Speaker box calculations step by step: the formulas with an example

Behind every loudspeaker simulator there are a handful of formulas you can work through on a pocket calculator. They are worth knowing: you see straight away whether a design makes sense, and you understand why the app recommends a particular cabinet. We will work everything out on one real driver, the Tang Band W5-1138SMF woofer from my build diary.

1. The parameters you need

Take the Thiele-Small parameters from the catalogue or from a measurement:

Parameter	Meaning	W5-1138SMF
fs	free-air resonance	45 Hz
Qms, Qes	mechanical and electrical Q	3.56 and 0.57
Vas	volume of air with the same stiffness as the suspension	4.81 l
Sd	effective cone area	94 cm²
Xmax	linear excursion, one way	9.25 mm
Re	DC resistance of the voice coil	3.4 Ω

The total Q is Qts = Qms · Qes / (Qms + Qes), here 3.56 · 0.57 / 4.13 = 0.49. The displacement volume is Vd = Sd · Xmax = 94 cm² · 0.925 cm = 87 cm³. That is how much air the cone pushes at full excursion, and it sets how loud the driver can play in the bass.

A rough rule: a Qts below 0.4 suits a bass reflex, 0.4 to 0.7 suits both a sealed box and a bass reflex, above 0.7 a sealed box or an open baffle. It is a guide, not a law.

2. Sealed box

The air in the box adds stiffness to the cone. The compliance ratio is α = Vas / Vb, and from it:

Qtc = Qts · √(α + 1), the Q in the box,
fc = fs · √(α + 1), the resonance in the box,
and the other way round, the volume you need: Vb = Vas / ((Qtc / Qts)² − 1).

For our woofer at different target Qtc values:

Qtc	Volume Vb	fc	−3 dB point	Character
0.577	12.7 l	53 Hz	67 Hz	gentle roll-off, best transients
0.707	4.5 l	65 Hz	65 Hz	Butterworth, flattest response
0.8	2.9 l	73 Hz	66 Hz	slight hump, “fuller” bass
1.0	1.5 l	92 Hz	72 Hz	+1.2 dB hump, boomier bass

Interestingly, a box three times larger (12.7 l instead of 4.5 l) does not play deeper, it just rolls off more gently. In a room that can be an advantage: the room lifts the bass below 50 Hz, and a gentle roll-off blends well with that. Damping material increases the “acoustic” volume by roughly 15 to 25 %, so a stuffed box behaves as if it were a little larger.

3. Bass reflex

The port together with the air in the box forms a Helmholtz resonator. Its tuning is

fb = c / (2π) · √(Sp / (Vb · Leff)),

where Sp is the port area, Leff its length including the end correction, and c = 343 m/s. In practice a form that gives the tube length directly is used:

Lv = 23,562.5 · D² · N / (fb² · Vb) − 0.732 · D (Lv and D in cm, Vb in litres, N the number of ports).

For a 9.5 l box tuned to 43 Hz this gives:

Port	Length	Air speed*	Pipe resonance
1 × 50 mm	29.9 cm	12 m/s	about 570 Hz
1 × 40 mm	18.5 cm	19 m/s	about 930 Hz
2 × 25 mm	14.9 cm	24 m/s	about 1,150 Hz

* A worst-case estimate: v ≈ 2π · fb · Vd / Sp, the cone's whole displacement volume pushed through the port. The safe limit is about 17 m/s, or 5 % of the speed of sound. Above it the port chuffs and hisses at higher levels.

This shows the classic dilemma of a small box: a wide port that stays quiet is almost 30 cm long and will not fit into a 286 mm deep cabinet, even folded. Narrow ports fit but will chuff. The pipe's own resonance, roughly c / (2 · L), also leaks through the cone into the midrange. That is why I used a passive radiator in this speaker.

4. Passive radiator

A passive radiator is a cone without a magnet that replaces the air in the port. Its tuning is set by its mass: the heavier, the lower fb. Two rules:

The radiator's displacement volume should be at least 1.5 to 2 times the woofer's. The SB15SFCR-00 has 196 cm³ against the woofer's 87 cm³, 2.25 times as much.
Below fb the response falls more steeply than with a port, and just below fb the radiator has its own resonance, which creates a notch. Below its tuning the speaker therefore “cuts off” the bass more sharply.

The advantages: no flow noise and no pipe resonance, and the tuning can be fine-tuned by adding weights.

5. Quarter-wave line (TQWT, TL)

A line closed at one end and open at the other resonates when its length is a quarter of the wavelength:

f ≈ c / (4 · L)

The line in the Needle is 1,746 mm long, so 343 / (4 · 1.746) = 49 Hz. The actual tuning is shifted slightly by the taper, the driver's position along the line and the outlet. Martin J. King described these effects precisely in his line models. Loudsmith calculates the line section by section, with damping, and matches the ADW impedance measurement within 0.2 Ω.

6. Baffle step

At low frequencies a driver radiates in all directions, at high frequencies only forwards. The transition is around

f3 ≈ 115 / W (W is the baffle width in metres)

and below it the level drops by up to 6 dB. For a 190 mm baffle that is about 605 Hz, for the narrow Needle (160 mm) about 720 Hz. Near a wall the loss is smaller, which is why the step is usually compensated by only 2 to 4 dB in the crossover. Besides the step itself, a wave reflects off the edges and causes ripple in the mids and treble. Rounded edges and an off-centre tweeter reduce it.

7. Sensitivity from the parameters and Hoffman's law

The reference efficiency follows directly from the parameters:

η₀ = 4π² / c³ · fs³ · Vas / Qes and the sensitivity SPL = 112.1 + 10 · log₁₀(η₀) dB/1 W/1 m.

For the W5-1138SMF: η₀ = 0.075 %, which is 80.9 dB/1 W. At 2.83 V into 3.4 Ω the driver gets 2.35 W, which adds 3.7 dB, giving 84.6 dB. The manufacturer states 85 dB, so the calculation checks out.

The formula shows Hoffman's “iron law”: you cannot have a low fs, a small box and high sensitivity all at once. A long-throw, low-resonance woofer in a small box always pays in sensitivity.

What Loudsmith adds

The formulas above are a first approximation and are perfectly good for checking a design. The app goes further:

It calculates the whole electro-acoustic circuit with box losses, damping and voice-coil inductance, and so the impedance too.
It compares up to eight cabinet types at once, including transmission line, TQWT, ML-TL and open baffle.
It checks cone excursion, port air speed and bass delay at real power, not just at the tuning frequency.
It estimates the baffle step and edge reflections from the cabinet dimensions and driver positions.
It picks ports from tubes you can actually buy and checks that they fit.
And finally it shows how the speaker will behave in your room.