Speaker Box Calculator

1 — Compliance ratio is the master knob

Every datasheet gives you $f_s$, $Q_{ts}$, $V_{as}$. $V_{as}$ is the volume of air that has same compliance as the driver suspension. Put the driver in a box of volume $V_b$ and you add another spring in parallel. The key ratio:

$$\alpha = \frac{V_{as}}{V_b}$$

Big box → $\alpha$ small → soft extra spring, $f_c$ barely above $f_s$. Small box → $\alpha$ large → stiff spring, $f_c$ pushed up. You cannot cheat volume — smaller box always costs low-end unless you raise $Q$.

2 — Sealed box math: choose your Qtc

Sealed is a 2nd-order high-pass. Its entire character is $Q_{tc}$:

$$Q_{tc} = Q_{ts}\sqrt{1+\alpha} \quad\quad f_c = f_s\sqrt{1+\alpha}$$

$$V_b = \frac{V_{as}}{(Q_{tc}/Q_{ts})^2 - 1}$$

$$|H(f)| = \frac{(f/f_c)^2}{\sqrt{(1-(f/f_c)^2)^2 + (f/(f_c Q_{tc}))^2}}$$

$Q_{tc}=0.5$ critically damped transient-perfect, lean. 0.577 Bessel — best phase linearity. 0.707 Butterworth — maximally flat amplitude, the universal default. 1.0 adds +1.5 dB peaking, warmer but rings. The slider recomputes $V_b$ for each target — that is why hitting Butterworth shrinks or grows the box in real time.

Box losses $Q_a$ (absorption) and $Q_l$ (leakage) enter as $1/Q_{tc}=1/Q_{tc,ideal}+1/Q_a+1/Q_l$. Typical stuffed box: $Q_a\approx 30–100$, leaky unsealed: $Q_l\approx 10–20$.

3 — Vented box: 4th-order Helmholtz bargain

Ported adds a Helmholtz resonator $f_b$. Below $f_b$ port and driver cancel, above both sum. Transfer becomes 4th-order:

$$|H|^2 = \frac{\Omega^8}{(\Omega^4 - B\Omega^2 + D)^2 + (A\Omega^3 - C\Omega)^2}$$

where $\Omega=f/f_s$, $h=f_b/f_s$, and $A,B,C,D$ are Small's 1973 coefficients containing $\alpha$ and $Q_l$. QB3 is quasi-Butterworth 3rd-order approximation — punch over extension. SBB4 super 4th-order boombox, best for low $Q_{ts}$ drivers in large boxes. SC4 sub-Chebyshev 4, ripple allowed for deepest $f_3$ at expense of group delay.

Port length: $$L_p = \frac{c^2 S_p}{4\pi^2 f_b^2 V_b} - 0.732 r_p$$

That end-correction $0.732 r_p$ is flanged. This tool uses exact acoustic mass calc from $S_p$.

4 — Hoffman's Iron Law & why specs lie

Reference efficiency for half-space:

$$\eta_0 = \frac{4\pi^2}{c^3}\frac{f_s^3 V_{as}}{Q_{es}} \quad Q_{es}= \frac{1}{1/Q_{ts}-1/Q_{ms}}$$

$$Sens = 112 + 10\log_{10}\eta_0 \quad dB @ 1W/1m$$

Hoffman's Iron Law says you only get two of three: small box, deep bass, high efficiency. Formally:

$$\eta_0 \cdot V_b \cdot f_3^3 = constant$$

Shrink $V_b$ 50%, you lose 3 Hz extension or 3 dB sensitivity. No port or DSP escapes it long-term — EQ restores response but not $\eta_0$, so power demand and $X_{max}$ blow up. Max SPL here min() of thermal power $Sens+10\log_{10}P$ and excursion limit $SPL_{Xmax}=20\log_{10}( \rho_0 2\pi f^2 V_d/ p_0)$ where $V_d=S_d X_{max}$.