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# Sequential RC Filters

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 .

Thus

For large (ie. and .

And thus

Doug Gingrich
Tue Jul 13 16:55:15 EDT 1999