Wind Turbine Power Calculator (P = ½ρAv³Cp, Betz limit and tip speed ratio)

Understanding the wind turbine power calculator — concept and principle

This calculator estimates how much electrical power a wind turbine produces at a given wind speed. It finds the power carried by the wind through the rotor, applies the turbine's power coefficient and the Betz limit, caps the result at the generator rating and gives the energy per day and the tip speed ratio. It is used in renewable-energy courses and early project sizing.

Power grows with wind speed cubed; 40 % of 1,738 kW is 695 kW.

How it is calculated

ρ = p ÷ (287.05 × T)

A = π R², with R = blade length + hub radius

Pwind = ½ ρ A v³

P = Pwind × Cp, capped at the rated power

Betz maximum = Pwind × 16/27

Tip speed ratio = 2π R × rpm ÷ 60 ÷ v

p air pressure in Pa, T temperature in K, v wind speed at hub height in m/s, Cp power coefficient in %.

The principle

Moving air carries kinetic energy ½mv². The mass of air passing through the rotor each second is ρAv, so the power is ½ρAv³. The cube is the key: doubling the wind speed gives eight times the power. A turbine cannot capture all of it, because the air must keep moving away behind the rotor. Betz showed that the best possible rotor slows the wind to one third of its original speed downstream and extracts 16/27, about 59.3 %. Real turbines lose more to blade drag, tip losses and the drivetrain. Above the rated wind speed, the blades pitch to spill excess wind and keep the generator at its rating.

Worked example

Default inputs: blade 40 m, hub 2 m, wind 8 m/s, Cp 40 %, 15 °C, 101.325 kPa, rated 3,000 kW, 12 rpm.

ρ = 101,325 ÷ (287.05 × 288.15) = 1.225 kg/m³.
R = 42 m; A = π × 42² = 5,542 m².
Pwind = 0.5 × 1.225 × 5,542 × 8³ = 1,737.9 kW; power density = 313.6 W/m².
P = 1,737.9 × 0.40 = 695.2 kW (No, below rated power); Betz maximum = 1,029.9 kW.
Energy per day at this steady wind = 695.2 × 24 = 16,684 kWh.
Tip speed ratio = 2π × 42 × 12 ÷ 60 ÷ 8 = 6.6.

Each point is the calculator re-run with only “Wind speed at hub height (m/s)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Wind speed at hub height (m/s)	Turbine power output (kW)
2	10.9
4	86.9
6	293.3
8	695.2
10	1,357.7
12	2,346.2
16	3,000

How to read the result

Modern large turbines typically reach peak Cp of roughly 40–50 %. Three-blade rotors work best at tip speed ratios around 6–8. Because power varies with v³, a site's yearly energy cannot be found from the average wind speed; you need the wind speed distribution, and the capacity factor of real wind farms is well below 100 %.

Common mistakes

Using the average wind speed to estimate yearly energy.
Using wind speed at ground level instead of at hub height.
Entering a Cp above the Betz limit.
Forgetting the hub radius when computing swept area.

Keep in mind: This is a steady-wind estimate. Real output follows the manufacturer's power curve, availability and losses; project decisions need measured site data and professional assessment.

Frequently asked questions

What is the Betz limit?

The theoretical maximum fraction of wind power a rotor can capture, 16/27 or about 59.3 %.

Why does power rise with the cube of wind speed?

Faster wind brings both more air mass per second and more energy per kilogram.

What is a good tip speed ratio?

About 6–8 for modern three-blade turbines.

Why does cold air give more power?

Cold air is denser, so the same wind speed carries more mass.

Sources: University of Illinois ECE 333 "Green Electric Energy", Lecture 6 "Limits on Conversion of Wind Into Electricity" (G. Gross)

Checked 9 Nov 2026. Power in the wind ½ρAv³ and the 59.3% maximum conversion efficiency (downstream speed ratio 1/3), tip speed ratio, practical efficiency below 50%: University of Illinois ECE 333 “Green Electric Energy”, Lecture 6 “Limits on Conversion of Wind Into Electricity” (G. Gross). Ideal-gas air density with R_d = 287.05 J/(kg·K) — standard value.

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