Finite Element Method Magnetics : DriverDynamics

Loundspeaker Driver Dynamics

David Meeker
dmeeker@ieee.org

07Sep2015

Introduction

The purpose of this page is to describe how FEMM can be used to analyze the small signal properties of a loudspeaker driver. With FEMM, the effects of eddy currents and/or shorting rings can be included in the small signal model.

This note will first discuss a model of a loudspeaker that is consistent with the Thiele/Small Model. Then, the note will discuss modifications to this model to include more complicated electrical behavior identified by incremental permeability AC simulations in FEMM.

Baseline Loudspeaker Model

It is assumed that the loudspeaker is driven by a voltage source with no output impedance. This is a reasonable assumption for audio amplifiers--the load impedance is typically at least 10X the output impedance of the amplifier (see wikipedia/Output_impedance).

The loudspeaker's mechanical dynamics might be simply modeled as the spring-mass-damper system pictured in Figure 1. It is assumed that the mass, \(m\) has the mass of the air moved by the cone rolled in, along with the moving mass of the loudspeaker (cone, voice coil, dust cap). The stiffness, \(k\), is the spring constant of the spider. The damping, \(c\), is mostly due to the surround and to the spider itself--damping due to the air is usually neglected, because it is comparatively small.



In Figure 1, force \(F\) is due to the action of the voice coil. For small displacements, the force due to the voice coil is proportional to the current carried by the coil:

\[ \large F = K_F i \]

(1)

Incorporating (1) as the definition of force, the equation of motion of the mechanical system can be formulated as a balance of forces on the mass, i.e. sum of inertial, damping, and stiffness forces equal the force applied by the voice coil, where \(x\) and \(i\) are the displacement and current of the voice coil, respectively:

\[ \large m \ddot{x} + c \dot{x} + k x = K_F i\]

(2)

The simplest representation of the electrical system is shown as Figure 2.


Figure 2 shows amplifier voltage \(v\) driving the inductance, \(L\), and resistance, \(R\) of the voice coil, pushing against the back-EMF voltage, \(v_B\), that arises due to the motion of the voice coil. From the mechanical system, the power applied to the mechanic system from the voice coil is the product of force and velocity:

\[ \large P_{mech} = F \dot{x} = k_F i \dot{x}\]

(3)

The power supplied from the electrical system is the product of current and back-EMF voltage:

\[ \large P_{elect} = v_B i\]

(4)

Applying the conservation of energy, the sum of the energy applied to the mechancial system and the energy supplied by the electrical system must add up to zero:

\[ \large P_{mech} + P_{elect} = 0\]

(5)

Solving the conservation of energy equation for the back EMF voltage shows that the back EMF uses the same proportionality constant as the force expression:

\[ \large v_B = - K_F \dot{x} \]

(6)

The electrical equation of the system can then be written by as a balance of voltages around the electrical circuit:

\[ \large L \frac{di}{dt} + R i = v - K_F \dot{x}\]

(7)

Equations (2) and (7) represent the mechanical and electrical models of the system. However, two different equations are more commonly used to characterize the performance of a loudspeaker: the electrical impedance of the loudspeaker, and the frequency response of the loudspeaker. As the first step to converting the electrical and mechanical equations to these forms, it is first useful to solve (2) and (7) for current \(i\) and displacement \(x\) in terms of applied voltage \(v\) to have two equations that appear to be purely electrical and purely mechanical. This task is much easier if (2) and (7) are transformed into the Laplace domain, where differentiation in time is replaced by multiplication by the Laplace variable, \(s\). The electrical and mechanical equations, re-written in the Laplace domain, are:

\[ \large \left(s L + R \right) i = v - s K_F x\]

(8)

\[ \large \left( s^2 m + s c + k \right) x = K_F i\]

(9)

Solving for \(x\) and \(i\) yields:

\[ \large x = \left( \frac{k_F}{s k_F^2 + \left( s L + R \right) \left( s^2 m + s c + k \right)} \right) v \]

(10)

\[ \large i = \left( \frac{s^2 m + s c + k}{s k_F^2 + \left( s L + R \right) \left( s^2 m + s c + k \right)} \right) v\]

(11)

The definition of impedance, \(Z\) is the ratio of voltage to current. The impedance follows directly from (11):

\[ \large Z = \frac{v}{i} = \frac{s k_F^2}{s^2 m + s c + k} + \left( s L + R \right) \]

(12)

The first term in the impedance is the mechanical system, as seen through the back EMF. The second and third terms are the coil's electrical impedance.

Connection to Thiele-Small Parameters

Additional References

A. N. Thiele, Loudspeakers in vented boxes: Part I, Journal of the Audio Engineering Society, 19(5):382-392, 1971.

A. N. Thiele, Loudspeakers in vented boxes: Part II, Journal of the Audio Engineering Society, 19(6):471-483, 1971.

https://en.wikipedia.org/wiki/Thiele/Small

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