SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Figure 3.12 – Simplified developed view of a four pole motor.
This notion leads to the development of two systems of angular measure. The relationship
between electrical and mechanical measurement is shown in Equation (3.8), where P indicates the
number of poles. 16 Taking the time derivative yields Equation (3.9), which is the relationship
between electrical and mechanical velocities; ωe is usually called the synchronous speed. To
avoid having to memorize the formulas, reconsider the gear ratio analogy, where P/2 is the ratio.
A free rotating piece of steel or magnet situated inside the stator can do nothing other than follow
the rotating field tied to the currents’ electrical position and velocity, thus the mechanical velocity
can never be faster than the electrical velocity. This means the ratio has a lower limit of unity,
hence the ratio is always multiplied by the mechanical measure.
P
2
elec mech
P
2
elec mech
It is important to make the distinction between a mechanical or electrical angle (that is, an angle
describing a mechanical or electrical displacement) and an angle measured in mechanical or
electrical degrees. For example, in Equation (3.4) and Equation (3.6) t is an electrical angle
and θ is a mechanical angle but both are measured in electrical degrees. The derivative of a
16 Some texts use P to indicate pairs of poles to eliminate the factor of two in equations. However, industry
specifies motors almost exclusively by the number of poles (not pole pairs) so this report uses P to indicate
the number of poles. The reader is cautioned that many authors use a lowercase p to indicate poles (or pole
pairs) which can also be used to denote the time derivative operator d/dt. The two are used interchangeably
without distinction, often within the same formula. This is unfortunate, as the correct meaning in an
equation cannot always be discerned from context without studying the derivation, which is often not
given. This report uses rho ( ) to denote the time derivative operator d/dt.
(3.8)
(3.9)
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