  Fringe field becomes important when one requires more accurate modeling of a
ring lattice to study the long-term beam dynamics in storage rings and deal
with large aperture magnets in high-intensity proton synchrotrons or
accumulator rings. In this paper, a simple expression to calculate the tune
shifts due to quadrupole fringe fields is derived by using Lie algebra
technique. With higher-order terms included, this method is more accurate
compared with the linear fringe field model used in SAD code. The method is
also applied to a BEPCII lattice. Also based on the Lie algebra technique and
an inverse series technique, an equivalent hard-edge model for quadrupoles is
proposed, in which the model parameters are derived analytically. The model has
the advantages of the direct calculation of the equivalent length and strength
of a quadrupole and the easy adaptation when the strength changes. The validity
of the model and its simplified version has been checked with a numerical
method, and they show very good agreements.
