SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

In the previous chapter the per-phase model was expanded to produce the phase variable
electromechanical model. That model explicitly accounts for the component torque produced by
each phase; it is capable of describing motors with non-sinusoidal torque functions and it
correctly accounts for transient conditions. Similarly, the applied voltage, winding current, and
phase bEMF need not be sinusoidal functions; it is a general time-domain model. In contrast, Part
I treats only the sinusoidal motor under balanced conditions; it is a phasor-domain model. The
torque production was developed using phasors (albeit implicitly) and the SPE circuit developed
in this section uses phasors as well. Whereas the phase-variable model is electromechanical, the
single-phase equivalent model treats the circuit and torque production separately. Essentially, the
torque expressions and electrical circuits of Part I (SPE) are a simplification of the phase-variable
model that result when electrical quantities are restricted to sinusoids and the motor is operating
in a mechanical steady-state condition. Since SPE assumes the motor is balanced and in steadystate,
the electrical power and mechanical torque/power are found by multiplying a phase quantity
by a factor of three. In contrast, the phase-variable model explicitly accounts for each phase,
whether the quantities of the phases are equal or not. This is true for space vector analysis as well
and in fact, space vector analysis is essentially a compact encapsulation of the phase-variable
model with only one limitation: it cannot directly describe nonsinusoidal motors (but the currents
may be arbitrary).
The SPE model represents one phase but it is known that there is mutual inductive coupling
between phases. The SPE model accounts for this in the same way that the phase-variable model
does: by modifying the simple self-inductance to include the mutual inductance due to current in
the other phases (Appendix B). The resulting “effective” inductance is called the synchronous
inductance Ls and is given by Equation (3.22), where L is the phase leakage inductance and Lmag
is the magnetizing inductance. For simplicity the leakage inductance is ignored in these
discussions.
3
Ls Ll Lmag
(3.22)
2
The factor 3/2 is the same factor encountered in the MMF derivation of the previous chapter.
The SPE circuit shown in Figure 3.16 is therefore similar to the per-phase circuit from the
previous chapter (Figure 2.17) except it is in the phasor domain. The KVL equation is Equation
(3.23). (The voltage a G~ is the phasor representation of g(t) from the previous chapter and will be
discussed at the end of Part I.)
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