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Old 05-09-2009, 09:44 PM   #1
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Post Loudspeaker Primer

LOUDSPEAKERS PRIMER by William K. McFadden [billmc@agora.rdrop.com]

1. Introduction

For the purposes of this discussion, an optimum enclosure is defined as one that is maximally flat (i.e., has no peak or droop in the passband of the frequency response). However, this is only one way to optimize a design. Garry Margolis, co-author of [10], has this to say:

"To paraphrase the introduction to the paper Small and I wrote, maximally flat is not necessarily maximally desirable. Allowing minor amounts of ripple and/or droop can result in significantly extended low frequency output."

2. Small Signal Parameters

The three parameters that primarily determine the frequency response of a loudspeaker are compliance, free-air resonance, and Q. The compliance, Vas, is a measure of the overall stiffness of the cone, surround (the part the attaches to front of the cone), and spider (the part that attaches to the rear of the cone). It is specified as the volume of air having the same compliance as the driver. A small number corresponds to a small volume of air, which is stiffer than a larger volume of air. Thus, compliance and stiffness are inversely proportional. Optimum enclosure volume is proportional to Vas.

Free-air resonance, Fs, is the resonant frequency of the driver's voice coil impedance with the driver suspended in free air (no enclosure). The -3 dB frequency (F3) of an enclosure is proportional to Fs. The Q, Qts, is a measure of the sharpness of the driver's free-air resonance. It is defined as (Fh-Fl)/Fs, where Fh and Fl are the upper and lower -3 dB points of the driver's voice coil impedance in free air. Optimum enclosure volume is related to Qts but is not directly proportional. It is accurate to say that the volume gets larger as Qts gets larger. Likewise, F3 gets smaller as Qts gets larger, and for the sealed box enclosure, F3 is inversely proportional to Qts.

3. Efficiency & Loudness

The efficiency of a driver is given in decibels of sound pressure level (SPL). 0 dB SPL is defined as 2.0E-10 bar (2.0E-5 N/m^2), which is the lowest level of 1 kHz tone the average person can detect. A 10dB increase in SPL results in an apparent doubling of the loudness and requires 10 times as much power. Accordingly, a 10 dB decrease halves the loudness and reduces the power requirement by a factor of 10.

Most driver manufacturers specify the SPL of the driver with a one watt input measured at a distance of one meter. To calculate the SPL at other power levels, add the following number to the SPL rating: 10*log(POWER), where POWER is in watts, and the log is base 10. This equation is derived from the fact that a doubling of electrical power produces an doubling of acoustic power. To calculate the SPL at other distances, subtract the following number from the SPL rating: 20*log(DISTANCE), where DISTANCE is in meters. This equation is derived from the inverse square law of wave propagation.

One watt of acoustic power is equal to 112 dB SPL at one meter. To calculate the efficiency of the speaker in percent, use the following: %EFFICIENCY = 100*(10^((SPL - 112)/10)), where SPL is the driver's SPL rating in dB, at one watt, measured at one meter. For example, a driver with a 92 dB SPL rating @ 1W/1m is 1% efficient.

4. Power Handling

The power rating of a driver is usually (but not always) specified in watts RMS by the manufacturer. This is the continuous thermal power rating of the driver. Exceeding this rating for more than a moment will cause voice coil overheating, which can result in warping or burn-out.

Speaker systems also have a displacement-limited power rating (Per). This is the amount of power the system can take without exceeding the absolute maximum voice coil displacement. Per is a function of frequency and depends on the design of the enclosure in addition to the peak displacement limit of the driver, xmax. Thus, it is meaningless for manufacturers to specify peak power handling without also specifying the enclosure and the frequency range. At some frequencies, Per will exceed the thermal RMS power rating. For continuous tones, the smaller of the two ratings applies. For signals with large crest factors or low duty cycles, Per applies, providing the average power does not exceed the thermal rating. Per is calculated for sine waves, which have a 3 dB crest factor. The peak power rating at a given frequency is therefore 2*Per.

5. Sealed Box Enclosures

For the sealed box enclosure, the optimum volume can be determined. Many designers like to use a 0.62:1:1.62 ratio for the interior cabinet dimensions. This is known as the golden ratio. A box designed to this ratio will have smaller resonant peaks than one whose dimensions are equal. Another ratio sometimes used is 0.8:1:1.25. You can determine the middle dimension by taking the cube root of the enclosure volume. (Keep in mind this is the inside volume and doesn't take into account the volume taken up by bracing materials and the drivers.)

The box will have a resonant frequency and a Q. For an optimum sealed box, the resonant frequency is equal to the -3 dB point, and the Q is 0.707. The -3 dB frequency is also known as the half-power point, because it is the frequency at which the acoustic output power drops by half. Below this frequency, the response will have a second order roll off, e.g., the output decreases 12 dB for every halving of the frequency below the -3 dB point.

6. Ported Box Enclosures

The ported enclosure is a little more complicated. As with the sealed box, the ported enclosure has an optimum volume and -3 dB point.

The enclosure also has an optimum tuning frequency, Fb, which is the resonant frequency of the enclosure's duct. The tuning frequency is determined by the cross sectional area and length of the duct. For a tubular duct, the following equation applies, LENGTH = 2118*DIAMETER^2/(Fb^2*Vb) - 0.73*DIAMETER, where LENGTH is the length of the duct in inches, DIAMETER is the inside diameter of the duct in inches, Fb is the tuning frequency in Hz, and Vb is the box volume in cubic feet.

Ported enclosures have a steeper roll off than sealed boxes. The roll off is fourth order, or 24dB for every halving of the frequency below the -3dB point. Below Fb, the displacement-limited power rating will be very low because the driver is essentially operating in free air. It is therefore wise to roll off the signal below the -3dB frequency to avoid damage. This constraint does not usually apply to sealed boxes, which dampen cone movement at all frequencies.

7. Additional Resources

A lot of loudspeaker software and other information is available on the internet. Here are a few locations known to the author:

http://www.rdrop.com/users/billmc/
http://www.diyloudspeakers.org/
http://www.muohio.edu/~bullocrm/
http://www.hi-fi.com/speaker/
http://www.speakerbuilding.com/
http://www.spiceisle.com/homepages/brian/audiodiy/| ftp://snippets.org/pub/snippets/ldsg.txt
ftp://ftp.uu.net/usenet/rec.audio.high-end/Software

8. References

[1] David Weems, Building Speaker Enclosures (Tab Books, 1981).
[2] Gordon McComb, Building Speaker Systems (Master Publishing, Richardson, TX, 1988).
[3] Vance Dickason, The Loudspeaker Design Cookbook, Fourth Edition (Audio Amateur Press, Peterborough, NH, 1991).
[4] L.L. Beranek, Acoustics (McGraw-Hill, New York, 1954).
[5] J.F. Novak, "Performance of Enclosures for Low-Resonance High-Compliance Loudspeakers," J. Audio Eng. Soc., vol. 7, p 29 (Jan. 1959).
[6] A.N. Thiele, "Loudspeakers in Vented Boxes, Parts I and II," J. Audio Eng. Soc., vol. 19, pp. 382-392 (1971 May); pp. 471-483 (1971 June).
[7] R.H. Small, "Direct-Radiator Loudspeaker System Analysis," J. Audio Eng. Soc., vol. 20, pp. 383-395 (1972 June).
[8] R.H. Small, "Closed-Box Loudspeaker Systems," J. Audio Eng. Soc., vol. 20, pp. 798-808 (1972 Dec.); vol. 21, pp. 11-18 (1973 Jan./Feb.).
[9] R.H. Small, "Vented-Box Loudspeaker Systems," J. Audio Eng. Soc., vol. 21, pp. 363-372 (1973 June); pp. 438-444 (1973 July/Aug.); pp. 549-554 (1973 Sept.); pp. 635-639 (1973 Oct.).
[10] G. Margolis and R. H. Small, "Personal Calculator Programs for Approximate Vented-Box and Closed-Box Loudspeaker System Design," J. Audio Eng. Soc., vol. 29, pp. 421-441 (1981 June); corrected on p. 824 (1981 Nov.).
[11] W.M. Leach, Jr., "A Generalized Active Equalizer for Closed-Box Loudspeaker Systems," J. Audio Eng. Soc., Vol. 38, pp. 142-145 (March 1990).

[1] and [2] are useful as an introduction and contain a lot of construction tips. [3] is vastly improved over [1] and [2] and is probably the best overall source of practical speaker design information. [4] is a the industry bible on acoustics. [5] is historically significant, and is the foundation for [6]. [6] and [8] are the landmark works on loudspeaker systems (you can't consider yourself knowledgeable without having read them). [7] is background for [8] and [9]. [9] updates the original work of [6]. [10] revises the equations of [6] through [9] and presents them in a form suitable for programmable calculators. [11] is a more recent paper that shows how to equalize closed-box systems to any desired F3. [5] through [9] are reprinted in the AES two-part "Loudspeakers" anthology. [2] is sold at Radio Shack stores. [3] and [4] are available from Old Colony Sound Lab [603-924-6371]. AES reprints are available from Audio Engineering Society [212-661-8528].

The latest version of this document can be accessed via my web page at:http://www.rdrop.com/users/billmc
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Old 01-14-2010, 05:45 AM   #2
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who the hell is this guy and where the hell did he go?
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Old 07-25-2013, 01:36 PM   #3
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this is one of the best sub/box design threads I have ever read...
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