VIP Access

Loudspeaker System Modelisation

Final student paper of Luc Lemaire, electronics engineer

Introduction

This paper attempts to model a complete loudspeaker system, e.g.: a two or three way system with the filters' action taken into account. The calculation is based on two different models for each type of driver. The first one is the theory of analogous circuits and the second one uses mathematical integration.

The work will be presented in three parts. The first one deals with the analogous circuits, the second one with the mathematical modelisation and the third one will present the results obtained.

First Part - Modelisation with the analogous circuits

The fundamentals of the analogous circuit theory can be found in many books. Beranek [1] used this theory to model the different enclosures encountered in practice. First Thiele [2], then Small [3 & 4] presented their analysis with their own parameters. Today every manufacturer refers to these parameters for the drivers. The theory presented here is based on the papers from Thiele-Small [2-4].

Low frequency model

The two most used enclosures are the closed and the bass-reflex boxes. But the only one implemented in the final program for the low frequency driver within the enclosure is the bass-reflex model. The acoustical equivalent circuit is presented in figure 1a. The left part represents the voice coil. The centre part represents the volume of the enclosure and the right part represents the vent. The transfer function (1) of such an enclosure is a fourth order high pass filter (slope of 24 dB per octave). Thiele presented a few alignments (Butterworth, Chebyshev,...) in [2].
(a)
The frequency responses are given in figure 1b for the most used alignments.

Fig. 1 - Bass-reflex systems:
a) acoustical analogous circuit
b) frequency response (1)
(b)

Mid and high frequency model

The mid-range and high-range drivers into the enclosure can be modelled by the closed box. In fact, mid-range drivers are generally inserted in a small closed volume into the enclosure, and behind the membrane's high-range drivers, there is often a low volume, so in all cases it is very similar to a closed box.

The acoustical equivalent circuit is the one presented in figure 2a. The transfer function (2) is a second order high pass filter (slope of 12 dB per octave). The frequency responses are presented in figure 2b.

(2) These two models are all limited to the piston range of the drivers, so it will be necessary to find another modelisation. This one is presented in the second part.

Fig. 2 - Closed-box systems:
a) acoustical analogous circuit
b) frequency response
(a)

(b)

Second Part - Mathematical Modelisation

Mechanical modelisation of loudspeaker cones

To model the cone of a loudspeaker mechanically, the model described by Frankort in [5] will be used. The goal of this modelisation is to calculate the axial impedance Za of the cone. Once this axial impedance is known, it is possible to integrate the radiated sound by the loudspeaker's cone.

As explained in [5], a few simplifications are made. The alternative current supplying the loudspeaker voice coil is supposed to be independant of the frequency. The study is limited to the frequency range above the fundamental resonance frequency : the behavior under this frequency is well known and can be modelled with the analoguous circuits as explained in Modelisation with the analogous circuits.

A real cone is far from being rigid in all the frequency range. Above a given frequency, resonance phenomenoms known as nodal lines (fig. 3a) and nodal circles (fig. 3b) appear on the cone's surface. Only the latter are taken into account in the following modelisation. In fact, asymetrical waves which produce the nodal lines don't influence the radiated sound as much as the axisymetrical waves (nodal circles) do. The cone is considered to be a set of conical rings and its outer edge is assumed to be free (a greater loss factor is used in the calculations to take this into account). In [5], it is also shown that the fact that the cone is not rigid increases its bandwidth.


(a)

(b)

Fig. 3 - Resonance phenomenoms in loudspeaker cones :
a) nodal lines b) nodal circles

Two kinds of waves appear on the cone's surface : longitudinal and bending waves. These waves are dependant on each other as opposed to the case of a plate where longitudinal and bending waves may exist independantly. Due to this dependancy, standing waves appear only above a given frequency which is called the antiresonant frequency. Below the antiresonant frequency, the cone motion is nearly uniform. For further details, see [5]. So the cone geometry is presented in figure 4.


Fig. 4 - The cone geometry

The equation (3) presents the system for the axisymetrical waves without losses. With losses taken into account, the system's size doubles (12x12).


(3)

Behavior of the cone in the frequency range


The frequency range can be divided in three parts. The first part (fig. 5a) is caracterized by proportional longitudinal and transverse displacements below the antiresonant frequency. The second part is caracterized by the presence of a point called the transition point (x). This point moves from the outer edge of the cone to the inner edge. Between the cone's base and the transition point, the transverse displacement is determined by a longitudinal wave of high wavelength, and between the transition point and the outer edge, there is only a bending wave of short wavelength. The cone is submitted to a succession of tranverse resonance phenomenoms. As illustrated in figure 5c, the cone is entirely covered by bending waves in the third part. The cone behaves like a semi-infinite plate.
(a)

(b)

(c)

Fig. 5 - Tansverse (left) and longitudinal (right) displacements at :
a) low frequencies
b) medium frequencies
c) high frequencies

Mathematical modelisation of loudspeaker domes


The integration of the radiated sound produced by loudspeaker domes is based only on the diffusor's shape. The simple model described in [6] is used.

Third Part - The results

The results are presented in three categories :

Modelisation with the analoguous circuits

The first curve (figure 6) presents the frequency response of a woofer placed in a bass-reflex box. A value of 3 has been used for the quality factor Ql (eq. 1).


Fig. 6 - Frequency response of the neoflex 5" woofer placed in the CR-10S enclosure


Figure 7 presents the frequency response of a tweeter dome with the closed box model. The loss factor in eq. 2 has been neglected.


Fig. 7 - Frequency response of the vifa 1" dome tweeter placed in the CR-10S enclosure

Mechanical modelisation of loudspeaker cones

The mechanical parameters of the loudspeaker cone had to be measured before the simulation. A value of 0.15 has been used for the loss factor. The frequency characteristic of the reduced axial admittance of the cone is illustrated in figure 8. From this graph, the three regions can be clearly found. The first region is situated below 2.5 kHz. The second one between 2.5 kHz and 10 kHz, and the third one, above 10 kHz.


Fig. 8 - Reduced axial admittance of the neoflex 5" woofer


Figure 9 presents the frequency response of the same woofer as in figure 6, but this time, based on the mechanical modelisation.


Fig. 9 - Frequency response of the neoflex 5" woofer based on mechanical modelisation

A complete two ways louspeaker system

Finally, complete loudspeaker systems have been simulated. One is presented below. The system is composed of the two preceding components : the neoflex 5" woofer and the vifa 1". The simulated curve is the "0 dB" labelled one. The real curve is well simulated. The deep at 2 kHz found in the measurements is also present in the simulation. The only difference is that the simulated curve starts to decrease before the measured one. This is mainly due to the fact that asymetrical waves (nodal lines) have been neglected. The slope of 24 dB per octave in the low frequencies isn't visible in the measured curve due to non-perfect conditions during the measurement session in that frequency range.


Fig. 10 - Calculated frequency curve of the CR-10S two ways system



Fig. 11 - Measured frequency curve of the CR-10S two ways system

Conclusion

The final program is a mathematical modelisation of the entire behavior of a complete loudspeaker system (woofer, medium, and tweeter placed in an enclosure with filters). The program permits visualization and therefore to predict the behavior of future developments in loudspeaker systems. It is then easier to determine which driver and also which kind of filters have to be used. One possibility of improvement would be to use a more accurate model for louspeaker domes (medium and tweeter).

References

[1] L. L. BERANEK, Acoustics (McGraw-Hill, New York, 1954)
[2] A. N. THIELE, Loudspeakers in Vented Boxes, Journal of the Audio Engineering Society.(Loudspeakers Vol.1 pg.281)
[3] R.H. SMALL, Closed-Box Loudspeaker Systems, Journal of the Audio Engineering Society (Loudspeakers Vol.1 pg.285)
[4] R.H. SMALL, Vented-Box Loudspeaker Systems, Journal of the Audio Engineering Society (Loudspeakers Vol.1 pg.316)
[5] F.J.M. FRANKORT, Vibration and sound radiation of Loudspeakers cones Thesis, Philips Res. Repts Suppl. 1975, No.2.
[6] J.M. KATES, Radiation from a Dome, Journal of the Audio Engineering Society.(Loudspeakers Vol.1 pg.413)

Copyright 1995-2007, Fundamental Acoustic Research - All Rights Reserved