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Chapter 15: Active Filters
A filter is a circuit that passes certain frequencies and
rejects or attenuates all others.
The passband is the range of frequencies allowed to pass
through the filter.
The critical frequency , f c , defines the end (or ends) of the
passband and is normally specified at the point where the
response drops -3dB (70.7%) from the passband response.
15.1: Basic filter Responses
Basic filter responses are:
f
Gain
f
Gain
f
Gain
f
Gain
Low-pass High-pass Band-pass Band-stop
15.1: Basic filter Responses
Low-Pass Filter Response
The low-pass filter allows frequencies below the critical frequency to pass (from dc to f (^) c ) and rejects other. The simplest low-pass filter is a passive RC circuit with the output taken across C. Æ The bandwidth of an ideal low-pass filter is
Ideal response (shaded area): ideal low-pass filter; no response for frequencies above fc
Actual response (curved line): the gain drops rapidly after fc with a rate decided by number of poles (number of RC circuits contained in the filter)
The critical frequency of a low-pass RC filter occurs when XC = R where
15.1: Basic filter Responses
Low-Pass Filter Response
The -20dB roll-off rate is not a particularly good filter characteristic (far from ideal filter) because too much of the unwanted frequencies (beyond the passband) are allowed through the filter In order to produce a more effective filter that has a steeper transition region, it is necessary to add additional poles ( RC circuits) combined with op-amps that have frequency-selective feedback circuits Æ filters can be designed with roll-off rates of -40dB, -60dB or more dB/decade as shown
Filters that include one or more op-amps in the design are called active filters. These filters can optimize the roll-off rate or other attribute (such as phase response) with a particular filter design.
15.1: Basic filter Responses
Band-Stop Filter Response
A band-stop filter rejects frequencies between two critical frequencies; the bandwidth is measured between the critical frequencies. The band-pass filter can be obtained by joining the low-pass filter with high-pass filter or by RLC circuit (not described in this chapter)
The bandwidth is
The center frequency f 0 is
quality factor ( Q )
15.2: Filter Response Characteristics
Active filters: include one or more op-amps in the design. These filters can provide much better responses than the passive filters illustrated befor. Active filter designs optimize various parameters such as amplitude response, roll-off rate, or phase response. Each type of filter response (low-pass, high-pass, band-pass, or band- stop) can be tailored by circuit component values to have either a Butterworth, Chebyshev, or Bessel characteristic.
Av
f
Butterworth: flat amplitude response
Chebyshev: rapid roll-off characteristic
Bessel: linear phase response
15.2: Filter Response Characteristics
The Damping Factor
The damping factor primarily determines if the filter will have a Butterworth, Chebyshev, or Bessel response. The damping factor in the shown general diagram of active filter is determined by the feedback resistors R 1 and R 2 and is defined by:
The value of the damping factor required to produce a desired response characteristic depends on the order (number of poles) of the filter. A pole is simply a circuit with one resistor and one capacitor. The more poles a filter has, the faster its roll-off rate is.
Every filter type (Butterworth, Chebyshev, or Bessel response) has it’s own damping factor table derived using a advanced mathematics (not covered)
15.2: Filter Response Characteristics
The Damping Factor
Because of its maximally flat response, the Butterworth characteristic is the most widely used Æ we will limit our coverage to the Butterworth response Parameters for Butterworth filters up to four poles are given in the following table. (See text for larger order filters).
(^4) − 80 2 1.848 0.152 2 0.765 1.
3 − 60 2 1.00 1.00 1 1.00 1.
2 − 40 2 1.414 0.
1 − 20 1 Optional
Order Poles DF R 1 /R 2 Poles DF R 1 /R (^2)
Roll-off 1 st^ stage 2 nd^ stage
Table for dB/decade Butterworth filter values
For example, To achieve a second-order Butterworth response Æ damping factor must be 1.414. Æ
Æ The gain which is 1 more than
the resistor ratio
15.3: Active Low-Pass Filters
The Sallen-Key Low-Pass Filter (Double-Pole Low-Pass Filter)
It is an active filter with a two low-pass RC circuits that provides a roll-off of -40 dB/decade
The critical frequency
The closed-loop voltage gain
The Sallen-Key is one of the most common configurations for a second-order (two-pole) filter.
If we choose RA = RB = R and CA = CB = C.
Æ critical frequency simplifies to
15.3: Active Low-Pass Filters
Cascaded Low-Pass Filters
Third-order or higher low-pass response (- dB/decade or lower) can be done by cascading a single pole and/or two-pole low- pass filter
Third order configuration; 3-poles (2-poles stage1 + 1-pole stage 2)
Fourth order configuration; 3-poles (2-poles stage1 + 1-pole stage 2)
15.3: Active Low-Pass Filters: Example
Determine the critical frequency of the Sallen-Key low-pass filter in Figure, and set the value of R 1 for an approximate Butterworth response.
15.3: Active Low-Pass Filters: Example
For the four-pole filter in Figure before in cascaded filters determine the capacitance values required to produce a critical frequency of 2680 Hz if all the resistors in the RC low-pass circuits are 1.8 kΩ. Also select values for the feedback resistors to get a Butterworth response
15.5: Active Band-Pass Filters
As mentioned, band-pass filters pass all frequencies bounded by a lower-frequency limit and an upper-frequency limit and reject all others lying outside this specified band
Cascaded High-Pass and Low-Pass Filters
implementing a band- pass filter can be done by cascading arrangement of a high-pass filter and a low-pass filter, as shown in Figure
