Propulsion and Battery Model Theory
This document defines the core mathematical model for the electric propulsion system (propeller + brushless motor + battery/ESC).
1) Symbols and Units
| Symbol | Description | SI Unit |
|---|---|---|
| \(V\) | Flight airspeed | \(\text{m/s}\) |
| \(D\) | Propeller diameter | \(\text{m}\) |
| \(n\) | Propeller shaft speed | \(\text{1/s}\) (with \(n = \text{RPM}/60\)) |
| \(\rho\) | Air density | \(\text{kg/m}^3\) |
| \(J\) | Advance ratio | Dimensionless |
| \(C_t, C_p\) | Thrust and power coefficients | Dimensionless |
| \(T\) | Thrust force | \(\text{N}\) |
| \(Q\) | Propeller torque | \(\text{N}\cdot\text{m}\) |
| \(P_{\text{shaft}}\) | Shaft mechanical power | \(\text{W}\) |
| \(K_v\) | Motor speed constant | \(\text{RPM/V}\) |
| \(K_t\) | Motor torque constant | \(\text{N}\cdot\text{m/A}\) |
| \(I\) | Motor winding current | \(\text{A}\) |
| \(I_0\) | Motor no-load current | \(\text{A}\) |
| \(R\) | Motor internal winding resistance | \(\Omega\) |
| \(R_{\text{system}}\) | Lumped transmission system resistance | \(\Omega\) |
| \(V_m\) | Motor terminal voltage | \(\text{V}\) |
| \(V_{\text{back}}\) | Back-EMF voltage | \(\text{V}\) |
| \(V_{\text{pack}}\) | Battery terminal pack voltage | \(\text{V}\) |
| \(x\) | Battery depth of discharge | Dimensionless |
| \(N_s, N_p\) | Pack cells in series and parallel | Dimensionless |
2) Propeller Aerodynamics
Propeller coefficients \(C_t(J, \text{RPM})\) and \(C_p(J, \text{RPM})\) are loaded from an empirical database and evaluated using bilinear interpolation.
Aerodynamic Equations
3) Brushless DC Motor Model
The relationship between torque, back-EMF, and speed is defined below:
where \(\tau\) is the magnetic lag time constant (set to \(0\) for first-order models), and \(\omega = n \cdot 2\pi\).
4) Battery Model
The fixed-voltage battery model uses a constant terminal pack voltage:
The rate-map battery model uses a cell open-circuit voltage curve and a cell resistance curve indexed by depth of discharge:
For a pack topology:
During mission stepping, state advances by charge conservation:
5) System Electrical Resistance & Power Chain
The voltage drops across ESC MOSFETs, battery internal resistance, cables, and connectors are modeled as a lumped transmission system resistance (\(R_{\text{system}}\)):
The overall electrical power drawn from the battery pack is:
For RateMapBattery, the equivalent-circuit terminal voltage already captures
the cell-level voltage sag from the battery curves. R_system should still be
used for wiring, ESC, connector, or other losses outside the battery model.
6) Coupled Equilibrium Condition
For a given throttle setting and airspeed, the equilibrium shaft speed (RPM) is the solution of the coupled electrical and aerodynamic torque equilibrium equation:
A root-finding method (e.g., Brent's method) solves \(F(\text{RPM}) = 0\) for RPM. Once the equilibrium RPM is determined, \(T, Q, P_{\text{shaft}}, I\), and efficiency parameters are calculated.