# Stepper Motor Max RPM

I dabble in CNC as a hobby. Like all physics based hobbies noobs frequently do not understand why their projects perform poorly and generally refuse to believe the answer is as simple as component matching. A constantly recurring scenario is poor stepper motor performance....Stepper motors are typically marketed, sold and purchased almost exclusively on "Holding Torque", a specification that is all but meaningless in determining the performance of any given stepper motor in any particular system....but is none-the-less the guiding star that drives the market.

In the past year or two the proliferation of various NEMA23 425 oz-in Stepper Motors from China has caused a seemingly endless barrage of forum posts and PMs from DIYers who are desperate to "fix" their systems. In an effort to help some of these people understand WHY they are getting abysmal performance from their 425oz-in stepper motors I invariably end up attempting to give a physics lesson....but sadly simply stating that L= di/dt never seems to help; so, I end up having to play 50 questions with them to determine exactly which of the 425oz-in variants they have, what type of driver they are using and then finally what voltage power supply they are using.....All too frequently the answer is a > 4mH motor with a china-cheap TB6600 based driver and a 24V or 36V power supply.... as an example:

http://www.ebay.com/itm/US-Ship-3-A...r-Hobby-CNC-/231603783895?hash=item35eca948d7

In this case the motors are 5mH with 1.2ohms Resistance....there are several others, (one common one is 6.8mH) ....in any case, attempting to explain to people that the inductance of the motor AND the voltage of the power supply impose limits on the maximum theoretical RPM is generally met with skepticism....Presenting them with a link to HyperPhysics ( http://hyperphysics.phy-astr.gsu.edu/hbase/electric/indtra.html ) has yet to have the intended effect, so I have come up with an alternative solution that a few people have at least looked at before they dismiss the notion that physics is not just theory.....The assumption in the following is that the motor RPM is limited by the time it takes to reach the motor's rated current at any given supply Voltage.... and is solved by a simple re-arrangement of the formula given in the HyperPhysics link:

Code:
Given:
i = Vb/R * ( 1 - e^(-tR/L)
Where:
i = Motor Current (Amps)
Vb = Power Supply Voltage (Volts)
R = Coil Resistance (Ohms)
L = Motor Inductance (Henrys)
t = Time (Seconds)........(we will solve for t = Sec/Step)

AND assuming we know:

Steps/Rev
Motor Inductance
Motor Design Current
Coil Resistance
Power Supply Voltage
Seconds/Minutes

We can Solve for the maximum Theoretical motor RPM:

RPM = (Rev/200 Steps) * (Step/Sec) * (60 Sec/Min) ==>  60/(200 * t )

AND putting them together and solving for t

t = -L/R * ln(1-iR/Vb)

We Get:

RPM = 0.3 * 1 / (-L/R * ln(1-iR/Vb)

For the specific Case of:
Steps/Rev = 200
L = 5.0mH
i = 3.5A
Vb = 36V
R = 1.2 ohms

==> 0.3 *  1/((-0.005/1.2) * ln(1 - 3.5 * 1.2/36)) = 580 RPM

So the Theoretical maximum RPM for the Stepper using a 36V Power Supply AND reaching the rated current of 3.5A is only 580 RPM.....I am NOT saying a 36V supply will not spin one of these motors faster than 580RPM, but the torque will drop-off very rapidly above the 580RPM point. Same math for a 48V supply yields ~770RPM, and a 60V supply yields 975RPM while a 24V supply only yields 362RPM.

Other factors that limit the RPM of stepper motors include Rotor/Winding heating and magnetic/iron losses in the rotor core.....my point is not to suggest that the only limitation on stepper RPM is Voltage, but rather to demonstrate how using a low Voltage supply can severely limit stepper performance long before the other limiting factors come into play.

Please feel free to comment on ways to further clarify the relationship of Supply Voltage on Stepper performance....or point out any errors in my math/logic.

Thanks!

Fish
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Fish4Fun
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