Wind

Wind turbines convert the kinetic energy of moving air into electricity. Learn the power equation and v³ law, the Betz limit, tip-speed ratio, power curves, wind shear, capacity factor and generator/converter types.

A wind turbine's rotor slows the air passing through it and extracts part of its kinetic energy. Modern turbines have 3 blades up to 120 m long, hubs over 150 m high, and ratings from 3 MW onshore to 15+ MW offshore.

Because power rises with the cube of wind speed, a site's wind resource dominates its economics — small differences in average speed mean large differences in energy.

An analogy

A wind turbine is like a sail that spins instead of pulling: double the wind speed and you get eight times the push-power, which is why a slightly windier hill is worth so much more.

The cube law

Power in the wind is ½ρAv³. Doubling wind speed gives 8× the power; a site averaging 7.5 m/s offers ~95 % more power than one at 6 m/s. Doubling rotor diameter gives 4× the area and 4× the power.

Betz limit and C_p

A turbine can't stop the air completely (it would pile up), so at most 16/27 ≈ 59.3 % of the wind's power can be extracted. Modern turbines reach C_p ≈ 0.45–0.50 at their optimal tip-speed ratio (λ ≈ 7–9 for three-bladed rotors).

Power curve and control

Below cut-in (~3 m/s) nothing is generated; between cut-in and rated speed (~11–13 m/s), variable-speed control keeps λ optimal for maximum C_p; above rated speed, blade pitch control spills excess power to hold rated output; above cut-out (~25 m/s) the turbine feathers and stops for safety.

Generators and grid connection

Type 3 turbines use doubly-fed induction generators with a partial-rated converter on the rotor; Type 4 use full-converter PMSG or induction generators, fully decoupling rotor speed from grid frequency and providing strong grid-support functions. Offshore farms connect via HVAC or HVDC export cables.

Resource assessment

Wind speeds follow a Weibull distribution; annual energy = ∫ power curve × probability. Wake losses inside farms, turbulence, air density at altitude and availability all feed the P50/P90 energy estimates that finance projects.

Formulas

• Power in the wind: P_wind = ½ ρ A v³
• Turbine power: P = ½ ρ A v³ · C_p (C_p,max = 16/27 = 0.593, Betz)
• Tip-speed ratio: λ = ω R / v
• Wind shear: v₂ = v₁ (h₂/h₁)^α, α ≈ 0.14 (open land)

Worked example: Power of a 100 m rotor

Given: D = 100 m → A = π × 50² = 7854 m²; v = 10 m/s, ρ = 1.225; C_p = 0.45

Find: Electrical power (ignoring drivetrain losses).

• P_wind = 0.5 × 1.225 × 7854 × 1000 = 4.81 MW
• P = 4.81 × 0.45 = 2.16 MW

Answer: ≈ 2.2 MW

Worked example: Effect of hub height

Given: v = 5 m/s at 10 m; Hub at 100 m, α = 0.14

Find: Hub-height speed and power gain.

• v₁₀₀ = 5 × (100/10)^0.14 = 6.90 m/s
• Power ratio = (6.90/5)³ = 2.63

Answer: 6.9 m/s → 2.6× more power

Common mistakes

• Wrong: Using average wind speed cubed to estimate energy. — Right: Average the cube (use the Weibull distribution and power curve) — mean(v³) > mean(v)³.
• Wrong: Expecting C_p above 0.593. — Right: The Betz limit caps extraction at 59.3 % of the wind's power.

Applications

• Onshore wind farms: Some of the cheapest bulk electricity where wind resources are good.
• Offshore wind: Stronger, steadier winds and room for 15 MW+ turbines; floating foundations open deep waters.
• Hybrid plants: Wind + solar + storage on one connection smooths output because the resources are partly complementary.

Key takeaways

• P = ½ρAv³C_p; power scales with v³ and D².
• Betz limit: C_p ≤ 0.593; real turbines ≈ 0.45–0.50.
• Pitch control above rated speed, optimal λ below it.
• Hub height, wind shear and the speed distribution drive energy yield.

Going further

• Blade-element momentum theory: BEM divides the blade into sections and balances aerodynamic forces with momentum change, predicting C_p, loads and optimal twist/chord distributions.
• Wake steering: Deliberately yawing upstream turbines deflects their wakes away from downstream ones, raising total farm output by a few percent.
• Grid services from wind: Synthetic inertia, fast frequency response and grid-forming control let wind farms support grid stability, not just supply energy.