The Subwoofer DIY Page
4th Order Bandpass Systems - Power Response Calculations
last updated: 15 April 2006
The Subwoofer DIY Page

To calculate the power response of a 4th order bandpass system, you will need to know the following:
Vas = equivalent compliance of driver (litres)
Qes = electrical Q of driver at resonance
Fs = resonance frequency of driver (Hz)
PEmax = maximum input power for driver
Fb = resonance frequency of the system
F3 = frequency at which response is down by -3dB
Dia = effective driver diameter driver (cm)
Xmax = peak linear displacement of driver cone (mm)
Vr = net volume of sealed section (litres)
Vf = net volume of vented section (litres)

then,

Qts' = Qts*((Vas+Vr)/Vr)^0.5
Fs' = Fs*Qts'/Qts
Qes' = Qes*Qts'/Qts
Vas' = Vas*Vr/(Vas+Vr)
Vb' = Vf

Sd = pi*(Dia/100)^2/4
Vd = Sd*Xmax/1000
n0 = 9.64*10^(-10)*Fs'^3*Vas'/Qes'
SPL = 112 + 10*LOG(n0)
Par = 3*F3^4*Vd^2
Per = Par/n0
PeakSPL = SPL+10*LOG(PEmax)

K1 = (4*pi^3*Ro/c)*Fs'^4*Vd^2

where,

pi = 3.14159265359
c = speed of sound in air (345 m/s)
Ro = density of air (1.18 kg/m^3)
n0 = free-air efficiency of driver
SPL = SPL of driver @1W/1M
Par = maximum linear power output
Per = electrical input required to produce Par
PeakSPL = Thermally-limited SPL in passband

Maximum output at a given frequency F can be calculated as follows:

Fn2 = (F/Fs')^2
Fn4 = Fn2^2
A = (Fb/Fs')^2
B = A/Qts'+Fb/(Ql*Fs')
C = 1+A+(Vas'/Vb')+Fb/(Fs'*Qts'*Ql)
D = 1/Qts'+Fb/(Fs'*Ql)
E = (97/49)*A
dBmag = 10*LOG((A*Fn2)^2/((Fn4-C*Fn2+A)^2+Fn2*(D*Fn2-B)^2)))
Pmax = (K1/n0)*((Fn4-C*Fn2+A)^2+Fn2*(D*Fn2-B)^2)/(Fn4-E*Fn2+A^2)
SPLmax = SPL+dBmag+10*LOG(Pmax)
SPLtherm = PeakSPL+dBmag

where,

SPLmax = displacement-limited SPL at F (dB/1M)
SPLtherm = thermally-limited SPL at  F (dB/1M)

My thanks to Ron Ennenga, for providing these equations for calculating the power response of 4th order bandpass systems.