bill@flutter.tv.tek.com (William K. McFadden) (03/12/91)
The speaker design equations I posted recently are only useful for
people with an HP48SX and an equation library card. A lot of people
don't have these, so here are the equations in ASCII form. See the
original article "Speaker Design Equations -- Part 1/3" in
comp.sys.handhelds for an explanation.
The special sequences \v/, \Gn , and \pi represent the square root
operation, the greek character eta, and pi, respectively. SQ() is the
x^2 operation. INV() is the 1/x operation. LOG() is the base 10
logarithm. UVAL(UBASE(Par)) is a fancy way of saying "return the
numerical value of Par in watts." You can omit UVAL(UBASE()) as long
as Par is in watts. In the equations below, IFTE(a,b,c) means "IF a,
THEN return b ELSE return c."
_m/s, _kg/m^3, _kg*s/m^4, _m^2/s^2 and _s/m are used to reconcile the
units in some of the equations. If you are not using an HP calculator
that supports unit conversion, you can ignore the unit objects as long
as you use SI units throughout. This means all lengths, areas, and
volumes are in units of meters, square meters, and cubic meters,
respectively; all frequencies are in Hz; and all powers are in watts.
CLOSED BOX DESIGN
Design Equations
DESIGN.EQ BYTES #2729h 741
%%HP: T(3)A(R)F(.);
{ 'Vb=Vas/Vr' 'Fb=
Qr*Fs' 'F3dB=Qr*Fs*
\v/((1/Qb^2-2+\v/((1/Qb
^2-2)^2+4))/2)' '
dBPEAK=20*LOG(AMAX)
' 'AMAX=IFTE(Qb>INV
(\v/2),SQ(Qb)/\v/(SQ(Qb
)-.25),1)' 'Vr=Qr^2
-1' 'Qr=Qb/Qts' '
Par=(4*\pi^3*1.18_kg/
m^3)*Fb^4*Vd^2/((
345_m/s)*AMAX^2)' '
PeakSPL=112_dB+10*
LOG(UVAL(UBASE(Par)
))' 'Per=Par/\Gno' '
\Gno=10^((SPL-112_dB)
/10)' 'Vd=Sd*xmax'
'Sd=\pi*(DIA*.83)^2/4
' }
Frequency Response
RESPONSE BYTES #6FFFh 112
%%HP: T(3)A(R)F(.);
'dBMAG=20*LOG(SQ(F/
Fb)/\v/(SQ(SQ(F/Fb)-1
)+SQ(F/Fb/Qb)))'
PORTED BOX DESIGN
Design Equations:
DESIGN.EQ BYTES #CB31h 733.5
%%HP: T(3)A(R)F(.);
{ 'Vb=15*Qts^2.87*
Vas' 'Fb=(Vas/Vb)^
.32*Fs' 'F3dB=\v/(Vas
/Vb)*Fs' 'dBPEAK=20
*LOG(2.6*Qts*(Vas/
Vb)^.35)' '\Gno=10^((
SPL-112)/10)' 'Sd=\pi
*(DIA*.83)^2/4' 'Vd
=Sd*xmax' 'Par=(3_
kg*s/m^4)*F3dB^4*Vd
^2' 'PeakSPL=112_dB
+10*LOG(UVAL(UBASE(
Par)))' 'Per=Par/\Gno
' 'Lv=(2362_m^2/s^2
)*Dv^2/(Fb^2*Vb)-
.85*Dv' 'Dmin=\v/(Fb*
Vd*1_s/m)' }
Frequency Response:
RESPONSE BYTES #D7AFh 314
%%HP: T(3)A(R)F(.);
'dBMAG=20*LOG(SQ(SQ
(F/Fs))/\v/(SQ(SQ(SQ(
F/Fs))-(1+SQ(Fb/Fs)
+Fb/(7*Fs*Qts)+Vas/
Vb)*SQ(F/Fs)+SQ(Fb/
Fs))+SQ((SQ(Fb/Fs)/
Qts+Fb/(7*Fs))*(F/
Fs)-(1/Qts+Fb/(7*Fs
))*(F/Fs)^3)))'
Here is a list of values that can be used to verify the equations:
CLOSED BOX DESIGN:
Vas 0.3030_m^3
Qts 0.3300 (unitless)
Fs 30_Hz
SPL 95_dB (unitless)
DIA 0.4572_m
xmax 0.005486_m
Vb 0.08437_m^3
Qb 0.7071 (unitless)
Fb 64.28_Hz
F3dB 64.28_Hz
dBPEAK 0_dB
Par 2.789_W
PeakSPL 116.5_dB (unitless)
Per 139.8_W
\Gno 0.01995 (unitless, 1.995%)
Sd 0.1131_m^2
Vd 0.0006205_m^3
AMAX 1 (unitless)
Vr 3.591 (unitless)
Qr 2.143 (unitless)
PORTED BOX DESIGN:
Vas 0.3030_m^3
Qts 0.3300 (unitless)
Fs 30_Hz
SPL 95_dB (unitless)
DIA 0.4572_m
xmax 0.005486_m
Vb 0.1886_m^3
Fb 34.91_Hz
F3dB 38.02_Hz
dBPEAK 0.1102_dB (unitless)
Par 2.413_W
PeakSPL 115.8_dB (unitless)
Per 121.0_W
\Gno 0.01995 (unitless, 1.995%)
Dmin 0.1472_m
Dv 0.1472_m
Lv 0.09743_m
Sd 0.1131_m^2
Vd 0.0006205_m^3
--
Bill McFadden Tektronix, Inc. P.O. Box 500 MS 58-639 Beaverton, OR 97077
bill@videovax.tv.tek.com, {hplabs,uw-beaver,decvax}!tektronix!videovax!bill
Phone: (503) 627-6920 "SCUD: Shoots Crooked, Usually Destroyed"