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]

A loudspeaker’s diaphragm may be thought of as a piston that pushes and pulls on the air facing it, converting mechanical force and velocity into acoustic pressure and volume velocity. The equations are as follows:

 \tilde{P_d} = \dfrac{\tilde{f_c}}{\tilde{S_d}}
 \tilde{U_c} = \tilde{u_c}{S_D}

These equations can be modeled by a transformer.

MechanicalToAcousticalLoudspeaker.png
Figure 5 The transformer modeling transduction from mechanical mobility to acoustical mobility analogy

Acoustical subsystem[edit]

The impedance presented by the air load on the loudspeaker's diaphragm is both resistive due to sound radiation and reactive due to the air mass that is being pushed but does not contribute to sound radiation. The air load on the diaphragm can be modeled as an impedance or an admittance. Specific values and approximations can be found in [1] or [2]. Note that the air load depends on the mounting conditions of the loudspeaker. If the loudspeaker is mounted in a baffle, the air load will be the same on each side of the diaphragm. If the air load on one side is  Y_{AR} , then the total air load is  Y_{AR}/2 as both loads are in parallel.

Complete electro-mechano-acoustical equivalent circuit[edit]

Using electrical impedance, mechanical mobility and acoustical admittance yield the following equivalent circuit, modeling the entire loudspeaker drive unit.

LoudspeakerEquivalentCircuit.png
Figure 6 A complete electro-mechano-acoustical equivalent circuit of a loudspeaker drive unit

This circuit can be reduced by substituting the transformers and connected load by an equivalent loading that would present the same impedance as the transformer. An example of this is shown on figure 7, where acoustical and electrical loads have been "brought over" on the mechanical side.

LoudspeakerEquivalentCircuitMechanical.png
Figure 7 Mechanical equivalent circuit modeling of a loudspeaker drive unit

The advantage of doing such manipulation is that we can then directly relate electrical measurements with elements in the circuit. This will later allow us to obtain values for the different components of the models and match this model to real loudspeaker drivers. We can further simplify this circuit by using Norton's theorem, and converting the series electrical components and voltage source into an equivalent current source and parallel electrical components. Then, using a technique called the Dot method, we can obtain a single loop series circuit which is the dual of the parallel circuit obtained with Norton's theorem. Additionally, we neglect the effect of the voice coil inductance, which has an effect only at high frequencies, where lumped element modeling is usually not valid. Furthermore, the air load impedance at low frequencies, is mass-like and has been modeled by an inductance. This results in the low frequency model equivalent circuit of figure 8, which is somewhat easier to manipulate than the circuit of figure 7. Note that the analogy used for this circuit is of the impedance type.

LowFrequencyLoudspeakerEquivalentCircuitMechanical.png
Figure 8 Low frequency approximation mechanical equivalent circuit of a loudspeaker drive unit

Mass elements, in this case the mass of the diaphragm and voice coil M_{MS} and the air mass loading the diaphragm 2M_{M1} can be regrouped in a single element :

 M_{MS} = M_{MD}+2M_{M1}

Thiele-Small Parameters[edit]

The complete low frequency behavior of a loudspeaker drive unit can be modeled with just 6 parameters, the Thiele-Small parameters. Most of these parameters result from algebraic manipulation of the equations of the circuit of figure 8. Loudspeaker driver manufacturers seldom provide electro-mechano-acoustical parameters directly and rather provide Thiele-Small parameters, but conversion from one to the other is quite simple. The Thiele-Small parameters are as follow :

1.  R_e , the voice coil DC resistance;

2.  Q_{ES} , the electrical Q factor;

3.  Q_{MS} , the mechanical Q factor;

4.  f_s , the loudspeaker resonance frequency;

5.  S_D , the effective surface area of the diaphragm;

6.  V_{AS} , the equivalent suspension volume: the volume of air that has the same acoustic compliance as the suspension of the loudspeaker driver.

These parameters can be related directly from the low frequency approximation circuit of figure 8, with  R_e and  S_D being explicit.

 Q_{MS} = \dfrac{1}{R_{MS}} \sqrt{\dfrac{M_{MS}}{C_{MS}}} ;  Q_{ES} = \dfrac{R_g + R_e }{(Bl)^2} \sqrt{\dfrac{M_{MS}}{C_{MS}}}  ; f_s= \dfrac{1}{2\pi\sqrt{M_{MS}C_{MS}}}; V_{AS}= C_{MS}S_D^2\rho c^2


Where \rho c^2 is the characteristic impedance of air. It follows that, if given Thiele-Small parameters, one can extract the values of each component of the circuit of figure 8 using the following equations :

 C_{MS} = \dfrac{V_{AS}}{S_D^2 \rho c^2} ; M_{MS}= \dfrac{1}{(2 \pi f_s)^2 C_{MS}}

References[edit]

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