A further complication comes from the brushless motor controller software. Unsensored brushless motor contollers need a bit of time to figure out the power pulse timing they must send to the motor to get it spinning in the right direction and speed. During this period the controller delivers a limited amount of current to the motor, which greatly reduces low speed motor torque and prolongs the spin up time.
I'll use the PropDrive 2826 1000KV Brushless Outrunner Motor as an example. The motor is rated 235 watts continuous output on a 4cell LiPo battery and has a stator resistance of 0.140 ohm. The estimated 'peak' power output is very close to 400 watts:
Peak Output Watts
= Voltage^{2} ÷ Resistance × 0.25
= 15 × 15 ÷ 0.14 = ~400 Watts
Using the PropDrive 2826 motor with a 2:1 belt reduction to spin a steel weapon bar 41mm x 12mm x 240mm (0.0045 kgm^{2} Moment of Inertia) to 7,000 RPM (95% of peak motor RPM) gives very close to 1200 joules of stored kinetic energy.
OK, how do you track the effect of these fluctuating power levels to find out how long it takes to spin up the weapon? I've written a pair of tools to do the calculations for you! Your choice of a sophisticated Excel spreadsheet or a simplified online javascript calculator: Click Here.
The spinner spreadsheet calculates the spinup time within a blink of six seconds  remarkably close to twice the simple 'joules ÷ peak watts' figure.
Q: Wait... What's that formula you used to calculate Peak Output Watts? Where did that come from?
A: At stall a Permanent Magnet Direct Current (PMDC) motor produces maximum torque and will consume:
Power Consumption at Stall (Watts) = Voltage × Stall Amperage
The mechanical power output of a motor is a product of torque and RPM. At stall, a PMDC motor produces zero mechanical power: the torque is at maximum but RPM is zero. PMDC motor torque and amperage consumption decrease linearly to approach zero at noload RPM ^{1}. At maximum noload RPM the motor again produces zero mechanical power: the RPM is at maximum, but the torque is zero.
The product of torque and RPM is power, and power reaches maximum at 50% of the noload RPM where the motor will consume very close to 50% of the stall amperage:
Power Consumption at Peak Output (Watts)
= Voltage × (50% of Stall Amperage)
The efficiency of a PMDC motor in converting electrical power to mechanical power varies with motor design, materials, and RPM. Approximately 50% efficiency at peak output is typical. This gives an estimate of power output as:
Peak Output (Watts) = Voltage × (50% of Stall Amperage) × (50% Efficiency)
The above equation simplifies to the equation I use to estimate peak motor output power:
Peak Output (Watts)
= Voltage × Stall Amperage × 0.25
Brushless motors do not specify a stall amperage figure because it is the brushless motor controller that determines current flow at low motor speeds. However, brushless motor specs do commonly provide a Terminal Resistance that allows you to calculate a theoretical stall amperage:
Stall Amperage
= Voltage ÷ Terminal Resistance
A little algebraic substitution combines the two equations above to provide a onestep shortcut that estimates peak output power from just voltage and terminal resistance:
Peak Output (Watts)
= (Voltage^{2} ÷ Terminal Resistance) × 0.25
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