Engineering Acoustics/Transducers - Loudspeaker

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Part 1: Lumped Acoustical Systems1.11.21.31.41.51.61.71.81.91.101.11

Part 2: One-Dimensional Wave Motion2.12.22.3

Part 3: Applications3.13.23.33.43.53.63.73.83.93.103.113.123.133.143.153.163.173.183.193.203.213.223.233.24

Acoustic Transducer[edit]

The purpose of the acoustic transducer is to convert electrical energy into acoustic energy. Many variations of acoustic transducers exists, such as electrostatic or balanced armature drivers, but this article focuses on moving-coil loudspeakers since they are the most commonly available type of acoustic transducer.


The classic moving-coil loudspeaker driver can be divided into three key components:


1) The magnet motor drive system, comprising the permanent magnet, the center pole and the voice coil act together to produce a force on the diaphragm;


2) The loudspeaker cone system, permitting force to be translated into pressure;


3) The loudspeaker suspension, preventing the diaphragm from breaking due to over excursion, allowing only translational movement and tending to bring the diaphragm back to its rest position.


The following illustration shows a cut-away of the moving coil-permanent magnet loudspeaker. A coil is mechanically coupled to a diaphragm, also called cone, and rests in a fixed field produced by a magnet. When an electrical current flows through the coil, a corresponding magnetic field is emitted, interacting with the field of the magnet thus applying a force to the coil, pushing it away or towards the magnet. Since the cone is mechanically coupled to the coil, it will push or pull the air it is facing, causing pressure changes and emitting a sound wave.

MovingCoilLoudspeaker.png
Figure 1 A cross-sectional view of a typical moving-coil loudspeaker

An equivalent circuit can be obtained to model the loudspeaker as a lumped system. This circuit is often the basis of the design of a complete loudspeaker system, including an enclosure and sometimes an amplifier that is matched the the driver. The following section shows how this circuit can be obtained.

Electro-mechano-acoustical equivalent circuit[edit]

Electro-mechanico-acoustical systems such as loudspeakers can be modeled as equivalent electrical circuit as long as each elements moves as a whole. This is usually the case at low frequencies or at frequencies where the dimensions of the system are small compared to the wavelength. To obtain a complete model of the loudspeaker, the interactions and properties of electrical, mechanical, and acoustical subsystems of the loudspeaker driver must each be modeled. The following sections detail how the circuit may be obtained starting with the amplifier and ending with the acoustical load. A similar development can be found in [1] or [2].

Electrical subsystem[edit]

The electrical part of the system is composed of a driving amplifier and a voice coil. Most amplifiers can be approximated as a perfect voltage source in series with the amplifier output impedance. The voice coil has an inductance and a resistance that may directly be modeled in a circuit.

ElectricalSubsystemLoudspeaker.png
Figure 2 The amplifier and loudspeaker electrical elements modeled as a circuit

Electrical to mechanical subsystem[edit]

When the loudspeaker is fed an electrical signal, the voice coil and magnet convert current to force. Similarly, voltage is related to the velocity. This relationship between the electrical side and the mechanical side can be modeled by a transformer.

 \tilde{f_c} = Bl \tilde{i}
 \tilde{u_c} = \dfrac{\tilde{e}}{Bl}


ElectricalToMechanicalLoudspeaker.png
Figure 3 The transformer modeling transduction from the electrical side to mechanical mobility analogy

Mechanical subsystem[edit]

In a first approximation, a moving coil loudspeaker may be thought of as a mass-spring system where the diaphragm and the voice coil constitute the mass and the spider and surround constitute the spring element. Losses in the suspension can be modeled as a resistor.

MechanicalSubsystemModelingLoudspeaker.png


Figure 4 Mass spring system and associated electroacoustic analogies of the impedance and mobility (admittance) type.

The equation of motion gives us :

 \tilde{f_c} = R_m \tilde{u_c} + \dfrac{\tilde{u_c}}{ j \omega C_{MS}} + j \omega M_{MD} \tilde{u_c}
 \dfrac{\tilde{f_c} }{\tilde{u_c}}= R_m+\dfrac{1}{ j \omega C_{MS}}+ j\omega M_{MD}

Which yields the mechanical impedance type analogy in the form of a series RLC circuit. A parallel RLC circuit may also be obtained to get the mobility analog following mathematical manipulation :

 \dfrac{\tilde{u_c} }{\tilde{f_c}}= \dfrac{1}{R_m+\dfrac{1}{ j \omega C_{MS}}+ j\omega M_{MD} }
 \dfrac{\tilde{u_c} }{\tilde{f_c}}= \dfrac{1}{\dfrac{1}{G_m}+\dfrac{1}{ j \omega C_{MS}} + \dfrac{1}{\dfrac{1}{j \omega M_{MD}}}}

Which expresses the mechanical mobility type analogy in the form of a parallel RLC circuit where the denominator elements are respectively a parallel conductance, inductance, and compliance.

Mechanical to acoustical subsystem[edit]

Acoustical subsystem[edit]

Complete electro-mechano-acoustical equivalent circuit[edit]

References[edit]

[1] Kleiner, Mendel. Electroacoustics. CRC Press, 2013. [2] Beranek, Leo L., and Tim Mellow. Acoustics: sound fields and transducers. Academic Press, 2012.