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What is a brushless DC motor and what are the differences with a synchronous AC motor

Update:06-11-2020
Summary:...
The short answer is: Brushless DC motors and synchronous AC motors are very similar in terms of construction and operation. Some manufacturers can even group them together under the permanent magnet synchronous motor section. The key difference is, however, the stator coil windings and the corresponding back EMF for each motor. This gives them different performance characteristics and stipulates their own drive technology.

Structural similarity
Despite the peculiarities of their names, both brushless DC and synchronous AC motors are brushless and both operate at synchronous speeds. Brushless means that they rely on electronic devices (typically Hall sensors) instead of mechanical carbon brushes to control current to the windings. And synchronization means that their rotor and stator magnetic windings rotate at a synchronous frequency or a synchronous speed.
Both brushless DC and synchronous AC motors have permanent magnets embedded in the rotor (typically 4 or more). The rotor magnet can be ferrite, which is cheaper but the magnetic flux density is relatively low. Or a rare earth alloy (such as neodymium), which has a high magnetic flux density, but in some references, its price is very high. The stator is composed of iron laminations, and the windings (usually three) are placed in axially cut slots.
The rotor permanent magnets create a rotor magnetic flux, and the current applied in the stator windings creates electronic magnetic poles. When the position of the stator is such that the N pole of the rotor is close to the N pole of the stator, the two poles repel each other and torque is generated.

Difference in operation and performance
In a brushless DC motor, the stator coil is wound in a trapezoid shape, and the generated back electromotive force has a trapezoidal waveform. Because of the trapezoidal waveform, the required DC is obtained in order for better performance. On the contrary, synchronous AC motors are sinusoidally wound and generate a sinusoidal back electromotive force. Therefore, they require sinusoidal current to obtain better performance.
This type of current will have an impact on the overall noise generated by the motor. The trapezoidal current used in brushless DC gear motors tends to produce a huge auditory and electronic noise, compared to synchronous AC motors with sinusoidal drives.
Commutation, which is to convert the phase current of the motor to drive the appropriate electronic coil, which is determined by the stator position. In a brushless DC motor, the rotor position is typically monitored by three Hall sensors. And the commutation is through six steps, or every 60 electronic angles. Because the commutation is discontinuous, a torque fluctuation will be generated during each commutation (every 60 degrees).
Through a single Hall sensor or rotary encoder, combined with control logic, synchronous AC motors can benefit from constant monitoring of the rotor position. Because the commutation is continuous, the synchronous AC motor can operate without torque fluctuations. However, sine commutation requires more complicated control algorithms than trapezoidal commutation.
Although the construction is very consistent, the difference between DC and back EMF in brushless DC and permanent magnet AC motors is an important difference. In terms of control and performance, the application of appropriate DC and control is a very important factor.