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Here is a list of some basic Antenna Types.  This is not exstensive by any means, but a good start on some of the basics.

The Wire Dipole Antenna
     This is the simplest, and most basic of Antennas.  Usually this antenna consists of a wire, and a connector in the middle.  It is very easy to build, and is the first antenna for most hams.  You can take a female coax connector, and solder one wire to the "center pin," and the other wire to "ground."  The ends should have some type of insulator with some rope attached to secure.  It may be smart to attach some rope to the connector, and use this to hang on something non-metallic (tree limb) as a center point.  Also this antennas needs to be "Tuned" to the frequency it will opperate on.  This is a simple 1/2 wave formula ( 468 divided by F = in decmimal feet) ( F= frequency) is for the entire length of the wire portion of the antenna, and does not consider the rope.  As an example, lets say you wanted to cut an antenna for the middle of the 20 Meter band on 14.150Mhz.  Take 468 divided by 14.150, and you will get 33.07 feet.  Also  for best performance, this also would be the best minimal height above ground for this antenna if hung as a horizontal.  This same antenna can also be hung vertically.  If you can't hang the antenna a full 1/2 wave above ground, just hang it as high as you can.  The antenna is bi-directional.  as an example if you hang with the ends pointing north, and south, the pattern will best send/receive from the east, and west or vise versa.

Yagi / Log Periodic Antenna
     The Yagi or Log Periodic both work in a similliar way.  These antena work by cancelling RF energy from other directions, and reinforcing all the RF energy in one direction.  This is why these antennas are also known as directional antennas.  A directional antenna will have a radiating element and a reflecting element as perhaps one or more directing elements.  These antennas are tuned for a frequency, or a range of frequencies.  The formulas for these are more complicated.  The spacing between elements is a fraction of the tuned wave.  The Radiator is  tuned, to the operating frequency, while the Reflector is roughly 5% larger, and each Director is about 5% smaller from the previous element.  This antenna can be vertically or horizontally polarized.  It is great for communications in one directions, while muting noise from other directions.

Vertical Antenna
     This is also another very popular antenna.  It is an omnidirectional antenna meaning that it sends, and receives in all directions equally.  These can be made, or can be commercially bought.  These antennas are commonly made at 1/4 wave, 1/2 wave or 5/8 wave. (wave is the wave length of a given frequency)  The higher frequency the shorter the wave.  As with all antennas, this comes with advantages, and disadvantages.  It may be simple to hang, does not need a antenna rotator, and radiates in all directions.  however it does not reject noise from unwanted directions, and does not have much gain.  Also this antenna requires radials to some degree based on what frequency it is tuned to.  This Antenna is used by Hams, Police, Fire, CBers, Ambulancce, and other first responcance.  It is a great choice for simple short to medium range line of site communications.  One important note, each time you double the height placement of this antenna, you will extend your range about 25%.  An antenna placed on a 100 foot tower will get a whole county, while the same antenna on a satellite in spaces will get about a 500 mile range.  These figures are a ballpark range.  Most hams will have several types of antennas.  Each chosen for a specific application.  In addition to the antenna, power, and coax should be taken into concideration to determin the Effective Radiated Power of a chosen antenna system.  With ANY antenna, please be safe.  Always take the height of any antenna system, and double it.  This should be the minimum distance from powerlines.  Also NEVER attempt to catch a falling antenna.  Many people have been electricuted this way.  Antenas are cheeper to replace than you are.

Dish Antennas
     The Dish Antenna is becomming a common antenna today.  A lot of people use these antennas for satellite TV.  The dish is circular polarited ( helps with Faraday Rotation noise ) and has high gain. ( amplifies RF)  These atributes makes it the perfect canidate for satellite communications.  All will contain a parabolic dish, and some type of Feed Horn.  Some will have additional equipment such as a preamplifier, and or a downconverter to minimize feedline loss.  Some of these Dish antenas are large while other are very small.  They are very useful when a very directional high gain antenna is desired.

Cubical Quad Antenna
     These are one of the best types antennas on may want to put up.  You can put several antenna bands on the same frame without compramising antenna performance.  This is a directional antenna that does not take up as much room.  One interesting note, if the feed point is on the side, the Antenna is vertically polarized.  If the feed point is on the Top or Bottom, it will be horizontally polarized.  Personally I think this antenna has some characteristics of both.  This seems to help with the shifting polarization of radio signals in the HF bands, and minimizes the fluttering noise this creates.  It is recommened to be put on a tower at least 1/2 wave above ground of the lowest band.  however the higher the better, as is the case most times.  This is for several reasons.  It brings the antenna higher above people allowing higher power to be used and still be safe concerning RF exsposure.  It helps with getting above some forms of RF noise, and hopefully gets you above the tree line for clearer signals.

Circular Polarized Yagi Antenna
     This is a odd breed of antenna, however it is popular because of it's  many applications.  Depending on how it it phased, it can be set as vertical, horizontal, lefthand circular, or righthand circular polarization.  This means with the right switching, this antenna is a good choice for VHF/ UHF  distant FM, SSB, and satellite.  for satellite use, it is best to use with a antenna elevator / rotator.  This is not a complete listing of the many different types of antennas available.  Just for starters there is Cantenna, Eggbeaters, Quagis, Corner Reflector, Spiral, Butterfly, NVIS, Hybrid, and experimental antennas.  One can use a antenna tuner, and tune many metallic objects.  People have been known to run Ham radio contests with highchairs, car / truck bumpers, yard fences, and umbrellas.