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dB
V o = 0.707V i
0
V o
Pass-band
fc Stop-band
FIG. 23.11
V o versus frequency for a low-pass R-C filter.
f (log scale)
For filters, a normalized plot is employed more often than the plot of
Vo versus frequency of Fig. 23.11.
Normalization is a process whereby a quantity such as voltage,
current, or impedance is divided by a quantity of the same unit of
measure to establish a dimensionless level of a specific value or
range.
A normalized plot in the filter domain can be obtained by dividing
the plotted quantity such as Vo of Fig. 23.11 with the applied voltage Vi for the frequency range of interest. Since the maximum value of Vo for
the low-pass filter of Fig. 23.8 is Vi, each level of Vo in Fig. 23.11 is
divided by the level of Vi. The result is the plot of Av � Vo/Vi of Fig.
23.12. Note that the maximum value is 1 and the cutoff frequency is
defined at the 0.707 level.
1
0.707
0
A v =
V o
V i
Pass-band
fc Stop-band
FIG. 23.12
Normalized plot of Fig. 23.11.
f (log scale)
At any intermediate frequency, the output voltage V o of Fig. 23.8
can be determined using the voltage divider rule:
or
and
V o
V o �
XC ��90°
Av ����� �
Vi R � j XC
V o
X C
X C ��90°V i
��
R � j XC
X C ��90°
���
�R� 2 ��� X� 2 C� / �tan�1 (X C/R)
X C
� �
Av � � ��90° � tan �1
� ��
�
2 2
Vi �R� ��� X� C�
R
R-C LOW-PASS FILTER ⏐⏐⏐ 1029