SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Expressing bEMF in terms of the bEMF function allows easy manipulation of the expression,
even if the bEMF function is not easily described mathematically. This is demonstrated in
Equation (2.39) and the corresponding simulation diagram is shown in Figure 2.25.
e e r
( t)
k ( ) (
t)
(2.39)
Figure 2.25 – Simulation diagram for per-phase bEMF function.
The torque and bEMF functions will be examined together in the next two subsections.
Equivalence of Per-Phase Back-EMF and Torque Functions
Now that the per-phase bEMF generation and torque production have been examined a very
important fact is stated. The per-phase bEMF and torque functions have the same waveshape. In
addition, when SI units are used, the per-phase bEMF and torque functions are numerically equal
and the units of each function (volt s/(rad) and N m/A)
are dimensionally equivalent
2 -2 -1
(
kg msA ). For the simple coil studied earlier, the expression for the conservation of
energy was given as Equation (2.13). For the rotational case this becomes Equation (2.40), which
can be derived using several different methods of differing rigor [68], [69], [27]. For example, an
intuitive method is to solve the BLi and BLv laws for the product BL and equate the result.
e i
(2.40)
Equation (2.40) holds for transient and steady state conditions, but it is important to note that the
equation only governs the part of energy conversion process that involves flux that links both the
field and the armature. It describes only the energy that crosses the airgap. When the
instantaneous values of bEMF and torque are replaced with their general descriptions given by
Equation (2.34) and Equation (2.39), the result is Equation (2.41).
( ) i
k ( ) i
(2.41)
ke t
Regardless of the type of motor or motor controller, the current and velocity on each side of the
equation will be the same and may be cancelled, leaving the all-important result shown as
Equation (2.42). Naturally, any equation is only true if the variables are expressed in compatible
units and in SI base units this is automatically the case.
k ( ) k ( )
(2.42)
e t
41