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P q
s
j
I 2 X L = Q L
I 2 X C = Q C
S = I 2 Z
P R = I 2 R
Therefore, the third power can always be found if the other two are
known.
It is particularly interesting that the equation
(19.29)
will provide the vector form of the apparent power of a system. Here,
V is the voltage across the system, and I * is the complex conjugate of
the current.
Consider, for example, the simple R-L circuit of Fig. 19.15, where
V 10 V �0° 10 V �0°
I ����� ��� �2 A ��53.13°
ZT 3 ��j 4 � 5 � �53.13°
The real power (the term real being derived from the positive real axis
of the complex plane) is
and the reactive power is
P � I 2 R � (2 A) 2 (3 �) � 12 W
Q L � I 2 X L � (2 A) 2 (4 �) � 16 VAR (L)
with S � P � j Q L � 12 W � j 16 VAR (L) � 20 VA �53.13°
as shown in Fig. 19.16. Applying Eq. (19.29) yields
S � VI * � (10 V �0°)(2 A ��53.13°) � 20 VA �53.13°
as obtained above.
The angle v associated with S and appearing in Figs. 19.10, 19.11,
and 19.16 is the power-factor angle of the network. Since
P � VI cos v
or P � S cos v
Q (resultant) = Q L – Q C = I 2 (X L – X C )
FIG. 19.14
The result of multiplying each vector of Fig. 19.13 by I 2 for a series R-L-C
circuit.
S � VI *
then Fp � cos v � � (19.30)
P
S �
THE POWER TRIANGLE ⏐⏐⏐ 859
+
I
V = 10 V 0°
–
R
3 �
X L
4 �
FIG. 19.15
Demonstrating the validity of Eq. (19.29).
S = 20 VA
v = 53.13°
P = 12 W
Q L = 16 VAR
FIG. 19.16
The power triangle for the circuit of
Fig. 19.15.