Vented-Box Loudspeaker Systems Part I: Small-Signal Analysis

Vented-Box Loudspeaker Systems Part II: Large-Signal Analysis

Vented-Box Loudspeaker Systems Part III: Synthesis

Vented-Box Loudspeaker Systems Part I: Appendices

Loudspeakers in Vented Boxes Part I

Loudspeakers in Vented Boxes Part II

The Audio Engineering League

The magnitude of Vb is the internal volume minus the
sum total of the volumes associated with the vent duct(s), bracing and
speaker. By assuming a magnitude for Vb,
the compliance ratio, alpha, is then determined (=Vas/Vb). The magnitude of Ql is
a function of the total
enclosure losses associated with absorption, leakage and the vent(s).
Of the three, leakage losses are significant and Ql values
between 5-20 are typical. By using the spreadsheet calculator, a
range of enclosure Ql values, for a given set
of system parameters is easily model as shown below (left). The
impedance modulus, as a function of the electrical equivalent
resistance, Rel, associated with the given
Ql, is shown below (right). As is evident in the graph, the
motional impedance of the driver increases as the resistive losses associated with leakage decrease
(green Ql=5,
fuscia Ql=10,
red Ql=20).

The electrical equivalent circuit of a loudspeaker mounted in a vented enclosure ignoring mutual coupling between the diaphragm and vent is modeled in LTSPICE IV and shown below. The additional components Lceb, Rel and Cmep are the electrical equivalent inductance of the enclosure compliance, the electrical equivalent resistance of the enclosure leakage losses and the electrical equivalent capacitance of the air mass present in the vent, respectively. An Excel file named

The vented-box equivalent circuit is a 4th-order high-pass filter function, G(s), where s is the complex frequency variable (=

Examples of the spreadsheet plots of magnitude versus frequency response derived from the general form of the high-pass transfer function, G(

The equations and coefficient shown in the Thiele-Small Vented Box spreadsheet are consistent in form with those published by Small.
The file is the the only source on the internet, we're aware of, that shows how the equations are solved.

We have provided copies of Small's papers in the dropdown menu below and, for comparison purposes, included the earlier papers of A. N. Thiele. Both papers examine the 4th order high-pass filter transfer function that is derived from equivalent circuits. For example, multiplying the top and bottom of Thiele's Eq. (21) by*p*^{4}*T*_{o}^{4}
(i.e. multiplication by unity), and setting the coefficients
x1, x2 and x3 to a1, a2 and a3, respectively, the result is Small's Eq. (57).
Small extends Thiele's work by considering efficiency,
large-signal behavior, enclosure losses and interactions
between the diaphragm and vent.

We have provided copies of Small's papers in the dropdown menu below and, for comparison purposes, included the earlier papers of A. N. Thiele. Both papers examine the 4th order high-pass filter transfer function that is derived from equivalent circuits. For example, multiplying the top and bottom of Thiele's Eq. (21) by