Simple array of two antennas Consider two identical ...

1 antennas Simple array of two antennas Consider two identical parallel hertzian dipoles separated by a distance d i . 1. i i 2 . z r2 r r1. 2 1.. x I2 d 2 d 2 I1. Amanogawa, 2006 Digital Maestro Series 64. antennas The dipole currents have the same amplitude and total phase difference . I1 ( t ) = Io cos( t + 2) I1 = Io e j 2. phasor I2 ( t ) = Io cos( t 2) I 2 = I o e j 2. phasor The electric far field components at the observation point are j r1 + j 2. j Io z e E1 i 1 sin 1. 4 r1. j r2 j 2. j Io z e E2 i 2 sin 2. 4 r2. Amanogawa, 2006 Digital Maestro Series 65. antennas At long distance, we have r >> d 1 2 . i 1 i 2 i . d d r1 r cos r2 r + cos . 2 2. and the field components can be written as j Io z e j ( r ( d 2) cos ) + j 2. E1 i sin . 4 ( r ( d 2) cos ). j Io z e j ( r + ( d 2) cos ) j 2. E2 i sin . 4 ( r + ( d 2) cos ). Amanogawa, 2006 Digital Maestro Series 66. antennas After applying the approximations, the two components can be combined to give the total electric field j Io z sin j r E = E1 + E2 i e 4 r e ( (.))

2 J ( d 2) cos + 2 ) j ( d 2) cos + 2 ). +e ( ). The final result is j Io z sin j r d cos + . E i e 2 cos 4 r 2 . field of a hertzian dipole located array factor at the center of the array Amanogawa, 2006 Digital Maestro Series 67. antennas The resultant radiation pattern of the electric field is proportional to d cos + . sin cos 2 . unit pattern group pattern 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 x y plane, coincides with the azimuthal angle . Following are examples of two antenna arrays with specific values of dipole distance and current phase difference. Amanogawa, 2006 Digital Maestro Series 68. antennas d = /2 =0 Broad-side pattern z z z x x x Unit Pattern Group Pattern Resultant Pattern y y y x x x Amanogawa, 2006 Digital Maestro Series 69.

3 antennas d = /2 =0 Broad-side pattern Radiation Pattern for E and H Power Radiation Pattern Amanogawa, 2006 Digital Maestro Series 70. antennas d = /2 = 180 End-fire pattern Radiation Pattern for E and H Power Radiation Pattern Amanogawa, 2006 Digital Maestro Series 71. antennas d = /4 = 90 Cardioid pattern Radiation Pattern for E and H Power Radiation Pattern Amanogawa, 2006 Digital Maestro Series 72. antennas d = /4 = 90 Cardioid pattern Radiation Pattern for E and H Power Radiation Pattern Amanogawa, 2006 Digital Maestro Series 73. antennas d = /2 = 90. Radiation Pattern for E and H Power Radiation Pattern Amanogawa, 2006 Digital Maestro Series 74. antennas d= =0. Radiation Pattern for E and H Power Radiation Pattern Amanogawa, 2006 Digital Maestro Series 75. antennas d= = 180. Radiation Pattern for E and H Power Radiation Pattern Amanogawa, 2006 Digital Maestro Series 76. antennas CASE STUDY - 1) A radio broadcast transmitter is located 15 km West of the city it needs to serve.

4 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 j e j r Imax cos90 . E = i cos / 2 dipole 2 r sin 90 2 . Imax E = 120 = V/m Imax = A. 2 15, 000. 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. Amanogawa, 2006 Digital Maestro Series 77. antennas Monopole P. /4. GROUND PLANE. 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: 1 2 2 Ptot = Imax / 2 = = 2 2.

5 Req Amanogawa, 2006 Digital Maestro Series 78. antennas 2) Improve the design by using a two antenna array . A good choice of array parameters is d = /4. phase(antenna B) phase(antenna A) = = 90 . which gives a cardioid pattern d A B. 15 km Amanogawa, 2006 Digital Maestro Series 79. antennas The Poynting vector is given by 2. Imax 2 cos 2 d cos + . P( t ) = i r cos 4 cos 8 r sin . 2 2 2 2 2 . 2 dipole Poynting vector ( array factor ) 2. 2. Imax 2 cos 2 . = ir cos 4 cos cos . 8 r sin . 2 2 2 2 4 4 . ( array factor ) 2. Note cos = sin cos R cos . R sin sin . R. R sin cos . R sin . Amanogawa, 2006 Digital Maestro Series 80. antennas The total radiated power is Only /2 because it is a monopole 2 2 2 . Ptot = r 0 . sin d . 0. P( t ) d . 2 2. 2 I max 2 cos . Ptot = r cos sin d . 0 8 2 r 2 sin 2 2 . 2 2 . 0 4cos 4 sin cos 4 d . ( array factor )2. Amanogawa, 2006 Digital Maestro Series 81. antennas The integral over the azimuthal angle gives 2 2.

6 4 . 0. cos sin cos d . 4 4 . 2 1 1 . =4 .. 0 2 2. + cos sin cos d . 2 2 . 2 1 1 d = 4 . =4 +. 0 2 2. sin . 2. sin cos .. For a monopole, we only have the integral 2 . 0. d = 2 . Amanogawa, 2006 Digital Maestro Series 82. antennas In the direction of maximum = 90 & = 90 array factor = 2. The same field strength (25 mV/m) is obtained by applying half the current of the original monopole to the array elements (also monopoles). Imax = = A. 2. 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 714. Ptot = = 357 W. 2. Amanogawa, 2006 Digital Maestro Series 83. antennas For a monopole Radiated Power = = [ W ]. Rad. Resistance = = [ ]. SAME FEEDING CURRENT. FOR BOTH DIPOLE AND. MONOPOLE. Amanogawa, 2006 Digital Maestro Series 84.

7 antennas For monopoles Radiated Power = = [ W ]. SAME FEEDING. CURRENTS FOR. BOTH dipoles . AND MONOPOLES. Amanogawa, 2006 Digital Maestro Series 85. antennas 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 r ( N 1) d cos .. 1 23 N. d I (1) = Io ; I (2) = Io e j ; ; I ( N ) = Io e j( N 1) . Amanogawa, 2006 Digital Maestro Series 86. antennas The electric field at the observation point (r, ) is of the form E( r, ) = Eo e j r + Eo e j ( r d cos ) e j + . + Eo e j ( r ( N 1) d cos ) e j ( N 1) . = Eo e j r 1 + e j( d cos + ) +.. + e j( N 1)( d cos + ) .. jN ( d cos + ). j r 1 e = Eo e 1 e j( d cos + ). N 1. 1 e jN ( d cos + ). We have used e jn( d cos + ) =. n= 0 1 e j ( d cos + ). Amanogawa, 2006 Digital Maestro Series 87.

8 antennas The magnitude of the electric field is given by 1 e jN ( d cos + ). E( r, ) = Eo 1 e j ( d cos + ). sin[ N( d cos + ) / 2]. = Eo sin[( d cos + ) / 2]. array factor We have used jx x jx 2 x 1 e = 2 j sin e = 2 sin 2 2. Amanogawa, 2006 Digital Maestro Series 88. antennas We can rewrite 1 sin[ N ( d cos + ) / 2]. E( r, ) = N Eo N sin[( d cos + ) / 2]. group pattern The group pattern has Maxima when d cos + = 0, 2 , 4 . Nulls when N ( d cos + ) = 2 m . for m = integer 0, N , 2 N , . Amanogawa, 2006 Digital Maestro Series 89. antennas Examples Broadside array (plots on the azimuthal plane, with = 90 ).. 2 dipoles 4 dipoles Amanogawa, 2006 Digital Maestro Series 90. antennas 8 dipoles 16 dipoles Amanogawa, 2006 Digital Maestro Series 91. antennas End-fire array (plots on the azimuthal plane, with = 90 ). 2 dipoles 4 dipoles Amanogawa, 2006 Digital Maestro Series 92. antennas 8 dipoles 16 dipoles Amanogawa, 2006 Digital Maestro Series 93.

9 antennas Cardioid array (plots on the azimuthal plane, with = 90 ). 2 dipoles 4 dipoles Amanogawa, 2006 Digital Maestro Series 94. antennas 8 dipoles 16 dipoles Amanogawa, 2006 Digital Maestro Series 95.