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The theory and the practice of short antennas are quite different from each other. In practical Part 15 AM, you determine what the total system loss resistance is, find the transmitter efficiency, and then determine the efficiency of the entire system. If the radiation resistance is .13 ohms, such as with a 3 m antenna at 1.7 MHz, the system loss resistance is 50 ohms, and the transmitter efficiency is 25% (much more likely with Part 15 AM transmitters than the often-assumed 75%) the system efficiency is only .065%, allowing a tiny radiated power of 65 uV. I agree that the system described here cannot produce anywhere near 17.47 mW of radiated power.
In the theoretical study of short antennas, however, the approach is much different from the practical approach shown above. The objective is to discover the fundamental limitations of short antennas, considering only the limitations imposed by the laws of physics, and not any technical constraints.
In principle, a short antenna of any length can have 100% efficiency. Consider, for example, a radiator above ground 3 m long an 3″ in diameter, operating at 1.7 MHz. The radiation resistance is .13 ohms and the capacitive reactance is 2048 ohms. Assume no losses in the system, other than the radiation resistance. Because there are no other losses, the system is 100 % efficient. Of course, the superconductive tecnology does not exist to actually make this system. But even if it did, it could not be used for AM. The Q is 2048/.13 = 15754. The RF bandwidth would be 1,700,000/15754 = 108 Hz. The audio bandwidth is 108/2 = 54 Hz. This system could be used for slow-speed telegraphy.
Now, let’s suppose that we want to use this antenna with an audio bandwidth of 5 kHz. The RF bandwidth is 10 kHz. The required Q is 1700/10 = 170. This means that the system must have a resistive loss of 2048/170 = 12 ohms. The efficiency would be .13/(12 + .13) = 1%. This is a rather disturbing calculation. This is the best efficiency that can be obtained for the system described, under ideal conditions. You simly can’t do better, no matter how good your design is.
As I mentioned in my previous post, intelligible audio may be obtained with only 2 kHz RF bandwidth. In that case, the Q would be 1700/2 = 850. The loss resistance of the system would be 2048/850 = 2.4 ohms. The efficiency would be .13/(2.4 + .13) = 5.1 %. This is the best efficiency that can be obtained for the given monopole with reduced bandwidth.
A considerable improvement in theoretical efficiency can be obtained by adding a top hat. A 1 foot radius top hat would increase the radiation resistance and decrease the capacitive reactance enough to increase the low-bandwidth efficiency to about 12 %. A 2 foot radius top hat would increase the efficiency to about 20%.
There is a question as to whether a Part 15 AM operator can get away with using a top hat. A top hat does not radiate, but the FCC might consider the diameter to be part of the 3 m allowed length of the antenna. Also, the top hat increases the effective length of the antenna. I don’t know if this issue has been addressed in any enforcement action. So, I am not sure if an antenna with a top hat would be considered to be “compliant.”