Vented Loudspeaker Bass-Reflex Enclosure - 2D Cross-Section with Thiele-Small Response, Port Tuning and Group Delay

Presets

Playback

Time speed

Choose a driver (generic examples)

Driver parameters (Thiele-Small)

Fs, Qes, Qms, Vas, Sd, Re and Xmax come from a driver data sheet.

Enclosure and port

Sealed vs vented, box volume, port size, leakage.

Test tone and amplifier

The yellow marker in every graph and the 3D animation follow this tone.

Display

Animation runs at a slow visual rate; amplitudes are exaggerated so you can see them.

Layers

Box air particles (compress with pressure) Box pressure colour (red = compression) Port air plug Radiation arcs at the cabinet Radiated pressure field (vertical slice) Labels, mic and scale

Model checks

The science

Three coupled resonators

The driver is a mass-spring system (cone mass M_d, suspension compliance C_ms, loss R_ms) driven by the force B·l·E/R_e and damped electrically by (B·l)²/R_e. The box is an air spring, C_ab = V_b/(ρc²). The port is an air slug of mass M_p = ρ L_eff S_p. With x the cone displacement into the box, y the slug displacement out of it and p the box pressure:

M_d x'' + (R_ms + Bl²/R_e) x' + x/C_ms = Bl E/R_e − S_d p
M_p y'' = S_p p
C_ab p' = S_d x' − S_p y' − p/R_al

The sound radiated into the room follows the net volume acceleration a_v = S_p y'' − S_d x''. In a half space the pressure at distance r is p = ρ a_v/(2π r), and SPL = 20 log₁₀(p_rms/20 µPa).

Port tuning

Box air and port slug form a Helmholtz resonator: f_b = (c/2π) √(S_p/(V_b L_eff)) with L_eff = L + 1.463 a (end correction for one flanged and one open end). At f_b the cone is nearly still and the port does the work, so the excursion graph has a dip there. Below f_b the port and cone radiate in anti-phase and the output falls at 24 dB/octave (fourth-order high-pass); a sealed box falls at 12 dB/octave but starts higher up.

Small's normalised response

With T_s = 1/ω_s, T_b = 1/ω_b, α = V_as/V_b, Q_T the total driver Q and Q_L the box leakage Q, the lossless-box response is G(s) = s⁴T_b²T_s² / D(s) with D(s) = s⁴T_b²T_s² + s³(T_b²T_s/Q_T + T_bT_s²/Q_L) + s²((α+1)T_b² + T_bT_s/(Q_LQ_T) + T_s²) + s(T_b/Q_L + T_s/Q_T) + 1. The graphs use the full coupled equations above (which also handle the box loss exactly); the closed form is used as an independent check.

Limits

The "max SPL" curve scales the drive until the first limit is met: cone excursion reaching X_max, the port air reaching 17 m/s (about 5 % of the speed of sound, where chuffing noise sets in), or the voice coil reaching the assumed power handling. All three scale linearly with voltage in this linear model.

Honest simplifications

Linear small-signal model: no suspension non-linearity, voice-coil inductance, thermal compression, cone break-up or cabinet-wall vibration; the box is lumped (valid when its size is much smaller than a wavelength, i.e. below a few hundred Hz); the port is a plug of air with a fixed end correction and no flow-dependent losses other than the leak Q_L; the radiation is half-space (2π) from two point sources. The wave-field slice uses an adjustable wavenumber so that the pattern is visible in a 4-metre scene - at 1x it is real scale. Drivers are generic examples, not real products.

What to try

Press "B4 flat alignment", then "Over-tuned boomy" and watch the response get a hump and the group delay peak rise.
Set the test tone to the box tuning fb: the cone dip and the port air-speed peak appear at the same frequency, and the field shows the port radiating alone.
"Narrow port chuffing": the port air speed passes 17 m/s (red line) at moderate volume.
Turn the port off ("Sealed") and compare the roll-off and the transient group delay.
Raise the voltage until the max-SPL curve shows which limit you hit first.