|
I
a
as if it is a real number with positive or negative sign.
In the case of
I
r
, it can be specified as a real number with positive or negative sign. Or else, one may
add the qualifiers ‘capacitive’ or ‘inductive’ and then skip the sign. The j in
I
r
is taken to be implied by
the word ‘reactive’ in ‘reactive component’.
Thus, if V
S
=
V
m
∠
0
°
and
I
=
I
m
∠-
q
then, active component
of current is I
m
cos
q
(I
rms
cos
q
A rms) and reactive component of current is
-
I
m
sin
q
(
-
I
rms
sin
q
A rms). If
the negative sign is skipped, then we have to state that inductive component of current is I
m
sin
q
(I
rms
sin
q
A rms). For instance, a 5 A reactive current implies 5 A of a capacitive current, a
-
5 A reactive current
implies 5 A of an inductive current, a 5 A inductive current implies 5 A of an inductive current and a
–5 A inductive current implies 5 A of capacitive current. Similarly a 5 A capacitive current implies
5 A of a capacitive current and –5 A capacitive current implies 5 A of an inductive current. Inductive
current component lags voltage phasor by 90
°
and capacitive current component leads voltage
phasor by 90
°
.
7.9.2
reactive power and the power triangle
We restate the concepts we developed in Section 7.9-1 before we continue with the concept of reactive
power that tends to be a confusing one to beginners in Electrical Engineering.
Apparent Power, Active Power, Reactive Power and Power Factor
7.47
• Let
v
(
t
)
=
V
m
cos
w
t
V and
i
(
t
)
=
I
m
cos(
w
t
-
q
) A be the steady-state voltage and current
at a pair of load terminals as per passive sign convention. Then apparent power which
is the power that a DC circuit with same effective values of voltage and current would
have delivered is
V
rms
I
rms
=
0.5
V
m
I
m
VA. The average power which is also called
active
power
is
P
=
V
rms
I
rms
cos
q
W
=
0.5
V
m
I
m
cos
q
W. The power factor of the circuit which
is defined as the ratio of active power to apparent power is cos
q
.
• The minimum magnitude of current required to deliver a given amount of power
P
is
given by
I
rms
=
P
/
V
rms
with cos
q
=
1. This happens when the driving-point impedance of
the load at
w
rad/sec is effectively a resistance. The circuit has
q
=
0 then, and, draws
power at unity power factor with minimum magnitude of current.
• Reactive component in the driving-point impedance of the load circuit makes
q
non-
zero and increases the current magnitude for a given amount of active power. Thus
current is underutilized as far as active power delivery is considered. The power
factor of the circuit will be less than unity.
• The current phasor can be resolved into active component (
=
I
rms
cos
q
A rms) and
reactive component (
=
-
I
rms
sin
q
A rms) by finding its projection along voltage phasor
and along a perpendicular to voltage phasor respectively. The active component
may be thought of as the current drawn by the resistor in a parallel connected
resistance–reactance combination that has same impedance as that of the load
circuit. The reactive component of current is the current drawn by the reactance
in that equivalent parallel circuit. The active current component carries the entire
active power.
The utilisation of load current in its role as a vehicle to carry active power can be judged from
the relative proportion of its active and reactive current components. It can be shown easily that
I
I
I
I
I
I
a
r
am
rm
rms
rms
rms
m
and
=
+
=
+
,
,
2
2
2
where I
a,rms
and I
r,rms
indicate the rms values of the
components, whereas I
am
and I
rm
indicate their amplitudes. The following relations also hold between
various quantities.
cos
sin
t
,
,
q
q
=
=
=
= −
= −
power factor
rms
rms
m
rms
rms
m
I
I
I
I
I
I
I
I
a
am
r
rm
aan
,
,
q
= −
= −
I
I
I
I
r
a
rm
am
rms
rms
I
r,rms
and I
rm
contain the sign of reactive component. Therefore, while power factor is always
positive, sin
q
and tan
q
are positive for a lagging load and negative for a leading load. Note that
q
is
the angle by which voltage phasor leads the current phasor. Power factor of a load is independent of
sign of
q
and hence a qualifier lag or lead is to be appended to the number representing power factor
to distinguish between positive and negative values of
q
. Thus, if
q
=
45
°
, the power factor is ‘0.7 lag’
and if
q
is –45
°
the power factor is ‘0.7 lead’.
Another method to describe the utilisation of load current in carrying active power will be to
compare the active and reactive rms components of current after scaling the active rms current
component by
+
V
rms
and reactive rms current component by
-
V
rms
. But then, if the active component
of current is multiplied by
+
V
rms
, the result is active power. Then, it will be tempting to call the
7.48
The Sinusoidal Steady-State Response
product of the reactive component of current multiplied by
-
V
rms
a power – but not a real average
power, since this component of current produces only V
rms
I
r,rms
sin2
w
t term in instantaneous power.
Electrical Engineers yielded to this temptation long back and they called it reactive power in contrast
to active power.
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