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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

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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:

enter image description here

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.

enter image description here

and it you plot the whole thing, you have:

enter image description here

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

http://cbasso.pagesperso-orange.fr/Downloads/PPTs/Chris%20Basso%20APEC%20seminar%202016.pdf

but also at the numerous transfer functions derived in the book

http://cbasso.pagesperso-orange.fr/Downloads/Book/List%20of%20FACTs%20examples.pdf

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

  • Start with the general formula for feedback:

    H(s)=Hf/(1/Acl+Hb)

    with Hf=Forward function, Hb=backward function (feedback factor) and Acl=closed loop gain of the opamp.

  • Hf=Vp/V1 for Vout=0 and Hb=Vp/Vout for Vin=0. (Vp=Voltage at pos input).

  • Acl=(1+R6/R7) (in your case: Acl=+2)

Using this method, you have split the whole task into two parts. Finally, you have to rearrange the function with the aim to get a second-order polynominal in the denominator.

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