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1028 ⏐⏐⏐ DECIBELS, FILTERS, AND BODE PLOTS
+ R
–
+ R
V i
–
FIG. 23.8
Low-pass filter.
+
C V o
+
–
V o = V i
FIG. 23.9
R-C low-pass filter at low frequencies.
+ R
V i
–
+
–
V o = 0 V
FIG. 23.10
R-C low-pass filter at high frequencies.
–
dB
Since this text is limited to passive devices, the analysis of this chapter
will be limited to passive filters. In addition, only the most fundamental
forms will be examined in the next few sections. The subject of
filters is a very broad one that continues to receive extensive research
support from industry and the government as new communication systems
are developed to meet the demands of increased volume and
speed. There are courses and texts devoted solely to the analysis and
design of filter systems that can become quite complex and sophisticated.
In general, however, all filters belong to the four broad categories
of low-pass, high-pass, pass-band, and stop-band, as depicted in Fig.
23.7. For each form there are critical frequencies that define the regions
of pass-bands and stop-bands (often called reject bands). Any frequency
in the pass-band will pass through to the next stage with at least 70.7%
of the maximum output voltage. Recall the use of the 0.707 level to
define the bandwidth of a series or parallel resonant circuit (both with
the general shape of the pass-band filter).
For some stop-band filters, the stop-band is defined by conditions
other than the 0.707 level. In fact, for many stop-band filters, the condition
that V o � 1/1000V max (corresponding with �60 dB in the discussion
to follow) is used to define the stop-band region, with the passband
continuing to be defined by the 0.707-V level. The resulting
frequencies between the two regions are then called the transition frequencies
and establish the transition region.
At least one example of each filter of Fig. 23.7 will be discussed in
some detail in the sections to follow. Take particular note of the relative
simplicity of some of the designs.
23.5 R-C LOW-PASS FILTER
The R-C filter, incredibly simple in design, can be used as a low-pass or
a high-pass filter. If the output is taken off the capacitor, as shown in
Fig. 23.8, it will respond as a low-pass filter. If the positions of the
resistor and capacitor are interchanged and the output is taken off the
resistor, the response will be that of a high-pass filter.
A glance at Fig. 23.7(a) reveals that the circuit should behave in a
manner that will result in a high-level output for low frequencies and a
declining level for frequencies above the critical value. Let us first
examine the network at the frequency extremes of f � 0 Hz and very
high frequencies to test the response of the circuit.
At f � 0 Hz,
1
XC ���∞� 2pfC
and the open-circuit equivalent can be substituted for the capacitor, as
shown in Fig. 23.9, resulting in Vo � Vi. At very high frequencies, the reactance is
1
XC �� � 0 �
2pfC
and the short-circuit equivalent can be substituted for the capacitor, as
shown in Fig. 23.10, resulting in Vo � 0 V.
A plot of the magnitude of Vo versus frequency will result in the curve
of Fig. 23.11. Our next goal is now clearly defined: Find the frequency
at which the transition takes place from a pass-band to a stop-band.