# Butterworth filter with non arbitrary component

How do I design a filter with certain cutoff frequency and as flat pass-band as possible, given that:

1. one of the filter components is non arbitrary, and
2. that filter component does not match the corresponding Butterworth prototype (i.e., the impedance conditions and cutoff frequency)?

For this question, I understand that performance will be degraded compared to real Butterworth filter. Still, is there known analytical or numerical method for designing this to match the Butterworth filter as closely as possible?

As you see in the schematic below, I can arbitrarily choose values for $L_{1}$ and $C_{2}$. However, I cannot change the source or load impedances (since they do not change with frequency), nor can I change $C_{fixed}$.

simulate this circuit – Schematic created using CircuitLab

• What are the values for R_source and R_load? Jun 21, 2017 at 18:08
• Can you use active filters? What's the frequency of the pole introduced by your C_fixed compared to the bandwidth of your desired response? What order filter do you want to end up with? Jun 21, 2017 at 18:46
• If that C_fixed must stay onboard, can you add something as parallel or series with it? This way you could get any effective C value. What makes it fixed? Is it some kind of cornerstone that is soldered , raffled or calculated by some VIP that you cannot insult? Or is this a homework? Jun 21, 2017 at 20:23
• Can you add zeroes? If yes, then go for inverse Chebyshev, it has more flatness than Butterworth. Jun 23, 2017 at 4:54
• Actually the point was just to come up with a rationalization for selecting the values I do. It is easy enough to simulate or calculate the response of the filter with certain values to see if it is satisfactory but when designing product or writing a paper I think it would be better if the values were based on something no really matter if it is butterworth or chebysev or what. Jun 25, 2017 at 12:12

Like the Butterworth itself, this is an optimization problem.

Pick a way to parametrize "flatness in the Pass Band" as an error term (call it E) -- maybe rms deviation from flat, or some such -- figure out the closed form solution E in terms of your fixed and floating parameters, and minimize it.

The quickest method is plug the circuit into a simulator like LTSpice to find values for the unknown components that produce what you feel to be the best response. Sims like this are free but have a steep learning curve but with something as simple as this is probably worth the effort and you'll never look back once you have got the hang of it because you'll be able to use the sim for all manner of electronic circuits.

Algebraically, this could be an awkward problem to solve but numerically it'll be a breeze to a sim.

There is a free filter design tool called Elsie that allows you to fix components or find closest commercial values while showing you the effect on the passband, stop band, etc. It saves me a lot of back and forth calculations in situations like yours.

You probably can find proper source and load impedances that allow the fixed capacitor (= previously selected) and still give to you the wanted frequency response. By adding 2 resistors to both input and output you can fool the LC-circuit to see the right source and load impedances, but you must accept the caused losses.

You can prevent the losses if you can insert buffer amps just after the source and just before the final load. Probably only one amp is needed because there are available prototype filters with different impedance ratios.

Finally there's allways possible to use circuit analysis to find if the result still is acceptable regardless some wrong values. Nearly any AC analysis capable program can be used, But high end circuit analysis programs surely have tools for tuning the circuit without compromising too much the frequency response.

ADDENDUM: User Glenn W9IQ just showed us one such tool in his answer. Still better: A free version for simple circuits is available. Upvote to him is well used!