How to convert Sallen-Key Low pass filter to Signal Flow Graph

[Edward1212]

 

[LvW]

The circuit consists of a fixed-gain amplifier (opamp with R3, R4) and a positive feedback function. Hence, where is the problem to find the signal-flow graph?

 

[Audioguru]

There are two RC lowpass networks, R1 and C1 and R2 and C2. Each RC network has an output of -3dB at the cutoff frequency.
Actually, the positive feedback occurs only near the cutoff frequency so that the -6dB response is boosted to be -3dB and the cutoff is sharp instead of droopy.

 

[LvW]

There are two RC lowpass networks, R1 and C1 and R2 and C2. Each RC network has an output of -3dB at the cutoff frequency.
.

No - it is obvious that the feedback network consists of a band pass (highpass C1-R1, lowpass R2-C2).
As a consequence, there is no lowpass -6dB response.

 

[Audioguru]

No - it is obvious that the feedback network consists of a band pass (highpass C1-R1, lowpass R2-C2).
As a consequence, there is no lowpass -6dB response.

I disagree. C1-R1 is a lowpass, not a highpass.
If C1 connects to ground instead of the the opamp output then there is no boost and the output at the cutoff frequency will be -6dB and the slope will gradually be 12dB per octave.

 

[LvW]

We are discussing the FEEDBACK network. Hence, you have to look into the circuit from the opamp output.
You cannot deny that - in this case - the feedback network resembles the well known highpass-lowpass CR-RC bandpass characteristic. This is a well-known property of a positive-gain Sallen-Key lowpass.

 

[Audioguru]

The bandpass circuit occurs only near the cutoff frequency to boost the response so that it has flat levels at low frequencies, -3dB at the cutoff frequency instead of a droopy -6dB and a sharp 12dB per octave slope at higher frequencies. Above the cutoff frequency both RC networks are lowpass filters.

 

[LvW]

The bandpass circuit occurs only near the cutoff frequency to boost the response so that it has flat levels at low frequencies, -3dB at the cutoff frequency instead of a droopy -6dB and a sharp 12dB per octave slope at higher frequencies. Above the cutoff frequency both RC networks are lowpass filters.

I don`t understand your position.
A bandpass is a bandpass - full stop.
Just one question, which can be answered with yes/no:
Do you agree that - between the ouput pin and the non-inv. input of the opamp - there is the classical four-element ladder network C1-R1-R2-C2 ? And this is the well known RC-bandpass.

How can you say that "Above the cutoff frequency both RC networks are lowpass filters"?

At first, we don`t have two RC-networks because you are not allowed to separate them, because they influence each other.
Secondly, the series cap C1 - of course - has highpass properties. I don`t think that you will argue against this.

 

[Audioguru]

I see the circuit as two lowpass RC networks at frequencies above cutoff like this:

 

[LvW]

OK - I know what you mean. But that does not answer the question.
The question concerns the corresponding signal-flow diagram (which shows forward and backward ways.).
That is the background we are speaking about FEEDBACK.
And - it does not matter how the circuit looks like without feedback.
It is a fact that the Sallen-Key lowpass has a feedback function that resembles a bandpass.
This is the background for realizing a complex pole pair.