Hover power: momentum theory and figure of merit
A rotor of disc area A that holds a weight W = mg must push air down. Momentum theory gives the ideal induced velocity v_h = sqrt(T / 2ρA) and ideal power:
P_ideal = T^1.5 / sqrt(2 rho A) (T = W in hover)
P_shaft = P_ideal / FM FM = figure of merit, 0.6 to 0.75 for real props
P_elec = P_shaft / eta_motor+ESC + P_avionics
Adding rotors with the same total disc area costs nothing in ideal power, but more rotors give redundancy. Thrust is T = kT ω², torque Q = cQ T and shaft power P = Qω. Air density ρ falls with altitude (ISA troposphere), so the same rpm gives less thrust and more power is needed: the mountain preset shows this.
Forward flight
The disc tilts by θ so that T sinθ = D and T cosθ = W, with D = ½ρV²CdA for the frame and payload. The induced velocity is the Glauert fixed point vi = T / (2ρA sqrt((V cosθ)² + (V sinθ + vi)²)), solved by iteration. Induced power falls as speed rises, blade profile power grows with advance ratio μ (1 + 4.65 μ²), and parasite power grows as V³. The sum has a minimum: the best-range speed minimises energy per kilometre, found here by golden-section search. A head wind shifts that optimum to a higher air speed.
Battery
The pack is an open-circuit voltage source Voc(s) (12 cells in series, 4.2 V full to about 3.3 V) behind a resistance R = 0.05 Ω (1000 Wh / E). Power balance P = V I − I²R gives the current, so heavy loads waste more energy as heat and the terminal voltage sags. State of charge falls with charge drawn, ds/dt = −I / Q. Pack-level specific energy (Wh/kg) is much lower than the cell figure because of casing, wiring and the BMS. The mission needs the reserve (default 20 %) left at touchdown.
Payload against range
Total mass m = mframe + mbattery + mpayload. More payload raises hover power as m1.5, so range falls faster than linearly. There are two limits: the thrust limit (rotors cannot lift more than about Tmax / 1.35) and the energy limit. At short range the thrust limit wins and the payload curve is flat, at long range energy wins. Delivering supplies carries the load on the way out, evacuation carries it on the way back; a head wind out is a tail wind home.
Redundancy and control
The 8 rotors alternate spin direction so yaw torques cancel. A control-allocation matrix B maps rotor thrusts to [T, τx, τy, τz], and the commanded thrusts are the minimum-norm solution u = BT(B BT)-1 v over the rotors that still work. With one rotor out the other seven still give full control when the thrust margin allows it (7 Tmax > W, plus authority). Attitude uses PD control on the rotation error of the body axes, τ = I (−Kp eR − Kd ω) + ω×Iω. Linearised it is a second-order system with ωn = sqrt(Kp) and ζ = Kd / (2 sqrt(Kp)), overshoot exp(−πζ/sqrt(1−ζ²)). The motors add a first-order lag, which makes the real response slightly more oscillatory. A cascaded position loop turns position error into a desired thrust vector and so a desired tilt.
Downwash and ground effect
The wake leaves the discs at about 2 v_h. At 33 kg and 3.1 m² of rotor area that is a 7 m/s induced speed and a 14 m/s far wake, enough to throw dust and spray, which is why landing zones need clearing. Close to the ground (below about 2 rotor radii) the thrust at a given power rises, modelled with 1 / (1 − (R / 4z)²).
What is simplified
Figure of merit and profile drag are lumped constants; no rotor-rotor interference, blade flapping, inflow dynamics or vortex-ring dynamics in the 6-DOF model (only a power estimate in the mission descent); the atmosphere is ISA with no humidity; battery temperature, ageing and cut-off protection are ignored; wind is a mean plus first-order turbulence and a 1-cos gust. The 3D scene is a compressed visual of a 4 to 30 km mission: the energy and distance numbers use the real distance, the drone is only shown flying a 370 m route between the two pads. A generic, unbranded airframe; for education only, no flight advice.