6.3 Enclosure Calculations

Sealed Box — System Parameters

System resonance:

F_c = F_s × √(1 + Vas / Vb)

System Q (Qtc):

Q_tc = Q_ts × √(1 + Vas / Vb)

F3 (−3 dB frequency):

For Qtc ≥ 0.577:

F3 = F_c × √((1/(2 × Q_tc²) − 1) + √((1/(2 × Q_tc²) − 1)² + 1))

Simplified for Q_tc = 0.707 (Butterworth):

F3 = F_c

Optimal volume for Butterworth alignment:

Vb_butterworth = Vas × ((Q_ts / 0.577)² − 1)⁻¹

Worked example:

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

Target Qtc = 0.707:

Q_tc = Q_ts × √(Vas/Vb + 1) = 0.707
√(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 = 246L

That's enormous — 8.7 cubic feet. Not practical.

Try smaller box: Vb = 30L:

Q_tc = 0.65 × √(45/30 + 1) = 0.65 × √2.5 = 0.65 × 1.58 = 1.03

Overdamped — tight bass, limited output. More realistic for a car build.

F_c at 30L:

F_c = 35 × √(45/30 + 1) = 35 × 1.58 = 55.4 Hz

F3 ≈ F_c for Qtc near 1.0: ~55 Hz

This is why many car subwoofers have a reduced bass extension compared to home audio — real-world box sizes force higher Qtc.

Ported Box — Helmholtz Resonance

Tuning frequency:

F_b = (c / 2π) × √(S_p / (V_b × L_eff))

Effective port length:

L_eff = L_physical + k₁ × √(S_p) + k₂ × √(S_p)

Where: - k₁ = 0.732 (outer end, open) - k₂ = 0.732 (inner end, if chamfered) or 0.850 (if square)

Solving for port length:

L_physical = (c² × S_p) / (4π² × F_b² × V_b) − correction_terms

Illustration in preparation Description: Graphical calculator — three axes: box volume, port area, tuning frequency — with lines showing port length at intersection

Worked example:

Box volume: 2.0 ft³ = 56.6L = 0.0566 m³ Target tuning: 35 Hz Port: 4" diameter round (area = π × 2² = 12.57 in² = 0.00811 m²)

F_b = (343 / 2π) × √(0.00811 / (0.0566 × L_eff))

Solving for L_eff:

35 = 54.6 × √(0.1432 / L_eff)
35 / 54.6 = √(0.1432 / L_eff)
0.641 = √(0.1432 / L_eff)
0.411 = 0.1432 / L_eff
L_eff = 0.1432 / 0.411 = 0.348 m = 13.7 in

Subtract end corrections:

L_physical = 13.7 − (0.732 × √(12.57)) − (0.732 × √(12.57))
L_physical = 13.7 − 2.59 − 2.59 = 8.5 inches

Port length: 8.5 inches for a 4" round port in a 2 ft³ box tuned to 35 Hz.

Verify port velocity (at rated power):

V_port = (S_d × X_max × F_b) / S_p

If Sd = 90 cm², Xmax = 12mm, Fb = 35 Hz, Sp = 12.57 in² = 81 cm²:

V_port = (90 × 0.012 × 35) / 81 = 0.47 m/s

Well below 30 m/s limit — no port noise.

Box Volume from Dimensions

Rectangular:

V_gross = L × W × H
V_net = V_gross − V_driver − V_port − V_bracing

Rule of thumb for deductions: - 10" woofer: 0.1 ft³ displacement - 12" woofer: 0.15 ft³ - 15" woofer: 0.25 ft³ - 18" woofer: 0.35 ft³ - 4" round port × 12" long: 0.06 ft³

Unit conversion:

1 ft³ = 1728 in³ = 28.317 L
1 L = 61.02 in³ = 0.0353 ft³

Bandpass Enclosure

4th-order bandpass — two chamber volumes:

Sealed chamber: Vs (treats the driver as if in sealed box) Ported chamber: Vp (resonates at passband center)

Bandwidth (−3 dB points):

f_upper / f_lower = Q_bp²  [approximately]

Peak frequency:

f_peak ≈ √(f_upper × f_lower)

Design equations:

The simplest approach for a 4th-order bandpass targeting a single frequency fâ‚€:

Vs = 0.7 × Vas
Vp = 1.5 to 2.5 × Vas
F_b_port = fâ‚€ (tune ported chamber to target frequency)

Then design ported chamber exactly as a ported enclosure (use Helmholtz formula above).

Illustration in preparation Description: Frequency response curves for 4th-order bandpass with three Vp/Vs ratios shown, demonstrating trade-off between bandwidth and peak efficiency