SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

common schemes in use and they differ significantly from one another. But one thing they have
in common is that each they seek to drive current in a winding with the same polarity as the
bEMF/torque function.
Equation (2.40) describes the energy balance for one phase acting alone. Since the three-phase
motor has three phases and each phase contributes to torque (per Figure 2.28), the energy balance
for the three-phase motor is Equation (2.54), where all quantities are instantaneous functions of
time. This equation is valid for transient and steady-state conditions and is valid for any bEMF
and current waveshape; it holds even if the phase bEMFs are asymmetrical or displaced by an
angle other than 120°.
ei ei ei
(2.54)
A A B B C C
If the bEMF terms are expanded according to Equation (2.39), each side of Equation (2.54) could
be divided by ω, leaving the bEMF function on the right-hand side, Equation (2.55).
k ( ) i k ( ) i k ( ) i
(2.55)
eA r A eB r B eC r C
Each winding of a sinusoidal motor will have a torque function like the one shown in Figure 2.22,
but since each winding is displaced from the other windings by 120°, each winding’s torque
function will be similarly displaced by 120°. Assuming balanced bEMF waveforms, this is
expressed as Equation (2.56) and is shown in Figure 2.29. 8
K sin( ) i K sin( 120 ) i K sin( 120 ) i
e r A e r B e r C
(2.56)
8 The math is less cumbersome if cos(θr) or +sin(θr) is used and the figures are clearer if an arbitrary rotor
angle is used. Most of the literature takes advantage of this and as a result, plots such as Figure 2.29 (and
especially Figure 2.31) look very different in each reference. Additionally a single article will often use
several different references for θr and never mention the change nor discuss its importance. While each
approach is perhaps acceptable it has been the author’s experience in learning that using arbitrary
references and divorcing the plots from the actual rotor position is a detriment to properly visualizing
synchronous motor operation and FOC in particular. Thus it is strictly avoided in this report in faith that a
less-simple introduction can be endured in exchange for a more accurate understanding in the long run.
Finally, there are two common choices in defining an absolute θr and they are 90° apart, thus there are two
valid versions of the “actual” waveforms shown here; the alternate version will be discussed in Appendix
E.
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