Most commercially available PMBDC drives are sinusoidal – not six-step.
Select figure to enlarge.

Using the trigonometric identity sin(a + b) = sina·cosb + cosa·sinb twice, simplifying, and recognizing that sin2(θ)+cos2(θ) = 1 gives:

and

Where R = Motor resistance, Ω

L = Line-neutral inductance, henry

P = Number of rotor poles

ω = Motor's velocity (radian/sec)

Continuous torque output and power efficiency are maximized when each phase current is kept in phase with the motor's corresponding back emf/torque function — for example, when ϕ = 0. Second, for a truly sinusoidal motor-drive combination, the phase difference between each phase current and its corresponding back emf/torque function causes the motor to output less than maximum continuous torque — for example, when ϕ is greater than 0 and cosine(ϕ) is less than 1. That said, there is no drive-induced torque ripple.

This differs significantly from an ideal trapezoidal motor controlled by a six-step drive — because a phase difference between the phase current and its corresponding torque function causes position-dependent torque ripple. Furthermore, a six-step drive powering a real PMBDC motor always produces drive induced torque ripple — even if the phase current is kept in phase with its corresponding torque function as shown in Figs. 4, 5, and 6.

Hence, if a design requires ripple free torque output (neglecting rotor magnet cogging) then the best choice is a true sinusoidal motor-drive combination.

Finally, the equation for ϕ shows that a motor's electrical inductance L, resistance R, and number of magnet poles on its rotor P along with motor velocity all combine to produce phase shift between a phase current and its corresponding back emf/torque function. The equation also shows that the phase shift is an inverse tangent function that increases with increasing motor velocity ω.

For this reason, maximum continuous torque output and power efficiency can be a major issue for a high-inductance, high pole count PMBDC motor operating at high velocity — unless the drive electronics incorporate current phase advance into control algorithms.

The first installment of this series can be found at motionsystemdesign.com. For more information, visit www.renco.com or call (805) 968-1525.

In this installment

• Real dc motors and output
• Models and ideal operation
• Feedback and sensor options