| State | P (kPa) | V (L) | T (K) |
|---|
—
Look at state 4 on the Otto loop: the exhaust valve opens while the gas is still well above atmospheric pressure. That pressure was paid for and is thrown away — the blowdown you hear as the exhaust note. Atkinson keeps expanding until the pressure has fallen back to where it started, and pushes the exhaust out at constant pressure instead.
The expansion ratio is not something you choose. Requiring the pressure to arrive back at atmospheric fixes it: e = (P₃/P₁)1/γ. At a compression ratio of 8 that comes out near 15:1 — almost double — and the efficiency goes from 56.5% to 62.5%. Switch between the two cycles above and watch the extra area appear under the expansion curve.
Compression ratio and expansion ratio are two different numbers, and only one of them is limited by knock. That is the entire insight. A petrol engine cannot compress harder without the mixture igniting itself, but nothing stops it expanding further — so Atkinson takes the compression it is allowed and the expansion it wants.
At the same compression ratio the gain is real at every setting, and it shrinks as the compression ratio rises: +6.7 points at r = 6, +6.0 at 8, +4.8 at 14. The harder you already compress, the less pressure is left over at the bottom for the longer stroke to recover.
And it is bought entirely with heat. Turn the heat input down and the advantage goes with it: at no heat addition at all the expansion ratio collapses back to the compression ratio and the Atkinson efficiency becomes exactly Otto's. That is the clearest statement of where the gain comes from — it is recovered pressure, so with no pressure left over there is nothing to recover.
The cost is power density. V₄ comes out larger than V₁ — the gas is expanded further than it was drawn in — so for a given engine size less air is inducted per cycle and the peak power falls. That trade is unacceptable in a car that has only an engine, and perfectly acceptable in one with an electric motor to cover the gaps. It is why the cycle was patented in 1882, went almost unused for a century, and is now in millions of cars.
The mechanism, not the thermodynamics. A true Atkinson engine uses a linkage that makes the expansion stroke physically longer than the compression stroke. A Miller engine keeps the ordinary crank and holds the intake valve open into the compression stroke, pushing some of the charge back out so that compression effectively starts late. The loop drawn here is the same either way; almost every modern “Atkinson-cycle” engine is mechanically a Miller.
It is the point. The gas is expanded to a larger volume than it was drawn in at, which is exactly what “over-expanded” means. In a true Atkinson that extra volume comes from a longer expansion stroke; in a Miller it comes from having inducted less than the full swept volume in the first place. Either way the cycle ends further right than it began, and the area under that last stretch of expansion curve is the efficiency gain.
Because efficiency is not the only thing being bought. Over-expansion means less air per cycle, so an Atkinson engine of a given size makes less power than the Otto version of itself — and a car with nothing but an engine has to size that engine for its worst moment. Add an electric motor to cover acceleration and the compromise disappears, which is why the cycle waited a century for the hybrid.
A vertical line: P/T constant, and no work at all, since W = ∫P dV = 0 when V never changes.
A horizontal line: real work W = P·ΔV, with V/T constant.
Steeper than isothermal, PVγ = constant: the work comes from, or goes into, the gas’s own internal energy.
A cycle returns to its start, so net work is the area enclosed. Clockwise means net work done by the gas: an engine.
Diesel engines run much higher compression ratios (16–22:1 against roughly 8–12:1 for petrol) because they inject fuel only at the top of the stroke, avoiding the pre-ignition (“knock”) that limits how far a petrol and air mixture can be compressed. At the same compression ratio the Diesel cycle is actually the less efficient one: the higher ratio, not the cycle shape, is what gives the Diesel its edge in practice.
The Carnot cycle is built from reversible isothermal and adiabatic steps, and for those the Second Law fixes the efficiency by the hot and cold reservoir temperatures alone: the ceiling no real heat engine between those temperatures can exceed, whatever the working fluid or mechanism.
Gas turbines and jet engines: continuous-flow compression, combustion and expansion rather than a piston’s stop-start strokes, which is why its efficiency depends on pressure ratio instead of the volumetric compression ratio.
Real gases are not ideal, combustion is not instantaneous heat addition, compression and expansion are neither frictionless nor adiabatic, and every real cycle loses heat through the cylinder walls: each gap costs real efficiency the air-standard analysis cannot see.
It does not, really: time is stepped evenly, but volume does not change at a constant rate along every process. An isochoric stroke has zero piston movement despite pressure changing rapidly; an adiabatic stroke moves the piston fastest where the curve is steepest. The animation is honest about that.
Diesel engines run much higher compression ratios in practice (16–22:1 vs. roughly 8–12:1 for petrol) precisely because they inject fuel only at the top of the stroke, avoiding the pre-ignition ("knock") that limits how far a petrol/air mixture can be compressed — the higher ratio, not the cycle shape itself, is what usually gives Diesel engines their efficiency edge in practice. At the same compression ratio the Diesel cycle is actually the less efficient of the two.
Carnot's cycle is built entirely from reversible isothermal and adiabatic steps, and for those the Second Law fixes the efficiency purely by the hot and cold reservoir temperatures — it's the theoretical ceiling no real heat engine operating between those two temperatures can exceed, regardless of working fluid or mechanism.
Gas turbines and jet engines — continuous-flow compression, combustion and expansion rather than a piston's stop-start strokes, which is exactly why its efficiency depends on pressure ratio instead of the volumetric compression ratio the piston cycles use.
Real gases aren't ideal, combustion isn't instantaneous heat addition, compression and expansion aren't frictionless or adiabatic, and every real cycle loses heat through the cylinder walls — each of these gaps costs real efficiency the air-standard analysis can't see.
It doesn't, really — time is stepped evenly, but volume doesn't change at a constant rate along every process. An isochoric stroke has zero piston movement at all despite pressure changing rapidly; an adiabatic stroke moves the piston fastest where the curve is steepest. The animation is honest about that unevenness rather than smoothing it into a constant-speed loop.
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