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.

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 .

Thus

For large (ie. and .

And thus

Tue Jul 13 16:55:15 EDT 1999