The Subwoofer DIY Page v1.1
4th Order Bandpass Systems: Box Calculations
last updated: 7th May 1998

To use the following calculations, you will need to know the following: 
Vas = equivalent air compliance for the 
      driver (litres) 
Fs  = driver resonance frequency
Qts = driver Q at Fs

The following equations will allow you to design a 4th order bandpass system with a desired low frequency limit or a desired gain. You will need to choose a value for "S" that suits your requirements. 4th order bandpass systems where S is less than 0.7 will have a degraded transient response, but wider bandwidth and smaller box requirements. 

if S = 0.7, then
 b = 0.7206, passband ripple = 0.00 dB

if S = 0.6, then
 b = 0.9560, passband ripple = 0.35 dB 

if S = 0.5, then
 b = 1.2712, passband ripple = 1.25 dB

4th order bandpass system with desired low frequency limit
Choose a value for Fl, the lower 3dB cutoff frequency,

then, 

Fl' = (Fl*Qts)/Fs 
Fh  = (Fl'+b)*Fs/Qts 
Qbp = (Fl'*(Fl'+b))^0.5 
Fb  = Qbp*Fs/Qts
Vf  = (2*S*Qts)^2*Vas 
Vr  = Vas/((Qbp/Qts)^2-1) 
Pa  = -40*LOG(1/(Qbp*2*S))

where, 

Fh  = upper -3dB cutoff frequency (Hz) 
Qbp = Qtc of sealed chamber 
Fb  = resonance frequency 
     of vented chamber(Hz) 
Vf  = net volume of vented chamber (litres) 
Vr  = net volume of sealed chamber (litres) 
Pa  = gain (dB)

4th order bandpass system with desired gain
Choose a value for Pa, the gain in efficiency,

then, 

Qbp = ((10^(-Pa/40))*2*S)^-1 
Fl  = ((-b+(b^2+4*Qbp^2)^0.5)/2)*(Fs/Qts) 
Fh  = Fl+(b*Fs/Qts) 
Fb  = Qbp*Fs/Qts 
Vf  = (2*S*Qts)^2*Vas 
Vr  = Vas/((Qbp/Qts)^2-1)

where, 

Fl  = lower -3dB cutoff frequency (Hz) 
Fh  = upper -3dB cutoff frequency (Hz) 
Qbp = Qtc of sealed chamber 
Fb  = tuning frequency of 
      vented chamber (Hz) 
Vf  = net volume of vented chamber (litres) 
Vr  = net volume of sealed chamber (litres) 
Pa  = gain (dB)
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