An analogy for the difference between squarewave (or Trapezoidal wave) and sinewave is like pushing a bowling ball down the street. Sinewave uses a smooth constant push the whole way, while squarewave attempts to simulate the same forces but uses constant jabbing/nudging instead of an even "push".
The loss of energy to sound (harmonic losses) in square wave controllers which is caused by the transitions between each of the 120 deg phase differences makes your regular cheap square wave controller less efficient.
Sinewave (or Field Oriented Control FOC) controllers are generally more expensive because it requires alot more computational power to output a sine waveform to drive the motor efficiently.
Square wave on the other hand is simple in its computation requiring just the 3 phase on/off signal to drive the motor in a time sequence. Here is how it works for a square waveform driven motor as it goes through its rev.
Here are the pros and cons of a sinewave vs square wave controller:
|Sinewave FOC Controller||Squarewave Controller|
|Pros||1. Lower noise and heat generated
2. Smoother fine speed control. Allows riders to maneuver turns without braking too hard
3. More efficient at lower speeds going up slopes
4. Smoother acceleration
1. Cheaper due to less computational power needed
|Cons||1. Higher price due to more sophisticated motor control
2. Inefficient at high speeds
3. Motors and controller needs to be paired and tuned
1. Noisier motor with more heat generated
So for workhorse controllers that require high efficiency for normal operations like cruising or slope climbing, sinewave FOC is the better option as they will not require such high power consumption.
On the top end, for high speed transactions, square wave controllers with its high torque and punchy drive will be more efficient but you have to accommodate for the high voltage drop by having a higher discharge and higher capacity battery.
So it seems there could be room for a best in class universal controller that will be able to be efficient at both low speeds and high speeds. At low speeds, it will output sinewave and at higher speeds it will configure to square wave. That would be the most ideal controller design but do not expect it to come cheap!
Here is an actual demonstration of 2 equivalent electric scooter setups. 60V 100A output with a 60V 21Ah battery. The only difference is that one is equipped with a sinewave FOC controller and the other with a square wave punchy controller.
This is only a drag race test to see which controller would come out on top over 30m. You would expect the square wave punchy controller to be much faster off the blocks every time which is the case but the FOC controller always catches up within a short distance due to its efficiency.