Solar & Wind Power formula solvers

Angle of Incidence on a Tilted Surface

\theta = \lvert \phi - \delta - \beta \rvert

Solar & Wind PowerHow far off square the noon sun strikes an equator-facing collector tilted at angle β, at latitude φ, on a day of declination δ. Set the tilt to φ − δ and the answer is zero: the rays arrive dead on.

Betz Limit — Maximum Extractable Wind Power

P_{max} = \frac{16}{27} \cdot \tfrac{1}{2} \rho A v^{3}

Solar & Wind PowerThe most power any device can take from an open stream of moving air: 16/27, or 59.26%, of the kinetic flux through its swept area. Betz proved it in 1920 from momentum and mass conservation alone, with no reference to blades.

Capacity Factor

CF = \frac{E}{P_r \, t}

Solar & Wind PowerEnergy actually generated over a period, divided by the energy the plant would have made running flat out at its rated power for the whole of it. The number that separates a nameplate from a power station.

Peak Sun Hours from Insolation

PSH = \frac{E}{G_{ref} A}

Solar & Wind PowerA day's solar energy on a surface, re-expressed as the number of hours of full 1000 W/m² sun that would have delivered the same total. The tidy fiction that turns a curve across the sky into a single number a designer can multiply by.

PV Array Energy Yield

E = P_{dc} \, H \, PR

Solar & Wind PowerThe energy a photovoltaic array delivers over a period: its DC rating, times the peak sun hours the site receives, times a performance ratio covering every loss between the glass and the meter. NREL's PVWatts model in one line.

PV Temperature Derate

P = P_{stc} \left[ 1 + \gamma \left( T_c - 25^\circ\text{C} \right) \right]

Solar & Wind PowerWhat a photovoltaic module actually delivers once its cells are hotter than the 25 °C the nameplate was measured at. The correction that explains why a panel makes less on the hottest afternoon of the year than on a bright cold morning.

Rotor Swept Area

A = \frac{\pi d^{2}}{4}

Solar & Wind PowerThe area of the disc a horizontal-axis rotor sweeps, from its diameter. Every wind power equation on this site takes this area, and it is the circle the blade tips trace — not the surface of the blades themselves.

Solar Declination Angle (Cooper's Equation)

\delta = 23.45^\circ \sin\!\left( \frac{360^\circ \, (284 + n)}{365} \right)

Solar & Wind PowerThe angle between the sun's rays and the plane of the equator on a given day of the year. Cooper's 1969 approximation, the one line of solar geometry every collector calculation begins from.

Solar Panel Power Output

P = G A \eta

Solar & Wind PowerThe electrical power a photovoltaic module delivers: the irradiance falling on it, times its area, times its conversion efficiency. The definition that lies behind every nameplate wattage on the back of a panel.

Tip-Speed Ratio

\lambda = \frac{\omega R}{v}

Solar & Wind PowerHow fast the blade tips travel compared with the wind coming at them. The single number that decides whether a rotor is running at its best power coefficient, and the reason large turbines turn slowly while small ones spin fast.

Wind Power Density

\frac{P}{A} = \tfrac{1}{2} \rho v^{3}

Solar & Wind PowerThe power carried past every square metre of open air by a wind of speed v. The figure that ranks one site against another before any turbine is chosen, and the place the cube law first bites.

Wind Shear Power Law

v_2 = v_1 \left( \frac{h_2}{h_1} \right)^{\alpha}

Solar & Wind PowerWind speed at one height from a measurement at another. Ground friction slows the lowest air, so the wind at a 100 m hub is meaningfully stronger than the wind at the 10 m mast that measured it — and the cube law turns that difference into a large one.

Wind Turbine Power Output

P = \tfrac{1}{2} \rho A v^{3} C_p

Solar & Wind PowerThe mechanical power a rotor of swept area A takes from a wind of speed v, where the power coefficient Cp is the fraction of the available kinetic flux it actually captures.