Why do AC motors have various back EMF shapes?
Just to lay a correct foundation. They are Synchronous machines &amp; the machine analysis is the same for all types.

A synchronous machine is a type of machine that has AC flux in the stator &amp; DC flux on the rotor (inside out machines aside). They generate torque only at synchronous speed - The rotor freq and the stator freq match, hence the name.

They have wound stators connected to an AC source with a wound rotor to produce a DC field, connected via sliprings ( some use mercury or graphite powder).  These are usually the large national grid type machines.

There are then the rotating diode rectifier Main exciter type to facilitate a "brushless" rotor field excitation.

You then have the Permanent Magnet rotor type where surface magnets on the rotor to produce the DC flux needed for synchronous motor-generating operation. These are Permanent Magnet Synchronous machines.

There are two types that exist


Permanent Magnet Alternating Current: PMAC
Permanent Magnet Direct Current:  PMDC 


Just to be clear both types produce an AC backEMF if they are back-driven. They both need their stator excited with an AC field (and thus need something to generate an AC current/voltage). What is important is the type of control &amp; the shape of the flux.

PMDC, as the name implies is DC. As I previously stated, they are not driven by DC but AC. The controller however will operate with a DC quantity and a final commutation stage will switch such a waveform through 60degree conduction points.

PMAC, as the name implies is AC. The core of the controller will more than likely be some form of Space vector modulation controller that utilises Clark &amp; Park (to then produce a DC representation to control against).

Why the difference? Well for the same shaft characteristics (torque, speed) and for the same volume &amp; weight a BLDC will produce higher torque &amp; it is has a very simple control.
The downside is the higher backEMF that is produced &amp; the torque ripple that is generated.

To get the most out of a BLDC control the BackEMF must be "shaped" to maximise the flux linkage. With DC current being applied in 60degree electrical sections the BackEMF needs to closely resemble this and thus it is shaped to be trapezoidal in shape as opposed to being sinusoidal.

How is this done though?  The usual method is via a fatter stator tooth, stumpier tooth tip &amp; the rotor magnets are not a full pitch (ie a 4 pole pair rotor with surface magnets would not have them covering 90deg but say... 87deg). This produces a period of VERY low flux linkage which shapes the BackEMF to be trapezoidal.