This paper describes a radial open stub and its application to a simple design of a four-element planar Butler matrix. In the first stage of our work, we propose a 45-degree phase shifter composed of an eighth-wavelength delay line and a serial connection of a quarter-wavelength straight line and a quarter-wavelength straight open stub. Next, in order to improve relative-phase characteristics between output ports, we propose a 45-degree phase shifter configuration using a quarter-wavelength radial open stub instead of using a quarter-wavelength straight open stub. It is shown by simulation that relative-phase characteristics of the configuration using the radial open stub are better than that using the straight open stub at the high frequencies. Finally, an experimental UHF-band four-element planar Butler matrix is presented. Over the frequency range from 0.83 to 0.92 GHz, the experimental four-element planar Butler matrix exhibits power splits of -6.510.29 dB, return losses of greater than 13 dB, errors in the desired relative-phase difference between output ports of less than 2 degrees.