Understanding bootstrapping in analog active filters

Question

"Bootstrapping" is a common technique in (analog) active filters, where the output of the filter is fed back to a node that would otherwise be connected to ground.

For example, the Sallen Key topology begins with two RC L-sections followed by a unity-gain buffer. Then the output of the buffer is also connected to the base of the first L-section (C3 in the following schematic). This is said to "bootstrap" that node.

Here is another example, from an old National Semiconductor / Texas Instruments app note (LB-5 / snoa690), in which the ground node of a Twin-T notch filter is 'bootstrapped' in order to greatly increase the Q of the filter:

How can I understand the operation of the 'bootstrap' intuitively?

Answer 1

A sine signal going through a T notch filter is phase shifted.

So bootstrapping means connecting the point normally connected to GND to the buffered phase shifted signal.
So the signal at notch frequency (or near it) not only has to work against GND but against its own opposite (negative signal).

This improves the filter (higher Q) as described in the app note.

Answer 2

A simpler example would be a transistor voltage follower in which bootstrapping is used to increase the input impedance. Normally, the input impedance is significantly affected (reduced) by the bias resistors R1 & R2.

By applying some of the output back to a series resistor R3 the input impedance is increased since the same voltage (or nearly) appears on either side of R3 so little current flows through it and the effect of the bias resistors is reduced.

With active filters, the situation is more complicated due to phase shift. Curd gives an explanation which could also be applied to bandpass filters but I'm not sure that there is any reasonable way of intuitively understanding how this works in the case of the S&K LPF. Perhaps others know different!

Answer 3

I wonder if you have found the term "bootstrapping" in connection with this filter circuit.

As shown in the previous post (MikeJ-UK) this method is used for the purpose of increasing the input impedance of an amplifier circuit.

However, for filtering purposes the method of feedback is used exclusively for the purpose of increasing the quality factor (pole-Q) of the transfer function (thus allowing Butterworth and Chebyshev responses).

Without such feedback, any RC-filter circuit would be passive with a maximum of Q=0.5.

As can be shown, the input impedance of the shown S&K filter is not drastically influenced by the feedback path.

For DC the input current is (nearly) zero and for very high frequencies the input current is determined by the resistors only (with and without feedback)