# Acoustics/Acoustic Loudspeaker

The purpose of the acoustic transducer is to convert electrical energy into acoustic energy. Many variations of acoustic transducers exist, although the most common is the moving coil-permanent magnet transducer. The classic loudspeaker is of the moving coil-permanent magnet type.
The classic electrodynamic loudspeaker driver can be divided into three key components:
* The Magnet Motor Drive System
* The Loudspeaker Cone System
* The Loudspeaker Suspension

## The Magnet Motor Drive System

The main purpose of the Magnet Motor Drive System is to establish a symmetrical magnetic field in which the voice coil will operate. The Magnet Motor Drive System is comprised of a front focusing plate, permanent magnet, back plate, and a pole piece. In figure 2, the assembled drive system is illustrated. In most cases, the back plate and the pole piece are built into one piece called the yoke. The yoke and the front focusing plate are normally made of a very soft cast iron. Iron is a material that is used in conjunction with magnetic structures because the iron is easily saturated when exposed to a magnetic field. Notice in figure 2, that an air gap was intentionally left between the front focusing plate and the yoke. The magnetic field is coupled through the air gap. The magnetic field strength (B) of the air gap is typically optimized for uniformity across the gap. [1]
When a coil of wire with a current flowing is placed inside the permanent magnetic field, a force is produced. B is the magnetic field strength, $l$ is the length of the coil, and $I$ is the current flowing through the coil. The electro-magnetic force is given by the expression of Laplace :
$d{\underline {F}}=I{\underline {dl}}\times {\underline {B}}$
${\underline {B}}$ and ${\underline {dl}}$ are orthogonal, so the force is obtained by integration on the length of the wire (Re is the radius of a spire, n is the number of spires and ${\underline {e}}_{x}$ is on the axis of the coil):
${\underline {F}}=2\pi R_{e}BnI{\underline {e}}_{x}$
This force is directly proportional to the current flowing through the coil.
The coil is excited with the AC signal that is intended for sound reproduction, when the changing magnetic field of the coil interacts with the permanent magnetic field then the coil moves back and forth in order to reproduce the input signal. The coil of a loudspeaker is known as the voice coil.

## The loudspeaker cone system

On a typical loudspeaker, the cone serves the purpose of creating a larger radiating area allowing more air to be moved when excited by the voice coil. The cone serves a piston that is excited by the voice coil. The cone then displaces air creating a sound wave. In an ideal environment, the cone should be infinitely rigid and have zero mass, but in reality neither is true. Cone materials vary from carbon fiber, paper, bamboo, and just about any other material that can be shaped into a stiff conical shape. The loudspeaker cone is a very critical part of the loudspeaker. Since the cone is not infinitely rigid, it tends to have different types of resonance modes form at different frequencies, which in turn alters and colors the reproduction of the sound waves. The shape of the cone directly influences the directivity and frequency response of the loudspeaker. When the cone is attached to the voice coil, a large gap above the voice coil is left exposed. This could be a problem if foreign particles make their way into the air gap of the voice coil and the permanent magnet structure. The solution to this problem is to place what is known as a dust cap on the cone to cover the air gap. Below a figure of the cone and dust cap are shown.
The speed of the cone can be expressed with an equation of a mass-spring system with a damping coefficient \xi :
$m{\frac {dv}{dt}}+\xi v+k\int {v\,dt}=Bli$
The current intensity $i$ and the speed $v$ can also be related by this equation ($U$ is the voltage, $R$ the electrical resistance and $L_{b}$ the inductance) :
$L_{b}{\frac {di}{dt}}+Ri=U-Blv$
By using a harmonic solution, the expression of the speed is :
$v={\frac {Bli}{\xi +j\left(m\omega -{\frac {k}{\omega }}\right)}}$
The electrical impedance can be determined as the ratio of the voltage on the current intensity :
$Z={\frac {U}{i}}=R+jL\omega +{\frac {B^{2}l^{2}}{\xi +j\left(m\omega -{\frac {k}{\omega }}\right)}}$
The frequency response of the loudspeaker is provided in Figure 7.
A phenomena of electrical resonance is observable around the frequency of 100 Hz. Besides, the inductance of the coil makes the impedance increase from the frequency of 400 Hz. So the range of frequency where the loudspeaker is used is 100 – 4000 Hz

## The loudspeaker suspension

Most moving coil loudspeakers have a two piece suspension system, also known as a flexure system. The combination of the two flexures allows the voice coil to maintain linear travel as the voice coil is energized and provide a restoring force for the voice coil system. The two piece system consists of large flexible membrane surrounding the outside edge of the cone, called the surround, and an additional flexure connected directly to the voice coil, called the spider. The surround has another purpose and that is to seal the loudspeaker when mounted in an enclosure. Commonly, the surround is made of a variety of different materials, such as, folded paper, cloth, rubber, and foam. Construction of the spider consists of different woven cloth or synthetic materials that are compressed to form a flexible membrane. The following two figures illustrate where the suspension components are physically at on the loudspeaker and how they function as the loudspeaker operates.

## Modeling the loudspeaker as a lumped system

Before implementing a loudspeaker into a specific application, a series of parameters characterizing the loudspeaker must be extracted. The equivalent circuit of the loudspeaker is key when developing enclosures. The circuit models all aspects of the loudspeaker through an equivalent electrical, mechanical, and acoustical circuit. Figure 9 shows how the three equivalent circuits are connected. The electrical circuit is comprised of the DC resistance of the voice coil, $R_{e}$, the imaginary part of the voice coil inductance, $L_{e}$, and the real part of the voice coil inductance, $R_{evc}$. The mechanical system has electrical components that model different physical parameters of the loudspeaker. In the mechanical circuit, $M_{m}$, is the electrical capacitance due to the moving mass, $C_{m}$, is the electrical inductance due to the compliance of the moving mass, and $R_{m}$, is the electrical resistance due to the suspension system. In the acoustical equivalent circuit, $M_{a}$ models the air mass and $R_{a}$ models the radiation impedance[2]. This equivalent circuit allows insight into what parameters change the characteristics of the loudspeaker. Figure 10 shows the electrical input impedance as a function of frequency developed using the equivalent circuit of the loudspeaker.

## The acoustical enclosure
### Function of the enclosure

The loudspeaker emits two waves : a front wave and a back wave. With a reflection on a wall, the back wave can be added with the front wave and produces destructive interferences. As a result, the sound pressure level in the room is not uniform. At certain positions, the interaction is additive, and the sound pressure level is higher. On the contrary, certain positions offer destructive interaction between the waves and the sound pressure level is lower.
The solution is to put a baffle round the loudspeaker in order to prevent the back wave from interfering with the front wave. The sound pressure level is uniform in the room and the quality of the loudspeaker is higher.

### Loudspeaker-external fluid interaction

The external fluid exerts a pressure on the membrane of the loudspeaker cone. This additive force can be evaluate as an additive mass and an additive damping in the equation of vibration of the membrane.
$-(m+MS)\omega ^{2}\xi +i\omega (C+RS)\xi +K\xi =fe^{i\omega t}$
When the fluid is the air, this additive mass and additive damping are negligible. For example, at the frequency of 1000 Hz, the additive mass is 3g.

### Loudspeaker-internal fluid interaction

The volume of air in the enclosure constitutes an additive stiffness. This is called the acoustic load. In low frequencies, this additive stiffness can be four times the stiffness of the loudspeaker cone. The internal air stiffness is very high because of the boundary conditions inside the enclosure. The walls impose a condition of zero airspeed that makes the stiffness increase.
The stiffness of the internal air (in red) is fourth time higher than the stiffness of the loudspeaker cone (in blue). That is why the design of the enclosure is relevant in order to improve the quality of the sound and avoid a decrease of the sound pressure level in the room at some frequencies.

## References

* The Loudspeaker Design Cookbook 5th Edition; Dickason, Vance., Audio Amateur Press, 1997.
* Beranek, L. L. Acoustics. 2nd ed. Acoustical Society of America, Woodbridge, NY. 1993.
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