The loudspeaker Thiele/Small parameters can be used with the Spice analog electronic circuit simulator to provide behavioral analysis of loudspeaker enclosures. Of particular interest is the frequency response, phase response, and group-delay of loudspeaker enclosures, each of which can be optimized for maximum performance.

As an example, the Dayton Audio ND90-4 will be utilized in a design suitable for high quality computer monitors. The ND90-4 is rather unique in that the optimal, (optimized for low frequency response,) ported enclosure is nearly the same size as the optimal, (maximally flat midrange,) sealed enclosure, permitting an empirical comparison of the sound produced by both optimizations using the same enclosure for both. The ND90-4 driver is readily available, for under US $25, from Parts Express, Amazon, Ebay, etc.

The Design

gEDA Electronic Design Automation tools, particularly, Gschem(1), was used for schematic capture and Spice electronic circuit simulation netlist extraction. Ngspice was used for the Spice circuit simulations

The design work sheets and data bases for the speakers are available in a single tape archive format, (TAR file, using RCS for version control,) from nd90-4.tar.gz.

Optimal Designs

The optimization programs are avalable from the Subwoofer Speaker Design project as Calc(1) scripts:

• ported-box.calc, which is for optimizing ported/vented speaker systems.
• sealed-box.calc, which is for optimizing sealed speaker systems.
• acoustic-suspension-box.calc,, which is for optimizing acoustic suspension speaker systems.
• bessel-box.calc,, which is for optimizing bessel speaker systems.

The ported-box.calc script indicates that a ported enclosure volume of 10 liters, (which may be too large for most computer stations-the box optimal internal dimensions are 13.7" X 8.5" X 5.2",) which would have a low frequency -3 dB of 32 Hz., and the sealed-box.calc script indicates that the 10 liter sealed enclosure would have a low frequency -3 dB of 75 Hz., and a Q of 0.78, (a Q of 1 / sqrt(2) = 0.707 is optimal maximally flat, which is fairly close.)

The Characteristics

Figure I shows the frequency response of the various optimal ported and sealed enclosure sizes utilizing the ND90-4 driver.

Now that the optimal enclosures have been determined, the analysis can begin-remember, two designs, ported and sealed, are being developed.

The Ported Design

Figure II, (1600X1200),(3200X2400), is the ND90-4 symbol used for the ported design.

And descending down the hierarchy of the ND90-4 symbol:

Figure III, (1600X1200),(3200X2400), is the schematic of the ND90-4 ported design using the Thiele/Small parameters.

Figure IV is the frequency response, simulated using Spice, of the ported design.

Figure V is the phase response, simulated using Spice, of the ported design.

Figure VI is the transient response, simulated using Spice, of the ported design. Note the ringing at the enclosure's resonant frequency, Fb ~ 42 Hz.. This is a result of optimizing for maximum low frequency response. The enclosure volume is about 10 liters-about twice what the manufacturer recommends, (most computer sound systems available at the big-box stores use the same optimization.)

Figure VII is the group delay, simulated using Spice, of the ported design.

Note that the group-delay of the ported design is acceptable, (at least by ported enclosure standards-but it is a controversial issue,) and is at the threshold of audibility.

One of the advantages of using Spice simulations including the Thiele/Small parameters is the ability to electronically modify an enclosure's responses, (i.e., analog, DSP, etc.) For example, consider, (as a thought experiment,) using another ported ND90-4 driver's characteristics to modify the frequency response curves of the ND90-4.

Figure VIII, (1600X1200),(3200X2400), is the implementation of using a second ND90-4 driver to modify the frequency response curves of the ported ND90-4.

Notice the subtle point: the second ND90-4, X2, is connected backwards, and since the negative input of the operational amplifier, X3, is a virtual ground, and summing the currents to the virtual ground for the X3's output voltage, Vox2:

          Vox2 * H(s) / R2 = -Vin / R1

          Vox2 / Vin = - R2 / (R1 * H(s))

          R1 = R2

          Vox2 / Vin = - 1 / H(s)

        

Where H(s) is the transfer function of the second ND90-4, X2, and 1 / H(s) is the inverse transfer function. The Thiele/Small parameters were used to construct a prototype software filter that provides an overall gain of unity, flat frequency response, and with a phase shift of zero as the output of the X1 ND90-4. Remember that this is a thought experiment, (negative group delay?) and this circuit-and variants thereof-are unstable in the real world; however, small signal simulation can be used for verification with the theoretical operational amplifiers used, etc.

Figure IX is the frequency response using a second ND90-4 driver to modify the response curves of the ported ND90-4. Note that the ouput of the X1 ND90-4 is unity, and not frequency sensitive.

Figure X is the phase response using a second ND90-4 driver to modify the response curves of the ported ND90-4. Note that the output of the X1 ND90-4 phase is zero, and not frequency sensitive. It is not clear that a Bilinear Transform is applicable to the circuit due to stability issues, and frequency warping may limit the applicability of the circuit to woofers at digital music sampling frequencies.

Note that this was a thought experiment, but we can approximate the response of the second ND90-4 with two all-pass filters to shift the phase of the ported ND90-4, addressing the issues of group delay.

Figure XI, (1600X1200),(3200X2400), is the ported ND90-4 with two all-pass filters to shift the phase of the ported ND90-4, addressing the issues of group delay.

Figure XII is the frequency response of the ported ND90-4 design with two all-pass filters to shift the phase of the ported ND90-4, addressing the issues of group delay.

Figure XIII is the phase response of the ported ND90-4 design with two all-pass filters to shift the phase of the ported ND90-4, addressing the issues of group delay.

Figure XIV is the group delay of the ported ND90-4 design with two all-pass filters to shift the phase of the ported ND90-4, addressing the issues of group delay.

Note that the all-pass filters are a simplified model of the ported ND90-4 and enclosure.

A more complex model would use high-pass filters, with Q's other than maximally flat, where Q = 1 / sqrt(2) = 0.707, or Bessel, where Q = 0.577215665, e.g., Euler's Constant.

Figure XV, (1600X1200),(3200X2400), is the ported ND90-4 with two Sallen-Key high-pass filters to shift the phase of the ported ND90-4, addressing the issues of group delay.

Figure XVI is the ported ND90-4 frequency response of the ported ND90-4 with two Sallen-Key high-pass filters to shift the phase of the ported ND90-4, addressing the issues of group delay.

Figure XVII is the ported ND90-4 phase response of the ported ND90-4 with two Sallen-Key high-pass filters to shift the phase of the ported ND90-4, addressing the issues of group delay.

Figure XVIII is the ported ND90-4 group delay of the ported ND90-4 with two Sallen-Key high-pass filters to shift the phase of the ported ND90-4, addressing the issues of group delay.

Note that the high-pass filters are a simplified model of the ported ND90-4 and enclosure.

There are limits to the gain that can be used in compensating circuits-specifically, the maximum power that the driver can handle.

Figure XIX is the maximum output power, at a specific frequency, that the ND90-4 ported design can handle.

The Sealed Design

Figure XX, (1600X1200),(3200X2400), is the ND90-4 symbol used for the sealed design.

And descending down the hierarchy of the ND90-4 symbol:

Figure XXI, (1600X1200),(3200X2400), is the schematic of the ND90-4 sealed design using the Thiele/Small parameters.

Figure XXII is the frequency response, simulated using Spice, of the sealed design. Note that the ported design has a substantially lower cutoff frequency.

Figure XXIII is the phase response, simulated using Spice, of the sealed design.

Figure XXIV is the transient response, simulated using Spice, of the sealed design. Note that the ported design has substantially more ringing.

Figure XXV is the group delay, simulated using Spice, of the sealed design. Note the the ported design has substantially more group delay.

One of the advantages of using Spice simulations including the Thiele/Small parameters is the ability to electronically modify an enclosure's responses.

Figure XXVI, (1600X1200),(3200X2400), is the sealed ND90-4 design with a two pole shelving filter addressing the issue of low frequency response.

Figure XXVII is the gain response, simulated using Spice, of the sealed design with a two pole shelving filter addressing the issue of low frequency response. Note that the filters, (which have similar characteristics to the tone controls on power amplifiers,) have two zeros at the same frequency as the ND90-4's two low frequency poles-but the frequency response shapes around the zeros and poles are different, creating a bump in the overall frequency response.

Note that extending the frequency response down to 40 Hz., (competitive with the ported design,) is not practicable due to power limitations of the ND90-4.

Figure XXVIII is the phase response, simulated using Spice, of the sealed design with a two pole shelving filter addressing the issue of low frequency response.

What Was This Project For?

It was to investigate the issue of group delay in ported speaker systems. Both the sealed and ported designs are optimal, (or nearly so,) using the same driver in a simple enclosure box, allowing one to place/remove a hand over the port and listen to the difference under different program scenarios. Various simple schemes to improve the ported design's poor group delay, (that's why the components were chosen,) were implemented, also.

License

A license is hereby granted to reproduce this software for personal, non-commercial use.

THIS PROGRAM IS PROVIDED "AS IS". THE AUTHOR PROVIDES NO WARRANTIES WHATSOEVER, EXPRESSED OR IMPLIED, INCLUDING WARRANTIES OF MERCHANTABILITY, TITLE, OR FITNESS FOR ANY PARTICULAR PURPOSE. THE AUTHOR DOES NOT WARRANT THAT USE OF THIS PROGRAM DOES NOT INFRINGE THE INTELLECTUAL PROPERTY RIGHTS OF ANY THIRD PARTY IN ANY COUNTRY.

So there.

Copyright © 1992-2015, John Conover, All Rights Reserved.

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