# How to find transfer function for this pi-filter using FACTS Method with 6 reactive elements?

Am trying to apply the FACTS method to find out the transfer function for this circuit as shown below. This is a Pi-filter circuit with all its component parasitics and also including the source and load resistances. I would like to find out the transfer function of this circuit and match its plot from Mathcad with simulation. First, to find out the zeros of the this transfer function by inspection, I placed all circuit elements in its high frequency state. I can observe the response Vout is still present. In that case, can I assume that this circuit has 6 zeros associated with it?

But as per the answer provided in this link, we have to place the other associated circuit element in its DC state and observe if the response is still present. Since this circuit has around 6 reactive elements, how do I decide which circuit element should be in DC State and which circuit element should be in high frequency state?

I was able to follow some examples done based on 2nd and 3rd order circuits shared in the above links. But with this circuit configuration and so many reactive elements, frankly am lost.

It would be great if you could share some insight on how to derive the transfer function for this circuit including its poles and zeros.

• Did you buy Linear Circuit Transfer Functions: An Introduction to Fast Analytical Techniques yet? Or are you only using EESE posts for your education? It matters. If you have the book and you are not understanding it well enough yet, I'd say that Basso needs to consider writing a more accessible one. If just EESE, then I've no problem understanding the difficulties. What's the situation? (Helps me to know what to write, should I have time today to do so.) Also, have you just tried to develop the TF in the usual way? Do you have a result, if so? And finally where's the input node reference? Sep 11 at 16:32
• The zeros will be found from: (r5/r3)*(c1*l1*s**2 + c1*r2*s + 1)*(c3*l3*s**2 + c3*r4*s + 1)*(c2*l2*s**2 + l2/r3*s + 1). Just FYI. You can spot the taus there pretty easily. Sep 11 at 17:31
• Am yet to get the book. Am learning about FACTS based on some answers shared in EESE and also some articles published by Basso on various topics where he uses this method to get the transfer function. But I have not come across an example where it has 6 reactive elements. By the way what is the method that you applied to find out the zeros of this transfer function. Sep 12 at 3:19
• Hello @Verbal Kint Could you help share some insight on how this circuit could be solved using FACTs? Sep 12 at 9:33

My calculation of the transfer function of the given circuit is based on the ratio of the impedances of a voltage divider. • I think this is a better approach. +1 Sep 12 at 10:29

If you are trying to analyze this in Mathcad, analyze the following simple circuit: If the source/load resistors are different you can account for that, it's just a little more complex. I used nodal analysis with only two nodes: and then use Mathcad to compute:

$$\Z1 = R_2+j\omega L_1 + 1/{j\omega C_1}\$$

Similarly for Z2 and Z3, then substitute in the expression for $$\V_o/V_i\$$