Sealed Box - System Parameters

System resonance:

F_c = F_s x sqrt(1 + Vas / Vb)

System Q (Qtc):

Q_tc = Q_ts x sqrt(1 + Vas / Vb)

F3 (-3 dB frequency):

For Qtc >= 0.577:

F3 = F_c x sqrt((1 / (2 x Q_tc^2) - 1) + sqrt((1 / (2 x Q_tc^2) - 1)^2 + 1))

Simplified for Q_tc = 0.707 (Butterworth):

F3 = F_c

Optimal volume for Butterworth alignment:

Vb_butterworth = Vas x ((Q_ts / 0.707)^2 - 1)^-1

Worked example:

Driver: Fs = 35 Hz, Qts = 0.65, Vas = 45 L

Target Qtc = 0.707:

Q_tc = Q_ts x sqrt(Vas/Vb + 1) = 0.707
sqrt(Vas/Vb + 1) = 0.707 / 0.65 = 1.088
Vas/Vb + 1 = 1.183
Vas/Vb = 0.183
Vb = Vas / 0.183 = 45 / 0.183 = 246 L

That is enormous - about 8.7 cubic feet. Not practical for most car installs.

Try a smaller box: Vb = 30 L.

Q_tc = 0.65 x sqrt(45/30 + 1) = 0.65 x sqrt(2.5) = 1.03

Qtc = 1.03 is an underdamped, slightly peaky alignment. It gives more output around system resonance, less-flat response, and a much more realistic box size for a car build.

F_c at 30 L:

F_c = 35 x sqrt(45/30 + 1) = 35 x 1.58 = 55.4 Hz

Using the full F3 expression for Qtc = 1.03 gives F3 ~= 0.776 x 55.4 = 43 Hz.

This is why many car subwoofers trade a textbook Butterworth alignment for practical box size. Smaller boxes push Qtc upward and raise system resonance, then in-car cabin gain often helps rebuild some of the low-end balance.