Great post LC. I've read this same section of the ARRL book about a dozen times also. It's amazing how some people can make it infer something it does not, although it is a technical subject so it's easy to misconstrue things.
When you add the stinger on top of that, it will reduce the number of turns necessary in that loading coil. Put an analyzer or SWR meter inline and you'll see it immediately. If you add more radials to the hat, it will add capacitance and cause the same thing. Rather than add elements (which is OK, BTW), it might be fun to try to make it more like the cap hats shown here:
The problem with the hat shown in your pic is that it is not symmetrical, which will probably result in some pattern distortion and some radiation from the hat itself which you don't want.
I've quoted some material from the hiqantennas.com site from their cap hat installation instructions below that is relevant to the discussion. Notice that the entire purpose of doing this is where it is impractical to use a full length quarter wave antenna:
When a quarter-wave antenna is mounted near the rear of a vehicle, the back of the vehicle and capacitance to the roadway act as the other required quarter-wave of a half-wave dipole. An antenna which uses the ground (earth/vehicle) as part of its resonant circuit is a Marconi antenna. An example of a Marconi antenna is a quarter-wave antenna, where the ground operates as the missing quarter wavelength. Most mobile antennas are a variation of the Marconi antenna.
At High Frequencies (HF) it becomes impractical to use full length quarter wave or half wave antennas for mobile operation. Therefore, it is often desirable to top or center load a mobile antenna. This top or center loading lengthens the antenna, but more importantly, it materially increases the capacitance between the top or center of the antenna and ground. As a result of this lengthening and the increased capacitance, a shorter antenna can be made to resonate at a relatively lower frequency than that of the unloaded antenna.
One method of achieving the loading of an antenna is to add a hat-type structure to the top or center of the antenna. Since this structure increases the capacitance and originally was shaped to resemble the rim of a hat, they were referred to as “Capacitance Hats”. This term has since been used to describe many shapes of load devices that have the effect of increasing the relative capacitance of the antenna with respect to the ground. A Capacitance Hat may appear as an “X”, a butterfly, a circular hat or a sphere. Basically the idea of a capacitance hat is to make up the length needed for resonance while limiting the vertical height.
Normally a vertical antenna has the lowest RF voltage/highest RF current flowing near the base, where most of the radiation occurs, and the highest RF voltage/lowest RF current at the top of the antenna, where the least radiation occurs. A capacitance hat placed at the top of an antenna will be exposed to the highest RF voltage/lowest RF current. Therefore due to the low RF current, the capacitance hat does not radiate a significant amount of signal. However, due to the increased effective height of the vertical radiator, the radiated energy is more evenly distributed along the length of the vertical radiator, resulting in an increase of RF radiation from the top of the vertical radiator with respect to a vertical radiator without the capacitance hat. Increasing the effective height of the antenna has the beneficial side-effect of reducing the losses caused by nearby shrubs and buildings, assuming your antenna is taller than these objects.
The symmetry of the hat causes any radiation from the hat to cancel and multiple elements are used to reduce the RF current flowing in each individual element. (If you use 1 radial element, it's called an inverted L, the radiation does not cancel and the radiation pattern is distorted.) The effective length of the vertical is then approximately the vertical length plus the length of the capacitance hat radials. It is also advantageous to use a “skirt”; joining the ends of the hat to maintain symmetry and ensure that a more balanced RF current flows in the hat section. In that case, the effective length is approximately the vertical length, plus the radial length of the hat, plus 1/2 of the length of the skirt joining the radials.
Adding a large capacitance hat at either end of a coil also reduces coil Q, since a large portion of the hat’s capacitance directly shunts the inductor. It will act like a big capacitor in lowering the resonant frequency of the antenna. The Q will decrease and your bandwidth will widen. Also, if we add a large capacitance hat at the top of the coil, (bottom of the whip), we change nothing below the coil.
A capacity hat will also raise the radiation resistance (the resistance that represents energy radiated from the antenna) for shortened antennas. If the radiation resistance is increased, it will increase the efficiency of the antenna system assuming other losses are kept the same.
