by Brad Olson
Dr. Earl Geddes from GedLee Associates spoke to the Chicago Section of the AES on Wednesday January 28, 1998. He discussed some of the analysis he has done in the field of loudspeaker distortion modeling. According to Earl, this work has interested him for a very long time, but he could never finish it due to the enormously complex equations that had to be manipulated by hand in the days prior to computerized equation solving. He found that attempting these long algebraic calculations by hand made errors almost a certainty. Even with PCs, the exact and all-encompassing approaches of many distortion researchers are very complex to implement. Earl's model is based on the premise that for a driver to be useful it needs to be nearly linear. He simplifies the equations to eliminate many higher order terms, giving a simpler, faster but admittedly inexact approach. He used MathCAD to do algebraic manipulations, and ran the actual number crunching in his own program, Speak32.
He looked
at the basic equation relating cone velocity to input voltage in terms of
fundamental speaker parameters RE, MM,
RM, CM, ZM, Bl and frequency. The problem
with setting the input equal to a single complex exponential is that it does
not give the proper static displacement terms seen with second order
nonlinearities. He instead set the input equal to a sum of two complex
exponentials. This gave a very messy result when a series expansion was
performed. However, he broke things apart and observed certain symmetries,
namely the pairs of complex conjugates. He then treated the single equation
as a series of equations where the first order terms on the left were set
equal to the first order terms on the right, etc. He stated that the proof
that this works would be far too complex for his speech, but thank goodness
it does! Then terms with squares of C, x2, and x3
(x2=displacement due to 2nd harmonic, x3= due to 3rd,
etc…) were eliminated on the assumption that they are significantly
lower in magnitude than the fundamental. This simplifies the equation to
a manageable result. Using this result along with a few assumptions regarding
the logical behavior of Bl versus displacement and stiffness (inverse
of compliance) versus displacement, he generated results for a real driver
using the Speak32 program.He showed
these results by comparing several enclosure configurations, including closed
box, ported box, infinite baffle, and bandpass designs. His models showed
that the closed and bandpass designs produce much better low frequency
performance, especially with respect to distortion. He explained that this
is due to the fact that these designs exert more control over the cone motion
at low frequencies. In particular, the popular vented box configuration produces
surprisingly high levels of distortion unless a high pass filter is used
to limit driver displacement below resonance. He then proceeded to mention
that other non-traditonal configurations could be easily modeled and gave
one example. He said that his system correlates well with many real measurements,
but that it still may be best for relative comparisons, not absolute
measurements. Finally, he mentioned the fact that he has developed a proprietary
new enclosure design with better THD performance than any of these, which
we are all eager to see. Unfortunately, details were not revealed because
a patent is in progress.The section
would like to thank Earl for his enlightening and enjoyable discussion. Although
he lost a few people in the complex algebra, everyone learned something!
