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Low-Pass Filter Design
The multiplication of the denominator terms with each other yields an n th order polynomial
of S, with n being the filter order.
While n determines the gain rolloff above f C with n·20 dBdecade, a i and b i determine
the gain behavior in the passband.
In addition, the ratio
bi
a i
more a filter inclines to instability.
Q is defined as the pole quality. The higher the Q value, the
16.3 Low-Pass Filter Design
Equation 16–1 represents a cascade of second-order low-pass filters. The transfer function
of a single stage is:
A i
(s)
A 0
1 ai s b i
s 2
(16–2)
For a first-order filter, the coefficient b is always zero (b 1 =0), thus yielding:
A(s) A 0
1 a 1
s
(16–3)
The first-order and second-order filter stages are the building blocks for higher-order filters.
Often the filters operate at unity gain (A 0 =1) to lessen the stringent demands on the op
amp’s open-loop gain.
Figure 16–11 shows the cascading of filter stages up to the sixth order. A filter with an even
order number consists of second-order stages only, while filters with an odd order number
include an additional first-order stage at the beginning.
Active Filter Design Techniques
16-11