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When the exhaust valve of a conventional spark ignition engine opens at the end of the expansion stroke, a large quantity of high pressure exhaust gas is freed to the atmosphere, without using its availability. An engine that could use this lost energy should have a better efficiency. The equations for an over-expanded cycle (Miller cycle) are developed in this paper, together with equations for the Otto cycle, diesel cycle and dual cycle, all at part load, so they can be compared. Furthermore, indicated cycle thermodynamical comparisons of a S.I. engine at part load (Otto cycle at half load), a S.I. engine at WOT (with half displacement) and two over-expanded S.I. engines (with different compression strokes) are examined and compared, with the aim of extending the referred theoretical cycle comparisons.