Visualise the ideal cycle of the over-expanded engine, where the expansion stroke is longer than the compression stroke. Change the compression ratio, specific heat ratio, intake conditions and heat input to see the thermal efficiency, state temperatures and pressures, expansion ratio and net work update in real time, with a complete-expansion P-V diagram animation and an efficiency comparison against the Otto cycle.
Parameters
Compression ratio r
Volume ratio of the compression stroke V₁/V₂
Specific heat ratio γ
c_p/c_v of air. About 1.40 for air at room temperature
Intake temperature T₁
K
Intake pressure P₁
kPa
About 100 kPa for natural aspiration. Expansion continues down to this pressure
Heat input q_in
kJ/kg
Heat added to the air at constant volume (the combustion equivalent)
Results
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Thermal efficiency η (%)
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Peak temperature T₃ (K)
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Peak pressure P₃ (kPa)
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Expansion / compression
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Net work (kJ/kg)
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Otto comparison
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P-V diagram — cycle animation
1→2 isentropic compression, 2→3 constant-volume heat addition, 3→4 complete expansion (isentropic expansion down to intake pressure), 4→1 constant-pressure heat rejection. The expansion sweeps a larger volume than the compression, and the enclosed area is the net work.
Thermal efficiency of the Atkinson cycle. Heat rejection happens at constant pressure, so c_p is used for the rejected heat. q_in is the heat input, c_p the constant-pressure specific heat.
End-of-over-expansion temperature T₄. The gas expands fully back down to the intake pressure P₁, so the expansion ratio exceeds the compression ratio. γ is the specific heat ratio, P₃ the peak pressure.
State temperatures and expansion ratio. 1→2 is isentropic compression, 2→3 constant-volume heating, r_exp=V₄/V₂ the expansion ratio. c_v is the constant-volume specific heat.
What is the Atkinson Cycle Simulator?
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I heard that the engine in a hybrid car like the Prius runs on the "Atkinson cycle". How is it different from an ordinary petrol engine?
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The biggest difference is that it makes the expansion stroke longer than the compression stroke. In an ordinary petrol engine — an Otto cycle — the piston travels the same distance for compression and for expansion. So at the moment the exhaust valve opens, the burnt gas is still at a pressure far above atmospheric. That "still able to push" energy is dumped out as exhaust without being used. The Atkinson cycle is a cycle that goes back to collect it.
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"Goes back to collect it" — how exactly does it do that?
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It does not stop expanding; it keeps expanding the gas completely, until the pressure has fallen all the way down to the intake pressure. Look at the P-V diagram on the left. The 3→4 expansion curve stretches far past V₁, the volume where compression began, well over to the right. That long extended part is the region recovering work that an Otto cycle would have thrown away. That is why it is also called the "complete-expansion" or "over-expanded" cycle. James Atkinson came up with it in the 1880s.
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I see! So the efficiency must come out better than the Otto cycle. The "Otto comparison" card does say it is more efficient.
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Right — compared at the same compression ratio, the Atkinson cycle has higher thermal efficiency. On the "efficiency vs compression ratio" chart on the right, you can see the blue Atkinson line sits above the orange Otto line. With the default settings, Atkinson is about 69% and Otto about 60% — nearly a 9-point gap. That gap is the exhaust pressure that used to be thrown away, now turned into work.
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If it is that much more efficient, why don't we make every engine an Atkinson engine?
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Good question. There is a price. Because the expansion is longer, the amount of fresh charge the engine can effectively draw in is reduced. So even with an engine of the same size, the power output is smaller. For nearly a century this "efficient but weak" trade-off held back its adoption. An engine that gives no push when you press the accelerator is hard to use on its own.
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Ah, so that is where the hybrid comes in. The motor supplies the missing power, so the engine can focus purely on efficiency?
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Exactly that. The electric motor takes on the power demand during acceleration, so the petrol engine can run as an efficiency-first Atkinson-cycle engine. In real engines it is not done with Atkinson's complex linkage — it is achieved by holding the intake valve open part-way through the compression stroke to shorten the effective compression, the "Miller cycle" approach. Thanks to variable valve timing, the engine can even switch between Atkinson-like behaviour at low load and Otto-like behaviour at high load.
Frequently Asked Questions
The defining difference is that the expansion (power) stroke is longer than the compression stroke. In the Otto cycle the compression and expansion volume ratios are equal, so when the exhaust valve opens the burnt gas is still well above atmospheric pressure, and that pressure energy is simply thrown away. The Atkinson cycle instead lets the gas expand completely, all the way down to intake pressure, converting that extra pressure into useful piston work. The result is higher thermal efficiency than an Otto engine of the same compression ratio, at the cost of lower power for a given engine size.
In the ideal Atkinson cycle the gas is compressed isentropically from state 1 (intake) by the compression ratio r, heated at constant volume, then expanded isentropically until the pressure returns to the intake pressure P1, and finally cooled at constant pressure back to state 1. The heat rejected is q_out = c_p(T4 - T1) and the thermal efficiency is η = 1 - q_out/q_in. Heat rejection occurring at constant pressure rather than constant volume is the calculation difference from the Otto cycle, and it raises the efficiency. This tool computes η in real time as you change the compression ratio, specific heat ratio, intake conditions and heat input.
The Atkinson cycle trades higher thermal efficiency for lower power per unit of engine displacement, because the effective intake charge is reduced. That power penalty held back its adoption for a century, but in a hybrid car the missing power can be supplied by the electric motor. This frees the petrol engine to run as an efficiency-focused Atkinson-cycle engine. In practice it is implemented by holding the intake valve open part-way through the compression stroke to lower the effective compression ratio — the Miller-style approach using variable valve timing.
The expansion ratio r_exp is the volume at the end of expansion V4 divided by the volume at the end of compression V2, and it measures how far the burnt gas expands. In the Atkinson cycle the gas expands until the pressure drops to the intake pressure, so the expansion ratio is always larger than the compression ratio. When this tool's expansion/compression ratio is greater than 1, that is direct evidence that the expansion stroke is longer than the compression stroke — that is, over-expansion. The larger this ratio, the greater the fraction of work recovered instead of being thrown away.
Real-World Applications
Generator and drive engines in hybrid cars: The Atkinson cycle is most widely used in hybrid vehicles. Starting with the Toyota Prius, the petrol engines of many hybrids run on the Atkinson (Miller) cycle. Because the electric motor covers the power shortfall, the engine can stay in its high-efficiency region, greatly improving the fuel economy of the system as a whole. The way η rises when you increase the compression ratio in this tool is exactly that pursuit of efficiency.
Implementation via variable valve timing: The complex linkage Atkinson originally conceived was too heavy and intricate for a practical engine. Modern production engines instead keep the intake valve open part-way through the compression stroke, pushing some of the charge back out, which shortens the "effective compression stroke". The expansion stroke stays long, so the expansion ratio ends up larger than the compression ratio and the same effect as the Atkinson cycle is obtained. This method is called the Miller cycle.
High-efficiency non-hybrid engines: With the increasing freedom of variable valve timing, even ordinary non-hybrid petrol cars now use engines that continuously shift between Atkinson-like operation at low load and Otto-like operation at high load. By taking efficiency at small throttle openings and power when the pedal is pressed, even a standalone engine can save fuel.
Thermodynamics education and cycle comparison: The Atkinson cycle is an ideal subject for learning, alongside the Otto and Diesel cycles, how the choice of heat-transfer process affects efficiency. Understanding deepens once you organise it as: Otto rejects heat at constant volume, Diesel adds heat at constant pressure, Atkinson rejects heat at constant pressure. Overlaying the efficiency curve with the Otto cycle in this tool gives an intuitive grasp of the efficiency gain from over-expansion.
Common Misconceptions and Pitfalls
A common misconception is that the Atkinson cycle is a complete upgrade over the Otto cycle. It is true that at the same compression ratio the thermal efficiency is higher, but the price is lower power per unit of engine size, because the effective intake charge is reduced for the sake of complete expansion. Efficiency and power are a trade-off, and which is "better" depends on the application. The Atkinson cycle is chosen in hybrid cars precisely because the motor can make up the missing power. It is not suited to a sports engine where power is the top priority.
Next, assuming the air-standard cycle efficiency is the real engine efficiency. The η this tool computes is an ideal value, treating the working fluid as ideal-gas air, combustion as external heating, and the specific heats as constant. In a real engine, the time combustion takes, heat loss to the walls, intake and exhaust throttling losses, and friction all drag the net thermal efficiency well below the ideal value. Furthermore, this tool assumes the ideal case where expansion ends exactly at the intake pressure, while a real engine cannot expand quite that completely. Use this tool's numbers as an "upper-bound guide" and a teaching aid for the influence of each parameter.
Finally, assuming the Atkinson cycle and the Miller cycle are completely different things. Historically, Atkinson physically changes the stroke length with a special linkage, while Miller shifts the timing of intake valve closing to lower the effective compression ratio. But as a thermodynamic cycle, both are "over-expanded cycles where the expansion ratio exceeds the compression ratio" — they are essentially the same. Almost every engine that today's automakers call an Atkinson engine is, internally, a Miller-style implementation. Keep the textbook's strict distinction and the industry's loose usage separate when you reason about them.
How to Use
Set compression ratio (rComp) between 8:1 and 14:1 typical for gasoline engines, accounting for the Atkinson's longer expansion phase.
Input heat capacity ratio (gamma) for your working gas—air at 1.40, or adjusted values for combustion products.
Define initial intake conditions: T₁ in Kelvin (typically 300–350 K ambient) and P₁ in kPa (101.3 kPa standard).
Observe thermal efficiency, peak temperature T₃, peak pressure P₃, and work outputs; compare expansion and compression strokes against Otto cycle baseline.
Worked Example
Simulate a hybrid vehicle engine: compression ratio 10:1, gamma 1.40 (air), T₁=298 K, P₁=101.3 kPa. The Atkinson cycle delivers thermal efficiency ~42%, with T₃≈2520 K and P₃≈2860 kPa. Expansion stroke generates 520 kJ/kg; compression consumes 380 kJ/kg, yielding net work 140 kJ/kg. Contrast this against equivalent Otto cycle (η≈35%, net work 155 kJ/kg), demonstrating the efficiency gain despite lower peak pressures.
Practical Notes
Atkinson engines sacrifice peak power density (lower P₃ and net work) to reduce fuel consumption—ideal for cruise efficiency in Toyota Prius and Honda Insight hybrid powertrains.
Increase compression ratio cautiously above 12:1; combustion temperatures rise dramatically, requiring fuel with high octane rating (98+ RON) to prevent knock.
Gamma varies with combustion chemistry; use 1.35 for rich mixtures and 1.38 for lean burn conditions to refine accuracy.