SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

If a ZS component could possibly be present in the ABC quantities (such as the measurement of
voltages in a three-phase system), it could be calculated by adding a row to the matrix in Equation
(D.26), as in Equation (D.29). This is the original transformation used by Edith Clarke [11,
p.308], although it had been presented and used by different authors prior to that, at least as early
as 1917 [11, p.310]. The transform was used in a fashion similar to that in the MSC (which is not
what we have been using it for here). More information about the original Clarke transform and
its application to synchronous machines can be found in [11], [12].
x 1 1/2 1/2
xA
x
k 0 3/2 3/2
x
B
x
0 1 1 1
x C
(D.29)
In the popular literature it is common to call Equation (D.26) (or any of its 0 variants) the
Clarke transform and this report adopts that usage. The academic literature is more technical and
generalized, studying circuits in which a ZS component is assumed to be present. Equation (D.29)
or a variant (discussed below) is used, which in this report is called the αβ0 transform. These
names are used interchangeably in the literature and it may also be called a Concordia transform
(after the pioneer Charles Concordia ([9], [10]). In addition, the axis along which the ZS
component acts may be called the γ- or Z- axis. Some variations will now be discussed.
We have seen the usage of the scaling constant k in regards to the αβ components. The same
remarks apply to the ZS component, meaning that there is a choice of k to make the ZS part of the
transform in Equation (D.29) be magnitude- or power- invariant just like the αβ part. Since the ZS
coefficients are scaled by k, one would expect to find three matrices that match the three varieties
of the Clarke transform. However, no such standardization exists because authors often choose
scaling between the αβ and 0 components to be different. This is handled in this report by
introducing a separate constant c to describe the ZS component scaling. The general αβ0
transform is given by Equation (D.30) (hence, the original Clarke transform used c 2 ).
x 1 1/2 1/2
k xA
x
k
0 3/2 3/2
x
B
x0 c 1/2 1/2 1/2
x C
(D.30)
Since the αβ0 transform is the Clarke transform plus the ZS (0) component, the matrix will be
denoted C 0 . The “raw,” magnitude-invariant, and power-invariant αβ0 transform matrices are
given by Equations (D.31), (D.32), and (D.33), respectively [87, p.34], [88, p.87]. Note that in the
323