Environmental Engineering Reference
In-Depth Information
Figure 5.17
Series Connection of Resistance and Inductance with
Vector Diagram
The vector of the voltage
V
R
has the same direction as the vector of the curre
n
t
I
. The voltage across an inductance
L
with reactance
X
=
L
and j = e
j
2
,
ω
becomes:
(5.56)
The voltage
V
2
can now be calculated as:
(5.57)
The translation of the voltage vectors
V
R
and
V
L
provides the vector
V
2
that
closes the loop in the vector diagram.
The instantaneous power
p
(
t
) is calculated similarly to the DC power as:
(5.58)
The
active power P
is the difference between the positive and negative areas
defined by the curve
p
(
t
) and the horizontal time axis. Using
(5.59)
the active power
P
of a harmonic voltage and current curve with the phase
angle
ϕ
becomes:
(5.60)
If the phase displacement between current and voltage is ±
/2 (±90°), the
positive areas are equal to the negative areas. Thus, the active power becomes
zero. However, this does not mean that there is really no power. The remaining
power oscillates between consumer and generator. A lower level of oscillating
π
