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a c
0°
–30°
–45°
–60°
–90°
θL
of IL θL = θs
5 10 20 30 40 f (kHz)
Network inductive
θ L ( f )
Network resistive
θR ≅ θs
≅ 0°
FIG. 15.84
The phase angle of the current I L versus frequency for the parallel R-L network
of Fig. 15.79.
lower-resistance path of the resistor. The magnitude of the phase angle
between I L and I is approaching 90° with an increase in frequency, as
shown in Fig. 15.84, leaving its initial value of zero degrees at f � 0Hz
far behind.
At f � 1 kHz, the phasor diagram of the network appears as shown
in Fig. 15.85. First note that the magnitude and the phase angle of I L are
very close to those of I. Since the voltage across a coil must lead the
current through a coil by 90°, the voltage V s appears as shown. The
voltage across a resistor is in phase with the current through the resistor,
resulting in the direction of I R shown in Fig. 15.85. Of course, at
this frequency R > X L, and the current I R is relatively small in magnitude.
At f � 40 kHz, the phasor diagram changes to that appearing in Fig.
15.86. Note that now I R and I are close in magnitude and phase because
X L > R. The magnitude of I L has dropped to very low levels, and the
phase angle associated with I L is approaching �90°. The network is
now more “resistive” compared to its “inductive” characteristics at low
frequencies.
The analysis of a parallel R-C or R-L-C network would proceed in
much the same manner, with the inductive impedance predominating at
low frequencies and the capacitive reactance predominating at high frequencies.
15.11 SUMMARY: PARALLEL ac NETWORKS
The following is a review of important conclusions that can be derived
from the discussion and examples of the previous sections. The list is
not all-inclusive, but it does emphasize some of the conclusions that
should be carried forward in the future analysis of ac systems.
SUMMARY: PARALLEL ac NETWORKS ⏐⏐⏐ 673
I R
V s
83.49°
–6.51°
FIG. 15.85
The phasor diagram for the parallel R-L network
of Fig. 15.79 at f � 1 kHz.
I L
I L
12.35° IR Vs –77.65°
I
FIG. 15.86
The phasor diagram for the parallel R-L network
of Fig. 15.79 at f � 40 kHz.
I