What Yagi driven element feed point impedance is required to use a beta or hairpin matching system?

A beta match, commonly known as a hairpin match, consists of a small U-shaped piece of wire or tubing connected directly across the feed point of the driven element. This physical shape essentially forms a shunt inductor, which introduces inductive reactance into the antenna system. For an antenna to present a purely resistive load (a perfect match), its overall reactance must be zero. Because the hairpin match adds inductance, the driven element itself must be designed to have an equal and opposite capacitive reactance to cancel it out. To make a driven element capacitive, it must be physically tuned to be slightly shorter than a standard resonant half-wavelength. If the element were inductive (longer than a half-wavelength), adding the hairpin's inductance would only compound the mismatch.