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For an ideal working gas and $r=V_2/V_1>1$, two isotherms at $T_h,T_c$ produce net work $nR(T_h-T_c)\ln r$. Without a regenerator, isochoric heating requires additional external heat $C_V(T_h-T_c)$, giving the displayed thermal efficiency, strictly below Carnot efficiency. A perfect reversible regenerator recovers this heat and attains the bound instead. Reversing the unregenerated cycle gives cooling coefficient of performance $T_c/(T_h-T_c)-C_V/(nR\ln r)$; positive cooling requires a sufficiently large expansion ratio.

For an ideal Diesel cycle with compression ratio $r=V_1/V_2>1$, cutoff ratio $\rho =V_3/V_2>1$ and constant heat-capacity ratio $\gamma$, its thermal efficiency is $\eta =1-({\rho }^{\gamma }-1)/[\gamma r^{\gamma -1}(\rho -1)]$. As $\rho \to 1^{+}$ the quotient tends to $\gamma$, giving $\eta \to 1-r^{1-\gamma }$. The maximum-to-minimum temperature ratio tends to $r^{\gamma -1}$, so the limiting efficiency equals that of a Carnot cycle between those temperatures. Both heat input and net work vanish in this limit: the result is a limit of their ratio, not a finite-work engine at exactly zero cutoff interval.