Most simply, a piece of wire connected to a radio or television to pick up transmissions. A tuned electrical conductor used to transmit or receive broadcasts.

There are many types of antennas, from whips, dipoles, beams or Yagis, to the directional antenna arrays used by radio stations that you see on the highway at night. Cell-phone antennas on towers have proliferated in recent years.

How worketh an antenna, ask you?

Consider a guitar. Pluck the low E string and a pitched tone is created. If another guitar is lying nearby, the incoming sound energy will cause its strings to vibrate sympathetically. The low E string of the second guitar, in particular, will vibrate strongly. This is because its length is resonant with the wavelength of the incoming sound energy. The other, non-resonant strings will vibrate less, although the high-E string will vibrate quite a bit.

Much like the second low-E string, an antenna will most efficiently "pick up" signals whose wavelength is closest to its physical length. So an antenna 2 meters long will most strongly resonate with radio signals with a frequency around 150 megahertz. The signal induces an electrical voltage in the antenna, causing a current to flow in the circuit attached to the antenna.

Also like a guitar string, the antenna will still "pick up" signals of other nearby frequencies (and those which are harmonic or subharmonic multiples of its resonant frequency. For example a half- or quarter- wavelength antenna is still quite efficient.

But as the signal gets farther and farther from the antenna's natural or resonant frequency, the induced current is less and less. The antenna is acting as a bandpass filter.

In electronics, an antenna is a device for receiving or transmitting of electromagnetic radiation. The simplest form of antenna would be a rod of a electrically leading material. The basic physical principles of antennas are

Transmitting antenna: An alternating current in the antenna couples with an external electromagnetic field
Receiving antenna: An external electromagnetic field induces an alternating current in the antenna. 

There is no difference in principle between a transmitting and a receiving antenna. For instance, cellular phones use the same antenna for both. However, there are great differences in what you want to optimize for either type, so if you want high performance you would use separate antennas. For transmitting antennas you want to maximize gain, and the antenna must be able to handle high effects and large currents. On the receiving antenna, on the other hand, it is paramount to maximize the signal to noise ratio. The reason that we know how to construct antennas at all, is because we know about electromagnetism. More specifically, we know about Maxwell's equations, which describe time-variant electrical and magnetic fields, and how electricity, magnetism and light interact. They give us the current distributions, and with these we can calculate the radiating fields for any antenna, in theory. 

Microwave antennas
When the feeding current of the antenna reaches microwave frequencies (> 1 GHz), several possibilities open. First, the antenna itself becomes several wavelengths in size, which makes the analysis of it similar to optics, which in turn makes things easier. Also, it is a lot easier because of less diffraction and refraction, to make a highly directional beam. Examples of these are satellite dishes where a 1 meter antenna is about 30-40 wavelengths in size, giving excellent sensitivity in a very specific direction, preferably towards the satellite. Another important feature of microwave antennas is that you can build advanced array antennas, where each radiating element is fed at a certain phase, which gives the possibility of directing the beam without moving the antenna. Commonly used for radars. 

Important concepts

Radiation direction:
There is no antenna that transmits equally in all directions. This is because the electromagnetic field have perpendicular components of the electric field E and the magnetic field H, and the emitted transverse electromagnetic wave moves perpendicular to both of them. Trying to construct an antenna that transmits uniformly in all directions, called an isotropic radiator, is sometimes referred to as the hairy ball problem

Imagine a ball covered with hair. Try to comb the hair so that there is no parting anywhere on the ball.

This is as you realize impossible, and the same applies to antennas. All antennas will have discontinuity somewhere, because of how the E- and H-field interact in creating the emitted wave. Therefore the emitted signal will differ depending on direction, and radiation direction is always specified with the horizontal angle φ (0 - 360 degrees) and the vertical angle θ (-90 to +90 degrees).  

Antenna diagram
The antenna diagram is a representation of the radiation pattern, due to the above mentioned directivity of antennas. It shows in what direction there's most and least power radiated. Ideally, this would be a 3-dimensional representation, but generally these are drawn 2-dimensional. The further from the axis the diagram goes, the more power in that direction. Examples:

                                                       . <--- main lobe
                                                      . .
      .  .             .  .                   .      .   .      . <--- side lobe
   .        .   |   .        .               . .    .     .    . .    
 .            . | .            .         .  .   .  .       .  .   .  .
.              .|.              .       . ..     ..         ..     .. .
----------------|---------------        -------------------------------
.              .|.              .         
 .            . | .            .          HHHHHHHHHHHHHHHHHHHHHHHHHHH
   .        .   |   .        .            (Slotted waveguide array)
      .  .             .  .                  
          Rod antenna                            Slot antenna

Typical for microwave antennas is that they're highly directional, as the aperture antenna above. It usually has a main lobe, which points in the desired direction. The side lobes however are desired to be as low as possible, since this signal, received or transmitted, will interfere with the main lobe signal.

Antenna gain, directivity:
The term gain of an antenna is a measurement of the power density in a certain direction (φ,θ)  at a certain distance (R), compared to an ideal isotropic radiator. The term directivity is similar, as it is the power density in a certain direction (φ,θ) compared to the average power density of all directions for that antenna. The power density is measured in watts per square meter, W/m2. The antenna gain includes internal effects such as power loss in circuits as well as external losses due to interfering objects in the near field. An example of the latter would be a bird sitting on the antenna. The "near field" is usually defined as R < 2 ¬∑ d2 / λ , where d is the maximum aperture, and λ is the antenna wavelength.

An ideal isotropic antenna has the antenna gain of 1. A simple half wave dipole antenna, just a rod of certain length, has the antenna gain of 1.65, which means that it has 1.65 times higher intensity in its most favorable direction, at the same feeding effect. Usually, antenna gain is expressed in dB, which gives 2.15 dB for the half wave dipole. Antennas for radio astronomy can have a gain of 60-80 dB in certain directions, more than a million times the gain of the dipole.

Types of antennas

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Ok, that's a brief introduction without digging too deep in the mathematics.

Reference: ne.se, Antenna Theory and Design (Stutzman, Thiele)
Allow me to get slightly mathematical for a moment.

An antenna's radiation is related to the spatial convolution of the 3D free space Green's Function and the electric currents residing on the antenna.

Consider the 3D free space Green's Function G:

G = e -jkR/R

where k is the wavenumber, and R is the distance between the observation point r' and the source point r, R = |r - r'|

The radiated magnetic field H due to a vector electric current J on the antenna is:

H = x A

where A is the vector magnetic potential involving the convolution mentioned earlier:

A = (μ / 4 π) ∫s J (e -jkR/R) ds

The radiated electric field E can be obtained from H by using the frequency-domain Maxwell's Equations thusly:

E = x H / jw ε

If you've studied fourier analysis before you might have noticed that the antenna radiation pattern is actually a spatial fourier transform of the antenna currents! That is so cool ...

An*ten"na (#), n.; pl. Antennae (#). [L. antenna sail-yard; NL., a feeler, horn of an insect.] Zool.

A movable, articulated organ of sensation, attached to the heads of insects and Crustacea. There are two in the former, and usually four in the latter. They are used as organs of touch, and in some species of Crustacea the cavity of the ear is situated near the basal joint. In insects, they are popularly called horns, and also feelers. The term in also applied to similar organs on the heads of other arthropods and of annelids.

 

© Webster 1913.

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