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dB
90°
45°
0°
–45°
–90°
v (A v )
continued negative drop due to the f c asymptote, resulting in the leveling
response indicated. Due to the equal and opposite slopes of the
asymptotes in the midregion, the angles of f 1 and f c will be the same,
but note that they are less than 45°. The maximum negative angle will
occur between f 1 and f c. The remaining points on the curve of Fig. 23.67
can be determined by simply substituting specific frequencies into Eq.
(23.45). However, it is also useful to know that the most dramatic (the
quickest) changes in the phase angle occur when the dB plot of the
magnitude also goes through its greatest changes (such as at f 1 and f c).
23.13 HIGH-PASS FILTER WITH
LIMITED ATTENUATION
The filter of Fig. 23.68 is designed to limit the low-frequency attenuation
in much the same manner as described for the low-pass filter of the
previous section.
At f � 0 Hz the capacitor can assume its open-circuit equivalence,
and V o � [R 2/(R 1 � R 2)]V i. At high frequencies the capacitor can be
approximated by a short-circuit equivalence, and V o � V i.
The resistance to be employed when determining f c can be found by
finding the Thévenin resistance for the capacitor C, as shown in Fig.
23.69. A careful examination of the resulting configuration will reveal
that R Th � R 1 � R 2 and f c � 1/2p(R 1 � R 2)C.
A plot of V o versus frequency is provided in Fig. 23.70(a), and a
sketch of A v � V o /V i appears in Fig. 23.70(b).
An equation for A v � V o/ V i can be derived by first applying the
voltage divider rule:
V o �
1
R 2V i
��
R2 � R 1� �jX C
V o
f c /10
R 2
and Av � � �––––
R2 �� R
� ��
Vi R2 � R1� �jXC 1 ( �j XC)
�
R � j X
R2(R1 � jXC) R1R2 � jR2XC ���� ����
R2(R1 � jXC) � jR1XC R1R2 � jR2XC � jR1XC R 2
1
C
HIGH-PASS FILTER WITH LIMITED ATTENUATION ⏐⏐⏐ 1061
2 3 4 5
f 1 /10
f c
f 1
10 f c
FIG. 23.67
Phase angle for the low-pass filter of Fig. 23.62.
+
V i
–
10 f 1
R 1
C
R 2
+
–
f
V o
FIG. 23.68
High-pass filter with limited attenuation.
V i = 0 V
R 1
R Th
FIG. 23.69
Determining R for the f c calculation for the
filter of Fig. 23.68.
R 2