**The twin-T filter**

The twin-T network is composed of two T-networks:

- The RCR network is formed by two resistors and one
capacitor .
- The CRC network is formed by two capacitors and one
resistor .

When the output is open-circuit, i.e., , the frequency
response function of the twin-T network can be found to be
(see here):

where

is the quality factor, and is the bandwidth of the filter. This twin-T network is a band-stop filter (notch filter) which attenuates the frequency to zero:

When this notch filter is used in a negative feedback loop of an amplifier, it becomes an oscillator.

**The active twin-T filter**

The bandwidth may not be narrow enough for most applications due to the small quality factor . To overcome this problem, an active filter containing two op-amp followers (with unity gain ) can be used to introduce a positive feedback loop as shown below:

Now the common terminal of the twin-T filter is no longer grounded,
instead it is connected a potentiometer, a voltage divider composed
of and , to form a feedback loop by which a fraction of the
output is fed back:

where , i.e., .

The input and output of the twin-T network are respectively and
, and they are now related by the frequency response function
of the twin-T network:

Rearranging and substituting , we get

Now the frequency response function of this active filter with feedback can be found to be

Substituting we get

where

are respectively the quality factor and the bandwidth of the active filter with feedback.

It can be shown (see here)
that the frequency response function of this active twin-T filter is

where

are respectively the quality factor and the bandwidth of the active filter with feedback. By changing and , the bandwidth can be adjusted. In particular,

- when , (no feedback), , ;
- when , (one hundred percent feedback), , .

**The bridged T filter**

If in the RCR T-network the vertical capacitor branch is dropped,
i.e., , the twin-T network becomes a bridged T network. Now
we have , while the CRC T-network is still the same with
, we get:

The frequency response function of this bridged T network (a voltage divider) is:

We let , and express both the numerator and the denominator in the canonical form as

where

are the bandwidth of the 2nd-order systems of the numerator and the denominator, respectively.

- If ,
- If ,
- If ,