Prof. D. R. Wilton Notes 22 Antennas and Radiation Antennas and Radiation ECE 3317 [Chapter 7]

Antenna Radiation We consider here the radiation from an arbitrary antenna. The far-field radiation acts like a plane wave going in the radial direction. + - x y z r "far field" S

Antenna Radiation (cont.) The far-field has the following form: x y z E H S x y z H E S

Antenna Radiation (cont.) The far-field Poynting vector is now calculated:

Antenna Radiation (cont.) Hence we have or Note: in the far field, the Poynting vector is purely real (no reactive power flow).

Radiation Pattern The far field always has the following form: In dB:

Radiation Pattern (cont.) The far-field pattern is usually shown vs. the angle  (for a fixed angle  ) in polar coordinates. 0 dB 30° 60° 120° 150° 120° 60° -10 dB -20 dB -30 dB

Radiated Power The Poynting vector in the far field is The total power radiated is then given by Hence we have

Directivity In dB, The directivity in a particular direction is the ratio of the power density radiated in that direction to the power density that would be radiated in that direction if the antenna were an isotropic radiator (radiates equally in all directions). The directivity of the antenna in the directions ( ,  ) is defined as Note: The directivity is sometimes referred to as the “directivity with respect to an isotropic radiator.”

Directivity (cont.) The directivity is now expressed in terms of the far field pattern. Therefore, Hence we have

Directivity (cont.) Two Common Cases Short dipole wire antenna ( l << 0 ): D = 1.5 Resonant half-wavelength dipole wire antenna ( l = 0 / 2 ): D = y +h z x -h feed

Beamwidth The beamwidth measures how narrow the beam is. (The narrower the beamwidth, the higher the directivity). HPBW = half-power beamwidth

Gain and Efficiency The radiation efficiency of an antenna is defined as P rad = power radiated by the antenna P in = power input to the antenna The gain of an antenna in the directions ( ,  ) is defined as The gain tells us how strong the radiated power density is in a certain direction, for a given amount of input power. In dB, we have

Infinitesimal Dipole The infinitesimal dipole current element is shown below. x y z I l The dipole moment (amplitude) is defined as I l. From Maxwell’s equations we can calculate the fields radiated by this source (see chapter 7 of the textbook). The infinitesimal dipole is the foundation for many practical wire antennas.

Infinitesimal Dipole (cont.) The exact fields of the infinitesimal dipole in spherical coordinates are

Infinitesimal Dipole (cont.) In the far field we have: Hence, we can identify

Infinitesimal Dipole (cont.) The radiation pattern is shown below dB 30° 60° 120° 150° 120° 60°

Infinitesimal Dipole (cont.) The directivity of the infinitesimal dipole is now calculated Hence

Infinitesimal Dipole (cont.) Evaluating the integrals, we have Hence, we have

Infinitesimal Dipole (cont.) dB 30° 60° 120° 150° 120° 60° The far-field pattern is shown, with the directivity labeled at various points.

Wire Antenna A good approximation to the current is: A center-fed wire antenna is shown below. y +h z I (z)I (z) x -h feed I 0I 0

Wire Antenna (cont.) A sketch of the current is shown below. resonant dipole ( l = 0 / 2, k 0 h =  / 2 ) +h -h l short dipole ( l << 0 / 2 ) +h -h l

Short Dipole short dipole ( l << 0 / 2 ) +h -h l Wire Antenna (cont.) The average value of the current is I 0 / 2. Infinitesimal dipole: Short dipole:

Wire Antenna (cont.) For an arbitrary length dipole wire antenna, we need to consider the phase radiated by each differential piece of the current. Far-field observation point Wire antenna: Infinitesimal dipole: y +h z x -h feed r R dz' z'z' I (z')I (z')

Wire Antenna (cont.) Far-field observation point z y +h x -h feed r R dz'

Wire Antenna (cont.) Far-field observation point z y +h x -h feed r R dz' Note: 

Wire Antenna (cont.) Far-field observation point z y +h x -h feed r R dz'

Wire Antenna (cont.) We define the array factor of the wire antenna: We then have the following result for the far-field pattern of the wire antenna:

Wire Antenna (cont.) Using our assumed approximate current function we have The result is (derivation omitted)

Wire Antenna (cont.) In summary, we have Thus, we have

Wire Antenna (cont.) For a resonant half-wave dipole antenna The directivity is

Wire Antenna (cont.) Results

Wire Antenna (cont.) Radiated Power: Simplify using

Wire Antenna (cont.) Performing the  integral gives us The result is then

Wire Antenna (cont.) The radiation resistance is defined from Z0Z0 Z in Circuit Model For a resonant antenna ( l  0 /2 ), X in = 0. y +h z I (z)I (z) x -h feed

Wire Antenna (cont.) The radiation resistance is now evaluated. This yields the result    2 Dipole:

Wire Antenna (cont.) The result can be extended to the case of a monopole antenna h Feeding coax

Wire Antenna (cont.) This can be justified as shown below. + - dipole V dipole I0I0 Virtual ground + V monopole I0I0 -

Receive Antenna The Thévenin equivalent circuit of a wire antenna being used as a receive antenna is shown below. + - V Th Z Th + - V Th E inc l = 2h