Environmental Engineering Reference
In-Depth Information
A
I
AB
S
=+
PQ
V
AB
Element
B
Figure A.20
Reference directions for
V, I, S
, P
and
Q
where
α
and
β
are the angles by which
V
AB
and
I
AB
lead a reference phasor. Assume that
α
).
It would seem sensible that the multiplication of these two phasors gives the complex
power. This task will now be carried out:
>
β
. The angle by which
V
AB
leads
I
AB
(the power factor angle) is (
α
−
β
( )
cos sin
Unfortunately, this product gives the wrong result! In fact, there is no good reason why an
algebraic manipulation would invariably produce a result that complies with the adopted
conventions. An alternative product will be tried using the
conjugate
of the current phasor
I
A
*
in which the sign of the angle is negated:
V
e
j
α
I
e
j
b
=
VI
e
j
α
β
=
VI
(
αβ
+
)
+
j
VI
(
αβ
+
)
AB
AB
AB
AB
AB
AB
AB
AB
*
SV
=
I
=
VI
e
j
α
e
−
j
b
=
VI
e
j
(
α
−
β
)
=
VI
cos
(
αβ
−
)
+
j
VI
AB
sin
(
αβ
−
)
AB
AB
AB
AB
AB
AB
AB
AB
AB
This gives the correct result; hence it can be confi rmed that in general
SVI
==+
*
PQ
(A.28)
The accepted reference directions of voltage, current,
P
and
Q
are shown in Figure A.20,
which depicts any power network element. If the complex product of
V
AB
and
I
A
*
results in
positive
P
and
Q
(i.e. the active and reactive power fl ow into the element terminals), the
element is a consumer of
P
and
Q
and operates in the fi rst quadrant of Figure A.18. In Chap-
ters 4, 5 and 6 it is shown that, depending on its nature, an element could operate in any of
the four quadrants.
A.13 Conservation of Active and Reactive Power
Unlike voltage and current in AC systems, active power is a scalar quantity. If a heating
element and an induction motor are connected at the terminals of a consumer, the total active
power absorbed from the mains is the scalar sum of the two active powers associated
with the two components. This is, of course, demanded by the conservation of energy
principle.
By analogy to active power conservation, a similar notion applies to reactive power. An
element consisting of an inductor and a capacitor absorbs a total reactive power equal to the
sum of the component reactive powers. If the
Q
of the inductor is larger than the
Q
of the
capacitor, the grid supplies reactive power to the
L - C
combination. If the capacitive
Q
is
