Environmental Engineering Reference
In-Depth Information
This coincides with the recommendation in (Weiss and Hafele, 1999, Section 2). This recom-
mendation is good when
ω
c
τ
is very large, which is normally the case. When
ω
c
τ
is not so
large,
τ
d
should be chosen according to (2.10) with the plus sign.
2.3 Reference Frames
For a three-phase system, the voltages and currents can be described in different reference
frames (also called coordinate systems). As a result, the controller can be designed in different
coordinates. Here, voltages are taken as an example to describe the different reference frames
but the analysis can be applied to currents as well, even to some other quantities; see (Bose,
2001, 2009). The case with the phase sequence of
abc
will be discussed first in detail and the
case with the phase sequence of
acb
will be discussed briefly.
2.3.1 Natural (abc) Frame
The voltages of a balanced three-phase system in the natural frame, also called the
abc
frame,
can be represented as
⎡
⎤
V
m
cos(
θ
)
⎡
⎤
V
m
cos
⎣
⎦
2
3
v
a
v
b
v
c
θ
−
⎣
⎦
=
,
(2.11)
V
m
cos
2
3
θ
+
where
V
m
is the peak value of the voltage.
is the phase of Phase
a
voltage and it changes
with time
t
. Hence, the voltages are functions of time
t
(and frequency). Note that the phase
sequence here is
abc
.
The three phases (coordinates)
a
,
b
and
c
in the natural frame can be regarded as spatially
distributed away from each other by
θ
3
rad, as shown in Figure 2.4(a). It is drawn to be
consistent with the phase sequence
abc
. The three-phase voltages
2
v
a
,
v
b
and
v
c
at a particular
θ
time
t
or phase
can be expressed as vectors according to their values spatially distributed
on the corresponding coordinates, as shown in Figure 2.4(a). That is,
v
a
is on the horizontal
e
−
j
2
3
e
−
j
4
3
. The
resulting diagrams are referred to as spatial diagrams in the sequel. Note that this is different
from the widely used phasor diagrams, where the length of a vector is the amplitude of the
voltage and the angle of the vector is the phase of the voltage. Here, the length of a vector on
a spatial diagram is the instantaneous value of the voltage and the angle of the vector is fixed
as the angle of the coordinate, i.e. 0 for Phase
a
,
2
line;
v
b
is in line with the vector
α
=
and
v
c
is in line with the vector
α
=
2
3
radians for Phase
b
and
2
3
radians for
Phase
c
in Figure 2.4(a). Moreover, a phasor diagram is independent of time but the spatial
diagram is dependent on time (and frequency). When
−
θ
changes with time
t
, the vectors
v
a
,
v
b
and
v
c
on the spatial diagram shown in Figure 2.4(a) change their lengths (and direction)
but do
not
rotate.
It is worth noting that
v
a
+
v
b
+
v
c
=
0
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