import aerosandbox.numpy as np
from aerosandbox.tools import units as u
from aerosandbox.performance.operating_point import OperatingPoint
from typing import Union, Dict
[docs]def electric_propeller_propulsion_analysis(
total_thrust: float,
n_engines: int,
propeller_diameter: float,
op_point: OperatingPoint,
motor_kv: float,
motor_no_load_current: float,
motor_resistance: float,
wire_resistance: float,
battery_voltage: float,
propeller_tip_mach: float = 0.50,
gearbox_ratio: float = 1,
gearbox_efficiency: float = 1,
esc_efficiency: float = 0.98,
battery_discharge_efficiency: float = 0.985,
) -> Dict[str, float]:
"""
Performs a propulsion analysis for an electric propeller-driven aircraft.
May be used for single-engine or multi-engine aircraft, so long as all engines / propellers are identical.
Args:
total_thrust: Total thrust force produced by all engines at the cruise operating point [N].
n_engines: Number of engines on the aircraft.
propeller_diameter: Diameter of each of the propellers [m].
op_point: The cruise operating point. Must be an AeroSandbox OperatingPoint object.
motor_kv: Motor voltage constant [rpm/volt].
motor_no_load_current: Motor no-load current [amps].
motor_resistance: Motor resistance [ohms].
wire_resistance: Round-trip resistance of the wires connecting the ESC to the battery [ohms].
battery_voltage: Battery voltage [volts].
propeller_tip_mach: Mach number at the propeller tip. Defaults to 0.50. From a propulsive efficiency
perspective, you want this to be as high as possible while still keeping the tip speed (hypotenuse of the
velocity triangle) below the critical Mach number of the propeller blade airfoil. This is because motor
efficiency and specific power tend to be better at high-speed low-torque conditions, and also the propeller
aerodynamics tend to be better at low solidity. But there may be reasons to lower this, such as propeller
structural considerations or noise considerations (with noise being a *strong* function of tip Mach).
gearbox_ratio: Gearbox reduction ratio. Defaults to 1 (no gearbox). For example, a `gearbox_ratio` of 5 is a 5:1
reduction, meaning that the propeller turns 5 times slower than the motor.
gearbox_efficiency: Gearbox efficiency. Defaults to 1, only because the `gearbox_ratio` defaults to 1 (no
gearbox), and so this represents no losses. If you have a gearbox, you should probably use a value of 0.98 or
so.
esc_efficiency: Efficiency of the electronic speed controller (ESC), sometimes called the inverter. Defaults to
0.98, which is a reasonable value for a high-quality ESC at a large (>5 kW) scale. Small components will
lower efficiencies than this.
battery_discharge_efficiency: Coulobmic efficiency of the battery in discharge only (i.e., not round-trip).
Defaults to 0.985, which is a reasonable value for a high-quality lithium-polymer battery. Other battery
chemistries will have different values.
Returns: A dictionary of various parameters of the propulsion analysis. Of particular note are the following keys:
* "air_power": The power delivered to the air (thrust * velocity) [W]
* "shaft_power": The power at the propeller shaft (after the gearbox; rotational speed * torque) [W]
* "motor_electrical_power": The electrical power input to the motor [W]
* "esc_electrical_power": The electrical power input to the ESC [W]
* "battery_power": The power draw from the battery [W].
* "propeller_efficiency": The propulsive efficiency of the propeller, defined as (air_power / shaft_power).
* "motor_efficiency": The efficiency of the motor, defined as (shaft_power / motor_electrical_power).
* "overall_efficiency": The overall efficiency of the propulsion system, defined as (air_power / battery_power).
"""
### Propeller Analysis
propulsive_area_per_propeller = (np.pi / 4) * propeller_diameter ** 2
propulsive_area_total = propulsive_area_per_propeller * n_engines
propeller_wake_dynamic_pressure = op_point.dynamic_pressure() + total_thrust / propulsive_area_total
propeller_wake_velocity = (
# Derived from the above pressure jump relation, with adjustments to avoid singularity at zero velocity
2 * total_thrust / (propulsive_area_total * op_point.atmosphere.density())
+ op_point.velocity ** 2
) ** 0.5
propeller_tip_speed = propeller_tip_mach * op_point.atmosphere.speed_of_sound()
propeller_rads_per_sec = propeller_tip_speed / (propeller_diameter / 2)
propeller_rpm = propeller_rads_per_sec * 60 / (2 * np.pi)
propeller_advance_ratio = op_point.velocity / propeller_tip_speed
air_power = total_thrust * op_point.velocity
from aerosandbox.library.propulsion_propeller import propeller_shaft_power_from_thrust
shaft_power = propeller_shaft_power_from_thrust(
thrust_force=total_thrust,
area_propulsive=propulsive_area_total,
airspeed=op_point.velocity,
rho=op_point.atmosphere.density(),
propeller_coefficient_of_performance=0.90,
)
propeller_efficiency = air_power / shaft_power
### Motor Analysis
motor_rpm = propeller_rpm / gearbox_ratio
motor_rads_per_sec = motor_rpm * 2 * np.pi / 60
motor_torque_per_motor = shaft_power / n_engines / motor_rads_per_sec / gearbox_efficiency
motor_parameters_per_motor = motor_electric_performance(
rpm=motor_rpm,
torque=motor_torque_per_motor,
kv=motor_kv,
no_load_current=motor_no_load_current,
resistance=motor_resistance,
)
motor_electrical_power = motor_parameters_per_motor["electrical power"] * n_engines
motor_efficiency = shaft_power / motor_electrical_power
### ESC Analysis
esc_electrical_power = motor_electrical_power / esc_efficiency
### Wire Analysis
wire_power_loss = (esc_electrical_power / battery_voltage) ** 2 * wire_resistance
wire_efficiency = esc_electrical_power / (esc_electrical_power + wire_power_loss)
### Battery Analysis
battery_power = (esc_electrical_power + wire_power_loss) / battery_discharge_efficiency
battery_current = battery_power / battery_voltage
### Overall
overall_efficiency = air_power / battery_power
return locals()
[docs]def motor_resistance_from_no_load_current(
no_load_current
):
"""
Estimates the internal resistance of a motor from its no_load_current. Gates quotes R^2=0.93 for this model.
Source: Gates, et. al., "Combined Trajectory, Propulsion, and Battery Mass Optimization for Solar-Regen..."
https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=3932&context=facpub
Args:
no_load_current: No-load current [amps]
Returns:
motor internal resistance [ohms]
"""
return 0.0467 * no_load_current ** -1.892
[docs]def mass_ESC(
max_power,
):
"""
Estimates the mass of an ESC.
Informal correlation I did to Hobbyking ESCs in the 8S LiPo, 100A range
Args:
max_power: maximum power [W]
Returns:
estimated ESC mass [kg]
"""
return 2.38e-5 * max_power
[docs]def mass_battery_pack(
battery_capacity_Wh,
battery_cell_specific_energy_Wh_kg=240,
battery_pack_cell_fraction=0.7,
):
"""
Estimates the mass of a lithium-polymer battery.
Args:
battery_capacity_Wh: Battery capacity, in Watt-hours [W*h]
battery_cell_specific_energy: Specific energy of the battery at the CELL level [W*h/kg]
battery_pack_cell_fraction: Fraction of the battery pack that is cells, by weight.
* Note: Ed Lovelace, a battery engineer for Aurora Flight Sciences, gives this figure as 0.70 in a Feb.
2020 presentation for MIT 16.82
Returns:
Estimated battery mass [kg]
"""
return battery_capacity_Wh / battery_cell_specific_energy_Wh_kg / battery_pack_cell_fraction
[docs]def mass_motor_electric(
max_power,
kv_rpm_volt=1000, # This is in rpm/volt, not rads/sec/volt!
voltage=20,
method="hobbyking"
):
"""
Estimates the mass of a brushless DC electric motor.
Curve fit to scraped Hobbyking BLDC motor data as of 2/24/2020.
Estimated range of validity: 50 < max_power < 10000
Args:
max_power (float): maximum power [W]
kv_rpm_volt (float): Voltage constant of the motor, measured in rpm/volt, not rads/sec/volt! [rpm/volt]
voltage (float): Operating voltage of the motor [V]
method (str): method to use. "burton", "hobbyking", or "astroflight" (increasing level of detail).
* Burton source: https://dspace.mit.edu/handle/1721.1/112414
* Hobbyking source: C:\Projects\GitHub\MotorScraper,
* Astroflight source: Gates, et. al., "Combined Trajectory, Propulsion, and Battery Mass Optimization for
Solar-Regen..." https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=3932&context=facpub
* Validity claimed from 1.5 kW to 15 kW, kv from 32 to 1355.
Returns:
Estimated motor mass [kg]
"""
if method == "burton":
return max_power / 4128 # Less sophisticated model. 95% CI (3992, 4263), R^2 = 0.866
elif method == "hobbyking":
return 10 ** (0.8205 * np.log10(max_power) - 3.155) # More sophisticated model
elif method == "astroflight":
max_current = max_power / voltage
return 2.464 * max_current / kv_rpm_volt + 0.368 # Even more sophisticated model
[docs]def mass_wires(
wire_length,
max_current,
allowable_voltage_drop,
material="aluminum",
insulated=True,
max_voltage=600,
wire_packing_factor=1,
insulator_density=1700,
insulator_dielectric_strength=12e6,
insulator_min_thickness=0.2e-3, # silicone wire
return_dict: bool = False
):
"""
Estimates the mass of wires used for power transmission.
Materials data from: https://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity#Resistivity-density_product
All data measured at STP; beware, as this data (especially resistivity) can be a strong function of temperature.
Args:
wire_length (float): Length of the wire [m]
max_current (float): Max current of the wire [Amps]
allowable_voltage_drop (float): How much is the voltage allowed to drop along the wire?
material (str): Conductive material of the wire ("aluminum"). Determines density and resistivity. One of:
* "sodium"
* "lithium"
* "calcium"
* "potassium"
* "beryllium"
* "aluminum"
* "magnesium"
* "copper"
* "silver"
* "gold"
* "iron"
insulated (bool): Should we add the mass of the wire's insulator coating? Usually you'll want to leave this True.
max_voltage (float): Maximum allowable voltage (used for sizing insulator). 600 is a common off-the-shelf rating.
wire_packing_factor (float): What fraction of the enclosed cross section is conductor? This is 1 for solid wire,
and less for stranded wire.
insulator_density (float): Density of the insulator [kg/m^3]
insulator_dielectric_strength (float): Dielectric strength of the insulator [V/m]. The default value of 12e6 corresponds
to rubber.
insulator_min_thickness (float): Minimum thickness of the insulator [m]. This is essentially a gauge limit.
The default value is 0.2 mm.
return_dict (bool): If True, returns a dictionary of all local variables. If False, just returns the wire
mass as a float. Defaults to False.
Returns: If `return_dict` is False (default), returns the wire mass as a single number. If `return_dict` is True,
returns a dictionary of all local variables.
"""
if material == "sodium": # highly reactive with water & oxygen, low physical strength
density = 970 # kg/m^3
resistivity = 47.7e-9 # ohm-meters
elif material == "lithium": # highly reactive with water & oxygen, low physical strength
density = 530 # kg/m^3
resistivity = 92.8e-9 # ohm-meters
elif material == "calcium": # highly reactive with water & oxygen, low physical strength
density = 1550 # kg/m^3
resistivity = 33.6e-9 # ohm-meters
elif material == "potassium": # highly reactive with water & oxygen, low physical strength
density = 890 # kg/m^3
resistivity = 72.0e-9 # ohm-meters
elif material == "beryllium": # toxic, brittle
density = 1850 # kg/m^3
resistivity = 35.6e-9 # ohm-meters
elif material == "aluminum":
density = 2700 # kg/m^3
resistivity = 26.50e-9 # ohm-meters
elif material == "magnesium": # worse specific conductivity than aluminum
density = 1740 # kg/m^3
resistivity = 43.90e-9 # ohm-meters
elif material == "copper": # worse specific conductivity than aluminum, moderately expensive
density = 8960 # kg/m^3
resistivity = 16.78e-9 # ohm-meters
elif material == "silver": # worse specific conductivity than aluminum, expensive
density = 10490 # kg/m^3
resistivity = 15.87e-9 # ohm-meters
elif material == "gold": # worse specific conductivity than aluminum, very expensive
density = 19300 # kg/m^3
resistivity = 22.14e-9 # ohm-meters
elif material == "iron": # worse specific conductivity than aluminum
density = 7874 # kg/m^3
resistivity = 96.1e-9 # ohm-meters
else:
raise ValueError("Bad value of 'material'!")
# Conductor mass
resistance = allowable_voltage_drop / max_current
area_conductor = resistivity * wire_length / resistance
volume_conductor = area_conductor * wire_length
mass_conductor = volume_conductor * density
# Insulator mass
if insulated:
insulator_thickness = np.softmax(
4.0 * max_voltage / insulator_dielectric_strength,
insulator_min_thickness,
softness=0.005 * u.inch,
)
radius_conductor = (area_conductor / wire_packing_factor / np.pi) ** 0.5
radius_insulator = radius_conductor + insulator_thickness
area_insulator = np.pi * radius_insulator ** 2 - area_conductor
volume_insulator = area_insulator * wire_length
mass_insulator = insulator_density * volume_insulator
else:
mass_insulator = 0
# Total them up
mass_total = mass_conductor + mass_insulator
if return_dict:
return locals()
else:
return mass_total
if __name__ == '__main__':
print(motor_electric_performance(
rpm=100,
current=3
))
print(motor_electric_performance(
rpm=4700,
torque=0.02482817
))
print(
mass_battery_pack(100)
)
[docs] pows = np.logspace(2, 5, 300)
mass_mot_burton = mass_motor_electric(pows, method="burton")
mass_mot_hobbyking = mass_motor_electric(pows, method="hobbyking")
mass_mot_astroflight = mass_motor_electric(pows, method="astroflight")
import matplotlib.pyplot as plt
import aerosandbox.tools.pretty_plots as p
fig, ax = plt.subplots(1, 1, figsize=(6.4, 4.8), dpi=200)
plt.loglog(pows, np.array(mass_mot_burton), "-", label="Burton Model")
plt.plot(pows, np.array(mass_mot_hobbyking), "--", label="Hobbyking Model")
plt.plot(pows, np.array(mass_mot_astroflight), "-.", label="Astroflight Model")
p.show_plot(
"Small Electric Motor Mass Models\n(500 kv, 100 V)",
"Motor Power [W]",
"Motor Mass [kg]"
)
print(mass_wires(
wire_length=1,
max_current=100,
allowable_voltage_drop=1,
material="aluminum"
))