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
(
f¼
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
f¼
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

and

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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