28.14. Useful equations

This is a section with various bits of math which are useful for calculating various parameters of helicopters.

28.14.1. Calculating headspeed

There are three steps to calculating the headspeed:

  1. Calculate the motor rpm

  2. Check the motor's maximum rated RPM.

  3. Calculate the gear ratio

In order to calculate the motor RPM, you need to know two things: the motor Kv constant, and the battery voltage.

The Kv constant is the no-load RPM/V, or basically the RPM of the motor at a given voltage with no load. If you multiply this value by the voltage, then the result is the motor speed at the voltage with no load.

As the helicopter takes off, the load on the motor will increase, so the motor RPM will drop by about 15%. So this unloaded RPM should be multiplied by about 0.85 to calculate the loaded motor RPM.

For example, a Hacker B50-18S has a Kv of 2006, so the no-load motor speed on 8 cells (9.6 volts) is: 2006 * 9.6 = 19257.6 RPM. Under load the motor RPM will drop to about 19257.6 * 0.85 = 16369 RPM.

The second step is to check the motor's maximum rated RPM. For the Hackers this is about 60,000 rpm, so 19257.6 RPM is well below the motor's maximum rated RPM.

The third step is to calculate the gear ratio. The ECO 8 has a 180 tooth main gear and if the motor pinion is 18 tooth, then the gear ratio will be 180:18 or 10:1.

The final headspeed will be the motor RPM multiplied by this gear ratio, multiplied by about 0.85 to simulate the loading effect of the main rotor blades.

So: 2006 (motor Kv) * 9.6 (battery voltage) * 0.85 = 16369 RPM motor speed

16369 * 18 (motor pinion) / 180 (main gear teeth) = 1636.9 RPM head speed

For a fixed pitch helicopter, you should plan to have double the hovering headspeed at full throttle so the helicopter will hover at approximately half stick.

[Note]Note

The 0.85 multiplier is the speed drop under load, which is a rough estimate. It is NOT an efficiency factor. If a motor is inefficient, it will usually consume more current rather than dropping RPM.

28.14.2. Estimating power consumption

R/C helicopters need about:

  • 100 watts/kg for hovering

  • 150 watts/kg for entering/exiting forward flight

  • 200 watts/kg for light aerobatics (loops/rolls)

  • 300+ watts/kg for serious 3D flight

(add up to 50% for higher headspeeds)

Since watts = voltage * current, we can divide the watts by voltage to estimate the amps required. For example, an ECO 8 weighs about 1.5 kg. From this, we can estimate the power at hover:

100 watts/kg * 1.5 kg = 150 watts

For a 10 cell nicad pack, this would be:

150 watts / 12 volts = about 12.5 amps

Note that the 100 watts/kg value is an average power draw during hovering. It is not safe to hover a heli with a 100 watt/kg power system, because extra power will be required to recover the heli safely if a gust of wind blows, etc. A realistic minimum power system would be about 150-200 watts/kg.

Power consumption is very dependent on motor and headspeed. For example:

  • ECO 8 w/Mega 16/25/3 at 1400 rpm HS: 105 watts/kg to hover

  • ECO 8 w/Mega 22/20/3 at 1600 rpm HS: 174 watts/kg to hover