Butterworth filter with non arbitrary component

Question

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}

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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!