The X-Philes (2nd Revision)/The X-Philes Number 1 (1995).iso/xphiles/hp48hor1/spkr.doc

Text File | 1995-03-31 | 12KB | 227 lines

(Comp.sys.handhelds)
Item: 2002 by bill at flutter.tv.tek.com
Author: [William K. McFadden]
  Subj: Speaker Design Equations
  Date: Fri Feb 08 1991

I have put together a library of equations for designing ported and
closed-box speaker enclosures.  The basic design equations were taken from
a couple of hobbyist-oriented speaker design books.  The power rating
equations were taken from papers by Richard Small (see references below).

The equations are intended to be used with the multiple equation solver in
the equation library ROM card.  I welcome any comments or refinements.

The main directory is called SPKR and consists of two subdirectories:

CB      Closed Box Design
PORTED  Ported Box Design

Running the multiple equation solver from either subdirectory will produce
a menu of variables:

Vas     Volume of air having same acoustic complaince as driver suspension
Qts     Total driver Q at Fs
Fs      Resonant frequency of driver
SPL     Efficiency of driver in dB SPL at 1W/1m
DIA     Diameter of driver
xmax    Peak displacement limit of driver diaphragm (1/2 of "throw")
Vb      Inside volume of enclosure
Fb      Resonance frequency of enclosure
F3dB    Half-power (-3 dB) frequency of loudspeaker system response
dBPEAK  Maximum peak or dip of loudspeaker system response
Par     Acoustic power rating
PeakSPL Equivalent sound pressure level (at 1m) of acoustic power rating
Per     Electrical power rating (worst case)
\Gno    Percent driver efficiency (\Gn is greek character eta)
Sd      Effective projected surface area of driver diaphragm (approximated)
Vd      Peak displacement volume of driver diaphragm

In addition, the following variables are defined for the closed box case:

Qb      Total Q of system at Fb
AMAX    Maximum amplitude of loudspeaker system response: 10^(dBPEAK/20)
Vr      Ratio of Vas to Vb
Qr      Ratio of Qb to Qts and Fb to Fs

For the ported box case, the following apply:

1. Fb is the tuning frequency for the vent.
2. Most of the results are approximate.

To use, run MSOLVR in either directory.  Enter the speaker parameters into
the variables Vas, Qts, Fs, SPL, DIA, xmax.  (If you don't have all the
parameters available, purge the ones you don't know, so they'll be
undefined and the solver won't attempt to use them.)  For the closed-box
case, define one of Vb or Qb and solve for the other (or make it a
calculated value with MCALC).  Pressing <- ALL will solve for all the
unknowns for which a solution exists (indicated by a small box in the
menu).  This takes about 2.5 minutes for the closed box and about half as
long for the ported box.

To find the optimum box size for the closed box system, set Qb=0.707
(=1/sqrt(2)) and solve for Vb.  Solving for Vb for the ported box always
finds the optimum box size.  The optimum box size is defined as the size
which produces no peak or dip in the frequency response (e.g., dBPEAK=0).
(A B2 response is used for the optimum closed box, and B4 for the ported
box.)

To solve for given box size, for the closed box system, enter a value for
Vb, type 'Qb' MCALC, and solve for any or all unknowns.  For the ported
box, enter a value for Vb and solve for the unknowns.  To return to the
optimum enclosure, for the closed box, set Qb = 0.707 and type 'Vb' MCALC.
For the ported box, type 'Vb' MCALC.

To run a frequency response plot, press -> PLOT.  The X axis is frequency,
and the Y axis is the magnitude of the response in dB. Change the ranges,
if desired, and press ERASE and DRAW.  It takes about a minute for the
closed box, and four minutes for the ported box.

You can also use the built-in solver to locate points of interest in the
frequency response by pressing -> SOLVE.

If you get curious, the design equations are in a list called DESIGN.EQ,
and the frequency response equation is in a variable called RESPONSE.

There is a subdirectory in CB called EQUALIZER that will find the component
values for an active equalizer that can extend F3dB of any closed box
system to any desired lower limit (at the expense of efficiency and power
handling--watch out!)  See pp. 142 of the March 1990 AES Journal for theory
and circuit details.

First, use the multiple equation solver in the CB directory to solve for
the system as shown above.  Next, enter the EQUALIZER subdirectory.  Enter
the new desired cutoff frequency into F3dB, and press CIRCUIT.  The
component values will appear in the display.  The values of R, C, N are
chosen by the user to make the remaining component values realistic (see
article).

You can run a response plot of the equalizer with -> PLOT.  It's pretty
interesting, but takes FOREVER (like 20 min.).  The reason is I copied the
equations right out of the article without any optimization for speed.  (If
anybody wants to tackle this, be my guest.)  Wherever possible, I left out
the units so it would run faster.  You can also solve for points of
interest with -> SOLVE.  The point where maximum boost occurs is at F3dB.
If you put this in for f and solve for dB, you will see how much boost is
needed without having to wait all day. (Don't enter values for Fb and Qb;
they are defined in the parent directory, and entering values will redefine
them locally.  If you do this by mistake, purge Fb and Qb.)  Efficiency and
power handling of the system at this frequency will be degraded by this
amount if the equalizer is used.  This gives a pretty good worst case
scenario.

Don't be surprised if more than 20 dB of boost is needed to get down to 20
Hz, even for large drivers.  "There ain't no such thing as a free lunch."
If you don't need the equalizer program, just PGDIR the EQUALIZER
subdirectory.  Doing so will save about 1.6K.

By the way, the default speaker parameters when you first download the file
are for the Eminence 18029 18" driver.

 The following is a small tutorial on speaker enclosures.



An optimum enclosure is defined as one that has no peak or droop in the
passband.

The power rating of each driver is given in watts RMS.  This is the
continuous thermal power rating of the speaker.  Most speakers can handle
two to four times as much power for brief periods without overheating.

The efficiency of the speaker 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 ear can detect.  A 10 dB increase in SPL
results in an apparent doubling of the loudness and requires 10 times as
much acoustic power.  Accordingly, a 10 dB decrease halves the loudness and
reduces the acoustic power by a factor of 10.

Most driver manufacturers specify the SPL of the driver with a one watt
input measured at a distance one meter away.  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 about 112 dB SPL at one meter. To
calculate the efficiency of the speaker in percent, use the following:
%EFFICIENCY = 100*(10^((RATING - 112)/10)), where RATING 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.

For the sealed box enclusure, the optimum volume in cubic feet can be
determined.  Many designers like to use a 0.62:1:1.62 ratio for the cabinet
dimensions.  This is known as the golden ratio.  A box designed to this
ratio will be less peaky 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 driver itself.)  The box will have a resonant frequency
and a Q.  For an optimum sealed box, the resonant frequency is equal to the
-3dB point, and the Q is 0.707. The frequency (in Hz) at which the
speaker's response is 3 dB down can be found.  This 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.

The ported enclosure is a little more complicated.  As with the sealed box,
the ported enclosure has an optimum volume (stated in cubic feet) and -3 dB
point (stated in Hz).  The speaker also has a tuning frequency, called Fb.
This is the resonant frequency of the enclosure's duct.  The tuning
frequency is determined by the cross sectional area and length of the duct.
You may consult a book on speaker design to determine the proper duct size.
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.  At very low frequencies, the driver will be undamped, hence the
speaker could be damaged by excessive cone movement.  It is therefore wise
to roll off the signal below the -3dB frequency to avoid damage. This
constraint does not apply to sealed boxes, which damp cone movement at all
frequencies.

REFERENCES:

[1] Hobbyist speaker building books, such as the one sold at Radio Shack.

[2] L.L. Beranek, Acoustics (McGraw-Hill, New York, 1954).

[3] J.F. Novak, "Performance of Enclosures for Low-Resonance
High-Compliance Loudspeakers," J. Audio Eng. Soc., vol. 7, p 29 (Jan.
1959).

[4] 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).

[5] R.H. Small, "Direct-Radiator Loudspeaker System Analysis," J. Audio
Eng. Soc., vol. 20, p. 383 (June 1972).

[6] R.H. Small, "Closed-Box Loudspeaker Systems," J. Audio Eng.  Soc., vol.
20, p. 798 (Dec. 1972), and vol. 21, p. 11 (Jan/Feb 1973).

[7] R.H. Small, "Vented-Box Loudspeaker Systems," J. Audio Eng.  Soc., vol.
21, (four parts, starting in the June 1973 issue).

[8] 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] is useful as an introduction and has a lot of construction tips.
[2] is a standard reference text that seems to be the industry bible.
[3] is historically significant, and is the foundation for [4].
[4] and [6] are the landmark works on loudspeaker systems (you can't
consider youself knowledgeable without having read them).
[5] is background for [6], and [7].
[7] updates the original work of [4].
[8] is a recent paper that shows how to equalize closed-box systems to any
desired low-frequency cutoff.  [3], [4], [5], [6], and [7] are reprinted in
the AES two-part "Loudspeakers" anthology.
--
Bill McFadden    Tektronix, Inc.  P.O. Box 500  MS 58-639  Beaverton, OR
97077
[e-mail],
{hplabs,uw-beaver,decvax}!tektronix!videovax!bill
Phone: [phone]             "SCUD: Shoots Crooked, Usually Destroyed"