In the last installment of this two-part series, we explored how permanent-magnet brushless dc servomotors (PMBDCs) provide the highest continuous torque density and efficiency of any electric motors.

To actually get top performance from these motors, though, and keep each phase-current input coordinated with the motor's corresponding back emf/torque function, proper feedback is required.

### Real PMBDC motors

Until now, we have assumed that our motor is an ideal three-phase, wye connected PMBDC motor, with perfect back emf/torque functions. In reality, it is difficult (if not impossible) to construct a motor with ideal back emf. Manufacturers can only construct motors that produce phase-neutral back emf that, when plotted, looks trapezoidal — but not with 120 electrical-degree flat areas at the top and bottom.

Instead, real PMBDC motor plots have 60 to 90 electrical-degree flats at top and bottom — and when two phases are combined, the resulting line-line back emf looks more sinusoidal than trapezoidal. See *Figs. 4* and *5*. This type of PMBDC motor output is often called quasi-sinusoidal *K _{E}*.

In addition to quasi-sinusoidal motors, it is quite simple to construct motors that output true sinusoidal back emf with less than 1% total harmonic distortion. Today, the majority of commercially available PMBDC motors output this sinusoidal *K _{E}* — shown in

*Fig. 6*.

Both sinusoidal and quasi-sinusoidal motors can be operated using a six-step drive. However, this doesn't output normalized, constant torque output. Instead, the output includes what's called **drive-induced torque ripple**, shown at the bottom of **Figs. 4**, **5**, and **6**. If an application can tolerate torque ripple, then a six-step drive may be most suitable — as they are less costly than other options.

### True sinusoidal K_{E} motors

Most commercially available PMBDC drives are sinusoidal - not six step. Traditionally, Hall switches provide commutation signals to these drives. However, by the 1980s, commutating optical encoders proliferated; now, their incorporated Hall signals are generally used for motor startup only. Why?

Until a drive senses an index pulse, it doesn't know exact rotor position. For this reason, optical Halls are still used for initial startup information.

Once a PMBDC motor is up and running, its drive servo electronics only need an encoder's line count per revolution and index pulse to control motor position and velocity. As with ideal trapezoidal motors (theoretically) driven by a six-step drive, true sinusoidally driven *K _{E}* motors output maximum continuous torque and efficiency when each phase current is kept in phase with the motor's corresponding back emf/torque function.

Let us explore this mathematically.

As with all PMBDC motors, *K _{E}*(θ) (in V/rad/sec) equals

*K*(θ) (in Nm/A) — so that the three line-neutral back emf/torque functions

_{T}*R*,

*S*, and

*T*for a true sinusoidal

*K*motor are:

_{E}With these three line-neutral torque functions, three corresponding phase currents supplied to the motor by the sinusoidal drive are:

As is customary in sinusoidal ac circuit analysis, each phase current is allowed to have a different phasing ϕ with respect to the corresponding back emf/torque function.

Multiplying each phase current by its corresponding back emf/torque function and then adding all three gives the total torque produced by a true sinusoidal motor/drive combination:

*T _{R}* +

*T*+

_{S}*T*

_{T}Using the trigonometric identity *sin*(a + b) = *sin***a**·*cos***b** + *cos***a**·*sin***b** 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