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1040 ⏐⏐⏐ DECIBELS, FILTERS, AND BODE PLOTS
+
–
R l
2 �
V i = 20 mV ∠ 0°
L
1 mH
C
0.01 mF
R 33 �
FIG. 23.35
Series resonant pass-band filter for
Example 23.8.
+
V o
–
For the parallel resonant circuit
Qp � � XL
�
R
(23.28)
and BW � �� (23.29)
Q
As a first approximation that is acceptable for most practical applications,
it can be assumed that the resonant frequency bisects the bandwidth.
EXAMPLE 23.8
a. Determine the frequency response for the voltage V o for the series
circuit of Fig. 23.35.
b. Plot the normalized response A v � V o /V i.
c. Plot a normalized response defined by A′ v � A v/A vmax .
Solutions:
a. fs � � � 50,329.21 Hz
Qs � � �9.04
BW � � �5.57 kHz
At resonance:
Vomax � � �0.943V 1
1
� ���
2p�L�C� 2p�(1� m�H�)(�0. �01� m�F) �
XL 2p(50,329.21 Hz)(1 mH)
� ���
R � Rl 33 ��2 �
fs 50,329.21 Hz
� ��
Qs 9.04
RVi 33 �(Vi) � ��
i � 0.943(20 mV)
R � Rl 33 ��2 �
� 18.86 mV
At the cutoff frequencies:
Vo � (0.707)(0.943Vi) � 0.667Vi � 0.667(20 mV)
� 13.34 mV
Note Fig. 23.36.
18.86 mV
13.34 mV
0
V o
l
f p
p
BW = 5.57 kHz
f s ≅ 50.3 kHz f (log scale)
FIG. 23.36
Pass-band response for the network.
dB
b. Dividing all levels of Fig. 23.36 by V i � 20 mV will result in the
normalized plot of Fig. 23.37(a).