transfer function of active filters

I've been looking for some straightforward method or trick to obtain the transfer functions of active filters (like the Sallen-Key filter , the butterworth or Cauer topology etc...) since KCL or KVL requires a lot of algebraic manipulations . I've heard that SFG (signal flow graphs) could be very helpful , but with my minimum of experience i can not found any worth reference that treat the subject , so what i am looking for is some brief explanation of SFG with introductory applications to the basic models and topologies of active filters and thanks.

• Usually the design flow goes the other way -- design filter topology, figure out where your poles need to be, pick an active filter topology, and then calculate passive component values to put the poles where you want them. Yes, the algebra and simultaneous equations you need to solve can be a bit tedious -- that's why we often stick to canned topologies for active filters. – Scott Seidman Mar 26 '13 at 12:25

I can't really speak for signal flow graphs, simply because I don't use them. I've always seem them as useful for planning an algorithm in assembly, but for non discrete time signals, they're more confusing than helpful.

It is possible to extract transfer functions directly from a Sallen-Key circuit, but you need to know what you're looking for, and what the basic transfer functions look like. For example:

Here we have an assumed Sallen-Key second order low pass, where $R=R_1=R_2,\ C=C_1=C_2$. The generic transfer function is known, as well as $\alpha=3-\dfrac{R_4}{R_3}$, and $\omega_c=\dfrac{1}{RC}$. From there, it can be reconstructed.

The SFG is useful. And you can read these paper.

1、Feedback Analysis of Transimpedance Operational Amplifier Circuits

２、'Signal Flow Graphs for Getting Transfer Functions of Active Filters' Maybe this paper is better except for its being written in Latin. But It's so simple that you can get it by the figures in the paper.

I know this may sound unkosher, but why simply not run it on spice of your choosing?

If you need to do algebraic manipulations, it's time to learn a symbolic algebra package. Maxima is free and works reasonably well.