i am very interested in how different antennas matching systems work and how it affects things like bandwidth etc
Most matching systems will narrow the bandwidth by nature, although their are exceptions. The reason for this is the antenna has a bandwidth, and the matching circuit also has a bandwidth. When both are in tune you have a match, however, as you move away from that tuned point both systems are working against having a good bandwidth.
Their are some electrical lengths the provide better natural bandwidths than others, for various reasons. These lengths are easily found, namely, any electrical length that is a multiple of 1/4 wavelength. In the case of this antenna, we have a 3/4 electrical wavelength element that is physically shortened to the 5/8 wavelength length with a loading coil. Some people will tell you that all 5/8 wavelength antennas are this way, however that is not a true statement, and is actually mostly false. Most 5/8 wavelength antennas, instead of shortening a longer electrical length, simply use a matching circuit to correct the phase (some people will call in an impedance) mismatch that is naturally present with a 5/8 wavelength antenna. This is not the same as electrically shortening an antenna.
Their is actually an antenna out their that is an electrically lengthened 1/2 wavelength antenna to a physical 5/8 wavelength. That is the imax 2000, and they do that to essentially use the same matching system as their half wavelength a99 antenna.
Back to matching systems and bandwidths. As I said, their are exceptions to the normal matching system lessening the bandwidth, but these are few and far between. The Gainmaster, for example, uses an anomaly in the impedance curve to create a large bandwidth. However, most other antennas with a wide bandwidth essentially use losses to achieve their wide bandwidths.
it must mean a 3/4 wave will match up with no matching device like a 1/4 wave
The 3/4 wavelength element will create a higher impedance than a 1/4 wavelength section, however, this antenna also has several devices that lower that impedance back down to compensate built in as well. For example, the loading coil will lower the impedance of the antenna. The four elevated radials will also, by nature, lower the antenna's impedance as well. Essentially, when it comes to antennas like this, it is all a balancing act.
The DB