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I have an in-wheel BLDC motor with 46 poles (permanent magnet) on the rotor and 51 slots in the stator. When I used an external machine to rotate my in-wheel BLDC motor, then measuring across two phase back EMF I got sinusoidal back EMF instead of trapezoidal.

Can any one explain why this happened?

enter image description here

For the measurement of back EMF constant, how one can calculate if back EMF is sinusoidal?

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You are measuring the line-line voltage and not the phase voltage.

The characteristic "trapezoidal" backEMF is only present at the phase voltage. When you view this line-line the waveform appears like a pointy sinewave. Then there are specifics of the build: magnet span, width of stator tooth foot... all these things influence the shape of the backEMF. With a high polecount the arc needed for a magnetic gap will be very small and probably non-existent.

enter image description here

And thus the line-line at the terminals would appear:

enter image description here

in practice you won't see this and it will naturally be rounder as this is the mathematical concise result. The point still stands. Line-Line is more sinus than most appreciate.

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    \$\begingroup\$ In this situation how one can find Back EMF constant? \$\endgroup\$
    – aman2909
    Commented Apr 21, 2018 at 10:14
  • \$\begingroup\$ you make a distinction between phase voltage and line-line voltage, I think you perhaps mean phase-neutral voltage vs phase-phase voltage? any how if the phase-phase voltage is sinusoidal then the phase-neutral voltage will also be sinusoidal. \$\endgroup\$ Commented Nov 22, 2023 at 9:33
  • \$\begingroup\$ you have drawn the trapezoidal wave form incorrectly, the high or low values are maintained for 120 degrees not 60. \$\endgroup\$ Commented Nov 22, 2023 at 9:40
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It seems like you have a PMSM and not a BLDC. There is a lot of confusion about how to identify them, but the easiest way is exactly what you did and since you have a Sinusoidal bEMF your motor must be a PMSM.

In this link you will find a nicely elaborated explanation.

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