A rotor's aerodynamic efficiency does not just depend on wind speed — it depends on how fast the blade tips move relative to the wind, the tip-speed ratio:
λ = ω·R / v
where ω is the rotor's angular speed (rad/s), R the blade radius and v the wind speed. The fraction of kinetic power the rotor actually extracts is the power coefficient Cp(λ, β), a function of λ and the blade pitch angle β. This simulator uses the standard empirical model from wind-turbine control literature:
Cp(λ,β) = 0.5176·(116/λi − 0.4β − 5)·e^(−21/λi) + 0.0068·λ
1/λi = 1/(λ + 0.08β) − 0.035/(β³ + 1)
No real rotor can beat the theoretical Betz limit, Cp ≤ 0.593 — the dashed line on the graph — because slowing the wind down completely would stop it from flowing through the rotor at all. The curve peaks at one particular λ* for each pitch angle; below it the blades "let wind through" without capturing it, above it they act like a wall and stall the flow. Delivered electrical power then follows:
P = ½·ρ·A·v³·Cp(λ,β)·cos³γ (A = πR², ρ = 1.225 kg/m³)
The cos³γ term is this 2D version's extra dimension: when the rotor's own axis is misaligned from the wind by a yaw angle γ (drag the top-down inset, or use the slider), only the wind component along the rotor axis, v·cosγ, drives the rotor — that same cosine also shrinks the effective swept area facing the wind, so the two factors compound into the widely used cosine-cubed yaw-loss law from wind-energy engineering. λ is computed from that same reduced axial component, v·cosγ, so the operating point on the Cp curve also shifts as you yaw off-axis.
- Wind speed and pitch angle reshape the Cp(λ) curve and shift its peak.
- Rotor speed slider moves the operating point (white dot) left/right along the curve — drag it off the peak and watch power drop even though the wind hasn't changed.
- Yaw derates power smoothly with no change to λ's curve shape — a real turbine's yaw controller exists exactly to keep this angle near zero.
- MPPT mimics a real turbine controller: it continuously adjusts rotor speed to sit at the peak λ*, the maximum-power-point-tracking strategy every grid-connected turbine actually runs below rated wind speed.