R.H. Small Vented-Box Papers
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

A.N. Thiele Vented Box Papers
Loudspeakers in Vented Boxes Part I
Loudspeakers in Vented Boxes Part II

Main Pages
North Reading Engineering
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 Thiele-Small Vented Box has been developed to calculate the necessary values of Lceb, Rel and Cmep from the TS parameters that are routinely provided by manufacturers.  Note that the units must be the same as those shown in the spreadsheet.  Magnitudes for the resonance frequency of the vented enclosure, Fb, the net enclosure volume, Vb, and the estimated enclosure Quality factor, Ql, must be provided (the Excel ANALYSIS TOOLPAK must be installed to solve the transfer functions). 

The vented-box equivalent circuit is a 4th-order high-pass filter function, G(s), where s is the complex frequency variable (=jw).  The Thiele-Small Vented Box spreadsheet calculates and plots the transfer function log modulus and phase angle for the driver and enclosure parameters provided by the user.  The coefficients a1, a2 and a3 (shown as a_1, a_2 and a_3 in the spreadsheet) determine the response characteristics of the function and, for a desired filter type (Butterworth, Bessel, Chebyshev, etc.), have specific values (see Small's Vented-Box Loudspeaker Systems Part I: Small-Signal Analysis and
Vented-Box Loudspeaker Systems Part IV: Appendices below)).

Examples of the spreadsheet plots of magnitude versus frequency response derived from the general form of the high-pass transfer function, G(jw), and the phase-angle, f, are shown below.  As a check for consistency, the spreadsheet calculates the magnitude-squared form of the transfer function, IG(jw)I^2, and compares it to the result obtained by solving for the log modulus directly using complex algebra. 
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 p4To4  (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.  

Accurate simulations of amplitude response, phase-angle, impedance modulus, group delay, impulse response and other parameters are possible using the general purpose, open source circuit simulator SPICE (= Simulation Program with Integrated Circuit Emphasis).  Here, we are using a version provided by Linear Technologies called LTSPICE IV.  By example, we demonstrate how the package can be used to model the motional impedance of a low frequency loudspeaker in free-air and installed in a vented enclosure.  We provide two Excel spreadsheets, one that calculates the equivalent circuit loudspeaker parameters necessary to perform the impedance and phase simulations and a second that does the same with the loudspeaker mounted in a vented-enclosure.  The vented enclosure simulation spreadsheet also provides solutions to the general form of the 4th order (24dB/octave) high-pass filter function and outputs the coefficients that determine the characteristic shape of the response.  Note that the figures and graphs are best viewed in either MS Internet Explorer 9 or Google Chrome. 

The electrical equivalent circuit of a loudspeaker in air or mounted on an infinite baffle is modeled as a parallel RLC resonant circuit.  The amplifier is a voltage source component.  The Thiele-Small parameters, Cmes, Lces and Res correspond to the electrical capacitance associated with the driver moving mass, the electrical inductance of the suspension compliance (the reciprocal stiffness) and the electrical resistance of the suspension frictional losses, respectively.  The voice coil impedance is modeled as a resistor with magnitude Re in series with an inductor, Le (the actual impedance is somewhat more complicated).  The radiation impedance is ignored . 

Since the equivalent circuit parameters are not always provided in specification sheets.  An Excel file named Thiele-Small Parameters was developed and calculates the necessary values of Cmes, Lces and Res from the TS parameters that are routinely provided by manufacturers.  Note that the units must be the same as those shown in the spreadsheet.  The electrical equivalent circuit of a loudspeaker in LTSPICE IV and corresponding impedance modulus (right) are given below.  The impedance modulus is derived by dividing the source voltage V(n001) by the magnitude of the current passing through the voice coil, I(Re).