Drone & eVTOL Hover: Altitude, Cold Battery and Thrust Margin

Atmosphere. The tropospheric ISA relations use $T_{ISA}=288.15-0.0065h$ and $p=101325(T_{ISA}/288.15)^{g/(R_a L)}$. Actual dry-air density is $\rho=p/(R_a T)$; $\Delta T_{ISA}=T-T_{ISA}$. Density altitude inverts the standard density relation, switching to the isothermal 11–20 km ISA layer when needed. Altitude is geopotential pressure altitude, not a weather-corrected geometric elevation (COESA 1976).

Hover and rotor limits. For total separate disk area $A$, thrust $W=mg$ gives $v_i=\sqrt{W/(2\rho A)}$, $P_i=Wv_i$, shaft power $P_s=P_i/FM$, and pack demand $P=P_s/\eta+P_{aux}$. Figure of merit includes rotor profile and nonideal induced losses; do not add profile power again. Solidity is $\sigma=Bc/(\pi R)$ and $C_T=W/[\rho A(\Omega R)^2]$. The propeller convention $T/(\rho n^2D^4)$ converts to this rotor coefficient by multiplication by $4/\pi^3$. Fixed-pitch operation solves the RPM at the entered $C_T$; collective operation evaluates $C_T/\sigma$ at governed RPM. The supplied blade-loading boundary and local tip Mach limit are enforced, along with maximum RPM. A fixed-pitch coefficient above its stall boundary has no modeled feasible thrust.

Cold battery. $R(T_b)=R_{25}\exp[(E_a/R_g)(1/T_b-1/298.15)]$. Low-rate available charge follows an empirical reciprocal-temperature exponential, equal to the entered reference capacity at 25 °C and the entered cold fraction at −20 °C; it is capped at reference capacity above 25 °C. These two anchors define a surrogate, not a universal lithium-ion chemistry model. OCV varies linearly with effective SOC between the supplied cell endpoints.

Power and duration. For pack OCV $U$, admissible current is $\min[I_{cont},\max((U-V_{cut})/R,0),U/(2R)]$, and available terminal power is $I(U-IR)$. This enforces current, loaded cutoff and the maximum-power knee. The stable constant-power branch is $I=2P/[U+\sqrt{U^2-4RP}]$. The exact electrical stopping SOC follows from $I_b=\min[I_{cont},\sqrt{P/R},P/V_{cut}]$ and $U_{req}=P/I_b+RI_b$. Charge integration $t=Q(T_b)\int dz/I(z)$ uses 401 SOC samples, stopping at reserve or the first electrical limit. Total maximum thrust is the minimum of rotor, motor and battery ceilings; power ceilings are inverted through the same hover-power law.

• Identical, isolated lifting rotors share thrust equally; level out-of-ground-effect hover, no airframe download.
• Constant figure of merit, drive efficiency and fixed-pitch thrust coefficient across each comparison.
• The blade-loading threshold is a rotor-specific test input, not a universal airfoil stall coefficient. Equivalent chord includes the actual blade planform and root cutout.
• Pack temperature stays fixed during discharge. The cold-soaked choice sets it equal to air temperature; a held setpoint represents thermal conditioning.
• Pack resistance is an effective DC value independent of SOC. Cells are balanced; all cells share the entered OCV and loaded cutoff. SOC refers to the temperature-derated available charge.
• The continuous current limit is the user-supplied value at operating temperature and stays fixed across sweeps. Capacity derating must come from low-rate data so load-induced cutoff loss is not counted twice.

Pressure altitude −500 to 11,000 m and entered air temperature −50 to 50 °C; dry ideal air with standard tropospheric pressure. Local humidity, pressure weather and geometric-altitude corrections are excluded. The 121-point altitude curve holds the present battery temperature and ISA offset fixed; the 141-point temperature curve follows the selected pack temperature mode at fixed altitude. Density altitude uses the ISA isothermal layer above the 11 km tropopause when required. The sweep curves are computed directly at every sample.

This is a hover screening model. A positive reserve checks only the entered rotor, drive and battery constraints; it is not an aircraft operational envelope. Use manufacturer propulsion data and measured cold-discharge behavior. Set maximum RPM to a speed actually achievable at loaded pack voltage, and set shaft rating to power available at that RPM. Motor torque, back-EMF, ESC duty limits, Reynolds-number changes, radial stall, compressibility losses below the chosen Mach threshold, duct/coaxial interference, wind and icing require more detailed models.

The illustrative battery parameters are not a product specification. A single Arrhenius slope may fail below 0 °C; fit it over the relevant temperature range. Linear OCV and SOC-independent DC resistance omit polarization dynamics and weak-cell cutoff. Fixed temperature omits self-heating, cooling and thermal limits. Low-rate capacity loss and high-load sag are modeled separately. Hover duration is to reserve or first limit, not usable mission flight time, and does not preserve the requested excess-thrust target throughout that duration. A reserve-charge margin below target requires an earlier operational stop or a design change. Undefined operating current/voltage and an absent discharge trace mean initial hover is infeasible. The 401-point charge quadrature resolves the smooth constant-temperature model; it does not reproduce measured electrochemical transients.