Atkinson cycle

The ideal Atkinson cycle considered here has reversible adiabatic compression, isochoric heat addition, a longer reversible adiabatic expansion and isobaric heat rejection. Its expansion ends at the initial pressure rather than at the initial volume. With constant ideal-gas heat capacities, $r=V_1/V_2$ and $r_p=p_3/p_2$, the temperatures are $T_2=T_1{r}^{\gamma -1}$, $T_3=r_p{T}_2$, and $T_{4^{\prime }}=T_1{r}_p^{1/\gamma }$. This ideal cycle is distinguished from the detailed valve timing and losses of a practical engine.

For the same initial state, compression ratio and constant-volume heat input, the ideal Atkinson cycle adds net work during the extended expansion and subsequent constant-pressure return. The extra work is the integral of $p_{\mathrm{expansion}}-p_1$ over the added volume, which is positive. Equivalently, concavity of $x^{1/\gamma }$ gives $\gamma (r_p^{1/\gamma }-1)<r_p-1$ when $r_p>1$ and $\gamma >1$, so its efficiency exceeds the corresponding Otto cycle efficiency. The comparison fixes these shared conditions rather than asserting universal superiority under arbitrary engine constraints.