Voltage and Stepper motors, Thoughts and why.

Part 1

Working with CNC and 3D printing I have often run into folks who disregard the need of proper use of voltage. It is a lesson I learned while refining machine function in CNC and I find that it applies to not only CNC but 3D printing as well.

So why is voltage important at all? I have heard time and again from online experts state that only current is important for stepper motors. That too much voltage is more trouble then it is worth.  While high voltage is something one needs to be aware of and deal with safely the greater risk is the current in a line for health and life reasons. Voltage on the other hand is very important to a stepper motor.

The rating on a Stepper motor has the base voltage which is the voltage needed to rotate the stepper on step. It has the current for the stepper, which will be different may be different if it is possible to wire it in series or parallel. The resistance will be per phase and at what temperature that resistance was measured (and the tolerance of the measurement). Lastly but not least it will have the inductance per phase (again with tolerance for measurement).

So going over basic electrical theory we have these laws to deal with:

V=IR   Where V = Voltage, I = Current in Amps, and R = resistance in Ohms

That allows us to extract other needed equations dealing with current and resistance. The next law also has a very important effect on the motor and is what causes the direct link between I and V in functional usefulness.

The next equations allow us to deal with the inductance and how that effects the voltage and current usable to the motor. First is getting the back emf off of a motor for a set value:

V_l = -L(di/dt) Where V_L equals the back emf voltage and -L equals the negative effect in Henries multiplied by the change in current (di) divided by the time it takes (dt)

From here we can add the following equation that will effect what voltage we need.

L = V_l /(di/Dt) Which is an extraction of the equation above.

Now inductance is an issue for a stepper motor as it the RESISTANCE to CHANGE in CURRENT. That will directly effect how well a stepper motor is able to deal with steps per second and the amount of torque that can be applied by the stepper via said force.

Now it is also known that a stepper motor has a decay rate and a maximum rpm. That there is a point at approximately 40 to 60% of maximum rpm where you lose enough torque to significantly increase the risk of stalling and loss steps. So the inductance tells us what Voltage a given motor needs to be able to get the maximum effective torque for the minimum amount of time. As this voltage will inhibit loading of the coils and insure that the current arrives and drops as needed. The equation for that comes in two forms depending on where in the world you learned it. They are as follows

32 * √L (disregarding the mH and using the number without accounting for the mH). An example is a stepper with 3.2 µH, the equation would read 32*√3.2.

The other option with this equation is:

1000*√L (accounting for the micro). So this equation would look like 1000*√0.0032.

Realize that any voltage above to the base and below the optimum can be used and the motors will function. What will happen is you will lose torque and have a quicker decrease in the amount of micro stepping that can be effectively achieved.

For CNC this directly effects the ability to cut and effectively use tooling at optimum speeds and feeds. Which directly effects what materials you can work and the finish you will from the cutting tools.

For 3D printing it directly effects micro stepping and accuracy/smoothness in the motion of the printer. Which directly effects surface finish and accuracy of material placement. So both sides stand to win a good bit by taking the time to give the motors the voltage that will allow them to run smoother. the interesting point is that running a motor at this voltage I have found that unless under heavy or sustained loads they tend to run cooler than running at lower voltages. The most defined difference I have seen is going from less than half optimum voltage to within 10% of optimum.

Let's look at three different stepper motors and the equations that will govern optimum voltage and what that means to either CNC or 3D printing.

Motor number one is common to CNC and has the following information:

3Nm Nema 23 Bipolar motor

Bipolar wiring with 2 phases
Degree per step: 1.8°
Rated voltage: 2.73V (Bipolar, Parallel), 5.46V (Serial Bipolar), 3.9V (Unipolar) (Minimum Voltage to cause function of the motor)
Rated current: 2.1 Amps per Phase (4.2 Amps when wired in parallel bipolar) 3A Unipolar
Resistance: 0.65 Ohms
Inductance: 12.8 microHenries (Serial Bipolar) and  3.2 microHenries (Unipolar and Parallel Bipolar)
Nm : 2.15 when in serial and 3.0 in parallel.

Now comes an interesting point on Voltage by numerous sellers is recommend to be between 48-80V.

Let's get into the why of that and from their we can build a picture of the uses intended and shift that to a 3D printer which will have some different demands but a stepper motor still deals with step and direction as it's main function.

Using either 32*√(L) or 1000*√(L). Voltage of a stepper motor) we get the following optimum voltages.
Serial Bipolar then has a required Voltage of 32*√(12.8) or 1000*√(0.0128). Which gives us the spread of 113.137 to 114.4867 Volts.  With a current of 2A So that sets up one option.

Now for Parallel Bipolar or Unipolar set up we have a different set of numbers 32*√3.2 or 1000*√0.0032.  This gives us a spread of 56.57 to 57.24 Volts. with a current of 3A for Unipolar and 4.2A for bipolar parallel. Remember this current is per phase so you are shifting through the arch when dealing with current, this is addressed below.

The most popular option in small CNC and DIY is running in parallel bipolar as it give you the most torque for the energy supplied.  Many in 3D printing will tell you that is not needed for 3D printing as you are not dealing with the resistive forces that cutting creates.  I would point out that same torque allows smoother changes in direction and real access to micro stepping without the increased risk of loss steps which translates into misalignment of layers loss of integrity of the model at worse. 

Now let's look at how that Voltage effects both systems directly. That area is steps per second available at a given voltage with a constant current and impedance. The equation for that is T = IL/V.

Where: T = time per step in seconds or parts of.
             I = 2*Imax (Imax is supplied current and doubled to account for a full cycle)
             L = Inductance measured in microHenries
             V = Voltage applied

T1 = 1/T and gives you Steps per second. So let's for now work with the 56 to 57 V power range and see where using different voltages will get us.

12V which many folks entering DIY or trying to get into light industrial on the cheap will play with, can be our starting point for this exercise. So here it goes:

T = (8.4*3.2 E-3)/12; T = 2.24 E-3; T1 = 446.43 Steps/sec = 892.86 Pulses/sec
T = (8.4*3.2 E-3)/24; T = 1.12 E-3; T1 = 892.86 Steps/sec = 1,785.71 Pulses/sec
T = (8.4*3.2 E-3)/48; T = 560.0 E-6; T1 = 1,785.71 Steps/sec = 3,571.43 Pulses/sec
T = (8.4*3.2 E-3)/56; T = 480.00 E-6; T1 =2,083.33 Steps = 4,166.66 Pulses/sec

Now how does the above directly effect production and movement?

Available motion is measured as a percentage of maximum rpm. Due to the decay and torque cut off point. Also at this point the mass of the drive system comes into play and the effects of backlash start becoming much more an issue

At this point I am not going to go into the physics side of the mass of the screw and/or loading stretch with belts and the different levels of backlash each system has. That is for another discussion and is widely debated on the internet.  Though among mechanical engineers and those who work with drive systems daily the normally accepted order of hierarchy for tolerances and accuracy in descending order is Ballscrews, Leadscrews (trapezoid/ACME not threaded rod), Belt drives. Realize that all of these drive trains have their uses and are used as different system benefit from each depending on cost, accuracy required, levels of maintenance and lifespan desired, etc.

Now take the fact that most motors work best between 40-60% of their maximum rpm and you can rapidly see why using voltages closer to optimum yield much better control and finer motion. as the difference in real motion ability between 12V and 48V is 357.144 steps/sec or 833.33 steps/sec. Consider that a stepper has 200 steps per rotation and you can see that lower voltage directly effects both speed and micro stepping. 


The next motor has this information (which is a very common motor in 3D printing):

54.374 Ncm Nema 17 motor
2 phase
Degree per step: 1.8°
Rated voltage: 2.77 Volts (V)
Rated current: 1.68 Amps (A)
Resistance: 1.65 Ohms
Inductance: 2.8 mH/Phase

Optimum voltage is between 52.915 and 53.546V. From this it is easy to see that 12V to 24V PSU to the motors is going to cripple the motors in not only removing the ability to functionally do microstepping effectively, but also increasing the risk of step loss at any serious level of motion. A 48V power supply with a healthy smoothing ability to insure constant and clean current to the motors would more than pay for itself in improvement in speed and accuracy.

The last motor is to help show that this is not only 1.8° step motors that have this issue.

40 Ncm Nema 17 Motor
2 Phase
Degree per step: 0.9°
Rated voltage: 3.1 V
Rated current: 1.7 A
Resistance: 1.8 Ohms/Phase
Inductance: 4.0 mH/Phase

Optimum voltage for this motor ranges between 63.25 an 64 Volts. Add in that you have 400 steps per rotation and the need to get closer to the optimum voltage becomes even more important.  With even higher benefits as well as the decay arch is sharper with faster loss as these motors often have a torque curve that stops below 40% of maximum.

Also remember that you extruder motors are using torque to move the filament and that those with reduction gears are even more dependent on being at or near optimum voltage as that is where the greatest amount of torque for them to the job is located.

Hopefully this has helped folks and will go into some more application for both 3D printing and CNC in Part 2

Additional reading for those who are interested:
From Gecko Drivers