Environmental Engineering Reference
In-Depth Information
V
V
AB
VV
δ
-V
-V
AB
V
V
I
I
Figure 5.10
Phasor diagram illustrating voltages and current shown in Figure 5.9
There will be a voltage across this impedance, according to Ohm's law:
(5.3)
VIZ
=
AB
and, by Kirchoff's law,
=−
(5.4)
VVV
B
A
B
The phasor diagram of Figure 5.10 represents pictorially all the variables of Figure 5.9. Here
the current
I
at end A is shown lagging the voltage
V
A
by about 30 °. Hence the transmission
line at A is a sink of active (+
P
) and reactive (+
Q
) power and
j
Combining Equations (5.2) and (5.3) and taking
V
A
as the reference phasor with an angle
of zero gives
SP Q
=+
SZ
*
V
=
AB
V
PQRX
V
A
(
)
(
)
−
j
+
j
=
A
PR
+
QX
PX
−
QR
=
+
j
(5.5)
V
V
A
A
5.4.4 Simplifi cations and Conclusions
The use of complex numbers, as exemplifi ed above, provides a systematic and accurate means
of calculating magnitudes and phase angles of all quantities. Good engineering calculators
and many software packages can manipulate complex numbers directly and make such cal-
culations straightforward.
Nonetheless, the following approximations to Equation (5.5) are often very helpful, both
for rough calculations and to get a better feel for the interdependencies. First, there is often
interest in the scalar difference between the magnitudes of voltages
V
A
and
V
B
. Referring to
Figure 5.10, it can be seen that this difference is contributed mainly by the real part of
V
AB
(its horizontal component). Therefore Equation (5.5) can be simplifi ed to give a scalar
relationship:
PR
+
QX
(5.6)
Δ
V
≈
V
where
Δ
V
= |
V
A
|
−
|
V
B
|. The denominator
V
should really be |
V
A
|, but |
V
B
| can be used,
provided that
Δ
V
is small. Equation (5.6) provides a convenient way to estimate the voltage
