Wind Turbine Power Calculator (Betz Limit)
Work out the power a wind turbine makes from its rotor size and the wind speed. Check it against the Betz limit, size a rotor for a power, correct the air density for altitude, and estimate the energy per year from a capacity factor or a Rayleigh wind distribution.
How to Use
- Choose what to work out: the Power a turbine makes, the rotor Diameter a power needs, the wind Speed it needs, or the Energy per year.
- Enter the rotor diameter and the wind speed, each with its unit, and the power coefficient Cp: 0.35 to 0.45 is typical for a large modern turbine, less for small ones, and it can never pass the Betz limit of 0.593.
- Leave the air density at the sea-level standard of 1.225 kg/m³, or work it out from the altitude and temperature, or type your own.
- For energy, either give the rated power and a capacity factor (25 to 45% is typical), or a mean wind speed and the cut-in, rated and cut-out speeds for a Rayleigh wind distribution.
- Add the rotor speed in rpm to get the tip speed and tip-speed ratio. The plot shows power against wind speed, with the Betz limit dashed.
- Press a preset to load a rooftop turbine, a 100 m utility turbine, an energy estimate, a rotor sizing or a turbine at altitude.
Worked Example
A utility turbine. A 100 m rotor sweeps A = π × 100² ÷ 4 = 7,854 m². In an 11 m/s wind at sea level the power flowing through it is ½ × 1.225 × 7,854 × 11³ = 6.403 MW. The Betz limit allows 16/27 of that, 3.794 MW, and with a realistic Cp of 0.45 the turbine makes 2.881 MW. Turning at 15 rpm its blade tips move at 78.54 m/s, a tip-speed ratio of 7.14.
A rooftop turbine. A 1.5 m rotor sweeps 1.767 m². At 5 m/s, with Cp = 0.30, it makes ½ × 1.225 × 1.767 × 5³ × 0.30 = 40.59 W, under 1 kWh a day. Rooftops are sheltered and turbulent, so small turbines often see less wind than the figure on the box assumes.
The common mistake: putting the average wind speed into the formula. Because power goes with v³, the fast hours count far more than the slow ones. A 3 MW, 100 m turbine makes 742.5 kW at exactly 7 m/s, but at a site whose wind averages 7 m/s with a Rayleigh spread it averages 1.042 MW, 40% more, for 9.126 GWh a year. Using the mean speed alone would understate the energy by 28.7%.
Show Work
Formulas
From Brush’s Windmill to the Betz Limit
In the winter of 1887–88 Charles F. Brush built a wind turbine behind his house in Cleveland, Ohio, to charge the batteries that lit it: a rotor about 17 m across with 144 cedar blades, making around 12 kW. In Denmark in the 1890s Poul la Cour tested rotors in a wind tunnel and found that fast-turning rotors with few blades were far more efficient, the shape every modern turbine follows.
How much a rotor could take from the wind was settled in the years around 1920. Frederick Lanchester in Britain (1915), Albert Betz in Germany (1920) and Nikolai Zhukovsky in Russia reached the same answer independently: 16/27 of the power passing through the rotor. The result is usually called the Betz limit.
The first megawatt machine, the 1.25 MW Smith–Putnam turbine on Grandpa’s Knob in Vermont, fed the grid in 1941 until a blade broke off in 1945. Today’s turbines have rotors well over 100 m across, and their power coefficients of 0.45 or more reach three quarters of what Betz allows.
About This Tool
This calculator solves P = ½ρAv³Cp for the power, the rotor diameter or the wind speed, and refuses a power coefficient above the Betz limit. Air density is the International Standard Atmosphere value of 1.225 kg/m³ at sea level, or comes from the ISA pressure at an altitude and the air temperature (dry air, 287.05 J/(kg·K)). Annual energy comes from a capacity factor you give, or from a Rayleigh wind distribution with a mean speed run through a simple power curve: nothing below cut-in, the cubic law up to the rated power, a flat top, and nothing above cut-out.
Real turbines have measured power curves, and real sites have their own wind distributions (often a Weibull shape) and lose a few percent to wakes, availability and the grid connection, so treat the energy figure as a first estimate. Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Work & Power Calculator, Kinetic Energy Calculator, and Bernoulli Equation Calculator.
Frequently Asked Questions
How do you calculate wind turbine power?
P = ½ ρ A v³ Cp. A 100 m rotor sweeps π × 100² ÷ 4 = 7,854 m². At 11 m/s in sea-level air (1.225 kg/m³) the wind carries 6.403 MW through that circle; with a power coefficient of 0.45 the turbine takes 2.881 MW.
What is the Betz limit?
The most any turbine can take from the wind, 16/27 or 59.3% of the power passing through its rotor. Taking it all would mean stopping the air dead, and then no more could flow through. For the 100 m rotor at 11 m/s that ceiling is 3.794 MW; real turbines reach about three quarters of it.
Why does wind speed matter so much?
Power goes with the cube of the speed, so doubling the wind gives 8 times the power. A 1.5 m rooftop turbine with Cp 0.30 makes 40.59 W at 5 m/s but 324.7 W at 10 m/s. That is also why averaging the wind speed misleads: a site averaging 7 m/s gives a 100 m, 3 MW turbine 1.042 MW on average, 40% more than its 742.5 kW output at exactly 7 m/s.
How much energy will a wind turbine make in a year?
Multiply the rated power by the 8,760 hours in a year and the capacity factor. A 2 MW turbine at 35% makes 2 × 8,760 × 0.35 = 6,132 MWh (6.132 GWh). From the wind itself, a 100 m, 3 MW turbine at a site with a 7 m/s mean (Rayleigh distribution, cutting in at 3 m/s and out at 25 m/s) makes about 9.126 GWh, a capacity factor of 34.7%.
How does altitude affect wind turbine power?
Power is proportional to air density. At 2,000 m in the standard atmosphere (79.5 kPa, 2 °C) air is 1.0065 kg/m³, 82.2% of sea level, so the 100 m turbine at 11 m/s makes 2.367 MW instead of 2.881 MW.
How do I use the Wind Turbine Power Calculator (Betz Limit)?
Just type your numbers. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Do I need to install or sign up for anything?
Not at all — it runs in the browser with nothing to install and no account. After it loads once, it even works without an internet connection.
Is my information private?
Yes. Everything happens in your browser. Nothing you type is sent to a server or saved anywhere.
Common Use Cases
Small and rooftop turbines
A 1.5 m rotor at 5 m/s with Cp 0.30 makes 40.59 W, under 1 kWh a day; it needs 10.72 m/s to make 400 W.
Utility turbines
A 100 m rotor at 11 m/s with Cp 0.45 makes 2.881 MW, 75.9% of its 3.794 MW Betz limit; at 15 rpm its blade tips move at 78.54 m/s, a tip-speed ratio of 7.14.
Site energy estimates
A 3 MW, 100 m turbine at a 7 m/s mean site generates for 86.6% of the year and makes about 9.126 GWh.
Sizing a rotor
Making 1 kW at 10 m/s with Cp 0.35 needs 4.665 m² of swept area, a rotor 2.437 m across.
Mountain and high sites
At 2,000 m the thinner air (1.0065 kg/m³) cuts the 100 m turbine’s output at 11 m/s from 2.881 MW to 2.367 MW.
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