phase shifts and 1st-order crossovers

[keto]

High all. If a high pass is realized using the RC and or LC coupling within a power amp, the phase shift at the cutoff frequency is 90 degrees, right? I'm assuming that one of the advantages to doing this is that the cap is going to be there anyway, influencing he sound, and that going to a smaller value might also allow for a higher quality cap and so maybe a better sound. If a cap was added at the input, and both the RC interstage cap and parafeed cap were made smaller, and all three were "tuned" to the same frequency, it would seem like in addition to the 6dB/ octave high passes summing to 18dB/octave, the phase shifts would also sum to 270 degrees. If this is true and only a 1st-order high pass is required, and realized by resizing the parafeed cap, how much lower in frequency should the other 1st-order high passes be so as not to interfere with the simpler 90 degree phase shift at cutoff, but also improve linearity in the preceding stages by reducing part of the LF signal? For example, would sizing the input cap high pass one octave below the parafeed cap one serve to improve linearity without effecting the desired 6dB/octave roll-off and 90 degree phase shift? Thanks for any light being shed on this! --keto

[Paul Joppa]

Just to clarify, a first-order filter has a 45 degree phase shift at the corner frequency, increasing to approach 90 degrees as you get further into the stop-band and decreasing to approach zero phase shift in the pass band. It never actually reaches 90 degrees.

The choice depends on what error you are willing to tolerate. As a rough guide, I use a level that is 20dB down, which is a bit over 3 octaves for a first-order filter. That should keep you within a dB or so. I'd rather see 4 octaves, but I'll compromise down to 2 octaves if it's necessary.

For high-pass filters such as you describe (series capacitors) you must remember to take into account the speaker driver as well. If you cross over first order at 2kHz to a tweeter, then the tweeter must be flat down to 250Hz in order to maintain a 3-octave margin. In the vast majority of cases, the actual acoustic response function is a composite of the electronic crossover plus the driver's own response, and a first-order electrical crossover is often at least third order acoustically.

Doc B's big system uses this combination extensively. For example, the tweeter has a first-order response itself, another first-order is done with the parafeed cap, and a second-order filter is used in the electronic crossover to obtain a net fourth-order Linkwitz/Riley highpass function. The midrange is quite flat to beyond 10kHz so an active fourth-order lowpass is used there.

[keto]

Very helpful. Thanks! I now see how I could include the 2nd-order Marchand passive 300hz line-level low pass with a 2nd-order passive speaker-level low pass, to achieve a 4th order low-pass (after checking the "type" of slopes used by Marchand). For the high-pass, the mids are 1st-order acoustic, so I could add the 1st-order Marchand high pass, and tune the interstage coupling cap and parafeed cap to 300hz for a net 4th order acoustic high-pass. Lots-o-caps-n-coils :-( but I will give it a try. Does a strictly acoustic high- or low- pass effect phase?

[Paul Joppa]

Couple of points-

1) Most individual loudspeaker drivers are "minimum-phase." In a minimum-phase system, the phase is determined by the frequency response, or vice-versa - they are Hilbert transforms of each other. So yes, the measured acoustic amplitude response function determines the phase response as well. (This is only true for individual drivers; most crossover networks introduce an additional phase shift so the whole-system response is not minimum-phase.)

2) Higher-order responses that are useful for crossovers cannot be implemented by simply cascading first-order responses; they require some controlled resonance as well. This can only be accomplished with either feedback (in an active crossover) or with LC filters.

This is no place for a tutorial on crossover design - way too complicated. But I'll mention one specific example that might be relevant, a fourth-order Linkwitz/Riley filter. A standard fourth order Linkwitz-Riley is composed of two second-order Butterworth filters, each of which has a Q of 0.707. You cannot get a Q greater than 0.5 using cascaded first-order filters. However, if you cascade a second order filter with a Q of 0.5 with another having a Q of 1.0, the result is very close to the standard version, and you can get two of the four poles using simple first-order filters. Thus you can use a second-order Marchand highpass (re-tuned for a Q of 1) along with the driver's first order plus either a modified parafeed cap or a modified interstage cap. You'll have to consult Marchand on how to get the Q=1 response.

Hope that helps. Crossover design, like reducing hum, is a can of worms - or maybe a nest of snakes!?

[keto]

Okay, it looks like I should concentrate on a Linkwitz-Riley 4th order XO at about 300hz, and that means I'll first have to measure the acoustic roll-off of the mids in the Lambda Unity Horn, to find their -3dB point and their basic slope, and then calculate what additional filters will be needed to get to LR-4. Then, I'll get a pair of LR-4 PLLXO's Marchand at that -3dB point for the bass bins. I now see why what I was trying before didn't work, and the options for moving forward are narrowing, nicely. Thanks!