= Atkinson efficiency advantage from extended expansion
{c}
{title2=$\eta_A=1-\gamma r^{1-\gamma}(r_p^{1/\gamma}-1)/(r_p-1)$}
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 $p_{\rm expansion}-p_1$ 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.
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