the lower wire is substituted by the ground. The equivalent radiation resistance is half that of the corresponding dipole. Therefore, the total radiated power is half the power radiated by the half-wavelength dipole, for the same current.

max

maxmax

/ 2 dipolecos90E cos2 sin90 2

E 120 0.025 V/m 6.25 A2 15, 000

j rj e Iir

I I

=

= =

=

• Antennas

Amanogawa, 2006 Digital Maestro Series 78

A perfect ground would act like a metal surface, reflecting 100% of the signal. The ground creates an image of the missing wire delivering to a given point above the ground the same signal as a complete dipole. The transmission line connected to the antenna sees only half of the radiation resistance, with total radiated power:

2 2max

1 73.070.193978 0.5 6.25 713.622

W2

/

eqR

totP I

= = =

P

GROUND PLANE

Monopole

/4

• Antennas

Amanogawa, 2006 Digital Maestro Series 79

2) Improve the design by using a twoantenna array.

A good choice of array parameters is

which gives a cardioid pattern

dphase(antenna B) phase(antenna A) 90

/ 4

= =

=

15 km

BA

d

• Antennas

Amanogawa, 2006 Digital Maestro Series 80

The Poynting vector is given by

( )

( )

r

r

I dP t i

r

Ii

r

2

2

2 dipole Poynting vecto array factor

array fa

22 2max

2 2 2

22 2max

ct

2

r

r

o

2 2

cos cos( ) cos 4 cos

2 28 sin

coscos 4 cos cos

2 48 sin 4

+=

=

R

sinR sin cosR

cosR

Nocos sin cos

te =

sin sinR

• Antennas

Amanogawa, 2006 Digital Maestro Series 81

2 220 0

sin ( )totP r d P t d

=

Only /2 because it is a monopole

( )2

22max

2 2 2

array facto

2

r

2 cos2 cos sin0 28 sin2

4cos0 4sin cos

4

I dtotr

P r

d

=

• Antennas

Amanogawa, 2006 Digital Maestro Series 82

The integral over the azimuthal angle gives

For a monopole, we only have the integral

2 20

20

20

4 cos sin cos4 4

1 14 cos sin cos2 2 2 21 14 sin sin cos 42 2 2

d

d

d

= + = + =

20 2d =

• Antennas

Amanogawa, 2006 Digital Maestro Series 83

In the direction of maximum

The same field strength (25 mV/m) is obtained by applying half the current of the original monopole to the array elements (also monopoles)

The total radiated power is proportional to the square of the current, and the integral over gives a factor 4 instead of 2 for the array. Overall, the total radiated power needed by the array, to produce the same electric field, is half that of the individual monopole

90 & 90 array factor 2 = = =

max6.25 3.125 A

2I = =

714 357 W2tot

P = =

• Antennas

Amanogawa, 2006 Digital Maestro Series 84

For a monopole Radiated Power = 0.51428.3127 = 714.155635 [ W ] Rad. Resistance = 0.573.129616 = 36.564808 [ ]

SAME FEEDING CURRENT FOR BOTH DIPOLE AND MONOPOLE

• Antennas

Amanogawa, 2006 Digital Maestro Series 85

For monopoles Radiated Power = 0.5714.1532 = 357.0766 [ W ]

SAME FEEDING CURRENTS FOR BOTH DIPOLES AND MONOPOLES

• Antennas

Amanogawa, 2006 Digital Maestro Series 86

N-Element Antenna Array

Assume a uniform array of N identical antennas. The elements are fed by currents with constant amplitude and with phase increasing by an amount from one to the other. The spacing d between the antennas is uniform.

r ( 1) cosr N d

d

1 2 3 N

( 1)(1) ; (2) ; ; ( )j j No o oI I I I e I N I e = = =

• Antennas

Amanogawa, 2006 Digital Maestro Series 87

The electric field at the observation point (r,) is of the form

j r j r d jo o

j r N d j Noj r j d

o

j N d

jN dj r

o j d

r E e E e e

E e e

E e e

e

eE ee

( cos )

( ( 1) cos ) ( 1)

( cos )

( 1)( cos )

( cos )

( cos )

E( , )

1

11

+

+

+

+

= + +

+

= + ++

=

We have used( cos )1

( cos )( cos )

0

1 1

jN dNjn d

j dn

eee

++

+=

=

• Antennas

Amanogawa, 2006 Digital Maestro Series 88

The magnitude of the electric field is given by We have used

ar

( cos )

( cos

ray facto

)

r

1E( , )1sin[ ( cos ) / 2]sin[( cos ) / 2]

jN do j d

o

er EeN dE

d

+

+

=

+=

+

21 2 sin 2 sin2 2

jx j xx xe j e = =

• Antennas

Amanogawa, 2006 Digital Maestro Series 89

We can rewrite The group pattern has

group pattern

1 sin[ ( cos ) / 2]sin[( cos

E( ,) / 2]

) oN d

N dr N E

+

=+

Maxima when

Nulls when

for

( cos ) 2

cos

integer

0,

0,

2

, 2 ,

, 4

NN

d

d mm N

+ = ##### Antenna design 3: RFID tag antennas using AMCs tag antennas...Antenna design 3: RFID tag antennas using AMCs Dongho Kim, Professor, Ph/D Antennas EM applications Lab., Sejong University
Documents ##### Two Small Helical Antennas for 2 Meters - w6nbc.com turn spacing for VHF are measured in inches ... Two Small Helical Antennas for 2 Meters ... Monopole â€” Single vertical antenna
Documents ##### Consider the region bounded by the curve r = f ( ¸ ) and the two rays
Documents ##### Antennas 1 Antennas - œ‹ç«‹è‡Œ—ç§‘€¤§­¸Taipei juiching/ Circular - Elliptical Impedance Antennas 4 Bandwidth Antennas 5! Antenna types Electrically
Documents ##### Expansion of an Interacting Boson Condensate: the We consider two diï¬€erent kinds of 1D expansion:
Documents ##### 1 Projektseminar Antennas. 2 AntennasSimulationVersuchsaufbauAblaufInfos Einf¼hrungsveranstaltung Antennas Grundlagen Versuchsaufbau Simulation Ablauf
Documents ##### 1 ï¶ The Problem: Consider a two class task with ‰ 1, ‰ 2 ïƒ ïƒ LINEAR CLASSIFIERS
Documents ##### 1 ï¶ Problem: Consider a two class task with ‰ 1, ‰ 2 ïƒ ïƒ LINEAR CLASSIFIERS
Documents ##### Indoor antennas ××•×™××™×‘ ×‌×™× ×¤ ××•× ×× × antennas-lh.pdf15 Indoor antennas - ××•×™××™×‘ ×‌×™× ×¤ ××•× ×× ×
Documents ##### Optical Antennas and Microwave Antennas - Zhejiang Antennas versus...¢  Optical Antennas versus Microwave
Documents ##### Purely pathwise probability-free It^o call \purely pathwise", let us consider two examples in which
Documents Documents