Re: Phase, delays and offset baffle spacing

Thanks for an interesting article. May I make a couple of comments?

The apparent offset of the driver is proportional to phase shift divided by frequency. Phase shift is proportional to the arctan of the measurement frequency divided by the crossover frequency. When this ratio is small (i.e., when f << fc), the arctan (and therefore the phase shift) varies almost linearly with frequency. You can see this on the graph. The result, as you say, is that the woofer offset is nearly constant. Taking the speed of sound to be 1130 ft/sec, I calculate a 1.35" offset from DC to around 100Hz. After that, the rate of change of phase with respect to frequency begins to slow down and the driver offset goes down as well. At the crossover frequency, the offset is almost exactly 1". It decreases to about 0.30" at 10kHz, and 0.16" at 20kHz.

I was puzzled for a while that the tweeter offset, unlike the woofer offset, varies so widely, especially since the phases of the two drivers track each other exactly, being ninety degrees apart at all frequencies. Looking at the math, I realized the 90 degree phase lead of the tweeter introduces an offset term propertional to 1/f. This explains why the tweeter offset changes so rapidly at low frequencies. At zero frequency, in fact, the offset is theoretically infinite.

The delay associated with phase shift is somewhat abstract. If you apply an inpulse to both drivers at the same time, they will both start producing an output signal immediately (ignoring a tiny delay due to the finite speed of light). The risetimes will vary due to the differing bandwidths, but the delay introduced by the phase shift is not the same as a pure delay. If it were, the 90 degree phase lead of the tweeter would cause the signal to start coming out before the arrival of the input signal.

Time aligning drivers does introduce a pure time delay. It can be useful to correct a tilt in the speakers polar response, or to add a real delay term to simplify crossover design.

Finally, I believe speaker designers are accustomed to thinking in terms of idealized target crossover transfer functions. It follows as a matter of course that simple coils and capacitors connected to real-world drivers will not result in optimum system crossover performance. That's part of the black art of crossover design.

Please accept my apologies for going on like some kind of expert. Thanks again for an interesting article and website.


Follow Ups: