# Butterworth passive filters normalized values equation

Recently I'm reading in-depth about butterworth filters, more specifically about the normalized values the tables give. Most of the tables give you the normalized values for the case of RL/RS=1 or RL/RS=∞ but I have found a table that gives the values for other ratios too.

Now the equation that these values come from (maybe this not the only way) is the Cauer topology equation that is shown here (equation 10.5.1): But this equation is valid only for RS=RL=1Ohm so my goal is to find out the general equation that counts in the RS and RL to be able to calculate the normalized values for any resistors ratio.

For some reason I can't find where this equation came from or how it can be changed to include the resistors as parameters.

How can I generalize this equation?

Start here with the Cauer topology:

https://en.wikipedia.org/wiki/Butterworth_filter

Notice that these are the same as your equations. Also notice that the equations don't specify the frequency either, it is assumed to be 1 rad/sec. The diagram doesn't show the source and load resistor, but the text says that they are unity (1).

Now go here to the Impedance Scaling section:

https://en.wikipedia.org/wiki/Prototype_filter

This shows you how to scale to your resistance and cutoff frequency.

• Hello Mattman, the impedance denormalization requires that your new desired resistors are equal. i.e. for a second order low pass passive filter if your desired resistors are RL=RS=2 you pick 1.414 from the table and you multiply-devide this value by 2. The question in the end is how you can denormalize the impedance if your desired RL, RS resistors are unequal by using the normalized values for RL=RS=1. i.e. for the above example I want to denormalize for RS=2 and RL=4 using the normalized values 1.414. Any help would be appreciated, thank you. May 4 '19 at 9:35

Replying to my own question in case anyone has the same wondering... the answer to the problem "How to denormalize the impedance for any ratio of terminal resistors is to use Bartletts’ Bisection Theorem or the Darlington realization procedure (of course these may not be the only ways to achieve this).