HF Mobile Antenna Performance Concepts1
The mobile HF antenna is usually limited to a relatively modest length of around 10 feet (in the United States, the height limit on major highways is 13-1/2 feet). On the 10 m band, a 10 foot vertical antenna is actually a bit more than a quarter-wavelength long, while on the 80 m band the same antenna is only a thirty-second of a wavelength long.
The mobile vehicle provides the bulk of the counterpoise, or "ground", for the "missing half" of the antenna, but everything that is within a horizontal radius of (at least) the antenna length will carry some amount of induced current back to the base of the antenna. Some of this return current flows on the surface of the vehicle, some through the earth and then capacitively-coupled to the vehicle, some through surrounding objects, etc. All of these items that the return current flows through have some resistance, and this resistance is observable at the base of the antenna. To simplify matters, these various induced-current resistance contributions are lumped together and referred to as "ground resistance", and considered to act in series with the impedance of the antenna itself.
This simple ground resistance model lends itself to an easy formula for antenna efficiency:
%efficiency = Rant/(Rant+Rg)
where Rant is the radiation resistance of the antenna. For a quarter-wave vertical the radiation resistance is one-half that of the half-wave diploe, or 37 ohms. For a short vertical, the radiation resistance is
394*(height in wavelengths)2.
So on the 10 m band, the antenna resistance is about 37 ohms. On the 80 m band, it's much smaller: around 394*(1/32)2 or 0.385 ohms.
1Yes I know very well that the simple mobile antenna model is a gross oversimplification of about a hundred factors. Thanks for not pointing that out.
From the %efficiency and radiation resistance equations, it's apparent that efficiency is worst at the lowest frequency. When comparing different HF mobile installations, the comparison is usually done at the 80 m band for exactly that reason. If you're serious about 80 m mobile operation, you must work to minimize your total ground resistance. The primary means of doing so is to make sure all the metal pieces of the mobile vehicle are well bonded to one another.
If you had a perfect ground on 80 m, you would still have another problem. The 50 ohm output of your transceiver/amplifier has to try and feed the 0.385 ohm resistance of your antenna. Some sort of matching network is needed. In practical installations, the matching problem isn't so severe but still needs to be addressed.
The easy solution (which works pretty well) is to add a shunt inductance to the base of the antenna. If you adjust the antenna so it's just a bit shorter than resonance (a bit capacitive), the correct value of shunt inductor will get you right to 50 ohms. For higher frequency bands, the shunt reactance increases and therefore becomes less of a factor. Usually the shunt inductor is a compromise value for the 80 m and 40 m bands, as it has nearly zero effect above 40 m.
Even if you don't have an antenna analyzer, you can make a pretty good guess at your antenna resistance (and therefore efficiency) by the size of the shunt inductor that yields a good match. The smaller (in uH) the inductor, the lower the input resistance of the antenna, and (likely) the more efficient it is. This can easily be illustrated with a bit of complex algebra.
For the shunt inductor to yield a match,
Yant = 0.02 + jBant
and the shunt coil adds a susceptance of
Ycoil = -jBcoil, where Bcoil = 1/Xcoil = 1/(2*pi*f*L)
Admittances add in parallel, so the result is Y = 0.02. Inverting, Z = 1/Y = 50 ohms. Now, back to the antenna alone:
Zant = 1/Yant = (0.02 - jBant)/(0.022+Bant2)
Rant = 0.02/(0.022+Bant2)
So: the larger Bant is, the smaller Rant is, and also the larger Bcoil is (because they have to be equal to get the match). Large Bcoil is obtained by small L. There ya go!
An example: your base coil for a match at 3.5 MHz is 1 uH. Then Bcoil = -j0.0455. If just the shunt coil alone gets you to 50 ohms, then the antenna admittance at 3.5 MHz is 0.02+j0.0455. Converting to impedance, Zant = 8.1 -j18.4 ohms. As mentioned earlier, the antenna is a bit capacitive where the match is achieved.
An example: your base coil for a match at 3.5 MHz is 2 uH. Then Bcoil = -j0.02275. If just the shunt coil alone gets you to 50 ohms, then the antenna admittance at 3.5 MHz is 0.02+j0.02275. Converting to impedance, Zant = 21.8 -j24.8 ohms. The larger coil is needed because Rant is higher.
A 1 uH coil could be 0.5" diameter, 1-1/4" long, 14 close-wound turns.
A 2 uH coil could be 1.0" diameter, 1.2" long, 12 close-wound turns.