Propulsion matching

Why this matters

The propulsion set (motors, propellers, ESCs and battery) uses almost all of a drone’s energy. Good matching can greatly extend flight time without adding battery; poor matching overheats motors, burns out ESCs or leaves too little thrust to fight wind. This module ties modules 1–3 together.

Propeller coefficients

The University of Illinois propeller database (UIUC Propeller Database) describes performance with the dimensionless $C_T$ (thrust) and $C_P$ (power), where $n$ is revolutions per second and $D$ is diameter (m):

$T=C_T\rho n^2{D}^4,P=C_P\rho n^3{D}^5$

Thrust grows with the square of speed but power with the cube, so a large, slow propeller gives the same thrust for less power than a small, fast one.

import math

g, rho, mass = 9.81, 1.225, 3.2
thrust_each = mass * g / 4
ct, cp, eta, avionics_w, usable_wh = 0.10, 0.045, 0.80, 26, 222 * 0.8
for name, d in (("15 in", 0.381), ("13 in", 0.3302)):
    n = math.sqrt(thrust_each / (ct * rho * d ** 4))      # rev/s
    shaft_w = cp * rho * n ** 3 * d ** 5
    total_w = 4 * shaft_w / eta + avionics_w
    print(f"{name}: {n * 60:.0f} rpm, shaft {shaft_w:.1f} W per motor, total {total_w:.0f} W, "
          f"hover time ≈ {usable_wh / total_w * 60:.1f} min")

15 in: 3308 rpm, shaft 74.2 W per motor, total 397 W, hover time ≈ 26.8 min
13 in: 4405 rpm, shaft 85.6 W per motor, total 454 W, hover time ≈ 23.5 min

Checking the whole set

Criteria to check:

• Thrust reserve: total maximum thrust should exceed weight enough to climb and fight wind (see TWR in UAT 321)
• Speed headroom: hover speed should be well below the maximum the motor can reach at operating voltage. A 400 KV motor at 22.2 V reaches about 8,880 rpm without load, against 3,308 rpm needed to hover on the 15-inch propeller, leaving plenty of headroom
• Current: the maximum current of that motor and propeller must be below the ESC’s continuous rating with margin, and the pack must supply the total
• Heat: motors and ESCs need airflow in their actual mounting position

Tools such as eCalc xcopterCalc calculate the whole set from databases of motors, propellers and batteries. The developer states an accuracy of about ±15%, so use it to shortlist options, then confirm with a thrust stand and real flight logs.

Module lab

Common mistakes

Summary

• Thrust grows with $n^2{D}^4$ and power with $n^3{D}^5$, so large slow propellers save power
• Calculate the required speed from thrust, then power and flight time
• Check thrust reserve, speed headroom, current and heat before concluding
• Calculators have errors; confirm with real measurements

Check your understanding

1. If speed doubles, by how many times do thrust and power increase?
2. If diameter doubles at the same speed, by how many times does thrust increase?
3. A 2.0 kg quadcopter hovers. How much thrust per motor is needed?
4. A system uses 400 W in total with 160 Wh usable. About how many minutes can it hover?
5. What accuracy does eCalc state, and how should it be used?