Appendix A: Formulæ

Radio waves travel more slowly in a conductor than in space. One result
is that practical antennas can be 3 to 5 percent smaller than the free-space
wavelength size. For example, a half wavelength at 1 MHz is about 150 meters,
but a half-wave dipole made for that frequency only needs to be 142 to 145
meters long. The exact electrical length of an antenna depends on the
ratio of wavelength to conductor diameter, and on the effect of supporting
insulators at the ends.

radiation resistance and efficiency

"Radiation resistance" (not to be confused with impedance)
refers to the energy absorbed into free space from the antenna circuit. A higher
radiation resistance indicates more energy is being radiated into space.

The approximate radiation resistance (R_R) of a short vertical antenna (not endowed with
loading coils or capacitance hat) can be found by this equation:

 

Example: An antenna for a frequency of 1700 kHz (wavelength = 176
meters) is 5 meters tall. Radiation resistance is about 0.32 ohms.

If the antenna is 1/16 wavelength tall, the radiation resistance will
be around 2 ohms. If 1/8 wavelength tall, 7.5 ohms.

The efficiency of a vertical antenna system is given by:

Example: Radiation resistance is 2 ohms, ground loss is 3 ohms,
and loading coil loss is 2 ohms. Efficiency is 0.285 or about 29 percent. If
10 watts were input to this antenna, 2.9 watts would be radiated into space
and the rest would be dissipated in the system losses.