Shortening ratios of modified dipole antennas

Two types of modified dipole antennas, zigzag and meander-line types, are analyzed and the shortening ratios are calculated. A zigzag dipole antenna with a wire length of 0.58 wavelengths has a shortening ratio of 24 percent with a resonant resistance of
46 \Omega
. A meander-line dipole antenna with a wire length of 0.70 wavelengths has a shortening ratio of 30 percent with a resonant resistance of
43 \Omega
. It is found that the radiation patterns of these two types of antennas are similar to the radiation pattern of a conventional half-wave linear dipole antenna.