# How can one achieve sinusoidal back EMF in PMSM and trapezoidal back EMF in BLDC? What are the winding differences?

How can one achieve sinusoidal back EMF and trapezoidal back EMF in the motor?

What is the winding differences in those motors?

I have takeN an in-wheel BLDC exterior motor which is used in e-bike. I get sinusoidal shape instead of a tapezoidal shape.

How can I calculate back the EMF constant?

• Is this the voltage across the motor terminals when the motor is energized by a controller? If so, it is determined mostly by the design of the controller. I think the back emf would be more like the voltage produced when the motor is operated as a generator.
– user80875
Commented Jan 2, 2018 at 14:28
• This is achieved by rotate the motor by some other external motor and getting the voltage of two phase. Commented Jan 3, 2018 at 6:15

The magnets and pole faces can be shaped and positioned to achieve a more 'trapezoid' back-emf. Different winding patterns may also have an effect.

However I suspect that matching back-emf to the drive waveform is often not done, because real BLDC back-emf waveforms are all over the place. Here are some scope traces showing the phase-to-phase waveforms of 3 motors that I tested (vertical pulses are PWM drive, back-emf is the middle waveform that occurs when the phases are not driven):-

These are all small 'in-runner' BLDC motors rated for 100-300 Watts, designed to power RC model aircraft. The first two motors have slotted iron stators. One produces close to trapezoid back-emf, the other nowhere near it.

The last trace is from a coreless ironless motor, which explains its almost perfect sine wave back-emf. Despite having a 'suboptimal' back-emf waveform, this motor (which only weighs 28 grams) produces 90W at 60,000rpm with 83% efficiency.

• that's line-line potentials... the trapezoidal shape is apparent at phase voltage. I have a post on this somewhere with the shape of the line-line. Even then the flux (1st integral) is more triangular & is closer to the line-line voltage shape
– user16222
Commented Jan 2, 2018 at 10:55
• Wouldn't the output voltage when operating the motor as a generator be a better representation of the back emf?
– user80875
Commented Jan 2, 2018 at 14:24
• @CharlesCowie yup, not sure why a switching waveform is shown. Sure the underlying backEMF can be seen, but it just confuses those that are not aware
– user16222
Commented Jan 2, 2018 at 14:42
• @CharlesCowie you are right. I have used same method. But I got sinusoidal back emf instead of trapezoidal in BLDC motor. Sinusoidal waveform occur with PMSM Motor. Commented Jan 3, 2018 at 6:18

The general design configuration of the motor must first be considered. That includes whether the air gap is radial or axial, whether the motor has an interior or exterior rotor and whether the claw-pole or conventional structure is used.

For a conventional motor with an interior rotor, the following design features would be considered: The influence of the stator winding on the shape of the back emf waveform is determined by the way that the stator windings are distributed among the rotor slots, the number of rotor slots per pole, the slot diameter and the skew angle of the slots. The rotor design also influences the shape of the back emf waveform. The relevant factors include the use of interior permanent magnets vs. surface permanent magnets, the skew angle of the magnets and the geometry of the magnets.

Re question revision

The design configuration in question seems to have an outer rotor that is a ring of homogeneous material that is magnetized in an alternating N-S pattern. That would result in magnets do not have distinct edges. That would tend to soften the edge of the resulting bemf waveform making it more sinusoidal.

The inner stator could have windings distributed to some extent, but not enough space for a lot of options in slot number or winding pattern.

The flux distribution is probably far from sinusoidal, but there are enough factors in the design suppressing the higher order harmonics to make the current look somewhat more sinusoidal than trapezoidal.

• I have used "In-wheel" exterior motor which is used in e-bike. As e-bike manual suggested tha this is a BLDC trapezoidal motor.But when I have measured its back emf then it is sinusoidal. That is why I am asking that which thing make its back emf trapezoidal or sinusoidal. Commented Jan 2, 2018 at 12:38
• I have revised my answer.
– user80875
Commented Jan 2, 2018 at 14:16

Most motors are closer to sinusoids than trapezoidal waveforms. (if you find one that is heavily trapezoidal, please let me know!) The difference is the spatial distribution of windings. The manufacturer can engineer the stator windings in a BLDC to be sinusoidal or nonsinusoidal.

In this case how can I calculate back emf constant.

Very easy. With motor disconnected from drive electronics:

• Spin a motor at constant speed (typically 1000RPM but it can be anything as long as it's below the rated motor speed but not too far below it)
• Measure RMS voltage between any two terminals with a DMM
• Measure the speed with a tachometer, or by taking the electrical frequency of backemf on an oscilloscope and dividing by the # of pole pairs (see motor datasheet, or put DC current into one pair of terminals and count the number of equilibrium points per mechanical rotation)
• Calculate voltage divided by speed, e.g. 10Vrms / 2000 RPM = 5Vrms/KRPM line-line.
• The Astroflight 010 gives very trapezoidal waveforms. Commented May 10, 2019 at 7:05
• Commented May 16, 2019 at 4:02
• Yep. Although i might have the older low-kv 14t version. dropbox.com/s/6jll8cwprxqcstg/20190509_204031.jpg?dl=0 Commented May 17, 2019 at 5:43
• I know this post is old, but I want to add that line-line back emf = sqrt(3) * back emf because I was having trouble with this and found this calculation on page 6 of this ocw.mit.edu/courses/mechanical-engineering/… Commented Jul 23, 2020 at 18:35
• line-to-line back emf = sqrt(3) * line-to-neutral back emf. The term "back emf" by itself is ambiguous. Commented Jul 23, 2020 at 18:43

The magnetic field for a sinusoidal motor is profiled such that the field strength with respect to angular position is sinusoidal. A magnet in a BLDC motor is generally uniform in cross section. This results in more of a square response. In practice, the square response has fairly linear rise and fall characteristics, resulting in a trapezoidal response.

This graphic is from TI / Texas Instruments