SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

In Equation (3.135) Z has elements identical to those of Z abc (eigenvalues are invariant under
a linear transformation) but is only 2x2 (owing to the fact that we used the Clarke instead of αβ0
transform). This result means that the three-phase circuit can be modeled as a two-phase
equivalent with the same values for the impedance elements (unlike the three-phase circuits, these
two-phase circuits are independent). Expanding the impedance matrix yields Equation (3.136).
v Ri L i
e
(3.136)
s
By the relationship between matrix and SV representations (Equation 3.72), Equation (3.136) is
equivalent to Equation (3.137).
v Ri Lsi e
(3.137)
i ji SV diagram for an arbitrary lagging power
Figure 3.41 shows the steady-state
factor; the bEMF is used as the reference and the diagram is shown for t 0 . This is similar to
the phasor diagram of Figure 3.17 and all of the relationships that held for the SPE analysis
(Figure 3.19 to Figure 3.23) hold for the SV analysis as well, giving the same clear picture of how
the physical quantities relate to the electrical quantities. But since the phasors are replaced with
space vectors which are instantaneous quantities that represent the contribution of all phase
variables, the SV model is valid for nonsinusoidal and transient conditions. The two diagrams are
the same only because we assumed balanced sinusoidal conditions at steady state, because we
chose e as the reference, and because we drew the diagram at t 0 . The alignment with the
real axis actually has no meaning just as in phasor analysis—it simply shows the relationship
between the SVs.
Figure 3.41 – SV diagram of synchronous motor in stationary frame (steady state).
In traditional analysis Equation (3.136) is used but its SV equivalent (Equation 3.137) is not used
as often in SV analysis. Instead, the variant that incorporates the electromagnetic quantities is
most common. Rewriting the stator flux linkage (as done in Chapter 2, in Appendix B, and in the
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