# Sallen-Key active bandpass first order filter

I have an assignment to manually derive transfer function for this active filter. I tried 2 ways. First I did it with kirchoff and substitution, and I get wrong function, and then I tried control systems approach by dividing it into blocks with feedback, but the function takes almost half a page. Is there an easy way to derive it? I found only provided ffunctions for this and that wont do.

Thanks!

Update: I added link here to my calculations. Va, Vb and Vc denote the nodes. I checked twice, could not find mistakes. http://imgur.com/a/xT8Qj

I am not using classical node-mesh analysis here but rather the fast analytical circuits techniques (FACTs). I am not saying they are the best path here but I could derive the transfer function of this second-order filter quite quickly with the help of a few individual sketches I could fix in case of problems. The generalized transfer function of a 2nd-order filter is defined as

$H(s)=\frac{H_0+s(H^2\tau_2+H^3\tau_3)+s^2H^{32}\tau_3\tau_{32}}{1+s(\tau_2+\tau_3)+s^2(\tau_3\tau_{32})}$

The time constants in the denominators are found by setting the excitation to 0 (reduce $V_{in}$ to 0 V and replace it by a short circuit) and "looking" at the individual resistance driving each capacitor. The resistance multiplied by the capacitor it drives forms the time constants $\tau$ we want to determine the denominator $D(s)$. Then, we look for the gain of this circuit in dc, when $s=0$ and it gives us $H_0$. Finally, we determine the gain of this filter when some of its capacitors are set in their hi-frequency state. For instance, if I replace $C_3$ by a short circuit (its hi-frequency state), the gain $H^3=0$ and you continue with the rest of the gains to form $N(s)$. The sketches appear below:

When you have determined all these time constants (you obviously need KVL and KCL here), you gather all the elements in Mathcad to obtain the final expression that I put in a low-entropy form, making the band-pass gain clearly appearing.

and it you plot the whole thing, you have:

The FACTs are an excellent way of deriving transfer functions in a swift and efficient manner. I have gone through individual sketches to determine the $a_i$ and $b_i$ of $N(s)$ and $D(s)$ meaning that if I observe a deviation at the end between simulation and the Mathcad plot, I can fix the guilty coefficient immediately, something I could not do with the classical approach. If you want to know more about FACTs, have a look at the seminar taught at APEC 2016

but also at the numerous transfer functions derived in the book

Here is the way I would choose for manually derive the transfer function: