The wind turbine calculator above applies the standard power equation to a rotor of a given diameter in air of a given density at a given wind speed, and then applies whatever power coefficient you supply to get an electrical output. It also shows the Betz limit alongside — the theoretical maximum fraction of the wind's kinetic energy any open-rotor turbine can extract — so that any coefficient you enter can be checked against physics rather than against optimism.
Arb Digital publishes free energy and environmental tools that separate the parts of a calculation that are settled physics from the parts that are site-specific measurements. Here the power equation is settled. The power coefficient, the air density, and above all the capacity factor are not: they belong to a particular machine at a particular place, and this page takes them as inputs rather than pretending to know them. The US Department of Energy's explanation of how wind turbines work covers the aerodynamics behind the equation.
What This Wind Turbine Calculator Does
It computes the swept area from the rotor diameter, the total power carried by the wind passing through that area, the Betz-limited maximum any rotor could extract from it, and the electrical power your stated coefficient and efficiency would actually deliver. It reports the fraction of the Betz maximum you are claiming, and it rejects a coefficient above the limit rather than quietly computing an impossible number.
Separately, it converts a rated power and a capacity factor into an annual energy figure. Those two calculations are deliberately kept apart, because the instantaneous power at one wind speed tells you almost nothing about the annual yield: real output depends on the whole distribution of wind speeds across the year, on cut-in and cut-out thresholds, on availability, and on wake and array losses. Multiplying one wind speed's power by 8,760 hours is the most common error in this area and it overstates output badly.
How to Use It
- Enter the rotor diameter and wind speed. Use the wind speed at hub height, not at ground level or at a nearby weather station, unless you have adjusted for the height difference.
- Set the air density for your site. Standard sea-level density is 1.225 kg/m³; high-altitude and hot sites are meaningfully lower.
- Take Cp from the manufacturer's power curve at the wind speed you are modelling. It is not a constant across the operating range.
- Get the capacity factor from measured data for comparable machines in a comparable region rather than assuming one.
- Read the Betz fraction. If your inputs claim more than about three-quarters of the Betz limit, one of them is wrong.
The Power Equation and the Betz Limit
The power carried by wind through a swept area is P = ½ ρ A v³, where ρ is air density in kg/m³, A is the swept area in m², and v is wind speed in m/s. The swept area of a rotor is A = π(D/2)². A turbine cannot capture all of that, because doing so would require bringing the air to a complete stop behind the rotor, which would prevent any more air from passing through. Betz's analysis puts the maximum extractable fraction at 16/27 ≈ 0.5926, and the electrical output is P = ½ ρ A v³ Cp η, where Cp is the power coefficient and η any additional efficiency you apply.
Work the defaults through. A 20 m rotor sweeps π × 10² = 314.16 m². At 1.225 kg/m³ and 8 m/s, the power in the wind is 0.5 × 1.225 × 314.16 × 512 = 98,520 W, or 98.52 kW. The Betz ceiling is 98.52 × 0.5926 = 58.38 kW. With a power coefficient of 0.40 the electrical output is 39.41 kW, which is 67.5% of the Betz maximum — a plausible figure for a good modern machine near its best operating point, and one that would be a red flag if it came out above about 85%.
Why the Cube Law Dominates Everything
Power scales with the cube of wind speed and only with the square of rotor diameter. Doubling the diameter quadruples the power; doubling the wind speed multiplies it by eight. That asymmetry is why wind resource assessment is worth more than almost any engineering decision: a site with a mean speed of 7.2 m/s carries roughly 73% more power than one at 6.0 m/s, which no amount of blade design can recover.
It is also why hub height matters so much. Wind speed increases with height above ground, so a taller tower reaches faster air, and the cube law converts a modest speed increase into a large power increase. And it is why a mean wind speed is a poor summary: because power depends on v³, the average of the cubes is much larger than the cube of the average, so an annual energy estimate built from a mean speed will understate a gusty site and overstate a steady one. Proper assessment uses the full speed distribution, not a single number.
Why This Tool Will Not Promise You an Output
The instantaneous calculation above is exact arithmetic on your inputs. An annual energy figure is not, and no honest calculator can produce one from a diameter and an average wind speed. Real annual output depends on: the frequency distribution of wind speeds through the year, the cut-in speed below which the machine produces nothing, the rated speed above which output is capped, the cut-out speed at which it shuts down for safety, turbine availability after maintenance and faults, wake losses where several machines share a site, electrical losses to the point of connection, and curtailment when the grid cannot take the power.
The capacity factor bundles all of that into one measured ratio: actual energy produced divided by what the machine would have produced running at rated power for every hour of the period. It is a retrospective measurement, not a design parameter, so it has to come from data. The EIA publishes monthly and annual capacity factors for utility-scale generators using non-fossil fuels, wind included, which is the right kind of source. Small turbines in built-up areas typically perform far below utility-scale figures, because buildings and trees both slow and disrupt the flow.
Air Density Is Not a Constant
Density appears linearly in the power equation, so a 10% drop in density is a 10% drop in power. The standard 1.225 kg/m³ applies at sea level at 15 °C. Air at 2,000 m of altitude is roughly a fifth less dense, and hot air is less dense than cold at the same pressure — which is why a turbine on a cold winter day produces more power at the same wind speed than on a hot summer afternoon.
For most sites, using the standard value overstates output slightly. If you have local pressure and temperature, density follows from the ideal gas law: ρ = p ÷ (R·T), with p in pascals, T in kelvin, and R about 287 J/(kg·K) for dry air. Humidity lowers density slightly further, though the effect is small compared with altitude and temperature. Our energy converter is useful for moving between the joule, watt-hour and kilowatt-hour figures this calculation produces.
Where the Betz Limit Comes From, and What It Does Not Cover
Betz's result follows from conservation of mass and momentum applied to an idealised actuator disc. If the rotor slowed the air completely, no air would flow through and no energy would be extracted; if it slowed the air not at all, no energy would be extracted either. The optimum sits between, at the point where the downstream wind speed is one third of the upstream speed, and it yields 16/27 of the available power. It applies to any open-rotor device in a free stream, regardless of blade count or design.
Two caveats are worth stating. The limit applies to the kinetic energy passing through the swept area, so a ducted or shrouded rotor is sometimes described as exceeding it — but that is a bookkeeping artefact of measuring the area at the rotor rather than at the duct inlet, not a violation of physics. And the limit is an upper bound, not a target: real machines lose further energy to blade drag, tip vortices, wake rotation and drivetrain friction, which is why a Cp above roughly 0.5 is not achievable in practice even though the limit sits at 0.59. Our kinetic energy calculator handles the underlying ½mv² relationship that the whole derivation rests on.
Turning Energy Into Emissions or Money
Once you have an annual kWh figure grounded in a measured capacity factor, converting it into avoided emissions requires a grid emission factor, and that factor is regional and changes year to year as the generation mix changes. The EPA documents the approach in its greenhouse gas equivalencies calculations and references, including how regional grid data is applied. Multiplying kWh by a national average will be wrong in either direction depending on where you are.
Converting to money is a separate question again, involving tariffs, export rates, capital cost, financing and maintenance, none of which this page models. Price the energy at your own rate with the electricity bill calculator, and if you are comparing generation options, the solar panel calculator takes the same approach of asking for site-specific inputs rather than assuming them. For storage sizing, the battery life calculator works from load and capacity.
Arb Digital builds content that makes engineering credible to buyers without overstating what the numbers support.
See Content Marketing Talk To Our TeamCommon Mistakes to Avoid
- Multiplying instantaneous power by 8,760 hours to get annual energy, which ignores the entire wind speed distribution.
- Using a power coefficient above 0.5926, which is physically impossible for an open rotor.
- Taking wind speed from ground level or a distant station instead of hub height at the site.
- Assuming standard air density at a high-altitude or hot site, where it can be a fifth lower.
- Borrowing a utility-scale capacity factor for a small rooftop machine, where turbulence and obstruction cut output sharply.
Related Free Tools From Arb Digital
Convert between energy units with the energy converter, work the underlying physics with the kinetic energy calculator, price output with the electricity bill calculator, compare with the solar panel calculator, or size storage with the battery life calculator. Everything else is on the free online tools hub.
Frequently Asked Questions
Power equals one half times air density times swept area times wind speed cubed, multiplied by the power coefficient. Swept area is pi times the square of half the rotor diameter.
The theoretical maximum fraction of the wind's kinetic energy an open rotor can extract, equal to 16 divided by 27, or about 59.3%. Extracting more would require stopping the air completely, which would block further flow.
No. Blade drag, tip losses, wake rotation and drivetrain friction all take a further share, so practical power coefficients sit below about 0.5 even for well-designed modern machines.
Because power depends on the cube of wind speed. Two cubed is eight, which is why site wind resource matters more than almost any design decision.
Not reliably. Annual output depends on the full distribution of wind speeds, on cut-in and cut-out limits, on availability and on wake losses. A measured capacity factor is the only sound route from rated power to annual energy.
Actual energy produced over a period divided by the energy the machine would have produced running at rated power for every hour of it. It is a retrospective measurement, not a design assumption.
Yes. Density enters the equation linearly, so a site a fifth less dense because of altitude produces a fifth less power at the same wind speed.
This tool is provided for educational use. It reports arithmetic on the inputs you supply and does not predict the output, yield or financial return of any real installation.
