Why do VR motors like sinusoidal current?

I have a question regarding powering electrical motors, variable reluctance motors to be more precise (ie. Stepper motors). This kind of motor, in a perfect situation would like to be powered with current thats shape is a sin wave and a cosine wave (for stepper). Even when doing microstepping, one should try to follow the shape of sin and cos to have the smoothest rotor rotation. Also if VRM was used as generator, it would generate sinusoidal current. This all seems logical, looking at the circle construction of the machine. I believe if one was to unroll the vr motor and make it drive in a plane motion way, it wouldnt have to be powered with sin/cos and also wouldnt generate current in that shape. My question is- would someone please explain me in a more "fact and proof" way why does the current shape has to be sinusoidal for a vr motor? Also, please correct me if anything i wrote was not true. I would apreciate all help!

• To be short: when you sum the magnetic fields created from 3 phases you get a constant rotating field that it's amplitude is 1.5x of phase amplitede. – Marko Buršič May 27 '16 at 8:46

Because a sinusoidal current will make the torque applied to the motor constant, whetever the position of the shaft.

Let's say we have winding A and B, and supply current I. If you apply a square current to both windings, you'll have:

• a phase where a current I is applied to winding A, and no current in winding B
• a phase where a current I is applied for both A and B, resulting in more power being applied to the motor then in the previous phase
• a phase where a current I is applied to winding B, and no current to winding A

So you see, the power given to the motor varies (sometimes I, sometimes 2*I), depending on the position of the shaft, so it will not run smoothly.

To solve this, you have to apply sinusoidal currents. Then, even when both winding are powered, the torque is kept constant.

Why a sine and not a triangle wave ?

Think of what you need to provide when the shaft is at 45° (both windings powered): the power provided to the motor is twice V*I (once for each winding). V and I are linked by Ohm's law (if we assume steady state and rule out inductance). So the power is twice RI². Whereas when the shaft is at 0°, the power is only once RI². Now, how to make this constant ? By setting the I at 45° to sqrt(0.5)*I at 0°. Indeed : 2*R*(sqrt(0.5)I)² = RI². Now, what is sqrt(0.5) ? It is cos(45°). So you can start visualising the sine wave. Basically, what is required is that I² in winding A + I² in winding B is constant. That is the definition of sin and cos.

And yes, basically, it comes from the fact that the motor turns in circle.

• I understand the concept of commutation, but the question is why does it have to be sine? I can apply triangle the same way as sine and cosine but the motor movement wont be const. I believe it has to be sine because of the circle construction of the motor. I just dont know how to describe it. – Łukasz Przeniosło May 27 '16 at 6:55
• @ŁukaszPrzeniosło I edited the answer to explain this – dim May 27 '16 at 7:25
• Thank you for edit. At the moment I am driving a stepper motor in the exact way you have described: sin(I^2) + cos(I^2) = 1. Using this equasion current will always be "constant". But one also has to distiguish magnetic field angle and physical angle, which also is a bit confusing for me. What we talk about here is in regard of magnetic angles and physically motor make only 4 steps in a full sine wave coarse. Could you reffer to that? – Łukasz Przeniosło May 27 '16 at 7:36
• I'm sorry, I don't really understand what you asked here. What is true is that the 45° / 0° angle is not actually real. Because there are much more than just two windings in the motor (a stepper does not do a full rotation in 4 coarse steps, it usually has much more steps), it is just a simplified view. But the principle is the same, you just have to divide the angle by the actual number of A/B windings. Is that what you asked ? – dim May 27 '16 at 7:47
• I understand this as well. By saying that full sine wave is needed to do 4 steps, I means 4 out of 200 total steps of the motor. Sorry, I forgot to add that. In this case to make a full physical rotation of the rotor, one has to provide 50 full sine waves (if the motor has 200 steps in total). So full sine wave gives 360 degrees of magnetic rotation, but only 7.2 degrees of physical rotation. Even though the physical rotation is very little, current is still aplied as full sine wave- Full magnetic rotation for only little physical rotation, how to explain this? – Łukasz Przeniosło May 27 '16 at 7:54