Memo: Active frequency filters

Frequency filters attenuate signals outside its band-pass thresholds. There are two flavors of frequency filters: active and passive. Active filters have op-amps and offer amplification, as well as impedance matching functionality, of the output signal. Passive filters are simple R-C-L networks without any op-amp.

Band-pass and Band-stop construction

A band-pass filter is defined as follow:

band-pass filter is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range.

Band-stop filter is the opposite of Band-pass filter.

Both band-pass and band-stop filters can be created from low-pass and high-pass filters via the corresponding arrangement as shown below:

band stop filter configuration
Band-stop filter construction: Summing low-pass and high-pass filter
band pass filter design
Band-pass filter construction: cascading low-pass and high-pass filter

On a related note, there exists a special form of band-stop filter called notch filter and it is defined as follow:

notch filter is a band-stop filter with a narrow stopband (high Q factor).

with Q factor is, in turn, defined as:

The Q-factor is the reciprocal of the fractional bandwidth. A high-Q filter will have a narrow passband and a low-Q filter will have a wide passband. These are respectively referred to as narrow-band and wide-band filters.

Frequency filters conversion

There are a total of four frequency filters and they can be combined with other electronics and one another to create a new filter. Their second order relationships are described by the following equations:

(1) Low-pass filter = ∫(Band-pass filter)

(2) Band-stop filter = X(t) – Band-pass filter

(3) Band-pass filter = ∫(High-pass filter)

For first-order systems, derivations from Band-stop and Band-pass constructions permit the following:

(4) High-pass filter = Low-pass filter – Band-stop filter

(5) Band-pass filter = Low-pass filter (High-pass filter)

IMPORTANT: Anything that works in first order also works in second order.

First order active filters

For the first order filters, a resistor and capacitor bridge is employed. The use of feedback eliminates the need for inductors as used in first order passive filters.

Low-pass filter has the resistor near the source while high-pass filter has the capacitor. In high-pass filters, a small resistor might be included between the source and the capacitor to prevent overloading the capacitor.

Non-inverting low-pass filter
Non-inverting high-pass filter

The difference between inverting and non-inverting filters is the input terminal the input source is connected to. If Vin is connected to the inverting terminal, it’s an inverting filter.

Inverting low-pass filter
Inverting high-pass filter

In many cases, the feedback should always be negative for filters and amplifiers alike. Positive feedback will result in hysteresis circuits (Schmitt triggers).

Second order active filters

Second order filters offer more drastic attenuation (steep roll-off). The simplest kinds are based on Sallen-Key topology

Generic Sallen-Key topology

Second order filters are designed around a non-inverting amplifier with equal resistor and capacitor values. The specific values are determined by the cut-off frequency desired by the designer. As with first-order filters, low-pass filters have a resistor near the source and high-pass filters have a capacitor.

second order low pass filter
Second order active low-pass filter
second order high pass filter
Second order active high-pass filter

Second order universal active filter (Conglomerate filters)

The all the filters above are “section”, meaning, they stand alone and are not dependent on any other filters. The opposite of section filters is “conglomerate” filters. Every component of the conglomerate filter must be in order for the entire filter to work correctly. They offer reduced circuitry at the cost of reliability due to high dependency on other components of the system.

At the core, the universal filter is one such conglomerate. Each output of the four op-amps provides a different second order filter behavior:


2nd order high-pass filter

Band-pass filter

2nd order low-pass filter

The universal filter is based on cascading two inverting amplifiers blocks and two inverting integrators blocks with additional, second-order feedback loops from each integrator back to the input of the amplifier furthest from it.

Second order universal active filter circuit

Tow-Thomas biquad filter is another conglomerate filter offering low-pass and band-pass characteristics depending on where the input is taken.

Read more (Passive low pass) (Passive high pass) (Passive bandpass) (Active low pass) (Active high pass) (Active bandpass)

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