SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

fundamental of stator MMF (produced by any type of winding). The MMF created by balanced
sinusoidal currents is in principle no different than that produced by a single winding save that it
is 3/2 larger and it rotates as a traveling wave whereas the per-phase MMF is a pulsating
standing wave. In fact, with some manipulation (involving the removal of the factor 3/2 and
finding that r
) it could be shown that Equation (3.18) is essentially the same as the torque
expression found in Chapter 2 for the simple sinusoidal motor and CFP winding:
T r
p r
( ) N
D Y
B sin(
) i(
t)
.
Finally, Equations (3.18) and (3.19) can be written in terms of rotor-stator flux linkage. From
Appendix C, the peak value of flux linkage of a sinusoidal winding with a sinusoidal rotor flux is
given by Equation (3.20).
R NeDY Bp
(3.20)
4
Comparing this with c3 in Equation (3.18) (which is defined by Equation 3.19) it is seen that the
torque can be rewritten as Equation (3.21). This version will be encountered in the SV model of
Part III.
3
T RIpsin( )
(3.21)
2
Phasor Diagram and Single-Phase Equivalent (SPE) Circuit
Now that the rotating field and torque production have been discussed the electrical model for a
synchronous machine will be developed in order to complete the standard treatment. Traditional
analysis of the three-phase synchronous machine employs the use of phasors to describe the
electrical circuit using the “single-phase equivalent” (SPE) concept. 19 In SPE a symmetrical
sinusoidal machine is assumed, driven by balanced sinusoidal currents and operating in steadystate.
Thus the machine can be sufficiently analyzed by examining a single phase (typically
phase-A); all similar phase-B or phase-C quantities are thus unnecessary but may be found simply
by adding a –120° or +120° phase shift to the phase-A quantities.
19 Sometimes this is called the “per-phase equivalent” but that usage is avoided here because the threephase
“phase-variable” model can be analyzed on a per-phase basis and that is not the same as SPE.
82