SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Figure 3.42 – Coupling between d- and q- axes in voltage equations.
From Chapter 2 onward we have routinely encountered the fact that the stator flux linkage is
composed of its self flux linkage and the component due to the rotor (Equation 3.139). When
expressed in the rotor frame (Equation 3.148) it is clear that the flux linkage the rotor contributes
appears constant from the rotor’s perspective. This is emphasized by Equation (3.150), which
confirms what we already know: the d axis is defined as being aligned to the rotor flux, thus all of
that flux (and its flux linkage) should be along the d axis. At this point it would be helpful to
draw the SV diagram in the rotor frame. To do this we need the full voltage equations, which can
be written either in SV for or dq-component form (which are equivalent via Equation 3.106). To
get the SV form, substitute Equation (3.148) into (3.147); to get the component form, substitute
Equation (3.150) into (3.149). Due to the coupling, the component form is simpler and the result
is given by Equation (3.153). The peak value of the rotor-stator flux linkage is by definition
constant thus its derivative is zero and the final result is rearranged as Equation (3.154).
d d
v Ri L i L i
dt dt
d
vq Riq Ls iq Lsid R
dt
d d s d R s q
d
vd Rid Ls id Lsiq dt
d
v Ri L i L i
dt
q q s q s d R
(3.153)
(3.154)
In Equation (3.154) the time derivative operates on the dq components of the current SV. As will
be shown in the chapter on FOC, in the normal situation a BPMS motor drive will keep the stator
current SV a fixed angular displacement ahead of the rotor flux, in which case these components
are constant and only change during electrical or mechanical transients (electrical transients
because a time-varying SV will have time-varying components; mechanical transients because
149