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Can I feed multiple different band antennas with one coax to the transceiver?

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Yes. The device you want is known as a diplexer (or triplexer, etc, for more than two bands; also sometimes duplexer).

Electronically, it is just a combination of a low-pass filter and a high-pass filter, which will route the signals to/from the correct antenna based on their frequency.

You can obtain diplexers for various band combinations from amateur radio parts suppliers. With some electronics skill it is also feasible to make your own, since they are straightforward filter circuits made with inductors and capacitors.

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Kevin's answer about diplexers/triplexers is certainly correct especially in the general case. However, sometimes a separate diplexer is not really needed in practice!

For example you gave an example elsewhere of "different bands like 20m + 15m + 10M" [bold added].

It's possible to run one coax to a multiband dipole made ± simply by attaching multiple antennas to the same feedpoint. It's often called a "fan dipole" nowadays (although that originally referred to something else almost more related to a discone).

The idea is that a center-fed dipole antenna that readily accepts 20m signals will eagerly reject 10m signals! (Since the frequencies are approximately double, the 20m quarter wave element with an impedance not terribly far from 50Ω looks like an end-fed half wave element to a 10m signal — an impedance perhaps more in the 500–5000Ω range!) And perhaps somewhat similar for a band like 15m in between. At the feedpoint the antennas each sort of "self-filter" so that a signal on a particular band ends up radiating mostly from the elements best for that band.

Now it's not quite as simple as taking antennas perfectly tuned for each band and hooking them together. The wires nearby mutually affect each other and there must be some effect not unlike "resistors in parallel" that effects the impedance at the combined feedpoint. So it can take a bit of careful adjusting to get everything to work out right, but once it's tuned can be a reasonably efficient setup.

I'd recommend starting with an established design such as:

The trick might be to not try squeeze too many bands into one antenna. E.g. a 40/20 or 20/10 setup is probably fairly easy to get right, whereas a 30/20/17/15/12/10 combination might be pretty finicky, and the set of lengths that end up working for one ham's soil and surroundings might not even work there reliably, when the weather changes!

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