## Coupled Controlled Sources

The controlled sources present in SPICE type software are very useful and flexible tools and, in this particular context, allow to circumvent in a rather simple way the restrictions imposed by transformers.

Figure 7 shows the electrical-mechanical coupling of the loudspeaker using a transformer (a) and the relative electrical circuit implemented with coupled controlled sources (b); in this case these are two current controlled voltage sources. The voltage source Bli replaces the secondary of the transformer; the current i flowing in the voice coil now generates a current representing a speed (instead of a force) in the mechanical part of the system. The correct transformation ratio (Bl) is guaranteed by the gain of the controlled sources.

Figure 7. Electrical-mechanical sections coupling. a) Transformer (Mobility).
b) Controlled sources (Impedance). c) Mechanical representation.

Figure 8 shows the same principle applied to mechanical-acoustic transformation.

Figure 8. Mechanical and acoustic sections coupling. a) Transformer (Impedance)
b) Controlled sources (Impedance). c) Mechanical representation.

In this circuit the coupling is provided by a voltage controlled voltage source and a current controlled current source. In this case the transformation ratio is proportional to the cone area.

## The Equivalent Electrical Circuit

To create the complete circuit, in addition to the two current controlled voltage sources (HX), the voltage controlled voltage source (EX) and the current controlled current source (FX), two constant voltage sources (VX) are also required; the latter, if set to zero, in SPICE, are used as ammeters.

To understand how electro-mechanical transduction takes place, let’s look at the equivalent electrical diagram of the entire system, divided into its three components, in Figure 9. The voltage (force) source H2 is controlled by the current, detected by V2, flowing in the voice coil of the loudspeaker. In the same way the current (speed) flowing in the mechanical circuit, detected by V3, controls the voltage source H1, giving rise to the back EMF (counter-electromotive force) in the voice coil. Since e=(Bl)u and f=(Bl)i, the gain of both controlled sources corresponds to the driver’s Bl parameter.

From the same figure we can understand how the mechanical-acoustic transduction takes place: the current of the mechanical circuit, detected by V3, modulates the current of the current source F1, transforming, in a proportional way to the surface of the cone Sd (which in fact represents the gain of the source), a mechanical quantity (speed) into an acoustic quantity (volume speed). The return pressure, a sort of acoustic back EMF, is modulated by the voltage (pressure) at the terminals of F1, which controls, always in a proportional way to Sd, the return voltage (force) applied to the mechanical circuit. It is easy to guess that F1 represents to all effects the membrane of our loudspeaker. The 50 RLC sections model the load that the transmission line exerts on the rear surface of the diaphragm. It is a distributed parameter model and the general rule is that each element cannot be longer than 1/10 of the wavelength; the subdivision of the TL impedance on 50 elements, consequently, allows an analysis of the frequency response accurate up to about 1700 Hz divided the length of the line in meters.

The values of L (air acoustic mass of air; MAA in the circuit) and C (air acoustic compliance; CAA in the circuit) are calculated as follows: where SL is the surface area of the transmission line in m2 and Δz represents the length of the line segment (the full length of the TL divided by the number of elements, in this case 50) expressed in metres.

The circuit composed of MA1, RA1, RA2 and CA1 models the radiation impedance due to the air load on the front part of the cone (we find it also at the open end of the TL to model the radiation impedance of the opening). In  we find the formulas to define the value of the circuit components in the case of a loudspeaker mounted at the end of a pipe: where a is the radius of the loudspeaker diaphragm (or duct opening) in metres, ρ0 is the air density in kg/m3 and c is the speed of sound in m/s.

V4 and V5 are two voltage sources set to 0 (ammeters) which are used to measure the SPL of the speaker and the opening respectively.

Andrea Rubino