Closed-box alignment: Qtc decides everything, volume is the only control

A sealed box does one thing to a drive unit: it traps air, and that air acts as a spring in parallel with the driver's own suspension. Nothing else changes — not the motor, not the mass, not the cone. Every sealed-box alignment in existence follows from that single addition of stiffness.

The consequence is both convenient and unforgiving. Convenient, because three data-sheet numbers are enough to size the box. Unforgiving, because a box can only add stiffness, so the system Q can never be lower than the driver's own Qts — and no volume, however large, will get you below it.

One addition, one free parameter

The driver's suspension has a compliance, and the air in the box has one too. They add, and the ratio of the second to the first is what the whole design turns on. It is written α = Vas/Vb, where Vas is the volume of air with the same compliance as the driver's suspension — a number the data sheet already gives you.

From that one ratio everything follows: the system Q becomes Qts · √(α+1) and the resonance becomes Fs · √(α+1). Both are raised by the same factor, which is why the shape of a closed-box response changes with Q while the corner frequency moves with it.

There is exactly one free parameter, the box volume. Choose a Q and the volume is determined; choose a volume and the Q is determined. Nothing else is available, which is the main structural difference between a sealed box and a ported one, where the tuning frequency is a second knob.

Qtc, and what the values mean

• 0.50 — critically damped. No overshoot at all, and a very gentle corner: f3 is 1.55 times Fc. Used when the box will be equalised anyway.
• 0.577 — Bessel. Maximally flat group delay, so the transient response is as clean as a second-order system allows. Slightly soft in the low bass.
• 0.707 — Butterworth. Maximally flat amplitude, and the point where f3 equals Fc. The default for good reason.
• 0.9 to 1.0 — a deliberate 1 to 1.3 dB of lift at the corner. Common in small monitors, where the lift compensates the baffle step and the shelf in the mid-bass that would otherwise make the box sound thin.
• Above 1.2 — audible ringing on transients and a peak that dominates the sound. Occasionally useful for a subwoofer with a high-pass filter above it, rarely otherwise.

Fc is not f3

The response of a sealed box is a second-order high-pass, so the corner frequency and the resonance are only the same point for one particular Q. Solving |H| = 1/√2 for the normalised frequency gives a quartic whose positive root is

At Q = 0.7071 the middle coefficient vanishes and the root is exactly 1, which is the coincidence that makes Butterworth worth naming. Away from it the two diverge quickly: at Q = 0.5 the -3 dB point is 1.55 Fc, and at Q = 1.5 it is 0.70 Fc.

Reading f3 off a plot instead of assuming f3 = Fc is worth the trouble, because a design quoted at "Fc = 45 Hz" can easily have its -3 dB point at 70 Hz, and the difference is the whole bottom octave of a bass guitar.

The lift has a cost elsewhere

A peak at the corner is not free energy. It comes with group delay that rises towards Fc, and with a driver that is being asked to produce more output at the frequency where its excursion is already largest. Below Fc the cone is unloaded by the box and excursion rises at 12 dB per octave for a constant input, which is why a sealed subwoofer needs a high-pass filter even though the box itself is a high-pass.

The rolloff is also only 12 dB per octave — half the slope of a ported box. That is the trade in the other direction: less output per watt at the bottom, but a gentler phase response and a group delay that stays small, which is why sealed boxes integrate with rooms more easily than ported ones.

Where the model is wrong

The data sheet Qts is measured with the driver in free air and driven from a source with negligible output impedance. Add the series resistance of a crossover inductor and a cable and the electrical damping falls, which raises Qts — often by 0.05 to 0.15, which is enough to move the box volume by 30 per cent. Model with the Qts you will actually have, not the one printed.

Vas has a tolerance of ±20 per cent on most production drivers, and it is not published per unit. Filling the box makes the trapped air behave 10 to 15 per cent larger. Leakage lowers the effective Q. Between those four, a hand calculation is a starting point within a few litres, not a specification — which is why measuring the finished box is the only step that settles it.

Five mistakes worth avoiding

1. Treating Fc as f3. They agree only at Q = 0.7071; elsewhere the gap is tens of per cent.
2. Using the free-air Qts. Series resistance from the crossover and the cable raises it, and the box you build from the printed figure will be too small.
3. Targeting Q below Qts. Impossible, and the calculator will refuse it rather than return a negative volume.
4. Forgetting the high-pass filter. Below Fc the driver is unloaded, and a sealed box does not protect the cone from a rumble or a dropped needle.
5. Ignoring the baffle step. The box's low-frequency lift is often compensating a 6 dB shelf above the baffle frequency, and neither is visible in the alignment numbers.

Summary

Vas and Qts give the ratio, the ratio gives the volume, and the volume gives Qtc, Fc and f3 — but only if the Qts you used is the one the driver will have in the finished system. Build to the measured value, and measure the box once it exists.

Values shown are engineering aids rather than measurements; see the disclaimer and the tool index.