Environmental Engineering Reference
In-Depth Information
be clear from Figure A.17 that as the phase-shift angle
of the current sine wave
i
AB
increases,
the average power
P
will decrease as the power waveform
p
becomes increasingly symmetri-
cal about the time axis. When
θ
90 ° the average
power will reverse sign and the element will generate rather than consume power. It can then
be postulated that for
θ
reaches 90 ° , cos
θ
= 0 and
P
= 0. For
θ
>
−
90 °
<
θ
<
90 ° the element is a consumer of power and that for
90 °
90 ° , the element acts
as a pure inductor or capacitor respectively and it is neither generating nor consuming active
power.
<
θ
<
270 ° the element is a generator of power. When
θ
= 90 ° or
−
A.11 Reactive Power
Equation (A.12) gives the instantaneous power for an inductor as
p
= (
ω
L
)
I
2
sin 2
ω
t
. It is
known that
ω
L = X
L
and therefore the peak of the power variation is given by
2
V
X
I
V
I
IX
2
==
IZ
2
sin
θ
=
2
sin
θ
==
QVI
sin
θ
(A.26)
L
The quantity
Q
is known as reactive power because of its similarity to the active power
P = I
2
R
. It is measured in
reactive volt-amperes
denoted as VAR, pronounced ' vars ' . The
reactive power
Q
is of considerable importance in power systems. Notwithstanding the fact
that it has zero average value it represents real reciprocating energy transfers between
storage elements that are unavoidably parts of power system networks. These transfers are
important because they result in extra energy loss in transmission lines (affecting effi ciency)
and in network voltage changes (affecting adversely, at times, the voltage at consumer
terminals).
As with Equation (A.25) for active power, the sign of sin
θ
in Equation (A.26) indicates
whether
Q
is positive or negative. For 0 °
is
positive and, by analogy to the active power notation, the element is said to be a consumer
or a sink of reactive power. For 0 °
<
θ
<
180 ° the current lags the voltage, sin
θ
>
θ
>
180 ° , the current leads the voltage, sin
θ
is negative
and the element is said to be a generator of
Q
. For a capacitive reactance,
V
X
2
QI X
=−
2
=−
C
C
The whole concept is illustrated diagrammatically in Figure A.18, which is similar to the
complex number plane of Figure A.14. With the current as the reference phasor the position
of
V
on the diagram defi nes the nature of a power system element in terms of
P
and
Q
. The
fi rst quadrant represents elements that are consumers of
P
and
Q
, i.e. consist of
resistive-inductive components, while the fourth quadrant is for resistive-capacitive ele-
ments. Quadrants 2 and 3 require generation of active power and are reserved for 'active'
elements such as AC generators. In Chapter 4 it was shown that such a device, known as a
synchronous machine, is capable of operating in any of the four quadrants.
A.12 Complex Power
The
V
and
I
phasors in Figure A.18 are shown again in Figure A.19(b) in the form of a tri-
angle. With the current as reference, the voltage
V
applied across the circuit is equal to the
