In this paper, a composite planar spiral antenna consisting of an eight-arm equiangular spiral antenna and eight Archimedean spiral antennas has been designed to radiate electromagnetic wave carrying tunable angular momenta in a wide band. A tunable eight-way Wilkinson power divider network is used to offer three kinds of feeding modes for the equiangular spiral antenna, and thus the composite antenna can radiate the electromagnetic waves with angular momenta of the modes l=1, 2, and 3, respectively. The Archimedean spiral is introduced to improve the gain of the composite antenna in the case of the angular momentum of l=3. By analyzing axis ratio (AR) of the proposed antenna, the generated angular momentum of l=1 is spin angular momentum (SAM), and the angular momenta of both l=2 and 3 include SAM and orbital angular momentum (OAM). Simulated and measured results are given to demonstrate good performance including tunable modes, good purity and wide band.