Environmental Engineering Reference
In-Depth Information
The situation for reactive power Q is less clear. The sign of Q depends entirely on
howweviewthesignof f ,sincesin( f ) ¼ sin( f ). This reflects the nature of reac-
tive power, which describes the oscillating component of power. There is no net flow of
Q during the electrical cycle. Hence the sign allocated to reactive power is optional.
When deriving the impedances of resistance, inductance and capacitance ear-
lier, we took current as reference and measured the angle of the resulting voltage
anti-clockwise from this datum. In the case of inductance, the voltage led the
current by 90 ( 90 ) and the reactive power was positive. However, we could
equally have taken voltage as reference; in that case the current would have lagged
by 90 , giving 90 and negative reactive power.
Hence we need to decide on the sign of reactive power, based on whether the
current is leading or lagging the voltage. As it happens, most loads take a lagging or
inductive current. The associated reactive power manifests itself as extra current
and losses in the utility's cables. Thus utilities tend to charge the consumer for
reactive as well as for active power and energy. By assigning a positive sign to
lagging current or inductive reactive power, utilities avoid the embarrassment of
charging for a negative quantity. This supports the convention that
reactive power is deemed to be positive for a lagging/inductive current
The question now arises: how is complex power related to complex voltage
and current? It is tempting to set
S ¼ V I
Assume that the voltage phasor leads the current phasor by the angle f , and
that the current phasor has an arbitrary angle q . Taking V ¼ Ve j( qþf )
and I ¼ Ie j q ,
the complex power will then be
S ¼ VIe j ð 2 qþfÞ ¼ VI cos ð 2 q þ fÞþ j VI sin ð 2 q þ fÞ
This bears no obvious relationship to power and reactive power. On the other
hand, suppose we set
S ¼ VI
This equation uses the complex conjugate of the current, I *. The complex
conjugate of a complex quantity is obtained by reversing its angle, or by changing
the sign of its imaginary part. Thus
I ¼ Ie j q
S ¼ VIe j f
¼ VI cos f þ j VI sin f ¼ P þ j Q
as required.
Thus complex power is given by
S ¼ VI
ð 2 : 17 Þ
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