When rapid acceleration is important, synchrotrons with very short pulse times can be used to accelerate particle beams. We will describe rapidly pulsed synchrotrons and their distinction from ordinary synchrotrons. We will introduce a hybrid synchrotron which interleaves pulsed magnets with superconducting dipoles to allow rapid acceleration while still maintaining a high average bending field. We will describe particular characteristics of the lattice design for these machines. We will describe how to design magnets to limit power consumption while still maintaining high fields. We will discuss the impact of the choice and properties of magnetic materials on the magnet performance. We show a magnet design that limits losses in the core while giving a high field by using multiple materials: 6.5% silicon steel for the back yoke due to its low losses at high frequencies, and 3% silicon steel in the pole for its high saturation field. The magnet has a unique coil configuration that minimizes eddy current losses. We compute losses and field quality for this design.