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    Definition Closed-Box



Definition of terms:
f-3 -3dB half-power frequency.
fs resonance frequency of driver
fc resonance frequency of the closed box system
Q ratio of reactance to resistance (series circuit) or resistance to reactance (parallel circuit)
Qts total Q of driver (woofer) at fs, considering all driver resistance.
Qtc total Q of speaker system at fc, including all system resistance.
Vas volume of air having the same acoustic compliance as the driver suspension.
Vab volume of air having the same acoustic compliance as the enclosure.
Xmax peak linear displacement of driver cone.
Sd effective surface area of driver cone.
Vd peak displacement volume of driver cone.
Vb net internal volume of enclosure.
a compliance ratio (alpha).
no reference efficiency
Cas acoustic compliance of the driver suspension.
Cab acoustic compliance of the air in the enclosure.




Driver "Q" and Enclosure response
The point of fitting a woofer to a box is to control the response characteristics.
The objective method is to measure and align de Q-factor. Q is a composite term, used to describe resonant magnification in speaker boxes.
It represents the degree to which the electrical, mechanical and pneumatic circuits of the woofer/box combination interact to control resonance.
Certain values of Qtc have specific response characteristics, which can be categorized as follows :
 
      Qtc = 0,5 critically damped -- transient perfect
      Qtc = 0,707 Butterworth-response(B2)-- max falt amplitude response with minimum cutoff
      Qtc > 0,707 Chebychev-response -- equal Ripple response - nax power handling and max efficiency
Qtc = 0,5 is usually regarded as excessivly taut and overdamped.
Some authorities still consider this value (0.5 - 0.6) to be optimal.
      Peak(db) = 20 * Log (Qtc4 / Qtc2 - 0,25)0,5
      fgmax = 1 / (1 - 1 / 2 * Qtc2)0,5
The frequency of this peak (fgmax) is given as a ratio with the box resonance (fc).




Closed-Box Design Equations

Qtc, Qts, A(lpha), Vas, fs Qtc=0,707 , Qts=0,37 , Vas = 293 Ltr , fs = 20Hz
A = (Qtc/Qts)2 -1 (0,707/0,37)2 -1 = 2,65
Vab = Vas/A 293/2,65 = 110,56 Liter
Vb = Vab/1.2 110,56/1,2 = 92 Liter
fc = fs*(Qtc/Qts) 20*(0,707/0,37) = 38,2 Hz


Check EBP

Firstly, calculate the Efficiency Bandwidth Product: EBP = fs/Qe.

Fs [Hz]
Qe []
EBP[]
 
  • 50 or less = best used in a sealed enclosure.

  • 50 - 90 = flexible enclosure options.

  • 90 or greater = best used in ported enclosure.

If you have Qe and Qm, you can take external resistance into account when calculating Qts.

Qts = 1/((1/Qm) + Rs/((R+Rs)Qe)) where R is the resistance of the wiring, typically 0.5 ohm.

Alternatively, use Qts as specified in the TS parameters.


Sealed Cabinet (Butterworth)

Fs [Hz] : Qts : Vas [l] : Qtc :


Vb [l] : Fc [Hz] : F3 [Hz] :
Fmax [Hz] : Amax [dB] :




Fc [Hz] - Resonant Frequency of driver in box
F3 [Hz] - -3dB Frequency
Fmax [dB] - Frequency of the maximum amplitude increased height (only for Qtc > 0.71)
Amax [dB] - maximum amplitude increased height at Fmax (only for Qtc > 0.71)
Vb [l] - Net volume of the box in liter (Acceptance: 10% volume increase by filling with absorption material)


Source: The Loudspeaker Design Cookbook, Vance Dickason