## The Simulation Model

The theory of electrical circuits has been applied to the study of electromechanical vibrating systems since the early 1900s, but its use for the design of loudspeaker systems was first described in 1952 by Bart N. Locanthi [9]. The great diffusion of this method occurred shortly afterwards, probably after the publication in 1954 of Acoustics [13] by Leo Beranek, considered the reference text on acoustics still today. Beranek described in detail the two types of analogies (mobility and impedance) that allow the use of resistors, inductors and capacitors as basic mechanical or acoustic elements and illustrated the transformers technique (and their transformation ratios) to couple the electrical, mechanical and acoustic section of the loudspeaker into a single equivalent electrical circuit from which the transfer function of the entire system can be derived. This approach was also used by Thiele and Small to identify and describe the fundamental parameters of the loudspeakers (which still bear their name) and to analyse the behaviour of closed and bass-reflex enclosures.

At a practical level the transformer technique obliges the designer to use the mobility analogy, however, when it comes to modelling an acoustic device, it is preferable to use the impedance analogy, where the sound pressure is the equivalent of the electric voltage in the analogous circuit: to measure the sound pressure in an acoustic circuit it is sufficient to place a microphone probe in the desired point without making any modification to the system; similarly we can easily measure the voltage in a point of the electric circuit by applying a probe without modifying or interrupting it. If we decided to use this model of analogy, particularly suitable for acoustic circuits, applying the transformer technique, since we could not use two different types of analogy in the same system at the same time, we would be obliged to apply it to the whole equivalent circuit and a big difficulty would arise in coupling the mechanical part of the loudspeaker with its electrical part: the mechanical force moving the loudspeaker (the voltage in the equivalent circuit, using the impedance analogy) comes from the current flowing in its electric circuit (f=Bli, where Bl, a parameter usually supplied by the loudspeaker manufacturer, represents the motor force factor). In order to obtain the correct conversion from current to voltage we would then be forced to apply the principle of duality of the electric circuits to the electric section of the driver, complicating the model considerably (especially if we wanted to include a crossover filter in the simulation).

In 1991 W. Marshall Leach, Jr. [14] described the use of the circuit simulation program SPICE applied to the design of electroacoustic systems. The main advantage introduced by the use of SPICE is the possibility to use controlled-sources, instead of transformers, for the coupling of the three sections of the electro-mechanical-acoustic system. The controlled sources allow the use of impedance analogy without upsets in the electrical section of the system. Potentially, it would also be possible to use both types of analogy simultaneously, however, as the problem is essentially an acoustic one, it is convenient for practical purposes to use only the impedance analogy.

The powerful graphical interface of SPICE also allows a quick analysis of the main features of the system such as frequency response, impedance curve and speaker cone excursion.

In his paper Leach provided the equivalent circuits for various types of acoustic transducers, microphones and closed and reflex speaker systems, taking into account for the latter also the mutual coupling between woofer and duct.

## Impedance Analogy

As mentioned above, by constructing the analogous electrical circuit, two types of analogy can be used: mobility or impedance. The first is usually preferred for mechanical systems, while the second for acoustic systems. To deepen and better understand this subject I recommend reading Chapter 3 of Acoustics by Leo Beranek. Since a loudspeaker includes electrical, mechanical and acoustic parameters, you should now ponder and make a choice.

Locanthi in [9] uses the mobility analogy to design circuits suitable to simulate the three most popular speaker types at that time: infinite baffle, bass-reflex and horns.This was an obligatory choice since he actually built an electric circuit with real components. Locanthi’s model omits the coupling transformer between the electrical and mechanical section of the driver and uses it only in the mechanical-acoustic coupling to match the horn throat area with the cone area. It is important to underline that, since there is no transformer to couple the electrical and mechanical part of the system, the values of the RES and LES components (see Figure 4), in order to obtain the correct interfacing, must be divided by (Bl)2.

Augspurger, as mentioned above, in [10] assimilates the TLs to leaky horns and makes only one major modification to Locanthi’s work, adding a (variable) shunt resistor to represent the losses due to the absorbent material. Although he uses computer simulations, he also uses the more “familiar” mobility analogy. The few other modifications concern the addition of an optional line section to simulate an offset TL and the omission of the coupling transformer also on the mechanical-acoustic side (with the appropriate adjustments in case the area of the initial section of the line is different from the cone area).

Leach, in [14], while not treating TLs, provided enough information to develop a loudspeaker-transmission line system model based on impedance analogy. Having a circuit simulation program such as SPICE at my disposal, I consider without doubt the latter the most elegant choice according to the problem. The following circuits use impedance analogy only. Omitting the acoustic load and considering only the mechanical section, the equivalent loudspeaker circuit is visible in Figure 6b.

In this analog circuit the LMS inductance, expressed in henry, has the numerical value of the moving mass (m) of the loudspeaker in kg. CMS and RMS represent respectively the compliance (1/k), expressed in farad, and the mechanical damping (d) of the suspension, expressed in ohms. The voltage generator e, finally, represents the force (f), in Newton, applied to the cone. From a mechanical point of view the driver can be represented as in Figure 6a.

Table 1 lists the electrical quantities and the equivalent mechanical and acoustic quantities of similar circuits with the relative symbols. The table shows why the impedance analogy is also called voltage-force-pressure analogy.

Andrea Rubino

Table 1. Quantity equivalence, impedance analogy

## References

[1] M. Colloms, “High Performance Loudspeakers”, Pentech Press Ltd., Fourth Edition; 1991

[2] J. Backman, “A Computational Model of Transmission Line Loudspeakers”, 92nd AES Convention, 1992, preprint no. 3326.

[3] G. L. Augspurger, “Loudspeaker on Damped Pipes”, JAES Volume 48 Issue 5 pp. 424-436; May 2000.

[4] B. Olney, “A Method of Eliminating Cavity Resonance, Extending Low Frequency Responce and Increasing Acoustical Damping in Cabinet Type Loudspeakers”, J. Acoust. Soc. Amer. Volume 8; October 1936.

[5] A. R. Bailey, “A Non-resonant Loudspeaker Enclosure”, Wireless World; October 1965.

[6] A. R. Bailey, “The Transmission Line Loudspeaker Enclosure”, Wireless World; May 1972.

[7] L. J. S. Bradbury, “The Use of Fibrous Materials in Loudspeaker Enclosures”, JAES Volume 24 Issue 3 pp.162-170; April 1976.

[8] R. M. Bullock and P. E. Hillman, “A Transmission-line Woofer Model”, 81st AES Convention, 1986, Preprint no. 2384.

[9] B. N. Locanthi, “Application of Electric Circuit Analogies to Loudspeaker Design Problems”, JAES Volume 19 Issue 9 pp. 778-785; October 1971.

[10] G. L. Augspurger, “Transmission Lines Updated, Part 1”, Speaker Builder; 2/00.

[11] G. L. Augspurger, “Transmission Lines Updated, Part 2”, Speaker Builder; 3/00.

[12] G. L. Augspurger, “Transmission Lines Updated, Part 3”, Speaker Builder; 4/00.

[13] L. L. Beranek, “Acoustics”, New York: McGraw-Hill; 1954.

[14] W. M. Leach, Jr., “Computer-Aided Electroacustic Design with SPICE”, JAES Volume 39 Issue 12; December 1991.

[15] V. Landi, “La linea di trasmissione”, AUDIOreview, n.104-106-112-132-133; 1991-1993.