Brushless motor selection for combat robot weapon and drive.

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Weapon Spin-up Time  

I frequently see over-simplified methods for calculating spinner weapon spin-up time given on forums and social media sites. Builders constructing combat weapons based on these methods would be disappointed in the performance of the resulting system.

I've written the following detailed explanation of the factors involved in estimating spinner weapon performance as a handy reference to which I can point interested builders.

Q: A joule of kinetic energy is equal to a watt-second of electrical output, so if I have a spinner weapon that stores 1200 joules of energy and I power it with a 400 watt brushless outrunner motor it should take 1200 joules ÷ 400 watts = 3 seconds to spin up, right? [Social Media]

A: This would be true if your '400 watt brushless motor' actually produced 400 watts of output power over the full speed range from zero to maximum RPM needed to spin-up a rotary weapon -- but this is not the case.

Some background:

Hobby brushless outrunner motors are typically intended for model aircraft use. They are often rated rated by their Maximum Continuous Power which relates to how large a propeller they can spin at full throttle without failing from thermal overload. This power level has the motor operating up high in its RPM range, close to the motor's peak efficiency. Spinning up a kinetic energy weapon on a combat robot places much different demands on a motor; a heavy load for a short period to bring the weapon up to speed but very little continuing load once it has reached top speed.
The Peak Power available from the motor is considerably greater than the Maximum Continuous Power, but can be maintained for only a short period without thermal damage to the motor. Peak power output is available when the load on the motor limits speed to 1/2 the unloaded motor RPM - see chart below. The power available from the motor at the high and low ends of the RPM range is much less than peak power. The average power available to spin up the weapon as the motor climbs from a dead stop to full speed is only a fraction of the 'peak power'. 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 4-cell 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
= Voltage2 ÷ 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 kg-m2 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 on-line javascript calculator: Click Here. The spinner spreadsheet calculates the spin-up 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 no-load RPM 1. At maximum no-load 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 no-load 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 one-step shortcut that estimates peak output power from just voltage and terminal resistance:
Peak Output (Watts)
= (Voltage2 ÷ Terminal Resistance) × 0.25

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