Environmental Engineering Reference
In-Depth Information
A
I
B
B
X
S
P
B
o
V
B
=V
B
0
I
f
V
A
=V
A
δ
Q
B
Q
t
Figure 4.7
Equivalent circuit of a synchronous machine
of a generator as complex as an alternator. The steps in this transformation are not given here
but can be found in any topic on electrical machines (for example see Reference [2]). In what
follows it is assumed that the reader is familiar with the use of phasors to represent AC
quantities. Readers not familiar with this concept should, at this stage, study the material in
the Appendix.
In Figure 4.7 the electrical generator has been reduced into a single-phase (the relationship
with 3-phase is dealt with later) Thevenin equivalent circuit consisting of a voltage source
V
A
=
V
A
(the generated or 'internal' voltage of Equation (4.3)) and a source impedance
X
s
, known as the
synchronous reactance
. The synchronous reactance represents in one lumped
element all the internal complex interactions between the rotor and stator magnetic fi elds,
which are not of concern here. To maximize conversion effi ciency, synchronous machines
are designed to have as low winding resistance as possible; hence the source resistance rep-
resenting the ohmic value of the stator winding is omitted here with little loss in accuracy.
The equivalent circuit is shown connected to an infi nite bus, i.e. a network of fi xed frequency
f
and of fi xed voltage
V
B
=
V
B
∠
δ
0 ° where its 0 ° angle defi nes it as the reference voltage.
An investigation will explore how the two available external control parameters, namely
the fi eld current
I
f
and the shaft torque
Q
t
, infl uence the synchronous machine and conse-
quently the equivalent circuit behaviour. Equation (4.3) shows that |
V
A
| depends on the fi eld
current, which is the source of the magnetic fl ux. It is also known that the angular disposition
of the rotor magnetic axis depends on the direction and magnitude of the torque applied to
the shaft. Angle
∠
(the load angle) is defi ned as the angle by which the axis of the rotor fl ux
space vector that induces
V
A
leads the axis of the net fl ux space vector in the machine that
induces
V
B
. The load angle in the spatial disposition of rotating vector fi elds is the same as
that in the phasorial disposition of voltages in the equivalent circuit. An accelerating or
'generating' torque will result in a positive
δ
δ
and in
V
A
leading
V
B
. A decelerating or 'motor-
ing' torque will result in a negative
δ
and in
V
A
lagging
V
B
.
4.2.5 Power Transfer Equations
There is interest in exploring the mechanism by which power is injected into the grid by a
synchronous generator. This can be done by means of the concept of complex power devel-
oped in Appendix A. The grid connected synchronous generator of Figure 4.7 will be con-
sidered. The complex power at end B of the line is given by
*
SVI
BBB
=
(4.6)
