Single-pole filters are rather limited (6 dB/octave slope). For better band-pass and band-reject filters we require more poles and zeros and thus more reactive circuit elements. A simple solution is to connect two or more single-pole RC filters in sequence. If filter draws no current from filter , the transfer function for the combined filter is
One way to do this is to choose a large impedance for . Hence has more poles and zeros than or .
The corner frequency for a high-pass filter is and the transfer function may be written as
For a low-pass filter the corner frequency is and
We may build a two-section low-pass filter by requiring and , as shown in figure 3.7, so that
Figure 3.7: Two-section low-pass filter.
A special case occurs when and we obtain one corner frequency but the slope of the filter is .
The results are similarly for a two-section high-pass filter.
A band-pass filter can be built from one low-pass filter and one high-pass filter, as shown in figure 3.8. The order of the filter sections does not matter as long as the impedance rule is obeyed.
Figure 3.8: Band-pass filter.
If we have only two straight regions and the band-pass frequency range degenerates to zero.
Example: Show that the magnitude of the transfer function
falls off -6 dB/octave at both the low- and high-frequency extremes.
For small (ie. and .
For large (ie. and .