Any mismatch between the receiver, the transmission line and associated components (e.g. connect-or, switches) and the antenna will have some impact on how much the signal from the antenna is attenuated, before it is at the receiver antenna input., but it does not determine if a antenna is providing you with good reception .
Even very short vertical antenna with a top capacitance and good ground plane, will provide similarly good reception (~4.8 dBi) as a ¼ lambda antenna. The drawback is the low impedance, which will get lower the lower the operating frequency is, in average <<1 Ohm.
The antenna efficiency for transmit and receive is, despite being very short, is still similar to that of ¼ Lambda ground plane. But such short antenna requires matching of the low antenna impedance.
Again my example of connecting a non radiating termination instead of an antenna, you will have no mismatch to worry about but will not receive much since the antenna efficiency for reception is 0%.
Antenna pattern gain and directivity: The antenna pattern and gain towards the signal source is important. The main beam directivity of the azimuth and elevation toward the signal source is important.
The optimal elevation of the main beam depends on how high the signal source is (reflected) above ground. For SW reception via reflections at the atmosphere, or VHF or higher frequency reception e.g. of aircraft or satellites, a main beam close to the horizon will produce only weak signals, The same applies in azimuth if the main gain is not towards the signal source.
Unfortunately The antenna azimuth and elevation of the antenna lobes will vary in gain, azimuth and elevation with frequency, even for broad and or broadband antenna designs
Also the higher the gain will increase above the gain of an isotropic antenna the more directive and the narrower the azimuth and/or elevation beamwidth of an antenna will become.
So a lower gain antenna might serve SWL better than trying to increase gain at the cost of not beeing able to receive all possible sources, unless you need a high gain to receive a sources you cannot receive with low gain.
At 50Mhz a whip style antenna 82" long worked against ground has a radiation resistance of 50 ohms. If this same antenna was say 12" it would have a radiation resistance of 1 ohm.
A 12 inch wire will receive a signal! The front end mismatch along with the fact that such a short fraction of a wavelength not being efficient will seriously degrade the signal strength.
Do you mean a whip of 1/4 Lambda length for 50 MHz?
A 82 " (~2.08m) whip, without accounting for thickness and the corresponding effects that require it to be shortened somewhat, would be resonant at 36 MHz.
For 50 MHz a 1/4 wavelength antenna would be ~1.5 m (59 "),
A 1/4 Lambda above a ground plane/counterpoise requires it to have a groundplane/counterpoise to form a ground plane. The impedance is just about ~36 Ohm, or half of the impedance of a 1/2 Lambda Dipole.
A dipole antenna has 3db gain in free space over a narrow frequency range. Outside of this fre-quency range the impedance mismatch makes the antenna inefficient.
Did you mean Fullwave or ½ wave dipole?
A fullwave dipole has ~3.8 dBi, a ½ dipole ~2.15 dBi, and ¼ groundplane ~5.2 dBi gain.
With increased diameter of the antenna element in reference to the wavelength the bandwidth will increase, so only thin dipoles will be narrowband.
If you further increase the thickness by widening the antenna elements toward the end, you’ll get a wideband dipole (Bi-cone), which is a very broadband dipole.
You received a signal at .0775 MHZ with a 20 meter wire. You had a radiation resistance of .1 ohm. This obviously worked for you. It would have worked much better if you could have gotten the resistance up by using a wire closer to resonance
This would have given you a larger capture area along with an input impedance closer to that required.
The point I was trying to make was, since we want to receive only, we do not have to worry about matching the antenna to the transmitter to avoid missmatch. Unless we are interested only in very narrow band-segements, there are ver few options to optimize a broadband reception.
So we have to and IMHO can live with in relation to the wavelength very small/short antennas like e.g. the 20 m wire for a wavelength of 3871 m. I don't know about you, but I am limited to about 40m in horizonal length for antenna where I live today. 20m of cooper wire allowed me not onlay recepetion down to 77.7 kHz but also to transmitt on most shortwave bands between 80 m and 10 m available to hams. It is a compromise, but what in life is optimal?