[Last modified 25-Jan-99]
The latest version of this file can be accessed via my web page at:
http://www.rdrop.com/users/billmc
LOUDSPEAKERS PRIMER
by William K. McFadden
[billmc@agora.rdrop.com]
(Changes preceeded by "|")
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 10
dB 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].