# Need some clarification on a transfer function.

#### awesomeamy

Joined Oct 21, 2015
1
Hi there, I'm new so be nice. I have a active filter and its transfer function but I want to learn how to derive an active filter. So if I could get some help working this out it would be very helpful.

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#### StayatHomeElectronics

Joined Sep 25, 2008
1,073
This looks very much like a homework problem. Please show your work first so others can help point you in the right direction.

#### Uul Sid

Joined Dec 26, 2015
2
Yeah, you have to show your work first. I can give u some hints to solve ur problem.
First, u need to convert the capacitor into impedance ( I bet u know the formula). After that, just use a simple node analysis between input and outputs.
Second, u have to have a math sense in doing ur problem. It's not too difficult to gain that sense of math. U will get used to it.
That's all from me.

#### sailorjoe

Joined Jun 4, 2013
365
I don't know if this is homework for you or not, Amy, but I do know that learning to derive an active filter is at least a half a semester of study just to get started. After that, you could study it for years and still keep learning.

If memory serves, you have the circuit and transfer function for a Sallen-Key low pass filter. It's described in Wikipedia, among other places. It's popular because it gives you a second order filter with a single op amp. There are other ways to do that, too.

Is there a specific issue you're struggling with, Where we could help?

#### MrAl

Joined Jun 17, 2014
11,583
Hi there, I'm new so be nice. I have a active filter and its transfer function but I want to learn how to derive an active filter. So if I could get some help working this out it would be very helpful.

View attachment 93280

Hi there,

I somehow missed this thread and i see now it is about 2 months old, so i'll provide a little information and if you find it sounds interesting we can continue.

First, for the pure theoretical circuit the analysis is fairly simple, and that is what you seem to be after so we will start with that. If you want to add practical aspects later we will also be able to do that.

The simplest way to proceed is to replace the op amp with a voltage controlled voltage source. There are other approaches, but this one seems the simplest for filters that act on a voltage and provide a voltage output. In case you dont know that a VCVS (votlage controlled voltage source) is, i suggest you start by reading about that and work out some simple examples first.

When using a VCVS, there is a gain associated with it which could be 1, 10, 100, 1000, or even infinite. For the purely theoretical op amp circuit we use a variable gain which could simply be "A" or more common "Aol" (which stands for the open loop gain) and we later allow this gain to go towards infinity. That gives us the theoretical analysis of the circuit because the theoretical version is almost always the one where the gain is considered to be infinite.

Once the op amp is replaced with the VCVS, the circuit is analyzed in the usual way using whatever kind of analysis you like to use such as nodal analysis. The result of this analysis is the transfer function sometimes called H(s) or T(s). H(s) or T(s) will depend on the open loop gain which we will call "A" for now, and the next step would be to take the limit of H(s) as A goes to positive infinity. The result is then the theoretically exact transfer function which will no longer depend on A.

The next step would be to decide if you want the frequency response or the time response. From your diagram you appear to be looking for the frequency response. To get the frequency response you then replace every ocurrance of 's' with "w*j" where j is the complex operator. Then using complex algebra, reduce the equation to the simplest form. That gives you the frequency response as a function of 'w' which is 2*pi*f. Replacing 'w' with 2*pi*f then gives you the frequency response as a function of the frequency 'f'.

If you want to find the time domain solution that requires some other skills such as being able to find the Inverse Laplace Transform, or calculus. You can always leave this for later in the future.

For the final step you might check to see that the circuit does not oscillate or latch up for certain values.

That's the procedure in a nut shell.

Prerequisites:
1. Circuit analysis skills that include dependent sources.
2. Complex algebra.

There's not much to it really so after you do a few examples you'll be turning these out in no time.
If you dont have the prerequisites listed above then you will do well to look into them first or else you'll be stuck with being able to only solve SOME circuits like this.