Several factors affect the voltage at the end of an antenna even when the
power level is held constant:
absolute voltage actually isn't the important thing here.. it's the field. A 1 foot diameter sphere at 10kV isn't a problem. A needle point at 10kV is.
1) the wire diameter: a fat wire (like a cage) will have lower voltage
Not all cages have lower field. A fat *solid* conductor with a rounded end will have a lower field. A solid metal tube that just "ends" will not. Some cages will have a lower field, some won't: a lot depends on the details of the construction. Are the ends of the wires in the cage connected with a ring of some sort? what's the radius of curvature, etc.
A fat dipole (or bicone, or fan, etc.) has a wider SWR bandwidth, but that doesn't change the voltage as much.
2) the radiation resistance: a shortened antenna (coil loaded, folded, or
matched with an antenna tuner) will have higher voltage at the ends than
a full-sized one.
I'm not sure about that one. A physically small antenna often has a large reactive component to the feedpoint Z, and a lot of stored energy in any case. Stored energy makes for high voltage in the C and high current in the L. But it's not because of the low radiation resistance. I would say that small antennas have low radiation resistance AND high stored energy: more correlation than causation.
3) The capacitance at the end of the antenna. Adding a capacity hat or
something like the old copper toilet ball will lower the voltage.
It's not the capacitance that is changing the voltage: The voltage is the same (for the most part).. what you're really doing is increasing the radius of curvature, so for the same voltage, the field is smaller (volts/meter), and volts/meter is what causes sparks. Actually, having a capacity hat or ball or toroid will *increase* the peak voltage. A sharp point will have corona and arc, which limits the voltage, while the gently curved surface of the ball will support a much higher voltage before breakdown. Van deGraaff generators and Tesla Coils are fine examples of this.
So a 20m dipole fed with ladder line will have higher end voltages when
operated on 40m than on 20m. Since many attic antennas are folded or
otherwise shortened, voltages may be higher than expected.
How much voltage are we talking about? If we assume a standard dipole
has a 50 ohm radiation resistance and the wire represents a 600 ohm
transmission line, then we'd expect an impedance around 7200 ohms at
I'm not sure where that 600 ohms comes from. And how you get from 50 to 7200. It's actually pretty hard to calculate what the voltage at the end of an antenna is, especially with a simple equation. Trying to back into it by looking at the feedpoint Z of a doublet running at twice the resonant frequency isn't a particularly good way.
The only real way to do it is with a finite element model, and integrate along the conductor. NEC can do this, but it won't give you total voltage: you'd have to add up the voltages along each segment. Nor does NEC give you the E-field at the surface of the wire (because NEC is a method of moments code, and it is worried more about accurately solving for the *current* in each segment). One of the other 3-D FEM codes would probably give you an accurate E-field, but most hams aren't running HFSS or similar. And even then, for resonant structures, it's hard to predict breakdown.
Practical experiments with end-fed antennas suggest that
something in the 2000 to 5000 ohm range may be more typical. (The
higher the impedance, the higher the voltage.) Let's assume 5000 ohms
for a start. Running 100W the voltage across 5000 ohms would be
707V RMS or 1000V peak. At 1.5kW it would be almost 4000V peak.
Basically similar to what we used to encounter in the plate circuit of
tube transmitters at similar power levels.
1000V isn't too big of a deal - the insulation on standard house wiring
is usually rated at 600V. An inch of dry air is more than necessary -
we often use variable capacitors with far less spacing than that in
antenna tuners, where they may be subject to high voltages as well.
But providing a few inches of space around the end of the wire is
probably sufficient in most cases.
This is where the problem is. The deal isn't so much the absolute voltage, it's all about the field strength in V/m. If the field is bigger than 3kV/mm (70kV/inch), air breaks down. What you need to do is know the radius of the conductor. A bare 1mm radius conductor (AWG 18) will break down at 3kV. Or, more important, a 10mm radius conductor ( a bit less than 1/2 inch), but with a blob of solder that has a sharp point that's less than a mm across will break down.
Or some semiconductive dust with sharp corners.
If the voltage is less than about 350 V, it can't break down (called the "minimum sparking voltage") regardless of how close the gap or small the radius.
A foot of clearance is probably enough at high power, though you
will need to give better consideration to the types of insulators you
use, especially with shortened antennas: ceramic would be better
than some scrap plastic that you don't know the properties of.
Designing your clearances for 10kV should give enough of a safety
factor to handle most types of antennas, though I'd still be wary
of any sort of grossly shortened antenna.
The rule of thumb is 1" clearance per 10kV (factor of 7 safety), and 1/4" radius of curvature per 10kV (factor of about 2). That's surprisingly fat: 1/2" diameter for 10kV.
For breakdown along an insulator surface, the distance needs to be at least 3 times the free air breakdown (this is why insulators have grooves or fins). So, for 10kV, free air breakdown is 1/7th inch, so you'd want the surface distance on a *clean* insulator to be about 1/2".
A real issue you're neglecting here, though, is that an attic antenna is next to other conductors. And while your antenna may be nice and smooth, and suitably curved, it could be inducing a high voltage somewhere near by. Like that roofing nail sticking through. Or that sheet metal screw holding the AC duct, etc. There are lots of well documented stories of people having problems with this running Tesla Coils in their garage, and wondering why the ceiling caught on fire or smoldered, when none of the sparks went anywhere near there.
If you go see the HV display at the Boston Museum of Science, you can see this effect demonstrated really, really well, as the big Van deGraaff charges, and you see the corona on all the other stuff around it. You see it with their Tesla Coils too, but it's harder to spot.
(To be honest, I'd be wary about running a kW to an attic antenna
anyway for other reasons.)
yeah, the RF exposure issue is a dicey one, for sure.
You also might be cooking the electronics in things like your solid state fluorescent ballasts (been there, done that) or smoke alarms (not a good thing) or garage door opener (killed one of those)