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ƒ r
X L
2pfsL 2p(15,915.49 Hz)(60 mH)
d. Qs � � ������ � 6.67
R R
900 �
f2 � f1 BW 1 1
e. � ��� � ��� 0.15
fs fs Qs 6.67
PARALLEL RESONANCE
20.8 PARALLEL RESONANT CIRCUIT
The basic format of the series resonant circuit is a series R-L-C combination
in series with an applied voltage source. The parallel resonant
circuit has the basic configuration of Fig. 20.21, a parallel R-L-C combination
in parallel with an applied current source.
For the series circuit, the impedance was a minimum at resonance,
producing a significant current that resulted in a high output voltage for
VC and VL. For the parallel resonant circuit, the impedance is relatively
high at resonance, producing a significant voltage for VC and VL through the Ohm’s law relationship (VC � IZT). For the network of Fig.
20.21, resonance will occur when XL � XC, and the resonant frequency
will have the same format obtained for series resonance.
If the practical equivalent of Fig. 20.21 had the format of Fig. 20.21,
the analysis would be as direct and lucid as that experienced for series
resonance. However, in the practical world, the internal resistance of the
coil must be placed in series with the inductor, as shown in Fig. 20.22.
The resistance Rl can no longer be included in a simple series or parallel
combination with the source resistance and any other resistance
added for design purposes. Even though Rl is usually relatively small in
magnitude compared with other resistance and reactance levels of the
network, it does have an important impact on the parallel resonant condition,
as will be demonstrated in the sections to follow. In other words,
the network of Fig. 20.21 is an ideal situation that can be assumed only
for specific network conditions.
Our first effort will be to find a parallel network equivalent (at the
terminals) for the series R-L branch of Fig. 20.22 using the technique
introduced in Section 15.10. That is,
ZR-L � Rl � j XL and YR-L � � � � j
1
� �—— �
j �� R2l � X
X
2 1
—
L
� �� R2l � X 2 XL �� 2
R l � X
L
�
2 Rl �2 R l � X L
2 1 1
� �
ZR-L Rl � j XL L
1 1
�� � �� Rp
jXLp
R l
with Rp � � (20.24)
R2l � X 2 L
�
and XLp � � (20.25)
R2l � X 2 L
�
as shown in Fig. 20.23.
L
R l
X L
PARALLEL RESONANT CIRCUIT ⏐⏐⏐ 901
I
Z T
R
FIG. 20.21
Ideal parallel resonant network.
R l
X L
L
X C
FIG. 20.22
Practical parallel L-C network.
+
C VC –