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Design of modern race-track microtrons (RTMs) requires better understanding of the longitudinal beam dynamics in these machines, in particular of the phase slip effect which is important for low energy beams. We generalize an analytical approach for the description of the synchronous particle motion and synchrotron oscillations, developed in our previous papers, by including the fringe fields of the RTM end magnets. Explicit, though approximate, formulas are derived and an algorithm for improving their accuracy is formulated. The efficiency of the analytic description is checked numerically, in particular by tracking simulations using the RTMTRace code. Explicit examples of low energy injection schemes and applications of this formalism for the injection phase fixing are given.
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