Resonance Characteristics Analysis and Formula Derivation in Loudspeaker Design

Published by IWISTAO

In loudspeaker design, the system composed of the speaker driver and the enclosure exhibits specific resonance characteristics. These resonance properties directly affect the low-frequency response and overall sound quality of the loudspeaker.

This article explores the resonance behavior of sealed and bass-reflex (ported) enclosures and provides detailed calculation formulas and derivations.

 

1. Basic Parameters of the Speaker Driver

The performance of a loudspeaker driver is usually characterized by several key parameters:

• Fs — Free-air resonance frequency: the natural resonant frequency of the driver when vibrating in free air.

• Vas — Equivalent compliance volume: the volume of air that has the same compliance as the driver’s suspension system.

• Qts — Total quality factor: represents the damping characteristics of the driver, combining the mechanical quality factor (Qms) and electrical quality factor (Qes) in parallel:

where:

and:

• Mm — moving mass (including diaphragm and voice coil)

• Rm — mechanical resistance

• Re — DC resistance of the voice coil

• Bl — product of magnetic flux density (B) and voice coil length (l)

 

2. Resonance Frequency of Sealed Enclosures

In a sealed-box loudspeaker, the driver is mounted in a completely airtight cabinet.

The air inside the box provides an additional stiffness that, together with the driver’s suspension, forms a new resonant system.

The system’s resonance frequency (fc) can be expressed as:

where:

• fc — resonance frequency of the sealed enclosure

• fs — free-air resonance frequency of the driver

• Vas — equivalent compliance volume of the driver

• Vb — effective internal volume of the enclosure

 

It can be observed that a smaller box volume raises the resonance frequency, degrading low-frequency performance,

while a larger box brings fc closer to fs, improving bass extension.

However, overly large enclosures may reduce driving force and cause poor low-frequency control.

• As V_b increases, f_c drops toward f_s (here from 86.6 Hz → 61.2 Hz), giving deeper bass—exactly as predicted by the formula and design intuition.

 

3. Resonance Frequency and Port Design in Bass-Reflex Enclosures

A bass-reflex (ported) loudspeaker introduces a vent (or port) in the enclosure, forming a Helmholtz resonator with the air inside.

This structure enhances bass efficiency and extends low-frequency response.

The enclosure’s resonance frequency (fb) is given by the Helmholtz resonance formula:

where:

• fb — resonance frequency of the ported enclosure

• c — speed of sound (≈ 343 m/s at room temperature)

• S — cross-sectional area of the port (m²)

• Lport — effective length of the port (m)

• Vb — internal volume of the enclosure (m³)

 

For optimal performance, fb is typically set slightly below the overall system resonance frequency determined by the driver and box.

A practical alignment based on Thiele–Small parameters can be obtained using the empirical relationship:

During real-world tuning, adjusting the port length (Lport) or area (S) allows precise control over fb.

 

4. Port Air Velocity and Distortion Control

To avoid air turbulence and chuffing noise, the maximum air velocity in the port should be limited according to:

where:

• vport — maximum air velocity inside the port (m/s, typically < 17 m/s)

• Xmax — maximum linear excursion of the driver (m)

• Sd — effective diaphragm area of the driver (m²)

• f — frequency corresponding to maximum excursion (Hz)

• S — port cross-sectional area (m²)

By increasing the port area or adjusting its length, designers can reduce air velocity and distortion, improving bass clarity.

 

5. Summary and Design Recommendations

From the above derivations:

• Sealed Enclosures — Larger volume yields deeper bass and lower resonance frequency, but requires more space.

• Bass-Reflex Enclosures — Offer better low-frequency extension in smaller boxes through port tuning, but require careful optimization of port dimensions and damping.

In practice, designers must balance driver parameters, box volume, and port geometry to achieve an optimal resonance frequency and smooth, accurate bass response.