1
Real, reactive, and apparent power
In a DC circuit, the power is:
P = VI
(2.34.1)
In an AC circuit, an instantaneous power is a
product of an instantaneous voltage and
an instantaneous current.
The voltage applied to an AC load is:
v(t ) = 2 V cos ωt
(2.34.2)
Where V is the root mean square (rms) value of a voltage applied to the load.
2
Real, reactive, and apparent power
The current flowing through the AC load will be
i (t ) = 2 Ι cos(ωt − θ )
(2.35.1)
Where I is an rms value of the current and θ is the impedance angle.
Therefore, the instantaneous power will be:
p (t ) = v(t )i (t ) = 2VI cos ωt cos(ωt − θ )
(2.35.2)
For inductive loads, θ is positive and the current lags the voltage.
p (t ) = VI cos θ (1 + cos 2ωt ) + VI sin θ sin 2ωt
(2.35.3)
3
Real, reactive, and apparent power
The first component of (2.35.3) is in
phase with the voltage while the second
term is due to the current that is 900 out
of phase.
The average value of the first term is
the average or real power:
P = VI cos θ [W ]
(2.36.1)
The average power supplied by the second term is zero.
The second term represents an reactive power: power that is transmitted to the
load and then reflected back to the power source either through electric energy
stored in a capacitor or through magnetic energy stored in an inductor.
4
Real, reactive, and apparent power
The reactive power of a load is given by:
Q = VI sin θ
(2.37.1)
By the convention, Q is positive for inductive loads and negative for capacitive
loads. Units are volt-amperes reactive (var).
The power that “appears” to be supplied to the load is called the
apparent power:
S = VI [VA]
(2.37.2)
5
Real, reactive, and apparent power
Alternative forms of power equations
The magnitude of voltage across the load Z is:
Therefore:
V = IZ
(2.38.1)
P = I 2 Z cos θ
(2.38.2)
Q = I 2 Z sin θ
(2.38.3)
S = I 2Z
reactance
Z is the magnitude of the impedance
Since the impedance of the load Z is
Z = R + jX = Z cos θ + Z cos θ
(2.38.4)
P=I R
(2.38.5)
Q = I2X
(2.38.6)
2
6
Real, reactive, and apparent power
Complex power
For a simplicity of computer calculations, real and reactive power are
sometimes represented together as a complex power S:
S = P + jQ = VI*
(2.39.1)
We assume that V = Ve jα and Ι = Ie − j β and θ = α − β
and denote:
S = VI* = (V ∠α )( I ∠ − β ) = VI ∠ (α − β )
(2.39.2)
7
Real, reactive, and apparent power
The relationship between impedance angle, current angle, and power
V Ve j 0 V − jθ V
I = = jθ = e = ∠ − θ
Z
Z
Z Ze
(2.40.1)
If the impedance angle of the load is positive,
the phase angle of the current flowing through
the load will lag the phase angle of voltage.
If the impedance angle is positive, the load is
said to consume both real and reactive power
from the source (the reactive power is positive)
Capacitive loads have a negative impedance
angle and are said to consume real power
and supply reactive power to the source.
Inductive
load
Capacitive
load
8
Real, reactive, and apparent power
The power triangle
The real, reactive, and apparent power supplied to a load are related
by the power triangle.
cos θ is usually called a power factor of the load:
PF = cos θ
Since the sign of the impedance angle does not affect the PF, it is usually
specified whether the current is leading or lagging the voltage.
(2.41.1)