Five (5) Audio Myths Dispelled - page 3
5. The Perfect Driver....
… would be infinitely stiff and completely massless.
I think this myth has survived as long as it has because, at first glance, the assertion seems to make sense; after all a driver that is infinitely stiff and has no mass should have zero problem responding perfectly and instantly to every electrical impulse fed it's inputs, right? Let's see what a bit of examination behind the scenes of the typical bandpass response of moving coil driver turns up.
We can divide the response curve roughly into 3 regions, each demarcated by, fl, lower resonance and, fh, upper resonance. (Would that all drivers had such well-behaved response curves!)
Below fl, stiffness (or its inverse, compliance - as supplied by the driver's suspension - expressed as Cms) predominates in limiting the motion of the cone and output is proportional to frequency ^ 4. Which, below resonance, means a roll off of 12dB/octave as frequency drops. In this region the driver functions as a constant excursion device.
At fl, output depends on resistance, including Rsource and Revc. The frequency at which fl occurs is determined by the moving mass of the driver. And as the theoretical moving mass of our driver → 0, fl approaches ∞ Hz. As shown by:
fs = .5π * (1/(Cms * Mms))^.5 (Hz)
fs = Free air resonance of driver, Hz
Cms = Mechanical compliance of driver, m/N
Mms = Effective mechanical mass of driver diaphragm, in kg
Between fl and fh is the mass-controlled reference region of the driver. In this portion of its output spectrum, the cone is mass-limited and the driver functions as a constant acceleration device, producing constant volume velocity. Radiation remains essentially hemispherical and directivity, constant.
At fh, like fl, resistance plays the controlling role. Fh can be determined by:
fh = ω*Levc*B^2l^2
ω^2*Levc^2 + (Rg + Revc)^2
and total resistance at fh can be found by:
Rtotal = (Rg + Revc) * B^2l^2
(Rg + Rc)^2 + ω^2* Levc^2
Figure 20: Bandpass response of a generic electromagnetic driver
So where's the problem with the infinitely stiff/massless driver?
By setting stiffness to ∞, fl is pushed up to ∞ Hz (besides sounding painful in an abstract sort of way), means that since response rolls off at 12/dB per octave, by the time frequency has dropped to within the audible range response is -∞ dB down, hence no output.
Above fl = ∞, our incredible driver is, predictably, functioning as a constant acceleration device generating a flat frequency response, but as its all happening at a frequency of ∞ Hz, there's no audible output. Our perfect driver remains, as far as human hearing goes, silent. There exists a whole host of other technical issues that would contribute to the driver's zero output (e.g. out of phase cancellation), but just focusing on Cms = 1/∞ and Mms = 0 are, in and of themselves, sufficient to silence the myth.
Ferreting out the facts behind a myth - or any facet of loudspeaker design, for that matter - can be enjoyable, satisfying and enlightening. Like a detective story where every clue is important, you can find yourself developing an almost child-like curiosity when investigating the facts using whichever tool you have at hand: mathematics, CAD, your calculator or a workbench with a bunch of analog test gear on it. The Internet, while providing forums for the propagation of audio myths is also a formidable tool in providing many resources useful to sorting the facts from fiction. Perhaps the greatest benefit of all such research is that, in the end, you can only gain in understanding. In this article I've shared what I've uncovered in the course of my examinations and would hope you too have gained in your understanding by reading it.
Ashley, Robert j. and Saponas, Thomas A, Wisdom and Witchcraft of Old Wives' Tales about Woofer Baffles , Journal of the Audio Engineering Society, Vol 18, November 5, October 1970
Benson, J. E., Theory & Design Of Loudspeaker Enclosures, Howard W. Sams & Company, Indianapolis , IN., 1996
Colloms, Martin, High Performance Loudspeakers, John Wiley & Sons, 1978, New York
D'Appolito, Joseph, Testing Loudspeakers, Audio Amateur Press, Peterborough, New Hampshire, USA, 1998
Dickason, Vance, The Loudspeaker Design Cookbook, 5th Edition, Audio Amateur Press, Peterborough, New Hampshire, USA , 1998
Faucett, Max A., Kraehenbuehl, John O., Circuits And Machines In Electrical Engineering , John Wiley & Sons, New York, 1939
Greiner, R. A., Amplifier-Loudspeaker Interfacing, Journal of the Audio Engineering Society, Vol. 44, Number 12, December 1996
Griffiths, David J. Introduction To Electrodynamics, 2nd Edition, Prentice Hall, Englewood Cliffs, NJ USA 1989
Harrison, Jeffrey, An Integral Limitation upon Loudspeaker Frequency Response for a Given Enclosure Volume , Journal of the Audio Engineering Society, Vol. 44, Number 12, December 1996
King, John, Loudspeaker Voice Coils, Journal of the Audio Engineering Society, Vol. 18, Number 1, February 1970
Locanthi, Bart, Application of Electric Circuit Analogies to Loudspeaker Design Problems, Journal of the Audio Engineering Society, Vol. 19, Number 9, October 1971
Magnetic Shielding Lab Kit With AC Probe, Brochure, Magnetic Shield Corp, Bensenville , Illinois , USA
Novak, James F., Performance of Enclosures for Low Resonance High Compliance Loudspeakers, Journal of the Audio Engineering Society, Vol. ??, Number ?, January 1959
Sanfilipo, Mark, Inductor Coil Crosstalk, Speaker Builder
Small, Richard H., Closed-Box Loudspeaker Systems Part I: Analysis , Journal of the Audio Engineering Society, Vol. 20, Number 10, December 1972
Small, Richard H., Closed-Box Loudspeaker Systems Part II: Synthesis , Journal of the Audio Engineering Society, Vol. 21, Number 1, Jan. /Feb. 1973
Small, Richard H., Direct-Radiator Loudspeaker System Analysis , Journal of the Audio Engineering Society, Vol. 20, Number 5, June, 1972
Small, Richard H., Vented-Box Loudspeaker System Part l: Small-Signal Analysis , Journal of the Audio Engineering Society, Vol. 21, Number 5, June, 1973
Small, Richard H., Vented-Box Loudspeaker System Part ll: Large-Signal Analysis , Journal of the Audio Engineering Society, Vol. 21, Number 6, July/August, 1973
Thiele, A. N., Loudspeakers in Vented Boxes: Part l , Journal of the Audio Engineering Society, Vol. 19, Number 5, May, 1971
Thiele, A. N., Loudspeakers in Vented Boxes: Part ll , Journal of the Audio Engineering Society, Vol. 19, Number 6, June, 1971
Weidner, Richard T., Physics Allyn & Bacon, Inc, Needham Heights, MA, 1989
Yarbrough, Raymond B., Electrical Engineering Reference Manual , 5th Edition, Professional Publications, Belmont, CA, 1990