Propulsion Solver User & API Guide

This guide describes how PyThrust solves a motor-propeller operating point and how to interpret the result.

The solver answers one core question:

Given a motor, propeller, battery, throttle, and airspeed, what shaft RPM makes the electrical motor model and propeller aerodynamic load agree?

Solver Inputs

The operating-point solver combines five inputs:

Input	Class or value	Purpose
Motor	`MotorSpec`	Kv, winding resistance, no-load current, current limit, optional higher-order loss terms
Battery	`BatterySpec`	Pack voltage and discharge efficiency
System	`SystemSpec`	Lumped resistance for ESC, battery internal resistance, wiring, and connectors
Propeller	`PropellerSpec` + `PropellerEntry`	Geometry plus empirical aerodynamic coefficients
Flight condition	`rho`, `airspeed_mps`, `throttle`	Air density, freestream speed, and commanded voltage fraction

Symbols

Symbol	Meaning
\(\text{RPM}\)	Propeller shaft speed being solved
\(n\)	Shaft speed in revolutions per second
\(V\)	Flight airspeed
\(D\)	Propeller diameter
\(J\)	Propeller advance ratio
\(C_t\), \(C_p\)	Propeller thrust and power coefficients
\(T\)	Thrust
\(Q\)	Propeller shaft torque
\(K_v\)	Motor speed constant
\(K_t\)	Motor torque constant
\(I\)	Motor winding current
\(I_0\)	No-load current
\(R_m\)	Motor winding resistance
\(R_{\text{system}}\)	Lumped system resistance
\(V_{\text{back}}\)	Motor back-EMF voltage
\(V_m\)	Motor terminal voltage
\(V_{\text{pack}}\)	Battery pack voltage

What the Solver Does

Step 1: Evaluate the propeller at a candidate RPM

For each candidate RPM, PyThrust converts shaft speed to revolutions per second:

\[ n = \frac{\text{RPM}}{60} \]

Then it calculates the propeller advance ratio:

\[ J = \frac{V}{nD} \]

The propeller database returns empirical coefficients at that RPM and advance ratio:

\[ C_t = C_t(\text{RPM}, J) \]

\[ C_p = C_p(\text{RPM}, J) \]

From those coefficients, PyThrust computes thrust, torque, and shaft power:

\[ T = C_t \rho n^2 D^4 \]

\[ Q = \frac{C_p \rho n^2 D^5}{2\pi} \]

\[ P_{\text{shaft}} = C_p \rho n^3 D^5 \]

Step 2: Evaluate the motor state

By default, PyThrust uses a first-order brushless DC motor model. The torque constant is:

\[ K_t = \frac{30}{\pi K_v} \]

The motor current needed to drive the propeller torque is:

\[ I = \frac{Q}{K_t} + I_0 \]

The back-EMF voltage is:

\[ V_{\text{back}} = \frac{\text{RPM}}{K_v} \]

The motor terminal voltage is:

\[ V_m = V_{\text{back}} + I R_m \]

Drela / QPROP Motor Model Notes

MotorSpec also includes optional parameters inspired by Mark Drela's QPROP motor models:

Field	Effect
`torque_constant_kv_ratio`	Adjusts the relation between speed constant and torque constant
`magnetic_lag_tau`	Adds magnetic lag to the back-EMF relation
`no_load_current_linear`	Adds a linear speed term to no-load current
`no_load_current_quadratic`	Adds a quadratic speed term to no-load current
`resistance_quadratic`	Adds current-dependent winding resistance
`iron_loss_exponent`	Scales no-load current with RPM using a power law

When these fields are left at their defaults, the model reduces to the simpler first-order motor equations above. See Mathematical Model for the full equations and Drela references.

Equilibrium RPM

The throttle command defines the average voltage available from the battery:

\[ V_{\text{applied}} = \text{throttle} \cdot V_{\text{pack}} \]

The solver searches for the RPM where the motor voltage demand plus system voltage drop equals the applied voltage:

\[ g(\text{RPM}) = V_m(\text{RPM}) + I(\text{RPM}) R_{\text{system}} - V_{\text{applied}} \]

The equilibrium point is:

\[ g(\text{RPM}) = 0 \]

PyThrust solves this scalar root-finding problem with Brent's method through scipy.optimize.root_scalar.

RPM Bracket

Before solving, PyThrust builds a search interval:

Bound	How it is chosen
Lower RPM	At least `SolverConfig.rpm_min`; raised if airspeed and propeller `J` range require a higher RPM
Upper RPM	Estimated from `motor.kv_rpm_per_v * battery.voltage_v * throttle`, then expanded by `rpm_max_margin`

If the residual does not change sign inside the bracket, the point is returned as infeasible with reason no_bracket.

Solver Configuration

The numerical behavior of the root finder is controlled by SolverConfig:

Parameter	Type	Default	Description
`rpm_min`	`float`	`100.0`	Lower bound limit for RPM
`rpm_max_margin`	`float`	`1.1`	Safety scaling factor applied to the upper RPM estimate
`eps_rpm`	`float`	`1e-8`	Convergence tolerance for shaft speed
`eps_v`	`float`	`1e-8`	Voltage residual tolerance
`max_iter`	`int`	`100`	Maximum root-finder iterations

Result Fields

solve_operating_point(...) returns an OperatingPoint:

Field	Meaning
`rpm`	Solved shaft speed
`advance_ratio`	Propeller advance ratio at the solved point
`ct`, `cp`	Interpolated thrust and power coefficients
`thrust_n`	Thrust force
`torque_nm`	Propeller shaft torque
`shaft_power_w`	Mechanical shaft power
`motor_power_w`	Electrical power at motor terminals
`battery_power_w`	Battery-side power including system losses
`motor_current_a`	Motor winding current
`motor_voltage_v`	Motor terminal voltage
`propeller_efficiency`	Propulsive efficiency based on thrust power
`motor_efficiency`	Shaft power divided by motor electrical power
`system_efficiency`	Thrust power divided by battery power
`is_feasible`	Whether the point passed feasibility checks
`infeasible_reason`	Reason string when `is_feasible` is false

Feasibility Rules

An operating point is marked as infeasible when one of these checks fails:

Reason	Meaning
`throttle<=0`	No positive throttle command
`no_bracket`	The RPM search interval does not contain a valid voltage-balance root
`no_convergence`	The root finder did not converge
`current_limit`	`motor_current_a` exceeds `current_max_a`
`invalid_coefficients`	Propeller coefficient lookup produced non-physical values
`invalid_efficiency`	Computed efficiency is outside the physically expected range

Example Usage

Here is a complete example showing how to load a propeller dataset, define specifications, and solve for an operating point:

from pathlib import Path

from pythrust.propellers import PropellerDatabase
from pythrust.propulsion import (
    BatterySpec,
    MotorSpec,
    PropellerSpec,
    PropulsionSolver,
    SystemSpec,
)

db = PropellerDatabase()
db.load(Path("data/propellers/apc_202602"), strict=False)
prop_entry = db.get("APC_13x6.5E")

motor = MotorSpec(
    kv_rpm_per_v=860.0,
    resistance_ohm=0.0258,
    no_load_current_a=1.3,
    current_max_a=65.0,
)

battery = BatterySpec(
    voltage_v=14.8,
    discharge_efficiency=1.0,
)

system = SystemSpec(resistance_ohm=0.05)
propeller = PropellerSpec(diameter_m=0.3302, blade_count=2)

solver = PropulsionSolver()
point = solver.solve_operating_point(
    motor=motor,
    battery=battery,
    system=system,
    propeller=propeller,
    prop_entry=prop_entry,
    rho=1.225,
    airspeed_mps=15.0,
    throttle=0.7,
)

print(f"RPM             : {point.rpm:.1f}")
print(f"Thrust          : {point.thrust_n:.2f} N")
print(f"Motor Current   : {point.motor_current_a:.2f} A")
print(f"Battery Power   : {point.battery_power_w:.1f} W")
print(f"Feasible        : {point.is_feasible}")
print(f"Reason          : {point.infeasible_reason}")