Hello everyone,

I was trying to design a circuit where I split an audio signal in two channels: one which is a low pass filtered version of the original and the other which is a high pass filtered one. Both LPF and HPF are 2nd order and have same fc.
The idea was to merge the two signals back at the end of the circuit, after the high end signal gets minor level adjustments. Though when I do this, using a simple summing opamp, I happen to have a notch in the frequency response precisely at fc, which I realised it's where the phase of the two signals is +90 and -90 respectively.

I tried inverting one of the signals 180 degrees to make the frequencies at fc in phase, but there's still some weird modulation due to phase happening at fc which also compromises the flatness of the frequency response. And also the signal's low end would be out of phase with the rest of the audio band which I'd rather avoid.

I was wondering if anyone knew how would I be able to restore the filtered low end in the high end signal, archieving a flat response at the end?


Thank you in advance for any help, stay safe!

A

 

Yes, a 2nd order Butterworth lowpass and a 2nd order Butterworth high pass (and all other even-order Butterworth filters) make a phase-cancellation notch at their crossover frequency. If you invert one side then you have a +3dB boost at the crossover frequency. Using a Linkwitz-Riley alignment (less Q than a Butterworth) eliminates the +3db boost when one side has its phase reversed.

I use 3rd order Butterworth crossovers instead.

 

Thanks for the suggestion! I've just simulated on LTspice a 3rd order butterworth crossover and the signals summed up nicely with a minimal bump at fc.
I'll try to see if I can just add an additional RC stage in front of the discrete 2nd order HPF to turn it into a 3rd order filter. Hopefully that will work, I'm expecting the real life version not to be as linear as the simulation.

 

If you get a flat enough response by experimenting then fine.

There is a theory that will give you a flat response as Audioguru said, use Linkwitz-Riley theory, just duplicate your butterworth filters in each channel. This was discussed in this thread (it was a very long thread so I linked part way through):
https://forum.allaboutcircuits.com/threads/q-of-3rd-order-filters.168519/page-9#post-1517446

Typically if you sum butterworth filters you get a bump as shown, using L-R theory you get a flat response.

 

Thank you for the link, it looks very tempting. I'm only hesitating on the idea of having to add another two stages so I think I'll try to see how the 3rd order Butterworth configuration works in real life.

For what I've understood, each configuration as in Butterworth or Chebyshev must be about 2nd and higher order filters as the configuration itself is about the positioning of poles which you can't do if you only have one. So by seeing the slope on the graph above I'd suppose that's a 2nd order filter? Wouldn't a double Butterworth filter be 4rd order from the sum of two 2nd order filters? I was wondering then what the graph would look like with the sum of two 3rd order Butterworths.

 

Two 2nd-order butterworth filters in series made a 4th-order filter with a low Q which is what Linkwitz-Riley uses to make a flat frequency response of a crossover.
Two 3-order Butterworth filters each are down -3dB at the crossover frequency so they add to produce no gap.

 

I've just tried this crossover circuit in real life but I'm having a 2dB dip at fc. It has a 3rd order butterworth LPF and a 3rd order HPF which match well in the LTspice simulation (small 150mdB dip at fc) but works horribly in real life. The high pass section is a discrete 2nd order filter with an extra RC section for an extra pole.
Could it be just a case of the HPF poles not being in the right place?
I'm not sure how to make the calculations for RC values in order to have the correct butterworth poles as most resources online cover the basic opamp filters.
Cheers!

 

you seem to have a fundamental bi here,
which to be honest having built a few graphic equalisers a few decades back off plans,
I had not considered.

analog filters are going to drift with time , voltage , process variants.

QED, if you have two filters with an identical Fc, then they are not going to be identical for ever.

I'd say the way to do what you want to do is to filter without phase shift,
and / or with consistence / tracking Fc,
which implies digital filtering,,,

One thing, if you want to just boost the upper frequencies,
would an amplifier of everything then a low pass filter be sufficient,

 

I made some speakers using 3rd-order Butterworth passive filters and they sounded and measured awful. I bought the inductors from a local high-end speaker manufacturer so I listened to and measured the speakers in their demo room and they also sounded and measured awful!
I measured the inductance of their labelled inductor and the number was WRONG! They sold pretty speakers but with the wrong inductor values.
With the correct inductor values my speakers sound great.

 

Two 2nd-order butterworth filters in series made a 4th-order filter with a low Q which is what Linkwitz-Riley uses to make a flat frequency response of a crossover.
Two 3-order Butterworth filters each are down -3dB at the crossover frequency so they add to produce no gap.

The LInkwitz-Riley is 6dB down at the crossover frequency, that is why they add up to a flat response, because the filtered signals are coherent so one has to add the voltages. (6dB down is half the voltage)
If two filters which were 3dB down at the crossover frequency are fed with two different (incoherent) white noise signals, then one has to add the power, and so the overall response will be flat. (3db down is half the power)

 

The LInkwitz-Riley is 6dB down at the crossover frequency, that is why they add up to a flat response, because the filtered signals are coherent so one has to add the voltages. (6dB down is half the voltage)

\
Coherent as in coming from the same source?

Is the reason why the different crossover types give different bumps at fc a matter of the phase difference at fc? I was trying to understand how two even or odd order butterworth interact differently.

I've just analized the two signals before entering the summing amplifier and they seem around 81 degrees out of phase at fc (1.5ms dt on a 150hz sine).
I'm not too sure why if I add this sines in a DAW with the above phase shift and at -3 they sum up to very close to 0dB, while in real life trying to tweak the two filters doesn't get me better than a -2dB dip. Probably something wrong with the schematic?

 

Coherent as in coming from the same source?

Yes. If you imagine a loudspeaker with bass and treble units, at the crossover frequency, both bass unit and treble unit should give exactly the same pressure pulse. The two pressure pulses add up to give a signal which would be equal to an unfiltered signal.

Is the reason why the different crossover types give different bumps at fc a matter of the phase difference at fc? I was trying to understand how two even or odd order butterworth interact differently.

I've just analized the two signals before entering the summing amplifier and they seem around 81 degrees out of phase at fc (1.5ms dt on a 150hz sine).
I'm not too sure why if I add this sines in a DAW with the above phase shift and at -3 they sum up to very close to 0dB, while in real life trying to tweak the two filters doesn't get me better than a -2dB dip. Probably something wrong with the schematic?

Have a look at the Wikipedia article on Linkwitz-Riley filters.
https://en.wikipedia.org/wiki/Linkwitz–Riley_filter

A single-order filter has a phase shift of 45° at the 3dB frequency, low pass and high pass shift the phase in opposite directions, so the total at the crossover frequency is 90°.

If you want filters that definitely add up to the original, you could always derive the high-pass response by subtracting the low-pass response from the original signal.

 

Thanks Ian, I'll definitely have to learn more about the maths. I managed to get the dip down to 0.2dB which is amazing but I would have loved having been able to do some calculations instead of messing around for two days till it worked right!

 

Also have a look at the theory behind the Kerwin–Huelsman–Newcomb state variable filter.
https://en.wikipedia.org/wiki/State_variable_filter
It produces high-pass, band-pass and low-pass from the same circuit.